193 10.3 lnstability of the "Fermi Sea" and Cooper Pain exponentiallywith the interior of the superconductoragainst extelnal fields, alsodecay London penetlatlon distanceinto the solid. For an order of magnitudeestimate of the density' i'e'' J.pth, *" set m equal to the electronmass and take nsto be the atomic Sn' for example *. urru-" that ea;h atom providesa superconductingelectron' For of super- one obtainsAr= 260A. An important elementof the microscopictheory pairs, which conductivityisihat it is not electronsbut electronpairs, so-calledCooper if, insteadof are the current carriers.This can be includedin the London equations describe n,, on" ut.t half of the electrondensity n"/2' Becartsethe London equations of superconduc- ttreMeissner-Ochsenfeld effect particularlywell, a rnicroscopictheory (10'10b)' iivity shouldbe ableto provide' amongother things, an equationof the tvpe temperature ihis, utta an explanationof the vanishingresistance at sufficiently low super- (T< f.), is the cetttral concernof the BCS theory mentionedabove' Because alsobe sufficient- conductivityis evidentlya widespreadphenomenon, the theory must sec- ly general;it should not hittg" on one specialmetallic property' In the following will be treated iioi, tii" t"ri" aspectsof thi microscopictheory of superconductivity in a somewhatsimPlified form'

10.3 Instability of the "Fermi Sea" and Cooper Pairs

phenomenonof From the previously mentioned fundamental experimentson the phase electrongas superconductivity,ii is clearthat we are dealingwith a new of the An in a metal which displaysthe unusualproperty of "infinitely high" conductivity' important contributi;n 10 our understandingof this new phasewas madebv Cooper ground (T=0K) of an electrongas ttO.Ol wtro, in 1956,recognised, that the state interactionbetween each pair of iS..t. O.Z)it unstabieif one adds a weak attractive in the form el""trorrr. sr"h an interactionhad alreadybeen discussedby Fr,hlich [10.7] on ac- of the phonon-mediatedinteraction. As it passesthrough the solid' an electron' positions if itt negative charge, leavesbehind a deformation trail affecting the positive charge ".r"tof the ion Thi, trail is associatedwith an increaseddensity of duetotheioncores,andthushasanattractiveeffectonasecondelectron'Thelattice"o..i. (Fig' 10'7a)' deformationtherefore causes a weak attractionbetween pairs of electrons of This attractiveelectron-electron interaction is retardedbecause of the slow motion theionsincomparisonwiththealmostinstantaneousCoulornbrepulsionbetweenelec- pull only after itorr; instant when an electronpasses, the ions receivea which' polarizationof the the eiectron"t,ft" has passed,leads to a displacementand therebyto a maximum at a distancefrom iutti.. €ig. 10.7b). The lattice deformation teachesits (Fe^rmivelocity the first electron which can be estimatedfrom the electronvelocity tit. ','*i."- prt"non vibration period (2n/ae-r0-13 9' The t*o- ;;i".;7;;;; of eiectronscorielated by the lattice deformationthus havean approximatesbparation (whichis also esti- uUou,1OOO A. Thi, then to the "size" of a Cooperpair "orresponds of matedin sect. 10.7using a different method).The extremelylong interactionrange repul- the two electronscorrelated by a lattice defolmation explainswhy the coulomb just a few sion is insignificant; it is completely screenedout over distancesof Angstroms.-Quantum mechanically,the lattice deformationcan be understoodas the superposi- tionif phononswhich the electron,due to its interactionwith the lattice,continuously emitsand absorbs.To comply with energyconservation, the phononsconstituting the t94 10. Superconductivity 10.3 Instal

