THRESHOLD FIELD IN NUCLEATION PROCESS FOR AN INHOMOGENEOUS DEFORMABLE NONLINEAR KLEIN-GORDON SYSTEM

IC/2005/099 Available at: http://www.ictp.it/~pub− off United Nations Educational Scientific and Cultural Organization and International Atomic Ener...
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IC/2005/099

Available at: http://www.ictp.it/~pub− off

United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THRESHOLD FIELD IN NUCLEATION PROCESS FOR AN INHOMOGENEOUS DEFORMABLE NONLINEAR KLEIN-GORDON SYSTEM

Rosalie Laure Woulach´e1 Laboratoire de M´ecanique, D´epartement de Physique, Facult´e des Sciences, Universit´e de Yaound´e I, B.P. 812, Yaound´e, Cameroun and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, David Yem´el´e D´epartement de Physique, Facult´e des Sciences, Universit´e de Dschang, B.P. 67, Dschang, Cameroun and Timol´eon C. Kofan´e2 Laboratoire de M´ecanique, D´epartement de Physique, Facult´e des Sciences, Universit´e de Yaound´e I, B.P. 812, Yaound´e, Cameroun and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE – TRIESTE October 2005

1 2

[email protected]; [email protected] Senior Associate of ICTP.

Abstract We investigate the effect of local inhomogeneity on the nucleation process of kink-antikink pairs in the driven nonlinear Klein-Gordon model with the Remoissenet-Peyrard substrate potential whose shape can be varied as a function of the deformability parameter and which has the sine-Gordon shape as a particular case. In particular, the expression of the depinning threshold field of kink-antikink pairs, which is the value of the applied field at which the process of the nucleation of kink-antikink pairs takes place, is calculated. The dependence of this depinning threshold field on the deformability parameter r shows that it strongly increases when the shape of the substrate and/or the impurity potential deviate from the sinusoidal one.

1

1

Introduction It is well known that many physical properties of real systems are directly related to non-

linear effects produced by anharmonic interactions. The governing equations frequently admit large-amplitude localized wave profiles which are physically distinct from those obtainable by superposition of small-amplitude or linearized profile waves. These localized excitations, which can propagate through the system without important distortion of shape, are commonly referred to as solitary waves or solitons. The propagation of these solitons through homogeneous and disordered media has been extensively investigated during the past decade [1-4]. The main issue is how nonlinearity can qualitatively modify the effects of disorder on transport properties, and conversely how disorder may change the steady-state motion of solitons in nonlinear systems. Since in nonlinear systems disorder generally creates Anderson localization, which means that the transmission coefficient of a plane wave decays exponentially with the system length [5, 6]. As a step towards understanding this propagation of soliton through disordered media, it was necessary to study in detail the scattering of soliton by a local or single impurity where impurity may mean spatial modulation, quasiperiodicity or disorder of several kinds. Thus, a number of works devoted to the soliton-impurity interaction has been done in the framework of the sineGordon and φ4 models [7-16]. Most of these works are related to the fluxon dynamics in the presence of local inhomogeneities (microshunts and microresistors) in long Josephson junctions [7-15], where inhomogeneities are installed into the junction during fabrication [17] while other are related to coumpounds where the electrical properties are subjected to the existence of charge density waves (CDW) [18-20]. Although these results are quite interesting, they are nevertheless limited to the effect of inhomogeneities on the soliton motion. However, these inhomogeneities may also act on the creation process of solitons, that is, on the nucleation process. In fact, nucleation is generally defined as a phenomenon where a new phase appears locally in space. It is one of the most drastic phenomenon in various fields of physics, chemistry, biology, and also in engineering [21]. In condensed matter physics, this phenomenon of nucleation is most interesting in the sense that it can be controlled by external drives such as pressure, temperature, electric and magnetic fields and so on. One can usually distinguish homogeneous and heterogeneous nucleation. In the first case, embryos of a stable phase emerge from a matrix of a metastable parent phase due to spontaneous thermodynamic fluctuations. Droplets, which in the magnetic compounds are known as magnetic bubbles, larger than a critical size will grow while smaller ones decay back to the metastable phase [22-24]. In the second case, impurities or inhomogeneities catalyze the transition by making growth energetically favorable [21]. The study of nucleation has been nevertheless carried out in a number of papers [20, 25, 26] taking into account the presence of these localized impurities. The central point being how impurities can quantitatively modify the nucleation rate of kink-antikink pairs in the system where this nucleation rate designates the number of kink and antikink thermally activated per

