Outlines Cooper-Pairs BCS Theory Finite Temperatures
BCS Theory of Superconductivity
Thomas Burgener Supervisor: Dr. Christian Iniotakis
Proseminar in Theoretical Physics Institut f¨urtheoretische Physik ETH Z¨urich
June 11, 2007
1 / 52 Outlines Cooper-Pairs BCS Theory Finite Temperatures What is BCS Theory?
Original publication: Phys. Rev. 108, 1175 (1957)
2 / 52 Outlines Cooper-Pairs BCS Theory Finite Temperatures 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 Outlines Cooper-Pairs BCS Theory Finite Temperatures 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 Outlines Cooper-Pairs BCS Theory Finite Temperatures 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 Outlines Cooper-Pairs BCS Theory Finite Temperatures Outline
1 Cooper-Pairs Formation of Pairs Origin of Attractive Interaction
2 BCS Theory The model Hamiltonian Bogoliubov-Valatin-Transformation Calculation of the condensation energy
3 Finite Temperatures Excitation Energies and the Energy Gap Determination of Tc Temperature dependence of the energy gap Thermodynamic quantities
4 / 52 Outlines Cooper-Pairs BCS Theory Finite Temperatures Outline
1 Cooper-Pairs Formation of Pairs Origin of Attractive Interaction
2 BCS Theory The model Hamiltonian Bogoliubov-Valatin-Transformation Calculation of the condensation energy
3 Finite Temperatures Excitation Energies and the Energy Gap Determination of Tc Temperature dependence of the energy gap Thermodynamic quantities
4 / 52 Outlines Cooper-Pairs BCS Theory Finite Temperatures Outline
1 Cooper-Pairs Formation of Pairs Origin of Attractive Interaction
2 BCS Theory The model Hamiltonian Bogoliubov-Valatin-Transformation Calculation of the condensation energy
3 Finite Temperatures Excitation Energies and the Energy Gap Determination of Tc Temperature dependence of the energy gap Thermodynamic quantities
4 / 52 Outlines Cooper-Pairs Formation of Pairs BCS Theory Origin of Attractive Interaction Finite Temperatures Outline
1 Cooper-Pairs Formation of Pairs Origin of Attractive Interaction
2 BCS Theory The model Hamiltonian Bogoliubov-Valatin-Transformation Calculation of the condensation energy
3 Finite Temperatures Excitation Energies and the Energy Gap Determination of Tc Temperature dependence of the energy gap Thermodynamic quantities
5 / 52 Outlines Cooper-Pairs Formation of Pairs BCS Theory Origin of Attractive Interaction Finite Temperatures Formation of Pairs
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 Outlines Cooper-Pairs Formation of Pairs BCS Theory Origin of Attractive Interaction Finite Temperatures Formation of Pairs
Look for the groundstate wavefunction for the two added electrons, which has zero momentum: X ik·r1 −ik·r2 Ψ0(r1, r2) = gke e (|↑↓i − |↓↑i) k 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.
! ⇒ gk = g−k.
7 / 52 Outlines Cooper-Pairs Formation of Pairs BCS Theory Origin of Attractive Interaction Finite Temperatures Formation of Pairs
Inserting this into the Schr¨odingerequation of the problem leads to the following equation for the determination of the coefficients gk and the energy eigenvalue E:
X (E − 2k)gk = Vkk0 gk0 ,
k>kF
where Z 1 i(k0−k)·r V 0 = V (r)e dr kk Ω (r: distance between the two electrons, Ω: normalization volume, k: unperturbated plane-wave energies).
8 / 52 Outlines Cooper-Pairs Formation of Pairs BCS Theory Origin of Attractive Interaction Finite Temperatures Formation of Pairs
Since it is hard to analyze the situation for general Vkk0 , assume: −V , EF < k < EF + ~ωc V 0 = kk 0 , otherwise
with ~ωc a cutoff energy away from EF .
9 / 52 Outlines Cooper-Pairs Formation of Pairs BCS Theory Origin of Attractive Interaction Finite Temperatures Formation of Pairs
With this approximation we get:
1 X 1 Z EF +~ωc d = = N(0) V 2k − E EF 2 − E k>kF 1 2E − E + 2 ω = N(0) ln F ~ c . 2 2EF − E If N(0)V 1, we can solve approximativly for the energy E
− 2 E ≈ 2EF − 2~ωc e N(0)V < 2EF .