Latticeplanes

z + Distanceutstance fromtrom electron each of t Fig. 10.7a,b. Schematicrepresentation of the electron-phononinteraction that leadsto the formation of The two- Cooper pairs. (a) An electron(e- ) travellingthrough the crystallattice leaves behind a deformationtrail, which can be regardedas an accumulationof the positivelycharged ion cores,i.e., as a compressionof the /t Iatticeplanes (exaggerated here for clarity). This meansthat an areaof enhancedpositive charge (compared t_ to th€ otherwise neutral crystal) is created behind the electron, and exerts an attractive force on a second \,2, electron.(b) Qualitativeplot of the displacementof the ion coresas a function of their distancebehind the first electron.Compared with the high electronvelocity uF (- 108cmls). the tatticefollows only very slowly We note and hasits ma{mum delormationat a distanceuF2n/(DDbehind, the electron,as determinedby the typical Sect.8.3) phonon frequencycoD (Debye frequency). Thus the couplingol the two electronsinto a Cooperpak occurs vanishing over distancesol more than 1000A, at which thei Coulomb repulsionis completelyscreened v(\- lattice deformationmay only existfor a time t = 2t/a determinedby the uncertainty relationl thereafter,they must be absorbed.One thus speaksof "virtual" phonons. which de1 The ground state of a non-interactingFermi gas of electronsin a potential well to pairs \i (Chap.6) correspondsto the situation where all electron stateswith wave vector ft hrttp (Fig within the Fermi sphere[,Eg(f = 0K1= n2tc2rtZml are filled and all stateswith,E>Eg are unoccupied.We now perform a Gedankenexperiment.andadd to this system two .ei<- electrons[ft1, -E(ft1)] and, lk2,E(k)l in statesjust aboveE$. A weak attractiveinterac- tion betweenthese two electronsis switchedon in the form of phonon exchange.All other electronsin the Fermi seaare assumedto be non-interacting,and, on account The quan of the Pauli exclusionprinciple, they exclude a further occupation of stateswith in - ft, thr l/c]

Sincethe interactionin ft-spaceis restrictedto a shellwith an energythickness of haD Insertingr (with (,D = Debye frequency)above E$ the possibleft-states are given by the shaded malizatior areain Fig. 10.8. This areaand thereforethe number of energy-reducingphonon ex- changeprocesses - i.e. the strengthof the attractiveinteraction - is maximum for h2k2 1( = 0. It is thereforesufficient in what follows to considerthe caseft1 = - kz: k, i.e. m electronpairs with equal and oppositewave vectors. The associatedtwo particlewave function y/(/r,/2) must obey the Schrddingerequation The in

- =- {z t + z t) v {n, r) + V (r y r) ry(r e r) = E ry(r y 12\= @+ 2E!) y Qr, rr) . zm (10.18) describesr the simple € is the energyof the electronpair relativeto the interaction-freestate ( r/ = 0), in which tractive, tl 10.3 Instability of the "Fermi Sea" and Cooper Pairs 195

Fig.10.8. Representation(in re- ciprocal space) of electron pair collisionsfor vthich k1+ k2= k\ +tri = If remainsconstant. Two spherical shells with Fermi mdius tF and thickness/,t desc be the pairs of wav€ vectors*t and ,.2. AII pairs for whi,ch4+ k2= K end in the shadedvolume (rota- tionally symmetric about -a<).The number of palrs ,it,/iz is propor- tional to this volume in /. space and is maximum for,K= 0 eachof the two electronsat the Fermi level would possessan eneryyEqF: h2tc2F72n' The two-particlefunction in this caseconsists of two plane waves =L q,-,s. (10.1e) (# "'.')G"*,) "* We note that (10.19) implies that the two electronshave opposite spin (compare Sect.8.3). The most generalrepresentation of a two-particlestate for the caseof a non- vanishinginteraction (l/+ 0) is given by the series (10.20)

which dependsonly on the relativecoordinate r : \- rz. The summationis confined to pairs with k = h = -kz, which, becausethe interactionis restrictedto the region i a.ro€ig. 10.9),must obey the condition

r-2 t-2 nf;

The quantity lg(ft) l2 is the probability of finding one electronin stateft and the other i" - L, ttrat is, ihe eiectronpair in (ft, - ft). Due to the Pauli principleand the condition (10.21a)we have for (10.21b) s@\=o ft:n *.rr* (10.20)in (10.1S),multiplving by exp (-ik' 'r) and integratingover the nor- Inserting Fig.10,9. Diagram to illustrate malization volume yields the simplest attmctive interaction between two electrons which t2 12 | " ^ gk'\Vkk=@+2Ei1gG) (10.z2) leads to Cooper pairing within g(k)+- L BCS theory. The interaction m L- k, potential is assumedto be con- - The interaction matrix element stant (= I/o) in ihe dashed region of ,{ space,i.e., between - k')' the energy shellstg+ri@D and Vkk,- Y(r) e-l(k dr (10.23) I Eg - h(DDknD is the Debye fre- quency In of the matedal). For the describesscattering ofthe electronpair from (k,-k)to (k''-k') and viceversa. case of one extra Coopet pair the simplestmodel this matrix element211, is assumedto be independentof ft and at- considered here only the k-space tractive, that is, Vpp,<0: with -E>EF is of i[terest 10.Superconductivity 10.4The B

