Introduction

The response of a metal to a magnetic field, $H$, in the $z$-direction say, has given considerable information about the Fermi surface, because electrons follow a trajectory on the Fermi surface in a plane of constant $k_z$. In these notes we discuss how free electron states are modified by a magnetic field, and how this affects the energy of the system in an oscillatory manner, provided certain conditions, given in Eqs. (10) and (11) below, are satisfied.

As is shown in standard texts on quantum mechanics, the energy levels of free electrons in a magnetic field are modified from

$$\epsilon({\bf k}) = {\hbar^2 k^2 \over 2 m}$$ (1)

to

$$\epsilon(n_L, k_z) = {\hbar^2 k_z^2 \over 2m} + \left( n_L + \mbox{\small$1 \over 2$} \right) \hbar \omega_c ,$$ (2)

where $n_L = 0, 1, 2, \cdots$ and

$$\omega_c = {e H \over m c}$$ (3)

is the cyclotron frequency, i.e. the frequency of the classical motion. The states with a given $n_L$ are known collectively as a Landau level. The degeneracy of a Landau level is

$${\cal N} = 2 {e \over h c} A H = {A H \over \phi_0} ,$$ (4)

where $A$ is the area of the sample in the plane perpendicular to $H$, the factor of two in the first expression comes from spin degeneracy, and

$$\phi_0 = { hc \over 2 e} = 2.07 \times 10^{-7} \mbox{\rm G-cm}^2$$ (5)

is the flux quantum for a pair of electrons.