Theory of Diamagnetism of Bismuth

Abstract

The susceptibility formula for bismuth is derived starting with the general expression for the susceptibility of Bloch electrons. By applying approximations consistent with the Lax two-band model for the energy-band structure near the symmetry point $L$ of the Brillouin zone, the numerous terms in the susceptibility expression were condensed into a very simple form. The "induced diamagnetism" cancels both the Landau-Peierls term and the "crystalline paramagnetism," leaving a very simple and compact expression. The simple expression clearly reveals the interband origin of the large diamagnetism of bismuth and accounts for the experimental results on bismuth and bismuth-antimony alloys. The result can be expressed as the sum of a large background diamagnetism which depends on the direct-energy gap, plus carrier paramagnetism. If the Fermi level is near a band edge, the carrier paramagnetism is equal to the sum of the Landau-Peierls diamagnetism and the Pauli paramagnetism calculated using the effective $g$ factor. For the conduction band (the contribution from valence band differs by sign) the induced diamagnetism in the general formulation for the susceptibility is made up of a paramagnetic contribution due to the second-order influence of the effective $g$ factor in bismuth plus a diamagnetic contribution similar to the standard atomic diamagnetism using the cyclotron effective mass (with the spread of the charge distribution measured by a quantity which plays an analogous role to the Compton wavelength for free electrons). The exact expression (beyond the usual power expansion in $B$ of the one-band-effective-Hamiltonian function (valid near the $L$ point) which yields the correct magnetic energy levels for the Lax model is also obtained.