Energy-level structure of zinc in magnetic fields using a single-band effective Hamiltonian. II. WKB treatment

A map of the zero-magnetic-field electronic-energy-band structure of the midplane of the second band of zinc contains four types of energy contours: (1) circular, (2) lens, (3) triangle, and (4) concave hexagon. The semiclassical theory of Onsager predicts that with a magnetic field imposed the allowed energy levels (Landau levels) correspond to contours whose areas are $\frac{(n+\Theta )eB}{hc}$, where $n$ is an integer and $\Theta$ is a constant. In a recent paper we developed a magnetic effective Hamiltonian $H\left(\overrightarrow{\Pi }\right)={\Sigma }_R\mathcal{E}\left(\overrightarrow{\mathrm{R}}\right)e^{\frac{i\overrightarrow{\mathrm{R}}·\overrightarrow{\Pi }}{\hslash }}(\overrightarrow{\Pi }=\overrightarrow{\mathrm{p}}+\frac{e\overrightarrow{A}}{c})$ where the $\mathcal{E}\left(\overrightarrow{\mathrm{R}}\right)$ are the Fourier coefficients of the zero-magnetic-field energy band. Using exact numerical techniques we computed the energy levels of this effective Hamiltonian for a range of magnetic fields. The results essentially agreed with Onsager's predictions except that the lens and triangle Landau levels were split into groups of broadened bands. In the present paper by allowing one of the components of $\overrightarrow{\Pi }$ to be complex we construct a type of WKB solution to the effective Hamiltonian which involves paths in the complex variable between the energy contours in $\overrightarrow{K}$ space. We compare the WKB results with the results of the exact treatment of the previous paper. For the lens levels the WKB width is consistently about 30% greater than that predicted by the exact treatment. For the triangle levels the WKB widths and the exact widths usually agree within 30 or 40% but there is no definite pattern to the deviation as there was with the lens levels. There are also several interesting qualitative differences—most notably that the WKB method predicts that triangle levels should split into triplets whereas the exact treatment shows these levels split into sextuplets.