Effective Field Approach to Phase Transitions And Some Applications to Ferroelectrics (World Scintific Lecture Notes in Physics)

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World Scientific Lecture Notes in Physics - Vol. 76

Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (2nd Edition) Julio A Gonzalo

Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (2nd Edition)

World Scientific Lecture Notes in Physics Published Vol. 55: Lectures in Particle Physics D Green Vol. 56: Chaos and Gauge Field Theory TSBiro, etal. Vol. 57: Foundations of Quantum Chromodynamics (2nd ed.): An Introduction to Perturbative Methods in Gauge Theories TMuta Vol. 58: Concepts in Solids, Lectures on the Theory of Solids P. W. Anderson and H. Bunke Vol. 59: Lattice Gauge Theories: An Introduction (2nd ed.) HJRothe Vol. 60: Massive Neutrinos in Physics and Astrophysics (2nd ed.) R N Mohapatra and P B Pal Vol. 61: Modern Differential Geometry for Physicists (2nd ed.) C J Isham Vol. 62: ITEP Lectures on Particle Physics and Field Theory (In 2 Volumes) M A Shifman Vol. 64: Fluctuations and Localization in Mesoscopic Electron Systems MJanssen Vol. 65: Universal Fluctuations: The Phenomenology of Hadronic Matter R Botet and M Ploszajczak Vol. 66: Microcanonical Thermodynamics: Phase Transitions in "Small" Systems DHE Gross Vol. 67: Quantum Scaling in Many-Body Systems M A Continentino Vol. 69: Deparametrization and Path Integral Quantization of Cosmological Models C Simeone Vol. 70: Noise Sustained Patterns: Fluctuations & Nonlinearities Markus Loecher Vol. 71: The QCD Vacuum, Hadrons and Superdense Matter (2nd ed.) Edward V Shuryak Vol. 72: Massive Neutrinos in Physics and Astrophysics (3rd ed.) R Mohapatra and P B Pal Vol. 73: The Elementary Process of Bremsstrahlung W Nakel and E Haug Vol. 74: Lattice Gauge Theories: An Introduction (3rd ed.) H J Rothe

World Scientific Lecture Notes in Physics - Vol. 76

Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics (2nd Edition)

Julio A Gonzalo Universidad Automa de Madrid, Spain

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EFFECTIVE FIELD APPROACH TO PHASE TRANSITIONS AND SOME APPLICATIONS TO FERROELECTRICS (2nd Edition) World Scientific Lecture Notes in Physics — Vol. 76 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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To the Virgin Mary, Queen of all things created by God, including beautiful phase transitions. To my parents

Preface

The effective field approach to a phase transition, used by Pierre Weiss at the beginning of this century to describe theoretically ferromagnetic transitions in iron, nickel, and cobalt, is the second oldest (surpassed in this respect only by Van der Waals theory of liquid-vapor transitions) and certainly the simplest approach to investigate cooperative phenomena of this kind. Early in the effort to describe satisfactorily the rich variety of physical phenomena accompanying a phase transition, it was realized that the role of fluctuations, introduced by Einstein in 1905 (the same year in which he introduced special relativity and the first successful explanation of the photoelectric effect), was crucial to investigate the subtleties of the behavior of systems undergoing phase changes at temperatures very close to the transition temperature. Of course, the concept of effective field, being an average or mean field, leaves out completely thermodynamic fluctuations. Nevertheless, the effective field approach gives in most cases a good qualitative description of the cooperative phenomena, and in some cases, ferroelectric transitions being an outstanding example, it even gives a fairly good quantitative description. It took some time for prominent theorists to realize that deviations of "classical" (effective field) behavior in ferroelectrics and dipolar ferromagnets should show up in the form of logarithmic corrections, which are very hard to detect experimentally. This work, which could as well be entitled "Pedestrian approach to Phase Transitions," obviously does not aim at a comprehensive discussion of the effective field approach to phase transitions in general and less so at a discussion that goes deeply beyond this approach to such a broad and active field of contemporary research. It attempts at a simple presentation of the approach, at a very elementary level in most cases, to a number of interesting phase transitions, mostly solid-state transitions, with special attention,

vii

viii

Effective Field Approach to Phase

Transitions

in the second half of the book, to some work in ferroelectric systems inflecting, unavoidably, the research interest of the author. This material formed the basis for a short graduate course on "Phase Transitions" imparted by the author at the Universidad Autonoma de Madrid, in 1989. I would like to thank Professor J. Palacios and Professor S. Velayos, former teachers of mine at the University of Madrid, who introduced me to this field; Drs. B.C. Prazer, K. Okada, M. Ray, and G. Shirane, from whom I learned much on ferroelectrics and phase transitions at Brookhaven National Laboratory and at Puerto Rico Nuclear Center; I. Lefkowitz and K.A. Miiller to whom I am also indebted; and many of my former graduate students from whom I have also learned a good deal. Madrid, January 15, 1990 Julio A. Gonzalo

Preface to the Second Edition

This second edition of Effective Field Approach to Phase Transitions includes additional sections, Parts III-IV, in which additional papers involving the efFecitve field approach to Ferroelectric Transitions published in the period 1991-2005 are included. It is for me a pleasure to give proper credit to the various main authors of these papers: to my younger Spanish colleagues R. Ramirez, G. Lifante, M. de la Pascua, B. Noheda, T. Iglesias, J.R. Fernadez del Castillo, N. Duan, C. Arago, M.I. Marques, and J. Garcia, some of them at UAM or other Universities in Madrid; and to my long time friends and colleagues from abroad, including G. Shirane (BNL, recently deceased), D. Cox (BNL), C.W. Garland (MIT), J.O. Tocho (La Plata), W. Windsch (Leipzig), M. Koralewski (Poznan), Y.L. Wuag (Shanghai), B. Mroz (Poznan), L. Cross (Penn State), R. Guo (Penn State), and C.L. Wang (Sandong). Madrid, November 15, 2005 Julio A. Gonzalo

Contents

Preface

vii

Preface to the Second Edition

ix

Part 1 Mean Field Approach to Cooperative Phenomena 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

An overview Liquid-vapor transitions Ferromagnetic transitions Superconductive transitions Order-disorder transitions in alloys Ferroelectric transitions Superfluid transitions Ferroelastic transitions Landau theory and effective field approach. Role of fluctuations 1.10 Equation of state and the scaling function Appendix: Effective field approach to superconductors

Part 2 Some Applications to Ferroelectrics: 1970-1991 Behavior at T = Tc of pure ferroelectric systems with second order phase transition 2.2 Effects of dipolar impurities in small amounts 2.3 Mixed ferro-antiferroelectric systems and other mixed ferroelectric systems 2.3.1 Comment on "Ferroelectricity in zinc cadmium telluride"

1 3 11 21 31 37 45 53 61 67 81 91 101

2.1

xi

103 119 129 137

xii


2.4 2.5 2.6 2.7 2.8 2.9 2.10

Transitions

Relaxation phenomena near Tc 141 Polarization reversal in ferroelectric systems 147 Polarization switching by domain wall motion 155 Switching current pulse shape 163 Elementary excitations in ferroelectrics: Dipole waves . . . 169 Low-temperature behavior of ferroelectrics 179 Logarithmic corrections 183

Part 3 Some Applications to Ferroelectrics: 1991-1997 3.1

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14

189

Pressure dependence of the free energy expansion coefficients in PbTi03 and BaTi03, and tricritical point behavior 191 Ultrasonic study of the ferroelectric phase transition in R b D 2 P 0 4 197 New technique for investigating ferroelectric phase transitions: The photoacoustic effect 207 Tricritical point behavior and quadrupole interactions in ferroelectrics 215 Frequency and temperature dependence of sound velocity in TGS near Tc 221 Dipolar and higher order interactions in ferroelectric TSCC 229 Thermal hysteresis and quadrupole interactions in ferroelectric transitions 237 Specific heat and quadrupole interactions in uniaxial ferroelectrics 243 Field-dependent temperature shift of the dielectric losses peak in TGS 249 Discontinuity and quasitricritical behavior near Tc in ferroelectric triglycine selenate 255 Scaling equation of state for ferroelectric triglycine selenate atr«Tc 265 Composition dependence of the ferroelectric-paraelectric transition in the mixed system PbZri-^TiÔs 273 Observations of two ferroelectric response times in TGSe at T ro it should have a certain dependence with the distance, r, to the center of the molecule, but for the sake of simplicity it is considered to be averaged out throughout the space occupied by the gas. Consequently, the value of U(r) must be li

12


U( r)

Transitions

a

1

0 (r ^

Fig. 1.2.1. Effective interaction potential of an individual molecule in a gas. Note that u(r) depends on the density of molecules in the gas.

dependent on the density of molecules within the volume occupied by the gas. Figure 1.2.1 shows the shape of U(r). The pressure is given by {dA/8V)T

= kBT

\d(lnZ)/dV],

(2)

where A is the free energy, V the volume, kB Boltzmann's constant, T the temperature, and Z the partition function, and we have used the fact that A = kB In Z. The global partition function Z for the gas can be thought of as the product of the N single-particle partition functions, defined by ZSP

ex

/ * /

dr exp

\2m

U(r)

0 = 1/feT,

(3)

where we omit a proportionality factor, the integrals are extended to momentum (p) and position (r) space, as usual, m is the mass of a single particle, and T is the temperature. The integral / ( . . . ) dp does give a factor that is independent of V, and, therefore, does not contribute to P. The integral / ( . . . ) dr, on the other hand, can be written as Kxcie-°° + {V~ Kxci)e

-0U

(V - Kxd)e

-/3u

(4)

Liquid- Vapor

13

Transitions

where Vexci is the "excluded volume" corresponding to the "hard cores" of all molecules in the gas interacting with a single molecule. Then, taking into account that Zsp = Z1' , P = NkBT—

[\n(V - K x c l ) - 0u].

(5)

Now we make the reasonable assumptions that (i) Vexc\ is proportional to the total number of molecules in the gas and (ii) u, which as mentioned before corresponds to an attractive interaction, is proportional to the density of molecules in the gas. Therefore, Kxci « N,

i.e., Kexci = (b/NA)N,

u « (-N/V),

(6)

2

i.e., u = -(a/N A)(N/V).

(7)

Here, b and a are constants whose physical meaning will become apparent later on, defined in terms of Avogadro's number NA, which is the number of molecules in a mole of gas. Making use of Eqs. (6) and (7) in Eq. (5) we get P = NkBT which, for n = N/NA

1 _V-(N/NA)b

^

pa(N/NA)^ 2

NV

(8)

= 1 (i.e., one mole of gas), leads to

(P + a / K L ) (^moi -b) = NAkBT

= RT,

(9)

which is the Van der Waals equation, where a = a/NA has the dimensions of attractive energy (PV) times volume (V) per mole of gas and b those of excluded volume per mole. It may be noted that for P » a/V^ ol and Vmo\ 2> b, i.e., a mole of a gas occupying a sufficiently large volume at a moderately high temperature, Eq. (9) reduces to the ideal gas equation, PVmol = RT.

(10)

To investigate in more detail the Van der Waals equation, we simplify the notation by writing V instead of Vmo\ and a instead of a, implicitly assuming

14


Transitions

that we are referring in what follows to one mole of gas (i.e., n = N/N& = 1). Equation (9) can then be written as (PV2 + a){V - b) = RTV2 or, equivalently,

v-r^wsv-^-o, p j

\p

\pj

mi

which reflects in a visible way the cubic character of the Van der Waals equation. Each pair of values (T, P) corresponds, in general, to three different solutions for (V"). We may note that for small T the three solutions are real; for a certain T = Tc, the three solutions become a single solution with V = Vc, P = P c ; finally, for large T a pair of roots become complex and a single real solution remains. This implies that there is a critical point, characterized by Pc, Vc, Tc, which can be used as a point of reference to describe the behavior of the gas. Let us write (V - Vc)3 = V3 - 3VCV2 + 3V2V - yc3 = 0 &tV = Vc.

