10.3 lnstability of the "Fermi Sea" and Cooper Pain 193

the interior of the superconductor against extelnal fields, also decay exponentially with

distance into the solid. For an order of magnitude estimate of the London penetlatlon

J.pth, *" set m equal to the electron mass and take ns to be the atomic density' i'e''

*. urru-" that ea;h atom provides a superconducting electron' For Sn' for example

one obtains Ar= 260 A. An important element of the microscopic theory of super-

conductivity isihat it is not electrons but electron pairs, so-called Cooper pairs, which

are the current carriers. This can be included in the London equations if, instead of

n,, on" ut.t half of the electron density n"/2' Becartse the London equations describe

ttre Meissner-Ochsenfeld effect particularly well, a rnicroscopic theory of superconduc-

iivity should be able to provide' among other things, an equation of the tvpe (10'10b)'

ihis, utta an explanation of the vanishing resistance at sufficiently low temperature

(T< f.), is the cetttral concern of the BCS theory mentioned above' Because super-

conductivity is evidently a widespread phenomenon, the theory must also be sufficient-

ly general; it should not hittg" on one special metallic property' In the following sec-

iioi, tii" t"ri" aspects of thi microscopic theory of superconductivity will be treated

in a somewhat simPlified form'

10.3 Instability of the "Fermi Sea" and Cooper Pairs

From the previously mentioned fundamental experiments on the phenomenon of

superconductivity, ii is clear that we are dealing with a new phase of the electron gas

in a metal which displays the unusual property of "infinitely high" conductivity' An

important contributi;n 10 our understanding of this new phase was made bv Cooper

ttO.Ol wtro, in 1956, recognised, that the ground state (T=0K) of an electron gas

iS..t. O.Z) it unstabie if one adds a weak attractive interaction between each pair of

el""trorrr. sr"h an interaction had already been discussedby Fr,hlich [10.7] in the form

of the phonon-mediated interaction. As it passes through the solid' an electron' on ac-

".r"t if itt negative charge, leaves behind a deformation trail affecting the positions

of the ion "o..i.

Thi, trail is associated with an increased density of positive charge

duetotheioncores,andthushasanattractiveeffectonasecondelectron'Thelatticedeformation therefore causes a weak attraction between pairs of electrons (Fig' 10'7a)'

This attractive electron-electron interaction is retarded because of the slow motion of

theionsincomparisonwiththealmostinstantaneousCoulornbrepulsionbetweenelec-itorr;

"t,ft" instant when an electron passes, the ions receive a pull which' only after

the eiectron has passed, leads to a displacement and thereby to a polarization of the

iutti.. €ig. 10.7b). The lattice deformation teaches its maximum at a distance from

the first electron which can be estimated from the electron velocity (Fe^rmi velocity

;;i".;7;;;; tit. ','*i."- prt"non vibration period (2n/ae-r0-13 9' The t*o-

eiectrons corielated by the lattice deformation thus have an approximate sbparation of

uUou, 1OOO A. Thi, then "orresponds

to the "size" of a Cooper pair (which is also esti-

mated in sect. 10.7 using a different method). The extremely long interaction range of

the two electrons correlated by a lattice defolmation explains why the coulomb repul-

sion is insignificant; it is completely screened out over distances of just a few

Angstroms.-Quantum mechanically, the lattice deformation can be understood as the superposi-

tionif phonons which the electron, due to its interaction with the lattice, continuously

emits and absorbs. To comply with energy conservation, the phonons constituting the

t94

Lattice planes

z + utstance trom electron

Fig. 10.7a,b. Schematic representation of the electron-phonon interaction that leads to the formation ofCooper pairs. (a) An electron (e- ) travelling through the crystal lattice leaves behind a deformation trail,which can be regarded as an accumulation of the positively charged ion cores, i.e., as a compression of theIattice planes (exaggerated here for clarity). This means that an area of enhanced positive charge (comparedto th€ otherwise neutral crystal) is created behind the electron, and exerts an attractive force on a secondelectron. (b) Qualitative plot of the displacement of the ion cores as a function of their distance behind thefirst electron. Compared with the high electron velocity uF (- 108 cmls). the tattice follows only very slowlyand has its ma{mum delormation at a distance uF2n/(DDbehind, the electron, as determined by the typicalphonon frequency coD (Debye frequency). Thus the coupling ol the two electrons into a Cooper pak occursover distances ol more than 1000 A, at which thei Coulomb repulsion is completely screened

lattice deformation may only exist for a time t = 2t/a determined by the uncertaintyrelationl thereafter, they must be absorbed. One thus speaks of "virtual" phonons.

