Fig. 1. Dirty and clean optical responses.

a Band structure in the Bogoliubov de-Gennes (BdG) formalism. Superconducting pairing opens the bandgap at the Fermi level E_F by mixing electron (red) and hole (blue) bands as shown in (b). c_k and $c_{-\mathbf{k}}^{†}$ indicate the electron and hole annihilation operators, respectively. b Optical excitations in dirty and clean limits by spatially uniform light (q = 0) across the superconducting gap 2Δ. The momentum transfer p − k in the left figure is supplied from the impurity potential. The crossover from dirty to clean optical responses occurs when the mean free path l exceeds the superconducting coherence length ξ₀ by a factor $x_c={\left(k_F{\xi }_0\right)}^2{\alpha }^{-2}$. Here, k_F is the Fermi wave number, and α is a degree of multiband pairing (see the text above Eq. (6) for its more precise definition). The red and blue colors schematically represent the electron–hole band mixing in the superconducting state. c Real part of the optical conductivity in clean systems (x > 1). σ_s = σ_dis + σ_int in the superconducting state is the sum of disorder-mediated and intrinsic parts. σ_int ∝ ω⁻¹ is insensitive to x while σ_dis ∝ x⁻¹ω⁻² depends significantly on x. The Drude conductivity in the normal state σ_n is also shown as dashed lines. d, e Ratio of superconducting and normal state conductivities. d Disorder-mediated part. e Intrinsic part.