Kubo Formula — Optical Conductivity

The Kubo formula expresses conductivity as the retarded current-current correlation function. σ(ω) features a Drude peak at ω=0 from free carriers and interband absorption at ω≥2Δ. The f-sum rule ∫σ₁(ω)dω = πne²/2m is exact.

Material

σ₁(ω)

σ₂(ω)

Sum rule

Drude weight D (plasma) 1.00 Scattering rate Γ 0.15 Interband gap 2Δ 2.00 Interband strength σ_inter 0.40

Kubo formula:
σ(ω) = (i/ω)∫₀^∞ ⟨j(t)j(0)⟩eⁱᵒᵗdt

Drude: σ_D = D/(Γ−iω)
f-sum: ∫σ₁dω = πne²/2m

Kramers-Kronig connects σ₁,σ₂.

Kubo Formula — Optical Conductivity σ(ω)

Material