Kinetic roughening of vicinal surfaces

Kardar-Parisi-Zhang roughening associated with nucleation-limited steady crystal growth

The roughness of crystal surfaces and the shape of crystals play important roles in multiscale phenomena. For example, the roughness of the crystal surface affects the frictional and optical properties of materials such as ice or silica. Theoretical studies on crystal surfaces based on the symmetry principle proposed that the growing surfaces of crystal growth could be classified in the universal class of Kardar-Parisi-Zhang (KPZ), but experiments rarely observe KPZ properties. To fill this the gap, extensive numerical calculations of the crystal growth rates and the surface roughness (surface width) have been performed for a nanoscale lattice model using the Monte Carlo method. The results indicate that a (001) surface is smooth within the single nucleation growth region. In contrast, the same surface is atomically smooth but thermodynamically rough in the poly-nucleation growth region in conjunction with a KPZ roughness exponent. Inclined surfaces are known to become Berezinskii-Kosterlitz-Thouless (BKT) rough surfaces both at and near equilibrium. The two types of steps associated with the (001) and (111) terraces were found to induce KPZ surface roughness, while the interplay between steps and multilayered islands promoted BKT roughness.

Faceted-rough surface with disassembling of macrosteps in nucleation-limited crystal growth

To clarify whether a surface can be rough with faceted macrosteps that maintain their shape on the surface, crystal surface roughness is studied by a Monte Carlo method for a nucleation-limited crystal-growth process. As a surface model, the restricted solid-on-solid (RSOS) model with point-contact-type step-step attraction (p-RSOS model) is adopted. At equilibrium and at sufficiently low temperatures, the vicinal surface of the p-RSOS model consists of faceted macrosteps with (111) side surfaces and smooth terraces with (001) surfaces (the step-faceting zone). We found that a surface with faceted macrosteps has an approximately self-affine-rough structure on a 'faceted-rough surface'; the surface width is strongly divergent at the step-disassembling point, which is a characteristic driving force for crystal growth. A 'faceted-rough surface' is realized in the region between the step-disassembling point and a crossover point where the single nucleation growth changes to poly-nucleation growth.

Chiral Active Hexatics: Giant Number Fluctuations, Waves, and Destruction of Order

Active materials, composed of internally driven particles, have properties that are qualitatively distinct from matter at thermal equilibrium. However, the most spectacular departures from equilibrium phase behavior are thought to be confined to systems with polar or nematic asymmetry. In this Letter, we show that such departures are also displayed by more symmetric phases such as hexatics if, in addition, the constituent particles have chiral asymmetry. We show that chiral active hexatics whose rotation rate does not depend on density have giant number fluctuations. If the rotation rate depends on density, the giant number fluctuations are suppressed due to a novel orientation-density sound mode with a linear dispersion which propagates even in the overdamped limit. However, we demonstrate that beyond a finite but large length scale, a chirality and activity-induced relevant nonlinearity invalidates the predictions of the linear theory and destroys the hexatic order. In addition, we show that activity modifies the interactions between defects in the active chiral hexatic phase, making them nonmutual. Finally, to demonstrate the generality of a chiral active hexatic phase we show that it results from the melting of chiral active crystals in finite systems.

Crossover from BKT-rough to KPZ-rough surfaces for interface-limited crystal growth/recession

The crossover from a Berezinskii-Kosterlitz-Thouless (BKT) rough surface to a Kardar-Parisi-Zhang (KPZ) rough surface on a vicinal surface is studied using the Monte Carlo method in the non-equilibrium steady state in order to address discrepancies between theoretical results and experiments. The model used is a restricted solid-on-solid model with a discrete Hamiltonian without surface or volume diffusion (interface limited growth/recession). The temperature, driving force for growth, system size, and surface slope dependences of the surface width are calculated for vicinal surfaces tilted between the (001) and (111) surfaces. The surface velocity, kinetic coefficient of the surface, and mean height of the locally merged steps are also calculated. In contrast to the accepted theory for (2 + 1) surfaces, we found that the crossover point from a BKT (logarithmic) rough surface to a KPZ (algebraic) rough surface is different from the kinetic roughening point for the (001) surface. The driving force for crystal growth was found to be a relevant parameter for determining whether the system is in the BKT class or the KPZ class. It was also determined that ad-atoms, ad-holes, islands, and negative-islands block surface fluctuations, which contributes to making a BKT-rough surface.

