Condensed Matter II
Problem set #6
Spring 2023
1 Cyclotron resonance of electrons and holes
In this section, we extend the description of cyclotron resonance to an anisotropic
band structure with the following dispersion relation, with ml and mt the longitudinal
and transversal effective masses.
E(⃗k) =
ℏ2
2
k2
x + k2
y
mt
+
k2
z
ml
The magnetic field is assumed to lie in the (x, z) plane, making an angle θ with the
longitudinal axis:
⃗B = B0(sin θ, 0, cos θ)
(i) Give the expression of the effective Hamiltonian, as a function of ml and mt.
(ii) Working in the semiclassical framework, establish the equation of motion obeyed by
the particles, assuming that the fields evolve as exp (iωt).
(iii) Deduce a condition on ω, under which a finite value of ⃗P may exist in the material.
(iv) Discuss how this could be used in order to determine ml and mt experimentally.
2 Donor states in III-V compounds
In the litterature, you may find the following values for the effective mass, the dielectric
constant and the experimental binding energy of shallow donors:
• GaAs: m = 0.07m0, ϵ = 12.5, Ee = 5.8 meV
• InSb: m = 0.014m0, ϵ = 16.8, Ee = 0.6 meV
Use these values in the hydrogenic donor state model, and compute the ionization
energy of the ground state in this model. Compare with experimental data.
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