ELECTRON CORRELATION IN METALS
CONTENTS FERMI GAS 1 1 METALS 1 2 FREE FERMI GAS 1 3 ELECTRONIC SPECI?C HEAT AND PAULI SUSCEPTIBILITY 1 4 MANY-BODY EFFECT OF ELECTRON GAS 1 5 EXCHANGE ENERGY 1 6 SCREENING EFFECT 1 7 PLASMA OSCILLATION 1 8 GROUND STATE ENERGY 1 9 WIGNER CRYSTAL FERMI LIQUID THEORY 2 1 PRINCIPLE OF CONTINUITY 2 2 LANDAU’S FERMI LIQUID THEORY ANDERSON’S ORTHOGONALITY THEOREM 3 1 FRIEDEL SUM RULE 3 2 ORTHOGONALITY THEOREM ON LOCAL PERTURBATION 3 3 PHOTOEMISSION IN METALS AND THE ORTHOGONALITY THEOREM 3 4 DIFFUSION OF CHARGED PARTICLES IN METALS S–D HAMILTONIAN AND KONDO EFFECT 4 1 SPIN SUSCEPTIBILITY OF CONDUCTION ELECTRONS 4 2 S–D EXCHANGE INTERACTION AND SPIN POLARIZATION 4 3 KONDO EFFECT 4 4 GROUND STATE OF DILUTE MAGNETIC ALLOY SYSTEM 4 5 SCALING LAW OF THE S–D SYSTEM 4 6 WILSON’S THEORY 4 7 FERMI LIQUID THEORY BY NOZI`ERES ANDERSON HAMILTONIAN 5 1 HARTREE–FOCK APPROXIMATION 5 2 PERTURBATION EXPANSION WITH RESPECT TO VKD 5 3 GREEN’S FUNCTION 5 4 PERTURBATION EXPANSION WITH RESPECT TO U 5 5 EXACT SOLUTION OF THE ANDERSON HAMILTONIAN HUBBARD HAMILTONIAN 6 1 BASIC PROPERTIES 6 2 THEORY OF ELECTRON CORRELATION 6 3 IN?NITE-DIMENSIONAL HUBBARD HAMILTONIAN 6 4 MOTT TRANSITION 6 5 ONE-DIMENSIONAL HUBBARD MODEL 6 6 FERROMAGNETISM OF TRANSITION METALS 6 7 SUPERCONDUCTIVITY IN THE HUBBARD HAMILTONIAN FERMI LIQUID THEORY OF STRONGLY CORRELATED ELECTRON SYSTEMS 7 1 HEAVY FERMION SYSTEMS 7 2 HEAVY FERMIONS 7 3 KONDO TEMPERATURE IN CRYSTALLINE ?ELDS 7 4 FERMI LIQUID THEORY ON HEAVY FERMION SYSTEMS 7 5 SPIN ?UCTUATION AND FERMI LIQUID TRANSPORT THEORY BASED ON FERMI LIQUID THEORY 8 1 CONDUCTIVITY 8 2 OPTICAL CONDUCTIVITY 8 3 HALL CONDUCTIVITY 8 4 CYCLOTRON RESONANCE 8 5 MAGNETORESISTANCE 8 6 CONCLUDING REMARKS SUPERCONDUCTIVITY IN STRONGLY CORRELATED ELECTRON SYSTEMS 9 1 CUPRATE HIGH-TEMPERATURE SUPERCONDUCTORS 9 2 ANISOTROPIC SUPERCONDUCTIVITY DUE TO COULOMB REPULSION 9 3 ELECTRON-DOPED CUPRATES 9 5 CONCLUSION I 2M + I 2M + |RI ?R J|, (1 3) HE?I = I J V(RI ?R J) (1 4) HERE, HI IN (1 2) REPRESENTS THE POSITIVE ION SYSTEM; WE ASSUME ONE KIND OF POSITIVE ION WITH MASS M PI IS THE MOMENTUM OF ION I AND V (RI ?R J) IS THE POTENTIAL BETWEEN IONS, WHICH DEPENDS ONLY ON THEIR DISTANCE HE IN (1 3) IS THE HAMILTONIAN FOR THE ELECTRON SYSTEM WITH ELECTRON MASS M; THE ?RST AND SECOND TERMS DENOTE THE FERMI GAS KINETIC ENERGY AND THE COULOMB INTERACTION BETWEEN ELECTRONS, RESPECTIVELY HE?I IN (1 4) REPRESENTS THE POTENTIAL BETWEEN ELECTRONS AND POSITIVE IONS THE HAMILTONIAN