Effects of short-range order on the electronic structure of disordered metallic systems

For many years the Korringa-Kohn-Rostoker coherent-potential approximation (KKR-CPA) has been widely used to describe the electronic structure of disordered systems based upon a first-principles description of the crystal potential. However, as a single-site theory the KKR-CPA is unable to account for important environmental effects such as short-range order (SRO) in alloys and spin fluctuations in magnets, among others. Using the recently devised KKR-NLCPA (where NL stands for nonlocal), we show how to remedy this by presenting explicit calculations for the effects of SRO on the electronic structure of the bcc ${\mathrm{Cu}}_{50}{\mathrm{Zn}}_{50}$ solid solution.

Figure 1

Example (bcc lattice): (a) Cross section of the conventional real-space tiling with Wigner-Seitz cells. The shaded sites lie outside of the page. (b) Cross section of the conventional reciprocal-space tiling with Brillouin zones. The shaded sites lie outside of the page. (c) Cross section of the real-space tiling (dashed lines) with $N_c=2$. (d) Cross section of the reciprocal lattice points and reciprocal-space tiles (dashed lines) of the coarse-grained system. (e) Cross section of a real-space tile (dashed line) for a $N_c=2$ cluster containing the points ${\mathbf{R}}_1=(0,0,0)$ and ${\mathbf{R}}_2=(a∕2,a∕2,a∕2)$. The shaded sites lie outside of the page. (f) Cross section of the corresponding reciprocal-space tiles (dashed lines) for the $N_c=2$ cluster, with ${\mathbf{K}}_1=(0,0,0)$ and ${\mathbf{K}}_2=(2\pi ∕a,0,0)$ at their centers. The shaded points lie outside of the page and the solid line denotes a cross section of the first BZ in the $(k_x,k_y)$ plane. The BZ can be visualized as a cube with a pyramid attached to each of the six faces, and the dotted line shows a projection of such a pyramid into the $k_z$ plane.

Figure 2

(a) Cross section of a real-space tile (dashed line) for a $N_c=4$ cluster on the fcc lattice containing the points ${\mathbf{R}}_1=(0,0,0)$, ${\mathbf{R}}_2=(a∕2,0,a∕2)$, ${\mathbf{R}}_3=(a∕2,a∕2,0)$, and ${\mathbf{R}}_4=(0,a∕2,a∕2)$. The shaded sites lie outside of the page. (b) Cross section of the corresponding reciprocal-space tiles (dashed lines) for the $N_c=4$ cluster, with ${\mathbf{K}}_1=(0,0,0)$, ${\mathbf{K}}_2=(2\pi ∕a,0,0)$, and ${\mathbf{K}}_3=(0,2\pi ∕a,0)$ shown as the $\Gamma$ point and the two $X$ points. The fourth tile is centered at the $X$ point ${\mathbf{K}}_4=(0,0,2\pi ∕a)$ and is situated out of the page vertically above $\Gamma$. Again the shaded points lie outside of the page and the solid line denotes a cross section of the first BZ in the $(k_x,k_y)$ plane.

Figure 3

(Color online) (a) DOS for pure Cu and pure Zn. (b) DOS for ordered ${\mathrm{Cu}}_{50}{\mathrm{Zn}}_{50}$.

Figure 4

(Color online) (a) Total average DOS for disordered bcc ${\mathrm{Cu}}_{50}{\mathrm{Zn}}_{50}$ using the KKR-CPA. Also shown are the contributions from the Cu and Zn components (site-restricted average DOS). $E_F$ is the Fermi energy. (b) Total average DOS for bcc ${\mathrm{Cu}}_{50}{\mathrm{Zn}}_{50}$ using the KKR-NLCPA with $N_c=2$, along with the contributions from the four possible cluster configurations (cluster-restricted average DOS) measured at the first site, i.e., Cu for Cu-Cu, Cu-Zn, and Zn for Zn-Cu, Zn-Zn. (Owing to the translational invariance, contributions measured at the second site would give the same results with a simple reversal of the labels). Also shown are total DOS results for $N_c=16$. (c) Plot of the difference in the total DOS between the $N_c=1$ and $2$ calculations, i.e., $(\text{total}{N}_c=1)-(\text{total}{N}_c=2)$.

Figure 5

(Color online) (a) Total and cluster component DOS for bcc ${\mathrm{Cu}}_{50}{\mathrm{Zn}}_{50}$ using the KKR-NLCPA with $N_c=2$ and $\alpha =0$. (b)–(f) Same as (a) but with increasing values of $\alpha$, corresponding to short-range clustering.

Figure 6

(Color online) (a) Total and cluster component DOS for bcc ${\mathrm{Cu}}_{50}{\mathrm{Zn}}_{50}$ using the KKR-NLCPA with $N_c=2$ and $\alpha =0$. (b)–(f) Same as (a) but with decreasing values of $\alpha$, corresponding to short-range ordering.