Theory of disordered alloys in the tight binding approximation. I

This paper gives a theoretical treatment of the wave functions of electrons in disordered binary alloys in the tight binding approximation (TBA). For simplicity only s-type atomic wave functions are assumed though the results can be qualitatively applied to the d-bands of transition metals. The most important result is that the constituents in the alloy have in general a different number of TBA electrons than in the pure metals. This charging effect depends on the composition of the alloy and the parameter ${\mathrm{\delta }}_0={\mathrm{\epsilon }}_{21}{}^{\prime }/\mathrm{\Delta }$, where ${\mathrm{\epsilon }}_{21}{}^{\prime }$ is the magnitude of the difference of the atomic energies of the two constituents in the alloy and $\mathrm{\Delta }$ is the half-width of the band. The charging effect, in turn, changes the atomic energies of the constituents in the alloy from their values in the pure metals re-quiring a self-consistent solution. We solve in the TBA the simplest possible case which shows a charging effect, that of an ordered alloy with equal composition of the two constituents. The charging effect and ${\mathrm{\delta }}_0$ are numerically found self-consistently for the two cases of equal valency and non-equal valency constituents in the ordered alloy. In this alloy a band gap always appears for any finite value of ${\mathrm{\delta }}_0$. This is not true for a disordered alloy; for this we use a wave function for which scattering can be neglected in the two extreme cases when ${\mathrm{\delta }}_0$ is small or large compared to unity. In the former case there is no band gap while in the latter the band splits into two, each sub-band depending on the properties of only one type of atom. In the former case, with $N_1$ type one atoms and $N_2$ type two atoms, the amplitude of the wave function is about equal on the two types of atoms throughout the band, differing only of order ${\mathrm{\delta }}_0$, and the rigid band model applies. In the latter case the amplitude is always mainly on either one or the other type of atom. These conclusions follow if the values of the band widths of the pure metals are approximately equal. When they differ, the character of the band near the top is determined only by the properties of the atom for which the band is wide in the pure State. This has application to alloys of Pd with Ag and Pt with Au where it is shown that for small concentrations of Ag and Au the specific heat of the alloy is a correct measure of the density of states of the d-band of the pure transition metal. For large concentrations, scattering becomes important, changing the shape of the d-band in the alloy, and explaining the tail found in the experimental results. Orthogonalization of the alloy wave functions insures that for a completely filled band all atoms have the same average electronic charge. However, when the band is not completely full there is the above mentioned net electronic charge difference between the two types of atoms which for small ${\mathrm{\delta }}_0$ is proportional to ${\mathrm{\delta }}_0$. This charge difference is measured by $\mathrm{\Delta }{\mathrm{\rho }}_i$, the difference between the average charge and the charge on the i-th type atom (cf. equation (A10)), and is the only way to produce a difference in electronic charge on the two constituents as may be required by different numbers of d-band electrons per atom. The constant of proportionality between $\mathrm{\Delta }{\mathrm{\rho }}_i$ and ${\mathrm{\delta }}_0$ is largest when the Fermi level of the alloy is near the middle of the band tending to zero as the band empties or fills. Thus alloys whose Fermi level is near the middle of the band are more likely to have small ${\mathrm{\delta }}_0$ and satisfy the rigid band model than those whose Fermi levels are near the top or bottom of the band. It is thïs $\mathrm{\Delta }{\mathrm{\rho }}_i$ which must be solved self-consistently with ${\mathrm{\delta }}_0$ and in a general solution one expects deviations from charge neutrality on a given atom type ($\sigma$ in equation 13') which changes the atomic energy of the TBA electrons. The variation of the atomic energy with $\sigma$ must include, in any real transition metal, effects of screening by the s-electrons, which, as recent investigations show, has a major effect on the energies concerned. The theory neglects all spin dependent interactions. Although only the TBA is treated here, general physical reasoning indicates that the charging effect is present in all types of alloys, and must be included in a self-consistent manner when determining the lattice potential seen by an electron.