Physical Properties of Nanosystems

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Author: Janez Bonca | Sergei Kruchinin

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Physical Properties of Nanosystems

NATO Science for Peace and Security Series This Series presents the results of scientific meetings supported under the NATO Programme: Science for Peace and Security (SPS). The NATO SPS Programme supports meetings in the following Key Priority areas: (1) Defence Against Terrorism; (2) Countering other Threats to Security and (3) NATO, Partner and Mediterranean Dialogue Country Priorities. The types of meeting supported are generally “Advanced Study Institutes” and “Advanced Research Workshops”. The NATO SPS Series collects together the results of these meetings. The meetings are coorganized by scientists from NATO countries and scientists from NATO’s “Partner” or “Mediterranean Dialogue” countries. The observations and recommendations made at the meetings, as well as the contents of the volumes in the Series, reflect those of participants and contributors only; they should not necessarily be regarded as reflecting NATO views or policy. Advanced Study Institutes (ASI) are high-level tutorial courses intended to convey the latest developments in a subject to an advanced-level audience Advanced Research Workshops (ARW) are expert meetings where an intense but informal exchange of views at the frontiers of a subject aims at identifying directions for future action Following a transformation of the programme in 2006 the Series has been re-named and re-organised. Recent volumes on topics not related to security, which result from meetings supported under the programme earlier, may be found in the NATO Science Series. The Series is published by IOS Press, Amsterdam, and Springer, Dordrecht, in conjunction with the NATO Public Diplomacy Division. Sub-Series A. B. C. D. E.

Chemistry and Biology Physics and Biophysics Environmental Security Information and Communication Security Human and Societal Dynamics

http://www.nato.int/science http://www.springer.com http://www.iospress.nl

Series B: Physics and Biophysics

Springer Springer Springer IOS Press IOS Press

Physical Properties of Nanosystems

edited by

Janez Bonˇca J. Stefan Institute Ljubljana, Slovenia and

Sergei Kruchinin Bogolyubov Institute for Theoretical Physics Kiev, Ukraine

123 Published in cooperation with NATO Public Diplomacy Division

Proceedings of the NATO Advanced Research Workshop on Physical Properties of Nanosystems Yalta, Ukraine 28 September–2 October 2009

Library of Congress Control Number: 2010938720

ISBN 978-94-007-0129-8 (PB) ISBN 978-94-007-0043-7 (HB) ISBN 978-94-007-0044-4 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

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All Rights Reserved c Springer Science + Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Preface

These proceedings of the NATO-ARW “Physical properties of nanosystems” held at the “Yalta” Hotel, Yalta, Ukraine from 28 September–2 October 2009 emmerged as a result of many presentations and discussions between workshop participants. Yalta workshop focused on several open problems in the theory of correlated electron systems, nanophysics and sensor technology. The programme of the workshop allowed presentations and opened discussions on several emerging modern research topics, such as iron pnictides. Theoretical advances were tested against major experimental and technological achievements in related materials.In the session on novel superconductors, the physical properties of F e-based and M gB2 and cuprate superconductors were discussed. Recent advances in nanoscience have demonstrated that fundamentally new physical phenomena are found when systems are reduced in size down to dimensions, comparable to the fundamental microscopic length scales of the investigated material. Latest developments in nanotechnology and measurement techniques facilitate experimental investigation of the transport properties of nanosystems. Special focus sessions were devoted to contemporary topics in nanophysics, such as carbon nanotubes, graphene, magnetic nanostructures, quantum dot, spintronics, molecular electronics, and quantum information processing. We are grateful to members of the International Advisory Committee: F.Peeters and D.Logan for their consistent help and suggestions. We would like to thank the NATO Science Committee for the essential financial support, without which the meeting could not have taken place. We also acknowledge the National Academy of Science of Ukraine, Ministry of Ukraine for Education and Science, J.Stefan Institute and Faculty of Mathematics and Physics, University of Ljubljana, Slovenia for their generous support. Ljubljana, Kiev, June 2010

Janez Bonˇca Sergei Kruchinin

Contents

Part I Electron transport in nanosystems 1 OPTICS OF FLAT CARBON – SPECTROSCOPIC ELLIPSOMETRY OF GRAPHENE FLAKES V. G. Kravets, R. R. Nair, P. Blake, L. A. Ponomarenko, I. Riaz, R. Jalil, S. Anisimova, A. N. Grigorenko, K. S. Novoselov, and A. K. Geim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2 RENORMALIZED PERTURBATION APPROACH TO ELECTRON TRANSPORT THROUGH QUANTUM DOT A. C. Hewson, A. Oguri, and J. Bauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 LOW TEMPERATURE TRANSPORT IN TUNNEL JUNCTION ARRAYS: CASCADE ENERGY RELAXATION N. M. Chtchelkatchev, V. M. Vinokur, and T. I. Baturina . . . . . . . . . . . . 25 4 ELECTRON TRANSPORT THROUGH MOLECULES IN THE KONDO REGIME: THE ROLE OF MOLECULAR VIBRATIONS J. Mravlje and A. Ramˇsak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 WAVE DELOCALIZATION IN NONLINEAR DISORDERED MEDIA S. Flach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6 PHYSICAL LIMITS FOR SCALING OF ELECTRONIC DEVICES IN INTEGRATED CIRCUITS W. Nawrocki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7 SYNCHRONIZED ANDREEV TRANSMISSION IN CHAINS OF SNS JUNCTIONS N. M. Chtchelkatchev, T. I. Baturina, A. Glatz, and V. M. Vinokur . . . . . . 87

VIII

Contents

Part II Superconductivity 8 JOSEPHSON EFFECT IN POINT CONTACTS BETWEEN TWO-BAND SUPERCONDUCTORS A. Omelyanchouk and Y. Yerin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9 SUPERCONDUCTING NANOWIRES: NEW TYPE OF BCS-BEC CROSSOVER DRIVEN BY QUANTUM-SIZE EFFECTS A. A. Shanenko, M. D. Croitoru, A. Vagov, and F. M. Peeters . . . . . . . . 119 10 TRANSIENT RESPONSE OF A SUPERCONDUCTOR IN AN APPLIED ELECTRIC FIELD M. N. Kunchur and G. F. Saracila . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 11 INTERBAND NODAL-REGION PAIRING AND THE ANTINODAL PSEUDOGAP IN HOLE DOPED CUPRATES N. Kristoffel and P. Rubin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 12 ELECTRONIC STRUCTURE OF CUPRATE SUPERCONDUCTORS IN THE PRESENCE OF OUT-OF-PLANE IMPURITIES Z. Wang and S. Feng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 13 MULTIPLE QUASIPARTICLE PAIRS IN THE BCS MODEL J. D. Fan and Y. M. Malozovsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 14 CRITICAL AND NON-CRITICAL CHANNEL IN THE DAMPING OF SUPERCONDUCTING FLUCTUATIONS IN TWO-BAND SYSTEMS ¨ T. Ord, K. R¨ ago, and A. Vargunin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 15 PHASE SLIP PHENOMENA ONE AND TWO DIMENSIONAL SUPERCONDUCTING RING M. Lu-Dac and V. V. Kabanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 16 LOCUS OF THE SUPERCONDUCTIVITY IN THE CUPRATES J.D. Dow and D.R. Harshman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 17 AN APPROXIMATING HAMILTONIAN METHOD IN THE THEORY OF IMPERFECT BOSE GASES N. N. Bogolyubov, Jr. and D. P. Sankovich . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Contents

IX

Part III Spintronics 18 MAGNETIC NANOSTRUCTURES K. Bennemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 19 HEAVY FERMIONS AND SUPERCONDUCTIVITY IN THE KONDO-LATTICE MODEL WITH PHONONS ˇ O. Bodensiek, R. Zitko, R. Peters, and T. Pruschke . . . . . . . . . . . . . . . . . . 233 20 MAGNETIZATION CURVES FOR ANISOTROPIC MAGNETIC IMPURITIES ADSORBED ON A NORMAL METAL SUBSTRATE ˇ R. Zitko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 21 CONDUCTIVITY OF LAYERED SYSTEMS WITH PLANAR AND BULK DISORDERS D.L. Maslov, V.I. Yudson, A.M. Somoza, and M. Ortu˜ no . . . . . . . . . . . . . 259 22 SUPERPOSITION OF FLUX-QUBIT STATES AND THE LAW OF ANGULAR MOMENTUM CONSERVATION A.V. Nikulov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 23 CREATION AND CONTROL OF ORDERED NANOSTRUCTURES IN SPIN-GLASS MEDIA A.S. Gevorkyan, A.A. Gevorkyan, and K.B. Oganesyan . . . . . . . . . . . . . . . 281

Part IV Sensors 24 BIOSCOPE: NEW SENSOR FOR REMOTE EVALUATION OF THE PHYSIOLOGICAL STATE OF BIOLOGICAL SYSTEMS R. Sh. Sargsyan, A. S. Gevorkyan, G. G. Karamyan, V. T. Vardanyan, A. M. Manukyan, and A. H. Nikogosyan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 25 THERMOELECTRICITY IN DOUBLE-BARRIER RESONANT TUNNELING STRUCTURES V.N. Ermakov, S.P. Kruchinin, A. Fujiwara, and S.J. O’Shea . . . . . . . . . 311 26 LIGHT-EMITTING DIODES AND OPTICAL FIBERS J.D. Dow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 27 GAS SENSING PROPERTIES OF THE NANOSIZED Pd AND Cu x Pd LAYERS V.G. Litovchenko, T.I. Gorbanyuk, Yu.G. Ptushinskii, and O.V. Kanach 325 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Part I

Electron transport in nanosystems

1 OPTICS OF FLAT CARBON – SPECTROSCOPIC ELLIPSOMETRY OF GRAPHENE FLAKES V. G. Kravets, R. R. Nair, P. Blake, L. A. Ponomarenko, I. Riaz, R. Jalil, S. Anisimova, A. N. Grigorenko, K. S. Novoselov, and A. K. Geim School of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, UK [email protected] Abstract. We present ellipsometric spectra of a graphene flake placed on a surface of oxidized silicon wafer. Our measurements demonstrate that spectroscopic ellipsometry can be successfully used to count the number of graphene layers in a flake. We also show that visible transparency of any two-dimensional system with a symmetric electronic spectrum is governed by the fine structure constant and derive an expression for the absorption coefficient of such a system.

Key words: graphene, spectroscopic ellipsometry, absorption, spectra.

1.1 Introduction Graphene, a recently discovered two-dimensional carbon-based material [1], attracted a lot of attention due to unique physical, chemical and mechanical properties [2]. Its low energy excitations, known as massless Dirac fermions [2], specify distinctive and unexpected properties of integer and fractional quantum Hall effect [3, 4], Schubnikov-de-Gaas oscillations [1], electronic transport [2]. Recently, it was shown that graphene also possesses remarkable optical properties: its visible transparency is determined by the fine structure constant only [5]; quantum efficiency of light absorption in graphene is close to one [6]; infrared transmission of suspended graphene membrane can be modulated by a gate voltage [7]. This opens prospects of graphene applications in optics and optoelectronics, e.g., in solar cells, filters and modulators. Despite a rapid progress in studying infrared properties of graphene layers [7, 8], works on optical properties of graphene in visible are scarce [9]. The aim of this letter is to elucidate optical properties of graphene layers in visible light usingvariable angle spectroscopic ellipsometry and large (typical size of

J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 1,

3

4

V. G. Kravets et al.

50 μm × 100 μm) high quality flakes deposited on an oxidized silicon substrate. We also demonstrate that absorption coefficient of any two-dimensional system with a symmetric electronic spectrum is governed by the fine structure constant.

1.2 Spectroscopic ellipsometry A schematic representation of our experimental set-up is presented on Fig. 1.1(a). Graphene flakes have been prepared on a silicon wafer covered with ≈ 300 nm layer of silicon oxide in order to improve graphene visibility [1]. The ellipsometric parameters Ψ and Δ (defined as tan(Ψ ) exp(iΔ) = rp /rs , where rp and rs are the reflection coefficients for the light of p- and s-polarizations [10]) have been measured with the help of a focused beam variable angle Woollam ellipsometer with a focal spot of 30 μm. Measurements have been performed both on the bare substrate and on the large graphene flakes (see an example of a studied flake on Fig. 1.1(b)) and subsequently modeled with Wvase32 software based on Fresnel coefficients for multilayered films. First, we tested our installation by studying thick flakes of highly ordered pyrolytic graphite (results are not presented here). The fit resulted in the reconstructed optical constants shown in Fig. 1.2. (For simplicity, the in-plane optical constants of graphite and graphene are referred to as x-constants, while the perpendicular to the graphene layer constants – as z-constants.) We found that there is a coupling between reconstructed x- and z-components which is most likely caused by small variations in flatness of graphite surface and hence deviations of in-plane currents. This coupling might be even stronger for graphene due to intrinsic or extrinsic ripples present in graphene sheets [11]. To avoid the coupling we choose to model z-response of graphite flakes as a Cauchy material [10], which is a good approximation for mostly dielectric response of s-electrons [12]. The extracted spectral dependences for

a

b

Ep

100×

q Es

Substrate Graphene

SiO2

Graphene 20 µm

Si

Fig. 1.1. Samples for spectroscopic ellipsometry measurements. (a) Sample geometry. (b) A photograph of a graphene flake.

1 Optics of flat carbon

5

5 nx kx nz kz

4

n,k

3

2

1

0 200

400

600

800

1000

Wavelength (nm)

Fig. 1.2. Variable angle spectroscopic ellipsometry of graphite: Reconstructed optical constants of graphite withz-component being treated as a Cauchy material.

in-pane optical constants are close to the reported in literature [12] and for perpendicular components nz – slightly larger [12]. Next we performed the spectroscopic ellipsometry of graphene flakes. Figure 1.3(a) demonstrates spectral dependence of Ψ measured at the incidence angle of 45◦ for the substrate (a green curve), a graphene monolayer (dark yellow), bilayer graphene (maroon), and triple layer graphene (brown). We note an excellent contrast in Ψ at the position of the reflection peaks ( ∼ 320 nm and ∼ 510 nm) as a function of the number of layers. The ratio of the maximal Ψ change due to the presence of one graphene layer ( ∼ 3˚) to the Ψ noise ( ∼ 0.02˚) is large. As a result, ellipsometric measurements provide a fast and accurate way for finding the number of graphene layers in the flake immobilized on any reasonably reflective and flat substrate. This can be achieved by modeling the graphene flake as a graphite material and extracting the optical thickness of the layer from the spectra. Although the effective optical thickness of thin layers can deviate from the “geometrical” thickness of the sample [14], this effect should not be large for flat graphite flakes. In principle, spectroscopic ellipsometry has a potential to replace Raman spectroscopy for the purpose of counting graphene layers in a flake [15], especially for in-situ control of graphene growth. The angle dependence of the ellipsometric spectra of a graphene monolayer is shown in Fig. 1.3(b) and (c). We discuss extraction of the optical constants of graphene layers from spectroscopic measurements elsewhere. We found that indeed the absorption coefficient of graphene in visible is close to πα, where α is the fine structure constant. Here we show that such behavior is not unique and generally the absorption of any sufficiently thin

6


a Trilayer Graphene Bilayer Graphene Monolayer Graphene Substrate

ψ(deg)

60

50

40

300

400 500 Wavelength (nm)

600

b 49 deg 55 deg 61 deg 67 deg Fresnel Fit

80

ψ(deg)

60

40

20

200

400

600 Wavelength (nm)

800

1000

Fig. 1.3. Variable angle spectroscopic ellipsometry of graphene. (a) Ψ spectra measured at θ = 45◦ for the bare substrate, the substrate with a graphene monolayer, a graphene bilayer and a graphene triple layer. (b) and (c) Ψ and Δ spectra for different angles of incidence θ measured on a graphene flake.

electronic system with symmetric energy spectrum is governed by the fine structure constant. Indeed, let us consider optical absorption of a generic thin 2D system witha symmetric energy spectrum of particles described by ˆ0 = ε(p)a†p ap . In electromagnetic field the momentum is Hamiltonian H p

gauged as p → p − ec A, which gives a semiclassical interaction Hamiltonian ˙ = E is the real ˆ int = − e ∂ Hˆ 0 (A exp(ιωt) + A exp(−ιωt)), where 1 A as H 2c ∂p

c

1 Optics of flat carbon

7

c

Δ(deg)

250

200

150

49 deg 55 deg 61 deg 67 deg Fresnel Fit

100

200

400

600 Wavelength (nm)

800

1000

Fig. 1.3. Contd.

vector potential of electromagnetic wave of frequency ω with electric field E (E0 being a real amplitude of the electric field). The interaction Hamiltonian ˆ int = Vˆ exp(ιωt) + Vˆ exp(−ιωt) and hence the energy absorption has a form H of normal incident light per unit area is given by the Fermi golden rule: W = ω

2 2πg f | Vˆ |i (n(εi ) − n(εf )) δ(pf − pi )δ(εf − εi − ω), (1.1) i,f

where g is the degeneracy of states, i, f are the initial and final states, respectively, and n(ε) is the occupancy number. For a symmetric spectrum we have εf = −εi = ω/2. This gives ge2 W = 8π2 ω

2 f | E ∂ε |i (n(−ε) − n(ε)) δ(ε − ω/2)pdpdφ, ∂p

(1.2)

where ϕ is the polar angle in the plane. Assuming that the energy spectrum dε is also symmetric in the plane ε(p) = ε(p), we can write ∂ε(p) = dp pˆ (with ∂p pˆ = p/p and p = |p|) and get ge2 E02 d ln(ε) W = S (n(−ε) − n(ε)) , 8 d ln(p) ε=ω/2

(1.3)

2 1 e is the unit polariza|f | eˆpˆ |i| dφ is the form-factor (ˆ where S = 2π tion vector). For linear polarization (S=1/2) and a sufficiently thin 2D

8


layer the electromagnetic energy that falls onto the system per unit area is c approximately W0 = 8π E02 and the absorption ratio is gπα d ln(ε) . (1.4) Abs(ω) = W/W0 = · (n(−ε) − n(ε)) 2 d ln(p) ε=ω/2 This expression connects the experimentally measured spectrum of light absorption Abs(ω) with the electronic dispersion ε(p). It is worth noting that ln(ε) dε (n(−ε) − n(ε)) dd ln(p) = (n(−ε) − n(ε)) pε dp is a number, which implies that absorption of any 2D material is governed by the fine structure constant. Simple calculations give for Fermi distribution: n(−ε) − n(ε) = sinh(ε/T )/ (cosh(μ/T ) + cosh(ε/T )) ,

(1.5)

where μ is the chemical potential and T is measured in the energy units. It is easy to check that n(−ε) − n(ε) ≈ 1 for optical frequencies (ε >> T ) and small doping (μ Teff (the emission of the excitations with the energy larger than V is forbidden), and Teff ≡ lim ωNω = 0.5V coth(V /2T ) .

(3.8)

ω→0

This result shows that the system with the environment well isolated from the bath cannot be cooled below Teff . Note that the coth-expression for Teff obtained in the first approximation in ρ. In a general case, Teff depends on ρ. Equations (3.1)–(3.6) give the full description of the kinetics of the tunnel junction in a nonequilibrium environment. To derive the I-V characteristics we (12) find Nω ∼ = nω and plug it into Eqs. (3.1)–(3.4). Introducing the parameters −1 = ρ(0) and Λ, the characteristic frequency of the ρ(ω) decay [for the g Ohmic model [28], ρ = g −1/{1 + (ω/Λ)2 } and Λ/g is of the order of the charging energy of the tunnel junction], we find: I∼

V Λ ln ; RT V

(3.9)

in the interval T V Λ, where Teff V . Note that I(V ) given by Eq. (3.9) differs from the power law dependence obtained in [28] for Te = Teff = 0. This shows that tuning the environment one can control the I(V )-characteristics of the tunnel junction (the gating effect). At high voltages, V Λ, one finds

Δ∞ = iJ (0) = 2

0

I(V ) (V − Δ∞ )/RT , ∞

(3.10)

dωρω [1 + Nω(out) − Nω(in) ] Δ(0) ∞ ln(Λ/ min{Te , Tenv }), (3.11)

(0)

(out)

where Δ∞ = Δ∞ [N (out) = N (in) ] ∼ Λ/g, since at V Λ, Nω (in) Nω .

Λ/ω

3.3 Arrays of tunnel junctions Extending Eq. (3.2) onto an array comprised of N junctions one finds

N RQ → − 2 S dd f1 ()[1 − f2 ( )]P ( − ), (3.12) Γ = 2R 4π i i=1

3 Low temperature transport in tunnel junction arrays

31

where

∞

P (E) =

dt exp(iEt)

−∞

∞

dω 0

ρ(ω) ω

(in) (out) . (3.13) Nω,j eiωt + (1 + Nω,j )e−iωt × j≤N −1

−(N −1)

Here S = Ec N N /(N − 1)!, and Ec = e2 /2C is the Coulomb charging energy of a single junction (C is a single junction capacitance) and for the Cooper pair transport e → 2e. Equations (3.12) and (3.13) were derived in a first order in tunneling Hamiltonian. Shown in Fig. 3.2 is a diagrammatic representation of Eq. (3.13) for N = 3. A generalization of the results obtained for a single junction including the structure of the collision integral and the concept of the effective temperature Eq. (3.8), onto large arrays is straightforward. As long as temperatures are not extremely low [23], the charge transfer in large arrays is dominated by the inelastic cotunneling and the cascade energy relaxation. The tunneling

a

V

b c

d

Ν(out)(ω)+1 f(Œ)

1−f(Œ−ω−ω’)

Ν(out)(ω)+1 f(Œ)

f(Œ)

Ν(out)(ω’)+1 1−f(Œ+ω−ω’)

Ν(in)(ω) f(Œ)

Ν(in)(ω’) 1−f(Œ−ω+ω’)

Ν(in)(ω)

e

Ν(out)(ω’)+1

Ν(in)(ω’) 1−f(Œ+ω+ω’)

Fig. 3.2. (a) The single electron two-islands’ circuit. (b)–(e) Diagrams describing the forward inelastic cotunneling rate. The “up” arrows stand for the e-h pairs excited during the cotunneling and the “down” arrows correspond to the recombination of the e-h pairs. The vertices shown by boxes are proportional to the probability of an elemental e-h pair excitation, ρ(ω)/ω.

32

N. M. Chtchelkatchev et al.

carriers generate e-h pairs [23, 24] serving as an environment exchanging the energy with the tunneling current and then slowly losing it to the bath.

3.4 Cascade two-stage relaxation: general applications In the preceding sections we have formulated a general approach to description of electronic transport in mesoscopic tunnel junctions mediated by the energy exchange between the charge carriers and the environment, which, in its turn, relaxes the energy to thermostat. In what follows we will show that there is a rich variety of the seemingly disparate kinetic phenomena in tunnel junction arrays that allow for a natural and transparent description within the unique framework of this hierarchical or cascade relaxation concept. We illustrate the power of our approach applying it to phenomena of overheating [35, 36], the Coulomb anomaly [11, 28], the transport in the granular systems governed by the cotunneling processes [28, 37] and mediated by the electron-hole environment. The concept of the cascade two-stage energy relaxation appears to be of the fundamental importance to revealing the microscopic mechanism of the insulator-to-superinsulator transition and the nature of low temperature transport in the superinsulating state [38, 39]. We show, finally, that the concept of the cascade energy relaxation resolves the long standing puzzle of temperature-independent pre-exponential factor in variable range hopping conductivity [16–18, 40–44]. We demonstrate that the underlying physical mechanism behind all the above is the imbalance between the intense energy exchange of the charge carriers with the (nonequilibrium) environment and the comparatively weak coupling of the environment to the phonon bath. Nonequilibrium Coulomb anomaly Consider the quasi-2D disordered metallic film attached at the edges to two electrodes, see Fig. 3.3(a). The Coulomb anomaly arises due to electromagnetic fluctuations associated with the electron-electron interaction in the wire (electron-hole environment excitation) and manifest itself as the suppression of the local tunnel density of states (TDOS) at small energies [45]. Experimentally the density of states can be determined by means of tunneling transport measurements with the scanning tunnel microscope (STM) tip serving as one of the electrodes, see Fig. 3.3(a) and Refs. [11, 22]. The corresponding equivalent circuit is shown in Fig. 3.3(b). The resistor characterizes the contact between the tip and the quasi-2D metallic film. This is the same resistor that appears in Fig. 3.1. In the absence of the current, i.e. in an equilibrium, the distribution function of electrons at any point of the film is the Fermi function. Then the current voltage characteristics of the junction, I(Vtip ), where Vtip is the potential of the tip, can be found following the recipes of Refs. [3, 45]: we have to


a

33

f

fF (Œ)

f(Œ)

F(

V

Œ− V)

fF(Œ-Vtip)

Vtip

metallic film b fŒ(1)=fF(Œ−Vtip)

(2)=f(e) fŒ

Fig. 3.3. (a) The quasi-2D disordered metallic film attached at the edges to two electrodes. The tunneling density of states (TDOS) can be determined using the transport measurement done with the help of the Scanning Tunnel Microscope (STM) tip. (b) Equivalent circuit that describes transport between the STM-tip and metal (in nonequilibrium state).

use Eqs. (3.1)–(3.2) with the equilibrium environment distribution function ˜ /(Dq 2 − iω)2 , where U ˜ is the dynamically-screened and take ρ(ω) = 2 Im q U Coulomb interaction in the metal and D is the diffusion coefficient. Then from the differential conductance, dI/dVtip , we get the local density of states (the same as in Ref. [46]) and find all the standard Coulomb anomaly features, see Refs. [45, 46]. As the current starts to flow through the metallic film, then the distribution function of electrons, f (, r), at low temperatures (where the phonon bath is frozen out) becomes nonequilibrium and should be found from the kinetic equation [11, 22], f = 0 with the edge conditions relating electron distribution function in the metal with the electron distribution functions at the reservoirs. The solution of the kinetic equation at the center of the metallic film can be approximated as follows: f () =

1 [fF () + fF ( − V )] , 2

(3.14)

where V is the electrical potential of the left electrode that pushes the current through the metal, see Fig. 3.3(a). Here it was assumed that the electron inelastic scattering length, lin , is larger than the separation between the electrodes attached to the metallic film. The electron-hole environmental modes in the metallic film, which are responsible for the Coulomb anomaly, become now nonequilibrium. Since the phonon bath is frozen, then τenv−bath τe−env , so the stationary distribution function Nω is to be found from Eq. (3.6). Then we obtain,

34


Nω = nω ,

nω = (2ω)−1

df+ (1 − f− ) .

(3.15)

We calculate the tunneling rate between for electrons passing from the tip to the metallic film using Eq. (3.2): → − 1 Γ = f (tip) (1 − f )P < ( − ) , (3.16) RT (tip)

= fF ( − Vtip ) and f is given in Eq. (3.14). Note that when evalwhere f uating P < , we used the nonequilibrium environmental bosonic distribution function Nω as given by Eq. (3.15). Equations (3.1) and (3.16) yield the current, flowing through the tip, I(Vtip ), from which one determines the TDOS as: dI(Vtip ) νT () ∝ . (3.17) dVtip Vtip → One finds after some algebra that the TDOS acquires the ‘superstructure’ on the energy scale of the order of Teff (V ): two dips develop at the energies corresponding to the positions of the Fermi level in the tip and in the film, where the electronic distribution function experiences discontinuous jumps (i.e. at = 0 and = V ) [11, 47]. Negative differential resistance and overheating If the rate of the energy supply from the external bias to the charge carriers exceeds the rate of energy losses to the environment, the phenomenon of overheating takes place and the energy distribution function of the current carriers noticeably deviates from the equilibrium distribution function [35,36]. One of the characteristic manifestations of the overheating effect is the onset of the “falling” region of the I-V curve where the differential conductivity G = ∂I/∂V < 0. The corresponding I(V ) characteristics is referred to as that of the S-type if the current is the multi-valued function of the voltage, and the I-V curve of the N -type corresponding to the case where the current is nonmonotonic but still remains a single valued function of the voltage. The phenomenon of overheating has been a subject of the incremental interest and extensive studies during the decades, see the detailed Volkov and Kogan review [35], and the impressive progress in understanding of the underlying mechanisms was achieved. Recently, for instance, the ideas of [35] were applied to description of nonlinear low temperature current-voltage characteristics in disordered granular metal [48]. The concept of the two-stage relaxation of the present work enables us to construct a general approach to overheating taking into the consideration also non-phonon mechanisms of relaxation. A general scheme is as follows. Let us consider the far-from-equilibrium tunneling transport mediated by an environment. Let the environmental excitations emitted by tunneling carriers be “hot” (“up” lines in Figs. 3.1–3.2) whereas the


35

environmental excitations absorbed by tunneling carriers are “cold” (“down” lines in Figs. 3.1–3.2). This implies that there exist two distinct energy scales (in) (out) characterizing the frequency distributions, Nω and Nω , corresponding to emitted and absorbed environment excitations, respectively: (out)

Teff

= lim ωNω(out) ,

(3.18)

ω→0

(in)

Teff = lim ωNω(in) .

(3.19)

ω→0

These two temperature scales are, in general, different provided the effective inelastic length, Lφ , describing thermalization of the excitations with the phonon bath (thermostat) is of the same order or smaller than the hot region where tunneling electrons strongly interact with the environment. Then the absorbed environmental excitations come from the outside of the hot (in) region and have the temperature of the bath, Teff ≈ Tbath . The emitted excitations are hot, and their effective temperature is the function of the applied voltage (current). The temperature of the hot excitations is to be found from the heat balance equation following the recipes of Refs. [35, 36]. Namely, it is obtained by integrating the product of the kinetic equation for electron distribution function [see, e.g., Eq. (3.5)] and the electron energy , over the volume and energy: (in)

P

E(Teff )

(in) τe−env (Teff )

(out)

−

E(Teff

)

(out) τe−env (Teff )

,

(3.20)

where τe−env (T ) is the temperature dependent electron-environment inelastic scattering rate, E(T ) is of the order of total the energy of electrons having the (out) temperature T in the volume involved and P V 2 /R(Teff ) is the (Joule) heat produced by the passing current. The I-V characteristics comes out as solutions to Eq. (3.20) with the specific relations between E and the applied voltage and current. The resulting I-V curves may come up as the so called S- or N -type of the I-V characteristics depending on the specific tempera(out) ture dependencies R(Teff ), see Ref. [35]. Making use of Eq. (3.5) we can estimate τe−env (T ). It is proportional to the coefficient standing in the scat(1) tering integral by f1 :

ρ 1 ∼ dωνω ν2 d2 W12 τe−env ωνω × δ(12 − ω)[(Nω + 1)(1 − f(2) ) + Nω f(2) ] 2 2 ) + (Nω + 1)f(2) ] . (3.21) + δ(12 + ω)[Nω (1 − f(2) 2 2 The equation (3.21) is derived for a single tunnel junctions. To describe the overheating effects in arrays of many junctions one needs an appropriate generalization of a single junction approach. An important example of

36


such a multi-junction system is an array of metallic granules in the insulating state. In granular metals the role of environment is taken by the electron(in) hole pairs. In this case τe−env is electron inelastic relaxation time, Teff = T is the bath temperature. The energy scale E can be roughly approximated by the energy of the quasiparticles in the bulk of the sample of the volume Ω as E(T ) ∼ π 2 νΩT 2 /6. Employing the standard theory of the tunneling conductivity in the granular materials [24] one then can arrive at the S-type I-V -characteristics for granular metals. Electron-hole environment in a disordered metal To gain an insight into the behavior of the electron-hole environment in mesoscopic tunnel junctions and reveal the role of Coulomb interactions, which are instrumental to formation of the environment properties, let us employ the Finkelstein’s theory of Fermi-liquid fluctuations in a disordered granular metal [49]. The theory treats the Coulomb potential as the contact interaction. This offers a pretty good description of disordered metals, where Coulomb potential is well screened to δ-function. Of course, in the close vicinity of the disorder-driven metal-insulator (or superconductor-insulator) transition where the real Coulomb fields come into play, the orthodox Finkelstein’s theory is not valid, and one needs the approach capable to accommodate the long-range Coulomb effects. We adopt the Keldysh representation which is most adequate for discussing dynamic effects. The action describing Fermi-liquid fluctuations acquires the form: → − → − 1 4 iΓρ iSF = − σN tr (∂ˇr Q)2 − ∂ˇt Q − {(ˆ ρ1 )tx (ˆ ρ2 )tx + 2iν tr[ φ τ σx φ ]}. 4 D 4ν tx (3.22) Here ρˆ1(2) is the operator of the Fermion fluctuations density in the classical (quantum) sector, 2πν φ1 2πν φ2 tr(σx Qt,t ) + tr(σx Qt,t ) + , ρˆ2 = − , ρˆ1 = − 1 + Fρ 2π 1 + Fρ 2π (3.23) → − φ are the conjugated fields, Γρ = Fρ /(1 + Fρ ), Fρ is the contact amplitude modeling the screened Coulomb interaction in the singlet channel. The Fourier transform of the typical retarded fluctuation propagator describing fluctuations has the form: DR =

Dq 2

1 , − iω(1 + Γρ )

(3.24)

where ω are the Matsubara frequencies. The diffusion poles of the propagators thus possess the structure, Dq 2 − iω, Dq2 − iω(1 + Γρ ). One sees that this


37

description applies and the spectrum of the excitations is stable only as long as the single granule mean level spacing remains the smallest energy parameter, i.e. δ < Dq 2 , ω, ωΓρ . This implies the development of the excitations spectrum instability and opening the energy gap at temperatures T < T∗ =

δ . Γρ

(3.25)

Formation of the energy gap in the spectrum of the environmental excitations is a phenomenon that tremendously influences the tunneling electronic transport. One sees straightforwardly that as the energy gap, T ∗ , in the spectrum of electron-hole excitations appears at T < T ∗ , the spectral probability ρ(ω) for the electron-environment interaction vanishes in the interval 0 < ω < T ∗ . Then Eq. (3.12) yields the suppression of the tunneling current to the practically zero magnitude. The physical significance and meaning of this result is merely that as the gap in the excitation spectrum opens, the environment ceases to efficiently mediate the energy relaxation from the tunneling carriers, impeding thus the tunneling current. Interestingly, the similar temperature that marked vanishing of the conductivity by the e-e interactions in the absence of coupling of electronic system to phonons in the model of disordered quantum wire was found in Refs. [50,51] by reducing the electron conductivity to the Anderson model on the Bethe lattice. This suppression of conductivity was interpreted as the Anderson localization in Fock space. We reiterate here that models based on the contact e-e interactions hold only as long as the long-range Coulomb effects are effectively screened. As a result its applicability, for example, to the description of physical phenomena in the vicinity of metal-insulator or superconductorinsulator transition needs special justification. In the next section we propose an approach capable to explicitly account for the long-range Coulomb interactions in the critical region of disorder-driven superconductor-insulator transition. Superinsulating behavior Consider charge transfer in a two-dimensional array of superconducting tunnel junctions (or, equivalently, Josephson junction network) in the insulating state, i.e. under the conditions that EJ < Ec , where EJ is the Josephson coupling energy, and Ec is the charging energy related to the capacitance C between the two adjacent granules (or the capacitance of a single Josephson junction). We focus on the limit C C0 , where C0 is the capacitance of a single junction to the ground. The electric properties of the array are quantified by the screening length λ = a C/C0 , where a is the size of the elemental unit of the Josephson junction network. At distances R < λ, the electric charges interact according the logarithmic law, the energy of the interaction being ∝ ln(R/λ), at larger distances Coulomb interaction

38


is exponentially screened. If the linear dimension of an array, L does not exceed λ, the charge plasma, comprising the environment in the superconducting array, experiences the charge binding-unbinding Berezinskii-KosterlitzThouless (BKT) transition [52–57] at T = TBKT Ec [56, 57]. In the low temperature BKT phase positive and negative charges are bound into the dipole pairs, whereas above the transition the charges form a free plasma. Binding positive and negative charges into pairs implies that at T Ec the energy gap Ec opens in the spectrum of unbound charges. As we have discussed in the preceding section, opening the energy gap in the environmental excitation spectrum results in vanishing ρ(ω) in the interval 0 < ω < Ec . Following further the same line of reasoning and using again Eq. (3.12), we arrive at the conclusion that at T < TBKT , the charge transfer in the array of the superconducting tunnel junctions is suppressed. Remembering now, that at low temperatures tunneling current in an array of superconducting junctions is governed by the cotunneling processes [23, 24] described in Sect. 3.3. Analyzing the contribution from higher orders into the cotunneling process, one finds that the current suppression holds in all orders. On a qualitative level the effects of the gap opening in the charge plasma excitation spectrum on the charge transfer in the superconducting tunneling array can be described as follows. Starting with Eq. (3.1) and using Eqs. (3.12) and (3.13), one can find I ∝ exp(−E/W ) ,

(3.26)

where E is the characteristic energy for the charge transfer between the granules. In the system with the linear size L < λ, E = Ec ln(L/a), where Ec = 2e2 /C for the Cooper pair, and Ec = e2 /2C for the quasiparticle. The quantity W is the energy scale associated with the rate of the energy exchange between the tunneling charges and the environment. While the rigorous derivation for W is not available at this point, the estimates suggest the interpolation formula W

Ec . exp(Ec /T ) − 1

(3.27)

The meaning of this formula is that the relevant energy scale characterizing the tunneling rate is the energy gap that enters with the weight equal to the Bose-kind filling factor describing the probability of exciting the unbound charges. Well above the charge BKT transition Eq. (3.27) gives W = T (this reflects the fact that the number of the unbound charges is determined by the equipartition theorem and is proportional to T /Ec ). This yields I ∝ exp[−Ec ln(L/a)/T ] , T TBKT .

(3.28)

One has to bear in mind that this formula holds only at temperatures not too high above the charge BKT transition, where Coulomb interactions are


39

not completely screened. At T TBKT , E = Ec ln(λ/a) and as the screening length becomes of the order of a, E Ec . In this temperature region the system exhibits the ‘bad metal’ behavior. Notably, Eq. (3.28) looks like a formula for the thermally activated current, however one has to remember that the physical mechanism behind the considered charge transfer is quantum mechanical tunneling process which can take place only if the mechanisms for the energy relaxation are switched on. At low temperatures, T TBKT Ec , the characteristic energy from Eq. (3.27) is W = Ec exp(−Ec /T ). The unbound charges that have to mediate the energy relaxation from the tunneling carriers are in an exponentially short supply, and one finds I ∝ exp[− ln(L/a) exp(Ec /T )] , T TBKT ,

(3.29)

reproducing the results of [38, 39] for the Cooper pair conductivity. The additional suppression of the electronic transport in Josephson junction arrays as compared to the activation regime was indeed experimentally observed in Refs. [58, 59]. The outlined picture of the tunneling current in the superconducting junction array applies to thin superconducting films close to superconductorinsulator transition (SIT) [16–18]. Indeed, in the close vicinity of the SIT, the disorder-induced spatial modulations of the superconducting gap give rise to the electronic phase separation in a form of the texture of weakly coupled superconducting islands [60] with the characteristic spatial scale of the order of several superconducting coherence lengths ξ. The consequences of this phase separation are two-fold. First, the fine balance between disorder and Coulomb forces near the SIT results, on one hand, in the effective reduction of the disorder strength, and, on the other hand, in the emergence of the strong non-screened Coulomb field in the spaces between the islands. This field can be argued to reconstruct the island texture into a nearly regular array of weakly coupled superconducting islands, i.e. into an array of mesoscopic superconducting junctions. Note that the regular lattice of the superconducting islands can be also induced by nonlocal elastic fields due to coupling between the film and the substrate [61]. Second, in the critical region near the SIT this island array is on the verge of the percolation-like transition between the superconductor and insulator (the cluster of the islands touching each other and traversing the sample means the emergence of global superconductivity). Since in such two-phase systems the dielectric constant diverges on approach the percolation transition [40, 62–66], the thin films in the critical region of the SIT develop a huge global dielectric permeability. This means that on the distances L < εd, where d is the film thickness, all the electric fields, emerging due to fluctuations in the local charge of the 2D environment, remain trapped within the film, and, therefore, the charge environment is nothing but the 2D neutral plasma, where charges interact with each other according to the logarithmic law. Therefore the 2D charge environment experiences

40


the binding-unbinding charge BKT transition at T = TBKT , and the energy gap opens in the environmental excitations spectrum at T ≤ TBKT , giving rise to suppression of both Cooper pair- and normal quasiparticle currents, since both tunneling processes – irrespectively to whether it is transfer of Cooper pairs or quasiparticles across the junction – can occur only if the energy exchange with an environment is possible. Thus opening the gap in an environmental spectrum due to long-range Coulomb effects and the resulting suppression of the tunneling current offers a microscopic mechanism behind the insulator-to-superinsulator transition and the conductivity in the superinsulating state [38, 39]. Note in this connection, that in the recent work [67] the dc conductivity of an array of Josephson junctions in the insulating state is discussed in terms of transport of Cooper pairs through the narrow band formed by the Josephson coupling and distorted by weak disorder. This model for the Cooper pair transfer is in a sense complimentary to that of the present work where we discuss the opposite limit of sequential tunneling between the granules with the essentially different tunneling levels. Variable range hopping conductivity The notion of the cascade two-stage relaxation is a key to resolving the controversy of the variable range hopping (VRH) conductivity. Many experimental studies of the hopping conductivity in doped semiconductors [41–43], in arrays of quantum dot [44], and in disordered superconducting films [16–18,40], revealed the temperature independent pre-exponential factor, indicating the non-phonon mechanism of relaxation. What more, it had the universal values of the integer multiples of e2 /h. According to the early ideas by Fleishman, Licciardello, and Anderson [68], the universal pre-exponential factor may evidence that the energy relaxation is due to electron-electron (e-e) rather than the electron-phonon interactions. On the other hand, according to Refs. [50,51] the e-e relaxation alone (i.e. in the absence of the coupling to phonons, which are in any case present in a real physical system) cannot ensure a finite conductivity below the certain localization temperature. Namely, in the absence of phonons the system of interacting electrons subject to disorder undergoes a localization transition (‘many-body localization’) in Fock space. The concept of the cascade energy relaxation developed in the present work resolves this controversy. The first stage is the fast energy exchange between the hopping electrons and the electron-hole environment (electrons do not see sparse phonons). Since this process involves only electron-electron interactions, the resulting pre-exponential factor does not depend on temperature. The secondstage process is the transfer of the energy from the electron-hole environment to the phonon thermostat. This process which is of course necessary to ensure the current, does not influence the pre-exponential factor.


41

3.5 Conclusions In conclusion, we have developed a theory of the far from the equilibrium tunneling transport in arrays of tunnel junctions in the limit 1/τe−env 1/τenv−bath. We have demonstrated that the energy relaxation ensuring the low-temperature tunneling current occurs as a cascade two-stage process: the tunneling charges lose their energy to an intermediate agent, environment, and the latter relaxes the energy further to the thermostat. We have derived the tunneling I-V characteristics and shown that they are highly sensitive to the structure of the spectrum of the excitations of the environmental modes. In particular, opening the energy gap in the excitation spectrum suppresses the tunneling current. As an important example we have considered a two-dimensional array of the superconducting junctions where the two-stage relaxation occurs via the two dimensional Coulomb plasma of positive and negative charges with logarithmic interaction, which experiences the charge binding-unbinding BKT transition at T = TBKT . We argued that the gap in the plasma excitation spectrum that emerges at T < TBKT gives rise to suppression of both, tunneling Cooper pair- and quasiparticle currents, thus offering the possible microscopic mechanism for the insulator-to-superinsulator transition and the low temperature transport in the superinsulating state. We considered applications of our general approach to several physical systems and low temperature transport phenomena, including Coulomb anomaly in 2D disordered metals, overheating effects, electron-hole environment in disordered metals, and the origin of temperature independent pre-exponential factor in hopping conductivity.

Acknowledgments We are grateful to R. Fazio, A. Shytov, A. Gurevich, I. Burmistrov and Ya. Rodionov for useful discussions. This work was supported by the U.S. Department of Energy Office of Science under the Contract No. DE-AC0206CH11357, by the Programs of the Russian Academy of Sciences, and by the Russian Foundation for Basic Research (Grant Nos. 09-02-01205 and 09-02-12206).

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4 ELECTRON TRANSPORT THROUGH MOLECULES IN THE KONDO REGIME: THE ROLE OF MOLECULAR VIBRATIONS J. Mravlje1 and A. Ramˇsak2,1 1 2

Joˇzef Stefan Institute, Ljubljana, Slovenia [email protected] Faculty of Mathematics and Physics, University of Ljubljana, Slovenia [email protected]

Abstract. We discuss the electronic transport through molecules in the Kondo regime. We concentrate here on the influence of molecular vibrations. Two types of vibrations are investigated: (i) the breathing internal molecular modes, where the coupling occurs between the molecular deformation and the charge density, and (ii) the oscillations of molecule between the contacts, where the displacement affects the tunneling. The system is described by models which are solved numerically using Sch¨ onhammer-Gunnarsson’s projection operators and Wilson’s numerical renormalization group methods. Case (i) is considered within the Anderson-Holstein model. Here the influence of the phonons is mainly to suppress the repulsion between the electrons at the molecular orbital. Case (ii) is described by a two-channel Anderson model with phonon-assisted hybridization. In both cases, the coupling to electrons softens the vibrational mode and in the strong coupling regime makes the displacement unstable to perturbations that break the symmetry of the confining potential. For instance, in case (ii) when the frequency of oscillations decreases below the magnitude of perturbation breaking the left-right symmetry, the molecule will be abruptly attracted to one of the electrodes. In this regime, the Kondo temperature increases but the conductance through the molecule is suppressed.

Key words: electronic transport, molecules, Kondo temperature, AndersonHolstein model.

4.1 Introduction The fast pace of the computer industry is mainly driven by the miniaturization of elements in microprocessors. The ultimate limit of the miniaturization is to control the current through individual molecules and it is remarkable that transistors based on single molecules bridging metallic electrodes have already been produced and their current-voltage characteristics have been J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 4,

45

46

J. Mravlje and A. Ramˇsak

measured [1–4]. Such molecular junctions are produced using mechanical breaking or electromigration techniques which currently do not allow for scaling up to larger circuits, but they already provide information on the electron transport on the nanoscale that could be essential to the circuitry of tomorrow [5]. Moreover, because the transmission through a molecule is sensitive to its immediate electro-chemical (and also magnetic) environment such devices could work as molecular sensors. For instance, the binding of guest species to a single host molecule bridging two electrodes has already been discerned in conductance measurements [6]. The studies of conductance could thus enable recognition of single molecules and thereby realize the ultimate limit of analytical chemistry. In certain regimes the molecular junctions exhibit the Kondo effect [1, 7–9]: the anomalous behavior of conductance due to the increased scattering rate driven by the residual spin (i.e., the quantum impurity) localized at the molecular orbital. The molecular transistors thus provide the nanoscopic realization of quantum impurity models and can be used thus also as a laboratory to investigate many-particle physics, for instance the quantum phase transitions [10]. The transport through molecules is affected by molecular vibrations (MV). The molecular internal vibrational modes and oscillations of molecules with respect to the electrodes explain the side-peaks observed in the non-linear conductance [1, 8, 9, 11]. In this article we are interested in the effects of coupling to the MV at low temperatures and at a small bias (i.e. in the Kondo regime) and their signature in the dependence of the conductance on the gate-voltage. We consider two different types of molecular vibrations. (i) In the case of breathing molecular modes, i.e., when the MV couple to the electron density, the electron-electron repulsion is effectively diminished and the electron effective mass is enhanced. (ii) When the molecule itself oscillates between the contacts, i.e., the MV modulate the tunneling, the effective repulsion is unmodified but the asymmetrical part of the modulation introduces the charge fluctuations in the odd conduction channel, which leads to the competition between odd and even channel that can result (albeit with some artificial finetuning of the models, as we discuss) in the ground state with the non-Fermi liquid 2 channel Kondo fixed point. Although the influence of the MV on the electrons differs profoundly in these two cases, the back-action of electrons to molecular vibrations is universal. The coupling to electrons tends to soften the molecular modes (diminish their effective frequencies). This softening is related to the increased charge-susceptibility in case (i) or increased susceptibility to breaking of inversion symmetry in case (ii). The result is a suppressed conductance with simultaneous increase of the Kondo temperature. This contribution provides an overview of our work on quantum impurity models coupled to phonons [12–15]. Because of lack of space we here develop only the main ideas and refer the reader to these articles and the references

4 Phonons and Kondo effect

47

therein. For background on the Anderson-Holstein model we specifically refer also to [16–18] and for the work on the oscillating molecules we refer also to [19, 20].

4.2 Coupling of vibration to charge: Anderson-Holstein model Consider a molecule with a breathing mode, trapped between two electrodes as depicted schematically on Fig. 4.1. Assuming that a single molecular orbital is relevant for the electron transport (experimentally this assumption is supported by wide inter-orbital energy spacings [8]), the system can be described by the Anderson-Holstein Hamiltonian, H=

kα

k nkα +

Vkα c†kασ dσ + h.c. + n + U n↑ n↓ + M (n − 1)x + Ωa† a kασ

(4.1) describing bands of noninteracting electrons in the left (α = L) and right (α = R) electrodes, with energies k , nkα = nkα↑ + nkα↓ , which are counted by nkασ = c†kασ ckασ . Likewise n = n↑ + n↓ with nσ = d†σ dσ counts the electrons at the molecular orbital with the single-electron energy denoted by ; ckσ , dσ are the electron annihilation and c†kσ , d†σ the electron creation operators. The tunneling matrix element between k-state in the electrode α and the molecular orbital is given by Vkα . The electrons in the electrodes are assumed noninteracting, the electron repulsion between electrons at the molecular orbital is U . The charge on the molecular orbital couples to the displacement of the phonon mode x = a + a† [a(†) is the phonon annihilation (creation) operator] via a Holstein coupling of strength M , while the frequency of the internal vibrational mode of isolated molecule is Ω. Assuming (for simplicity) that the system is inversion symmetric (meaning that the tunneling to the left is equal to the tunneling to the right electrode, VkL = VkR ) it is convenient to define operators in the electrodes which are even/odd upon inversion 1 ce,o = √ (cL ± cR ) . 2

Ω l+x

Fig. 4.1. Transport through a molecule with a breathing mode.

(4.2)

48


By rewriting the Hamiltonian √ in the new basis, likewise, the coupling to even channel is given by√Ve = (1/ 2)(VL +VR ) and the coupling to the odd channel vanishes Vo = (1/ 2)(VL −VR ) = 0. It is thus sufficient to retain only the even operators explicitly and describe the system by a single channel AndersonHolstein model. H= k nke + V fσ† dσ + h.c. + n + U n↑n↓ + M (n − 1)x + Ωa†a, (4.3) σ

k

where fσ is the linear combination of the conduction electrons to which the molecular orbital (i.e., the impurity) couples directly, fσ =

1/2 √ 2 Vke ckeσ / |Vke | = (1/ N ) ckeσ .

k

k

(4.4)

k

Analytical considerations A convenient starting point for the analysis of the model is to perform the unitary displaced oscillator transformation. One obtains: †

M

H = e Ω (a−a

)(n−1)

†

M

)(n−1) He− Ω (a−a =

fσ† dσ + h.c. + k nk + Ωa† a, (4.5) = eff n + Ueff n↑ n↓ + V σ

where

M

k

†

d = e− Ω (a−a ) d; Ueff = U − 2M 2 /Ω; eff = + M 2 /Ω.

The transformed Hamiltonian H is of the same form as H, but with a reduced repulsion Ueff . The coupling to the phonons is hidden in the transformed operator d . The transformed boson operators read, a = a −

M (n − 1). Ω

(4.6)

The displacement is shifted depending on the occupancy of the molecular orbital x = x − 2M (n − 1)/Ω.

(4.7)

There is an interesting large-frequency limit M/Ω → 0, but M 2 /Ω finite, where the coupling to phonons is entirely described in terms of the effective parameters of the model. The same result follows from considering the EP interaction perturbatively [16]. A pair of EP vertices and a phonon propagator can be formally substituted by a frequency-dependent point electron-electron interaction vertex, The influence of the phonon mode is then retained in the frequency dependence of the interaction


U (ω) = U + M 2 D0 (ω) = U −

2M 2 Ω , Ω2 − ω2

49

(4.8)

because D0 (ω) = 2Ω/(ω 2 − Ω 2 ). At low frequencies the interaction is screened due to the formation of bipolarons U (ω → 0) = Ueff , at high frequencies the bare interaction is recovered. If Ω is large, then the low-energy behavior is given entirely by the Anderson model with Ueff . Note that effective repulsion can become negative; that provides the motivation for studies in the U < 0 regime. Numerical results The results presented here are discussed in more detail in [12]. The results for repulsive U as a function of gate voltage + U/2 are presented in Fig. 4.2 with full lines. On the top panel the conductance, in the middle panel the average charge and on the bottom panel the fluctuations of charge are plotted. In the Kondo regime, where the average charge n ∼ 1, the conductance is enhanced towards the unitary limit G → G0 . The maximal conductance is given by the quantum of conductance G0 = 2e2 /h (h is the Planck’s constant, e the electron charge) and corresponds to the unitary transmission [21,22]. Actually, the average charge and conductance are related by the Friedel sum rule G = G0 sin2

π n. 2

(4.9)

The Friedel sum rule in this form holds for single-impurity parity symmetric models. The generalization to non-symmetric case is possible [13]. The fingerprint of the Kondo physics is also the appearance of the local 1/2 moment Mloc = (n↑ − n↓ )2 , presented in the inset of Fig. 4.2(b) and 2 the suppression of the charge fluctuations Δn2 = (n − n) , Fig. 4.2(c). In the inset the corresponding charge susceptibility, χc = −∂n/∂ is given. In agreement with the fluctuation-dissipation theorem, the charge fluctuations are similar to the charge susceptibility, Δn2 ∼ (πΓ/4)χc . Strictly, n2 is given with the integral of the imaginary part of the dynamic charge susceptibility, χc (ω), therefore the relation to static χc is only qualitative. In Fig. 4.2(a) the conductance for various U < 0 is presented with dashed lines. The first observation is a narrowing of the conductance curve and the corresponding enhanced charge fluctuations (Fig. 4.2(c)), consistent with a sharp transition in the local occupation and a suppression of the local moment, Fig. 4.2(b). For increasing |U |, the charge susceptibility diverges and overshoots the charge fluctuations. For general values of parameters, i.e., for moderate Ω, the problem with EP coupling cannot be mapped onto the Anderson model. However, the behavior is still to the largest extent determined by Ueff , and similar to the above discussed results, Fig. 4.2. For example, the result of the Schrieffer-Wolff transformation is now an anisotropic Kondo model [23], where the anisotropy stems

50


a G / G0

0.5

b

0

1

1

c

0

U> 0 U= 0 U< 0

U = –6 Γ

Mloc

0.5 U = 10Γ

0

0

5

(πΓ / 4)χc

Δn2

0.4 0.4

0

0

–5

0 5 (ε+U / 2) / Γ

0

5 10

Fig. 4.2. (a) Conductance for the Anderson model with −6Γ ≤ U ≤ 10Γ in increments of 2Γ (full lines for U > 0, dashed lines for U < 0 and a thicker full line for U = 0). (b) Local occupancy n and local moment Mloc (inset). (c) Charge fluctua2 tions Δn2 = 2n − n2 − Mloc . Inset: renormalized charge susceptibility (πΓ/4)χc .

from the fact that the phonon displacement couples only to the z-component of the isospin Tz . In addition to the renormalization of U now also the hybridization is renormalized as shown on Fig. 4.3, where the results for bare U = 5Γ case are compared to the U = 10Γ, Ueff = 5Γ case for Ω = 10Γ , Ω = Γ and Ω = Γ/100. The smaller the Ω, the sharper the jump in the conductance corresponding † M to an enhanced effective mass due to the larger effect of e− Ω (a−a ) . The results for very soft phonons Ω = Γ/100 can also be understood in an alternative manner. In the Kondo regime the conductance is close to the bare Anderson model result with U = 10Γ . In the mixed valence regime the curve is much steeper, due to a strong renormalization [24] of the hopping. In the empty-orbital regime the conductance approaches the result obtained with a doubly reduced electron-electron interaction Ueff = U −

4M 2 , Ω

(4.10)

which can be understood as follows. First the oscillator displacement is shifted, x→x ˜ + 2λ thus the Hamiltonian is transformed into ˜ = ( + 2λM ) n + x˜ [M (n − 1) + Ωλ] + Ω˜ ˜ + ··· , H a† a

(4.11)


G / G0

a

Δn2

b

Δx2

c

1

U=0 U=5Γ U=10Γ

Ω = Γ / 100

0.5

Ω=Γ

51

Ω = 10 Γ

0 0.4

0.2

0 1.6

3

Ω=Γ/100

2

1.4

Ω=Γ

1

0

2

4

6

8

1.2 Ω=10Γ

1

0

2

4 (ε+U/2)/Γ

6

Fig. 4.3. A fixed U = 10Γ and Ueff = 5Γ with for Ω = Γ , Ω2 = 10Γ and Ω2 = 10Γ . Also plotted are the results for a bare Anderson model with U = 10Γ , U = 5Γ and U = 0 (dotted, short-dashed and dashed-dotted, respectively). (a) Conductance, (b) occupation fluctuations and (c) deformation fluctuations. In the inset, the deformation fluctuations for a softer mode are shown.

where λ = −M (n − 1) /Ω, with vanishing transformed displacement. This Hamiltonian can be solved with trial wave functions with no phonons. The renormalized local energies are then + 2M 2 /Ω, , and − 2M 2 /Ω for n = 0, 1, 2, respectively. The shifts of where n = 0, 2 in turn correspond to ˜ = U − 4M 2 /Ω and to U ˜ = U for n = 1. reduced U Softening of the phonon mode Phase transitions are ubiquitously related to the instability of the symmetry restoring modes. In ferroelectrics, for instance, the para→ferro phase transition will occur when the unstable phonon is frozen-in to one of the equivalent configurations [25, 26]. As the temperature is tuned towards the transition, T → Tc the related static temperature-dependent susceptibility will diverge, χ(T ) = C/(T − Tc ). According to the Kramers-Kronig relation

2 ∞ χ (ω , T ) dω (4.12) χ(T ) = χ (0, T ) = π 0 ω

52


this will occur when the dissipative imaginary part of the susceptibility χ (ω) has a peak at low frequencies. As poles of χ (ω) indicate the normal modes of the system, the frequency of the normal mode ω0 should vanish at the transition. There is a remarkable analogy to this behavior in the strong-coupling regime of the Anderson-Holstein model (although there is no phase transition; we are dealing with a single degree of freedom here). As shown in the inset of Fig. 4.2(c), the charge susceptibility ∂n/∂ diverges for large M . The charge-charge correlation function should also increase there according to the fluctuation-dissipation theorem. Due to the Holstein coupling of charge to the displacement it seems plausible that the phonon correlation function should also be influenced. Indeed, in the Anderson-Holstein model the charge-charge and the displacement-displacement correlation functions are directly related D(ω) = D0 (ω) + M 2 D0 (ω) (n − 1), (n − 1) ω D0 (ω),

(4.13)

as can easily be proved by considering the equation of motion. The phonon propagator must thus develop a low frequency component. The phonon mode is softened as M grows large. On Fig. 4.4 the NRG results for imaginary part of the phonon propagator

∞ 1 1 A(ω) = − Im x, x ω = − Im (−i)[x(t), x(0)]eiωt dt (4.14) π π 0 are plotted. The oscillations which occur at frequency Ω for the uncoupled oscillator become with increasing electron phonon coupling softer and their characteristic frequency diminishes. The spectral functions are broadened on the logarithmic scale as in [15]. The softening can also be related to the change in the shape of the effective potential the oscillator experiences due to the coupling to the electrons. 40

M=0 M=0.1 M=0.15

Ueff < 0

A(ω)

30 Ueff > 0

20 10 0

0

0.05

0.1 ω=Ω

0.15

0.2 ω

Fig. 4.4. The displacement-displacement spectral function for Ω/Γ = 5, U/Γ = 15, M/Γ = 0, 5, 7.5.


53

Ueff > 0 effective potential

Ueff ~ 0

–2

Ueff < 0

–1

0 x

1

2

Fig. 4.5. Effective potential for U/Γ = 7.5, Ω/Γ = 2.5, and M/Ω = 1.15, 1.25, 1.35.

The effective potential can be extracted using the SG method as explained in [15]. The results are shown in Fig. 4.5. We see that when the sign of the effective repulsion is changed, the oscillator potential will evolve to a double well form. The displacements of large magnitude (corresponding to displaced oscillator transformation for states of double and zero occupancy) will be preferred. The low frequency component of the propagator corresponds to slow fluctuations of the oscillator between the degenerate minima of the effective potential, the high-frequency component corresponds to fast oscillations within the wells.

