Vacancy Hopping & Ionic Conductivity (2D)

Lattice Conditions

Temperature 650 K

Vacancy concentration 10%

Applied field E 1.0 MV/m

Migration barrier E_a 0.60 eV

Lattice size 18 × 18

Simulation speed 1.0×

Live Readouts

Hop rate (lattice)

— Hz

Total hops

Ionic conductivity σ (Nernst–Einstein)

— S/m

Net drift current density J

— A/m²

How it works

A Schottky vacancy in an ionic crystal is an empty lattice site. A neighboring ion can jump into it if it clears the migration energy barrier E_a — a thermally activated, Arrhenius-rate process, simulated here as a kinetic Monte Carlo walk on a 2D square sublattice (the field axis is the horizontal, +x, direction):

Γ = ν₀·exp(−Ea / kBT)     (unbiased hop rate)
Γ± = ν₀·exp(−(Ea ∓ qEa₀/2) / kBT)   (field-biased, ± = with/against E)
D  = a₀²·ν₀·exp(−Ea / kBT)          (hopping diffusivity)
σ  = n·q²·D / (kBT)                  (Nernst–Einstein conductivity)

ν₀ ≈ 10¹³ Hz — the lattice's phonon "attempt frequency"; every ion rattles in its site this many times per second, and each rattle is one chance to clear the barrier.
Temperature — raises kBT, exponentially increasing the fraction of attempts that succeed. This is why solid electrolytes conduct far better hot than cold.
Vacancy concentration — sets how many empty sites exist for ions to hop into; conduction needs both a mobile ion and an adjacent vacancy.
Applied field E — tilts the energy landscape (see the middle panel), lowering the barrier for hops along the field and raising it against, biasing the random walk into a net drift current (glowing amber ions on hop, drifting toward the field arrow).
Migration barrier E_a — an intrinsic property of the ion/lattice pair (≈0.3 eV for fast Na⁺-β-alumina-type conductors, ≈1 eV+ for sluggish oxide conductors).

This vacancy-hopping mechanism is the microscopic basis of solid-state ionic conductivity — the same physics behind solid electrolytes in batteries, oxygen-ion conduction in fuel cells, and the Schottky-defect diffusion described in classic solid-state chemistry. Drag the lattice panel to pan, scroll to zoom; the tilted-barrier diagram and the hop-rate history update live as you move the sliders.

Mobile ion Just hopped Vacancy

Field E →

drag to pan · scroll to zoom

Tilted energy landscape

Hop-rate history