The ground state of a non-interacting Fermi gas of electrons in a potential well(Chap.6) corresponds to the situation where all electron states with wave vector ftwithin the Fermi sphere [,Eg(f = 0K1= n2 tc2rtZml are filled and all states with,E>Egare unoccupied. We now perform a Gedankenexperiment.and add to this system twoelectrons [ft1, -E(ft1)] and, lk2,E(k)l in states just above E$. A weak attractive interac-tion between these two electrons is switched on in the form of phonon exchange. Allother electrons in the Fermi sea are assumed to be non-interacting, and, on accountof the Pauli exclusion principle, they exclude a further occupation of states withl/c]<tp. Due to phonon exchange the two additional electrons continually changetheir wave vector, whereby, however, momentum must be conserved:

k t + k 2 = k i + k i : K (10.17)

Since the interaction in ft-space is restricted to a shell with an energy thickness of haD(with (,D = Debye frequency) above E$ the possible ft-states are given by the shadedarea in Fig. 10.8. This area and therefore the number of energy-reducing phonon ex-change processes - i.e. the strength of the attractive interaction - is maximum for1( = 0. It is therefore sufficient in what follows to consider the case ft1 = - kz: k, i.e.electron pairs with equal and opposite wave vectors. The associated two particle wavefunction y/(/r,/2) must obey the Schrddinger equation

- =- {z t + z t) v {n, r) + V (r y r) ry (r e r) = E ry (r y 12\ = @ + 2 E!) y Q r, rr) .zm (10.18)

€ is the energy of the electron pair relative to the interaction-free state ( r/ = 0), in which

10. Superconductivity

Distance from electron

10.3 Instal

each of tThe two-

We noteSect .8.3)vanishing

v(\ -

which de1to pairs \ihrttp (Fig

. e i < -

The quanin - ft, thr(10.21a) ,

s(k) =

Inserting rmalizatior

h 2 k 2

m

The in

describes rthe simpletractive, tl

/ tt _\,2,

10.3 Instability of the "Fermi Sea" and Cooper Pairs

each of the two electrons at the Fermi level would possess an eneryy EqF: h2 tc2F72n'

The two-particle function in this case consists of two plane waves

(10.1e)

(10.20)

(10.21a)

(10.z2)

(10.23)

195

(# "'.') G"*,) =L "* q,-,s .

Fig.10.8. Representation (in re-ciprocal space) of electron paircollisions for vthich k1+ k2= k\+tri = If remains constant. Twospherical shells with Fermi mdiustF and thickness /,t desc be thepairs of wav€ vectors *t and ,.2.AII pairs for whi,ch 4+ k2= Kend in the shaded volume (rota-tionally symmetric about -a<). Thenumber of palrs ,it,/iz is propor-tional to this volume in /. spaceand is maximum for,K= 0

Fig.10,9. Diagram to illustratethe simplest attmctive interactionbetween two electrons whichleads to Cooper pairing withinBCS theory. The interactionpotential is assumed to be con-stant (= - I/o) in ihe dashedregion of ,{ space, i.e., betweenthe energy shells tg+ri@D andEg - h(DD knD is the Debye fre-quency of the matedal). For thecase of one extra Coopet pairconsidered here only the k-spacewith -E>EF is of i[terest

