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
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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
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 ) e
ik r u
n.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
Phonon Coupling The Cooper Pair Problem
+ + + +
+ + + + Analogy
+ + + + 2 Bowling Balls on a
- + + + + MATTRESS
Cooper Pairing
Many electron system
+ + _ + +
† †
1122
21 ,kqkkqk
kkqCCCCVH
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 ,,,xkxkxxkk
i
e
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
)(
2122211
xxX
xxkk xkxk
i
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'H'
- , 0
4
11
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),,,(
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)(
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ieg
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and
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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
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) Cooper the in (as -k'k which for sonly state consider us Let
the are s' s,Here indices. spin
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
kC
kC
kH
0 taken and
†
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:
kc
kc
kc
kc
kkkk
cvcu
cvcu
kkkkk
kkkkk
,,,
0''
for solve and (2.)invert then We
i.e. ate,anticommut also s' theshown that becan It
and
The Bogoliubov/ Valatin
transformation. (2.)
†
†
†
†
†
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
km
km
vummuvvH
and Here
)(22 222
2
1
2
1
2
1 ,
2
1
' '''
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2
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