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↓