2

unit time and per unit length. For example, in the presence of impurity, the current carried by the CDW may increase as a result of the increase in the rate of generation of solitons due to fluctuations in random field of defects [25]. Similarly, the mean velocity of particles in the nonlinear Klein-Gordon (NKG) system with Remoissenet-Peyrard substrate potential depends strongly on the intensity of impurities as well as on the shape of the substrate potential [26]. Finally, the question of the effect of a local inhomogeneity on the nucleation process and specially the case of CDW depinning has been recently work out by Yumoto et al. [20]. They found that the CDW starts to move on the system only above some critical field called threshold field. In addition, this threshold field is lowered by a finite amount compared to that in the absence of an impurity. In this paper, we develop this idea with the approach closely related to the concept developed in the dislocation literature. Thus, we consider the NKG model. In fact, introduced to model the dynamics of dislocation in crystals, in a rather general context a NKG-type model describes a 1D chain of interacting particles subjected to a periodic substrate (on-site) potential. This model may also describe, for example, a closely packed row of atoms in crystals, a chain of ions in a channel of quasi-1D conductor, hydrogen atoms in hydrogenbonded systems [27], concentration dependence of the conductivity and diffusivity in several quasi-one-dimensional systems [28], to name only a few. In all the case mentioned above, the chain of interacting particles is a part of the whole physical system under consideration, and the remainder is modelled as an external substrate potential. The organization of the paper is as follows. Section 2 deals with the description of the model, which takes into account the deformability of the substrate. The basic depinning field is also presented. In section 3, we derive the configuration of the ground-state of the system, i.e., the configuration of the system with the lowest energy (nucleus). The depinning field of this configuration is then calculated in section 4. In section 5, the model is modified by extending the deformability of the substrate potential to the impurity potential. The new depinning field of the system is then evaluated. Finally, section 6 provides a summary and concluding remarks.

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Model and basic depinning field To begin, we consider a system of particles of mass m placed on an infinite one-dimensional

(1D) lattice of lattice spacing a, oriented in the direction of the x axis, including a localized inhomogeneity (impurity). The system is also subjected to an external constant field F . In the continuum or ”displacive ” limit, the energy functional of the commensurate system may be written as follows: H = Aa

Z

dx a

(

1 K 2



∂φ ∂x

2

)

˜ + µVsub (qφ) + vi Vimp (χ + φ)δ(x − xi ) . − fφ

(1)

The first term in Eq.(1) is the elastic energy, where K is a characteristic parameter related to the elastic constant k by K = ka2 . The second term is the energy associated with the applied 3

field F which may be due to mechanical stress or to an electric field if we are in the presence of charged particles f˜ = (2π/a)F . In this latter case, the external constant field F and the electrical field Ee are related through the equation F = e ∗ Ee ,

(2)

where e∗ is the coupling constant or the effective charge of each particle. The third term is the substrate potential energy where µ is a characteristic parameter related to the amplitude V0 of the substrate potential by µ = (2π/a)V 0 . This nonlinear on-site potential represents the combined influence of the surrounding crystal or macromolecule and external effects. We concentrate our attention on this potential introduced by Remoissenet and Peyrard (RP) [29] Vsub (qφ) = (1 − r)2

1 − cos(qφ) . 1 + r 2 + 2r cos(qφ)

(3)

Here the shape parameter r, satisfies the constraint |r| < 1 . At r = 0, the RP potential reduces

to the well-known sine-Gordon (sG) potential. As r varies, the amplitude of the potential remains

constant with degenerate minima 2πn/q and maxima (2n + 1)π/q , while its shape changes from a potential with sharp bottoms separated by flat tops to a potential with flat bottoms separated by sharp peaks (see Fig.1). The integer q = a/a s is related to the atomic concentration θ = 1/q, where as is the dimension of the period of the substrate potential. In fact, the parameter r depends on the physical characteristics of each system. For example, in quasi-1D compounds whose electrical properties are due to the existence of CDW, the substrate potential which corresponds to the interaction of CDW with host atom may be calculated up to higher order of the perturbation theory. Up to the first order of this perturbation theory, we obtain the sG potential which is a good approximation only in the weak-and strong coupling cases. Thus, at higher order, in addition to the first harmonic which describes the sG potential, one also obtains the second, the third and higher harmonics [30]. The compact form of this interaction between CDW and host atoms may then be approximated by the RP-type function where the parameter describing the shape of the substrate potential depends on the amplitude of the CDW gap, the Fermi velocity and the quasi-particle energy. Similarly, for the adatomic systems, the parameter r of the substrate potential is related to the frequency ω0 of the oscillations of an isolated adatom at the bottom of the adsorption site, the adatom mass ma and the period as of the substrate potential [31] where r = (1 − κ)/(1 + κ), with

κ = ω0 (as /2π)(2ma /V0 )1/2 . Note that the above parameters, for the adatomic systems, are related to the characteristic parameters of the system described by the Eq.(1). The last term of Eq.(1) is the interaction energy of an impurity particle located at site i with amplitude v i .

