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Lecture 8: BCS theory --- Attractive interaction and the BCS wavefunction and ground state
Lecture 7: BCS theory --- Clues to the mechanism and the Cooper instability problem
Next time
Today
Discussion the BCS theory in four parts:
1. Clues to the mechanism and the Cooper instability problem
2. Attractive interaction and the BCS wavefunction and ground state
3. Self-consistent solution and quasiparticles
4. Thermodynamics, electrodynamics, and the coherence factors

Microscopic Clues
Things seemed to be understood thermodynamically trend was to make up phenomenological theories and
study thermal and electrodynamical properties 1930
But, clues to the microscopic origins emerged that ultimately led to the microscopic BCS model
(1) Phase Transitions
𝛻 𝐺=𝐺𝑁 −𝐺𝑆=𝐸𝑐=
1
8 𝜋
𝐻𝑐
2
¿
1
2 𝜇0
𝐵𝑐
2
𝐻𝑐 100 𝐺/ 𝐾 =10−2
𝑇 / 𝐾
𝑛=1022
/𝑐𝑚3
𝐸𝑐 10− 7
𝑒𝑉 /𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛
𝐸𝐹 10 𝑒𝑉
𝑘𝐵𝑇 𝑐 10−3
𝑒𝑉
Very small energy per electron
- or -
few electrons involved
condensation energy
𝐺
𝐻
𝐸𝑐
(MKS)
(cgs)
Type I SC:
Compare to Fermi energy
and thermal energy
1950
𝐸𝑐=
10−4
𝑇 2
2(4 𝜋 ×10
−7
𝐻 /𝑚)
=10
2
𝐽 /𝑚
3
¿ 1021
𝑒𝑉 /𝑚3
¿1015
𝑒𝑉 /𝑐𝑚3
For :
𝑇 𝑐=1𝐾

(2) Existence of an Energy Gap of the charge carriers
𝐶
𝑇
𝑇 3
𝑛𝑇 𝐶 𝑒 𝛾 𝑇 +𝛽 𝑇3
electrons lattice No latent heat 2nd order phase transition
 1st
derivative of is discontinuous
Low - fit to
Implies existence of energy gap – must excite excitations above gap
∆ 1.5𝑘𝐵 𝑇𝑐
Later we will see that BCS predicts:
Boltzman factor:
Jump of x 2-3 in specific heat
∆ 1.76 𝑘𝐵 𝑇𝐶
2nd
clue: Low temperature specific heat --- Satterwaithe (1950) at UIUC
1st
clue: Absence of thermoelectric effects --- Daunt & Mendelsohn (1946)
𝑒
−𝑏𝑇 𝑐 / 𝑇

3rd
clue: Electromagnetic absorption – Tinkham (Beasley, Ginsberg UIUC)
REFLECTIVITY (far-infrared and microwaves) which depends on surface impedance
Details depend on
- supercurrent screening
- quasiparticles
- coherence factors (selection rules)
𝑁𝑏
𝑁
𝑓
𝑓 =
2 ∆
h
𝐻𝑔
𝐼𝑛
𝑇𝑎
4th
clue: Quasiparticle tunneling (Giaever) tunneling spectroscopy
𝑅
Reflectivity changes above a given frequency ---
attribute to an energy gap
-3 -2 -1 0 1 2 3
0
1
2
3
eV/
G
/G
(e
V
>
>

)
MOST DEFINITIVE EVIDENCE

(4) Isotope effect
(3) Non-local effects impurity dependence of properties
frequency dependence (screening length)
implies long-range order
𝑌𝐵𝐶𝑂 : 𝑂
16
↔ 𝑂
18
𝛼=0 −0.05
non transition metals
dependence of on isotope mass ions, phonons involved in the superconducting mechanism
1st
experiment: Kamerlingh Onnes 1922
𝛼=
1
2
≡−
𝑑 ℓ 𝑛𝑇𝑐
𝑑 ℓ 𝑛𝑀
Important clue but not definitive --- not seen is all superconductors (even conventional ones)
Weakly observed in some HTSC superconductors that are not thought to be conventional BCS superconductors

Microscopic Theory
Pre- BCS: (1) Perfect Conductivity
(2) Meissner Effect
(3) Second order transition small energy scale of
(4) Energy Gap
(5) Does not occur in best metals
(6) Isotope effect -
electrons
involved
phonons
involved
BCS - 1957
Steps in the development of a microscopic theory:
1950 FrölichNature of attractive electron-phonon interactions (refined by Bardeen, Pines UIUC)
1956 Cooper Mechanism to get phase transition from electron-phonon coupling
1957 BCS Full theory of wavefunction  SC properties
∗

Cooper Instability Problem (1956) Attractive force  new state
NORMAL
STATE
 
0
T 
,

k
k
  B
k T
k
f

k

  
 
k k Occupational probability =
1
1


k
B
k T
e
1

k
Add two electrons-interacting with each other
Normal State:
Will find that (normal state unstable)
2 F
E E
 
2 F
E E
 
Expect ground state to be 1 2
0
k k k k

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
 
1 2
F
k k
k k
a r r 

 
    
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
is the spin state
where
1 1
1 1
  

 
 
k k
B B
k
k T k T
f
e e
2

k
SUPERCONDUCTING
STATE
 
1 2
2
1 ~
k
ik r r
k k e

  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
and 

 
0 1 2
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
H H V r r
zero momentum
~ F
E


