Theory Of Superconductivity, A Primer

THEORY OF SUPERCONDUCTIVITY A Primer by

Helmut Es hrig

Dire tor of the Institute for Solid State and Materials Resear h and

Professor for Solid State Physi s, Dresden University of Te hnology

These le ture notes are dedi ated, on the o

assion of his eighties birthday on 4 July 2001, to Musik Kaganov who has done me the honor and delight of his friendship over de ades and whom I owe a great part of an attitude towards Theoreti al Physi s.

2

Contents

1 INTRODUCTION

4

2 PHENOMENA, LONDON THEORY

5

2.1 Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 London theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Gauge symmetry, London gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 THE THERMODYNAMICS OF THE PHASE TRANSITION

3.1 3.2 3.3 3.4

The Free Energy . . . . . . . . . The Free Enthalpy . . . . . . . . The thermodynami riti al eld Heat apa ity jump . . . . . . . .

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4 THE GINSBURG-LANDAU THEORY; TYPES OF SUPERCONDUCTORS

4.1 4.2 4.3 4.4 4.5

The Landau theory . . . . . . . . . . The Ginsburg-Landau equations . . The Ginsburg-Landau parameter . . The phase boundary . . . . . . . . . The energy of the phase boundary .

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13

13 14 15 16

17

17 18 20 21 23

5 INTERMEDIATE STATE, MIXED STATE

25

6 JOSEPHSON EFFECTS

32

5.1 The intermediate state of a type I super ondu tor . . . . . . . . . . . . . . . . . . . . 25 5.2 Mixed state of a type II super ondu tor . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3 The ux line in a type II super ondu tor . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.1 The d. . Josephson ee t, quantum interferen e . . . . . . . . . . . . . . . . . . . . . . 33 6.2 The a. . Josephson ee t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 MICROSCOPIC THEORY: THE FOCK SPACE

7.1 7.2 7.3 7.4

Slater determinants . . . . . . . . . . The Fo k spa e . . . . . . . . . . . . O

upation number representation . Field operators . . . . . . . . . . . .

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8 MICROSCOPIC THEORY: THE BCS MODEL

8.1 8.2 8.3 8.4

The normal Fermi liquid as a quasi-parti le gas . The Cooper problem . . . . . . . . . . . . . . . . The BCS Hamiltonian . . . . . . . . . . . . . . . The Bogoliubov-Valatin transformation . . . . .

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9 MICROSCOPIC THEORY: PAIR STATES

37

37 38 39 40

42

42 44 46 47

51

9.1 The BCS ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 9.2 The pair fun tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 9.3 Non-zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

10 MICROSCOPIC THEORY: COHERENCE FACTORS

10.1 10.2 10.3 10.4

The thermodynami state . . . . . . . . The harge and spin moment densities . Ultrasoni attenuation . . . . . . . . . . The spin sus eptibility . . . . . . . . . .

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3

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55 55 56 57

1

INTRODUCTION

These le ture notes introdu e into the phenomenologi al and qualitative theory of super ondu tivity. Nowhere any spe i assumption on the mi ros opi me hanism of super ondu tivity is made although on a few o

asions ele tron-phonon intera tion is mentioned as an example. The theoreti al presuppositions are ex lusively guided by phenomena and kept to a minimum in order to arrive at results in a reasonably simple manner. At present there are indi ations of non-phonon me hanisms of super ondu tivity, yet there is no hard proof up to now. The whole of this treatise would apply to any me hanism, possibly with indi ated modi ations, for instan e a symmetry of the order parameter dierent from isotropy whi h has been hosen for the sake of simpli ity. This is a primer. For ea h onsidered phenomenon, only the simplest ase is treated. Referen es are given basi ally to the most important seminal original papers. Despite the above mentioned stri t phenomenologi al approa h the te hni al presentation is standard throughout, so that it readily

ompares to the existing literature.1 More advan ed theoreti al tools as eld quantization and the quasi-parti le on ept are introdu ed to the needed level before they are used. Basi notions of Quantum Theory and of Thermodynami s (as well as of Statisti al Physi s in a few o

asions) are presupposed as known. In Chapter 2, after a short enumeration of the essential phenomena of super ondu tivity, the London theory is derived from the sole assumption that the super urrent as an ele tri al urrent is a property of the quantum ground state. Thermoele tri s, ele trodynami s and gauge properties are dis ussed. With the help of simple thermodynami relations, the ondensation energy, the thermodynami

riti al eld and the spe i heat are onsidered in Chapter 3. In Chapter 4, the Ginsburg-Landau theory is introdu ed for spatially inhomogeneous situations, leading to the lassi ation of all super ondu tors into types I and II. The simplest phase diagram of an isotropi type II super ondu tor is obtained in Chapter 5. The Josephson ee ts are qualitatively onsidered on the basis of the Ginsburg-Landau theory in Chapter 6. Both, d. . and a. . ee ts are treated. The remaining four hapters are devoted to the simplest phenomenologi al weak oupling theory of super ondu tivity on a mi ros opi level, the BCS theory whi h gave the rst quantum theoreti al understanding of super ondu tivity, 46 years after the experimental dis overy of the phenomenon. For this purpose, in Chapter 7 the Fo k spa e and the on ept of eld quantization is introdu ed. Then, in Chapter 8, the Cooper theorem and the BCS model are treated with o

upation number operators of quasi-parti le states whi h latter are introdu ed as a working approximation in Solid State Physi s. The nature of the harged bosoni ondensate, phenomenologi ally introdu ed in Chapter 2, is derived in Chapter 9 as the ondensate of Cooper pairs. The ex itation gap as a fun tion of temperature is here the essential result. The treatise is losed with a onsideration of basi examples of the important notion of oheren e fa tors. By spe ifying more details as lower point symmetry, real stru ture features of the solid (for instan e

ausing pinning of vortex lines) and many more, a lot of additional theoreti al onsiderations would be possible without spe ifying the mi ros opi me hanism of the attra tive intera tion leading to super ondu tivity. However, these are just the notes of a one-term two-hours le ture to introdu e into the spirit of this kind of theoreti al approa h, not only addressing theorists. In our days of lively spe ulations on possible auses of super ondu tivity it should provide the new ommer to the eld (again not just the theorist) with a safe ground to start out.

1 Two lassi s are re ommended for more details: J. R. S hrieer, Theory of Super ondu tivity, Benjamin, New York, 1964, and R. D. Parks (ed.), Super ondu tivity, vol. I and II, Dekker, New York, 1969.

4

Figure 1: Resistan e in ohms of a spe imen of mer ury versus absolute temperature. This plot by Kamerlingh Onnes marked the dis overy of super ondu tivity. (Taken from: Ch. Kittel, Introdu tion to Solid State Physi s, Wiley, New York, 1986, Chap. 12.) 2

PHENOMENA, LONDON THEORY

Helium was rst lique ed by Kammerling Onnes at Leiden in 1908. By exhausting the helium vapor above the liquid the temperature ould soon be lowered down to 1.5K. Shortly afterwards, in the year 1911, it was found in the same laboratory1 that in pure mer ury the ele tri al resistan e disappeared abruptly below a riti al temperature, T = 4.2K. Deliberately in reasing ele tron s attering by making the mer ury impure did not ae t the phenomenon. Shortly thereafter, the same ee t was found in indium (3.4K), tin (3.72K) and in lead (7.19K). In 1930, super ondu tivity was found in niobium (T = 9.2K) and in 1940 in the metalli

ompound NbN (T = 17.3K), and this remained the highest T until the 50's, when super ondu tivity in the A15 ompounds was found and higher T -values appeared up to T = 23.2K in Nb3Ge, in 1973. These materials were all normal metals and more or less good ondu tors. In 1964, Marvin L. Cohen made theoreti al predi tions of T -values as high as 0.1K for ertain doped semi ondu tors, and in the same year and the following years, super ondu tivity was found in GeTe, SnTe (T  0.1K, ne  1021 m 3 ) and in SrTiO3 (T = 0.38K at ne  1021 m 3, T  0.1K at ne  1018 m 3). In the early 80's, super ondu tivity was found in several ondu ting polymers as well as in \heavy fermion systems" like UBe13 (T  1K in both ases). The year 2000 Nobel pri e in Chemistry was dedi ated tho the predi tion and realization of ondu ting polymers (syntheti metals) in the late 70's. 1 H. K. Onnes, Commun. Phys. Lab. Univ. Leiden, No124 (1911); H. K. Onnes, Akad. van Wetens happen (Amsterdam) 14, 818 (1911).

