Thermally activated escape rate and dynamics of a particle under a harmonic potential

In this paper, we study the dynamics of particles along a semiconductor layer by imposing a confinement potential assisted by both thermal noise strength D and trap potential ϕ. By applying a nonhomogeneous cold temperature alongside the uniform background temperature, the system is driven towards a phase transition. When a weak signal is pass across a semiconductor layer, the thermally activated particles become easily hop from one lattice site to another lattice site. We perform a numerical simulation of the trajectory of a particle under a harmonic potential represents a bistable and tristable effective potential as a function of thermal noise. As a result, at an optimal level of noise, the particle synchronizes with a weak periodic signal.


Introduction
Noise from a few years back can be considered as a determinants and degrading factor for systems performance.However, noise can also play a constructive role to enhance performance of nonlinear system, called Stochastic resonance (SR).SR is a phenomenon where in the response of a nonlinear system to the weak input signal can be amplified with an appropriate amount of the noise level.As a result, its integral system properties such as periodic response, signal-to-noise ratio, and spectral amplification can be described as a function of noise intensity at an optimal noise level [1].
The phenomenon of SR was clearly introduced four decades years back by Benzi et al [2] as an explanation of the observed periodicity in the ice ages on earth.Various works have resulted since then [3][4][5][6][7] to name a few.SR has become an attractive research topic in the field of nonlinear science and has been implemented in confined systems [8], complex systems [9][10][11][12] and Brownian particle moving across a porous membrane [13] as well as widely adopted in amplification of weak signals in different research fields.Researchers started addressing the stochastic resonance effect in a bistable system theoretically described based on a two-state model [3,14] and its effect could be explained in a consideration of the motion of over-damped particle in symmetric double-well potential subject to both noise source and periodic driving force [2].Therefore, the mobility of the dopants in a semiconductor can also be enhanced by applying time-varying signals, and in this particular case, the system may lead to stochastic resonance.
In the field of semiconductor physics has received a considerable attention due to the problem of how to adjust the conductivity of the semiconductor by adding the impurities into their crystal lattice and manipulating the level of conductivity on the type and amount of impurities.Recent studies showed that the conductivity of the semiconductor is controlled by also changing the strength of the background temperature along the semiconductor layer [15][16][17][18][19][20][21][22][23][24][25].The background temperature in a semiconductor affects the concentration and diffusivity of impurities which are responsible for conduction.Researchers clearly described that the nonlinear potential produced by the movable split gate along the semiconductor layer [26] and neglecting the interaction between the dopants at low impurity density, the diffusion of impurities can be controlled not only by varying the external potential but also by altering the intensity of the temperature and trap depth of the impurities in the semiconductor layer [18][19][20][21][22][23].They proposed the way of eradicating the unwanted impurities from the certain region of the semiconductor by varying the different model parameters and studying the stochastic resonance when the time varying signal applied to the semiconductor, that may induce the system into stochastic resonance (SR).
In Refs.[20], [21], they have designed a semiconductor layer subjected to an external quartic and double well potential under a uniform trap depth potential, respectively.By applying a non-homogeneous cold temperature in addition to a background uniform temperature, the system undergoes a phase transition to effective bistable and tristable potential, respectively.In this paper, we consider a bistable and tristable symmetrical effective potential.We explore the thermally activated crossing barrier rate with different model parameter of both models.Furthermore, we also study the numerical simulation of input/output synchronization with a timevarying weak signal.
The rest of the paper is organized as follows: in section 2, we present the model and the non-equilibrium phase transition is introduced.In section 3, we study the thermally activated barrier crossing rate of the impurities under both effective bistable and triple well potentials behaves as a function of the model parameter.
In section 4, we study trajectory of an overdamped Brownian particle in both a symmetrical double well and triple well effective potential.Section 5 deals with the conclusion.