(12)

We can compare Eq. (12) with Eq. (11) and equate the respective coefficients of V3, V2, V, and V° at the critical point. Thus, (6 + i£r c /Pc) = 3Vrc,

(13)

2

a/Pc = 3VC ,

(14)

3

ab/Pc = Vc .

(15)

From Eqs. (14) and (15), Vc = Zb.

(16)

Pc = a/3V2 = a/276 2 .

(17)

From Eqs. (14) and (16),

Finally, from Eqs. (13), (16), and (17), RTC - (3VC - b)Pc = (9b-b)(a/27b2)

=~ .

(18)

Liquid-Vapor

15

Transitions

These results give a direct physical meaning to the constants a and b characteristic of a given mass (a mole) of a gas, in terms of its volume Vc and its pressure Pc at the critical temperature Tc at which the phase transition between the liquid state and the vapor state takes place. The same relationships for Vc, Pc, and RTC can be alternatively obtained taking as a starting point

(d2p/dv2)Tc=o,

(dP/dv)Tc = a,

PC = P(VC),

which indicates that the function P(V) has an inflection point precisely at T = TC. We may define z = PV/RT and note that, for a Van der Waals gas, making use of Eqs. (16)-(18), zc = PCVC/RTC = 3/8,

(19)

which is lower but of the same order of magnitude than z = PV/RT

= 1 (ideal gas).

(20)

Real gases, or vapors, may differ considerably one from another in their critical pressure (Pc), critical volume (Vc), and critical temperature (T c ). However a law of corresponding states can be established measuring P, V, and T, in units of Pc, Vc, and T c , respectively. Dividing Van der Waals equation (n = 1) (p+±yV-b) = RT

(21)

by (1/3)PCVC, one obtains [(P/Pc) + (a/PcVc2) (Vc/V)2]

[3(V/VC) - 3(6/14)] = 3RT/PCVC,

(22)

which reduces to (P + 3 / V 2 ) ( 3 F - l ) = 8 T ;

P = P/PC,

V==V/VC,

T = T/TC, (23)

taking into account that, using Eqs. (16)-(18), (a/PcV2)

= 3,

3b/Vc = l,

3RTC/PCVC = 3(8/3).

(24)

Equation (23) clearly implies a single function P(V, T) for all gases, as far as they can be considered Van der Waals gases, and therefore a law

16


Transitions

V=V/Vc Fig. 1.2.2.

Shape of surface P(V,T)

for a Van der Waals gas.

of corresponding states using the dimensionless units (P/Pc), (V/Vc), and (T/Tc) as illustrated in the introductory section. Figure 1.2.2 depicts the shape of the surface P(V,T) and shows cross sections for constant T. On this surface, at constant T such that T/Tc 3> 1 the cross-sectional curves correspond to hyperboles (PV = constant) indicating ideal gas-like behavior. At constant T = Tc, the curve passes through C, the critical point, which is an inflection point. At constant T < Tc, the curves contain maxima and minima that enclose a region of coexistence of phases between them, indicated by a dotted line. Table 1.2.1 gives experimental data of the constants a, b, and zc = RTC for a few representative fluids, and gives also the Van der Waals value for zc. It can be seen that, in spite of differences of more than an order of magnitude (related to the attractive interaction energy between molecules), the values of zc remain close to each other and to the Van der Waals value z = 3/8 given by Eq. (19).

Liquid-Vapor Table 1.2.1.

Transitions

17

Parameters a, b, and z c for some real fluids.

Fluid

a (l 2 atm/mol 2 )

6 (1/mol)

zc = PCVC/RTC

5.464 3.592 1.345 0.341

0.0305 0.0427 0.0322 0.0237

—

—

0.230 0.273 0.291 0.308 0.375

H20 C02 A He Van der Waals

As mentioned in the introductory section, the behavior near the phase transition is usually described by the various critical exponents. To this end it is convenient to rewrite the Van der Waals equation in terms of the new variables p = P - 1 = (P - Pc)/Pc,

(25)

= V-l

= (V-

Vc)/Vc,

(26)

t = T-l

= (T-

Tc)/Tc.

(27)

v

Substituting these definitions in Eq. (23), we get P

= 8(1 +t) ~ (2 + 3v)

3 (1 + vf

'

(28)

which is particularly useful to describe the behavior of the Van der Waals gas near the critical point, where we know that p < l , « > 1 , and t Tc, and Eq. (37) for T < Tc. Then, for T < Tc, A « A0(T) + ~ 1 2 § ( T - TC)(V - VC)2A, b

~

8A_ dA0(T) IP dT~ dT + 6 K K ( dA0(T) t &PCVC (V-VcV dT

c) L

Tc V Vc

and using PCVC/TC = (3/8)i? = (3/8)NkB, from Eq. (19), and v2 = {V Vc)2/V2 = (4/3)(T c - T)/Tc, from Eq. (31), we get C ( T < T

C

) = T ( | | ) « C O + 3JV*BQ^...

(40)

20


Transitions

On the other hand, for T > Tc the order parameter v is equal to zero, and C(T>Tc)^r(g)«Co-

(41)

Co = —(dAo(T)/dT) is the non-singular part of the specific heat and is expected to be nearly the same at T < Tc as at T > T c . Therefore, the jump in specific heat from a finite value at T < Tc to another, different finite value at T > Tc is given by AC(T C ) = 3NkB

(42)

and the specific heat critical exponent for a Van der Waals gas must be zero. References 1. T. Andrews, Phil. Trans. 159, 575-591 (1869). 2. J.D. Van der Waals, On the Continuity of the Liquid and Gaseous States, Doctoral dissertation (1873). 3. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford, 1971).

Chapter 1.3

Ferromagnetic Transitions

A ferromagnetic material is characterized by presenting a "spontaneous" magnetization (i.e., a magnetization under zero external magnetic field) at temperatures below a certain critical temperature Tc. In ferromagnets, the magnetization versus magnetic field curve shows hysteresis, as shown in Fig. 1.3.1. Hysteresis means memory, i.e., the ability to "remember" in which direction a sufficiently strong external magnetic field last pointed before dropping to zero field. That is why ferromagnets are widely used in computer memories, among other things. There are "hard" ferromagnets (large Hc) and "soft" ferromagnets (small Hc), which are useful for one type of application or another. Of course, the phenomenon of magnetism has been known to man from ancient times, but only toward the end of the 19th century was the ferromagnetic transition investigated quantitatively as a function of temperature by P. Curie at the University of Paris. He found experimentally that the susceptibility x = 4ir(dM/dH) of a ferromagnet depends on temperature according to the simple law x(T) = C/(T — 6) where C is a constant (the Curie constant), and 6 is the critical temperature below which spontaneous magnetization shows up. We can distinguish two main types of ferromagnets: metallic (for instance, Fe, Ni, Co, and many alloys containing these elements) and oxides (e.g., Fe 2 03). P. Weiss1 put forward in 1907 the famous "molecular field" hypothesis, which assumes that within each "domain" (homogeneously magnetized region of a ferromagnetic sample) the existence of an effective field, HeS = H + 7 M tends to produce a parallel arrangement of the atomic dipoles. This effective or molecular field is made up of two contributions: the external field and 21

22


Transitions

1

B =H + 4*M

H

' 0

1

Fig. 1.3.1. Typical hysteresis loop for a ferromagnet as X < T c . At H = 0, B$ = 47rMs, M s = spontaneous magnetization. At B = 0, H = Hc = coercive field (cgs units).

an additional field that is proportional to the magnetization, the coefficient 7 being a constant. Consider a system formed by N elementary magnetic moments (per unit volume) on a three-dimensional lattice. If the dipoles can be oriented only in either one of two possible (up or down) directions, corresponding, for instance, to spin s = ±Y2, or, more generally, to what is called the Ising case (one-dimensional order parameter), there are only two possible values for the interaction energy of every single dipole and the effective field f +u = (H + 7 M ) M lî = \

(dipole "down"), (1)

\-u = -(H + 7M)/i (dipole "up"), where /i is the atomic dipole magnetic moment. Figure 1.3.2 shows a planar arrangement of dipoles predominantly oriented in the "up" direction. At T = 0 K, the tendency for the dipoles is to attain the lowest energy, with all dipoles pointing "up." (We ignore, for the moment, the zero point energy, which, because of the uncertainty principle, precludes a perfect alignment of all dipoles in one direction.) At T > OK, the ther nal energy tends to produce some misalignment of the dipoles, which imp] es a lowering of the effective field (through a lowering of M) and, correspondingly, a lowering of the interaction energy \ui\, which favors further the misalignment.

Ferromagnetic

23

Transitions

Fig. 1.3.2. Planar arrangement of elementary magnetic moments predominantly oriented in the "up" direction (energy + w) with a single moment oriented in the "down" direction (energy — ui).

The partition function for a dipole is given by Z = J2 exp(-w i /fc B r) = exp(w/fcBT) + exp(-uj/kBT).

(2)

i

Therefore, the number of "up" and "down" dipoles, according to elementary statistical mechanics, is given by N! = TV (up) =

(N/Z)exp(u/kBT),

N2 = JV(down) = (N/Z)

exp(-oj/kBT),

(3)

and the net magnetization (per unit volume) is given by y

ZJH- êxp(w/kBT)

+

exp(-u/kBT)

= %tanh^-ltt,

(4)

where Eqs. (l)-(3) have been used. Equation (4) involves, in implicit form, an equation of state for the ferromagnet in question, which can be rewritten as — =tanh I — ^ 7 — - £ ) ,

wthTc = - V - -

(5)

Equation (5) implies that, for T = OK, H — 0, the spontaneous magnetization is given by MB(0) = M s 0 = Nfi,

(6)

24


Transitions

corresponding to the fact that tanh(oo) = 1. It also implies that, for T = Tc, H = 0, the spontaneous magnetization becomes zero, MS(TC) = 0 due to the fact that the only solution of Ms/Nfi = tanh((M s /A//x)(T c /T)) is zero. This confirms the definition of Tc in terms of 7, N, /z, and fce given with Eq. (5). As mentioned before, Eq. (4) gives an implicit equation of state M = M(H,M,T). It is easy to get directly from Eq. (4) an explicit equation of state H = H{M, T) as follows ^ = ( | ) which, expanding tanh

H

% t a n h - ( ^ ) -

7

M ,

(7)

(M/Nfi) in a power series, becomes

/ M_\

7./V/Z

7

1 (MX 5 \NVLJ

•JNH

M

'Nil

(8) The power series expansion makes clear the analogy of this equation of state with the phenomenological equation of state in Landau's theory of phase transitions (see Chapter 1.9). The critical exponents for a Weiss ferromagnet can be easily obtained from the equation of state as follows. (a) Spontaneous magnetization. For H = 0, we can obtain MS(T), by means of Eq. (7), solving (Ms/Ms0) t a n h - 1 (M s /M s 0 )

(9)

Since 1 < (Ms/Ms0) < 0 for T < Tc, we can choose values of (Ms/Ms0) as closely spaced as desired and tabulate the corresponding values of (T/Tc). Figure 1.3.3 depicts the behavior of the (reduced) spontaneous magnetization as a function of (reduced) temperature calculated in this way. At T < Tc, (Ms/Ms0) < 1, Eq. (9) becomes (Ms/Msi

T (Ms/Msi

1 + i(M s /M s0 ) 2 +

(10)

Ferromagnetic

25

Transitions

Ms/Mso

T / Tc Fig. 1.3.3.

