Configuration and self-averaging in disordered systems

Abstract

The main aim of this work is to present two different methodologies for configuration averaging in disordered systems. The Recursion method is suitable for the calculation of spatial or self-averaging, while the augmented space formalism averages over different possible configurations of the system. We have applied these techniques to a simple example and compared their results. Based on these, we have reexamined the concept of spatial ergodicity in disordered systems. The specific aspect, we have focused on, is the question “Why does an experimentalist often obtain the averaged result on a single sample?” We have found that in our example of disordered graphene, the two lead to the same result within the error limits of the two methods.

Appendix

It is interesting to understand why different methods have different energy resolutions. Periodicity and the Bloch Theorem leads to the labeling of quantum states by a real vector $\vec {k}$ and a supplementary band label ‘n’.

$$\begin{aligned} \langle \vec {k},n\vert G(z) \vert \vec {k},n\rangle = \frac{1}{z-E_n(\vec {k})} \end{aligned}$$

So that:

$$\begin{aligned} -(1/\pi ) \sum _n Im G(E+i0,\vec {k},n) = \sum _n \delta (E-E_n(\vec {k})) \end{aligned}$$

For a fixed $\vec {k}$ this is a set of delta functions. If we abandon periodicity and adopt the augmented space formalism:

$$\begin{aligned} -(1/\pi ) \sum _n Im \ll G(E+i0,\vec {k},n)\gg\,= -(1/\pi ) \sum _n Im \frac{1}{E-E_n(\vec {k})- \Sigma _r(E,\vec {k})-i\Sigma _{im}(E,\vec {k})} \\= \sum _n \frac{[1/\pi \tau (E,\vec {k},n)]}{[(E-E'_n(\vec {k})]^2 + [(1/\tau (E,\vec {k},n)]^2} \end{aligned}$$

This spectral density comes out to be much smoother and the smoothening lifetime has its origin in scattering by disorder fluctuations. Moreover, the lifetime emerges out of the calculations, and no external broadening or smoothening factors are required.