For the sake of convenience, we make the Hamiltonian dimensionless by setting X = x/(K/µ) 1/2 and H = A(Kµ)1/2 H, where ) (   Z 1 1 ∂φ 2 − f φ + 2 Vsub (qφ) + vVimp (χ + φ)δ(X − Xi ) , H = dX 2 ∂X q 4

(4)

˜ and v = vi /(µK)1/2 . Here, to concentrate our attention on the qualitative effect with f = f/µ of the shape of the substrate potential on the kink-antikink pairs depinning, we first consider the form of the impurity potential limited up to the first harmonic Vimp (χ + φ) = − cos(χ + φ),

(5)

where χ may be viewed as a parameter characterizing the location of an impurity relative to the site where the energy gained by commensurability is maximum. To obtain the stable configuration of the displacement field, we minimize the energy functional of the system described by Eq.(4). In other words, this stable configuration is solution of the Euler-Lagrange equation δH/δφ = 0, leading to −φXX − f +

1 dVsub (qφ) + vsin(χ + φ)δ(X − Xi ) = 0. q2 dφ

(6)

In the homogeneous system, the stable configuration of the displacement field φ(X) verifies the following equation −φXX − f +

1 dVsub (qφ) = 0, q2 dφ

(7)

obtained by setting in Eq.(6), v = 0. In the absence of the applied field f , two kinds of solutions can be obtained: The uniform solutions φ 0 = 2πn/q, with n an integer, and the non-uniform solutions ( α∗ X − X0 ∗ −1 = sign(π − qφ) (α /α) tan ± 1/2 L(1) [α2 + tan2 (qφ/2)] 1/2 )  2 α + tanh−1 2 α + tan2 (qφ/2)

(8)

for −1 < r ≤0, and ( X − X0 α∗ ∗ −1 ± = sign(qφ − π) α tanh 1/2 L(2) [1 + α2 tan2 (qφ/2)]  1/2 ) 1 −1 − tanh 1 + α2 tan2 (qφ/2)

(9)

for 0 ≤ r < 1, with L(1) = α,

L(2) = 1/α,

α = (1 − |r|)/(1 + |r|)

and

α∗ =

p 1 − α2 ,

(10)

where X0 designates the center of mass of these solutions. Eqs.(8) and (9) are the well-known soliton solutions of the RP model [29] centered at X 0 , where the configuration with ”+” sign is the kink, while the configuration with the ”-” sign designates the antikink. The energy of the uniform solution is Esc = 0 and the energy of the non-uniform solution is given by Es(`) = (8/q)G(`) , 5

` = 1, 2,

(11)

with G(1) = (1/α∗ ) tan−1 (α∗ /α), G(2) = (α/α∗ ) tanh−1 (α∗ ).

(12)

Hereafter, the upper-script ` = 1 stands for r ≤ 0, while ` = 2 stands for r ≥ 0. Fig.2 shows the shape of the non-uniform solutions: the kink and the antikink solitons. In the presence of the applied field f , Eq.(7) admits also the uniform solution φ u verifying the following relation 0

Vsub (qφu ) = q 2 f,

(13)

where the prime designates the derivative with respect to the argument. This solution corresponds to the pinning of the particles in the substrate potential well of the system. On the contrary to the preceding case which where characterized by the absence of the applied field, the potential well in this case will be raised with respect to the adjacent potential maximum, due to the presence of the applied field, while the adjacent potential well will be lowered. These maxima and minima of the potential shown in Fig.3, which are known as Peierls valleys and Peierls hills, respectively, will be shifted and eventually disappear when the applied field approaches the threshold value Fm defined by : √ i q√ √ h 2 2 (3α2 − 1) + ∆ ∆ − 3(1 − α2 ) Fm = ,  √ 2 5α2 − 3 + ∆

