Thermal nucleation of kink-antikink pairs in the presence of impurities: The case of a Remoissenet-Peyrard substrate potential

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  • Thermal nucleation of kink-antikink pairs in the presence of impurities:The case of a Remoissenet-Peyrard substrate potential

    Rosalie Laure Woulach,1,4 David Yeml,2 and Timolon C. Kofan1,3,41Laboratoire de Mcanique. Dpartement de Physique. Facult des Sciences. Universit de Yaound I. B.P. 812, Yaound, Cameroun

    2Dpartement de Physique. Facult des Sciences. Universit de Dschang. B.P. 67, Dschang, Cameroun3Max-Planck-Institute for the Physics of complex systems, Noethnitzer Strasse 38, D-01187 Dresden, Germany

    4The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34014 Triste, ItalyReceived 11 August 2004; revised manuscript received 10 January 2005; published 19 September 2005

    Thermal nucleation of kink-antikink pairs in a nonlinear Klein-Gordon NKG model with a Remoissenet-Peyrard RP substrate potential in the presence of impurities and coupled to an applied field is analyzed in thelimits of moderate temperature and strong damping. Using the Kolmogorov method, the average velocity ofparticles of the lattice is calculated and its dependence on the intensity of impurities is discussed in connectionwith the deformability parameter or the shape of the RP substrate potential. Numerical values are carried outby making use of parameters of the hydrogen atom adsorbed in the tungsten and ruthenium substrates. Weshow that, for large values of the applied field, the presence of impurities in the system makes the nucleationprocess of kink-antikink pairs more favorable in the high-temperature regime while they contribute to make itless favorable in the low-temperature regime.

    DOI: 10.1103/PhysRevE.72.031604 PACS numbers: 64.60.Qb, 05.45.Yv, 89.90.n

    I. INTRODUCTION

    Nucleation is generally defined as a phenomenon where anew phase appears locally in space. It is one of the mostdrastic phenomena in the various fields of physics, chemistry,biology, and also engineering 1. More precisely, the nucle-ation in condensed matter physics is most interesting in thesense that it can be controlled by parameters such as pres-sure, temperature, electric and magnetic fields, and so on.One usually distinguishes homogeneous and heterogeneousnucleation. In the first case, embryos of a stable phaseemerge from a matrix of a metastable parent phase due tospontaneous thermodynamic fluctuations. Droplets largerthan a critical size will grow while smaller ones decay backto the metastable phase 24. In the second case, randomforces catalyze the transition by making growth energeticallyfavorable 1.

    The study of the nucleation connected to the formation ofsolitary structures in spatially one-dimensional 1D andmultistable systems is well developed theoretically 420,experimentally, and numerically 2125. These studies offera fundamental understanding of nucleation in a homoge-neous medium. More specifically, theoretical analysis ofnucleation was introduced four decades ago by Seeger andSchiller 10 to describe the kinetic process of dislocationand a few years later by Langer 4 to investigate the prob-lem of reversing the direction of magnetization in a ferro-magnetic system. The same ideas, but where the approach isclosely related to the concepts already developed in the dis-location literature, were also developed by Bttiker and Lan-dauer 16 to present a detailed calculation of the nucleationrate of thermal kink-antikink pairs in the overdamped sine-Gordon SG chain and by Yeml and Kofan 5 to presentthe calculation of the nucleation rate of kink-antikink pairs ina driven and overdamped deformable chain. This theory waslater improved by Marchesoni et al. 20 when analyzing thethermal nucleation of kink-antikink pairs in an elastic string.

    The above studies deal with nucleation in homogeneoussystems. However, most of the realistic physical systemspossess impurities which may influence the nucleation pro-cess and disturb the newly formed kink-antikink pairs. Inho-mogeneity may mean spatial modulation, quasiperiodicity, ordisorder of several kinds. For example, local inhomogene-ities microshunts and microresistors may be installed intothe long Josephson junction during fabrication see Ref.26. Neutron scattering experiments by Boucher et al. 27on quasi-1D magnetic compounds magnetic chains, whichhave revealed that the crossover from ballistic to diffusivebehavior of solitons is driven by the impurity concentration,evidences the fact that these materials contain impurities. Incompounds whose electrical properties are due to the exis-tence of charge density waves CDWsthat is, interactingelectron gasimpurities may also be present and representthe sites or atoms where electrical properties are differentfrom those of the host atomfor example, Br disorder inK2PtCN4Br0.3nH2OKCP. In adatomic systems, impuri-ties are also present due to the geometrical imperfections ofthe adsorbed surfaces which in general are at the origin ofthe spatial deformation of the nucleus, to name only a few.

    The nucleation in condensed matter physics may be con-sidered in the framework of the Frenkel-Kontorova FKmodel 28, which describes the behavior of an harmonicchain of atoms in the periodic substrate potential, known asthe nonlinear Klein-Gordon NKG model with SG potential.This model can be generalized by considering another formof potential and by taking into account inhomogeneities inorder to go beyond the mathematical problem and to obtainresults that may be useful for real materials that undergostructural changes such as shape distortion, variations ofcrystalline structures, or conformational changes in some re-gions of their physical parameters. By the way, the study ofthe effect of local inhomogeneity or single impurity on thenucleation in the case of CDWs shows that the CDWs can

