Typical-medium multiple-scattering theory for disordered systems with Anderson localization

The typical medium dynamical cluster approximation (TMDCA) is reformulated in the language of multiple scattering theory to make possible first-principles calculations of the electronic structure of substitutionally disordered alloys including the effect of Anderson localization. The TMDCA allows for a systematic inclusion of nonlocal multisite correlations and at the same time provides an order parameter, the typical density of states, for the Anderson localization transition. The relation between the dynamical cluster approximation and the multiple scattering theory is analyzed, and is illustrated for a tight-binding model.

Figure 1

The DCA self-consistency loop. The arrows correspond to the steps taken in the DCA algorithm described in Sec. 2b.

Figure 2

An alternative DCA-II self-consistency algorithm with separate local and nonlocal contributions to the self-energy.

Figure 3

The self-consistency loop of the KKR-NLCPA algorithm I presented in Sec. 2c1.

Figure 4

The KKR-NLCPA algorithm II self-consistency loop applied to the tight-binding model.

Figure 5

Multiple-scattering DCA: a Green's function based multiple scattering algorithm where we calculate the disorder-averaged Green's function instead of the scattering-path operator.

Figure 6

MS-TMDCA: a Green's function based typical medium multiple scattering algorithm.

Figure 7

Comparison of the imaginary parts of the cluster excluded Green's function $\mathcal{G}(\mathbf{K},\omega )$ from the DCA procedure and ${\overline{G}}_0^{{}^{\prime }}(\mathbf{K},\omega )+{\overline{\mathrm{\Delta }}}^{{}^{\prime }}(\mathbf{K},\omega )$ calculated in the KKR-NLCPA procedure for $N_c=38$ at disorder strength, ${\mathrm{V}}_A=0.7$ $\left(V_B=-V_A\right)$ and concentration, ${\mathrm{c}}_A=0.5$. The solid lines depict the DCA results while the dash lines are the corresponding KKR-NLCPA and their associated momenta $\mathbf{K}$ correspond to each of the eight distinct cells obtained using the point-group symmetry of the cluster.

Figure 8

A numerical comparison of the local self-energy $\frac{1}{N_c}{\sum }_{\mathbf{K}}\overline{\mathrm{\Sigma }}(\mathbf{K},\omega )$ obtained from the DCA procedure and the local KKR-NLCPA self-energy $\frac{1}{N_c}{\sum }_{\mathbf{K}}{[t_l^{-1}\left(\omega \right)-\overline{\alpha }\left(\mathbf{K}\right)]}^{-1}$. Here, solid lines are the DCA results and dash lines are their KKR-NLCPA counterparts.

Figure 9

A numerical comparison of the $\mathbf{K}$-resolved self-energy $\overline{\mathrm{\Sigma }}(\mathbf{K},\omega )$ obtained from the DCA procedure and the KKR-NLCPA self-energy ${[{\overline{t}}_l^{-1}\left(\omega \right)-\overline{\alpha }\left(\mathbf{K}\right)]}^{-1}$ at $V_A=0.7$ $\left(V_B=-V_A\right)$ and $c_A=0.5$. We observe again that two procedure are numerically equivalent. Here, the solid lines are the DCA results while the dashed lines depict their KKR-NLCPA counterparts. Note, the labels A–D and their associated momenta $\mathbf{K}$ depict each of the four distinct cells obtained using the point-group and particle-hole symmetry $[\overline{\rho }(\mathbf{K},\omega )=\overline{\rho }(\mathbf{Q}-\mathbf{K},-\omega )$, with $\mathbf{Q}=(\pi ,\pi ,\pi )]$ of the cluster.

Figure 10

The arithmetically averaged density of states (ADOS) obtained from the DCA and KKR-NLCPA calculations. We present data for the binary alloy disorder with $c_A=0.5$, for cluster size $N_c=38$ at small disorder $V_A=0.2$ and larger disorder $V_A=0.7$ $\left(V_B=-V_A\right)$. The results show that the two procedures are equivalent when applied to the tight-binding model.

Figure 11

ADOS calculated in DCA and KKR-NLCPA procedures and TDOS calculated in MS-TMDCA and TMDCA procedures. (a) For $V_A=0.6,0.65,\text{and}0.7$ at $c_A=c_B=0.5$. (b) For $c_a=0.1,0.25,\text{and}0.5$ at $V_A=-V_B=0.7$