KPZ Model Research Articles

New approach to KPZ models through free fermions at positive temperature

We give a short account of our new approach to study models in the Kardar–Parisi–Zhang universality class by connecting them to free fermions at positive temperature. Our ideas and methods are explained mainly for the semi-discrete directed polymer model due to O’Connell and Yor.

Upper Tail Bounds for Stationary KPZ Models

We present a proof of an upper tail bound of the correct order (up to a constant factor in the exponent) in two classes of stationary models in the KPZ universality class. The proof is based on an exponential identity due to Rains in the case of last passage percolation with exponential weights, and recently re-derived by Emrah–Janjigian–Seppäiläinen (EJS). Our proof follows very similar lines for the two classes of models we consider, using only general monotonocity and convexity properties, and can thus be expected to apply to many other stationary models.

KPZ models: height gradient fluctuations and the tilt method

When an interface model belonging to the KPZ universality class is tilted with average slope m, its average velocity increases in . The coefficient Λ can only be related to the non-linear coefficient λ from the KPZ equation if the mean square of the height gradient also increases in bm 2 when the interface is tilted. For the continuous KPZ equation, b = 1 and the relation Λ = λ is achieved. In this work, we study the local fluctuations of the height gradient through an analysis of the values of b. We show that, for one-dimensional discrete KPZ models, , where s is the discretization step chosen to calculate the height gradient. The exponent γ b that we measure matches the power-law exponent associated with the finite-size corrections of the interface average velocity, i.e. γ b = 2(ζ − 1), where ζ is the global roughness exponent. Lastly, we show how, for restricted (unrestricted) growth models, the value of b goes to 1 from below (above) as s increases.

Surface growth on tree-like lattices and the upper critical dimension of the KPZ class

Aiming to investigate the upper critical dimension, d u , of the KPZ class, in Saberi A. A., EPL, 103 (2013) 10005, some growth models were numerically analyzed using Cayley trees (CTs) as substrates, as a way to access their behavior in the infinite-dimensional limit, and some unexpected results were reported: logarithmic roughness scaling, differing for EW and KPZ models (indicating that even at the KPZ nonlinearity is still relevant); beyond asymptotically rough EW surfaces above the upper critical dimension of the EW class. Motivated by these strange findings, I revisit these growth models here to show that such results are simple consequences of boundary effects, inherent to systems defined on CTs. In fact, I demonstrate that the anomalous boundary of the CT leads the growing surfaces to develop curved shapes, which explains the strange behaviors previously found for these systems, once the global “roughness” was analyzed for non-flat surfaces in the study above. Importantly, by measuring the height fluctuations at the central site of the CT, which can be seen as an approximation for the Bethe lattice, smooth surfaces are found for both EW and KPZ classes, consistently with the behavior expected for growing systems in dimensions . Interesting features of the 1-point height fluctuations, such as the possibility of non-saturation in the steady state regime, are also discussed for substrates in general.

Upper tail decay of KPZ models with Brownian initial conditions

In this paper we consider the limiting distribution of KPZ growth models with random but not stationary initial conditions introduced in [Chhita-Ferrari-Spohn 2018]. The one-point distribution of the limit is given in terms of a variational problem. By directly studying it, we deduce the right tail asymptotic of the distribution function. This gives a rigorous proof and extends the results obtained in [Meerson-Schmidt 2017].

Circular Kardar-Parisi-Zhang interfaces evolving out of the plane.

Circular KPZ interfaces spreading radially in the plane have Gaussian unitary ensemble (GUE) Tracy-Widom (TW) height distribution (HD) and Airy_{2} spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a mountain, or any surface of revolution? To give an answer to this, we report here extensive numerical analyses of several one-dimensional KPZ models on substrates whose size enlarges as 〈L(t)〉=L_{0}+ωt^{γ}, while their mean height 〈h〉 increases as usual [〈h〉∼t]. We show that the competition between the L enlargement and the correlation length (ξ≃ct^{1/z}) plays a key role in the asymptotic statistics of the interfaces. While systems with γ>1/z have HDs given by GUE and the interface width increasing as w∼t^{β}, for γ<1/z the HDs are Gaussian, in a correlated regime where w∼t^{αγ}. For the special case γ=1/z, a continuous class of distributions exists, which interpolate between Gaussian (for small ω/c) and GUE (for ω/c≫1). Interestingly, the HD seems to agree with the Gaussian symplectic ensemble (GSE) TW distribution for ω/c≈10. Despite the GUE HDs for γ>1/z, the spatial covariances present a strong dependence on the parameters ω and γ, agreeing with Airy_{2} only for ω≫1, for a given γ, or when γ=1, for a fixed ω. These results considerably generalize our knowledge on 1D KPZ systems, unveiling the importance of the background space on their statistics.

