Interaction of intrinsic localized modes with impurities in the classical anisotropic Heisenberg spin model

15 February

1999

PHYSICS

EISBVIER

Physics Letters A 252

LETTERS

A

( 1999) 77-82

Interaction of intrinsic localized modes with impurities in the classical anisotropic Heisenberg spin model S.V. Rakhmanova Department Received 4 September

of Physics and Astronomy,

’

Universi@ of California, Irvine, CA 92697, USA

1998; revised manuscript received 4 December 1998; accepted Communicated by L.J. Sham

for publication

5 December

1998

Abstract

We study moving intrinsic localized modes on a spin chain with point impurities. It is shown that a stationary localized mode can be excited when a moving wave collides with the impurity. The frequency of this stationary mode depends on the strength of the defect perturbation. We also consider interaction of moving intrinsic localized waves with an already excited impurity mode. We find that when the frequencies of the two modes are close to each other they interact strongly forming a metastable resonance state. @ 1999 Published by Elsevier Science B.V. PACS: 7S.IO.Hk;

75.30.-m;

75.40.Gb

Spatially localized nonlinear excitations in perfectly periodic low-dimensional lattices, often called intrinsic localized modes, have been extensively studied during the last several years [ l-41. The interest in this type of excitations is explained by their unusual nature as well as by the important role they play in condensed matter physics, molecular biology, fiber optics and other areas of science. There are indications that localized modes contribute significantly to the properties of semiconductor superlattices, biological molecules and conjugated polymers [5]. The amazing feature of these modes is that the localization occurs in ideal lattices due to the interplay between discreteness and nonlinearity. However, in order to build an adequate nonlinear model corresponding to a real physical system, it is essential to study effects of point impurities on nonlinear localization. It is important to investigate

whether defects destroy the localization or, on the contrary, serve as a catalyst for the formation of highly localized excitations. This issue has already been addressed in a number of papers [ 6,7], where moving and stationary localized excitations in anharmonic lattices with defects have been studied. In view of the many complications in dealing with discrete nonlinear models, due to the lack of analytical expressions for localized solutions, one is forced to make approximations and use perturbative methods. Like in Ref. [ 61 a nonlinear lattice is approximated by a completely integrable Ablowitz-Ladik model [ 81 in order to generate moving discrete sohtons in the framework of this model. The authors, then, consider defects as a perturbation. In Ref. [ 71 the equation of motion of a discrete breather on the anharmonic lattice with Morse potential was solved with the aid of collective coordinate analysis. Regardless of the chosen approach, the results of these studies show a distinct similarity.

’ E-mail: [email protected]. 0375-9601/99/$ - see front matter @ 1999 Published PIISO375-9601(98)00933-5

by Elsevier Science B.V. All rights reserved.

78

S. V Rakhmanova / Physics Letters A 252 (I 999) 77-82

Namely, they establish that, depending on the impurity strength, the traveling localized mode can be either trapped by the impurity or escape it. It was also found that for a certain range of the defect strengths the scattering of a soliton by an impurity exhibits a resonant structure associated with an energy exchange between the moving wave and the stationary impurity mode. In the present paper we investigate the interaction between moving intrinsic localized spin modes (ILSM) and a point defect in a classical anisotropic spin chain with ferromagnetic exchange by directly solving equations of motion numerically. The ILSMs are qualitatively different from the discrete breathers mentioned above in that each mode is characterized by an internal frequency 0 which is the frequency of precession of spins within the localized excitation. We have already established the existence of moving ILSMs in ferromagnetic spin chains and discussed the methods of their numerical generation [ 93. Here we intend to study the scattering of ILSMs by impurities. Rather than considering a mass defect as is usually done in the case of anharmonic lattices, we introduce an impurity in our model by changing the on-site anisotropy constant of one particular spin. We do not treat this defect as a small perturbation and, therefore, we allow the existence of highly localized nonlinear impurity modes in the system. As the previous studies [9] suggest, a moving ILSM interacts with a defect in one of the following three ways. It can pass through, be trapped or be reflected with or without exciting an impurity mode. In fact, one can view the trapping of a moving ILSM on a defect as the excitation of a large-amplitude nonlinear impurity mode so that the last two scenarios are somewhat the same. Our goal in this paper is to determine the conditions which favor the excitation of an impurity mode, find how the frequency of the impurity mode is related to the impurity strength, and study the interaction of moving ILSMs with an already excited defect. We will demonstrate that resonant features in the interaction between an intrinsic localized mode and an impurity mode exist for a relatively broad range of the impurity strengths, unlike the situation described in Ref. [7], where resonances were found only for very narrow windows of parameters. We use the model that has been studied in Ref. [4] and is described by the Hamiltonian, which, normalized to the exchange, takes the form

H(B) ~=--cs,.s,+~+C(B-ABS,,.,)(~~)’ n n

-H0CS:.