4.3 Oscillations with respect to the leads We now turn to the case where the molecule oscillates between the electrodes. We model the system with the Hamiltonian, Eq. (4.1) for M = 0, but with phonon-assisted hopping induced by the displacement dependent tunneling matrix elements Vkα → Vkα (x), as schematically presented on Fig. 4.6. The full Hamiltonian thus reads, H= k nkα + Vkα (x)c†kασ dσ + h.c. + n + U n↑ n↓ + Ωa† a. (4.15) kα

kσ

From now on, we are here interested in the particle-hole symmetric point = −U/2, where the molecule is on average singly occupied. We use U = 15Γ . Other details of calculation can be found in [15]. Again, it is practical to define even and odd combinations of states in the electrodes, and in this basis the tunneling part of the Hamiltonian reads, Ve (x) ve + Vo (x) vo , where Ve,o (x) =

VL (x) ± VR (x) √ , 2

(4.16)

(4.17)

54


VL(x)

VR(x)

x

Fig. 4.6. (Color online) Schematic plot of the model device.

modulate the tunneling to even and odd channels. Hybridization operators † are vα = fσα dσ + h.c. for α = e, o, respectively, where fe is the even combination of electrode orbitals and fo is the odd combination of electrode orbitals. Note that |Ve (x)| > |Vo (x)|,

(4.18)

if VL,R (x) are both positive or both negative for all x. Two-channel Kondo model The odd channel is coupled to the molecule only due to the asymmetric modulation of tunneling. For example, in the linear approximation VL,R (x) = V (1 ∓ gx) the even channel is coupled to the molecule directly and the odd channel is coupled to the molecule via a term proportional to gx. Unlike in the Anderson-Holstein model, the attempts to eliminate the coupling to phonons using a variant of Lang-Firsov transformation fail. Note that the coupling to phonons as considered in this section does not affect the effective repulsion but it affects the hybridization and therefore the Kondo temperature can be enhanced [15]. As a consequence of the coupling the molecular orbital to two channels the low-energy behavior is that of the two-channel Kondo (2CK) model [19, 27]. The screening of the spin occurs in the channel with the larger coupling constant. If the couplings match, an overscreened, i.e., a genuine 2CK problem with a non-Fermi liquid behaviour results. For a linearized model to be introduced below such a fixed point has indeed been found at an isolated value of the electron-phonon coupling with simulations based on numerical renormalization group [14, 15, 28]. Overlap integrals The calculations are performed using several functional forms of Vα (x) depicted in Fig. 4.7. In a realistic experimental situation the tunneling between the molecule and the tip of an electrode will be saturated at small distances and it will progressively decrease with increasing distance of the molecule from the electrode. The precise functional dependence of overlap integrals will in general depend on details of the molecule and the tips of the electrodes, but the overall behavior should be as shown in Fig. 4.7(a) with dotted line.


overlap integrals

4

ex 1+x

3

e–x

VR

2

1–x

1

e x/ cosh x

VL

0 –1

55

e–x / cosh x

0

0.5

1

1.5

x

Fig. 4.7. (Color online) Various forms of the tunneling-modulation. The unphysical regime of LM where the tunneling starts to increase with increasing distance to the electrode is indicated by dashing.

Linearized modulation The simplest form of overlap integrals is obtained by the expansion to lowest order in displacement resulting in linear modulation (LM) VL,R (x) = V [1 ∓ (gx + ζ)] .

(4.19)

The tunneling matrix element, constant V for g = 0, is linearly modulated by displacement for g > 0. We assume the system is almost inversion symmetric. A small ζ ≥ 0 is the magnitude of the symmetry breaking perturbation. In the symmetrized basis the overlap integrals take on the following form √ √ Ve = 2V, Vo = 2V (gx + ζ). (4.20) Note that Eq. (4.20) does not satisfy the requirement Eq. (4.18) for gx > 1−ζ, because the overlap to the left electrode becomes negative and its absolute value starts to increase with increasing x (dashed region in Fig. 4.7). Regularized modulation A more realistic approximation to overlap integrals is the exponential dependence on displacement but it breaks down at small distance to the electrodes as discussed exhaustively in [15]. The modulation should therefore at large displacements be regularized and for the rest of this paper we use VL,R (x) = V [exp(∓gx)/ cosh(gx) ∓ ζ] ,

(4.21)

or in the symmetrized basis √ √ Ve = 2V, Vo = 2V [tanh(gx) + ζ] .

(4.22)

The inequality Eq. (4.18) is satisfied (the 2CK fixed point is thus inaccessible) and the normalization with the cosh function ensures that the hybridization saturates at small distances to the electrodes.

56


effective potential

0.04 g=0 0.2 0.3 0.4

0.03 0.02 0.01 0

–2

0 x

2

Fig. 4.8. Semi-classical estimate of effective oscillator potential for tanh modulation. Parameters Ω = 0.01, Γ = 0.02 are in units of D (half-width of the band).

Effective potential At U = 0 and replacing operators a, x by real valued quantities, the model Eq. 4.15 is solvable exactly and the energy of the ground state as a function of x provides the estimate of the effective oscillator potential. This simple estimate agrees qualitatively with the results of more elaborate methods [15]. We plot the results on Fig. 4.8. Initially harmonic potential softens with increasing g and at a certain point a double well potential develops. The softening thus occurs similarly as in the case of Anderson-Holstein model but here it is related to the dynamical breaking of inversion symmetry [19]. Due to the softening, the instability towards perturbations breaking the symmetry (degeneracy between the two minima of the double-well potential) can be expected. On the mean field level [13], the instability is indeed seen as a tendency towards an asymmetric ground state with large average x in systems with inversion symmetry.

4.4 Numerical results The development of the double well potential induces fluctuations of displacement and its influence can also be seen in the NRG results of static quantities shown on Fig. 4.9 for U = 0.3, Γ = 0.02, Ω = 0.01 In these results an inversion symmetry breaking perturbation of strength ζ = 0.01 is included. The average displacement presented on Fig. 4.9(a) increases as the electron-phonon coupling is increased. The fluctuations of displacement initially increase then they diminish, as the oscillator gets trapped in the lower of the two-well potential (also this behavior is discussed in more detail in [15]). At large electron-phonon coupling also the average displacement starts to diminish. This happens because for tanh-form of the hybridization for large g the hybridization is maximal already for small displacements and therefore the system can minimize the elastic energy without cost in the kinetic energy.

4 Phonons and Kondo effect 2

0.8

a

b

0.6

x Δx 0

hopL hopR

0.4

1

0

0.5

57

Δ n2

0.2 0

1

0

1

0.5 g

g

Fig. 4.9. (Color online) (a) Displacement and displacement fluctuations. (b) Average hopping to left and right; fluctuations of charge. 1000

A(ω)

g=0 g=0.25 500

g=0.5 g=0.75 g=1

0

0

0.01

ω

0.02

0.03

Fig. 4.10. (Color online) Displacement spectral functions for parameters as in Fig. 4.9.

On Fig. 4.9(b) also the electronic expectation values are shown. At large g, the molecule is near the right lead as signified by increased hopping to the right. The total hybridization and, correspondingly, the charge fluctuations are also increased there. The Kondo temperature is increased, but the conductance is suppressed due to the asymmetric configuration [15]. The softening of the potential is seen also in the displacement spectral function shown on Fig. 4.10. Similar to the behavior discussed in the previous chapter, the frequency of vibration diminishes with increasing g, because the confining potential is softened. At large g the molecule is trapped to the lower of the two wells, the oscillations between the two-wells become unfavourable and the spectral weight is again transferred to high frequencies corresponding to oscillations within the lower of the two wells.

4.5 Conclusion We invoked Anderson model coupled to phonons to describe the transport through the molecule coupled to molecular vibrations. While the influence the MV exhibit on electron depends on the details of the coupling,

58


the coupling to electrons tends to soften the MV. In the strong coupling regime of the Anderson-Holstein model (Ueff < 0), a perturbation of the orbital energy drives the system from the particle-hole symmetric point characterized with zero average displacement of the oscillator. Likewise, in the Anderson model with asymmetrically modulated hybridization, a perturbation of the left-right symmetry results in a state with large average displacement (the molecule is attracted to one of the electrodes). Discussion In measurements of conductance through molecular junctions the side-peaks pertaining to the excitation/annihilation of vibrational quanta are clearly discerned at a finite bias. The influence of phonons in the equilibrium at a small bias is less investigated, because it does not appear to affect the measured conductance significantly. Why is this so? The answer is that systems with electron-phonon (EP) coupling to internal molecular modes can usually be reformulated in a manner that refers to the EP coupling only by a redefinition (renormalization) of the original (bare) parameters, which themselves are not, unfortunately, known to start with. For instance, such a parameter is the repulsion, which is effectively diminished due to the influence of the EP coupling. But because the repulsion is diminished already for a decoupled molecule any attempt to discern the effects of the Holstein phonon by comparing to the data for isolated molecules will likely prove in vain. If one was interested in discriminating the effects of the coupling to phonons nevertheless, a convenient quantity to look at would be the frequency dependence of the local charge susceptibility. In the regime of reduced repulsion due to the EP coupling, the charge would be susceptible to the oscillations of gate voltage only below the phonon frequency. In order to investigate this experimentally, one would need to be able to measure the time dependence of molecular charge. While this currently seems a formidable task, we note that in quantum dot (QDs) the time-resolved measurements of charge using quantum-point contacts have already been demonstrated, e.g. in [29]. In near future, there is more hope to discern the effects of the EP coupling to the contacts. However, even qualitative effects – such as the breaking of the particle-hole (PH) symmetry, which occurs when the breathing modes couple to the electron charge and modulate the tunneling at the same time – can be dominated by non-perfectly symmetric contacts. But still, measuring the linear conductance to a better precision and comparing the data for rigid molecules to the data for softer molecules could unravel the influence of the breathing oscillations and the type of their coupling to the electron transport in the equilibrium. The case of molecules oscillating between the electrodes is a bit different. One particular effect of the EP coupling is obvious. Imagine a break junction with a single molecule bridging the two contacts. In principle, one can


59

d

Fig. 4.11. Scheme of the proposed device.

slowly increase the tension between the contacts, until they separate. Any tiny perturbation of the parity will choose a contact to which the bridging molecule will be attracted. This is precisely the physics analyzed here: the tension pulls the contacts apart and the separation sets the modulation of tunneling by induced displacement of the molecule. With increasing tension also the modulation increases. The potential confining the molecule to the center is softened, and instantly, the molecule is attracted to one of the contacts. One might argue that the idea is rather to get as close to the breaking point to observe the onset of the non-Fermi liquid (NFL) fixed point. But because NFL fixed point corresponds to the case when the molecule fluctuates between far left and far right, the conductance is zero as is also if the molecule is far left or far right. It is worth calculating the temperature dependence of conductance and hope for some anomalies due to the NFL formation energy scale (for a very recent work in this direction see [28]), but we anticipate that it will be difficult to distinguish the results from these corresponding to the displacement of the molecule towards one of the contacts only. On the other hand, one could observe the effects of the phonon-mode softening directly in nanoelectromechanical suspended cantilevers. Consider the setup, depicted in Fig. 4.11. The cantilever (blue) oscillates between the auxiliary contacts (gold). The frequency of oscillations can be detected independently, e.g. by a quantumpoint contact shifted perpendicularly with respect to the plane defined by the cantilever and the contacts [30]. We predict that the frequency of oscillations will decrease when the distance between the contacts d is diminished provided that the device will operate in the regime of coherent tunneling. It would be interesting also to investigate the softening by suppressing it with the magnetic field. For magnetic fields above the Kondo temperature the spin at the orbital is frozen. According to the Pauli principle the charge fluctuations and the transport of electrons are blocked. Simultaneously, also the related kinetic energy gain vanishes and the softening is suppressed [31].

References 1. J. Park, A.N. Pasupathy, J.I. Goldsmith, C. Chang, Y. Yaish, J.R. Petta, M. Rinkoski, J.P. Sethna, H.D. Abrunas, P.L. McEuen, D.C. Ralph, Nature 417, 722 (2002) 2. A. Nitzan, M.A. Ratner, Science 300, 1384 (2003)

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3. N.J. Tao, Nature Nanotech. 1, 173 (2006) 4. M. Galperin, M.A. Ratner, A. Nitzan, J. Phys.: Condens. Matter. 19, 103201 (2007) 5. D. Natelson, Phys. World 22, 26 (2009) 6. X. Xiao, B. Xu, N. Tao, Angew. Chem. Int. Ed. 43, 6148 (2004) 7. A.C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, 1993) 8. L.H. Yu, D. Natelson, Nano Lett. 4, 79 (2004) 9. A.N. Pasupathy, J. Park, C. Chang, A.V. Soldatov, S. Lebedkin, R.C. Bialczak, J.E. Grose, L.A.K. Donev, J.P. Sethna, D.C. Ralph, P.L. McEuen, Nano Lett. 5, 203 (2005) 10. N. Roch, S. Florens, V. Bouchiat, W. Werndorfer, F. Balestro, Nature 453, 633 (2008) 11. N.B. Zhitenev, H. Meng, Z. Bao, Phys. Rev. Lett 88, 226801 (2002) 12. J. Mravlje, A. Ramˇsak, T. Rejec, Phys. Rev. B 72, 121403(R) (2005) 13. J. Mravlje, A. Ramˇsak, T. Rejec, Phys. Rev. B 74(20), 205320 (2006) ˇ 14. J. Mravlje, A. Ramˇsak, R. Zitko, Physica B 403, 1484 (2008) 15. J. Mravlje, A. Ramˇsak, Phys. Rev. B 78, 235416 (2008) 16. A.C. Hewson, D. Meyer, J. Phys: Condens. Matter 14(3), 427 (2002) 17. P.S. Cornaglia, H. Ness, D.R. Grempel, Phys. Rev. Lett. 93(14), 147201 (2004) 18. P.S. Cornaglia, D.R. Grempel, H. Ness, Phys. Rev. B 71(7), 075320 (2005) 19. C.A. Balseiro, P.S. Cornaglia, D.R. Grempel, Phys. Rev. B 74(23), 235409 (2006) 20. P. Lucignano, G.E. Santoro, M. Fabrizio, E. Tosatti, Phys. Rev. B 78, 155418 (2008) 21. S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, 1995) 22. H. van Houten, C.W.J. Beenakker, Phys. Today 49, 22 (1996) 23. T.A. Costi, Phys. Rev. Lett. 80(5), 1038 (1998) 24. K. Schönhammer, O. Gunnarsson, Phys. Rev. B 30(6), 3141 (1984) ˇ s, Soft Modes in Ferroelectrics and Antiferroelectrics 25. R. Blinc, B. Zekˇ (North-Holland, Elsevier, 1974) 26. P.W. Anderson, Basic Notions of Condensed Matter Physics (The Benjamin/Cummings Publishing Company, Inc., 1984) 27. P. Nozières, A. Blandin, J. Phys. (France) 41, 193 (1980) 28. L.G.G.V.D. da Silva, E. Dagotto, Phys. Rev. B 79, 155302 (2009) 29. J.R. Petta, A.C. Johnson, J.M. Taylor, E.A. Laird, A. Yacoby, M.D. Lukin, C.M. Marcus, M.P. Hanson, A.C. Gossard, Science 309(5744), 2180 (2005) 30. M. Poggio, M.P. Jura, C.L. Degen, M.A. Topinka, H.J. Mamin, D. GoldhabedGordon, D. Rugar, Nature Phys. 4 (2008) 31. M. Fabrizio and E. Tossatti, internal communication (2009)

5 WAVE DELOCALIZATION IN NONLINEAR DISORDERED MEDIA S. Flach Max-Planck-Institut f¨ ur Physik komplexer Systeme, N¨ othnitzer Str. 38, 01187 Dresden, Germany [email protected] Abstract. We analyze mechanisms and regimes of wave packet spreading in nonlinear disordered media. We predict that wave packets can spread in two regimes of strong and weak chaos. We discuss resonance probabilities, nonlinear diffusion equations, and predict a dynamical crossover from strong to weak chaos. The crossover is controlled by the ratio of nonlinear frequency shifts and the average eigenvalue spacing of eigenstates of the linear equations within one localization volume. We consider generalized models in higher lattice dimensions and obtain critical values for the nonlinearity power, the dimension, and norm density, which influence possible dynamical outcomes in a qualitative way.

Key words: nonlinear media, chaos, dynamical crossover, diffusion.

5.1 Introduction In this paper we will discuss the mechanisms of wave packet spreading in nonlinear disordered systems. More specifically, we will consider cases when (i) the corresponding linear wave equations yield Anderson localization, (ii) the localization length is bounded from above by a finite value, (iii) the nonlinearity is compact in real space and therefore does not induce long range interactions between eigenstates of the linear equations. There are several reasons to analyze such situations. First, wave propagation in spatially disordered media has been of practical interest since the early times of studies of waves. In particular, it became of much practical interest for the conductance properties of electrons in semiconductor devices more than half a century ago. It was probably these issues which motivated P. W. Anderson to perform his groundbreaking studies on what is now called Anderson localization [1]. With evolving technology, wave propagation became of importance also in photonic and acoustic devices [2,3]. Finally, recent advances in the control over ultracold atoms in optical potentials made it possible to observe Anderson localization there as well [4]. Peter H¨ anggi and J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 5,

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collaborators studied properties of wave propagation in disordered media by phase space visualization of the underlying dynamical system in high dimensional phase spaces [5]. Second, in many if not all cases wave-wave interactions are of importance, or can even be controlled experimentally. Screening effects can reduce the long range character of these interactions considerably for electrons. Short range interactions also hold for s-wave scattering of atoms. When many quantum particles interact, mean field approximations often lead to effective nonlinear wave equations. As a result, nonlinear wave equations in disordered media become of practical importance. Third, there is fundamental interest in understanding, how Anderson localization is modified for nonlinear wave equations. All of the above motivates the choice of corresponding linear wave equations with finite upper bounds on the localization length. Then, the linear equations admit no transport. Analyzing transport properties of nonlinear disordered wave equations allows to observe and characterize the influence of wave-wave interactions on Anderson localization in a straightforward way. A number of studies was recently devoted to the above subject [6–14]. In the present work we will present a detailed analysis of the chaotic dynamics which is at the heart of the observed destruction of Anderson localization. In particular, we will show that an optional intermediate strong chaos regime of subdiffusive spreading is followed by an even slower subdiffusive spreading process in the regime of weak chaos.

5.2 Wave equations We will use the Hamiltonian of the disordered discrete nonlinear Schr¨ odinger equation (DNLS) HD =

l

l |ψl |2 +

β ∗ ψl ) |ψl |4 − (ψl+1 ψl∗ + ψl+1 2

(5.1)

with complex variables ψl , lattice site indices l and nonlinearity strength The β ≥W0. W random on-site energies l are chosen uniformly from the interval − 2 , 2 , with W denoting the disorder strength. The equations of motion are generated by ψ˙ l = ∂HD /∂(iψl ): iψ˙ l = l ψl + β|ψl |2 ψl − ψl+1 − ψl−1 .

(5.2)

Equations (5.2) conserve the energy (5.1) and the norm S = l |ψl |2 . We note that varying the norm of an initial wave packet is strictly equivalent to varying β. Equations (5.1) and (5.2) are derived e. g. when describing two-body interactions in ultracold atomic gases on an optical lattice within a mean field approximation [15], but also when describing the propagation of light through networks of coupled optical waveguides in Kerr media [16].

5 Wave delocalization in nonlinear disordered media

63

Alternatively we also refer to results for the Hamiltonian of the quartic Klein-Gordon lattice (KG) HK =

p2 l

l

2

+

1 ˜l 2 1 4 u l + ul + (ul+1 − ul )2 , 2 4 2W

(5.3)

where ul and pl are respectively the generalized coordinates and momenta, and ˜l are chosen uniformly from the interval 12 , 32 . The equations of motion are u ¨l = − ∂HK /∂ul and yield u ¨l = −˜ l ul − u3l +

1 (ul+1 + ul−1 − 2ul). W

(5.4)

Equations (5.4) conserve the energy (5.3). They serve e.g. as simple models for the dissipationless dynamics of anharmonic optical lattice vibrations in molecular crystals [17]. The energy of an initial state E ≥ 0 serves as a control parameter of nonlinearity similar to β for the DNLS case. For small amplitudes the equations of motion of the KG chain can be approximately mapped onto a DNLS model [18]. For the KG model with given parameters W and E, the corresponding DNLS model (5.1) with norm S = 1, has a nonlinearity parameter β ≈ 3W E. The norm density of the DNLS model corresponds to the normalized energy density of the KG model. The theoretical considerations will be performed within the DNLS framework. It is straightforward to adapt them to the KG case.

5.3 Anderson localization For β = 0 with ψl = Al exp(−iλt) Eq. (5.1) is reduced to the linear eigenvalue problem λAl = l Al − Al−1 − Al+1 . (5.5) The normalized eigenvectors Aν,l ( l A2ν,l = 1) are the NMs, and the eigenvalues λν are the frequencies of the NMs. The width of the eigenfrequency spectrum λν of Eq. (5.5) is Δ = W + 4 with λν ∈ −2 − W ,2 + W 2 2 . The asymptotic spatial decay of an eigenvector is given by Aν,l ∼ e−l/ξ(λν ) 2 where ξ(λν ) is the localization length and ξ(λν ) ≈ 24(4 − λ2ν )/W 4for weak disorder W ≤ 4 [1, 19]. The NM participation number pν = 1/ l Aν,l is one possible way to quantize the spatial extend (localization volume) of the NM. The localization volume V is on average of the order of 3ξ(0) for weak disorder, and tends to V = 1 in the limit of strong disorder. The average spacing d of eigenvalues of NMs within the range of a localization volume is therefore of the order of d ≈ Δ/V , which becomes d ≈ ΔW 2 /300 for weak disorder. The two scales d ≤ Δ are expected to determine the packet evolution details in the presence of nonlinearity.

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Due to the localized character of the NMs, any localized wave packet with size L which is launched into the system for β = 0, will stay localized for all times. If L V , then the wave packet will expand into the localization volume. This expansion will take a time of the order of τlin = 2π/d. If instead L ≥ V , no substantial expansion will be observed in real space. We remind that Anderson localization is relying on the phase coherence of waves. Wave packets which are trapped due to Anderson localization correspond to trajectories in phase space evolving on tori, i.e. quasiperiodically in time.

5.4 Adding nonlinearity The equations of motion of (5.2) in normal mode space read iφ˙ ν = λν φν + β Iν,ν1 ,ν2 ,ν3 φ∗ν1 φν2 φν3

(5.6)

ν1 ,ν2 ,ν3

with the overlap integral Iν,ν1 ,ν2 ,ν3 =

Aν,l Aν1 ,l Aν2 ,l Aν3 ,l .

(5.7)

l

The variables φν determine the complex time-dependent amplitudes of the NMs. The frequency shift of a single site oscillator induced by the nonlinearity is δl = β|ψl |2 . If instead a single mode is excited, its frequency shift can be estimated by δν = β|φν |2 /pν . As it follows from (5.6), nonlinearity induces an interaction between NMs. Since all NMs are exponentially localized in space, each normal mode is effectively coupled to a finite number of neighbouring NMs, i.e. the interaction range is finite. However the strength of the coupling is proportional to the norm density n = |φ|2 . Let us assume that a wave packet spreads. In the course of spreading its norm density will become smaller. Therefore the effective coupling strength between NMs decreases as well. At the same time the number of excited NMs grows. One possible outcome would be: (i) that after some time the coupling will be weak enough to be neglected. If neglected, the nonlinear terms are removed, the problem is reduced to the linear wave equation, and we obtain again Anderson localization. That implies that the trajectory happens to be on a quasiperiodic torus. Then it has to be on that torus from the beginning. Another possibility is: (ii) that spreading continues for all times. That would imply that the trajectory evolves not on a quasiperiodic torus, but in some chaotic part of phase space. A third possibility is: (iii) that the trajectory was initially strongly chaotic, but manages in the course of spreading to get trapped between denser and denser torus structures in phase space after some spreading, leading again to localization as an asymptotic outcome.


65

Consider a wave packet with size L and norm density n. Replace it by a finite system of size L and norm density n. Such a finite system will be in general nonintegrable. Therefore the only possibility to generically obtain a quasiperiodic evolution is to be in the regime where the KAM theorem holds. Then there is a finite fraction of the available phase space volume which is filled with KAM tori. For a given L it is expected that there is a critical density nKAM (L) below which the KAM regime will hold. We do not know this L-dependence. Computational studies may not be very conclusive here, since it is hard to distinguish a regime of very weak chaos from a strict quasiperiodic one on finite time scales. The above first possible outcome (i) (localization) will be realized if the packet is launched in a KAM regime. Whether that is possible at all for an infinite system is an open issue. The second outcome (ii) (spreading) implies that we start in a chaotic regime and remain there. Since the packet density is reduced and is proportional to its inverse size L at later times, this option implies that the critical density nKAM (L) decays faster than 1/L, possibly faster than any power of 1/L. The third possibility (iii) (asymptotic localization) should be observable by some substantial slowing down of the spreading process. Measuring properties of wave packets We order theNMs in space by increasing value of the center-of-norm coordinate Xν = l lA2ν,l . We analyze normalized distributions nν ≥ 0 using the second moment m2 = ν (ν − ν¯)2 nν , which quantifiesthe wave packet’s de2 gree of spreading and the participation number P = 1/ ν n ν , which measures the number of the strongest excited sitesin nν . Here ν¯ = ν νnν . We follow norm density distributions nν ≡ |φν |2 / μ |φμ |2 . The second moment m2 is sensitive to the distance of the tails of a distribution from the center, while the participation number P is a measure of the inhomogeneity of the distribution, being insensitive to any spatial correlations. Thus, P and m2 can be used to quantify the sparseness of a wave packet through the compactness index ζ=

P2 . m2

(5.8)

A thermalized wave packet yields ζ = 3. Distributions with larger gaps between equally excited isolated sites attain a compactness index ζ < 3. Expected regimes of spreading Previous studies suggested the existence of various dynamical regimes of spreading of wave packets [7,10,11]. Some of these definitions were contradictory. Below we will resolve this.

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Consider a wave packet at t = 0 which has norm density n and size L. If βn ≥ Δ, a substantial part of the wave packet will be selftrapped [9,11]. This is due to the above discussed nonlinear frequency shifts, which will tune the excited sites immediately out of resonance with the nonexcited neighborhood. As a result, discrete breather like structures will be formed, which can persist for immensely long times. While selftrapping and discrete breather formation are interesting localization phenomena at strong nonlinearity, they are very different from Anderson localization since they require the existence of gaps in the spectrum of the linear wave equations [20]. If now βn < Δ, selftrapping is avoided, and the wave packet can start to spread. For L < V and β = 0, the packet will spread over the localization volume during the time τlin . After that, the new norm density will drop down to n(τlin ) ≈ n VL . For L > V the norm density will not change appreciably up to τlin , n(τlin ) ≈ n. The nonlinear frequency shift βn(τlin ) can be now compared with the average spacing d. If βn(τlin ) > d, all NMs in the packet are resonantly interacting with each other. This regime will be coined strong chaos. If instead βn(τlin ) < d, NMs are weakly interacting with each other. This regime will be coined weak chaos. To summarize: βn(τlin ) < d : weak chaos βn(τlin ) > d : strong chaos βn > Δ : selftrapping In terms of the above wave packet characteristics n, L it follows ˜ < Δ : weak chaos βnL ˜ > Δ : strong chaos βnL βn > Δ : selftrapping

(5.9)

˜ = L for L < V and L ˜ = V for L > V . It follows that the regime where L of strong chaos can be observed only if L > 1. For L = 1 we expect only two regimes – selftrapping and weak chaos. Furthermore, we obtain that the regimes of strong and weak chaos are separated by the quantity βn = d, i.e. the average spacing d is the only characteristic frequency scale here. Discussion of numerical results Let us discuss the above in the light of published computational experiments. We show results for single site excitations from [11] in Fig. 5.1 with W = 4, L = 1 and n = 1. For the DNLS model (left plots in Fig. 5.1) with β = 4.5 it follows βn = 4.5. Already at these values selftrapping of a part of the wave packet is observed. Therefore P does not grow significantly, while the second moment m2 ∼ tα with α ≈ 1/3 (red curves). A part of the excitation stays highly localized [9], while another part delocalizes. For β = 1 selftrapping is avoided since βn < Δ. With V ≈ 20 and d ≈ 0.4 it follows that τlin ≈ 16

5 Wave delocalization in nonlinear disordered media 105 ζ

4

m2

104

103

102

ζ

4 2

2 0

67

104

t

0

108

g

105 t

b

r

b

r

1010

g o

o

101 102

g

g

P

b b

101

r

o

o

r 102

104

t

106

108

104

106

t

108

1010

Fig. 5.1. (Color online) Single site excitations. m2 and P versus time in log–log plots. Left plots: DNLS with W = 4, β = 0, 0.1, 1, 4.5 [(o), orange; (b), blue; (g) green; (r) red]. Right plots: KG with W = 4 and initial energy E = 0.05, 0.4, 1.5 [(b) blue; (g) green; (r) red]. The orange curves (o) correspond to the solution of the linear equations of motion, where the term u3l in (5.4) was absent. The disorder realization is kept unchanged for each of the models. Dashed straight lines guide the eye for exponents 1/3 (m2 ) and 1/6 (P ) respectively. Insets: the compactness index ζ as a function of time in linear–log plots for β = 1 (DNLS) and E = 0.4 (KG). Adapted from [11].

and βn(τlin ) ≈ 0.05 d. Therefore we expect to observe the regime of weak chaos. It is characterized by subdiffusive spreading with m2 ∼ tα and P ∼ tα/2 (green curves). For β = 0.1 we will remain in the regime of weak chaos, however the time scales for observing spreading grow. Therefore one finds no visible spreading up to some time τd which increases with further decreasing nonlinearity. For t < τd both m2 and P are not changing. However for t > τd the packet shows visible growth with the characteristics of weak chaos (blue curves). The simulation of the equations of motion in the absence of nonlinear terms (orange curves) shows Anderson localization. Since L = 1 in the above numerical data, strong chaos has not been observed. Notably, the authors of Ref. [11] also considered single mode excitations with total norm S = 1. Using the above terminology, n ≈ 1/V and L = V with

68

S. Flach

W = 4 and therefore again V ≈ 20. For the case β = 6.5 the authors detected a growth of m2 which was subdiffusive but faster than t1/3 . We think that these observations are a clear hint towards the realization of strong chaos, which should be observable for 5 · · · 10 < β < 30 · · · 40 in these cases. The time evolution of ζ for excitations in the regime of weak chaos is shown in the insets of Fig. 5.1. As one can see the compactness index oscillates around some constant nonzero value both for the DNLS and the KG models. This means that the wave packet spreads but does not become more sparse. The average value ζ of the compactness index over 20 realizations of single mode excitations at t = 108 for the DNLS model with W = 4 and β = 5 was found to be ζ = 2.95 ± 0.39 [11]. The norm density distribution for the DNLS model at t = 108 is plotted in Fig. 5.2. The distribution is characterized by a flat plateau of almost ideally thermalized NMs. The width of this plateau is more than an order of magnitude larger than the localization volume of the linear equations. Therefore Anderson localization is destroyed. The plateau is bounded by exponentially

|ϕν|2

0.04 0.03 0.02 0.01 0

|ϕν|2

10–6 10–12 10–18 p

p

10–24

0

100

200

300

N

400

500

750

1000

1250

N

Fig. 5.2. Norm density distributions in the NM space at time t = 108 for the initial excitations in the regime of weak chaos of the DNLS model. Left plots: single site excitation for W = 4 and β = 1. Right plots: single mode excitation for W = 4 and β = 5. |φν |2 is plotted in linear (logarithmic) scale in the upper (lower) plots. The average localization volume V ≈ 20 (shown schematically in the lower plots) is much smaller than the length over which the wave packets have spread. Adapted from [11].


69

decreasing tails, with exponents corresponding to the localization length of the linear equations. With growing time the plateau widens, drops in height, and is pushing the tails to larger distances. Another remarkable feature are the huge fluctuations of norm densities in the tails, reaching 4-6 orders of magnitude. Such fluctuations are observed even in the case β = 0. They are due to the fact, that NMs are ordered in space. Neighbouring NMs in space may have different eigenfrequencies, and therefore different values of their localization length. Tail NMs are excited by the core. The further away they are, the weaker the excitation. But within a small tail volume, NMs with larger localization length will be more strongly excited than those with smaller localization length, hence the large observed fluctuations, which on a logarithmic scale are of the order of the relative variation of the localization length. The remarkable observation is, that these fluctuations in the tails persist for the nonlinear case. Anderson localization is destroyed in the core (plateau) of the wave packet due to mode-mode interactions. The tail NMs are slaved to the core and excited by it. The interaction between neighbouring tail NMs is negligible, and the huge fluctuations persist. Therefore, Anderson localization is preserved in the tails of the distributions over very long times (essentially until the given tail volume becomes a part of the core). For single site excitations in the regime of weak chaos the exponent α of subdiffusive spreading does not appear to depend on β in the case of the DNLS model or on the value of E in the case of KG. We find no visible dependence of the exponent α on W . Therefore the subdiffusive spreading is rather universal and the parameters β (or E) and W are only affecting the prefactor. Excluding selftrapping, any nonzero nonlinearity appears to completely delocalize the wave packet and destroy Anderson localization. Fittings were performed by analyzing 20 runs in the regime of weak chaos with different disorder realizations. For each realization the exponent α was fitted, and then averaged over all computational measurements. We find α = 0.33 ± 0.02 for DNLS, and α = 0.33 ± 0.05 for KG [10, 11]. Therefore, the universal exponent α = 1/3 [10] appears to explain the data. Another intriguing test was performed on the same disorder realizations and single site initial conditions, by additionally dephasing the NMs in a random way every hundred time units [11]. In that case, subdiffusion speeds up, and m2 grows as t1/2 implying αdeph = 1/2. This regime of complete decoherence of NM phases exactly corresponds to the above anticipated one of strong chaos, but here enforced by explicit dephasing.

5.5 From strong to weak chaos, from resonances to nonlinear diffusion We can think of two possible mechanisms of wave packet spreading. A NM with index μ in a boundary layer of width V in the cold exterior, which borders the packet, is either incoherently heated by the packet, or resonantly excited by some particular NM from a boundary layer with width V inside the packet.

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S. Flach

For heating to work, the packet modes φν (t) should contain a part φcν (t), having a continuous frequency spectrum (similar to a white noise), in addition to a regular part φrν (t) of pure point frequency spectrum: φν (t) = φrν (t) + φcν (t).

(5.10)

Therefore at least some NMs of the packet should evolve chaotically in time. The more the packet spreads, the less the mode amplitudes in the packet become. Therefore its dynamics should become more and more regular, implying limt→∞ φcν (t)/φrν (t) → 0. Strong chaos Let us assume that all NMs in the packet are strongly chaotic, and their phases can be assumed to be random on the time scales of the observed spreading. According to (5.6) the heating of the exterior mode should evolve as iφ˙ μ ≈ λμ φμ + βn3/2 f (t) where f (t)f (t ) = δ(t − t ) ensures that f (t) has a continuous frequency spectrum. Then the exterior NM increases its norm according to |φμ |2 ∼ β 2 n3 t. The momentary diffusion rate of the packet is given by the inverse time T it needs to heat the exterior mode up to the packet level: D = 1/T ∼ β 2 n2 . The second moment is of the order of m2 ∼ 1/n2 , since the packet size is 1/n. The diffusion equation m2 ∼ Dt yields m2 ∼ βt1/2 . This agrees very well with the numerical results for dephasing in NM space. Moreover, we expect it to hold also without explicit dephasing, provided the initial wave packet satisfies the above conditions for strong chaos (5.9). First numerical tests show that this is correct [21], but it contradicts the observations of the numerical data in the regime of weak chaos without additional dephasing. Thus, in the regime of weak chaos, not all NMs in the packet are chaotic, and dephasing is at best some partial outcome. Resonance probability Chaos is a combined result of resonances and nonintegrability. Let us estimate the number of resonant modes in the packet for the DNLS model. Excluding secular interactions, the amplitude of a NM with |φν |2 = nν is modified by a triplet of other modes µ ≡ (μ1 , μ2 , μ3 ) in first order in β as (5.6) |φ(1) ν |

λν,µ √ −1 , = β nμ1 nμ2 nμ3 Rν,µ , Rν,µ ∼ Iν,μ1 ,μ2 ,μ3

(5.11)

where λν,µ = λν + λμ1 − λμ2 − λμ3 . The perturbation approach breaks down, √ (1) and resonances set in, when nν < |φν |. Since all considered NMs belong to the packet, we assume their norms to be equal to n for what follows. Then the


71

resonance condition for a given NM with index ν is met if there is at least one given triplet of other NMs µ such that βn < Rν,µ .

(5.12)

If three of the four mode indices are identical, one is left with interacting NM pairs. A statistical analysis of the probability of resonant interaction was performed in Ref. [10]. For small values of n (i.e. when the packet has spread over many NMs) the main contribution to resonances are due to rare multipeak modes [10], with peak distances being larger than the localization volume. However pair resonances are expected not to contribute to the spreading process [22]. When distances between the peaks of multipeak modes are larger than the localization volume, the time scale of excitation transfer from one peak to another will grow exponentially with the distance. Such processes are too slow in order to be observed in numerical experiments [22]. If two or none of the four mode indices are identical, one is left with triplets and quadruplets of interacting NMs respectively. In both cases the resonance condition (5.12) can be met at arbitrarily small values of n for NMs from one localization volume. For a given NM ν we define Rν,µ0 = minµ Rν,µ . Collecting Rν,µ0 for many ν and many disorder realizations, we can obtain the probability density distribution W(Rν,µ0 ). The probability P for a mode, which is excited to a norm n (the average norm density in the packet), to be resonant with at least one triplet of other modes at a given value of the interaction parameter β is therefore given by βn W(x)dx. (5.13) P= 0

The main result is that W(Rν,µ0 → 0) → C(W ) = 0 [11]. For the cases studied, the constant C drops with increasing disorder strength W . This result of nonzero values of C is not contradicting the fact of level repulsion of neighbouring NMs, since triplet and quadruplet combinations of NM frequencies can yield practically zero values of λν,µ with finite distances between the eigenfrequencies. For the case of strong disorder (W 1) the localization volume tends to one, and quadruplet resonances are rare. Excluding also pair resonances for the above reasons, we are left with triplet resonances. A given mode may yield a triplet resonance with its two nearest neighbours to the left and right. Replacing the overlap integrals by some characteristic average, and assuming that the three participating modes have essentially uncorrelated eigenfrequencies, it follows that 2 CR W(R) ≈ C 1 − . (5.14) 3

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Due to the nonnegativity of P it would follow that P = 0 for R ≥ 3/C. In reality we expect an exponential tail for large R. As a simple approximation, we may instead use W(R) ≈ Ce−CR

(5.15)

which in turn can be expected to hold also for the case of weak disorder. It leads to the approximative result P = 1 − e−Cβn .

(5.16)

P ≈ Cβn.

(5.17)

For βn → 0 it follows

Therefore the probability for a mode in the packet to be resonant is proportional to Cβn in the limit of small n [10,11]. However, on average the number of resonant modes in the packet is proportional to the product of P and the total number of modes in the packet. Since the total number is proportional to 1/n, the average number of resonant modes in a packet is constant, proportional to Cβ, and their fraction within the packet is ∼ Cβn [10, 11]. Since packet mode amplitudes fluctuate in general, averaging is meant both over the packet, and over suitably long time windows. A detailed numerical analysis of the statistical properties of resonances and related issues is in preparation [23]. Finally we consider the process of resonant excitation of an exterior mode by a mode from the packet. The number of packet modes in a layer of the width of the localization volume at the edge, which are resonant with a cold exterior mode, will be proportional to βn. After long enough spreading βn 1. On average there will be no mode inside the packet, which could efficiently resonate with an exterior mode. Resonant growth can be excluded [10,11]. Thus, a wave packet is trapped at its edges, and stays localized until the interior of the wave packet decoheres (thermalizes). On these (growing) time scales, the packet will be finally able to incoherently excite the exterior and to extend its size. A conjecture leading to the correct asymptotics We assume, that the continuous frequency part of the dynamics of a packet mode is P(βn). It follows that φcν (t)/φrν (t) ∼ P(βn). As expected initially, the chaotic part in the dynamics of packet modes becomes weaker the more the packet spreads, and the packet dynamics becomes more and more regular in the limit of large times. Therefore the chaotic component is conjectured to be a small parameter φcν (t) φrν (t). Expanding the term |φν |2 φν to first order in φcν (t), the heating of the exterior mode should evolve according to iφ˙ μ ≈ λμ φμ + βn3/2P(βn)f (t). It follows |φμ |2 ∼ β 2 n3 (P(βn))2 t, and the rate D = 1/T ∼ β 2 n2 (P(βn))2 .

(5.18)


73

With (5.15), (5.16) and m2 ∼ 1/n2 the diffusion equation m2 ∼ Dt yields 1 ∼ β(1 − e−Cβn )t1/2 . n2

(5.19)

The solution of this equation yields a crossover from subdiffusive spreading in the regime of strong chaos to subdiffusive spreading in the regime of weak chaos: m2 ∼ (β 2 t)1/2 , strong chaos, Cβn > 1, m2 ∼ C 2/3 β 4/3 t1/3 , weak chaos, Cβn < 1. The crossover from strong chaos to weak chaos According to (5.13) the probability of resonance for a packet NM will be practically equal to one, if βn is sufficiently larger than 1/C. Such a situation can be generated for packets with large enough βn, and should yield spreading, provided one avoids selftrapping βn ≤ 4 + W [9, 11]. This spreading will be different from the asymptotic behaviour discussed above over potentially large time scales. Let us use as an example W = 4 and β = 1, with the constant C ≈ 6.2 [11]. Single site excitations with norm S = 1 lead after very short times to a spreading of the excitation into the localization volume of the linear wave equations, which is of the order of 10–20. The attained norm density is therefore of the order of n ≤ 0.1. The observed spreading is the asymptotic one since P ∼ Cβn. However, if we choose a packet size L to be of the order of the localization volume, and the norm density n of the order of n = 1, initially P ≈ 1. Thus every mode in the packet will be resonant, and the condition for strong chaos should hold. At the same time βn = 1 is far below the selftrapping threshold 4 + W = 8. For strong chaos we derived m2 ∼ t1/2 . With spreading continuing, the norm density in the packet will decrease, and eventually βn ≤ 1/C. Then there will be a crossover from strong chaos to weak chaos, and m2 ∼ t1/3 for larger times. This crossover happens on logarithmic time scales, and it will be not easy to confirm it numerically [21]. In Fig. 5.3 we show the resulting time dependence of m2 on t from (5.19) in a log-log plot, where we used β = 1, C = 6.2, L = 20 and n(t = 102 ) = 1. With x = log10 (t) and y = log10 (m2 ) it is straightforward to calculate the zero of the third derivative d3 y/dx3 = 0 to obtain the crossover position: Cβnc ≈ 1.86.

(5.20)

Therefore the only characteristic frequency scale here is 1/C. From the above discussion of the different spreading regimes (5.9) it follows, that this scale is corresponding to the average spacing d: 1 ≈ d. C

(5.21)

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S. Flach 10

log10 m2

weak chaos

t1 / 2

8

6

t1 / 3 4 strong chaos

2

0

5

15

10

20

log10 t Fig. 5.3. m2 (t) in a log-log plot according to (5.19) (black solid line). Dashed lines – power laws for strong and weak chaos.

Then 100 , W ≤ 4, W2 1 , W 4. C≈ W

C≈

(5.22) (5.23)

Our results can be used to predict the critical value of the norm density nc at which the crossover should take place. For W = 4 and β = 1 it follows nc ≈ 0.3.

5.6 Generalizations Let us consider D-dimensional lattices with nonlinearity order σ > 0: iψ˙ l = l ψl − β|ψl |σ ψl − ψm . (5.24) m∈D(l)

Here l denotes a D-dimensional lattice vector with integer components, and m ∈ D(l) defines its set of nearest neighbour lattice sites. We assume that (a) all NMs are spatially localized (which can be obtained for strong enough disorder W ), (b) the property W(x → 0) → const = 0 holds, and (c) the probability of resonances on the edge surface of a wave packet is tending to zero during the spreading process. A wavepacket with average norm n per


75

excited mode has a second moment m2 ∼ 1/n2/D . The nonlinear frequency shift is proportional to βnσ/2 . The typical localization volume of a NM is still denoted by V , and the average spacing by d. Consider a wave packet with norm density n and volume L < V . A straightforward generalization of the expected regimes of spreading leads to the following: σ/2 L βnσ/2 V < Δ : weak chaos V σ/2 L σ/2 βn V > Δ : strong chaos V βnσ/2 > Δ : selftrapping The regime of strong chaos, which is located between selftrapping and weak chaos, can be observed only if 2/σ V d L > Lc = V 1−2/σ , n > nc = . (5.25) L β For σ = 2 we need L > 1, for σ → ∞ we need L > V , and for σ < 2 we need L ≥ 1. Thus the regime of strong chaos can be observed e.g. in a one-dimensional system with a single site excitation and σ < 2. If the wave packet size L > V then the conditions for observing different regimes simplify to βnσ/2 < d : weak chaos βnσ/2 > d : strong chaos βnσ/2 > Δ : selftrapping The regime of strong chaos can be observed if 2/σ d n > nc = . β

(5.26)

Similar to the above we obtain a diffusion coefficient D ∼ β 2 nσ (P(βnσ/2 ))2 .

(5.27)

In both regimes of strong and weak chaos the spreading is subdiffusive [10]: 2

m2 ∼ (β 2 t) 2+σD , strong chaos, 4

m2 ∼ (β t)

1 1+σD

, weak chaos.

(5.28) (5.29)

Let us calculate the number of resonances in the wave packet volume (NRV ) and on its surface (NRS ) in the regime of weak chaos:

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S. Flach

NRV ∼ βnσ/2−1 , NRS ∼ βn

D(σ−2)+2 2D

.

(5.30)

We find that there is a critical value of nonlinearity power σc = 2 such that the number of volume resonances grows for σ < σc with time, drops for σ > σc and stays constant for σ = σc . Therefore subdiffusive spreading will be more effective for σ < σc . We also find that the number of surface resonances will grow with time for 1 D > Dc = , σ < 2. (5.31) 1 − σ/2 Therefore, for these cases, the wave packet surface will not stay compact. Instead surface resonances will lead to a resonant leakage of excitations into the exterior. This process will increase the surface area, and therefore lead to even more surface resonances, which again increase the surface area, and so on. The wave packet will fragmentize, perhaps get a fractal-like structure, and lower its compactness index. The spreading of the wave packet will speed up, but will not anymore be due to pure incoherent transfer, instead it will become a complicated mixture of incoherent and coherent transfer processes. Mulansky computed spreading exponents for single site excitations with β = 1, W = 4, L = 1, D = 1 n = 1 and σ = 1, 2, 4, 6 [13]. Since for σ = 2, 4, 6 strong chaos is avoided, the fitting of the dependence ms (t) with a single power law is reasonable. The corresponding fitted exponents 0.31 ± 0.04 (σ = 2), 0.18 ± 0.04 (σ = 4) and 0.14 ± 0.05 (σ = 6) agree well with the predicted weak chaos result 1/3, 1/5, 1/7 from (5.29). For σ = 1 the initial condition is launched in the regime of strong chaos. A single power law fit will therefore not be reasonable. Since the outcome is a mixture of first strong and later possibly weak chaos, the fitted exponent should be a number which is located between the two theoretical values 1/2 and 2/3. Indeed, Mulansky reports a number 0.56 ± 0.04 confirming our prediction. Veksler et al. [12] considered short time evolutions of single site excitations (up to t = 103 ). While the time window may happen to be too short for conclusive results, the observed increase of fitted exponents with increasing β for σ < 2 is possibly also influenced by the crossover from weak to strong chaos. Note that Skokos et al. [24] performed a more detailed analysis for the KG lattice, which confirm many the above results.

5.7 Discussion and conclusions Let us summarize the findings. If the strength of nonlinearity is large enough, a wave packet (or at least an appreciable part of it) is selftrapped due to the finite bounds for the spectrum of the linear equation. If the nonlinearity is weak enough so as to avoid selftrapping, two possible outcomes are predicted, which now depend also on the volume L of the packet. If L > Lc and n > nc , the NMs in the packet will be all resonant, strongly interacting with each


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other and quickly dephase. That leads to a regime of strong chaos. As time grows, the norm density n drops below nc , and the spreading continues in the regime of weak chaos. If however either L < Lc or n < nc , strong chaos is avoided, and the packet will spread in the regime of weak chaos. Lowering β or n further will keep the spreading in the regime of weak chaos, but time scales of subdiffusion will grow, and the process will not be observable on the finite time window currently accessible by computational experiments. The above holds if D < Dc which implies that Anderson localization is preserved in the tails and destroyed in the wave packet core. In other words, the time scales for destroying Anderson localization in the tails are much larger than the time scales which lead to a thermalization of the core and the corresponding subdiffusive spreading. In order to observe the crossover from strong to weak chaos, one has to carefully choose the system parameters. In particular, it is desirable to make the crossover region more narrow. If D > Dc then the spreading process will be different from the above predictions, because resonant interaction in the surface and the tails of the wave packet will destroy Anderson localization as well. The spreading will presumably stay subdiffusive. But we do not know currently how to estimate and characterize the details of this process. Our results rely on a conjecture of the dependence of a diffusion coefficient on the probability of resonances. Future investigations may consider the connection between this conjecture and the dependence of Lyapunov coefficients, relaxation times of correlation functions, and detrapping times on the system parameters.

Acknowledgments I thank I. Aleiner, B. Altshuler, S. Aubry, J. Bodyfelt, S. Fishman, D. Krimer, Y. Krivolapov, T. Lapteva, N. Li, Ch. Skokos, and H. Veksler for useful discussions.

References 1. P. W. Anderson, Phys. Rev. 109, 1492 (1958). 2. T. Schwartz, G. Bartal, S. Fishman and M. Segev, Nature 446, 52 (2007). 3. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008). 4. D. Clement A. F. Varon, J. A. Retter, L. Sanchez-Palencia, A. Aspect and P. Bouyer, New J. Phys. 8, 165 (2006); L. Sanches-Palencia D. Clement, P. Lugan, P. Bouyer, G. V. Shlyapnikov and A. Aspect, Phys. Rev. Lett. 98, 210401 (2007); J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer and A. Aspect, Nature 453, 891 (2008); G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno and M . Inguscio, Nature 453, 895 (2008).

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5. D. Weinmann, S. Kohler, G.-L. Ingold and P. H¨ anggi, Ann. Phys. (Leipzig) 8, 277 (1999); C. Aulbach, A. Wobst, G.-L. Ingold, P. H¨ anggi and I. Varga, New J. Phys. 6, 70 (2004). 6. M. I. Molina, Phys. Rev. B 58, 12547 (1998). 7. A. S. Pikovsky and D. L. Shepelyansky, Phys. Rev. Lett. 100, 094101 (2008). 8. I. Garc´ıa-Mata and D. L. Shepelyansky, Phys. Rev. E, 79, 026205 (2009). 9. G. Kopidakis, S. Komineas, S. Flach and S. Aubry, Phys. Rev. Lett. 100, 084103 (2008). 10. S. Flach, D. Krimer and Ch. Skokos, Phys. Rev. Lett. 102, 024101 (2009). 11. Ch. Skokos, D. O. Krimer, S. Komineas and S. Flach, Phys. Rev. E 79, 056211 (2009). 12. H. Veksler, Y. Krivolapov and S. Fishman, Phys. Rev. E 80, 037201 (2009). 13. M. Mulansky, Localization Properties of Nonlinear Disordered Lattices, Diplomarbeit Universit¨ at Potsdam (2009); http://opus.kobv.de/ubp/volltexte/2009/ 3146/. 14. M. Mulansky, K. Ahnert, A. Pikovsky and D. Shepelyansky, Phys. Rev. E 80, 056212 (2009). 15. O. Morsch and M. Oberthaler, Rep. Prog. Phys. 78, 176 (2006). 16. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, Amsterdam, 2003). 17. A. A. Ovchinnikov, N. S. Erikhman and K. A. Pronin, Vibrational-Rotational Excitations in Nonlinear Molecular Systems, Kluwer Academic/Plenum Publishers (New York) 2001. 18. Yu. S. Kivshar and M. Peyrard, Phys. Rev. A 46, 3198 (1992); Yu. S. Kivshar, Phys. Lett. A 173, 172 (1993); M. Johansson, Physica D 216, 62 (2006). 19. B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1469 (1993). 20. S. Flach and C. R. Willis, Phys. Rep. 295, 181 (1998); S. Flach and A. V. Gorbach, ibid. 467, 1 (2008). 21. J. Bodyfelt and T. Lapteva, private communication. 22. H. Veksler, Y. Krivolapov and S. Fishman, Phys. Rev. E 81, 017201 (2010). 23. D. O. Krimer and S. Flach, Cond- mat. arXiv:1005.4820 (2010). 24. Ch. Skokos and S. Flach, Phys. Rev. E 82, 016208 (2010).

6 PHYSICAL LIMITS FOR SCALING OF ELECTRONIC DEVICES IN INTEGRATED CIRCUITS W. Nawrocki Faculty of Electronics and Telecommunications, Poznan University of Technology, ul. Piotrowo 3, 60-965 Poznan, Poland [email protected] Abstract. In the paper physical limits for scaling of electronic devices in integrated circuits are discussed. The quantization of both electrical and thermal conductance in nanostructures is considered and estimated numerically. Problems of heat exchange in nanostructures, spread of doping atoms in a semiconductor material, a loss of electrostatic control of the current in a MOSFET, and electrons tunneling between a source and a drain inside a MOSFET are also discussed in the paper.

Key words: integrated circuits, scaling, quantization.

6.1 Introduction In this paper we discuss some physical limits for scaling of devices and conducting paths inside of semiconductor integrated circuits (ICs). Since 40 years only a semiconductor technology, mostly the CMOS and the TTL technologies, are used for fabrication of integrated circuits in the industrial scale. Miniaturization of electronic devices in integrated circuits has technological limits and physical limits as well. Probably the CMOS technology will be used at least in the next 10–15 years. In 2010 best parameters of commercial ICs shown the dual-core Intel Core i5-670 processor manufactured in the technology of 32 nm. Its clock frequency in turbo mode is 3.73 GHz. The other example is the quad-core Intel Core i7-975 manufactured in the technology of 45 nm. A forecast of the development of the semi-conductor industry (ITRS 2009) predicts that sizes of electronic devices in ICs circuits will be smaller than 10 nm in the next 10 years [1]. The physical gate length in a MOSFET will even amount 7 nm (see Table 6.1) in the year 2024. One can notice that the previous forecast (ITRS 2007) gave different values of the predicted length of a physical gate in ICs: 9 nm in 2016 and 4 nm in 2022.


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Table 6.1. Data of integrated circuits (IC) according to the Report of The International Technology Roadmap for Semiconductors (Edition 2009) Year 2009 2012 2015 2018 2021 2024 Physical gate length in nm 29 22 17 12.8 9.7 7.4 a FET transistor inside of ICs (microprocessor unit – MPU) Clock frequency (on GHz 5.45 6.82 8.52 10.6 13.3 16.6 chip MPU) Functionality of IC mln 2212 4424 8848 17696 35391 70782 (number of transistors) Supply voltage V 1.0 0.9 0.81 0.73 0.66 0.60 Dissipated power W 143 158 143 136 133 130 (cooling on)

At least five physical effects should be taken into account if we discuss limits of miniaturization of integrated circuits: – quantization of both electrical and thermal conductance in narrow and thin transistors’ channels and in conducting paths; – spread of doping atoms in a semiconductor material; each dopant would induce a relatively high potential bump; – propagation time of electromagnetic wave along and across a chip (IC); – electrostatics; a loss of electrostatic control of the drain current vs the gate voltage; – electron tunneling between a source and a drain inside a MOSFET through a insulation (oxide). Below we discuss quantization of electrical conductance and thermal conductance in nanostructures. Spread of doping atoms in a semiconductor material is discussed as well. We mention only the other physical effects important for scaling of integrated circuits.

6.2 Quantization of electrical conductance in nanostructures Electric and thermal properties of electronic devices or paths with nanometer sizes are not more described by a classical theory of conductance but by quantum theories. The theoretical quantum unit of electrical conductance G0 = 2e2 /h was predicted by Landauer in his theory of electrical conductance [2]. Parameters characterizing the system are a Fermi wavelength gλF (λF = 2π/k F , where kF is the Fermi wavevector) and a mean free path gΛ. For metals like gold λF ≈ g 0.5 nm is shorter than free electron path Λg(ΛAu g = 14 nm). If a length of the system is shorter than the free electron path,

6 Physical limits for scaling

81

b a

Fig. 6.1. Conductance quantization in a nanowire (conductor with length L < Λ and width W comparable with the length of Fermi wave λF ): (a) nanowire outline (the third dimension is not considered); (b) conductance quantization G versus width W .

the impurity scattering is negligible, so the electrons transport is ballistic. If a wire has outside diameter comparable with the Fermi wavelength gλF , and its length L is less than Λ, the system can be regarded as one-dimensional (1-D), the electron – as a wave, and one can expect quantum effects – see Fig. 6.1. The total electrical conductance of a wire is given by formula (6.1). G=

2e2 N h

(6.1)

where N is the number of transmission channels. For 1-D system, with thickness H ≤ λF , N depends on the width of the wire, N = int (2W /λF ). For 2-D system, with H, W ≥ λF , N = int (W × H/λ2F ), where int (A) means the integer of A. However, defects, impurities and irregularities of the shape of the conductor can induce scattering, then conductivity is given by the Landauer equation: G=

N 2e2 tij h i,j=1

(6.2)

where tij denotes probability of the transition from jth to ith state. In the absence of scattering tij = δ ij thus Eq. (6.2) is reduced to Eq. (6.1). Figure 6.1 presents a picture of a path (nanowire) – the constriction in an electrical conductor with dimension W (width), H (thickness) and L (length) W . A set-up for measurements of electrical conductance in nanowires formed by mechanical contact between two macrowires is shown in Fig. 6.2. The quantization of electric conductance depends neither on the kind of element nor on temperature. For conductors and semiconductors the conductance quantization in units of G0 = 2e2 /h = (12.9kΩ)−1 was measured in many experiments. We measured nanowires formed by pure metals and by metallic alloys. In our experiments the quantization of conductance was evident [3].

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Fig. 6.2. Set-up for measurements of electrical conductance in dynamically formed nanowires.

In Fig. 6.3 are presented plots of electrical conductance versus time for Au, Co nanowires and for a junction: cobalt tip – germanium plate. On this picture (Fig. 6.3) an intensity of dynamic formation process of nanowires is shown as histograms. Let’s consider a silicon path with length L = 20 nm (L < ΛSi ), width W = 3 nm, thickness H = 3 nm and λF = 1 nm. Such path is a nanowire with the conductance: GP = (2e2 /h)× N ; N = int (W×H /λ2F ) = 9; GP =7.75×10−5× 9 = 7×10−4 [A/V]. A resistance of the path RP = 1/GP = 1.43 kΩ, thus it is surprisingly high. An electrical capacity of a path is 2 pF/cm [1], so for L = 20 nm CP = 4×10−18 F. The path inside an IC forms a low-pass RC filter with the upper frequency fu = 1/(2πRP CP ) ≈ 3×1013 Hz, thus the path filters of signals.

6.3 Quantization of thermal conductance in nanostructures It is generally known that limits for speed-up of digital circuits, especially microprocessors, are determined by thermal problems. There are several analogues between the electrical GE and the thermal GT conductance of a nanostructure. However an analyze of thermal conductance is more complex than electrical conductance because of contribution either phonons or electrons in heat exchange. Quantization of thermal conductance in one-dimensional systems was predicted theoretically by Greiner [4] and by Rego [5] for ballistic transport of electrons and phonons. Quantized thermal conductance GT and

83

H(G)

H(G)


G [G0] 4 3.5 3 2.5 2 1.5 1 0.5 0

H(G) Co tip - Ge sample traces from 1 to 500

0.2 0.15 1

40 80 120160 200 240 280 320 380420 460500

0.1 0.05

0

0.02

0.04

0.06

0.08

0.1

0.12 t [s]

0 0

0.5

1

1.5

2

2.5

3

3.5

4

G [G0]

Fig. 6.3. Conductance quantization in nanowires (left) and quantization intensity – histograms (right): in gold nanowires (upper graphs), in cobalt nanowires (central graphs) and in cobalt-germanium junction (bottom graphs); all measurements made by M. Wawrzyniak, PTU Poland. Table 6.2. Electrical resistance R and thermal conductance with diameter D (λF = 0.5 nm) Diameter D nm 0.5 1.0 1.5 Number of channels – 1 4 9 Ω 12 903 3226 1434 R = 1/GE 10−9 [W/K] 0.285 1.14 2.56 GT (at 300 K)

GT of gold nanowire 2.0 16 806 4.56

2.5 25 516 7.12

3.0 36 358 10.26

its quantum (unit) GT 0 was confirmed experimentally by Schwab [6]. The quantum of thermal conductance 2 /3h)T = 9.5×10−13T GT 0 [W/K] = (π 2 kB

depends on the temperature. At T = 300 K value of GT 0 = 2.8 × 10−10 [W/K]. The thermal conductance of a nanowire is very small. We can compare the thermal conductance in a nanowire with the thermal conductance of Au microwire which connects a silicon part and a metallic terminal (pin) in a transistor with a length of a wire L = 1.5 mm and a diameter D = 25 μm is much larger: GT ≈10−4 [W/K].

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A single nanowire should be consider together with its terminals. Electron transport in the nanowire is ballistic itself, it means the transport without scattering of electrons and without energy dissipation. The energy dissipation occurs in terminals. Because of the energy dissipation the local temperature Tterm in terminals is higher then the temperature Twire of nanowires itself.

6.4 Spread of doping atoms in a semiconductor material Classical theories of electrical and thermal conductance assume a huge number of atoms and free electrons. Let’s assume a silicon cube with one side dimension of a and with common doping of 1016 cm−3 . In a n-doped silicon cube with the size of (100 nm)3 there are 5×107 atoms and 10 free electrons at 300 K, but in the Si cube with the size of (10 nm)3 there are 5×104 atoms and 1% chance only to find one free electron (Fig. 6.4). Free electrons are necessary for electrical conductance as charge carriers. It means not only that classical theories of conductance are not valid for nanostructures. It means that common doping in semiconductor material is not sufficient for electronic devices of nanometric size. In order to keep the conductive properties of the semiconductor material one should apply more intensive doping, e.g. 1020 cm−3 . However such intensive doping decreases resistivity of the material dramatically from 2×10−3 Ωm to 10−5 Ωm, respectively (for n-type Si, at 300 K). Low number of free electrons should be scattered evenly in whole volume of a material.

a

b c

Fig. 6.4. Free electrons in a silicon cube; (a) silicon cube of (1 mm)3 includes 5 × 1019 atom, 1013 free electrons at T = 300 K (doping: 1016 cm3 ); (b) silicon cube of (100 nm)3 includes 50 ml atom, 10 free electrons; (c) silicon cube of (10 nm)3 includes 50,000 atom, 1% chance to find one free electrons.


