Stochastic model showing a transition to selfcontrolled particledeposition state induced by optical nearfields
Abstract
We study a stochastic model for the selfcontrolled particledeposition process induced by optical nearfields. This process was experimentally realized by Yukutake et al. on an electrode of a novel photovoltaic device as Ag deposition under light illumination, in which the wavelength of incident light is longer than the longwavelength cutoff of the materials composing the device. Naruse et al. introduced a stochastic cellular automaton model to simulate underlying nonequilibrium processes which are necessary to formulate unique granular Ag film in this deposition process. In the present paper, we generalize their model and clarify the essential role of optical nearfields generated on the electrode surface. We introduce a parameter indicating the incident light power per site and a function representing the resonance effect of optical nearfields depending on the Agcluster size on the surface. Numerical simulation shows a transition from a trivial particledeposition state to a nontrivial selfcontrolled particledeposition state at a critical value , and only in the latter state optical nearfields are effectively generated. The properties of transition in this mesoscopic surface model in nonequilibrium are studied by the analogy of equilibrium phase transitions associated with critical phenomena and the criteria of transition are reported.
1 Introduction
Nanophotonics, which investigates lightmatter interactions on the nanometer scale, has been intensively studied from a variety of aspects ranging from fundamental interests, such as atom and optical nearfield interactions [1, 2, 3], to applications including environment and energy [4, 5], healthcare [6], solidstate lighting [7], information and communications technologies [8, 9], among others. Precision control of the geometrical features, such as the size, layout, morphology, at the nanometer scale are important in realizing valuable functionalities [1, 3]. In this context, the nanophotonics fabricationprinciples and techniques have been providing interesting and important lightassisted, selfcontrolled nanostructures; nanoparticle array formation [10], light emission from indirecttransitiontype semiconductors (such as silicon) [7], appearances of photosensitivity at wavelengths longer than the longwavelength cutoff [5], etc. In these lightassisted material formations, while elemental physical processes indeed involve optical nearfield interactions at the nanometerscale, the system is open to environment and thus is accompanied with energy flow and environmental fluctuations. For these systems, stochastic modeling [11, 12] is very useful in order to obtain deeper understandings of the underlying physical processes as well as to gain design principles for future devices and systems [3]. In the present paper, we develop the cellular automaton model [13] proposed by Naruse et al. [14], which was introduced to simulate selfcontrolled pattern formation of Ag film reported by Yukutake et al. [5] on the electrode of their photovoltaic device.
Figure 1 shows the crosssectional structure of the photovoltaic device of poly(3hexyl thiophene) (P3HT) and ZnO sandwiched by Ag and indium tin oxide (ITO) electrodes. A P3HT layer (about 50 nm thick) and a ZnO layer (about 100 nm thick) are used as ptype and ntype semiconductors, respectively, and an ITO film (about 200 nm thick) and an Ag film (a few nanometers thick) are used as two electrodes. This multilayered film with an area of 30 mm is deposited on a sapphire substrate in series. At the last stage of the fabrication process of the multilayered device, Ag was deposited on the Ag thin film. The Ag is deposited by radio frequency (RF) sputtering under light illumination while applying a reverse bias DC voltage, V, to the P3HT/ZnO pn junction. The wavelength of the incident light is 670 nm, which is longer than the cutoff wavelength nm of P3HT. Under light illumination, optical nearfields are locally generated on the Ag surface. They induce coherent phonons and form a coupled state with them, which is called a virtual excitonphononpolariton [15] or the dressedphotonphonon (DPP) [3]. If the DPP field extends to the pn junction, the twostep excitation process of electrons occurs (see [5] and Sec.7.2.1 of [3]) and then electronhole pairs are created at the pn junction. As illustrated by Fig. 2, by the reverse bias voltage, the electrons and holes are separated from each other. The positive holes are attracted to the Ag film, which make the Ag film be positively charged. It was argued in [5, 14] that due to randomness in Ag deposition process by RF sputtering, the density of Ag deposits will spatially fluctuate. Since the optical nearfields are generated by shortranged lightmatter interaction, the optical nearfields will also become spatially inhomogeneous in the film. In the local area of the Ag surface, where the optical nearfield is effectively induced, more positive holes are generated and transferred into the area. In such an area, which is more positively charged than other areas, the subsequent deposition of Ag will be suppressed, since the sputtering Ag is positively ionized by passing through the argon plasma used for RF sputtering. Such a feedback mechanism induced by optical nearfields will lead to the unique and nontrivial granular structure of the Ag film and such a selfcontrolled pattern formation was indeed observed in experiments [5].
