Transitions from low-density state towards high-density state in stochastic bistable plasma-condensate systems

In this article we study transitions from low-density states towards high-density states in bistable plasma-condensate systems. We take into account an anisotropy in transference of adatoms between neighbour layers induced by the electric field near substrate. We derive the generalized one-layer model by assuming that the strength of the electric field is subjected to both periodic oscillations and multiplicative fluctuations. By studying the homogeneous system we discuss the corresponding mean passage time. In the limit of weak fluctuations, we show the optimization of the mean passage time with variation in the frequency of periodic driving in the non-adiabatic limit. Noise induced effects corresponding to asynchronization and acceleration in the transition dynamics are studied in detail.


I. INTRODUCTION
Plasma-condensate systems serve an useful technique to produce well structured thin films with separated multi-layer adsorbate islands of nano-meter size of semiconductors and metals [1,2]. Nowadays such nanostructured thin films attract increasable interest because of their technological application in modern nanoelectronic devices due to exceptional functionality [3][4][5][6]. Adsorptive bistable systems manifest stochastic resonance phenomenon under conditions of periodically varying pressure of gaseous atmosphere (see, for example, [7]), or chemical potential [8]. In the technological applications this stochastic resonance effect is used to optimize the output signal-to-noise ratio, when fluctuations (noise) play a constructive role and enhance a response of a nonlinear dynamical system subjected to a weak external periodic signal [9][10][11][12][13][14][15].
Previously it was shown, that one can control the adsorbate concentration on the substrate and the corresponding first-order phase transitions between lowdensity and high-density states by varying temperature, adsorption and desorption rates in adsorptive systems [16][17][18][19][20][21][22][23]. At the same time it was shown that multilayer systems manifest cascade of first-order phase transitions when a new additional layer of adsorbate is formed [24,25]. Such systems mainly were studied under the assumption of the equiprobable transference of adatoms between neighbor layers according to the standard verti- * Electronic address: vasiliy@ipfcentr.sumy.ua cal diffusion mechanism [26,27].
A fabrication of nanostructured thin films with a help of plasma-condensate devices is governed by the following mechanism. Ions, sputtered by magnetron, attain growing surface, located in a hollow cathode due to presence of the electric field nearby it and become adatoms. During exposing under the influence of electric field near the substrate the main part of adatoms are re-evapourated to be later ionized again and returned back onto the upper layers of the growing surface [28]. Hence plasma-condensate system are characterized by the anisotropy in the transitions of adatoms between neighbor layers, induced by the electric field, with preferential motion from bottom layers towards top ones. We have shown previously, that in such bistable system the anisotropy strength, related to the strength of the static electric field near substrate, controls dynamics and morphology of the growing adsorbate structures [29]. In Ref. [30] we have derived a reduced one-layer model describing pattern formation on the intermediate layer of multi-layer system.
In the present study we will focus on transitions from low-density states towards high-density states in the effective reduced model of plasma-condensate system, derived in Ref. [30], by taking into account both periodic oscillations and fluctuations of the strength of the electric field. The main aim of the work is to define an influence of such an external influence onto the mean passage time from low-to high-density state in a homogeneous model of multi-layer plasma-condensate systems.
We organize our work in the following manner. In the next section we discuss the stochastic model of plasmacondensate system. In section 3 we analyze an influence of the periodic driving and stochastic force onto mean arXiv:1806.08526v1 [cond-mat.stat-mech] 22 Jun 2018 passage time. Main conclusions and prospects for the future are collected in the last section.