Spontaneous magnetization as a function of temperature.

and, consequently, T

1 / M.

Tc ~

3 VMs0

(11)

which implies Ms Ms0

1/2

V3

1-

for T < Tc.

(12)

Therefore, the critical exponent describing the temperature dependence of the spontaneous magnetization just below Tc is (see Table 1.1.2) /? = 0 1 n M 8 / 0 1 n | T c - T | =

2'

(13)

which is the same as that describing the coexistence curve for a Van der Waals gas. We note that this result is approximately valid only for dipolar uniaxial ferromagnets (to be distinguished from the normal exchange ferromagnets like Ni or Fe 2 03). Dipolar ferromagnets have low transition temperatures of the order of IK, while exchange ferromagnets have usually much higher transition temperatures, up to 103 K. For instance, for Fe,

26


Transitions

which is an "exchange ferromagnet," the critical exponent is j3 « 1/3, due to critical fluctuations near Tc, and is not taken into account in the Weiss model. (b) Critical isotherm. Likewise, for T = Tc, Eq. (8) gives

Iffi S ...

(14)

which yields a critical exponent 5 = d\n\H\/dln\M\

=3

(15)

again identical to that for a Van der Waals gas. (c) Magnetic susceptibility. For T > T c , Ms = 0, 1

- J_ dH ~

X ~ 47T 8M

l

T

*r (

1

•7A/ju 47T

~ 47T

4TTT C

(T-rc),

(16)

which is the Curie-Weiss law,

H, where 1 C

7

(17)

4TTTC '

9 = TC

(18)

(Curie temperature).

For T > Tc, M s = 0, and taking into account Eq. (12), 1

1

iV^y

X _ 1 ~ 4TT 7

3 1

1 /Ms\2 1

J_\

T

T 2;

+

3 \Mv) T

Nn 27 (T - r ) , 47rrc c

••yNfi

A-K

(19)

which is fully analogous to Eq. (16) but has a coefficient twice as large as before (T — T c ). Then, the critical exponent describing the temperature dependence of the low amplitude susceptibility, both for T > Tc and for

Ferromagnetic

27

Transitions

T (£) the electronic density of states at e = £, £ being the chemical potential. In the weak coupling limit (1/VD 3> 1), Eq. (5) can be used to obtain approximations for the gap as » 2fc B e D e- 1 / v ' 1 3 ,

A 0 « hojB/smh(l/VD)

for T ^ OK,

(6)

0 D being the Debye temperature, and as A«2\/2A;BTC[1-(T/TC)]1/2,

for T < T c ,

(7)

which describes approximately the behavior of the order parameter in the vicinity of the transition temperature for type I (second order transition) superconductors, under zero external magnetic field. The behavior of the specific heat near Tc can also be obtained from C =

d(HBCS)/dT

= dJ2 Ek-(Êk-

^ j

(1 - 2/(exP(£fe/fcBT) + l))

AT.

(8)

This expression, which is non-zero but finite at T < Tc since Tc, leads to a finite jump in specific heat at T = Tc AC « 27VfcB, which implies a zero specific superconductors.

heat

critical exponent

(9) for

type

I

34


Transitions

Conventional metallic superconductors have been the subject of intensive research for several decades, but the efforts to push up the transition temperature, Tc, in systems of this class have found only modest rewards, since it has proved almost impossible to increase the transition temperature beyond a few years after the discovery by Muller and Bednorz 2 of superconducting oxides with Tc above 30 K, several families of compounds with transition temperatures ranging from 30 to 125 K have been discovered and the expectations of getting compounds with even higher Tc are still high. These new compounds are in general very anisotropic, and they differ considerably from the metallic superconductors in several important respects (see Table 1.4.1). Perhaps the first question that comes to mind in considering the new oxide superconductors is the following: is their pairing mechanism the same, i.e., electron-phonon interaction, compared to the older, conventional superconductors? And, if this is not the case: which is the pairing mechanism in the new superconductors? In answering the first question, it seems to be clear that the upper theoretical limit in Tc for strong electronphonon coupled superconductors is about 40 K, well below the observed Tc = 125 K. Therefore, electron-phonon coupling alone does not seem sufficient to explain Cooper pairing in the new superconducting materials. In addition, no isotope effect on Tc has been unambiguously observed in them, indicating that the value of the Debye temperature, which enters the expression of Ts in the electron-phonon picture, does not influence appreciably the strength of the attractive interaction responsible for the electron pairing. Several mechanisms5 have been proposed as candidates for the pairing mechanism, including (i) charge density waves, (ii) spin density waves, (iii) excitons, and (iv) resonating valence bonds. The theoretical situation is far from clear for the time being, but some proposals (e.g., the resonating valence bond picture) seem to be losing ground, due to the fact that the onsite Coulomb correlation energy seems to be large enough in these materials. Others seem to be gaining support from inelastic neutron diffraction Table 1.4.1. Superconductive parameters for some bcc elements (cgs-emu units). Superconductor

Tc

Bso

fj, = ksTc/Bso

N = B s0 /47r/i

V Nb Ta

5.3 9.26 4.48

1020 1980 830

0.71 x 10~ 1 8 0.64 x . 1 0 - 1 8 0.74 x 1 0 " 1 8

1.14 x 10 2 0 2.44 x 10 2 0 0.88 x 10 2 0

Superconductive

35

Transitions

data, which appear to support a significant role for magnetic pairing. In any case, it appears not unlikely that once a realistic expression for the effective interaction V between two electrons is obtained, the observed behavior might be understood within the framework of a generalized BCS theory, which properly takes into account the highly anisotropic character of these materials. For a three-dimensional crystal, one can write, as can be seen from Eq. (4), Afc = ^ ^ A , t a n h ( f / 2 f c B r c ) ,

(10)

^k'

k'

where Vkk' is the electron pairing potential

Vkk, =V(q = k- #,&,&)

^2J2 T c ), in which every point in the lattice is occupied randomly by Cu and Zn atoms with 50% probability for either one of these atoms. At T = OK, if the temperature has been lowered down sufficiently slowly so that the system has remained in intermediate states of quasi-thermal-equilibrium, the structure of the alloy is one of almost perfect

37

38


TT C

f^TT^T --6./ OZn

£zn,Cu

#Cu

0Cu,Zn

Ordered phase Disordered phase Fig. 1.5.1.

Unit cell of CuZn in the ordered and disordered phases.

order. (Note, on the other hand, that a rapid quenching of the system from T > Tc to T « OK will produce a state largely disordered, because the characteristic relaxation times at very low temperatures become very large, and the disordered structure, which is far from the one with minimum free energy at these temperatures, becomes relatively stable.) If the temperature of the ordered structure is slowly increased again toward Tc, the degree of order of the sample will decrease gradually first, and more rapidly as Tc is approached, showing the characteristic behavior of a cooperative effect: an increase of disorder gives rise to more and more disorder. Figure 1.5.2 shows, schematically, anomalies in physical properties [specific heat, CP(T), electrical resistivity, p(T)], which give indirect evidence of the order-disorder transition in a metallic alloy, as well as direct experimental evidence [X-ray or neutron diffraction intensity IhkiiT) corresponding to superlattice reflections] of the transition. As mentioned earlier, the Bragg-Williams (1934) theory assumes that the degree of order (and disorder) in the alloy sample is homogeneously distributed over long distances. Since the order-disorder process is driven by the difference between local potential energy between pairs of unlike (AB) neighboring atoms and like (AA, BB) neighboring atoms, assuming a homogeneous distribution of order is equivalent to assuming an effective potential, which depends on the average degree of order at a given

Order-Disorder

Cp(T)

Transitions

/•(T)

39

in Alloys

Ihki(T)

Jc

Jc

Fig. 1.5.2. Anomalies in physical properties, CP(T) = specific heat, p(T) = electrical resistivity, Ihkl(T) = diffraction intensity of superlattice reflection, accompanying orderdisorder transition.

temperature. Thus, this assumption is in complete analogy with the effective field assumption in Weiss's theory of ferromagnetism, for instance. Consider a simple case such as that of /3-brass, whose unit cell is shown in Fig. 1.5.1. In the ordered phase there are two interpenetrating simple cubic lattices in which each atom A is surrounded by eight nearest neighbor atoms B and vice versa. In the disordered phase each atomic position can be occupied by either A or B atoms with equal probability (50%), and the unit cell of the crystal becomes body centred cubic. In the fully ordered state all A atoms occupy a positions and all B atoms occupy /3 positions. In a partially ordered state there will be R (right) A atoms in a positions and W (wrong) A atoms in j3 positions. Consequently, there will be also R atoms B in j3 positions and W atoms B in a positions. Then we can define an "order parameter" as

C

N

~

N

'

(1)

where N = R + W is the total number of A or B atoms in the crystal. The order parameter ( represents correctly the degree of order of the system and can take any value from ( = 1 for R = N (fully ordered state) to C = 0 for R = 1/2 (fully disordered state). Evidently, values of R < 1/2 should be excluded because in that case one would recover some order in the system by interchanging the labels of R and W. Our aim is to investigate the temperature dependence C(^) °f the order parameter. In thermal equilibrium, the free energy is a minimum, and we

40


Transitions

can write AF » AE - TASc{ ss 0,

(2)

where AE is any small change in internal energy associated with small changes in R and W, T the temperature, and A5cf the corresponding small change in configurational entropy. We have neglected the small change in thermal entropy A5th corresponding to changes in lattice vibrations. If we now take a minimum change in R (and W) corresponding to the interchange of one A atom initially in a "right" position by a B atom also initially in a "right" position we have AW = -AR

= 2

(3)

and we can write Eq. (2) as $(R, W) - kBTA ln[(N\/R\W\)]

« 0.

(4)

Therefore, taking into account that N is large (of the order of 1022 a t / cm ) and so are R and W, we can use Stirling's formula to approximate 3

Aln[(N\/RlW\)]

« AfTVlnJV - RlnR - Win W] « 2[lni? - InW]

using Eq. (3) and taking into account that N = R + W is a constant. Then we have, from Eq. (4) $(R, W) - 2kBTA \n[(R/W)} * 0

(5)

and, consequently, [R/W) = exp[${W, R)/2kBT],

(6)

which, when substituted into Eq. (1), provides a relationship between C(.R, W) and $(R, W). On the other hand, we can get another expression of $(R,W), taking into account the assumed large-scale homogeneity of the system, as a properly averaged function of $ A B , $AA, $ B B , which are the possible pair potentials between a given atom and its nearest neighbors. Taking into account that the respective probabilities for atoms to be in "right" or "wrong" positions are {R/N) and (W/N), respectively, the change in internal energy corresponding to an interchange of two neighboring atoms

41

Order-Disorder Transitions in Alloys

located at "right" positions is given by &{R,W) = -z

($AA + $ B B - 2$AB)

W ($AA + $BB - 2$AB) ( -Jj = z ( 2 $ A B - *AA - 2$BB)

R-W_ N

= $oC,

(7)

where z is the number of nearest neighbors, $ 0 = Z{2$AB — $ A A — $ B B ] and ( — (R — W)/N; according to Eq. (1) $o means the change in internal energy corresponding to interchanging two unlike atoms in the perfectly ordered state (i.e., at very low temperature) where this change has the largest value. It may be noted that, actually, Bragg and Williams assumed the relationship between &(R, W) and £ given by Eq. (7) rather than deducing it from the assumed large-scale homogeneity of the degree of order in the system. Consequently, we can write

c=

R-W _ (R/W)1'2 R + W ~ (R/W)1/2

(R/W)-1'2 + (R/W)-1/2 ~

1/2' tan

* Tc the order parameter is zero, and, accordingly, ACp = 0

(T>Tc).