COMBINEDWITHTHESETERMSCANDESCRIBEVARIOUSPROPERTIESINCLUDINGMAGNETISMAND SUPERCONDUCTIVITY IN THIS BOOK WE DISCUSS MAINLY THE SECOND TERM OF (1 3), ELECTRON INTERACTION OWING TO THE COULOMB INTERACTION, ELECTRONS MOVE SO AS TO AVOID EACH OTHER THIS KIND OF CORRELATED ELECTRON MOTION IS CALLED ‘ELECTRON CORRELATION’ WHEN WE ARE MAINLY INTERESTED IN ELECTRON INTERACTION, WE SIMPLIFY THE POSITIVE IONS BY REPLACING THEM WITH A UNIFORM POSITIVE CHARGE DISTRIBUTED OVER THE CRYSTAL AND DISCUSS ONLY THE TERM HE BY THIS REPLACEMENT, WE CAN AVOID THE DIF?CULTY ARISING FROM A PERIODIC POTENTIAL AND LATTICE VIBRATIONS 1 2 FREE FERMI GAS AN ELECTRON IS A FERMI PARTICLE WITH A SPIN OF 1/2 THERE EXIST 1022–1023 CONDUCTION ELECTRONS PER CUBIC CENTIMETRE IN MOST METALS FOR SIMPLICITY, LET US IGNORE ELECTRON INTERACTIONS AMONG THESE, AND ALSO THE PERIODIC POTENTIAL DUE TO THE POSITIVE IONS, AND CONSIDER THE SYSTEM COMPOSED OF FREE ELECTRONS WE ASSUME A SYSTEM WITH N ELECTRONS IN A CUBE OF SIDE L = 1/3 THE WAVE-FUNCTION ?K(R) FOR A FREE ELECTRON WITH WAVE-VECTOR K IS GIVEN BY ?K = 2M 2? L 3) (1 9) LET US CONSTRUCT THE GROUND STATE COMPOSED OF N FREE ELECTRONS EACH ONE-ELECTRON STATE SPECI?ED BY WAVE-VECTOR K AND SPIN QUANTUM NUMBER ? CAN BE OCCUPIED BY ONLY ONE ELECTRON BECAUSE OF THE PAULI EXCLUSION PRINCIPLE IN THE GROUND STATE ELECTRONS OCCUPY THE N STATES FROM THE LOWEST ENERGY STATE TO THE NTH LOWEST STATE FREE FERMI GAS 2M KF 2 (1 10) THUS, WE HAVE THE GROUND STATE IN WHICH TWO ELECTRONS WITH UP- AND DOWN-SPINS OCCUPY THE STATE K WITHIN THE SPHERE WITH RADIUS KF SINCE THE WAVE-VECTOR K IS SITUATED AT A LATTICE POINT SPACED 2?/L APART, WAVE-VECTORS DISTRIBUTE IN UNIFORM DENSITY (L/2?)3 = /(2?)3 AS A RESULT, THE ELECTRON NUMBER N IS RELATED TO KF BY N = 4? 3 KF 3, (1 11) WHERE FACTOR 2 REPRESENTS THE DEGREE OF FREEDOM ARISING FROM ELECTRON SPIN FROM (1 11), KF IS GIVEN BY ELECTRON DENSITY N = N/ AS KF = (3?2N)1/3 (1 12) LET US CALCULATE THE NUMBER OF ELECTRON STATES BETWEEN ENERGY E AND E + E, E BEING AN IN?NITESIMAL ENERGY ASSUMING THE TOTAL NUMBER OF STATES BELOW ENERGY E AS N(E), WE OBTAIN N(E + E) ?N(E) = DN(E) DE E (1 13) HERE, DN(E)/DE = ?(E) (1 14) IS THE ENERGY DENSITY OF STATES USING (1 10) AND (1 11), WE OBTAIN ?(E) = DN(K) DK DK DE = ? 2ME (1 15) THE DENSITY OF STATES FOR A FREE ELECTRON SYSTEM IS PROPORTIONAL TO ? E FERMI GAS NOW LET US CONSIDER REAL METALS WE ASSUME N = 1022 CM?3 AND OBTAIN KF ? 