(14)

with ∆ = 9α4 −14α2 +9. This particular value of the applied field is known as the basic depinning

field of the uniform displacement field φ u . Eq.(7) admits also the non-uniform displacement field verifying the relation φnu (X) = − {2 [f (φu − φnu (X)) + Vsub (qφnu ) − Vsub (qφu )]}1/2 , 0

(15) 0

obtained by integrating Eq.(7) with the boundary conditions φ(±∞) = φ u and φ (±∞) = 0. This solution presented in Fig.4 is known as the critical nucleus [32]. The amplitude φ m and the energy of creation of this nucleus ∆E N depend on two parameters : The applied field f and the deformability parameter r. Fig. 5 shows the variation of this energy as a function of the applied field for few values of the deformability parameter. In fact, to understand the mechanism of the creation of kink-antikink pairs in the system, it is necessary to know the key role of this critical nucleus. Due to the application of the field f , the energy of the system between two adjacent Peierls valleys evaluated from the energy functional Eq.(1), is 2πlf , where l is the length of the chain. The first uniform state φN u is metastable towards the adjacent state φ (N +1)u . The segment of the system which was initially at the state of higher energy φ N u is likely to jump to the next state φ(N +1)u with lower energy triggered off by stochastic forces. This segment of the system with lower energy is connected to the segment in Peierls valley with higher energy 6

through kink-antikink pairs or nucleus. The transitions described above will be possible only if the fluctuations produce within the system a minimum of energy ∆E N >> kB T necessary to create a critical nucleus.

3

Ground state of the inhomogeneous system In many cases, the dynamical behavior of the inhomogeneous system is studied in terms of the

interactions of solitons with separated localized inhomogeneities (impurities). Such impurities break the translational invariance of the original unperturbed system, so the soliton can no longer propagate with a constant velocity and/or a constant amplitude because one of a general effect in the soliton scattering by local impurities is energy loss through radiation. Furthermore, the impurity may frequently give rise to an effective potential to a soliton; leading to the pinning or the reflexion of soliton. This means that, the impurities act on the system after the creation of solitons. Here, we are looking for the case where impurities act at the beginning of the process of creation of solitons, that is, on the nucleation process of kink-antikink pairs. In the presence of localized impurity, the dynamics of the system is described by Eq.(6). In this case the non-uniform solution φ s (X), which is the stable configuration of the displacement field, satisfies the boundary conditions lim φs (X) = φu

and

X→±∞

lim dφs (X)/dX = 0.

(16)

X→±∞

The first derivative of the stable configuration exhibits a discontinuity, following from Eq.(6), given by 0

0

φ+ − φ− = v sin(χ + φi ),

(17) 0

0

where φi is the value of the displacement field at the impurity site X i and φ+ and φ− are its 0

0

0

0

first derivatives near Xi , i.e. φ+ = limX→X + φ (X) and φ− = limX→X − φ (X). i

i

The solution of Eq.(6) with the boundary conditions Eq.(16) can be obtained analytically in the implicit form φs (X) = φnu (|X − Xi | + C0 ),

(18)

where φnu is the non-uniform solution of the unperturbed homogeneous system, while C 0 is a constant verifying the continuity condition given by Eq.(17). It is not straightforward to show 0

0

that φ+ = −φ− and consequently 0

φ+ = (v/2) sin(χ + φi ).

(19)

By combining Eq.(16) and the first integration resulting from Eq.(7) we obtain 1 2



dφ dX

2

− fφ +

1 1 Vsub (qφ) = −f φu + 2 Vsub (qφu ), 2 q q

7

(20)

and the following implicit formula for C 0 is derived 1/2   1 = 2 f (φu − φnu (C0 )) + 2 [Vsub (qφnu (C0 )) − Vsub (qφu )] q (v/2) sin(χ + φnu (C0 )).

(21)

The above relation Eq.(21) gives some informations about the displacement field φ at the impurity site. Due to the complexity of this relation, it is not possible to obtain the analytical expression of this displacement field. However, one can obtain it numerically. The knowledge of this value allows us to determine completely the configuration of the ground-state of the 0

0

inhomogeneous system. This configuration is also determined numerically since φ i , φ+ and φ− are determined numerically. Fig.6 shows these configurations for different values of the deformability parameter and for a given value of the applied field f . It appears that the amplitude of above configurations increases with the deformability parameter r while its width decreases.