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  • be pinned by an impurity if the external applied field is lessthan a threshold field 29. Similarly, it has been demon-strated that the current carried by CDWs may rise as a resultof the increase in the rate of generation of solitons on fluc-tuations in the random field of defects 30. In quasi-1Dmagnetic compounds, it has also been shown that stochasticmotion of SG solitons in a random potential can be used tomodel their statistical properties. This random potential isgenerated by the presence of impurities and explains the ob-served crossover from the ballistic to the diffusive behaviorof spin correlations 31. Although these results are quiteinteresting, they are limited to the rigid substrate potential.Thus at this stage of research one may wonder what is theinfluence of the shape of the substrate potential on the nucle-ation process in these inhomogeneous systems. The answerto this question is the main objective of the present work. Inthis paper we focus our attention on the Remoissenet-Peyrard RP substrate potential whose shape can be variedcontinuously as a function of a deformability parameter andwhich has the SG shape as a particular case 32,33. In ad-dition, it can be successfully used to model the substratepotential along the surface of adsorbed layers in adatomicsystems see, e.g., Ref. 34 and references therein.

    The organization of the paper is as follows: In Sec. II, wepresent the generalized NKG model under consideration inthe presence of impurities. In Sec. III, we reformulate thebasic results on the nucleation rate of kink-antikink pairs inthe homogeneous system 5 by taking into account the non-Gaussian correction in the spirit of Marchesoni et al. 20. InSec. IV, we focus our attention on the influence of impuritieson the nucleation rate of kink-antikink pairs. The mean timefor a transition of an arbitrary point on the chain to a neigh-boring valley of the Peierls distribution is calculated bymeans of the Kolmogorov method in order to obtain themean velocity or average velocity of particles from one siteto an adjacent one, due to the passing of kinks triggered bystochastic forces. In Sec. V, experimental values of the latticeparameters for H/W and H/Ru adsystems are used as a nu-merical application to quantify the correction factor of themean velocity of particles, due to the presence of impuritiesin the system. Finally, Sec. VI is devoted to concluding re-marks.

    II. MODEL DESCRIPTION

    We consider a generalized NKG model describing the dy-namics of a chain of particles in a periodic nonsinusoidalsubstrate potential in the presence of external forces and im-purities. The dynamical behavior of the system is governedby the nonlinear Langevin equation NLE

    Mutt kuxx + V0dVRPu,r

    du= ut + F + x,t

    dVimpudu

    ,

    1

    where u is the longitudinal dimensionless displacement ofthe particles from their equilibrium position along the x axis.The subscripts x and t denote the derivative with respect tospace and time, respectively. V0 is the amplitude of the sub-

    strate potential. The constant force F is related to the appliedfield f through the relation F= f /2. To model the on-sitepotential VRPu ,r, we shall use the nonsinusoidal substratepotential introduced by Remoissenet and Peyrard 32,33:

    VRPu,r = 1 r21 cos u

    1 + r2 + 2r cos u, 2

    where r is the shape parameter, r1. As this parametervaries, the amplitude of the potential remains constant withdegenerate minima 2n and maxima 2n+1 while itsshape changes. When r0, it has flat bottoms separated bythin barriers, while for r0, it has the shape of sharp wellsseparated by flat wide barriers see Fig. 1. At r=0, the RPpotential reduces to the well-known SG potential. This pa-rameter depends on the physical characteristics of each sys-tem. For example, in quasi-1D compounds whose electricalproperties are due to the existence of CDWs, the substratepotential which corresponds to the interaction of CDWswith the host atom may be calculated up to higher order ofthe perturbation theory. Up to the first order of this perturba-tion theory, we obtain the SG potential which is a good ap-proximation only in the weak- and strong-coupling cases.Thus, at higher order, in addition to the first harmonic whichdescribes the SG potential, one also obtains the second, third,and higher harmonics 35. The compact form of this inter-action between CDWs and host atoms may then be approxi-mated by the RP-type function where the parameter describ-ing the shape of the substrate potential depends on theamplitude of the CDW gap, the Fermi velocity, and the qua-siparticle energy. Similarly, for the adatomic systems, theparameter r of the substrate potential is related to the fre-quency 0 of oscillation of an isolated adatom at the bottomof the adsorption site, the adatom mass ma, and the period asof the substrate potential 36; more precisely, r= 1 / 1+, with =0as /22ma /V01/2. Note that the above pa-rameters for the adatomic systems are related to the charac-teristic parameters of the system described by the NLE 1.

    The coupling of the scalar field ux , t to the heat bath atabsolute temperature T is described by a viscous term utand a zero-mean Gaussian noise source x , t. At Boltzmannequilibrium, the damping constant =M0, where 0 corre-sponds to the rate of the energy exchange with the substrate,and the noise intensity are related through the fluctuation-dissipation relationship

    x,tx,t = 2kBTx xt t . 3

    Finally, the last term of the NLE 1, Vimpu, is the potentialenergy density of impurities and its analytical expression de-pends on the nature of these impurities since various types ofimpurities may exist such as the local variations of masses ofparticles, of elastic constants, and of substrate potential bar-riers, respectively. It has been shown that in the presence ofimpurities, the nonlinear waves may be trapped, reflected, ortransmitted with more or less distortion of their structureaccording to the intensity of the impurities 37,38. We as-sume here that impurities are randomly distributed in thesystem and then cause the deformation of any spatially lo-calized structure as a main effect. A simple realization of the

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  • proposed model is obtained by considering the analytical ex-pression