Naruse et al. [14] introduced a stochastic cellular automaton model on a square lattice [13] such that each cell has one of the two values 0 and 1, which represents a vacancy and an occupation by an Ag grain, respectively. Time evolution of Ag clusters in random deposition of Ag grains on a lattice has been numerically simulated. It was assumed in their model that the repulsive force acting on an Ag grain when it is deposited on a site in the lattice is simply proportional to the total number of occupied sites in the eight neighbors (i.e. the Moore neighborhood in the square lattice). This model works well as a minimal model for the inhomogeneous pattern formation of Ag film on an electrode of the photovoltaic device, while a role of spatially inhomogeneous chargedistribution due to the heterogeneous effect of optical nearfields was not clarified.
In the present paper, we propose a nonequilibrium statisticalmechanics model on an square lattice , in which two kinds of stochastic variables at each site evolve in discrete time ; the number of deposited grain on the site and the amount of charge per site . These two variables are dynamically coupled as explained below. It is known that the effective potential of optical nearfield is welldescribed by a Yukawa function of a distance , . Here the interaction range is proportional to the size of the matter which generates the optical nearfield under light illumination [3]. In the present situation, the range will be approximately equal to the size of Agclusters. Hence growth of sufficiently large clusters is required in order to realize the situation such that the optical nearfields generated on the Ag surface reach the pn junction and electronhole pairs are created (see Fig.2). As a result, the charge increment of a cluster of Agdeposited sites depends on the cluster size , which is a functional of . We introduce a characteristic size of the cluster at which the charge increment of a cluster caused by the optical nearfields is maximized. Moreover, we assume existence of a characteristic sizevariation such that if the cluster size deviates from by much larger than , that is, if , then the effect of the optical nearfields for charging cluster becomes very small. These assumptions are due to the basic property of DPPs, the optical nearfields coupled with phonons, called the sizedependent resonance, and here we use the Gaussian function
(1.1) 
to represent such size dependence (see Fig.2.6 in Sec.2.2.2 of [3]). Furthermore, in our numerical simulation, we perform exact calculation of longranged repulsive Coulomb interaction between the Ag grain being deposited on a site and positively charged Ag clusters formed on the surface.
The main result of our numerical simulation of the model is that, when the lattice size is large but finite, there are following two distinct mesoscopic states.

State A : If Ag clusters on the surface are formed with a proper size such that , then the optical nearfields are induced effectively to increase the charge of clusters and hence the total positive charge on the surface increases monotonically in time. It causes strong suppression of further deposition of Ag grains on the surface and then the deposition will be stopped. As a result, a nontrivial and unique Ag film is formed.

State B : If sizes of Ag clusters on the surface tend to deviate from by more than , the optical nearfields are not induced effectively. In this case the charge on surface will be saturated at a low level, and the suppression of deposition of Ag grains on the surface remains weak. Hence the random deposition process will continue without any selfcontrol and Ag film formation is rather trivial; as long as laser light is irradiated the density of Ag grains on the surface gradually increases.
We will show that if we change some parameters of model, for instance, a parameter which denotes the incident light power per site, then there occurs a transition from State B to State A, and then the selfcontrolled formation of Ag file is realized. We remark that in the real experiments the pn junction was reversely biased with a fixed voltage ( V). Therefore, the positive charge on the Ag surface is bounded and it will be saturated. In order to simplify the model, however, we do not take into account such an effect in the present study.
The paper is organized as follows. In Sec.2 the setting of the present model, the algorithm of processes, and the quantities which we will calculate are explained. The results of numerical simulations are given in Sec.3. There properties of the dynamical transition of surface state are studied by the analogy of equilibrium phase transitions associated with critical phenomena [11, 12, 13]. Sec. 4 is devoted to concluding remarks.
2 DiscreteTime Stochastic Model on a Lattice
Let be an integer and consider an square lattice . For two sites , with distance , we say that they are the nearestneighbor sites. We consider a stochastic process with a discrete time . Here a particle will represent a positively charged Ag grain. At each time , the following two kinds of stochastic variables will be defined,
(2.1) 
, where and (the set of nonnegative real numbers). At each time , a collection of pairs of these stochastic variables over the lattice gives a configuration of the process, which is denoted by
(2.2) 
In a given configuration , if and for a pair of the nearestneighbor sites , we say that these two sites are connected. Moreover, for a pair of sites and with , if there is a sequence of sites in with , such that any pair of successive sites are connected, then we also say that and are connected. For each site with , a twodimensional cluster including the site is defined by
(2.3) 
By definition, if and are connected, then . The total number of sites included in is denoted by , which represents an area on of the cluster of deposited particles.