We note that (10.19) implies that the two electrons have opposite spin (compare

Sect. 8.3). The most general representation of a two-particle state for the case of a non-

vanishing interaction (l/+ 0) is given by the series

which depends only on the relative coordinate r : \- rz. The summation is confined

to pairs with k = h = -kz, which, because the interaction is restricted to the region

i a.ro €ig. 10.9), must obey the condition

r -2 t -2

nf;<!-L<n9+nao

The quantity lg(ft) l2 is the probability of finding one electron in state ft and the otheri" - L, ttrat is, ihe eiectron pair in (ft, - ft). Due to the Pauli principle and the condition(10.21a) we have

s@\=o for ft:n *.rr*(10.21b)

Inserting (10.20) in (10.1S), multiplving by exp (-ik' 'r) and integrating over the nor-

malization volume yields

t 2 1 2 |" ^ g ( k )+ - L gk ' \Vkk =@+2E i1gG)

m L - k ,

The interaction matrix element

Vkk, - I Y (r) e-l(k - k')' dr

describes scattering ofthe electron pair from (k,-k)to (k''-k') and vice versa. In

the simplest model this matrix element 211, is assumed to be independent of ft and at-tractive, that is, Vpp,<0:

vo(vo>ot ror t I . (3. 4k ' ) . r ! *n,o\ z m 2 m /

otherwise

It thus follows from (10.22) that

/ t 2 t 2 \

| -L" - t -z r l l s$ t - -e ,\ / 1 t /

e=4 | o* ' tL t

- a " '

is independent of &.After summing (10.25a) over ft and comparing with (10.25b), consistency demands

that

, - zo t - I' - ' ?We replace the sum over ft by the integral

z' I-=f iatk l z t t ) -

(10,27)

and keep in mind that the sum as well as the integral extends only over the manifoldof states of o/,e spin type. We have already encountered such summations in the con-text of exchange interaction between electrons (Sect.8.4). With reference to Sect.7.5we split the integral over the entire ft-space into an integral over the Fermi sphere andthe energy and obtain from (10.26)

, r r = { ,

where

dEr r ^ r r 0

h 2 k zwith E = :--::-

2 m

10. Superconductivi ty

(r0.24)

(10.25 a)

(10.25 b)

(10.26)

(10.28)

(10.29 a)

(10.29b)

1 rc dse'= 'o

enf JJ

lerad4k) l

Since the integral over the energy extends only over the narrow interval between ,E$and EY + h@D, the first part of the integral can be considered as a conslant. This con-stant is the density of states for one spin type (7 .41) at the Fermi level which we shalldenote as Z(E$) here and in the following parts of the chapter. By performing the in-tegration we obtain

Eg+h@n

1= voz(Eg) j-Eg

dE ^ = ! v^z t r l l r n t -2hao

2E- t 2E i 2 " e

^ 2 h a p"- l-"-pz/vrz@n

For the case of a weak interaction, VyZ(EY)<1, it follows that

E= - 2h aDe-2 / vaz (E l ) (10.30)

10.4 The B

There thrfully occrthe non-irminute a1that the (Fermi-seathe attra(treated. Iltron pairslower-enephase, as

In theelectrons,(r1, 12 ) unspins musspin partmetric. Tland oppolshould beparallel spperrmentahave howea degeneri

10.4 Tht

In Sect. 10interactionmi sea dueagined as(kr, - k!),the pair frron the fornCooper paiis achievedreduction irpairs. Themust thus sfigurationsreduction dcitation ablenergy. Thtthe pair sta

The total en

l9'7

E ,o1o '

10.4 The BCS Ground State

There thus exists a two-electron bound state, whose energy- is lower than that of thefully occupied Fermi sea (Z= 0) by an amount e = E-2EY<0. The ground state ofthe non-interacting free electron gas, as treated in Sect.6.2, becomes unstable when aminute attractive interaction between electrons is "switched on". It should be notedthat the energy reduction e^ (10.30) results ftom a Gedankenexperiment in which theFermisea for states with h'k'/2m<E$ is assumed to be fixed and only the effect ofthe attraction between two additional electrons in the presence of the Fermi-sea istreated. In reality the instability leads to the formation of a high density of such elec-tron pairs, so called Cooper pairs (k, -k), via which the system tries to achieve a newlower-energy ground state. This new ground state is identical to the superconductingphase, as we shall see.