    Vimpu = Vfxu

    x, 4

    where Vfx is a random function of the spatial coordinate.Thus, the quantity dVimpu /du in the NLE 1 is then equalto dVfx /dx. In addition, we restrict our analysis to the casewhere impurities are weak and where their mean separationis less than the characteristic length of the system or the sizeof kink solitons, 0= k /V01/2, which enables us to describethe statistical properties of this random field by

    VfxVfxV = x x , 5

    with zero mean VfxV=0, where describes the intensityof impurities and V denotes the average over the differentrealizations of the random potential Vfx. Note also thatexpression 4 can be successfully used to describe forwardscattering of CDWs in the quasi-1D compounds whose elec-trical properties are due to the existence of CDWs 30. In

    the next sections, we shall have occasion to use the Hamil-tonian derived from the NLE 1 which is given by

    H = dxaM

    2ut

    2 +k

    2ux

    2 + V0VRPu,r Fu + Vimpu ,6

    where a is the lattice constant. In this expression, since u isthe dimensionless displacement of particles, the parametersM, k, and V0 have the dimension of mass length,energy length, and energy length1, respectively.

    Before ending this section, we would like to mention thatthe NLE 1 with the Hamiltonian 6 is well known as thegeneralized NKG model. This NKG model has been success-fully used in investigations of a number of physical phenom-ena such as CDWs, adsorbed layers of atoms, domain wallsin ferromagnetic and antiferromagnetic systems, crowdionsin metals, and hydrogen-bonded systems see, e.g., the re-view paper in 39 and references therein for applications ofthe NKG model. The use of the RP potential as a substratepotential is justified by the fact that it can be invoked to

    FIG. 1. Substrate potential VRPu ,r as a function of u /2 for a few values of the deformability parameter: 1 r=0.3, 2 r=0.0 sGcase, 3 r=0.3, and 4 r=0.9.

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  • describe a large amount of physical systems. As a result, anappropriate choice of the shape parameter enables us to em-ploy a suitable form of the shape of the potential close to thesystem under consideration such as epitaxial or incommen-surate structure 36 in crystals and other various systems.

    III. NUCLEATION OF KINK-ANTIKINK PAIRS IN THEHOMOGENEOUS SYSTEM

    The dynamics of the pure system obtained from the NLE1 by setting its right-hand side equal to zero is dominatedby elementary excitations: phonons and solitons kink andantikink. While phonons are extended modes of the system,solitons are localized modes and may be viewed as effectiveparticles characterized by a mass and an energy. In a numberof situations, kink dynamics may be described by equationsof its collective coordinatesnamely, the kink center ofmass. If one assumes periodic boundary conditions on thechain of length L, ux , t=ux+L , t, kinks are only presentas a result of thermal activation. These thermal kinks arecreated in pairs involving a kink and an antikink. On theother hand, if the system is not subjected to periodic bound-ary conditions or, in other words, if the ends of the string arefree, the so-called geometric solitons of the same sign ap-pear in the system. Characteristic parameters of kink solitonsin the pure system governed by the NLE 1 are well known5. For example, the pseudokink width d, the static kinkantikink energy Es, and the rest mass Ms are given by

    d1 = 0/, d2 = 0, =

    1 r1 + r

    , 7a

    Es = 8kV01/2Gr, Ms

    = 8/0Gr , 7b

    with =1,2, and

    G1r = /* tanh1 *, 7c

    G2r = * tan1*/ ,

    * = 1 2,where the superscripts =1 and =2 stand for 0r1and 1r0, respectively, and 0 designates the character-istic length of the system. For r=0, the above equations re-duce to those of the usual SG kink soliton.

    In the presence of an applied field, the total on-site poten-tial energy, given by

    Vu,F = V0VRPu,r Fu , 8

    is the sum of the substrate potential energy V0VRPu ,r andthe energy due to the applied field, Fu. The minima usnand the maxima uin of the above on-site potential energy8 which are known as Peierls valleys and Peierls hills, re-spectively, are different from those of the substrate potentialenergy and may disappear if the applied field F is greaterthan the threshold value Fm 5:

    Fm/V0 =22232 1 + 31 2

    52 3 + , 9

    with =94142+9. This means that kink solutions of theNLE 1 can only exist if FFm. Note that these Peierlsvalleys and Peierls hills obey dVu ,F /du=0. More specifi-cally, the NLE 1 describing the configuration of the nucleusin a pure system may also be viewed as the equation ofmotion of a classical particle with mass k and time x in apotential Vu ,F, where Vu ,F is given by Eq. 8. Thecritical nucleus of amplitude um will be a configurationwhich deviated only in a localized region from the uniformstate usn followed by motion to the right until the turningpoint usn+um is reached. Then, the particle again returnsasymptotically to the local maxima at usn. The correspondingstationary solution of the NLE 1 is the saddle-point con-figuration or the critical nucleus which departs from the sta-tionary uniform state usn at x= . Its amplitude umstrongly depends on the applied field. Furthermore, um de-creases monotonically with respect to F :um=2 for F=0and um=0 for F=Fm. The transition between two adjacentPeierls valleys due to thermal fluctuations called the criticalnucleus is the newly formed kink-antikink pairs, whose sizedepends on the applied field F. This transition is possibleonly if the fluctuations produce, within the system, a mini-mum of energy ENkBT necessary to create a criticalnucleus uNx ,X, where X designates the position of thenewly formed kink which in the continuum limit is linearlytime dependentthat is, Xt=X0+vt ,X0 being the criticalinitial position of the kink center of mass and v the kinkvelocity. For FFm, the nucleus uNx ,X can be well ap-proximated by the linear superposition of a kink and an an-tikink centered at X, respectivelythat is,

    uNx,X = u+x + X,0 + ux X,0 , 10

    where the solutions ux , t satisfy the NLE 1 without theright-hand side.