(19)

Fluctuations, and other complications not taken into account in the simple Bragg-Williams theory, contribute to produce a more pronounced temperature dependence of the specific heat peak accompanying orderdisorder phase transitions in alloys.

Order-Disorder

Transitions

43

in Alloys

The transition heat per atom, given by the integral of the extra specific heat is, using Eqs. (15) and (10),

L o

^ ,

ACp(T)dT=^ 4

(20)

JO

a result that is fairly well fulfilled by the experimental data. Transition temperatures for order-disorder in alloys range in hundreds of degrees Kelvin. For instance, for (/3-brass *CuZn), the transition temperature is Tc = 740 K. Therefore, according to Eq. (10), $ 0 = z [2$ A B - *AA - 2 $ B B ] = 4fcBrc « 0.25 eV.

(21)

As mentioned before, the theory of Bragg and Williams does not take into account the possible existence of short-range order. It only considers long-range order, homogeneously distributed throughout the crystal. It can be easily illustrated with an example that overlooking short-range order leads to trouble. For instance, consider the case depicted in Fig. 1.5.3. The planar lattice shown would appear as disordered in the BraggWilliams theory, because each point in the lattice has equal probability of being occupied by A or B atoms. On the other hand, from a local point of view, it is clear that this arrangement possesses a high degree of order, almost perfect if one does not consider the discontinuity indicated by the vertical dotted line. These considerations led Bethe 4 and others to introduce a short-range order parameter, a, defined as follows. Let us call P(AnnA) the probability that an A atom has an A atom as nearest neighbor, and P(AnnB) the probability that an A atom has a B atom as its nearest neighbor. Then, we

o o o o •

•

•

#

•

•

0

0

• 0

0

o o o o • • • • • • • o o o o o o o o • • • • • • • o o o o o o o o • • • • • • • o o o o Fig. 1.5.3. Planar lattice showing zero order from the long-range order point of view, and a high degree of order from the short-range order point of view.

44


Transitions

define a in such a way that P(AnnA) = (1 + a)/2,

(22)

P(AnnB) = (1 - a)/2.

(23)

This implies that a = 1 corresponds to perfect order and a = 0 to perfect disorder. Then, dividing Eq. (22) by Eq. (23) and writing in the appropriate Boltzmann's factors,

which, to simplify matters, can be written, assuming that $AA = $ B B , as {TZ~)

=exp($SR/fcBT).

(25)

This implies In ( ^ f )

= $ S R /fc B T,

(26)

which can be compared with the result in Eq. (11) t a n h ' 1 C _ $o/4 which gives the long-range order parameter £ (in the Bragg-Williams theory). The concept of short-range order can be incorporated into the basic framework of the long-range order theory leading to improvements in the calculated specific heat peak, which in this way comes closer to the experimentally observed peak. References 1. See, e.g., G. Tamman, The States of Aggregation (Van Nostrand, New York, 1925). 2. See, e.g., A.J. Dekker, Solid State Physics (Prentice Hall, Englewood Cliffs, NJ, 1962). 3. W.L. Bragg and E. Williams, Proc. Roy. Soc. (London) A145, 699 (1934). 4. H.A. Bethe, Proc. Roy. Soc. (London) A150, 552 (1935).

Chapter 1.6

Ferroelectric Transitions

In 1921, J. Valasek,1 an American investigator, discovered anomalous dielectric properties in crystals of Rochelle salt, or "sal de la Seignette" (sodium potassium tartrate tetrahydrate) in the vicinity of 24°C, which were correctly interpreted as analogous to those accompanying a ferromagnetic transition. This led him to name the temperature at which the dielectric constant presented a sharp peak as the Curie temperature. Some years later, in the United States of America, C.B. Sawyer and C.H. Tower2 observed for the first time hysteresis loops in the P (electric polarization) versus E (electric field) dependence for Rochelle salt. This confirmed the analogy with ferromagnetic transitions. In 1943, G. Busch and P. Scherrer3 (working in Switzerland) discovered peaks in the dielectric constant of KDP (potassium dihydrogen phosphate) and several isomorphs at temperatures below room temperature. In 1943, during the Second World War, Wainer and Salomon (in the United States), Ogawa (in Japan), and Whul and Goldman (in Russia) found anomalous dielectric properties 4 in ceramic samples of barium titanate, which were similar in many respects to the anomalous properties previously found in Rochelle salt and KDP. After the war, several investigators, especially B. Matthias at Bell Labs. and R. Pepinski at the University of Pennsylvania, and their coworkers, discovered a multitude of new materials with phase transitions accompanied by dielectric anomalies. The use of the term "ferroelectrics," instead of the original term "seignetteelectrics" became widespread for these materials, and their number has kept growing. At present, the number of known ferroelectrics is in several hundreds and new ferroelectrics are often added to the list. In part, the initial drive to find new ferroelectrics was prompted by the excellent piezoelectric properties of these materials, which make them useful in "sonar" devices for submarine detection.

45

46


Transitions

A ferroelectric crystal can be defined as a piezoelectric possessing a spontaneous electric polarization that is reversible under the action of an external electric field. Out of the 32 existing crystal classes, 20 are piezoelectric (i.e., non-centrosymmetric). Twenty out of these 32 are pyroelectric (i.e., they possess a temperature-dependent spontaneous polarization). Those pyroelectrics in which the polarization can be switched back and forth along the polar axis under the action of an external electric field are called ferroelectrics. It may be noted that at T < T c the coercive field, i.e., the minimum value of the external electric field sufficient to reverse the spontaneous polarization, may become very large, even larger than the threshold dielectric breakdown field. On the other hand, in some crystals the molecular units can decompose chemically or can become altered at temperatures well below the expected Curie temperature for the phase transition, where its estimated coercive field is still too large. In this case, according to the previous definition, the crystal would not be ferroelectric, but a "frustrated" ferroelectric, in spite of the fact that it has a spontaneous polarization. So, the definition is artificial to some extent, because sample perfection and experimental conditions can be determinant, in many respects, of the realization or not of polarization reversal. One of the main characteristics of a ferroelectric is its nonlinear response to an external electric field. Figure 1.6.1 shows P vs. E curves for a typical ferroelectric at temperatures below the transition, close to the transition

D=E+4itP

Fig. 1.6.1. crystal.

P vs. E curves and Ps(T) for a typical second order transition ferroelectric

Ferroelectric

Transitions

47

and above it, and the characteristic temperature dependence of the spontaneous polarization. These curves can be easily displayed on the screen of an oscilloscope by means of a simple Sawyer-Tower circuit if the sample shows little electrical conductivity. If the conductivity is appreciable, the P vs. E curves would become distorted and it would be necessary to introduce a variable resistor at the reference capacitor in the Sawyer-Tower circuit to be able to get phase compensation and recover good looking hysteresis loops. The nonlinear ferroelectric polarization of the crystal in response to the external field is manifest, except at T > T c . The dielectric constant, which is denned as e = AndP/dE (esu-cgs units), shows also a markedly nonlinear behavior and a strong temperature dependence in the vicinity of the transition. This is shown in Fig. 1.6.2. Ferroelectric transitions are usually accompanied by pronounced anomalies near Tc in many other physical properties: structural properties (unit cell dimensions, atomic positions), thermal properties (specific heat, thermal conductivity), elastic properties (sound velocity and attenuation, elastic constants), optical properties (refractive indices, birefringence, optical activity), etc. This fact makes ferroelectric crystals useful in a variety of applications. In the case of uniaxial systems with rigid elementary dipoles reorientable in either one or the other of two opposite directions, the effective field approach to ferroelectric transitions 5 is completely analogous to the Weiss theory for ferromagnets described in one of the preceding chapters. The

Fig. 1.6.2. Dielectric response (e) as a function of field and low-amplitude dielectric constant e(T) for a typical second order transition ferroelectric crystal.

48


Transitions

effective field is given by EeS = E + (3P,

(1)

where E is the external field and /3P (with /? a generalized Lorentz factor depending on the geometry of the dipole lattice and P the electric polarization per unit volume) is the cooperative field due to partially ordered system of dipoles, which gives rise to a non-zero dipolar field on any point of the lattice. The energies associated with the two possible orientations of a given dipole are, therefore, w = ±(E + (3P)/J,, where fi is the elementary dipole moment. The partition function is the sum of only two Boltzmann factors with Wi — (+w) or (—w), i = 1,2, and the number of dipoles pointing in the direction favored and opposed by the effective field is given, respectively, by Ni = (N/Z) exp(w/kBT),

N2 = (N/Z) exp(-oj/kBT).

(2)

The polarization is then given by P=(N1-

N2)n = N^ tanh

{E

+ ^

.

(3)

Taking into account that as T approaches Tc from below at E = 0, P = Ps approaches zero, one gets • Tc = / ? V A B

(4)

P ~TCE + (3P~ = tanh T pNp Nft~

(5)

and, therefore,

The equation of state obtained in Eq. (5) in implicit form can be made explicit as ^ = | ^

t a n h

" ( i | ) -

/ 3 P

-

^

It is easy from here to get the behavior of the spontaneous polarization (E = 0) and the polarization along the critical isotherm (T = Tc) in the vicinity of Tc, as was done in the ferromagnetic case. Spontaneous polarization:

A = V3[1-(T/T C )] 1/2 . PsO

(7)

Ferroelectric

Transitions

49

Hence, d In Ps {N — No), and we take into account commutation relations for the boson creation and annihilation operators, [ak,ak'] = Sk,k',

[ainat

J = lak,ak>] = 0,

(13)

Superfluid

57

Transitions

we can write Hy « ^V0[aââoa0]

+ - ^ ] V0 [a% aka£a0] + - ^ ] V0[aâÎQa0] k,0,0

+ - Y^ Vk [aoataoak]

0,fc',0 V k

+ 2 S

fc,0,fc

~'

[at'aoak'ao\

0,k',-k'

+ 2 1 3 Vq[a±qa+aoa0] + ^ Y 0,0,9

V-q[aoaoaqa-q]'

(14)

q,-q,q

where, under the assumption that N0 > (N-N0), terms in the sum

we have kept only those

V a

i t-qat'+q - ak'ak

Y k,k' ,q

that contain at least one pair of factors aoao or a J a j corresponding to the densely populated ground state. Since we expect that iVo is a macroscopic number we can make the substitutions ao"(ao"a0)a0 = aJ(a 0 ao" - l)a0 = ajaoaôo - aâo —> NQ - N0 « iVg, a 0 ao -> VNOT/NQ

- 1 « AT0,

a+a+ -» v / T V o T l / N o T 2 « JV0, and taking into account that Vk = V-k> we get

Hv ~ ^N0V0

+ 2 N V

2 ° ° Y akak + 22No Y

V a a

ktk

k

+ -N0^2vq[aqa-gatqa+].

(15)

q

Then, we can rewrite the total hamiltonian as follows: H « -N0V02 + Y (&k + N0Vk)a+ak

+ ^ Y N0Vk(aka„k

+ a+ fe a fc ),

k

(16) where we can now make use of Bogolubov's procedure to transform it into a simplified hamiltonian. To this end we define the new operators ak = (cosh6k)ak - (sinh0fe)a+fc;

a £ = (cosh#fc)a£ - (sinh0fc)a_fc, (17)

58


Transitions

with 9k being a free parameter to be specified later. Here, the new operators ak and a_fc fulfill the commutation relation [ak,a^,\

=

fa,k'-

(18)

We can write, making use of the definitions in Eq. (17), = ^/iw f c [(cosh 20fc)aâfc + (sinh20fc)]

^hcukOtlak k

k

- - ^ M f e ( s i n h 26k)(aka-k

+ atkal)

(19)

k

and notice that the right-hand sides of Eqs. (16) and (19) have the same form, provided that we identify htok cosh 20k = Ek + N0Vk,

(20)

hu!k sinh 26k =

(21)

-N0Vk.

