Xiao-Ping Yuan¹, Li-Yan Qiao², Xia Huang^,31Information Engineering School, Hangzhou Dianzi University, Hangzhou 310018, China
²Department of Physics, Hangzhou Dianzi University, Hangzhou 310018, China
³School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

Received:2019-11-27Revised:2020-01-21Accepted:2020-02-26Online:2020-06-24


Abstract
The transport of a chain of charged particles with a transverse degree of freedom is investigated in a 2D asymmetric potential. Here, the energy of the periodic driving force is converted into motion in the vertical direction. The analysis exhibits a transition from stick–slip motion to periodic oscillation. The chain velocity can be controlled to an optimized value by adjusting system parameters, such as the amplitude and frequency of the periodic force. The existence of a resonance platform indicates resonance between the motion of the chain and the periodic force as coupling strength increases adiabatically. The atomic configuration and the transverse degree of freedom also play key roles in the control.
Keywords： collective transport;driven Frenkel–Kontorova model;cooperative ratchet


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Xiao-Ping Yuan, Li-Yan Qiao, Xia Huang. A dynamical transition of a chain of charged particles in a 2D substrate potential. Communications in Theoretical Physics, 2020, 72(7): 075602- doi:10.1088/1572-9494/ab7ecc

1. Introduction

The problem of dynamical transitions in a periodic substrate potential influences several fields in physics, chemistry and biology. The problem is especially applicable to solid friction [1–4], charge-density wave condensates [5, 6] and the Josephson junction [7, 8], to name a few. As a means of gaining insight into the physics of complex macroscopic multibody systems with competing interactions, the driven Frenkel–Kontorova (FK)-type model has been studied to represent a chain of harmonically interacting particles in a sinusoidal substrate potential that is driven by an external force [9–13]. Although the model is simple, the two inherent locking frequencies that come from the external force and from the particles’ motion in the periodic potential continue to exhibit very rich dynamics, and they account for many nonlinear problems [14–16]. This model has recently received increasing interest because it may useful for understanding solid friction phenomena and staircase macroscopic response [17, 18]. In both theoretical and experimental studies, system parameters such as winding number, external driving force, interaction between atoms, damping, pinning and geometry of the substrate play a crucial role in the transition phenomena and the properties of the dynamical phase [19–22].

The dynamics of the driven FK model is characterized by the collective phenomenon induced by multiparticle coupling [23–26]. This interaction-induced transport is based on the collective behavior of interacting particles that yield a large amount of new spatiotemporal nonequilibrium. The breaking behavior related to parity and time symmetries and system nonlinearity become the new mechanisms for inducing directional transport, e.g., controlling the interaction potential by time-dependent length modulation between particles [27] and an asymmetrically coupled lattice in a symmetric potential without external force [25, 28]. Inspired by how molecular motors convert chemical energy into directional motion [29, 30], interest in these studies has now focused on driving energy that can be converted into directional motion in another direction. In Hamiltonian systems, current can be induced dynamically from diffusive particle motion in one direction into ballistic motion in the opposite direction [31]. Zheng et al explored the ratchet motion of interacting particles with the aid of external rocking forces applied transversely in single- and double-channel potentials [32, 33]. This cooperative ratchet effect may open ways to investigate 2D collaborative ratchet dynamics that can be implemented in many different experiments, such as on optical lattices [34, 35], superlattices [36] and some biological locomotion processes [37].

The staircase macroscopic response, also called Shapiro steps, as a response function of a system, appears to be another main characteristic dynamics of the driven FK model and thus is of greater significance for technical applications [38, 39]. These steps are the result of mode-locking solutions. References [40, 41], which presented a detailed analysis of the amplitude dependence of the Shapiro steps, revealed that the relative sizes of the steps follow a Farey sequence only for a purely sinusoidal substrate. In nonsinusoidal potentials, the symmetry of the Stern–Brocot tree was increasingly broken. New subharmonic steps, which became separated by chaotic windows, appeared in the overdamped regime, even in the commensurate structure [42].