(1)

n

where J is the exchange interaction constant, S is the spin magnitude, B is the anisotropy constant, ~5,,~, is the Kronecker delta, and Ho is an external magnetic field directed along the anisotropy axis. Each vector s, has a unit length and describes the orientation of a spin at site n. To obtain the equation of motion for the nth spin one needs to evaluate the commutator of the Hamiltonian ( 1) with the operator s,. The ferromagnetic chain of spins represented by the Hamiltonian (1) supports small-amplitude linear spin waves s,’ = SC+ isi = SOexp i (kn - UT), where 7 is the real time normalized to the exchange, 7 = t( JS/fi), with their frequency being related to the wave vector by (2) with 0 = w - HO.Therefore, the extended wave spectrum is bounded by -2B < 0 < 4 - 2B. If the anisotropy constant of one spin at site no is changed from B to B - AB ( AB > 0), there exists a linear localized mode centered at the impurity site. The frequency of this impurity mode is readily calculated by substituting the expression

s,’ = sge -+-WI

e-iw,n7

=

so

e-yln-n~l

,-i(fJh+H0)r

(3) for the localized wave into the equation of motion for s,‘. After linearizing the thus obtained equation with respect to SO,one finds that G,(AB) with Aa

= 0,~ + AR(AB), = 2dl+

(AB)2 - 2 > 0,

(4)

independent of the amplitude SO. Here 0~ = 4 - 2B is the upper limit of the linear spin wave spectrum. It is clearly seen from Eq. (4) that the linear impurity mode exists above the linear spin wave band and is separated from it by a gap of width An. In addition to the linear impurity mode with frequency & ( AB) , a large-amplitude stable stationary localized excitation can be realized at the impurity site. We call this excitation the nonlinear impurity mode

S.V Rakhmanova/Physics Lefrers A 252 (1999) 77-82

and denote its frequency by Qnimpur.The main difference from the linear case is that the frequency of the localized mode now depends not only on defect size, AB, but also on the amplitude of the mode. There is no exact analytical expression for Qmpur as a function of mode’s amplitude and defect size. However, the dependence of fZimpuron A B has been investigated numerically in Ref. [ 91. It has been shown there that the frequency of a nonlinear localized mode increases as the value of the defect perturbation becomes larger, provided that the amplitude of the mode is kept constant. On the other hand, if the frequency ampur of the nonlinear mode is fixed, the amplitude decreases with increasing AB until it becomes zero at some value of the defect size. That value of AB corresponds to having firi, = Gmpur. Therefore, the condition for a nonlinear impurity mode to exist on the defect of a given size AB is Qmpur > firin ( AB) . For the frequencies Qmpur much higher than J&i”, the structure of the nonlinear impurity mode resembles that of an ILSM on a perfect chain. But as G,,,rur approaches & from above, the amplitude of the nonlinear mode decreases and its profile begins to look like an exponentially decaying linear mode (3). If the frequency falls into the gap between the linear spectrum and J&n, no localization on the defect can be observed. Armed with this knowledge about linear and nonlinear impurity modes, we proceed to study numerically the propagation of an ILSM through a chain of 50 1 spins with an impurity placed at site 25 1. We assume that at time r = 0 we have a moving ILSM at one end of the chain, far away from the defect. We follow the evolution of this initial condition according to the equations of motion resulted from the Hamiltonian ( 1) . The strength of the impurity is determined by AB - the deviation of the anisotropy constant at site 25 1 from the value B = 4 for the rest of the chain. It is varied from one simulation to another, while the frequency of the incident ILSM, J&c, is kept fixed. The results reported in this paper are obtained using Qn,,, = -3.85. Starting with small defects, AB < 0.1, we find that the ILSM easily passes over the defect, leaving the impurity unexcited. When we increase AB up to about 0.1, the moving mode gets captured by the impurity. As a result of this ILSM trapping, a huge nonlinear stationary excitation settles down at the defect site. After increasing AB even further, the interaction of the ILSM with the impurity takes the form

