Charge ordering and self-assembled nanostructures in a fcc Coulomb lattice gas

The compositional ordering of Ag, Pb, Sb, Te ions in ${\left(\mathrm{Ag}\mathrm{Sb}{\mathrm{Te}}_2\right)}_x{\left(\mathrm{Pb}\mathrm{Te}\right)}_{2(1-x)}$ systems possessing a NaCl structure is studied using a Coulomb lattice gas (CLG) model on a face-centered cubic (fcc) lattice and Monte Carlo simulations. Our results show different possible microstructural orderings. Ordered superlattice structures formed out of $\mathrm{Ag}\mathrm{Sb}{\mathrm{Te}}_2$ layers separated by ${\mathrm{Pb}}_2{\mathrm{Te}}_2$ layers are observed for a large range of $x$ values. For $x=0.5$, we see an array of tubular structures formed by $\mathrm{Ag}\mathrm{Sb}{\mathrm{Te}}_2$ and ${\mathrm{Pb}}_2{\mathrm{Te}}_2$ blocks. For $x=1$, $\mathrm{Ag}\mathrm{Sb}{\mathrm{Te}}_2$ has a body-centered tetragonal (bct) structure which is in agreement with previous Monte Carlo simulation results for restricted primitive model (RPM) at closed packed density. The phase diagram of this frustrated CLG system is discussed.

Figure 1

Energy and heat capacity per particle versus temperature for $x=0.75$. Phase transitions occur at $T=0.106$ and 0.21 which are first-order and second-order transitions, respectively. There is hysteresis associated with the low $T$ transition.

Figure 2

Energy and heat capacity per particle versus temperature for $x=0.25$. Phase transition occurs at $T=0.08$ which is first-order. There is hysteresis associated with this transition.

Figure 3

Concentration $\left(x\right)$ versus temperature $\left(T\right)$ phase diagram constructed from the loci of the heat capacity maxima. There are first (solid lines) and second (dotted line) order transitions with two possible tricritical points $(x_t,T_t)$: A at $\approx (0.50,0.088)$ and B at $\approx (0.875,0.29)$.

Figure 4

(Color online) Energy as a function of Monte Carlo steps for $x=0.75$. The system is quenched from $T=0.3$ to 0.24, then from $T=0.24$ to 0.18, and so on.

Figure 5

(Color online) A low temperature configuration for $x=0.25$ [created using XCrySDen (Ref. 30)]. Dark layers are for $\mathrm{Ag}∕\mathrm{Sb}$, grey layers are for Pb; Te sublattice is not shown. This typical configuration showing a ordered superlattice structure formed out of $\mathrm{Ag}\mathrm{Sb}{\mathrm{Te}}_2$ layers separated by ${\mathrm{Pb}}_2{\mathrm{Te}}_2$ layers.

Figure 6

(Color online) A low temperature configuration for $x=0.5$ [created using XCrySDen (Ref. 30)]. Connected balls are for $\mathrm{Ag}∕\mathrm{Sb}$, unconnected balls are for Pb; Te sublattice is not shown. Checkerboard pattern formed by $\mathrm{Ag}\mathrm{Sb}{\mathrm{Te}}_2$ and ${\mathrm{Pb}}_2{\mathrm{Te}}_2$ blocks.