Electron Energies in a 1D Crystal

Electron energies in a 1D crystal

Teach me

E–k curve free-electron parabola

Guided Inquiry

1 · A wire, atoms off

Let’s get our bearings. On the left: a row of atoms with the periodic potential beneath. On the right: energy E versus wavevector k. With the wells off (P = 0), the E–k curve is exactly the smooth free-electron parabola you’d expect.

2 · Switch the wells on

Now let’s switch the atoms on. Suppose you raise the well strength P from zero — what do you think happens to the allowed electron energies?

3 · Drag P up

Let’s see how your prediction holds up. Drag the well-strength slider up from 0. Where do forbidden strips appear — and do they appear anywhere else? Keep an eye on the Gap-width panel as you go.

4 · Gaps at the zone boundaries

Here’s what you just found. Gaps open only where k = ±nπ/a — the Bragg planes — and nowhere in between. Deeper wells (larger P) widen them. Between the gaps, allowed bands remain, and the gap’s lower edge stays pinned on the parabola.

5 · The two edges of one gap

Now zoom in on a single gap. The states just below and just above gap 1 are both standing waves of the SAME wavelength λ = 2a. So how do you think the upper state earns its extra energy? Make your prediction.

6 · Click both edges

Let’s take a look. Click the E–k curve just below gap 1, then just above. The left panel fills with each state’s |ψ|². Compare where the density piles up relative to the atom row.

7 · Same wave, different place

Did you spot it? Lower edge: density sits ON the atoms, where the potential is low. Upper edge: density sits BETWEEN them, on the barriers, where it’s high. Same λ, same k — the gap IS that potential-energy difference.

That’s also why the lower edge stays pinned to the free-electron parabola at every P: with a node on every barrier, the sine wave never feels the barriers at all.

8 · Why a gap can exist

So here’s why a gap can exist at all. It equals the cost of relocating the same standing wave from atoms to barriers — so it grows with P. No third placement exists, so no in-between energy is allowed.

Now try squeezing the atoms closer with the spacing slider, and watch the axis. What happens to every gap energy?

One last thought to take away. A semiconductor is this crystal with a small gap 1 — so which placement do its highest electrons occupy, and what must a photon supply to move one across?

Listen

Well strength P0.0

0 (free electron)10 (deep wells)

Atom spacing a0.40 nm

0.20 nm0.60 nm

Kronig–Penney (delta-comb). P is the dimensionless barrier strength; α is the wavenumber inside a cell. Allowed α are those where cos(αa) + P·sin(αa)/(αa) lies in [−1, 1] — only there does a real k exist.

Energy of a state with cell-wavenumber α. Zone boundaries live at k = nπ/a.

The lower band edge of gap n, at the zone boundary k = nπ/a.

The gap-edge wavelength for gap n.

The Physics

Kronig–Penney bands · what this sim shows, the physics behind it, and every control

At a glance

This sim follows one electron along a row of identical atoms and plots which energies it may have at each wavevector k. That plot is the starting point for why a solid conducts, insulates, or sits in between.

What you'll learn

How one condition ties an allowed energy to a wavevector k in a periodic potential.
How the barrier strength P and the atom spacing a set the energies the model allows.
How to read a state's probability density |ψ|² against the row of atoms and barriers.
Which parts of this one-dimensional model carry over to real crystals, and what it leaves out.

The model: an infinite row of atoms, spacing a, with a thin barrier of strength P between neighbours.

What's on screen

Teal curve (right panel): energy E against wavevector k for every allowed state.
Slate parabola: E = ħ²k²/2m, the energy of a free electron.
Rose strips, labelled gap 1, 2, 3: the forbidden energy intervals. Dashed vertical lines mark k = ±π/a, ±2π/a, ±3π/a.
Left panel: grey atoms, rose barrier spikes whose height follows P. A clicked state adds its |ψ|² there, green for a lower edge and amber for an upper edge.
Readouts: Gap width, and the Selected state's energy and wavelength.

See itSwitch the wells off, P = 0, and look at the right panel. Watch for: only the teal curve visible, because it covers the slate parabola exactly.