    In the overdamped limit V01/2, where the inertial term

    Mutt is neglected, the substitution of Eq. 10 into the NLE1 in the absence of the impurity leads to the followingequation for the nucleus:

    dR

    dt=

    dVN

    dR+ Rt , 11

    with the reduced coordinate R=2X, where the potential ofthe critical nucleus is given by

    VNR =

    2F

    MsR

    4Es

    Ms e

    R/d, 12

    with

    1 = *exp 2* tanh1 *

    2 tanh1 *, 13

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  • 2 =

    *exp2*/tan1*/

    tan1*/.

    The noise Rt associated with Eq. 11 for the nucleus veri-fies the fluctuation-dissipation relationship

    RtRt = 22DRt t, DR

    =2kBT

    Ms . 14

    Following the procedure outlined in Ref. 20 useful forthe calculation of the nucleation rate of kink-antikink pairs, itis necessary to determine the size of the critical nucleus, RN

    ,and the negative eigenvalue 0

    N of the nonuniform state.Thus, the nucleus size is set by the condition that

    VNRRN =0, leading to

    RN = d ln2Es

    Fd

    , 15

    and the negative eigenvalue of the nonuniform state, whichis the eigenvalue of the RP scattering potential in the pres-ence of the applied field defined asd2Vu ,F /d2uuN / d

    2Vu ,F /du2usn, is given by

    0N = d2VNR

    dR2

    RN

    = 2F

    Msd

    . 16

    In the limit where the shape parameter r0, Eq. 13 re-duces to 1 and, then, Eq. 16 reduces to that obtained for theSG systems.

    With the results above stated, in the Gaussian approxima-tion, the improved formula of the number of kink-antikinkpairs per unit time and length is then given by

    J0 = KFexp EN

    , 17

    with =1/kBT and the prefactor

    = 2

    3/2k1/2 0N

    1/2

    n1

    p1 n

    N1/2ENkBT 1/2

    Q , 18

    where nN are the eigenvalues of the nonuniform state,

    = V0 /d2Vu ,F /du2usn, and Q the product of the eigen-values of the localized eigenmodes of the critical nucleus. Inaddition, p is the number of localized modes and stronglydepends on the shape parameter r. In fact, when r0, thesystem possesses two bound states p=2, with n

    N=0 and0

    N, given by Eq. 16. Moreover, internal modes appearwhen r decreases from 0 to 1for example, p=5 for r=0.5 and p=21 for r=0.9.

    The non-Gaussian correction KF to the nucleation rateformula of kink-antikink pairs obtained through the Gaussianapproximation is given by 20

    KF =

    exp 0N2DR

    R2dR

    0

    expVNRDR

    dR .19

    In the absence of these correction termsthat is, KF1Eq. 17 reduces to that obtained by Yeml and Ko-fan 5. The presence of these factors gives rise to a betterestimation of the nucleation rate of kink-antikink pairs in thesystem. The quantity EN

    interfering in Eq. 17 designatesthe energy of the critical nucleus whose accurate value at agiven field FFm is evaluated numerically through the rela-tion

    ENl =

    dxkduNxdx

    2, 20where uNx satisfies the NLE 1 without the right-handside. However, for some particular cases, an explicit analyti-cal expression of EN

    may be obtained.For small F values FFm, the amplitude of the critical

    nucleus is large and very close to 2: um1=2

    4F /V01/2 and um2=2 1/4F /V01/2. This

    nucleus is called the large-amplitude nucleus LAN withenergy

    EN 2Es

    1 dFEs

    dF

    Es ln

    2Es

    Fd

    , 21

    where Es designates the static kink energy defined in Eq.

    7.For large F values FFm, the amplitude of the critical

    nucleus is close to zero. This critical nucleus solution of theNLE 1 is called the small-amplitude nucleus SAN whoseanalytical expression is given by

    uNx = b sec h2x/2 , 22

    with amplitude

    b = 31 + r21 r2

    tan usn1 2 cos usn + 4/cos usn1 5 cos usn 23a

    and size

    2 = 021 + r2

    1 r22 1 + 2 cos usn3

    cos usn + 21 + sin2 usn

    , 23b

    where =r / 1+r2. The energy of this SAN is also given by

    EN = 24/5kV01/2bF/V01/21 + r21 r2

    1/tan usn + 41 + 2 cos usnF/V01 + r21 r222.24

    whereIn the presence of random fields, this critical nucleusmay always exist in the system even at T0 and resultingfrom the combined effects of the energy fluctuations and theapplied field F. At high temperatures, the thermal nucleus

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  • will play the main role. In what follows, we focus our atten-tion on the thermally activated kink-antikink pairs. Theabove results constitute the starting point of the treatment ofthe inhomogeneous system. In order to relate the results ofthe nucleation rate of kink-antikink pairs to an easily acces-sible physical parameter we will evaluate, in the next section,the mean velocity of a particle in the chain which from amacroscopic viewpoint accounts for this microscopic phe-nomenon of the nucleation of kink-antikink pairs. Note thatthis question has been of interest in the theory of dislocationfor more than four decades 6,8. One should keep in mindthat at low temperatures and in the absence of fluctuations,the particles undergo small-amplitude oscillations aroundtheir equilibrium position. From a macroscopic viewpoint,the system is at equilibrium. A remarkable displacement ofparticles comes from its transition from one site to an adja-cent one due to the expansion of the newly formed kink-antikink pairs triggered by stochastic forces. The mean ve-locity of this displacement is thus determined by the numberof kink-antikink pairs created per unit time and length. Inother physical systems, such as compounds whose electricalproperties at low temperatures are due to the existence ofCDWs, the above mean velocity can be interpreted as theelectric current passing through the physical systems 30,40.