85

Fig. 6.5. MOSFET transistor – the simple model.

6.5 Other physical limits The channel length LE of Si MOSFET, limited by a degradation of electrostatic control in the transistor, was analyzed by Frank [7] and Likharev [8]. The shortest channel length LE (see Fig. 6.5) depends on thickness of channel Hch , thickness of insulation layer Hi , dielectric constants of a channel ε and insulation εi – formula (6.3) [8]. 1/2 εHch Hi (6.3) LE = 2εi If we take the ratio εi /ε ≈ 0.3 (for silicon oxide and silicon), thickness Hch = 2 nm, Hi = 1.5 nm – the estimated minimal length of channel is LE ≈ 3 nm. The channel length LE can be shorter if a better insulator than silicon oxide (SiO2 ) would be applied. Following materials are tested as an insulator for MOSFETs: silicon nitride, hafnium oxide and zirconium silicate. The next limit to miniaturization of electronic devices comes from source to drain tunneling through the potential barrier along the channel. The tunneling effect depends on the channel length L and the supply voltage. Because of tunneling the minimal channel length for silicon device is around 2 nm [8]. High-tech microprocessors have dimensions over 30 nm. A new Intel product the quad-core Intel Core i5-670 has a width a = 37.5 mm and a length b = 37.5 mm. The pins of the IC are distributed on the bottom plate, along its four sides too. The period of the highest clock frequency (3.73 GHz) is Tck = 270 ps. But the propagation time for an electromagnetic wave on the way of (a + b) is: Tp = (a + b)/v= 75 mm/2.5×108 m/s = 300 ps (v – speed of an electromagnetic wave in silicon). Thus, the propagation time of the clock signal is longer than the period of the clock signal.

6.6 Conclusions Several physical effects must be taken into account for miniaturization of electronic circuits. Most important are: quantization of electrical and thermal conductance in nanostructures, a degradation of electrostatic control

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in the channel of MOSFET and tunneling along the channel of MOSFET. Conductance quantization has proved to be observable in a simple experimental setup, giving opportunity to investigate subtle quantum effects in electrical conductivity. The energy dissipation in nanowires takes part in their terminals. Other physical effects important for scaling were analyzed theoretically and simulated. According to the state of the art the minimal length of gate in MOSFET in silicon integrated circuits is around 3 nm.

References 1. International Technology Roadmap for Semiconductors, Edition 2009, www.itrs. net/reports, published in January 2010. 2. R. Landauer, J. Phys.: Cond. Matter 1, 8099 (1989). 3. B. Susa, M. Wawrzyniak, J. Barna, and W. Nawrocki, Materials Science. Poland 25, 305 (2007). 4. A. Greiner, L. Reggiani, T. Kuhn, and L. Varani, PRL 78, 1114 (1997). 5. L. G. C. Rego and G. Kirczenow, PRL 81, 232 (1998). 6. K. Schwab, E. A. Henriksen, J. M. Worlock, and M.L. Roukes, Nature 404, 974 (2000). 7. D. J. Frank, R. H. Dennard, E. Nowak, P. M. Solomon, Y. Taur, and H. S. P. Wong, Proc. IEEE 89, 259 (2001). 8. K. Likharev, Chapter 4 in Advanced Semiconductor and Organic Nanotechniques, Marko¸c H. (Ed.), Elsevier (2003).

7 SYNCHRONIZED ANDREEV TRANSMISSION IN CHAINS OF SNS JUNCTIONS N. M. Chtchelkatchev1,2,3 , T. I. Baturina3,4, A. Glatz3 , and V. M. Vinokur3 1 2

3 4

Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk 142190, Moscow region, Russia L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Akademika Semenova av. 1-A, Chernogolovka 142432, Moscow Region, Russia [email protected] Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA [email protected] Institute of Semiconductor Physics, 13 Lavrentjev Avenue, Novosibirsk 630090, Russia

Abstract. We construct a nonequilibrium theory for the charge transfer through a diffusive array of alternating normal (N) and superconducting (S) islands comprising an SNSNS junction, with the size of the central S-island being smaller than the energy relaxation length. We demonstrate that in the nonequilibrium regime the central island acts as Andreev retransmitter with the Andreev conversions at both NS interfaces of the central island correlated via over-the-gap transmission and Andreev reflection. This results in a synchronized transmission at certain resonant voltages which can be experimentally observed as a sequence of spikes in the differential conductivity.

Key words: nonequilibrium superconductivity, mesoscopic superconductivity, Josephson junction arrays, hybrid structures, Andreev reflection, macroscopic quantum phenomena, synchronization effects.

7.1 Introduction An array of alternating superconductor (S) – normal metal (N) islands is a fundamental laboratory representing a wealth of physical systems ranging from Josephson junction networks and layered high temperature superconductors to disordered superconducting films in the vicinity of the superconductorinsulator transition. Electronic transport in these systems is mediated by Andreev conversion of a supercurrent into a current of quasiparticles and vice versa at interfaces between the superconducting and normal regions [1]. J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 7,

87

88


A fascinating phenomenon benchmarking this mechanism is the enhancement of the conductivity observed in a single SNS junction at matching voltages constituting an integer (m) fraction of the superconducting gap, V = 2Δ/(em) [2–11] due to the effect of multiple Andreev reflection (MAR) [12–14]. The current-voltage characteristics of diffusive SNS junctions were discussed in great detail in Refs. [15, 16]. In this work superconductors were considered to be in a local equilibrium and the relation Ie (ε) = Ih (ε − V ) was satisfied. This approach was further developed by N. M. Chtchelkatchev, [22]; however in case of geometrically non-symmetric SNS arrays, it results in an equivalent circuit with the ennumerable number of elements. Further developments were obliged to studies of large arrays comprised of many SNS junctions [17–21]. Experimental results, especially those obtained on the multiconnected arrays [17, 18, 21], indicated clearly that singularities in transport characteristics cannot be explained by MARs at individual SNS junctions only and that there is evidently a certain coherence of the Andreev processes that occur at different NS interfaces. These findings call for a comprehensive theory of transport in large SNS arrays. In this article we develop a nonequilibrium theory of electronic transport in a series of two diffusive SNS junctions, i.e. an SNSNS junction and derive the corresponding current-voltage characteristics. We demonstrate that splitting the normal part of the SNS junction into two normal islands that have, in general, different resistances and are coupled via a small superconducting granule, SC leads to the nontrivial physics and emergence of a new distinct resonant mechanism for the current transfer: the Synchronized Andreev Transmission (SAT). The main component of our consideration is a nonequilibrium circuit theory of the charge transfer across SC . [The symmetric case with the equal resistances of the normal parts was discussed in detail in [22]. Unfortunately the technique developed there does not allow straightforward generalization onto a nonsymmetric case.] In the SAT regime the processes of Andreev conversion at the boundaries of the central superconducting island are correlated: as a quasiparticle with the energy ε hits one NSC interface, a quasiparticle with the same energy emerges from the other SC N interface and enters the bulk of the normal island (and vice versa, see Fig. 7.1). This energy synchronization is achieved via over-the-gap Andreev processes [19], which couple MARs occurring within the each of the normal islands and make the quasiparticle distribution at the central island essentially nonequilibrium. Effectiveness of the synchronization is controlled by the value of the energy relaxation lengths of both, the quasiparticles crossing SC with energies above Δ, and the quasiparticles experiencing MAR in the normal parts. The SAT processes result in spikes in the differential conductivity of the SNSNS circuit, which appear at resonant values of the total applied voltage Vtot defined by the condition 2Δ Vtot = (7.1) en with integer n, irrespectively of the details of the distribution of the partial voltages at the two normal islands.

7 Synchronized Andreev transmission in chains of SNS junctions

89

n = 1: eVtot = 2Δ

a

eV1 = 6Δ / 7

b

eV2= 8Δ / 7

SL

SR B

2Δ A

SC

SL

D C

N2

N1

N2

N1

SR

A'

e

h

2Δ

SC B' C'

e

h

D'

n = 2: eVtot = Δ

c

d eV1= 3Δ / 7

SL

N1

SC

eV2= 4Δ / 7 N2

SR

SL

2Δ

e SL 2Δ

N1

SC

N2

SR

2Δ

N1

SC

f

N2 SR

N1 SL

SC

N2 SR

2Δ

Fig. 7.1. Diagrams of the SAT processes for the first, n = 1, (a)–(b), and second, n = 2 (c)–(f), subharmonics of the resonant singularities in dI/dV described by Eq. (7.1) for the SNSNS junction with the normal resistances ratio R1 /R2 = 3/4 (depicted through 3/4 ratio of the respective lengths of the normal regions). The thick solid lines represent the quasiparticle paths starting and/or ending at the points of singularity in the density of states at energies ε = ±Δ at the electrodes SL and SR . The dashed solid lines show paths starting and/or ending at the edges of the gap of the central island SC . The circle denotes the over-the-gap Andreev reflections at the electrodes. The paths ABCD [panel (a)] and D C B A [panel (b)] correspond to the electron- and hole trajectories, respectively. Synchronization of the energies of the incident and emitted quasiparticles at points B and C (B and C ) is shown by arrows. SAT is realized by trajectories passing through the singular points ε = ±Δ of the central island SC and including over-the-gap transmissions and Andreev reflections. Trajectories synchronizing other transmissions across SC and those of higher orders are not shown. Note, that voltage drops eV1 = 6Δ/7 and eV2 = 8Δ/7 [panels (a)–(b)] (eV1 = 3Δ/7 and eV2 = 4Δ/7 [panels (c)–(f)]) are not MAR matching voltages of individual SL N1 SC or SC N2 SR parts.

90


The article is organized as follows. In the Sect. 7.2 we define the system, a diffusive SNSNS junction which will be a subject of our study. Section 7.3 is devoted to introduction and description of the employed theoretical tools: the electronic transport of the system in the resistive state is given by the LarkinOvchinnikov equation in a form of matrix equations for the Green’s functions taken in Keldysh representation. In Sect. 7.4–7.8 we construct an equivalent circuit theory for an SNSNS junction resulting in the recurrent relations for the spectral current flow in the energy space. In Sect. 7.9 we present the original numerical method enabling us to solve the recurrent relations for the spectral current and obtain the I-V characteristics for the SNSNS junction. The obtained results are discussed in Sect. 7.10, where we demonstrate, in particular, that the SAT-induced features become dominant in large arrays consisting of many SNS junctions.

7.2 The system We consider charge transfer across an SL N1 SC N2 SR junction, where SL , SC , and SR are mesoscopic superconductors with the identical gap Δ; the ‘edge’ superconducting granules SL and SR play the role of electrodes, and SC is the central island separating the two normal parts with, in general, different normal resistances. We discuss the common experimental situation of a diffusive regime where the most of the energy scales are smaller than /τ , where τ is the impurity scattering time. We assume the size, LC , of the central island to be much larger than the superconducting coherence length ξ, hence processes of subgap elastic cotunneling and/or direct Andreev tunneling [23] do not contribute much to the charge transfer. In general, this condition ensures that LC is large enough so that charges do not accumulate in the central island and Coulomb blockade effects are irrelevant for the quasiparticle transport. At the same time LC is assumed to be less than the charge imbalance length, such that we can neglect the coordinate dependence of the quasiparticle distribution functions across the island SC . Additionally, the condition ε LC , where ε is the energy relaxation length, implies that quasiparticles with energies ε > Δ traverse the central superconducting island SC without any noticeable loss of energy. The normal parts N1 and N2 are the diffusive normal metals of length L1,2 > ξ, and L1,2 > LT , LT = DN /ε, where DN is the diffusion coefficient in the normal metal. We assume the Thouless energy, ETh = DN /L21,2 , to be small, ETh Δ, and not to exceed the characteristic voltage drops, ETh < eV1,2 . If these conditions that define the so called incoherent regime [15] are met, the Josephson coupling between the superconducting islands is suppressed. And, finally, we let the energy relaxation length in the normal parts N1 and N2 be much larger than their sizes, so that quasiparticles may experience many incoherent Andreev reflections inside the normal regions.


91

7.3 Theoretical formalism The current transfer across the SNSNS junction is described by quasiclassical Larkin-Ovchinnikov (LO) equations for the dirty limit [24, 25]: ˇ eff ◦, G] ˇ = ∇J, ˇ −i[H

ˇ·n = J

1 ˇ ], ˇ ,G [G N 2σS R S

(7.2)

ˇ eff = ˇ1(iˆ ˆ J ˇ = DG ˇ ◦ ∇G ˇ is the matrix current, where H σz ∂t − ϕˆ σ0 + Δ), the subscripts “S” and “N” denote the superconducting and normal materials, respectively, “◦” stands for the time-convolution, σ î (i = x,y,z) are the Pauli matrices, operating in the Nambu space of 2 × 2 matrices denoted by ˆ = iˆ ‘hats’, Δ σx Im Δ + iˆ σy Re Δ, and R is the resistance of an NS interface. The diffusion coefficient D assumes the value DN in the normal metal and the value DS in the superconductor, and ϕ is the electrical potential which we calculate self-consistently. The unit vector n is normal to the NS interface and is assumed to be directed from N to S. The momentum averaged Green’s ˇ t, t ) are 2 × 2 supermatrices in a Keldysh space. Each element functions G(r, of the Keldysh matrix, labelled with a hat sign, is, in its turn, a 2 × 2 matrix in the electron-hole space: ˇ = G

ˆR G ˆK G Â 0 G

;

ˆ R(A) = G

G R(A) F R(A) F˜ R(A) G˜R(A)

,

(7.3)

r is the spatial position, t and t are the two time arguments. The Keldysh ˆK = G ˆ R ◦ fˆ − fˆ ◦ component of the Green’s function is parametrized as [24]: G A ˆ ˆ G , where f is the distribution function matrix, diagonal in Nambu space, fˆ ≡ diag [1 − 2ne , 1 − 2nh ], ne(h) is the electron (hole) distribution function. In equilibrium ne(h) becomes the Fermi function. And, finally, the Green’s ˇ 2 = ˇ1. function satisfies the normalization condition G The edge conditions closing Eqs. (7.2) are given by the expressions for the Green’s functions in the bulk of the left (L) and right (R) superconducting leads: ˇ 0 (t − t )eiμL(R) t τˆ3 / , ˇ L(R) (t, t ) = e−iμL(R) tˆτ3 / G G ˇ 0 (t) is the equilibthe chemical potentials are μL = 0 and μR = eV . Here, G rium bulk BCS Green’s function. ˇ as The current density is expressed through the Keldysh component of J 1 πσN I(t, r) = Tr σ ˆz JˆK (t, t; r) = 4 2

dε [Ie (ε) + Ih (ε)] ,

(7.4)

where the spectral currents Ie and Ih representing the electron and hole quasiparticle currents, respectively, are the time Wigner-transforms of top- and ˇ(K) . bottom diagonal elements of the matrix current J

92


On the normal side of the superconductor-normal metal interface, the Keldysh component of Eqs. (7.2) yield the conservation conditions: ∇Ie(h) = 0,

(7.5)

Ie (ε) = σN {Dp (ε + u)∇ ne (ε) − Dm (ε + u)∇nh (ε + 2u)} ,

(7.6)

Ih (ε) = σN {Dp (ε − u) ∇nh (ε) − Dm (ε − u)∇ne (ε − 2u)} ,

(7.7)

where u is the electrical potential of the adjacent superconductor, Dp(m) = (D− ± D+ )/2, D+ (ε) =

1 ˆ R (ε) G ˆ A (ε)] , Tr[ˆ 1−G 4

D− (ε) =

1 ˆ R (ε) σz G ˆ A (ε)] , Tr[ˆ 1 − σz G 4

(7.8)

and the trace is taken over components in the Nambu-space. In the bulk of ˆ R(A) (ε) → ±ˆ a normal metal, G σz and D+ ≈ D− ≈ 1, so Ie = σN ∇ne and Ih = σN ∇nh .

7.4 Circuit representation of the boundary conditions We start the construction of the circuit theory with the corresponding formulation of the boundary conditions for the distribution functions at the interface between the normal parts and the central superconducting island. We consider a stationary situation where the applied voltage does not depend on time. Then the Green’s functions can be parameterized near an NS interface as follows: ˆ R ]j (ε, ε ) = σ [G ˆz δε−ε cosh θj (ε) +ˆ σ+ δε−ε +2u sinh θj (ε) − σ ˆ− δε−ε −2u sinh θj (ε) , GA = −ˆ σz (GR )† σ ˆz ,

(7.9)

ˆx ± iˆ σy , j = S, N, and u is the electrochemical potential. where σ ˆ± = σ The effective diffusion coefficients are correspondingly D+ = cos2 Im θ and D− = cosh2 Re θ. When deriving Eq. (7.9), we have used the condition that the Josephson coupling between the superconducting islands in the junction is suppressed. The proximity effect results in an additional term in Eq. (7.9) proportional to δ(ε − ε − 2(u − u)), where u is the potential of the adjacent superconductor involved. Taking the Keldysh component of the boundary term in Eq. (7.2) we derive the boundary conditions for the currents Ie(h) at the NS interface, which assume the form of Kirchhoff’s laws for the circuit shown in Fig. 7.2(a). The


(1C)

RP (ε+u)

Ih(ε+2u)

Ie(ε)

(1C)

RQ

(ε+u)

nh(C)(ε+2u)

–R (1

)

(1C

–R P

ne(1)(ε)

(ε+

(ε+

u)

SC

(1C)

nh(1)(ε+2u)

Ih(ε+2u)

P C)

u)

b

RQ (ε+u)

ne(C)(ε)

R+ (1C) (ε+ u)

(1C)

nh(1)(ε+2u)

RP (ε+u)

a

Ie(ε)

) (1C) (ε+u

ne(1)(ε)

93

SC n(C) F (ε+u)

R+

Fig. 7.2. Effective circuit for the boundary between the normal metal and the superconductor. Kirchhoff laws where the role of the potential in the nodes is taken by the electron- and hole distribution functions give the boundary conditions for the LO equations. (a) A general nonequilibrium case. (b) Equivalent circuit for an equilibrium case where quasiparticle distribution functions in the superconductor are the Fermi-functions, nF , then ne (ε) = nF (ε + u) = nh (ε + 2u), used in [15]. The superconductor electrical potential at the boundary is equal to u.

electron and hole distribution functions take the role of voltages at the nodes. The equation for an electronic spectral current flowing into the lower left corner node: Ie (ε) =

(1) (1) n(C) n(C) e (ε) − ne (ε) h (ε + 2u) − ne (ε) + (1C) (1C) RQ (ε + u) [−RP (ε + u)]

+

(1) n(1) h (ε + 2u) − ne (ε) . (1C) RP (ε + u)

(7.10)

The equation for the hole current going into the top left node of the circuit of the Fig. 7.2(a) is easily obtained analogously to (7.10) with the aid of the additional transformation ε → ε − 2u i.e. by shifting all the energies over −2u: Ih (ε) =

(1) n(C) (ε − 2u) − n(1) n(C) h (ε) − nh (ε) h (ε) + e (1C) RQ (ε − u) [−RP(1C) (ε − u)]

+

(1) n(1) e (ε − 2u) − nh (ε) . (1C) RP (ε − u)

(7.11)

In an equilibrium the quasiparticles in the superconductor follow the Fermi (C) distribution, then n(C) e (ε) = nF (ε + u) = nh (ε + 2u) and Eqs. (7.10)–(7.11) reduce to Ie (ε) =

n(1) (ε + 2u) − n(1) nF (ε − u) − n(1) e (ε) e (ε) + h , (1C) R+ (ε + u) RP(1C) (ε + u)

(7.12)

−1 −1 where the interjacent resistances are defined as RQ (P ) (ε) = {R− (ε) ± ¯ −1 (ε)}/2; here 1/R± (ε) = [N2 N1 ∓ M ± M ± ]/R, Nj (ε) = Re cosh θj , R + 2 1 Mj+ (ε)+i Mj−(ε) = sin θj and j = 1, 2 labels the different sides of the interface.

94


The circuit representation of Eq. (7.12) is shown in Fig. 7.2(b). This is the boundary conditions and the corresponding circuit used in Ref. [15].

7.5 Conductance renormalization procedure We consider the normal metal between the left superconducting lead SL and the superconducting island, SC , see Fig. 7.3. The boundary conditions, Eqs. (7.10)–(7.11), relate electron and hole distribution functions at the right NS and left NS interfaces. Below we relate electron and hole distribution functions at x = 0 and x = d building the effective circuit, where d is the length of the normal layer. At the first step we neglect the proximity effect change of the junction resistance and take D± = 1 everywhere in the normal layer. Then, Ie(h) = σ1 ∇ne(h) , ne(h) = 0 and therefore Ie(h) (ε) = [ne(h) (d, ε) − ne(h) (x = 0, ε)]/R1 ,

(7.13)

where R1 = σ1 /d is the normal resistance of the N1 -layer. Equation (7.13) resembles the Ohm law for the resistor R1 , but where the role of the voltages play the distribution functions at the ends of the resistor. Equation (7.13) is approximate because it neglects the proximity renormalization of the normal layer conductivity [26]. It was shown in Ref. [15] for a SNS junction that the replacement of ne(h) (x = {0, d}) by the properly chosen proximity renormalized distribution functions makes Eq. (7.13) accurate. We show below that this idea is applicable when electron and hole distribution functions in the superconductors essentially deviate from the Fermi functions and when the electron and hole currents can not be in general related by a shift of the energy like in SNS junction. At the left NS-interface the spectral currents, Ie (ε) and Ih (ε + 2u) are related by the Andreev process, see Fig. 7.2a. It follows from Eqs. (7.5)–(7.7) (1C) that the combination of the quasiparticle currents, I± (ε) = Ie (ε) ± Ih (ε + Ih(ε+2u) SL

SC

Ie(ε)

u

Ih(ε+2u') Ie(ε−2u')

u'

Ih(ε) 0

ε

Ie(ε−2u) d

x

Fig. 7.3. Illustration of the spectral currents flow in the normal metal (white area) surrounded by the superconductors (grey area). The black boxes at the interfaces encode the boundary conditions picture like it is shown in Fig. 7.2.


95

(1C) (1C) 2u) = σ1 D± (ε + u)∇n(1C) = 0. ± (ε + u), conserve in the normal metal: ∇I± Integrating the last equation over x we get,

d

(1C)

I±

0

dx = σ1 [n± (d) − n± (x)], D± (x) (1C)

(7.14)

where n(1C) ± (ε) = ne (ε) ± nh (ε + 2u). Equation (7.14) can be equivalently rewritten: (1C) (1C) I± (d − x) = σ1 [¯ n(1C) ± (d) − n± (x)], (1C) (1C) (1C) n ¯ (1C) ± (d) ≡ n± (d) − m± I± (ε + u),

(7.15) (7.16)

where m(1C) = ±

1 σ1

d

1 (1C) D± (x)

0

− 1 dx.

(7.17)

Here the variable x occupies the domain ξN x d − ξN where the Cooper pair wave functions from the left and right superconductors, see Fig. 7.3, do not overlap. At these values of x, the angle θ(x) → 0, D± (x) → 1 and we can therefore substitute x by 0 in the integral written in Eq. (7.14). (1C) (ε + u) = Ie (ε) ± Ih (ε + 2u) we finally get Taking into account that I± the following important result: (1C) (d − x) Ie (ε) = σ1 [¯ n(1C) e (d) − ne (x)],

(7.18)

where (1C) (d) = n(1C) (ε + u) − Ih (ε + 2u) m(1C) n ¯ (1C) e e (d) − Ie (ε) me h (ε + u) .

(7.19)

(1C) (1C) Here m(1C) e(h) = [m+ ± m− ]/2. Applying the procedure, Eqs. (7.14)–(7.19), to the left NS interface in Fig. 7.3, we find for ξN x d − ξN :

x Ie (ε) = σ1 [n(1L) ¯ (1L) e (x) − n e (x = 0)] ,

(7.20)

where (1L) (1L) n ¯ (1L) e (x = 0) = ne (x = 0) + Ie (ε) me (ε + u )

+ Ih (ε + 2u ) m(1L) h (ε + u ).

(7.21)

(1L) (1L) Here m(1L) e(h) = [m+ ± m− ]/2,

m(1L) ± =

1 σ1

0

d

1 (1L) D± (x)

− 1 dx.

(7.22)

96

N. M. Chtchelkatchev et al. (1)

Equations (7.18)–(7.20) show how ne depends on x in the central part of the normal layer in Fig. 7.3. Equation (7.18) must be consistent with Eq. (7.20). The only way to satisfy this condition is the following one: Ie (ε) =

n ¯ (1C) ¯ (1L) e (d) − n e (x = 0) , R1

(7.23)

where R1 = d/σ1 is the normal resistance of the N-layer and we used that (1C) n(1L) e (x) = ne (x). The condition Eq. (7.23) resembles the Ohm law. It allows to relate the distribution functions at x = 0 and x = d. Similar condition holds for Ih : Ih (ε) =

−n ¯ (1L) n ¯ (1C) h h , R1

(7.24)

where (1C) (1C) = n(1C) n ¯ (1C) h h (d) − mh (ε − u) Ie (ε − 2u) − me (ε − u) Ih (ε) , (7.25)

(1L) n ¯ (1L) = n(1L) + m(1L) h h h (ε − u ) Ie (ε − 2u ) + me (ε − u ) Ih (ε)] .

(7.26)

The last step is the formulation of the boundary conditions at the NS interfaces in terms of the distribution functions with bars. Using the I± notations we can rewrite the boundary conditions, Eqs. (7.10)–(7.11), in the compact form (1C) I± (ε) =

(1C) n(C) ± (ε) − n± (ε) . (1C) R± (ε + u)

(7.27)

Then it follows from Eq. (7.15) that we can write: n(C) − n ¯ (1C) (1C) ± (ε + u) = ¯ ± I± , (1C) R± (ε + u) ¯ (1C) (ε) = m(1C) (ε) + R(1C) (ε). R ± ± ±

(7.28) (7.29)

The same form has the boundary condition at x = 0: (L) n ¯ (1L) (1L) ± − n± , (ε + u ) = ¯ (1L) I± R± (ε + u ) (1L) (1L) ¯± (ε) = m(1L) R ± (ε) + R± (ε).

(7.30) (7.31)

It follows that the physical meaning of m± terms is the proximity effect contribution to the NS interface resistance, see [15]. It is more convenient to work with the boundary conditions for Ie(h) rather then with those for I± . Then one can use Eqs. (7.10), (7.11) but with n(1L) e(h) → (1C) (1C) ¯ (1C) (1C) ¯ (1C) , where, for example, R ¯ (1C) = 2R ¯− ¯ R and R → R /[ R n ¯ (1L) + + ± e(h) Q(P) Q(P) Q(P) (1C) ¯− R ].


97

7.6 Retarded and advanced Greens functions evolution in normal metals and superconductors Having formulated the boundary conditions for the distribution functions we turn now to advanced and retarded Greens functions behaviors. Normal layers in experimental SNS junctions and SNS arrays, see Ref. [17, 19], connect with superconductors like it is shown in Fig. 7.4. The junctions of this type are usually referred to as “weak-links”. [27, 28] Boundary conditions for retarded and advanced Greens functions, Eq. (7.2), can be simplified in this case: retarded and advanced Greens functions at superconducting sides of NS boundaries can be substituted by Greens function from the bulk of the superconductors. These “rigid” boundary conditions approximation is reasonable because the magnitude of the current is much smaller than the critical current of the superconductor [this is assumed] and the current entering the superconductor from narrow normal metal wire with the width comparable with the Cooper pair size. There are also other cases when the rigid boundary conditions are correct, for example, if the NS boundary has the small transparency due to, e.g., an insulator layer at the NS interface. The recipe telling how one should evaluate θ(x) in the normal metal near the NS boundary, where the rigid boundary conditions hold, can be taken from e.g. Ref. [15], and we reproduce briefly their result for the completeness. We will write down θ(x) near the right NS boundary (see Fig. 7.3) taking u = 0. In the superconductor, θS = atanh (Δ/ε), where Δ is the gap. The value of θN = θ(x = 0) in the normal metal side should be found from the equation: θN iΔ = 0, (7.32) W sinh(θN − θS ) + 2 sinh ε + i/2τσ 2

I

I w

S

N

Fig. 7.4. Typical experimental array of SNS junctions [17, 19]. This type of the link enables us to use the rigid boundary conditions for the retarded and advanced Greens functions.

98


where W = RΔ /RNS , RΔ = ξΔ /σN , (ξΔ = D/Δ) is the resistance of the normal metal layer with the width ξΔ . Here τσ is the pair breaking rate[29] [e.g., induced by electron-phonon or electron-electron interactions] and RNS is the normal resistance of the interface. Then the solution for θ(x > 0) is the following: x θ θN (7.33) tanh = exp − √ tanh , 4 4 ξε i where ξε = D/[2(ε + i/2τσ )]. The effective conductances g¯± should be expressed through θ found from Eqs. (7.32)–(7.33).

7.7 Spectral current flow through the superconducting grains The circuit shown in Fig. 7.5a is the graphic representation of the boundary conditions to Eq. (7.2) at the edges of the superconducting island. It is constructed from the circuit units shown in Fig. 7.2a. We consider the case where the size of the superconducting island is less than the charge imbalance length, and therefore the coordinate dependence of the quasiparticle distribution functions at the island can be neglected. Solving the Kirchhoff equations RQ (ε+u) –R

Rp(1C)(ε+u)

Ih(ε+2u) )

Ie(ε)

RB(ε+u) (1)

ne (ε)

)

)

(2C

(ε+

u)

–R P

u)

(ε+

SC

(1C)

RQ (ε+u)

)

–R

)

(2)

ne (ε)

Ie(ε)

c

2)

u)

Ih(ε+2u)

(ε+ u)

nh (ε+2u)

B (ε+

RD(ε+u)

(2C

(2C) RQ (ε+u)

(C)

ne (ε)

–R

ε+u)

–R B(

nh (ε+2u)

u (ε+ P

RD(ε+u)

(1)

nh (ε+2u) Ih(ε+2u)

(1C

(2)

(2C)

RQ (ε+u)

–R P

Ie(ε) n (1)(ε) e

b

(1C

P

(C)

nh (ε+2u)

Rp(2C)(ε+u)

(1C)

nh(1)(ε+2u)

Ie(ε)

Ih(ε+2u)

RB(ε+u)

a

Ie(ε) (2)

ne (ε)

Ih(ε+2u)

Ie(ε)

Ih(ε+2u)

Fig. 7.5. (a) Effective circuit representing current conversion at the interfaces of the central superconducting island SC . Resistors, RP and RQ stand for an Andreevand a normal processes respectively. The role of voltages at the nodes is played by the electron and hole distribution functions. (b) An illustration of the boundary conditions Eqs. (7.34)–(7.36) in terms of a pyramid-circuit is given in this figure. Electron and hole currents entering the left side of the pyramid flow in one normal ¯D layer, the right currents flow in the other normal layer. The effective resistance R describes the “direct” quasiparticle transmission from one normal layer to the other ¯ B describes Andreev processes. (c) through the superconductor and the resistance R Equivalent 3D-sketch of the circuit (b).


99

for the circuit shown in Fig. 7.5a we exclude the quasiparticle distribution functions corresponding to the superconducting island and express the spectral currents through the quasiparticle distribution functions in the normal layers: (2)

Ie (ε) =

(1)

(2)

(1)

¯ (ε + 2u) − n ¯ h (ε + 2u) n ¯ e (ε) − n ¯ e (ε) n + h , ¯ ¯ RD (ε + u) RB (ε + u) (2)

(1)

(2)

(7.34)

(1)

¯ h (ε) n n ¯ h (ε) − n ¯ e (ε − 2u) − n ¯ e (ε − 2u) + , ¯ ¯ RD (ε − u) RB (ε − u) −1 1 1 = 2 ¯ (1C) ¯ (2C) ± ¯ (1C) ¯ (2C) . R+ + R+ R− + R−

Ih (ε) =

(7.35)

¯ D(B) R

(7.36)

¯ D describes the “direct” quasiparticle transmission The effective resistance R from one normal layer to the other through the superconductor and the resis¯ B describes the Andreev processes, see Fig. 7.5b. Note that the direct tance R and indirect transmissions here are different from the so-called “elastic cotunneling” and “crossed Andreev tunneling” [23, 30] processes where Bogoliubov quasiparticles tunnel below the gap through a thin (with the width of the order of the Cooper pair size) superconducting layer. The probability of these tunneling processes decreases exponentially if the width of the superconducting layer exceeds the Cooper pair size. and they occur without generating supercurrent across a superconductor (the supercurrent flows “virtually”). The size of superconducting islands of the SNS arrays that we consider here exceed well the Cooper pair size, and the current of the quasiparticles with the energies below the hap converts at the NS interface into the supercurrent across the S-islands and then transforms again into the quasiparticle current at the opposite SN-interface.

7.8 Recurrent relations We have demonstrated that there is a direct correspondence between the effective electric circuit and the solution of the Usadel equations with the appropriately chosen boundary conditions. The effective circuit describing transport in SNSNS-array is shown in Fig. 7.6. We choose the direction of the current flow in such a way that the electron, Ie , and the hole, Ih , currents go in opposite directions. The expression for the total current then assumes the form: 1 dε (Ie + Ih ) . I(V ) = − (7.37) 2e The spectral currents Ie and Ih satisfy in general the relation: Ie (ε) = − Ih (ε)|V →−V . Similarly, ne (ε) = nh (ε)|V →−V , ensuring the identity I(−V ) = − I(V ).

N. M. Chtchelkatchev et al. RQ(1C)(ε+u)

nh(1)(ε+2u)

–R

Ih(ε+2u)

Rp(1C)(ε+u)

P

Ie(ε)

b

)

(1C

Ie(ε)

RB(ε+u) ne(1)(ε)

) ε (2C ( –R P – R

)

(ε+

u) (ε+

nh(2)(ε+2u)

) +u

u)

SC

P

ne(1)(ε)

RQ(1C)(ε+u)

ne(C)(ε)

RD(ε+u)

Ih(ε+2u)

(2C

(2C) RQ (ε+u)

)

(ε+ u)

B (ε+

u)

(2)

ne (ε)

RD(ε+u)

Ie(ε)

(2)

ne (ε)

Ie(ε)

c

nh(2)(ε+2u)

–R

)

+u R B(ε

–

RQ(2C)(ε+u)

(1C

–R P

nh(1)(ε+2u) Ih(ε+2u)

nh(C)(ε+2u)

Rp(2C)(ε+u)

a

RB(ε+u)

100

Ih(ε+2u)

Ie(ε) Ih(ε+2u) Ie(ε)

Ih(ε+2u)

Fig. 7.6. MAR in a SNSNS array. The graph shows the effective circuit for quasiparticle currents Ie and Ih in the energy space. The role of voltages here play quasiparticle distribution functions. Boxes, triangles and ovals play the role of effective resistances that come from Usadel equations and their boundary conditions.

Writing down the Kirchhoff’s equations for potential distribution at the circuit in Fig. 7.6 we arrive at the recurrent relations, see Appendix A: R(ε, −u, −V )Ih (ε) − ρ(◦) (ε − u)Ie (ε − 2u) −ρ() (ε)Ie (ε) − ρ() (ε − V )Ie (ε − V ) = nF (ε) − nF (ε − V ),

(7.38)

R(ε, u, V )Ie (ε) − ρ(◦) (ε + u)Ih (ε + 2u) −ρ() (ε)Ih (ε) − ρ() (ε + V )Ih (ε + 2V ) = nF (ε + V ) − nF (ε). (7.39) Here the effective resistance R = R1 + R2 + ρ(◦) , where ¯ (1L) + R ¯ (1C) + R ¯ (2C) + R ¯ (2R) }, {R ρ(◦) = (1/2) α,ε+u α,ε+u α,ε+V α,ε

(7.40)

α=± (1C)

(2C) (1C) (2C) ¯+ ¯− ¯− +R −R −R },

(7.41)

(2R)

(2R) ¯− −R },

(7.42)

(1L)

¯ (1L) }. −R −

(7.43)

¯+ ρ(◦) = (1/2){R ¯+ ρ() = (1/2){R ¯ ρ() = (1/2){R +

In the normal state of the array (or if |ε| Δ) R reduces to a normal resistance of the array whereas ρ() and ρ() vanish. Then we find from Eqs. (7.38)–(7.39) that Ih (ε) = [nF (ε) − nF (ε − V )]/R, and Ie (ε) = [nF (ε + V ) − nF (ε)]/R that with Eq. (7.37) reproduces the Ohm’s law, I = V /R.


101

It is easy to find the island potential in the case of symmetrical array when the transitivity of the island-normal metal interfaces are equal as well as the transitivity of the lead-normal metal interfaces and R1 = R2 . Then the ¯ (1R) , R ¯ (1C) = R ¯ (2C) and for the symmetry reasons, u = V /2. ¯ (1L) = R resistances R ± ± ± ± At the same time the recurrent relations Eqs. (7.38)–(7.39) become invariant under the substitution Ie (ε − V ) = Ih (ε) and reduce to the relation: R(ε, V )Ie (ε) − ρ() (ε)Ie (ε − V ) − ρ() (ε + V )Ie (ε + V ) = nF (ε + V ) − nF (ε), (7.44) where R(ε, V ) ≡ R(ε, V /2, V ) − ρ(◦) (ε + V /2) ¯ (1L) + R ¯ (2R) }, = RN (ε) + (1/2) {R α,ε+V α,ε

(7.45)

α=±

¯ (1C) ¯ (2C) where RN (ε) = R1 + R2 + R −,ε+V/2 + R−,ε+V/2 . The recurrent relation, Eq. (7.44), is similar to that of a (symmetric) SNS junction, see Refs. [15, 22] and Appendix B, but in our case the normal resistance RN (ε) becomes energy dependent [22]. In other words, a symmetric SNSNS array has the same transport properties as a single (symmetric!) SNS junction, but with the energy dependent resistance of the normal layer. The ¯ (1C) has singularities at the energy corresponding to the imbalance resistance R −,ε gap edges of the superconducting island in the center of our SNSNS array. This is the origin of the subharmonic singularities in the current-voltage characteristics at voltages 2Δ/V = n/2, n = 1, 2, . . . , contrasting the “conventional values” in an SNS junction determined by the relations 2Δ/V = n. It follows from Eq. (7.45) the unusual subharmonic singularities should disappear if the resistance of the normal layer greatly exceeds the resistance of the SN (1C) ¯ −,ε+ ¯ (2C) interfaces. Then R1 R V/2 + R−,ε+V/2 and the central superconducting island of the SNSNS array effectively “disappears” and the array completely transforms into a SNS junction [22].

7.9 Results and discussion Calculation of the current-voltage characteristics I(V ) requires numerical solving of the recurrent relations, Eqs. (7.38)–(7.39). To accomplish the numerical task, we have developed a computational scheme allowing to bypass instabilities caused by the non-analytic behavior of the spectral currents Ie(h) (ε), which poses the major computational challenge. The procedure is as follows: first, we fix certain chosen energy ε and identify the set of energies connected through the equations in the given energy interval, solving afterwards the resulting subsystem of equations. We then repeat the procedure, until the

102

N. M. Chtchelkatchev et al. 1.4 ’testf.dat’ u 1:10 1.2

a(E,V=1.0)

1 0.8 0.6 0.4 0.2 0 –5

–4

–3

–2

–1

0 E

1

2

3

4

5

Fig. 7.7. Left panel: Differential resistances as functions of the applied voltage Vtot (around n = 1 in Eq. (1)) for the SN1 SN2 S junction. The fractions 3/4, and 4/5, and 9/10 represent the ratios of resistances of the normal regions, R1 /R2 . The differential resistance dV /dI of the SN1 SN2 S junction demonstrates the pronounced SAT spike at Vtot = 2Δ/e, irrespectively to the partial voltage drops. The SAT spike is sandwiched between the two additional spikes corresponding to individual MAR processes occurring at junctions SN1S and SN2S for m1 , m2 = 2. The voltage positions of these features depend on R1 /R2 . Right panel: The corresponding dV /dI(V1 + V2 ) for the two SN1 S and SN2 S junctions in series as they would have appeared in the absence of the synchronization process, i.e. in the case where LC > ε . These dV /dI dependencies were calculated following [15] (with transmissivity W=1).

required energy resolution of δε = 10−5 Δ is achieved. Typically, up to 106 linear equations had to be solved for every given voltage, but the complexity of the coupled subsystem depends on the commensurability of u and V . Figure 7.7 shows the comparative results for the SNSNS junction and two SNS junctions in series. The latter corresponds to the case where the size of the central island well exceeds the energy relaxation length, LC > ε . We display the differential resistances as functions of the applied voltage, which demonstrate the singularities in Andreev transmission more profoundly than the I-V curves. There is a pronounced SAT spike in the dV /dI for an SNSNS junction at Vtot = 2Δ/e. The spike appears irrespectively to the partial voltage drops in the normal regions and is absent in the corresponding curves representing two individual MAR processes at the junctions SN1 S and SN2 S. The resonant voltages of the SAT singularities can be found from the consideration of the quasiparticle trajectories in the space-energy diagrams. Such a diagram for the first subharmonic, n = 1 and ratio R1 /R2 = 3/4 is given in Fig. 7.1. A quasiparticle starts from the left superconducting electrode with the energy ε = −Δ to traverse N1 , and the quasiparticle that starts from


103

the central island Sc with the same energy as the incident one to take up upon the current across the island N2 , and hit SR with the energy ε = Δ (the ABCD path, the corresponding path for the hole is D C B A ). In general, relevant trajectories yielding resonant voltages of Eq. (7.1) have the following structure: they start and end at the BCS quasiparticle density of states singular points (ε = ±Δ), contain the closed polygonal path, which include MAR staircases in the normal parts and over-the-gap transmissions and Andreev reflections, and pass the density of states singular points at the central island. Apart from the main singularities (Eq. (7.1)), additional SAT satellite spikes appear at V = (2Δ/e)(p + q)/n, where p/q is the irreducible rational approximation of the real number r = R1 /R2 , (we take R1 < R2 ), and n (p + q). The achieved qualitative understanding enables us to observe that the manifestations of the SAT mechanism in an experimental situation becomes even more pronounced with the growth of the number of SNS junctions in the system. To see this, let us assume that the resistances of the normal islands in a chain of SNS junctions are randomly scattered around their average value R0 and follow Gaussian statistics with the standard deviation σR = σR0 , where σ is dimensionless. Accordingly, the dispersion of the distribution of the MAR resonant voltages is characterized by the same σ, and the MAR features get smeared. Let us distribute the voltage drop 2Δ/e among the n successive islands. Then the quasiparticle SAT path starts at the lower edge of the superconducting gap at island j, traverses n − 1 intermediate superconducting islands and hits the edge of the gap at the j + n-th island in the chain. √ n islands √ The standard deviation of the voltage drop on the grows as n resulting in a voltage deviation per one island ∝ 1/ n, i.e. √ the dispersion of the distribution of Vn drops with increasing n: σSAT = σ/ n. In contrast to the MAR-induced features, with an increase of n, the subharmonic spikes at voltages Vn per junction due to SAT processes become more sharp and pronounced.

7.10 Conclusions In conclusion, we have developed a nonequilibrium theory of the charge transfer across a central superconducting island in an SNSNS array and found that this island acts as Andreev retransmitter. We have shown that the nonequilibrium transport through an SNSNS array is governed by the synchronized Andreev transmission with the correlated conversion processes at the opposite NS interfaces of the central island. The constructed theory is a fundamental building unit for a general quantitative description of a large array consisting of many SNS junctions.

104


Acknowledgments We thank A. N. Omelyanchuk for helpful discussions. The work was supported by the U.S. Department of Energy Office of Science under the Contract N◦ . DE-AC02-06CH11357, and partially by the RFBR 10-02-00700, Russian President Science Support foundation mk-7674.2010.2, the Dynasty, Russian Federal Programs and the Programs of the Russian Academy of Science.

A Recurrent relations for the quasiparticle currents and distribution functions The Kirchhoff’s laws applied for the circuit in Fig. 7.6 generate the following linear system of equations: ¯ (2R) n ¯ (2R) ¯ (2R) nF (ε − V ) − n h,ε − n e,ε−2V e,ε−2V + , Ie (ε − 2V ) = (2R) (2R) ¯ ¯ R R+,ε

(7.46)

P,ε−V

Ih (ε) =

(4) −n ¯ (2R) n ¯ (2R) n ¯ h − nF (ε − V ) h e,ε−2V + , ¯ (2R) ¯ (2R) R R

(7.47)

+,ε−V

P,ε−V

(2)

Ih (ε) =

¯ (2R) n ¯h − n h , R2 (1)

(7.48)

(2)

(1) (2) ¯ h,ε n ¯ h,ε − n n ¯ e,ε−2u − n ¯ e,ε−2u Ih (ε) = ¯ + , ¯ B,ε−u RD,ε−u R (2)

(7.49)

(1)

(2) (1) ¯ h,ε − n ¯ h,ε n ¯ e,ε−2u − n ¯ e,ε−2u n + ¯ , ¯ RD,ε−u RB,ε−u

(7.50)

Ih (ε) =

n ¯ (1L) −n ¯ (1C) h h R1

(7.51)

Ih (ε) =

−n ¯ (1L) ¯ (1L) n ¯ (1L) nF (ε) − n e h h + , (1L) (1L) ¯ P,ε ¯ R R +,ε

(7.52)

¯ (2R) n ¯ (2R) n ¯ (2R) e h,ε+2V − n h,ε+2V − nF (ε + V ) + , (2R) ¯ ¯ (2R) R R

(7.53)

¯ (2R) n ¯ (2R) ¯ (2R) nF (ε + V ) − n e,ε h,ε+2V − n e,ε + , ¯ (2R) ¯ (2R) R R

(7.54)

n ¯ (2R) ¯ (2C) e,ε − n e,ε , R2

(7.55)

Ie (ε − 2u) =

and Ih (ε + 2V ) =

P,ε+V

Ie (ε) =

P,ε+V

Ie (ε) =

P,ε+V

P,ε+V


n ¯ e,ε+2u − n ¯ e,ε+2u n ¯ (2C) ¯ (1C) e,ε − n e,ε + , (12) (12) ¯ ¯ R R (2C)

Ie (ε) =

D,ε+u

Ih (ε + 2u) =

n ¯ (1C) ¯ (2C) h,ε+2u − n h,ε+2u ¯ (12) R D,ε+u

105

(1C)

(7.56)

B,ε+u

+

n ¯ (1C) ¯ (2C) e,ε − n e,ε , (12) ¯ R

(7.57)

B,ε+u

Ie (ε) =

¯ (1L) n ¯ (1C) e,ε − n e,ε , R1

(7.58)

Ie (ε) =

¯ (1L) n ¯ (1L) n ¯ (1L) e,ε − n h,ε e,ε − nF (ε) + . (1L) ¯ ¯ P(1L) RP,ε R ,ε

(7.59)

Equations (7.46)–(7.59) are the recurrent relations (i.e. the relations coupling the functions at energy ε with the functions at ε ± V ) for the currents and the distribution functions. It follows from Eqs. (7.48), (7.51) that Ih,ε [R1 + R2 ] = n ¯ (2C) −n ¯ (2R) +n ¯ (1L) −n ¯ (1C) h h h h .

(7.60)

The distributions functions entering Eq. (7.60) we can express below through the currents. Combining Eqs. (7.49)–(7.50) we get,

¯ I,ε−u + Ie,ε−2u R ¯ D,ε−u R ¯ B,ε−u ¯ D,ε−u R Ih,ε R (1C) (2C) n ¯h − n ¯h = . (7.61)

2

2 ¯ D,ε−u ¯ B,ε−u − R R At the same time from Eqs. (7.52), (7.59) follows that (1L)

(1L) (1L) ¯ P,ε + R ¯ +,ε ¯ +,ε I R − Ih R (1L) (1L) e ¯ , n ¯ h,ε = nF (ε) + R+,ε ¯ (1L) + R ¯ P(1L) 2R ,ε +,ε

(7.62)

and finally from Eqs. (7.46)–(7.47) we get ¯ (2R) n ¯ (2R) h,ε = nF (ε − V ) + R+,ε−V

¯ (2R) ¯ (2R) + Ih R ¯ (2R) + R −Ie,ε−2V R P,ε−V +,ε−V +,ε−V ¯ (2R) + R ¯ (2R) 2R P,ε−V +,ε−V

.

(7.63) Combining Eq. (7.60) and Eqs. (7.61)–(7.63) we find the recurrent relation for the currents, Eq. (7.38). Similar procedure helps to derive Eq. (7.39).

B Charge transport in SNS junctions We discuss below the transport properties of SNS and SNN junctions to make a mapping between our technique and the well-known results obtained before us.

106


The recurrent relations, Eqs. (7.38)–(7.39), solve the transport problem in a SNS junction in the incoherent regime. Then there is no island, so ρ(◦) = 0 and we should remove the island resistances with the indices (1C) and (2C) from the coefficient functions of the recurrent relations. So, R(ε, −V )Ih (ε) − ρ() (ε)Ie (ε) − ρ() (ε − V )Ie (ε − V ) = nF (ε) − nF (ε − V ), (7.64) R(ε, V )Ie (ε) − ρ() (ε)Ih (ε) − ρ() (ε + V )Ih (ε + 2V ) = nF (ε + V ) − nF (ε). (7.65) where, for example, R(ε, V ) = R1 + R2 + (1/2)

(1L) ¯ α,ε ¯ (2R) }. {R +R α,ε+V

(7.66)

α=±

Equations (7.64)–(7.65) are invariant under the following transformation, ¯ (1L) ↔ Ie (ε − V ) → Ih (ε), if at the same time we exchange the resistances, R ±,ε ¯ (1R) . Thus the relation, Ie (ε − V ) = Ih (ε) and the reduction of the recurrent R ±,ε relations to the one equation for Ie or for Ih as it was done in Ref. [15]: R(ε, V )Ie (ε) − ρ() (ε)Ie (ε − V ) − ρ() (ε + V )Ie (ε + V ) = nF (ε + V )−nF (ε), (7.67) (1L) (1R) ¯ ±,ε ¯ ±,ε holds only for a symmetric SNS junction with R =R . To summarize here our consideration [summarized by the recurrent relations Eqs. (7.64)–(7.65)], reduces to that presented in Ref. [15] only in the case where the contacts are symmetric and the assumption Ie (ε − V ) = Ih (ε) holds.

References 1. A. F. Andreev, Zh. Eksp. Teor. Fiz. 46 (1964) 1823 [Sov. Phys. JETP 19 (1964) 1228]. 2. J. M. Rowell and W. E. Feldmann, Phys. Rev. 172, 393 (1968). 3. P. E. Gregers-Hansen, E. Hendricks, M. T. Levinsen, and G. R. Pickett, Phys. Rev. Lett. 31, 524 (1973). 4. W. M. van Huffelen, T. M. Klapwijk, D. R. Heslinga, M. J. de Boer, and N. van der Post, Phys. Rev. B 47, 5170 (1993). 5. A. W. Kleinsasser, R. E. Miller, W. H. Mallison, and G. B. Arnold, Phys. Rev. Lett. 72, 1738 (1994). 6. E. Scheer, P. Joyez, D. Esteve, C. Urbina, and M. H. Devoret, Phys. Rev. Lett. 78, 3535 (1997). 7. J. Kutchinsky, R. Taboryski, T. Clausen, C. B. Sørensen, A. Kristensen, P. E. Lindelof, J. Bindslev Hansen, C. Schelde Jacobsen, and J. L. Skov, Phys. Rev. Lett. 78, 931 (1997).


107

8. A. Frydman and R. C. Dynes, Phys. Rev. B 59, 8432 (1999). 9. T. Hoss, C. Strunk, T. Nussbaumer, R. Huber, U. Staufer, and C. Sch¨ onenberger, Phys. Rev. B 62, 4079 (2000). 10. T. I. Baturina, Z .D. Kvon, R. A. Donaton, M. R. Baklanov, E. B. Olshanetsky, K. Maex, A. E. Plotnikov, J. C. Portal, Physica B 284, 1860 (2000). 11. Z. D. Kvon, T. I. Baturina, R. A. Donaton, M. R. Baklanov, K. Maex, E. B. Olshanetsky, A. E. Plotnikov, J. C. Portal, Phys. Rev. B 61, 11340 (2000). 12. T. M. Klapwijk, G. E. Blonder, and M. Tinkham, Physica B+C (Amsterdam) 110, 1657 (1982). 13. M. Octavio, M. Tinkham, G. E. Blonder, and T. M. Klapwijk, Phys. Rev. B 27, 6739 (1983). 14. K. Flensberg, J. Bindslev Hansen, and M. Octavio, Phys. Rev. B 38, 8707 (1988). 15. E. V. Bezuglyi, E. N. Bratus’, V. S. Shumeiko, G. Wendin, H. Takayanagi, Phys. Rev. B 62, 14439 (2000). 16. J. C. Cuevas, J. Hammer, J. Kopu, J. K. Viljas, and M. Eschrig, Phys. Rev. B 73, 184505 (2006). 17. T. I. Baturina, Z. D. Kvon, and A. E. Plotnikov, Phys. Rev. B 63, 180503(R) (2001). 18. T. I. Baturina, Yu. A. Tsaplin, A. E. Plotnikov, and M. R. Baklanov, JETP Lett. 81, 10 (2005). 19. T. I. Baturina, D. R. Islamov, and Z. D. Kvon, JETP Lett. 75, 326 (2002). 20. J. Fritzsche, R. B. G. Kramer, and V. V. Moshchalkov, Phys. Rev. B 80, 094514 (2009). 21. T. I. Baturina, A. Yu. Mironov, V. M. Vinokur, N. M. Chtchelkatchev, A. Glatz, D. A. Nasimov, A. V. Latyshev, Physica C (2009) doi:10.1016/j.physc. 2009.11.107. 22. N. M. Chtchelkatchev, JETP Lett. 83, 250 (2005). 23. G. Deutscher and D. Feinberg, Appl. Phys. Lett. 76, 487 (2000). 24. A.I. Larkin and Yu.N. Ovchinnikov, Sov. Phys. JETP 41, 960 (1975); ibid, 46, 155 (1977). 25. M. Yu. Kupriyanov and V. F. Lukichev, Zh. Eksp. Teor. Fiz. 94, 139 (1988) [Sov. Phys. JETP 67, 1163 (1988)]. 26. A.F. Volkov and T.M. Klapwijk, Phys. Lett. A 168, 217 (1992). 27. K.K. Likharev, Rev. Mod. Phys. 51, 101 (1979). 28. C.W.J. Beenakker, Phys. Rev. Lett. 67, 3836 (1991). 29. M.Tinkham, Introduction to superconductivity, McGraw-Hill, New York, 1996. 30. N.M. Chtchelkatchev, JETP Lett. 78, 230 (2003).

Part II

Superconductivity

8 JOSEPHSON EFFECT IN POINT CONTACTS BETWEEN TWO-BAND SUPERCONDUCTORS A. Omelyanchouk1 and Y. Yerin2 1

2

B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine [email protected] B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine [email protected]

Abstract. The microscopic theory of Josephson effect in point contacts between two-band superconductors is developed. The general expression for the Josephson current, which is valid for arbitrary temperatures, is obtained. We considered the dirty superconductors with interband scattering, which produces the coupling of the Josephson currents between different bands. The influence of phase shifts and interband scattering rates in the banks is analyzed near critical temperature Tc . It is shown that for some values of parameters the critical current can be negative, which means the π-junction behavior.

Key words: two-band superconductor, Usadel equations, Josephson effect, critical current, π-contact.

8.1 Introduction Discovery of high-temperature superconductivity in iron-based compounds [1] have expanded a range of multiband superconductors besides well-known magnesium diboride M gB2 with Tc = 39 K [2]. Two-band superconductivity proposes new interesting physics. In a large number researches the specific effects in temperature behavior of the first and upper critical fields [3–5] and London penetration depth [3, 6, 7] were demonstrated. Mixed state and peculiar vortex core structure were studied in [8]. The coexistence of two distinctive order parameters |Φ1 | = Δ1 exp(iϕ1 ) and Φ2 = |Δ2 | exp(iϕ2 ) renewed interest in phase coherent effects in superconductors. In the case of two order parameters the question arises, what is the phase shift ϕ1 − ϕ2 between Φ1 and Φ2 ? From the minimization of free energy it follows that in homogeneous equilibrium state this phase shift is fixed to 0 or π, depending on the sign of interband J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 8,

111

112

A. Omelyanchouk and Y. Yerin

coupling. The phases ϕ1 and ϕ2 can be decoupled as small plasmon oscillations (Leggett mode [9]) or due to formation of phase slips textures in strong electric field [10]. The coherent current states and depairing curves have been calculated in [11], where it was shown the possibility of phase shift switching in homogeneous current state with increasing of the superfluid velocity vs . Such switching manifests itself in the dependence j(vs ) and also in Little-Parks effect [12]. The Josephson effect in superconducting junctions is the probe for research of phase coherent effects. The stationary Josephson effect in tunnel S1 -I-S2 junctions (I – dielectric) between two- and one-band superconductors have been studied recently in a number of articles [13–15]. Another basic type of Josephson junctions are the junctions with direct conductivity, S-C-S contacts (C – constriction). As was shown in [16–18] the Josephson behavior of S-C-S structures qualitatively differ from properties of tunnel junctions. In this paper we generalize KO theory [16, 17] of stationary Josephson effect in S-C-S point contacts for the case of two-band superconductors. Within the microscopic Usadel equations we calculate the Josephson current and study its dependence on the mixing of order parameters due to interband scattering and phase shifts in the contacting two-band superconductors.

8.2 Model and basic equations Consider the weak superconducting link as a thing filament of length L and diameter d, connecting two superconducting banks (Fig. 8.1). Such model describes the S-C-S (Superconductor-Constriction-Superconductor) contacts with direct conductivity (point contacts, microbridges), which qualitatively differ from the tunnel S-I-S junctions. On condition that d L and d min[ξ1 (0) , ξ2 (0)](ξi (T ) – coherence lengths) we can solve inside the filament (0 ≤ x ≤ L) a one-dimensional problem with “rigid” boundary conditions. At x = 0, L all functions are assumed equal to the values in homogeneous no-current state of corresponding bank. We investigate a case of two-band superconductor with strong impurity intraband scattering rates (dirty limit) and weak interband scattering. L SL

|Δ1|exp(if1R)

|Δ1|exp(if L) 1

|Δ2|exp(if2L)

d

SR

|Δ2|exp(if2R)

Fig. 8.1. A model of Superconductor-Constriction-Superconductor contact (S-C-S contact). The right and left banks are massive two-band superconductors connected by the thing filament of length L and diameter d.

8 Josephson effect in point contacts

113

In the dirty limit superconductor is described by the Usadel equations [19] for normal and anomalous Green’s functions g and f , which for two-band superconductor take the form [3]: ωf1 − D1 g1 ∇2 f1 − f1 ∇2 g1 = Δ1 g1 + γ12 (g1 f2 − g2 f1 ) , (8.1) ωf2 − D2 g2 ∇2 f2 − f2 ∇2 g2 = Δ2 g2 + γ21 (g2 f1 − g1 f2 ) .

(8.2)

Usadel equations are supplemented with self-consistency equations for order parameters Δi : ωD λij fj , (8.3) Δi = 2πT j

ω>0

and with expression for the current density j = −ieπT

ωD i

Ni Di (fi∗ ∇fi − fi ∇fi∗ ).

(8.4)

ω

Index i = 1, 2 numerates the first and second bands. Green’s functions gi and 2 fi are connected by normalization condition gi2 + |fi | = 1 and depend on x and the Matsubara frequency ω = (2n + 1) πT . Di are the intraband diffusivities due to nonmagnetic impurity scattering, Ni are the density of states on the Fermi surface for the electrons of the i-th band, electron-phonon constants λij take into account Coulomb pseudopotentials and γij are the interband scattering rates. There are the symmetry relations λ12 N1 = λ21 N2 and γ12 N1 = γ21 N2 . In considered case of short weak link L min[ξ1 (0) , ξ2 (0)] we can neglect all terms in the Eqs. (8.1), (8.2) except the gradient one. And using the normalization condition we have equations for f1,2

2

1 − |f1 |

d2 d2 f 1 − f1 2 2 dx dx

2 1 − |f1 | = 0,

(8.5)

2 d2 2 d 2 1 − |f2 | f2 − f2 2 1 − |f2 | = 0. (8.6) dx2 dx The boundary conditions for Eqs. (8.5) and (8.6) are determined by solutions of equations for Green’s functions in the banks:

⎧ 2 2 2 ⎪ L(R) L(R) L(R) L(R) L(R) L(R) L(R) ⎪ ⎪ , ωf = Δ1 1− f1 +γ12 1− f1 f2 − 1− f2 f1 ⎪ ⎨ 1

⎪ ⎪ L(R)2 L(R) L(R)2 L(R) ⎪ωf L(R) = ΔL(R) 1− L(R)2 +γ ⎪ 1− f2 f1 − 1− f1 f2 . f2 ⎩ 2 21 2 (8.7)

114


Introducing the phases of order parameters in banks L(R)

Δ1

L(R)

= |Δ1 | exp(iϕ1

L(R)

), Δ2

L(R)

= |Δ2 | exp(iϕ2

),

(8.8)

and writing fi (x) in Eqs. (8.5) and (8.6) as fi (x) = |fi (x)| exp(iχi (x)) we have R |fi (0)| = |fi | , |fi (L)| = |fi | , χi (0) = χL i , χi (L) = χi , L(R)

(8.9)

L(R)

are connected with |Δi | and ϕi through the where |fi | and χi Eq. (8.7). The solution of Eqs. (8.5)–(8.9) determines the Josephson current in the system. It depends on the phase difference on the contact L R L ϕ ≡ ϕR 1 − ϕ1 = ϕ2 − ϕ2 and from possible phase shifts in each banks L L L R R δϕ = ϕ1 − ϕ2 and δϕ = ϕR 1 − ϕ2 . The phase shift δϕ between the phases of the two order parameters in two-band superconductor can be 0 or π, depending on the sign on the interband coupling constants [20] and the values of the interband scattering rates. Equations (8.5), (8.6) with boundary conditions Eq. (8.9) admit simple analytical solution, and for the current (Eq. 8.4) we obtain j=

4eπT L

ω D ω

R χL 1 −χ1 2 χL −χR χL −χR 1−|f1 |2 sin2 1 2 1 +cos2 1 2 1 L R χ −χ |f1 | sin 1 2 1 L χ −χR χL −χR 1−|f1 |2 sin2 1 2 1 +cos2 1 2 1 χL −χR |f2 | cos 2 2 2

|f1 | cos

N 1 D1

(

× arc tan

( +

4eπT L

ω D ω

N 2 D2

× arc tan

)

)

R L χL −χR 2 +cos2 χ2 −χ2 2 2 2 χL −χR |f2 | sin 2 2 2 χL −χR χL −χR 1−|f2 |2 sin2 2 2 2 +cos2 2 2 2

(1−|f2 |2 )sin2

.

