Bloch Oscillations — An Electron That Turns Around

Bloch Oscillations

Electric field E 1.20

Hopping amplitude J 1.00

Playback speed 1.0×

Bloch period T_B

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Amplitude x_max

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Crystal momentum k/π

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Velocity v

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How it works

An electron in a periodic crystal lives on a tight-binding band E(k) = -2J·cos(ka), where k is its crystal momentum and J sets the bandwidth. A constant electric field E exerts a constant force, so semiclassically k itself climbs uniformly with time: ħ·dk/dt = -eE. Because k lives on a ring — the Brillouin zone — it doesn't run away, it wraps around at the zone boundary k = ±π/a (a Bragg reflection).

k(t) = k₀ + eEt/ħ   (mod 2π/a)
v(k) = (1/ħ)·dE/dk = 2Ja·sin(ka)/ħ
x(t) = (2J/eE)·[1 - cos(eEt/ħ)]
T_B  = 2πħ / (eEa)

Since the group velocity v(k) is a periodic function of k, real-space position x(t) is periodic too — the electron accelerates, decelerates, reverses, and returns, rather than accelerating without limit the way a genuinely free electron would under the same constant force. This is a Bloch oscillation: a DC field produces an AC current inside a perfect lattice.

Electric field E — stronger field sweeps k across the zone faster, shortening the Bloch period T_B and shrinking the real-space swing.
Hopping amplitude J — sets the bandwidth 4J and hence how far the electron travels before turning around; wider bands give bigger swings.
Free electron toggle — adds a ghost particle under an identical force but with no lattice, so it never Bragg-reflects and just keeps accelerating away.

Real-world relevance: true Bloch oscillations are almost never seen in ordinary bulk crystals because electrons scatter off phonons and defects long before completing one cycle. They were first observed in semiconductor superlattices (with an artificially long period a) and are now a routine tool in ultracold-atom optical lattices, where they provide extremely precise measurements of gravity and other forces.

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

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