There thr vo(vo>otror tI.(3. 4k').r! *n,o fully occr ,rr={, \zm 2m / (r0.24) the non-ir otherwise minute a1 that the ( It thus follows from (10.22)that Fermi-sea the attra( / t2t2 \ -L" -t-zrll -e treated.Il | s$t- , where (10.25a) tron pairs \/1t/ lower-ene phase,as e=4 | o*'t In the -a "' (10.25b) Lt electrons, (r1,12 un is independentof &. ) spinsmus After summing(10.25a) overft and comparingwith (10.25b),consistency demands spin part that metric. Tl I '-',-zo t- (10.26) and oppol ? shouldbe parallel sp We replacethe sum over ft by the integral perrmenta havehowe z'I-=fiat (10,27) a degeneri k lztt )- and keepin mind that the sum as well as the integralextends only over the manifold 10.4 Tht of statesof o/,e spin type. We havealready encountered such summationsin the con- text of exchangeinteraction between electrons (Sect.8.4). With referenceto Sect.7.5 we split the integral over the entireft-space into an integralover the Fermi sphereand In Sect.10 the energyand obtain from (10.26) interaction mi seadue 1 rc dse dE h2kz agined as '= 'o JJ with E = :--::- (10.28) (kr,- enf lerad4k)l rr ^ rr0 2m k!), the pair frr Sincethe integral over the energyextends only over the narrow interval between,E$ on the forn and EY+ h@D,the first part of the integralcan be consideredas a conslant.This con- Cooperpai stant is the densityof statesfor one spin type (7.41) at the Fermi level which we shall is achieved denoteas Z(E$) hereand in the following parts of the chapter.By performingthe in- reductionir tegration we obtain pairs.The must thus s Eg+h@n dE figurations 1= voz(Eg)j ^ = !v^ztrllrnt-2hao (10.29a) -Eg 2E-t 2Ei 2 " e reductiond citation abl 2hap energy.Tht ^ (10.29b) "- l-"-pz/vrz@n the pair sta

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

E= - 2h aDe-2/vaz(El) (10.30) The total en 10.4 The BCS Ground State l9'7

There thus existsa two-electronbound state, whoseenergy- is lower than that of the fully occupiedFermi sea(Z= 0) by an amount e = E-2EY<0. The ground stateof the non-interactingfree electrongas, as treatedin Sect.6.2,becomes unstable when a minute attractive interactionbetween electrons is "switched on". It should be noted that the energyreduction e^(10.30) results ftom a Gedankenexperimentin which the Fermisea for stateswith h'k'/2m

10.4 The BCS Ground State

In Sect.10.3 we sawhow a weak attractiveinteraction, resulting from electron-phonon interaction,leads to the formation of "Cooper pairs". The energyreduction of the Fer- mi seadue to a singlepair was calculatedin (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 opposedk-vectors and spin. The scatteringof the pair from (/r1, - ft.|) to (k'l, - k'I) mediatedby Vpp,leadsto an energyreduction on the formation of a Cooperpair (Fig. 10.10).Due to this energyreduction ever more Cooper pairs are formed. The new ground stateof the Fermi seaafter pair formation is achievedthrough a complicatedinteraction between the electrons.The total energy reductionis not obtainedby simply summingthe contributions(10.29) of singleCooper pairs. The effect of each single Cooper pair dependson those aheady present.One must thus seekthe minimum total energyof the whole systemfor all possiblepair con- figurations, taking into accountthe kinetic one-electroncomponent and the energy reduction due to "pair collisions", i.e., the electron-phononinteraction. Sincean ex- citation above .E$ is necessary,the pairing is associatedwith an increasein kinetic energy.The kinetic componentcan be givenimmediately: If n& is the probability that the pair state (kI, - kI) is occupied,the kinetic componentEkin is

