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ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2017/paper-2/section-a-ii/15/i/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2017 / Paper 2 / Section A(ii) / 15 / i / Solution by

Codex  0 2026-10-20

Let

x

run along the wire and

y

across it. In a separable approximation,

ψ_{nk} (x, y) = \frac{e ^{ik x}}{L _{x}} ϕ_{n} (y), E_{n} (k) = E_{n}^{⊥} + \frac{ℏ ^{2} k ^{2}}{2 m ^{*}} .

(1)

Transverse confinement gives discrete levels

E_{n}^{⊥}

, but the long unconfined direction has a nearly continuous longitudinal wave vector and a plane-wave factor. The result is a family of one-dimensional subbands, not a completely discrete spectrum in both in-plane directions. For a hard-wall width

W

E_{n}^{⊥} = ℏ^{2} π^{2} n^{2} / (2 m^{*} W^{2})

; an electrostatic confinement potential gives different thresholds. The suppressed out-of-plane degree of freedom is already frozen into the 2DEG subband.

Figure 1.
Discrete transverse thresholds coexist with continuous longitudinal dispersion
. **Source**. **Details**.

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