108 CM?1 FROM (1 12) THE VALUE OF 1/KF IS AROUND 1 Å = 10?8 CM, CORRESPONDING TO ATOMIC DISTANCE INSERTING THIS VALUE INTO (1 10) AND USING ¯H = 1 × 10?27 ERG S AND M = 9 × 10?28 G, WE OBTAIN ?F ?6 × 10?12 ERG ?4 EV SINCE 1 EV IS THE ENERGY CORRESPONDING TO 104 K, THE ROOM TEMPERATURE, 300 K, IS SUF?CIENTLY LOW COMPARED WITH THE FERMI TEMPERATURE, TF = ?F/KB THIS FACT IS VERY IMPORTANT IN UNDERSTAND- ING THE ELECTRONIC SPECI?C HEAT DISCUSSED IN THE NEXT SECTION THE LEVEL SPLITTING OF ONE-ELECTRON ENERGY NEAR THE FERMI SURFACE IS GIVEN BY ? = ?F/N ?10?22 EV ?10?18KB K AS A RESULT, ELECTRON–HOLE PAIR EXCITATIONS NEAR THE FERMI ENERGY CAN BE CREATED WITH VANISHINGLY SMALL EXCITATION ENERGY THESE ELECTRON–HOLE PAIR EXCITATIONS EXIST IN AN IN?NITE NUMBER THUS, THE CONDUC- TION ELECTRON SYSTEM IN THE FERMI DEGENERACY IS DEGENERATE IN THE IN?NITE NUM- BER OF STATES IN GENERAL, THE DEGENERATED STATES SUFFER A STRONG EFFECT EVEN UNDER SMALL PERTURBATIONS THE SPECIAL NATURE OF THE FERMI SURFACE, WHICH CAN BE CALLED ‘FRAGILITY’, PLAYS AN IMPORTANT ROLE IN THE ORTHOGONALITY THEOREM AND THE THEORY OF SUPERCONDUCTIVITY 1 3 ELECTRONIC SPECI?C HEAT AND PAULI SUSCEPTIBILITY IF WE APPLY THE PRINCIPLE OF EQUIPARTITION TO THE FREE ELECTRON SYSTEM TO CALCULATE THE SPECI?C HEAT, WE OBTAIN THE FOLLOWING RESULT THE INTERNAL ENERGY W IS GIVEN BY W = 2 NKBT, (1 16) KB BEING THE BOLTZMANN CONSTANT THE SPECI?C HEAT CV TAKES A CONSTANT VALUE, CV = DW DT = (1 18) ELECTRONIC SPECI?C HEAT AND PAULI SUSCEPTIBILITY IN THIS CASE THE INTERNAL ENERGY OF THE ELECTRON SYSTEM W IS GIVEN BY W = K ?K F (?K), (1 19) WHERE THE FACTOR 2 ARISES FROM SPIN DEGENERACY USING THE DENSITY OF STATES ?(?K) INCLUDING BOTH SPINS, WE OBTAIN W = ? ??(?) F (?)D? (1 20) THE TOTAL NUMBER OF ELECTRONS N IS GIVEN BY N = K? F (?K) = ? ?(?) F (?)D? (1 21) THIS EQUATION AT T = 0 BECOMES N = ?F ?(?)D? (1 22) NOW LET US CONSIDER THE FOLLOWING INTEGRAL I TO ACCOMPLISH THE CALCULATION AT LOW TEMPERATURES: I = ? G(?) F (?)D? (1 23) HERE, G(?) IS A SMOOTH FUNCTION OF ENERGY ? USING A PARTIAL INTEGRATION, WE OBTAIN I = [G(?) F (?)]? 0 ? ? G(?)?F ?? D?, (1 24) G(?) = ? G(?)D? (1 25) THE ?RST TERM OF (1 24) VANISHES BECAUSE F (?) = 0 TO CALCULATE THE SECOND TERM, WE EXPAND G(?) AROUND ? = µ WRITING THE NTH DERIVATIVE OF G AS G(N), WE OBTAIN G(?) = G(µ) + (? ?µ)G?(µ) + 2(? ?µ)2G??(µ) + (1 26) INSERTING THIS INTO (1 24), WE GET I = G(µ) ? ??F ?? D? + G?(µ) ? (? ?µ) ??F ?? D? + + G(N)(µ) N! ? (? ?µ)N ??F ?? D? + (1 27) FERMI GAS THE ?RST TERM GIVES G(µ) THE GENERAL TERMS ARE GIVEN BY N! ? (? ?