In the aforegoing treatment it was impodant that the Pauli principle applies to bothelectrons. The two-padicle wavefunction (10.19) was symmetdc in spatial coordinates(r1,r2) under exchange of electrons 1 and 2, but the whole wavefunction includingspins must be antisymmetric (the most general formulation of the Pauli principle). Thespin part of the wave-function, not indicated in (10.19), must therefore be antisym-metric. The Cooper pair therefore compdses two electrons with opposite wave vectorsand opposite spins (/r1, -kl). In this connection one often speaks of singlet pairs. Itshould be noted that a more complicated electron-electron coupling could lead toparallel spin pairs, so-called triplets. Models of this type have been discussed, but ex-perimental proof of the existence of such states has not yet been found. Triplet pairshave however been found in liquid'He. At low temperatures this system behaves likea degenerate Fermi gas.

10.4 The BCS Ground State

In Sect. 10.3 we saw how a weak attractive interaction, resulting from electron-phononinteraction, leads to the formation of "Cooper pairs". The energy reduction of the Fer-mi sea due to a single pair was calculated in (10.29). Such a Cooper pair must be im-agined as an electron pair in which the two electrons always occupy states(kI , - kl),(k't , - ft'J ) and so on, with opposed k-vectors and spin. The scattering ofthe pair from (/r1, - ft.|) to (k'l, - k'I) mediated by Vpp,leads to an energy reductionon the formation of a Cooper pair (Fig. 10.10). Due to this energy reduction ever moreCooper pairs are formed. The new ground state of the Fermi sea after pair formationis achieved through a complicated interaction between the electrons. The total energyreduction is not obtained by simply summing the contributions (10.29) of single Cooperpairs. The effect of each single Cooper pair depends on those aheady present. Onemust thus seek the minimum total energy of the whole system for all possible pair con-figurations, taking into account the kinetic one-electron component and the energyreduction due to "pair collisions", i.e., the electron-phonon interaction. Since an ex-citation above .E$ is necessary, the pairing is associated with an increase in kineticenergy. The kinetic component can be given immediately: If n& is the probability thatthe pair state (kI, - kI) is occupied, the kinetic component Ekin is

with 1: h2 k2/2m - Eg (10.31) Fig. 10.10. Representation in ,rspace of th€ scatte ng of an elec-tron pair with wave vectors(*, -t) to the state (k', k')The total energy reduction due to the pair collisions (ftl, - kI) .2 (k'I , k'I) can be most

198 10. Superconductivity

easily calculated via the Hamiltonian .i4 which explicitly takes account of the fact thatthe "annihilation" of a pair (kI,-kI) and the ..simultaneous creation', of a pair(k'I , - k'I), i.e. a scattering from (ftf, - ftJ ) to (ft1, - ft'J ), leads, to an energy reduc-tion of Vpp, (Sect. 10.3 and Fig. 10.10). Since a pair state ft can be either occupied orunoccupied, we choose a representation consisting of two orthogonal states l1)r andl0)1, where l1)1is the state in which (ftl, -ftJ) is occupied, and l0[ is rhe correipon-ding unoccupied state. The most general state of the pair (kI , - k!) is thus given by

lV)* = u* l0)k + uk | 1)k (10.32)

This is.an alternative representation of the Cooper pair wavefunction (10.19). Hencewk = ui and 1 - wo = y"o 4ys the probabilities that the pair state is occupied and unoc-cupied, respectively. We shall assume that the probability amplitudes u1, and u1, arereal. It can be shown in a more rigorous theoretical treatment that this restdction isnot important. In the representation (10.32) the ground state of the many-body systemof all Cooper pairs can be approximated by the product of the state vectors of the sinslepairs

ldscs)= II (u1,10)o+ uo)1)1,)

