A metal conducts through one electron band and one hole band

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Electronic transport in solids · Electronic transport in solids

Solid State Physics, Ch The Boltzmann equation and electrical transport

Problem

A metal conducts through one electron band and one hole band, each an isotropic Boltzmann band with its own density and mobility,

n_e,\mu_e

and

n_h,\mu_h

. With

\mathbf{B}

along

z

, each band on its own has the conductivity tensor of a single band,

\sigma^{(i)}=\frac{\sigma_i}{1+\beta_i^2}\begin{pmatrix}1&\mp\beta_i\\ \pm\beta_i&1\end{pmatrix},\qquad\sigma_i=n_ie\mu_i,\quad\beta_i=\mu_iB,

with the upper signs for the electron band and the lower signs for the hole band. ((a)) Add the two tensors, invert the sum, and show that the transverse magnetoresistance is

\frac{\Delta\rho}{\rho}\equiv\frac{\rho_{xx}(B)-\rho_{xx}(0)}{\rho_{xx}(0)}=\frac{n_en_h\mu_e\mu_h(\mu_e+\mu_h)^2B^2}{(n_e\mu_e+n_h\mu_h)^2+(n_h-n_e)^2\mu_e^2\mu_h^2B^2}.

((b)) Find the high-field limit of

\Delta\rho/\rho

when

n_h\neq n_e

, and show that it grows as

B^2

without limit when

n_h=n_e

. ((c)) Aluminium, treated as in the worked example on its Hall coefficient, has

1.06

holes and

0.06

electrons per atom, with

\mu_e=5\mu_h

. Evaluate its saturated

\Delta\rho/\rho

at high field.

Browse the problem library