Acoustic Phonon Thermal Conductivity & Umklapp Scattering

Debye Temp θ_D (K) 450 K Sound velocity v_s 5000 m/s Grüneisen γ 2.0 Sample size L (mm) 5 mm Current Temp T (K) 300 K

κ (W/m·K)—

τ_U (ps)—

τ_B (ns)—

MFP λ (nm)—

Regime—

κ = ¹⁄₃ C_v v_s λ_eff

Umklapp rate: τ_U⁻¹ ∝ γ² T exp(−θ_D/3T)

Boundary scattering: τ_B = L/v_s

Matthiessen's rule: τ⁻¹ = τ_U⁻¹ + τ_B⁻¹

At T ≫ θ_D/3: Umklapp dominates, κ ∝ 1/T.
At low T: boundary scattering, κ ∝ T³.

Total κ

Umklapp

Boundary

Current T