BCS Theory of Superconductivity
BCS Theory of Superconductivity
Thomas Burgener
Supervisor: Dr. Christian Iniotakis
Proseminar in Theoretical Physics
June 11, 2007
1 / 52
What is BCS Theory?
Original publication: Phys. Rev.
108 , 1175 (1957)
2 / 52
What is BCS Theory?
First “working” microscopic theory for superconductors.
It’s a mean-field theory.
In it’s original form only applied for conventional superconductors.
3 / 52
What is BCS Theory?
First “working” microscopic theory for superconductors.
It’s a mean-field theory.
In it’s original form only applied for conventional superconductors.
3 / 52
What is BCS Theory?
First “working” microscopic theory for superconductors.
It’s a mean-field theory.
In it’s original form only applied for conventional superconductors.
3 / 52
Outline
1
Origin of Attractive Interaction
2
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
3
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
4 / 52
Outline
1
Origin of Attractive Interaction
2
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
3
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
4 / 52
Outline
1
Origin of Attractive Interaction
2
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
3
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
4 / 52
Origin of Attractive Interaction
Outline
1
Origin of Attractive Interaction
2
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
3
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
5 / 52
Formation of Pairs
Origin of Attractive Interaction
Let’s assume the following things:
Consider a material with a filled Fermi sea at T = 0.
Add two more electrons that interact attractively with each other but don’t interact with the other electrons except via
Pauli-prinziple.
6 / 52
Formation of Pairs
Origin of Attractive Interaction
Look for the groundstate wavefunction for the two added electrons, which has zero momentum:
Ψ
0
( r
1
, r
2
) =
X k g k e i k · r
1 e
− i k · r
2 ( |↑↓i − |↓↑i )
The total wavefunction has to be antisymmetric with respect to exchange of the two electrons. The spin part is antisymmetric and therefore the spacial part has to be symmetric.
⇒ g k
!
= g
− k
.
7 / 52
Formation of Pairs
Origin of Attractive Interaction
the following equation for the determination of the coefficients g k and the energy eigenvalue E :
( E − 2 k
) g k
=
X
V kk 0 g k 0
, k > k
F where
1
V kk 0
=
Ω
Z
V ( r ) e i ( k
0
− k ) · r d r
( r : distance between the two electrons, Ω: normalization volume, k
: unperturbated plane-wave energies).
8 / 52
Formation of Pairs
Origin of Attractive Interaction
Since it is hard to analyze the situation for general V kk
0
, assume:
V kk 0
=
− V , E
F
0
< k
, otherwise
< E
F
+
~
ω c with
~
ω c a cutoff energy away from E
F
.
9 / 52
Formation of Pairs
Origin of Attractive Interaction
With this approximation we get:
1
V
=
=
X
1
2 k
− E k > k
F
1
2
N (0) ln
= N (0)
Z
E
F
+
~
ω c
E
F
2 E
F
2
− E + 2
~
ω c
E
F
− E
.
d
2 − E
If N (0) V 1, we can solve approximativly for the energy E
E ≈ 2 E
F
− 2
~
ω c e
−
2
N (0) V < 2 E
F
.
10 / 52
Origin of Attractive Interaction
Origin of Attractive Interaction
Negative terms come in when one takes the motion of the ion cores into account, e.g. considering electron-phonon interactions.
The physical idea is that the first electron polarizes the medium by attracting positive ions; these excess positive ions in turn attract the second electron, giving an effective attractive interaction between the electrons.
11 / 52
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
Outline
1
Origin of Attractive Interaction
2
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
3
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
12 / 52
BCS Theory
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
Having seen that the Fermi sea is unstable against the formation of a bound Cooper pair when the net interaction is attractive, we must then expect pairs to condense until an equilibrium point is reached.
We need a smart way to write down antisymmetric wavefunctions for many electrons. This will be done in the language of second quantization.
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Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
BCS Theory
†
Introduce the creation operator c k σ
, which creates an electron of momentum k and spin σ , and the correspondig annihilation operator c k σ
. These operators obey the standard anticommutation relations for fermions:
{ c k σ
†
, c k
0
σ
0
} ≡ c k σ
† c k
0
σ
0
+ c k
†
0
σ
0 c k σ
= δ kk
0
δ
σσ
0
{ c k σ
, c k
0
σ
0
} = 0 = { c
† k σ
, c
† k
0
σ
0
} .
Additionally the particle number operator n k σ is defined by n k σ
≡ c
† k σ c k σ
.