5

September, 1993 (under pressure)

160 Liquid CF4

Hg−1223

140 Tl−2223

[

[

August, 1993 (under pressure) April, 1993 September, 1992 February, 1988

Tc(K)

120 100

Shuttle

Bi−2223

January, 1988

Y−123

January, 1987

Liquid N2

80 60

(La−Ba)−214

40

[

December, 1986 (under pressure) December, 1986 (under pressure) April, 1986

Liquid H2

20

Liquid He

1900

NbC

Pb Hg

Nb3Ge

Nb3Sn

NbN V3Si

Nb−Al−Ge

Nb

1920

1940

1960

1980

2000

YEAR

Figure 2: The evolution of T with time (from C. W. Chu, Super ondu tivity Above 90K and Beyond in: B. Batlogg, C. W. Chu, W. K. Chu D. U. Gubser and K. A. Muller (eds.) Pro . HTS Workshop on Physi s, Materials and Appli ations, World S ienti , Singapore, 1996.). In 1986, Georg Bednorz and Alex Muller found super ondu tivity in (La,Sr)2CuO4 with T = 36K, an in redible new re ord.1 Within months, T -values in uprates were shooting up, and the re ord is now at T  135K. 2.1 Phenomena

Zero resistan e2

No resistan e is dete table even for high s attering rates of ondu tion ele trons. Persistent urrents magneti ally indu ed in a oil of Nb0:75 Zr0:25 and wat hed with NMR yielded an estimate of the de ay time greater than 105 years! (From theoreti al estimates the de ay time may 10 10 be as large as 10 years!) Absen e of thermoele tri ee ts3 No Seebe k voltage, no Peltier heat, no Thomson heat is dete table (see next se tion). Ideal diamagnetism m = 1. Weak magneti elds are ompletely s reened away from the bulk of a super ondu tor. Meissner ee t4 If a super ondu tor is ooled down in the presen e of a weak magneti eld, below T the eld is ompletely expelled from the bulk of the super ondu tor. Flux quantization5 The magneti ux through a super ondu ting ring is quantized and onstant in time. This phenomenon was theoreti ally predi ted by F. London in 1950 and experimentally veri ed 1961. 1 J. G. Bednorz and K. A. M uller, Z. Phys. B64, 189 (1986). 2 J. File and R. G. Mills, Phys. Rev. Lett. 10, 93 (1963). 3 W. Meissner, Z. Ges. K alteindustrie 34, 197 (1927). 4 W. Meissner and R. O hsenfeld, Naturwiss. 21, 787 (1933). 5 B. S. Deaver and W. M. Fairbank, Phys. Rev. Lett. 7, 43 (1961);

51 (1961).

6

R. Doll and M. Nabauer, Phys. Rev. Lett. 7,

2.2 London theory1

Phenomena (a) and (b) learly indi ate that the super urrent (at T = 0) is a property of the quantum ground state: There must be an ele tri ally harged ( harge quantum q), hen e omplex bosoni eld whi h ondenses in the ground state into a ma ros opi amplitude: (1) nB = j j2 ; where nB means the bosoni density, and is the orresponding eld amplitude. Sin e the eld is ele tri ally harged, it is subje t to ele tromagneti elds (E ; B) whi h are usually des ribed by potentials (U; A):  A U ; (2a) E= t  r B=

  A: r

(2b)

The eld amplitude should obey a S hrodinger equation 1  ~  qA2 + qU = E B
; (3) 2mB i  r where the energy is reasonably measured from the hemi al potential B of the boson eld, sin e what is measured in a voltmeter is rather the ele tro hemi al potential  = B + qU (4) than the external potential U , or the ee tive ele tri eld  A 1  : (5) Ee = t

q r

As usual in Quantum Me hani s, i~= r is the anoni al momentum and ( the me hani al momentum. The super urrent density is then 



i~= r qA) = p^m is

2

= 2iqm~  r r  mq  A: (6) B B It onsists as usual of a `paramagneti urrent' ( rst term) and a `diamagneti urrent' (se ond term).2 In a homogeneous super ondu tor, where nB = onst., we may write (7) (r; t) = pnB ei(r;t); and have js = ~q r A;  = nmBq2 : (8) B Sin e in the ground state E = , and E = i~ =t, we also have js = q

pm n mB B

=

q  p^m
< mB

 t

= : (9) The London theory derives from (8) and (9). It is valid in the London limit, where nB = onst. in spa e an be assumed. ~

1 F. London and H. London, Pro . Roy. So . A149, 71 (1935); F. London, Pro . Roy. So . A152, 24 (1935); F. London, Super uids, Wiley, London, 1950. 2 These are formal names: sin e the splitting into the two urrent ontributions depends on the gauge, it has no deeper physi al meaning. Physi ally, paramagneti means a positive response on an external magneti eld (enhan ing the eld inside the material) and diamagneti means a negative response.

7

The time derivative of (8) yields with (9)
 js = or

A t

t

 js t




1  ; q r

= Ee

(10)

This is the rst London equation: A super urrent is freely a

elerated by an applied voltage, or, in a bulk super ondu tor with no super urrent or with a stationary super urrent there is no ee tive ele tri eld ( onstant ele tro hemi al potential). The rst London equation yields the absen e of thermoele tri ee ts, if the ele tro hemi al potentials of ondu tion ele trons, el, and of the super urrent, , are oupled. The thermoele tri ee ts are sket hy illustrated in Fig. 3. The rst London equation auses the ele tro hemi al potential of the super urrent arrying eld to be onstant in every stationary situation. If the super urrent arrying eld rea ts with the ondu tion ele tron eld with n ele trons forming a eld quantum with

harge q, then the ele tro hemi al potentials must be related as nel = . Hen e the ele tro hemi al potential of the ondu tion ele trons must also be onstant: no thermopower (Seebe k voltage) may develop in a super ondu tor. The thermoele tri urrent owing due to the temperature dieren e is an eled by a ba k owing super urrent, with a ontinuous transformation of ondu tion ele trons into super urrent density at the one end of the sample and a ba k transformation at the other end. If a loop of two dierent normal ondu tors is formed with the jun tions kept at dierent temperatures, then a thermoele tri urrent develops together with a dieren e of the ele tro hemi al potentials of the two jun tions, and several forms of heat are produ ed, everything depending on the

ombination of the two metals. If there is no temperature dieren e at the beginning, but a urrent is maintained in the ring (by inserting a power supply into one of the metal halfs), then a temperature dieren e between the jun tions will develop. This is how a Peltier ooler works. In a loop of two super ondu tors non of those phenomena an appear sin e a dieren e of ele tro hemi al potentials

annot be maintained. Every normal urrent is lo ally short- ir uited by super urrents. If, however, a normal metal A is ombined with a super ondu tor B in a loop, a thermoele tri

urrent will ow in the normal half without developing an ele tro hemi al potential dieren e of the jun tions be ause of the presen e of the super ondu tor on the other side. This yields a dire t absolute measurement of the thermoele tri oeÆ ients of a single material A. The url of Eq. (8) yields (with r  r = 0)     js = B : r