The model and effective potential
We consider a doped semiconductor layer which contains uniformly distributed traps and lattice sites.The impurities at low density, where the Coulomb-Coulomb interaction neglected, can be hop from one lattice site to another site by the help of thermal noise and an external potential.The dynamics of impurities can be governed by modified Fokker-Planck equation [27][28][29] where P(x, t) is the probability density of a particle at position x at time t, f denotes the potential energy of the particle at a lattice site or trap and V x ¢( ) is the force of the particle at site x due to an external potential.
The steady-state probability distribution P ss (x) of equation (1) can be written in the form Where c 0 is a normalization constant.
In this section, we obtain the effective potential from both quartic and double well potentials due to position dependent temperature and uniform traps of a semiconductor layer.

The quartic potential
We consider an external confining quartic potential energy is given by [20] where V 0 denotes the potential energy amplitude, x is the position of lattice site in the range between x max to x max .Due to this potential profile, the impurities accumulated around the potential minimum.When the nonhomogeneous temperature added to the system, which is position dependent, that can be expressed as [20] T x T induces the system undergoes a phase transition from a single well to double well potential.As a result, the positively charged particle pile up around two distinct position of the effective potential minima while the negative charged particle migrates mostly around the edge of the semiconductor.Here the parameter T 0 , x and σ are the background temperature, the spatial length of the excess temperature and the width of the laser beam's radiation (light), respectively.
By substituting the expression of V(x) from equation (3) and T(x) from equation (4) into equation (2) we get where the effective potential energy, V eff (x), which is expressed as The effective potential of equation ( 6) is plotted in figure 1.The figure shows that the potential barrier increase with the trap potential depth increases.
We want also to stress that the dopants that are exposed to the quartic potential and non homogeneous background temperature undergo a non equilibrium phase transition.On the other hand, the regime where σ ?x 0 and f = V 0 , the system does not occur phase transition.

The double well potential
The external confining double well potential is expressed as [21], eV nm 4 respectively and x is the position of impurities in the unit of nm.As a result, the impurities are resides around the potential minima as long as the background temperature is homogeneous.When a system subjected into a nonhomogeneous temperature [21], the system may undergoes a phase transition from a bistable to a tristable effective potential.As a result, the impurities migrates into three locations.The parameter T 0 and σ denotes the background temperature and standard deviation of a laser coolant, respectively.By substituting equations (7) and (8) into (2), we get the expression of equations ( 5), where an effective potential V eff (x) is given by: From equation (9), the effective potential has three stable minimum and two unstable maximum when f > 0 and 2 a b s < are satisfied.The effective potential of equation ( 9) is plotted in figure 2. The figure shows that the potential barrier increase with increase the trap potential depth.
In section 3, we explore how the thermally activated barrier crossing rate of impurities in a harmonic potential under different model parameter.

Thermally activated escape rate
The effective double and triple well potential energies at the saddle V eff (0) and V eff ( ± x 1 ), respectively and stable points are given as V eff ( ± x m ) and V eff ( ± x 2 , 0), respectively.The barrier height of the two minima of the effective bistable and tristable potentials are given by respectively.The thermally activated barrier crossing rate of the impurities can be described by considering the non-interacting impurities initially situated at one of the potential minima.Due to thermal fluctuation, the particles cross the potential barrier assisted by the thermal kicks they encounter along the reaction coordinate.The crossing rate for the impurities in a high barrier limit In the next sections, we investigate the thermally activated escape rate of an effective harmonic potential under different model parameter.

Effective bistable potential
Figures 3 and 4 shows that how the thermally activated barrier crossing rate behaves as a function of noise strength (D).When the barrier height of the effective potential increase that means the particles takes a longer time and oscillates around one of a potential minima, the transition rate is also decreased.Figure 3 plots for fixed σ = 1 nm(dotted line), σ = 0.5 nm(solid line), V 0 = 0.3 eV, f = 0.1 eV and x 0 = 1.5 nm. Figure 4 is plotted for parameter choice f = 0.1 eV(solid line), f = 0.5 eV(dotted line), V 0 = 0.3 eV, σ = 1 nm and x 0 = 1.5 nm.Both figures 3 and 4 depicts thermally activated rate is increasing with increasing thermal noise, and maximum when the potential barrier is smaller.Since σ increase and f decrease, the potential barrier height is decrease.Note that, all results is valid at high barrier limit.= and σ = 0.3 nm.

