The effective bandstructure (EBS) method as introduced by, amongst others, V Popescu and A Zunger (Phys. Rev. Lett. 104, 236403 (2010)), amounts to an attempt to preserving some characteristics of periodic systems in the cases where the system under consideration has no translational symmetry, as in amorphous solids, whereby momentum is not a good quantum number. The effective bandstructure calculated on the basis of this method is described in terms of a single-particle spectral function A(k,ε), which depends on the reduced wave vector k and the energy ε. This description applies also to ordinary bands in that as, for a given k, one moves along the ε axis, one encounters a sequence of Dirac δ-functions, each δ-function signifying a band energy at the given wave vector k. By tracing these δ-functions on moving from one k point in the Brillouin zone to another, one recovers the ordinary bandstructure.

In the cases where one has no translational symmetry but calculates the function A(k,ε) along the lines described in the above-mentioned reference for k varying over the first Brillouin zone corresponding to the primitive unit cell from which the underlying large super cell consists, one tends to get clusters of δ-functions centred around the δ-functions associated with the periodic crystal whose primitive unit cell one considers. Ideally, if the amorphous material were to be simulated on an infinitely large super cell, the above-mentioned clusters of δ-functions corresponding to an arbitrary k would transform into a continuous distribution function which consists of prominent peaks of finite widths. In practice, where calculation on an infinitely large super cell is not feasible, one can arrive at a continuous distribution function by artificially broadening each δ-function peak at any given k. By tracing out for instance the expectation values of the positions of the isolated clusters of the δ-functions in A(k,ε) along the ε axis for a dense set of k points, one arrives at an effective bandstructure.

In calculations on alloys, one calculates the average of the spectral functions {Aι(k,ε) | i=1,2,...} corresponding to different realisations of the alloy under consideration. This averaging results in a that consists of a more dense set of δ-functions along the ε axis for any given k in comparison with the density of the δ-functions associated with one Aι(k,ε). Consequently, for obtaining a continuous distribution function from the ensemble average corresponding to a finite super cell, the amount of artificial broadening of each δ-function is smaller than the amount required for the Ai(k,ε) corresponding to one realisation of the alloy.

Naturally, the above formalism can also be applied to super cell calculations, where the finite super cell has the appropriate size so that its finiteness is not an approximation. By calculating the effective bandstructure over the first Brillouin zone corresponding to the primitive unit cell, one obtains a less complicated bandstructure, the complexity of the actual bandstructure having been accounted for by the finite widths of the peaks in the calculated A(k,ε).  □

Dear Dr. Farid,

Thank you very much for your detailed answers. It is clear to me now.

Best,

Raihan