(11)

This is the se ond London equation. It yields the ideal diamagnetism, the Meissner ee t, and the

ux quantization. Take the url of Maxwell's equation (Ampere's law) and onsider r  r  B
= r Br
r22 B:
  B = 0 js + j ; r
    B r r 2 B  r2 2 B  r2

= = = 8

B r

= 0;


  js + j ; r
  0  js + j ; r 0   B 0  r  j :

0

(12)

normal ondu tor

super ondu tor

 T1 1;el


T (t > 0) T

= T (t = 0)

T < T (t < 0)

0110 0min n1

j j2

B;

Figure 11: The Free Energy fun tion. Close to the transition, for

t=

T

T

T

;

jtj  1;

the order parameter j j2 is small, and F may be Taylor expanded (for xed V ): F (t; j j2 ) = Fn (t) + A(t)j j2 + 21 B (t)j j4 +


From the gure we see that A(t) T 0 for t T 0; B (t) > 0: 1 V. L. 2 L. D.

Ginsburg and L. D. Landau, Zh. Eksp. Teor. Fiz. (Russ.) 20, 1064 (1950). Landau, Zh. Eksp. Teor. Fiz. (Russ.) 7, 627 (1937).

17

(47) (48)

Sin e jtj  1, we put Then we have and

A(t)  tV; B (t)  V:

(49)

Fn (t) = Fn (t) for t  0;

(50)

1 F V  j j2

= t + j j2 = 0; that is,

2 t2 ; Fs (t) = Fn (t) j j2 = t 2 V for t < 0:

(51)

Re alling that small hanges in the Free Energy and Free Enthalpy are equal and omparing to (35) yields r 2 t2 B 2 0 (52) 2 = 20 =) B (t) = jtj : From (43),


Cp =

T V B 0 T t

!2

2



= TV

(53)

follows. While
Cp an be measured, this is not always the ase for the thermodynami riti al eld, B , as we will later see. Eqs. (51) and (52) may be rewritten as nB (t) =

hen e,

=

 2 t2 0 j tj; B 2 (t) = ;

B 2 (t) B 2 (t) ; = : 2 0 nB (t) 0 jtjnB (t)

Sin e a

ording to (37) B  t, it follows

nB  t:

The bosoni density tends to zero linearly in T

T.

(54) (55)

4.2 The Ginsburg-Landau equations

If we want to in orporate a magneti eld B into the \Free Energy fun tion" (48), we have to realize that B auses super urrents js   = r, and these reate an internal eld, whi h was alled Bm in (26). The energy ontribution of must be related to (3). Ginsburg and Landau wrote it in the form F (t; B ; ) = Fn (t) + ) (   2 Z Z 1 2 2 2 ie B ~  + d3r 2m0 + d3r 4m  r + ~ A + tj j2 + 2 j j4 ; (56a) V

where also (21) was onsidered. The rst orre tion term is the eld energy of the eld Bm reated by , in luding the stray eld outside of the volume V while 6= 0 inside V only. A is the ve tor potential of the total eld a ting on :   A = B + Bm : r

18

(56b)

The Free Energy is obtained by minimizing (56a) with respe t to (r) and (r). To prepare for a variation of , the se ond integral in (56a) is integrated by parts: "  #"  # Z  2 ie 2 ie  d3 r + A A  = V

r

r

~

2



 2ie d3 r  + A r ~ V

Z

=

~

+



Z V



 2ie + A d2 n  r ~

:

(56 )

From the rst integral on the right we see that (56a) indeed orresponds to (3). The preferen e of the writing in (56a) derives from that kineti energy expression being manifestly positive de nite in any partial volume. Now, the variation  !  + Æ  yields ( )   Z ~2  2 ie 2 ! 3  2 0 = ÆF = d rÆ 4m  r + ~ A + t + j j + V   Z 2 ie ~2  2  + d nÆ 4m  r + ~ A : V

F is stationary for any variation Æ (r), if

1  ~  + 2eA2 4m i  r and



jtj + j j2 = 0

(57)



~ 

+ 2eA = 0: (58) The onne tion of with Bm must be that of Ampere's law: (= r)  Bm = 0js with js given by (6). Sin e in thermodynami equilibrium there are no urrents besides js in the super ondu tor, (= r)  B = 0 there. Hen e, we also have n

i r

  Btot = 0 js ; r  ie~   js = 2m  r

 Btot = B + Bm =  r  2 e2    r A : m

 A;

(59)

It is interesting to see that (59) is also obtained from (56a), if , and A are varied independently: The variation of A on the left hand side of (56 ) yields 2ie Z d3rÆA
   2ie A    + 2ie A : ~

r

V

r

~

~

With ÆBm = (= r)  ÆA the variation of the rst integral of (56a) yields Æ

Z

1

= =

2 =2 d3 rBm 1

z

Z

1

d3 rÆBm
Bm = 2

}| {

Z

1





 d3 r  Æ A
Bm = r

1 2 d3r r
ÆA Bm = 2 d3 rÆA
r  Bm   Z 2 d3 rÆA
r  Btot +


: V Z



Z





= (60)

In the fourth equality an integration per parts was performed, and a
(b  ) = b
(a  ) was used. (The over bra e indi ates the range of the dierential operator.) Finally, the integral over the in nite 19

spa e is split into an integral over the super ondu tor (volume V ), where (= r)Bm = (= r)Btot, and the integral over the volume outside of the super ondu tor, indi ated by dots, sin e we do not need it. Now, after adding the prefa tors from (56a) we see that stationarity of (56a) with respe t to a variation ÆA inside the volume V again leeds to (59). This situation is no a

ident. From a more general point of view the Ginsburg-Landau fun tional (56a) may be onsidered as an ee tive Hamiltonian for the u tuations of the elds and A near the phase transition.1 This is pre isely the meaning of relating (56a) to (3). Eqs. (57) and (59) form the omplete system of the Ginsburg-Landau equations. The boundary ondition (58) omes about by the spe ial writing of (56a) without additional surfa e terms. This is orre t for a boundary super ondu tor/va uum or super ondu tor/semi ondu tor. A

areful analysis on a mi ros opi theory level yields the more general boundary ondition n






i + 2 eA = ; i r b

~ 

(61)

where b depends on the outside material: b = 1 for va uum or a non-metal, b = 0 for a ferromagnet, b nite and non-zero for a normal metal.2 In all ases, multiplying (61) by  and taking the real part yields n
js = 0 (62) as it must: there is no super urrent passing through the surfa e of a super ondu tor into the nonsuper ondu ting volume. Btot must be ontinuous on the boundary be ause, a

ording to  Btot = r = 0 and (59), its derivatives are all nite. 4.3 The Ginsburg-Landau parameter

Taking the url of (59) yields, like in (13),  2 Btot  r2

= Btot2 ;

m m (63) 20e2j j2 = 20e2jtj ; where (51) was taken into a

ount in the last expression.  is the Ginsburg-Landau penetration depth; it diverges at T like   jtj 1: if T is approa hed from below, the external eld penetrates more and more, and eventually, at T , the diamagnetism vanishes. Eq. (57) ontains a se ond length parameter: In the absen e of an external eld, A = 0, and for small , j j2  jtj= , one is left with 2 2 = ~2 : ;  = (64)  r2  2 4mjtj This equation des ribes spatial modulations of the order parameter j j2 lose to T .  is the GinsburgLandau oheren e length of su h order parameter u tuations. It has the same temperature dependen e as , and their ratio, s 2m2 ;  (65) = =  ~2  e2 2 =

0

is the elebrated Ginsburg-Landau parameter. 1 L. 2 P.