,o1o' with 1: h2k2/2m - Eg (10.31) Fig. 10.10. Representationin ,r E spaceof th€ scatte ng of an elec- tron pair with wave vectors The total energyreduction due to the pair collisions(ftl, - kI) .2 (k'I , k'I) can be most (*, -t) to the state(k', k') 198 10. Superconductivity

easilycalculated via the Hamiltonian .i4 which explicitlytakes account of the fact that ..simultaneous the "annihilation" of a pair (kI,-kI) and the creation', of a pair - (k'I , k'I), i.e. a scatteringfrom (ftf, - ftJ ) to (ft1, - ft'J ), leads,to an energyreduc- tion of Vpp,(Sect. 10.3 and Fig. 10.10).Since a pair stateft can be either occupiedor unoccupied, we choosea representationconsisting of two orthogonal statesl1)r and -ftJ) l0)1,where l1)1is the statein which(ftl, is occupied,and l0[ is rhecorreipon- ding unoccupiedstate. The most generalstate of the pair (kI , - k!) is thus given by

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

This is.an alternativerepresentation of the Cooper pair wavefunction(10.19). Hence wk = ui and 1- wo = y"o4ys the probabilitiesthat the pair stateis occupiedand unoc- cupied, respectively.We shall assumethat the probability amplitudesu1, and u1,are real. It can be shown in a more rigorous theoreticaltreatment that this restdction is not important. In the representation(10.32) the ground stateof the many-bodysystem of all Cooperpairs can be approximatedby the productof the statevectors of the sinsle pairs

ldscs)=II (u1,10)o+uo)1)1,) (10.33)

This approximatibnamounts to a descriptionof the many-bodystate in termsof non- interactingpairs, i.e., interactionsbetween the pairs are neglectedin the statevector. In the two dimensionalrepresentation ,,r:(;)_ , ,_= 0_ (10.34) one can use the Pauli matrices t "r,=(? ;)_ , "r,=(i ;), (10.35) to describethe "creation" or "annihilation" of a Cooper pair: The operator

of =i@p +iop) (10.36 a)

transformsthe unoccupiedstate ]0)k into the occupiedstate ] 1)0,while

o; =+@P-ioP) (10.36b)

transformsthe state ]1)* into l0)k. From the representations(10.34-36) one can deducethe following properties: T 01 l1)ft=0, of10\=11)*, (10.37a) u p o; l>k= 0>k, ofl0[=0. (10.37b) rl o The matriceso/ and,op are formally identical to the spin operatorsintroduced in ..creator" ,,annihilator,, Sect.8.7. Their physicalinterpretation as and of Cooperoairs 10.4 The BCS Ground State 199 is howevercompletely different from that in Sect.8.7, wherethe reversalof a spin was described. Scatteringfrom (ftl, - ftJ ) to (ft'1, - &'J ) is associatedwith an energyreduction by an amount Vkk,.In the simpleBCS model of superconductivity(as in Sect.10.3) this interaction matrix element 211,is assumedto be independentof k,k', r.e. constant. We relate this to the normalizationvolume of the crystal, Zr, by settingit equal to Vn/L3. The scatteringprocess is describedin the two-dimensionalrepresentation as annihilation of k(op ) and creation of k'(of,). The operator that describesthe cor- respondingenergy reduction is thus found to be - (Vo/ L')o ['o I . The total energy reductiondue to pair collisionsft - ft' andft' - ft is givenby surnmingover all collisions and may be expressedin operatorterminology as

_l v" .-l -;\ VoL :tol.ok io1o1\= L ol o; (10.38) kk'l L- kk'

SinceZ6 is restrictedto the shell afrtue aroundE9, the sum overk,k'likewise in- cludesonly pair statesin this shell. In the sum, scatteringin both direction is con- sidered;the right-hand side of (10.38)follows when one exchangesthe indices. The energyreduction due to collisionsis given from perturbationtheory as the ex- pectationvalue of the operator/l(10.38) in the state | @Bcs)(10.33) of the many-body svstem

(@scsf Otrr>=-# (upp<01+ uee<11) T oI ok, f! 1uoo>o+uoll>o)) . kk' [+ (10.39)

In evaluating(10.39), one should note that the operator 6l @;) acts only on the state I 1)r ( | 0)r). The explicit rulesare given in (10.37).Furthermore, from (10.34)we havethe orthonormaliryrelations

1\11170=1 , *(010)o=t , k(1lg)fr= 0 (10.40)