4

Depinning field of the critical nucleus in the inhomogeneous system As previously mentioned in Sec.3, the application of the field f to the system can lead to

the generation of the critical nucleus if the fluctuations of the temperature produce within the system a minimum of energy ∆EN corresponding to the energy of this critical nucleus. In the homogeneous system, this critical nucleus which is the configuration relevant to the saddle point of the surface energy expands, leading to the creation of a kink and an antikink. This process is possible for all magnitude of the driven field f . However, in the inhomogeneous systems, the process of the nucleation of kink-antikink pairs is slightly different because there exists a threshold field fc for which the process of expansion of the critical nucleus takes place. The calculation of this threshold field, which corresponds to the depinning field of the critical nucleus in the system, is the aim of this section. For this purpose, we use the procedure outlined in ref.[20]. In fact, the depinning threshold field, f c , is defined as the field at which the stable solution becomes unstable. The instability of this solution being triggered by the onset of a zero eigenvalue of the fluctuation mode around the stable configuration. This method has been successfully used in other context namely the π junction [33] and in the charge density waves depinning [20]. To test this stability, we add a small perturbation δφ(X, t), which can be assumed to have the form δφ(X, t) = δφλ (X) exp(−λt), to φnu in the equation derived from the energy functional, that is, Eq.(4) [δH/δφ = −∂φ/∂t ]. After linearisation which respect to δφ λ , this leads to the

Schr¨odinger-type eigenvalues equation   1 ” ∂2 + V (qφnu ) + v cos(χ + φs )δ(X − Xi ) δφλ = λδφλ − ∂X 2 q 2 sub

(22)

where λ is the eigenvalue. As mentioned above, we consider the translational mode, λ = 0, to determine the threshold field fc . In the homogeneous system (absence of impurity), the 8

translational mode solution of Eq.(22), with v = 0, is given by δφλ (X) = Λ

dφnu (X) , dX

(23)

where Λ is the normalization coefficient. However, in the inhomogeneous system, v 6= 0, the translational mode solution of Eq.(22) is given by

0

δφλ (X) = −Λ [2θ(X − Xi ) − 1] φs (X),

(24)

where θ(X) is the step function. As can be easily seen, the translational modes in both sides of the impurity site are not connected. Hence, the threshold f c is determined as the particular value of the applied field f at which these fluctuations are connected at the impurity site. At the threshold field fc , 0

0

φs (X) = [2θ(X − Xi ) − 1] φnu (X).

(25)

Substituting this relation into Eq.(24) gives 0

δφλ (X) = −φnu (|X − Xi | + C0 ),

(26)

which is in fact the solution of Eq.(22) at the threshold field. The substitution of Eq.(26) into Eq.(22) leads to the constraint verified by the displacement field and its derivatives at the impurity site 00

0

2φnu (C0 ) − v cos(χ + φnu (C0 ))φnu (C0 ) = 0.

(27)

By combining Eq.(7) and Eq.(19), Eq.(27) becomes −f +

v 1 Vsub (qφnu (C0 )) − sin (2(χ + φnu (C0 ))) = 0. q2 8

(28)

As expected, the threshold field fc obtained by solving numerically Eq.(28) is a decreasing function of the amplitude of the impurity potential v (see Fig.9). In addition, this threshold field has a finite value f∞ even if v → ∞ . In this limit, the finite value f ∞ satisfies the relation f∞ (φu + qχ) +

1 [Vsub (qχ) − Vsub (qφu )] = 0. q2

(29)

Fig.7 shows the variation of f∞ as a function of the deformability parameter r. It is obvious that, it is an increasing function of r. These observations are also evident on the plot of the threshold field in the presence of an impurity as a function of the amplitude v of the impurity potential. For the better understanding of the physical meaning of the critical field on the depinning process of the critical nucleus, it is necessary to know the shape of the potential energy of the inhomogeneous system in the presence of the critical nucleus. Let us mention first that, the critical nucleus obtained here is viewed as the configuration of the ground-state of the inhomogeneous system, since as we shall see below, it corresponds to the configuration of the system with lower energy. Next, we mention also that in the inhomogeneous system, the most 9

suitable variable to represent the potential energy of the system is the displacement field at the impurity site φ(Xi ) = φi , which corresponds to the maximum displacement of the particle in the system. Finally, with the fixed values of f , v and χ, we can determine the lowest energy configuration by solving Eq.(6) under the condition of continuity φ(Xi ) = φi

and

0

φ (Xi ) =





1 2 f (φu − φi ) + 2 [Vsub (qφi ) − Vsub (qφu )] q

1/2

,

(30)

and obtain the corresponding energy for φ i varying between 0 and φi−max . The results of this calculation are plotted in Fig.8, for some values of the deformability parameter r.

