An Electron in a Perfect Crystal — What a Constant Field Actually Does

Electron in a perfect crystal

Teach me

Speed

electron ghost — if the band never bent

Guided Inquiry

1 · The setup

Here’s our playground: a perfect 1‑D crystal of 200 atoms, spacing a = 0.5 nm. A wavepacket electron rests at the band bottom, right at the origin — take a moment to find it.

2 · Release it

Let’s make the first prediction. The electron rests at the band bottom, and the field is already on. Once time starts running, how do you think its position will move?

3 · Watch them move

Now let’s see how your guess holds up. Release it — press ▶ Play and let at least one full cycle run. Follow the terracotta electron against the grey ghost, and keep an eye on the x–t trace too. What did you see?

4 · Why it turns

So the electron returned to x = 0 while the ghost ran off. Why? The band panel shows you: velocity is the slope at the dot, and that slope reverses each time the dot wraps at ±π/a. Watch v in the Live packet panel — you’ll see it flip sign at each wrap.

5 · Push harder

Now let’s put our foot down. The field is the accelerator pedal — suppose you double it, 1 → 2 MV/cm. What do you think happens to the full swing?

6 · Faster and smaller

Time to check that prediction. Drag the field 1 → 2 MV/cm, reading the Bloch clock and Excursion panels as you go, then push on toward 5 MV/cm. What happened as E grew?

7 · Widen the band

Now let’s try the other slider. Drag W: 2 → 4 eV and watch both the Excursion and the Bloch clock. Which one changed?

8 · Where it hides

One last puzzle. Real wires obey Ohm’s law, yet you’ve just watched a DC field make AC motion. So what do you think must interrupt the cycle before one T_B completes?

Listen

Field strength E 1.0 MV/cm

0.55 MV/cm

Band width W 2.0 eV

14 eV

Time to walk one Brillouin zone

Period T_B 82.7 fs

Frequency f_B 12.09 THz

Where the electron is right now

Position x 0.0 nm

Velocity v 0.000 nm/fs

Peak-to-peak swing of x(t)

Full swing W/eE 20.0 nm

Tight‑binding band: bottom at k = 0, top at the zone edge.

A constant force feeds k at a steady rate — not v. E here is the applied field, not the band energy E(k).

Velocity is the band slope; it flips sign at each wrap.

Position oscillates between 0 and W/eE — back to 0 each period.

Faster with stronger field; swing shrinks as 1/E; period independent of W.

The Physics

Bloch oscillations · what this sim shows, the physics behind it, and every control

At a glance

A free electron pushed by a steady force keeps speeding up. This sim follows an electron in a perfect one-dimensional crystal beside a free-particle prediction under the same force, so you can see what the lattice does.

What you'll learn

How the shape of an energy band enters the electron's motion.
How a steady force and the band shape together decide velocity and position.
How the Bloch period T_B and the excursion depend on the field E and the band width W.
Why ordinary metals hide this behaviour, and where it can be seen.

Start: x = 0, bottom of the band (dot). Dashed: the ghost.

What's on screen

Electron (terracotta): the wave packet, on a 100 nm strip of atoms. The arrow shows the field E.
Ghost (dashed grey): a free particle of the band-bottom mass m* (0.31 mₑ at the defaults) under the same force.
Band panel: the energy band E(k) in teal, with a terracotta dot at the electron's k and its slope.
Position chart: x against time for the electron (terracotta) and the ghost (dashed grey).
Readouts: Bloch clock, Live packet and Excursion (next section).

See itRelease the electron and the ghost together at E = 1 MV/cm, W = 2 eV. Watch for: both leaving x = 0, and the dot starting to move along the band curve.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Field strength E	The electric field along the crystal. The arrow on the strip gets thicker as E grows. Moving it restarts the run.	0.5–5 MV/cm opens at 1.0
Band width W	The energy width of the band: its bottom sits at −W/2 and its top at +W/2. Moving it restarts the run.	1–4 eV opens at 2.0
Bloch clock (Readouts)	T_B, the time for k to cross one Brillouin zone, and its frequency f_B.	fs and THz
Live packet (Readouts)	The electron's position x and velocity v right now.	nm and nm/fs
Excursion (Readouts)	The peak-to-peak size of x(t), as a distance.	nm
▶ Play / ⏸ Pause	Starts the run, or freezes it so you can read the electron where it stands.	top bar
Speed	How fast the clock runs on screen: 25 fs of crystal time per second at 1×. The physics does not change.	0.25×–4× opens at 1×
↻ Reset	Restarts the run from t = 0. Your slider settings stay as they are.	top bar

Fixed: lattice spacing a = 0.5 nm, a perfect crystal with no scattering, one band and no jumps to other bands. The packet is drawn at a fixed width.

A steady force makes k grow

In a crystal, an electron's state is labelled by its crystal momentum ħk. A constant force changes ħk at a constant rate:

E here is the applied field, not the band energy E(k).

The force has size eE. The electron's charge is −e, so the force points opposite to the field: the E arrow on the strip points left and the electron is pushed right, toward larger k.

The crystal's own forces are not missing. They are already built into the band E(k), the subject of the next section. In one Bloch period T_B, k advances by 2π/a, the width of one Brillouin zone.

Key idea. The force sets how fast k changes. It does not, by itself, set the velocity.