We thus obtain v"_ (Qecsltrlbscs)---+ I upupulul (10.41) L- kk,

According to (10.31,41) the total energyof the systemof Cooperpairs can therefore be reDresentedas

wscs:2 ,? n-# l,u1,u2uyu1,, (10.42) |

The BCS ground stateat T = 0K of the systemof Cooper pairs is given by the mini- mum, IZf,6s,of the energydensity I/BCS.By minimizing (10.42)as a function of the probability amplitudesup and u1,we obtain the energyof the ground stateJ,tr{6s and the occupationand non-occupationprobabilities w* = utr and (1- w) = rf. Because of the relationshipbetween ,& and.up the calculationis greatly simplified by setting

u1,= llw1, = s6 Qo , (10.43a) 200 10. Superconductivity

u1,= {1 -w1, = si11go , (10.43b) which guaranteesthat

ul+ ul = cos2d* + sin2dr I (10.43c)

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

\' ., ^^-2a - -]Vo t llecs | 2(1cos- 01, |. cos 0l sin d1,cos da sin do

= f. _*__-zt,cos2 "*o,-! \ f, sin2o1,sin2o1. (10.44) 7 4 Lt *t

The condition for the minimum of l4ls6sthen reads

AII' I/ "':-!". -Z1osin201-'_\| cos2dlsin200 =g , or (10.45a) dAt L' t, y $,tan201=-!vo sin29p,. (10.45b) 2L3 7t We let n -vo t ., .. zo r. ^, O =; L, ur,ut I sindlcosdl . (10.46) i Eo:{4+V (10.47)

and obtain from standardtrigonometry

sin20, A ' (10.48) cos29p 1o A ZUkDk = Sln Zt& = -- , (10.4e) Lk u?,-u'o = -1*/E* . (10.50)

Thus the occupationprobability, w*=u?, of a pair state (ftl,-ftJ) in the BCS ground stateat 7= 0K is given by

' / ,\ , / .' \ w*--u1-^( t-i )-. ( r- -ir . I (10.51) 2\ E*/ 2\ /ei+t/

This function is plotted in Fig. 10.11.At f : 0 K(!) it has a form similar to the Fermi function at finite temperatureand a more exactanalysis shows that it is like the Fermi distribution at the finite critical temDerature7:. It should be noted that this form of 10.4 The BCS Ground Stat€ 201

- wk:v?atT=O ; --- Fermifunction atT=Tc

(L

- h(,)o h (rlo Electron energy 4k fig. 10.11,The BCS occupationprobability ,? for Cooperpails in the vicinity of the Fermi energyE$. The energyis givenas (r : t(/r) - E$, i.e., the Fermi energy({i = 0) seryesas a referencepoint Also shownfor comparisonis the Fermi-Dilacdistribution lunction for normally conductingelectrons at the citical temper- atnreT.(dashed ltue\. The curvesare relatedto one anotherby the BCS relationshipbetween / (0) and Tc (10.67) uf resultsfrom the representationin terms of single-particlestates with well-defined quantumnumbers ft. This representationis not particularlyappropriate for the many- body problem; for example,one doesnot recognizethe energygap in the excitation spectrumof the superconductor(see below). On the other hand, it can be seenin the behaviorof u2othat the Cooper pairs which contributeto the energyreduction of the ground state are constructedfrom one-particlewavefunctions from a particular ft- region, correspondingto an energyshell of t/ around the Fermi surface. The energyof the superconductingBCS ground state lft6s is obtainedby insert- ing in Zs6s - (0.42 or 10.44) - relationships(10.48-51) which follow from the minimization. The result is

.42 tf"." = L &( - €k/E) - L3 (10.52) k vo

The condensationenergy of the superconductingphase is obtainedby subtractingfrom lr\ss the energyof the normal conductingphase, i.e., the energyof the Fermi-sea without the attractiveinteraction W,= L *1.+21*..II analogyto-(10.26, 28) we now go from a sum in ft-spaceto an intilral'(L''Lo-latt}z3;. After some calculation,the specificcondensation energy density is obtained as tt ;_tr_l_ 12'?_2')\_! z'0,)a2= _! zrro,tz, (10.53) L' \zo vo/ 2 2 Thus, for finite /, thereis alwaysa reductidnin energyfor the superconductingstate, whereby/ is ameasureof the sizeof the reduction.One can visualize(10.53) by imag- ining that Z(Ei)/ electronpairs per unit volume from the energyregion 1 below the Fermi level all "condense"into a stateat exactly/ below Ei. Their averagegain in energyis thus /,/2. The decisiverole of the parameter/ is clearfrom the following: The first excilation stateabove the BCS ground stateinvolves the breakingup of a Cooperpair due to an externalinfluence. Here an electronis scatteredout of (ftl) leavingbehind an unpaired electronin (ftJ). In order to calculatethe necessaryexcitation energy, we rewrite the ground stateenergy II{6s (10.52)as follows 10. Superconductivity

