Screening effects in Coulomb-frustrated phase separation

We solve a model of phase separation among two competing phases frustrated by the long-range Coulomb interaction in two and three dimensions, taking into account finite compressibility effects. In the limit of strong frustration in two dimension, we recover the results of Jamei et al. [Phys. Rev. Lett. 94, 056805 (2005)] and the system always breaks into domains in a narrow range of densities, no matter how big is the frustration. For weak frustration in two dimension and for arbitrary frustration in three dimension, the finite compressibility of the phases is shown to play a fundamental role. Our results clarify the different roles of screening in two-dimensional and three-dimensional systems. We discuss the thermodynamic stability of the system near the transition of the phase-separated state and the possibility of observing it in real systems.

Figure 1

(Color online) Behavior of the free energy density for the uniform phases with $s=\pm 1$ in the present model (thick lines) compared with the $\kappa \to \infty$ model discussed in Ref. 13 (thin full lines). The dashed lines are the energy density of the mixed state solutions in 2D systems for different values of the frustrating parameter defined in Sec. 3. The energy density is measured in unit of the characteristic phase separation energy gain $e_0\equiv {\left(\delta n^0\right)}^2∕\kappa =2\pi Q^2{\left(\delta n^0\right)}^2{l}_s$. The dot-dashed line corresponds to the Maxwell construction $(\lambda =0)$.

Figure 2

(Color online) The phase diagram for the smectic stripe solution in two dimension. The central region corresponds to the mixed state (MS). The thick line is the exact solution of the present model with finite compressibility; the thin full line is the $\kappa \to \infty$ limit taking the short distance cutoff of Ref. 13 equal to the present screening length $l_s$. The dashed line is the UDA. The inset shows an enlargement near the crossing density $n_c$ where the UDA predicts a critical value of the frustrating parameter.

Figure 3

(Color online) Charge density modulation for a cut perpendicular to a stripe in two dimension at $\overline{n}=n_c$, $R_d=R_c∕2$, and three different values of the screening length corresponding to $\lambda \sim 0.2$ $(R_c=l_s)$, $\lambda \sim 0.7$ $(R_c=10l_s)$, and $\lambda \sim 1.1$ $(R_c=100l_s)$. In the case $R_c∕l_s\to \infty$, the charge density diverges at the boundary. A finite $l_s$ removes the divergence.

Figure 4

(Color online) Behavior of $R_c$ (logarithmic scale) at the crossing density as a function of the frustrating parameter. The thick line is the solution of the present model, while the thin line is the infinite compressibility limit (Ref. 13). The UDA is the dashed line.

Figure 5

(Color online) Behavior of the width (top panel) and periodicity (bottom panel) of ↑ stripes in 2D systems for different values of $\lambda <1$. The dashed lines correspond to the uniform density approximation, whereas the full line is the exact solution. Note that the ↑ stripes are the minority phase for $n<n_c$ and the majority phase for $n>n_c$.

Figure 6

(Color online) Behavior of the period $R_c$ (logarithmic scale) vs $\lambda$ at the crossing density in 3D (thick line) and 2D systems (dashed line).

Figure 7

(Color online) Charge density modulation for a cut perpendicular to the layers in 3D systems (thick line) and stripes in 2D systems (dashed line) at $\overline{n}=n_c$ and $R_c=20l_s$.

Figure 8

(Color online) The phase diagram in the $\lambda -(\overline{n}-n_c)∕\delta n^0$ plane for three-dimensional systems. The thick lines correspond to the second-order transitions from the homogeneous phase to the layered mixed state characterized by a divergent periodicity and a finite minority phase layer size. The thin line represents the first order transition between the two homogeneous phases that occurs for $\lambda >{\lambda }_c$.