Nanostructure and velocity of field-driven solid-on-solid interfaces moving under a phonon-assisted dynamic

The nanoscopic 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, are studied by 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 the case that the system is coupled to a two-dimensional phonon bath. In the resulting dynamic [K. Saito et al., Phys. Rev. E 61, 2397 (2000); K. Park and M. A. Novotny, Comput. Phys. Commun. 147, 737 (2002)], transitions that conserve the system energy are forbidden, and the effects of the applied force and the interaction energies do not factorize (a so-called hard dynamic). In full agreement with previous general theoretical results, we find that the local interface width changes dramatically with the applied force. However, in contrast with other hard dynamics, this change is nonmonotonic in the driving force. 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. However, significant differences between theory and simulation are found near two special values of the driving force, where certain transitions allowed by the solid-on-solid model become forbidden by the phonon-assisted dynamic. Our results show 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. Thus, they represent a significant step toward providing a solid physical foundation for kinetic Monte Carlo simulations.

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 Table ) are marked with the notation $jks$, explained in the text. Sites in the uniform bulk phases are $00-$ and $00+$. This interface was generated with a symmetric step-height distribution, corresponding to $\varphi =0$.

Figure 2

(Color online) (a) The transition rates for the $d$-dimensional phonon-assisted dynamic, $W_{\mathrm{PB}}$, shown scaled by $T^d$ vs the energy difference $\Delta E$ scaled by $T$. (b) The transition rates for the standard Glauber dynamic, $W_G$, shown vs $\Delta E∕T$.

Figure 3

(Color online) MC (data points) and analytical (solid lines) results for the stationary single-step pdf, shown on a logarithmic scale vs $\delta$, for $\varphi =0$ and the values of $H∕J$ given in the legend. (a) $T=0.2T_c$ and (b) $T=0.6T_c$. The symbols (and colors) have the same interpretations in (a) and (b). Note the nonmonotonic field dependence near $H∕J=2$.

Figure 4

(Color online) Average stationary step height $⟨\mid \delta \mid ⟩$ vs $H∕J$ for $\varphi =0$ at $T=0.2T_c$ and $0.6T_c$. The curves represent the theoretical results. Curve with circles (black): $T=0.2T_c$. Curve with squares (gray, red online): $T=0.6T_c$. These data refer to interfaces that started from a microscopically flat initial state. We also include values near $H∕J=0$, calculated by the phonon-assisted dynamic, using as a starting state the thermalized interface obtained with the standard Glauber dynamic. Black asterisks: $T=0.2T_c$. Gray asterisks (red online): $T=0.6T_c$. The differences are evident only near $H∕J=0$. In this and all the following figures, the statistical uncertainty is much smaller than the symbol size. The inset shows a magnified view of the region around $H∕J=2$. Note the disagreement between the theoretical and the simulation values at $T=0.6T_c$ when $H∕J=2$.

Figure 5

(Color online) Short segments of thermalized interfaces for $\varphi =0$. (a) $H∕J=0$, $T=0.6T_c$. The three graphs show an equilibrium interface created with the standard Glauber dynamic over ${10}^4$ UPS (medium gray, red online), an interface created by the phonon-assisted dynamic over ${10}^6$ UPS, using the equilibrium interface as a starting state (dark gray, blue online), and an interface created by the phonon-assisted dynamic over ${10}^{10}$ UPS, using a microscopically flat interface as starting state (black). (b) $H∕J=2$, $T=0.6T_c$. The jagged interface (medium gray, red online) is in the statistically stationary state, propagating in the direction of the arrow under the standard Glauber dynamic. At a given time, the dynamic is switched to the phonon-assisted transition rates, using the Glauber interface as initial state. The lagging parts rapidly catch up with the absolute maximum of the Glauber interface, where the interface gets permanently stuck in an almost perfect, microscopically flat configuration with a very small density of “backward” $12-$ notches (circled). At this field, the transition forbidden by the phonon-assisted dynamic is the nucleation of “forward” $12+$ notches, which are needed to nucleate propagation of a microscopically flat interface.

Figure 6

(Color online) The average stationary normal interface velocity $⟨v_{\perp }⟩$ vs $H∕J$ 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. The inset shows a magnified view of the region around $H∕J=2$. Note again the disagreement between the theoretical and the simulation values at $T=0.6T_c$ when $H∕J=2$.

Figure 7

(Color online) Average stationary normal interface velocity at $T=0.2T_c$, obtained by coupling the system to a one-dimensional phonon bath. MC data are represented by the symbols, and analytical results by the solid curves. (a) Velocity vs $H∕J$ for $\varphi =0$. (b) Velocity vs $\tan \varphi$ for several values of $H∕J$.

Figure 8

(Color online) The average stationary normal interface velocity $⟨v_{\perp }⟩$ vs $\tan \varphi$, for several values of $H∕J$. The symbols represent MC data, and the solid curves analytical results. (a) $T=0.2T_c$ and (b) $T=0.6T_c$. The symbols have the same interpretations in (a) and (b), given by the legend in (a). Online, the colors of the curves and symbols match.

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. MC data are represented by data points and analytical results by solid curves. Online, the colors of the curves and symbols match. The agreement between simulation and theory is quite good, except for $H∕J=2$.

Figure 10

(Color online) Mean stationary class populations $⟨n\left(jks\right)⟩$ vs $H∕J$ for $\varphi =0$. The simulation results are indicated by symbols, and the analytic approximations by solid curves. (a) $T=0.2T_c$ and (b) $T=0.6T_c$. At $T=0.6T_c$, we also include some values [asterisks for $⟨n(jk+)⟩$ and pluses for $⟨n(jk-)⟩$] calculated with the interface created by the phonon-assisted dynamic, using the thermalized interface obtained by the Glauber dynamic as starting state. Note that in this case, there is excellent agreement between theory and simulations at $H∕J=0$. The other symbols have the same interpretations in (a) and (b), given by the legend in (a). The insets in both (a) and (b) show $⟨n\left(11s\right)⟩$ and $⟨n\left(21s\right)⟩$ near $H∕J=2$.

Figure 11

(Color online) The two relative skewness parameters $\rho$ (circles, black) and $ϵ$ (squares, red online), defined in Eqs. (9, 10), respectively. The parameters are shown vs $H∕J$ for $\varphi =0$, at $T=0.2T_c$ (empty symbols) and at $T=0.6T_c$ (filled symbols). The inset shows a magnified view of the region around $H∕J=2$.