10 / 52 Outlines Cooper-Pairs Formation of Pairs BCS Theory Origin of Attractive Interaction Finite Temperatures 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 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures Outline
1 Cooper-Pairs Formation of Pairs Origin of Attractive Interaction
2 BCS Theory The model Hamiltonian Bogoliubov-Valatin-Transformation Calculation of the condensation energy
3 Finite Temperatures Excitation Energies and the Energy Gap Determination of Tc Temperature dependence of the energy gap Thermodynamic quantities
12 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures BCS Theory
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.
13 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures BCS Theory
† Introduce the creation operator ckσ, which creates an electron of momentum k and spin σ, and the correspondig annihilation operator ckσ. These operators obey the standard anticommutation relations for fermions:
† † † {ckσ, ck0σ0 } ≡ ckσck0σ0 + ck0σ0 ckσ = δkk0 δσσ0 † † {ckσ, ck0σ0 } = 0 = {ckσ, ck0σ0 }.
Additionally the particle number operator nkσ is defined by
† nkσ ≡ ckσckσ.
14 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures The model Hamiltonian
We start with the so-called pairing-hamiltonian
X X † † H = knkσ + Vklck↑c−k↓c−l↓cl↑, kσ 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 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures The model Hamiltonian
We then add a term −µN , where µ is the chemical potential, leading to
X X † † H − µN = ξknkσ + Vklck↑c−k↓c−l↓cl↑. kσ kl The inclusion of this factor is mathematically equivalent to taking the zero of kinetic energy to be at µ (or EF ).
16 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures Bogoliubov-Valatin-Transformation
Define:
bk ≡ hc−k↓ck↑i Because of the large number of particles involved, the fluctuations of c−k↓ck↑ about these expectations values bk should be small. Therefor express such products of operators formally as
c−k↓ck↑ = bk + (c−k↓ck↑ − bk) and neglect quantities which are bilinear in the presumably small fluctuation term in parentheses.
17 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures Bogoliubov-Valatin-Transformation
Inserting this in our pairing Hamiltonian we obtain the so-called model-hamiltonian
X † X † † ∗ ∗ HM − µN = ξkckσckσ + Vkl(ck↑c−k↓bl + bkc−l↓cl↑ − bkbl) kσ kl
where the bk are to be determined self-consistently.
18 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures Bogoliubov-Valatin-Transformation
Defining further
X X ∆k = − Vklbl = − Vkl hc−k↓ck↑i l l leads to the following form of the model-hamiltonian
X † X † † ∗ ∗ HM − µN = ξkckσckσ − (∆kck↑c−k↓ + ∆kc−k↓ck↑ − ∆kbk) kσ k
19 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures Bogoliubov-Valatin-Transformation
This hamiltonian can be diagonalized by a suitable linear transformation to define new Fermi operators γk: Bogoliubov-Valatin-Transformation
∗ † ck↑ = ukγk↑ + vkγ−k↓ † ∗ † c−k↓ = −vk γk↑ + ukγ−k↓
2 2 with |uk| + |vk| = 1. Our “job” is now to determine the values of vk and uk.
20 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures Bogoliubov-Valatin-Transformation
Inserting these operators in the model-hamiltonian gives
X 2 2 † † HM − µN = ξk (|uk| − |vk| )(γk↑γk↑ + γ−k↓γ−k↓) k 2 ∗ ∗ † † +2|vk| + 2ukvk γ−k↓γk↑ + 2ukvkγk↑γ−k↓ X ∗ ∗ ∗ † † + (∆kukvk + ∆kukvk)(γk↑γk↑ + γ−k↓γ−k↓ − 1) k ∗2 ∗ ∗2 +(∆kvk − ∆kuk )γ−k↓γk↑ ∗ 2 2 † † ∗ +(∆kvk − ∆kuk)γk↑γ−k↓ + ∆kbk .