(

)

(8.10)

This general expression together with Eqs. (8.7) describes the Josephson current as function of gaps in the banks |Δi | and phase difference on the contact L R L ϕ ≡ ϕR 1 − ϕ1 = ϕ2 − ϕ2 . If we neglect the interband scattering γik , the equations (8.7) for f1,2 become decoupled and the current (Eq. 8.10) consists of two independent inputs from transitions 1 → 1 and 2 → 2. Thus in this case we have for each components the Josephson currents in KO theory with corresponding values of Δ1,2 . The interesting case is the mixing of different contributions due to the interband scattering. For arbitrary values of γik and arbitrary temperature T Eqs. (8.7) can be solved numerically. To study the effects of interband scattering we consider the Josephson current near critical temperature Tc .

8.3 Josephson current near critical temperature Tc Near Tc Eqs. (8.5)–(8.7) are considerably simplified and we have equations for f1,2 (x)

8 Josephson effect in point contacts

d2 d2 f1 = 0, 2 f2 = 0, 2 dx dx

115

(8.11)

with boundary conditions f1

L(R)

f2

L(R)

= =

L(R) L(R) +γ12 |Δ2 | exp iφ2 (ω+γ21 )|Δ1 | exp iφ1 ω 2 +(γ12 +γ21 )ω L(R) L(R) (ω+γ12 )|Δ2 | exp iφ2 +γ21 |Δ1 | exp iφ1 ω 2 +(γ12 +γ21 )ω

, (8.12) .

The current density (Eq. 8.4) with solutions of Eqs. (8.11) takes the form: ωD ∗ ∗ ieπT Ni Di fiL fiR − fiL fiR , j=− L ω i

(8.13)

L(R)

are related to the order parameters in the banks by where functions fi the expressions (Eqs. 8.12). The current density j (Eq. 8.13) consists of four partials inputs produced by transitions from left bank to right bank between different bands 2 L j11 ∼ |Δ1 | sin(ϕR 1 − ϕ1 ), L j22 ∼ |Δ2 |2 sin(ϕR 2 − ϕ2 ),

(8.14)

L j12 ∼ |Δ1 | |Δ2 | sin(ϕR 2 − ϕ1 ), L j21 ∼ |Δ1 | |Δ2 | sin(ϕR 1 − ϕ2 ).

The relative directions of components jik depend on the intrinsic phase shifts in the banks δϕL,R (Fig. 8.2). Introducing resistance of contact in a normal state: 2Se2 1 = (N1 D1 + N2 D2 ) , RN L

(8.15)

where S is the contact cross-section area, for the current components we have 1. For δϕL = 0 and δϕR = 0: πT I= × eRN (N1 D1 + N2 D2 ) + N2 D2

N 1 D1

ωD 2 (|Δ1 | (ω + γ21 ) + |Δ2 | γ12 ) ω

ωD (|Δ2 | (ω + γ12 ) + |Δ1 | γ21 )2 ω

(ω 2 + (γ12 + γ21 ) ω)

2

(ω 2 + (γ12 + γ21 ) ω) sin ϕ = Ic sin ϕ,

2

(8.16)

116


a

1

j11 j21

2

j12

SL

SR

j22

b

SL

j11 j21

1 2

c

SR

j12 j22

SL

j11

SR

j21

1 2

j12 j22

Fig. 8.2. Current directions in S-C-S contact between two-band superconductors. (a) – there is no shift between phases of order parameters in the left and right superconductors; (b) – π-shift is present in the right superconductor and is absent in the left superconductor; (c) – there is the π-shift of order parameters phases at the both banks.

2. For δϕL = π and δϕR = π: πT × I= eRN (N1 D1 + N2 D2 ) + N2 D2

N 1 D1

ωD 2 (|Δ1 | (ω + γ21 ) − |Δ2 | γ12 ) ω

ωD 2 (|Δ2 | (ω + γ12 ) − |Δ1 | γ21 )

(ω 2 + (γ12 + γ21 ) ω)

ω

2

(ω 2 + (γ12 + γ21 ) ω)

2

sin ϕ = Ic sin ϕ,

(8.17)

3. For δϕL = 0 and δϕR = π: πT × I= eRN (N1 D1 + N2 D2 ) + N2 D 2

N1 D 1

(ω 2 + (γ12 + γ21 ) ω)

1

ω

ωD 2 Δ21 γ21 − Δ22 (ω + γ12 )2 ω

ωD 2 Δ2 (ω + γ21 ) − Δ2 γ 2

2

2 12

(ω 2 + (γ12 + γ21 ) ω)2

sin ϕ = Ic sin ϕ.

(8.18)

From expressions (Eq. 8.18) it follows that at certain values of constants γ12,21 , ratio N2 D2 /N1 D1 and of the values of gaps |Δ1 | and |Δ2 | the critical

8 Josephson effect in point contacts I

117

dfL = 0 dfR = 0

Ic

f I p - junction f

Ic

dfL = 0 dfR = p

Fig. 8.3. Current-phase relations for different phase shifts in the banks.

current Ic can be negative. It means that in this case we have the so-called π-junction [21, 22] (see illustration in Fig. 8.3).

8.4 Conclusion The microscopic theory of Josephson effect in point contacts between twoband superconductors is developed. The general expression for the Josephson current, which is valid for arbitrary temperatures, is obtained. We considered the dirty superconductors with interband scattering. Interband scattering in the contacting superconductors produces the coupling of the currents between different bands. The influence of phase shifts and interband scattering rates in the banks is analyzed near critical temperature Tc . It is shown that for some values of parameters the critical current can be negative, which means the π-junction behavior.

Acknowledgment We acknowledge partial support from the FRSF (grant F28.21019) and NASU program nanostructures, nanomaterials and nanotechnologies.

References 1. Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, Iron-based layered superconductor La[O(1-x)F(x)]FeAs (x = 0.05-0.12) with T(c) = 26 K, J. Am. Chem. Soc. 130, 3296(2008).

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2. J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, Superconductivity at 39 K in magnesium diboride, Nature 410, 63 (2001). 3. A. Gurevich, Enhancement of the upper critical field by nonmagnetic impurities in dirty two-gap superconductors, Phys. Rev. B 67, 184515 (2003). 4. A. A. Gaolubov, A. E. Koshelev, Upper critical field in dirty two-band superconductors: Breakdown of the anisotropic Ginzburg-Landau theory, Phys. Rev. B 68, 104503 (2003). 5. M. Mansor, J.P. Carbotte, Upper critical field in two-band superconductivity,Phys. Rev. B 72, 024538 (2005). 6. A. A. Golubov, A. Brinkman, O.V. Dolgov, J. Kortus, and O. Jepsen, Multiband model for penetration depth in MgB2 , Phys. Rev. B 66, 054524 (2002). 7. V. G. Kogan and N. V. Zhelezina, Penetration-depth anisotropy in two-band superconductors, Phys. Rev. B 69, 132506 (2004). 8. A. E. Koshelev and A. A. Golubov, Mixed state of a dirty two-band superconductor: application to MgB2 , Phys. Rev. Lett. 90, 177002 (2003). 9. A. J. Leggett, Number-phase fluctuations in two-band superconductors, Progr. Theor. Phys. 36, 901 (1966). 10. A. Gurevich and V. M. Vinokur, Phase textures induced by dc-current pair breaking in weakly coupled multilayer structures and two-gap superconductors, Phys. Rev. Lett. 97, 137003 (2006). 11. Y. S. Yerin, A.N. Omelyanchouk, Coherent current states in a two-band superconductor, Low Temp. Phys. 33, 401 (2007). 12. Y.S. Yerin, S. V. Kuplevakhskii, and A. N. Omelyanchuk, Little–Parks effect for two-band superconductors, Low Temp. Phys. 34, 891 (2008). 13. D. F. Agterberg, Eugene Demler, and B. Janko, Josephson effects between multigap and single-gap superconductors, Phys. Rev. B 66, 214507 (2002). 14. Y. Ota, M. Machida, T. Koyama, and H. Matsumoto, Theory of heterotic superconductor-insulator-superconductor Josephson junctions between singleand multiple-gap superconductors, Phys. Rev. Lett. 102, 237003 (2009). 15. T. K. Ng, N. Nagaosa, Broken time-reversal symmetry in Josephson junction involving two-band superconductors, EPL 87, 17003 (2009). 16. I. O. Kulik, A. N. Omelyanchouk, Microscopic theory of Josephson effect in superconducting microbridges, Pis’ma Zh. Eksp. Teor. Fiz. 21, 216 (1975). 17. I. O. Kulik, A. N. Omelyanchouk, Josephson effect in superconducting bridges: microscopic theory, Fiz. Nizk. Temp. 4, 296 (1978). 18. S. N. Artemenko, A. F. Volkov, A. V. Zaitsev, Theory of nonstationary Josephson effect in short superconducting junctions, Zh. Eksp. Teor. Fiz. 76, 1816 (1979). 19. C. Usadel, Generalized Diffusion Equation for Superconducting Alloys, Phys. Rev. Lett. 25, 507 (1970). 20. M. E. Zhitomirsky and V.-H. Dao, Ginzburg-Landau theory of vortices in a multigap superconductor, Phys. Rev. B 69, 054508 (2004). 21. L. N. Bulaevskii, V. V. Kuzii, A. A. Sobyanin, Superconducting system with weak coupling to the current in the ground state, Pis’ma Zh. Eksp. Teor. Fiz. 25, 314 (1977). 22. A. A. Golubov, M. Yu. Kupriyanov, E. Il’ichev, The current-phase relation in Josephson junctions, Rev. Mod. Phys. 76, 411 (2004).

9 SUPERCONDUCTING NANOWIRES: NEW TYPE OF BCS-BEC CROSSOVER DRIVEN BY QUANTUM-SIZE EFFECTS A. A. Shanenko1 , M. D. Croitoru2 , A. Vagov2 , and F. M. Peeters3 1 2 3

TGM, Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium [email protected] University of Bayreuth, Institute of Theoretical Physics, D-95440, Bayreuth, Germany TGM, Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium [email protected]

Abstract. We show that a superconducting quantum nanowire undergoes a new type of BCS-BEC crossover each time when an electron subband approaches the Fermi surface. In this case the longitudinal Cooper-pair size drops by two-three orders of magnitude down to a few nanometers. This unconventional BCS-BEC crossover is driven by quantum-size effects rather than by tuning the fermion-fermion interaction.

Key words: nanowire, Cooper pairs, BCS-BEC crossover, quantum-size effects.

9.1 Introduction Present-day advances in nano-fabrication and nano-detection techniques bring experimental physics to a new level of sophistication, which makes it possible to revisit old classical problems. One of such problems is a crossover of a system of interacting fermions from the Bardeen-Cooper-Schrieffer (BCS) regime to the Bose-Einstein condensation (BEC), i.e., the celebrated BCSBEC crossover (see, for details, [1–3] and the recent review [4]). On the BCS side there are loosely bound extended Cooper pairs with characteristic size (i.e., the BCS coherence length) several orders of magnitude larger than the interparticle spacing. While on the BEC side, there are tightly confined fermion pairs, i.e., molecule-like states. Interest in the BCS-BEC crossover was renewed in the early 90’s due to the discovery of high-Tc superconductors with extremely small coherence length. Then, a decade later, successive cooling of trapped superfluid Fermi gases resulted in experimental observation of such a J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 9,

119

120

A. A. Shanenko et al.

crossover (see, e.g., [4]) realized by externally tuning the fermion interaction through a Feshbach resonance. There is another classical problem, dating back to the pioneering paper of Blatt and Thompson [5] and related to quantum-size oscillations of physical properties in quasi-1D (e.g., nanowires) and quasi-2D (e.g., nanofilms) superconductors. The physics behind such oscillations is as follows. Due to quantization of the transverse electron motion (for nanofilms, quantization of the perpendicular motion), the conduction band splits up into a series of subbands so that superconductivity is supported by a set of quantum channels. Such single-electron subbands move in energy while changing the nanowire/nanofilm thickness. Each time when a discrete electron level (i.e., the bottom of a subband) passes through the Fermi surface, we have an abrupt increase in the density of single-electron states at the Fermi level and, in turn, an enhancement of the basic superconducting characteristics (i.e., a superconducting resonance, see, e.g., [5, 6]). Recently, ultrathin single-crystalline Pb films were fabricated with atomically uniform thickness, where the quantumsize oscillations of the critical superconducting temperature and the second upper critical magnetic field were observed [7–9]. In the present paper we report about an interesting and unexpected link between the BCS-BEC crossover and quantum-size oscillations. Based on a numerical self-consistent solution of the Bogoliubov-de Gennes equations for a clean metallic cylindrical nanowire, we show that it undergoes a new type of BCS-BEC crossover each time when a transverse discrete single-electron level approaches the Fermi surface. In this case the longitudinal Cooper-pair size drops by two-three orders of magnitude down to a few nanometers. This unconventional BCS-BEC crossover is driven by quantum-size effects rather than by tuning the fermion interaction.

9.2 Quasiparticle spectrum in a quantum wire We start with preliminary remarks on the form of the quasiparticle spectrum in a superconducting quantum wire. Due to the splitting of the conduction band into a series of single-electron subbands, such an energy spectrum is very different from that in bulk: for any subband we get the corresponding quasiparticle branch. In particular, the single-electron states in a nanocylinder are governed by the set of quantum numbers ν = {j, m, k}, with j the radial quantum number, m the azimuthal quantum number and k the wavevector of the quasi-free electron motion along the nanowire. Longitudinal dispersion of the single-electron energy ξjmk (measuredfrom the Fermi level) and of the corresponding quasiparticle energy Ejmk =

2 ξjmk + Δ2jm [6] are sketched in

Fig. 9.1. Here Δjm (chosen as real) is the subband-dependent energy gap, as seen from Fig. 9.1(b). Recall that only the single-electron states situated in the Debye “window”, i.e., |ξjmk | < ωD , contribute to the superconducting

9 New type of BCS-BEC crossover driven by quantum-size effects

121

b

a +hwD

0,±4

0 0,±3

Ejmk (a.u.)

ξjmk (a.u.)

0,±4

–hwD 0,±2

0,±3

0,±2 0,0

0 ,±1

0,±1 j=0 ,m=0

0

Wavevector k > 0 (a.u.)

k0, ± 2 0 0

Wavevector k > 0 (a.u.)

Fig. 9.1. (Color online) Cylindrical superconducting nanowire: (a) sketch of single-electron energies ξjmk (measured from the Fermi level EF ) versus the wavevector of the longitudinal motion k for the three relevant subbands (j,m)=(0,0), (0, ±1) and (j, ±2); (b) quasiparticle energies Ejmk = ξjmk + Δjm as function of k for the same subbands.

quantities (see, e.g., Ref. [10]). In Fig. 9.1(a) the bottoms of the two subbands with (j, m) = (0, ±3) are in the vicinity of the Fermi level. These subbands make a major contribution to the order parameter (due to an enhanced density of states) and, so, the sketch in Fig. 9.1 illustrates the situation of the superconducting resonance governed by (j, m) = (0, ±3). For any subband we can introduce the longitudinal BCS coherence length or, in other words, the (jm) longitudinal subband-dependent Cooper-pair size ξ0 . Based on Fig. 9.1(b) and keep in mined the well-known formula for the BCS coherence length (see, (jm) e.g., [11]), we can get the estimate ξ0 = vjm /Δjm , where vjm = kjm /me and kjm plays the role of the subband longitudinal Fermi wavevector defined by the equation ξjmk = 0. As seen from Fig. 9.1, kjm depends strongly on the position of the lower edge (bottom) of a single-electron subband εjm (i.e., the transverse electron spectrum) with respect to the Fermi level (zero in Fig. 9.1). When εjm EF [e.g., (j, m) = (0, 0) in Fig. 9.1], the corresponding kjm approaches the bulk Fermi wavevector kF and ξ0jm is close to the bulk BCS coherence length ξ0,bulk = vF /Δbulk . When μjm > ωD and μjm /EF 1 [e.g., (j, m) = (0, ±2) in Fig. 9.1], one can still use the above estimate, which results (jm) (jm) ξ0,bulk . A question arises what happens to ξ0 in kjm kF and ξ0 when εjm approaches the Fermi level [e.g., (j, m) = (0, ±3) in Fig. 9.1]. And, in addition, what can be expected for negative μjm [e.g., for the subband with (jm) (j, m) = (0, ±4) in Fig. 9.1]? Here the above oversimplified estimate of ξ0 is not relevant any more because of the absence of a solution of ξjmk = 0 (kjm is not defined).

122


9.3 BCS-BEC crossover driven by quantum confinement To go in more detail about ξ0jm for the regime of μjm ≤ 0, we should consider the anomalous Green’s function (see, e.g., [12, 13]) 1 T ψ↑ (x1 t1 )ψ↓ (x2 t2 ), ı

F (x1 t1 , x2 t2 ) =

(9.1)

where x1 = {ρ, ϕ, z1 } and x2 = {ρ, ϕ, z2 }, expressed in the cylindrical coordinates involved. Notice that the radial and azimuthal coordinates of two electrons with the opposite spin projections are taken the same in Eq. (9.1): we are only interested in the longitudinal superconducting spatial correlations. Based on Eq. (9.1), we can construct the off-diagonal superconducting order parameter (t = t1 − t2 , z = z1 − z2 ) Δ(ρ, z) = g lim ı F (x1 t1 , x2 t2 ),

(9.2)

t→0+

where g > 0 stands for the Gor’kov coupling constant. Setting z = 0, we arrive at the usual (diagonal) order parameter, i.e., Δ(ρ) = Δ(ρ, z)|z=0 . We remark that Δ(ρ, z) does not depend on ϕ due to axial symmetry. Quantum confinement for the transverse electron motion does not break the translational invariance in the longitudinal direction and, so, F (x1 t1 , x2 t2 ) depends only on z = z1 − z2 . A straightforward way to calculate Δ(ρ, z) is to express it in terms of the particle-like and hole-like wave functions ujmk (ρ, ϕ, z) and vjmk (ρ, ϕ, z) that obey the Bogoliubov-de Gennes (BdG) equations (see, e.g., Ref. [10]) Ejmk

ujmk vjmk

=

H Δ(ρ) Δ∗ (ρ) −H ∗

ujmk vjmk

,

(9.3)

1 where H = 2m ( i ∇− ec A)2 +Vconf (ρ)−EF is the single-electron Hamiltonian, e with me the band effective mass (set to the free-electron mass below) and Vconf (ρ) the transverse confining interaction. For the sake of simplicity we take Vconf = λθ(R − ρ), λ → ∞. In the present paper A = 0. Using the Bogoliubov canonical transformation (see, e.g., Ref. [10]) we can find (for the ground state)

Δ(ρ, z1 − z2 ) = g

∗ ujmk (ρ, ϕ, z1 )vjmk (ρ, ϕ, z2 ).

(9.4)

jmk

The next step is to take into account that

ujmk (ρ, ϕ, z) vjmk (ρ, ϕ, z)

eımϕ eıkz = √ √ 2π L

u ˜jmk (ρ) v˜jmk (ρ)

,

(9.5)


123

with L ∼ 10 μm the unit cell in the longitudinal z direction (the quantum -confinement boundary conditions in the transverse direction means u ˜jmk |ρ=R = v˜jmk |ρ=R = 0, with R = d/2). Inserting Eq. (9.5) into Eq. (9.4), we get Δ(jm) (ρ, z), (9.6) Δ(ρ, z) = jm

with Δ

(jm)

g (ρ, z) = (2π)2

+∞ ∗ dk u ˜jmk (ρ)˜ vjmk (ρ) eıkz ,

−∞ (jm)

which controls the subband-dependent longitudinal coherence length ξ0 . It can be extracted numerically as the decay length of Δ(jm) (ρ, z) along z. Numerical investigations of the BdG equations for a superconducting nanocylinder supports our expectations based on Fig. 9.1. From Fig. 9.2 one can see how the longitudinal correlations of electrons in a Cooper pair change when passing from the superconducting resonance developing at d = 4.22 nm (a,b,c) to the off-resonant diameter d = 4.35 nm (d,e,f). Calculations were performed for the same set of parameters as in papers [6]:

b 1.0

8

0.8

6 1, ±4

4

0, ±7 1

0

3 2 k (nm–1)

4

0.6 0.4

e

d = 4.35nm 1.0

1.0

0.8

0.8

20

0.0 0

1

2

3

–1 k (nm )

4

5

0.4

f

(j, m) = (0,±7)

0.5

0.6 0.4

0.0 –30 –20 –10 0 10 z (nm)

1.0

0 10 z (nm)

20

30

(j, m) = (1, ±4)

0.5

0.8

0.2

0, ±7 1, ±4

0

0.6

0.0 –30 –20 –10

30

0 ρ/R

0.4

0.8

0.2

0

0.6

( j, m) = (1, ±4)

0.8 0

0.0 –30 –20 –10 0 10 z (nm)

5

ρ/R

Ejmk (meV)

d

0.2

1.0

0.8

0.2

2 0

c

( j, m) = (0, ±7)

ρ/R

d = 4.22 nm 10

ρ/R

Ejmk (meV)

a

0.6 0.4 0.2

20

30

0.0 –30 –20 –10

0 10 z (nm)

20 30

Fig. 9.2. (Color online) The resonant diameter d = 4.22 nm: (a), the relevant quasi¯ particle energies Ejmk versus k > 0; (b) and (c), the contour plots of Δ(0,±7) (ρ, z)/Δ ¯ respectively. Panels (d), (e), and (f) are the same but for the and Δ(1,±4) (ρ, z)/Δ, off-resonant diameter d = 4.35 nm.

124


ωD = 32.31 meV, gN (0) = 0.18, EF = 0.9 eV, where N (0) is the bulk density of states at the Fermi level and EF is the effective Fermi level for the parabolic band approximation (see, for details, discussion about an effective Fermi level in the second paper of [6]). Notice that the particular choice of these parameters is not important for our results. In Fig. 9.2(a) the quasiparticle energy Ejmk is given versus k as calculated from the BdG equations for d = 4.22 nm. There are two single-electron subbands whose bottoms are situated just below the Fermi level, i.e., with (j, m) = (0, ±7) and (1, ±4). They control the superconducting resonance developing at d = 4.22 nm (compare the subband energy gaps Δjm ∼ 2 meV to the bulk zero-temperature gap Δbulk = 0.25 meV). Panels (b) and (c) show the contour-plots of the modulus of the quantum-channel correlation function Δ(jm) (ρ, z) [given in ¯ = 22 R dρΔ(ρ)] for units of spatially averaged diagonal order parameter Δ R 0 (j, m) = (0, ±7) and (1, ±4), respectively. We find a significant decrease of (0,±7) (1,±4) ξ0 and ξ0 at d = 4.22 nm. We note that the transverse particle- and hole-like wave functions can be well approximated as (Anderson’s approximate solution to the BdG equations, see, for details, Ref. [14]) u ˜jmk (ρ) = Ujmk ϑjm (ρ), v˜jmk (ρ) = Vjmk ϑjm (ρ), √

(9.7)

ρ where ϑjm (ρ) = RJm+12(αjm ) Jm (αjm R ), with Jm (x) the Bessel function of the first kind of the m-order and αjm its jth zero. Equation (9.7) gives a good feeling about the ρ-dependence of Δ(0,±7) (ρ, z) and Δ(1,±4) (ρ, z). Figure 9.2(c–f) show what happens when we are in the off-resonance regime. From panel (c) one can see that the bottoms of the single-electron subbands (0, ±7) and (1, ±4) are shifted down (k0,±7 and k1,±4 moved to the right) and, as a result, the quasiparticle gaps Δjm dropped by an order of magnitude down to Δbulk = 0.25 meV. Due to such a drop and an increase of k0,±7 and k1,±4 by (0,±7) (1,±4) an order of magnitude, we find that ξ0 and ξ0 increase by two orders of magnitude up to ≈400 nm at d = 4.35 nm. We note that our particular choice of d = 4.22 and 4.35 nm is not of importance. The same picture is found for the resonant subbands at any superconducting resonance. For sufficiently small diameters, i.e., d < 10−15 nm, resonant subbands make as a rule a major contribution to the superconducting characteristics. In this case we get a pronounced width-dependent drop of the total longitudinal coherence length ξ0 [it is associated with Δ(ρ, z)] each time when a new superconducting resonance develops (see Fig. 9.3). For d > 10−15 nm the total number of relevant subbands increases abnormally and, so, superconducting resonances and width-dependent drops in the BCS coherence length become less important. For d > 30−40 nm the effect is washed out. We remark that inevitable width fluctuations in the superconducting nanowires smooth the quantum-size oscillations of superconducting properties. However, when averaged over the possible fluctuation interval d ≈ 1 nm, d+d 1 i.e., ξ¯0 = d dx ξ0 (x), the total coherence length deviates from its bulk d

125

|


average length,

x0

x0 (nm)

103

102

101

3.8

4.0

4.2

4.4

z (nm)

4.6

Fig. 9.3. Width-dependent drops in the longitudinal BCS coherence length (squares), and its averaged value over the interval 3.8−4.6 nm (dashed line).

value ξ0,bulk (1.5 μm for the chosen parameters). In particular, for d ∼ 4−5 nm this deviation is about an order of magnitude, i.e., ξ¯0 ≈ 170 nm as seen from Fig. 9.3. Now let us return to the subbands with (j, m) = (0, ±7) and (1, ±4) and (0,±7) (1,±4) consider what happens to ξ0 and ξ0 when ε0,±7 and ε1,±4 become larger than EF (i.e., d goes below 4.22 nm). To have an idea about this regime, it is convenient to use Anderson’s approximate solution to the BdG equations (see Eq. (9.7)). Within this approximation we get (for the ground state) (jm)

Δ

2 g ϑjm (ρ)Δjm (ρ, z) = 8π 2

+∞ dk

−∞

where Δjm is taken real and positive, ξjmk = and εjm =

2 2 αjm 2me R2 .

eıkz 2 ξjmk + Δ2jm

2 k2 2me

− μjm =

,

2 k2 2me

(9.8) − (EF − εjm )

From Eq. (9.8) it follows that

(jm) ξ0

=

me

μ2jm + Δ2jm − μjm .

(9.9)

This can be derived by contour integration in the complex plane. The integrand in Eq. (9.8) has four singular points (the square-root branch points)

with the same absolute value of the imaginary part, i.e., [me ( μ2jm + Δ2jm − μjm )]1/2 /. For, say, positive z, the contour is closed in the upper half plane and distorted to encircle the cut between the two upper singular

126

A. A. Shanenko et al. (jm)

points. Their imaginary part controls ξ0 , which results in Eq. (9.9). When (jm) ≈ vjm /Δjm , which was used in the above μjm Δjm , Eq. (9.9) gives ξ0 discussion of Fig. 9.1. Such an approximation is not relevant for μjm close to (jm) zero or negative. When μjm = 0, Eq. (9.9) reduces to ξ0 = / me Δjm . (jm) For μjm < 0, |μjm | Δjm , we get ξ0 = / 2me μjm . We also remark that when μjm approaches −ωD , the cut-off |ξjmk | < ωD implied in Eqs. (9.6) and (9.8), distorts this picture: Eq. (9.9) does not work any more, and the corresponding single-electron subband stops to make a contribution to the superconducting characteristics. This gives the lower bound for the (jm) longitudinal Cooper-pair size, i.e. ξ0 = /(2me ωD ) ∼ 1 nm. The minimum value of the excitation energy Ejmk for μjm < 0 is not Δjm but

μ2jm + Δ2jm ≈ |μjm |. This allows one to treat |μjm | as half of the binding energy of a strongly-confined molecular state appearing in a resonant channel for 0 > μjm > −ωD . These picture is very similar to the results of the so-called extended BCS description at the BEC side of the conventional BCS-BEC crossover in atomic superfluid Fermi gases [4]. Hence, our results can be interpreted as a new type of BCS-BEC crossover. However, there are important differences to highlight. First, in our case the crossover occurs only in one (resonant) subband: μjm is the subband-dependent “longitudinal chemical potential” rather than the real chemical potential in the extended BCS model. Second, μjm becomes negative not due to a change in the interparticle interaction but because of the quantum-size effects.

9.4 Conclusions Concluding, we have found that the longitudinal BCS coherence length of a superconducting metallic nanowire undergoes size-dependent giant drops down to a few nanometers. This occurs each time when a transverse discrete single-electron level is located in the vicinity of the Fermi surface so that the longitudinal motion in the corresponding single-electron subband is significantly suppressed. As a result, the longitudinal Cooper-pair size in this quantum channel and, in turn, the total longitudinal size of a Cooper pair drops by several orders of magnitude. This phenomenon is similar to what is found in the BCS-BEC crossover in superfluid Fermi gases. However, an important difference with the latter system is that the collapse of the longitudinal Cooper-pair size in a quantum superconducting wire is due to the transverse quantization and not due to an increase in the coupling strength. Notice that the same qualitative behavior can be found for different values of the material parameters and a sharp Debye window is not essential. Therefore, such a phenomenon is a generic feature present in other low-dimensional systems, where the condensate is formed via multiple subbands, e.g., in superconducting nanofilms and ultracold Fermi gases confined in a quantum-wire/well geometry.


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Acknowledgments This work was supported by the Flemish Science Foundation (FWO-Vl), the Belgian Science Policy (IAP) and the ESF-network: INSTANS. M.D.C. acknowledges support from the Alexander von Humboldt Foundation.

References 1. 2. 3. 4. 5. 6.

7.

8. 9. 10. 11. 12. 13. 14.

D. M. Eagles, Phys. Rev. 186, 456 (1969). A. J. Leggett, J. Phys. (Paris), Colloq. 41, 7 (1980). P. Nozieres and S. Schmitt-Rink, J. Low. Temp. Phys. 59, 195 (1985). I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008). J. M. Blatt and C. J. Thompson, Phys. Rev. Lett. 10, 332 (1963). A. A. Shanenko, M. D. Croitoru, M. Zgirski, F. M. Peeters, and K. Arutyunov, Phys. Rev. B 74, 052502 (2006); A. A. Shanenko, M. D. Croitoru, and F. M. Peeters, Phys. Rev. B 75, 014519 (2007). Y. Guo, Y.-F. Zhang, X.-Y. Bao, T.-Z. Han, Z. Tang, L.-X. Zhang, W.-G. Zhu, E.G. Wang, Q. Niu, Z. Q. Qiu, J.-F. Jia, Z.-X. Zhao, and Q. K. Xue, Science 306, 1915 (2004). X.-Y. Bao, Y.-F. Zhang, Y. Wang, J.-F. Jia, Q.-K. Xue, X. C. Xie, and Z.-X. Zhao, Phys. Rev. Lett. 95, 247005 (2005). D. Eom, S. Qin, M.-Y. Chou, and C. K. Shih, Phys. Rev. Lett. 96, 027005 (2006). P. G. de Gennes, 1966. Superconductivity of metals and alloys (W. A. Bejamin, New York, 1966). A. L. Fetter, and J. D. Walecka, Quantum theory of many-particle systems. (Dover, New York, 2003). A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinski, Quantum field theoretical methods in statistical physics (Pergamon, Oxford, 1965). A. M. Zagoskin, Quantum theory of many-body systems: techniques and applications (Springer, New York, 1998). A. A. Shanenko, M. D. Croitoru, and F. M. Peeters, Phys. Rev. B 78, 024505 (2008); M. D. Croitoru, A. A. Shanenko, C. C. Kaun, and F. M. Peeters, Phys. Rev. B 80, 024513 (2009).

10 TRANSIENT RESPONSE OF A SUPERCONDUCTOR IN AN APPLIED ELECTRIC FIELD M. N. Kunchur and G. F. Saracila Department of Physics and Astronomy University of South Carolina, Columbia, SC 29208, USA [email protected] Abstract. A fundamental property of a superconductor in the Meissner state is its ability to carry a current without dissipation. As a result, upon the first application of an electric field one expects to observe an inductive like transient rise in the current until the latter reaches its critical value and dissipation sets in. The present work provides a quantitative real-time experimental demonstration of this transient phase, which yields information on the inertia of the superfluid and shows a signature of superfluid density suppression at high current densities.

Key words: transient, ballistic, acceleration, kinetic, inductance, supercurrent, superfluid, superconductor, superconductivity.

10.1 Introduction An electric field acting upon a normal conductor produces a constant current, whereas a superconductor should show a kinetic inductive response coming from the inertia of the superfluid as per the first London equation. This primitive growth transient lasts for a very brief duration, making it difficult to observe in real time. In the present work, a measurement system was developed that allowed a true four-probe current-voltage measurement with sub-nanosecond correlation accuracy, which facilitated the observation of the current growth transient in real time. Furthermore the techniques developed in this work provide a way to systematically probe effects such as superfluid density suppression with current (whose first signature was seen here) and time dependent phenomena in condensed-matter systems in general.


129

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M. N. Kunchur and G. F. Saracila

10.2 Experimental methods Measurement system The measurements were carried out in a pulsed-tube closed-cycle refrigerator in zero applied magnetic field (the sample space was surrounded by Mu-metal). Figure 10.1 shows the four-probe configuration used for measuring a current-voltage response. A power source supplies a current (which may be constant DC, AC, or pulsed) through the sample and a series standard impedance Zstd . Their respective voltages are measured differentially to obtain V (t) and I(t) as functions of time and of each other.

a

Vo(t)

or

Io(t)

Zstd

sample

differential preamps ch 2

ch 1 Oscilloscope

b Voltage coax~1 µV – mV + instead

-

Current coax~ 10 V

Fig. 10.1. (a) Electrical configuration for four-probe time dependent transport measurements. The time varying differential voltage Vstd (t), across the series standard impedance Zstd , indicates the I(t) function. This along with the differentially extracted sample voltage V (t) are measured and displayed on a digital storage oscilloscope. (b) Coaxial cables carry the source current to the sample and convey the time varying voltages V (t) and Vstd (t) to the differential preamplifiers feeding the oscilloscope channels. The buffered ground arrangement shown can improve the high-frequency isolation of the feeble signals in the sample-voltage cable from the current cable, which carries far higher voltages.

10 Transient response of a superconductor in an applied electric field

131

For pulsed measurements at short time scales, this process become extremely difficult as it is hard to ensure that the currents in Zstd and the sample are the same at a given instant in time. In the present arrangement, the standard and sample impedances are both much lower than the characteristic impedance of the coaxial cables that sense these voltages. This allows each cable to be terminated with matching impedances at both free ends to avoid reflections, without drawing parasitic currents. The lengths of all cables were carefully matched to minimize relative propagation delays between V (t) and Vstd (t). Another caution that must be exercised is to ensure that each signal being measured is the true differential component and that the common-mode component is fully rejected. Floating selected portions of the circuit, proper ground points, and the use of differential preamplifiers ensures this. Finally the quality of grounding becomes critical. The coaxial cable carrying the current down to the sample in the cryostat has a highfrequency signal amplitude of ∼10 V. This can induce noise into the coax carrying the much smaller (μV–mV amplitude) sample voltage. This problem can be alleviated by improving the effective shielding by using the buffered ground arrangement shown in Fig. 10.1(b) along with proper placement of the ground point. The coupling between current and voltage leads was < 1 nH. The aforementioned steps and precautions made it possible to simultaneously track and correlate current and voltage on sub-nanosecond time scales. In the two graphs shown in Fig. 10.2, test samples (a pure resistor and a pure inductor) were placed in the cryostat instead of a superconducting sample. In the first case we expect the sample V to be proportional to I and in the second case it should be proportional to dI/dt. As can be seen, this is indeed the case and the two pairs of curves track each other in instantaneous detail. The temporal tracking accuracy of the system is ∼100 ps. These checks ensure that the measurement system is performing correctly. While other types of techniques (e.g., laser pulses or pulses generated by Josephson-junction electronics) may offer even shorter time scales than were resolved here, those setups usually measure the impulse response which may involve non-equilibrium states. Also in those fast measurements the time dependence of both the stimulus and the resultant are usually not measured independently and tracked simultaneously. In the present setup both V (t) and I(t) are tracked at every instant in time rather than assuming some nominal pulse shape. The present experimental setup therefore offers a precise instantby-instant correlation between the stimulus and response, and provides complete control over the time scales so as to avoid undesirable non-equilibrium effects. Some additional information about the measurement system has been published previously [1–3]. Samples The samples used in this work were niobium films deposited on silicon substrates with DC magnetron sputtering. The films were patterned into long

132


a

b

Fig. 10.2. Measurement-apparatus temporal accuracy checks. (a) V (t) and I(t) (multiplied by a constant R = 62Ω) for a test resistor in place of a superconducting sample. (b) dI(t)/dt and V (t) (divided by a constant L = 15 nH) for a test inductor in place of a superconducting sample. The voltages across the purely resistive and inductive loads are seen to track their respective current and current-derivative functions with sub-nanosecond accuracy.

narrow meanders by electron-beam lithography using the lift-off technique. For the present experiment, it is necessary to make the current path in the superconductor as long as possible in order to prolong the transient phase during which the current rises after an electric field is applied. This phase is in effect while the acceleration is impeded only by the superfluid inertia and there is no dissipation. Once any resistance appears, the resistive drop completely overwhelms the intrinsic inductive signal being sought here. The growth transient lasts for a duration given by Δt ≈ jc μ0 λ2 l/V , where jc is the (conventional) critical current density that demarcates the onset of dissipation, λ is the magnetic penetration depth (discussed in more detail below), and l is the length of the superconducting path through which the current flows. In order to extend this duration and window of opportunity, l must be made as long as is practically possible, while keeping the self inductance to a minimum. This is best achieved with a meander geometry. Two samples were included in this study. Sample A had a thickness of t = 70 ± 8 nm, a width of w = 12.1 ± 0.6 μm, and a length (between voltage


133

Fig. 10.3. Optical micrograph of a sample; the meander geometry extends the length of the current path while minimizing self inductance. The light areas represent the superconductive film.

probes) of l = 4.80 ± 0.01 cm. Sample B had the dimensions t = 85 ± 8 nm, w = 8.9 ± 0.6 μm, and l = 4.53 ± 0.01 cm. Their respective superconducting transition temperatures were Tc = 6.74 K and Tc = 7.23 K. Figure 10.3 shows a photograph of one of the samples.

10.3 Theoretical background The acceleration of the supercurrent density js is given by (from the London equations [4, 5]) djs Ee2 ns dj E = = , ∗ dt dt m μ0 λ2L

(10.1)

where E is the local internal electric field, e is the electronic charge, m∗ is the effective mass and ns is the superfluid density (related to the number of electrons per volume participating in the condensate); the far right hand side of the equation relates ns to the London magnetic-field penetration depth λL ; we can take j = js + jn js because the normal current density jn is a negligible component of the total current density j. This supercurrent acceleration phase lasts for the duration Δt ≈ jc μ0 λ2L /E, where jc is the critical current density that marks the onset of resistance. The inductance-like proportionality between dj/dt and E in Eq. 10.1, arising from the inertia of the superfluid,

134


is referred to as the kinetic inductance Lk . In terms of the geometrical length l and cross sectional area A, it is given by Lk =

μ0 λ2 l , A

(10.2)

where λ is a more general penetration depth, which includes effects such as impurity scattering (λ ≥ λL ). Kinetic inductive effects are small except close to the transition temperature Tc , where their signatures have been seen in the high-frequency ac response or as non-equilibrium inductive voltage spikes during abrupt current steps [6–14]. In the present work, timescales were chosen to be short enough to have a sufficient magnitude of V while long enough (compared to characteristic timescales such the gap-relaxation and electronphonon scattering times) to avoid non-equilibrium effects. Variations in fields occurred at length scales that were long compared with both λ and the coherence length ξ, so as to avoid non-local effects. Thus the conditions were optimum for observing the simplest limiting behavior of an accelerating condensate as predicted by the London equations, i.e., Eq. 10.1.

10.4 Results and discussion Data and analysis Figure 10.4(a) and (b) shows V (t) (solid lines) across two niobium-meander samples in the superconducting state at one temperature. Panels (c) and (d) show the corresponding I(t) functions, which are seen to accelerate steadily during the plateaus in V (t). The dashed lines in panels (a) and (b) show dI/dt scaled by a constant (L=16.7 and 16.9 nH for samples A and B respectively) and are seen to track V (t) in instantaneous detail. Thus the response is purely inductive, with an inductance that is independent of I and dI/dt. The ratio between the dI(t)/dt and V (t) curves in the top panels of Fig. 10.4 gives the time and current dependent inductance: L(t) = V (t)/[dI(t) /dt]. Figure 10.5(a) and (b) plots this L(t) versus time for various values of T , for each of the samples. The plateau value of L is seen to increase steadily with temperature as is expected because of the declining superfluid density and consequent rising λ. Another interesting trend is that the curves at highest temperatures show L(t) functions that rise with time (i.e., current). This happens because the current suppresses the superfluid density through its pair-breaking action, a regime not seen before in any other kind of measurement. Note that continuous-ac probes of ns cannot endure high enough excitation levels to explore this regime because of Joule heating; and tunneling measurements reveal the spectral gap Ωg rather than ns . A systematic study of the suppression of ns by j will be the subject of a future investigation, since the optimum sample geometry for studying this effect is almost opposite to the sample geometry required for the present experiment.


a

b

c

d

135

Fig. 10.4. Time dependencies of voltage and current in niobium meanders at T = 3.84 K. Panels (a) and (b) represent V (t) and dI/dt (scaled by a constant L) for samples A and B respectively. Panels (c) and (d) show the corresponding I(t) functions, which rise steadily during the voltage plateaus of the panels above; the time axes for panels (a) and (c) and for panels (b) and (d) are the same.

Figure 10.5(c) and (d) plots L (as measured above) versus T for each sample. This total inductance L = Lk +Lg has components corresponding to the kinetic inductance Lk as well as a geometrical inductance Lg (from magnetic flux change). The temperature dependence arises chiefly from Lk ; the changes in Lg with temperature – arising from changes in the current-density profile across the cross section – are relatively small, as discussed below. From Eq. 10.2 and the empirical temperature dependence of the penetration depth [11–13] we have Lk (T ) ≈

Lk (0) , [1 − (T /Tc )2 ]

(10.3)

where Lk (0) = μ0 λ2 (0)l/A. The solid line curves in Fig. 10.5(c) and (d) correspond to a least-squares fit to the function L(T ) = Lg + Lk (0)/[1 − (T /Tc )2 ]. The values of Lk (0) and Lg obtained from this fitting (Tc is not a fitting parameter) are listed in columns 2 and 4 of Table 10.1. The coefficients of determination of the fits are R2 = 0.9989 and R2 = 0.9994 for samples A and B

136


b

a 32

L (nH)

32

24 24

16 16 11

12

13 t (ns)

14

13 t (ns)

15

14

d

c 25

L (nH)

12

15

25

20

20

15 15 0

1

2

3 T (K)

4

5

6

0

1

2

3 4 T (K)

5

6

Fig. 10.5. (a) Measured total inductance, L(t) = V (t).dt/dI(t), versus time for sample A. The curves correspond to the temperatures (from top to bottom): T = 6.10, 5.91, 5.54, 5.17, 4.73, 4.10, and 3.78 K. (Each plot symbol on these curves corresponds to a separately measured digital voltage sample. The period between samples is 100 ps.) (b) A similar L(t) plot for sample B. For this panel, the temperatures (from top to bottom) are: T = 6.51, 6.4, 6.22, 6.02, 5.84, 5.24, 4.64, and 3.84 K. Panels (c) and (d) show the temperature dependencies of the above measured L values (taken on the plateaus around t ∼ 11 ns) for samples A and B respectively. The symbols show the experimental data and the solid line represents the least-squares fit to the two-parameter function L(T ) = Lg + Lk (0)/[1 − (T /Tc )2 ] as discussed in the text.

respectively. The standard errors of the fit combined with the error in the inductance measurement gives the error bars for Lk (0) that are indicated in the table. Kinetic inductance calculation The third column of Table 10.1 shows for comparison the theoretical estimates of Lk (0). For finding Lk (0) we note that the effective penetration depth (λ) becomes lengthened with respect to its clean-limit value (λL ) in


137

the presence of scattering by static disorder. This effect of impurity scattering can be incorporated through the residual resistivity ρo and expressed in terms of the order parameter Δ as [15] ρo . (10.4) λ(0) = πμo Δ(0) Taking the measured values of ρo = 0.347 μΩ.m and 0.369 μΩ.m, and obtaining Δ(0) ≈ 2kB Tc from our measured values of Tc , we get λ = 223 nm and 222 nm, and Lk = 3.5 nH and 3.7 nH for samples A and B respectively. There is an uncertainty in the values of ρo because of the uncertainty in sample dimensions and an uncertainty in Δ(0) because of an uncertainty in the absolute value of Tc (this is roughly estimated to be around 200 mK). This gives rise to the error bars in the theoretical Lk values that are stated in the table. The calculated values of Lk are somewhat larger than the measured ones but of comparable magnitude. Geometrical inductance calculation The theoretical geometrical inductances for the meanders, tabulated in the last column of Table 10.1, were calculated from the flux linking the circuit between the voltage probes. An applied external voltage will produce an electric field that is uniform across the cross section if no current flows (the case of infinite superfluid inertia arising from λ → ∞). However, a finite accelerating current and its consequent changing magnetic field (Fig. 10.6(a)) will induce an internal electric field whose profile boosts the electric field value near the edges and reduces it in the center, pushing the magnetic flux out until a steady is achieved. This process was solved iteratively to obtain the current density profile across the cross section, shown in Fig. 10.6(b). (Note that unlike the familiar case when a constant dissipationless current flows in zero applied electric field, here the current is not pushed entirely to the periphery but flows, relatively uniformly, across the entire cross section.) It should be noted that the induced circulating electric fields only redistribute j without affecting the total current. Hence this does not affect the value of Lk ; however, it does lead to a small temperature dependent correction in Lg , which causes it vary by Table 10.1. Experimentally observed values and theoretically estimated values of the kinetic and geometrical inductances. Lk (0) in nH

Lg in nH

Sample

Expt.

Theor.

Expt.

Theor.

A B

2.8 ± 0.2 2.8 ± 0.2

3.5 ± 0.7 3.7 ± 0.7

12.2 ± 1 12.9 ± 1

15.4 ± 3 12.0 ± 3

138


Fig. 10.6. (a) The self magnetic field of the current links a variable flux (shaded region between dashed lines) to each the current path through a strip, causing an induced electric field that has a reverse direction in the center of the strip and forward direction near the edges. (b) This induced electric field adds to the externally applied electric field and distorts the current distribution as shown by the solid line; the dashed line shows the undistorted distribution. < j > = I/A is the average current density and j is the local value. The induction and flux expulsion does not alter < j > but only the j distribution. x is the distance along the width measured from one edge.

4% over the temperature range of interest. The error bars in the theoretical Lg arise from the uncertainties in the sample dimensions and the approximations inherent in the calculation. It can be seen that the theoretically estimated Lg values are in agreement with their measured counterparts.

10.5 Summary In summary, this work has conducted a detailed temporal study of the initial growth transient of a supercurrent in response to an applied electric field. The voltage and current curves of Fig. 10.4 represent the first clear and direct time-domain demonstration of this fundamental regime. The instrumentation developed for this experiment is unique and represents the first measurement of its kind where both V (t) and I(t) are tracked in a superconductor with subnanosecond timing accuracy. This technique can reveal more detailed information than just an impulse-response measurement, and it can be used to explore time-dependent and non-equilibrium phenomena in condensed-matter systems in a controlled way (some examples of such regimes in superconductors would be those related to phase slippage, glassy dynamics, and the nascent stage of a vortex right after its nucleation). The present work and its method should be


139

distinguished from past experiments in which an abrupt supercritical current pulse was applied [8,14] and only the subsequent V (t) response was measured without monitoring I(t). In those experiments the superconductor recoils in a highly non-equilibrium manner to the supercritical stimulus. In the present study, the superconducting system is always maintained close to equilibrium by keeping the experimental timescales well in excess of the gap-relaxation and electron-phonon scattering times, while keeping the timescales short enough to observe the inertial acceleration of the superfluid.

Acknowledgment The authors acknowledge useful discussions with J. M. Knight, B. I. Ivlev, R. A. Webb, R. J. Creswick, T. R. Lemberger, G. Simin, T. M. Crawford, and F. T. Avignone III. This research was supported by the U. S. Department of Energy through grant number DE-FG02-99ER45763.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15.

G. F. Saracila and M. N. Kunchur, Phys. Rev. Lett. 102, 077001 (2009). M. N. Kunchur, Mod. Phys. Lett. B. 9, 399 (1995). M. N. Kunchur, J. Phys.: Condens. Matter 16, R1183 (2004). F. London and H. London, Proc. Roy. Soc. (London) A149, 71 (1935). M. Tinkham, Introduction to Superconductivity (2nd edition) McGraw-Hill (New York, 1996). I. F. Oppenheim, S. Frota-Pessona, and M. Octavio, Phys. Rev. B. 25, 4495 (1982). A. Geier and G. Schon, J. Low Temp. Phys. 46, 151 (1982). D. J. Frank, M. Tinkham, A. Davidson, and S. M. Faris, Phys. Rev. Lett. 50, 1611 (1983). S. M. Anlage, H. S. Howard, J. Snortland, S. Tahara, B. Langley, C. B. Eom, and M. R. Beasley, Appl. Phys. Lett. 54, 2710 (1989). S. M. Anlage, B. W. Langley, G. Deutscher, J. Halbritter, and M.R. Beasley, Phys. Rev. B 44, 9764 (1991). J. Y. Lee, and T. R. Lemberger, Appl. Phys. Lett. 62, 2419 (1993). S. D. Brorson, R. Buhleier, J. O. White, I. E. Trofimov, H.-U. Habermeier, and J. Kuhl, Phys. Rev. B 49, 6185 (1994). S. Cho, J. of the Korean Phys. Soc. 31, 337 (1997). F. S. Jelila, J.-P. Maneval, F.-R. Ladan, F. Chibane, A. Marie-de-Ficquelmont, L. Mechin, J.-C. Villegier, M. Aprili, and J. Lesueur, Phys. Rev. Lett. 81, 1933 (1998). T. R. Lemberger, I. Hetel, J. W. Knepper, and F. Y. Yang, Phys. Rev. B. 76, 094515 (2007).

11 INTERBAND NODAL-REGION PAIRING AND THE ANTINODAL PSEUDOGAP IN HOLE DOPED CUPRATES N. Kristoffel and P. Rubin Institute of Physics, University of Tartu, Riia 142, 51014 Tartu, Estonia [email protected] Abstract. Recent experimental findings show that the pairing interaction in hole-doped cuprates resides in the nodal (FS arcs) region accompanied by the separate antinodal pseudogap. A corresponding multiband model of cuprate superconductivity is developed. It is based on the electronic spectrum evolving with doping and extends authors’ earlier approaches. The leading pair-transfer interaction is supposed between the itinerant (mainly oxygen) band and a nodal defect (polaron) band created by doping. These components are overlapping. The defect subband created in the antinodal region of the momentum space does not participate in the pairing. A supposed bare gap separating it from the itinerant band top disappears with extended doping. The corresponding antinodal pseudogap appears as a perturbative band structure effect. The low energy excitation spectrum treated in the mean-field approximation includes two nodal superconducting gaps and one pseudogap. The behaviour of these gaps and of other pairing characteristics agree qualitatively with the observations on the whole doping scale.

Key words: cuprates, multiband superconductivity, phase diagram, gaps, doping.

11.1 Introduction Cuprate superconductivity mechanism remains elusive. A huge package of information on cuprate low-energy excitations [1, 2] has been collected by diverse experiments. However, numerous improvements, corrections and precessions have been necessary to get comparable data for distinct conditions. Some essential results have been obtained quite recently, e.g. the coexistence of the pseudogap (PG) and of the superconducting gap (SCG) with the own energy scales at fixed doping [3–12]. The extraction of a true SCG from the background becomes possible [4,7,12–16]. The presence of the Fermi surface (FS) fragments in the form of arcs at underdoping [5, 8, 10, 17, 18]


141

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N. Kristoffel and P. Rubin

has beencontrolled by measuring the Shubnikov-de Haas oscillations [19–21]. The distinct reveal of the two SCG in cuprates succeed in [6] which was expected for long. Recently one converged to the knowledge that at underdoping the energy scales of the large PG and of the smaller SCG are different by genealogy [4, 5, 10, 16, 24]. These gaps develop from different momentum space regions: the PG is an antinodal, and the SCG a nodal event [3, 4, 8, 9, 22–24]. The PG is extrinsic with respect of the superconductivity, i.e. different mechanisms in the superconductivity playground CuO2 planes seem to be hidden behind these phenomena [23–26]. Contrary to the PG the superconducting pairing develops mostly on the FS arc segments around the nodal π2 , π2 -type Brillouin zone positions [6, 8, 14, 16, 17, 20, 26–28]. The formation and expansion of these segments is due by doping. The introduced carriers concentrate first here [5]. The superconductivity created on arcs is accompanied by a sharp spectral coherence peak corresponding to the “arc metal” quasiparticles (QP) [3,8,10]. Extended doping brings wider momentum space regions into the game and a common FS will be built up. The SCG rises and the PG diminishes with doping before the maximal Tc is reached. In the underdoped regime the energy scales of these gaps are remarkably different. The PG disappears with overdoping. At this it does not transform into a SCG with entering the superconductivity dome [3]. The QP-s corresponding to these two type of gaps coexist [10] below the superconducting transition temperature (Tc ). The PG QP-s remain here non-coherent. The PG is not very material sensitive, the SCG is [17]. The PG fills in with thermal excitation and fits a temperature driven metal-to-insulator transition under the doping concentration where it vanishes [29, 30]. One has usually classified cuprates as two-gap systems having in mind the presence at least of one SCG and one PG. At present a long list of real two- (or multi-) gap superconductors which show multiple SCG-s is known. Cuprates seem to enter this community also. In fact, numerous direct and indirect experimental and theoretical results crowned by the observation of two SCG-s [6] positioned the cuprates into the class of multigap superconductors with a multiband superconductivity mechanism. We refer here to the reviews [31–34] and some recent theoretical approaches [35–45]. Authors multiband approaches to cuprate superconductivity are based on the nonrigid nature of the electron spectrum reorganized by the necessary doping [43–50]. Interband pairing channels are opened on this background including the appeared defect-polaron type new bands. The present work takes into account the mentioned dichotomy in cuprate momentum space functioning, especially the exposed “arc superconductivity”. The corresponding model includes only one antinodal nonmagnetic PG besides two SCG-s, cf. [43–45, 50].

11 Interband nodal-region pairing

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11.2 The model Strong electronic correlations cause nonrigid behaviour of cuprate electron spectrum with doping. Perturbed distribution of states can include new band components and spectral gaps. Besides the itinerant carriers the defectpolaron type particles have been identified in doped cuprates. A corresponding two-component scenario of cuprate superconductivity has been formulated [51–53]. On these two sorts of carriers a phase separation in the material becomes possible [54]. Sample regions rich of doped carriers create new defectpolaron type bands, the regions poor of them support the weakened itinerant band(s). As the result, the energetic spectrum near the FS is essentially changed and acquires multiband nature [55–62]. In the case of hole doping the actual FS evolution embraces the region near the top of the itinerant (mainly oxygen) band. Electron doping comprises the whole parent Hubbard antiferromagnetic spectrum. The defect bands are then induced under the bottom of the UHB. The present work uses a modified model (cf. [45]) of hole doped cuprate defect subbands to follow more closely the novel findings mentioned in the Introduction. We use largely the foundation arguments and ideas from our earlier approaches [43–50] in coming to the model illustrated in Fig. 11.1. The itinerant valence band (γ) with nodal top (energy zero) symmetry overlaps the nodal defect β-band states. The latter occupy energies from zero to (−βc2 ). At this c is a measure of doped hole concentration. The itinerant band states of weight (1–c) end at (–D). The loosed weight charges equally by c/2 the defect β- and α-bands. The latter evolves in antinodal region between d and d−αc2 . The 2D densities of states (CuO2 plane) of defect subbands (ρα,β ) 1/(2αc) and 1/(2βc) depend on doping; ργ = (1–c) D−1 . The bare β − γ overlap and the ρβ (c) dependence take account of the residing “arc superconductivity”.

Fig. 11.1. The proposed energetic scheme for a hole-doped cuprate. In the nodal region there are the itinerant band γ (0> E > −D) and the defect band β (0>E> −βc2 ). The antinodal defect band (α) occupies states between d and d-αc2 . Note the defect bands bottom descending with doping.

144


The supposed bare gap between the α-band bottom and the top of the itinerant band (ZRS binding energy?) closes at cα = (d/α)1/2 . This gap acts as an extrinsic source for the antinodal PG as the result of the perturbation excerted by doping. The quadratic lowering of the defect band bottoms has been chosen by a sublinear fall-off of the PG with doping, as, e.g. has been indicated in [23, 62]. A further original effect of the doping with central importance for our model consists in the appearance of the interband pairing channel in the reorganized spectrum. The “arc superconductivity” is ascribed to the pairtransfer interaction [32] between the overlapping β- and γ-bands. The supposition that the defect β-subsystem borns as non-gapped from the itinerant states is an essential difference of the present model as compared with [43,45]. The corresponding characteristic interaction constant W = Wβ−γ can involve contribution of Coulombic and interband electron-phonon nature [32]. The analogous α − γ antinodal-nodal pairing will be supposed to be absent by the discussed cuprate superconducting properties. This second difference with [43,45] excludes the antinodal superconductivity and its contribution into the PG QP energy at Tcα , all the bands overlap. The α-band bottom reaches μ1 at c1 (μ1 = d − αc21 ) and for larger dopings −1 2αβc . (11.2) μ2 = −β(2αc2 − d) α + β + (1 − c) D This regime can end when the situation μ2 (c2 ) = −β c22 is reached where the chemical potential leaves the β-band, and is given by −1 2αc . (11.3) μ3 = (d − αc2 ) 1 + (1 − c) D Now the effective β −γ pairing disappears as also the resonance of corresponding FS-s. The composite trend of μ shown in Fig. 11.2 follows the data for hole doped cuprates [63, 64]. The nearly quadratic behaviour of μg with doping has also supported the chosen slipping down of the defect band bottoms with doping.

11 Interband nodal-region pairing

145

p

1

2

0.0

0.1

0.2

0.3

p Fig. 11.2. The calculated antinodal pseudogap (11.1) and the Tc dome (11.2) on the doping scale. The insert shows the behaviour of the chemical potential.

