Soft versus Hard Dynamics for Field-driven Solid-on-Solid Interfaces

Analytical arguments and dynamic Monte Carlo simulations show that the microscopic structure of field-driven Solid-on-Solid interfaces depends strongly on the details of the dynamics. For nonconservative dynamics with transition rates that factorize into parts dependent only on the changes in interaction energy and field energy, respectively (soft dynamics), the intrinsic interface width is field-independent. For non-factorizing rates, such as the standard Glauber and Metropolis algorithms (hard dynamics), it increases with the field. Consequences for the interface velocity and its anisotropy are discussed. Submitted to: J. Phys. A: Math. Gen. PACS numbers: 68.35.Ct 75.60.Jk 68.43.Jk 05.10.Ln Letter to the Editor 2 Surfaces and interfaces moving under far-from-equilibrium conditions are important in the formation of patterns and structures. For example, domain growth by the motion of defects such as grain boundaries or dislocations influences the mechanical properties of metals [1] and the structures formed during phase transformations in adsorbate systems [2], while the propagation of domain walls influences the switching dynamics of magnetic nanoparticles and ultrathin films [3]. Recently, interface dynamics was even used to analyze the scalability of discrete-event simulations on parallel computers [4]. The importance of moving interfaces in a wide variety of fields has inspired enormous interest in their structure and dynamics [5]. However, despite the fact that many interface properties, such as mobility and chemical activity, are determined by the microscopic structure, interest has mostly focused on the large-scale interface structure and its universal properties. In this Letter we show that the microstructure of a moving interface can be dramatically influenced by seemingly minor modifications of the growth mechanism, with measurable consequences for such macroscopic quantities as the interface velocity and its anisotropy. The detailed microscopic mechanism of the interface motion is usually not known, and it is therefore common to mimic the essential features in a dynamic Monte Carlo (MC) simulation of a model stochastic process [1]. In doing so, there are two important distinctions that must be made. One is between those transition probabilities that only depend on the energy difference between the initial and final states (often referred to as Metropolis, Glauber, or heat-bath rates [6, 7, 8]), and those that involve an activation barrier between the two states (often referred to as Arrhenius rates) [2, 6, 8, 9]. Another distinction is between dynamics that do not conserve the order parameter, such as the Metropolis and Glauber single-spin flip algorithms, and conservative dynamics, such as the Kawasaki spin-exchange dynamic [7]. Once it is decided to which of these categories the system belongs, the dynamic is often chosen on the basis of convenience. It is well known that different microscopic dynamics can yield different equilibration paths and equilibrium fluctuations [10] (cluster vs local MC algorithms being the most extreme example [7]) and even noticeable differences in the steady-state microstructure [6, 8]. Nevertheless, the general expectation is that, if no additional parameters (such as an activation barrier or a diffusion rate) is introduced into the physical model, observables are only affected quantitatively . However, here we demonstrate striking qualitative differences between the microstructural consequences of two different stochastic dynamics that do not involve a transition barrier, even though both obey detailed balance and neither involves order-parameter conserving moves. For this demonstration we use an unrestricted Solid-on-Solid (SOS) interface [11] separating uniform spin-up and spin-down phases in a ferromagnetic S = 1/2 Ising system on a square lattice of unit lattice constant. The interface consists of integer-valued steps parallel to the y-direction, δ(x) ∈ {−∞,+∞}, where the step position x is also an integer. Its energy is given by the nearest-neighbour Ising Hamiltonian with anisotropic ferromagnetic interactions, Jx and Jy in the x and y direction, respectively: H = − ∑ x,y sx,y (Jxsx+1,y + Jysx,y+1 +H). The two states at site (x, y) are denoted Letter to the Editor 3 by the spin sx,y = ±1, ∑ x,y runs over all sites, and H is the applied field. We take sx,y = −1 on the side of the interface corresponding to large positive y, so that the interface on average moves in the positive y direction for H > 0. In the equivalent lattice-gas language, sx,y = +1 corresponds to solid and sx,y = −1 to gas or liquid, and H is proportional to a chemical-potential imbalance. The interface evolves through single spin flips (in lattic-gas language: adsorption/desorption events) that occur with probability W [β∆E] where ∆E is the energy change that would result from the transition, constrained by the requirement that the probabilities obey detailed balance (see [12, 13] for details). To restrict the accessible configurations to a simple SOS interface without bubbles or overhangs, transitions are allowed only for spins that have one single broken bond in the y-direction. Regardless of the details of the transition probabilities, which lead to dramatic microscopic structural differences as demonstrated below, the macroscopic structure of the interfaces belongs to the Kardar-Parisi-Zhang (KPZ) dynamic universality class [5, 14]. We emphasize that the absence of energy barriers and order-parameter conserving diffusion (spin-exchange) moves sets our model clearly apart from commonly studied models of Molecular Beam Epitaxy (MBE) [6, 8]. It is closely similar to the diffusion-free dynamical model studied in three dimensions in Ref. [15] which, however, does not consider the microscopic interface structure in detail. Recently [12] we introduced a mean-field approximation for the driven-interface microstructure, in which the probability density function (pdf) for the height of a single step takes the Burton-Cabrera-Frank (BCF) form [11], p[δ(x)] = ZX |δ(x)| e , (1) where the Lagrange multiplier γ(φ) enforces 〈δ(x)〉 = tanφ independent of x, corresponding to an overall angle φ between the interface and the x-axis. The width parameter X, which for a driven interface can depend on both H and the temperature T , is discussed below. The partition function in (1) is