In contrast to most previous works on the driven FK model, which were performed to determine the impact of linear or short-range interactions on noise, we study it in the statistical context, with particular focus on the particles’ cooperation. Moreover, the cooperative directed transport among particles remains a significant topic. In this study, we examine simultaneously the motion of charged particles with multiple symmetry breaks to describe a chain of charged atoms subject to a 2D potential that is periodic in one direction and parabolic in the transverse direction. The hopping process is made possible with the aid of repulsive interaction among adjacent charged particles in the electric field. In the previous work, the ground state of the model transitioned from a 2D configuration to a 1D configuration when system parameters varied, which indicates that the particles are more likely to be moving over the potential with less depinning force [43]. We concentrate on how the ac electric driving force can be shifted to the vertical direction. The coupling of particles may enhance the transport current and the resonance steps by manifesting in the response function $\bar{v}(K)$ when coupling strength overcomes a threshold. The influence of the atom’s configuration on the transport of the chain is also investigated.

2. Modified Frenkel–Kontorova model

We consider N charged particles in two dimensions with real coordinates r_i=(x_i, y_i) and unit mass m=1 that are exposed to a spatially periodic and laterally parabolic substrate potential. As shown in figure 1(a), the 2D substrate potential V_s(r_i) is assumed to be(1)$ \begin{eqnarray}{V}_{{\rm{s}}}({r}_{i})=\displaystyle \frac{1}{2}\varepsilon [1-\cos (2\pi x/{a}_{{\rm{s}}})]+\displaystyle \frac{1}{2}{\omega }_{0y}^{2}{y}^{2},\end{eqnarray}$where ϵ and a_s are real constants that represent the amplitude and the distance between two adjacent minima of the potential along the x direction, respectively, and ${\omega }_{0y}$ is the vibration frequency of a single atom in the transverse y direction. As long as frequency ${\omega }_{0y}$ is large enough, the confining potential in the transverse direction will contribute to the total potential and promote the movement of particles. When ${\omega }_{0y}\to \infty $, the standard FK model is recovered.

Figure 1.

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Figure 1.(a) Schematic diagram of the two-dimensional substrate potential. (b) Schematic structure of the effective tilted ratchet potential. The tilted ratchet potential is represent by the black solid line. The red dashed sinusoid is the effective potential field force.


The interaction among particles plays a significant role in the dynamics of collective transport. For simplicity, the interatomic interaction is assumed to be a generalized elastic coupling, in which the nearest-neighbor interaction is denoted by W_int:(2)$ \begin{eqnarray}{W}_{{\rm{int}}}({{\boldsymbol{r}}}_{i+1}-{{\bf{r}}}_{i})=\displaystyle \frac{1}{2}K{\left(| {{\boldsymbol{r}}}_{i+1}-{{\boldsymbol{r}}}_{i}| -{a}_{{\rm{A}}}\right)}^{2},\end{eqnarray}$where K is coupling strength and a_A is the equilibrium spacing distance of each two neighboring atoms.

Without loss of generality, ϵ=2 and a_s=2π are always chosen in this work. In this manner, the longitudinal vibration frequency ${\omega }_{0x}={(\varepsilon /2m)}^{1/2}(2\pi /{a}_{{\rm{s}}})$ of an individual atom is equal to 1. Thus, we have a system size of $L={{Na}}_{{\rm{A}}}={{Ma}}_{{\rm{s}}}$, where M is the number of minima of the potential in the x direction. The commensurability of the system is characterized by the dimensionless concentration $\theta =N/M={a}_{{\rm{s}}}/{a}_{{\rm{A}}}$, which is fixed in the limit of $N,M\to \infty $. A rational value of θ defines a commensurate structure, and vice versa.