Topological Defects in Anisotropic Driven Open Systems

We study the dynamics and unbinding transition of vortices in the compact anisotropic Kardar-Parisi-Zhang equation. The combination of nonequilibrium conditions and strong spatial anisotropy drastically affects the structure of vortices and amplifies their mutual binding forces, thus stabilizing the ordered phase. We find novel universal critical behavior in the vortex-unbinding crossover in finite-size systems. These results are relevant for a wide variety of physical systems, ranging from strongly coupled light-matter quantum systems to dissipative time crystals.

Keldysh field theory for driven open quantum systems

Recent experimental developments in diverse areas-ranging from cold atomic gases to light-driven semiconductors to microcavity arrays-move systems into the focus which are located on the interface of quantum optics, many-body physics and statistical mechanics. They share in common that coherent and driven-dissipative quantum dynamics occur on an equal footing, creating genuine non-equilibrium scenarios without immediate counterpart in equilibrium condensed matter physics. This concerns both their non-thermal stationary states and their many-body time evolution. It is a challenge to theory to identify novel instances of universal emergent macroscopic phenomena, which are tied unambiguously and in an observable way to the microscopic drive conditions. In this review, we discuss some recent results in this direction. Moreover, we provide a systematic introduction to the open system Keldysh functional integral approach, which is the proper technical tool to accomplish a merger of quantum optics and many-body physics, and leverages the power of modern quantum field theory to driven open quantum systems.

Strong-coupling phases of the anisotropic Kardar-Parisi-Zhang equation

We study the anisotropic Kardar-Parisi-Zhang equation using nonperturbative renormalization group methods. In contrast to a previous analysis in the weak-coupling regime, we find the strong-coupling fixed point corresponding to the isotropic rough phase to be always locally stable and unaffected by the anisotropy even at noninteger dimensions. Apart from the well-known weak-coupling and the now well-established isotropic strong-coupling behavior, we find an anisotropic strong-coupling fixed point for nonlinear couplings of opposite signs at noninteger dimensions.

Strong anisotropy in two-dimensional surfaces with generic scale invariance: nonlinear effects

We expand a previous study [Phys. Rev. E 86, 051611 (2012)] on the conditions for occurrence of strong anisotropy in the scaling properties of two-dimensional surfaces displaying generic scale invariance. In that study, a natural scaling ansatz was proposed for strongly anisotropic systems, which arises naturally when analyzing data from, e.g., thin-film production experiments. The ansatz was tested in Gaussian (linear) models of surface dynamics and in nonlinear models, like the Hwa-Kardar (HK) equation [Phys. Rev. Lett. 62, 1813 (1989)], which are susceptible of accurate approximations through the former. In contrast, here we analyze nonlinear equations for which such approximations fail. Working within generically scale-invariant situations, and as representative case studies, we formulate and study a generalization of the HK equation for conserved dynamics and reconsider well-known systems, such as the conserved and the nonconserved anisotropic Kardar-Parisi-Zhang equations. Through the combined use of dynamic renormalization group analysis and direct numerical simulations, we conclude that the occurrence of strong anisotropy in two-dimensional surfaces requires dynamics to be conserved. We find that, moreover, strong anisotropy is not generic in parameter space but requires, rather, specific forms of the terms appearing in the equation of motion, whose justification needs detailed information on the dynamical process that is being modeled in each particular case.

Universality for moving stripes: a hydrodynamic theory of polar active smectics

We present a theory of moving stripes ("polar active smectics"), both with and without number conservation. The latter is described by a compact anisotropic Kardar-Parisi-Zhang equation, which implies smectic order is quasilong ranged in d=2 and long ranged in d=3. In d=2 the smectic disorders via a Kosterlitz-Thouless transition, which can be driven by either increasing the noise or varying certain nonlinearities. For the number-conserving case, giant number fluctuations are greatly suppressed by the smectic order, which is long ranged in d=3. Nonlinear effects become important in d=2.

