Influence of moving reathers on vacancies migration J Cuevas a,1, C Katerji a, JFR Archilla a, JC Eileck, FM Russell a Grupo de Física No Lineal. Departamento de Física Aplicada I. ETSI Informática. Universidad de Sevilla. Avda. Reina Mercedes, s/n. 4112-Sevilla (Spain) Department of Mathematics. Heriot-Watt University. Edinurgh EH14 4AS (UK) Astract A vacancy defect is descried y a Frenkel Kontorova model with a discommensuration. This vacancy can migrate when interacts with a moving reather. We estalish that the width of the interaction potential must e larger than a threshold value in order that the vacancy can move forward. This value is related to the eistence of a reather centred at the particles adjacent to the vacancy. Key words: Discrete reathers, Moile reathers, Intrinsic localized modes, Defects PACS: 63.2.Pw, 63.2.Ry, 63..+, 66.9.+r 1 Introduction The interaction of moving localized ecitations with defects is presently a suject of great interest and can e connected with certain phenomena oserved in crystals and iomolecules. Recently, Sen et al [1] have oserved that, when a silicon crystal is irradiated with an ion eam, the defects are pushed towards the edges of the sample. The authors suggest that moile localized ecitations called quodons, which are created in atomic collisions, are responsile for this phenomenon. The interpretation is that the quodons are moving discrete reathers that can appear in 2D and 3D lattices and move following a 1 Corresponding author. E-mail: jcuevas@us.es Preprint sumitted to Physics Letters A 1 April 23
quasi-one-dimensional path [2]. The interaction of moving reathers with defects is currently of much interest [3,4] (for a review on the concept of discrete reather see, e.g. []). In this paper, we consider a simple one-dimensional model in order to study how a moving reather can cause a lattice defect to move. This study is new in the sense that most studies that consider the interaction of moving discrete reathers with defects, assume that the position of the latter are fied and cannot move through the lattice. In particular, the defect we consider is a lattice vacancy, which is represented y an empty well or discommensuration in a Frenkel Kontorova model [6]. The aim of this paper is to determine in which conditions the vacancy moves towards the ends of the chain. This is a previous step to reproduce the phenomenon oserved in [1] for higher dimensional lattices. 2 The model In order to study the migration of vacancies, we consider a Hamiltonian Frenkel Kontorova model with anharmonic interaction potential [7]: H = n 1 2ẋ2 n + V ( n )+C W( n n+1 ). (1) The dynamical equations are: F ({ n }) ẍ n + V ( n )+C [W ( n n+1 ) W (n n 1 n )], (2) where { n } are the asolute coordinates of the particles; V () is the on site potential, which is chosen of the sine-gordon type: ( V () = L2 1 cos 2π ), (3) 4π 2 L with L eing the distance etween neighoring minima of the on site potential. The choice of a periodic potential allows to represent a vacancy easily. Thus, if we denote the vacancy site as n v (see figure 1), the displacements of the particles with respect to their equilirium position are: u n = n nl n < n v (4) u n = n (n +1)Ln>n v. 2
V() n n v 1 v n v +1 Fig. 1. Scheme of the Frenkel Kontorova model with sine-gordon on site potential. The alls represent to the particles, which interact through a Morse potential. The vacancy is located at the site n v. The interaction potential W () is of the Morse type: W () = 1 2 [ep( ( a)) 1]2, () where a is the distance etween neighoring minima of the interaction potential. In order to avoid discommensurations, we choose L = a =1.is a parameter which is related to the width of the interaction potential, so that the interaction etween particles is stronger when decreases. The reason of the choice of this potential is twofold. On the one hand, it represents a way of modelling the interaction etween atoms in a lattice so that, if the distance etween particles is large, the interaction etween them is weak. On the other hand, if a harmonic interaction potential were chosen, apart from eing unphysical in this model, the movement of the reather would involve a great amount of phonon radiation, making it impossile to perform the study developed in this paper. The linearized dynamical equations are: ẍ n + n + C(2 n n+1 n 1 )=, (6) with C = C 2. Throughout this paper, the results correspond to a reather frequency ω =.9. Values of ω [.9, 1) lead to qualitatively similar results. Values of ω.9 are not possile as moving reathers do not eist [8]. 3