Each atom configuration is characterized by its total potential energy V(r_i), which is the sum of substrate potential V_s and interatomic interaction W_int, namely, $V({r}_{i})={V}_{{\rm{s}}}({{\bf{r}}}_{i})\,+{W}_{{\rm{int}}}({{\bf{r}}}_{i+1}-{{\bf{r}}}_{i})$. As such, the dissipative and overdamped dynamics is given by the equation(3)$ \begin{eqnarray}\dot{{{\boldsymbol{r}}}_{i}}=-\displaystyle \frac{\partial V({{\boldsymbol{r}}}_{i})}{\partial {r}_{i}}+{\boldsymbol{F}}.\end{eqnarray}$

The driving force can be written as $\vec{F}\,={F}_{{\rm{d}}}\vec{{e}_{x}}\,+{(-1)}^{i}E(t)\vec{{e}_{y}}$, where F_d is a constant and $E(t)=b\cos (\omega t)$ is a time-periodic driving. The expression (−1)ⁱ denotes that chain particles have a positive charge or a negative charge. Remarkably, particles with opposite charges are subject to a mutually repulsive interaction in the electric field, which renders the hopping process easy. The dc force F_d added in the x direction breaks the symmetry of the potential field. This phenomenon is shown in figure 1(b), in which the ratchet potential is ${V}_{{\rm{eff}}}=1-\cos (x)+{F}_{{\rm{d}}}x$ and is represented by the black solid line. The red dashed sinusoid is effective potential field force. Above (below) zero, the direction of the force ${f}_{{\rm{s}}}={{\rm{d}}V}_{{\rm{eff}}}/{\rm{d}}x$ is the same as (opposite to) the directed flow. The tilted potential allows for the direct transport of atoms. Two different frequency scales are present in this system: the frequency of the periodic driving force ω and the characteristic frequency of the motion of atoms over the substrate potential driven by F_d. The competition between the two frequency scales can lead to the dynamics mode locking at some critical value of the driving force. If the time for the phase of the periodic force to change by 2πm is the same as that for the lattice shifts, then the distance is $x={{la}}_{{\rm{s}}}+2\pi n$, that is, $2\pi m/\omega =x/v$. The directional current reaches the resonant values as follows:(4)$ \begin{eqnarray}v=\displaystyle \frac{{{la}}_{{\rm{s}}}+2\pi n}{2\pi m}\omega ,\end{eqnarray}$where l, m and n are integers. Then l=0 and $v=n\omega /m$. This scheme is called the Shapiro step in dc- and ac-driven Josephson junction experiments [38]. These steps are derived from the resonance of the coupled system with the periodic force. Multiple steps will be comprehensively discussed in a subsequent study. In this present work, the current of the directed motion caused by the following average drift velocity is defined as(5)$ \begin{eqnarray}\bar{v}=\displaystyle \frac{1}{N}\sum _{j=1}^{N}\mathop{{\rm{lim}}}\limits_{}\displaystyle \frac{1}{T}{\int }_{0}^{T}\displaystyle \frac{{\rm{\partial }}{x}_{i}}{{\rm{\partial }}t}{\rm{d}}t,\end{eqnarray}$where the averages include the long-time average and the number of particles is N=64 with open boundary conditions. The above equations of motion have been integrated numerically by the fourth-order Runge–Kutta algorithm with a time step of ${\rm{\Delta }}t={10}^{-3}$, which is sufficiently small to maintain numerical stability.

3. Numerical simulation

In the present work, we examine how to control direct transport in a ratchet potential V_eff. We briefly initially recall the structure of the ground states (GS) of the chain, as described in our previous work [43]. Then, the GS configuration experiences a transition from a 2D pattern to a 1D pattern with the changing of the system parameters, such as $K,{\omega }_{0y}$ and θ. When θ=1/q (q is an integer), called the commensurate configuration, the distance between atoms a_A is a multiple of the substrate potential a_s, indicating that the atoms occupy the minima of the substrate potential. Meanwhile, the incommensurate case expressed by θ=p/q (p and q are integers) is extremely complicated because q kinks are induced.