Effect of deposition rate on the growth mechanism of microcrystalline silicon thin films using very high frequency PECVD

The intrinsic microcrystalline silicon thin films were deposited by very high frequency plasma enhanced chemical vapor deposition (VHF-PECVD). Two series of films with different deposition rate 0.30 nm/s and 1.94 nm/s were prepared. The film surface and gas phase reaction growth processes were monitored with real-time spectroscopic ellipsometry and optical emission spectroscopy. The effect of deposition rate on the microcrystalline silicon thin film growth mechanism has been studied. The microcrystalline silicon surface growth was analyzed with KPZ model. The results show that the growth exponent of β is 0.448 for the films with low deposition rate, and the growth exponent of β is 0.302 for the films with high deposition rate. The growth exponent does not increase with deposition rate, but declines. And the reasons for this phenomenon were explained.

A combinatorial identity for the speed of growth in an anisotropic KPZ model

The speed of growth for a particular stochastic growth model introduced by Borodin and Ferrari in [5], which belongs to the KPZ anisotropic universality class, was computed using multi-time correlations. The model was recently generalized by Toninelli in [38] and for this generalization the stationarymeasure is known but the time correlations are unknown. In this note, we obtain algebraic and combinatorial proofs for the expression of the speed of growth from the prescribed dynamics.

Boron doping effects on microcrystalline silicon film roughness studied by spectroscopic ellipsometry

Microcrystalline silicon thin films were prepared in the p-type chamber using radio frequency plasma enhanced chemical vapor deposition. The surface roughness evolution of microcrystalline silicon thin films was investigated with spectroscopic ellipsometry (SE). The differences between the real-time SE and ex-situ SE have been studied. The effects of boron doping on the surface roughness have been analyzed using real-time SE. For the intrinsic microcrystalline silicon thin films, with an increase in the deposition time, the surface roughness exhibited the following behavior: (a) quickly increased, (b) gradually increased, (c) slightly increased. In the case of boron-doped microcrystalline silicon thin films, the surface roughness showed different behavior: (a) slightly increased, (b) quickly increased, (c) rapidly dropped, (d) increased again. Based on the KPZ model, the intrinsic silicon thin film growth exponent β is about 0.4, corresponding to limited diffusion model. In contrast, boron-doped silicon thin films behavior cannot be described by the KPZ model. Boron catalysis effects promote the reaction radical BHx aggregation, and result in shadowing. Boron doping changed the film growth process.

Random interface growth in a random environment: Renormalization group analysis of a simple model

We study effects of turbulent mixing on the random growth of an interface in the problem of the deposition of a substance on a substrate. The growth is modelled by the well-known Kardar--Parisi--Zhang model. The turbulent advecting velocity field is modelled by the Kraichnan's rapid-change ensemble: Gaussian statistics with the correlation function $\langle vv\rangle \propto \delta(t-t') \, k^{-d-\xi}$, where $k$ is the wave number and $0<\xi<2$ is a free parameter. Effects of compressibility of the fluid are studied. Using the field theoretic renormalization group we show that, depending on the relation between the exponent $\xi$ and the spatial dimension $d$, the system reveals different types of large-scale, long-time asymptotic behaviour, associated with four possible fixed points of the renormalization group equations. In addition to known regimes (ordinary diffusion, ordinary growth process, and passively advected scalar field), existence of a new nonequilibrium universality class is established. Practical calculations of the fixed point coordinates, their regions of stability and critical dimensions are calculated to the first order of the double expansion in $\xi$ and $\varepsilon=2-d$ (one-loop approximation). It turns out that for incompressible fluid, the most realistic values $\xi=4/3$ or 2 and $d=1$ or 2 correspond to the case of passive scalar field, when the nonlinearity of the KPZ model is irrelevant and the interface growth is completely determined by the turbulent transfer. If the compressibility becomes strong enough, the crossover in the critical behaviour occurs, and these values of $d$ and $\xi$ fall into the region of stability of the new regime, where the advection and the nonlinearity are both important.