We introduce the parameters in as
effective coupling constant of  
(2.4) 
If the kinetic energy of Ag grain injected to the surface by RF sputtering is relatively small, the deposition on the surface will be inhibited by the Coulomb repulsive force between the Ag grain and the positively charged surface. In the present stochastic model, we do not calculate such electrodynamical processes and simply introduce two variables and as parameters. We set a function , which specifies the increment ratio of charge per site and per timestep as a function of cluster size to which the site belongs. We assume that is given by (1.1) with positive constants and . Now we explain the elementary process, .
Assume that a configuration is given at time .

(i) Choose a site randomly in . The chosen site is denoted by .

(ii) Calculate the repulsive Coulomb potential at caused by all charges on by
(2.5) where , if and , if . (Here we consider a ‘coarse grain model’, and thus any singularity of the Coulombpotential function should be eliminated.)

(ii1) If , then a charged particle is deposited at the site ;
(2.6) 
(ii2) If , then let and calculate at every nearestneighbor site of , . If the site which attains is uniquely determined, let that site be . When the minimum is attained by plural sites, we choose one of them randomly and let it be . If , then we regard that the particle is repulsed from the charged surface and cannot be deposited. Hence the configuration is not changed at all,
(2.7) If , we set . If , calculate at every and follow the same procedure as above to determine a site . If , the deposition is not done and we have (2.7). If , set . If , we repeat the similar procedure to those explained above. When is determined, the charged particle is deposited at the site ;
(2.8) Each sequence with some simulates a path of drift process on the surface performed by a particle before it is deposited at

(iii) Clusters at time , are redefined for the configuration given by (ii). For each cluster , the accumulated total charge has been . In addition to that, the following amount of charge is added; . (See Eq.(1.1) and the explanation given above it.) Then the charge at site at time is given by
(2.9) We note that the charge distribution in cluster can be spatially inhomogeneous in general depending on the shape of cluster on the surface. We have assumed the uniform distribution in (2.9), however, since keeping the model simple we would like to report the advantage of introducing the variable , which was not considered in the previous cellular automaton model [14], in order to realize transitions between the two distinct particledeposition states.
We start at the empty configuration and observe the following quantities at times with a sufficiently large ,
(2.10) 
where is an indicator function of a condition ; if is satisfied, and otherwise. The quantity shows the total area of clusters at time and the total charge of particles deposited on the surface at time . We set
(2.11) 
which denotes the occupation ratio by deposition on the surface at time . We also define
(2.12) 
with a given .
3 Simulation Results
We performed computer simulation of the stochastic model. Here we fix the following parameters as
(3.1) 
and
(3.2) 
The parameters and are changed and dependence of the process on them is studied. (We will also discuss the results on systems with different sizes in Sec.3.3 and Sec.4.)
3.1 Two states
Figure 3 shows time dependence of the occupation ratios by deposition on the surface, (solid lines), and the total charges on the surface normalized by the values at , (broken lines), for , and , when we set . When is small ( and 0.30), increases monotonically and shows saturation after some timeperiod. When is large (), on the other hand, shows saturation in time, while increases monotonically. The results implies when is small the system is in State B, while is large in State A. The critical value of will be evaluated in the next subsection as in this case with .
3.2 Order parameter and critical exponent
When the system is in State B, increases monotonically and , that is, the surface will be almost covered by particles. On the other hand, in State A, the deposition is suppressed by which increases monotonically in time and the surface formation is selfcontrolled. As a result we will have a nontrivial steady state, in which .
The vacantsite density on the surface in the steady state is defined as
(3.3) 
It will play a role as an order parameter for the transition from State B to State A.
Here is approximated by the value at . For each choice of parameters, we performed ten independent simulations and studied the averaged values of . Figure 4 shows dependence of on for , and 50. There ’s seem to behave as continuous functions of . It implies that the transition is continuous and the critical value will be defined by
(3.4)  
Figure 5 shows the dependence of for , and 50. The result implies that, for , , while for , (see the remark given at the end of Sec.1). Thus the critical value is also characterized as
(3.5)  
As an analogy of the secondorder phase transitions associated with critical phenomena in the equilibrium statistical mechanics, singular behavior of the order parameter in the vicinity of transition point in State A is expected to be described by the following powerlaw,
(3.6) 
For each value of , we performed the numerical fitting of the data to the powerlaw (3.6) and evaluated values of and . For example, the curve in Fig. 6 shows the obtained curve (3.6) by fitting the data for , which gives and . As we will report in the next subsection, the critical value depends on , but we found that the dependence of evaluated on is very small; 0.39 for 50.
