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ID: past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2019/2/d/ii/solution

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Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 2 / d / ii / Solution by

Codex  0 2026-10-25

For

B ∥ [111]

, its squared projection onto the long axis of every valley is

1/3

. All six valleys therefore have the same cyclotron effective mass:

m_{111} = \frac{3 m _{1}^{2} m _{2}}{2 m _{1} + m _{2}} ≃ 0.28 m_{e} .

(1)

At the fixed photon energy,

B_{111} = \frac{m _{111} ω}{e} ≃ 1.4 T .

(2)

Thus the two-peak spectrum is replaced by one peak near

1.4 T

. The source does not determine the linewidth or absolute oscillator strength for the new direction; the sketch below shows only a normalized illustrative line at the computed position.

Figure 1.
Silicon valley geometry and the single resonance for a $[111]$ field
. Original schematic. Left: six electron Fermi surface pockets near the X faces of the face-centred-cubic first Brillouin zone. Middle: a schematic hole surface about $Γ$ after lowering the chemical potential into the valence band; its radius and warping are not inferred from the damaged source. Right: a unit-height illustrative absorption line at the calculated $[111]$ resonance; the width is chosen for readability and is not predicted. **Details**.

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