Effective tristable potential
In figure 5 shows how escaping rate r K behaves with noise strength D for different model parameter.Whatever the values of parameters varies, if D is very small the particle still remains at the potential minima where it was initially located.When D increases the particle has a tendency to cross the potential barrier occasionally.Overall we observe that r K is increase when D increase.When σ decrease the potential barrier is increase.Thus, the particle attains a difficulty when crossing the barrier height as shown in figure 5.In figure 6 both r K increase with increasing D.Here we take two different f values.We observe that when f increase r K is decrease because the potential barrier increase.
In the next section, we study the trajectory of a particle in both double well and triple well effective potential with the noise strength by fixing angular frequency Ω.

Dynamics of a particle in a symmetrical double and triple well effective potential
In this section, we examine the noise induced dynamics for the double well and triple well potential in the presence of a weak sinusoidal signal.In this case, the interaction between noise and weak sinusoidal signal in the system can drive the system into stochastic resonance as long as the random kicks are optimally matched to the recurring external force.
In the presence of a time-varying periodic signal A t cos 0 W ( ), the time evolution of a particle moving in both a symmetric double well and a triple well effective potential subjected to an external periodic driving force is determined by the Langevin equation

dx t dt dV
Here, A 0 and Ω are the amplitude and the driving frequency, respectively, while ξ(t) represents the Gaussian white noise with 〈ξ(t)〉 = 0 and autocorrelation function 〈ξ(t)ξ(0)〉 = 2Dδ(t).The noise intensity D can be expressed in the unit of eV.The trajectory of a particle in a both symmetrical double and triple well effective potential of equation (12) can be determined by using a fourth-order Runge-Kutta method [32] for a single realization at three different values of noise level strength.To make a system synchronization with the input weak periodic signal, we tune the noise levels and we let the amplitude A 0 = 1. Figure 7 shows a numerical simulation result of input/output synchronization in a symmetrical double well effective potential.Here, we can see that the three different figures, which are labels by A, B, and C. Due to a very small noise strength, the particle spend more time in one of the potential minima and make a transition to the other potential minima shown in figure 7(A).Further tuning the noise level until it reaches to optimal level as see in figure 7(B), the particle synchronizes with the input period, which is the manifestation of stochastic resonance phenomenon, and then increasing the noise level D, the particle looses of synchronization as shown in figure 7(C).
Besides, the numerical simulation of input/output synchronization in a symmetrical triple well effective potential as shown in figure 8.The behavior of the trajectory under a triple well potential land escape has the similar property that we mentioned in the case of symmetrical double well potential.

Conclusion
All of these results indicate that by varying the noise strength D, the impurities in a semiconductor layer can be surmount easily from the potential barrier.The thermally activated rate (r k ) at a high barrier limit ΔV eff ?k B T shows how much the particle spend their time in a potential minima.This implies that at an optimal level of noise strength, a particle moves unidirectionally over a half periodic forcing from one metastable potential minimum to the other stable potential minima.In general, this study show as the distribution of impurity and serves us the basic paradigm in which to understand a dynamics of overdamped Brownian particle and noiseinduced non-equilibrium phase transition in discrete systems.

Acknowledgments
YA, TB and YB would like to thank Mulugeta Bekele.
V(x) has two minimum x a b =  and one maxima x = 0.Where a and b are a constant parameters in the unit of eV nm 2 and

Figure 5 .
Figure 5.The barrier crossing rate r K versus D in the units of electron volt (eV ) for a parameter choice σ = 0.3 nm(black), σ = 0.3 nm (red), f = 0.09 eV.

Figure 6 .
Figure 6.The barrier crossing rate r K versus D in the units of electron volt (eV ) for a parameter choice f = 0.07 eV (red), f = 0.09 eV (black), σ = 0.3 nm.

Figure 8 .
Figure 8.The trajectory of a particle in a symmetrical three well effective potential by varying noise strength D and fixed angular frequency Ω with the following parameters: A 0 = 0.1, Ω = 0.00005 Hz, a = 0.0625, b = 0.03125, f = 0.09, σ = 0.3.