Since 8k was taken to be an adjustable parameter in the definitions of a~l and ak we can choose it in such a way that Eqs. (20) and (21) are satisfied. Consequently, (hwkf

= (Ek + N0Vk)2

- (N0Vk)2

(22)

2EkN0Vk]1/2.

(23)

and, therefore, uj{k) =wk = ±[El+

The elementary excitations of the system described by Eq. (23) are boson, or phonon-like, -, 1/2

Wfc «

-

h

^ l " ' " '

T.

/

N

1/2

-(*£) "

(24)

for Ek < 2N0Vk;

2 „ 4MN0Vk i.e., k2
3. The transition can be investigated in systems like this by a variety of techniques: optical spectroscopy, Raman and Brillouin scattering, X-ray and neutron diffraction, strain-stress characterization, EPR, etc. We will outline, first, the basis of the basic features of the formal theory worked out by Eliot 4 and Pytte 5 for these systems and then we will see that the simple effective field approach can also be satisfactorily applied to ferroelastic transitions of this kind. The mechanism that drives a cooperative Jahn-Teller transition is that of electron-phonon interaction. To understand it one may consider the fact that a lattice distortion splits the levels of a degenerate electronic state. This is accompanied by a decrease in electronic energy if the temperature is low enough to favor a substantial difference in population between the lower energy split levels and the higher energy ones. In this case, the overall energy of the system may become lower when the decrease in energy due to population of lower energy electronic states overcomes the increase in energy due to the lattice distortion. Then, the distortion becomes energetically favorable, and a ferroelastic phase transition takes place precisely at the temperature at which the gain in free energy in the electron system is equal to the loss in energy in the phonon system associated with the lattice distortion. The simplest conceivable case is that of the interaction of a doublet with a non-degenerated phonon mode. The corresponding hamiltonian is H = ^hw(q) q

+ êxp(iqRna(q)Sz(n)(aq

a+aq + L

J

+ o+)),

(1)

q,n

where a+, aq are respectively the annihilation and creation operators for the phonon with wave vector q, Rn is the position vector for the nth ion of the unit cell, Sz(n) is the pseudo-spin operator that determines the electronic state of the ion located at Rn, and a(q) is a coupling coefficient between the pseudo-spin and the vibrational state specified by q. Introducing the displaced "mixed mode" annihilation operator 7+

•

»(g)

it = TC,T
CT-rc)

forT>Tc

(772 = 0)

(35)

= A0(TC-T)

for T Tc 1 2

+

1 2

= So-Â0(A0/C) / (Tc-T) /

A(dr,2/dT)]h=0 (37) for T < Tc

(38)

for T > T27r rZIT

$ « $01 + kBT / / 7o Jo

ffk, Kmax

/ Jo

ln[a(T - Tc) + df?)(A; dp) dfc,

(64)

where (v/8ir3) is the density of points in the reciprocal lattice, 0 < 6 < TT, 0 <
)

(65)

The second integral can then be easily carried out, resulting in

•-*- mî: tan

cos e/y/{a{T

S0-

.-|1

- Tc) +

y/(a{T - Tc) + =

A;2 dfc 6k2/top2)

5k2/top2)

1^) (£) l ^ fc2d^4^2/(a(T-Tc) + ^ )

x tan - 1 ^ ^ / ( a ^ - T J + Jfc2).

(66)

Landau Theory and Effective Field

77

Approach

The region of interest is T = Tc, where k = 0; therefore, one can make the approximation t a n - 1 y/4np2/(a(T

- Tc) + 5k2) « tan _ 1 (oo)

(67)

and then write „

„

fkBTv\

( a \

/ - — ^ n fkm™

k2 dk y/a{T - Tc) + 6k2

(68)

k2dk • (a(T - Tc) + 5k2)3/2

(69)

and, for the specific heat,

ACp

=

vTdf

~

Cp0

kBT2a2 Sn3/2p J JQ

+

Sufficiently close to Tc, a(T — Tc) becomes much smaller than -) \dMW)k

(10) = constant,

=yV™1/0=x-3(6-1)*®w } dk/dt h'{xy

(11) (

}

In a certain close vicinity of the critical point where M1^ -C 1, 0 < k < l , a very weak dependence of $ on M 1 ^ and k may be expected (in analogy with the mean-field case). Hence, taking $ « $o along x = 0, we can integrate Eq. (11), obtaining X [* !£*idx = 3(8 1) [ ^—dx, Jo h(x) 'Jo ^ - * o i.e.,

h(x) h(0)

( x -$ V -*o

0(6-1)

(12)

or taking h(0) = ( - $ 0 ) w _ 1 ) , h(x)=(x-^0fS-1) = (x-%y.

(13)

84


Transitions

The integration has been performed along lines of constant k with origin in the critical isotherm (x = 0) either toward the low-temperature phase (x < 0) or toward the high-temperature phase (x > 0). Then, we should expect less accurate results as we approach x = 3>o and x « +oo, which are farther away from x = 0, where 0 (at T > Tc) going through x = 0, where

x = t{e)/M1/0(e) = (i-b2e2)/Ke = o, ec = r 1 = 0.7543.

(31)

Equation of State and the Scaling Function

89

(b) At the same time, h(x) = H/M5 goes from h(x) < h(0) to h(x) > h(0) crossing the critical point (see below) from T < Tc to T > Tc. In the linear model, on the other hand, h(x) goes through a maximum at 2 d \ H(9) 1 d A9(l-9 ) d9 M5{9) ~ d9

= 0,

9m=

(j~)

= 0-7969.

(32)

Then for 9 > 9C, up to 9 — 9m, h{x) is still increasing in spite of the fact that already T > Tc, and for 9 = 9m, h(x) decreases instead of increasing, reaching values lower than that corresponding to 9 — 9C. This shows that, while this model produces a good overall fit to the data, it leads to inconsistencies precisely nearest the critical point. Changing the value of 9C toward that of 9m would probably worsen the overall fit to the data. On the other hand, the simpler scaling function given by Eq. (13) gives an overall fit to the data as good as that for the linear model, and a better fit in the close neighborhood of the critical point, being free from the inconsistencies mentioned above. It is fairly obvious that h(x) = H/M goes over from h(x) < h(0) to h{x) > h(0) in crossing the critical point from T < Tc to T > Tc. Indeed, at the H = 0 line (which is the one that crosses the critical point) we have lim

(H/M5)

=0

for t < 0 {x < 0)

(33)

and, for t > 0 (x > 0), hin (H/M*) t 7 i < \\mo(H/M*)tn+v since M n + 1 < M„ for the same H. the critical point (H = 0, t = 0, M dummy variable r, which does not of values H{9) = 0, t(9) = 0, M(9) model.

0max- The relative speed of two helium atoms at this point may be estimated as vp = (vmax/R)r0,

(27)

where r0 is the nearest-neighbor separation between helium atoms. Since all bound Cooper pairs may be thought of as acting cooperatively to oppose a pair breaking, the energy necessary to break a pair may be assumed to be 2w = {Fs)vp = (NsMvP)vpj

(28)

where Fs is a generalized "effective field," conjugated with the velocity v, given in terms of the density of Cooper pairs present at NS(T) and the momentum of each helium atom within a pair. Prom here we can proceed like in previous sections to get

^

N

^ Ns0

= tanh (J?-)

= tanh (»«MW»*

\kBTj

\

T

£_)

,

iVs0y

(29)

v

;

which determines, in this simplified picture, the temperature dependence of the order parameter (superfluid Cooper pair density). Equation (29) can be written as Ns

^

= t a n (T

c

Ns \

Hr^J'

(30)

where Tc = Ns0Mvl/2kB.

(31)

Thus, we can get for the superfluid energy gap ^ A = Ns0Mv2p = 2kBTc,

(32)

which is analogous to the expression obtained in Chapter 1.4 for the superconducting gap. The maximum number of Cooper pairs in 4 He is given by N = p/2M = 1.09 x 10 22 pairs/cm 3

(p = 0.145g/cm 3 ),

(33)

which can be compared to ATs0 from Eq. (31) using the experimental values for Tc = 2.18K and for vp = (vmax/R)r0, where vmax = 0.9cm/s, R = 0.15cm, and VQ = 2.51 x 10 _ 8 cm may be used, resulting in vp — 1.5 x 10 _ 7 cm/s. 3

Appendix

99

W i t h these d a t a Ns0 = 2kBTc/Mvl

« 0.40 x 10 2 2 p a i r s / c m 3 ,

(34)

which is not far from t h e experimental maximum number of possible Cooper pairs at OK in 4 H e given by Eq. (33).

References 1. N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976). 2. J.P. Carney, A.M. Guenault, G.R. Pickett and G.F. Spencer, Phys. Rev. Lett. 62, 3042 (1989). 3. This can be extrapolated from data for noble gas crystals. See table in N.W. Ashcroft and D. Mermin, Solid State Physics, p. 401 (Holt, Rinehart and Winston, New York, 1976).

Part 2

Some Applications to Ferroelectrics: 1970-1991

Chapter 2.1

Behavior at T = Tc of P u r e Ferroelectric Systems with Second Order Phase Transition*

The second order transition in ferroelectric triglycine sulfate (TGS) seems to be a very good test case for the mean field theory. Previous work 1 ' 2 has shown that the behavior of the dielectric constant and the spontaneous polarization is in agreement with the mean field predictions. The present investigation, partially reported in a previous letter, 3 aimed at a more complete analysis of the order-disorder cooperative transition by means of a detailed study of the variation of polarization with electrical field, as well as with the temperature near the critical point. Accurate data of P versus E near Tc allow the determination of critical exponents through log-log graphic representations. In addition, once the two fundamental parameters /? and d, defined by (-PS)TRSTC = const, x [1 — (T/Tcyf and 1 6 (P)T^TC = const, x E / , are determined, the way is open for the search of a "law of corresponding states" in terms of the properly "scaled" variables. The sample preparation and experimental procedure were described in previous communications. The determination of the Curie temperature was done in two different ways. First, a plot of the squared spontaneous polarization (Ps2) vs. temperature was made, which showed an almost perfect linear behavior yielding Tc by extrapolation to Ps2 = 0. Alternatively, the method described by Kouvel and Fisher 4 was used, yielding the same result within experimental accuracy. It has been noted by Reese5 that corrections due to the electrocaloric effect should be considered. While the accurate

*Work previously published under the title "Equation of state for the cooperative transition of triglycine sulfate near T," J.A. Gonzalo, Phys. Rev. B 1, 3125 (1970). Copyright © 1970. The American Physical Society. 103