This approximatibn amounts to a description of the many-body state in terms of non-interacting pairs, i.e., interactions between the pairs are neglected in the state vector.In the two dimensional representation

(10.34)

one can use the Pauli matrices

(10.35)

to describe the "creation" or "annihilation" of a Cooper pair:The operator

of = i@p + iop)

transforms the unoccupied state ]0)k into the occupied state ] 1)0, while

o; =+@P-ioP)

transforms the state ]1)* into l0)k. From the representations (10.34-36) one candeduce the following properties:

0 1 l 1 ) f t = 0 ,

o; l>k = 0>k ,

The matrices o/ and, op are formally identical to the spin operators introduced inSect. 8.7. Their physical interpretation as ..creator" and ,,annihilator,, of Cooper oairs

,,r: (;)_ , ,_= 0_

(10.33)

( 10.36 a)

(10.36b)

(10.37 a)

(10.37 b)

t

"r,= (? ;)_ , "r,= (i ;),

of10 \=11) * ,

o f l0 [=0 .

Tuprlo

10.4 The BCS Ground State

is however completely different from that in Sect. 8.7, where the reversal of a spin wasdescribed.

Scattering from (ftl, - ftJ ) to (ft'1, - &'J ) is associated with an energy reduction byan amount Vkk,. In the simple BCS model of superconductivity (as in Sect. 10.3) thisinteraction matrix element 211, is assumed to be independent of k,k', r.e. constant.We relate this to the normalization volume of the crystal, Zr, by setting it equal toVn/L3. The scattering process is described in the two-dimensional representation asannihilation of k(op ) and creation of k'(of,). The operator that describes the cor-responding energy reduction is thus found to be - (Vo/ L')o ['o I . The total energyreduction due to pair collisions ft - ft' and ft' - ft is given by surnming over all collisionsand may be expressed in operator terminology as

199

Since Z6 is restricted to the shell afrtue around E9, the sum over k,k'likewise in-cludes only pair states in this shell. In the sum, scattering in both direction is con-sidered; the right-hand side of (10.38) follows when one exchanges the indices.

The energy reduction due to collisions is given from perturbation theory as the ex-pectation value of the operator /l(10.38) in the state | @Bcs) (10.33) of the many-bodysvstem

(@scs f Otrr>= -# (upp<01 + ue e<11)Tk k '

oI ok ,

In evaluating (10.39), one should note that the operator 6l @;) acts only on thestate I 1)r ( | 0)r). The explicit rules are given in (10.37). Furthermore, from (10.34) wehave the orthonormaliry relations

. - l _ l v "V o L : t o l . o k i o 1 o 1 \ = - ; \ L o l o ;

k k ' l L - k k '

1\11170= 1 , * (010)o= t , k(1 lg) f r= 0

We thus obtain

v"_(Qecsltr lbscs) ---+ I upup ululL - k k ,

wscs:2 |

,? n-# l ,u1,u2uyu1,,

f ! 1uo o>o+uoll>o)) .

(10.39)[+

(10.38)

(10.40)

(10.41)

According to (10.31, 41) the total energy of the system of Cooper pairs can thereforebe reDresented as

(10.42)

The BCS ground state at T = 0K of the system of Cooper pairs is given by the mini-mum, IZf,6s, of the energy density I/BCS. By minimizing (10.42) as a function of theprobability amplitudes up and u1, we obtain the energy of the ground state J,tr{6s andthe occupation and non-occupation probabilities w* = utr and (1- w) = rf. Becauseof the relationship between ,& and. up the calculation is greatly simplified by setting

u1, = llw1, = s6 Qo , (10.43 a)

200

u1, = {1 -w1, = si11 go ,

which guarantees that

u l+ u l = cos2 d* + s in2 dr I

The minimizing is then with respect to 9k.The quantity to be minimized can be written

\ ' . , ^ ^ -2a Vo tl lecs | 2(1cos- 01, - - ] | . cos 0l sin d1, cos da sin do

= f . z t ,cos2 o, - ! \ f , s in2o1,s in2o1.7

_ * _ _ - " * 4 L t * t

The condition for the minimum of l4ls6s then reads

AII' I/

" ' : - ! " . - Z 1 o s i n 2 0 1 - ' _ \ | c o s 2 d l s i n 2 0 0 = g , o rdAt L' t ,

10. Superconductivi ty

(10.43 b)

(10.43 c)

(10.44)

(10.45 a)

(10.45 b)

(10.46)

(10.47)

(10.48)

(10.4e)

(10.50)

the BCS

( 1 0 . 5 1 )

s i n 2 9 p , .