Starting from the different initial configurations is significant in the evolutionary dynamics of the chain. A critical depinning force F_c≈0.38 exists for the GS configuration in an incommensurate structure for a given set of parameters, such as when atoms cannot overcome the potential barrier and pin. With the help of an appropriate periodic driving electrical field E(t) in the y direction, the average velocity becomes greater than zero at F_d=0.28<F_c. As shown by the whole parameter plane in figure 2, depinning regions alternate with pinning ones. For a very low frequency ω, the role of the periodic ac force is equivalent to that of a constant force due to the period length. As such, the system can easily convert horizontal energy into movements in the x direction, as shown by the thin strip-like region in the left part of figure 2. Furthermore, the amplitude of ac driving must be large enough to help the particles cross the barrier, i.e., $b\gt {b}_{0};$ however, a bigger value does not necessarily mean a better method. When the frequency ω is increasing, two radial channels of the directed transport in the b−ω plane exist. The darker the color, the greater the speed in the figure. This scheme is a consequence of the cooperation between neighboring particles induced by mutual driving. Due to the coupling effect, an appropriate driving force in the transverse direction can be transformed into the interaction force between the particles. In this manner, the particles can collectively overcome the substrate potential force f_s and successfully cross the barrier to generate a directional flow.

Figure 2.

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Figure 2.Dependence of the average velocity of the system on amplitude b and frequency ω of the ac force. The grayscale indicates the value of the average velocity. The parameters are F_d=0.28, K=0.51, $\theta =(\sqrt{5}-1)/2,{\omega }_{0y}=0.5$.


We analyze in detail the dynamics of the two points at ω =1.56π, which are marked in figure 2, to gain insight into the different dynamical regimes. Figure 3 shows the time evolution of the average velocity of the chain and the corresponding center-of-mass displacement. This exhibits the transition process from stick–slip motion to periodic oscillation. The left panel with b=7 clearly shows a typical regular stick–slip motion and step dependence of x(t). In this case, each particle jumps from one well to the next, stays in the well at a short time, and then jumps again to the next well. Meanwhile, with b=19, the stick–slip motion is replaced by periodic oscillation in a potential well with zero displacements in the x direction. The particles oscillate in the potential well and fail to pass over successfully. This scenario indicates that the lateral periodic force should not be as large as possible, and the focus should be on how much energy can be converted into mutual cooperation between particles.

Figure 3.

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Figure 3.The time evolution of average velocity and displacement of the chain separately for (a) ac amplitude b=7, ω=1.56π, (b) ac amplitude b=19, ω=1.56π. The other parameters are the same as in figure 2.


Our next concern is the situation in which particles can overcome the barriers and produce direct current under the dc driving force F_d>F_c. What role will the ac force play in the transverse direction? Figure 4 presents the dependence of the average velocity of the system on amplitude b and frequency ω of the ac force with dc driving force F_d=0.5. In this case with F_d>F_c, the energy of the lateral driver cannot be effectively converted into a directional current in the x direction. As shown in figure 4, when the frequency ω is low, the velocity of the direct transport alternates as amplitude b increases. For larger ω, the velocity decreases instead as the amplitude b increases. The chain velocity can be easily controlled to a lower value or zero by properly adjusting the values of the amplitude and frequency to the gray or blank area in figure 4. This finding indicates that the critical forces that transform a locked state into a sliding one in the chain vary with the ac driving force, and an optimal electric field exists to control the particles. This scenario is very different from our intuition, which suggests that increasing the ac force would help the chain to collectively overcome the substrate. If the external force F_d increases continually, then it becomes difficult to keep system transport under control.

Figure 4.

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Figure 4.Dependence of the average velocity of the system on amplitude b and frequency ω of the ac force for F_d=0.5. The grayscale indicates the value of the average velocity. The other parameters are the same as in figure 2.


The mutual couplings among particles describe the relative strength of the two interaction types. When coupling strength K is below a critical value, the system is in a floating state with zero static friction. However, for K>K_c, the system enters a stick–slip state. This transition that involves the breaking of analyticity is known as an Aubry transition. Here, we are interested in the transport against the coupling strength under an ac force field. The parameter plane between coupling strength and amplitude of the ac driving is shown in figure 5. For weaker couplings in which K<K_c=0.1, no current exists regardless of whether the horizontal ac electric forcing is opposing. This finding can be explained by the loose connection between particles that may introduce phonon dispersion, which brings additional energy dissipation to the system. Only when the coupling strength exceeds the threshold K_c will the transport be enhanced. In this case, the particles cooperate with one another to produce high-efficiency directed transport. When the amplitude of the ac force becomes sufficiently strong, a very large resonance platform of velocity appears on the K−b parameter plane, where $\bar{v}=2\omega $ as seen in figure 5(a). The velocity resonance steps are even more clearly demonstrated in figure 5(b), in which the current is plotted against the coupling strength with different amplitudes of ac force. For the low amplitude of b=0.5, a gradual increase in velocity with K can be observed. For the intermediate value of b=1.0, a step starts to manifest itself. If the coupling strength is adiabatically increased, two resonance steps of 2ωand 4ωexist in the response function $\bar{v}(K)$ at b=1.3. These response steps indicate that a locking occurs between the internal frequency from the motion of the chain over the periodic potential and that of the periodic ac force. The steps are called harmonic (or subharmonic) if the locking appears at integer (or rational) multiples of the ac frequency. However, the resonance behavior is non-existent when ω is much larger because the movement of particles cannot keep up with the change in ω.

