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Driven intrinsic localized modes and their stability in anharmonic lattices with realistic potentials
ELSEVIER
Physica B 219&220 (1996) 387-389
Driven intrinsic localized modes and their stability in anharmonic lattices with realistic potentials T. R6ssler, J.B. Page * Department o f Physics and Astronomy, Arizona State University, Tempe, A Z 85287-1504, USA
Abstract Highly localized response to a spatially extended harmonic driving force is found in one-dimensional periodic lattices with realistic anharmonic interaction potentials. Using specific examples of these driven intrinsic localized modes, we describe some new aspects introduced by the driving force and briefly discuss the modes' stability properties.
The presence of interparticle anharmonicity in periodic one-dimensional lattices can lead to the existence of intrinsic localized modes (ILMs) [1-4]. These novel modes with frequencies outside the harmonic phonon bands have highly localized displacement patterns and can occur at any lattice site. Their movement [5-7], interaction with impurities [8, 9], stability [7, 10, 11], and interrelations with extended lattice modes [10-12] are just a few properties which have been investigated. Although most of the earlier work is restricted to the dynamics of u n d r i v e n nonlinear lattices, the experimental verification of ILMs requires an external probe. We consider here the case of an applied harmonic electric field, e.g. infrared radiation, and investigate how the presence of this spatially extended driving force affects ILM existence and properties. Recent related papers [13, 14] are restricted to weak anharmonicity and/or weak driving forces, whereas these restrictions are absent in our work. We find an interesting variety of driven ILMs in periodic lattices with realistic anharmonic interparticle potentials. These are obtained within a rotating wave approximation (RWA) outlined below and are verified by molecular dynamics simulations. Here we sketch only a few results; numerous additional aspects are detailed in Refs. [15, 16]. We focus on one-dimensional monatomic or diatomic lattices of N particles with masses mn at positions r,,(t) (n -1. . . . . N ), subject to periodic boundary conditions. The par* Corresponding author. 0921-4526/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSD1 0921-4526(95)00753-9
ticles interact via full realistic two-body central potentials V ( r ) , as well as their Taylor series expansions about the equilibrium configuration. Alternating charges qn = ( - 1)"q are assumed, and the lattice is externally driven by an applied harmonic electric field ~ cos(o)t). The equation of motion for particle n is m,?',~ = - ~ - ~ [ V ' ( r , - r, l
t) - V ' ( r , + l - r,~)]
I
+ ( - 1 )"qg cos(~t),
( 1)
where V'(r,, - r , - f ) = [~ V(r)/~r]r=r,_,. ~ and the summation is over the neighbors included in the interaction. For the steady state, we make an RWA by assuming that the particles oscillate harmonically at the driving frequency rn(t) = c, cos(~ot) + b, + na.
(2)
The b,'s describe the static distortions accompanying ILMs for asymmetric interparticle potentials, and a is the nearestneighbor distance [17]. Proceeding as in the undriven case [11], we insert Eq. (2) into Eq. (1), multiply the result by cos(o~t) or unity, and average over a period of the driving force. This leads to two sets of coupled nonlinear equations for the coefficients {c,, bn}, which are solved numerically for all particles in the lattice (N = 96 in this paper). Since a// particles are externally driven, it is important to allow for arbitrary nonzero displacements at every site. Recall that in the harmonic limit the only response to extended optical driving has the displacement pattern of the
388
72 ROssler. J.B. Page/ Physica B 219&220 (1996) 387 389
optically active mode in the undriven system. For a diatomic lattice this is the k = 0 optic mode, while for a monatomic lattice of alternating charges it is the zone boundary mode (ZBM). As expected, the corresponding driven anharmonic lattices can also respond with these displacement patterns, with the particles moving either in-phase or n out-of-phase with the external force. It is perhaps more surprising that localized response is also found in these anharmonic lattices, for full realistic potentials or their (k2, k3, k4) series expansions, at frequencies consistent with these systems' undriven ILMs. To illustrate, we consider a diatomic lattice with full Born-Mayer plus Coulomb (BMC) nearest-neighbor interactions. Undriven ILMs in the harmonic gap were reported previously for similar systems [9, 18]. Here we take Na! masses and BMC potential parameters obtained from shell-model harmonic force constants. The top panel of Fig. 1 shows the dynamic and static displacements near the center of a driven oddparity gap ILM. Despite the fact that all particles are driven, the response appears virtually identical with this system's
0.12 -
1.
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s~
~
I
0"010 A/a 0.2
0.00-
E
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65
particle Fig. 1. Driven odd-parity gap ILM in a 96-particle diatomic nearest-neighbor BMC lattice with NaI parameters, for a driving frequency of ~o = 0.85OJmax, where (.Omax is the maximum harmonic frequency of the undriven lattice. The magnitude of the applied force is qg = 0.001 eV/]~, and the mode is centered on a light mass. Top panel: dynamic displacements ({cn}, filled circles) and static displacements ({b~}, filled squares). The insert gives the driving frequency versus central particle amplitude. Middle panel: applied force pattern. Lower panel: magnification of the dynamic displacements (filled circles), revealing the extended lattice mode response. For comparison, the open circles show the corresponding undriven dynamic displacements.