11.3 Low-energy excitations The calculations on the model have been made using the mean-field version of the pair transfer mediated superconductor Hamiltonian H=

+ εσ (k)a+ σks aσks Δγ

+ a aγk↑ aγ−k↓ + a+ γ−k↓ γk↑ k

σ,k,s

−Δβ

aβk↑ aβ−k↓ + a

k

+

←

β− k ↓

a+ βk↑

,

(11.4)

with the SCG defined as

⎧ Δ = 2W aβk↑ aβ−k↓ ⎪ ⎨ γ k

⎪ ⎩ Δβ = 2W aγ−k↓ aγk↑

(11.5)

k

Usual designations are used with band energies counted from the chemical potential εσ = ξσ − μ. The coupled gap system looks as (θ = kB T) ⎧

−1 Eβ (k) = ⎪ ⎨ Δγ W Δβ Eβ (k)th 2Θ k

γ (k) ⎪ ⎩ Δβ = W Δγ Eγ−1 (k)th E2Θ k

(11.6)

146


with the Bogolyubov QP energies as Eσ (k) = ± supercarrier density is given by

ε2σ (k) + Δ2σ . The

Δ2β 1 Δ2γ 2 Eγ (k) 2 Eβ (k) ns = . th + th 2 Eγ2 (k) 2Θ Eβ2 (k) 2Θ k

(11.7)

k

Low-energy QP excitations of the present model are represented by two SCG-s Δβ and Δγ , and one PG Δpα . The SCG Δβ,γ are of defect and itinerant genealogy correspondingly. The detailed symmetry of them does not follows from the present model. In the case of two s-wave order parameters the gaps are of opposite signs at W>0 and the larger one corresponds to the component with smaller density of states [32]. Pure d and s, or mixed (d-s) ordering symmetries are allowed in two-band models according to the doping level and temperature. Extreme dopings favour separated d and s superconducting gap symmetries, cf. [65, 66]. The antinodal PG excitation energies lie in the interval between the α-band bottom and the chemical potential lying in the β − γ overlapping region, so that Δpα = |ξα − μ1 |min = |d − αc2 − μ1 |. This PG vanishes at c1 , where the α-band starts to participate in determining of the chemical potential inside of all the three overlapping α, β, γ bands. The α PG enters independently the energy region occupied at TkF

Eq. (13.11) in Eq. (13.10) instead of BCS Hamiltonian Eq. (13.1), we can derive the Schrodinger equation for two quasiparticles–quasiparticle pair as follows k2 2 E quasi + 2μ − · uk ·ψ quasi (k)− V (k, k ) · u2k · u2k ·ψ quasi (k ) = 0 m pair pair pair k (13.21)

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where ψ quasi pair

(k) = k|ψ quasi pair

BCS

+ = k| b+ k↑ b−k↓ |ψBCS

is the wave function for two quasiparticles. Equation (13.21) can be reduced to the form similar to equation for a single Cooper pair if we introduce the following notation: ˜ quasi (13.22) E quasi = u2k E pair

pair

˜ quasi is the renormalized energy of a pair of quasiparticles, and E quasi where E pair

pair

is the same as in Eq. (13.21). Equation (13.21) in terms of Eq. (13.22) can be written in the form similar to but different from for a single Cooper pair as follows 2 ˜ quasi + 2μ − k V (k, k ) · u2k ·ψ quasi (k ) = 0 (13.23) · ψ quasi (k) − E m pair pair pair k

Equation (13.23) has u2k under the symbol of summation whereas in equation for a single Cooper pair there is unity instead of u2k under the symbol of summation. In addition, in the equation for a single Cooper pair there is 2εF whereas in Eq. (13.23) it is 2μ twice the chemical potential and in the rest Eq. (13.23) is same as equation for a single Cooper pair. The chemical potential, μ, should be determined from the constraint condition 2 vk2 = N with N being the number of particles. Equation (13.23) in the k

→ limit of Δk approaching zero, when u→ k 1 for |k| > kF (ξk > 0) and uk 0 for |k| < kF (ξk < 0), takes on the form of regular Schrodinger equation for a single Cooper pair. Although Eq. (13.23) can be reduced to the equation for a Cooper pair yet it is quite different from it as it contains u2k factor that describes the smearing of the Fermi distribution due to Δk . Because of the ˜ quasi of a sinfactor u2k in Eq. (13.23) the state corresponding to the energy E pair

gle quasiparticle pair in Eq. (13.23) is no longer the real bound state, whereas equation for a regular single Cooper pair shows that an addition of a pair of particles with the energy of εF decreases the energy of the system by the amount given by the well-known equation for the bound state of a single Cooper pair as shown below 2ωc E Cooper = ≈ 2ωc exp −2/NF |V | , (13.24) pair exp 2/NF |V | − 1 where NF = N (εF ) = kF m/2π 2 is the density of states per spin orientation. Equation (13.24) is the solution of the two-particle problem in the presence of a quiescent Fermi sea (i.e., ideal Fermi gas) and gives a real energy corresponding to a stable bound state. The solution in the form of Eq. (13.24) exists

13 Multiple quasiparticle pairs in the BCS model

173

only if there is a sharp Fermi edge. This important result was discovered by Cooper [2] who suggested that pairs of electrons entering this type of bound state are associated with the occurrence of superconducting phase. Our anal˜ quasi is mainly the function of the degree of ysis of Eq. (13.23) shows that E pair

smearing of the Fermi distribution not the strength of the interaction as it is in the case of a sharp Fermi edge. Since u2k is sharp than the bound state for two quasiparticles is likely to exist. In addition, according to Eq. (13.2), the quasiparticle density of states is very large near the energy gap. This fact favors the formation of the bound state between two quasiparticles. Thus, we can conclude that the bound state for two quasiparticles is likely to exist in the BCS ground state. It is also interesting to notice that the energy for two quasiparticles in Eq. (13.22) is k-dependent due to u2k factor, this means that the bound state of two quasiparticles also exists for k < kF , well below the Fermi momentum, which is different from the case of regular Cooper pair. It is also interesting to notice that in the case of Δ = 0, the gap equal to zero, Eqs. (13.21)–(13.23) takes on the form of regular the equation for a single Cooper pair. Thus, we can conclude that a pair of quasiparticles can form a bound pair similar to the Cooper pair of regular particles with the bound state energy which may be significantly larger than that for a pair of particles in the BCS ground state since the quasiparticle density of states is very large, and the bound state for a pair of quasiparticles can extend well below the Fermi momentum. The bound state for two quasiparticles exists as a result of non-diagonal terms in the BCS Hamiltonian written in terms of quasiparticles and large quasiparticle density of states near the energy gap. Since the quasiparticles are regular fermions, then it is interesting to consider the case of more than two quasiparticle pairs. Next we consider the quartet of quasiparticles. The state ket for two quasiparticle pair follows from Eq. (13.8). For the case when q stands just for one wave vector there are actually two values of q = ±k. Thus, according to Eq. (13.8) we have four quasiparticles corresponding to two quasiparticle pairs in the same state. Thus, for two quasiparticle pairs – the quartet of quasiparticles the state ket according to Eq. (13.8) can be written as + + + ψ quasi = b+ (13.25) k↑ b−k↓ b−k↑ bk↓ |ΨBCS . quartet

BCS

Using Eq. (13.25) in Eq. (13.10) we can derive the Schrodinger equation for four quasiparticle pairs in the same state. The equation for the quartet of quasiparticles pairs can be written as below 2k2 2 E4quasi + 4μ− · uk ψ4quasi (k) −2 V (k, k ) · u2k ·u2k ·ψ4quasi (k ) = 0 m k (13.26) + + + + ψ4quasi (k) = k | ψ4quasi BCS = k| bk↑ b−k↓ b−k↑ bk↓ |ΨBCS where is the wave function for two quasiparticle pairs, and E4quasi is the energy of a quartet of

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J. D. Fan and Y. M. Malozovsky

quasiparticles. Using the same notation as in Eq. (13.22), Eq. (13.26) can be written in the form similar to Eq. (13.23) for the case of four particles in the BCS ground state. The Schrodinger equation for the quartet of quasiparticles can be written as follows 2 ˜4quasi + 4μ − 2k ψ4quasi (k) − 2 E V (k, k ) · u2k ·ψ4quasi (k ) = 0 m k (13.27) ˜4quasi is the renormalized energy of a quartet of quasiparticles, and it where E is related to the E4quasi by the relationship similar to Eq. (13.22), E4quasi = ˜4quasi . Comparing Eq. (13.27) with Eq. (13.21) we can conclude that u2k E E4quasi = 2E2quasi . This result is similar to the result for two Cooper pairs in a quiescent Fermi sea. Thus, binding of four quasiparticles in the BCS ground state depends on weather there is a binding of two quasiparticles or not, and the bound state of a quartet of quasiparticles is just the double of the bound state for a pair of quasiparticles. The bound state for a quartet of quasiparticles exists as a result of a bound state for a pair of quasiparticles and the Pauli principle since quasiparticles are fermion like excitations. It is interesting to notice that although there is no interaction between different quasiparticle pairs in terms of the BCS Hamiltonian, yet there is bound state for two quasiparticle pairs occupying the same state. In addition, the wave function for four fermions does not depend on spins of particles, and thus represents the real boson wave function. The wave function of the multiple quasiparticle pairs can be written as Ψ multiple (k1 , k2 , k3 , ....kn ) = k1 , k2 , k3 , ....kn Ψ multiple quasipairs quasipairs + = k1 , k2 , k3 , ....kn | b+ q↑ b−q↓ |ΨBCS |q|H0 (see e.g. [9]) determines the model mean-field free energy of a two-band superconductor F (H) = F (H0 )+ < H1 >H0 , where F (H0 ) =

α,k

Eα (k) , ˜α (k) − 2kB T ln 2ch 2kB T

(14.5)

(14.6)

14 Damping of superconducting fluctuations in two-band systems

Eα (k) = ˜2α (k) + |δα |2

179

(14.7)

and < H1 >H0 means averaging over H0 , + ∗ a + δ a a δ α a+ H1 H0 = − α−k↓ αk↑ α αk↑ α−k↓ H0 H0

α,k

+ +2 Wαα, (k, k, ) a+ αk↑ aα−k↓

H0

α,α, k,k,

aα −k ↓ aα k ↑ H0

.

(14.8) Since

−δαk th (Eα (k)/2kB T ) (14.9) 2Eα (k) we obtain the model mean-field free energy in the following form 1 F = Jα (T, δα ) + δα2 ξα (T, δα ) + Wαα, δα δα, ξα (T, δα )ξα, (T, δα, ), 2 α,α, α α < aα−k↓ aαk↑ >H0 =

(14.10) where Jα =

k

Eα (k) ˜α (k) − 2kB T ln 2ch , 2kB T

ξα =

k

Eα−1 (k)th

Eα (k) , 2kB T

(14.11)

(14.12)

and the non-equilibrium superconducting order parameters δα are supposed to be real quantities. We assume that electron-electron interactions are non-zero only in the layer μ ± ωc and that the interaction constants are independent of electron wave vector in this layer. By replacing the summation over wave vector k with the integration over energy, the model free energy (14.10) can be written as F (δ1 , δ2 ) = Fn + ρ1 I(T, δ1 ) + ρ2 I(T, δ2 ) + δ12 ρ1 η(T, δ1 ) + δ22 ρ2 η(T, δ2 ) 1 + [W11 δ12 ρ21 η 2 (T, δ1 ) + W22 δ22 ρ22 η 2 (T, δ2 ) 2 +2W12 δ1 δ2 ρ1 η(T, δ1 )ρ2 η(T, δ2 )] .

(14.13)

Here, W12 = W21 , ρα are the densities of electron states taken to be constant in the narrow integration layers, and

ωc ch(Eα (δα )/2kB T ) d˜ α , I(T, δα ) = −4kB T ln (14.14) ch(˜ α /2kB T ) 0

ωc Eα (δα ) d˜ α , η(T, δα ) = 2 Eα−1 (δα )th (14.15) 2kB T 0 (14.16) Eα (δα ) = ˜2α + δα2 .

¨ et al. T. Ord

180

14.3 Phase diagram of a two-band superconductor Next we study the analytical properties of model free energy F (δ1 , δ2 ). From the conditions (∂F/∂δα )δ1,2 =Δ1,2 = 0, α = 1, 2, one finds the equations which determine the stationary points of F (δ1 , δ2 ), Δ1 + W11 Δ1 ρ1 η(T, Δ1 ) + W12 Δ2 ρ2 η(T, Δ2 ) = 0 , Δ2 + W22 Δ2 ρ2 η(T, Δ2 ) + W12 Δ1 ρ1 η(T, Δ1 ) = 0 .

(14.17)

In the limit, where interband interaction is absent, i.e. W12 = 0, the system (14.17) splits into two independent equations which describe autonomous phase transitions in two bands. The corresponding critical temperatures Tcα , α = 1, 2, are determined by the conditions Wαα ρα η(Tcα , 0) = − 1 where Wαα < 0 (intraband attraction). However, for arbitrary weak interband in2 teraction W12 < W11 W22 , the temperatures of independent phase transitions Tc1,2 transform into critical points Tc± : Tc± = 1.13

ωc exp −Θ± , kB

(14.18)

with −1

2 ρ ρ ) . Θ± = − W11 ρ1 + W22 ρ2 ± (W11 ρ1 − W22 ρ2 )2 + 4(W12 1 2 (14.19) It follows from the expressions (14.19) that Θ− < Θ+ .

(14.20)

Tc− > Tc+ .

(14.21)

Consequently

It can be shown that the temperature Tc+ has physical meaning only if 2 W11 , W22 < 0 and W12 < W11 W22 . The dependencies of the critical temperatures Tc± on interband interaction constant W12 are shown in Fig. 14.1. There exist two classes of non-trivial solutions of the system (14.17): + − + Δ− 1,2 = 0 if T < Tc and Δ1,2 = 0 if T < Tc . One can find that in the case of interband repulsion (W12 > 0) − sgn(Δ− 1 ) = −sgn(Δ2 ), + sgn(Δ+ 1 ) = sgn(Δ2 ),

and in the case of interband attraction (W12 < 0) − sgn(Δ− 1 ) = sgn(Δ2 ),

(14.22)


181

Fig. 14.1. The dependencies of Tc− and Tc+ on W12 . + sgn(Δ+ 1 ) = −sgn(Δ2 ).

(14.23)

Note, that the rules (14.22), (14.23) do not determine the signs of Δ± 1,2 uniquely. The different solutions of the Eqs. (14.17) Δ± 1,2 are related to the phase diagram of a two-band superconductor (see Fig. 14.1) as following: (i) In the region N, the superconductor is in the normal state, i.e. Δ± 1,2 = 0. (ii) In the domain SCI we have a stable superconducting state given by Δ− 1,2 = 0. (iii) = 0 of Eqs. (14.17) In the region SCII, an additional non-zero solution Δ+ 1,2 appears which corresponds to a metastable superconducting phase (see also [10]) or, at least, to the saddle-points of the model free energy F (δ1 , δ2 ).

14.4 Relaxation of order parameters fluctuations In the following we will investigate the relaxation of small fluctuations of order parameters in the normal and stable superconducting phases. We describe the relaxation of order parameters by means of Landau-Khalatnikov equations, −

2 − duα ∂F (Δ− 1 + u1 , Δ2 + u2 ) = γαα , α = 1, 2 , dt ∂uα

(14.24)

α =1

where uα are the fluctuations of the gap order parameters from their equilibrium values (δα = Δ− α + uα ) and γαα are the phenomenological kinetic coefficients. By expanding the model free energy in powers of fluctuations we linearize the kinetic equations (14.24). As a result we have −

2 duα = χαα uα , α = 1, 2 , dt α =1

(14.25)

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¨ et al. T. Ord

where 2

χαα =

γαα ϑα α ,

(14.26)

α =1

and

∂ 2 F (δ1 , δ2 ) = ∂δα ∂δα

ϑαα

.

(14.27)

δ1,2 =Δ− 1,2

Using a suitable orthogonal transformation the system (14.25) decouples into two independent equations −

dur,s = Ωr,s ur,s. dt

(14.28)

Here ur =

2

Kα1 uα , us =

α=1

2

Kα2 uα ,

(14.29)

α=1

are the new, non-critical (rigid) and critical (soft), variables, −k ⎞ 1 √ 2 ⎜ 1 + k2 ⎟ (Kαα ) = ⎝ 1 k+ k ⎠, 1 √ √ 1 + k2 1 + k2

(14.30)

k = (Ωr − χ22 )/χ12 = −(Ωs − χ11 )/χ12 , c

(14.31)

⎛

√

and Ωr,s are the eigenvalues of the matrix (χαα ) determined by Eq. (14.26), Ωr,s

1 2 2 χ11 + χ22 ± (χ11 − χ22 ) + 4χ12 . = 2

(14.32)

These eigenvalues determine the relaxation times which characterize the damping of the fluctuations ur,s : τr,s = 1/Ωr,s .

(14.33)

14.5 Results and discussion Now we present the results of the calculations for the temperature dependencies of the relaxation times determined by the eigenvalues (14.32). The following set of parameters was used: W11 = − 0.35 eV , W22 = − 0.15 eV , 1 = 0.5 (eV )−1 , 2 = 1 (eV )−1 , ωc = 0.05 eV γ11 = γ22 ≡ γ, γ12 = γ21 = 0.


183

In the absence of interband interaction each of two relaxation times exist in the autonomous one-band system. The critical slow-down appears in the damping of the superconducting fluctuations near the phase transitions with critical temperatures Tc1 and Tc2 induced by the intraband effective attractions W11 and W22 , respectively. However, if an arbitrary weak interband pair-transfer attractive or repulsive interaction is present, the new relaxation channels with characteristic times τs and τr are formed (see Fig. 14.2). The soft relaxation time τs as a function of temperature diverges at the critical point Tc− and passes through a maximum in the domain Tc+ < T < Tc− . The rigid relaxation channel, characterized by the time τr , do not exhibit critical slow-down at the phase transition point, but it has a maximum approximately at the same temperature T < Tc− where τs passes through a minimum. In Fig. 14.2 the temperature dependencies of superconducting gaps and the contour plots of model free energy F (δ1 , δ2 ) near its stationary points are also displayed. In particular, one can see here the formation of metastable states below Tc+ . The observed temperature behavior of relaxation channels in the superconducting phase (T < Tc− ) is sensitive to the strength of interband coupling. As W12 increases the maxima of τs,r (T ) are suppressed,

− Fig. 14.2. Left upper panel: the dependencies of Δ− 1 (solid) and Δ2 (dashed) on the reduced temperature T /Tc− for W12 = 0.00009 eV. Right upper panel: the corresponding dependencies of γτs (solid) and γτr (dashed) on T /Tc− . In these figures the critical temperatures Tc+ = 23.5 K and Tc− = 37.8 K are depicted by the vertical markers. In the lower panels the contour plots of model free energy (14.13) are presented for temperatures slightly below Tc+ (left) and slightly below Tc− (right).

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− Fig. 14.3. Left upper panel: the dependencies of Δ− 1 (solid) and Δ2 (dashed) on the reduced temperature T /Tc− for W12 = 0.003 eV. Right upper panel: the corresponding dependencies of γτs (solid) and γτr (dashed) on T /Tc− . In these figures the critical temperatures Tc+ = 23.4 K and Tc− = 37.9 K are depicted by the vertical markers. In the lower panels the contour plots of model free energy (14.13) are presented for the temperature slightly below Tc+ (left) and slightly below Tc− (right).

cf. Figs. 14.2 and 14.3. Finally one reaches to the characteristic times shown in Fig. 14.4 (see also [7]). One of these times (τs ) describes the slowly relaxing component of fluctuations near the phase transition point, decreasing monotonically if temperature moves away from Tc− . The other one (τr ) is practically independent of temperature and characterizes the rapidly relaxing component of fluctuations. In Figs. 14.2–14.4 one can also observe the evolution of the temperature behavior of superconducting gaps and the shape of model free energy F (δ1 , δ2 ) as interband coupling increases. In the vicinity of the transition temperature Tc− one can expand the model free energy (14.13) in a series in powers of δα and evaluate Ωr,s as a function of temperature. As a result near Tc− the leading contributions of the relaxation rates Ωr,s read as Ωr = 2γΘ− ρ1 (1 + 2ρ1 W11 Θ− ) + ρ2 (1 + 2ρ2 W22 Θ− ) , (14.34)

Ωs = 4Γ

T − T− ρ1 ρ2 (1 + (ρ1 W11 + ρ2 W22 )Θ− ) c , (14.35) ρ1 (1 + 2ρ1 W11 Θ− ) + ρ2 (1 + 2ρ2 W22 Θ− ) Tc−


185

− Fig. 14.4. Left upper panel: the dependencies of Δ− 1 (solid) and Δ2 (dashed) on the − reduced temperature T /Tc for W12 = 0.03 eV. Right upper panel: the corresponding dependencies of γτs (solid) and γτr (dashed) on T /Tc− . In these figures the critical temperatures Tc+ = 17.5 K and Tc− = 45.5 K are depicted by the vertical markers. In the lower panels the contour plots of the model free energy (14.13) are presented for the temperature slightly below Tc+ (left) and slightly below Tc− (right).

where

Γ =

2γ if T < Tc− γ if T > Tc−

.

(14.36)

In this approximation the relaxation rate of rigid order parameter is independent of temperature, whereas the damping of the fluctuations of soft order parameter exhibits usual critical slow-down near Tc− : Ωs ∼ |T − Tc− |. It can be found that Ωr increases and the coefficient before the factor |T − Tc− |/Tc− in Ωs increases if |W12 | increases. For critical fluctuations the Landau “law of two” is valid, i.e. the relaxation rate below Tc− is two times larger than the relaxation rate above Tc− . As a conclusion, we have found the temperature dependence of relaxation times of superconducting fluctuations in a two-band model which exhibit below the phase transition point non-monotonic behavior. The observation is caused by the competition between intra- and interband couplings and disappears if interband interaction is sufficiently strong. The finding may be of relevance for the interpretation of similar peculiarities in the experimental data on relaxation phenomena in magnesium diboride and copper-oxides (see [11, 12]).

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Acknowledgement The work was supported by the Estonian Science Foundation grant N◦ 7296.

References H. Suhl, B. T. Matthias, and L. P. Walker, Phys. Rev. Lett. 3, 552 (1959) V. A. Moscalenko, Fiz. Met. Metalloved. 8, 503 (1959) A. Bianconi, J. Supercond 18, 625 (2005) S. Tsuda et al., Phys. Rev. Lett. 91, 127001 (2003) D. V. Evtushinsky et al., New J. Phys. 11, 055069 (2009) R. Khasanov et al., Phys. Rev. Lett. 98, 057007 (2007) ¨ and N. Kristoffel, Physica C 331, 13 (2000) T. Ord J. Demsar et al., Phys. Rev. Lett. 82, 4918 (1999) S. V. Tjablikov, Methods of Quantum theory of Magnetism, Nauka, M. (1975) (In Russian) 10. T. Soda and Y. Wada Progr. Theor. Phys. 36, 1111 (1966) 11. Y. Xu et al., Phys. Rev. Lett. 91, 197004 (2003) 12. N. Cao et al., Chin. Phys. Lett. 25, 2257 (2008) 1. 2. 3. 4. 5. 6. 7. 8. 9.

15 PHASE SLIP PHENOMENA ONE AND TWO DIMENSIONAL SUPERCONDUCTING RING M. Lu-Dac and V. V. Kabanov Josef Stefan Institute 1001, Ljubljana, Slovenia [email protected] Abstract. We analyze phase slip phenomena in one and two dimensional superconducting rings by solving the time-dependent Ginzburg-Landau equation. In the one dimensional case we show that the phase slip kinetics occurs simultaneously and consecutively depending on the dimensionless parameter u in the equation. In two dimensions there are two values of critical currents jc1 and jc2 . When the local current is larger then jc1 the phase slip is similar to the one dimensional case. Kinetics is governed by kinematic vortices. When the local current exceeds jc2 value the vortex generation is governed by the Kibble-Zurek quench mechanism.

Key words: Ginzburg-Landau equation, vortices, kinematic vortices, kinetics.

15.1 Introduction Nonequilibrium phenomena in superconductors are a challenging area for both experimental and theoretical research. They are also crucial for the development of applications as they are the key to the appearance of resistive states in superconducting samples and to the possible use of vortex dynamics. In onedimension (1D), the resistive phase slip process foreseen by Little [1] was described quantitatively by the theory developed by Langer and Ambegaokar [2] and extended by McCumber [3] and Halperin [4]. The theory describes thermally activated phase slips. It evaluates the resistance of a 1D superconductor when driven out of thermodynamical equilibrium by a voltage or current source. Since then, this theory has been accepted in rather good agreement with experiments (for a more complete review of theoretical and experimental works, see Ref. [5]). More recently, simulations were carried out on different versions of the time-dependent Ginzburg-Landau (GL) (TDGL) equations in order to investigate the dynamics of the process. In particular, the case of the 1D ring has raised interest since it exhibits multiple metastable states which can be J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 15,

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reached by phase slip processes. In the work of Tarlie and Elder [6], the ring was submitted to an electromotive force which constantly accelerates the superconducting electrons. Another approach was used by Vodolazov and Peeters [7], who increased the magnetic field gradually. The latter simulations were confirmed by the experiments made in Refs. [8]. Nonequilibrium phenomena in 2D films are even more interesting because of vortex production during the transition. This problem is interesting not only in the field of standard solid state physics and represents a very good model to study topological phase transitions in cosmology and other branches of contemporary physics [9–11]. There are two general ways to produce topological defects in superconductors or in superfluids. The first is based on the fact that at certain critical velocity vc the barrier between homogeneous flow and flow with vortex lines disappears and vortices penetrate the sample through its boundaries. Thermal activation or quantum tunnelling assists in generation of the vortices in the close vicinity to the critical velocity vc because of the large value of the correlation length in comparison with the lattice constant [12]. Measurements of the vortex nucleations in the rotating superfluid Helium-3 have shown that vortices are created when the velocity reaches some critical value. The measured value of the critical velocity appears to be smaller than the theoretically predicted value. Therefore the vortex nucleation occurs locally in the regions where the velocity exceeds the local critical velocity [13]. This is consistent with the fact that the phase slip (PS) in two-dimensional superconducting film occurs in one dimensional manner [14]. Indeed the standard stability analysis of two-dimensional uniform current leads to the pure one-dimensional instability. The PS in that case occurs in two steps. The order parameter becomes small and finally reduces to 0 at the line perpendicular to the current. The PS occurs exactly at that moment along this line. Therefore the solution is uniform across the film [14]. The inhomogeneity caused by the current contacts leads to a certain inhomogeneity across the film, but the general picture of the PS phenomena remains the same [15]. The order parameter is almost equal to 0 along the PS line. Along this line appears two points where the order parameter becomes 0 and this “kinematic” vortex-antivortex (VaV) pair crosses the film in opposite directions very quickly. Therefore phase slips occur without formation of well defined VaV pairs [15]. Another mechanism of vortex production was proposed by Kibble [10] and Zurek [11] (KZ). When the sample is quickly quenched trough critical temperature, the nucleation of the low temperature phase starts in different places with uncorrelated phases of the order parameter. Then, domains grow and start to overlap. On the boundaries of different domains with uncorrelated phases, topological defects are formed. The density of topological defects is determined by the ratio of quench time to the intrinsic GL relaxation time. In Refs. [16, 17] it was proposed that the quench occurs not only due to fast change of the temperature but also due to temperature front propagation. Aranson, Kopnin and Vinokur considered the case of temperature quench in the presence of external current [18]. The growth of the new phase occurs

15 Phase slip phenomena superconducting ring

189

H

d R

Fig. 15.1. Geometry of the ring of radius R and thickness d with the magnetic field H.

into the state with lowest energy. This state corresponds to the state with the absence of the current. Therefore on the border of the quenched region superfluid velocity has tangential discontinuity, leading to the vortex formation, similar to the classical hydrodynamic Kelvin-Helmholtz instability [19] which also exists in superconductors [20]. The vortices are created here also when current density is much lower then critical current densities. Moreover, the instability near the interface with the tangential discontinuity of velocity suppresses the development of KZ vortices [18]. In the standard formulation of the KZ theory the control parameter is temperature. When the temperature is quickly quenched through the critical temperature, the VaV pairs are created spontaneously. Therefore, before quench, the order parameter is equal to 0 and only thermal fluctuations are present. In the case of superconductors the formation of topological defects may be induced by the time-dependent external fields. One possibility is the illumination of the sample with the short pulse of electromagnetic field in terahertz region of the spectrum. The irradiation may induce current which may lead to formation of non-thermal VaV pairs. Here we can consider the response of the superconducting wire and the superconducting film rolled on the cylinder to sudden step-like change of magnetic field parallel to the cylinder axis (Fig. 15.1). The thickness of the film and wire d are small d ξ eff. Here ξ is the superconducting coherence length and λeff is Pearl penetration depth. Therefore we can neglect all corrections to the external magnetic field H caused by the superconducting current. The radius of the cylinder R > ξ. External magnetic field is parallel to the cylinder axis and considered as an external variable. To model inhomogeneity of the film we assume that there is a thin stripe of superconductor along the film with different value of superconducting coherence length.

15.2 Formulation of the problem The time dependent Ginzburg-Landau equation in dimensionless units has the form: ∂ψ + iΦψ) = ψ − ψ|ψ|2 − (i∇ + a)2 ψ + η. (15.1) u( ∂t

190


Here ψ is dimensionless complex order parameter, the spacial coordinate r is measured in units of coherence length ξ and time is measured in units of phase relaxation time τθ = 4πλeff σn /c2 , λeff = λ2 /d, λ is the bulk penetration depth, σn is the normal state conductivity, and c is the speed of light. The parameter u = τψ /τθ is a material dependent constant, where τψ is the relaxation time of the amplitude of the order parameter. According to the microscopic theory u is ranging from 5 to 12, but we formally assume that this parameter does not have any restriction. The dimensionless vector potential a is measured in units of φ0 /2πξ and φ0 is the flux quantum. Equation (15.1) requires boundary condition at z = 0 and z = w (w is the width of the film). In the future analysis we use periodic boundary conditions and the standard boundary condition with vacuum [21]. The equation for the dimensionless electrostatic potential Φ, measured in units φ0 /2πcτθ , where e is electronic charge, and is Planck’s constant is derived assuming that film thickness is small and corrections to the magnetic field due to superconducting current may be neglected: i ∇2 Φ = −∇ (ψ ∗ ∇ψ − ψ∇ψ ∗ ) + a|ψ|2 . 2

(15.2)

We also add a Langevin noise during the simulations. The intensity of the noise may vary considerably from one material to another and depends on the experimental conditions. Nevertheless, for the case of the second-order phase 2 transition, the noise should be small and on the order of kB T /d2 Hc2 ξ ∼ 2 (T /Tc )(kB Tc /EF ) in dimensionless units when d ξ. Here, kB is the Boltzman constant, T is the temperature, Tc is the critical temperature, Hc2 is the second critical field, and EF is the Fermi energy. To model kinetics of the PS processes we assume that at time t < 0, the external magnetic field is absent (a = 0). At t = 0 suddenly appears external magnetic field which stays constant at t > 0 i.e. the tangential component of the vector potential is defined as aϑ(t), where ϑ(t) is the Heaviside step function. As a result we study the kinetics of the vortex formation as a function of a. √ The linear analysis [14] shows that for a < ac1 = 1/ 3√the solution is metastable. In the range of the fields ac1 < a < ac2 , ac2 = 2 the decay of the uniform solution is always longer than the growth rate of a new solution. Therefore we expect that kinetics in 2D will be similar to the 1D case. The qualitative difference in kinetics takes place when the decay rate of the uniform solution become faster then the growth of the new phase with finite paramagnetic current. This effect is similar to quench trough Tc in the KZ mecha√ nism [9, 10]. It happens when a > ac2 = 2, leading to the rapid growth of topological defects. The density of topological defects may be estimated using Zurek arguments where quench time should be replaced by τQ = (a2 − 1)−1 −1/4 leading to n ∝ τQ [22].


191

15.3 Results We simulate Eqs. (15.1, 15.2) using the fourth order Runge-Kutta (RK4) method. The spatial derivatives in Eqs. (15.1, 15.2) are evaluated using a finite difference scheme of second order or using fast fourier transform algorithm depending on the boundary condition. The choice of the algorithm is made to optimize the convergence and the calculation times. Usually we find the solution first applying the simplest Euler method to find the optimal parameters, time and step size and then performed more accurate calculations with RK4. The calculations are performed for the vector potential 0 < a < 5 and for the total flux φ through the ring ranging so that φ/φ0 varies from 0 to 50 for different values of u. 1D Ring The kinetics of the phase slip (i.e. flux penetration to the ring) strongly depends on the parameter u [14]. In the limit of large u (u 1) the kinetics is determined by the single phase slip center for the case of φ/φ0 > 1. The order parameter becomes inhomogeneous and approaches to 0 at one arbitrary point of the ring. The phase of the order parameter becomes also inhomogeneous. The largest gradient of the phase is observed exactly in the place where the order parameter is small. At a certain moment of time, the order parameter becomes 0 at one point of the ring and the phase becomes discontinuous displaying the 2π jump. This process is repeated several times until the ring reaches the stable or metastable state. The electric field and order parameter have oscillatory behaviour during the transition (Fig. 15.2(d)). This multiple phase slips process is called consecutive. In the opposite limit u 1 the situation is different. The multiple phase slips process is characterized by many phase slip centers. Therefore, the order parameter becomes simultaneously equal to 0 at several points of the ring. Exactly at that time, the phase displays several 2π jumps (Fig. 15.2(a,b,c)) [14]. This multiple phase slips process is called simultaneous and is characterized by small value of u. 2D Film We investigate the flux penetration to the homogeneous ring for two different boundary conditions. In the case of periodic boundary conditions we identify different regimes. In the small field limit a < ac1 the ring is in the metastable state and the penetration of the magnetic flux to the ring may be induced only by the strong noise η in Eq. (15.1). In the higher field ac1 < a < ac2 the PS kinetics is different depending on the flux φ/φ0 . When φ/φ0 < 10 the kinetics is very similar to 1D ring [14]. The transition is characterized by one or more lines along z-axis where the value of the order parameter decreases towards zero. Then order parameter reaches zero at some point on this line

192


a

b

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1.00 ρ

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t=0.00 t=3.00 t=3.26 t=3.35 t=3.55 t=5.00

0.25

t=0.00 t=12.5 t=13.0 t=13.5 t=14.0 t=15.8 t=50.0

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20

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t=0.00 t=20.0 t=21.9 t=22.6 t=23.3 t=24.0 t=100

0.25

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t=0 t=95 t=103 t=115 t=125 t=500

0.25

0.00

0.00 0

10

20 x

30

40

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20 x

30

40

Fig. 15.2. Evolution (a)–(d) of the order-parameter amplitude along the ring exhibiting the competition between simultaneous and consecutive phase slips for different values of parameter u

where kinematic VaV pair is created. These kinematic VaV pairs propagate quickly along lines creating 2π jump of the phase along these lines. The order parameter along this line is indeed very close to 0 and kinematic vortices are not very well defined. They crosse the film quickly and are sometime difficult to detect. These lines may appear simultaneously or consecutively in time depending on u as in the case of 1D ring [14]. In the case of large fluxes φ/φ0 > 10, the kinetics is more stochastic because of the interaction between different PS lines. In Fig. 15.3(a,b) we present the time evolution of the average value of the order parameter together with time dependence of the number of VaV pairs. As clearly seen from this figure the kinetics is characterized by series of consecutive PS phenomena well separated in time. All this PS are produced by kinematic VaV pairs propagating along the same line with reduced order parameter. Further increase of a results in replacement of straight lines by vortex rivers (Fig. 15.3(b)). Along one vortex river few VaV pairs are created. The kinetics is determined by the motion of these pairs along these rivers and finally by their annihilation.


a

1 1.0 0.9 0.8 0.7 0.5 0.4 0.3 0.2 0.1 0

200 160 120

Y 80

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160

6 V 4 2 V

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–1

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10 20 30 40 50 60 Z

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8

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60

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0.8

|ρ|

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200

300 t

400

0.0 500

|ρ|

0.4

20

0

193

10

0.2

0

0.0 0

250

500

750

t

Fig. 15.3. Kinetics of the flux penetration to the 2D ring for u = 10 and φ/φ0 = 20. Upper panels represent the spatial distribution of the order parameter. Lower panels show the time evolution of the average over volume value of the order and time dependence of the number of VaV pairs. a = 0.6 (a,b), a = 0.8 (c,d), and a = 1.8 (e,f). Panels (a,c,e) are made at t = 700,105, and 100 respectively.

Importantly, the average value of the order parameter never reaches 0, on the contrary to the case of large field a > ac2 when the real quench takes place. The to1tal number of VaV pairs is substantially larger than φ/φ0 (Fig. 15.3(c,d)). The propagation of vortices along the rivers is much slower than in the previous case which can be seen from the time evolution of the number of vortices (Fig. 15.3(d)). Nevertheless the vortex velocity is still much faster then in the case when the order parameter has recovered its equilibrium value. The last regime a > ac2 is presented in Fig. 15.3(e,f). Here the quench condition is satisfied and the order parameter decreases uniformly until it reaches zero. As a results the vortices are created randomly. The number of the VaV pairs is substantially larger than φ/φ0 . Most of these vortices recombine rapidly. The remaining vortices travel relatively slowly through the sample, propagating the 2π phase jump. This behaviour is demonstrated in Fig. 15.3(f). Very quickly the average amplitude of the order parameter is suppressed to 0. After that, the new phase starts to grow uncorrelated. As a results a large amount of vortices is produced. The random dispersion of these vortices is fingerprint of KZ mechanism. It is important to notice that the total net vorticity is strictly equal to 0 at any time in the case of periodic boundary conditions in the z direction. In the case of vacuum boundary conditions on the edge of the film [21], the kinetics is very similar. In the case when ac1 < a < ac2 and φ/φ0 < 10 one or more lines with reduced order parameter are formed. The kinematic vortices

194

M. Lu-Dac and V. V. Kabanov 0 0.20 0.40 0.60 0.80

Y

1.00

X

Fig. 15.4. Creation of vortices in 2D film with vacuum boundary conditions. Long time evolution is related to annihilation and diffusion of vortex-antivortex pairs. φ/φ0 = 11 a = 0.8.

move along these lines almost instantaneously, creating 2π discontinuity in the phase. The difference between two different boundary conditions is that the PS lines here have finite curvature. It is caused by the fact that the development of the PS lines starts from edges of the film. Then lines growing from one edge of the film connect with lines growing from the other edge of the film. Further increase of the flux, keeping a constant leads to the formation of the flux rivers. The most important difference is that not all “rivers” connect two edges of the film. As a result, some of them ended in the middle of the film, leading to relatively small finite net vorticity (Fig. 15.4). These remaining vortices and antivortices propagate slowly to the edges of the film and the kinetics is determined by this slow vortex motion. The case when a > ac2 is governed by KZ mechanism, as with periodic boundary conditions, but vorticity may be finite.

15.4 Conclusion We have considered the kinetics of the flux penetration to the superconducting ring caused by the sudden change in magnetic field. We have shown that in the case of 1D ring the penetration of flux (PS kinetics) is different for large and small values of u. If u 1 the kinetic is determined by many phase slip centers where order parameter reduces to 0. For large u 1 the phase slip occurs consecutively i.e. the phase slips several times in the same place.


195

In 2D, the kinetics is more complicated. At small external field and with relatively small fluxes kinetics is essentially one dimensional. There can be one (u 1) or more (u 1) phase slip center across the field. The kinetics is governed by kinematic kinetics which cross the film along the phase slip lines. For larger fluxes these phase slip lines have a more stochastic character and are no longer straight lines. The net vorticity of the sample may be finite during the relaxation. The strong field a > ac2 leads to the quench of the sample and the new phase starts to grow from vacuum in accordance with KZ mechanism. Vortices are generated uncorrelated and quickly recombine.

References 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

W. A. Little, Phys. Rev. 156, 396 (1967). J. S. Langer and V. Ambegaokar, Phys. Rev. 164, 498 (1967). D. E. McCumber, Phys. Rev. 172, 427 (1968). D. E. McCumber and N. B. Halperin, Phys. Rev. B 1, 1054 (1970). R. Tidecks, Springer Tracts Mod. Phys. 121, 1 (1990). M. B. Tarlie and K. R. Elder, Phys. Rev. Lett. 81, 18 (1998). D. Y. Vodolazov and F. M. Peeters, Phys. Rev. B 66, 054537 (2002). D. Y. Vodolazov, F. M. Peeters, S. V. Dubonos, and A. K. Geim, Phys. Rev. B 67, 054506 (2003). S. Michotte, S. M` atéfi-Tempfli, L. Piraux, D. Y. Vodolazov, and F. M. Peeters, Phys. Rev. B 69, 094512 (2004). W. H. Zurek, Physics Reports, 276, 177 (1996). T. W. B. Kibble, J. Phys. A, 9, 1387 (1976). W. H. Zurek, Nature, 317, 505 (1985). G. E. Volovik, Pisma ZhETF, 15, 116 (1972). U. Parts et al., Europhys. Letters, 31, 449 (1995). M. Lu-Dac and V. V. Kabanov, Phys. Rev. B 79, 184521 (2009); J. Phys.: Conf. Ser. 129, 012050 (2008). G. R. Berdiyorov, M. V. Milosevic, and F. M. Peeters, Phys. Rev. B 79, 184506 (2009). T. W. B. Kibble and G. Volovik, JETP Lett., 65, 102 (1997). N. B. Kopnin and E. V. Thuneberg, Phys. Rev. Lett., 83, 116 (1999). I. S. Aranson, N. B. Kopnin, and V. M. Vinikur, Phys. Rev. Lett., 83, 2600 (1999). L. D. Landau and E. M. Lifshitz, Hydrodynamics. A. M. Fridman, Uspehi Fizicheskih Nauk, 78, 225 (2008). G. E. Volovik, Pis’ma ZhETF, 75, 491 (2002). S. E. Korshunov, ibid, 496 (2002). P. G. de Gennes, Superconductivity of Metals and Alloys, Perseus Books Publishing, L.L.C. 1989. P. Laguna and W. H. Zurek, Phys. Rev. D, 58, 085021 (1998).

16 LOCUS OF THE SUPERCONDUCTIVITY IN THE CUPRATES J.D. Dow1,2 and D.R. Harshman1,3 1 2 3

Department of Physics, Arizona State University, Tempe, AZ 85287 USA [email protected] Institute for Post-doctoral Studies, 6031 E. Cholla Lane, Scottsdale, AZ 85253 USA Department of Physics, University of Notre Dame, Notre Dame, IN 46556 USA

Abstract. In YBa2 Cu3 Ox , muon spin rotation (µ+ SR) measurements were taken at applied magnetic fields along the c-axis of 0.05, 1.0, 3.0, and 6.0 Tesla. These measurements, which take into account the expected field-dependent and temperatureactivated flux-line disorder, are consistent with a nodeless (s-wave) superconducting order parameter, and inconsistent with order parameters possessing nodes, including those with dx2 −y2 symmetry. We conclude that the superconductivity is p-type and occupies the BaO layers of YBa2 Cu3 Ox . In other cuprates or ruthenates, the superconductivity resides in the BaO, SrO, or interstitial oxygen regions.

Key words: cuprate superconductors, ruthenates.

16.1 Potentially superconducting layers of YBa2 Cu3 Ox In YBa2 Cu3 Ox , μ+ spin resonance spectroscopy (μ+ SR) detects only superconducting layers (not normal layers). The CuO and CuO2 layers are not detected by μ+ SR, because those layers are the ones containing Cu, and presumably do not superconduct [1]. Clearly the cuprate-planes (CuO2 ) and the CuO chains do not superconduct in YBa2 Cu3 Ox . The layers that might superconduct p-type are either Y+3 or BaO, and Y+3 is too ionic. That leaves BaO as the layer that superconducts p-type in YBa2 Cu3 Ox (see Fig. 16.1). This analysis disagrees with the widespread conclusion the it is the cuprate-planes that superconduct p-type.

16.2 Hole superconductivity in BaO layers The hole superconductivity of YBa2 Cu3 Ox must occupy the BaO layers, not the cuprate-planes (which planes, according to the muons, do not superconduct). The widespread misimpression that the cuprate-planes superconduct in J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 16,

197

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J.D. Dow and D.R. Harshman

Fig. 16.1. The observed layer charges (in units of |e|) versus oxygen content x of YBa2 Cu3 Ox , for (a) the CuO chain layers (top line), (b) the BaO layers (middle line), and (c) the /CuO2 /Y/CuO2 / layers (bottom line).

the cuprates is based on data by Cava et al. [2] and by Jorgensen et al. [3, 4] but can be contradicted by simply counting charges: a cuprate-plane has a charge of approximately –2 |e|, a Y+3 ion has a net charge of +3 |e|, and the combination of planes /CuO2 /Y/CuO2 /has a charge near –1 |e|. Hence each cuprate-plane, CuO2 , has a net negative charge and does not have the free holes needed to superconduct p-type. The CuO chain layers have positive charges detected by neutron spectroscopy (Fig. 16.1) that decrease with alloy composition x in YBa2 Cu3 Ox , so those layers cannot be responsible for the onset of p-type superconductivity, and their charge decreases with x. Furthermore, the holes of CuO or CuO2 are not detected by muon spectroscopy (which detects superconducting layers only), because those planes do not superconduct. Hence the origin of p-type conductivity and superconductivity in YBa2 Cu3 Ox must be in the nearly neutral BaO layers. Similarly, in Bi2 Sr2 CaCu2 O8 , the p-superconductivity originates in the SrO layers.

16 Locus of the superconductivity in the cuprates

199

16.3 Cuprate-plane’s charge in YBa2 Cu3 Ox To check the nature of superconductivity in YBa2 Cu3 Ox , we first look at the measured CuO2 charge in the cuprate-plane, and the charge of the CuO chain layer in YBa2 Cu3 Ox (see Fig. 16.2). We also examine the charge of the BaO layer [5–8]. Cava et al. [2] and Jorgensen et al. [3, 4], using the bond-valence sum method [9] have reported Cu(2) charges (i.e., Cu charges in the cuprateplane) versus alloy composition x according to Refs. [10–13] (see Fig. 16.1). Note that the data of Cava et al. do not agree with the data of Jorgensen et al., and that the data of Jorgensen et al. were not published in graphical form by their authors. We conclude that the data of Cava et al. must be in error because they do not agree with the data of Jorgensen et al., and that the Cava datum interpreted as the threshold of superconductivity does not exist in the Jorgensen data, or in our muon data.

Fig. 16.2. Cuprate-planar Cu charge (in units of |e|) versus oxygen content x for YBa2 Cu3 Ox , showing the Cava data (red line with squares), and the Jorgensen data (blue line with diamonds). Note the systematic displacement of the red line from the blue line, and the single point at x ≈ 6.35, which disagrees with the blue line (and which we think is in error).

200


16.4 Fluxon de-pinning and muon spin rotation of YBa2 Cu3 O7 Once the effects of temperature activated fluxon de-pinning have been taken into account, muon spin rotation measurements reveal that a single-crystal of YBa2 Cu3 O7 has data taken at fields H = 0.05, 1.0, 3.0, and 6.0 Tesla, which can be fit with a self-consistent pinning model, with s-wave pairing. The resulting gap function is strong-coupling, s-wave, and nodeless. The zero-field penetration depth λab (T,H = 0) for such a model has a higher probability of being correct than a d-wave model and lacks a Cu d-wave signature. This means that the superconductivity resides neither in the CuO2 layers nor in the CuO layers, but in the BaO layers.

16.5 Charges in the layers of Ba2 YRu1−uCuuO6 To check the nature of superconductivity in YBa2 Cu3 Ox , it is probably best to examine the cuprate-plane model of high-temperature superconductivity in the similar compound Ba2 YRuO6 doped with Ru-site Cu [1]. Except for vibrations of Ru, which freeze out at 23 K [14], Ba2 YRuO6 doped with Cu has an onset of superconductivity at 93 K, essentially the same as TC for YBa2 Cu3 O7 [15]. This can be understood by considering the crystal structure of Ba2 YRu1−u Cuu O8 : (i) a YRuO4 layer (with Y or Ru ions on opposite diagonals of the corners of a YRuO4 rectangle), an O on the sides of each rectangle, and Ba ions on the face-center of the next layer’s rectangle with O ions on the corners.

16.6 Specific heat and thermal conductivity of YBa2 Cu3 O7 Moler et al. [16] find that YBa2 Cu3 O7 exhibits excess specific heat which can be explained if one (or more) plane of the crystal structure does not superconduct, and so that plane has normal carriers. The cuprate-plane does not superconduct, according to our muon analysis. There are no lines of nodes in our specific heat, a consequence of the zero-field, linear-in-temperature part, which implies that the superconductivity is s-wave. This was pointed out by Taillefer et al. [17], who also authored a d-wave explanation of the data. Consequently the correct explanation of the superconductivity and specific-heat data is either (i) that all layers of YBa2 Cu3 O7 superconduct the same, and the superconductivity is d-wave in character (and inconsistent with the observation of excess specific heat by Moler et al.); or (ii) that the BaO layer superconducts, and it superconducts s-like, while the CuO2 layers do not superconduct. The alternative hypothesis (iii) is that the superconductivity occupies the BaO conduction bands. This is the only hypothesis consistent with the data [18–22].

16 Locus of the superconductivity in the cuprates

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Acknowledgments We are grateful to Physikon Project No. PL-206 for support.

References 1. J. D. Dow, D. R. Harshman, and A. T. Fiory, “High-TC superconductivity of cuprates and ruthenates. In “Electron Correlations in New Materials and Nanosystems,” edited by K. Scharnberg and S. Kruchinin, Springer, (Berlin, Heidelberg, New York, 2007), 263–274 (2007). (NATO Advanced Research Workshop, Yalta, Crimea, Ukraine). 2. R. J. Cava, A. W. Hewat, E. A. Hewat, B. Batlogg, M. Marezio, K. M. Rabe, J. J. Krajewski, W. F. Peck, Jr., and L. W. Rupp, Jr., Physica C 165, 419 (1990). 3. J. D. Jorgensen, B. W. Veal, A. P. Paulikas, L. J. Nowicki, G. W. Crabtree, H. Claus, and W. K. Kwok, Phys. Rev. B 41, 1863 (1990). 4. J. D. Jorgensen, Phys. Today, 34 (June, 1991). 5. H. A. Blackstead, J. D. Dow, D. R. Harshman, M. J. DeMarco, M. K. Wu, D. Y. Chen, F. Z. Chien, D. B. Pulling, W. J. Kossler, A. J. Greer, C. E. Stronach, E. Koster, B. Hitti, M. Haka, and S. Toorongian. EPJ B 15, 649 (2000). When Cu-doped, Sr2 YRuO6 superconducts. 6. J. D. Dow and D. R. Harshman, J. Supercond.: Incorporating Novel Magnetism 15, 455 (2002). Sr2 YRuO6 is an s-wave superconductor when doped with Cu, whose superconductivity becomes complete at 23 K. 7. J. D. Dow and D. R. Harshman, J. Low Temp. Phys. 131, 483 (2003). The ruthenates and cuprates of Sr2 YRuO6 exhibit s-wave superconductivity. 8. I. D. Brown and D. Altermatt, Acta Crystallogr., Sect. B: Struct. Sci. B 41, 244 (1985); D. Altermatt and I. D. Brown, Acta Crystallogr., Sect. B: Struct. Sci. B41, 241 (1985); I. D. Brown, J. Solid State Chem. 82, 122 (1989). I. D. Brown and K. K. Wu, Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem. B 32, 1957 (1976); I. D. Brown, Structure and Bonding in Crystals, Vol. II, edited by M. O’Keefe and A. Navrotsky, pp. 1–20 (Academic Press, New York, 1980). 9. B. A. Hunter, J. D. Jorgensen, J. L. Wagner, P. G. Radaelli, D. G. Hinks, H. Shaked, R. L. Hitterman, and R. B. Von Dreele, Physica C 221, 1 (1994). 10. O. Chmaissem, Q. Huang, S. N. Putilin, M. Marezio, and A. Santoro, Physica C 212, 259 (1993). 11. S. M. Loureiro, E. V. Antipov, J. L. Tholence, J. J. Capponi, O. Chmaissem, Q. Huang, and M. Marezio, Physica C 217, 253 (1993). 12. O. Chmaissem, Q. Huang, E. V. Antipov, S. N. Putilin, M. Marezio, S. M. Loureiro, J. J. Capponi, J. L. Tholence, and A. Santoro, Physics C 217, 265 (1993). 13. H. A. Blackstead and J. D. Dow, Solid State Commun. 95, 613–617 (1995). 14. M. Tarascon, W. R. McKinnon, P. Barboux, D. M. Hwang, B. G. Bagley, L. H. Greene, G. W. Hull, Y. Le Page, N. Stoffel, and M. Giroud, Phys. Rev. B 38, 8885 (1988). 15. J. D. Dow and D. R. Harshman, Int. J. Mod. Phys. B 19, 37–42 (2005).

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16. K. A. Moler, D. L. Sisson, J. S. Urbach, M. R. Beasley, A. Kapitulnik, D. J. Baar, R. Liang, and W. N. Hardy, Phys. Rev. B 55, 3954 (1997). 17. L. Taillefer, B. Lussier, R. Gagnon, K. Behnia, and H. Aubin, Phys. Rev. Lett. 79, 483 (1997). 18. J. D. Dow and D. R. Harshman, J. Vac. Sci. Technol. B 24, 1977–1986 (2006). 19. D. R. Harshman, W. J. Kossler, X. Wan, A. T. Fiory, A. J. Greer, D. R. Noakes, C. E. Stronach, E. Koster, and J. D. Dow, Phys. Rev. B 69, 174505 (2004). This paper reports carefully done muon spectroscopy, which does not produce observable d-waves. 20. D. R. Harshman and J. D. Dow, Int. J. Mod. Phys. B 147 (2005). 21. H. A. Blackstead and J. D. Dow, In Quantum Confinement III: Quantum Wires and Dots, Ed. by M. Cahay, S. Bandyopadhyay, J.-P. Leburton, and M. Razeghi, (Electrochemical Society, Inc., Pennington, New Jersey, 1996), pp. 339–354. This paper shows that the superconductivity is remote from the cuprate-planes. 22. D. R. Harshman, J. D. Dow, and A. T. Fiory, Phys. Rev. B 77, 024523 (2008). Phonons do not cause the isotope effect in YBa2 Cu3 Ox , electronic excitations do.

17 AN APPROXIMATING HAMILTONIAN METHOD IN THE THEORY OF IMPERFECT BOSE GASES N. N. Bogolyubov, Jr. and D. P. Sankovich Steklov Mathematical Institute, Gubkin str. 8, 119991, Moscow, Russia nikolai [email protected] Abstract. We use the Bogolyubov approximating Hamiltonian method to rigorous study the equilibrium properties of imperfect Bose gases. We calculate the pressure of the mean field Bose gas model. This model in external potential is considered.

Key words: Bose-condensation, approximating Hamiltonian method.

17.1 Introduction The approximating Hamiltonian method (AHM) [1, 2] developed originally by Bogolyubov [3], provides a powerful and mathematically rigorous tool to study some classes of quantum many particle systems in the thermodynamic limit. This method, developed initially to investigate the thermodynamic behavior of Fermi systems, applies also to Bose systems. The AHM consists in constructing a mathematically tractable model, thermodynamically equivalent to the original one by replacing the Hamiltonian H of the initial system by a suitable energy operator H appr (approximating Hamiltonian), obtained from H according predetermined rules. A fundamental role in this approach is played by the derivation of upper and lower bounds for the difference between the thermodynamic pressures of both systems. The main goal of this work is to extent, in the framework of the AHM, some well-known results concerning with the mean field model of a Bose gas. In this sense, we have to point out that the AHM, combined with the concept of Bogolyubov quasi-averages, has been already used (Pulé–Zagrebnov strategy [4]) to derive the limit pressure of the mean field Bose model, covering the case of attractive boundary conditions. In the present work, using the so called Bogolyubov–Ginibre approximation, we derive the thermodynamic limit pressure of the mean field Bose gas by constructing a quite different kind of approximating energy operators. Unlike the approach in Ref. [4], our approach does not resort to the concept of J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 17,

203

204

N. N. Bogolyubov, Jr. and D. P. Sankovich

quasi-averages, covering also all types of elastic boundary conditions, including the attractive ones. Furthermore we study the mean field Bose gas in some model external potential. Consider a system of identical bosons of mass m confined to a d-dimensional cubical box Λ ⊂ Rd of volume V centered around the origin. . Let E0Λ < E1Λ ≤ E2Λ ≤ . . . be the eigenvalues of the operator hΛ = −Δ/(2m) (we assume = 1) on Λ with some linear boundary conditions and let {ΦΛ l } with l = 0, 1, 2, . . . be the corresponding eigenfunctions. Let FΛ be the sym. † . † Λ metric Fock space constructed from L2 (Λ). Let al = a(ΦΛ l ) and al = a (Φl ) be the boson annihilation and creation operators on FΛ . Denote by TΛ the Hamiltonian of the free gas on FΛ constructed from hΛ in usual Bose the ∞ ∞ manner, that is TΛ = l=0 ElΛ Nl , where Nl = a†l al . Let NΛ = l=0 Nl be the number of particles operator on FΛ . The Hamiltonian of the original system is given by HΛ = TΛ +

a 2 N , 2V Λ

(17.1)

where a is a positive coupling constant. Equation (17.1) is known as the energy operator of the mean field Bose gas [5]. . Let μ0 = limΛ↑Rd E0Λ . Denote by p0 (μ) and ρ0 (μ) the grand-canonical pressure and density respectively for the free Bose gas at chemical potential μ < μ0 , that is p0 (μ) = − ρ0 (μ) =

ln[1 − exp(−β(ν − μ))]F (dν), 1 F (dν), exp(β(ν − μ)) − 1

where F is the integrated density of states of hΛ in the thermodynamic limit . Λ ↑ Rd . Let ρc = limμ→μ0 ρ0 (μ). The grand-canonical pressure of the mean field Bose gas model at finite volume is given by pΛ (μ) =

1 ln Tr exp[−β(HΛ − μNΛ )], βV

(17.2)

where the trace is taken over the above mentioned Fock space. If HΛ (μ) = HΛ − μNΛ , the equilibrium Gibbs state (grand canonical ensemble) −HΛ (μ) is defined as AHΛ (μ) = (Tr exp [−βHΛ (μ)])

−1

TrA exp [−βHΛ (μ)]

for any operator A acting on the symmetric Fock space. The main goal of our work is to obtain an explicit expression for the limit pressure p(μ) = limΛ↑Rd pΛ (μ).

17 AHM in the theory of imperfect Bose gases

205

Proposition 1. The pressure (17.2) in the thermodynamic limit exists and is given by ⎧ 1 ⎪ ⎨ aρ2 (μ) + p0 (μ − aρ(μ)) if μ ≤ μc p(μ) = 2 (17.3) 2 ⎪ ⎩ (μ − μ0 ) + p (μ ), if μ > μc 0 0 2a where μc = μ0 + aρc and ρ(μ) is the unique solution of the equation ρ = ρ0 (μ − aρ). We prove this Proposition in Sect. 17.2. The main technical ingredient of the AHM is the Bogolyubov convexity inequalities. Let HΛa and HΛb be selfadjoint operators defined on D ⊂ FΛ . paΛ (β, μ), pbΛ (β, μ) represent the grand canonical pressures corresponding to the operators HΛa , HΛb . In this case the following well known Bogolyubov inequalities [6] a HΛ (μ) − HΛb (μ) ≤ pbΛ (β, μ) − paΛ (β, μ) V a HΛ (μ) a HΛ (μ) − HΛb (μ) (17.4) ≤ V H b (μ) Λ

hold, where −HΛa (μ) , −HΛb (μ) are the Gibbs states in the grand canonical ensemble associated with the Hamiltonians HΛa , HΛb , respectively.

17.2 Application of the AHM to the mean field Bose gas Initially we shall restrict our discussion to the cases of periodic, Dirichlet, Newmann and repulsive-wall boundary conditions. We shall consider the case of attractive boundary conditions separately. Following the so-called Bogolyubov’s approximation [3, 7] introduced in 1947, we replace the corresponding operators a0 , a†0 by c-numbers. Therefore we define HΛ (c) =

∞

Λ a 2 N El − μ + |c|2 a Nl + 2V Λ l=1

+

a|c|2 a|c|4 + + (E0Λ − μ)|c|2 V, 2 2V

(17.5)

∞ where c ∈ C, NΛ = l=1 Nl and the Hamiltonian HΛ (c) contains a term −μNΛ . The approximating model determined by Eq. (17.5) is thermodynamically equivalent to the original mean field model [8, 9] in the following sense c, μ), lim pΛ (μ) = lim sup pΛ (c, μ) ≡ p(¯

Λ↑Rd

Λ↑Rd

c

(17.6)

206


where pΛ (c, μ) =

1 ln Tr exp[−βHΛ (c)] βV

and |¯ c|2 is the density of the Bose condensate. The trace Tr is taken over FΛ , where FΛ is the boson Fock space constructed on the orthogonal complement of the one dimensional subspace of L2 (Λ) generated by ΦΛ 0. Starting from the operator (17.5), we introduce the following approximating Hamiltonian c, ρ ) = HΛ (¯

∞

Λ a|¯ c|4 a|¯ c| 2 El − μ + |¯ + c|2 a Nl + 2 2V l=1

+ (E0Λ − μ)|¯ c|2 V + aρ NΛ −

aρ2 , 2V

(17.7)

where ρ ∈ R. Applying the Bogolyubov convexity inequalities (17.4), we get 0 ≤ pΛ (¯ ρΛ , c¯, μ) − pΛ (¯ c, μ) ≤

1 Δ (¯ c), 2V 2 Λ

(17.8)

where 1 ln Tr exp [−βHΛ (c, ρ )] , ΔΛ (¯ c) βV = a(NΛ − V ρ¯Λ )2 HΛ (¯c,ρ¯Λ )

pΛ (ρ , c, μ) =

(17.9)

and ρ¯Λ satisfies the equation ρ¯Λ =

1 NΛ HΛ (¯c,ρ¯Λ ) . V

(17.10)

Straightforward calculations lead to c) = a ΔΛ (¯

∞

l=1

exp(βΛ l ) , 2 [exp(βΛ ) l − 1]

(17.11)

Λ ρΛ + |¯ c|2 ). Noting that coth x < 1 + x−1 for x ≥ 0, we where Λ l = El − μ + a(¯ get ∞

2 1 c) < a ΔΛ (¯ 1+ Λ . (17.12) exp(βΛ βl l )−1 l=1

The inequality ([6]) [A, [H, A]]H ≥ 0 is valid for any operator A and for any self-adjoint Hamiltonian H. Taking A = a†l , H = HΛ (¯ c, ρ ), it is obtained, Λ 2 that E0 + a(¯ ρΛ + |¯ c| ) − μ ≥ 0 for large V and Λ Λ Λ l > E1 − E0 ≥

π 2 −2/3 V 2m

for l = 1, 2, 3, . . . .