104


Transitions

determination of these corrections near Tc is not easy, reasonable estimates indicate that our results would not be substantially altered by them. It may also be noted that perfect compensation of the P vs. E hysteresis loops very near Tc could not be fully achieved with the Sawyer-Tower circuit, possibly due to a field dependence of the conductivity of the crystal. This behavior actually sets limits of AT = Tc - T at +0.22 and -0.04°C within which a reliable determination of P for very small E was not possible. As is well known, an increase of the amplitude of the ac field applied to the sample for displaying the P vs. E curve produces a relatively small increase of the absolute value of the polarization with respect to the corresponding values for our ac amplitude. However, the relative variation of the polarization as a function of temperature was checked for various field amplitudes and it was found to be the same, the absolute values being different only by a constant factor. This effect might be attributed to a consistently partial switching of the ferroelectric domains at low ac amplitudes. The constant ac field amplitude chosen in our case, E = 190V/cm, was relatively low, which helps to keep down the electrocaloric effect in the vicinity of Tc. The experimental results below T c , as described in a previous letter, 3 were shown to yield the value of the critical exponents S and (3, along with four other exponents indicating the field and temperature dependence of both derivatives of the polarization with respect to field and temperature. The experimental values obtained from log-log plots of the data are given in Table 2.1.1 and compared with those calculated from the mean field model, using the expression6 where EQ « 4.4 x 106 V/cm is the saturation internal field and N/u, = 4.3 /iC/cm 2 is the saturation polarization. It is interesting to note that, as it should be expected (see Appendix), the ratio of the two critical exponents relating the same derivative of the free energy to field and temperature is constant. In Table 2.1.2 a summary of data for polarization and field at various temperatures below and above Tc is given. These data were "scaled" to determine p = P/t^6 and e = E/t0S, and plotted using a log-log scale. It can be seen from Fig. 2.1.1 that the scaling of the data is quite good, giving the evidence of the existence of a law of corresponding states. From this, the sequence of critical exponents, found directly and reported in a previous short communication, 3 results in an automatic fashion. What is more important, however, is the fact that this log-log representation, which shows the critical behavior over a wide range of three decades in the reduced field e, shows clearly the asymptotic behavior of the equation of state for

Behavior at T = Tc of Pure Ferroelectric

Systems

105

Table 2.1.1. Experimental critical exponents from TGS compared with mean field theory predictions. Defining relationship

Mean field relationship

Experimental value

(i^teO-eTs

73

= 1/5 = 0.32 ± 0.02

( P ) e = 0 ~ ei*

74 = P = 0.50 ± 0.03

( — | ~e77 V9e/t=0

77 = 0.66 ± 0 . 0 5

dp

; e79

9tJt=o

'

"

-eTio

\dt)e=0

79 = - 0 . 3 3 ± 0.05

"

""

710 = - 0 . 4 5 ± 0 . 1 0

1

73

(P) e =o = ( 3 t ) 1 / 2

74 = / 3 = -

= 5 = 3

77 =

9e / t=o

)e=0

1

(P)t=o = p e ) 1 ^

1-P2)

~e~>s 78 = - i = 0.95 ±0.10

de

Theoretical value

-P2) f^) = £• \9eJe=0 (t-p2) _1 tanh p(l-p2) dp\

—

\dt)t=o 9p\

-tanh_1p(l-p2)

a«/ e =o'

(*- P 2 )

- -

78 = - 1

1 79 = — x

1 71

°

=

"2

Table 2.1.2. Polarization vs. field for T G S from hysteresis loops in the vicinity of the Curie temperature. Below Curie temperature 2

2

Above Curie temperature

P(/iC/cm )

AT ( x l O " °C)

E• ( V / c m )

P(/iC/cm2)

A T ( x l O - 2 °C)

E (V/cm)

0.217 0.217 0.217 0.244 0.244 0.244 0.271 0.271 0.271 0.271 0.298 0.298 0.298 0.298 0.326 0.326 0.326 0.326 0.326

4.2 10.7 17.1 4.2 10.7 17.1 4.2 10.7 17.1 30.1 4.2 10.7 17.1 30.1 4.2 10.7 17.1 30.1 43.0

49.7 27.7 9.3 73.0 46.9 24.5 101.8 72.4 46.4 7.0 137.9 104.5 74.6 25.4 182.7 144.5 104.6 53.8 9.9

0.1085 0.1085 0.1085 0.1085 0.163 0.163 0.163 0.163 0.217 0.217 0.217 0.217 0.298 0.298 0.298 0.298

2.3 8.7 15.2 21.7 2.3 8.7 15.2 21.7 2.3 8.7 15.2 21.7 2.3 8.7 15.2 21.7

12.7 21.1 30.3 40.2 33.1 49.3 60.6 71.9 69.8 88.1 104.3 117.7 131.8 148.7 166.4 186.1

(1 - t) t a n h _ 1 ( P ) -P,E

= E/E0,

t = 1 - (T/Tc),

P = P/Np,.

106

Effective Field Approach to Phase r—I

I I I rl

-I—I—I

Transitions

I I I 11

e. E/h-(T/T c |l^ 8

Fig. 2.1.1. Log-log plot of the scaled polarization vs. scaled electric field for ferroelectric TGS near the Curie temperature. The full line is renormalized mean field equation of state with m = 0.450, n = 0.139.

both small and large e, above and below T c . This asymptotic character is in complete analogy with the observations of Green et al.7 for liquid-vapor transitions in a good number of systems. We have also recently examined very accurate data 8 ' 9 from ferromagnetic transitions, and the asymptotic trend for small and large scaled magnetic fields is seen again to be fully analogous. The asymptotic behavior can be summarized as follows: Below Tc: (la)

pe—o = const., Pe-,00 = const, x e1/5.

(lb)

Above Tc: pe-»o = const, x e,

(2a)

1

(2b)

pe-,oo = const, x e ' .

Behavior at T = T c of Pure Ferroelectric

Systems

107

The implications of these expressions are obvious: (la) means that for E Tc. This equation can be expanded in the following way:

Efi=(p+

i p 3 + ^P5A\ id (P + i p 3 + ^ P 5 A ) - P

1

3 2 l 2 Putting e = eE/t / and p3= +tP/t Ml3 / + 5P 5 (1 ^ ) + A M = HP + gP

Obviously, if t < 1 (for instance, 1.0 x 10" 5 < t < 3.0 x 10" 3 in our experiment), this expression reduces itself to e_ = p f^p2

- 1J

for T < Tc,

(3a)

e+ = P (\p2

+ l)

for T > Tc.

(3b)

To check these equations against the experimental results it has been found necessary to introduce proportionality factors for e and p in the above equations. They become 1 2 2— me- = np If —np

me+ = np ( -n2p2 + 1 j . To introduce these proportionality factors is equivalent to modifying the normalization parameters in such a way that (Nfx)/n replaces (Nfi),


108

Transitions

and Eo/rn replaces EQ. The best fit to the data is obtained with m = 0.450,

n = 0.139.

Figure 2.1.1 shows a plot of the mean field equation of state in scaling form, along with the experimental data. The agreement is very good except for a few points for T < Tc in the intervening region between small and large e, which fall slightly above the theoretical curve. The estimated experimental errors go from 5 to 1% as p increases and from 10 to 2% as e increases. The realization that the asymptotic behavior specified by Eqs. (la)-(2b) is not only characteristic of our ferroelectric cooperative transition, but also of liquid-vapor and magnetic cooperative transitions, strongly suggesting the convenience of using it along with Widom's homogeneity requirements to specify the equation of state for the system under consideration throughout Tc and in its vicinity. Since the formulation of the homogeneity assumption 10 for the free energy of a cooperative system undergoing a second order phase transition, considerable progress has been made in the understanding of the critical phenomena. 11 Griffiths12 has studied the problem of constructing explicit analytic expressions for the equation of state relating the scaled extensive variable (polarization, magnetization, volume, etc.) to the intensive variable (electric field, magnetic field, pressure, etc.), respectively, for the case of rational critical exponents. Very recently, several empirical 9 ' 13 and parametric 14 expressions have been proposed to fit the equation of state of some real systems. We wish to construct a compact expression of the free energy in a simple way, matching the critical exponents sequence both above Tc and below it, as well as the asymptotic behavior indicated in the preceding paragraph, from the law of corresponding states. Let us assume, following Widom 10 and Griffiths,12 that the free energy about the critical point can be given simultaneously by

F{X,t) =

X^+^'s

^ + a^x^)+a4xWs)

+A

forX1//3,5»Tc),

h + hf^j+ht^)

(4a)

+A

for t0S > X (T < T c ).

(4b)

Behavior atT = Tc of Pure Ferroelectric

Systems

109

Here, X = X/X0 is the reduced intensive variable and t = 1 — (T/Tc) the reduced temperature. (It may be noted that while the expansion (4b) is very familiar in the literature since the introduction of the homogeneity assumption, relatively less attention has been paid to the complementary expansion (4a); Ho and Litster have made use of the latter in their recent work9 on CrBr3.) These expansions ensure a sequence of critical exponents of the expected form (F)t=0 «

X^)l\

C = (d2F/dt2)t=0

R

S =

X(6+l)/6-(l//36)

(dF/dt)t=o

(5a)

,X(S+l)/5-2(l/0S)^

(F)T=0«^+1),

Y =

t0(6+l)-06

(dF/dX)x=0

(5b)

(d2F/dX% ! = 0 w t«*+i)-2/M,

Yx =

where the "gap" exponents are (1/A) = 1/(36 and A = (36, respectively. The last two expressions of (5b) are easily recognizable as the defining equations for the indices (3 and —7 = (3 — A, respectively. Let us call Y the partial derivative of the free energy with respect to the variable X. Its meaning will be, of course, that of the respective extensive variable in the various cases (polarization, magnetization, volume, etc.). Prom (4a) we obtain OF dX

Y

Xl's

+

Xl's ai + a,2

x

- (36)

dx

t Xl/Ps

a\ + a 2 t 1

x /^

t

a3

+ 3a 3

x1/?6 t

+A

1 36

x //

t \Xl/Ps

(6a) X

&i+26 2

363

w]

+L

(6b)

or, in other words, y

= Y/t" = xlls[Cl + c2(x-1/(jS)

+ c 3 (ar 1 / / 3 5 ) 2 + L]

for x = X/tps y=[di+

d2(x) + d3(x)

2

+ L]

> 1 (T > T c ),

(7)

for x «; 1 (T < T c ).

(8)

Our aim is to get a single expression for y which combines Eqs. (7) and (8) at T < Tc, approaching each of them for the limiting cases of x » 1

110


Transitions

and x 1, and ^(x) for x -C 1. Above T c , it is clear that the spontaneous order ceases to be non-zero for x = 0, so it is reasonable to eliminate the contribution from ip2(x). At this point, the empirical asymptotic behavior indicated in the preceding paragraph should be incorporated. Below T c , for x -C 1, 4>2(x) should approach a constant, and for x 3> 1, ipi(x~1//3S) should also approach a constant, according to (la) and (lb), respectively. Above T c , ipi(x) does not exist and V'i(£~1/'8'5) should approach a value proportional to a;1_(1/'5)forI < 1 , remaining the same as below Tc for x 3> 1. One could try different functional expressions for tp2 and tp\, all of them susceptible to being expanded in the power series of the required form. In principle, a logarithm, a binomial, or an exponential would meet this requirement. However, after testing these three forms against the experimental data, not only for TGS but also for magnetic and liquid-vapor systems, one comes to the conclusion that the logarithm changes too slowly with x and the exponential, on the other hand, too rapidly, in order to satisfy the asymptotic behavior indicated. On this ground, only the binomial forms are left as satisfactory ones. The simplest binomial forms one can think of, meeting the above-mentioned requirements, are

1/0S

ip1(x-

)

= A

1+

fe) /

1

Mx) = B

-I/PS' -P6{l-1/S)

T \ "

if

r\i\

\

5

(11a)

-(l/06)(l-0)

(lib)

The exponents -08(1-1/5) inEq. (11a) and -(l/05)(l-0) inEq. (lib) are the simplest ones that keep the homogeneity of Eq. (9) from x/x\ <S 1 to x/x2 3> 1 through the whole range in x. Also, the former is automatically required by condition (2b).

