Condensed Matter II
Problem set #9
Spring 2023
Hall effect
Isotropic semiconductor
In this section, we study the Hall coefficient in a semiconductor in the presence of both
hole and electron charge carriers:
• electron density ne
• hole density nh
• electron mobility µe
• hole mobility µh
(i) Recall the expression of the mobility as a function of relaxation time and effective
mass, as well as the expression of the cyclotron frequency.
(ii) Assuming B = 0.1 T and m∗ = m0, What is the critical relaxation time τc below
which the weak magnetic field approximation is valid?
(iii) Assuming µe = 1000 cm2 V−1 s−1, is this approximation valid?
(iv) Derive the expression of the resistivity matrix, considering a single carrier species.
(v) Take into account the presence of electrons and holes, and deduce the expression for
the Hall coefficient in the material.
Anisotropic case
In this section a single kind of charge carriers (electrons) is present, but its effective
mass tensor is anisotropic. Consider a degenerate n-type material with 1017 electron
carriers per cm3 in the conduction band. The electrons occupy conduction states associated
with the 6 electron carrier pockets of Si. Such carrier pockets are characterized by the
mass components mt = 0.19m0 and ml = 0.98m0. ⃗B is along the z axis, ⃗j is along the x
axis, and the relaxation time is τ.
(i) By analogy with the derivation in the isotropic case, establish the expression of the
contribution of one electron pocket to the conductivity tensor.
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(ii) Deduce from the result of the previous question, the expression for the total conductivity
tensor.
(iii) Express the Hall coefficient as a function of the components of the total conductivity
tensor, and B.
(iv) Assume the weak field approximation is valid, and deduce the expression of the Hall
coefficient as a function of the effective masses, and of the isotropic Hall coefficient.
(v) Derive the expression of the transverse magnetoresistance, and comment on its mag-
nitude.
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