Microstructure and velocity of field-driven solid-on-solid interfaces moving under stochastic dynamics with local energy barriers

We study the microscopic structure and the stationary propagation velocity of $(1+1)$-dimensional solid-on-solid interfaces in an Ising lattice-gas model, which are driven far from equilibrium by an applied force, such as a magnetic field or a difference in (electro)chemical potential. We use an analytic nonlinear-response approximation [P. A. Rikvold and M. Kolesik, J. Stat. Phys. 100, 377 (2000)] together with kinetic Monte Carlo simulations. Here we consider interfaces that move under Arrhenius dynamics, which include a microscopic energy barrier between the allowed Ising or lattice-gas states. Two different dynamics are studied: the standard one-step dynamics (OSD) [H. C. Kang and W. Weinberg, J. Chem. Phys. 90, 2824 (1992)] and the two-step transition-dynamics approximation (TDA) [T. Ala-Nissila, J. Kjoll, and S. C. Ying, Phys. Rev. B 46, 846 (1992)]. In the OSD the effects of the applied force and the interaction energies in the model factorize in the transition rates (soft dynamics), while in the TDA such factorization is not possible (hard dynamics). In full agreement with previous general theoretical results we find that the local interface width under the TDA increases dramatically with the applied force. In contrast, the interface structure with the OSD is only weakly influenced by the force, in qualitative agreement with the theoretical expectations. Results are also obtained for the force dependence and anisotropy of the interface velocity, which also show differences in good agreement with the theoretical expectations for the differences between soft and hard dynamics. Our results confirm that different stochastic interface dynamics that all obey detailed balance and the same conservation laws nevertheless can lead to radically different interface responses to an applied force.

Figure 1

A short segment of an SOS interface $y=h\left(x\right)$ between a positively magnetized phase (or “solid” phase in the lattice-gas picture) below and a negative (or “fluid”) phase above. The step heights are $\delta \left(x\right)=h(x+1∕2)-h(x-1∕2)$. Interface sites representative of the different SOS spin classes (see Tables ) are marked with the notation $jks$ explained in the text. Sites in the uniform bulk phases are $00-$ and $00+$. This interface was randomly generated with a symmetric step-height distribution, corresponding to $\varphi =0$. From Ref. 8.

Figure 2

Schematic picture of the transition barrier in the symmetric Butler-Volmer approximation, used to calculate the TDA and OSD transition rates. After Ref. 23.

Figure 3

(Color online) MC (data points) and analytical (solid lines) results for the stationary single-step PDF calculated with the TDA dynamics, shown on a logarithmic scale vs $\delta$, for the values of $H∕J$ given in the legend. (a) $T=0.2T_c$. (b) $T=0.6T_c$. The symbols (and colors) have the same interpretations in (a) and (b).

Figure 4

(Color online) The stationary single-step PDF calculated with the OSD dynamics, shown on a logarithmic scale vs $\delta$. The data points indicate MC results, and the straight lines are the theoretical predicted values (independent of $H$). The values of $H∕J$ are given in the legend. (a) $T=0.2T_c$. (b) $T=0.6T_c$. The symbols and colors have the same interpretations in (a) and (b). Note the very different scales from Fig. 3.

Figure 5

Average stationary step height $⟨\mid \delta \mid ⟩$ vs $H$ for $\varphi =0$ at $T=0.2T_c$ and $0.6T_c$. The curves represent the theoretical results. The MC data were obtained directly by summation over the simulated single-step PDF’s (asterisks and crosses) and from the probability of zero step height (circles and squares). See the text for details. Curve with circles and asterisks: $T=0.2T_c$. Curve with squares and crosses: $T=0.6T_c$. (a) TDA dynamics, shown on a logarithmic vertical scale. (b) OSD dynamics, shown on a linear vertical scale. In this and all the following figures, the statistical uncertainty is much smaller than the symbol size.

Figure 6

The average stationary normal interface velocity $⟨v_{\perp }⟩$ vs $H$ for $\varphi =0$. The MC results are shown as data points, circles for $T=0.2T_c$ and squares for $T=0.6T_c$, and the theoretical results as solid curves. (a) TDA, shown on a linear vertical scale. (b) OSD, shown on a logarithmic vertical scale.

Figure 7

The average stationary normal interface velocity $⟨v_{\perp }⟩$ vs $\tan \varphi$, calculated with the TDA dynamics, for $H∕J=0.1$ (up triangles), 0.5 (circles), $1$ (squares), $1.5$ (diamonds), and $2.0$ (left triangles). The symbols represent MC data and the solid curves analytical results. (a) $T=0.2T_c$. (b) $T=0.6T_c$.

Figure 8

The average stationary normal interface velocity $⟨v_{\perp }⟩$ vs $\tan \varphi$, calculated with the OSD dynamics, for $H∕J=0.1$ (up triangles), $0.5$ (circles), $1$ (squares), $1.5$ (diamonds), and $2.0$ (left triangles). The symbols represent MC data and the solid curves analytical results. (a) $T=0.2T_c$. (b) $T=0.6T_c$. Note the difference in the vertical scales.

Figure 9

(Color online) The average stationary normal interface velocity $⟨v_{\perp }⟩$ vs $T$ for $\varphi =0$ and $H∕J$ between $0.5$ and $3.5$. MC data are represented by data points and analytical results by solid curves. From below to above, the values of $H∕J$ are 0.5 (circles), 1.0 (squares), 1.5 (diamonds), 2.0 (up triangles), 2.5 (left triangles), 3.0 (down triangles), and 3.5 (right triangles). Online, the colors of the curves and symbols match. (a) TDA, on a linear vertical scale. (b) OSD, on a logarithmic vertical scale.

Figure 10

Mean stationary class populations $⟨n\left(jks\right)⟩$ vs $H∕J$ for $\varphi =0$, calculated for the TDA dynamics. The simulation results are indicated by symbols and the analytic approximations by solid curves. (a) $T=0.2T_c$. (b) $T=0.6T_c$. The symbols have the same interpretations in (a) and (b). Note the different vertical scales in the two parts.

Figure 11

Mean stationary class populations $⟨n\left(jks\right)⟩$ vs $H∕J$ for $\varphi =0$ calculated for the OSD dynamics. The simulation results are indicated by the symbols, and the straight lines indicate the theoretical predicted values (independent of $H$). (a) $T=0.2T_c$. (b) $T=0.6T_c$. The symbols have the same interpretations in (a) and (b). Note the different vertical scales in the two parts.

Figure 12

The two relative skewness parameters $\rho$ (circles) and $ϵ$ (squares), defined in Eqs. (14, 15), respectively. The parameters are shown vs $H$ for $\varphi =0$, at $T=0.2T_c$ (open symbols) and at $T=0.6T_c$ (solid symbols). (a) TDA. (b) OSD. Note the different vertical scales in the two parts.

Figure 13

Contour plots of ${\log }_{10}p\left[\delta \right(x),\delta (x+1\left)\right]$ for $\varphi =0$ at $T=0.6T_c$ for the TDA dynamics. (a) $H∕J=0$. (b) $H∕J=1.0$. (c) $H∕J=2.0$. (d) $H∕J=3.5$. Note the different scales in the four parts. See discussion in the text.

Figure 14

Contour plots of ${\log }_{10}p\left[\delta \right(x),\delta (x+1\left)\right]$ for $\varphi =0$ at $T=0.6T_c$ for the OSD dynamics. (a) $H∕J=0$. (b) $H∕J=1.0$. (c) $H∕J=2.0$. (d) $H∕J=2.5$. See discussion in the text.