D. Landau and E. M. Lifshits, Statisti al Physi s, Part I, x147, Pergamon, London, 1980. G. De Gennes, Super ondu tivity in metals and alloys, New York 1966, p. 225 .

20

Introdu tion of dimensionless quantities x = r=;

=

,s

p

jtj ;

r   2B (t) = B jtj 2 0 ; .p   is = js 0 2B (t) ; .p  2B (t) a=A

b=B

.

(66)

yields the dimensionless Ginsburg-Landau equations 2  + a + j j2 = 0; i  x  i    b = is ; is = 2  x x





1

  x



(67)

a

whi h ontain the only parameter . 4.4 The phase boundary

We onsider a homogeneous super ondu tor at T . T in an homogeneous external eld B  B (T ) in z -dire tion. We assume a plane phase boundary in the y z -plane so that for x ! 1 the material is still super ondu ting, and the magneti eld is expelled, but for x ! 1 the material is in the normal state with the eld penetrating. z

b=0

=1

b = 1=

=0

b

super ondu ting

2

normal x

y

phase boundary

is

Figure 12: Geometry of a plane phase boundary. We put

p

= (x);

bz = b(x); bx = by = 0;

21

ay = a(x); ax = az = 0; b(x) = a0 (x):

Then, the super urrent is ows in the y-dire tion, and hen e the phase of depends on y. We

onsider y = 0 and may then hoose real. Further, by xing another gauge onstant, we may hoose a(0) = 0: Then, Eqs. (67) redu e to 1

00 + a2

2

+ 3 = 0;

a00 = a 2 :

(68)

Let us rst onsider   1: For large enough negative x we have a  0 and  1. We put = 1 (x), and get from the rst equation (68) 00  2

1

1 + 3
= 22;



p   e 2x ; x .  1 :

On the other hand, for large enough positive x we have b = 1=p2; a = xp2;  1; hen e, again from the rst equation (68), 2 2 00   x ;

2

e

x2 =2

p2

; x2  1:

1 0 ; where 0 denotes the value of

The se ond Eq. (68) yields a penetration depth  eld drops:

1

1

1

p

e 2x

p1

b

2

e

(x) where the

p

0 

0x

e

x2 =2

p2

1=p > 1

1= =  Figure 13: The phase boundary of a type I super ondu tor. In the opposite ase   1; falls o for x & 1; where b  1=p2; a  x=p2; and for x  1; 00  2 x2 =2 : 22

1

1

p1

b

2 e

x2 =2

p2

1=p < 1 1= Figure 14: The phase boundary of a type II super ondu tor. 4.5 The energy of the phase boundary

For B = B (T ); b = 1 in our units, the Free Energy of the normal phase is just equal to the Free Energy of the super ondu ting phase in whi h b = 0; = 1: If we integrate the Free Energy density variation (per unit area of the y z-plane), we obtain the energy of the phase boundary per area: s=n =

Z

1 1

(

dx

(B

B )2 20

~2

+ 4m

2



j 0 j2 + 4e2 A2 j j2 ~



)

jtjj j2 + j j4 :

2

(69)

Sin e the external eld is B , we have used Bm = Btot Bext = B B : In our dimensionless quantities this is (x is now measured in units of )  Z 4
2
2 1 B 2 1 1 2 0 2 s=n = dx b p + 2 + a 1 + 0  2 = 2 1  2Z 1 1
1 00 + a2 1
2 + 4 =

= B dx a0 p 2 0 2 2 2  1  Z 1 4
B 2 1 (70) =  dx a0 p 2 2 : 2 0 1 First an integration per parts of 0 2 was performed, and then (68) was inserted. We see that s=n an have both signs: 2  4 2
1 0 ? 2 or p 7 p1 a0 : s=n ? 0 for a p 2 2 2

Sin e b must de rease if 2 in reases and = 0 at b = 1=p2, (b 1=p2) = (a0 1=p2) and have opposite signs whi h leads to the last ondition. If 2 p1 a0 = p 2 2 would be a solution of (68), it would orrespond to s=n = 0: 23

2 must

We now show that this is indeed the ase for 2 = 1=2: First we nd a rst integral of (68): 2

=

2 a2

0 00

=

2

02

1 2 and have from (71)

2 = ;

p1

2



2





1
+ 0 a2

3 ; 0+2 3 0

2



=



2 0+2 3 0 = 0 by the se ond Eq. (67)   4 = 2 2 a2 a02 2 + 2 + onst. sin e 0 = = 0 for a0 = p12 =) onst. = 12 :

=

Now we use



00

2

a0 =

02 =

1 2 2

2

p

2

02

2

0 a2 + 2 2 aa0 2a0 a00 | {z }

)

a00 =

a0 2

1

p

p

2

0= a 2

2a0
+ p12

0 = ap

)

(71)

2




1 a0 2 + ;

2

whi h is indeed an identity. Sin e 02=2 > 0 enters the integral for s=n in the rst line of (70), it is lear that s=n is positive for 2 ! 0: Therefore, the nal result is 1 s=n ? 0 for  7 p : type III (72) 2 The names \type I" and \type II" for super ondu tors were oined by Abrikosov,1 and it was the existen e of type II super ondu tors and a theoreti al predi tion by Abrikosov, whi h paved the way for te hni al appli ations of super ondu tivity.

1 A.

A. Abrikosov, Sov. Phys.{JETP 5, 1174 (1957).

24

5

INTERMEDIATE STATE, MIXED STATE

In Chapter 2 we onsidered a super ondu tor in a suÆ iently weak magneti eld, B < B ; where we found the ideal diamagnetism, the Meissner ee t, and the ux quantization. In Chapter 3 we found that the dieren e between the thermodynami potentials in the normal and the super ondu ting homogeneous phases per volume without magneti elds may be expressed as ( f. (35)) 1 hGn(p; T ) Gs (p; T )i = 1 hFn (V; T ) Fs (V; T )i = B 2(T ) (73) V V 2 0

by a thermodynami riti al eld B (T ): (We negle t here again the ee ts of pressure or of orresponding volume hanges on B :) If a magneti eld B is applied to some volume part of a super ondu tor, it may be expelled (Meissner ee t) byR reating an internal eld Bm = B through super urrents, on the ost of an additional energy d3rBm2 =20 for the super ondu ting phase ( f. (56a)) and of a kineti energy density (~2 =4m)j(= r + 2ieA=~) j2 in the surfa e where the super urrents ow. If Bm > B ; the Free Energy of the super ondu ting state be omes larger than that of the normal state in a homogeneous situation. However, B itself may ontain a part reated by urrents in another volume of the super ondu tor, and phase boundary energies must also be onsidered. There are therefore long range intera tions like in ferroele tri s and in ferromagnets, and orresponding domain patterns

orrespond to thermodynami stable states. The external eld B at whi h the phase transition appears depends on the geometry and on the phase boundary energy. 5.1 The intermediate state of a type I super ondu tor

Apply a homogeneous external eld Bext to a super ondu tor. B = Bext + Bm depends on the shape of the super ondu tor. There is a ertain point, at whi h B = Bmax > Bext (Fig. 15). If Bmax > B ; the super ondu ting state be omes instable there. On ould think of a normal-state on ave island forming (Fig. 16).