(17.13)


Using this estimate into Eq. (17.12), we obtain 4m 2/3 ΔΛ (¯ . c) < a¯ ρΛ V 1 + 2 V π β Therefore

(17.14)

ΔΛ (¯ c) = 0, V →∞ V 2

(17.15)

ρΛ , c¯, μ). lim pΛ (μ) = lim pΛ (¯

(17.16)

lim

concluding that

207

Λ↑Rd

Λ↑Rd

Hence, the grand-canonical pressure of the mean field model is 2 a|¯ c|4 aρ 2 2 c| + inf c| + ρ)) . p(μ) = − + (μ − μ0 )|¯ + p0 (μ − a(|¯ ρ≥0 2 2

(17.17)

Thus, for μ ≤ μc we have |¯ c| = 0 and ρ (μ) is the unique solution of ρ = ρ0 (μ − aρ ); for μ > μc we have μ − a(|¯ c|2 + ρ ) = μ0 , proving the Proposition. We have been considering the case of a cubical box Λ. Taking the case of a parallelepiped of the same volume with sides of length Lj = V αj , j = 1, 2, 3, such that α1 ≥ α2 ≥ α3 > 0 and α1 + α2 + α3 = 1, the situation changes. The estimate (17.13) can be proved only for α1 < 1/2. In the direction j = 1 there is generalized condensation in the so-called Girardeau sense [10] and the proposed method must be improved. We shall restrict ourselves to the case of attractive walls in dimension d = 1. The generalization to the case d > 1 is long and tedious and do not give much more insight. 1 The spectrum of the one-dimensional Schr¨ odinger equation − 2m Φ = εΛ Φ with attractive boundary conditions Φ (−L/2) = σΦ(−L/2), Φ (L/2) = −σΦ(L/2), where σ < 0, consists of two negative eigenvalues tending to the same limit (when L → ∞) and an infinite number of positive eigenvalues (for L|σ| > 2), namely σ2 − O(e−L|σ| ), 2m σ2 + O(e−L|σ| ), εΛ 1 = − 2m 2 2 1 1 (k − 1)π kπ Λ < εk < 2m L 2m L εΛ 0 = −

for k ≥ 2.

Taking into account this fact in the Bogolyubov approximation and replac# ing by c-numbers not only the operators a# 0 but also the operators a1 , the approximating Hamiltonian (17.7) becomes

208

N. N. Bogolyubov, Jr. and D. P. Sankovich 2 HΛ (c) = (εΛ 0 − μ)|c| L +

∞

a|c|4 2 L+ εΛ l − μ + |c| a Nl 2 l=2

a 2 a|c|2 N + + ΔεΛ |c1 |2 L, + (17.18) 2L Λ 2 ∞ . Λ −L|σ| 2 Λ = ), N where c, c1 ∈ C, ΔεΛ = εΛ 1 − ε0 ∼ O(e l=2 Nl and |c| = † 2 2 2 |c0 | + |c1 | with |ci | = ai ai /LHΛ (c) , i = 0, 1. Obviously the last term in (17.18) is inessential in the thermodynamic limit. Therefore under the previous consideration, the Proposition follows in the case of attractive walls. Thus for l ≥ 2 the estimate (17.13) becomes Λ Λ Λ l > ε2 − ε0 >

σ2 . 2m

In Ref. [11] the case of the imperfect Bose gas with attractive boundary conditions, being ˜ 2, ˜ Λ = TΛ + a N (17.19) H 2V Λ is studied. The authors prove Bose–Einstein condensation in the onedimensional case, being the condensate fraction equally distributed over the two negative energy levels and localized in the same area as for the free Bose gas with attractive boundary conditions [12]. The Hamiltonian (17.19) can be treated by the standard AHM, provided that ˜Λ − H ˜ appr (¯ lim H ρΛ )H˜ appr (ρ¯Λ ) = 0, Λ V →∞

Λ

where

2 ˜Λ − aρ V ˜ appr (ρ) = TΛ + aρN H Λ 2 and ρ¯Λ satisfies the self-consistency equation

ρ¯Λ =

1 ˜ NΛ H˜ appr (ρ¯Λ ) . Λ V

Hamiltonian in Eq. (17.1) is superstable, i.e. the corresponding limit pressure exits for any value of μ. However Hamiltonian (17.19) is stable only for μ ≤ μ0 = −dσ 2 /(2m). Consequently, this restriction has to be taken into account in the expression for the pressure of the model (17.19), 2 aρ appr ρ, μ) = inf p˜(μ) = lim p˜Λ (¯ + p0 (μ − aρ) , ρ≥0 2 Λ↑Rd where

1 ˜ appr (ρ) − μNΛ . ln Tr exp H Λ βV We conclude that in the case of the mean field model with attractive boundary conditions, as in the case of the free Bose gas, the condensate has infinite density and occupies essentially zero volume near the walls. pãppr (ρ, μ) = Λ


209

17.3 Mean field Bose gas in external potential Now we give outline report on some new results concerning the BEC for a mean field Bose gas in external potential. By making a Fourier transformation of external potential we neglect the off-diagonal terms. Hence this approximation is poor when many particles are thermally excited. Nevertheless, this approximation is useful at low temperatures where the condensate density is almost independent of the temperature. The purpose of this study is not to construct a quantitative model for the experiment, but to illuminate some qualitative features rigorously. We start with the simplest soluble problem involving interparticle interactions – a mean field Bose gas model. Our model differs from the earlier version [13, 14], where the Bogolyubov’s model Hamiltonian was considered as initial. Consider a grand canonical ensemble with chemical potential μ, with Hamiltonian H given by H = H1 + H2 , where a 2 N , 2V

1 1 H2 = h0 a†p ap + √ a0 hp a†p + √ a†0 h∗p ap , V V p p=0 p=0

H1 = T +

T =

(εp − μ)a†p ap , N = a†p ap . p

p

The first term of the Hamiltonian is the mean field Hamiltonian, whereas the second term refers to the approximating external potential. We assume that Fourier transform of the external potential is in L2 . Proceeding in a standard fashion, we introduce the Bogolyubov’s approximating Hamiltonian H1 (c) =

(εp − μ)a†p ap − μ|c|2 V +

p=0

+

a|c|2 a†p ap +

p=0

H2 (c) = h0 |c|2 V + h0

a|c|4 a|c|2 + V, 2 2

a†p ap + c

p=0

where

N =

a 2 N 2V

p=0

p=0

a†p ap .

hp a†p + c∗

p=0

h∗p ap ,

210


Now let us use the AHM.

H1 (x, c) = (εp − μ + a|c|2 )a†p ap + ax a†p ap p=0

p=0

a|c|4 ax2 a|c|2 V − μ|c|2 V + + V, − 2 2 2 H2 (c) = H2 (c). So the approximating Hamiltonian is

√

√ ∗ Ep (x, ρ0 )a†p ap + ρ0 hp a†p + ρ0 h p ap H(x, ρ0 ) = p=0

p=0

p=0

+ H0 (x, ρ0 ), where ρ0 = |c|2 and Ep (x, ρ0 ) = εp − μ + aρ0 + ax + h0 , H0 (x, ρ0 ) =

aρ0 aρ2 ax2 + 0V − V + ρ0 (h0 − μ)V. 2 2 2

The approximating Hamiltonian is diagonalized by a Bogolyubov transformation √ hp ap = bp − ρ0 , p = 0. Ep The diagonalized Hamiltonian has the form H(x, ρ0 ) =

Ep (x, ρ0 )b†p bp − ρ0

p=0

p=0

h2p + H0 (x, ρ0 ). Ep (x, ρ0 )

The limit pressure of this Hamiltonian is lim pΛ (μ) = lim sup pΛ (μ, ρ0 ) ≡ p(μ, ρ0 )

V →∞

V →∞ ρ0 ≥0

⎡

⎤ 2

h ax ρ p + 0 + p0 (μ − h0 − aρ0 − ax)⎦ = inf ⎣ x≥0 2 V Ep (x, ρ0 ) 2

p=0

−

aρ20 2

− ρ0 (h0 − μ),

where p0 is the ideal Bose gas pressure. We have μ ≤ h0 + aρ0 + ax. The self-consistency equation for x and the self-consistency equation for nontrivial positive solution ρ0 are, respectively x=

h2p 1

ρ0

1 , + V V E 2 (x, ρ0 ) eβEp (x,ρ0 ) − 1 p=0 p=0 p

(17.20)


y=

1 h2p , V εp + y

211

(17.21)

p=0

where y = a(ρ0 + x) + h0 − μ ≥ 0. Equation (17.21) has a unique nontrivial positive solution δ. Substituting this solution in (17.20) we have the equation for ρ0 : ⎡ ⎤

h2p 1 1 ⎦ = δ + μ − h0 − 1 . ρ0 ⎣1 + 2 β(ε +δ) p V (εp + δ) a V e −1 p=0

For θ = 0 we have

p=0

⎡

⎤

h2p 1 ⎦ = δ + μ − h0 . aρ0 ⎣1 + V (εp + δ)2

(17.22)

p=0

The equation (17.22) has a positive solution if μ > h0 − δ ≡ μc . For μ ≤ μc there is not the Bose condensation. For θ > 0 the critical value of the chemical potential in our model is μc = h0 − δ +

a

1 , β(εp +δ) − 1 V e p=0

(17.23)

where δ is the unique root of the equation (21). For μ ≤ μc we have ρ0 = 0 and x is the unique root of the equation x=

1

1 . β(ε +ax+h p 0 −μ) − 1 V e p=0

If the external potential is zero the critical value of the chemical potential is μc (hp = 0) ≡ μ0c =

a

1 = aρideal . c V eβεp − 1

(17.24)

p=0

Comparing (17.23) and (17.24) we conclude that μc < μ0c for h0 ≤ δ.

References 1. N. N. Bogolyubov, Jr., A Method for Studying Model Hamiltonians (Oxford, New York: Pergamon Press, 1972). 2. N. N. Bogolyubov, Jr., E. N. Bogolyubova, and S. P. Kruchinin, Mod. Phys. Lett. B, 17, No. 10–12, 709–724 (2003). 3. N. N. Bogolyubov, J. Phys. (USSR) 11, 23 (1947). 4. J. V. Pulé and V. A. Zagrebnov, J. Phys. A: Math. Gen. 37, 8929 (2004). 5. K. Huang, Statistical Mechanics (New York: Wiley, 1963).

212 6. 7. 8. 9. 10. 11. 12. 13. 14.

N. N. Bogolyubov, Jr. and D. P. Sankovich N. N. Bogolyubov, Phys. Abh. Sow. Un. 6, 113 (1962). J. Ginibre, Commun. Math. Phys. 8, 26 (1968). D. P. Sankovich, J. Math. Phys. 45, 4288 (2004). A. Bernal, M. Corgini, D. P. Sankovich, Theor. Math. Phys. 13, 866 (2004). M. Girardeau, J. Math. Phys. 1, 516 (1960). L. Vandevenne and A. Verbeure, Rep. Math. Phys. 56, 109 (2005). D. W. Robinson, Commun. Math. Phys. 50, 53 (1976). K. Huang and H.-F. Meng, Phys. Rev. Lett. 69, 644 (1992). M. Kobayashi and M. Tsubota, Phys. Rev. B 66, 174516 (2002).

Part III

Spintronics

18 MAGNETIC NANOSTRUCTURES K. Bennemann Institute of Theoretical Physics FU-Berlin Arnimallee 14, 14195 Berlin [email protected]

Abstract. Characteristic results of magnetism in small particles are presented. Results are given for a lattice of anti-dots, tunneling between quantum dot and tunnel systems.

Key words: magnetic nanostructures, clusters, thin films, tunnel junctions.

18.1 Introduction Due to advances in preparing small particles, quantum dot and tunnel junctions the area of nanostructures in physics has received increased interest. Clearly engineering on an atomic scale condensed matter offers many possibilities regarding new physics and technical applications. Of particular interest is the occurrence of magnetism in nanostructures like quantum dot and tunnel junctions [1–3]. The latter are of interest, for example, regarding switching of electric, charge and spin currents. Tunnel junctions Tunnel junctions involving magnetism are interesting microstructures, in particular regarding quantum mechanical behavior, switching devices and charge– and spin currents and their interdependence. Coupling of the magnetic order parameter phases, for both ferromagnets and antiferromagnets, on both sides of the tunnel medium yields Josephson like (spin) currents, driven by the phase difference, for details see study by Nogueira et al. [2] and others. Obviously, this will depend on the magnetic state of the medium through which tunneling occurs, on spin relaxation. The effective spin coupling between the left and right side of the tunnel junction N1 and N3 , respectively,


215

216

K. Bennemann 2d N1

N2

N3

jT

Fig. 18.1. A tunnel junction N1 | N2 | N3 , where Ni refers to the state, magnetic, superconducting, normal. The current jT may refer to spin – or charge transport and may be driven by the phase difference Δϕ = ϕ3 − ϕ1 , for example. Both spin and charge current may couple in general. The continuity equation for the magnetization, →

→

→

M , (d M /dt + div j s = 0), indicates that magnetic dynamics M (t) may induce spin currents js . As a consequence interesting nonequilibrium behavior is expected. Note, applying a voltage V to the sketched tunnel may yield a two–level system (resembling a qubit, etc.).

depends on the spin susceptibility of N2 and Jeff = Jeff (χ). Note, from the continuity equation follows → − → − jT ∝ Jeff S1 × S3 + · · ·

(18.1)

The tunneling is sketched in Fig. 18.1. Clearly tunneling allows to study magnetic effects, ferromagnetic versus antiferromagnetic configurations of the tunnel system (↑| T |↑), or (↑| T |↓) and for example interplay of magnetism and superconductivity (T ≡ S.C.), for example junctions (FM/SC/FM) or (SC/FM/SC). Such junctions may serve as detectors for triplet superconductivity (TSC) or ferromagnetism (FM), antiferromagnetism (AF). Of course tunneling is different for parallel or antiparallel magnetization of M1 and M3 . Also importantly, tunnel junctions may serve to study Onsager theory on a molecular–atomistic scale. Electron and Cooper pair transfer between two quantum dot exhibits interesting behavior, for example v.St¨ uckelberg oscillations due to bouncing back and forth of electrons if an energy barrier is present and assistance of photons is needed to overcome the energy barrier. In summary, for illustrating reasons various interesting nanostructures like tunnel junctions with important magnetic effects are discussed. In the next chapter some theoretical methods useful for calculations are presented. Then results obtained this way are given. It is important to note that for illustrational purposes the physics has been simplified and that the analysis can be improved. However, likely this will not change the physical insights obtained from the simplified analysis. For more details see tunnel junctions studies by Nogueira, Morr et al. [2].

18 Magnetic nanostructures

217

18.2 Theory Tunnel junctions: spin currents Tunnel junctions (N1 | N2 | N3 ) are interesting nanostructures regarding (ultrafast) switching effects, interplay of magnetism and superconductivity, and in general quantum mechanical interference effects. Note, Ni may refer to material which is ferromagnetic or superconducting, for example. The situation is illustrated in Fig. 18.1. On general grounds one may get spin currents driven by phase difference between two magnetic systems, ferromagnets or antiferromagnets (which order parameter is also characterized by a phase) [1]. Interesting tunnel junctions are shown in Fig. 18.2. In Fig. 18.2(a) is shown a two quantum dot system niσ with energy iσ controlling the tunnel current jσiσ . The position of the energies i may depend on occupation (see Hubbard like Hamiltonian: εiσ = εoi + U ni¯σ + · · · ). Note, niσ and thus the current jσ can be manipulated optically. The tunnel system shown in Fig. 18.2(b) can be used as a sensor for triplet → − superconductivity. Then the configuration of the angular momentum d of the triplet Cooper pairs and of the magnetization of the tunnel medium control characteristically the tunnel current. The Josephson current exhibits interesting behavior, for example upon rotating the magnetization relative to the angular momentum of the Cooper pairs. In Fig. 18.2(c) is illustrated how tunneling can be used (with the help of a bias voltage) to control magnetoresistance and to determine the magnetization of a ferromagnet [3]. Spin currents: Using the continuity equation one can find the relationship between spin currents and magnetization dynamics for the magnetic tunnel junction illustrated in Fig. 18.1. One has ∂t Mi + ∂μ jiμ,σ = 0.

(18.2)

This may give under certain assumptions straightforwardly the connection between magnetization and spin polarized electron currents (induced by hot electrons, temperature gradients or external fields). Note, the magnetic i dynamics (characterized by dM dt ) may be described by the LL–equation. This yields generally a spin Josephson current between magnets. Also according to Kirchhoff the emissivity (e) of the junction is related to its (time dependent) magnetization, magnetic resistance. It is Δe a (GMR), (18.3) e where (GMR) denotes the giant magnetoresistance resulting for the junction if the configuration (↑| N2 |↑) changes to (↑| N2 |↓). This changes the emissivity to Δe. Viewing the phase of the magnetic order parameter φi (magnetization Mi ) M =| Mi | eiφi ,

(18.4)

218

K. Bennemann

a

pulsed excitation e–

e–

ε2

e–

current

ε1 μL

μR

τ pulsed radiation

b

xˆ

xˆ M

dL

xˆ

α

θ zˆ

dR

Δ0 yˆ

yˆ

Δ0eiφ

yˆ triplet SC

FM Barrier

triplet SC

z0

c SC

FM

FM V/2

–V/2 (a) normal state eV/2

δμ –δμ

–eV/2

(b) superconducting state eV/2

δμ –δμ

–eV/2

Fig. 18.2. Illustration of various tunnel junctions. (a) A two quantum dot system with spin-dependent levels iσ (niσ , t) is shown (for example εiσ = εoi + Ui ni¯σ ).Two levels are separated by an energy barrier which can be overcome using an external electric field. (b) A triplet superconducting Josephson junction (SC1 | F M | SC3 ) →

is sketched. Here, d refers to the angular momentum of the Cooper pairs which order parameter has the phase φ. (c) A junction (F M1 | SC | F M2 ) is illustrated. An external electrical field (potential V ) shifts the electronic energy levels (bands). Thus, the tunnel current (magnetoresistance) can be controlled and ferromagnetism be determined (see Takahashi et al. [3]).


219

similarly as the phase of the S.C. state, one gets for a junction (F M | N | F M ) a Josephson like current j J driven by the phase difference of the spin polarizations on both sides of the tunnel junction. Using the continuity equation for the spins, integrating using Gauss integral, and the Landau–Lifshitz equation, one gets for the spin current − → − → jσ = jσ1 (V )+j J , j J ∝ dM/dt ∝ M L ×M R + · · · ∝ |ML ||MR | sin(φL −φR )+· · · (18.5) Here, L and R refer to the left and right side of the tunnel junction, respectively. jσ1 (V ) refers to the spin current due to the potential V and may result from the spin dependent DOS. The current j J is in direction perpendicular − → −−→ to M and Heff . For details of the derivation of the spin Josephson current see Nogueira et al. [2]. , where F is the free–energy, one Note, using the general formula j = ∂F ∂φ gets j = −(e/)Σi∂Ei /∂φ tanh(Ei /kT ).

(18.6)

→ − − → Then using Ei ∝ Jeff S L · S R + · · · , one gets also jσJ ∼ Jeff sin(φL − φR ) + · · · .

(18.7)

Here, Ei gives the energy difference between opposite directions of the magnetization (molecular field). Of course, such a Josephson like spin current is expected on general grounds, since φ and S z are canonical conjugate variables and [φ, S z ] = i.

(18.8)

Note, this holds for the Heisenberg Hamiltonian as well as for itinerant magnetism described by the Hubbard Hamiltonian, for example. The commutator relationship suggests in analogy to the BCS–theory to derive the spin Josephson current from the Hamiltonian H = −EJ S 2 cos(φL − φR ) +

μ2B z z 2 (S − SR ) + ··· , 2Cs L

(18.9)

where again L, R refer to the left and right side of the junction and Cs denotes the spin capacitance. In general spin relaxation effects should be taken into account (magnetization dissipation, see LL–equation). Using then the (classical) Hamiltonian equations of motion (φ = ∂H/∂S z , S z = −∂H/∂φ) one gets Δφ˙ = 2μB Vs ,

jSJ = (2EJ S 2 /μB ) sin Δφ.

(18.10)

z ). That Δφ˙ = 0 if ML MR Here, Δφ = φ1 − ϕ2 and Vs = (μB /Cs )(SLz − SR ˙ and Δφ = o if ML is antiferromagnetically aligned relative to MR can be

220

K. Bennemann

checked by experiment. It is jsJ ∼ sin(Δφ0 +4M t). Note, details of the analysis for the a.c. like effect are given by Nogueira et al. [2]. Thus, interestingly the spin current in a F M | F M tunnel junction behaves in the same way as the superconductor Josephson current. Of course, as already mentioned magnetic relaxation (see Landau–Lifshitz–Gilbert Equation) affects jSJ . For further details see again Nogueira and Bennemann [2]. Clearly, also junctions involving antiferromagnets (af), (AF/F), (AF/AF) yield such Josephson currents, since using the order parameter for an AF one gets also that Sqz and φ are conjugate variables. Note, also Jeff = Jeff (χ) is a functional of the spin susceptibility χ, since the effective exchange coupling between the L and R side of a tunnel junction is mediated by the spin susceptibility (of system N2 , see Fig. 18.1). → − The analysis may be easily extended if an external magnetic field B is − → M present. Then from the continuity equation one gets js ∼ ∂M and ∂∂t = ∂t − → −− → −−→ → − − → − − → −→ − ∂M R aML × MR − gμB BL × SL + · · · (and similarly ∂t = aMR × ML + · · · ). Alternatively one may use the Hamilton–Jacobi equations with the canonical ∂H conjugate variables S z and φ (S˙z = ∂H , φ˙ = − ∂S z ) and changing the Hamil∂φ → − − → − → − → tonian H −→ H − gμB ( B · SL + B · SR ) to derive the currents js and jsJ . Note, according to Maxwell equations the spin current jS should induce an electric field Ei given by ∂x Ey − ∂y Ex = −4πμB ∂t S z = 4πjs .c

(18.11)

Here, for simplicity we assume no voltage and ∂t H = 0 for an external magnetic field. Tunnel junctions with spin and charge current: Generally one gets both z a charge current jC = −eN˙L and a spin current js = −μB (S˙Lz − S˙R ) and these may interfere. This occurs for example for SCM | SCM junctions, where SCM refers to nonuniform superconductors coexisting with magnetic order (see Larkin–Ovchinikov state). Then one gets after some algebra for the Josephson currents (see Nogueira et al. [2]) jcJ = (j1 + j2 cos Δϕ) sin(Δφ +

2πl Hy ) φ0

(18.12)

and jsJ = js sin Δϕ cos(Δφ +

2πlHy ) . φ0

(18.13)

Here, φ0 is the elementary flux quantum, Hy an external magnetic field in y–direction, l = 2λ + d, with λ being the penetration thickness, and d the junction thickness. Δϕ and φ refer to the phase difference of magnetism and superconductivity, respectively. The magnetic field Hy is perpendicular to the current direction.


221

(TSC/FM/TSC) Junction: Regarding switching of tunnel current and analysis of triplet superconductivity the tunnel junction T SC | F M | T SC is of interest. Here, T SC refers to triplet superconductivity. The Josephson current flows then through low–energy Andreev states. Relative orientation of − → the magnetization M of the ferromagnet (F M ) and d–vectors of the triplet superconductors, see Fig. 18.2(b), control the tunnel current. One gets, see Morr et al. [1], Ei e ∂Ei jcJ = − tanh( ), (18.14) ∂φ kT i

where φ is the phase difference between the two superconductors and Ei the energies of the Andreev states and which are calculated using Bogolinbov–de Gennes analysis. Thus one derives an unusual temperature dependence of the Josephson current on temperature and even that jcJ may change sign for − → certain directions of M , although Δφ did not change. (SC/FM/SC) Junction: As discussed recently by Kastening et al. [1] such junctions reflect characteristically magnetism. Again the tunnel current is carried by Andreev states. No net spin current flows from left to right, since the spin polarized current through the Andreev states is compensated by the tunnel current through continuum states. In case of strong ferromagnetism single electrons tunnel, while for weak ferromagnetism Cooper pair tunneling occurs. The Josephson current may change sign for increasing temperature without a change in the relative phase of the two singlet superconductors. Of course, dependent on coherence length, temperature and thickness of the FM and strength of FM one may get j J = 0 for the Josephson current. (FM/SC/FM) Junction: It is already obvious from Fig. 18.2(c) that in the presence of an applied voltage V junctions (F M1 | SC2 | F M3 ) carry currents which depend sensitively on the relative orientation of the magnetization of the two ferromagnets. If these are directed in opposite direction (AF configuration) one gets a maximal spin accumulation in the superconductor and thus one may suppress (at a critical voltage) superconductivity in singlet superconductors. As a consequence the magnetoresistance changes. Hence, such junctions exhibit currents −→ −→ jcσ = jcσ (M1 , M3 , SC2 , T )

(18.15)

which reflect sensitively superconductivity and ferromagnetism. Of course, the current jcσ is affected by temperature gradients ΔT between the ferromagnets and resulting for example at nonequilibrium from hot electrons in one F M may cause this interesting behaviour. Quantum dot systems Interesting current pattern and interferences are also expected for the quantum dot grain structure shown in Fig. 18.3. Assuming a mixture of

222

K. Bennemann

empty sites(0), n,s - grains

Fig. 18.3. Illustration of an ensemble of small particles (grains) arranged in a lattice. The lattice sites may be empty or occupied by ferromagnetic, paramagnetic, or superconducting clusters, for example. Removing irregularly clusters from the lattice sites creates all sorts of nanostructures.

ferromagnetic and superconducting grains (clusters, quantum dot) one has the Hamiltonian H = HT − (ze2 /C)

(Ni − Nj )2 + H .

(18.16)

i,j

Here, HT denotes the tunneling Hamiltonian. This includes also spin JosephJ son currents of the form Eij cos ϕij and also Josephson Cooper pair current contributions resulting from the phase difference ϕij = ϕi − ϕj of the phases of the S.C. order parameter of grains i and j. The electrostatic effects due to different charges of grains i, j are given by the second term. H denotes remaining affects, for example due to ferromagnetic grains. One expects characteristic differences for singlet and triplet superconductivity and interplay of spin currents jσ between magnetic grains and Josephson–currents. In grains the interplay of Cooper pair size and distance between Cooper pairs manipulated by confinement is expected to exhibit interesting behavior. For a lattice like array of quantum dot it is of interest to study phase ordering of the order parameter of the various dots i, its dependence on distance between dots, etc. (see related situation for superconductors when Tc ∝ ρs , here ρs is the superfluid density). As speculated for two–band superconductors and as assumed for sc q–bits (strong) phase coupling of two magnetic systems (quantum dots) may cause a covalent splitting like process (mode–mode coupling like process). Thus, for example, one expects for the combined, phase coupled system N1 and N3 , see Fig. 18.1, the two covalently like split states | N1 N3 1 and | N1 N3 2 . Light irradiation causes interesting responses, since occupation of electronic states (by single electrons, Cooper pairs) can be manipulated. These examples may suffice already to demonstrate the interesting behaviour displayed by nanostructures involving tunneling. This holds also for currents across multilayers of magnetic films.


223

Electronic structure of mesoscopic systems: Balian–Bloch type theory The important electronic structure (shell structure) for mesoscopic systems like spherical clusters, discs, rings, dots can be determined using a relatively simple theory developed by Stampfli et al. extending original work by Balian– Bloch (Gutzwiler) [5–7]. The dominant contribution to the electronic structure results from (interfering) closed electronic orbital paths. Then the key quantity of the electronic structure of a quantum dot system, the density of states (DOS), can be calculated from (n =number of atoms in nanostructure) from → → the Green’s function G. The Green’s function G(− r ,− r ) is derived by using multiple scattering theory, see Stampfli et al. Interference of different electron paths yields oscillations in the DOS etc. rather than et ce. Thus, for example, oscillations in the magnetoresistance and other properties of quantum–dot systems can be calculated. From the electronic Green’s function G one determines the density of states (DOS) → − → − 1 → → − → Nσ (E, n) = dd r{Gσ (− r , r , E + i)− Gσ (− r , r , E − i)}→ . (18.17) r =− r 2πi V One gets (N = average DOS) Nσ (E, n) = N σ (E, n) + ΔNσ (E, n),

(18.18)

where ΔNσ refers to the oscillating part of the DOS due to interference of dominating closed electron paths in clusters, thin films, and ensemble of repelling anti-dots, see the theory by Stampfli et al. [5–7]. Clearly this scattering by the dots can be spin-dependent and can be manipulated by external magnetic fields B (s. cyclotron paths, Lorentz–force etc.). Under certain conditions regarding the potential felt by the electrons in the nanostructures (square–well like dot potentials etc.) one gets the result (see Stampfli et al. [5–7] using extensions of the Balian–Bloch type theory) ΔNσ (E, n) Alσ (E, n) cos(kLl + φlσ ). (18.19) l

Here, l refers to closed orbits (polygons) of length L, k = | E | +iδ, and φlσ denotes the phase shift characterizing the scattering potential and the geometry of the system. An external magnetic field affects ΔNσ (E, n) via path deformation and phase shifts resulting from magnetic flux (see Aharonov– Bohm effect). Clearly, the DOS in particular ΔNσ (E, n) will be affected characteristically by the magnetism in various nanostructures, in quantum dot systems. For details of the Balian–Bloch like analysis see the theory by Stampfli et al. [5,6]. Note, the electronic structure due to the interferences of the dominant electronic states near the Fermi–energy described approximately by Eq. 18.19

224

K. Bennemann

results in addition to the atomic symmetry of the nanostructure (spherical one for clusters, 2d-symmetry for thin films, etc.). Thus, for example, one should find for magnetic clusters a phase diagram temperature vs. cluster size which dependent on magnetization exhibits phases where atomic (shell) structure and then spin dependent electronic structure dominates. Of course, for sufficiently large clusters these structures disappear. The interference of closed electron orbits in magnetic mesoscopic systems like spherical clusters, discs, rings, thin films and quantum dot lattices, causes spin dependently characteristic structure, oscillations in the DOS N (E, n) (yielding corresponding ones for example in the cohesive energy, occupation of d, s electron states, Slater–Pauling curve for magnetism, magnetoresistance, etc.).

18.3 Results Characteristic results are presented for quantum dot lattice, see Fig. 18.3, and tunnel junctions. Quantum dot Quantum dot lattices: Typical results for a lattice of quantum dot or anti-dots, see Fig. 18.4, are shown in the following figures. Of course, one expects that

Fig. 18.4. Magnetic field effects on polygonal electron paths in an anti-dot lattice. Here, a denotes the spacing of the anti-dots with radius d. The electrons are repelled by the anti-dot potential and thus selectively the polygonal paths 1, 2, 3 etc. yield the most important contribution to the spin dependent DOS, magnetoresistance, etc. Details of the theory determining the electronic structure of such mesoscopic systems are given by Stampfli et al.


225

Fig. 18.5. Dependence of the DOS oscillations on dimension and geometry of a mesoscopic system obtained by Stampfli et al. using an extension of the Balian– Bloch theory. Results for (a) sphere, (b) disc and (c) ring are given. The dashed curves refer for comparison to quantum–mechanical calculations.

the density of states ρ(E), generally if magnetism is involved spin dependent, reflects characteristically the nanostructure. This is demonstrated in Fig. 18.5. The oscillating part of the DOS due to the interference of dominant electronic paths is shown for various nanostructures. Results may be compared with quantum–mechanical calculations obtained using Schr¨ odinger equation and Green’s function theory ρ(E, ..) = (γ/π)

dE

ρ(E ) .c (E − E )2 + γ 2

(18.20)

Note, in case of magnetism ρσ (E) and to simplify one may assume that spin up and spin down states are split by the molecular field heff which may include an external magnetic field B. Also hollow particles (see coated nanoparticles,

226

K. Bennemann

Fig. 18.6. Oscillations of the DOS for the orbits 1, 2, 3 due to scattering by repelling anti-dots, see calculations by Stampfli et al. [5, 6]. The inset shows the lattice of anti-dots. Note, the scattering may be spin-dependent and the density of states is spin split.

for example) can be treated similarly as rings. This is of interest, for example, to study confined and bent 2d electronic systems. And films are calculated in a similar way, see Fig. 18.6. In Fig. 18.6 results are given for the oscillating part of the DOS as a function of energy and external magnetic field of a lattice of anti-dots which repel the electrons. The magnetic field B changes the orbits and thus their interference. As a result DOS oscillations occur. Note, only a few closed electron orbits (1, 2, 3, . . .) give the dominant contribution to the electronic structure, the DOS, see theory by Stampfli et al. The flux due to the external field B is φ = SB and for orbits 1, 2, 3 S is approximately independent of B for small field. The cyclotron orbit radius is Rc = Bk and we assume for simplicity 2Rc > (a − d) and Rc R. Then, Rc ∼ B1 and the DOS depends on (d/a) and Δρ ∼ cos SB. This controls height of oscillations (note S ∝ Rc2 ∝ B12 ). For increasing external magnetic field B the oscillations in the DOS change. Interestingly, the contributions of some orbits may be nearly eliminated. Oscillation period in B decrease for increasing B and decreasing lattice constant a. Note, an internal molecular field due to magnetism is expected to affect the DOS similarly as the external magnetic field. Note also, the above oscillations in the DOS result from the confinement (closed electronic orbits) and additional structure may result from detailed atomic structure yielding the well known electronic shell structure in clusters etc. For increasing radius the results for thin rings may be similar to the ones for planar surfaces of thin films. For thin films one may perform the Balian– Bloch type calculations, for illustration Fig. 18.7. Transport between quantum dot: An interesting case of electron transport (electron pump model) between two quantum dot involving possibly Coulomb– and spin–blockade (see Hubbard Hamiltonian with spin dependent on–site interaction) is illustrated in Fig. 18.2(a). Driving the current with a pulsed (polarized) external


227

T R Cu

Co

Fig. 18.7. Sketch of the important orbits for thin films. Here, one has to include transmission (T ) and reflection (R) coefficients at the corners of the paths. Magnetic films yield spin dependent DOS.

field to overcome an energy barrier one gets spin-dependent charge transport with v.St¨ uckelberg oscillations due to bouncing forth and back of the electrons between the two quantum dot [4]. In Fig. 18.2(a) results are given for the photon assisted tunneling between two quantum dot. The applied electromagnetic field is given by V (t) = V0 cos(ωt). Note the dependence of the charge transfer on the duration of the applied light pulse and the Rabi oscillations due to the bouncing back and forth of the electrons. The Rabi oscillation frequency is Ω = 2ωJN (E = V0 /ω). Here JN is a Bessel function of order N, where√N refers to number of photons absorbed to fulfill resonant condition N ω = Δε2 + 4ω 2 . Different frequencies are used which cause resonant absorption of one, two and three and possibly more photons. Of course this affects the charge transfer. The time resolved analysis of the occupation of the electronic states shows that system if connected to reservoirs acts as electron pump. Before action of the external electromagnetic field the initial state is n1 = 1 and n2 = 0. After the pulse is over oscillations disappear and one gets again the initial state via transferring one electron to the right quantum dot and the left reservoir donating one electron to the left quantum dot. Spin dependent quantum dot electron states cause spin currents. Of interest is also the coherent control of photon assisted tunneling between quantum dot and its dependence on the shape of the light pulse, see Grigorenko, Garcia et al. [4]. The shape of the external electric (or magnetic pulse) may be optimized to get a maximal charge or spin current between the quantum dot connected to two metallic contacts. Note, these results are also of interest for Fermion or Boson systems on optical lattices, its dynamics and interaction with external fields. Optical manipulation of molecular binding, in particular Boson formation induced by an electromagnetic field is an option. In intense fields nonlinear behavior may be particular interesting. Furthermore, on optical lattices one may study in particular the interplay of magnetism and lattice structure, the transition from local, Heisenberg like to itinerant behavior of magnetism. Important

K. Bennemann

transfered charge per pulse

2.0

1.0

a 1 photon

1.0

2 photons

3 photons 0.0 0

50

100

150

quantum dot occupation

228

pulse length = 40 h / w

b left QD

0.8 0.6 0.4

right QD

0.2 0.0 –200

pulse length in h / w

0

200 time in h / w

400

Fig. 18.8. Charge transfer between two quantum dot exhibiting Rabi (v.St¨ uckelberg) oscillations due to bouncing electrons between the quantum dot. (a) Dependence of transferred charge on pulse duration of external field and for different frequencies ω1 , ω2 , ω3 which cause resonant tunneling by absorption of one, two and three photons, respectively; (b) Time dependent occupation of quantum dot states, see results by Garcia et al. [4]. Note, for spin dependent quantum dot states (magnetic quantum dot), the results depend on the light polarization and may involve spin dependent currents [4].

parameter for the occurrence of ferromagnetism (antiferromagnetism) are Coulomb interactions and particle hopping (kinetic energy) between lattice sites, and possibly also spin–orbit coupling. Note, as indicated by Hund’s rules ferromagnetism could occur already for a Fermi–liquid with strong enough repulsive Coulomb interactions, quasi irrespective of the lattice structure, since spin polarization (together with the Pauli principle) may minimize the repulsive Coulomb interactions. In case where a.f. dimerization of neighboring lattice sites yields Boson formation and resulting BEC condensation optical influence, also of the magnetic excitations, is an important option (Fig. 18.8). Tunnel junctions: magnetic effects Nanoscaled tunnel junctions offer interesting physics in particular regarding quantum mechanical effects and spin dependent currents, separation of charge and spin currents and their interdependence. Also one may use such nanostructures (F M | M | F M ), M = normal state, superconducting state, etc.) as interesting fast switches and magnetoresistance devices. Note, the Josephson tunnel current is carried by Andreev–states, see Kastening, Morr et al. [1].

18.4 Discussion For a tunnel junction (T SC | F M | T S)

(18.21)

where TSC denotes a triplet superconductor and FM a ferromagnet, one gets interesting behavior of the Josephson current jJ as a function of


229

temperature, α and θ. The angles α and θ determine the direction of the magnetization of the ferromagnet and the direction of the TSC–order parameter, see Fig. 18.2) for illustration. The Andreev states are determined from Hψj = Eψj using Bogoliubov–de Gennes method. As shown by Fig. 18.9 the Josephson current may change sign as a function of temperature and may be switched between zero and finite value upon varying the angles α, θ. The unconventional change of sign of IJ (without change of phase) for increasing temperature results from the changing occupation of i the Andreev states and from the factor ∂E ∂φ in the expression for the tunnel

I J [e KFΔ0 / h]

0.18 0.16

a

α=0

0.14 0.12 0.10 0.08 0.06 0.04

α=0.15π α=0.25π

0.02 0.00 –0.02 –0.04 –0.06 0.0

0.2

0.4

0.6

0.8

1.0

T / Tc 0.8

b

0.7

φ=0.25π φ=0.90π

I J [e KFΔ0/ h ]

0.6

φ=0.99π

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0 α[π]

1.5

2.0

Fig. 18.9. Tunnel junction (T SC | F M | T SC): (a) Josephson current IJ as a function of temperature (changing sign), (b) IJ as a function of α, the angle between →

magnetization M and direction normal to the current. Note, IJ may be switched between zero and a finite value.

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current. Note, this sign change is different than the one observed for singlet superconductors with a ferromagnetic barrier in between. The behavior of IJ for rotating magnetization, as a function of α, suggests to use such junctions as switching devices. Note, for increasing φ the current changes more drastically. Of particular interest is the behavior of IJ shown in Fig. 18.9(b) as a function of phase φ (for T= 0). Then only the states Ei < 0 are occupied. For (1) M dL,R , α = π/2, the ferromagnet (FM) couples Andreev states, and (2) for M ⊥ dL,R the Andreev states are not coupled by the FM and spins are well defined. Clearly the above junction would be a sensitive probe to detect triplet superconductivity, see results by Morr et al. [1]. Another tunnel system is one involving singlet superconductors, SC | F M | SC.

(18.22)

I J[eΔ0/h]

I J[eΔ0/h]

I J[eΔ0/h]

IJ[eΔ0/h]

The FM is approximately represented by a barrier potential scattering spin dependently the electrons (g refers to the magnetic scattering strength, z to the non–magnetic one). Again, depending on nonmagnetic and magnetic scattering of the tunnelling electrons the Josephson current may change sign as a function of temperature T and furthermore as a function of the relative phase of the two superconductors, see Fig. 18.10 and Fig. 18.11. (Note, IJ = I↑ + I↓ , Is = I↑ − I↓ , the Josephson current IJ is solely carried by the Andreev states, the spin current Is = 0, since spin current through Andreev states is canceled by the one through the continuum states). φ For z g one gets IJ = (eΔ0 /) sin 2Z 2 and non magnetic current is domφ inated by Cooper pairs. For g z one has IJ = −(eΔ0 /) sin 2g2 and current is carried by single electrons. Note, then the phase shift by π. Results assume 1 0

b

–1 1 0 –1 1 0 –1 1

spinpolarized

0 s=1/2

–1 0

π φ

2π 0

π φ

2π 0

π φ

2π 0

π φ

2π

Fig. 18.10. Josephson current IJ at T = 0 for a junction (SC/FM/SC) as a function of phase φ for several values of the scattering strength g and z of the magnetic and nonmagnetic potential, respectively (g = 0, 1/3, 2/3, 1 from left to right, z = 0, 1/3, 2/3, 1 from bottom to top), see results by Kastening, Morr et al. [1]. Note, the ferromagnet is approximately represented by a potential barrier.


a

g=0.53 g=0.83 g=0.93 g=1.23

0.4 0.3

I J[eΔ0 / h]

231

Z=0.5

0.2 0.1 0 –0.1 –0.2 0

0.2

0.6

0.4

0.8

1

T / Tc

Fig. 18.11. Josephson current IJ of a tunnel junction (SC/FM/SC) as a function of TTc for φ = π/2 and nonmagnetic scattering strength z = 0.5 and several values of g, the magnetic scattering strength.

Fig. 18.12. Illustration of a tunnel junction which may act as an ultrafast switching device due to optical excitation of (hot) electrons. Magnetization changes as the electronic temperature changes.

for the thickness of the tunnel junction d to be smaller than the coherence length, otherwise the situation is more complicated. Note, the sign change of the Josephson current as a function of temperature shown in Fig. 18.11 results not from a transition of the junction from a π–state at low temperature to a 0–phase at high temperature, but from a change in the population of the Andreev states (while the relative phase between the superconductors remains unchanged). Also interestingly, the total spin polarization of the two superconductors ground state changes from Sz = 0 to Sz = 1/2. In Fig. 18.12 the possibility of an optically operated ultrafast switching device is sketched. Note, in general changing in nanostructures the magnetization generates in accordance with Maxwell–equations light. Via hot electrons fast changing temperature gradients may yield interesting and novel behavior of tunnel junctions. Possibly in such way Onsager theory can be tested for nanostructures. In Fig. 18.12 the possibility of operating optically an ultrafast switching device is sketched. Note, in general changing in nanostructures the magnetization generates in accordance with Maxwell–equations light.

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Via hot electrons fast changing temperature gradients may yield interesting and novel behavior of tunnel junctions. Possibly in such way Onsager theory can be tested for nanostructures. For further discussion of the interesting physics realized by tunnel systems, the interplay of spin and charge currents (see continuity equation for the spin density) and of accompanying light (see Maxwell equations), one should study the literature, see Nogueira, Bennemann et al. [2] and to be published, FU-Berlin.

18.5 Summary Magnetic effects in nanostructures have been discussed. Typical properties of microscopic tunnel systems are presented. Due to the reduced dimension and system size discretization of the electronic spectrum and external fields may change sensitively the behavior, see studies by Peeters, Kruchinin et al. Spin dependent transport on a nanoscale, involving optical manipulation, temperature gradients etc., see corresponding Onsager theory, offers interesting possibilities. Also (strong) nonequilibrium behavior of nanostructure needs be studied.

Acknowledgements I am grateful for essential general help to Christof Bennemann and F. Nogueira. This review was supported by D.F.G. through S.F.B.

References 1. Kastening B., Morr D.K., Alff L., and Bennemann K.H., Charge Transport and Quantum Phase Transition in Tunnel Junctions, Phys. Rev. B 79, 144508 (2009); Kastening B., Morr D.K., Manske D., and Bennemann K.H., Novel Josephson Effect in Tunnel Junctions, Phys. Rev. Lett. 96, 47009–1 (2006) 2. F. Nogueira and K.H. Bennemann, Spin Josephson effect in ferromagnetic/ferromagnetic tunnel junctions, Europhysics Lett. 67, 620 (2004) 3. Takahashi S., Imamura H., and Maekawa S., Spin Imbalance and Magnetoresistance of Tunnel Junctions, Phys. Rev. Lett.82, 3911 (1999) 4. Garcia M., Habilitation thesis, FU Berlin (1999); Speer O., Garcia M., and Bennemann K.H., Phys. Rev. 62, 2630 (1996) 5. Stampfli P., Bennemann K.H., Electronic Shell Structure in Mesoscopic Systems: Balian–Bloch type theory, Z. Physik D 25, 87 (1992); Phys. Rev. Lett.69, 3471 (1992) 6. Tatievski B., Diploma thesis, Balian–Bloch type theory for Spheres, Discs, Rings and Anti–Dot Lattices (F.U. Berlin, 1993) 7. Tatievski B., Stampfli P., and Bennemann K.H., Analytical results for the electron shell structure in Mesoscopic Systems, Ann. der Physik 4, 202 (1995)

19 HEAVY FERMIONS AND SUPERCONDUCTIVITY IN THE KONDO-LATTICE MODEL WITH PHONONS 2 ˇ O. Bodensiek1 , R. Zitko , R. Peters1 , and T. Pruschke1 1 2

Institute for Theoretical Physics, University of G¨ ottingen, Friedrich-Hund-Platz 1, 37077 G¨ ottingen, Germany [email protected] J. Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia

Abstract. We study the Kondo lattice model with an additional Einstein phonon mode coupled via a Holstein term to the electrons within the dynamical mean-field theory at T = 0. As impurity solver we use the numerical renormalization group. We present results for the paramagnetic case showing the anticipated heavy Fermion physics, including direct evidence for the appearance of a large Fermi surface for antiferromagnetic exchange interaction. By introducing a Nambu notation, we find that increasing electron-phonon coupling favors superconductivity, which however is not BCS like but shows additional structures in the density of states and the gap function.

Key words: heavy fermions, superconductivity, Kondo-lattice, renormalization group.

19.1 Introduction The Kondo-lattice model (KLM) applies to systems where delocalized quasiparticle states, describing the conduction electrons, interact with strongly localized electrons or spins via a weak hybridization or an exchange interaction. Such a situation typically appears in compounds involving elements from the rare-earths. The low-temperature physics of these compounds is strongly influenced by the local moment on the f shell, subject to an antiferromagnetic exchange to the conduction electrons. The resulting physical properties are in many cases again Fermi liquid like, however with extremely enhanced Landau parameters, in particular an effective mass up to three orders of magnitude larger than the one found in conventional metals [1–3]. This large effective mass is the reason why these systems are referred to as Heavy-Fermion materials (HF). Moreover, in addition to these extreme J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 19,

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Fermi liquid properties, the HF also show various phase transitions and the thermodynamics show that these transitions actually occur within the heavy Fermi liquid [1]. Finally, the appearance of superconductivity in a system with initially well-defined magnetic moments is a rather unconventional feature, and close investigation revealed early on that the nature of the ordered state may be rather unconventional [2]. This observation has been substantiated by the development over the past 15 years which showed that a larger number of these HF systems exhibit rather peculiar quantum-phase transitions, partially identified as the driving force behind the superconducting transitions [3, 4]. Other materials which show a coupling between itinerant quasi-particles and localized spins are certain transition metal oxides [5], magnetic semiconductors or semi-metals in the series of the rare earth monopnictides and monochalcogenides [6, 7], and diluted magnetic semiconductors such as Ga1−x Mnx As [8, 9]. Here, the coupling between local spin and conduction electrons usually mediated through Hund’s exchange and thus typically is ferromagnetic. A theoretical description of HF compounds is conventionally based on the Kondo-lattice model (KLM) HKLM = k cˆ†k,σ cˆk,σ − J si · S i . (19.1) k,σ

(†)

i

The operators cˆk,σ denote annihilation (creation) operators of itinerant quasiparticles with dispersion k , si is the operator for the conduction states’ spin density at lattice site Ri and S i describes a spin of magnitude S localized at site Ri . The interaction between the spin of the conduction states and the localized spin is modeled as conventional isotropic exchange interaction J. The dilute version of the KLM Eq. (19.1), the so-called single-impurity Kondo model (SIKM) where there exists only one additional spin at site Ri = 0, is well understood and shows for antiferromagnetic coupling J < 0 the Kondo effect [10], which precisely leads to the phenomena observed in HF systems, viz a strongly enhanced mass in a Fermi liquid ground state. There are nowadays several computational tools to treat the SIKM, for instance continuous-time Monte-Carlo [11] or Wilson’s numerical renormalization group [12]. In theoretical treatments one usually ignores the lattice degrees of freedom. On the other hand, all the above mentioned materials have a rather strong electron-phonon coupling [5, 13] and one can expect that the charge physics driven by phonons somehow competes with the spin physics due to the exchange interaction with the localized spin. Moreover, without exchange coupling, phonons will lead to conventional s-wave superconductivity. Thus the investigation of the interplay between a coupling to a spin and the lattice degrees of freedom is highly interesting. The paper is organized as follows. In the next section we discuss the model and the approximation used to solve it. The case of a paramagnetic metal

19 Kondo-lattice model with phonons

235

is the subject of Sect. 19.3. Introducing a Nambu formalism allows to study superconductivity, which will be discussed in Sect. 19.4. A short summary and outlook will close the paper.

19.2 Model and method The basic model Hamiltonian is again the KLM as introduced in Eq. (19.1). Except for one dimension, no analytical solution exists, and even conventional numerical tools such as Quantum Monte-Carlo (QMC) become rather cumbersome due to a severe sign problem away from particle-hole symmetry. Thus, a reliable approximate method is needed. If one is not interested in the properties too close to a phase transition or in the rather complicated, nonlocal ordering phenomena, a suitable tool is the dynamical mean-field theory (DMFT) [14]. Here, the lattice is mapped onto an effective single-impurity problem, which can be then solved using standard techniques. Here, we use the numerical renormalization group approach [12, 15]. One of its apparent advantages is the possibility to access small energy scales without problem and cover the whole range from T = 0 to finite temperatures of the order of the bare energy scales. Furthermore, it also allows to include phonons to a certain extent, namely an Einstein mode coupled through a Holstein term to the charge degrees of freedom. As is suggestive from effects like Kondo volume collapse [5, 13], such a term can well be rather important. We will thus work with a Hamiltonian (for a detailed introduction and further references see Ref. [16]) † † H = HKLM + ω0 bi bi + g ciσ ciσ − 1 b†i + bi (19.2) i

iσ

where ω0 is the frequency of an appropriate optical mode and g a measure of the electron-phonon coupling. How such an additional local coupling can be treated within NRG, is described in detail in Ref. [12]. To obtain reasonably accurate spectra also for higher energies, we use the broadening strategy introduced by Freyn et al. [17]. As is well-known, one major effect of the Holstein model is to introduce an effective attractive interaction to the electronic subsystem: If the electronphonon coupling g becomes large, keeping g 2 /ω0 constant, the phonons can be integrated out, yielding an attractive local Coulomb interaction Ueff = − 2g 2 /ω0 . Without explicit exchange interaction J, one will then obtain a Hubbard model with attractive U , which shows charge-density and superconducting ordering phenomena [18, 19]. From the point of view of Kondo physics, the negative U will lead to a Kondo-like behavior in the charge sector, strongly competing with the spin Kondo effect introduced by J. We thus can expect interesting physics to occur when both couplings are present.

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19.3 Paramagnetic metal A rather detailed discussion of the physics of electron-phonon coupling in a HF system has been given in Ref. [16]. Here we will concentrate on a comprehensive summary of the heavy-fermion physics and how phonons modify them. We will use a 2D square lattice with nearest-neighbor hopping for the conduction states. Note that the DMFT is rather insensitive to the dimensionality, and we chose the 2D lattice to facilitate visualization of the results. Calculations were done with an NRG discretization parameter Λ = 2, between 1000 and 5000 states kept per NRG step and, where applicable, 50 bosons kept initially. These values were systematically changed for selected calculations to ensure that the results are independent of these numerical parameters. No phonons Let us start with a comparison of the properties at finite J, but with g = 0. We can distinguish two cases, namely an antiferromagnetic exchange J < 0 and a ferromagnetic J > 0. The resulting density of states (DOS) for T = 0 and |J|/W = 0.25 at a filling nc = 0.8 of the conduction band is shown in Fig. 19.1. The quantity W denotes the bandwidth of the conduction band and will serve as energy scale hereafter. There are notable differences between the two cases J < 0 and J > 0. The DOS for J > 0 looks very much like DOS of the bare conduction band (dotted curve in Fig. 19.1), although it is somewhat broadened. For J < 0, however, the DOS is strongly modified, showing a pseudo gap close to the Fermi energy ω = 0. This latter feature is

W⋅ρ(ω)

2

AFJ FMJ J=0 1

0

–1

0 ω/W

1

Fig. 19.1. DOS at T = 0 for the KLM Eq. (19.1) with |J| = 0.25W , where W denotes the bandwidth of the conduction electrons. The filling of the band is nc = 0.8. The dotted curve displays the DOS at J = 0 as reference.


237

ω

2

0

–2

(0, 0)

(π, π)

(π, 0)

(0, 0)

k Fig. 19.2. Spectral functions for |J| = 0.25W , other parameters as in Fig. 19.1. High intensity is represented by red, low by violet.

a fingerprint of heavy Fermion physics, resulting from a picture of hybridized bands [2]. This interpretation becomes even more apparent when one looks at the spectral function along the standard k directions in the first Brillouin zone of the 2D square lattice in Fig. 19.2. For J > 0 (left panel), we basically see the band structure of the 2D nearest neighbor tight-binding band. There is a moderate broadening, which actually is to be expected, because from J > 0 the local spin effectively acts as a potential scatterer. For such a situation the DMFT is equivalent to a CPA calculation, which yields a constant broadening. For J < 0 (right panel), on the other hand, there is a flattening of the band structure close to the Fermi energy and a structure similar to a hybridization gap opens. The flat portion of the lower band corresponds to a large effective mass of the quasi-particles. Note that we have a rather sharp structure at the Fermi energy, i.e. one can indeed talk about quasi-particles here. Another remarkable difference between the cases J > 0 and J < 0 is observed when one looks at the momentum distribution function n(k) displayed in Fig. 19.3. As already noted for the spectral function in Fig. 19.2, the result for J > 0 resembles the Fermi function, slightly smeared out by incoherent scattering from the spins. In any case, the Fermi surface is located at the kF of a non-interacting 2D tight-binding band with a filling nc = 0.8. On the other hand, the momentum distribution for J < 0 in the right panel of Fig. 19.3 does not show any distinct features at this particular value of k. Instead, one notes a small jump in n(k) at a vector outside the square marking the Fermi surface of a half-filled system. A closer inspection shows that this k vector corresponds to a Fermi surface of a system with nc = 1.8, i.e. the system shows a “large” Fermi surface with the spin degrees of freedom contributing to the quasi-particles now. Note that the height of the jump in n(k) is directly related to the inverse effective mass of the quasi-particles.

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1.0

π/

n(kx, ky)

π

0.5

2 0

ky

−π

/2 0.0 − π

−π

/2

π/2

0

π

kx

Fig. 19.3. Momentum distribution n(k) for |J| = 0.25W , other parameters as in Fig. 19.1.

The results for J < 0 in Figs. 19.2 and 19.3 represent the essence of HF physics, namely the generation of heavy quasi-particles, represented by flat bands with a structure known from hybridized bands and a large Fermi surface.

Finite electron-phonon coupling An extended discussion of the effect of Einstein phonons on HF physics is given in Ref. [16]. We restrict the discussion here to J < 0 and fix the phonon frequency to ω0 = 0.5W . Calculations were done for J = − 0.5W to make structures better visible. The results are summarized in Fig. 19.4 for g = 0, g = 0.4W and g = 0.8W . The first thing to note is that the phonons lead to a reduction of the width of the pseudo gap close to the Fermi energy and also a reduction of the overall bandwidth. In addition there occur new structures at higher energy with increasing electron-phonon coupling. The reduction of the overall bandwidth is expected and can be interpreted as an increase of the effective mass of the bare conduction states. The reduction of the width of the pseudo gap, on the other hand, signals a likewise reduction of the low-energy scale generated by the Kondo effect. The structures at higher energies are bound electron-phonon states, i.e. signals of the formation of polarons. These feature becomes more apparent by inspecting the spectral functions in Fig. 19.5. Both effects, the overall reduction of the bare bandwidth and the increased HF mass, are clearly visible. Moreover, with increasing g one finds roughly k-independent structures representing the polaronic modes. Further increasing g, we find a rather sharp crossover around g ≈ ω0 to a completely incoherent behavior. The results of a calculation for g = ω0 are shown


239

2 g=0 g = 0.2 w g = 0.4 w

W⋅ρ(ω)

1,5

1

0.5

0 –2

–1,5

–1

– 0,5

0

0,5

1

1,5

2

ω/W

Fig. 19.4. DOS at T = 0 for the KLM Eq. (19.1) with J = − 0.5W and g = 0, g = 0.2W and g = 0.4W . The filling of the band is nc = 0.8.

ω

2

0

–2

(0, 0)

(π, π)

(π, 0)

(0, 0)

k Fig. 19.5. Spectral functions for J = −0.5W , other parameters as in Fig. 19.4. High intensity is represented by red, low by violet.

in Fig. 19.6. Note that there is no Kondo feature left either in the DOS or in the spectral function, and all structures are rather broad. We also would like to mention that for g > ω0 it becomes increasingly hard to stabilize a given occupation n = 0, 1, 2 of the conduction band. This indicates that the system is close to a Peierls instability, i.e. the formation of a charge density wave together with a lattice distortion. Since this feature goes beyond the present investigations, we postpone a detailed investigation to a forthcoming publication. Finally, it is quite instructive to include the dependence of the effective

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g = W/2

W.ρ(ω)

1,5

1

0,5

0

0

–2

2

ω/W

Fig. 19.6. DOS (left panel) respectively spectral function (right panel) for g = 0.5W = ω0 . Other parameters as in Fig. 19.4.

1 / m*

100

10–2

10– 4

0

0,2

0,4

0,6

0,8

1

g ⎯ ω0

Fig. 19.7. Dependence of the inverse effective mass of the quasi-particles on the electron phonon coupling g [16].

mass of the quasi-particles in the system as a function of g. The result is shown in Fig. 19.7. Initially, the mass does not depend very strongly on g, but close to g = ω0 starts to diverge (i.e. 1/m∗ goes to zero) very strongly [16]. Another interesting question is how a calculation with phonons compares to a calculation with attractive U = − 2g 2 /ω0 as expected from the antiadiabatic limit. The comparison is shown in Fig. 19.8. Quite apparently, the low-energy features are reproduced quite accurately by the model with attractive interaction.

19 Kondo-lattice model with phonons 2

241

g / W = 0.2, ω0 / W = 0.5 U = –2g2 / ω0

W⋅ρ(ω)

1,5

1

0,5

0 –2

–1,5

–1

– 0,5

0 ω/W

0,5

1

1,5

2

Fig. 19.8. DOS for J = − 0.5W , ω0 = 0.5W , g = 0.2W (left panel) respectively g = 0.4W (right panel). Other parameters as in Fig. 19.4. The dashed lines show results for a calculation without phonons, but an attractive interaction U = −2g 2 /ω0 .

However, the overall reduction of the bare bandwidth and, in particular, the additional polaronic modes are missing.

19.4 Superconductivity In order to allow the system to show superconductivity, we have to reformulate the DMFT equations in Nambu space by introducing the Green function matrix ⎞ ⎛ ck↑ ; c†k↑ z ck↑ ; c−k↓ z ⎠ Gk (z) = ⎝ † c−k↓ ; c†k↑ z c†−k↓ ; c−k↓ z =:

Gk (z)

Fk (z)

F−k (−z)∗ G−k (−z)∗

.

The quantities Fk (z) are called anomalous propagators and contain the actual important information about the off-diagonal long-range order. Of course, one has to reformulate the DMFT and extend the NRG accordingly to deal with this matrix structure. Both aspects have been accomplished some time ago already [12, 14] and the actual combination has been extensively discussed by Bauer et al. recently [19]. In the end of the previous section we have shown that, at least as long as one is only interested in the low-energy properties, one can trade the increased numerical effort necessary to include phonons by using the antiadiabatic limit

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and replace the phonons by an effective attractive interaction. For the time being, we will pursue this route to investigate the competition between Kondo effect and superconducting order and postpone the detailed investigation of this aspect with real phonons to a forthcoming publication. Furthermore, because we do not intend to present k resolved results here, we choose a numerically more convenient bare DOS for our DMFT calculations, namely the semi-circular one [14]. Last but not least we focus on antiferromagnetic exchange, as we expect this to be the more interesting case. For a very small Kondo exchange interaction J = − W/25, the ground state of the model is dominated by superconductivity. This becomes apparent from Fig. 19.9, where the DOS (upper panels) and the real part of the anomalous Green’s function (lower panels) is shown at half filling (full curves) as well as at finite filling n ≈ 0.75 (dashed curves). As the low-energy scale of the model with g = 0 is always largest at half filling [20], we can expect that this result remains stable for all fillings n ≤ 1. Note that in none of the cases does one observe a significant dependence of the gap on the filling, i.e. local correlations due to Kondo screening are frozen out here since TK 2g 2 /ω0 . Furthermore, 2g 2 /(ω0 W ) < 1 and consequently one expects and indeed observes a BCS like gap structure, only weakly smeared out by self-energy broadening. Increasing |J| has two effects. First, at half filling there appears a finite, critical combination 2g 2 /ω0 c below which no superconducting solution exists. This can be seen in Fig. 19.10a, where DOS and real part of the anomalous Green’s function are shown for J = − W/10 and a small U = − W/100. Note that at half filling we find a Kondo insulator, which

w⋅ρ(ω)

3

a

b

c

2 1 0

W⋅ℜF(ω+iδ)

0 –0,5 –1 –1,5 –2

–0,005 0 ω/W

0,005

–0,05

0 ω/W

0,05

–0,05

0

0,05

ω/W

Fig. 19.9. DOS (upper panels) and real part of the anomalous Green’s function (lower panels) for small Kondo exchange J = − W/25 and three different values of 2g 2 /ω0 = W/100 (a), W/10 (b) and W/5 (c) at half filling nc = 1 (full curves) and a filling nc ≈ 0.75 (dashed curves).


