Effect of the Surface on the Magnetic Properties of an Electron Gas

The energy levels of free electrons confined to a finite cylindrical box with a uniform axial magnetic field are obtained by the WKB approximation and used to compute the magnetic susceptibility with Fermi statistics. The usual treatments which neglect the effects of the walls are shown to be justified for both the steady susceptibility and the de Haas van Alphen terms provided the radius of the box is sufficiently large. In the case of the oscillatory terms it is only necessary that the radius of the box exceed the classical orbit radius $R_c$ of an electron having the Fermi energy $\zeta$ in the magnetic field. However, there exists a surface correction to the steady susceptibility whose magnitude relative to the Landau value is ${\left(\frac{\zeta }{\beta H}\right)}^{\frac{1}{3}}\left(\frac{R_c}{R}\right)$. This surface correction, the existence of which has been previously pointed out by Osborne and Steele, and by Dingle, is shown to be extremely sensitive to the exact boundary conditions at the surface, including both the abruptness of the surface jump in potential and the height of the barrier relative to the Fermi energy. Indeed, the correction term can be either paramagnetic or diamagnetic depending on these details. The form of the WKB approximation appropriate to different boundary conditions is discussed, and a modification of Dingle's theory is presented which may be used to calculate approximately the susceptibility of the system for any value of the ratio $\frac{R}{R_c}$ when the boundary conditions are known.