We let

n - v o t . , . . z o r . ^ ,O = ; L , u r , u t i

I s i n d l c o s d l .

Eo:{4+V

and obtain from standard trigonometry

sin 20, A

cos29p '

1o

AZUkDk = Sln Zt& = -- ,

Lk

u?,- u'o = -1*/E* .

Thus the occupation probability, w*=u?, of a pair state (ftl,-ftJ) inground state at 7= 0K is given by

' / , \ , / . ' \w*--u1-^ ( t - i ) - . ( r - - i r . I2 \ E*/ 2\ /e i+t /

$ , t a n 2 0 1 =- !vo y2 L 3 7 t

This function is plotted in Fig. 10.11. At f : 0 K(!) it has a form similar to the Fermifunction at finite temperature and a more exact analysis shows that it is like the Fermidistribution at the finite critical temDerature 7:. It should be noted that this form of

10.4 The BCS Ground Stat€ 201

- w k : v ? a t T = O--- Fermi function

a t T = T c

- h(,)o

Electron energy 4k

fig. 10.11, The BCS occupation probability ,? for Cooper pails in the vicinity of the Fermi energy E$. Theenergy is given as (r : t(/r) - E$, i.e., the Fermi energy ({i = 0) seryes as a reference point Also shown forcomparison is the Fermi-Dilac distribution lunction for normally conducting electrons at the citical temper-atnre T.(dashed ltue\. The curves are related to one another by the BCS relationship between / (0) and Tc(10.67)

uf results from the representation in terms of single-particle states with well-definedquantum numbers ft. This representation is not particularly appropriate for the many-body problem; for example, one does not recognize the energy gap in the excitationspectrum of the superconductor (see below). On the other hand, it can be seen in thebehavior of u2o that the Cooper pairs which contribute to the energy reduction of theground state are constructed from one-particle wavefunctions from a particular ft-region, corresponding to an energy shell of t/ around the Fermi surface.

The energy of the superconducting BCS ground state lft6s is obtained by insert-ing in Zs6s - (0.42 or 10.44) - relationships (10.48-51) which follow from theminimization. The result is

;

(L

h (rlo

tf"." = L &( - €k/E) - L3k

.42

vo(10.52)

The condensation energy of the superconducting phase is obtained by subtracting fromlr\ss the energy of the normal conducting phase, i.e., the energy of the Fermi-seawithout the attractive interaction W,= L *1.+21*..II analogy to-(10.26, 28) wenow go from a sum in ft-space to an intilral'(L''Lo-latt}z3;. After somecalculation, the specific condensation energy density is obtained as

(10.53)

Thus, for finite /, there is always a reductidn in energy for the superconducting state,whereby / is ameasure of the size of the reduction. One can visualize (10.53) by imag-ining that Z(Ei)/ electron pairs per unit volume from the energy region 1 below theFermi level all "condense" into a state at exactly / below Ei. Their average gain inenergy is thus /,/2.