W⋅ρ(ω)

3

a

2

U = –W / 100

b

c

U = –W / 10

243

U = –W / 5

1 0

W⋅ℜF(ω+iδ)

0 –0,5 –1 –1,5 –2

–0,005 0 0,005

–0,05 0

ω/W

0,05

–0,05 0

ω/W

0,05

ω/W

Fig. 19.10. Left: DOS (upper panels) and real part of the anomalous Green’s function (lower panels) for larger Kondo exchange J = − W/10 at half filling nc = 1. Other parameters as in Fig. 19.9. Right: Value of the superconducting gap extracted from the anomalous part of the self-energy. 2 ℜeΣ(ω) / W+Δsc / W

2

W⋅ρ(ω)

1

1

0

–0,04

0

0,04

–0,5

2g / ω0 = 0.08 W

2g / (ω0W Δsc / W

2g / ω0 = 0.12 W 2g2 / ω0 = 0.16 W

0

2g2 / ω0 = 0.2 W

–0,5

0,5

0,05

2

– 0,75

0

2

2

0

0

–0,25

0 ω/W

0

0,25

0,2

0,5

0,75

Fig. 19.11. Results for a filling nc ≈ 0.75. Other parameters as in Fig. 19.10.

has a gap in the DOS, too. From that perspective the result is actually indistinguishable from the superconducting phase. The anomalous part F (z), however, vanishes here, i.e. we have indeed a normal state and thus an insulator. For larger interactions, the superconducting phase reappears. Compared to the case with small J, we observe here a visible reduction of the gap and also a broadening of the singularities at the gap edges. We attribute this behavior to the correlations induced by the Kondo exchange. Away from half filling we did not succeed in stabilizing convergent solutions within the Nambu formalism for small values of 2g 2 /ω0 . The results obtained so far are summarized in Fig. 19.11. One observes, with increasing

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effective attraction, a more pronounced BCS-like gap structure and a nicely developed and rather symmetric gap in the spectrum. With decreasing coupling the gap value becomes smaller (see lower inset to Fig. but from 19.11), the available values it is not clear if it vanishes at a finite 2g 2 ω0 c as for half filling or not. We note, however, that with decreasing coupling the structure of the DOS close to the Fermi energy changes significantly: The gap becomes increasingly asymmetric with respect to the Fermi energy and an additional feature seems to develop at the lower gap edge. Note that for this particular filling we expect at g = 0 a behavior similar to the one in Fig. 19.1, i.e. even without superconductivity one will find a gap at half filling or, respectively, a pseudo-gap slightly above the Fermi energy away from half filling. How this hybridization gap due to Kondo effect and the superconducting gap merge for small g is not clear and surely requires more detailed investigations. Finally, we would like to point out that the DMFT solution for superconductivity is far from equivalent to a standard BCS mean-field. This becomes evident from the energy dependence of the gap function shown in the right inset to Fig. 19.11, where we display the real part of the anomalous part of the self-energy with the BCS value of the gap subtracted off. For large |ω| this additional part goes to zero, i.e. we recover the BCS gap. However, especially in the low-energy region there is a rather strong energy dependence to the anomalous self-energy. We did not yet analyze these structures in detail, but we expect them to influence properties like universal BCS ratios quite significantly.

19.5 Summary We have presented a summary of properties of the Kondo lattice model within dynamical mean-field theory at T = 0 using the numerical renormalization group as impurity solver. We have extended the Kondo lattice model by including an Einstein mode coupled to the electrons via a Holstein term. The importance of such modes for HF materials can be deduced from strong effects such as the Kondo volume collapse observed in Ce. Restricting the calculations to the paramagnetic case, we find for antiferromagnetic coupling the expected heavy-fermion behavior, with hybridized bands appearing in the spectrum and a large Fermi surface. For ferromagnetic coupling on the other hand, the bare band structure is only weakly modified due to incoherent scattering from the local degrees of freedom. Adding phonons, we find a general narrowing of the bare band, which also leads to a reduction of the Kondo scale. Eventually, when the coupling becomes of the order of the phonon frequency, the electrons tend to localize and form polarons with the phonons. At that point, the effective mass diverges and the electronic spectrum becomes incoherent. As a side observation we note that in this region one sees a tendency of the system to form a charge density wave.


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When we allow for superconductivity via a Nambu formulation, we find a superconducting ground state at least for values 2g 2 /ω0 > T0 (g = 0), where T0 (g = 0) represents the Fermi liquid scale of the system without phonons. This superconducting ground state is not BCS-like in the sense that the gap function, represented by the anomalous part of the self-energy, shows rather strong energy dependence. Moreover, the structure of the gap in the DOS is rather peculiar, showing asymmetries and additional features close to the gap edges. A similar behavior is also observed in tunneling experiments on high-Tc compounds [21]. The results for T = 0 presented here strongly motivate further investigations, in particular at finite T , searching for the critical temperature and looking into ratios like Δ(0)/Tc . Here, Δ(0) can either be the limiting BCS value or it can be extracted from the gap in the DOS. Moreover, the present calculations were done for the antiadiabatic limit for numerical reasons. It is easy to extend them to real phonons, thereby including the additional tendency towards polaron formation and charge ordering.

Acknowledgments We want to thank helpful discussions with Andreas Honecker, Akihisa Koga, Achim Rosch, Fakher Assaad and Dieter Vollhardt. This work was supported by the DFG through PR298/10. Computer support was provided by the Gesellschaft f¨ ur wissenschaftliche Datenverarbeitung in Göttingen and the Norddeutsche Verbund f¨ ur Hoch- und H¨ ochstleistungsrechnen.

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20 MAGNETIZATION CURVES FOR ANISOTROPIC MAGNETIC IMPURITIES ADSORBED ON A NORMAL METAL SUBSTRATE ˇ R. Zitko Joˇzef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia [email protected] Abstract. Magnetization curves Sz (B, T ) for magnetic impurities which couple to the host medium via exchange interactions are computed using the numerical renormalization group. Deviations from the ideal paramagnetic behavior (as described by the Brillouin function) is discussed for various T /TK ratios, where TK is the Kondo temperature, focusing on the implications for the interpretation of experimental X-ray magnetic circular dichroism (XMCD) results. The case of anisotropic impurities is also considered.

Key words: electronic structure, cuprate superconductors, kinetic energy driven superconducting mechanism, out-of-plane impurities, d-wave superconducting gap.

20.1 Introduction A recent trend in the field of surface magnetism is to study magnetic properties of single adsorbed atoms on various surfaces [1]. Particularly interesting systems consist of magnetic impurities (transition metals, lanthanides) on noble metal surfaces where many-particle physics such as the Kondo effect [2] play an important role [3, 4]. In addition to low-temperature scanning tunneling microscopy (STM) which allows to probe the adsorbate electronic and magnetic properties at the level of single atoms [3, 5–8], a commonly applied technique is the X-ray magnetic circular dichroism (XMCD) [9–11]: the absorption coefficient μ(E) for X-rays with energy E is different in magnetic materials for right-circular and left-circular polarization [12,13]. XMCD allows to identify the magnetization direction and strength, magnetocrystalline anisotropy, and magnitudes of the spin (S) and orbital (L) magnetic moments separately [14–16]. While XMCD does not allow to probe individual atoms, but provides sample-averaged results, it is nevertheless an elementselective technique which can be therefore used to study dilute concentrations J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 20,

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of adsorbed atoms [10]. Since dilute magnetic impurities typically do not order magnetically, the experiments are performed by applying an external magnetic field and measuring the magnetization curves, Sz (B). By fitting the results using Brillouin or Langevin functions, the magnetic moments are then extracted. Magnetic anisotropy effects may also be studied by applying the magnetic field in various spatial directions [15]. Recently, it has also been demonstrated that the magnetization curves can be measured using an STM with spin-polarized tip [7]. When the magnetic impurity couples to the host, the description in terms of ideal paramagnetic moment is not appropriate. In particular, the Kondo effect significantly modifies the magnetization curves and introduces a new energy scale TK , the Kondo temperature, to the problem. These effects are most pronounced at low temperatures and magnetic fields, T, B TK . The purpose of the present work is to calculate the magnetization curves for simple impurity models using a numerically exact non-perturbative method (numerical renormalization group, NRG) in order to determine how the impurity-host coupling may manifest in experimental XMCD magnetization curves.

20.2 Kondo model For simplicity, we first consider the pure-spin Kondo model. Two aspects of the full problem are therefore neglected: (1) all features related to orbital magnetism, and (2) the reduction of the moment due to the hybridization with the conduction-band electrons. This implies that the expectation value S 2 will always be equal to S(S + 1), irrespective of the strength of the exchange coupling of the impurity spin with the host material (i.e. charge fluctuations are fully neglected). While this is clearly an oversimplification of the full problem, the Kondo model is sufficient to study the main effects of the exchange coupling to the substrate electrons. The additional effects of the hybridization are considered later in Sect. 20.6. We thus study the Kondo model as given by the following Hamiltonian [2]: H=

k c†kσ ckσ + JS · s + gμB S · B,

(20.1)

k,σ

where S is the quantum-mechanical spin-S operator which describes the magnetic impurity, while s is the spin-density of conduction-band electrons at the impurity position. The gyromagnetic ratio g is assumed to be isotropic, μB is the Bohr magneton and B the external magnetic field. We assume a constant density of states, ρ = 1/2W , where W is the half-width of the conduction band (which is taken as the energy unit in the following). The strength of the exchange coupling is given by ρJ, √ in terms of which the Kondo temperature is given approximately as TK ≈ D ρJ exp(−1/ρJ). For anisotropic impurities, we add an additional term to the Hamiltonian [5, 17, 18]:

20 Magnetization curves

Haniso

= DSz2 + E Sx2 − Sy2 .

249

(20.2)

Here D is the axial magnetic anisotropy and E the transverse magnetic anisotropy. By convention, the z axis is chosen so that |D| is maximal, while axes xy are rotated so that E > 0. If D < 0, the z-axis is an easy-axis; if D > 0, the anisotropy is planar. For a decoupled impurity, the impurity is an ideal paramagnetic spin whose magnetization curve Sz = SBS (x) is given exactly by the Brillouin function BS (x) =

2S + 1 1 2S + 1 1 cotanh x − cotanh x , 2S 2S 2S 2S

(20.3)

where x = gμB BS/kB T is the rescaled magnetic field. Experimentalists commonly extract parameters using a classical-spin approximation, taking an expression for the energy of the form [7, 10] E = −mB cos θ − K cos2 θ,

(20.4)

where B is taken as the z axis, θ is the angle between the field and the magnetic moment, and K is the magnetic anisotropy energy. The magnetization is then given by dφd(cos θ) cos θe−E/kB T , (20.5) M = Msat dφd(cos θ)e−E/kB T where Msat is the saturation magnetization. By fitting to experimental results, one can extract Msat , m, and K. In the isotropic K = 0 limit, Eq. (20.5) reduces to the Langevin function, M = Msat L(x), L(x) = cotanh(x) −

(20.6) 1 , x

(20.7)

with x = mB/kB T . The rationale behind this approach is that the hybridization of the impurity levels with the host states produces broad resonances, thus a continuum description was judged more appropriate [10]. In the following we show that neither Brillouin nor Langevin function fits the results in an acceptable way.

20.3 Numerical renormalization group method The Kondo model requires the use of non-perturbative techniques for correct description of its properties. The magnetization curves for the isotropic Kondo

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model (i.e. the D = E = 0 limit) can be calculated exactly using the BetheAnsatz technique [19]. In general, however, the problem needs to be address with numerical techniques, such as the NRG [20–22]. This approach consists of a discretization of the continuum of the conduction-band electrons into finite intervals with increasingly small width near the Fermi level, followed by a transformation into an effective one-dimensional tight-binding-chain model which is then diagonalized iteratively. After each iteration step, the exponentially growing number of many-particle levels is truncated to some relatively small number; this is a good approximation since the matrix elements coupling excitations with very different characteristic energies are very small (this property is called the “energy-scale separation”). All calculations presented in this work were performed using the “NRG Ljubljana” code with the discretization parameter Λ = 3 and the truncation cutoff set at Ecutoff = 10ωN . No z-averaging was used. Expectation values were calculated with the parameter β¯ = 1. A test calculation for a decoupled S = 1/2 Kondo impurity has shown that the numerically calculated magnetization differs from the exact result as given by the corresponding Brillouin function by less than 0.001; this is thus an approximate upper bound for the error in all results presented in this work. The magnetic moment is given by m = μB < S > .

(20.8)

At zero field, the model is symmetric with respect to the spin inversion S → −S, thus m ≡ 0. It is important to note that this is the case even in the presence of magnetic anisotropy of type Eq. (20.2).

20.4 Isotropic Kondo impurity For a decoupled isotropic spin, the magnetization curve depends universally solely on x = gμB BS/kB T , a feature which is reproduced by NRG with very high numerical accuracy at all temperatures (this is a good test of the numerical procedure). When the impurity couples to the host (i.e. J = 0), the magnetization curves depend on the ratio T /TK , see Fig. 20.1. When the temperature is much higher than TK , the magnetization curve strongly resembles the Brillouin function, although even for very high ratio T /TK = 72 the discrepancies are in the few percent range and, in particular, the saturation to full spin polarization is (logarithmically) slow as the magnetic field is increased. For strong Kondo exchange interaction or, equivalently, for low temperature so that T TK , the magnetization curves are clearly very different from the free-paramagneticimpurity results and a large magnetic field of the order of TK is required to polarize the spin.

20 Magnetization curves 0.5

251

S=1 / 2, ρJ=0.1 TK=1.8 10–4

0.4

B1 / 2(x) free spin T/TK = 72 T/TK = 8 T/TK = 0.9 T/TK = 0.1 T/TK = 0.006

〈Sz〉

0.3 0.2 0.1 0 10–2

10–1

100 x=gμBSB/kBT

101

102

Fig. 20.1. Magnetization curves for an isotropic S = 1/2 Kondo impurity.

2.5

〈Sz〉

2

ρJ = 0.1 TK = 1.8 10–4 T/TK = 0.9

1.5 1

BS(x) S = 1/2 S=1 S = 3/2 S=2 S = 5/2 L5/2(x)

0.5 0 10–2

10–1

100 x = gμBSB/kBT

101

102

Fig. 20.2. Magnetization curves for isotropic spin-S Kondo impurities at a fixed exchange coupling and at constant temperature.

For large impurity spin the behavior remains similar, see Fig. 20.2. It is worth pointing out that even for relatively high S = 5/2, the continuous (classical) description in terms of the Langevin function differs significantly from the discrete (quantum) description in terms of the Brillouin function. Furthermore, the exchange coupling to the host does not make the field-dependence of the magnetization behave in a more “classical way” as might be erroneously expected since the impurity couples to a continuous (“decohering”) environment. A correct description of the impurity magnetization critically depends on the proper inclusion of the quantum many-particle effects, in particular the Kondo effect.

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ˇ R. Zitko 0.5 S=1/2 0.4

〈Sz〉

0.3 0.2

T=1.6 10–4 B1/2(x) free spin AFM: ρJ = 0.1 [TK=1.8 10–4] FM: ρJ=–0.1

0.1 0 10–2

10–1

100 x=gμBSB/kBT

101

102

Fig. 20.3. Magnetization curves for an isotropic S = 1/2 Kondo impurity: comparison of antiferromagnetic and ferromagnetic exchange coupling to the host.

It is also instructive to compare the magnetization of impurities which couple to the host via antiferromagnetic (AFM) or via ferromagnetic (FM) exchange coupling, Fig. 20.3. The host has a larger effect in the first case due to the presence of the Kondo effect, yet a deviation from free-spin Brillouin function is also found for FM coupling, in particular at large magnetic fields. The magnetization curves (scaled by x ∝ B/T ) in this case depend only little on the temperature, since no new dynamically-generated temperature scale emerges.

20.5 Anisotropic Kondo impurity We now consider an anisotropic Kondo impurity in an external magnetic field. For Hamiltonian of a decoupled impurity (J = 0) with the anisotropic term of the form DSz2 + E(Sx2 − Sy2 ), the vector S will not point along the magnetic field B, unless the magnetic field is applied along one of the principal axes (xyz). The expectation value S can be easily derived for arbitrary spin S and field B, however the expressions are lengthy. As an illustration, we consider a spin-3/2 impurity with a planar anisotropy with D = 0.01 and transverse anisotropy E/D = 0.1. In this case, the lowenergy Sz = ± 1/2 multiplet behaves in many aspects as a S = 1/2 impurity [17], however the anisotropic nature of the system is fully revealed by its behavior in the magnetic field, see Fig. 20.4. For a decoupled impurity (J = 0), the behavior of Sz for a field along the z-axis is rather well described by the Brillouin function B1/2 . For a field along a transverse direction (x or y), the field dependence is more complex and the moment does not saturate,


〈Sα〉(Bα)

1.5

253

S = 3/2, D = 0.01, E = 0.001 T = 1.03 10–5 B3/2

1

0.5

0 10–2

ρJ=0-x ρJ=0-y ρJ=0-z ρJ=0.05-x ρJ=0.05-y ρJ=0.05-z ρJ=0.1-x ρJ=0.1-y ρJ=0.1-z

10–1

B1/2

100

101

102

x=gμBSBα/kBT

Fig. 20.4. Magnetization curves for an anisotropic S = 3/2 Kondo impurity with both longitudinal and transverse magnetic anisotropy.

since the Sz = ± 3/2 states start to play a role for fields approaching the scale of the magnetic anisotropy energy. For weak coupling to the host (ρJ = 0.05), the magnetization curves are only moderately modified since the Kondo temperature is much lower than the experimental temperature, Fig. 20.4. For stronger coupling (ρJ = 0.1) such that the Kondo temperature is higher than the experimental temperature (here the Kondo temperature of the effective spin-1/2 Kondo effect [17, 18] is TK (S = 1/2) = 9 × 10−4, therefore T /TK = 0.01), the magnetization curves are strongly affected and the moment becomes well developed only for magnetic field much higher than TK , as expected.

20.6 Effect of the hybridization When the effect of the hybridization is taken into account by describing the impurity interms of the Anderson impurity model, the impurity local moment defined as S2 will be reduced due to charge fluctuations. For this reason, at moderately high magnetic fields the impurity spin polarization will not saturate at 1/2, but at some reduced value (for simplicity, we discuss here the non-degenerate single-orbital model at half-filling, i.e. at the particle-hole symmetric point). In addition, as in the previously discussed case of the Kondo impurity, at low temperatures the Kondo effect will be at play, thus there will be further deviation from the free-spin behavior. Both features are well illustrated in Fig. 20.5, where we consider an Anderson impurity with a very high U/πΓ ratio of 12.7, so that the systemis well in the Kondo limit. The local moment fraction flm = n − 2n↑ n↓ = 2n − n2 is then approximately 0.97 at low temperatures and low magnetic field. Accordingly, we find that the spin polarization saturates at approximately 0.97/2 = 0.485, rather than 1/2.

254

ˇ R. Zitko 0.5 0.4

U=0.2, ε =–0.1, Γ=0.005 TK=1.4 10–9

〈Sz〉

0.3 0.2 0.1 0 10–2

B1/2(x) T=1.4 10–3 T=1.4 10–4 T=1.4 10–5 T=1.4 10–6 T=1.4 10–7 T=1.4 10–8

10–1

100 x=gμBSB/kBT

101

102

Fig. 20.5. Magnetization curves for the Anderson impurity model with parameters chosen so that the system is in the Kondo regime at low enough temperatures.

For low U/πΓ ratio, charge fluctuations become larger and the Kondo temperature TK eventually increases to the scale of U , in which case there is no Kondo effect in the usual sense (conventionally, the limit for the existence of the local moment is considered to be U/πΓ = 1, the point where the Hartree-Fock approximation breaks down, although the transition is in fact continuous and in the correct solution nothing particular happens at the said point). In the U/πΓ 1 limit, the only relevant scales are bare U and Γ . The magnetization is then low as long as B U , i.e. it is not possible to polarize the impurity using laboratory magnetic fields. The impurity is then non-magnetic for all practical purposes. Similar deviations from the Kondo-limit behavior may be found in the valence fluctuation regime when the impurity orbital is away from half-filling, i.e. for |d | Γ . We thus conclude that in all parameter ranges, the magnetization of an Anderson impurity cannot be described by the Brillouin function nor, in fact, by the Langevin function as it is sometimes done. This result clearly carries over to the multi-orbital Anderson model and it is general. It is thus expected that fitting XMCD results by Brillouin or Langevin function may lead to significant systematic errors. The only regime where reasonably good curve fitting by a modified (rescaled and shifted) Brillouin function can be performed is in the strong Kondo limit (large U/πΓ ratio, i.e. for low charge fluctuations, when the Anderson model maps onto an effective Kondo model by the Schrieffer-Wolff transformation) for temperature well above the Kondo temperature, as illustrated in Fig. 20.6.


255

0.5 U=0.2, ε =–0.1, Γ=0.005 0.4

T =1.4 10–6 B’1/2(x)

〈Sz〉

0.3

TK= 1.4 10–9

0.2 0.1 0 10–2

10–1

100 x=gμBSB/kBT

101

102

Fig. 20.6. Magnetization curve for the Anderson impurity model for T TK fitted using a shifted and rescaled Brillouin function B (x) = aB(bx).

20.7 Conclusion It was shown that the magnetization curves for magnetic impurities adsorbed on a metallic substrate strongly depend on the strength of the exchange coupling with the substrate conduction-band electrons. If the Kondo effect is well developed (i.e. if T TK ), the magnetization is strongly suppressed unless the magnetic field becomes comparable to the scale of the Kondo temperature. It is important to note, however, that even for T /TK as small as 0.1, the moment is not fully Kondo screened and that at moderate magnetic fields gμB SB ∼ kB T the magnetization is still sizeable. Only for weak exchange coupling (i.e. TK T ) are the magnetization curves qualitatively comparable to Brillouin functions, although quantitative deviations are still very pronounced even for T /TK as large as 10. We conclude by noting that in principle it is possible to extract all experimental parameters (spin S, Kondo temperature TK , magnetic anisotropies D and E) from known magnetization curves Sz (B), although this would constitute a difficult non-linear curve fitting problem involving repeated numerical calculations of the magnetization curves using NRG. Nevertheless, this approach would be more reliable than the extraction of anisotropy parameters by fitting XMCD results using the magnetization curves obtained from classical spin models (i.e. in the S → ∞ limit). The results presented in this work should serve as a caveat against such over-simplified parameter extraction procedures.

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References 1. H. Brune and P. Gambardella. Magnetism of individual atoms adsorbed on surfaces. Surf. Sci., 602:1812, 2009. 2. A. C. Hewson. The Kondo Problem to Heavy-Fermions. Cambridge University Press, Cambridge, 1993. 3. A. F. Otte, M. Ternes, K. von Bergmann, S. Loth, H. Brune, C. P. Lutz, C. F. Hirjibehedin, and A. J. Heinrich. The role of magnetic anisotropy in the kondo effect. Nature Physics, 4:847, 2008. 4. M. Ternes, A. J. Heinrich, and W. D. Schneider. Spectroscopic manifestations of the kondo effect on single adatoms. J. Phys.: Condens. Matter, 21:053001, 2009. 5. C. F. Hirjibehedin, C.-Y. Lin, A. F. Otte, M. Ternes, C. P. Lutz, B. A. Jones, and A. J. Heinrich. Large magnetic anisotropy of a single spin embedded in a surface molecular network. Science, 317:1199, 2007. 6. Y. Yayon, V. W. Brar, L. Senepati, S. C. Erwin, and M. F. Crommie. Observing spin polarization of individual magnetic adatoms. Phys. Rev. Lett., 99:067202, 2007. 7. F. Meier, L. H. Zhou, J. Wiebe, and R. Wiesendanger. Revealing magnetic interactions from single-atom magnetization curves. Science, 320:82, 2008. 8. T. Balashov, T. Schuh, A. F. Takacs, A. Ernst, S. Ostanin, J. Henk, I. Mertig, P. Bruno, T. Miyamachi, S. Suga, and W. Wulfhekel. Magnetic anisotropy and magnetization dynamics of individual atoms and clusters of fe and co on pt(111). Phys. Rev. Lett., 102:257203, 2009. 9. P. Gambardella, S. S. Dhesi, S. Gardonio, C. Grazioli, P. Ohresser, and C. Carbone. Localized magnetic states of fe, co, and ni impurities on alkali metal films. Phys. Rev. Lett., 88:047202, 2002. 10. P. Gambardella, S. Rusponi, M. Veronese, S. S. Dhesi, C. Grazioli, A. Dallmeyer, I. Cabria, R. Zeller, P. H. Dederichs, K. Kern, C. Carbone, and H. Brune. Giant magnetic anisotropy of single cobalt atoms and nanoparticles. Science, 300:1130, 2003. 11. P. Gambardella, S. Stepanow, A. Dmitriev, J. Honolka, F. M. F. de Groot, M. Lingenfelder, S. Sen Gupta, D. D. Sarma, P. Bencok, S. Stanescu, S. Clair, S. Pons, N. Lin, A. P. Seitsonen, H. Brune, J. V. Barth, and K. Kern. Supramolecular control of the magnetic anisotropy in two-dimensional high-spin fe arrays at a metal interface. Nature Materials, 8:189, 2009. 12. G. Sch¨ utz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller, R. Frahm, and G. Materlik. Absorption of circularly polarized x-rays in iron. Phys. Rev. Lett., 58:737, 1987. 13. C. T. Chen, F. Sette, Y. Ma, and S. Modesti. Soft-x-ray magnetic circular dichroism at the l2,3 edges of nickel. Phys. Rev. B, 42:7262, 1990. 14. B. T. Thole, P. Carra, F. Sette, and G. van der Laan. X-ray circular dichroism as a probe of orbital magnetization. Phys. Rev. Lett., 68:1943, 1992. 15. P. Carra, B. T. Thole, M. Altarelli, and X. Wang. X-ray circular dichroism and local magnetic fields. Phys. Rev. Lett., 70:694, 1993. 16. C. T. Chen, Y. U. Idzerda, H.-J. Lin, N. V. Smith, G. Meigs, E. Chaban, G. H. Ho, E. Pellegrin, and F. Sette. Experimental confirmation of the x-ray magnetic circular dichroism sum rules for iron and cobalt. Phys. Rev. Lett., 75:152, 1995.


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21 CONDUCTIVITY OF LAYERED SYSTEMS WITH PLANAR AND BULK DISORDERS D.L. Maslov1 , V.I. Yudson2 , A.M. Somoza3 , and M. Ortu˜ no3 1 2 3

Department of Physics, University of Florida, P. O. Box 118440, Gainesville, FL 32611-8440, USA [email protected] Institute for Spectroscopy, Russian Academy of Sciences, Troitsk, 142190 Moscow Region, Russia [email protected] Departamento de F´ısica-CIOyN, Universidad de Murcia, Murcia 30.071, Spain

Abstract. Conductivity of many layered materials is more anisotropic than it is predicted by the band theory. To understand this anomaly, we consider a simple model of randomly spaced “wrong” planes (this “planar disorder” is mimicked by a random potential U (z), z being the coordinate along the c-axis perpendicular to the planes) with isotropic impurities (“bulk disorder”) located randomly in the bulk of the material. This model has been solved numerically and analytically, with the use of an exact solution for the conductivity of a strictly one-dimensional (1D) disordered system. Bulk disorder destroys 1D localization along the c-axis which would take place for only planar disorder. The conductivity along the c-axis is finite and is proportional to the weak scattering rate by bulk impurities until planar and bulk disorder become comparable. Thus, the out-of-plane conductivity is of a nonDrude form, and this may result in a much stronger anisotropy of the conductivity than it follows from the band theory for systems with only a bulk disorder.

Key words: layed metals, conductivity, anisotropy, disorder, localization.

21.1 Introduction It is commonly believed that Anderson localization of electrons in disordered systems can be destroyed only by inelastic scattering destroying the phase coherence of carriers. Here we propose and analyze a simple model with two types of disorder where an increase in one type of disorder leads to a destruction of the Anderson-localized state and, consequently, to a increase in the conductivity in one direction. Our consideration is motivated by the phenomenon of an anomalously strong anisotropy of conductivity observed in some layered materials, the anisotropy being much greater than expected from the usual Drude description of electron scattering by isotropic impurities. The observed ratio of the in-plane and out-of-plane conductivities may J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 21,

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y

szz

s||

bulk disorder

Fig. 21.1. (Color online) Left: a system of randomly spaced parallel potential barriers and randomly distributed isotropic impurities. Right: expected dependences of the in- and out-of-plane conductivities on bulk disorder.

exceed the Drude one (given by (inverse) ratio of the effective masses) by several orders of magnitude. An example of such an anomaly is graphite, where the conductivity ratio exceeds the mass ratio by 2–3 orders of magnitude [1], but other materials, e.g., NaCo2 O4 [2], cuprates [3], etc., also provide examples of this behavior. Our model consists of planar potential barriers located at random spacings to each other and isotropic impurities distributed randomly in between the barriers (see Fig. 21.1). This model is motivated by the existence of stacking faults, e.g., “wrong” planes violating Bernal stacking of graphene sheets in graphite. These violations have been proposed to be responsible for abnormally large conductivity anisotropy long time ago [4]; however, little attention has been paid to localization of electrons by an array of faults. Another related motivation is localization of light in photon crystals [8] and carriers in superlattices [5] by random layer thickness variations. We consider a system of electrons with separable but otherwise arbitrary spectrum ε(k || , kz ) = ε|| (k|| ) + εz (kz ), subject to two types of random potential: the 1D potential of the barriers, U (z), and the 3D potential of isotropic impurities, V (r). [The assumption of separability of ε(k|| , kz ) is not necessary but helps to clarify the physical picture of localization.] In the absence of bulk disorder, the in- and out-of-planes degrees of freedom Accordingly, separate. the electron wave function is factorized as Ψ r || , z = ϕ r|| χ (z), with χ(z) satisfying aneffectively 1D Schrodinger equation [εz (−i∂z ) + U (z)] χ (z) = E − ε|| k|| χ (z), where k|| is the (quasi) momentum along the planes. All states of such a system are localized in the z-direction by infinitesimally weak disorder. Therefore, the dc conductivity across the planes, σzz , is zero. On the other hand, since barriers do not affect the electron motion along the planes, the in-plane conductivity, σ|| , is infinite. Bulk disorder mixes the inand out-of-planes degrees of freedom, so that the separation of variables is no longer possible. Therefore, 1D localization in the z direction is destroyed, and σzz increases with bulk disorder, as long as it remains weaker than the planar one. When two disorders become comparable, σzz reaches a maximum

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and decreases upon a further increase in bulk disorder in accord with the Drude formula. (A further increase in bulk disorder will lead eventually to a 3D Anderson localization [6], but we are not studying this regime here). At the same time, σ|| decreases monotonously with bulk disorder. A sketch of expected dependences of σzz and σ|| on 3D disorder is presented in Fig. 21.1 (right). In the rest of the paper, we confirm this simple picture numerically – by calculating σzz in the Anderson model, and analytically – by exploiting the Berezinskii solution of the 1D localization problem. Some of the results of this paper were presented earlier in [9].

21.2 Numerical analysis of the Anderson model Numerically, we study the Anderson model with nearest-neighbor hopping (set to unity to fix the energy scale) for a cubic lattice (of unit spacing) † † H=− aj ai + i ai ai . (21.1) i,j

i

Here, i = (ix , iy , iz ) and i = φi +ηiz is the on-site energy. The first term, φi , is the standard (bulk) disorder term which is chosen independently for each site in the interval (−WB /2, WB /2) with uniform probability. The second term, ηiz , describing planar disorder, is chosen as −W with probability p and as W with probability 1 − p. For all results reported in this paper, p = 1/2. The simulations are done at the energy equal to 0.1, to avoid the center of the band. We employ the recursive Green’s function technique [7] with periodic boundary conditions in the directions transverse to the z-axis. The out-of-plane conductance Gzz is equal to 2e2 T /h, where T is the transmission coefficient between two wide leads. The simulations were performed for cubic samples of sizes L up to 35 lattice spacings. The bandwidths of planar disorder W were chosen as 1, 1.5, 2, 2.5 and 3, which corresponds to localization lengths between roughly 2 and 15 lattice spacings, in the absence of bulk disorder. The bandwidth of bulk disorder WB ranged in between 0 and 18. We have averaged ln Gzz for 103 samples for each set of parameters [10]. Crystalline anisotropy can be readily accounted for; however, the conductance is anisotropic due to anisotropy of disorder even on a cubic lattice. Figure 21.2 shows G ≡ exp{ln Gzz } as a function of bulk disorder for several values of planar disorder. As expected, an increase in bulk disorder leads first to an in increase in G followed by a subsequent decrease. The position of the peak depends on planar disorder but is almost independent of L. We checked that the conductance scales linearly with L for most of the range of parameters represented in Fig. 21.2, so that we are in the diffusive regime. Specifically, the diffusive regime begins when the conductance becomes larger than 2e2 /h and continues up to the 3D Anderson transition (not shown in Fig. 21.2).

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Fig. 21.2. (Color online) Out-of-plane conductance versus the bandwidth of bulk disorder WB for a range of values of planar disorder W , as shown in the figure, and L = 30.

Fig. 21.3. (Color online) Out-of-plane conductivity versus the bandwidth of bulk disorder WB on a double logarithmic scale for a range of system sizes, as shown in the figure, and three values of planar disorder: W = 1.5 (upper set), W = 2 (middle set), and W = 2.5 (lower set).

Figure 21.3 shows the collapse of the data for the conductivity, σzz = G/L, on a double-logarithmic plot. Three sets of curves corresponds to three values of planar disorder: W = 1.5 (upper set), W = 2 (middle set) and W = 2.5 (lower set). Within each set, the conductivity was computed for different values of L, as indicated in the legend. The straight line has a slope equal to two. This scaling is confirmed by the analytic solution of the model, described below.

21.3 Analytical solution To treat the problem analytically, we take the delta-correlated forms for both types of disorder U (z) U (0) = γz δ (z) and V (r) V (0) = γδ (r), and assume that bulk disorder is weaker than planar one, i.e., 1/τ ≡ 2πν3 (EF )γ 1/τz = 2πν1 (E, k|| )γz , while planar disorder is weak in a sense that EF τz 1.


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Here, ν3 is the 3D density of states and ν1 is the 1D density of states at fixed value of k|| per one spin orientation. In the absence of bulk disorder, our problem reduces to the 1D case with the velocity vz = |∂ε(k|| , kz )/∂kz |kz =kzF (k|| ) , where kzF (k|| ) is a positive root of the equation ε(k|| , kz ) = EF , and the scattering time τz being functions of k|| . The result for the ac conductivity of a strictly 1D disordered system, surmised first by Mott [11] and derived rigorously by Berezinskii [12] reads σ 1D (ω) =

32e2 vz τz −iζ(3)ωτz + 2τz2 ω 2 ln2 (ωτz ) , π

(21.2)

for 0 < ωτz 1. (The numerical coefficient in the imaginary part was corrected in Refs. [13, 14]). The out-of-plane conductivity of a 3D sample V = 0 is obtained from Eq. (21.2) by summing over k|| : σzz (ω) = with d2 k|| σ 1D (ω) /(2π)2 . As expected, σzz (0) = 0. In the presence of both types of disorder, σzz is given by the Kubo formula σzz (ω) =

e2 1 dz 2π A3 k|| , k||

R A ×vz G+ (k|| , z; k|| , z )vz G− (k|| , z ; k|| , z)p b , (21.3)

where G± = G R(A) (k|| , z; k|| , z ; EF ± ω/2) is an exact retarded (advanced) electron Green’s function in the mixed k|| − z representation for a given disorder realization, A is the sample area in the lateral direction, and . . . b,p denotes averaging over bulk and planar disorders, correspondingly. The diagram for σzz is shown in Fig. 21.4 on the left. To leading order in γ, the conductivity (1) σzz averaged over bulk disorder is given by the sum of the two diagrams in the first row of Fig. 21.4, where thick solid lines denote Green’s functions R(A)

k ||

=

k ||

p || k ||

+ k ||

k || k ||

=

p || k ||

k ||

p ||

k ||

k ||

+ k ||

p ||

k ||

Fig. 21.4. (Color online) Diagrams for the out-of-plane conductivity to leading order in bulk disorder. Thin lines: exact Green’s functions in the presence of both types of disorder; thick solid lines: Green’s functions in the presence of planar disorder only; dashed lines: Green’s functions averaged over planar disorder; zigzag: correlator of bulk disorder; solid and dashed brackets: averaging over bulk and planar disorder, correspondingly.

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in the absence of bulk disorder, Gp (z, z ; k|| ; E), and zigzags denote the correlation function of bulk disorder. There are no vertex corrections for the case of delta-correlated bulk disorder. The first (second) diagram in the first row of Fig. 21.4 is obtained by replacing the exact Green’s function by GpR(A) (z, z1 ; k|| )GpR(A) (z1 , z1 ; p|| )GpR(A) (z1 , z ; k|| ) . γ R(A)

z1 ,p||

Subsequent averaging over planar disorder is simplified dramatically by noticing that the effective energies E − ε|| (k|| ) of the Green’s functions depend on a particular value of k|| . For short-range bulk disorder, the momentum p|| of the Green’s function below the zigzag line differs considerably from the momentum k|| in the rest of the diagram. This means that the typical difference of corresponding energies is of order EF , i.e., much greater than 1/τz . In this situation, one can safely neglect correlations between the Green’s functions with different momenta and average Gp (z1 , z1 ; p|| , E) over planar disorder independently from the rest of the diagram. [The contribution of the region where p|| ≈ k|| and, therefore, this procedure does not work, is small and can be neglected.] As a result, we arrive at the diagrams in the second row of Fig. 21.4, where thick dashed lines denote Green’s functions averaged over planar disorder. For weak planar disorder (EF τz 1), these Green’s function are 1 dkz R,A z, z; k|| ; E p = Gp 2π E − εz (kz ) − ε|| k|| ± i/2τz and the corresponding self-energy insertion reduces to Σ R(A) (z, z ; k|| ; E) = ∓ (i/2τ ) δ(z − z ).

(21.4)

Expanding GpR,A over the basis of exact eigenstates of the 1D problem, we reduce the convolution of two Green’s functions, sharing the point z1 , to ∂ R,A Gp z, z ; k|| ; E . GpR,A z, z1 ; k|| ; E GpR,A z1 , z ; k|| ; E = − ∂E z1 (1)

Consequently, σzz (ω) is obtained from the exact 1D result via (1) σzz (ω) =

i τ

d2 k|| ∂σ 1D (ω) . 2 ∂ω (2π)

(21.5)

To obtain the dc conductivity, one needs to differentiate only the imaginary (1) part of Eq. (21.2). This gives σzz (0) = 2e2 ν3 (EF )Dzz , where Dzz = 16ζ(3)

2 vz4 (k|| )|| lz,max . 4 vz,max τ

(21.6)


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Here, we introduced lz,max and vz,max as the maximum values of lz (k|| ) ≡ vz τz and vz (k|| ), attained for k|| = 0, and −1 d2 k|| ν1 (EF , k|| )f (k|| ) . f (k|| )|| = 4π 2 ν3 (EF ) The diffusion coefficient Dzz is proportional to the ratio of the square of the localization length in the 1D system to the bulk scattering time. Summing up higher-order diagrams with self-energy insertions due to bulk disorder amounts to replacing the exact Green’s functions in Eq. (21.3) by G R(A) (z, z ; k|| ; E ± ω/2 ± 2τi ), which can be viewed as functions of a complex frequency. One can verify that all intermediate steps in Refs. [12] and [14] are valid for complex ω as well. Therefore, the general result for the conductivity of our model is obtained from the Berezinskii’s solution as

2 d k|| 1D i . (21.7) σzz (ω) = σ ω+ τ (2π)2 To lowest order in 1/τ , Eq. (21.7) reduces back to Eq. (21.5). Within the logarithmic accuracy, we obtain

2 16lz 2τz 2 2 2 Re σzz (ω) = 2e ν3 ; (21.8) ζ (3) + ω τ −1 L τ τ ||

2 2τz 16lz 2 Im σzz (ω) = −2e ν3 ωτ ζ (3) − L , (21.9) τ τ || where L ≡ ln2 ω 2 τz2 + τz2 /τ 2 . These formulas are valid for an arbitrary value of ωτ but only for ωτz 1 and τz /τ 1. From Eq. (21.9), we see that 1/2 Reσzz (ω) is almost constant for ω ωcr ≡ 1/ (τz,max τ ) and increases 2 2 with ω in a Mott way, as ω ln ω, for ω ωcr . At higher frequencies, ω 1/τz , σzz (ω) can be found perturbatively in 1/τz : the leading order result is simply a Drude formula σzz (ω) ∝ 1/ω 2 τz − i/ω. Therefore, both Reσzz (ω) and −Imσzz (ω) have maxima at ω ∼ 1/τz . This prediction is amenable to a direct experimental verification.

21.4 Discussions and conclusions Equations (21.6)–(21.8) allow for a simple physical interpretation. Bulk scattering weakly couples 1D channels of localized electrons with different k|| . Each scattering event results in a random displacement of order lz in the z-direction, which leads to diffusion with the coefficient Dzz ∼ lz2 /τ and to the non-zero conductivity related with the diffusion coefficient by the Einstein relation. Notice that bulk disorder acts very similarly to the electron-phonon (e–ph) interaction in a strictly 1D system, where σ 1D (0) ∝ 1/τe−ph [13].

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The difference between the two cases is that σ1D (0) scales with 1/τe−ph only at temperatures higher than the single-level spacing within the localization length, i.e, for T τz 1, while at lower temperatures σ1D is of the hopping form. The condition T τz 1 allows one to neglect correlations between the Green’s functions in the self-energy insertions and in the rest of the diagram. In our case, these correlations can be always neglected for short-range bulk disorder, i.e., in contrast to phonon-activated transport, there is no “hopping” regime for disorder-activated transport. Coming back to the issue of anomalously large conductivity anisotropy, it is easy to show that the in-plane conductivity is given by the usual Drude formula σαβ = 2δαβ e2 ν3 (EF )vα vβ || τ . The conductivity ratio can be then estimated as σ|| /σzz ∼ v||2 || /vz2 || (τ /τz,max )2 .

As an example, we consider the case of graphite with σ|| /σzz = 104 at low temperatures. A realistic band structure model of graphite [1] gives v||2 /vz2 ∼ 140, thus τz /τ ∼ 0.12. Taking τ = 4 × 10−12 s from Ref. [15] and estimating vz2 1/2 ∼ 2 × 106 cm/s, we obtain for the mean free path due to planar disorder (stalking faults) lz ∼ 120 ˚ A. This means that stalking faults are separated by about a hundred perfect planes, which is quite a realistic assumption. To summarize, we have shown that a system with two types of disorder– randomly spaced planar barriers and bulk impurities–exhibits quite unusual transport properties. In the absence of bulk disorder, it behaves as a 1D insulator in the out-of-plane direction and as an ideal metal in the in-plane direction. Bulk disorder renders both conductivities finite; however, σzz increases with bulk disorder until two disorders become comparable. For weak bulk disorder, the ratio of the conductivities may exceed the ratio of the effective masses by orders of magnitude. The ac out-of-plane conductivity has a manifestly non-Drude frequency dependence with a maximum at intermediate frequencies. Finally, we notice that the predictions of our model are equally well applicable to a two-dimensional (2D) case, e.g, for line barriers crossing the plane. Such a system can be realized in a 2D electron gas with an array of randomly spaced stripe-like gates.

Acknowledgement We thank S. Blundell, H. Bouchiat, S. Brazovskii, K. Efetov, A. Hebard, S. Gueron, D. Gutman, N. Kirova, V. Kravtsov, I. Lerner, G. Montambaux, ´ Rashba, A. Schofield, S. Tongay, and I. Yurkevich H. Pal, P. Hirschfeld, E. for stimulating discussions. We acknowledge hospitality of the Abdus Salam International Center for Theoretical Physics (ICTP, Trieste) where a part of the work was done. D.L.M. acknowledges the financial support from RTRA


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Triangle de la Physique and hospitality of the Laboratoire de Physique des Solides, Université Paris-Sud. V.I.Y. acknowledges support from the RFBR (project 09-02-01235). A.M.S and M.O. acknowledge financial support from the Spanish DGI, project FIS2006-11126, and Fundacion Seneca, project 08832/PI/08.

References 1. N. B. Brandt, S. M. Chudinov, and Ya. G. Ponomarev, Semimetals: I. Graphite and its Compounds (North-Holland, Amsterdam, 1988). 2. D. J. Singh, Phys. Rev. B 61, 13397 (2000). 3. W. E. Pickett, Rev. Mod. Phys. 61, 433 (1989). 4. S. Ono, J. Phys. Soc. Jap. 40, 498 (1976). 5. S. Das Sarma, A. Kobayashi, and R. E. Prange, Phys. Rev. Lett. 56, 1280 (1986); A. Chomete, B. Deveaud, A. Regreny, and G. Bastard, ibid., 57, 1464 (1986); M. Lee, S. A. Solin, and D. R. Hines, Phys. Rev B 48, 11921 (1993). 6. A. A. Abrikosov, Phys. Rev. B 50, 1415 (1994); N. Dupuis, Phys. Rev B 56, 9377 (1997). 7. A. MacKinnon, Z. Phys. B 59, 385 (1985). 8. S. Zhang, J. Park, V. Milner, and A. Z. Genack, preprint ArXiv: 0904.1905. 9. D. L. Maslov, V. I. Yudson, A. M. Somoza, and M. Ortuno, Phys. Rev. Lett. 102, 216601 (2009). 10. Although averaging of ln Gzz and Gzz gives almost the same results in the diffusive regime, we prefer to average ln Gzz to account for 3D localization effects at larger WB . 11. N. F. Mott, Phil. Mag. 22, 7 (1970). 12. V. L. Berezinskii, Sov. Phys. JETP 38, 620 (1974). ´ I. Rashba, Sov. Phys. JETP 42, 168 13. A. A. Gogolin, V. I. Mel’nikov, and E. (1975). 14. A. A. Abrikosov and I. A. Ryzhkin, Adv. Phys. 27, 147 (1978). 15. X. Du, S.-W. Tsai, D. L. Maslov, and A. F. Hebard, Phys. Rev. Lett. 94, 166601 (2005).

22 SUPERPOSITION OF FLUX-QUBIT STATES AND THE LAW OF ANGULAR MOMENTUM CONSERVATION A.V. Nikulov Institute of Microelectronics Technology and High Purity Materials, Russian Academy of Sciences, 142432 Chernogolovka, Moscow Region, Russia [email protected] Abstract. Superconducting loop interrupted by one or three Josephson junctions is considered in many publications as a possible quantum bit, flux qubit, which can be used for creation of quantum computer. But the assumption on superposition of two macroscopically distinct quantum states of superconducting loop contradict to the fundamental law of angular momentum conservation and the universally recognized quantum formalism. Numerous publications devoted to the flux qubit testify to inadequate interpretation by many authors of paradoxical nature of superposition principle and the subject of quantum description.

Key words: superposition of macroscopically distinct quantum states, law of conservation, superconducting loop, quantum bit, quantum computation, foundation of quantum mechanics.

22.1 Introduction Quantum computation and quantum information is one of the most popular themes of the last decades [1, 2]. Many authors propose [3, 4] and make [5–9] quantum bits, main element of quantum computer, on base of different two-states quantum systems including superconducting one. The employees of D-Wave Systems Inc. claimed already that they have made the world’s first commercially viable quantum computer [10]. Main aim of this paper is to show that the assumptions by numerous authors on macroscopic quantum tunneling [11–13] and on superposition of two macroscopically distinct quantum states [14–16] of superconducting loop interrupted by Josephson junctions contradict to the fundamental law of angular momentum conservation and the universally recognized quantum formalism. It is important for the problem of practical realisation of the idea of quantum computation since many authors consider such loop as possible quantum bit, flux qubit. J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 22,

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Quantum bit is a quantum system with two permitted states, superposition of which is possible. Without superposition of states a quantum system with two permitted states is ordinary but no quantum bit. Therefore the contradiction between the assumption on superposition and fundamental physical laws casts doubt on numerous publication about “flux qubit”. These works may be unavailing. The contradiction between the assumption on “flux qubit” and the law of angular momentum conservation is obvious. It does not mean that this conservation law can be violated. No experimental result obtained for the present can give evidence of superposition of macroscopically distinct quantum states [17]. Many authors may interpret some experimental results as such evidence because of no enough profound understanding of paradoxical nature of the quantum principle of superposition [18]. Therefore before to consider the concrete problem of “flux qubit” I will touch “philosophical” problems of quantum foundation and the essence of controversy between creators of quantum theory about the subject of quantum mechanics description.

22.2 What is subject of quantum mechanics description? For centuries science had viewed its aim as the discovery of the real. Scientists believed that they investigate an objective reality as it exists irrespective of any act of observation. But on the atomic level physicists have come into collision with paradoxical phenomena which can not be described up to now as a manifestation of an objective reality. Therefore some creators of quantum theory, Heisenberg, Bohr and others were force to advocate positivism, the point of view according to which the aim of science is investigation no objective reality but only phenomena [19]. Other creators of quantum theory, Plank, Einstein, de Broglie, Schrodinger could not agree with this change of science aim. But no realistic description of quantum phenomena could be created. Therefore the quantum mechanics created at the cost of refusal of objective reality description dominates more than eighty years. It is important to understand that quantum mechanics, in contrast to other theories of physics, does not describe a reality. The basic principle of the idea of quantum computation was introduced in 1935 by opponents of the Copenhagen interpretation, Einstein and Schrodinger, who persisted in their opinion that the quantum theory which can describe only phenomena can not be considered as complete. Both Einstein, Podolsky, Rosen [20] and Schrodinger where sure that this principle, entanglement or Einstein– Podolsky–Rosen correlation, can not be real because of its contradiction with locality principle. Therefore Einstein, Podolsky, Rosen stated that quantummechanical description of physical reality can not be considered complete [20] and Schrodinger introduced [21] this principle as “entanglement of our knowledge” [22]. A “philosophical” question: “Could a real equipment be made on base of the principle which can not be describe reality?” forces to consider the essence of entanglement and history of its emergence.

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Two main paradoxes of quantum phenomena Two features of quantum phenomena are most paradoxical. The both were introduced into the consideration by Einstein, the principal opponent of the Copenhagen formalism as a complete theory. Wave-particle duality Bohr wrote in 1949 [23]: “With unfailing intuition Einstein thus was led step by step to the conclusion that any radiation process involves the emission or absorption of individual light quanta or “photons” with energy and momentum E = hν;

p = hσ

(22.1)

respectively, where h is Planck’s constant, while ν and σ are the number of vibrations per unit time and the number of waves per unit length, respectively. Notwithstanding its fertility, the idea of the photon implied a quite unforeseen dilemma, since any simple corpuscular picture of radiation would obviously be irreconcilable with interference effects, which present so essential an aspect of radiative phenomena, and which can be described only in terms of a wave picture. The acuteness of the dilemma is stressed by the fact that the interference effects offer our only means of defining the concepts of frequency and wavelength entering into the very expressions for the energy and momentum of the photon”. Indeterminism Further Bohr wrote in [23]: “In this situation, there could be no question of attempting a causal analysis of radiative phenomena, but only, by a combined use of the contrasting pictures, to estimate probabilities for the occurrence of the individual radiation processes”. Before [23] in 1924 [24] Bohr noted that Einstein was first who considered the individual radiation processes as spontaneous, i.e. causeless phenomenon. Superposition of states as a method of description duality and causeless phenomena Advocates of the Copenhagen interpretation believe that the principle of superposition can completely describe paradoxical nature of wave-particle duality and causeless phenomena. Double-slit interference experiment Indeed it seems that this principle can perfectly describe the duality observed in the double-slit interference experiment. If a particle with a momentum p

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and an energy E passes the double-slit as a wave Ψ = A exp i(pr−Et) describing an amplitude probability then the probability P (y) = |Ψ |2 = |Ψ1 + Ψ2 |2 = A21 + A22 + 2A1 A2 cos(

pdy ) L

(22.2)

to observe the arrival of the particle at a point y of a detecting screen placed on a distance L from two slits separated of a distance d is determined by the superposition Ψ1 + Ψ2 of possibilities to pass through first Ψ1 = A1 exp i(pr1−Et) or second Ψ2 = A2 exp i(pr2−Et) slit. In accordance with this prediction all experiments give the interference patter corresponding to the momentum p = mv of particles, electrons [25] with the mass m ≈ 9 10−31 kg, neutrons [26] with m ≈ 1.7 10−27 kg, atoms [27] with m ≈ 3.8 10−26 kg and even massive molecules [28, 29], for example C30 H12 F30 N2 O4 with m ≈ 1.7 10−24 kg and a size a ≈ 3.2 nm. The interference patter appears just as a probability when particles pass one by one through the two-slit system [25]. Probability of what? One may say that the wave-particle duality is observed in the double-slit interference experiment. Electron, for example, in the experiment [25] should pass the double-slit as a wave with the de Broglie wavelength λ = 1/σ = h/p = h/mv in order the interference patter of electrons distribution with a period Δy ≈ λL/d can emerge at the detecting screen. But each electron is detected as particle at a point of the detecting screen. What is the essence of the de Broglie-Schrodinger wave function Ψ ? According to the orthodox interpretation proposed by Born |Ψ (r, t)|2 is a probability density. But probability of what? There is possible a realistic or positivism interpretation. According to the first one |Ψ (r, t)|2 dV is a probability that the particle is in a vicinity dV of r at a time t. According to positivism point of view such statement has no sense since quantum mechanics can describe only results of observations. The interference observations [29] of molecules with the size a = 3.2 nm exceeding much its de Broglie wavelength λ = h/mv ≈ 0.004 nm corroborate this point of view. It is impossible to localize the molecule with the size a ≈ 3.2 nm in a volume with a size ≈ 0.1 nm. We must agree with the positivism point of view that the principle of superposition can describe only results of observations and nothing besides. Therefore it is important that we have not the ghost of a chance to observe the quantum interference of a particle larger ≈ 1 μm [19]. Radioactive decay of atom as classical example of causeless phenomena Bohr wrote in [23] that “in his famous article on radiative equilibrium” published in 1917 [30] “Einstein emphasized the fundamental character of the statistical description in a most suggestive way by drawing attention to the


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analogy between the assumptions regarding the occurrence of the spontaneous radiative transitions and the well-known laws governing transformations of radioactive substances”. Further Bohr quotes in [23] an opinion by Einstein about his theory of radiative equilibrium written at the end of the article [30]: “The weakness of the theory lies in the fact that, on the one hand, no closer connection with the wave concepts is obtainable and that, on the other hand, it leaves to chance (Zufall) the time and the direction of the elementary processes”. Thus, radioactive decay of atom may be considered as classical example of causeless phenomenon the time of which is left to chance. By 1928, George Gamow had solved the theory of the alpha decay via quantum tunneling. Following Gamow, as it was made by Einstein in [31], one can describe of uncertain state of radioactive atom with help of a superposition Ψatom = αΨdecay + βΨno

(22.3)

of decayed Ψdecay and not decayed Ψno atom. Who or what can a choice make? According to the positivism point of view of Heisenberg and Bohr the description of the double-slit interference experiment (22.2) and the radioactive decay (3) with help of the Ψ -function is complete. But one can agree with this point of view only if to avoid questions: “How can a particle make its way through two slits at the same time?” and “Who or what can choose result of observation?” Concerning the first question Heisenberg wrote in [32] “A real difficulty in the understanding of the Copenhagen interpretation arises, however, when one asks the famous question: But what happens ‘really’ in an atomic event?” The creators of the Copenhagen interpretation refused to answer on such question. Concerning the second question there was no agreement between they. Bohr wrote in [23] that at the Solvay meeting 1928 “an interesting discussion arose also about how to speak of the appearance of phenomena for which only predictions of statistical character can be made. The question was whether, as to the occurrence of individual effects, we should adopt a terminology proposed by Dirac, that we were concerned with a choice on the part of “nature” or, as suggested by Heisenberg, we should say that we have to do with a choice on the part of the “observer” constructing the measuring instruments and reading their recording. Any such terminology would, however, appear dubious since, on the one hand, it is hardly reasonable to endow nature with volition in the ordinary sense, while, on the other hand, it is certainly not possible for the observer to influence the events which may appear under the conditions he has arranged ”. Collapse of wave function – Von Neumann’s projection postulate The orthodox interpretation studied during last 80 years substitutes the answer on the second question with words on collapse of the wave function or

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a ‘quantum jump’ (according Heisenberg [32]) at observation. The necessity of the collapse postulated first by von Neumann in 1932 [33] reveals the incompleteness of the Copenhagen formalism even according to the positivism point of view. The problems of wave-particle duality and indeterminism were not solved but only taken away outside the theory. The two well known paradoxes, introducing entanglement, have demonstrated clearly this incompleteness. Entanglement of two particles states in the EPR paradox demonstrates incompleteness of quantum – mechanical description of physical reality Einstein et al. demonstrated in [20] paradoxical nature of the superposition collapse using the law of conservation. In the Bohm’s version [34] of the EPR paradox the spin states of two particles are entangled ΨEP R = αΨ↑ (rA )Ψ↓ (rB ) + βΨ↓ (rA )Ψ↑ (rB )

(22.4)

because of the law of angular momentum conservation. Any measurement of spin projection must give opposite results independently of the distance between the particles rA − rB since any other result means violation of this fundamental law. The description of this correlation with help of superposition and its collapse ΨEP R = Ψ↑ (rA )Ψ↓ (rB )

(22.5)

implies that a measurement of the particle A can instantly change a state of the particle B. This means the observation of real non-locality if superposition (22.4) is interpreted as description of a reality. Thus, the EPR paradox has prove unambiguously that quantum-mechanical description of physical reality can be considered complete only if non-local interaction is possible in this reality. Entanglement of atom and cat states by Schrodinger emphasizes incompleteness of causeless phenomena description In order to make obvious the incompleteness of causeless phenomena description with help of superposition (22.3) Schrodinger [21] has entangled the states of radioactive atom and a cat with unambiguous cause – effect connection Ψcat = αΨdecay Gyes F lyes Catdead + βΨno Gno F lno Catalive

(22.6)

If the atom decays Ψdecay then the Geiger counter tube Gyes discharges and through a relay releases a hammer which shatter a small flask of hydrocyanic acid F lyes . The hydrocyanic acid should kill the cat Catdead . In the opposite case Ψno the cat should still live Catalive . It is impossible logically to see that


275

the cat is dead Catdead and alive Catdead at the same time. When anyone will look on the cat he should see dead Ψcat = Ψdecay Gyes F lyes Catdead

(22.7)

Ψcat = Ψno Gno F lno Catalive

(22.8)

or alive cat The question: “Who or what can choose the cat’s fate?” reveals that even causeless phenomenon must have a cause in its complete description. We must choose between nature as proposed by Dirac or the observer as suggested by Heisenberg. In the first case the description with help of superposition (22.6) is obviously incomplete. A natural cause because of which the atom could decay is absent the left of Ψdecay and Ψno in (22.6). The suggestion of Heisenberg results to the conclusion that no reality can exist without an observer.

22.3 Can an experimental result be considered as a challenge to macroscopic realism? Heisenberg upheld just this absence of quantum objective reality [32]: “In classical physics science started from the belief – or should one say from the illusion? – that we could describe the world or at least parts of the world without any reference to ourselves”. How can one make a real equipment, which should operate without ourselves, using the quantum description, which has no sense without any reference to ourselves? Heisenberg stated in [32] that “there is no way of describing what happens between two consecutive observations” and “that the concept of the probability function does not allow a description of what happens between two observations”. According to this point of view quantum mechanics can not describe the process of quantum computation which should be between observations. Two different “Fathers” of quantum computing Thus, according to the point of view not only opponents, Einstein and Schrodinger, but also the creator of the Copenhagen formalism we have no description of the quantum computation process. Then why could this idea become so popular? The numerous publications about quantum computer result from the ideas of David Deutsch and Richard Feynman [35]. But it is important to note that Deutsch and Feynman have pointed different ways towards quantum computer. Deutsch invented the idea of the quantum computer in the 1970s as a way to experimentally test the “Many Universes Theory” of quantum physics – the idea that when a particle changes, it changes into all possible forms, across multiple universes [36]. This theory is one of the realistic interpretations [37] of quantum mechanics which allows to interpreted most

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paradoxical quantum phenomena as manifestation of real processes. But this processes should occur across multiple universes [38]. According to Deutsch, “quantum superposition is, in Many Universes terms, when an object is doing different things in different universes” [36]. The Many Universes interpretation allows to understand why quantum computer may excel the classical one. It can do “a number of computations simultaneously in different universes” [36]. But the idea of many Universes seems mad for most physicists. Therefore most authors follow to Richard Feynman who based the idea of quantum computing on the Copenhagen interpretation. They, as well as Feynman, have an illusion, in spite of opinion of the creators, that the probability function allows a description of what happens between observations. Moreover most modern physicists are sure that quantum mechanics is an universal theory of reality from elementary particles to superconductivity. What is the essence of Bell’s inequality violation in? Einstein foresaw possibility of such mass illusion. He wrote to Schrodinger in 1928 [39]: “The soothing philosophy-or religion?-of Heisenberg-Bohr is so cleverly concocted that it offers the believers a soft resting pillow from which they are not easily chased away”. Many modern authors are sure that the experimental evidence [40] of violation of the Bell’s inequality proves only that Einstein was not right, quantum mechanics is complete theory and we can continue to slip on the soft resting pillow proposed by Heisenberg and Bohr. But some experts understand that the experiments [40] rather cast doubt on very existence of physical reality. The violation of the Bell’s inequalities is sole experimental evidence of EPR correlation (entanglement) observation. In order to quantum computer could be a real equipment the entanglement must exist, but not only to be observed. But the entanglement, because of its very nature, contradict to realism, at the least local one and of single Universe. Doubtfulness of numerous publication about superposition and entangled states of superconductor structures The absence of comprehension of these internal conflicts of the idea of quantum computer results to illusion concerning possibility to make quantum bit. Many authors are sure that it is possible not only to make qubits [9, 15, 16] but even to entangle their states [8]. Modern physicists have already got accustomed to the principle of superposition in the course of eighty years history of quantum mechanics in its Copenhagen interpretation. Therefore the contradiction of the assumption on superposition of macroscopically distinct quantum states with macroscopic realism [14] can not trouble most of they. Many authors interpret thoughtlessly some experimental results obtained at measurements of the superconducting loop interrupted by Josephson junctions as evidence of macroscopic quantum tunneling [11–13] and superposition of states [9]. But this interpretation contradicts not only to macroscopic realism but also to fundamental law of angular momentum conservation.