19

of inelastic scattering, in which the energy of the incoming wave splits. Part of it is transferred into the moving localized excitation which is reflected off the defect. The other part is used to excite the impurity. If the defect is not very strong, the impurity mode, which is left after the scattering, has a large amplitude and resembles a stationary ILSM on a perfect chain 141. But, with increasing AB, the amplitude on the impurity mode decreases. In order to estimate the frequency of the impurity mode left on the defect after the ILSM scattering and to make some predictions about the character of this scattering we use the following reasoning. The frequency of the linear impurity mode flri” becomes equal to the frequency of the traveling ILSM &,, for A B x 0.4, as found from Eq. (4). It is natural to expect that if the excitation of an impurity mode occurs as a result of scattering of the ILSM by the defect, the frequency of the impurity mode fiimpur will be the same or close to the frequency of the incident wave, fiinc. Therefore, we predict that, as we increase the impurity strength, the excitation of the impurity is observed up to the value AB = 0.4. After that, Qni,, becomes less than 0rin, and, thus, moves into the gap between 0tin and the linear wave spectrum. The defect is now too strong to support a localized excitation with such a low frequency. However, in our simulations we observe that inelastic scattering with excitation of an impurity mode persists uptoAB M 0.6. It turns out that moving ILSM hitting a defect is able to excite an impurity mode with the frequency higher from its own. We measured the frequency of the stationary mode localized at the impurity by Fourier analyzing the precession of the defect spin for the whole range of AB values for which the excitation of the impurity mode is observed. The results are summarized in Table 1. For a special case of trapping, which occurs for AB = 0.1, the stationary defect mode has a frequency which is much higher than Qn,. Then, for a broad region of A B values 0.1 < AB < 0.4, the frequency Qmpur stays almost constant and is very close t0 Qni,, = -3.85. That region corresponds to the situation when a defect is not very strong, so that Qin < &,. But, when AB becomes greater than 0.4, fhnpur increases again. In this case the defect is strong enough, so that the frequency .nlin of the corresponding linear localized mode exceeds the frequency of the incident wave. Therefore, the incident ILSM cannot excite an impurity mode with a frequency equal to

80

S. K Rakhmanova/Physics

Letters A 252 (1999) 77-82

Table 1 Frequencies of the stationary localized mode excited on a defect site with various values of AB by a moving ILSM with R = -3.85

0.1”

-3.809 -3.846 -3.851 -3.854 -3.832 -3.760 -3.717

0.18 0.2 0.3 0.4 0.5 0.55 il Corresponds to trapping.

0.3

its own. The ILSM solves the problem

by leaving an higher than Qnincon the

excitation with the frequency defect. The interaction of a moving ILSM with an impurity can be modeled as the motion of a particle in an attractive potential [ lo]. To determine the structure of this potential numerically we introduce a quantity which we call the effective potential. We define it as the difference between the total energy of the chain with a defect and the total energy of the perfect chain, Av

=

H(B - AB) Js2

H(B)

-Tf--’

0.35

0.4

0.45

0.5

0.55

0.6

1o-3T Fig. 1. Effective potential AV versus time for AB = 0.1.

(5)

We find AV for each particular configuration of spins on the chain by substituting values s, into Eq. ( 1) for H(B) / JS*. Instead of traditionally regarding the potential as a function of distance between the moving ILSM and the impurity, we consider it as a function of time. Assuming that the ILSM moves uniformly through the chain, the two approaches are equivalent, but the second one simplifies computations. In Fig. I we plot AV versus time for AB = 0. I for the period of time which corresponds to the collision of an ILSM with the defect, with subsequent trapping of the ILSM. The collision occurs around time r = 350. The same event is illustrated in Fig. 2, where the energy density of a part of the chain, from site 220 to site 300, is shown in a contour plot as a function of time. As one sees from the figures, the captured ILSM executes small amplitude oscillations around the defect. This manifests itself by the appearance of two frequencies in the Fourier expansion of the motion of the defect spin. They are so close to each other that it superimposes long wave beats on a short wave

10-37

Fig. 2. ILSM-impurity collision with subsequent trapping of the ILSM. Energy density as a function of site number and time is shown in a contour plot. This collision is described by the effective potential illustrated in Fig. 1.