Figure 5.

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Figure 5.(a) Dependence of the average velocity of the system on the coupling strength K and the amplitude of the ac force b at ω=0.02π. (b) The variation of the current with the coupling strength for different amplitudes of the ac force for b=0.5, 1, 1.3, 3.5. The other parameters are F_d=0.8, $\theta =(1+\sqrt{5})/2$ and ${\omega }_{0y}=0.5$.


Incommensurability is a popular topic in the field of nonlinear dynamical systems with competing length scales. The critical force changes with the configuration of a system θ, which is in good agreement with an Aubry transition. In commensurate situations, because atoms always lie at the bottom of a potential well to obtain the lowest total potential, the critical force F_c reaches the maximum to depin the chain. Meanwhile, in incommensurate situations, the critical force may be much lower for the higher total potential of the system. As shown in figure 6, the response function $\bar{v}(1/\theta )$ is presented in three cases. At b=0, the response function appears as a regular arch curve, as shown in figure 6(a). The velocity reaches maximum values at the incommensurate configurations of 1/θ=q/p=1/2, 3/2, 5/2, ... in which two atoms occupy an odd number of minima of the substrate potential, that is 1, 3, 5, .... As for the commensurate case of 1/θ=1, 2, ..., the value is zero at F_d=0.8. When an external ac electric field is applied to the particles, the locking steps appear again. For small amplitude, such as b=0.5 in figure 6(b), the two steps of 2ω and 4ω can be found at 1/θ<1/2. As the amplitude increases to the higher values, such as b=5, only the staircase structure becomes present in the response function $\bar{v}(1/\theta )$. The velocity of the chain only appears on the response platforms of 2ω, 4ω and 6ω, indicating the resonance between the frequency of the chain movement and the ac force.

Figure 6.

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Figure 6.Dependence of the average velocity of the system on the atoms’ configuration 1/θ for three different ac forces. (a) b=0, (b) b=0.5, (c) b=5. The other parameters are ${F}_{d}=0.8,{\omega }_{0y}=0.5$ and ω=0.02π.

4. Conclusion

In this work, we present the overdamped dynamics of interacting charged particles subject to a 2D substrate potential in a dc- and ac-driven Frenkel–Kontorova model. The focus of our attention is the action of the ac force on the transport of the whole system. The mutually repulsive interaction between the nearest particles enhances the particles’ cooperation. When F_d<F_c, directional transport can be generated with the help of an appropriate driving energy of the periodic force in the transverse direction. The transition from stick–slip motion to periodic oscillation occurs in the process of modulation by the ac force. When F_d>F_c, as the energy of the ac field increases, the transport velocity decreases. In the examination of the dependence of system transport on coupling strength, resonance platforms are found under a moderate periodic force. The most striking and interesting behavior is the appearance of a whole staircase structure in the response function $\bar{v}(1/\theta )$ with a strong periodic external force. By adjusting the parameters to a proper interval, we can promote or inhibit the direct current to an optimized value. The system studied here thus represents a model for the transport of vortices in superconductors, and for surface electromigration and particle separation in 2D devices.

Acknowledgments

This work was supported by Zhejiang Province Commonweal Projects (Grant No. LGF18A050001), China Scholarship Council (Grant No. 201 708 330 401), National Natural Science Foundation of China (Grant No. 11 605 055) and Fundamental Research Funds for the Central Universities of China (Grant No. 2017MS054).

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