Why does k(t) look like a sawtooth?

In extended-zone language, ℏ dk/dt = eE makes k grow linearly. In the reduced zone, k is folded back into [−π/a, π/a] whenever it crosses a zone boundary, producing a periodic wrapped motion while the extended k(t) is a ramp.

What sets the Bloch period T_B?

T_B = 2π/(eEa) in our units: the time for k to advance by one reciprocal lattice vector 2π/a. Stronger field or smaller lattice constant shortens the period and makes x(t) oscillate faster.

What is the Wannier–Stark ladder?

In a finite chain with a linear potential from the field, exact eigenenergies cluster into “rungs” separated by approximately eEa, reflecting spatially localized Wannier–Stark states in an ideal infinite system.

Why might real electrons not oscillate forever?

Interband Landau–Zener tunneling, scattering, and finite sample size break the ideal single-band picture. This page illustrates the textbook band-theory limit, not room-temperature bulk transport.

Bloch Oscillations & Wannier–Stark Ladder

When a crystal electron experiences a uniform electric field, the semiclassical equations of motion predict that crystal momentum k increases linearly in time (ℏ dk/dt = eE) while the group velocity v = (1/ℏ) dE/dk oscillates as k sweeps through the Brillouin zone. The position x(t) = ∫ v dt therefore oscillates with the Bloch period T_B = 2π/(eEa) instead of accelerating without bound — the famous Bloch oscillation. This simulator uses the standard nearest-neighbor tight-binding dispersion E(k) = −2t cos(ka) in pedagogical units ℏ = e = 1. Four panels show the band structure with a moving k marker, the reduced-zone k(t) sawtooth (folded into the first Brillouin zone), the resulting x(t) over several Bloch periods, and the exact eigenvalue spectrum of a finite chain with onsite Stark potential −eE·a·n — the Wannier–Stark ladder with spacing Δ ≈ eEa. Open boundaries and finite N are explicit caveats for the ladder panel; the semiclassical trajectories assume a smooth single band without interband Zener tunneling.

Who it's for: Advanced undergraduate solid-state or condensed-matter students after band structure and before mesoscopic transport or ultracold atoms in optical lattices.

Key terms

Bloch oscillation
Crystal momentum
Wannier–Stark ladder
Tight-binding model
Brillouin zone
Group velocity
Bloch period

How it works

Bloch oscillations in a 1-D tight-binding crystal under a uniform electric field: watch k(t) accelerate through the Brillouin zone, x(t) oscillate instead of drifting, and the Wannier–Stark energy ladder emerge in a finite chain.

Frequently asked questions

Why does k(t) look like a sawtooth?: In extended-zone language, ℏ dk/dt = eE makes k grow linearly. In the reduced zone, k is folded back into [−π/a, π/a] whenever it crosses a zone boundary, producing a periodic wrapped motion while the extended k(t) is a ramp.
What sets the Bloch period T_B?: T_B = 2π/(eEa) in our units: the time for k to advance by one reciprocal lattice vector 2π/a. Stronger field or smaller lattice constant shortens the period and makes x(t) oscillate faster.
What is the Wannier–Stark ladder?: In a finite chain with a linear potential from the field, exact eigenenergies cluster into “rungs” separated by approximately eEa, reflecting spatially localized Wannier–Stark states in an ideal infinite system.
Why might real electrons not oscillate forever?: Interband Landau–Zener tunneling, scattering, and finite sample size break the ideal single-band picture. This page illustrates the textbook band-theory limit, not room-temperature bulk transport.

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