The convolution method for calculations of local densities of states

Abstract

The convolution method for the calculation of local densities of states is presented more thoroughly along with its expression in terms of Green functions. This constructive approach allows us to produce results for a higher dimensionality from lower-dimensional parts. Its applications and different aspects are discussed for some simple cases.

Local densities of states for some embedded structures

“…The convolution method is applicable for Hamiltonians that can be written in some representation as H = i H i (k i ). It was shown [1][2][3] that in such cases the LDOS, ρ(E), can 0953-8984/04/040605+07$30.00 © 2004 IOP Publishing Ltd Printed in the UK be obtained through the recurrence ρ (D) = ρ (D−1) * ρ (1) , (1) where D = 1, 2, 3 is the dimensionality of the system and ρ (0) is the delta function δ(E). This result can be reached directly, relying only on the form (i.e.…”

“…Writing an index 0, 1, 2 to it, we mean a single point impurity, a whole chain or a plane. Thus, for instance, we write ρ (3) (u 1 ) for the LDOS of a chain embedded in a crystal or ρ (2) (0) for a perfect lattice. In matrix notation u has to be interpreted with respect to the dimensionality of the perturbing heterogeneous potential.…”

“…The result of its convolution with ρ (D) (0) would be then where D = 1 or 2 and this suggests how the results would look: the usual curve with a displaced lower-dimensional one added. In the two-dimensional case ρ (2) (u 1 ) = ρ (1) (u 0 ) * ρ (1) (0), the result corresponds to a chain embedded in a lattice. Its plot in figure 2(b) has been obtained by convoluting the LDOS of a chain with an impurity (figure 2(a)) with an unperturbed LDOS.…”

“…The last possible combination here is a lattice embedded in a volume. Its LDOS is obtained either from ρ (3) (u 2 ) = ρ (2) (u 1 ) * ρ (1) (0) or, as the convolution is associative, from ρ (3) (u 2 ) = ρ (1) (u 0 ) * ρ (2)…”

“…Recently we proposed and demonstrated a convolution method for the calculation of the local density of states (LDOS) [1,2]. It has been shown to be applicable to relatively simple structures including some quasicrystals and other cases of disorder.…”