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Lecture 4 microscopic theory

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  • 1. Lecture 4 Microscopic Theory •The 2-Electron Problem •Second Quantization: •Annihilation and Creation Operators •Solution of the 2-electron Schroedinger Equation •Cooper Pairs •The many-electron problem-BCS Theory •Solution of the Many-particle Schroedinger Equation by the Bogoliubov-Valatin Transformation •The BCS Energy gap
  • 2. Even number of electrons/unit cell Band picture - electrons in momentum space electrons in a periodic potential form Bloch waves and energy bands Bloch waves n,k (r) eik r un.k (r) Energy eigenvalues n (k) Odd number of electrons/unit cell E metal insulator semiconductor E energy gap Repulsive interaction between electrons is a perturbation Fermi sea Fermi liquid of “independent” Quasiparticles (Landau, 1956) Insulator, Semiconductor Metal
  • 3. Phonon Coupling The Cooper Pair Problem + + + + + + + + Analogy + + + + 2 Bowling Balls on a - + + + + MATTRESS Cooper Pairing Many electron system + + _ + +
  • 4. † † 1122 21 , kqkkqk kkq CCCCVH  Consider a subset of the many – electron system , i.e. a Cooper pair, with 2 free electrons with antiparallel spins (for parallel spins, exchange terms reduce the phonon-mediated attractive electron-electron interaction). With no interaction,    2211 .. 2121 ,,, xkxk xxkk   i e
  • 5. Assume ϵF – ωD < ϵk , ϵk ± q < ϵF + ωD so that H ̎ is predominately attractive † † (here we have let k’ replace k2 and k replace k1). Consider two free electrons, and introduce center of mass coordinates: x = x1 – x2   q kk kqkkqk CCCCVH ' '''' )( 2 1 );( 21 )( 212 2211 xxX xxkk xkxk   i e1 xXx xXx 2 1 2 1 2 1  
  • 6. kKk kKk kkkkkK    2 1 2 1 )( 2 1 2 1 2121 '' 1 '')( 2 1 'H' -,0 4 11 22 ),,,( 22 2 2 1 21 22 2 2 2 1 )( ninteractioelectron-electrontheIntroduce thatsoConsider :isstatethisofenergyThe Hp m Hpp m H mm k m k ei            kkkkK kK xXkK xkXK 
  • 7.              ' ' ' )(' ' )( 21 0',''',)( 0)( 0)( 2121 21)( k kkk k ii Hgg egHedd H k k i e i eg i eg kkkk xx xxk k xxk xkxk k xk kx     formtheofoneigenfuntianforLook
  • 8.                 m F cgdVg V Hgdg DFmF k m k m and F and 2 2 )'()'(')()( ''H' )( 0''')'()'(')()( 22 2 1 :where, m K F Kbetweeni.e. DF betweenseafermitheoftopthetostateselectron-onetheConfine stateselectron-2ofdensity     
  • 9.                           Fm F m F F m F m F m F d V dV dV C g 22 ln 2 2 ln ' '1 ' 1 '1 0 ' )'( '1 )( 2 2 2 2 2 2        
  • 10. result.sobtain thinotcould ncalculatioonperturbatiVinseriespoweraaswrittenbenotmay pairbound 0e)(attractiv0 1 1 2 1 22 F2           V V Fe D VFe Fm    
  • 11. - - 2 1 The region of increased positive charge density propagates through the crystal as a quantized sound wave called a phonon The passing electron has emitted a phonon A second electron experiences a Coulomb attraction from the increased region of positive charge density created by the first electron
  • 12. BCS Theory – a Brief Treatment For many electrons, we need to make sure the many-particle wave function is anti symmetric. We can write in general that the Hamiltonian is: † † † † sksksqk qk sqkq sksksqk qsksk sqkq CCCCVHH CCCCVHH ,',', , ,0 ,','',' ,',',, ,0 2 1 2 1        :case)Cooperthein(as-k'kwhichforsonly stateconsiderusLet theares's,Here indices.spin