Bext

Bmax

Bext + Bm

SC SC

n

Bmax

FIG. 15: Total (external plus indu ed) magneti eld around a type I super ondu tor.

FIG. 16.

25

This, however, annot be stable either: the point of Bmax = B has now moved into the super ondu tor to a point of the phase boundary between the normal and super ondu ting phases, whi h means that in the shaded normal area B < B ; this area must be ome super ondu ting again (Fig. 16). Forming of a onvex island would ause the same problem (Fig. 17).

Bmax

n

SC

SC

FIG. 17.

What really forms is a ompli ated lamellous or lamentous stru ture of alternating super ondu ting and normal phases through whi h the eld penetrates (Fig. 18). The true magnetization

urve of a type I super ondu tor in dierent geometries is shown on Fig. 19. It depends on the geometry be ause the eld reated by the shielding super urrents does. In Se tion 3.C, for B 6= 0 the phase transition was obtained to be rst order. Generally, the movement of phase boundaries is hindered by defe ts, hen e there is hysteresis around B .

FIG. 18.

2 3 B

Bm

B

sphere

Bext

long rod rst order transition

0
fenergy of the stray eld +

+ energy of phase boundariesg Figure 19:

26

5.2 Mixed state of a type II super ondu tor1

If the phase boundary energy is negative, germs of normal phase may form well below B ; at the lower

riti al eld B 1 ; and germs of super ondu ting phase may form well above B ; at the upper riti al eld B 2 ; in both ases by gaining phase boundary energy. At B / B 2; in small germs  1 and b  onst. The dimensionless Ginsburg-Landau equations (67) may be linearized:  1  + a2 = : (74) i  x We apply the external B- eld in z-dire tion (Fig. 20), bx = by = 0; bz = b; ax = by; ay = az = 0; b and assume germ laments along the eld lines: = (x; y): a Then, (74) is ast into . " # y 1  by2 1  2 = : i x

2 y2

With (x;2y) =2 eipx2 (y); p=(b)2 = y0 this equation simpli es x to [ (1= )(d =dy )+ b(y y0 ) ℄ = ; or, after multiplying with =2 and by de ning u2  (y y0)2 : # " Figure 20: 1 d2 + 1 b2u2  =  : (75) 2 du2 2 2 This is the S hrodinger equation for the ground state of a harmoni os illator with p ! = b =  =) B = 2B : Hen e, p B 2 (T ) = 2B (T ): (76) If a germ lose to the surfa e of a super ondu tor at y= onst. is onsidered, then the u- oordinate must be ut at some nite value. There, the boundary ondition (58) yields d=du = 0 (sin e n
a = 0). Therefore, instead of the boundary problem, the symmetri ground state in a double os illator with a mirror plane may be onsidered (Fig. 21). 0 (0) = 0

u0

u

FIG. 21: The ground state of a double os illator.

1 A.

A. Abrikosov, unpublished 1955; Sov. Phys.{JETP 5, 1174 (1957).

27

~! min E = 0:59 u0 0 2

Hen e, at

B 2 0:59

= 1:7B 2 = 2:4B (77) super ondu tivity may set in in a surfa e layer of thi kness  . In a type I super ondu tor, B 2 < B . However, only below B 2 germs with arbitrarily small -values ould form where the super ondu ting phase is already absolutely stable in a type I super ondu tor. Here, for B 2 < B < B germs an only form with a non-zero minimal -value, whi h is inhibited pby positive surfa e energy. In this region the super ooled normal phase is metastable. For  > 0:59= 2; B 3 > B ; and surfa e super ondu tivity may exist above B : B 3 =

5.3 The ux line in a type II super ondu tor

To determine B 1; the opposite situation is onsidered. In the homogeneous super ondu ting state with = 1; the external magneti eld is tuned up until the rst normal state germ is forming. Again we may suppose that the germ is forming along a eld line in z-dire tion. Sin e the phase boundary energy is negative, there must be a tenden y to form many fa e boundaries. However, the normal germs annot form arbitrarily small sin e we know from (17) that the ux onne ted with a normal germ in a super ondu tor is quantized and annot be smaller than 2 p p (78) 0 = 22e~ = 2 2B  = 2 2B  : We onsider a ux line of total ux 0 along the z-dire tion (Fig. 22): Bz 

j j

= j jei ; B = ez B(); 2 = x2 + y2

 

FIG. 22: An isolated ux line.

One ould try to solve the Ginsburg-Landau equations for that ase. However, there is no general analyti solution, and the equations are valid lose to T only, where j j is small. Instead we assume   1; that is,   ; and onsider only the region   ; where j j = onst. Then, from Ampere's law and (6), 2   B = 0 js = 0 e~ j j2  20 e j j2 A = r

hen e,

m

r

 2 0e2 j j2 ~  = m 2e  r

A + 2

A



m

 1 = 2 20 r



A ;

  B = 20 r : r

We integrate this equation along a ir le around the ux line with radius    and use Stokes' theorem, H dsA = R d2n
( r  A) : Z

d2 n
B + 2

I

ds


28



 B r



= 0 :

(79)

The phase  must in rease by 2 on a ir le around one uxoid. Now, we take     : Then, the rst term in (79) may be negle ted. We nd = 0 22 dB d

or

(80)

   0 0 (81) B () = 22 ln  ; 0  : The integration onstant 0 was hosen su h that (81) vanishes for  & ; where a more a

urate analysis of (79) is ne essary to get the orre t asymptoti s. By applying Stokes' theorem also to the se ond integral of (79), we have for all    Z





  d2 n
B + 2  B r r



=

Z



d2 n
B  2

2 B  r2



= 0 :

Sin e the right hand side does not hange if we vary the area of integration, 2

B

must hold. In ylindri oordinates,

2  r2

hen e,

  1 2 2 = 1   + 2 2 + 2;    z B + 2

or

2 B=0  r2

1    B=0   

1

This equation is of the Bessel type.

1

B 00 + B 0 B = 0:  2 Its for large  de aying solution is

(82)

0 K0   ! p 0 e = ; (83) 22  !1 82 where K0 is M Donald's fun tion (Hankel's fun tion with imaginary argument), and the oeÆ ient has been hosen to meet (80) for   : The energy per length  of the ux line onsists of eld energy and kineti energy of the super urrent: B () =



=



= = = =

B 2 mB + 2 nB v2 ; js = 2enB v; nB = j j2; d2 r 2 0  2  Z b m 2 d2 r + j 20 4e2nB s =  2   2 Z m  B 2 dr 20 + 40e2j j2  rB =   2 Z B 2  d2 r +  B
r  B = 2 0 20  r 1 Z d2 rB
B + 2      B 20 r r !    I I  2  B : ds
B  20  !1 r Z





29

The rst integrand was shown to vanish for   ; and the last ontour integral vanishes for  ! 1. We negle t the ontribution from  . ; and nd with (80)) dB  B ( ) 0 B (0) 2 2 B  0 (84)  20 d 20  20 : With (81), the nal result with logarithmi a

ura y (ln(=)  1) is   20 20 ln :   =  ln (85) 402  402 This result also proves that the total energy is minimum for ux lines ontaining one uxoid ea h: For a ux line ontaining n uxoids the energy would be n2 while for n ux lines it would only be n ( > 0). In this analysis, B was the eld reated by the super urrent around the vertex line. Its intera tion energy per length with a homogeneous external eld in the same dire tion is Z B BB (86) d2 r ext = 0 ext : 0

0

(85) and (86) are equal at the lower riti al eld Bext = B 1:  ln  B 1 = 0 2 ln  = B p ;   1: (87) 4 2 The phase diagram of a type II super ondu tor is shown in Fig. 23. (There might be another phase with spatially modulated order parameter under ertain onditions, theoreti ally predi ted independently by Fulde and Ferell and by Larkin and Ovshinnikov; this FFLO phase has not yet been learly observed experimentally.) B

surfa e s

FFLO?