The decisive role of the parameter / is clear from the following: The first excilationstate above the BCS ground state involves the breaking up of a Cooper pair due to anexternal influence. Here an electron is scattered out of (ftl) leaving behind an unpairedelectron in (ftJ). In order to calculate the necessary excitation energy, we rewrite theground state energy II{6s (10.52) as follows

tt ;_tr_l _ 12'? _2')\_! z'0,)a2 = _! zrro,t z,L' \zo vo/ 2 2

t,4u.r: L<k(1-1k/il-L#

= L E1u.- ub - L Er@?-, i f -+

, L3 a2Eku iD i+ | Es lu i ( t - u i ) - u i ( t - u i , ) l - -

k v o

uuro -tff * l, Eku'zk@? - 1 - u'zk)

I f ( f t ' l , - f t 'J) is occupied, i . "^.u?, ' :breaking up the pair, i.e., ai, = 0,

wLcs = -2 L eou!, .k + k '

The necessary excitation energy is the difference between the enelgies of the initial and

final states

lE = wLcs- t4t$., = 2Bo, = 214f;+'E . (10.56)

The first term in the square-root, (|,. describes th-e k-inetic ene^rgy of the two electrons

"scattered" out of the cooper pair' Since (o' = h'zk'2/2m - Eul' this can be arbitrarilysmall, i.e., the excitation requires a minimum finite energy

: -2 L Eku lk

- r S

- , 4 \ '

10. Superconductivity

(10.54)

1, then the first excited state I'/;cs is achieved byand therefore

(10.5 5)

(10.57)

(10.58)

AEnin= 2A

The excitation spectrum of the superconducting state contains a gap of 2/, which cor-responds to the energy required to break up a Cooper pair. Equation (10'56) describesthe excitation energy of the two electrons that result from the destruction of a Cooperpair. If we imagine that a single electron is added to the BCS ground state, then it cannaturally find no partner for Cooper pairing. Which energy state can this electron oc-

cupy? From (10.56) we conclude that the possible states of this excited system are given

ay no - 1(o+ A2)1/2. If the unpaired electron is at (k = 0 it thus has an energy which

lies at least / above the BCS ground state (Fig. 10.12). However it can also occupy

states with finite (ft; for 120> Z2 tne one-electron energy levels

A

BCS Groundsta!e

- , I - t ) - n ^ -01 1 , k = V C i + a - - < k = - - E i

z m

Fig. 10.12. (a) Simplified rcpresentation of the excitatron spectrum of a supelconductol on the basis of one-

electron energies ,gr. ln the BCS ground state, the one-electron picture collapses: At I: 0 K all Cooperpairs occupy one and th€ same ground state (lik-e Bosons). This state is therefole eneBetically identical to

ihe chemical potential, i.e., the Fermi energy t9. The energy Ievel drawn here can thus be lormally int€r-preted as the "many-body energy of one electron" (total energy of all particles divided by the number of

;lectrons). Note that a minimum energy of 2/ is necessary to split up a Cooper pair. (b) Density of states

for excited electrons in a superconductor D" relative to that of a nolmal conductor. E/.: 0 corresponds to

tne rermr energy Ei0 l

Relative density of states Ds/Di

10.4 THe BCS Ground State

become occupied. These are exactly the levels of the free electron gas (of a normal con-ductor). Thus, for energies well above the Fermi energy ((f,> /2), the continuum ofstates of a normal conductor results. To compare the density of states in the energyrange 1 about the Fermi level for excited electrons D, (E1) in a superconductor withthat of a normal conductor Dn ((r) (Sect. 6.1), we note that in the phase transition nostates are lost, i.e.

Ds(Ek)dEk = D"(1r)d1* (10.59a)

Because we are only interested in the immediate region / around Eg, it is sufficientto assume that D|G)=DJEF) = const. According to (10.56) it then follows that

D"(E*)D"(E$)

a F _l " R

= , rF

= lvE i -a '- -K I

L0

t=!vo v a : !vo-

z L 1 i E k 2 L 3

for Ex>A

for E*</

(10.59 b)

This function possesses a pole above / and for Er>/ converts, as expected, to thedensity of states of a normal conductor; it is depicted in Fig. 10.12b.

It should again be emphasised that the illustration in Fig. 10.12 does not expressthe fact that the breaking up of a Cooper pair requires a minimum energy 2/.lt saysonly that the "addition" of an unpaired electron to the BCS gound state makes possi-ble the occupation of one-particle states which lie at least / above the BCS ground stateenergy (for one electron). The density of states in the vicinity of the minimum energysingle-particle states is singular (Fig. 10.12b).