21 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures Bogoliubov-Valatin-Transformation
† † Choose uk and vk so that the coefficients of γ−k↓γk↑ and γk↑γ−k↓ vanish. ∗ ∗ 2 2 ∆k ⇒ 2ξkukvk + ∆kvk − ∆kuk = 0 · 2 uk ∗ 2 ∗ ∆kvk ∆kvk 2 ⇒ + 2ξk − |∆k| = 0 uk uk ∗ q ∆kvk 2 2 ⇒ = ξk + |∆k| − ξk ≡ Ek − ξk uk
22 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures Bogoliubov-Valatin-Transformation
This gives us an equation for the vk and uk as 2 2 1 ξk |vk| = 1 − |uk| = 1 − . 2 Ek
23 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures The BCS ground state
BCS took as their form for the ground state
Y † † |ΨG i = (uk + vkck↑c−k↓) |0i k
2 2 where |uk| + |vk| = 1. This form implies that the probability of 2 the pair (k ↑, −k ↓) being occupied is |vk| , whereas the probability 2 2 that it is unoccupied is |uk| = 1 − |vk| . Note: |ΨG i is the vacuum state for the γ operators, e.g.
γk↑ |ΨG i = 0 = γ−k↓ |ΨG i
24 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures Calculation of the condensation energy
We can now calculate the groundstate energy to be
X 2 X hΨG | H − µN |ΨG i = 2 ξkvk + Vklukvkulvl k kl 2 2 X ξk ∆ = ξk − − Ek V k The energy of the normal state at T = 0 corresponds to the BCS state with ∆ = 0 and Ek = |ξk|. Thus X hΨn| H − µN |Ψni = 2ξk
|k| 25 / 52 Outlines The model Hamiltonian Cooper-Pairs Bogoliubov-Valatin-Transformation BCS Theory Calculation of the condensation energy Finite Temperatures Calculation of the condensation energy Thus, the condensation energy is given by 2 2 2 X ξk X ξk ∆ hEis − hEin = ξk − + −ξk − − Ek Ek V |k|>kF |k| 26 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Outline 1 Cooper-Pairs Formation of Pairs Origin of Attractive Interaction 2 BCS Theory The model Hamiltonian Bogoliubov-Valatin-Transformation Calculation of the condensation energy 3 Finite Temperatures Excitation Energies and the Energy Gap Determination of Tc Temperature dependence of the energy gap Thermodynamic quantities 27 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Excitation Energies and the Energy Gap With the above choice of the uk and vk, the model-hamiltonian becomes X ∗ X † † HM − µN = (ξk − Ek + ∆kbk) + Ek(γk↑γk↑ + γ−k↓γ−k↓). k k q 2 2 Ek = ∆k + ξk 28 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Excitation Energies and the Energy Gap 4 Eks Ekn 3.5 3 2.5 ∆ / k 2 E 1.5 1 0.5 0 -3 -2 -1 0 1 2 3 ξ ∆ k/ Figure: Energies of elementary excitations in the normal and superconducting states as functions of ξk. 29 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Excitation Energies and the Energy Gap Inserting the γ operators in the definition of ∆k gives X ∆k = − Vkl hc−l↓cl↑i l X ∗ D † † E = − Vklul vl 1 − γl↑γl↑ − γ−l↓γ−l↓ l X ∗ = − Vklul vl(1 − 2f (El)) l X ∆l βEl = − Vkl tanh 2El 2 l 30 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Excitation Energies and the Energy Gap Using again the approximated potential Vkl = −V , we have ∆k = ∆l = ∆ and therefor 1 1 X tanh(βEk/2) = . V 2 Ek k This formula determines the critical temperature Tc ! 31 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Determination of Tc The critical temperature Tc is the temperature at which ∆k → 0 and thus Ek → ξk. By inserting this in the above formula, rewriting the sum as an integral and changing to a dimensionless variable we find 1 Z βc ~ωc /2 tanh x 2eγ = dx = ln βc ~ωc N(0)V 0 x π (γ ≈ 0.577...: the Euler constant) 32 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Determination of Tc Critical temperatur Tc −1 −1/N(0)V kTc = βc ≈ 1.13~ωc e 33 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Determination of Tc For small temperatures we find 1 Z ~ωc dξ = 2 2 1/2 N(0)V 0 (ξ + ∆ ) ω ⇒ ∆ = ~ c ≈ 2 ω e−1/N(0)V , sinh(1/N(0)V ) ~ c which shows that Tc and ∆(0) are not independent from each other ∆(0) 2 ≈ ≈ 1.764 kTc 1.13 34 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Temperature dependence of the energy gap Rewriting again 1 1 X tanh(βEk/2) = . V 2 Ek k in an integral form and inserting Ek gives 1 Z ~ωc tanh 1 β(ξ2 + ∆2)1/2 = 2 dξ, 2 2 1/2 N(0)V 0 (ξ + ∆ ) which can be evaluated numerically. 