Nernst effect - 2D slab plan view

What happens

When the bar is hot at one end and cold at the other, charge carriers diffuse from hot to cold. A perpendicular magnetic field B deflects them with the Lorentz force; in the Drude picture with scattering time τ the steady drift has a transverse part, and the ratio of transverse to longitudinal drift is the Hall angle tanθ = ω_cτ = μB (μ = eτ/m). The simulation integrates dv/dt = a x̂ + sω_c(v_y, −v_x) − v/τ with thermal noise for hundreds of carriers.

Why a voltage appears

With energy-independent τ the deflected current is exactly cancelled by the Seebeck field (the Sondheimer cancellation), so the open-circuit Nernst signal needs an energy-dependent scattering time τ ∝ ε^r. For a degenerate conductor the Mott relation gives ν = (π²/3)(k_B/e)(k_BT/E_F) r μ and E_y = ν B (−dT/dx), so V_y = −ν B ΔT (w/L) in the plotted bar (w/L = 0.2). It grows linearly with B and ΔT, is odd in B, and vanishes for r = 0.

Carrier sign

Electrons and holes both diffuse hot to cold, so the field deflects them to opposite edges, but their charges are opposite, so the voltage keeps its sign in a single-band conductor. Real semimetals with both carrier types give much larger and more complicated signals.

Honest simplifications: the animation shows the Drude drift and the edge charge qualitatively (the open-circuit pile-up is a relaxation toward the Mott value, not a solution of Poisson equation); the Nernst voltage itself comes from the Mott formula, not from the carriers on screen; one parabolic band, constant E_F along the bar, no phonon drag, no magnon or vortex contributions, fixed bar geometry.

• Pick Constant tau (r = 0): the carriers still deflect, but the voltage is zero.
• Switch Electrons to Holes: the piled-up charge swaps edges, the voltage does not.
• Reverse the field: the voltage changes sign.
• Go to Cryogenic and watch the voltage fall with temperature in the last graph.