µ)N ??F ?? D? = (KBT )N N! ? ?? ZN (EZ + 1)(1 + E?Z)DZ = 2CN(KBT )N (N EVEN) (N ODD) (1 28) AS AN EXAMPLE, FOR N = 2, 2C2 = 6 (1 29) AS A RESULT, I IS GIVEN BY I = µ ?G(?) ?? ?=µ + (1 30) APPLYING THIS RESULT TO (1 20) AND (1 21), WE OBTAIN W = µ 3 KB2?(?F) (1 34) HERE, THE COEF?CIENT OF THE SPECI?C HEAT ? IS CALLED THE SOMMERFELD CONSTANT MANY-BODY EFFECT OF ELECTRON GAS LET US APPLY A WEAK MAGNETIC ?ELD H TO THE FREE ELECTRON SYSTEM AT LOW TEM- PERATURE AND OBTAIN THE EXPRESSION FOR THE PAULI SUSCEPTIBILITY USING A G-VALUE OF G = 2 AND THE BOHR MAGNETON µB, THE ZEEMAN ENERGY WITH SPIN ? IS GIVEN BY G?µBH/2 = ?µBH THE ZEEMAN ENERGY INDUCES THE MAGNETIZATION GIVEN BY M = µB(?N???N?) = µB2?(?F)H THUS, THE MAGNETIC SUSCEPTIBILITY ? IS GIVEN BY ? = M/H = µB 2?(?F) (1 35) THE PAULI SUSCEPTIBILITY IS PROPORTIONAL TO THE DENSITY OF STATES ?(?F) ON THE FERMI SURFACE; THIS IS COMMON TO THE COEF?CIENT OF SPECI?C HEAT ? 1 4 MANY-BODY EFFECT OF ELECTRON GAS THE EFFECTS OF COULOMB INTERACTION ON ELECTRON GAS HAD BEEN MADE CLEAR IN THE 1950S BY THE EFFORTS OF MANY PEOPLE, SUCH AS BOHM, PINES, NOZI`ERES, GELL-MANN, BRUECKNER AND SAWADA AS SHOWN BELOW, INITIALLY, BY A STRAIGHT PERTURBATION CAL- CULATION, THE ESSENTIAL POINTS OF THE PROBLEM WERE MADE CLEAR THEN, THE DIF?CULT KEY PROBLEMS WERE SOLVED BY PHYSICAL CONSIDERATIONS, CREATING NEW CONCEPTS AND DEVELOPING THE METHODS OF THE MANY-BODY PROBLEM SINCE THE HISTORY OF THE INVES- TIGATIONS ON ELECTRON GAS IS INSTRUCTIVE, WE DESCRIBE IT HERE IN DETAIL AS A STARTING POINT FOR THE MANY-BODY PROBLEM [1–6] IN ORDER TO DISCUSS MAINLY THE EFFECT OF ELECTRON INTERACTION, WE ASSUME THE ELECTRON GAS MODEL IN WHICH THE POSITIVE CHARGE DUE TO IONS IS REPLACED BY A UNIFORM ONE AS A RESULT, IN THIS MODEL THE UNIFORM DISTRIBUTION OF ELECTRONS IN SPACE CANCELS WITH THE POSITIVE BACKGROUND TO GIVE NO EFFECT THE DEVIATION FROM THE UNIFORM DISTRIBUTION GIVES RISE TO THE COULOMB INTERACTION AMONG ELECTRONS THE HAMILTONIAN OF THIS SYSTEM IS GIVEN BY HE IN (1 3) AND IS WRITTEN AS HE = I PI 2M + |RI ?R J|, (1 36) WHERE RI AND PI ARE THE POSITION AND MOMENTUM OF ELECTRON I, RESPECTIVELY HERE WE DE?NE ELECTRON DENSITY ?(R) AND ITS FOURIER TRANSFORM ?Q: ?(R) = I ?(R ?RI) = Q ?QEIQ R, (1 37) ?Q = I E?IQ RI = ??Q † (1 38) FERMI GAS THE Q = 0 COMPONENT OF ?Q, ?0 = NE/ = N, IS THE AVERAGE ELECTRON DENSITY AND CANCELS WITH THE UNIFORM POSITIVE CHARGE USING ?Q, WE CAN WRITE (1 36) AS HE = I PI Q2 (1 40) HERE IT IS NOTED THAT SINCE ?Q †?Q = |R1 ?R2|??2(R2)??1(R1) (1 42) BY EXPANDING THE WAVE-FUNCTION ??(R) WITH THE PLANE WAVES AS ??