    IV. INHOMOGENEOUS SYSTEMS

    A. Preliminaries

    For real physical systems inhomogeneous systems, thedynamics of the lattice may be described in terms of quasi-particles which, however, now interact with one another orwith impurities. The interaction of nonlinear excitations withimpurities plays an important role in transport properties andnucleation process of 1D systems. The kinks antikinks andbreathers may be trapped or reflected by local inhomogene-ities as in the case of a discrete lattice where the kink can betrapped in the Peierls-Nabarro energy increment of the en-ergy of the static kink due to the discrete character of thelattice 41. When the intensity of the random fields isweak 1/2V0, the impurity has little effect on the param-eters of the critical nucleus size, shape, and amplitude.However, the total energy EN

    * necessary to create thisnucleus is affectedthat is, EN

    * =EN+Ux, where Ux isthe increment on the energy of a nucleus due to the randomfields. From the Hamiltonian 6, we can define this incre-ment on the energy as

    Ux = dxa

    VfxNx x , 25

    where Nxx depends on the shape of the nucleus, withNxx =uNxx /x. From the statistical propertiesof the random function Vfx given by Eq. 5, it is easy toshow that this increment of energy verifies the correlator

    UxUy = dxa2Nx xNy x . 26

    Accordingly, the nucleation rate of kink-antikink pairs isgiven in the factored form as

    J = J0 exp Ux , 27

    where J0 is, in the first-order approximation in , the nucle-ation rate of kink-antikink pairs in the homogeneous systemdefined by Eq. 17. Thus, we are concerned here only withthe Arrhenius factor since is small.

    As mentioned in the preceding section, we focus our at-tention on the mean velocity of particles, u /t. In fact, akink passing the point x of the chain to the right reduces thedisplacement field u by 2 and the antikink passing x to theright advances u by 2. In the presence of an applied field,the kink current is jk=nk and the antikink current jk=nk,where v is the kink velocity and nk and nk are the kinkdensity and antikink density, respectively. Therefore, themean velocity of particles can then be written as u /t=2jk jk=4vn0, where n0= nk= nk is the averagekink antikink density in a chain. The steady-state density2n0 is maintained by balance of the annihilation recombina-tion and the nucleation of kink-antikink pairs. From theprobability that the kink encounters an antikink in the inter-val of time dt, one shows that the rate of recombination of n0kinks and n0 antikinks per unit length and time is 2n0

    2 and

    the balance for the steady-state density becomes J02n02

    =0, where J0 is the nucleation rate of kink-antikink pairs inthe homogeneous systems. We can then write the expression

    of the mean velocity of particle as a function of J0 as

    u/t = 2/t , 28

    where in the homogeneous system we have t= 2J01/2.This result takes into account the fact that in the limit ofheavy damping, the kink-antikink collision is destructive.The mean time t may be viewed as the time for the transi-tion of an arbitrary point on the chain to the neighboringminimum of the potential 8. In the inhomogeneous systemwhich is under consideration, the above mean time can begeneralized by means of the Kolmogorov method as

    t =0

    dt exp 0

    x

    Jzt zdz , 29where Jz is the nucleation rate of kink-antikink pairs whosecenter of mass lies at the point z and z is the travel time ofkinks initially located at the point z to reach the point ofobservation x. The exponential in the integrand 29 is theprobability that the point x=0 will be in the original mini-mum of the potential 8 at time t 42. This expression of themean time should take a simple particular form according towhether the intensities of the applied field F and of the im-purities are weak or not. As we shall see below, expression29 reduces to that obtained in the homogeneous systemwhen the intensity of the impurity potential takes the valuezero. This limiting case constitutes a proof that Eq. 29 takesinto account the annihilation of kink-antikink pairs due to thestrong dissipation of the system.

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  • B. Mean velocity of the chain in the threshold field FFm

    In the range FFm, the critical nucleus corresponds to aSAN defined in Eq. 22. Since the random fields are weak1/2V0, impurities have little effect on the motion ofkinks and their velocity v may be considered to be the sameas in pure systems. Accordingly, from Eqs. 27 and 29, themean time t is then given by

    t =

    v

    0

    dxexp J00

    x

    dzzeUz , 30with J0= 2 /vJ0, where we have transformed the integra-tion with respect to time t to integration with the dimension-less spaced variables x and z through the relation t=x /v,where x= x / and z= z /. After integration, we obtain

    t =

    2

    J0exp Uz. 31

    Substituting Eq. 31 into Eq. 28 yields

    u/t = 2J01/2W1/2, 32

    with

    W = exp Uz . 33

    Since in the homogeneous system the mean velocity is

    u /t= 2vJ01/2, it follows from Eq. 32 that the factorW1/2 designates the correction of this result when spatial in-homogeneities are taken into account. In order to evaluate W,we assume that the random field distribution is a Gaussian-type function since the impurity-assisted nucleation mecha-nism is local by definition 43. This assumption is justifiedby the fact that the random field can take positive and nega-tive values near zero and its intensity is weak. Thus, theprobability distribution of this random field tends to 1 whenUx is zero and decreases to zero in the case of high valuesof Ux. Accordingly, having in mind that UxEN thatis, EN

    * 0, the mean W is then defined as

    W = EN

    PUexp UdU , 34

    where

    PU =1

    2exp U2/2 35

    is the probability distribution with

    = U2 =15

    16a2b2, 36

    where b is the amplitude of the critical nucleus given by Eq.23a. Substituting Eq. 35 into Eq. 34 and integratingyields

    W =1

    2exp2/21 EN2 , 37

    where is the probability integral. The correction 37 isvalid for all absolute temperatures satisfying the constraint

    EN1. For certain regimes of temperature, the probabil-ity integral can be approximated by analytical expressions.