See itRelease the electron and follow the dot on the band panel. Watch for: the dot moving across the band at a steady rate, then reappearing at the left edge after reaching the right edge (+π/a).

Velocity follows the slope

The band gives the electron's energy for each k. This sim uses the simplest band for a chain of atoms, a cosine of width W with its bottom at k = 0:

Bottom −W/2 at k = 0, top +W/2 at the zone edge.

A wave packet moves at the group velocity, the slope of this curve divided by ħ:

The velocity is read from the slope at the dot.

The slope is zero at the flat bottom and the flat top, so v = 0 there.
The slope is steepest at ka = ±π/2, where |v| is largest: v_max = Wa/2ħ = 0.76 nm/fs at W = 2 eV. It does not depend on E.
The sign of v follows the sign of the slope.

Key idea. The force never appears in v(k). It only moves the dot along the curve.

See itRelease the electron and compare the Velocity readout with the slope at the dot. Watch for: v peaking where the tangent is steepest, and passing through zero where it is flat.

Why the motion turns back

The band repeats itself in k. The states k and k + 2π/a are the same state of the crystal, so when the dot reaches +π/a it re-enters at −π/a. Physically this is Bragg reflection: at the zone edge the electron's wavelength is just right for the lattice planes to reflect it.

The band is periodic in k. The effective mass m* changes with k.

Across the wrap the slope passes through zero and changes sign. v reverses and x turns back.
Where cos ka < 0 the effective mass is negative. The force is unchanged, but the acceleration points the other way.
After one period the electron is back at x = 0, so the force has done no net work. With no scattering, nothing is dissipated.

Key idea. A steady force on a band that repeats in k gives motion that repeats. The ghost has no such band, so nothing turns it back.

See itRelease the electron and let one full period run. Watch for: the Position readout rising, then returning to 0.0 while the ghost's label on the strip keeps counting up.

What sets period and swing

Adding up v(t) gives the position. It starts at 0 and never goes negative:

The electron swings out to W/eE and back.

The size follows from energy. Moving a distance x along the force changes the potential energy by −eEx, while the band energy can rise by at most W. The climb ends when eEx = W.

Period, frequency, full swing, and the energy eEa the field gives per lattice site.

E W

These sliders move the sim too.

T_B = h/eEa has no W in it. Doubling E halves the period.
The full swing W/eE grows with W and falls as E grows.

See itSet E to 2 MV/cm, with W = 2 eV. Watch for: the Period and the Full swing both at half their opening values (82.7 fs → 41.4 fs, 20.0 nm → 10.0 nm).

See itSet W to 4 eV, with E = 1 MV/cm. Watch for: the Full swing doubling to 40.0 nm while the Period stays at 82.7 fs.

Where it matters

Ordinary metals. In a wire, a field of about 0.1 V/m with a = 0.5 nm gives T_B ≈ 80 µs, but electrons scatter every ≈ 25 fs, about 3 × 10⁹ times sooner. They never finish a cycle. They drift instead, which is Ohm's law.
GaAs/AlGaAs superlattices. A layered crystal with period d ≈ 10 nm has T_B = h/eEd ≈ 1 ps at a few kV/cm. The oscillation was seen in the early 1990s as terahertz emission.
Ultracold atoms in laser lattices. Gravity makes the atoms oscillate, at a few hundred Hz for strontium. The frequency measures small forces.

The sim's fields are far above the breakdown field of bulk Si or GaAs (about 0.3–0.4 MV/cm). They are used here so that one period takes only 83 fs.

Go deeper: the Wannier–Stark ladder

In the quantum picture, the field turns the band into equally spaced levels ħω_B = eEa apart (50 meV at E = 1 MV/cm). The oscillation is the beating between neighbouring levels.
Each level is localised over about W/eEa lattice sites (40 at the defaults), the same distance as the full swing.

Summary

In one line. A steady force makes k grow steadily. Velocity follows the slope of a band that repeats in k, so x returns every T_B = h/eEa, over a full swing of W/eE.

Key points

ħ dk/dt = eE: k advances one zone, 2π/a, in each Bloch period.
v = (1/ħ) dE/dk: velocity follows the band's slope, peaking at Wa/2ħ.
The band repeats in k, so the dot wraps at ±π/a.
T_B = h/eEa contains no W. The full swing W/eE depends on both W and E.

Common mix-ups

"A constant force gives constant acceleration." Only for a free particle. In a band the effective mass changes with k.
"The electron turns back because it collides with something." There is no scattering here. The band repeats in k.
"A wider band means a slower cycle." T_B = h/eEa has no W. A wider band only makes the swing larger.

Things to try on your own

Set E = 5 MV/cm, W = 1 eV. Work out T_B and the swing, then check the readouts.
Find the largest swing the sliders allow, in lattice sites.
At 0.25× speed, pause at the largest Position and read Velocity.

Predict before reveal	Two gated predictions precede the motion and the field sweep.
Counterfactual by contrast	The ghost draws the "no‑band" runaway alongside the real oscillation.
Invariant made visible	Revealed after the inquiry.
Fixed honest scales	Strip (100 nm) and chart (0–80 nm) never autofit, so caging is visible.
One number, one meaning	Tangent slope, velocity chip and packet motion are the same v each frame.