3.3 Critical line between State A and State B
We have evaluated the critical values as , and , respectively. They are plotted by cross marks in Fig. 7. We can see that the critical line is well described by a straight line,
(3.7) 
As discussed in Sec.4, the critical value also depends on the lattice size . Because of the longranged Coulomb potential (2.5), decreases monotonically as increases. In other words, the coefficient in (3.7) is dependent; , and . In this sense, the transitions between State A and State B are not usual phase transitions, which should be defined in infinitesize (thermodynamic) limits. Therefore, Fig. 7 cannot be regarded as a phase diagram. In the present study, we are interested in transitions in nonequilibrium states found on a surface of device with a mesoscopic scale. The values themselves of as well as reported here are not so important, since they change depending on system size. Our preliminary study by numerical simulations for different lattice sizes imply, however, that the linear dependence of on is universal as far as the system size is large but finite as shown in Fig. 7. Note that validity of the powerlaw (3.6) has been numerically confirmed also for different sizes of with .
3.4 Cluster structures in State A
For , a configuration of clusters on the surface obtained in the steady state (at ) is plotted in Figure 8, where occupied sites by particles are dotted. For , we have estimated . Then this figure shows a selfcontrolled surfaceconfiguration realized in State A. Averaging over ten independent simulations, distribution of cluster size on the steadystate surface in State A is calculated. Figure 9 shows it for . A peak is found around , which is consistent with our choice of parameter in the simulation. The present results are consistent with the experimental observation reported by [5] as well as the simulation results by [14].
4 Concluding Remarks
In the present paper we generalized the cellular automaton model [13] by Naruse et al. [14] and proposed a new stochastic model to describe the nonequilibrium dynamics of Agfilm pattern formation on an electrode of the photovoltaic device of Yukutake et al. [5]. In the present new model, not only the number of deposited particle but also the amount of charge are considered as stochastic variables at each site in the square lattice at time . The previous model used the ‘pseudo footprint’ method to take into account the repulsive interaction between an Ag ion approaching the surface and positively charged Ag clusters on the surface. In the present algorithm of model, the repulsive Coulomb potential at each site caused by all charges on the lattice except the site is exactly calculated at each timestep, and drift and deposition processes are simulated. The essential improvement of the model is such that we include the effect of optical nearfields generated by lightmatter interaction by introducing a parameter and a function . There denotes the incident light power per site on the lattice and expresses the sizedependent resonance effect of optical nearfields on mattersize . We have regarded as an external control parameter of the model. We have shown that, as increases, there occurs a transition of the surface state of pattern formation with a critical value , such that if the optical nearfields are not induced effectively (State B), while if they are induced effectively and Ag deposition process is selfcontrolled (State A). A nontrivial and unique Ag film is formed in State A, while in State B random Agdeposition will be continued as long as the simulation is continued. We have discussed the transitions of surface processes from State B to State A by the analogy of equilibrium secondorder phase transitions with critical phenomena [11, 12, 13]. We also studied the dependence of on the threshold energy for elementary deposition of single Ag on the surface.
The transition of surface state discussed in this paper is not a phase transition in the usual sense, since our model is defined on a lattice with a finite size ; for the simulation data reported above. In equilibrium and hence in stable systems the limit will provide the thermodynamic limit and it describes macroscopic behavior of materials. In the present model, however, the direct limit may be meaningless, since the repulsive Coulomb repulsive potential (2.5) will diverge as in the late stage of process in which sufficient amount of charge is accumulated on the surface. (It implies that should be a decreasing function of , since given by (2.5) will take larger value as increases, and as for all .) In the present study, we are interested in nonequilibrium deposition dynamics of charged particles on mesoscopic materials. We should assume that the system size of model is sufficiently large but finite.
From the view point of study of nonequilibrium statistical mechanics [11, 12, 13], it is an interesting problem to find a relevant scaling limit which can realize the present transitions of surface state as nonequilibrium phase transitions. A preliminary study by changing the lattice size in numerical simulations implies that the parameter should be scaled as with a constant and a scaling exponent in the infinitesize limit . The dependence of on reported in Fig. 7 is consistent with it and gives a preliminary estimate . Such a mathematical physics aspect of the present model is also interesting and will be reported elsewhere.
Acknowledgements The present authors would like to thank S. Tojo for useful discussions on the present work. This study is supported by the GrantinAid for Challenging Exploratory Research (No.15K13374) of Japan Society for the Promotion of Science. MK is supported in part by the GrantinAid for Scientific Research (C) (No.26400405) of Japan Society for the Promotion of Science.
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