The controls

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

Control	What it changes	Range
Well strength P	The strength of the barrier between neighbouring atoms, and so the height of the rose spikes. P = 0 removes the barriers.	0–10 opens at 0 in the inquiry, 4 in free exploration
Atom spacing a	The distance between atoms. It rescales the energy axis; the shape of the picture stays the same.	0.20–0.60 nm opens at 0.40 nm
Click on the E–k curve	Selects the nearest gap edge, one of six (the lower and upper edge of gaps 1–3), and draws its \|ψ\|² in the left panel.	click once the inquiry has revealed it
Readouts	Gap width: the width of the selected state's gap in eV (gap 1 until you select another). Selected state: its energy and wavelength.	panel
↻ Reset	Restarts the mode you are in, with P and a back at their opening values.	top bar

Fixed: an infinite chain of identical delta-function barriers, one electron, no temperature. The plot covers k from −3π/a to 3π/a, so it shows three gaps. There is no Play or Speed control because nothing moves.

One condition sets the allowed energies

Between barriers the electron is free, with energy E = ħ²α²/2m, where α is its wavenumber inside one cell. The periodic barriers tie α to the Bloch wavevector k, which labels the state:

The Kronig–Penney condition for a delta-function comb. P is the dimensionless barrier strength.

A real k needs the right-hand side to lie between −1 and 1. So an energy is allowed only when:

Energy of a state with cell-wavenumber α, and the test the sim applies before it plots a point.

P = 0: the right side is cos αa, so every energy has a real k and the curve is the free-electron parabola.
P > 0: the right side can exceed 1 in size. Where it does, no real k exists and the infinite crystal has no state at that energy. No electron is removed; those energies simply cannot propagate.

See itSwitch the wells on at P = 3. Watch for: rose strips across the right panel, with the teal curve jumping across each one.

Gaps open at the zone boundaries

A lattice of spacing a can change a wave's k only by a reciprocal-lattice vector G_n = 2πn/a. Half of G_n is the zone boundary:

Zone boundary, reciprocal lattice, Bragg condition: three names for the same k.

At k = nπ/a the scattered wave lands on −nπ/a, with the same energy as the wave that came in. The two travelling waves cannot stay separate; they combine into two standing waves with different energies. The difference is the gap.
In wavelength terms this is Bragg reflection: 2a = nλ, so the echo from every cell returns in phase.
At other k the echo is off resonance. For weak wells the curve follows the parabola and splits only at the dashed lines.
The sim draws the extended zone: band n sits on (n−1)π/a < |k| < nπ/a, not folded into the first zone.

See itUse weak wells, P = 1. Watch for: the teal curve staying close to the slate parabola and breaking only where it crosses the dashed lines.

What sets the gap width

The lower edge of gap n sits at αa = nπ for every P. Only the upper edge moves, and the gap is the difference:

The upper edge α⁺ is the first allowed α above nπ/a: the condition from section 03, met with equality.

P a

These sliders move the sim too.

Weak wells. All gaps open to nearly the same width, 4PE₀ with E₀ = ħ²/2ma², because a delta comb has the same Fourier amplitude at every G. At P = 0.1 and a = 0.40 nm the three gaps are 0.094, 0.095 and 0.095 eV.
Stronger wells. Gaps grow more slowly than P, and the bands narrow. Gap 1 is 0.86 eV at P = 1 but only 4.36 eV at P = 10, not 8.6 eV. Band 1 shrinks from 0–2.35 eV at P = 0 to 1.64–2.35 eV at P = 10.
The spacing. Every energy is ħ²(αa)²/2ma², and each αa depends on P alone. At fixed P every energy and gap therefore scales as 1/a².

The weak-well limit, P ≪ 1.

See itTake gap 1 at P = 1 and then at P = 10. Watch for: the Gap width readout going from 0.863 to 4.357 eV, about five times larger for a tenfold stronger well.

See itTake gap 1 at a = 0.40 nm and then at a = 0.20 nm, with P = 3. Watch for: the Gap width readout going from 2.154 to 8.617 eV, while the picture keeps its shape and only the energy axis rescales.