Strong anisotropy in two-dimensional surfaces with generic scale invariance: Gaussian and related models

Among systems that display generic scale invariance, those whose asymptotic properties are anisotropic in space (strong anisotropy, SA) have received relatively less attention, especially in the context of kinetic roughening for two-dimensional surfaces. This is in contrast with their experimental ubiquity, e.g., in the context of thin-film production by diverse techniques. Based on exact results for integrable (linear) cases, here we formulate a SA ansatz that, albeit equivalent to existing ones borrowed from equilibrium critical phenomena, is more naturally adapted to the type of observables that are measured in experiments on the dynamics of thin films, such as one- and two-dimensional height structure factors. We test our ansatz on a paradigmatic nonlinear stochastic equation displaying strong anisotropy like the Hwa-Kardar equation [Phys. Rev. Lett. 62, 1813 (1989)], which was initially proposed to describe the interface dynamics of running sand piles. A very important role to elucidate its SA properties is played by an accurate (Gaussian) approximation through a nonlocal linear equation that shares the same asymptotic properties.

Dynamic effects induced by renormalization in anisotropic pattern forming systems

The dynamics of patterns in large two-dimensional domains remains a challenge in nonequilibrium phenomena. Often it is addressed through mild extensions of one-dimensional equations. We show that full two-dimensional generalizations of the latter can lead to unexpected dynamic behavior. As an example we consider the anisotropic Kuramoto-Sivashinsky equation, which is a generic model of anisotropic pattern forming systems and has been derived in different instances of thin film dynamics. A rotation of a ripple pattern by 90° occurs in the system evolution when nonlinearities are strongly suppressed along one direction. This effect originates in nonlinear parameter renormalization at different rates in the two system dimensions, showing a dynamic interplay between scale invariance and wavelength selection. Potential experimental realizations of this phenomenon are identified.

Epitaxial Growth and Recovery: an Analytic Approach

Most detailed studies of morphological evolution during epitaxial growth and recovery make use of computer-based simulation techniques. In this paper, we discuss an alternative, analytic approach to this problem which takes explicit account of the atomistically random processes of deposition and surface diffusion. Beginning with a master equation representation of the dynamics of a solid-on-solid model of epitaxial growth, we derive a discrete, stochastic equation of motion for the surface profile. This Langevin equation is appropriate for growth studies. In particular, we are able to provide a microscopic justification for a non-linear continuum equation of motion proposed for this problem by others on the basis of heuristic arguments. During recovery, the deposition flux and its associated shot noise are absent. We analyze this process with a completely deterministic equation of motion obtained by performing a statistical average of the original stochastic equation. Results using the latter compare favorably with full Monte Carlo simulations of the original model for the case of the decay of sinusoidally modulated initial surfaces.

Anisotropic model of kinetic roughening: the strong-coupling regime

We study the strong coupling (SC) limit of the anisotropic Kardar-Parisi-Zhang (KPZ) model. A systematic mapping of the continuum model to its lattice equivalent shows that in the SC limit, anisotropic perturbations destroy all spatial correlations but retain a temporal scaling which shows a remarkable crossover along one of the two spatial directions, the choice of direction depending on the relative strength of anisotropicity. The results agree with exact numerics and are expected to settle the long-standing SC problem of a KPZ model in the infinite range limit.

Nonlinear stochastic equations with calculable steady states

We consider generalizations of the Kardar-Parisi-Zhang equation that accommodate spatial anisotropies and the coupled evolution of several fields, and focus on their symmetries and nonperturbative properties. In particular, we derive generalized fluctuation-dissipation conditions on the form of the (nonlinear) equations for the realization of a Gaussian probability density of the fields in the steady state. For the amorphous growth of a single height field in one dimension we give a general class of equations with exactly calculable (Gaussian and more complicated) steady states. In two dimensions, we show that any anisotropic system evolves in long time and length scales either to the usual isotropic strong coupling regime or to a linearlike fixed point associated with a hidden symmetry. Similar results are derived for textural growth equations that couple the height field with additional order parameters which fluctuate on the growing surface. In this context, we propose phenomenological equations for the growth of a crystalline material, where the height field interacts with lattice distortions, and identify two special cases that obtain Gaussian steady states. In the first case compression modes influence growth and are advected by height fluctuations, while in the second case it is the density of dislocations that couples with the height.