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Superposition of quantum states with macroscopically different angular momentum is quite impossible according to the universally recognized quantum formalism Superposition and quantum tunneling are assumed between two permitted states n and n + 1 with equal energy but macroscopically different angular momentum. The angular momentum Mp = (2me /e)Ip S is connected with the persistent current circulating in the loop clockwise in the n permitted state and anticlockwise in the n + 1 one [16]. At the values Ip ≈ 5 10−7 A and loop area S ≈ 10−12 m2 of a typical “flux qubit” [9] the angular momentum equals approximately Mp,n ≈ 0.5 105 and Mp,n+1 ≈ −0.5 105 in the n and n + 1 state. At any transition between this states the angular momentum should change on the macroscopic value Mp,n −Mp,n+1 ≈ 105 . In spite of the obvious contradiction to the law of angular momentum conservation authors of many publications assume that this transition can be causeless, i.e. takes place through superposition of states or quantum tunneling. Such assumption can not be correct according to the universally recognized quantum formalism. Possible assumption about an EPR pair of macroscopic systems It may be that the authors of publications about “flux qubit” assume that superposition and quantum tunneling is possible thanks to a firm coupling with a large solid matrix that absorbs the change in the angular momentum, as it was made in [41]. Such fantastic assumption means that states of superconducting condensate are entangled (like in the relation (22.4)) with a large solid matrix, i.e. the loop, substrate and so forth, of uncertainly large mass. It is impossible to take seriously such fantasy about macroscopic EPR pair.

Acknowledgement This work has been supported by a grant “Possible applications of new mesoscopic quantum effects for making of element basis of quantum computer, nanoelectronics and micro-system technic” of the Fundamental Research Program of ITCS department of RAS and the Russian Foundation of Basic Research grant 08-02-99042-r-ofi.

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23 CREATION AND CONTROL OF ORDERED NANOSTRUCTURES IN SPIN-GLASS MEDIA A.S. Gevorkyan1, A.A. Gevorkyan2, and K.B. Oganesyan3 1 2 3

Institute for Informatics and Automation Problems, NAS of Armenia 0014, 1 P. Sevak St., Yerevan, Armenia; g [email protected] Yerevan State University, Armenia 0027, 1 A. Manoogian St., Yerevan, Armenia; [email protected] Yerevan Physics Institute, Yerevan 0036, 2 Alikhanian Brothers St., Yerevan, Armenia; [email protected]

Abstract. The dielectric medium is treated as a system of roughly polarized molecules which are distributed in 3D space randomly. Using ergodic hypothesis the initial 3D spin-glass problem on the space-time scale of external standing electromagnetic field is conditionally reduced to the two 1D problems. The first 1D problem describes disordered N -particle quantum system with taking into account relaxation in media which is being described by Langevin-Schrodinger type equation while the second one describes a statistical properties of the ensemble of 1D steric spin-chains. It is shown that in N -particles quantum systems at some critical value of amplitude of external field the phase transition of first order occurs in result of which in the system a partial ordering arises. The last in turn leads to formation of a superlattice of permittivity with characteristic space-time sizes of external field. The more interesting case when in the equation of Clausius-Mossotti at some critical polarization of media arises catastrophe. In this case values of permittivities of the neighboring layers can be essentially different that is sufficient for using media as a new type’s radiator for generation of anomalous intensive transition radiation.

Key words: spin-glass, external standing field, ergodic hypothesis, ClausiusMossotti relation, polarizability, collective orientational effects, phase transition, catastrophe in Clausius-Mossotti equation, permittivity.

23.1 Introduction The formation and governing of periodically modulated refractive index in media is a most important problem of solid state physics and material science. First of all it is related to the possibility of developing compact UV or X-ray J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 23,

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Free-Electron Lasers (FEL) based on emission of transition radiation (TR) [1]. Currently the following two problems are discussed intensely: 1. A gas-plasma medium with periodically varied ionization density [2–10], 2. A special periodical solid-state superlattice-like (SSL) structures composed of layers with different refraction indexes [11–19]. Recall that TR is generated at passing of electron beam via neighboring layers of media with different permittivities. The radiation power in this case R 2 is proportional to [R 1 (w) − 2 (w)] , where ω is a frequency of radiation and R 1,2 (w) = Re[1,2 (w)] designates the real parts of permittivities of corresponding layers [20]. This idea recently was implemented in TR generation experiments [21]. In particular, experimentally is shown, that the 20 Mev beam of electrons at the passage through the amorphous silicon dioxide SiO2 in the presence of the standing electromagnetic wave with the frequency of 10 GHz, generates an anomalously intensive short-wave radiation.The preliminary studies was shown that high intensity of radiation is a result of multiple passage of the electron beam through interfaces between regions with appreciably different permittivities. Theoretically the formation of super-lattice in disordered medium is explained by medium’s polarization due to the orientational relaxation of elastic molecular dipoles in the direction of external electromagnetic field’s propagation [22]. The main objective of our investigation to show a possibility of creation and control of 1D superlattice of permittivity 1 (w, g); 2 (w, g) in a dielectric media of type SiO2 by means of weak external field, where g designates the parameters of external field (the controlling parameters).

23.2 Formulation of the problem In isotropic amorphus media as well in crystals of cubic symmetry the static dielectric constant is well described by Clausius-Mossotti equation [23–25]: s − 1 4π 0 0 N α , = s + 2 3 m m m

(23.1)

0 where Nm is the concentration of particles (electrons, atoms, ions, molecules) with given m types of polarizability and α0m correspondingly are polarizability coefficients. In the external field the homogeneity and isotropy of the media in principle is being lost and obviously the formula (23.1) may be applicable after generalization. In this work we will investigates the solid states dielectrics of type of amorphous silicon dioxide a − SiO2 . According to numerical ab initio simulations [26], the structure of this compound may be well described by the model of 3D disordered spin system. In particular 3D spin system we can represent as 3D

23 Creation and control of ordered nanostructures in spin-glass media

283

lattice with the lattice’s constant d0 (T ) = {m0 /ρ0 (T )}1/3 , where m0 is the molecule mass, ρ0 is the density and T is the temperature. We will assume, that in each knot of a cell of a regular lattice there is only one spin(roughly polarized molecule) which has an random orientation. We will suppose that the media under the influence of external standing electromagnetic field the electrical part of which has a kind: E(x; E0 , Ω, λs , ϕ0 ) = E(x; g) = 2E0 sin(ϕ0 ) cos(kx),

(23.2)

where ϕ0 = Ωt0 and t0 are respectively the initial phase and time, Ω is a wave frequency, k = 2π/λs and λs is the wavelength, g describes the parameters of external standing electromagnetic field g ≡ (E0 , Ω, λs , ϕ0 ). Here arises a natural question, how is being changed the permittivity on the scale of space-time period (λs ∼ 10−4 cm; Ω −1 ∼ 10−9 sec) of external field? Note that such consideration of a problem is quite justified because the processes which form the transition (or Cerenkov) radiation in an environment are much faster (the characteristic time is lesser than 10−15 sec). Taking into account the external field, one can write the polarization of media at the arbitrary point in the following kind: p (l − r) = nm αm (l − r)Eloc (l − r), (23.3) P(r) = m

l

where l ≡ l(lx , ly , lz ) is 3D lattice vector, p is respectively the dipole moment of molecule. Note, that the number of the sorts of given polarization in an elementary cell is nm ∼ (d0 (T ))−3 , αm is the coefficient of the polarizability of corresponding sort with accounting of external field and Eloc is a local field, i.e. the effective field that induces the polarization at the site of an individual molecule. Recall that under the influence of external field the polarizabilities of various types arise in media. However a simple analysis shows, that the value of polarizability coefficient which is connected with an orientational effects essentially exceed the others. Note, that the coefficient of elastic orientational polarizability in amorphous media αdip (l − r) is a random function from a cell location. This fact is connected with the random orientation of local field strengths Eloc (l − r) with respect to the external field E(x; g). Recall that all terms in the right side of Eq. (23.1) are basically known and well studied in literature (see, e.g., [23–25]) except the terms which are connected with the orientational effects. The orientational effects have a collective nature and are being characterized by average value of random sum l αdip (l − r) (sum of random coefficients of orientational) polarizability. Multiplying both sides of equation (23.3) on the field (23.2), we can find: − δU (r, g) = nm αm (l − r)Eloc (l − r) E(x; g), (23.4) m

l

where −δU (r, g) = P(r, g)E(x, g) describes the potential energy of amorphous media in the external field.

284


Taking into account (23.4), one can obtain the following expression for the part of potential energy of 3D spin system which is related with the orientational effects of spins in an external field: −δUdip (r, g) = αdip (l − r)Eloc (l − r)E(x; g). (23.5) l

Let us separate a layer with volume V = Lx × Ly × Lz in the infinite crystal lattice, where Lx ∼ λs d0 (T ) and (Ly , Lz ) → (∞, ∞). Obviously that the mentioned volume is filled with the infinite number of 1D steric spin-chains (SSC) with length Lx . Now an important problem is to calculate the mean value of the interaction potential between the spin layer and the external field. Formally the following expression may be written for that: − δUV (r, g) = − δULx (l⊥ |r, g), l⊥ ≡ l⊥ (lx , ly ), l⊥

−δULx (l⊥ |r, g) =

αdip (l − r)Eloc (l − r)E(x; g),

(23.6)

lx

where −δULx (l⊥ |r, g) is the interaction potential between the 1D SSC and external field. At first we will obtain the mean value of the potential −δUV (r, g) on (y, z) plane: 1 δULx (x; g) = lim δUV (x, g) dS⊥ , S⊥ −→∞ S⊥ where S⊥ = Ly × Lz and ... is the averaging over all possible stable 1D SSC configurations. Assuming that the distribution of spins in (y, z) plane is random but isotropic it is simple to prove that in the limit of S⊥ → ∞ in the spin-system occurs the full self-averaging. This means that we can use Birgoff ergodic hypothesis [27] and to write the expression δULx (x, g) in the form: δULx (x, g) =

0

−∞

δULx (ε|x, g)P (ε; Lx )dε

0 −∞

P (ε; Lx )dε,

(23.7)

where −δULx (ε|x, g) denotes the interaction potential energy between 1D SSC with energy of ε and an external field (23.2), P (ε; Lx ) is an energy distribution function in ensemble of 1D SSCs. The definition of function P (ε; Lx ) will be given later in Sect. 23.4. Note that in Eq. (23.7) only the negative values of ε were taken into account, because only at these values 1D SSCs are stable. Now we can calculate the mean value of interaction potential in a unit of volume between standing wave and media: −δUV (r; g)V = −δULx (x, g)(x,) = αdip E 2 (x; g)x ,


δULx (x, g)(x,) =

1 Lx

+Lx /2 −Lx /2

δULx (ε|x, g)ε d x,

285

(23.8)

where the bracket ...x means the integration over x on a scale Lx , parameter αch will be named a collective polarization coefficient of 1D SSCs:

αch = −δULx (x, g)(x,) E 2 (x; g) x . (23.9) Note that αch is a complex value and describes the averaged 1D SSC polarizability with due regard for the lattice relaxation. It is simple to check when E0 → 0 then αch → 0. Now we can write expression for the sum of polarizabilities (see Eq. (23.1)) in which the orientational effects of spin-chains in the external field are taken into account:

δULx (x, g) (x,) 0 0 0 0

, (23.10) Nm αm = Nm αm + Nch αch = Nm αm − 3 2 d0 Nx E (x; g) x m m m −1 is the concentration of 1D SSCs. where Nch = d−3 0 Nx Now using (23.10) may be generalized the Clausius-Mossotti equation on a scale of space-time period of the standing electromagnetic wave with accounting the orientational effects: δULx (x, g(x,) 4π 0 0 1 + 2Λ(g) ,(23.11) st (g) = , Λ(g) N m αm − 3 1 − Λ(g) 3 m d0 Nx E2 (x; g)x

where st is the stationary dielectric constant. Now we may study the peculiarities of permittivity function. In the theory of dielectric relaxation, the frequency-dependent dielectric constant (permittivity) (ω) with the model of Williams-Watts dielectric relaxation [28] is being represented by the relation [29]: ∞ (ω) − ∞ = (σ, ω), (σ, ω) = − e−iωt d Fσ (t)/d t dt, (23.12) s − ∞ 0 where ∞ = (ω → ∞) is the high-frequency limit of the dielectric constant and s = (ω → 0), the static dielectric constant that can be defined from the Clausius-Mossotti equation (23.1). The permittivity of broad class of materials including polymeric systems and glasses may be interpreted in terms of the Williams-Watts [28] polarization decay function: Fσ (t) = exp[−(t/τ )σ ],

0 < σ < 1,

(23.13)

where the parameter σ and the relaxation time of media τ are depend on the type of material and the external conditions such as temperature T and pressure.

286


The relation (23.12) can be generalized for the case of external field as we are interested in the time scale much smaller than the time interval, during which the external standing electrical wave may be considered as stationary. After substitution (ω) → (ω, g) and s → st (g) in the Eq. (23.12) we can transform and write that in the form: ∞ σ (ω, g) − ∞ = (σ, ω), (σ, ω) = −σ e−iλs−s sσ−1 ds, (23.14) st (g) − ∞ 0 where λ = ωτ and s = t/τ . For example, in the classical Debye theory of dielectric relaxation σ = 1 [23], and the integral in the right side of Eq. (23.12) has the form (1 + iωτ )−1 .

23.3 The average interaction potential between 1D SSC and the external field Assuming, that the external field is weak i.e. |E(lx − x)| |Eloc (lx − x)| ∼ = |Eint (lx − x)|, we can expand the dipole angular momentum in a Taylor series: p (x) p0 (x) + δp (x), δp(x) ∼ E(x; g),

(23.15)

where |δp(x)| |p0 (x)|, as well E0int (x − lx ) and p0 (x − lx ) are respectively the field strength and dipole angular momentum of the molecule, located in the lx -th cell in the absence of the external field. When the coordinate x is outside of lx -th cell, the field vanishes. Their values inside the cell are constant. Based on the above discussion with due regard for Eq. (23.6) for interaction potential −δULx (ε|x, g)) one can write the following relation: −δULx (ε|x, g)) =

Lx

δp (lx − x)E(x; g).

(23.16)

lx =0

Now taking into account the above considerations one can describe the quantum motion of 1D SSC in the external field with accounting of the random environment by the stochastic Schrödinger equation: λ δULx (ε|x, g) = λ ε + Ψ −1 (dt )2 Ψ,

d2t = d2 /dt2 ,

(23.17)

where the following designations are made: t = x/d0 ,

λ = 2μ/(2 d20 ),

μ = m0 /Nx1/(Nx −1) ,

μ is a spin-chain effective mass, Nx is the number of particles ( molecules) in the chain, t denote the natural parameter of evolution along the spin-chain.


287

In Eq. (23.17) the interaction potential −δULx (ε|x, g) is a random complex function. Its detailed description is given below. Substituting:

t Ξ(t )dt , Ψ (t) = exp (23.18) 0

into Eq. (23.17) we obtain the following of Langevin type nonlinear complex stochastic differential equation (SDE) [30]: Ξt + Ξ 2 + λ ε − V + λf (t) = 0,

Ξt = d Ξ/dt,

(23.19)

where Ξ(t) = θ(t) + iϑ(t), where ϑ(t) ≥ 0, in addition: Lx

δp(lx − x)E(x; g) = V + f (t).

(23.20)

lx =0

In formulas (23.19) and (23.20) we denoted by V the mean value of the sum, and by f (t) its complex random part. Based on an analysis of different polarization mechanisms in the spin-glass media, we concludes that under the influence of external field (23.2) with frequency Ω ∼ 109 Hz the main contribution in the polarizability makes the orientational effects of elastic dipoles. Note that in this case the thermal polarization of dipoles is not essential due to the large relaxation time τ ∼ 10−4 ÷ 10−5 sec [23], [31]. The coefficient of elastic dipole polarization in low external fields is determined by [32]: 2 0 αdip (lx − x) = p0 /Eint sin [β(lx − x)], and correspondingly: δp (lx − x) = αdip (lx − x)E (x; g),

(23.21)

where β(lx − x) is an angle between the external E (x; g) and the internal 0 Eint (lx − x) fields. Following the heuristic reasonings of Debye [23,28,29] one can write down the expression for the elastic dipole polarization with accounting of dipolerelaxation process in the glass: α ¯ dip (lx − x) =

αdip (lx − x) ∼ p0 sin2 [β(lx − x)] , = 0 (1 − iΩτ ) Eint (1 − iΩτ )

(23.22)

where τ is the spin relaxation time in the glass. It is very small in the above media τ ∼ 10−11 ÷ 10−12 ([23]).

288


After substitution of Eq. (23.22) in Eq. (23.20) and simple calculations we obtain: 2 sin(2kLx ) 1 + iΩτ Nx p0 E 0 1 − , E 0 (ϕ0 ) = 2E0 sin ϕ0 . V =− 0 1 + (Ωτ )2 4d03 Eint 2kLx (23.23) Now we can investigate the properties of random function f (t). From relations (23.20) and (23.22) it is easy to find the random strength: 2 Lx 1 + iΩτ p0 E 0 f (t) = − · cos 2β(lt − t), 0 ξ(t), ξ(t) = 1 + cos(2kt t) 1 + (Ωτ )2 4d30 Eint lx =0

(23.24) where the following√ designations are mad: β(lt − t) = β(lx − x), √ kt = −1 2μ k, lt = l/ 2μ. If the phase β is homogeneously distributed in the interval [0, π] for the random function ξ, we can write the following relations: ξ(t) = 0, ξ(t)ξ(t ) = 4δ(t − t ), (23.25) For real and imaginary components of random function f (t) the following autocorrelation functions [33] may be written:

f r (t)f r (t ) = K ξ(t)ξ(t ) = 2Dr δ(t − t ), (23.26)

(23.27) f i (t)f i (t ) = K(Ωτ )2 ξ(t)ξ(t ) = 2Di δ(t − t ), 2 2 2 Nx p 0 E 0 1 where: K = 12 1+(Ωτ , in addition: f r (t) = Ref (t) and f i (t) = )2 4d 3 E 0 0

int

Imf (t): For further investigation the complex equation (23.20) it is useful to represent as a system of two real equations: (23.28) θ˙ + θ 2 − ϑ2 + λ ε − V r + f r (t) = 0, (23.29) ϑ˙ + 2ϑθ + λ −V i + f i (t) = 0, where θ˙ = dt θ, ϑ˙ = dt ϑ, V r = Re V and V i = Im V . Now the main problem is to find the evolution equation for the conditional probability: (23.30) Q(ε|θ, ϑ; t) = δ θ(t) − θ(t0 ) δ ϑ(t) − ϑ(t0 ) , which describes the probability that the trajectory θ ≡ θ(t), ϑ ≡ ϑ(t) leaving at the initial moment of natural parameter t0 from the point (θ0 , ϑ0 ), to come an arbitrary moment t in the vicinity of point (θ, ϑ).


289

Using the system of SDE (23.29) and (23.29) the Fokker-Plank equation is easily found [34] (see also [30]): ∂ ∂Q ∂ 2 2 ∂2 ∂2 θ −ϑ + λ ε − V r ) + 2θϑ − λV i ) Q, = Dr 2 + Di 2 + ∂t ∂θ ∂ϑ ∂θ ∂ϑ (23.31) where Q ≡ Q(ε|θ, ϑ; t). Note, that the solution of equation (23.31) must satisfy the initial condition: Q(ε|θ, ϑ; t) = δ(θ − θ0 )δ(ϑ − ϑ0 ), (23.32) t=t0

where initial phases θ0 and ϑ0 are equal to zero. However more interesting case for us is a stationary limit of solution (23.31) which is being received at the values t t = O(1), that is equivalent to condition t → ∞. In this case (23.31) to the following stationary kind is transformed:

2 ∂ 2 ∂ ∂2 i ∂ 2 r i θ −ϑ +λ ε−V ) + 2θϑ − λV ) Qs = 0, D +D + ∂θ2 ∂ϑ2 ∂θ ∂ϑ (23.33) r

where Qs ≡ Qs (ε|θ, ϑ) ≡ limt→∞ Q(θ, ϑ, t). The Eq. (23.33) is an elliptic differential equation which is being solved with the boundary conditions: ∂Qs Qs = = 0, ∂n S S

1/2

|n| = (θ2 + ϑ2 )

,

(23.34)

where n is the normal to curve S. Now we can calculate the mean value of interaction potential between 1D SSC with energy ε and the external field with accounting its relaxation into 3D lattice. In the stationary limit the interaction potential has a form: δU (ε|x, g)

x→∞

→ λε + θ2 − ϑ2 − i2θϑ,

(23.35)

(see Eqs. (23.17) and (23.19)). After averaging (23.35) over the stationary distribution Qs (ε|θ, ϑ) we find the mean value of interaction potential: 1 δULx (ε|x, g)x = λR where R =

+∞ +∞ −∞

0

+∞ +∞ Qs (ε|θ, ϑ) (λε + θ2 − ϑ2 ) − i2θϑ dθ dϑ, (23.36) −∞ 0

Qs (ε|θ, ϑ) dθdϑ denotes the normalization constant.

290


If to propose that the Dr >>D i (that is natural), then the quantum probability will be localized near the axis θ and is satisfied by equation: 2 2 ∂ r ∂ r D + θ +λ ε−V ) (23.37) + 2θ Qs = 0. ∂θ2 ∂θ The solution of equation (23.37) on the big distances has the following behavior: Qs (ε, g|θ) ∼ θ−3 . (23.38) Taking into account (23.38) it is easy to show that the average interaction potential (23.36) is diverging logarithmic δULx (ε|x, g)x ∼ ln θ. The nature of this type of divergence is well-known from field theory and is being named ultraviolet divergence [35]. Recall that the divergence is connected with the problem of infinite energy of vacuum, which includes the considered model of stochasticity. As will be shown below, this problem can be solved by the way of dimensional renormalization. Let’s consider the extended equation: ∂ ∂2 D r 2 + θ2 + λ ε − V r ) + (2 + η)θ Q = 0, ∂θ ∂θ

(23.39)

where η > 0 auxiliary small parameter. As shows a simple analysis, at the big distances the solution of equation (23.39) has a following behavior ∼ θ−3+η . The last fact mean that the integral: δULx (ε, η|x, g)x < ∞ now is converged. However it is obvious, that: limη→0 → ∞. Now our aim to make dimensional renormalization of integral (23.39). After differentiation of equation (23.39) by η we find: 2 2 ∂ r ∂ r + 2θ Qη + θQ = 0, D + θ +λ ε−V ) ∂θ2 ∂θ

(23.40)

where Qη (ε, η|g, θ) = ∂Q(ε, η|g, θ)/∂η. In the equation (23.40) the form of function Q(ε, η|g, θ) isn’t fixed. Representing the function in a kind: Q(ε, η|g, θ) = eη G(ε|g, θ),

(23.41)

in the limit of η → 0 from (23.41) we can find the following equation: 2 2 ∂ r ∂ r D + θ +λ ε−V ) + 3θ G = 0. (23.42) ∂θ2 ∂θ The solution of Eq. (23.42) on a big distances has a behavior G ∼ θ−4 which +∞ ∞ means that the integral −∞ 0 ...G(ε|g, θ)dθ < ∞ strictly is converged (compare it with (23.36)).


291

23.4 Energy distribution function of 1D SSCs ensemble We will consider non-perturbed 1D Heisenberg spin-glass Hamiltonian [36]: H(Nx ; g) = −

Nx

Jij Si Sj ,

(23.43)

ij

where spins Si ≡ p0i /p 0 and Sj ≡ p0j /p 0 are the vectors of the unit length. In addition Jij describes the interaction’s random constant (may be negative and positive) between the nearest-neighboring nodes. The disorder in choice of coupling constant J(i−1)i usually is introduced by means of GaussianEdwards-Anderson distribution model [37, 38]: W (J(i−1)i ) =

1 2 2πΔJ(i−1)i

2 J(i−1)i − J0 exp − , 2 2ΔJ(i−1)i

(23.44)

2 2 2 = J(i−1)i − J(i−1)i av . J0 = J(i−1)i av , ΔJ(i−1)i av In this model J0 and ΔJ(i−1)i are independent on the distance and scaled with spin number Nx as:

J(i−1)i av = J0 ∝ Nx−1 ,

ΔJ(i−1)i ∝ Nx−1/2 ,

(23.45)

to ensure a sensible thermodynamic limit. In Eqs. (23.45) and (23.45) the becket ... av describes the averaging procedure. Our purpose now is construction of the energy distribution function Z(ε; g) of ensemble 1D SSC. However, the difficulty here is that the time scale on which we study the statistical peculiarities of system, is very short ( ∼ 10−10 sec = 0, 1 nsec), while the characteristic thermal relaxation time in amorphous media [23] is of the order ΩT−1 ∼ 10−4 ÷ 10−5 sec, where ΩT is the frequency of thermal fluctuations. This means that in proposed problem the temperature as a thermodynamical parameter become meaningless. However, the spins system has a specific feature. The point is that the equilibrium state in gas is characterized by one temperature, whereas the spins system can be at various equilibrium states which by various negative energies are characterized. These energies coincide with local minima of the non-perturbed Hamiltonian [36, 39]. The Hamiltonian (23.45) in the spherical coordinate system may be rewritten as: H0 =

Nx {ij}=1, i=j

Jij cos ψi cos ψj cos(φi − φj ) + sin ψi sin ψj . (23.46)

292


For determination of local minimums one has to solve the following trigonometric equations obtained from conditions of definition of stationary points: x ∂H0 = Jij − sin ψi cos ψj cos(φi − φj ) + cos ψi sin ψj = 0, ∂ψi

N

j=1

Nx ∂H0 =− Jij cos ψi cos ψj sin(φi − φj ) = 0, ∂φi j=1

(23.47)

where Θi = (ψi , φi ) are angles of i-th spin (ψi is the polar and φi azimuthal angles), Θ = (Θ1 , Θ2 ....ΘNx ) respectively describes the angular part of spins configuration. Now we will assume, that the Hamiltonian of length Lx has n local minima εj (d0 (T )), to each of which corresponds the Mj spin’s configurations {Θ0i }, where i = 0, 1...Mj . Accordingly, the weight of every equilibrium state may be defined as: Pν (εν ; Lx ) = where Mf =

N j=1

Mν , Mf

N

Pν (εν ; Lx ) = 1,

(23.48)

ν=1

Mj is the number of all stabile 1D SSC configurations.

23.5 The critical properties of Nx particles system Let’s introduce Helmholtz free energy for local equilibrium state j: F εj , g = εj ln R εj , g , (23.49) where R εj , g describes the partition function of the quantum Nx particles system in equilibrium state (number of states with the energy in the range from −∞ to (εj − V R ), per unit interval ). Recall that all thermodynamic parameters of the statistical system may be obtained by means of derivative of free energy by an external field’s parameters g. Let’s consider the derivation: ∂F εj , g) εj ∂R r = , (23.50) q(εj , V , g) = r ∂V R ∂V r where value of V r has a sense of an average energy of external field which falls on one 1D SSC. Analysis of expression (23.50) in the general case is difficult problem, however in more interesting case when D r >> Di the partition function is being calculated exactly (see [27]): ∞ 3 R−1 (εj , V r , g) = exp − − λ0j χ −1/2 d , (23.51) 12 0


293

where χ = 1 − V r /εj and λ0j = λεj /(Dr )2/3 >> 1. Using (23.51) we can calculates the derivative of free energy: q(εj , V , g) = −λ0j R r

0

∞

3 exp − − λ0j χ 1/2 d . 12

(23.52)

Let’s calculate the derivative of free energy in two closely located points V−r = εj − δV r and V+r = εj + δV r . If to assume, what takes place the relations χ ∼ 1, then the integrals in Eq. (23.52) it is possible to calculate asymptotically by Laplas method [40]. In particular for V−r , conducting a simple calculation we can find: q− ≈

1 + O(λ−2 0j ), λ0j χ−

χ− =

δV r > 0. εj

(23.53)

In a similar way we can calculate derivative of free energy at value V+r : +

q ≈ 2π

4 3/2 λ0j + O(1), exp − −λ0j χ+ −χ+ 3

χ+ = −

δV r < 0. (23.54) εj

Comparing the values q(εj , V r , g) in two close points we are being convinced that at V r = εj there is an infinite jump (if to assume, that λ0j → ∞). In other words in the quantum system occurs the first-order phase transition.

23.6 Permittivity of neighboring layers As were being convinced above, under the influence of an external electromagnetic field in the spin-glass occur orientational effects that can lead to phase-transitions of first order and arising of order (arising of polarization). At the certain values of an external field the polarization can become critical and lead to catastrophe of Clausius-Mossotti equation (23.11) i.e. when ReΛ(g0 ) → 1, where g0 designates critical external field. This means that on the scale of space-time periods of external field arise a two regions with extremely different permittivities. Note, that the order of the layer thickness of media is defined by distance between two critical points where occur the first-order phase transitions. It is easy to see that the equation (23.11) has no singularities and is bounded by the imaginary part of the average potential (23.36). The important characteristic of the radiator for the transition radiation, it is a difference of dielectric constants of the neighboring layers. Taking into account the equation (23.11) we get: δst (g) = st (g) − st (0) = (1 − st (0)) +

3Λ(g) . 1 − Λ(g)

(23.55)

294


Correspondingly the permittivity of layer in the external field will be: st (g, ω) = ∞ + st (g) − ∞ (σ, ω). (23.56) Taking into account the Debye relaxation model in the X-rays region from Eq. (23.14) we find the following expression for permittivity of single layers: ωp ω p 2 + , (23.57) st (g, ω) 1 − st (g) − 1 i ω ω where ωp is the plasma frequency of media. Finally we can write the expression for difference of permittivities of the neighboring layers: ωp ωp 2 + δ(g, ω) −δst (g) i , (23.58) ω ω where δ(g, ω) = st (g, ω) − st (ω).

23.7 Concluding remarks In the present article a new microscopic approach has been developed for studying the properties of stationary dielectric constant and permittivity function in dielectric media under the influence of external standing electromagnetic field. The approach consists of the following two general steps: 1. Generalization of the Clausius-Mossotti equation for dielectric constant under the influence of external standing electromagnetic wave; 2. Generalization of the equation for permittivity function with taking into account the previous results. Mathematically the problem is solved as follows. The dielectric media in the external electromagnetic field is modeled as a 3Dspin-glass system. Taking into account the fact that on the infinite plane (x, y) the distribution of vector-spins are isotropic we use the Birgoff ergodic hypothesis (see Eqs. (23.6)–(23.7)) and reduce the initial 3D spin-glass problem on a two conditionally separated 1D problems. It allows to investigate each of the problem separately. In the work we have constructed all expressions which are necessary to allow for the contribution of orientational effects at calculation of stationary and frequency-depending dielectric constants. As was shown, in result of catastrophe of C-M equation, in the region of short wave-length, the difference between permittivities of neighboring layers may be essentially big. Last circumstance allows us in homogenous and the isotropic dielectrics of spin-glasses type, artificially to create a super-lattice from different permittivitys the parameters of which it is possible to control by external fields and use this structure for generation of extremely intensive transition radiation.


295

Acknowledgments This work was partially supported by ISTC grant N-1602.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

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Part IV

Sensors

24 BIOSCOPE: NEW SENSOR FOR REMOTE EVALUATION OF THE PHYSIOLOGICAL STATE OF BIOLOGICAL SYSTEMS R. Sh. Sargsyan1, A. S. Gevorkyan2, G. G. Karamyan1, V. T. Vardanyan1, A. M. Manukyan1 , and A. H. Nikogosyan1 1 2

Orbeli Institute of Physiology, NAS of Armenia 0028, 22 Orbeli Bros. St., Yerevan, Armenia; [email protected] Institute for Informatics and Automation Problems, NAS of Armenia, 0014, P. Sevak Str., 1, Yerevan, Armenia; g [email protected]

Abstract. A new device (BIOSCOPE) for noninvasive assessment of physiological statebiological systems of biological objects is created. The principle of operation of device (sensor) is based on the estimation of the intensity of laser light scattered from two interfaces. Note that from the first interface (lower bound of glass-vacuum) coherent light is reflected, while from the second interface which includes the upper bound of glass and an opaque hackly material covering it, a light is reflected diffusely. In the work are presented the results of various experiments with different kinds of biosystems (plants, animals, humans) which have been conducted on the distance on which is absent the electromagnetic or other interactions. The device does not respond to the inanimate objects having room temperature. The physiological significance of the device’s signals is discussed in detail. The device can be used as a new diagnosis tool in medicine and biology.

Key words: sensor, laser, diffuse refracted light, remote study, physiological state, biological systems, refraction indexes.

24.1 Introduction The problem of the usage of sensors for the control over the various physical processes in small spatial volumes under a variation of their characteristic parameters on the level of noise is one of the central problems of nanotechnologies. It is obvious that for the solution of assigned task, sensors should measure quantum changes in objects and correspondingly its should be arranged on the basis of the quantum phenomena (see for example [1]). In the end of 70th of last century at Princeton University the large-scale investigations have been performed the objective of which was consisted in an attempt experimental (device-based) revealing and the verification of a existence J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 24,

299

300

R. Sh. Sargsyan et al.

of special remote interactions between biological and physical systems [2]. In the succeeding years the existence of similar influences has been confirmed as well in other scientific centers (see for example [3]). However, despite of the existence of rich variety of actual evidences, a full and an unequivocal recognition of remote interactions has not been obtained for the present. On the one hand, this is connected with complexity and frequently uniqueness of the equipment which is used at the registration of such signals, and on the another hand, due to the absence of the comprehensible concept, allowing to coordinate the observable phenomena and modern scientific representations. Therefore today it is extremely urgent to verify the reality of similar interactions and to pass to the study of mechanisms of their formation, as well as to the revealing of their role in processes of organism functioning. In this connection the special attention attracts the hardware system BIOSCOPE developed in [4, 5]. BIOSCOPE does not react to the inanimate objects or more precisely to inanimate objects in which there are not changes of entropy. However at approaching of biological systems to the sensor the readings of device authentically vary.

24.2 The description of experimental device-BIOSCOPE The experimental device which later we will name BIOSCOPE schematically is represented as (see Fig. 24.1). The light emitted by incandescence lamp or laser is being reflected and scattered from the glass plate and opaque material, and gets back to a photo-detector. It should be mentioned that device is fully isolated from environment. In other words the scattering light don’t cross the borders of device. Photo-detector signals are being amplified (up to about 500 times) after the analogue-digital transformation arrive at the

biological object 5

2

1-10 cm

1 3 F

L

4

PC amplifier

power unit

Fig. 24.1. The general sketch of the hardware. 1 – glass plate; 2 – covering material; 3 – partition; 4 – metallic case; 5 – rack for the investigated object, L – ordinary incandescent lamp or laser; F – photo-detector; PC – recording computer.

24 BIOSCOPE: new sensor for remote evaluation

301

computer. The power supply provides stable operation of the lamp or laser. Initial background level of photo-detector signals makes 50 mV. The percentage deviation of BIOSCOPE’s readings from this level is estimated. Calculations have shown that the deviation of amplitude of a registered signal from initial level is already no more than 0.1%, i.e. is statistically reliable at significance value of P < 0.001. The analysis of the registered signals is carried out with software packages OriginPro and LabView.

24.3 Main results obtained by use of bioscope device In the case of non-coherent light source (incandescence lamp) the signal of BIOSCOPE presents almost smooth line Fig. 24.2(A). Figure 24.2(B) shows signals obtained from different objects located in vicinity of device. The distance between all objects and bioscope is 1 cm. Arrows show the moments of placing and removal of the objects on/from the sensor. One can see from this figure that the presence of biological objects near the BIOSCOPE brings to the formation of signals deflected above background line, however nonliving objects having environmental temperature do not influence the value of the registered signal, whereas the heated inanimate objects cause deflection in opposite direction. The magnitude of the effects differs for various biological objects. In the case of human palm, the increase of the reflected light intensity can amount to about 1% to 2% of the absolute value of the control level of the registered signal. After removing of a biological object from the BIOSCOPE’s sensor, the amplitude of the registered signal returns to the control level. When a distance between biological object and sensor is being increased, the time during which the effect occurs is being increased also. At this case the change of reflected light’s intensity is being diminished. A series of control experiments were carried out to verify the fact that measured signals are not attributed by any of the typical physical phenomena that would be expected to cause these results.

a 3min

b

300mv

3 0.3% 5min

2 1 4

5

Fig. 24.2. BIOSCOPE’s signals from different objects in the case of non-coherent light. 1 – apple; 2 – grapefruit; 3 – human palm; 4 – aluminum plate at ambient temperature; 5 – same plate heated to 40◦ C.

302


1. Heating of non-biological objects causes the intensity of the reflected light to decrease slightly (Fig. 24.2B(5)). Thus the observed biological influence registered by the sensor cannot be due to the heat radiation on the sensor. 2. Electromagnetic fields that typically arise around a biological system have extremely low intensities and do not have characteristics which would lead to the observed phenomena. 3. Owing to mechanical isolation of the system, the possibilities of chemical interactions have been experimentally eliminated as potential cause of the observed response of detector to biological systems (Fig. 24.3). The observed effects are reliable and replicable. The results underscore the ability of biological systems to exert distant influence on optical characteristics of the reflecting surface of the BIOSCOPE sensor, even through a metallic plate. The elaborated device demonstrates the ability of living systems to exert distant influence of an unknown nature on the non-biological objects. When the laser (coherent light) was used instead of incandescent lamp (noncoherent light) in the BIOSCOPE, the device sensitivity is greatly increased and signal form is changed. In the absence of studied object near the BIOSCOPE the background signals present irregular oscillations with frequency lower than 0.1 Hz (Fig. 24.4A). In experiments it was observed that the presence of experimenter renders influence on the registered signal of BIOSCOPE and can distort measurements results (Fig. 24.4B,C). It has been observed that approaching of biological objects to the BIOSCOPE’s sensor leads to the formation of characteristic oscillations (up to 10–15 Hz). The amplitude of these oscillations can reach 7–10% from absolute size of an initial signal of the photodetector (Fig. 24.5A,B). After removing of the living system from BIOSCOPE the oscillation frequency decreases and after 3–5 min signals return to the initial low-frequency and irregular form. Interesting phenomenon was found in experiments when after being in close proximity to biological objects for a few minutes, some non-living materials (e.g., paper, wood, glass), which at first did not cause any effect, temporarily acquired the possibility

a 5

human palm

1.5 cm

5 6 2

7

1.0 cm

b 1 0.2% 3 min

3 F

L

4

Fig. 24.3. Distant influence of the human palm on bioscope signals at the mechanical isolation of the sensor from an environment. A – scheme of experiment: labels 1–5, F, L are the same as in Fig. 24.1; 6 – metallic tube closely fitted to the body; 7 – thin metallic plate hermetically built in the tube. B – influence of the palm on the reflected light intensity. The brackets represent the time interval of the influence.


a

303

%0 –20 0

10

20

30

40

50 60 70 minutes

b

80

90

100 110 120

c

–mV 58

–mV 58

54

54

50

50

46

46

42

42

38

38 0

1

2

3 4 5 time in hours

6

7

0

1

2

3 4 5 time in hours

6

7

Fig. 24.4. Effect of experimenter presence on the BIOSCOPE signals. A – control signal in empty room (below 0.1 oscillation per minute), B – in presence of person in the room (about 0.2 oscillation per minute, 2–3 m from BIOSCOPE), C – after leaving of the person the experimental room. Arrows show the moments of coming (B) and leaving (C) the person in the lab.

a

b

1

2

3

1

2

3

1s

5%

Fig. 24.5. Frequency signals of BIOSCOPE device (with laser source) for various biological systems. A – human palm; B – apple. 1 – background signal before palm or apple approaching to BIOSCOPE sensor. 2 – palm/apple near the BIOSCOPE sensor; 3 – the response after removal of investigated object from device sensor of BIOSCOPE. Initial level of photo-detector signals is 50 mV. The distance between biological system and the sensor is 1 cm. The wide of square on the figure is 1 second and height corresponds to 5% deviation.

304


a

b 1 1

2 2

5 min

0.3 %

1s

5%

Fig. 24.6. Effect of temporary “biologization” of nonliving objects. A, B – Use of not-coherent (A) and coherent light (B) as a radiation sources inside the BIOSCOPE. 1 – control signal from piece of paper located near the BIOSCOPE; 2 – signal from the same piece of paper after it was kept 2 min between human palms; A – square brackets indicate the time of registration; B – line specifies the moment of approaching a piece of paper to BIOSCOPE. Distance between paper and sensor in both cases is 2 cm.

to change the intensity of the reflected light from the sensor. Figure 24.6A illustrates the results of the effect of temporary “biologization” of non-living objects. In Fig. 24.6, the numeral ‘1’ represents the control area for registration of influence of a thick piece of paper (the non-living material studied). The area designated by the numeral ‘2’ demonstrates the measurement of the exact same piece of paper after it has been held for 2 min between a human’s palms. It is clear from the figure that the character of the change of reflected light intensity is the same as in the case of biological objects. After close interaction between the biological object and a non-living object, certain changes occur in the condition of the non-living object. The non-living object became “biologized” in that manner that can be registered or measured by the device. The time during which the change entirely vanishes may amount to about 15–30 min. This effect is also observed even when there is no contact between the biological object and the non-biological object. The same effects were observed when laser was used as a light source (Fig. 24.6B). Moreover, it has been observed the effect of double “biologization” when the sheet of paper which is already “biologized” placed on the other piece of paper induces “biologization” of this sheet.

24.4 Distant evaluation of functional state of anaesthetized rat In series of experiments (when the laser was used in the BIOSCOPE) the rats were subjected to the action of some preparations. Under norm conditions the oscillation frequency of device signals from anaesthetized rats did


a

305

1

0

b

0

c

0.1

1

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0.1

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1

10

0.1

1

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0

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0 frequency (oscillation per minute)

0

15

30

45

60

min

Fig. 24.7. Influence of immobilization stress and of some pharmacological preparations on signals of BIOSCOPE device for the anaesthetized rats. A – control record; B, C – after 2 and 15 h immobilization before anesthesia of rats, correspondingly; D – after ten days introduction of Gentamycin; At the left – the power spectrum of registered signals is presented, on the right – the typical examples of hourly registration of rats anesthetized by Urethane. Relative scale on a vertical for all corresponding curves is identical. The sensor of BIOSCOPE is located on a distance of 1 cm from the rat back.

not exceed 1–2 oscillations per minute (Fig. 24.7A). If animals were exposed to 2 or 15 h immobilization before anesthesia than an essential change in character of registered signals (Fig. 24.7B–C) was observed in comparison with the control ones. The oscillation frequency increases in 10 times in the first case (Fig. 24.7B, 2 h immobilization), and in the second case (Fig. 24.7C, 15 h immobilization) it falls, both by frequency and amplitude of observable oscillations. Ten days introduction of Gentamycin also have led to a considerable (in comparison with the norm) reduction of amplitude and frequency of oscillations, however in spectral distribution of registered signals the frequencies of an order of 0.25 oscillations per minute prevail (Fig. 24.7D).

24.5 Mechanism of formation of equipment signals Simplicity of bioscope design and comparative clarity of processes which can take place at light propagation from source to photo-receiver allow to hope to reveal physical mechanisms which determine the formation of device’s signals in performed experiments. For this aim the researches have been carried out for estimation the role of functional parts of designed equipment in formation of observed effects. The overall assessment of these experimental results

306


a

covering material

F

F only glass plate

glass plate

F

L

PC

amplifier

L

power source

b

PC

amplifier

power source

PC

amplifier

power source

only covering material

covering material only glass plate

glass plate

F

amplifier

L

L

PC

F

power source

amplifier

L

PC

F

power source

amplifier

L

PC

power source

Fig. 24.8. Sketch of signal registrations at which the effects are not observed (A) and always observed (B).

leads to the conclusion that the remote influence of biological systems does no affect the operation of computer, amplifier electronics, power source and light propagating in front direction (Fig. 24.8A), however the necessary condition for appearance of signals characterizing the distant influence of biological objects is the scattering of light which is realized from the surface of glass plate and/or covering material (Fig. 24.8B). With the aim of excluding the role of glass plate and/or covering material in formation of observed effects the experiments were carried out when scattered light was created by passing of laser beam through diffraction slit (Fig. 24.8). The laser light is diffracted by slit, passes through metallic tube and arrive the screen placed on a distance of 60–70 cm from the slit. Here well distinguished diffraction zones of different orders are formed, for which incident light rays have different phases. Photo-receiver was placed sequentially in different diffraction zones and the level of illumination was measured. Then the experimenter has approached his palms to the metallic tube (without touching) for several minutes and the signal change was measured. As it is seen from Fig. 24.9, the photo-receiver readings in different zones change variously. In central zero zone the signal weakening amounts only 0.2%, but in zone ‘2’ it falls on 20%. From the other side, in the zone ‘1’ the signal increase on 3% is observed. Small by absolute value, but reliable changes take place in the ‘3’ and ‘4’ zones. However, here the absolute change of light intensity reaches 40–50% of signal initial level. The basic conclusions which can be derived from performed experiment are the followings: “The main function of materials used in BIOSCOPE design (glass plate, opaque covering material) consists in providing the scattering


a

307

208 0

207 206 33 32 31

1

b

30 11 2

mV

10

diffraction zones

diffraction slit

9 hotoreceiver

8 7 3

3

2

0 1 2 3 4

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1 3 4

2 1 0 0

100 200 300 400 500 600 700 800 sec

Fig. 24.9. Light scattering from diffraction slit. A – Effect of distant influence of human palm on the light intensity in different diffraction zones. (horizontal sections indicate the time of influence). Along the ordinate axis the photo-receiver readings before signal raise are presented. B – Sketch of experimental setup.

of radiation coming from the light source.” Biological systems are capable to influence directly on the phases of scattered light, which in turns results in their spatial redistribution. The light wave of radiation source can be presented as harmonic oscillation S0 = A0 sin(ωt+ϕ0 ), where A0 is the oscillation amplitude, ϕ0 is the frequency, t is the time and ϕ0 is the wave phase. During diffraction from different slit areas the light rays arrive at a photo-detector and summed up. For two such rays, S1 = A1 sin(ωt+ϕ1 ) and S2 = A2 sin(ωt+ϕ2 ), the resulting oscillation can be written as S = S1 +S2 , where S = A sin(ωt+ϕ) is also an harmonic oscillation. The light intensity which is actually registered by photo-detector is proportional to A2 = A21 + A22 + 2A1 A2 cos(ϕ1 − ϕ2 ) and its readings can be changed only in the case if diffracted light before arriving at the receiver will pass through optical active medium which will changes the values of amplitudes A1, A2 or phases difference (ϕ1 − ϕ2 ) (Fig. 24.10A). The distant influence of biological systems is not observed when light flux reaches directly the photo-receiver without scattering (see Fig. 24.8A). In this case the device readings are proportional to S02 = A20 sin2 (ωt + ϕ0 ). The light frequency is very high (5 × 1014 Hz) therefore in this case the photo-receiver signals are proportional only to the square of amplitude of light (S02 ∼ A20 ) and does not depends upon its phase ϕ0 (Fig. 24.10B). Hence the remote action of biological objects leads to the change of scattered light phases only, but does not affect on the light amplitude. Detailed studies have shown that the integral intensity of scattered light remains invariant in the presence of biological system near the sensor of BIOSCOPE, however relative phases of

308


a

diffraction slit

F S=S1+S 2

S1

photoreceiver’s signal

in(ωt+ = A 1s

L

ϕ 1) S0 = A0sin(ωt+ϕ0)

S2 = A2sin(ωt+ϕ2)

S2 = A12+A22+2A1A2 cos(ϕ1–ϕ2)

b F S = S0

photo-receiver’s signal

S0 = A0sin(ωt+ϕ0)

L

S2 ~ A02

Fig. 24.10. Registered signal at propagation of laser light through diffraction slit (A) and at its direct arriving at the photo-receiver (B). F, L – photo-receiver and laser correspondingly; S and A are amplitudes, ω, ϕ frequency and phase of light radiation, t is the time.

scattered light flux are changed which finally results in the change of the pattern of its angular distribution inside the BIOSCOPE’s case [4]. It is well known from the optics that such redistribution of light beam can be attributed by its propagation through some optically active phase medium [6]. Such medium can not be directly traced or photographed but it always leads to the phase change of passed light. It is obviously, that this optical active medium is formed on the path of light beam due to presence of biological system in the area of light scattering. Therefore it seems plausible that this substance primordially surrounds all biological systems.

24.6 Concluding remarks By means of various experiments it is proved that the device BIOSCOPE is reliably isolated from investigated biological objects. Then arises, a lawful question what measures and how measures BIOSCOPE? As was shown in the recent investigations of the author [7, 8], any system has very important quantum peculiarity which differs from de-Broglie wave. It is interesting that this peculiarity doesn’t disappear on distance, in some cases even for macroscopical bodies.


309

The idea consists in that, that every system interacting with vacuum forms in it the structures with certain sizes and topologies. These structures in result leads to changes of refraction indexes of vacuum around of body which may be also interpreted as a quantum halo. The changes of refraction indexes of vacuum at some cases (specially for biological systems) may be big and measurable. The role of an opaque hackly material consists in that the geometry of it surfaces is being changed at changing of environment refraction indexes. This changes the distribution of refracted diffuse light that in turn influences on intensity of measured light in BIOSCOPE.

References 1. V. N. Ermakov, S. P. Kruchinin, and A. Fujwara, Electronic nanosensors based on nanotransistor with bistability begaviour, J. Bonˇca, S. Kruchinin (eds.), Electron transport in Nanosystems, 341–349. 2. R. G. Jahn and B. J. Dunne: On the Quantum Mechanics of Consciousness, with Application to Anomalous Phenomena, Found. of Phys. 16(8), 721–772 (1986). 3. R. Bryan: Mental control of a single electron? Physics Essays, 19, 169–173, (2006). 4. R. Sh. Sargsyan: New aspects of functioning of biological systems, 2008, Doctor thesis, Orbeli Institute of Physiology NAS RA, Yerevan (in Russian). 5. J. P. Draayer, H. R. Grigoryan, R. Sh. Sargsyan, S. A. Ter-Grigoryan: Systems and Methods for Investigation of Living Systems, Patent Application, US 2007/0149866 A1. 6. M. Born and E. Wolf: Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Cambridge University Press, 836 (1997). 7. A. S. Gevorkyan, Exactly constructing model of quantum mechanics with random environment, Physics of Atomic Nuclei, 73(2), 876–885 (2010). 8. A. S. Gevorkyan, Nonrelativistic Quantum Mechanics With Fundamental Environment, DOI 10.1007/s10701-010-9446-y Found. of Phys. (2010).

25 THERMOELECTRICITY IN DOUBLE-BARRIER RESONANT TUNNELING STRUCTURES V.N. Ermakov1, S.P. Kruchinin2 , A. Fujiwara3 , and S.J. O’Shea4 1 2 3 4

Bogolyubov Institute for Theoretical Physics, NASU, Kiev 03680, Ukraine [email protected] Bogolyubov Institute for Theoretical Physics, NASU, Kiev 03680, Ukraine [email protected] Japan Advanced Institute of Science and Technology, Ishikawa 923-1292, Japan [email protected] Institute of Materials Research and Engineering, Singapore

Abstract. Resonant tunneling conception has been used to explain the thermoelectrical phenomenon in molecular nanosystems. Thermovoltage as a function of the temperature is calculated. The reverse of the thermovoltage sign under resonance tunneling through HUMO and LUMO levels is demonstrated. The dependence of the thermovoltage on the distance between electrodes is calculated. A result of calculations is in good agreement with experiment data.

Key words: thermoelectricity, molecular nanosystems, the resonance tunneling.

25.1 Introduction Studies of the origin of the voltage or current in nanosystems under conditions of a temperature gradient are the extremely interesting and promising trend in the development of nanotechnologies [1–3]. However, as distinct from the classical manifestation of thermoelectric phenomena, the researchers meet a number of difficulties related to the necessity to apply strictly quantummechanical methods in the description of this phenomenon. In fact, in order to realize the phenomenon of thermoelectricity in a nanostructure, it is sufficient to induce the tunnel current between electrodes possessing different temperatures. However, due to the smallness of the tunnel current and an insignificant thickness of a barrier, the thermoelectric effects are expected to be slight and be hampered by fluctuations. It seems to be more promising to use nanostructures with sufficiently wide potential barriers with various objects introduced to them which are, in essence, current conductors. The latter will J. Bonˇca and S. Kruchinin (eds.), Physical Properties of Nanosystems, NATO Science for Peace and Security Series B: Physics and Biophysics, c Springer Science+Business Media B.V. 2011 DOI 10.1007/978-94-007-0044-4 25,

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increase the electron conduction of a system, and thus we may expect that the thermoelectric effects will be significantly enhanced. Practically all experimental nanostructures intended for the realization of thermoelectric phenomena can be considered, at least formally, as systems consisting of two potential barriers with an active element between them. In fact, we mean nanostructures realizing the process of resonance tunneling which is a pure quantum phenomenon and has no classical analog. In the present work, we will develop the ideas of the phenomenon of resonance tunneling in thermally nonequilibrium nanostructures for the description of experimental regularities of thermoelectric phenomena. The obtained theoretical results are compared with experimental data.

25.2 The resonance tunneling under difference of temperatures Resonant electron tunneling of particles through a system of double potential barriers is very sensitive to a position of electronic states in a quantum well [4]. This circumstance can be used for the effective governing of the tunneling process. We confine ourselves by the consideration of one-dimensional case of tunneling [5–8]. This way to describe the resonant tunneling is legitimate expectations [9]. As a model of double-barrier tunneling system, we take a structure with the energy profile shown in Fig. 25.1. The Hamiltonian describing the tunneling of electrons through such a structure can be chosen in the form H = H0 + HW + HT . (25.1) The first term of this Hamiltonian is + H0 = εL (k)a+ εR (p)a+ pσ apσ . kσ akσ

b

ΔV T+ΔT (Hot)

V

BDT Molecula SH

SH

Voltage Amplifier

R Barrier

μ L Barrier

μ

T (Cold)

μ

ΔV

μ

T+ΔT (Hot)

V

EH BDT Molecula SH

SH

R Barrier

Em

L Barrier

a

(25.2)

pσ

kσ

T (Cold)

Voltage Amplifier

Fig. 25.1. STM sketch to study the thermoelectric phenomena and the diagram of energy potentials and energy levels for resonant tunneling are demonstrated. A is the case corresponds to a LUMO resonant level. B corresponds to a HOMO resonant level.

25 Thermoelectricity in double-barrier resonant tunneling structures

313

It describes electrons in the left electrode (source) and in the right electrode + (drain). Here, a+ kσ (akσ ) and apσ (apσ ) are the creation (annihilation) operators for the electrons in the source and the drain, respectively, εL (k) = εL + 2 k 2 /2mL is the energy of electrons in the source, k and mL are their quasimomentum and effective mass, respectively, and σ is the electron spin. For the drain with regard for the external potential V applied across the system, we haveεR (k) = εR + 2 k 2 /2mR − V , p being the momentum, and mR the effective mass. The Hamiltonian HW describes electronic states in the quantum well. It can be written in the form HW =

E 0 a+ α aα ,

(25.3)

α

where α = (l, σ), σ is the spin number, and l is a number of the quantum state. The state energy in the well with regard for the applied bias can be written as El = εl − γV , where εl is the energy of the resonant state in the quantum well, and γ is a factor depending on the profile of potential barriers (for identical barriers, γ = 0.5). The Hamiltonian HT describing the tunneling of electrons through the barriers has the conventional form HT =

Tkα a+ kσ aα .

(25.4)

kα

Here, Tkα and Tpα are matrix elements of tunneling through the emitter and the collector, respectively. In general case, they depend on the applied bias. When we apply a constant external bias across the system, a nonequilibrium steady-state electron distribution sets in. We assume that the electron distribution functions in the electrodes (source, drain) are equilibrium by virtue of their large volumes, but their chemical potentials and temperatures are different. The chemical potentials are connected by the relation μL − μR = V (where μL and μR are the chemical potentials of the source and the drain, respectively). The density state ρ(E) on local levels can be determined with help of the Fourier transformation of he retarded Green’s function G(α, α, E), ρ(E) = −

1 ImG(α, α, E), π α

(25.5)

where G(α, α, E) = −iθ(t) < [a+ α (t), aα (0)]+ >,

(25.6)

and θ(t) is a unit Heaviside function. The electron distribution function g(E) in the quantum well is essentially nonequilibrium. It can be determined from the condition of equality of the

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tunneling currents through the source and the drain. Then the distribution function has the form: g(E) =

1 [ΓL (E)fL (E) + ΓR (E)fR (E)], Γ (E)

Γ (E) = ΓL (E) + ΓR (E), (25.7)

where ΓL and ΓR are the rates of electron transmission (source-QD and QDdrain, respectively), 2 ΓL = |Tkα | δ(E − εL (k)) k , (25.8) 2 ΓR = |Tpα | δ(E − εR (p)) p

and fL and fR are electron distribution functions in the source and the drain, respectively. They have Dirac’s forms under a temperature difference −1 E − μL,R fL,R = exp +1 , (25.9) kB TL,R where kB is the Boltzmann constant, and TL,R are temperatures in the source and the drain, respectively. The occupancy of local states in the QD can be determined with help of the expression [5] 1 nα = − dE g(E) ImG(α, α, E). (25.10) π

25.3 Double barriers thermostructures for the resonant tunneling At the low transparency of the barriers and under a voltage V , the direct current Jcd is described by the equation [7] e G(E)(fL − fR )ρ(E)dE, Jcd = (25.11) where e is the elementary charge and is Planck’s constant. Conductance G(E) is expressed by G = ΓL (E)ΓR (E)/Γ (E). ΓR , ΓL are exponentially dependent on barriers widths, and ΓR ,ΓL , fL , fR depend on the energy (E) and the voltage (V ). Correlation effects of electrons in a QD can be taken into account by means of ρ, which is a harmless assumption in the case of low transparent barriers and a simple energy level structure. The density of states ρ depends on the energy structure of a QD ρ= Cm , ϕm (E), (25.12) m

25 Thermoelectricity in double-barrier resonant tunneling structures

ϕm (E) =

hm 1 , π (E − Em )2 + h2m

315

(25.13)

hm is a half-width of the energy level Em , and Cm is the weighting factor for the energy level Em . The typical sketch of a double-barrier nanodevice is demonstrated in Fig. 25.1, where the diagram of the potential energy is represented as well. The left electrode is hot, whereas the right electrode is cold. Thus, some part of electrons is transferred from the left to the right electrode. The electronic path is shown by dotted arrows. The total lengths of barriers and a BDT molecule are too long for a simple tunneling. Therefore, only the resonance tunneling is possible. The transfer of electrons will be such to reach the equilibrium. Thus, the current is zero under the condition of the difference in chemical potentials. In the case of a low barrier permeability (hm ≈ 0), we accept the approximation ϕm (E) = δ(E − Em ), where δ(E) is Dirac’s delta function. According to Eq. (25.1), this condition results in the equation

Cm G(Em ) (fL (Em ) − fR (Em )) = Jcd ,

(25.14)

m

where fL,R are Dirac’s functions for the electron distribution fL,R =

1 + exp

Em − μL,R kB TL,R

−1 .

(25.15)

Here, μL,R are the chemical potentials for the left and right electrodes, respectively, ΔV is the thermodynamic voltage, TR = T , and TL = T + ΔT . Here, ΔT is the temperature difference between the electrodes. For the schematic electric circuit of the experiment executed in [2], we have μL = μ − ΔV and μR = μ, where ΔV is positive. The distribution fL,R (Em ) is exponentially dependent on the energy Em . Thus, when b|Em − Em±1 | >> kB T , it is important one item with energy Em for one |Em − μ| is the least value. In this approximation, Eq. (25.14) can be reduced to the form G(Em )[fL (Em ) − fR (Em )] = 0.

(25.16)

Em is one of the energy levels of a BDT molecule which has the least distant from the chemical potential. That molecule level is shifted by the electric field V on by the value ΔVm = aR ΔV /(aL + aR ). In other words, we have Em = E0m − ΔVm . If G(Em ) = 0, then the solution of Eq. (25.16) leads to the relation ΔV =

(E0m − μ) ΔT . aR T 1 + ΔT T aR +aL

(25.17)

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Formula (25.17) can be compared with experiment data. In particular, such a comparison with experiment [1, 3] results in ΔV < 0. This means that E0m < μ. Thus, the energy level E0m is a HOMO level EH . In the case of the configuration of experiment [1, 3], we have the condition for Fig. 25.1B. In fact, the conductivity of the tunneling structure is determined by holes. The detailed comparison of the thermo-voltage dependence on ΔV is demonstrated in Fig. 25.2. In the case of ΔT

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