precession of the spin. That explains the high &,,rur in Table 1 for AB = 0.1. So far we have considered collisions of a moving ILSM with unexcited impurity. Due to the finite size of the chain, once reflected ILSM will come back to the impurity to interact with it again. If the impurity was excited during the first collision, then the second interaction is modified by the presence of the impurity mode. We looked at the subsequent interactions of the ILSM with aninc with already excited impurity. We found that for the defect strengths between 0.1 < AB < 0.4, for which the frequency of the impurity mode Gmpur is close to the frequency of the incident ILSM, according to our data form Table 1, the inter-

81

S. V RakJzmanova/Physics L.erters A 252 (1999) 77-82

0,200

0.190

‘~

>

d

0.180 0.170

. I I

O.l604j3-

----?4-

m--~

4.5

4.6

4.7

1 o-$ 0.400

f--

I

.

b

/-;

--

6.2

6.3

1 o-3T Fig. 3. Effective (b) AB = 0.4.

potential

AV versus time for (a) AB = 0.2 and ._..;

,/

240

action shows resonant features. To illustrate this resonant behavior we plot in Fig. 3a the effective potential during the time of the collision for the case of impurity strength AB = 0.2. The actual collision takes place in the time interval between r = 4400 and r = 4600. It is marked by several peaks in the effective potential symbolizing the energy exchange between the ILSM and the impurity. For comparison we plot the effective potential in Fig. 3b for a nonresonant interaction that occurs for AB = 0.4. The same two collisions are pictured in Fig. 4 where the energy density of the spin chain is shown in a contour plot as a function of site number and time. Fig. 4a shows that, after the ILSM hits the impurity, it stays there for some time, exchanging energy back and forth with the impurity mode, and finally escapes and travels in the opposite direction. One can say that during the interaction the ILSM and impurity mode form a metastable bound state which lives for the time about AT = 300. The resonant behavior of the interaction for AB = 0.2 is especially clearly seen when compared to the clean and quick reflection in Fig. 4b for AB = 0.4.

Fig. 4. Interaction of a moving ILSM with an excited impurity for (a) AB = 0.2. and (b) AE = 0.4. Energy density is illustrated in a contour plot.

In conclusion we would like to summarize the results reported in this paper. We established that when a moving ILSM strikes a defect, it can excite an impurity mode. The frequency of this impurity mode is generally depending on the size on the defect perturbation. However, in the case when the impurity is not very strong, so that the frequency of the incident ILSM is higher than the frequency of the linear localized defect mode, the defect begins to oscillate at the same frequency as the spins in the incident mode regardless of the defect strength. The excitation of a higher frequency impurity mode occurs if the defect is too strong to support a localized mode at the frequency of the incident ILSM. In this case, the frequency of the impurity mode increases with increasing the im-

82

S. V Rakhmanova/Physics

purity strength. When studying the collisions with an already excited impurity, we found that an ILSM and a defect mode interact resonantly if their frequencies are close to each other. During this interaction they form a me&stable state in which the energy is repeatedly transferred between the ILSM and the impurity mode. This resonant behavior is observed for a wide range of impurity strengths. The author would like to thank Y.S. Kivshar for useful discussions. The support from the Army Research Office under Contract CS0001028 is acknowledged. References [I] R. Lai. A.J. Sievers, Phys. Rev. B 57 (1998) 3433; 55

(1997) R11937.

Letters A 252 (1999) 77-82

[ 21 D. Bonart, T. Rossler, J.B. Page, Phys. Rev. B 55 ( 1997) 8829. [3] V. Bortolani, A, Fmnchini, RF. Wallis, Phys. Rev. B 56

(1997) 8047. [4] S. Rakhmanova, D.L. Mills, Phys. Rev. B 54 (1996) 9225; S.V. Rakhmanova, A.V. Shchegrov, Phvs. Rev. B 57 C1998) R14012. 151 K.H. Spatchek, F.G. Mertens, eds., Nonlinear Coherent Structures in Physics and Biology (Plenum, New York, 1994). 161Y.S. Kivshar, Phys. Rev. B 47 (1992) 11167; V.V. Konotop, D. Cai, M. Salerno, A.R. Bishop, N. Griinbech-Jensen, Phys. Rev. E 53 ( 1995) 6476. r71 K. For-mash, M. Peyrard, 9. Malomed, Phys. Rev. E 49 ( 1994) 3400. [81 M.J. Ablowitz, J.F. Ladik, J. Math. Phys. 17 (1976) 1011. [91 S.V. Rakhmanova, D.L. Mills, Phys. Rev. B 58 ( 1998) 11458. [lOI Y.S. Kivshar, F. Zhang, L. V&.quez, Phys. Rev. Lett. 67 (1991) 1177. and references therein.