5

Influence of the deformability of the impurity potential on the depinning field of the critical nucleus In the preceding sections, we have assumed that the inhomogeneity of the system results

from both the shape and the amplitude of the substrate potential. This is the reason why the two potentials have different analytical expressions. However, in most cases the inhomogeneity of the system results only from the amplitude of the substrate potential at the impurity site. These cases are known as the basic substrate impurity case. The underlying substrate potential and the impurity potential have the same analytical expression. Thus, we use here the RP potential to describe the impurity potential instead of the sG potential used in the preceding sections : Vimp (χ + φ) = (1 − r)2

1 − cos(χ + φ) , 1 + r 2 + 2r cos(χ + φ)

(31)

where r is the shape parameter defined in Sec.2. The functional energy of the system is defined by the Hamiltonian (1) except that the impurity potential is now given by Eq.(31). The aim of this section is to determine ,by means of the procedure presented in the preceding sections, the critical field or the depinning field of the critical nucleus in the driven RP system with localized substrate impurity. We minimize the energy functional Eq.(4) of the system by using Euler-Lagrange equation. This leads to the following equation for the displacement field −φXX − f +

1 dVsub (qφ) dVimp (χ + φ) −v δ(X − Xi ) = 0. q2 dφ dφ

(32)

In the homogeneous system (v = 0), this equation admits the non-uniform solution satisfying Eq.(15). In the inhomogeneous system, the stable configuration of the system called the nonuniform solution satisfies the boundary conditions (16) and the following condition of continuity at the impurity site Xi 0

0

φ+ = −φ− =

v dVimp (χ + φ) | φi , 2 dφ

(33) 0

0

where φi is the value of the displacement field at the impurity site while φ + and φ− are the upper and the lower first derivatives of φ at the impurity site, respectively. The value of φ i can 10

be obtained by combining Eq.(33) and the first integral resulting from Eq.(32), with v = 0. This leads to the following implicit relation for φ N (C) = φi :   1/2 1 2 (φu − φN (C))f + 2 [Vsub (qφN (C)) − Vsub (qφu )] = q v dVimp (χ + φ) |φN (C) . 2 dφ

(34)

The stable configuration of the system or the critical nucleus in the inhomogeneous system can then be obtained by solving numerically Eq.(32) for v = 0, with the condition at the impurity site on the displacement field φ and its first derivative obtained numerically, respectively from the two relations: (see Eq.(33) and Eq.(34)). As in the preceding section, the condition of the stability of this critical nucleus can also be well established using the zero-mode fluctuation solution of the linear equation resulting from the second variational equation of the Lagrangian given by   1 ” d2 Vimp (χ + φ) ∂2 + V (qφN ) + v δ(X − Xi ) δφλ = λδφλ − ∂X 2 q 2 sub dφ2

(35)

This stability is obtained at the threshold field f c where the following equation is satisfied : 00

2φN (C) − v

d2 Vimp (χ + φN (C)) 0 φN (C) = 0. dφ2

(36)

This equation can be written in the simple form by the use of Eq.(32) and Eq.(33) which yields :

 dVimp (χ + φ) v 2 d2 Vimp (χ + φ) 1 dVsub (qφ) |φN (C) − |φN (C) |φN (C) . 2 −f + 2 2 q dφ 2 dφ dφ 

(37)

The results plotted in Fig.9, based on a computational evaluation of Eq.(37), depend on the amplitude of the impurity potential v and on the deformability parameter of the system. The curves obtained here (dotted lines) have the same behavior as those previously obtained where the impurity potential were approximated by the sinusoidal (sG − type) potential (dashed lines).

However, there are a significative changes in the values of the threshold field obtained in these two cases.

