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Superconductivity

Field: Solid State Physics

A phenomenon of exactly zero electrical resistance and expulsion of magnetic flux fields occurring in certain materials when cooled below a characteristic critical temperature.

Sequence of Expressions

Definition

Cooper Pairs

Define the pairing potential Δk\Delta_{\mathbf{k}} in the context of the BCS mean-field approximation. The formation of Cooper pairs is characterized by the condensation of electron pairs with total momentum 0\mathbf{0} and spin 0\mathbf{0}. The gap parameter Δk\Delta_{\mathbf{k}} is defined by the expectation value of the pairing interaction: \nΔk=−∑k′Vk,k′⟨ck′,↑c−k′,↓⟩\Delta_{\mathbf{k}} = -\sum_{\mathbf{k}^{\prime}} V_{\mathbf{k}, \mathbf{k}^{\prime}} \langle c_{\mathbf{k}^{\prime}, \uparrow} c_{-\mathbf{k}^{\prime}, \downarrow} \rangle \nIn the weak coupling limit, this leads to the pairing gap equation, relating Δk\Delta_{\mathbf{k}} to the attractive interaction VV and the density of states N(0)N(0) at the Fermi level.
Definition

Energy Gap

Define the superconducting energy gap Δ\Delta using the BCS gap equation, which relates the gap magnitude to the coupling constant VV and the density of states N(0)N(0) at the Fermi level:\nΔ=12N(0)VN(0)∫02ωDdϵϵtanh⁡(ϵ2kBT)\Delta = \frac{1}{2} \frac{N(0)V}{N(0)} \int_{0}^{2\omega_D} \frac{d\epsilon}{\epsilon} \tanh\left(\frac{\epsilon}{2 k_B T}\right) \nAlternatively, the gap Δ(T)\Delta(T) is the minimum energy required to break a Cooper pair, measured relative to the normal state energy.
Let B(r)\mathbf{B}(\mathbf{r}) be the magnetic field and H(r)\mathbf{H}(\mathbf{r}) be the applied magnetic field in a superconductor. The Meissner effect dictates that the internal magnetic field Bint\mathbf{B}_{int} vanishes in the bulk material: \nBint(r)=0 for r∈Superconductor\mathbf{B}_{int}(\mathbf{r}) = 0 \text{ for } \mathbf{r} \in \text{Superconductor} \nThis implies that the magnetization M\mathbf{M} must satisfy M=−H\mathbf{M} = -\mathbf{H}, corresponding to a perfect diamagnetism, χ=−1\chi = -1.
Let Js\mathbf{J}_s be the supercurrent density and B\mathbf{B} be the magnetic field. The first London equation relates the supercurrent to the magnetic field: \nJs=−nse2mA=nse2mA\mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{A} = \frac{n_s e^2}{m} \mathbf{A} \n(where A\mathbf{A} is the magnetic vector potential, and nsn_s is the superfluid density). The second London equation describes the decay of the magnetic field inside the superconductor: \n∇2B=1λL2B\nabla^2 \mathbf{B} = \frac{1}{\lambda_L^2} \mathbf{B} \nwhere λL=mnse2\lambda_L = \sqrt{\frac{m}{n_s e^2}} is the London penetration depth.
Define the superconducting order parameter Δ(T)\Delta(T) via the BCS gap equation:\n1N(0)V=∫02ωD/(kBT)dϵϵ2−Δ(T)2tanh⁡(ϵ2−Δ(T)22kBT)\frac{1}{N(0)V} = \int_{0}^{2\omega_D/(k_B T)} \frac{d\epsilon}{\sqrt{\epsilon^2 - \Delta(T)^2}} \tanh\left(\frac{\sqrt{\epsilon^2 - \Delta(T)^2}}{2 k_B T}\right) \nThe critical temperature TcT_c is defined as the temperature where the non-trivial solution for the order parameter vanishes, i.e., Δ(Tc)=0\Delta(T_c) = 0.