t,4u.r:L

= L E1u.-ub - L Er@?-,if-+

L3a2 -r - - , S EkuiDi+| Eslui(t ui)- ui(t ui,)l- - k vo

-,4 \' uuro-tff * l, Eku'zk@?- 1- u'zk): -2 L Ekul (10.54) k

If (ft'l,-ft'J) isoccupied, i."^.u?,': 1, then the first excitedstate I'/;cs is achievedby breakingup thepair, i.e., ai, = 0,and therefore wLcs= -2 L eou!,. (10.55) k +k' The necessaryexcitation energy is the differencebetween the enelgiesof the initial and final states lE = wLcs- t4t$.,= 2Bo,= 214f;+'E . (10.56)

The first term in the square-root,(|,. describesth-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 arbitrarily small, i.e., the excitationrequires a minimum finite energy

AEnin= 2A (10.57)

The excitationspectrum of the superconductingstate contains a gap of 2/, which cor- respondsto the energyrequired to break up a Cooperpair. Equation (10'56)describes the excitationenergy of the two electronsthat result from the destructionof a Cooper pair. If we imaginethat a singleelectron is addedto the BCS ground state,then it can naturally find no partner for Cooperpairing. Which energystate can this electronoc- cupy?From (10.56)we concludethat the possiblestates of this excitedsystem are given A = ay no - 1(o+ A2)1/2.If the unpairedelectron is at (k 0 it thus has an energywhich lies at least/ abovethe BCS ground state (Fig. 10.12).However it can also occupy BCSGround levels sta!e stateswith finite (ft; for 120>Z2 tne one-electronenergy

- ,I-t ) - n ^ -0 (10.58) 11,k= VCi+a--

Fig. 10.12.(a) Simplifiedrcpresentation of the excitatronspectrum of a supelconductolon the basisof one- electronenergies ,gr. ln the BCS ground state,the one-electronpicture collapses:At I: 0 K all Cooper pairs occupyone and th€ sameground state(lik-e Bosons). This stateis therefoleeneBetically identical to ihe chemicalpotential, i.e., the Fermi energyt9. The energyIevel drawn herecan thus be lormally int€r- pretedas the "many-bodyenergy of one electron" (total energyof all particlesdivided by the number of ;lectrons).Note that a minimum energyof 2/ is necessaryto split up a Cooperpair. (b) Densityof states 0l for excitedelectrons in a superconductorD" relativeto that of a nolmal conductor.E/.: 0 correspondsto Relative density of states Ds/Di tne rermr energyEi 10.4 THe BCS Ground State

becomeoccupied. These are exactly the levelsof the free electrongas (of a normal con- ductor). Thus, for energieswell abovethe Fermi energy((f,> /2), the continuum of statesof a normal conductorresults. To comparethe densityof statesin the energy range1 about the Fermi level for excitedelectrons D, (E1) in a superconductorwith that of a normal conductorDn ((r) (Sect.6.1), we note that in the phasetransition no statesare lost, i.e.

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

Becausewe are only interestedin the immediateregion / around Eg, it is sufficient to assumethat D|G)=DJEF) = const. According to (10.56)it then follows that aF_ l"R D"(E*)= = for Ex>A D"(E$) --K,rF lvEi-a' (10.59b) I L0 for E*/ converts,as expected,to the densityof statesof a normal conductor; it is depictedin Fig. 10.12b. It should again be emphasisedthat the illustration in Fig. 10.12does not express the fact that the breakingup of a Cooperpair requiresa minimum energy2/.lt says only that the "addition" of an unpairedelectron to the BCS gound statemakes possi- ble the occupationof one-particlestates which lie at least/ abovethe BCSground state energy(for one electron).The densityof statesin the vicinity of the minimum energy single-particlestates is singular(Fig. 10.12b). The "addition" of electronsto the BCSground statecan be realizedin a experiment by the injection of electronsvia an insulatingtunnel barrier (panelIX). Such tunnel experimentsare today very common in superconductorresearch, They may be conve- niently interpretedwith referenceto diagramssuch as Fig. 10.12b We wish now to determinethe gap / (or 2/) n the excitationspectrum. For this we combine (10.49)with (10.46,47', and obtain a:!vo -t=!vo v rz (10.60) zL1 i Ek 2L3 7l1?+t'1

