Week 7: Boltzmann equation as particles (two potentials)

Boltzmann equation as particles: the two potentials (Datta Ch. 9, Figs. 9.1–9.3)

Each dot is an electron moving by Newton's laws, dz/dt = v, dp/dt = −dU/dz (Eq. 9.7), with random elastic backscattering (relaxation time τ). Following all of them is the same as solving the Boltzmann equation (Eqs. 9.5–9.6). Both contacts inject a thermal (Boltzmann) distribution set by their μ. The lower panel shows the band edge U(z) and the electrochemical potential μ(z) = U(z) + kT·ln(n/n_ref) read from the local density.

Built-in step ΔU Bias μ₁ − μ₂ = qV Scattering time τ

Net current, per unit time (as % of the one-way flux)–

Spread of μ(z) inside the channel–

Density right / left–

Electrons in the channel–