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ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2019/paper-2/section-b/21/v/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2019 / Paper 2 / Section B / 21 / v / Solution by

Codex  0 2026-10-20

With ballistic conductance quantization, as a gate opens the channel, conductance plateaux at $G = N_{b} (2 e^{2} / h)$ appear; successive transverse modes add one quantum, approximately

77.5 μ S

per spin-degenerate subband. Their appearance is the key low-temperature electrical signature.

Figure 1.
Ideal conductance plateaux sharpen as thermal and disorder broadening become small compared with subband spacing
. **Source**. **Details**.

Require transverse confinement with separated subbands,

k_{B} T

and measurement-bias energy well below the subband spacing, a mean free path exceeding the channel length, and smooth low-reflection coupling to reservoirs. Correct for appreciable external series resistance. Phase coherence is needed for additional interference oscillations, but perfect phase coherence across an entire macroscopic circuit is not a separate requirement for the basic mode-counting result. A field that resolves spin splitting can produce

e^{2} / h

increments.^[ref]

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