    1. Low temperature regime

    When the temperature satisfies the constraint EN,the probability integral is then given by

    EN2 = 1 1/exp EN2 2

    i=0

    n1

    1ii + 1/2 EN2 2i+1

    ,

    38

    where designates the gamma function. Limiting this seriesto first order leads to the following expression for the correc-tion factor:

    W =expEN2

    exp EN2 /2 . 39

    In Fig. 2a, we show that in this range of temperatures thecorrection factor W1/2 increases when the shape of the sub-strate potential deviates from the sinusoidal one r0. Fur-thermore, it appears that this factor is a decreasing functionof the applied field.

    2. High-temperature regime

    In the high-temperature regime, where the temperaturesatisfies the constraint EN, Eq. 37 can be reduced, ina first order approximation, to

    W = exp2/2 . 40

    The analysis of this result shows that W1/2 is an increasingfunction of F if r0 as well as for r0, as indicated in Fig.2b. Note also that from the above expression, it is possibleto recover the result obtained previously in the homogeneoussystem. In fact, in the limit 0e.g., 0the correc-tion factor W tends to 1, in accordance with the physicalexpectation, since, in this limit, the system is homogeneous.Finally, the correction factor strongly depends on the shapeof the substrate potential via the energy N

    and/or at lowtemperatures as well as at high temperatures.

    C. Mean velocity of the chain in subthreshold fields FFm

    In the low applied field FFm, two physical situationscan be obtained: the case where the applied field is greaterthan the intensity of impurities F1/2 and the oppositesituation where it is small compared to the intensity of im-purities 1/2F. For the general case FFm, the kink suf-fers the effects of thermal and stochastic fluctuations; whenthe temperature is lower than the specific temperature T0TT0= /kBEN

    , the kink motion has an activated characterwhereas the kink activated by impurities plays the major rolefor the high temperature TT0.

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  • 1. Case of 1/2FFm

    To evaluate the mean time t, here we have to take intoaccount the distance between the kink and impurities alongthe line. It is then convenient to rewrite Eq. 29 in the form

    t =dl

    vl

    0

    dxexp J00

    x

    dzx z

    exp

    dyNz y y

    , 41where x= xd is the dimensionless variable, d the kinkwidth defined in the preceding section, and y=Vfy isthe dimensionless random field whose properties are deter-mined by the correlator y y=2yy following

    from Eq. 5. The calculation of the mean time t after ex-panding the integrand of Eq. 41 into a series yields

    t =dl

    vl2/J0W1/2, 42

    with

    W =exp

    =1

    m

    z y ydy .As seen above, this quantity W describes the correction fac-tor to the mean velocity of the chain due to the presence ofimpurities in the system. Its calculation depends on the tem-perature regime.

    a. High temperature regime. To calculate the mean W, we

    FIG. 2. Correction factor to the mean velocity of particles W, induced by the presence of impurities in the system, as a function of theapplied field F: a The case of large values of the applied field FFm and in the regime of low temperatures T=280 K. The intensityof the impurity potential is taken to be =1.11102 eV/2. b The case of large values of the applied field FFm and in thehigh-temperature regime T=500 K, for =1.11103 eV/2. c The case of weak applied field satisfying the constraint 1/2FFm and in the high-temperature regime for example, T=400 K, =6.94106 eV/2. d The case of very weak applied field F1/2 and in the high-temperature regime. Here T=400 K and =6.94108 eV/2. Note that the choice of numerical values of theintensity of the impurity potential, in either case, is dictated by the condition EN or EN.

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  • must remember that the statistical properties of the randomfield Vfx are assumed to be correlated see Eq. 5. Thus,the distribution function P(Vfx) which satisfies this as-sumption is equivalent to a Gaussian probability density:

    PVfx = exp 12 Vf2xdx , 43where is the intensity of the impurity. Accordingly, in thehigh-temperature regime, the correction factor W may bewritten as

    W = D exp A D exp A0

    , 44

    with

    A =1

    22 2ydy +

    i=1

    m

    dyiNzi yi yi 45a

    and

    A0 =1

    22 2ydy , 45b

    where A may be viewed as the action. We can evaluate thiscorrection factor by minimizing the action A to obtain theextremal trajectory. The action corresponding to this particu-lar path is equal to

    Ac = 1

    22 dz

    i=1

    m

    Nz zi2. 46Next, we evaluated the series of these integrals. As pointedout in Ref. 30, to evaluate this series of integrals with re-spect to z1 ,z2 ,z3 , . . . ,zm, we can readily verify that the prin-cipal contribution come from the points lying close to thesurfaces zi=zjthat is, for Gzizj=G0G0, where G isrelated to the random field correlator as

    UxUy = 4d

    dz

    a2 uNx z

    x uNy z

    y 47a

    =4dGx y , 47b

    where uN is the shape of the critical nucleus in a pure system.Using Eqs. 4347, it follows that

