Bloch Oscillations: Electron on a Band (2D)

Field & Band

Electric field E 1.20

Hopping amplitude J 1.00

Playback speed 1.0×

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Bloch period T_B

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Amplitude x_max

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Crystal momentum k/π

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Group velocity v

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How it works

An electron in a 1D periodic crystal lives on a tight-binding band E(k) = −2J·cos(ka) (hopping amplitude J, lattice constant a=1, ħ=1, e=1 here). A constant field E exerts a constant force, so semiclassically the crystal momentum climbs uniformly, k(t) = k₀ + Et (mod 2π/a) — it wraps at the zone edge k=±π instead of growing forever, a Bragg reflection.

k(t) = k₀ + Et   (mod 2π)
v(k) = dE/dk = 2Ja·sin(ka)
x(t) = (2J/E)·[1 − cos(Et)]
T_B  = 2π / E

Because the group velocity is a periodic function of k, real-space position x(t) is periodic too: the electron speeds up, slows, reverses and returns instead of accelerating without limit the way a genuinely free electron under the same force would. A DC field produces an AC current inside a perfect lattice.

Band diagram — the electron's dot rides the cosine band E(k); its horizontal velocity readout is literally the local slope dE/dk.
k-dial — crystal momentum sweeping the Brillouin zone; the flash marks each Bragg reflection at the zone boundary.
x(t) graph — real-space position over time, periodic with the Bloch period T_B.
Free-electron ghost — same constant force, no lattice: it just keeps accelerating (x ∝ t²) and never turns around.

True Bloch oscillations are rarely seen in ordinary bulk crystals — electrons scatter off phonons and defects long before one cycle completes. They were first observed in semiconductor superlattices and are now routine in ultracold-atom optical lattices, giving extremely precise measurements of gravity and other weak forces.

Lattice ion Bloch electron Free electron (ghost) Crystal momentum k