(R) = K AK? ? EIK R, (1 43) WE WRITE THE COULOMB INTERACTION AS HC = †AK2?2AK1?1 (1 44) HERE AK? (AK? †) IS THE ANNIHILATION (CREATION) OPERATOR OF THE ELECTRON WITH WAVE- VECTOR K AND SPIN ? THE SUMMATION OVER Q IN (1 44) EXCLUDES THE PART OF Q = BECAUSE IT CANCELS WITH THE POSITIVE CHARGE THE DENSITY ?UCTUATION ?Q IS GIVEN BY ?Q = K? AK?Q? †AK? (1 45) MANY-BODY EFFECT OF ELECTRON GAS THUS, HE IS GIVEN IN THE SECOND QUANTIZATION AS HE = K? ?KAK? †AK? + 2M (1 47) OPERATORS AK? † AND AK? SATISFY THE COMMUTATION RULE FOR FERMI PARTICLES (SEE APPENDIX B): AK?, AK?? ? + = AK? †, AK?? ?† + = 0, AK?, AK?? ?† + = ?K,K???,? ? (1 48) NOW WE STUDY THE EFFECT OF THE COULOMB INTERACTION USING THE ABOVE RESULTS FIRST OF ALL, LET US CALCULATE THE GROUND STATE ENERGY BY REGARDING THE SECOND TERM, THE COULOMB INTERACTION, AS A PERTURBATION ON THE ?RST TERM, THE KINETIC ENERGY THE UNPERTURBED STATE GIVEN BY THE ?RST TERM IS THE FERMI SPHERE OCCUPIED UP TO KF BY TWO ELECTRONS WITH UP- AND DOWN-SPIN WE WRITE IT AS |0?AND OBTAIN NK? = ?0|AK? †AK?|0?= (K < KF) (K > KF) (1 49) THE AVERAGE KINETIC ENERGY PER ELECTRON IN THE UNPERTURBED STATE IS GIVEN BY ?KIN = NE ?0| K? ?KAK? †AK?|0?= 5?F, (1 50) WHERE ?F IS THE FERMI ENERGY GIVEN BY (1 10) THE RESULT OF (1 50) IS OBTAINED FROM ?KIN = KF 2M 4?K2DK KF 4?K2DK (1 51) NOW LET US INTRODUCE R0 REPRESENTING ELECTRON DENSITY AS NE = 3 (1 52) THE RADIUS R0 OF THE SPHERE FOR ONE ELECTRON IS REPLACED BY THE DIMENSIONLESS PARAMETER RS, WHICH IS GIVEN IN UNITS OF BOHR RADIUS AS RS = R0/AB (1 53) 2M (3?2N)2/3 = 9? 2/3 2AB ?13 5 EV = 2 17 × 10?11 ERG (1 55) AS A ?NAL RESULT, THE AVERAGE KINETIC ENERGY OF (1 50) IS GIVEN BY ?KIN = RS2 RY (1 56) 1 5 EXCHANGE ENERGY THE ?RST-ORDER PERTURBATION TERM WITH RESPECT TO THE COULOMB INTERACTION IN THE GROUND STATE ENERGY IS GIVEN BY E1 = KK? Q? VQ 2 ?0|AK+Q? †AK??Q? ?†AK?? ?AK?|0? = KQ? VQ 2 NK+Q?NK? = (2?)3 E2KF 4 (1 59) EXCHANGE ENERGY 2? 9? 1/3 2M + (1)(K) HOWEVER, FROM (1 61) THE DERIVATIVE D˜?K/DK DIVERGES ON THE FERMI SURFACE AND THE DENSITY OF STATES AT THE FERMI ENERGY VANISHES THIS RESULT GIVES RISE TO THE DIF?CULTY THAT THE ELECTRON SYSTEM DOES NOT GIVE THE T -LINEAR SPECI?C HEAT THIS DIF?CULTY IS SOLVED BY CUTTING THE LONG-RANGE PART OF THE COULOMB INTERACTION, AS WILL BE SHOWN IN THE FOLLOWING SECTIONS NOW WE CONSIDER THE PHYSICAL MEANING OF THE EXCHANGE TERM USING THE HARTREE– FOCK EQUATION [6] BY TREATING THE COULOMB INTERACTION WITH THE HARTREE–FOCK DR??K?? ?(R?)?K?(R?) |R ?R?|?(R ?R?), (1 66) WHERE ?(R ?R?) IS GIVEN BY ?(R ?R?) =