As in (10.26,28), the sum in ft-spaceis replacedby an integral (L-3 E k.-l dk/Bn3\. We note that we are summingagain over pair states,i.e., that insteadof the one-par- ticle densityof statesD(EF+ 6) we must take the pair densityof statesZ(,Eg+ O = +D(E"F+ t). Fudhermore, in contrast to Sect.10.3, the sum is taken over a sphericalshell aioD located symmetricallyaround E$. We than have h?, . v^ __ijz@g + t\ .- r =_:: J aq . (l0.6ta) 2 _h.^ Vt. + a'

In the region 1E$- hap,El+ hatpl where I/a doesnot vanish, Z(Eg+0 variesonly slightly, and, due to the symmetryabout E'$, it follows that

t ha" '" (10.61b) voz@g) ttri7' "' 10. Superconductivity

.. flan = aICSlnn - (10.62) voz(E:F) A

In the caseof a weak interaction,i.e., VyZ(EF)<\, the gap energyis thus

hao -1/voz(El) =2hrnoe . (10.63) sinhrl/voZ(E?)l

This resultbears a noticeablesimilarity to (10.30),i.e., to the binding energys of two electronsin a Cooper pair in the presenceof a fully occupiedsea. As in (10.30),one seesthat evena very small attractiveinteraction, i.e., a very small positive 20, results in a finite gap energy/, but that./ cannot be expandedin a seriesfor small 26. A perturbation calculationwould thus be unableto provide the result (10.63).For the sakeof completeness,it shouldalso be mentionedthat superconductorshave now been discoveredthat have a vanishinglysmall gap energy.

10.5 Consequencesof the BCS Theory and Compa son with ExperimentalResults

An important predictionof the BCS theory is the existenceof ^ gapA (or 2/ ) in the excitationspectrum of a superconductor.The tunnel experimentsdescribed in PanelIX providedirect experimentalevidence for the gap. Indicationsfor the existenceof a gap are also presentin the behaviorof the electronicspecific heat capacityof a supercon- ductor at very low temperature.The exponentialbehavior of this specificheat capacity (10.3)is readilyunderstandable for a systemwhose excited states are reached via excita- tion acrossan energygap. The probability that the excitedstate is occupiedis then pro- portional to an exponentialBoltzmann expression, which also appearsin the specific heat capacity(temperature derivative of the internal energy)as the main factor deter- mining its temperaturedependence. A further direct determinationof the energygap 2A is possibleusing spectroscopy with electromagneticradiation (optical spectroscopy).Electromagnetic radiation is on- ly absorbedwhen the photon energyha exceedsthe energynecessary to break up the Cooperpair, i.e., i (o must be largerthan than the gap eneryyZ/. Tlpical gap energies for classicalsuperconductors lie in the region of a few mev. The appropriate ex- periments must therefore be carried out with microwave radiation. The curves in Fig. 10.13result from an experimentin which the microwaveintensity 1is measured by a bolometerafter multiple reflectionin a cavity made of the materialto be exam- ined. With an externalmagnetic field, the matedal can be driven from the supercon- ducting state(intensity Is) to the normal state(intensity 1N). This allowsthe measure- ment of a differencequantity (,Is- 1N)/1N.At a photon energycorresponding to the gap energy2/, this quantity drops suddenly.This correspondsto an abrupt decrease in the reflectivity of the superconductingmaterial for ha>2/, whereasfor photon energiesbelow 2/ the superconductorreflects totally since there are no excitation mechanisms. At all temperaturesabove Z = 0 K there is a finite possibility of finding electrons in the normal state.As the temperaturerises, more and more Cooperpairs break up; thus a temperatureincrease has a destructiveeffect on the superconductingphase. The