II. MODEL OF ADSORPTIVE SYSTEM
By considering the adsorbate concentration x n ∈ [0, 1] on the selected n layer of a multi-layer system we will follow Refs. [24,29] and describe an evolution of the adsorbate on each n layer by the reaction force f (x n ) which includes adsorption, desorption and transference reactions between neighbor layers. Adsorption processes are governed by the term where adsorption is possible on free sites on the current n layer if there are both non-zero adsorbate concentration on the precursor (n − 1) layer and free space on the next (n + 1) layer exist; k a is the adsorption coefficient, proportional to density of the plasma. Desorption processes are described by the term Here we use the same limitations and admit desorption mediated by precursor layer; k d is the desorption coefficient and ε is the interaction strength of adsorbate, proportional to the inverse temperature T . Transference of adatoms between neighbor layers is described by the ordinary vertical dif- where the anisotropy strength k r is proportional to the strength of the electric field |E| nearby substrate: k r = |E|Ze/T , where Z is the valence of ion and e is the electron charge.
To define the adsorbate concentration on the (n − 1) and (n + 1) layers through x n we will exploit the receipt proposed in Ref. [30], by assuming that the adsorbate concentration decreases with the layer number growth. In the framework of the simple model we previously have shown [30] that the adsorbate concentration on the n-th layer, x n , can be defined as the ratio between square, occupied by the adsorbate on the n-th layer and on the substrate, that gives x n = (1 − n(d/R 0 )) 2 , where n is the layer's number, R 0 is the linear size of the substrate and d is the mean width of the terrace of the multi-layer (pyramidal) adsorbade structures. By defining x n−1 and x n+1 in the same manner and introducing the small parame- By combining all the terms and dropping index n we finally get the evolution equation of adsorbate concentration on the selected level of multi-layer plasma-condensate system in the following form [30]: where the following notations are used: Analysis of the stationary states of the deterministic system (1), defined from the condition d t x = 0 allows us to obtained the phase diagram, shown in Fig.1 the top insertion corresponds to the spinodal and plotted at α = 0.064, ε = 4.0 and u 0 = 0.7 (black dot inside the cusp in Fig.1).
The control parameter u defines the external conditions for layers growth. Generally, it can be a function of time and/or can be changed in stochastic manner. Next we assume that the anisotropy strength u is subjected to both periodic oscillations and fluctuations: u = u 0 + A sin(ωt) + ξ(t), where u 0 = u and ξ(t) is the Gaussian noise with zero mean, ξ(t) = 0 and correlation ξ(t)ξ(t ) = 2σ 2 δ(|t − t |); σ 2 is the fluctuation's intensity. In such a case the deterministic evolution equation (1) attains the form of the Langevin equation of the form with g(x) = γ(x). It follows, that if x = x 0 , where γ(x 0 ) = 0 then an influence of the electric field near substrate onto adsorbate concentration disappears leading to

III. MEAN PASSAGE TIME
Usually combined effect of periodic and stochastic driving forces in bistable potentials leads to stochastic resonance phenomenon. According to this scenario slow periodic force moves "Brownian particle", located in one minimum of the bistable potential towards its maximum. It the periodic driving synchronizes with the fluctuation force then last one throws the particle over a potential barrier and a switch between the two stable states occurs. Let us provide a detailed description of transitions from low-to high-density state by studying the passage time. To this end we fix α = 0.064, ε = 4.0 and u 0 = 0.7, corresponding to the black point on the phase diagram in Fig.1 on the spinodal and perform numerical simulations by solving numerically the Langevin equation (3) on graphical processor units (GPUs) with double precision. This technique provides an effective acceleration of numerics by a factor of about 500 over the standard CPUs computing for this special problem. We exploit the Heun method for numerical simulations with time step ∆t = 0.001.
In Fig.2 we present the time dependence of the concentration of the adsorbate (one realization shown by grey color) and mean adsorbate concentration, averaged by 10 4 realizations (black curve). In all simulations the initial condition for the adsorbate concentration was selected in the minimum of the potential U (x), corresponding to the low-density state. It follows, that during system evolution combined influence of periodic and stochastic driving leads to the transition towards high- density state, that in average occurs at time instant t p , when the dispersion (δx) 2 , averaged over an ensemble, falls to zero after its maximum (see insertion in Fig.2). Next we will study an influence of the periodic driving (amplitude A and frequency ω) and stochastic force (fluctuation's intensity σ 2 ) onto the mean passage time (mpt) from low-density state,corresponding to minimum of the bistable potential U (x) with small x, towards highdensity state (the minimum of U (x) with large x), defined as: mpt = N −1 N i=1 t p with N = 10 4 realizations.