    W = exp2G0 , 48

    where the quantity G0 is given by

    G0 = 41 dF

    EdF

    Eln

    2E

    Fd

    . 49

    The variation of W as a function of the applied field F isplotted in Fig. 2c. It appears that, in this range of tempera-tures, the correction factor is less sensitive to the variation ofthe applied field in the whole range of variation of the shapeparameter r.

    b. Low-temperature regime. In the range of low tempera-tures, it is necessary to take into account the fact that therandom field can be cut off. For this reason, we introduce, inthe expression of the mean W, the Heaviside function definedas

    !x =1

    2ilim0

    dq eiqxq i

    . 50

    For this purpose, the correction factor 42 is then given by

    W = D !EN + Ny Z ydyexp A D !EN + Ny Z ydyexp A0 .

    51

    Using the same procedure as before, we obtain, after somelengthy algebra,

    W = eENl+2G0

    l/2EN

    l G0l

    2G0l + 23ENl G0l2G0l

    3

    +

    4expENl G0l2G0l

    2

    ENl

    2G0l+

    2

    3 ENl2G0l3

    +

    4exp ENl2G0l

    2 , 52where G0

    is given by Eq. 49. This result is only qualitative since the perturbation approach is no longer valid. In fact, onecan easily show that in the low-temperature regime the main contribution to the mean velocity of particles in the chain is dueto random field fluctuations which are of the order of EN. Thus, impurities can produce an appreciable change in theequilibrium shape and size of the nucleus, and then the perturbation theory is no longer valid for a solution of the NLE 1.

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  • 2. Case of F1/2Fm

    When the field F is small compared to the intensity ofimpurities 1/2, the kink has to overcome the impurity poten-tial. In accordance with the activation-type formula, the mo-bility or the speed of the kink turns out to be exp2 /2. Taking into account this retardation of the kink byimpurities in the expression of the mean time t, we obtain

    t = exp 2/20

    dx exp J0e2/2

    0

    x

    dz exp yNy zdyz

    x

    dy

    exp ysx ydy

    exp ysy zdy . 53

    By expanding the exponential in series and performing theGaussian integration over D for high temperatures, we obtain

    t = exp 2/20

    x

    dxe2B0 exp J0e2/2x2/2

    exp2i,j=1

    n

    Gzi zj 2Czi zj + Bzi zjexp2

    i,j=1

    n

    2Bzi x Czi x , 54where

    Bzi zj = dzsz zisz zj, Czi zj = dzNz zisz zj . 55

    Integration of Eq. 54 can be easily performed if zi=zj, lead-ing to the following expression of the mean time:

    t = /21/2e2/2e

    2G0B0

    /2J01/2, 56

    and then the mean velocity of the chain

    u/t = 2uJ01/2W1/2

    , 57

    with the correction factor

    Wl = exp2/2exp2G0 B0

    , 58

    where B0=expRN

    /d. Figure 2d shows that the correc-tion factor W is an increasing function of the applied field Ffor r0 and is less sensitive for r0.

    V. APPLICATION TO THE DIFFUSION OF HYDROGENATOMS ON METALLIC SURFACES

    The question of surface diffusion of atoms and moleculesadsorbed on metallic surfaces is a long-standing problem

    which has recently attracted a renewal of interest with theintroduction of new ideas from the physics of nonlinear phe-nomena. The experimental investigations of this problem arebased on two essential classes of methods 44: the profileevolution methods such as electron beam scanning and theequilibrium methods such as the field ion microscopy. The-oretical works are outlined by experimental studies whichevidence 36,45,46 an important role of collective motion ofadsorbed atoms adatoms. According to these experimentalstudies, the diffusion of adatoms can be described by thenonlinear dynamics of the well-known FK model which isessentially a single model allowing an accurate descriptionof such a consistent motion of particles. In some cases, ada-toms may be treated as quasi-1D systems where a chain ofinteracting particles is placed in a channel. The atomicchain is subjected to a one-, two-, or three-dimensional sub-strate potential, which is periodic in one direction and un-bounded in transverse directions, so that atoms are confinedtransversally. However, when the concentration "C of ada-toms is weaki.e., closed to 1"C1one can ignoreatomic displacements in transverse directions and allow at-oms to move only along the direction of the chain, and themodel reduces to a well-known FK model. The concentrationof adatoms is characterized here by the dimensionless param-eter "C= p /q, the so-called coverage in surface physics,where p is the number of atoms and q is the number ofminima of the substrate potential. In addition, previous stud-ies see, e.g., Refs. 34,36 and references therein haveproved that the system of adsorbed atoms is subjected to anonsinusoidal substrate potential and that the RP potentialsee Eq. 2 provides an accurate description of such a sub-strate potential. According to these studies, the Hamiltonianmodel described by Eq. 6 may be successfully used tostudy the migration of atoms adsorbed on metallic surfaces.For the case of the H/W and H/Ru adsystems, an estimateparameter is r0.3 5. Thus, we apply the results of theabove analytical study to estimate the mean velocity of ahydrogen atom on a Ru and W substrates induced by theapplied field F. Note that geometrical imperfections of theadsorbed surfaces are considered here as impurities sincethey are at the origin of the spatial deformation of the newlycreated nucleus and consequently may be approximated bythe impurity potential given by Eq. 4. The model param-eters used in our numerical calculations are 5 V03.62102 eV 1 and k=3.57101 eV . The lattice constanta3 is taken to be the distance between the wells along afurrow on the W112 surface since "C1. In addition, theshape parameter of the substrate potential is taken to be r=0.3. The calculation of the correction factor from these nu-merical values of the characteristic parameters of the adsys-tem shows the following.