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The model Hamiltonian
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
We start with the so-called pairing-hamiltonian
H =
X k σ k n k σ
+
X
V kl
† c k ↑
† c
− k ↓ c
− l ↓ c l ↑
, kl presuming that it includes the terms that are decisive for superconductivity, although it omits many other terms which involve electrons not paired as ( k ↑ , − k ↓ ).
15 / 52
The model Hamiltonian
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
We then add a term − µ N , where µ is the chemical potential, leading to
H − µ N =
X
ξ k n k σ
+ k σ
X †
V kl c k ↑
† c
− k ↓ c
− l ↓ c l ↑
.
kl
The inclusion of this factor is mathematically equivalent to taking the zero of kinetic energy to be at µ (or E
F
).
16 / 52
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
Bogoliubov-Valatin-Transformation
Define: b k
≡ h c
− k ↓ c k ↑ i
Because of the large number of particles involved, the fluctuations of c
− k ↓ c k ↑ about these expectations values b k should be small.
Therefor express such products of operators formally as c
− k ↓ c k ↑
= b k
+ ( c
− k ↓ c k ↑
− b k
) and neglect quantities which are bilinear in the presumably small fluctuation term in parentheses.
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Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
Bogoliubov-Valatin-Transformation
Inserting this in our pairing Hamiltonian we obtain the so-called model-hamiltonian
H
M
− µ N =
X †
ξ k c k σ c k σ
+
X †
V kl
( c k ↑
† c
− k ↓ b l
+ b
∗ k c
− l ↓ c l ↑
− b
∗ k b l
) k σ kl where the b k are to be determined self-consistently.
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Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
Bogoliubov-Valatin-Transformation
Defining further
∆ k
= −
X
V kl b l
= −
X
V kl h c
− k ↓ c k ↑ i l l leads to the following form of the model-hamiltonian
H
M
− µ N =
X
ξ k c
† k σ c k σ k σ
−
X
(∆ k c
† k ↑ c
†
− k ↓ k
+ ∆
∗ k c
− k ↓ c k ↑
− ∆ k b k
∗
)
19 / 52
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
Bogoliubov-Valatin-Transformation
This hamiltonian can be diagonalized by a suitable linear transformation to define new Fermi operators γ k
:
Bogoliubov-Valatin-Transformation
† c k ↑ c
− k ↓
= u
∗ k
γ k ↑
+ v k
γ
†
− k ↓
= − v
∗ k
γ k ↑
+ u k
†
γ
− k ↓ with | u k
| 2 v k and u k
.
+ | v k
| 2
= 1. Our “job” is now to determine the values of
20 / 52
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
Bogoliubov-Valatin-Transformation
Inserting these operators in the model-hamiltonian gives
H
M
− µ N =
X
ξ k
( | u k
|
2
− | v k
|
2
)( γ
† k ↑
γ k ↑
+ γ
†
− k ↓
γ
− k ↓
) k
+2 | v k
|
2
+ 2 u
∗ k v
∗ k
γ
− k ↓
γ k ↑
+ 2 u k v k
†
γ k ↑
†
γ
− k ↓
+
X
(∆ k u k v k
∗
+ ∆
∗ k u k
∗ v k
)( γ
† k ↑
γ k ↑ k
+(∆ k v k
∗ 2
− ∆
∗ k u k
∗ 2
) γ
− k ↓
γ k ↑
+ γ
†
− k ↓
γ
− k ↓
− 1)
+(∆
∗ k v
2 k
− ∆ k u
2 k
) γ
† k ↑
γ
†
− k ↓
+ ∆ k b
∗ k
.
21 / 52
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
Bogoliubov-Valatin-Transformation
†
Choose u k and v k so that the coefficients of γ
− k ↓
γ k ↑ and γ k ↑ vanish.
†
γ
− k ↓
⇒
⇒
⇒
2 ξ k u k v k
+ ∆
∗ k v
2 k
− ∆ k u
2 k
= 0 ·
∆
∗ k u 2 k
∆
∗ k v k u k
∆
∗ k v k u k
=
2
+ 2 ξ k
∆ u
∗ k k v k q
ξ
2 k
+ | ∆ k
| 2 − ξ k
− | ∆ k
| 2
≡ E k
= 0
− ξ k
22 / 52
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
Bogoliubov-Valatin-Transformation
This gives us an equation for the v k and u k as
| v k
|
2
= 1 − | u k
|
2
=
1
2
1 −
ξ k
E k
.
