Density of states for a disordered system

Abstract

In this paper, the characteristic features of a disordered material is introduced, and multiple scattering theory and tight-binding theory are shown to be able to gives the good results for the density of states in a disordered material. To find the electronic density of states for the system, we use the various approximations such as quasicrystalline approximation (QCA) of Lax, selfconsistent approximation (SCA) of Schwartz and Ehrenreich, and effective medium approximation (EMA) of Roth. We first calculate the density of states of electron for random lattice (no short-range order). For the QCA, we have zero bandwidth: N(E) = $\delta$(E). For the SCA, we have two results depending on the choice of interaction forms, $\sin\; K_0r/r^3$ or $\exp(-\lambda{r}^2)$. The result corresponding to the former shows an asymetrical band form, $N(E) = [4n - (E+n)^2]^{\frac{1}{2}}/[n(1-E)]$, with finite bandwidth, while the result corresponding to the latter shows symmetrical band form, $N(E) = (4n-E^2)^{\frac{1}{2}}/n$, with finite bandwidth. For the EMA, we have a symmetical band,$N(E)=[4nv_o^2-(E - n v_o^2]^{\frac{1}{2}}/n\,v_o^2$, with finite bandwidth. Here, n refers to the density of atoms and $v_o$ to the potential magnitude such that $v(r)= v_o \delta (r)$. Next, we consider the correlation for the short range order system, taking the Gaussian correlation form. We derive the Dyson-like equation with $\vec{k}$-dependent self-energy. Finally we discuss the free electron model by perturbation method, which gives an asymmetrical band, $N(E)= (E - \Sigma)^{\frac{1}{2}}/(\tau^2 -\Sigma+ E)$, with unlimited upper bound. Here, $\Sigma$ refers to the self-energy and we have $\frac{\tau^2}{\tau^2+k^2}$ for cut-off factor with arbitrary number $\tau$.