B 3

mixed phase B 1

n

B 2

B

s

T

T

Figure 23: The phase diagram of a type II super ondu tor. A more detailed numeri al al ulation shows that in an isotropi material the energy is minimum for a regular triangular latti e of the ux lines2in the plane perpendi ular to them. From (87), at B p= B 1 the density of ux lines is ln =4 ; that is, the latti e onstant a1 is obtained from a21 3=2 = 42 = ln : s 8 (88) a1 = p 3ln   & : 30

The lines (of thi kness ) indeed form nearly individually (Fig. 24). Sin e B 2 = B 122= ln ; the latti e onstant a2 at B 2 is s s r  ln  4  4 a2 = a 1 (89) 22 = p3  = p3  & : The ores of the ux lines (of thi kness ) tou h ea h other while the eld is already quite homogeneous (Fig. 25). Sin e j j  1 in the ore, the Ginsburg-Landau equations apply, and j j may rise

ontinuously from zero: the phase transition at B 2 is se ond order. 

B

j j B

j j a2 & 

a1 &  FIG. 24: Mixed phase for Bext  B 1 .

solution of (4.3)

FIG. 25: Mixed phase for Bext

 B 2 .

For a long ylindri rod the stray eld reated by super urrents outside of the rod may be negle ted, and one may express the eld inside the rod as B = Bext 0 M by a magnetization density M: The hange in Free Energy at xed T and V by tuning up the external magneti eld is dF = MdBext Z B 2 B2 Fn F s = dBextM = : 2 0

0

0 M

B 1

B

B 2 Bext

111111111 000000000 00000000 11111111 000 111 000000000 111111111 00000000 11111111 000 111 000000000 111111111 000 111 000000000 111111111 000 111 000000000 111111111 jump or 000 111 000000000 111111111 in nite slope 000 111 000 111

Figure 26: Magnetization urve of a type II super ondu tor. The dierently dashed areas in Fig. 26 are equal. For type II super ondu tors, B is only a theoreti al quantity. 31

6

JOSEPHSON EFFECTS

1

The quantitative des ription of Josephson ee ts at T  T (the usual ase in appli ations) needs a mi ros opi treatment. However, qualitatively they are the same at all temperatures T < T , hen e qualitatively they may be treated within the Ginsburg-Landau theory. Consider a very thin weak link between two halfs of a super ondu tor (Fig. 27). 2 S1

S2 d x1

1

x2

2 =  1 2



(x1)  1 = j jei1 (x2)  2 = j jei2 x

2 = 1 +  FIG. 27: A weak link between two halfs of a super ondu tor.

The order parameter has its thermodynami value on both sides x < x1 ; x > x2; but is exponentially small at x = 0: Hen e, any super urrent through the weak link is small, and may be

onsidered onstant in both bulks of super ondu tor. In the weak link, not only j j is small, also its phase may hange rapidly (e.g. from 2 = 1 to 2 = 1 +  by a very small perturbation). Without the right half, the boundary ondition (58) would hold at x1 :    2ie + A = 0: x ~ x x1 In the presen e of the right half, this ondition must be modi ed to slightly depending on the value 2 :    2ie (90) + Ax = 2; x

1 B.

x1

~

D. Josephson, Phys. Lett. 1, 251 (1962).

32

where is a small number depending on the properties of the weak link. Time inversion symmetry demands that (90) remains valid for ! ; A ! A; hen e must be real as long as the phase of does not depend on A. For the moment we hoose a gauge in whi h Ax = 0: Then, the super urrent density at x1 is " # ie~     js;x (x1 ) = 2m 1 x x1 1 x x1 = h i = 2iem~ 1 2 1 2 = jm





= jm sin 2 1
: = 2 We generalize the argument of the sine fun tion by a general gauge transformation (22): 2e ;  ! 

2  1

ei(2 1 )

A+

A

!



! 

~



 ; r  t

! =  2 1

ei(1 2 )

Ax = 0 ! A x =

2e ~

2  1




=

2



 ; x Z 2 e 2 1 + dxAx ;

= 2~e 2 1
: Re all that  is the ele tro hemi al potential measured by a voltmeter. Now, the general Josephson equation reads js = jm sin : d dt

(91)

~

1

(92) (93) (94)

6.1 The d. . Josephson ee t, quantum interferen e

A

ording to (94), a d. . super urrent of any value between jm and jm may ow through the jun tion, and a

ording to (93) the potential dieren e 2 1 is zero in that ase. Now, onsider a jun tion in the y z-plane with a magneti eld applied in z-dire tion. There are super urrents s reening the eld away from the bulks of the super ondu ting halfs. y

b

S1

11 00 B 00 11

(y)11 00 4 11 3 00 00 1 0 11 00 2 11 d  2

S2 x

Figure 28: A Josephson jun tion in an external magneti eld B. 33

Let us onsider (y); and let (0) = 0 at the edge y = 0: We have Byd =

Z

1234

ds
A =

Z

1

2

dxAx +

Z

In the jun tion we hoose the gauge Ax =

2

3

dyAy +

4

Z

3

dxAx +

Z

4

1

dyAy :

By; Ay = 0: Then, the y-integrals vanish, and from (92), 2 Z 3 dxBy = 0 + 2Bd y:

(0) = 2 1 = 0 ; (y) = 0 + 0 4 0 The d. . Josephson urrent density through the jun tion os illates with y a

ording to !

2Bd y : js (y) = jm sin 0 + 0 In experiment, at B = 0 one always starts from a biased situation with js = jm; hen e 0 = =2; and 2Bd y: js (y) = jm os (95) 0 The total urrent through the jun tion is Z b 2Bd y; Is = jm dy os 0 0 where is the thi kness in z-dire tion. F = b is the area of the jun tion. With Z b Z b ei b 1 sin( b) dy os( y) = < dyei y = < = 0

i

0



we nd that, depending on the phase 0; the maximal urrent at a given eld B is j sin(2Bbd=0 )j : Is;max = F jm 2Bbd=0

(96)

An even simpler situation appears, if one splits the jun tion into a double jun tion: Now, H ds
A =  is the magneti ux through the ut-out, and I  2 

a b =  ds
A = 2  : 0

0

11 00 00 11 00 11 00 11 00B 11 00 11 00 11 00 11 00 11

b

S1

S2 s

a

Figure 29: A simple SQUID geometry. 34

Hen e, the maximal d. . Josephson urrent is "

F jm 2 = x

 2  sin a + sin a +  Is;max = F jm max

a 0

2=0

a

I s;m

#



a

F jm

Is;max = 2

Figure 30: Phase relations in Eq.(97).



 

os   : 0 

:

(97)

This is the basis to experimentally ount ux quanta with a devi e alled super ondu ting quantum interferometer (SQUID).

