General theory of magnetic-field-induced surface states

When an external magnetic field is applied to the surface of a normal metal, the group of electrons skipping along the surface by repeated specular reflections may be described as being bound in the triangular potential well formed by the surface potential barrier and the Lorentz force impelling the electrons to the surface. Previous determinations of the energy levels of an electron in this potential well by Prange and Nee (1968) have assumed an infinite surface potential barrier, V e , which requires the wavefunction to vanish (ψ=0) at the surface. The theory is generalized in this paper to include the effect of a finite V 0 , for which we find a different (gauge-invariant) boundary condition : (-ih Δ + eA/c)ψ. n=0, A being the magnetic vector potential and n the unit normal to the surface, to give a closer approximation to the true phase of ψ in a real crystal. Although the energy spectra given by the two boundary conditions are quite different, the predictions of the resonance peaks of the oscillatory microwave surface impedance associated with the transitions between the energy levels are close to each other and in good agreement with experiment in bismuth. The induced surface magnetization due to the surface states is also calculated.