3 Numerical Results 3.1 Preliminaries In order to investigate the migration of vacancies in our model, we launch a moving reather towards the vacancy located at the site n v.thismoving reather is generated using a simplified form of the marginal mode method [9,1], which consists asically in adding to the velocity of a stationary reather a perturation which reaks its translational symmetry, and letting it evolve in time. In these simulations, a damping term for the particles at the edges is introduced in order that the effects of the phonon radiation are minimized. The initial perturation, { V n } is chosen as V = λ(...,, 1/ 2,,1/ 2,,...), where the nonzero values correspond to the neighoring sites of the initial center of the reather. This choice of the perturation allows it to e independent on the parameters of the system or C. If the pinning mode were chosen as an initial perturation, it would depend on the parameters of the system. 3.2 Breather vacancy interaction When a moving reather reaches the site occupied y the particle adjacent to the vacancy, i.e., the location n v 1, it can jump to the vacancy site or remain at rest. If the former takes place, the vacancy moves ackwards. However, if the interaction potential is wide enough, the particle at the n v +1site,can feel the effect of the moving reather at the n v 1 site and it can also move towards the vacancy site. In this case, the vacancy moves forwards. Figures 2 and 3 illustrate oth phenomena. Numerical simulations show that the occurrence of the three different cases depends highly on the relative phase of the incoming reather and the particles adjacent to the vacancy. However, some conclusions can e etracted: 1) The incident reather always loses energy; 2) The reather can e reflected, trapped (with emission of energy) or refracted y the vacancy, in analogy to the interaction moving reather-mass defect [3]; 3) the refraction of the reather (i.e. the reather can pass through the vacancy) can only take place if the vacancy moves forwards. It is interesting that the vacancy can migrate along several sites if the interaction etween particles is strong enough (see Figure 4). 4
C=., =1, λ=.11 C=., =1, λ=.11723 4 4 4 4 3 3 3 3 Periods 2 2 2 2 1 1 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 Fig. 2. Energy density plot of the interaction moving reather vacancy. The vacancy is located at n v =. Note that the vacancy moves ackwards (left) and, in the case of the figure to the right, the reather passes through the vacancy. C=., =.8, λ=.966 C=., =., λ=.16727 4 4 4 4 3 3 3 3 2 2 Periods 2 2 1 1 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 Fig. 3. Energy density plot of the interaction moving reather vacancy. The vacancy is located at n v =. Note that, in the figure to the left, the reather is reflected and the vacancy remains at rest, while in the figure to the right, the vacancy moves forwards. 3.3 Numerical simulations As mentioned earlier, the moving reather vacancy interaction is highly phasedependent in a non ovious way. That is, the interaction depends on the velocity of the reather and the distance etween the reather and the vacancy. Consequently, a systematic study of the state of the moving reather and the vacancy after the interaction cannot e performed. Therefore, we have performed a simulation that consists in launching a large numers of reathers towards the vacancy site. In particular, we have chosen
C=., =., λ=.11829 4 4 3 Periods 3 2 2 1 1 2 2 1 1 1 1 2 2 Fig. 4. Energy density plot of the interaction moving reather vacancy. The vacancy is located at n v =. It can travel several sites along the lattice. 1 C=. 1 C=.4.9.9.8.8.7.7 Proaility.6..4 Proaility.6..4.3.3.2.2.1.1.1.2.3.4..6.7.8.9 1.1.2.3.4..6.7.8.9 1 Fig.. Proaility that the vacancy remains at its site (squares), moves ackwards (circles) or moves forwards (triangles), for a Gaussian distriution of λ. 1 reathers following a Gaussian distriution of the perturation parameter λ with mean value.13 and variance.3. Figure shows the proailities that the vacancy remains at its original site, or that it jumps ackwards, or forwards for C =.andc=.4. An important consequence can e etracted from this figure. There are two different regions for the parameter, separated y a critical value (C). For >(C), the proaility that the vacancy moves forwards is almost zero, whereas for > (C), this proaility is finite. For eample, (C =.).7 and (C =.4).. Figure 6 represents this dependence for a uniform distriution of λ (.1,.16), and shows the occurrence of the same phenomenon. Thus, this result seems to e independent on how λ is distriuted. 6