Behavior at T = TC of Pure Ferroelectric Systems

111

As a check of the "phenomenological" equation of state obtained, the principal critical exponents may be calculated. Below T c , from Eqs. (6)(10), one obtains x < 1, x>i,

x «i,

Y « fo^ » Btp, 1

y « V'IX /'

5

(12) 1

5

« AI / ,

(13)

1 s

1

ay/sx = x ' d^1/dx + (1/5) x^ /^-Vi + t0dil>2/dX = Cit13-^

= di"7'.

(14)

Similarly, above T c , a; < 1, i»l, a; « 1,

y « ^ i ^ 1 / 5 » 0, y » ^ 8Y/8X

1 / < 5

(15) 1 5

(16)

« AX / ,

= X^^/dX -

= C^ "

+ {1/5) X 4

(1/

*

)_

Vi

7

(17)

= C2^ .

The constants that appear in Eqs. (14) and (17) are, respectively, _

A

1-/3

B

A

It is interesting to note that 7 = 7', also supported by available experimental evidence. Figure 2.1.2 shows that the use of Eqs. (11a) and (lib) in the equation of state given by (9) and (10) leads to excellent agreement with the experimental data for ferroelectric TGS, below Tc as well as above Tc in this case, y = p (scaled polarization) and x = e (scaled electric field). Only four dimensionless numerical parameters have been used, their values being A = 2.12,

Xl=ei=

0.907,

B = 1.87,

x2 = e2 = 1.425.

(18)

For various liquid-vapor systems, it was earlier reported by Green et al.7 that the data suggest a scaling law asymptotic behavior as that indicated by Eqs. (1) and (2). Using Green and coworkers' critical exponents, j3 = 0.35 and 5 = 5.0, one could try to fit Eqs. (9) and (10) to the data, given in the chemical potential-density representation, in order to determine A, x\, B, and x2. Since the scattering of the experimental points (collected from many authors on many different systems and temperature intervals) is fairly high, it does not seem to be justified. However, accurate data 1 5 for He 4 are available. By using the tabulated data of Roach, 15 one can

112


Transitions

• • E/ll-(T/T e l|/3S

Fig. 2.1.2. Log-log plot of the scaled polarization vs. scaled electric field for ferroelectric TGS near T c . Pull line is the phenomenological equation of state with A = 2.12, B = 1.87, xi = 0.907, x2 = 1.425.

calculate the fundamental critical exponents in the pressure-volume representation. Since the transition occurs at a very low temperature, it is not surprising that the asymmetry of the experimental data is considerable. It is convenient to bypass this difficulty by using the following definitions: (^gas — Miq) cx AT13 V{-AP)

(along coexistence curve),

- V(AP) - V(AP) oc AP1/S

(along critical isotherm),

(19) (20)

where V is the volume, T is the temperature, and P is the pressure. In this way, a nicely defined straight line for two decades up to the vicinity of the critical point is obtained in the log-log plot, which yields /?. The analog plot for S is only approximately linear in the last decade up to the vicinity of the critical point, and since the trend suggests an increasing value of S, we extrapolate to the closest value that does not violate Griffith's inequality, taking a' w 0. This results in (3 = 0.411

and

5 = 3.84

(21)

Behavior at T = TC of Pure Ferroelectric

113

Systems

These numerical values are somewhat different from those obtained by Roach 15 and Vicentini-Missoni13 but it should be taken into account that they used the density instead of the volume and neglected the asymmetry. By using the exponents given by Eq. (21), one can scale the data corresponding to several isotherms above and below T c . The results are seen in Fig. 2.1.3(a) where the phenomenological equation of state is represented, with

y= v =

AV/Vc a— Ifi

, and x — p ~

AP/Pc t06

where V = 14.49cc (per gram), Tc = 5.193K, Pc = 1710.0Torr. The equation that best fits the data has been obtained by using the dimensionless

err T l

'•• "I

60 50

t

; l//3«5) ;

^

a-

£7fc+l

value. It was observed that there is a certain "thermal hysteresis" between the rising and lowering temperature data. Sufficiently away from Tc both sets of data for decreasing and increasing T coincide very well. This might be, in principle, attributed to a pronounced temperature dependence of the a-alanine polar centers' relaxation time near Tc. A similar, and even more pronounced "thermal hysteresis" effect has been observed in specific heat measurements 7 ' 8 and it has been seen that the rising temperature data approach more closely the successive equilibrium states, as inferred from the fact that, for cooling, if one waits for very long times at a fixed T, the value of E\, approaches asymptotically the corresponding i?b value for heating. This effect is still under investigation and eventually the results would be published elsewhere. Figure 2.2.2(b) shows the zero field polarization dependence with temperature. In this figure we can see clearly the smooth decrease of the polarization through T c , which is due to the presence of a-alanine polar centers in the crystal lattice. Figure 2.2.3 gives a representation of Eb (normalized to its maximum value) versus PQ (normalized to the zero field polarization at T = Tc) obtained also from the same hysteresis loops at various temperatures. Only data for rising temperature are shown. It can be seen that, at least for

| > H ^

y •

&S/

' / •

/ / occurs in the very close vicinity of Tc (see Fig. 2.2.1) where T w Tc, and that there PQ/NH < 1, we can approximate Eq. (12) by Eh

Eh tanh E\bM

^bM

I^Po_ NaNn

H

P^ Nil

Nn

N A /P0 Na3 \NIM

(13)

For T = Tc, we define Ehc = Eh (T = Tc) and P0c = P0 (T = Tc). Then, Ebc/Eh = Ebc/Ebu = 0.5 in all cases, and Po/N/x = P0c/N(j,, changing with the a-alanine concentration in such a way that the larger that concentration is, the larger POC/NJJL becomes. Then, right at the transition temperature, Eq. (13) reads, (0.5) tanh" 1 (0.5)

N_\ ATa3

Nn

(14)

and from Eqs. (13) and (14) we get 1/4

Pv_ PQC

( # - ) tanh" (0.5) t a n h - 1 (0.5)

(15)

This expression, obtained in a straightforward way from simple statistical physics considerations, fits very well the experimental data for all samples examined, as shown in Fig. 2.2.3. A good test for the results obtained from the statistical theory developed in the last paragraph consists in checking whether it is able to give estimates of the alanine concentration in the samples (not in the aqueous solutions from which they were grown, where the concentration is, of course, much higher) in agreement with the chemical analysis determinations previously made by other authors 9 on samples with similar values of internal bias.

128


Transitions

From Eq. (14) we get

Since we know P 0 c experimentally (see Fig. 2.2.2(b) for T = Tc) and Nfj, = ( P S ) M = 4 / i C / c m 2 = 1.2 x 10 3 esu, we can substitute in Eq. (15) the values of (Poc/Nfi), which are 0.04 for sample # 1 , 0.075 for sample # 2 , and 0.09 for sample # 3 . These lead to values of Na/N ranging from 10~ 4 to 10~ 5 , i.e., 10-100ppm, which is in good agreement with those quoted by Novik et al. for samples with room t e m p e r a t u r e internal bias in the same range as those of our samples.

References 1. M.E. Lines and A. Glass, Principles and Applications of Ferroelectric and Related Materials (Oxford University Press, Oxford, 1980). 2. K.L. Bye, P.W. Whipps and E.T. Keve, Ferroelectrics 4, 253 (1972). 3. K.L. Bye, P.W. Whipps, E.T. Keve and M.R. Josey, Ferroelectrics 7, 179 (1974). 4. E.T. Keve, Phillips Tech. Rev. 35, 247 (1975). 5. J.L. Martinez, A. Cintas, E. Dieguez and J.A. Gonzalo, Ferroelectrics Lett. 44, 221 (1983). 6. J.L. Martinez, J.E. Lorenzo, E. Dieguez and J.A. Gonzalo, Cryst. Lattice Defects Amorph. Mater. (1987). 7. J. del Cerro, Private communication. 8. F. Jimenez, Ph.D. Thesis, Universidad de Sevilla (1987). 9. V.K. Novik, N.D. Gavrilova and G.T. Galstyan, Sov. Phys. Crystallogr. 28, 684 (1983). 10. J.A. Gonzalo and J.R. Lopez-Alonso, J. Phys. Chem. Solids 25, 303 (1964). 11. J.A. Gonzalo, Phys. Rev. B9, 3149 (1974).

Chapter 2.3

Mixed Ferro-Antiferroelectric Systems and Other Mixed Ferroelectric Systems*

Considerable attention has been devoted in recent years to the investigation of phase transitions in mixed crystals with competing interactions and, in particular, to the mixed ferro-antiferroelectric system 1 (RDP)i_! (ADP)^ made up of rubidium and ammonium dihydrogen phosphate. The phase diagram of this system is shown in Fig. 2.3.1. In this chapter, we will investigate first a simple two-sublattice classical model for ferro- and antiferroelectricity, which will be shown to describe consistently the transition for pure RDP and pure ADP. Next, we will consider the behavior to be expected for the phase transition classically, i.e., in the absence of quantum fluctuations of the elementary dipole moments. Finally, we will introduce the appropriate quantum corrections in the expressions for Tc and TN and will show that satisfactory agreement with the observed phase diagrams is obtained. It is well known that crystals isomorphous with potassium dihydrogen phosphate (KDP) often show ferroelectric or antiferroelectric phases at low temperatures. The fact that an antiferroelectric arrangement of dipoles is sometimes realized suggests that a simple two-sublattice model, similar to the one for antiferromagnets, 2 may be useful to describe the phase transition in the KDP family. Since there are four chemical units (Z = 4) per (conventional) tetragonal unit cell in KDP crystals, the system may be thought of as a superposition of two interpenetrating bcc sublattices, each *Work previously published under the title "Quantum effects and competing interactions in crystals of the mixed rubidium and ammonium dihydrogen phosphate system," J.A. Gonzalo, Phys. Rev. B 39 (1989). Copyright © 1989. The American Physical Society. 129

130


150

Transitions

IROPVx lAOPl*

P£ (Telral

(Af£ JtOrtho)

'Dipolc glass phase reported 0 0.2 RDP

0.4

0.6

0.8

X

1.0 p

AD

Fig. 2.3.1. Phase diagram of (RDP)i_ ; E (ADP)x. Observed, solid line (see Ref. 1); theory (including quantum effects); crosses [Eqs. (22) and (23) with xc = 0.22 and yN = 0.26, respectively; see text].

containing two chemical units. Let us call A and B as these two sublattices, and take into account that we must consider intralattice (AA, BB) and interlattice (AB, BA) interactions. In the mean field approximation, the local effective field acting on unit dipoles at points in lattice A and B, respectively, will be (£ eff )a = £ + aPa + /3Pb,

(1)

(£eff)b - E + f3Pa + aPh,

(2)

where E is the external field, a(AA) = a(BB) and /3(AB) = /3(BA) are mean field coefficients, and Pa and Pb are the pertinent sublattice polarizations per unit volume. Using these effective fields in the standard 3 mean field expression for the polarization as a function of temperature we get

Nfia

tanh

kT

^..^(Ggte),

(3)

(4)

Mixed Ferro-Antiferroelectric

Systems

131

where N is the total number of sublattice sites (or unit cells) per unit volume, and /i a and /^b are the respective sublattice unit dipoles, which in the ordered phase can be pointing parallel (ferroelectric case: \ia = /Ub) or antiparallel (antiferroelectric case: \ia = —/^b) to each other, depending on the geometry of the lattice. Since the two sublattices are chemically identical, we take |/i a | = |/-*b|- From Eqs. (l)-(4), we get E=—

kT P t a n h - 1 - A - - aPa - (3Pb, Ma N/J,a kT

E=—

Pu tanh-1-^-/3Pa-aPb, Mb Nfih

(5) (6)

which, together, define an equation of state for E, P = P a + Pb and T. Let us consider first the ferroelectric case, in which at T < Tc we have a non-zero spontaneous polarization P s = P s a , Psb = 0 for E = 0. Adding Eqs. (5) and (6), and taking into account that t a n h - 1 Z± ± t a n h - 1 Zi = t a n h - 1 ( Z i ± ^2)/(l ± Z\Z-i) and that \i& — /J,\, = /z/2, we get 2£ = 2 - t a n h

\

+ A{Pa/NîPh/Nfl)-(Oc/2A;Tc(0) -C 1 and hu)o^/2kT^(0) <S 1. The last two equations yield Tc(0) = ( ^ 0 c / 2 f c ) / t a n h - 1 ( Z c ) ;

Zc = /iwoc/2fcTc*(0),

(20)

T N (0) = (/iw 0N /2fc)/tanh- 1 (Z N );

Z N = hu;ON/2kT^(0),

(21)

and, therefore, we finally get Tc(x) = T c ( 0 ) t a n h - 1 ( Z c ) / t a n h - 1 [ Z c / ( l - a;)],

(22)

T N (0) = r N ( 0 ) t a n h _ 1 ( Z N ) / t a n h - 1 Z N / ( 1 - j/)].

