Girish Mahajan (Editor)

Rigid band model

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The Rigid-Band Model (or RBM) is a model used by condensed matter physicists to describe the behavior of metal alloys. In some cases the model is even used for non-metal alloys such as Si alloys. The basic idea of the Rigid-Band model is to assume that the shape of the density of states curves and the shape of the constant energy surfaces (and therefore also the Fermi surface) of the alloy is the same as that of the solvent metal. The only change the solute makes, given that the valency of the solute metal is greater than that of the solvent, is to make the Fermi surface bigger and to fill the DOS to a higher level. Because of the simplicity of the Rigid-Band Model it is often the first theoretical framework physicists use to make estimates of the electric properties of an alloy. There are, however, many alloys for which the RBM fails to describe the electric properties.

Contents

Theory

The Rigid-band model makes a few important simplifying assumptions about the alloys it tries to describe. It first of all assumes that the shape of the Fermi surface and the shape of the density of states of the alloy are identical to that of the solvent metal. With other words, the solute doesn’t make any (local) perturbations in these two. It is however not uncommon to change the shape of the electronic structure to the weighted average of the constituents. The RBM also assumes that the volume of the material does not change significantly upon alloying.
Before giving a more mathematical outline of the RBM it may be convenient to give somewhat of a visualization of what happens to a metal upon alloying it. In a pure metal, say for example silver, all lattice sites are occupied by silver atoms. When you dissolve, say, 10% copper into it, some random lattice sites will now be occupied by copper atoms. Since silver has a valency of one and copper has a valency of 2, the alloy will now have a valency of 1.1. Since most lattice sites are still occupied by silver atoms, the changes in electronic structure will be minimal.
It turns out that under certain conditions the rigid-band model is very successful in predicting many features of the concerned alloys, such as the Hume-Rothery rules and the changes in axial ratios in hexagonal crystal structure. These conditions are:

  • The excess charge of the solute atoms remains in the vicinity of these atoms.
  • The mean free path of the electrons is much greater than the lattice spacing of the alloy.
  • The energy bands of the solvent near the Fermi energy are widely separated from all other bands.

    • Geometric structure

    In alloys, as long as the fractional of the solute, α , is small, the Bloch wave function ϕ k with energies ϵ k are continuously replaced by functions Ψ k with energies E k . These energies are given by

    E k = ϵ k + Δ E k   .

    As stated in the previous section, one of the assumptions made in the RBM is that Δ E k is not a function of k but only of ϵ k :

    Δ E k = Δ E k ( ϵ k )   .

    References

    Rigid-band model Wikipedia
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