    First, for high temperatures see Fig. 3a the correctionfactor increases with the applied field F and tends rapidly to1 when F becomes higher. This result is in accordance withphysical expectations since the increase of the intensity ofthe applied field results in the increase of the kinetic energyof the newly formed kink-antikink pairs. Consequently, im-purities have little effect on the nucleation rate of kink-antikink pairs W1. Note that the perturbation theory isnot valid in the case of low temperatures associated to theweak applied field.

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  • Second, for the value of the applied field F close to thethreshold field Fm, the correction factor W increases or de-creases according to whether the temperature is high seeFig. 3b or low see Fig. 3c. In fact, in the high-temperature regime, the correction factor increases with theapplied field. Thus, in this temperature regime, the disorderin the systems makes the nucleation of kink-antikink pairsmore favorable. However, in the low temperatures, disordercontributes to make it less favorable.

    Finally, it should be noted that the correction factor,which contains all the information concerning the effect ofimpurities, is less than 1 Figs. 3a and 3b, indicating thefact that the presence of impurities in the system makes theprocessing of nucleation of kink-antikink pairs less favor-able. However, for large values of the applied field, the cor-rection factor is greater than 1 Fig. 3c, indicating the factthat the impurities catalyze the transition from the criticalnucleus or saddle-point configuration to the newly formedkink-antikink pairs by making the growth energetically fa-vorable.

    It is important to mention that our model is valid when theconcentration of adatoms, "C, is close to 1. When "C isgreater than 1, the amplitude V0 of the substrate potential is afunction of adatom concentration "C and the compressionforces, in the adatomic chain, overcome the forces holdingthe adatoms in a given channel and adatoms will startcreeping out of the channel so that their motion will be-come more complex and can be described only in terms of atwo- or three-dimensional model.

    VI. CONCLUSION

    In this paper, we have investigated the influence of impu-rities on the nucleation of kink-antikink pairs in the nonlinearKlein-Gordon model with the Remoissenet-Peyrard substratepotential driven by an external constant field. We have fo-cused our attention on the mean velocity of particles of thisone-dimensional system, which is a physical parameterclosely related to the number of kinks and antikinks createdin the system per unit time and length. Moreover, in othersystems like compounds where the electrical properties aredirectly related to the existence of charge density waves, thismean velocity designates the electrical current carried by theCDWs.

    First, we have improved, by taking into account the non-Gaussian correction in our calculation, the analytical expres-sion of the nucleation rate of kink-antikink pairs in the ho-mogeneous system previously calculated by Yeml andKofan 5. This calculation is one step towards the study ofthe effects of impurities. Next, by means of the Kolmogorovmethod associated with the perturbation analysis, we haveshown that the dynamics of the system may be different ac-cording to whether the intensity of the applied field is weakor not compared to the intensity of the impurity potential andthe magnitude of the temperature. More precisely, we haveshown that, in the range of weak values of the applied field,the quantitative effects of impurities increase with the ap-plied field and temperature. Moreover, the presence of impu-rities in the system makes the nucleation process of kink-

    FIG. 3. Correction factor to the mean velocity of particles W,induced by the presence of impurities in the system of H/W, as afunction of the applied field F and for three values of temperature:a The case of weak applied field in the high-temperature regime,=6.94108 eV/2. b The case of large applied field in thehigh-temperature regime for =1.11102 eV/2. c The caseof large applied field in the low-temperature regime with =1.11102 eV/2. Note that the choice of the intensity of the impu-rity potential in the low- and high-temperature regimes is dictatedby the constraints verified by the quantity in these temperatureregimes. EN for the low-temperature regime or EN forthe high-temperature regime.

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  • antikink pairs less favorable. Furthermore and contrary to thepreceding case, for large values of the applied field, impuri-ties catalyze the transition from the saddle-point configura-tion of the system to the newly formed kink-antikink pairs bymaking the growth of the nucleus energetically favorable.

    Finally, we mention that our numerical applications arecarried out by making use of the parameters of H/W andH/Ru adsystems where available data exist. However, themodel may be applied to a number of various systems ofcondensed matter physics for which the substrate potential isused to describe its physical phenomenanamely, disloca-tion kinetic in crystals, electrical current carried by theCDWs in the compounds whose electrical properties are dueto the existence of this CDW, or the electrical current in thelong Josephson junctions, to name only a few. The perturba-tion analysis used here allows one to write down an analyti-cal expression of the nucleation rate of kink-antikink pairs inthe inhomogeneous 1D system, from which the quantitativeeffects of impurities on this quantity can be obtained. This

    calculation is one step towards a complete study of themodel. The method is valid only in the case of weak impurityfieldsthat is, in the case where impurities have little effecton the critical nucleus parameters and on its stability. An-other restriction of this study concerns the correlation lengthof the random field of impurities which has been taken equalto zero although the case of a spatially correlated field maybe of interest for applications to much of condensed mattersystems. These two limitations of our study are now underconsideration.

    ACKNOWLEDGMENT

    One of the authors T.C.K. appreciates the facilities pro-vided by the Max-Planck-Institute for the Physics of Com-plex Systems for the improvement of this work. Financialsupport from the Abdus Salam International Centre for The-oretical Physics and the Swedish International DevelopmentCooperation agency is acknowledged.

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