23 / 52
The BCS ground state
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
BCS took as their form for the ground state
| Ψ
G i =
Y
( u k
+ v k c
† k ↑
† c
− k ↓
) | 0 i k where | u k
| 2
+ | v k
| 2
= 1. This form implies that the probability of the pair ( k ↑ , − k ↓ ) being occupied is | v k
| 2
, whereas the probability that it is unoccupied is | u k
| 2
= 1 − | v k
| 2
.
Note: | Ψ
G i is the vacuum state for the γ operators, e.g.
γ k ↑
| Ψ
G i = 0 = γ
− k ↓
| Ψ
G i
24 / 52
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
Calculation of the condensation energy
We can now calculate the groundstate energy to be h Ψ
G
| H − µ N | Ψ
G i = 2
X
ξ k v
2 k
+
X
V kl u k v k u l v l k kl
=
X
ξ k
−
ξ
2 k
E k
−
∆
2 k
V
The energy of the normal state at T = 0 corresponds to the BCS state with ∆ = 0 and E k
= | ξ k
| . Thus h Ψ n
| H − µ N | Ψ n i =
X
2 ξ k
| k | < k
F
25 / 52
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
Calculation of the condensation energy
Thus, the condensation energy is given by h E i s
− h E i n
=
X
ξ k
−
ξ
2 k
E k
| k | > k
F
= 2
=
X
ξ k
−
ξ
2 k
E k
| k | > k
F
∆
2
V
−
1
2
N (0)∆
2
+
X
| k | < k
F
−
∆
2
V
−
∆
V
2
− ξ k
−
ξ
E
2 k k
= −
1
2
N (0)∆
2
−
∆
2
V
26 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Outline
1
Origin of Attractive Interaction
2
Bogoliubov-Valatin-Transformation
Calculation of the condensation energy
3
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
27 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Excitation Energies and the Energy Gap
With the above choice of the u k becomes and v k
, the model-hamiltonian
H
M
− µ N =
X
( ξ k
− E k k
+ ∆ k b k
∗
) +
X
E k
( γ
† k ↑
γ k ↑ k
+ γ
†
− k ↓
γ
− k ↓
) .
E k
= q
∆
2 k
+ ξ
2 k
28 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Excitation Energies and the Energy Gap
4
3.5
3
2.5
2
1.5
1
0.5
0
-3
E ks
E kn
-2 -1 0
ξ k
/
∆
1 2 3
Figure: Energies of elementary excitations in the normal and superconducting states as functions of ξ k
.
29 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Excitation Energies and the Energy Gap
Inserting the γ operators in the definition of ∆ k gives
∆ k
= −
X
V kl h c
− l ↓ c l ↑ i l
= −
X
V kl u l
∗ v l
D
1 − γ
† l ↑
γ l ↑ l
− γ
†
− l ↓
γ
− l ↓
E
= −
X
V kl u l
∗ v l
(1 − 2 f ( E l
)) l
= −
X
V kl
2 E l l
∆ l tanh
β E l
2
30 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Excitation Energies and the Energy Gap
Using again the approximated potential V kl
∆ k
= ∆ l
= ∆ and therefor
= − V , we have
1
V
=
1
X
2 tanh( β E k
/ 2)
.
E k k
This formula determines the critical temperature T c
!
31 / 52
Determination of T c
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
The critical temperature T c and thus E k
→ ξ k is the temperature at which ∆
. By inserting this in the above formula, k
→ 0 rewriting the sum as an integral and changing to a dimensionless variable we find
1
N (0) V
=
Z
β c ~
ω c
/ 2
0 tanh x dx = ln x
2 e
π
γ
β c ~
ω c
( γ ≈ 0 .
577 ...
: the Euler constant)
32 / 52
Determination of T c
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Critical temperatur T c kT c
= β
− 1 c
≈ 1 .
13
~
ω c e
− 1 / N (0) V
33 / 52
Determination of T c
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
For small temperatures we find
1
N (0) V
=
Z ~ ω c
0 d ξ
( ξ 2 + ∆ 2 ) 1 / 2
⇒ ∆ =
~
ω c sinh(1 / N (0) V )
≈ 2
~
ω c e
− 1 / N (0) V
, which shows that T c other and ∆(0) are not independent from each
∆(0)
≈ kT c
2
1 .
13
≈ 1 .
764
34 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Temperature dependence of the energy gap
Rewriting again
1
V
=
1
X
2 tanh( β E k
/ 2)
.
E k k in an integral form and inserting E k gives
1
N (0) V
=
Z ~ ω c
0 tanh
1
2
β ( ξ
2
( ξ 2
+ ∆
+ ∆ 2 ) 1 / 2
2
)
1 / 2 d ξ, which can be evaluated numerically.