6

Conclusion In summary, we have investigated the process of nucleation of kink-antikink pairs in the

nonlinear Klein-Gordon system with the localized impurity and subjected to the RemoissenetPeyrard substrate potential. We have shown that, on the contrary to the case of homogeneous system, this process of nucleation of kink-antikink pairs is characterized by the appearance of a threshold field for which the process of expansion of the critical nucleus can take place. Firstly, we consider the case where the impurity potential is described by the sinusoidal function, i.e. the deformability of the system is only described by the substrate potential. The configuration of this system is determined numerically for different values of the deformability 11

parameter. We have shown that the amplitude of this configuration increases with increasing values of the deformability parameter while its width decreases. This configuration, which describes the nucleus, may become unstable if the applied field is greater or equal to the threshold field fc . This instability leads to the creation of solitons with different polarities: a kink and an antikink. Secondly, we have determined the threshold field and shown that it is a decreasing function of the amplitude of the impurity potential and an increasing function of the deformability parameter. In fact, even when v tends to infinity, the threshold field has a finite value. Moreover, we have shown the physical meaning of the applied field on the nucleation process of kink-antikink pairs in the inhomogeneous system by evaluating the potential energy density of the system. it follows that the threshold field favored the depinning of the nucleus and then the nucleation process. Finally, we have studied the influence of the deformability of the impurity potential on the depinning field and show that it is a decreasing function of the amplitude of the impurity potential in the whole range of variation of the deformability parameter. This study may be applied to a number of various systems of condensed matter physics for which the substrate potential is used to describe its physical phenomena ,for example, the kinetic of dislocation in crystal, the electrical current carried out by the charged density waves (CDW), the crystal growth, to name only a few. Although the above results may be important for a wide class of physical systems, they are nevertheless limited to the strong damping case. This restriction is now under consideration.

Acknowledgements The authors are grateful to the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, where a part of this work was done during their visit under the Associate Federation Scheme and to the Swedish International Development Agency for financial support. We thank also Prof. S.Y. Mensah for reading through the paper.

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[6] ”Scattering and Localization of classical waves in Random Media” , Edited by Ping Sheng (World Scientific, Sigapore 1990). [7] J.F. Currie, S.E. Trullinger, A. R. Bishop, and J. A. Krumhansl, Phys. Rev. B 15, 5567 (1977). [8] D. W. Mclaughlin and A. C. Scott, Phys. rev. A 18, 1652 (1978). [9] B. A. Malomed, Physica D 15, 385 (1985). [10] Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989). [11] O. M. Braun and Yu. S. Kivshar, Phys. Rev. B 43, 1060 (1991). [12] Yu. S. Kivshar, Zhang Fei, and L. V´azquez, Phys. Rev. Lett. 67, 1177 (1991). [13] Zhang Fei, Yu. S. Kivshar, B. A. Malomed, and L. V´azquez, Phys. Lett. A 159, 318 (1991). [14] Zhang Fei, Yu. S. Kivshar, and L. V´azquez, Phys. Rev. A 45, 6019 (1992). [15] R. F. Tiwang, P. Woafo and T. C. Kofan´e, J. Phys: Condens. Matter 6, 9745 (1994). [16] Zhang Fei, Yu. S. Kivshar, and L. V´azquez, Phys. Rev. A 46, 5214 (1992). [17] A. A. Golubov, I. L. Serpuchenko, and A. V. Ustinov, Zh. Eksp. Teor. Fiz. 94, 297 (1988) [Sov. Phys. JETP 66, 545 (1987)]. [18] P. A. Lee and T. M. Rice, Phys. Rev. B 19, 3960 (1979). [19] T. M. Rice, S. Whitehouse and P. Littlewood, Phys. Rev. B 24, 2751 (1981). [20] M. Yumoto, H. Fukuyama and H. Matsukawa, J. Phys. Soc. Jpn 68, 170 (1999). [21] P. Hanggi, P. Talker and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990). [22] R. Becker and W. Doring, Ann. Phys. 24, 719 (1935). [23] F. F. Abraham in ” Homogeneous Nucleation Theory”, (Academic Press NY, 1974). [24] J. S. Langer, Ann. Phys. (NY) 54, 258 (1969). [25] V. M. Vinokur and M. B. Mineev, Sov. Phys. JETP 61, 1073 (1985). [26] R. L. Woulach´e, D. Yem´el´e and T. C. Kofan´e, Phys. Rev. E 2005(In press). [27] O.M. Braun and Yu. S. Kivshar, Phys. Rep. 306, 1 (1998). [28] O.M. Braun and Yu. S. Kivshar, Phys. Rev. B 50, 13388 (1994). [29] M. Remoissenet and M. Peyrard, J. Phys. C 14, L481 (1981). [30] I. T¨ utt¨o and A. Zawadowski, Phys. Rev. B 32, 2449 (1985). [31] O. M. Braun, Y. S. Kivshar and I. I. Zelenskaya, Phys. Rev. B 41, 7118 (1990). [32] D. Yem´el´e and T. C. Kofan´e, Phys. Rev. E 56, 1037 (1997). [33] T. Kato and M. Imada, J. Phys. Soc. Jpn 66, 1445 (1997).