1 C=..9.8.7 Proaility.6..4.3.2.1.1.2.3.4..6.7.8.9 1 Fig. 6. Proaility that the vacancy remains at its site (squares), moves ackwards (circles) or moves forwards (triangles), for an uniform distriution of λ..3 C=..3 C=.4.2.2 Jacoian eigenvalues.1.1.2 Jacoian eigenvalues.1.1.2.3.3.4.1.2.3.4..6.7.8.9 1.4.1.2.3.4..6.7.8.9 1 Fig. 7. Dependence of the Jacoian eigenvalues with respect to. It can e oserved that one eigenvalue changes its sign, and another is constant and close to zero. The first one is responsile for the ifurcation studied in the tet, while the second one indicates the quasi-staility necessary for reather moility [11]. 3.4 Analysis of some results. Vacancy reather ifurcation. The non-eistence of forwards vacancy migration can e eplained through a ifurcation. If we analyze the spectrum of the Jacoian of the dynamical equations (2) defined y J F ({ n }), ifurcations can e detected. A necessary condition for the occurrence of ifurcations is that an eigenvalue of J ecomes zero. Figure 7 shows the dependence of the eigenvalues closest to zero with respect to for C =.andc=.4. It can e oserved that, in oth cases, there is an eigenvalue that crosses zero in (.6,.7) for C =. and in (.,.) for C =.4. These values agree with the points where the proaility of the jump forward vanishes. These ifurcations are related to the disappearance of the entities we call 7
.16 C=..18 C=.4.14.16.12.14.12.1 n.8 n.1.8.6.6.4.4.2.2.1.2.3.4..6.7.8.9 1.1.2.3.4..6.7.8.9 1 Fig. 8. Amplitude maima of a vacancy reather versus. It can e oserved that the vacancy reather disappears at the ifurcation point (see figure 7). This is related to the vanishing of the forward movement proaility. vacancy reathers. These are defined as 1 site reathers centered at the site neighoring to the vacancy, e.g. the n v 1orn v + 1 sites. It can e oserved (figure 8) that, for elow the ifurcation value, vacancy reathers do not eist. 4 Conclusions In this paper, we have oserved that a moving reather con make a vacancy defect move forward, ackward or let it at its site. We have also analyzed the influence of the width of the coupling potential and the coupling strength on the possiility of movement of a vacancy through the collision with a moving reather. In particular, we have oserved that the width of the potential must e higher than a threshold for the vacancy eing ale to move forwards. This ehaviour is relevant ecause eperiments developed in crystals show that the defects are pushed towards the edges. We have also estalished that the non eistence of a reather centered at the sites adjacent to the vacancy is a necessary condition for the vacancy movement forwards. The incident reathers can e trapped, in the sense that the energy ecomes localized at the vacancy net neighors, which radiate and eventually the energy spreads through the lattice. It can also e transmitted or reflected. The transmission can only occur if the vacancy moves ackwards. The moving reather always losses energy ut there is not a clear correlation etween the vacancy and reather ehaviours. 8
Acknowledgements The authors acknowledge Prof. F Palmero, from the GFNL of the University of Sevilla, for valuale suggestions. They also acknowledge partial support under the European Commission RTN project LOCNET, HPRN-CT-1999-163. J Cuevas also acknowledges an FPDI grant from La Junta de Andalucía. References [1] P Sen, J Akhtar, and FM Russell. MeV ion-induced movement of lattice disorder in single crystalline silicon. Europhysics Letters, 1:41, 2. [2] JL Marín, JC Eileck, and FM Russell. Localized moving reathers in a 2-D heagonal lattice. Phys. Lett. A, 248:22, 1998. [3] J Cuevas, F Palmero, JFR Archilla, and FR Romero. Moving discrete reathers in a Klein Gordon chain with an impurity. J. Phys. A: Math. and Gen., 3:119, 22. [4] I Bena, A Saena, and JM Sancho, Interaction of a discrete reather with a lattice junction. Phys. Rev. E, 6:36617, 22. [] S Flach and CR Willis. Discrete reathers. Phys. Rep., 29:181, 1998. [6] LM Floría and JJ Mazo. Dissipative dynamics of the Frenkel-Kontorova model. Adv. Phys., 4:, 1996. [7] OM Braun and YuS Kivshar. Nonlinear dynamics of the Frenkel Kontorova model. Physics Reports, 36:1, 1998. [8] JL Marín. Intrinsic Localised Modes in Nonlinear Lattices. PhD dissertation, University of Zaragoza, Department of Condensed Matter, June 1997. [9] Ding Chen, S Aury, and GP Tsironis. Breather moility in discrete φ 4 lattices. Phys. Rev. Lett., 77:4776, 1996. [1] S Aury and T Cretegny. Moility and reactivity of discrete reathers. Physica D, 119:34, 1998. [11] J Cuevas, JFR Archilla, YuB Gaididei, and FR Romero. Moving reathers in a DNA model with competing short- and long-range dispersive interactions. Physica D, 163:16, 22. 9