6.2 The a. . Josephson ee t

We return to (93) and (94), and apply a voltage 2 An a. . Josephson urrent js (t) = jm sin !J t;

1 = V !J

to the jun tion, that is, = (2e=~)V t:

= 2eV=~

(98)

results, although a onstant voltage is applied. For a voltage of 10 V a frequen y !J =2 = 4.8 GHz is obtained: the a. . Josephson ee t is in the mi rowave region. If one overlays a radio frequen y voltage over the onstant voltage, 2  2 = V

+ Vr os(!r t);

(99)

one obtains a frequen y modulation of the a. . Josephson urrent: js

= =





2eV jm sin !J t + r sin(!r t) = ~!r ! 1 X 2 eVr sin(!J + n!r )t: jm Jjnj ~!r n= 1

(100)

is the Bessel fun tion of integer index. To interpret the experiments one must take into a

ount that a non-zero voltage a ross the jun tion

auses also a dissipative normal urrent In = V=R; where R is the resistan e of the jun tion for normal ele trons. The experimental issue depends on the oupling in of the radio frequen y.1 If the impedan e of the radio sour e is small ompared to that of the jun tion, we have a voltage-sour e situation, and the total urrent through the jun tion averaged over radio frequen ies is Jjnj

I = Is + V=R; Is = js F: 1 S.

Shapiro, Phys. Rev. Lett. 11, 80 (1963).

35

(101)

I ~!r =2e

d. . Josephson spike, Im

V=R

V

Shapiro spikes for !J + n!r = 0 Figure 31: Josephson urrent vs. voltage in the voltage-sour e situation. If the impedan e of the radio sour e is large ompared to the impedan e of the Josephson jun tion, as is usually the ase, we have a urrent-sour e situation, where the fed-in total urrent determines the voltage a ross the jun tion: V

= R(I = 2~e d dt

Is ) = R(I

Im sin ):

(102)

I

Shapiro steps

V =R

Im

V

Figure 32: Josephson urrent vs. voltage in the urrent-sour e situation. The a. . Josephson ee t yields a possibility of pre ise measurements of h=e. 36

7

MICROSCOPIC THEORY: THE FOCK SPACE

The S hrodinger wavefun tion of an (isolated) ele tron is a fun tion of its position, r; and of the dis rete spin variable, s :  = (r; s): Sin e there are only two independent spin states for an ele tron | the spin omponent with respe t to any (single) hosen axis may be either up (") or down (#) |, s takes on only two values, + and , hen e  may be thought as onsisting of two fun tions   (r; s) = ((rr;; +)) (103) forming a spinor fun tion of r: The expe tation value of any (usually lo al) one-parti le operator A(r; s; r0 ; s0 ) = Æ(r r0 )A^(r; s; s0) is XZ d3 r (r; s)A^(r; s; s0 )(r; s0 ): (104) hAi = s;s0

We will often use a short-hand notation x  (r; s). The S hrodinger wavefun tion (x1 : : : xN ) of a (fermioni ) many-parti le quantum state must be totally antisymmetri with respe t to parti le ex hange (Pauli prin iple): (: : : xi : : : xk : : :) = (: : : xk : : : xi : : :): (105) For a pie e of a solid, N  1023, this fun tion is totally in omprehensible and pra ti ally ina

essible: For any set of N values, + or for ea h si; it is a fun tion of 3N positional oordinates. Although by far not all of those 2N fun tions are independent | with the symmetry property (105) the +- and -values an always be brought to an order that all -values pre ede all +-values, hen e we have only to distinguish 0, 1, 2, . . . , N -values, that is (N + 1) ases | and although ea h of those (N + 1) fun tions need only be given on a ertain se tor of the 3N -dimensional position spa e | again with the symmetry property (105) ea h fun tion need only be given for a ertain order of the parti le

oordinates for all si = and for all si = + parti les |, it is lear that even if we would be ontent with 10 grid points along ea h oordinate axis we would need  101023 grid points for a very rude numeri al representation of that -fun tion. Nevertheless, for formal manipulations, we an introdu e in a systemati manner a fun tional basis in the fun tional spa e of those horrible -fun tions. 7.1 Slater determinants

Consider some (for the moment arbitrarily hosen) omplete orthonormal set of one-parti le spinor fun tions , l (x);

(l jl0 ) = X l

XZ s

d3 rl (r; s)l0 (r; s) = Æll0 ;

l (x)l (x0 ) = Æ(x x0 ) = Æss0 Æ(r

r0 ):

(106)

They are ommonly alled (spinor-)orbitals. The quantum number l refers to both the spatial and the spin state and is usually already a multi-index (for instan e (nlm) for an atomi orbital or (k) for a plane wave), and we agree upon a ertain on e and forever given linear order of those l-indi es. Choose N of those orbitals, l1 ; l2 ; : : : ; lN ; in as ending order of the li and form the determinant (107) L(x1 : : : xN ) = p1N ! det kli (xk )k: L = (l1 : : : lN ) is a new (hyper-)multi-index whi h labels an orbital on guration. This determinant of a matrix aik = li (xk ) for every point (x1 : : : xN ) in the spin-position spa e has the proper symmetry property (105). In view of (106) it is normalized, if all li are dierent, and it would be identi ally 37

zero, if at least two of the li would be equal (determinant with two equal raws): Two fermions annot be in the same spinor-orbital. This is the ompared to (105) very spe ial ase of the Pauli prin iple (whi h is the ommonly known ase). Now, given a omplete set of orbitals (106), we mention without proof that all possible orbital

on gurations of N orbitals (107) form a omplete set of N -fermion wavefun tions (105), that is, any wavefun tion (105) may be represented as X (x1 : : : xN ) = CLL(x1 : : : xN ) (108) L

jCL j2 = 1 (` on guration intera tion'). with ertain oeÆ ients For any operator whi h an be expanded into its one-parti le, two-parti le- and so on parts as for instan e a Hamiltonian with (possibly spin-dependent) pair intera tions, X 1 X w 0 0 (ri ; rj ); H^ = h^ si s0i (ri ) + (109) 2 i=6 j si si;sj sj i P CL ; L

the matrix with Slater determinants is HLL0 = hL jH^ jL0 i =

XX i

P

(lP i jhjli0 ) ( 1)jPj

Y j (6=i)

ÆlPj l0j +

XX Y + 21 ÆlPk l0k ; (lP ilP j jwjlj0 li0 ) ( 1)jPj i6=j P k(6=i;j )

(110)

where P is any permutation of the subs ripts i; j; k; and jPj is its order. The matrix elements are de ned as XZ 0 (li jhjli) = (111) d3 rli (r; s)h^ ss0 (r)l0i (r; s0 ); and (li lj jwjlj0 li0 ) =

ss0

X si s0i ;sj s0j

Z

d3 ri d3 rj li (ri ; si )lj (rj ; sj ) wsi s0i ;sj s0j (ri ; rj ) l0j (rj ; s0j )l0i (ri ; s0i ):

(112)

(Note our onvention on the order of indi es whi h may dier from that in other textbooks but leads to a ertain anoni al way of writing of formulas later on.) The rst line on the r.h.s. of (110) is non-zero only if the two on gurations L and L0 dier at most in one orbital, and the sum over all permutations P has only one non-zero term in this ase, determining the sign fa tor for that matrix element. The se ond line is non-zero only if the two on gurations dier at most in two orbitals, and the sum over all permutations has two non-zero terms in that ase: if P is a perturbation with lP k = lk0 for all k 6= i; j , then the orresponding ontribution is (1=2)[(lP ilP j jwjlj0 li0 ) ((lP j lP ijwjlj0 li0 )℄( 1)jPj. For L = L0 and P = identity (and sometimes also in the general ase) the rst matrix element is

alled dire t intera tion and the se ond one ex hange intera tion. 7.2 The Fo k spa e