Dynamics of GDOES-induced surface roughening in metal interfaces

The roughness induced during glow-discharge optical-emission spectroscopy (GDOES) measurements has been reported to cause a loss of resolution during GDOES depth-profiling analysis. In this paper, we undertake for the first time a study of the dynamics of the surface morphology of chromium and titanium thin films (designed in mono and multilayer structures) under the impinging of GDOES incoming ions. We performed this study under the theoretical framework of the dynamic scaling theory, by analysing surface morphology changes, as measured ex-situ by AFM, with irradiation time. For single metal layers it was found that, after an initial surface smoothening, the surface undergoes a rapid steep roughening for both systems, with quite similar quantitative dynamics. Once this roughening ends a second temporal scaling regime arises, operating for long length scales with dynamics depending on the sputtering rate of the material. For the chromium layer, with a very high sputtering rate of 5.5 μm min(-1), this regime is consistent with the KPZ model, whereas for the titanium layer an EW scaling regime is indicated. These different scaling regimes are consistent with the development of larger surface slopes for the Cr system. In the multilayer systems, the initial roughness induced on the top Cr layer by GDOES has similar dynamics to that for single-layer Cr. However, a clear decrease in the roughness was observed once the underlying Ti layer, with a lower sputtering rate, was reached. This decrease in the induced roughness is maintained while the Ti layer is eroded. Therefore, choice of appropriate material (i.e. sputtering yield values) combinations and of their depth of location can enable tuning of GDOES-induced roughness and achieve substantial control over the depth profiling process.

Comparison of different parallel implementations of the 2+1-dimensional KPZ model and the 3-dimensional KMC model

We show that efficient simulations of the Kardar-Parisi-Zhang interface growth in 2 + 1 dimensions and of the 3-dimensional Kinetic Monte Carlo of thermally activated diffusion can be realized both on GPUs and modern CPUs. In this article we present results of different implementations on GPUs using CUDA and OpenCL and also on CPUs using OpenCL and MPI. We investigate the runtime and scaling behavior on different architectures to find optimal solutions for solving current simulation problems in the field of statistical physics and materials science.

Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains

Consider the diffusive Hamilton-Jacobi equation u t = Δu + |∇u| p , p > 2, on a bounded domain Ω with zero-Dirichlet boundary conditions, which arises in the KPZ model of growing interfaces. It is known that u remains bounded and that ∇u may blow up only on the boundary ∂Ω. In this paper, under suitable assumptions on $${\Omega\subset \mathbb{R}^2}$$ and on the initial data, we show that the gradient blow-up singularity occurs only at a single point $${x_0\in\partial\Omega}$$ . This is the first result of this kind in the study of problems involving gradient blow-up phenomena. In general domains of $${\mathbb{R}^n}$$ , we also obtain results on nondegeneracy and localization of blow-up points.

The ideas behind self-consistent expansion

In recent years we have witnessed a growing interest in various non-equilibriumsystems described in terms of stochastic nonlinear field theories. In some of thosesystems, like KPZ and related models, the interesting behavior is in the strongcoupling regime, which is inaccessible by traditional perturbative treatments such asdynamical renormalization group (DRG). A useful tool in the study of such systems isthe self-consistent expansion (SCE), which might be said to generate its own‘small parameter’. The self-consistent expansion (SCE) has the advantage that itsstructure is just that of a regular expansion, the only difference is that the simplesystem around which the expansion is performed is adjustable. The purpose of thispaper is to present the method in a simple and understandable way that hopefullywill make it accessible to a wider public working on non-equilibrium statisticalphysics.

Pseudospectral versus finite-difference schemes in the numerical integration of stochastic models of surface growth

We present a comparison between finite differences schemes and a pseudospectral method applied to the numerical integration of stochastic partial differential equations that model surface growth. We have studied, in 1+1 dimensions, the Kardar, Parisi, and Zhang model (KPZ) and the Lai, Das Sarma, and Villain model (LDV). The pseudospectral method appears to be the most stable for a given time step for both models. This means that the time up to which we can follow the temporal evolution of a given system is larger for the pseudospectral method. Moreover, for the KPZ model, a pseudospectral scheme gives results closer to the predictions of the continuum model than those obtained through finite difference methods. On the other hand, some numerical instabilities appearing with finite difference methods for the LDV model are absent when a pseudospectral integration is performed. These numerical instabilities give rise to an approximate multiscaling observed in earlier numerical simulations. With the pseudospectral approach no multiscaling is seen in agreement with the continuum model.