The "addition" of electrons to the BCS ground state can be realized in a experimentby the injection of electrons via an insulating tunnel barrier (panel IX). Such tunnelexperiments are today very common in superconductor research, They may be conve-niently interpreted with reference to diagrams such as Fig. 10.12b

We wish now to determine the gap / (or 2/) n the excitation spectrum. For thiswe combine (10.49) with (10.46, 47', and obtain

(10.60)r z7 l1?+ t '1

As in (10.26, 28), the sum in ft-space is replaced by an integral (L-3 E k.-l dk/Bn3\.We note that we are summing again over pair states, i.e., that instead of the one-par-ticle density of states D(EF+ 6) we must take the pair density of states Z(,Eg+ O =+D(E"F + t).

Fudhermore, in contrast to Sect. 10.3, the sum is taken over a spherical shellaioD located symmetrically around E$. We than have

. v^ h?, z@g + t\ .-r =_: : J __i j aq . ( l0 .6ta)

2 _h.^ Vt . + a '

In the region 1E$ - hap,El+ hatpl where I/a does not vanish, Z(Eg+0 varies onlyslightly, and, due to the symmetry about E'$, it follows that

t h a "

voz@g) '" ttri7' "' (10.61b)

In the case of a weak interaction,

h a o =2hrnoe

10. Superconductivity

(10.62)

i.e., VyZ(EF)<\, the gap energy is thus

-1/voz(El) . (10.63)

voz(E:F). . f l a n

= aICSlnn -A

sinhrl/voZ(E?)l

This result bears a noticeable similarity to (10.30), i.e., to the binding energy s of twoelectrons in a Cooper pair in the presence of a fully occupied sea. As in (10.30), onesees that even a very small attractive interaction, i.e., a very small positive 20, resultsin a finite gap energy /, but that./ cannot be expanded in a series for small 26. Aperturbation calculation would thus be unable to provide the result (10.63). For thesake of completeness, it should also be mentioned that superconductors have now beendiscovered that have a vanishingly small gap energy.

10.5 Consequences of the BCS Theory and Compa sonwith Experimental Results

An important prediction of the BCS theory is the existence of ^ gap A (or 2/ ) in theexcitation spectrum of a superconductor. The tunnel experiments described in Panel IXprovide direct experimental evidence for the gap. Indications for the existence of a gapare also present in the behavior of the electronic specific heat capacity of a supercon-ductor at very low temperature. The exponential behavior of this specific heat capacity(10.3) is readily understandable for a system whose excited states are reached via excita-tion across an energy gap. The probability that the excited state is occupied is then pro-portional to an exponential Boltzmann expression, which also appears in the specificheat capacity (temperature derivative of the internal energy) as the main factor deter-mining its temperature dependence.

A further direct determination of the energy gap 2A is possible using spectroscopywith electromagnetic radiation (optical spectroscopy). Electromagnetic radiation is on-ly absorbed when the photon energy ha exceeds the energy necessary to break up theCooper pair, i.e., i (o must be larger than than the gap eneryy Z/. Tlpical gap energiesfor classical superconductors lie in the region of a few mev. The appropriate ex-periments must therefore be carried out with microwave radiation. The curves inFig. 10.13 result from an experiment in which the microwave intensity 1is measuredby a bolometer after multiple reflection in a cavity made of the material to be exam-ined. With an external magnetic field, the matedal can be driven from the supercon-ducting state (intensity Is) to the normal state (intensity 1N). This allows the measure-ment of a difference quantity (,Is - 1N)/1N. At a photon energy corresponding to thegap energy 2/, this quantity drops suddenly. This corresponds to an abrupt decreasein the reflectivity of the superconducting material for ha>2/, whereas for photonenergies below 2/ the superconductor reflects totally since there are no excitationmechanisms.

At all temperatures above Z = 0 K there is a finite possibility of finding electronsin the normal state. As the temperature rises, more and more Cooper pairs break up;thus a temperature increase has a destructive effect on the superconducting phase. The