A. Limit of quasi-deterministic driving
First we focus our attention onto an influence of the periodic driving in the limit of weak fluctuations with σ 2 = 10 −5 . Dependencies of the mean passage time mpt from the low-density state to the high-density state on amplitude of the periodic driving A at different values on the frequency ω and on frequency ω at different values of amplitude of the periodic driving A are shown in Figs.3a,b, respectively. From Fig.3a it follows, that at small values of the amplitude A the transition becomes impossible due to log(mpt) → ∞ independent on the frequency ω. With an increase in the amplitude A the value of the mpt abruptly decreases and remains constant at large values of A. An increase in the periodic driving frequency ω requires elevated values of the driving amplitude A for the transition from the one hand, and results to a decrease in the transition time at large A, from the other one.
The dependence mpt(ω), shown in Fig.3b has more complicated structure. Here we have the special kind of synchronization: at fixed value of the periodic force amplitude A an increase in the frequency leads to a decrease in the transition time, until the minimal value mpt min is reached; with further growth in ω the mpt increases. This minimal value of the transition time mpt min decreases with amplitude A growth. Hence, the mpt optimizes with the frequency of the periodic driving at nonadiabatic limit.

B. Noise-induced effects
Next we will analyze a change in the transition time mpt by varying the fluctuation's intensity σ 2 for different values of amplitude A and frequency ω of periodic driving. The dependencies mpt(σ 2 ) at A = 0.06 and different values of ω are shown in Fig.4a. It is seen that at small values of the driving frequency ω an increase in the noise intensity weakly decreases the transition time (see curve with filled squares at ω = 0.001 in Fig.4a). If σ 2 becomes large enough, σ 2 > σ 2 max , then the stochastic force starts to play a dominant role in the system dynamics and a further increase in its intensity significantly decreases the transition time from low-to high-density state. An increase in the periodic driving frequency at small σ 2 acts in the manner, presented in Fig.3b: mpt decreases, attains value mpt min and then increases. At σ 2 < σ 2 max the value mpt min weakly increases with σ 2 . At large values of the periodic force frequency the transition from low-to high-density state occurs at elevated values of the fluctuation's intensity, only. At narrow interval of frequency values the mean transition time manifests non-monotonic dependence on the noise intensity (see curves with filled and empty circles at ω = 0.005 and 0.007 and curve with filled triangles at ω = 0.0074). Here, with increase in the fluctuation's intensity the transition time increases, attains maximal value and then decreases. Hence, at small values of the noise intensity its increasing leads to the delay in the transition dynamics. It means, that in such conditions one has asynchronization periodic and stochastic driving: while periodic force moves the "Brownian particle" towards the maximum of the bistable potential, fluctuations return it back to the low-density state. With further increase in σ 2 the noise starts to play the dominant role in the system dynamics, that leads to the transition through the potential barrier.
In Fig.4b we present dependencies of the meant transition time on the noise intensity at fixed value of the periodic driving frequency ω = 0.01 and different values of the driving amplitude A. It follows, that with an increase in A the transition from low-to high-density state occurs faster. An increase in the noise intensity leads to (i) a decrease in the transition time at small values of the periodic driving amplitude; (ii) a delay in the transition dynamics at intermediate values of A; (iii) at elevated values of the periodic driving amplitude the noise acts the system dynamics at large values only.

IV. CONCLUSIONS
In this article we have provided the detailed study the transitions from low-density state to high-density state in multi-layer plasma-condensate systems in reduced onelayer model. By taking into account anisotropy in transference between layers, induced by the electric field near substrate, we assume periodic oscillations and fluctuations of the electric field strength. We discussed an influence of both periodic driving and stochastic force onto mean passage time needed for transition from the lowdensity state towards the high-density state. It is shown, that in the case of weak fluctuating force the mean passage time decreases with the periodic driving amplitude growth and optimizes with the frequency of periodic driving. An increase in the intensity of the electrical field fluctuations at small σ 2 delays the transition towards high-density state due to asynchronization in periodic and stochastic driving; and at large σ 2 leads to a decrease in the time needed to pass from low-density state towards high-density state.
It was shown that there is a special state x 0 in the derived model for the multi-layer plasma-condensate system, where the anisotropic transference reactions between layers, induced by the electric field presence near substrate, degenerate. It means that introduced fluctuations of the electric field strength die in this state due to g(x 0 ) = 0, whereas the deterministic force f (x 0 ) = 0 exists. Hence, the effective potential U ef f = − f /g 2 dx + σ 2 ln(g) for the stochastic system (3) in the state x 0 diverges. This singularity requires a detailed study which can be made elsewhere.