Studies on surface roughness and growth mechanisms of microcrystalline silicon films by grazing incidence X-ray reflectivity

The microcrystalline silicon films at different growth stages were deposited by plasma-enhanced chemical vapor deposition (PECVD). The reflectivity of grazing incidence X-ray from synchrotron radiation has been applied to investigate the evolution of surface roughness of these thin films. By study of surface morphology of microcrystalline silicon (μc-Si:H), we understand their growth kinetics and growth mechanism. The results show that the growth exponent β is 0.21±0.01 and 0.24±0.01 for μc-Si:H films deposited on glass substrate at fixed substrate temperature, Under the following condition of electrode distance, pressure, rf power density, H2 dilutied at 200 ℃ to be 2 cm, 6.66×102 Pa and 0.22 W/cm2, 99% and 98%, respectively. According to the KPZ model in the PECVD case the growth mechanism of the μc-Si:H films is a finite diffusion growth.

Numerical study of roughness distributions in nonlinear models of interface growth

We analyze the shapes of roughness distributions of discrete models in the Kardar, Parisi, and Zhang (KPZ) and in the Villain, Lai, and Das Sarma (VLDS) classes of interface growth, in one and two dimensions. Three KPZ models in d=2 confirm the expected scaling of the distribution and show a stretched exponential tail approximately as exp(-x0.8), with a significant asymmetry near the maximum. Conserved restricted solid-on-solid models belonging to the VLDS class were simulated in d=1 and d=2. The tail in d=1 has the form exp(-x2) and, in d=2, has a simple exponential decay, but is quantitatively different from the distribution of the linear fourth-order (Mullins-Herring) theory. It is not possible to fit any of the above distributions to those of 1/f(alpha) noise interfaces, in contrast with recently studied models with depinning transitions.

Relationship between particle deposit characteristics and the mechanism of particle arrival

An on-lattice Monte Carlo model is implemented for the simulation of particle deposit growth by advection and diffusion towards a flat surface. The particle deposit structure is characterized by its bulk (density) and interface (mean height and surface width) properties. Numerical correlations, fitted by simple expressions, are reported for these magnitudes, relating them to time (number of deposited particles) and Peclet number. Also a heuristic argument is presented which relates deposit density to local diffusion-limited-aggregation-like processes and interfacial dynamics to the KPZ model.

Cu/Si(0 0 1) epitaxial growth: role of the epitaxial silicide formation in the structure and the morphology

This communication deals with the early stage of the growth of Cu on Si(0 0 1)–H at room temperature. A copper silicide layer of about 1–2 nm of thickness is evidenced by reflection high-energy electron diffraction (RHEED), Auger electron spectroscopy and transmission electron microscopy (TEM). HRTEM experiments show that the silicide structure is in agreement with the bcc β-Cu 0.84Si 0.16 structure. With further deposition of Cu, TEM reveals a highly textured fiber structure with a weak in-plane misorientation (few degrees). Scanning tunneling microscope (STM) observations for different thicknesses show a quasi-continuous film structure with mounds of well-defined sizes and well-defined separation distances. The mean distance L deduced from the height–height autocorrelation function varies as L ( t ) ∼ t 0.5 with Cu thickness t. This particular growth morphology evolution does not correspond to the expected self-affine behavior supported by most kinetic models (based on KPZ model). As a matter of fact, the characteristic distance L is comparable with the size of the grains revealed by TEM. Therefore, it seems that silicide formation at the early stage of copper deposition play an important role in subsequent copper film microstructure.

Time evolution of growth phenomena in the KPZ model

Using a numerical approach based on parallel processing, the time evolution of a growing surface has been studied in the framework of the Kardar-Parisi-Zhang model. The growth term has been introduced as noise with a Gaussian distribution, a nonzero mean value and spatial Gaussian correlations. The procedure allowed us to demonstrate that both surface width and relative porosity evolve towards a steady state. The short-time behaviour and the dependence on the specimen dimension of the asymptotic value of these quantities have been investigated.