  • 13. Summing over s, it can be shown (using anticommutator relationships for the annihilation and creation operators) that: † † † † Here we have chosen S ↑ , S´↓ (to minimize the energy as before), and summed over S, We have also assumed that Vk,k’ = V-k,-k’ Note that the eigenstates for H0 are just the Block waves uk eik.x in the crystal.       k'k, - (1.) kkkkkk k kkkkk CCCCV CCCCH '''    k k C k C k H  0 takenand †
  • 14. Eq. (1.) is the BCS Hamiltonian There are in general 2 approaches to solve the many-particle Schroedinger equation (see, e.g. TINKHAM): 1. variational approach to minimize the energy 2. solution by a canonical transformation (the Bogoliubov/ Valatin transformation). We will illustrate the second approach here. Bogoliubov diagonalized the Hamiltonian for the liquid helium superfluid condensate by introducing 2 new operators:
  • 15. k c k c k c k c kkkk cvcu cvcu kkkkk kkkkk       ,,, 0 '' forsolveand(2.)invertthenWe i.e.ate,anticommutalsos'theshown thatbecanIt and     The Bogoliubov/ Valatin transformation. (2.) † † † † †
  • 16. Substituting these C’s into (1.) gives as the kinetic energy term HT (1st set of terms): † † † † Take mk = m-k = 0 for the ground state. Next we consider the potential energy term Hv (second set of terms with V)     kkkk k kkkkkkkkkkkkT k m k m vummuvvH          andHere )(22 22 2
  • 17.      2 1 2 1 2 1 , 2 1 ' ''' such that, k xvariablenewaintroduceweand sorder termth4eneglect thnowWe 0termsdiagonalofforderth4 ', 22 ''' 2 Then, . V Hintermsdiagonal-offingcorrespond the T Hintermsdiagonal-offt theinsist thaweH,ediagonalizTo                          kkkk xvxu kk k C k C k C k C kk VVH kk kkkkk v k u k v k u kk V k kkkkk v k u k  cancel † † † † † †
  • 18. This gives: 022 1 2 4 1 k 2 2 1 2 '4 1 ' (3.)0 4 1 2 4 1 2 toleads(4.)and(3.)from (4.)which, bygiven k quantitynewadefineweNow ' 2 1 2 '' 2 1 2                               kk x k x k k x kk V k xVxx k kkkkkk  
  • 19. constant).electronsofnumberthe(keepingpotential chemicalthebeenergy toofzerothechoosenowWe moment.ain casespecialaforthisdowillWesolved.becanthisknown,is kk' VIf 2 1 2 '' ' ' '2 1 give,now(5.)&(4.)and 22 2 givesforSolving (5.)           kk k k kk V k kk k k x k x   
  • 20.       2 1 , 21 (...)222 ,0 221              k x k k x k k v k u k v and k mSo k C k C k C kx or )degeneracy(Spin choose F EenergyFermi k For N :thatsons,interactioofabsenceintermslattertheneglectWe N isNofvaluenexpectatiothe k mstate,groundthein k k CNConsider  † †
  • 21.   involved. phonontheofenergythe, q thanless) F Eto(relative k choosen,interactiophoton-electronin theoriginitshaving kk' VFor root.squarethethereforechoosewe 0 kk' VwhencaseelectronfreethetoreduceTo 2 1 22 k 2 kwantWe 021 2 1 choose, k For      negative     k k x k x k x F E
  • 22. We take Vkk’ = V, constant if |εk|< ħωD = 0, otherwise Here ωD is the Debye Frequency Here ∆k can be evaluated as    2 1 '' 2 ' '' 2 1 kk k kkk dD        
  • 23. Here we will take the “density of states” D(εk’) as a constant, D(EF). Consider Vkk’ Vkk’ ħωD εk So we need only evaluate V     1 2 1 22 )( 1 sinh 2 1                       F D F EVD dEVD D D         givesThis
  • 24. This is the BCS gap energy in the density of states For weak coupling (V small), this can be written as: For the ground state wave function and finite temperature effects, See TINKHAM. )03.0~ 2 )( 1 eV e DD EVD D F     (of1%~Typically,   

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