35 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Temperature dependence of the energy gap
Figure: Temperature dependence of the energy gap with some experimental data (Phys. Rev.
122 , 1101 (1961))
36 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Temperature dependence of the energy gap
Near T c we get
Temperature dependence of ∆
∆( T )
∆(0)
T
≈ 1 .
74 1 −
T c
1 / 2
, T ≈ T c
, which shows the typical square root dependence of the order parameter for a mean-field theory.
37 / 52
Thermodynamic quantities
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
With ∆( T ) determined, we know the fermion excitation energies
E k
= q
ξ
2 k
+ ∆( T ) 2 . Then the quasi-particle occupation numbers will follow the Fermi-function f k
= (1 + e
β E k )
− 1
, which determine the electronic entropy for a fermion gas
S es
= − 2 k
X
((1 − f k
) ln(1 − f k
) + f k ln f k
) .
k
38 / 52
Thermodynamic quantities
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Figure: Electronic entropy in the superconducting and normal state.
39 / 52
Thermodynamic quantities
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Given S es
( T ), we find the specific heat
C es
= − β dS es d β
= 2 β k
X
−
∂ E k k
∂ f k
In the normal state we have
C en
=
2 π
2
N (0) k
2
T .
3
E
2 k
+
1
2
β d ∆
2 d β
40 / 52
Thermodynamic quantities
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
We expect a jump in the specific heat from the superconducting to the normal state:
∆ C = ( C es
− C en
) |
T c
= N (0)
− d ∆
2 dT
T c
≈ 9 .
4 N (0) k
2
T c
41 / 52
Thermodynamic quantities
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Figure: Experimental data for the specific heat in the superconducting and normal state (Phys. Rev.
114 , 676 (1959))
42 / 52
Type I superconductors
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
1
0.8
0.6
0.4
0.2
0
0
M-O-State
Normal-State
0.2
0.4
T/T c
0.6
0.8
1
Figure: Phase diagram of a Type I superconductor
43 / 52
Vortex-State
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Original publication: Zh. Eksperim. i Teor. Fiz.
32 , 1442 (1957)
(Sov. Phys. - JETP 5 , 1174 (1957))
44 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Type I and Type II superconductors
By applying Ginzburg-Landau theory for superconductors one finds two characteristic lengths:
1
2
The Landau penetration depth for external magnetic fields λ and the Ginzburg-Landau coherence length ξ , which characterizes the distance over which ψ can vary without undue energy increase.
45 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Type I and Type II superconductors
Define
Ginzburg-Landau parameter
κ ≡
λ
ξ
By linearizing the GL equations near T c one can find:
κ < √
2
: Type I superconductor
κ > √
2
: Type II superconductor
46 / 52
Type II superconductors
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
3
2.5
2
1.5
1
Vortex-State
0.5
0
0
M-O-State
0.2
Normal-State
0.4
T/T c
0.6
0.8
1
Figure: Phase diagram of a Type II superconductor
47 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
As a solution of the GL equation, one could find the following form of the orderparameter:
Ψ( x , y ) =
1
N
∞
X exp
π ( ixy − y
2
)
ω
1
= ω
2
+ i π n n = −∞
+ i π (2 n + 1)
ω
1
( x + iy ) + i π
ω
2
ω
1 n ( n + 1)
1 / 4
N =
ω
1
2 = ω
2 exp π
= ω
2
ω
1
48 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Vortex-State
2
1
0
-1
2
1
0
-1
-2
-2 -1 0 1 2
-2
-2 -1 0 1
Figure: Square and triangle symmetric state of the vortex lattice in a density plot of | Ψ | 2 .
2
49 / 52
Vortex-State
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
È Y È 2
1
0.5
0
-2
-2
1
-1
0 y
2
1
È Y È 2
0.5
0
-2
-2
1
-1
0 y
2
Figure: Square and triangle symmetric state of the vortex lattice in a 3D plot.
50 / 52
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Summary
An attractive interaction between electrons will result in forming bound Cooper pairs.
The model-hamiltonian can be diagonalized using a
Bogoliubov-Valatin-Transformation.
The order parameter in a superconductor is the energy-gap ∆.
BCS-Theory gives a prediction of the critical temperature T c and the energy gap ∆( T ).
Vortices will be observed in Type II superconductors.
51 / 52
The END
Excitation Energies and the Energy Gap
Temperature dependence of the energy gap
Thank you for your attention!
Are there any questions?
52 / 52
