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 Incoming links: Nearly-free electron dispersion near a one-dimensional band gap

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 36D / Solution Created 2026-09-23 Updated 2026-10-14

Let

G_{n} = 2 πn / a

. The free states

∣ k ⟩

and

∣ k - G_{n} ⟩

are degenerate at the Bragg point

k = G_{n} /2 = nπ / a

. Since

⟨ k ∣ V ∣ k - G_{n} ⟩ = V_{n},

degenerate perturbation theory restricts the Hamiltonian near this crossing to

(ℏ^{2} k^{2} / (2 m) + V_{0} V_{n}^{*} V_{n} ℏ^{2} (k - G_{n})^{2} / (2 m) + V_{0}) .

k = nπ / a + κ

and diagonalizing gives the nearly-free electron dispersion near a one-dimensional band gap

E_{\pm} (k) = V_{0} + \frac{ℏ ^{2}}{2 m} [(\frac{nπ}{a})^{2} + κ^{2}] \pm (\frac{ℏ ^{2} nπκ}{ma})^{2} + ∣ V_{n} ∣^{2} .

Thus the two continuous bands avoid crossing, and at

κ = 0

their separation is

Δ E_{n} = 2∣ V_{n} ∣ .

This is the nearly-free electron model. The dispersion relation is the relation

E = E (k)

between energy and Bloch wavevector.

For the specified potential,

\frac{8}{3} V_{0} cos^{4} (\frac{2 π x}{a}) = V_{0} + \frac{4}{3} V_{0} cos (\frac{4 π x}{a}) + \frac{1}{3} V_{0} cos (\frac{8 π x}{a}) .

Hence

V_{\pm 2} = \frac{2}{3} V_{0}, V_{\pm 4} = \frac{1}{6} V_{0},

and all other nonconstant Fourier coefficients vanish. The gaps are therefore

n 24 crossing wavevectors k = \pm 2 π / a k = \pm 4 π / a gap width 4∣ V_{0} ∣/3 ∣ V_{0} ∣/3.

Their centre energies are respectively

V_{0} + \frac{2 ℏ ^{2} π ^{2}}{m a ^{2}}, V_{0} + \frac{8 ℏ ^{2} π ^{2}}{m a ^{2}} .

The extended-zone sketch consists of the free-particle parabolas shifted upward by

V_{0}

, with avoided crossings of these two widths at the listed Bragg points; elsewhere they meet to this order because the corresponding Fourier coefficients vanish.

Figure 1.
Extended-zone nearly-free-electron bands and their two nonzero gaps
. Gray curves are the intersecting free-electron parabolas. Colored avoided crossings show the larger n equals 2 gap at ka over pi equals plus or minus 2 and the smaller n equals 4 gap at plus or minus 4. Details.

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Past exam of the physics A course of the University of Cambridge / 2021 / Paper 2 / Section C / C8 / c / Solution 2026-10-18

G = G_{1}

,

κ = Δ k

,

U = ∣ V_{1} ∣ > 0

and

E_{B} = ℏ^{2} G^{2} / (8 m)

. The two kinetic energies satisfy

\frac{ϵ _{1} + ϵ _{2}}{2} = E_{B} + \frac{ℏ ^{2} κ ^{2}}{2 m}, \frac{ϵ _{1} - ϵ _{2}}{2} = \frac{ℏ ^{2} G κ}{2 m} .

Consequently the nearly-free electron dispersion near a one-dimensional band gap becomes

E_{\pm} (\frac{G}{2} + κ) = V_{0} + E_{B} + \frac{ℏ ^{2} κ ^{2}}{2 m} \pm U^{2} + (\frac{ℏ ^{2} G κ}{2 m})^{2} .

The Taylor expansion

U^{2} + s^{2} = U + s^{2} / (2 U) + O (s^{4} / U^{3})

gives

E_{\pm} = V_{0} + E_{B} \pm U + (\frac{ℏ ^{2}}{2 m} \pm \frac{ℏ ^{4} G ^{2}}{8 m ^{2} U}) (Δ k)^{2} + O ((Δ k)^{4}) .

For the paper's energy convention, set

V_{0} = 0

. There is no linear or cubic term because each branch is an even function of displacement from the crossing in this two-wave approximation. Each coefficient multiplying

(Δ k)^{2}

has units of energy times length squared, as required.

The expansion requires

\frac{ℏ ^{2} G Δ k}{2 m} ≪ U,

a more restrictive condition than merely

∣Δ k ∣ ≪ G

when the band gap is small. If

U = 0

, the square root is proportional to

∣Δ k ∣

and this analytic expansion of the energy-ordered branches is unavailable.

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Past exam of the quantum condensed matter course of the University of Cambridge / 2019 / Section B / 4 / b / Solution 2026-10-18

Write

U = V_{g}

for the coupling Fourier coefficient, with

∣ U ∣ = ∣ V_{0} ∣

for the supplied cosine potential. Projecting the Schrödinger equation onto the two retained plane waves gives

(ϵ_{q + g /2} U^{*} U ϵ_{q - g /2}) (A B) = E (A B) .

The diagonal potential average is zero. The characteristic equation is

(E - ϵ_{q + g /2}) (E - ϵ_{q - g /2}) - ∣ U ∣^{2} = 0,

so the nearly-free electron dispersion near a one-dimensional band gap is

E_{\pm} (q) = \frac{ϵ _{q + g /2} + ϵ _{q - g /2}}{2} \pm \frac{1}{2} (ϵ_{q + g /2} - ϵ_{q - g /2})^{2} + 4 V_{0}^{2} .

Only the coupling magnitude enters the energies; its sign changes the standing-wave eigenvectors. The two-wave truncation is controlled near this crossing when

∣ V_{0} ∣ ≪ ϵ_{g /2}

, since the other plane waves are separated by much larger kinetic energies.

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