lecture 4 microscopic theory
TRANSCRIPT
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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
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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
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Phonon Coupling The Cooper Pair Problem
+ + + +
+ + + + Analogy
+ + + + 2 Bowling Balls on a
- + + + + MATTRESS
Cooper Pairing
Many electron system
+ + _ + +
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† †
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
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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
e1
xXx
xXx
2
1
2
1
2
1
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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
222
2
2
1
)(
ninteractioelectron -electron theIntroduce
thatsoConsider
: is state thisofenergy The
Hpm
Hppm
H
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k
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k
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kk kkK
kK
xXkK xkXK
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'
'
'
)('
'
)(
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k
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egHedd
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ie
ieg
ieg
kkkk
xx xxk
k
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xkxk
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kx
form the of oneigenfunti an forLook
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m
F
cgdVg
V
Hgdg
DFmFk
mk
m
andF
and
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)'()'(')()(
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)(
0''')'()'(')()(
22
2
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: where,m
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between sea fermi theof top the tostateselectron -one theConfine
stateselectron -2 ofdensity
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Fm
F
m
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m
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m
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d
V
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dV
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2
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'
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'
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0'
)'('1
)(
2
2
2
2
2
2
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result. sobtain thinot could
n calculatioon perturbati Vin seriespower a as written benot may
pair bound
0 e)(attractiv 0
1
1
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1
22
F2
V
VFe
D
VFe
Fm
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- - 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
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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
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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
†
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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:
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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.)
†
†
†
†
†
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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
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2
1
2
1
2
1 ,
2
1
' '''
such that, k
x variablenew a introduce weand
sorder term th 4 eneglect th now We
0 termsdiagonal offorder th 4
',
22'''
2
Then,
.V
Hin termsdiagonal-off ingcorrespond
the T
Hin termsdiagonal-off t theinsist tha weH, ediagonaliz To
kkkk xvxu
kk kC
kC
kC
kC
kkV
VH
kkkkkkk
vk
uk
vk
ukk
V
kkkkkk
vk
uk
cancel
†
†
†
†
† †
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This gives:
022
1
2
4
1
k2
2
1
2'4
1
'
(3.) 04
12
4
12
toleads (4.) and (3.) from
(4.) which,
by given k
quantity new a define weNow
'
2
1
2
''
2
1
2
kkx
kx
kk
xkk
Vk
xVxxk
kkkkkk
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constant). electrons ofnumber the(keeping potential
chemical thebe energy to of zero thechoose now We
moment. ain
case special afor thisdo will Wesolved. becan thisknown, is kk'
V If
2
1
2
''
'
''2
1
give, now (5.) & (4.) and
22
2
gives for Solving
(5.)
kk
k
k kk
Vk
kk
k
kx
kx
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2
1
,
21
(...)222
,0
221
kx
kk
x
kk
vk
uk
v
and
kmSo
kC
kC
kC
kx
or
)degeneracy (Spin
choose F
Eenergy Fermi k
For
N
:that sons,interactio of absence in terms latter the neglect We
N
isN of value nexpectatio the
km state,ground the in
kk
CN Consider
† †
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involved.
phonon theofenergy the,q
than less )F
E to(relative k
choose n,interactiophoton -electron in theorigin its having kk'
VFor
root. square the thereforechoose we
0kk'
V when caseelectron free the toreduce To
2
1
22k
2
k want We
021 2
1 choose ,
kFor
negative
k
kx
kx
kx
FE
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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
kkkk dD
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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
122
)(
1sinh
2
1
F
D
F
EVD
dEVDD
D
gives This
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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
DF
( of 1% ~ Typically,