35 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Temperature dependence of the energy gap Figure: Temperature dependence of the energy gap with some experimental data (Phys. Rev. 122, 1101 (1961)) 36 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Temperature dependence of the energy gap Near Tc we get Temperature dependence of ∆ ∆(T ) T 1/2 ≈ 1.74 1 − , T ≈ Tc , ∆(0) Tc which shows the typical square root dependence of the order parameter for a mean-field theory. 37 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Thermodynamic quantities With ∆(T ) determined, we know the fermion excitation energies q 2 2 Ek = ξk + ∆(T ) . Then the quasi-particle occupation numbers βE −1 will follow the Fermi-function fk = (1 + e k ) , which determine the electronic entropy for a fermion gas X Ses = −2k ((1 − fk) ln(1 − fk) + fk ln fk). k 38 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Thermodynamic quantities Figure: Electronic entropy in the superconducting and normal state. 39 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Thermodynamic quantities Given Ses (T ), we find the specific heat 2 dSes X ∂fk 2 1 d∆ Ces = −β = 2βk − Ek + β dβ ∂Ek 2 dβ k In the normal state we have 2π2 C = N(0)k2T . en 3 40 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Thermodynamic quantities We expect a jump in the specific heat from the superconducting to the normal state: 2 −d∆ 2 ∆C = (Ces − Cen)| = N(0) ≈ 9.4N(0)k Tc Tc dT Tc 41 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Thermodynamic quantities Figure: Experimental data for the specific heat in the superconducting and normal state (Phys. Rev. 114, 676 (1959)) 42 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Type I superconductors 1 0.8 Normal-State 0.6 H 0.4 M-O-State 0.2 0 0 0.2 0.4 0.6 0.8 1 T/Tc Figure: Phase diagram of a Type I superconductor 43 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Vortex-State Original publication: Zh. Eksperim. i Teor. Fiz. 32, 1442 (1957) (Sov. Phys. - JETP 5, 1174 (1957)) 44 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Type I and Type II superconductors By applying Ginzburg-Landau theory for superconductors one finds two characteristic lengths: 1 The Landau penetration depth for external magnetic fields λ and 2 the Ginzburg-Landau coherence length ξ, which characterizes the distance over which ψ can vary without undue energy increase. 45 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Type I and Type II superconductors Define Ginzburg-Landau parameter λ κ ≡ ξ By linearizing the GL equations near Tc one can find: κ < √1 : Type I superconductor 2 κ > √1 : Type II superconductor 2 46 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Type II superconductors 3 2.5 2 Normal-State H 1.5 Vortex-State 1 0.5 M-O-State 0 0 0.2 0.4 0.6 0.8 1 T/Tc Figure: Phase diagram of a Type II superconductor 47 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities As a solution of the GL equation, one could find the following form of the orderparameter: ∞ 1 X π(ixy − y 2) Ψ(x, y) = exp + iπn N ω =ω n=−∞ 1 2 iπ(2n + 1) ω + (x + iy) + iπ 2 n(n + 1) ω1 ω1 ω =ω 1/4 N = 1 exp π 2 2=ω2 ω1 48 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Vortex-State 2 2 1 1 0 0 -1 -1 -2 -2 -2 -1 0 1 2 -2 -1 0 1 2 Figure: Square and triangle symmetric state of the vortex lattice in a density plot of |Ψ|2. 49 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities Vortex-State 1 1 ÈYÈ2 2 ÈYÈ2 2 0.5 0.5 1 1 0 0 -2 0 -2 0 y y -1 -1 0 -1 0 -1 x 1 x 1 2-2 2-2 Figure: Square and triangle symmetric state of the vortex lattice in a 3D plot. 50 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities 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 Tc and the energy gap ∆(T ). Vortices will be observed in Type II superconductors. 51 / 52 Outlines Excitation Energies and the Energy Gap Cooper-Pairs Determination of Tc BCS Theory Temperature dependence of the energy gap Finite Temperatures Thermodynamic quantities The END Thank you for your attention! Are there any questions? 52 / 52