Diffusion

Modifications to improve the simulation of diffusion in AvalancheMC at low (or zero) electric field.

If the electric field or the drift speed are zero, diffusion steps are simulated by

• retrieving the low-field mobility μ from the Medium,
• calculating the diffusion coefficient D = μ k_BT/q,
• drawing Gaussian-distributed random steps in all three directions.
  • In case of fixed time steps Δt, the σ of the Gaussian distribution is given by (2DΔt)^1/2.
  • In case of fixed distance steps Δs, we take σ = Δs and calculate the corresponding time step using Δt = σ²/(2D).
  • In case of "collision" steps, we calculate σ using σ = nD/v_th, where v_th = (2k_BT/m)^1/2 is the thermal velocity, and subsequently determine the corresponding time step as in the case of fixed distance steps.

Note that the diffusion coefficient D used here has dimensions cm²/ns.

If the drift speed v_D is non-zero and fixed-distance (or "collision") steps are used, the time step is calculated using Δt = Δs/v_D. After making a step along the direction of the drift velocity, random diffusion steps in the longitudinal (L) and transverse (T) directions are added, with σ_L,T = (v_DΔt)^1/2D_L,T, where D_L,T are the longitudinal and transverse diffusion parameters. Note that the diffusion parameters used here have dimensions cm^1/2.

If the drift speed is small, this can lead to very large diffusion steps. A check has been added whether σ_L,T exceeds Δs. If it does, the time step is limited to Δt = Δs² / (D² v_D).