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And Cooper Pairs 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$. A weak attractiveinterac- tion betweenthese two electronsis switchedon in the form of phonon exchange.All other electronsin the Fermi seaare assumedto be non-interacting,and, on account The quan of the Pauli exclusionprinciple, they exclude a further occupation of stateswith in - ft, thr l/c]<tp. Due to phonon exchangethe two additional electronscontinually change (10.21a), their wave vector, whereby,however, momentum must be conserved: = kt+ k2=ki+ ki: K (10.17) s(k) Sincethe interactionin ft-spaceis restrictedto a shellwith an energythickness of haD Insertingr (with (,D = Debye frequency)above E$ 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-particle function 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;<!-L<n9+nao (10.21a) 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
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