Up to here we onsidered representations of quantum me hani s by wavefun tions with the parti le number N of the system xed. If this number is ma ros opi ally large, it annot be xed at a single de nite value in experiment. Zero mass bosons as e.g. photons may be emitted or absorbed in systems of any s ale. (In a relativisti des ription any parti le may be reated or annihilated, possibly together with its antiparti le, in a va uum region just by applying energy.) From a mere te hni al point of view, quantum statisti s of identi al parti les is mu h simpler to formulate with the grand anoni al ensemble with varying parti le number, than with the anoni al one. Hen e there are many good reasons to onsider quantum dynami s with hanges in parti le number. 38

In order to do so, we start with building the Hilbert spa e of quantum states of this wider frame: the Fo k spa e. The onsidered up to now Hilbert spa e of all N -parti le states having the appropriate symmetry with respe t to parti le ex hange will be denoted by HN . In the last subse tion we introdu ed a basis fLg in HN . Instead of spe ifying the multi-index L as a raw of N indi es li we may denote a basis state by spe ifying the o

upation numbers ni (being either 0 or 1) of all orbitals i: X ni = N: (113) jn1 : : : ni : : :i; i

Our previous determinantal state (107) is now represented as jL i = j0 : : : 01l1 0 : : : 01l2 0 : : : 01lN 0 : : :i: Two states (113) not oin iding in all o

upation numbers ni are orthogonal. HN is the omplete linear spa e spanned by the basis ve tors (113), i.e. the states of HN are either linear ombinations P jLiCL of states (113) (with the sum of the squared absolute values of the oeÆ ients CL equal to unity) or limits of Cau hy sequen es of su h linear ombinations. (A Cau hy sequen e is a sequen e fj n ig with limm;n!1 h m n j m n i = 0. The in lusion of all limits of su h sequen es into HN means realizing the topologi al ompleteness property of the Hilbert spa e, being extremely important in all onsiderations of limits. This ompleteness of the spa e is not to be onfused with the ompleteness of a basis set fi g. The extended Hilbert spa e F (Fo k spa e) of all states with the parti le number N not xed is now de ned as the ompleted dire t sum of all HN . It is spanned by all state ve tors (113) for all N with the above given de nition of orthogonality retained, and is ompleted by orresponding Cau hy sequen es, just as the real line is obtained from the rational line by ompleting it with the help of Cau hy sequen es of rational numbers. Note that F now ontains not only quantum states whi h are linear ombinations with varying ni so that ni does not have a de nite value in the quantum state (o

upation number u tuations), but also linear ombinations with varying N so that now quantum u tuations of the total parti le number are allowed too. (For bosoni elds as e.g. laser light those quantum u tuations an be ome important experimentally even for ma ros opi N .) 7.3 O

upation number representation

We now ompletely abandon the awful wavefun tions (105) and will ex lusively work with the o

upation number eigenstates (113) and matrix elements between them. The simplest operators are those whi h provide just a transition between basis states (113) whi h are as lose to ea h other as possible: those whi h dier in one o

upation number only. The de nition of these reation and annihilation operators for fermions must have regard to the antisymmetry of the quantum states and to Pauli's ex lusion prin iple following from this antisymmetry. They are de ned as P (114)

^i j : : : ni : : :i = j : : : ni 1 : : :i ni ( 1) j  an ex ited state is obtained with ex itation energy  . By removing a fermion with  <  | that is, reating a hole in the original ground state | an ex ited state is obtained with ex itation energy j j: rst lift the fermion to the level  and then remove it without hanging the hara ter of the state any more (Figs. 33 and 34).  

k

quasi-ele tron



jk j

0

k

kF

quasi-hole kF FIG. 33: Creation of an ex ited ele tron and of a hole, resp.

k

FIG. 34: Ex itation spe trum of a Fermi gas.

The ground state j0i of a normal metal has mu h the same properties: ondu tion ele trons with and holes with  <  may be ex ited with ex itation energies as above. These are not the original ele trons making up the metal together with the atomi nu lei. Rather they are ele trons or missing ele trons surrounded by polarization louds of other ele trons and nu lei in whi h nearly all the Coulomb intera tion is absorbed. We do not pre isely know these ex itations nor do we know the ground state j0i (although a quite elaborate theory exists for them whi h we ignore here). We just assume that they may be represented by fermioni operators with properties like those in the gas: (130) k <  :

^yk j0i = 0; ^yk ^k j0i = j0i;

>

k >  :

^k j0i = 0; ^k ^yk j0i = j0i;

42

(131)

k is the waveve tor and 

the spin state of the quasi-parti le. Sin e all intera tions present in the ground state j0i are already absorbed in the quasi-parti le energies k , only ex ited ondu tion ele trons or holes exert a remainder intera tion. Hen e, we may write down an ee tive Hamiltonian  >;  > ^H~ = X ^yk (k )^ k + 1 k Xk0 ^yk+q ^yk0 q0 wkk0 q ^k0 0 ^k + 2 k; k0 0 ; q k +

k >; k0 0 is stable. If at least one l -value is negative (at0 E tra tive intera tion), then 1 there is unavoidably a negl < 0 ative solution Elm of (146): Elm the `ex ited pair' has negative energy and the normal 2k ground state j0i is unstable against forming of pairs of bound quasi-parti les, no matter how small jl j is (how weak the attra tive intera tion is). Pairs are spontaneously formed and the ground state re onFigure 35: The fun tion F (E ) from Eq. (146). stru ts. This is the ontent of Cooper's theorem. If the intera tion is ut of at some energy ! ,  0 <  k < ! ; wkl = 10 for (147) elsewhere and the density of states for k is nearly onstant in this interval, N () = N (0); then, with negative Elm , 45

X

hen e,

k

j j2 wkl

Elm

1

2k =

N (0)

Z !

0

jElm j =

This yields

8 >
2! exp N (0)jl j for N (0)jl j   1 : N (0)jl j! in the weak and strong oupling limits. In this hapter we only onsider the weak oupling limit where jElm j is exponentially small. The whole analysis may be repeated for the ase where the pair has a non-zero total momentum q. In that ase the denominator of (146) is to be repla ed with Elm (q) k+q=2  k+q=2 where now jk  q=2j must be larger than kF . For small q; this ondition redu es the density of states in ee t in an interval of thi kness j= kjq=2 = vF q=2 at the lower -integration limit; vF is the Fermi velo ity. The result is Elm (q)  Elm + vF q=2: (150) In the weak oupling limit, Elm(q) an only be negative for exponentially small q. We performed the analysis with a pair of parti les. It an likewise be done with a pair of holes with an analogous result. 8.3 The BCS Hamiltonian

Frohli h1 was the rst to point out that the ele tron-phonon intera tion is apable of providing an ee tive attra tion between ondu tion ele trons in the energy range of phonon energies. From Cooper's analysis it follows that, if there is a weak attra tion, it an only be ee tive for pairs with zero total momentum, that is, between k and k. With the assumption that the attra tion is in the l = 0 spin singlet hannel, this led Bardeen, Cooper and S hrieer2 to the simple model Hamiltonian H^ BCS =

X

k

^yk (k

)^ k

g V

 !