Define the characteristic lengths λ\lambda (magnetic penetration depth) and ξ\xi (coherence length) in a superconductor. The Ginzburg-Landau parameter κ\kappa is defined as the ratio of these two lengths:\nκ=λξ\kappa = \frac{\lambda}{\xi} \nThis parameter determines the relative strength of magnetic field penetration versus the spatial variation of the superconducting order parameter Ψ\Psi.
Consider the BCS mean-field Hamiltonian for the pairing interaction: \nH=∑k,σξkck,σ†ck,σ+12∑k(Δkck,↑c−k,↓+Δ∗kc−k,↓†ck,↑†)+12∑k(Δk∗c−k,↓ck,↑+Δ∗kck,↑†c−k,↓†)\mathcal{H} = \sum_{\mathbf{k}, \sigma} \xi_{\mathbf{k}} c_{\mathbf{k}, \sigma}^{\dagger} c_{\mathbf{k}, \sigma} + \frac{1}{2} \sum_{\mathbf{k}} \left( \Delta \mathbf{k} c_{\mathbf{k}, \uparrow} c_{-\mathbf{k}, \downarrow} + \Delta^* \mathbf{k} c_{-\mathbf{k}, \downarrow}^{\dagger} c_{\mathbf{k}, \uparrow}^{\dagger} \right) + \frac{1}{2} \sum_{\mathbf{k}} \left( \Delta \mathbf{k}^* c_{-\mathbf{k}, \downarrow} c_{\mathbf{k}, \uparrow} + \Delta^* \mathbf{k} c_{\mathbf{k}, \uparrow}^{\dagger} c_{-\mathbf{k}, \downarrow}^{\dagger} \right) \nThe superconducting gap Δ\Delta is determined self-consistently by the gap equation:\n\Delta = \lambda \sum_{\mathbf{k}} \frac{1}{2 \sqrt{\xi_{\mathbf{k}}^2 + |\Delta \mathbf{k}|^2}} \tanh\left(\frac{\sqrt{\xi_{\mathbf{k}}^2 + |\Delta \mathbf{k}|^2}}}{2 k_B T}\right) \nwhere λ\lambda is the coupling constant, ξk\xi_{\mathbf{k}} is the electron dispersion, and TT is the temperature.
Let J\mathbf{J} be the current density and E\mathbf{E} be the electric field. The condition of zero electrical resistance (ρ=0\rho = 0) implies that the electric field vanishes for any steady current flow: \nE=0\mathbf{E} = 0 \nThis is consistent with the generalized Ohm's law, E=ρJ\mathbf{E} = \rho \mathbf{J}, requiring the resistivity ρ\rho to be zero, or equivalently, the conductivity σ\sigma to be infinite: \nσ=1ρ=∞\sigma = \frac{1}{\rho} = \infty
Let the full Hamiltonian be H=H0+VH = H_0 + V, where H0H_0 is the unperturbed Hamiltonian and VV is the interaction term. The effective Hamiltonian HeffH_{eff} is obtained via second-order perturbation theory:\nHeff=H0+Veff=H0+V12(E1−E2)E1−E2V21+…\mathcal{H}_{eff} = H_0 + V_{eff} = H_0 + \frac{V_{12} (E_1 - E_2)}{E_1 - E_2} V_{21} + \dots \nFor the superconducting context, VeffV_{eff} describes the renormalized electron-electron interaction, effectively simplifying the treatment by projecting out high-energy degrees of freedom.
Consider a particle with mass mm encountering a potential barrier V(x)V(x) of width LL and height V0V_0, where E<V0E < V_0. The transmission probability TT through the barrier is approximated by the WKB method:\nT≈e−2∫x1x22mℏ2(V(x)−E)dxT \approx e^{-2 \int_{x_1}^{x_2} \sqrt{\frac{2m}{\hbar^2} (V(x) - E)} dx} \nFor Cooper pairs, the tunneling current II is proportional to TT and the pair density.