Wavefunction contains a mixture of symmetric
and antisymmetric spatial wave functions
 
 
1 2
1 2
cos
, , sin
k r r
k r r










     
 
        














  
 
 
 
 
 
 
 
 
 
 
 
 
 
 














  
 
 
 
 
 
 
 
 
 
 
 
 
 
 


singlet (asymmetric)
triplet (symmetric)
symmetric
asymmetric
Assume singlet spin symmetric spatial wave function particles close together to take advantage of attraction
o
H H V r E
 
 
 
 
 
 
 
     















,
f
k
k k
a k k

  

 
' ' ' ' ' '
, , , , , ,
o
k
k k
a k k H k k k k V r k k E k k k k
 
       
 
 

'
2 
k kk    
'
'
1   
 















 
 
i k k r
kk
V V r e dr
r
 
  '
' '
2
k k k
kk
k
a E V a

 
', '
k k
 x

Overall state must be antisymmetric with exchange
due to Fermi statistics
'
kk
E

k

'
k

k
'
k
C D
 
 
  '
' '
2
k k k
kk
k
a E V a

 
Cooper approximation :
'
'
'
0 for ,
for 0 ,
k c
k
kk
k c
k
V
V
  
  




  




constant
k k
  
 
 
' '
( )
2 k
k k
k
ring
E a V a

  
'
'
2
k
k
k
k
V
a a
E

  
  
 
  
 

'
' ' '
'
2
k
k
k k k
k
V
a a
E

 
   
  
   
 

   
 
  
1
2
k k
V
E

 
  

 

Defines E in terms of V
0
V 
Debye energy
Cooper’s attractive interaction


Evaluate by connecting sum to an integral
 
N 
F
E

 
k
N d
 

 
 
1 0
2
F c
F
E
E
V
V N d
E








 
2 2
1
0
2 2
F c
F
E E w
N n
E E
 




For  
0 1:
N V   
2
0
2 2 2
F c F B
N V
E E e E E


   

binding energy
0, 2 F
V E E
 
   
3 3
0 0
2 2
 
    
 
F F
n V
N N V n
E E
“weak coupling”
 
2
0
2
B c
N V
E e


 
(normal state)
(superconducting state)
0, 2 F
V E E
 
The normal metallic ground state is unstable to excitations for any attractive interaction – we can excite two
electrons from the Fermi sphere and let them scatter into many available states lower energy state (SC)

Let’s look at what this calculation means:



(3) This is for one excited pair, but if it works for one pair, why not more?
There is a tradeoff between number of electrons
excited and number of scattering states available –
reach a point of diminishing return
(2) Attractive interaction lower energy state available
Repulsive interaction only gives higher energy states
Mixed interaction depends on the strength and spatial dependence of the potential
Two consequences of exciting a pair:
 shrinks as states near EF fill up and there are less states
available states to scatter into
(1) Not analytic at V=0 no power series expansion in (V/EF) so it is not possible treat this in perturbation theory


Gain of energy (to take advantage of )
Cost of energy (to raise )
𝜉𝑘
′
𝜉𝑘
increase as states fill up – need to reach deeper into Fermi sphere

2 F B
E E E
 
  1
2
1 E
V N d

 

 
 
 
1
2 B
E E
V N d

 
 
 
1
2
( )
F
B
E
V N E d

 

 

F
E
 
 
(4) The phenomenon requires a Fermi surface
Consider the integral relation we derived
( , )
B
V E d
 
 

  1
2
( ) F
B
E
N E 
  
  
where
𝑁 ( 𝜉)
𝜉
𝐸𝐵<0
0 ℏ 𝜔𝑐
𝐸𝐵>0
Fermi sphere
+ 2 electrons
For EB=0Integral(EB) 
For any V, can get 1 with EB<0
so always superconductivity

𝑁 ( 𝜉)
2 electrons (𝐸𝐹 ⟶ 0)
For EB=0Integral(EB) is finite
If V is too small, can’t get to 1
so no superconductivity
𝜉

Presence of filled Fermi sphere is essential to get lower energy state (Pauli exclusion principle)
Why? creates many available states for band of energy
B
Integral(E )
V


(7) Justifies Pippard’s idea that so that size
(5) We can estimate condensation energy
𝐸𝑐 (𝑁 (0)
2
𝐸𝐵)𝐸𝐵
1
2
𝑁 (0) 𝐸𝐵
2 𝐸 𝑐=
1
2
𝑁 (0)∆
2
ℏ𝑣 𝐹
𝑘𝐵 𝑇
(6) Note that choice of is not critical since when , consequence on system energy is small
BCS:
Number of pairs excited Energy gain per pair excited

Cooper Instability
Attractive for
𝐸𝐵=2 ℏ𝜔 𝑐 𝑒
−
2
𝑁 (0 ) 𝑉
weak coupling
Need filled FS to get lots of available states for scattering
(1) Details of the attractive interaction
(2) Nature of SC ground state (if N unstable)
(3) wavefunction
Ground state – band of scattering states of
energy widtharound Fermi surface
F
E
k


k

c


Next time:
(𝑁 (0)𝑉 ≪1)

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Physics 498 SQD -- Lecture 7 --- BCS 1 FINAL.pptx

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