Scaling and universality of self-organized patterns on unstable vicinal surfaces

We propose a unified treatment of the step bunching instability during epitaxial growth. The scaling properties of the self-organized surface pattern are shown to depend on a single parameter, the leading power in the expansion of the biased diffusion current in powers of the local surface slope. We demonstrate the existence of universality classes for the self-organized patterning appearing in models and experiments.

Kinetic roughening model with opposite Kardar-Parisi-Zhang nonlinearities

We introduce a model that simulates a kinetic roughening process with two kinds of particle: one follows ballistic deposition (BD) kinetics and the other restricted solid-on-solid Kim-Kosterlitz (KK) kinetics. Both of these kinetics are in the universality class of the nonlinear Kardar-Parisi-Zhang equation, but the BD kinetics has a positive nonlinear constant while the KK kinetics has a negative one. In our model, called the BD-KK model, we assign the probabilities p and (1-p) to the KK and BD kinetics, respectively. For a specific value of p, the system behaves as a quasilinear model and the up-down symmetry is restored. We show that nonlinearities of odd order are relevant in this low nonlinear limit.

Growth of a driven interface in isotropic and anisotropic random media

We introduce a simple stochastic model for a driven interface in a random medium, in which we can control the degree of the anisotropy of a random medium. When there is no anisotropy of a random medium, the motion of a growing interface in our model can be well described by the quenched Edwards-Wilkinson equation. When there is anisotropy of a random medium, however, the motion of a growing interface can be described by the quenched Kardar-Parisi-Zhang (KPZ) equation. In the two interfaces, apart from one growing in an isotropic medium and the other growing in an anisotropic medium, the growth rule of our model is the same. Our results support the fact that the anisotropy of a random medium is a source of the KPZ nonlinearity.

Renormalization group analysis of the anisotropic nonlocal kardar-parisi-zhang equation with spatially correlated noise

We study an anisotropic nonlocal Kardar-Parisi-Zhang (KPZ) equation with spatially correlated noise by using the dynamic renormalization group method. When the signs of nonlinear terms in parallel and perpendicular directions are opposite, the correlated noise coupled with the long ranged nature of interaction produces a stable non-KPZ fixed point for d<d(c). For the uncorrelated noise, the roughness and dynamic exponents associated with the stable fixed point are different from those of the isotropic nonlocal KPZ equation, while for the correlated noise the exponents are the same as those of the isotropic case.

Depinning of an anisotropic interface in random media: the tilt effect

We study the tilt dependence of the pinning-depinning transition for an interface described by the anisotropic quenched Kardar-Parisi-Zhang equation in 2+1 dimensions, where the two signs of the nonlinear terms are different from each other. When the substrate is tilted by m along the positive sign direction, the critical force F(c)(m) depends on m as F(c)(m)-F(c)(0) approximately -|m|(1.9(1)). The interface velocity v near the critical force follows the scaling form v approximately |f|(straight theta)Psi(+/-)(m(2)/|f|(straight theta+straight phi)) with straight theta=0.9(1) and straight phi=0.2(1), where f identical withF-F(c)(0) and F is the driving force.

Pattern formation in the instability of a vicinal surface by the drift of adatoms

We study the behavior of steps in a vicinal face with drift of adsorbed atoms (adatoms) by an external field. When the drift is in the downhill direction and its velocity exceeds critical values, v(x)(c) and v(y)(c), the vicinal face is linearly unstable to long-wavelength fluctuations parallel and/or perpendicular to the steps. By taking the continuum limit of the step-flow model, we derive an anisotropic Kuramoto-Sivashinsky equation with propagative terms, which describes the motion of an unstable vicinal face. Its numerical solution shows ripples or a zigzag pattern expected from the linear analysis. Nonlinearity becomes important in the late stage and, depending on the condition, various patterns are formed: regular step bunches, a hill and valley structure tilted from the initial step direction, mounds, and a chaotic pattern.