(23)

Since Tc(xc) = 0 for Zc = 1 — a;c and TN(T/N) = 0 for ZN = 1 — 2/N, it is sufficient to know the experimental values for xc and yN to completely specify the phase diagram. For the (RDP) x (ADP)i_ x system, xc ~ 0.22, 2/N ~ 0.26 (see Fig. 2.3.1). The agreement between theory and experiment for this mixed system is excellent, as seen in Fig. 2.3.1. References 1. For a recent review, see, for example, H. Terauchi, Phase Transitions A7, 315 (1986) and references therein. 2. See, for example, A. Dekker, Solid State Physics (Prentice Hall, Englewood Cliffs, NJ, 1962), pp. 484-488. 3. See, for example, J.A. Gonzalo and J.R. Lopez Alonso, J. Phys. Chem. Solids 95, 303 (1964); J.A. Gonzalo, Phys. Rev. B9, 3149 (1974).

Mixed Ferro-Antiferroelectric Systems

135

4. Ferroelectrics and Related Substances, edited by K.H. Hellmege and A.M. Hellwege, Landolt-Berstein, New Series, Group III, Vol. 16 (Springer-Verlag, Berlin, 1981 and 1982). 5. U.T. Hochli, Ferroelectrics 35, 17 (1981); see also K.A. Miiller, Jpn. J. Appl. Phys. (Suppl.) 24, 89 (1985). 6. W. Windsch, H. Braeter and B. Milsch, Ferroelectrics 47, 213 (1983).

Chapter 2.3.1

Comment on "Ferroelectricity in Zinc Cadmium Telluride"

Ferroelectricity in Zn^Cdi-^Te has recently been reported by R. Well et al.1 for Zn content from 4 to 45%. This is very interesting because, for the first time, ferroelectricity is found in materials crystallizing in the zincblende configuration, either of the binary constituents making up the mixed crystals being non-ferroelectric. Attempts by the authors 1 to fit the experimental curves giving the transition temperature as a function of Zn concentration, Tc(x), and the Curie constant as a function of Zn concentration, C(x), were unsuccessful because of the difficulty to match the rapid decrease of Tc(x) for x < 0.04. The aim of this comment is to show, hopefully, that the dipolar, effective field approach, 2 to ferroelectric transitions applied to this system can give a fair description, at least in semi-quantitative terms, of the dependence of transition temperature (Tc) upon concentration (x). The pure CdTe structure is made up of Cd tetrahedra with Te atoms at their center. Substitutional Zn atoms give rise to tetrahedra with 0, 1, 2, 3, and 4 zinc atoms, with different probabilities determined by the average concentration i in a homogeneous sample. Obviously, 0 and 4 zinc atoms produce symmetric tetrahedra with no dipole moment. It may be assumed also that 2 zinc atoms do not give rise to any dipole moment along (111). On the other hand, 1 and 3 zinc atoms produce asymmetric tetrahedra with possible non-zero dipole moments pointing in (111) directions. The probability of realization for tetrahedra with 1 zinc atom is p\x{l — x) 3 , and for tetrahedra with 3 zinc atoms it is p3 = x3(l — x). The effective field2 at a dipole site will be, therefore, Erf = E + frPi + 137

foPs,

(1)

138


Transitions

where E is the external field, (3\ and P3 are effective field coefficients, and P\ and P3 are polarizations due to tetrahedra with 1 and 3 zinc atoms, respectively. These polarizations are given2 by Pi/Nun P3/N3fi3

=tanh(EeStx1/kBT), =

(2)

tanh(EeSfjL3/kBT),

where Nx = (4/o~ 3 ) x (1 - x) 3 , i\r3 = (4/a~ 3 )x 3 (l - x), and Hi, /j,3 are the respective elementary dipole moments. Combining the relationships for P\ and P3 in Eqs. (2), one gets PiNi/n t a n h - 1

(PI/NKH)

= [(faNtvD/kBT]

+ p3N3^3 tanh" 1 ( A P i + p3P3),

which for T < T c , i.e., P i / A ^ i TO, optic, for regime (3) and VA, short wavelength acoustic, for regime (2)], we get reasonable estimates for t h e potential barrier heights. It may be pointed out, among other things, t h a t A t / from high-frequency relaxation d a t a 1 6 on e(w) at T < Tc gives A t / = 0.06 ev, in approximate agreement with A t / = O.lOev from (z m )3 as a function of field; Ecs(T) agrees well with the thermodynamic (Landau expansion of free energy) coercive fields, along with /? 3 ss 1 « 4wTc/C. Ec2 is two orders of magnitude lower t h a n Ec3, corresponding with the usual hysteresis loop coercive field value at room temperature.

References 1. See, e.g., M.E. Lines and A. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon Press, Oxford, 1977). 2. C.F. Pulvari and W. Kuebler, J. Appl. Phys. 29, 1742 (1958). 3. A.G. Chynoweth and W.L. Fledman, J. Phys. Chem. Solids 15, 225 (1960). 4. E. Fatuzzo and W.J. Merz, Phys. Rev. 116, 61 (1959). 5. B. Binggeli and E. Fatuzzo, J. Appl. Phys. 36, 1431 (1965). 6. M. Hayashi, J. Phys. Soc. Japan 33, 616 (1972). 7. A. Jaskiewicz and J. Przeslawski, Phys. Stat. Sol. (A) 56, 365 (1979). 8. R. Perez, E. Toribio, J.A. Gorri and L. Benadero, Ferroelectrics 74, 3 (1987). 9. R.C. Miller and G. Weinreich, Phys. Rev. 117, 1460 (1960). 10. J. Axe, Jpn. J. Appl. Phys. 24 (suppl. 24-2), 46 (1985). 11. See, e.g., A. Dekker, Solid State Physics (Prentice Hall, Englewood Cliffs, NJ, 1962). 12. F. Jona and G. Shirane, Ferroelectrics Crystals (Pergamon Press, New York, 1962) and references therein. 13. H.H. Wieder, J. Appl. Phys. 35, 1224 (1964). 14. G. Luther, Phys. Stat. Sol. A 20, 227 (1973).

Chapter 2.6

Polarization Switching by Domain Wall Motion*

Ferroelectric polarization reversal under external field pulses 1-9 of variable strength has been the subject of extensive research since the 1950s. It may be noted that most previous works 1-9 studied ferroelectric switching at room temperature. In this paper, switching measurements have been undertaken near T c , with the aim of obtaining a description of the field and temperature dependence of the maximum switching current (im) at low fields. We will summarize 11 and complement previous theoretical results for low field switching in ferroelectrics, which will be compared later with our experimental results. It is well known that TGS single crystals grown from water solution (usually at temperatures well above room temperature) suffer a thermal shock when they are taken out of the solution and it is also known that cooling shocks 11 give rise to many small spike-shaped microdomains (see Chapter 2.5). For simplicity, we may assume that the extended microdomains are equidistant parallelepipeds arranged periodically with an average distance 2r m between nearest neighbors. For t < tm, the rate equations then are 11 (similar equations can be obtained for tm < t < ts)

£--[(:H(i)~+-(0(f)-

(1)

*Work previously published under the title "Temperature dependence of polarization reversal in TGS family crystals near T c , " M. de la Pascua, P. Sanchez, J.E. Lorenzo, and J.A. Gonzalo, Ferroelectrics 94, 401 (1989). Copyright © 1989. Gordon and Breach Science Publishers S.A. 155


156

+n4

dt

Transitions

Pab - n4

u

Pba,,

(2)

where Na and ATb are t h e number of unit cells per unit volume with ± polarization, respectively, n is t h e number of microdomains per unit surface (n = l / 2 r ~ 2 ) , u is t h e lateral distance between neighboring unit cells (for T G S , u = (arcsin/3) 1 ' 2 , d is t h e sample thickness, b is t h e unit cell parameter along the ferroelectric axis, and pab, pb& are t h e transition probabilities (per dipole unit time): Pab=PO2[{E

+

Pba = m[-(E P02 =

02P)lM/kBT\, +

(hPMksT],

£l2exp[-Av2/kBT],

where v2 is a n a t t e m p t frequency a n d /32 t h e domain mean field coefficient. Subtracting Eq. (1) from Eq. (2), multiplying by /x, and taking into account t h a t N = Ns (d/b), we have: 1/2

16 ( n_

dPd dt

NfJL[l

+ - cosh Since Nfj, follows:

Tc(E

+ /32Pd

T

const.

{I + u+ f

(3)

esu 3> 1/2, we can rewrite this equation as 1/2

Pd Ps

1

sinh

T h e maximum value of (dPd/dt) vanishes: d2Pd dt 2

Tc (E + (32Pd T V EsQ

sinh

}

Eso

10->22 x 10

dPd dt

+

1/2

Pd

TCE + 02Pd E> sO

occurs when t h e second derivative -1/2

— sinh

const

(4)

Ps 1/2

T E,sO

cosh

TCE + 02Pd T

Es0

TCE + /3 2 P d E.sO

J dt

(5)

¥Es0

(6)

This implies, at t = t m , tanh

\TcE + foPd E,sO

-2P S

1+

Pdn T

Polarization

Switching by Domain

Wall

Motion

157

and, therefore, -2PS

1/2

I —•— I

dm

1+

= const. I 1 , „

1-

Pdm^\ Tc ft ' Ps J 1 T Es0 PdnO\TC (32~ Ps J ' T Es0

-2R 1

1/2-

(7) On the other hand, as shown below (see Eq. (13))

1+

E - (32PS

dm

(8)

3/?2Ps

Ps

and we can write: 1/2

N» (^j1'

im « 6Av2 exp(-AEyfcT) ( jfj" 1/2

E - (32PS 3-BsO

(^

^

x 2 ,

Tc E - f32Ps T 3Eso

1-4

-. 2 ^ - ! / 2

3ESQ

(9) 2

At low fields [ ( £ - / 3 2 P s ) / 3 £ s 0 ] < 1, where /32PS = Ec2 = Ec is the lowfield regime coercive field and including all constant factors in C (constant for each temperature), we get finally: = C(E -

Ecf'2.

(10)

We show next that Wieder's empirical relation 13 for the asymmetry of the switching current peak can be obtained from the present model. From Eq. (6), we can obtain ^

T

/?2Pdm=tanh-i

+

_22 PK [U1 ++ ^ ^ Z k A Ps )

3Eso

TEs0

(11)

and then E= — ^ o t a n h " 1 J- C.

2 1+

Pdm\ TC(32PS T E.sO

/?2Pdm.

(12)

Taking into account that f32Ps

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