Two edge states, one wavelength

The two states on either side of gap n both have k = nπ/a, so both repeat over λ = 2a/n. They differ in where their density sits relative to the barriers.

The two states bracketing gap n, for 0 ≤ x ≤ a with x measured from a barrier.

Gap 1 at P = 3. Spikes are the barriers, circles the atoms. At k = π/a, ψ changes sign from one cell to the next, so |ψ|² repeats every a.

Lower edge. A sine with a node on every barrier, so it never feels the potential. Its energy is exactly the free-electron value, which is why it stays on the parabola at every P. For gap 1, 81.8% of its probability (exactly ½ + 1/π) lies in the half-cell centred on an atom, at any P.
Upper edge. Its density does not vanish at the barriers. For gap 1 it has a node on every atom and 74% of its weight in the half-cell around a barrier (at P = 3), with a notch at the barrier itself.
Same k, same repeat length, different placement. The upper state pays for sitting where the potential is high; the lower state pays nothing.

See itMark both edges of gap 1 at P = 3. Watch for: the lower-edge and upper-edge markers sitting at the same k = π/a, one above the other.

See itSelect the lower edge of gap 1, then the upper edge. Watch for: the State line changing from "ψ on atoms" to "ψ on barriers" while λ stays 0.800 nm.

Where it matters

Measured 300 K gaps of real 3-D crystals beside this model's gap 1 at P = 3, a = 0.40 nm. The mechanism carries over; the number does not.

Semiconductors. The gaps of Ge (0.67 eV), Si (1.12), GaAs (1.42) and diamond (5.47) come from the same Bragg splitting at a zone boundary. Silicon's 1.12 eV matches a photon of about 1.1 µm, so silicon is transparent beyond that wavelength.
Metal or insulator. That depends on how many electrons fill the bands. This sim draws only the allowed energies.
Layered optics. A stack of alternating transparent layers is periodic too, and it reflects a band of wavelengths for the same Bragg reason.

Go deeper: what the gap is made of

The lower edge feels no potential, so the gap is the upper state's potential energy plus its extra kinetic energy. From the wavefunctions the sim draws, the potential share is about 95% at P = 0.5, 74% at P = 3 and 42% at P = 10.
A real crystal is three-dimensional. Gaps open at different energies in different directions, and a material is an insulator only if some energy range is forbidden in every direction.

Summary

In one line. A periodic potential splits the standing wave at each zone boundary into two states of the same wavelength but different energy; the gap is that difference, and no propagating state exists inside it.

Key points

An energy is allowed only if |cos αa + P sin αa / αa| ≤ 1.
Gaps open at k = ±nπ/a. The lower edge stays on the free-electron parabola; the upper edge rises with P.
Both edges share λ = 2a/n and differ in where the density sits. At fixed P, all energies scale as 1/a².

Common mix-ups

"A gap is a stretch of the curve where the electrons are missing." No electron is removed. There is no real k at those energies, so no propagating state exists.
"More energy must mean a shorter wavelength." Both edges of gap 1 repeat over λ = 2a. The extra energy comes from where the density sits.
"Stronger wells push both edges of a gap up." Only the upper edge moves; the lower edge stays on the parabola.

Things to try on your own

Set P = 0.1 and compare all three gap widths with 4PE₀.
Set a = 0.60 nm and P = 3. Predict gap 1 from its value at 0.40 nm, then check the readout.
Set P = 10 and read where band 1 starts and ends on the energy axis.

Predict before reveal	Gap-width and density readouts stay hidden until after each prediction is committed.
Commit before feedback	Each prediction locks in before the simulation responds; feedback addresses the choice actually made.
Read, don't assert	Selected states report their energy and wavelength exactly as computed, so comparisons are read off the screen, not claimed.
Controlled quantities respond	P grows the barrier spikes; a re-spaces the atoms and rescales the energy axis (∝ 1/a²).
Exact physics	Bands solve the transcendental equation; densities are the exact standing waves; numbers match the audit hook.