13

2 r=0.0 1.8 1.6 1.4

1

V

RP

(qφ)

1.2

0.8 0.6 0.4 0.2 0 −2

−1.5

−1

−0.5

0 qφ/2π

0.5

1

1.5

2

0.5

1

1.5

2

0.5

1

1.5

2

2 r=−0.3 1.8 1.6 1.4

1

V

RP

(qφ)

1.2

0.8 0.6 0.4 0.2 0 −2

−1.5

−1

−0.5

0 qφ/2π

2 r=0.5 1.8 1.6 1.4

1

V

RP

(qφ)

1.2

0.8 0.6 0.4 0.2 0 −2

−1.5

−1

−0.5

0 qφ/2π

Figure 1: Shape of the deformable Remoissenet-Peyrard substrate potential, for some values of the deformability parameter r, as a function of qφ.

14

1 0.9 kink

antikink 0.8 0.7

qφ/2π

0.6 0.5 0.4 0.3 0.2 0.1 0 −6

−4

−2

0 X−X0

2

4

6

Figure 2: Shape of the kink and antikink solutions of Eq.(6) with f = 0, for an arbitrary value of the deformability parameter of the substrate potential, for example, r = −0.3.

30 r=−0.3

25 20 15

5

t

V (qφ)

10

0 −5 −10 −15 −20 −25 −2

−1.5

−1

−0.5

0 qφ/2π

0.5

1

1.5

2

Figure 3: Schematic representation of the total potential energy resulting from the substrate 1 potential and the external constant applied field f , V t (qφ) = 2 Vsub (qφ) − f φ, for two different q situations, with q = 4. The curve with solid line corresponds to the case f = 0.1 < f m , while the dotted line stands for the case f = f m (note that, for r = −0.3, fm = 3.5). The Peierls valleys φpv and the Peierls hills φph are present for f < fm and disappear when f ≥ fm .

15

1 (a)

0.9 0.8 0.7

qφ/2π

0.6 0.5 0.4 0.3 0.2 0.1 0

−10

−5

0 X−X

5

10

0

1 (b) 0.9 0.8 0.7

qφ/2π

0.6 0.5 0.4 0.3 0.2 0.1 0 −10

−8

−6

−4

−2

0 X−X0

2

4

6

8

10

1 (c)

0.9 0.8 0.7

qφ/2π

0.6 0.5 0.4 0.3 0.2 0.1 0 −20

−15

−10

−5

0 X−X0

5

10

15

20

Figure 4: The shape of the critical nucleus in the homogeneous system, for three values of the deformability parameter of the substrate potential r: (a) r = 0.0 (sG case), (b) r = −0.3 and (c) r = 0.5. The intensity of the applied field is f = 0.1.

16

(a) r=0.0 (b) r=−0.3 (c) r=0.5

1 (b)

(a)

0.6

N

∆E /A√µk

0.8

0.4

(c)

0.2

0

0

0.2

0.4

0.6

0.8

1 F/Fm

1.2

1.4

1.6

1.8

2

Figure 5: Energy of the critical nucleus in unit of A(Kµ) (1/2) in the homogeneous system, versus the external constant applied field f. Three values of the deformability parameter of the substrate potential are considered: (a) r = 0.0 (sG case), (b) r = −0.3 and (c) r = 0.5.

17

0.4 (a) 0.35 0.3



0.25 0.2 0.15 0.1 0.05 0

−10

−5

0 X

5

10

0.25 (b)

0.2



0.15

0.1

0.05

0 −4

−3

−2

−1

0 X

1

2

3

4

0.55 (c)



0.5

0.45

0.4

−10

−5

0 X

5

10

Figure 6: Critical nucleus in the inhomogeneous system for three values of the deformability parameter of the substrate potential r: (a) r = 0.0 (sG case), (b) r = −0.3 and (c) r = 0.5. The following values of the impurity potential intensity and the applied field are considered: v = 1, f = 0.1 and for q = 4.

18

11 10 9 8 7

fC∞

6 5 4 3 2 1 0 −1

−0.8

−0.6

−0.4

−0.2

0 r

0.2

0.4

0.6

0.8

1

Figure 7: Lower limit of the critical depinning field, for higher value of the intensity of the impurity (v → ∞), versus the deformability parameter r, with χ = −π/q, where q = 4.

19

−1.7 f/fc

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