undriven gap ILM pattern at the same frequency. However, closer inspection reveals new features introduced by the driving force. Magnification of the dynamic displacements (bottom panel) shows clearly that the particles far from the mode center respond with the displacement pattern of the k = 0 optic mode, but at a very small amplitude. This "background k = 0 optic mode" does not occur in the undriven case (open circles). Furthermore, reference to the applied force pattern given in the middle panel reveals that the central particle of the driven ILM is moving n out-oJ: phase with the force, whereas the bottom panel shows that the particles in the background k = 0 optic mode move inphase with the force. At the same driving frequency, we also find a second odd-parity gap ILM, in which both the central particle and the background extended mode move in-phase with the force. Apart from this phase difference, this driven mode's dynamic and static displacements are almost identical in magnitude with those of the driven gap 1LM shown in Fig. 1. Similar results are obtained for the driven analogs of ILMs known to exist above the harmonic phonon band in undriven monatomic lattices with (k2, k3, k4) interactions [1-4]. For all of the systems studied, we thus find a wide variety of driven ILM types, corresponding to the relative phases between the driving force, the central particle, and the particles in the extended small-amplitude background. In monatomic (k2, k3, k4) lattices, the driven ILMs having different phases between the central and extended portions exhibit these phase changes abruptly, over just one or two sites. For diatomic lattices with full realistic potentials, these 'phase-domain walls' can be spread over several sites. In general, we find that the extended background and the presence of sharp phase-domain walls, when they occur, play a strong role in determining the driven ILMs' dynamical behavior as the magnitude and frequency of the driving force are varied. Detailed results are discussed elsewhere [ 15, 16]. We have verified our RWA predictions by molecular dynamics (MD) simulations, using the fifth-order Gear predictor-corrector method to integrate the exact equations of motion numerically [19]. Furthermore, we have applied a linear stability analysis following the approach introduced in Refs. [7, 11], in order to investigate the dynamical stability properties of the driven modes. The MD simulations reveal that the driven diatomic BMC odd-parity gap ILM shown in Fig. 1 and the well-localized driven even ILMs in monatomic (k2, k3,k4) lattices (not shown here) remain stationary for more than 200 oscillations, in qualitative agreement with the small instability growth rates predicted for these modes by our stability analysis. In contrast, we find that well-localized driven monatomic (k2, k3, k4) odd ILMs are stationary for at least 10 oscillations, after which they begin to move slowly from site to site, similar to the behavior of undriven odd-parity ILMs detailed in Ref. [7]. Our predicted instability growth rates for these modes are
72 Rdssler, J.R Page/ Physica B 219&220 (1996) 387 389 in quantitative agreement with rates measured in the M D simulations. This research was supported by NSF Grants D M R 9014729 and DMR-9510182.
References [I] A.S. Dolgov, Fiz. Tverd. Tela (Leningrad) 28 (1986) 1641 [Sov. Phys. Solid State 28 (1986) 907]. [2] A.J. Sievers and S. Takeno, Phys. Rev. Lett. 61 (1988) 970. [3] J.B. Page, Phys. Rev. B 41 (1990) 7835. [4] See, for instance, Section 3 of the review by A.J. Sievers and J.B. Page, in: Dynamical Properties of Solids, Vol. 7, eds. G.K. Horton and A.A. Maradudin (North Holland, Amsterdam, 1995), pp. 137-255. [5] S. Takeno and K. Hori, J. Phys. Soc. Japan 59 (1990) 3037. [6] S.R. Bickham, A.J. Sievers and S. Takeno, Phys. Rev. B 45 (1992) 10344. [7] K.W. Sandusky, J.B. Page and K.E. Schmidt, Phys. Rev. B 46 (1992) 6161, [8] Y.S. Kivshar, Phys. Lett. A 161 (1991) 80. [9] S.A. Kiselev, S.R. Bickham and A.J. Sievers, Phys. Rev. B 50 (1994) 9135.
389
[10] O.A. Chubykato, A.S. Kovalev and O.V. Usatenko, Phys. Lett. A 178 (1993) 129; A.S. Kovalev, O.V. Usatenko and O.A. Chubykalo, Phys. Solid State 35 (1993) 356. [11] K.W. Sandusky and J.B. Page, Phys. Rev. B 50 (i994) 866. [12] V.M. Burlakov, S.A. Kiselev and V.I. Rupasov, JETP Lett. 51 (1990) 544; Phys. Lett. A 147 (1990) 130; V.M. Burlakov and S.A. Kiselev, Sov. Phys. JETP 72 (1991) 854. [13] A.F. Vakakis, M.E. King and A.J. Pearlstein, Int. J. NonLinear Mech. 29 (1994) 429. [14] B.A. Malomed, Phys. Rev. B 49 (1994) 5962. [15] T. R6ssler and J.B. Page, Phys. Lett. A 204 (1995) 418. [16] T. R6ssler and J.B. Page, to be published. [17] The straightforward addition of a term s,~ sin(wt) in Eq. (2) would allow the treatment of cycle-average absorption when damping is included in Eq. (1). Interestingly, even without damping, we have found RWA solutions having nonzero s,,, in studies of the response of an integrable small anharmonic chain [T. R6ssler and J.B. Page, Phys. Rev. B 51 (1995) 11 382]. Here, we restrict our attention to zero damping and undriven ILMs having s,, - 0. [18] S.A. Kiselev, S.R. Bickham and A.J. Sievers, Phys. Rev. B 48 (1993) 13508. [19] See, for instance, M.P. Allen and D.J. Tildesley, Computer Simulations of Liquids (Clarendon, Oxford, 1987).
Physica B 219&220 (1996) 387-389
Driven intrinsic localized modes and their stability in anharmonic lattices with realistic potentials T. R6ssler, J.B. Page * Department o f Physics and Astronomy, Arizona State University, Tempe, A Z 85287-1504, USA
Abstract Highly localized response to a spatially extended harmonic driving force is found in one-dimensional periodic lattices with realistic anharmonic interaction potentials. Using specific examples of these driven intrinsic localized modes, we describe some new aspects introduced by the driving force and briefly discuss the modes' stability properties.
The presence of interparticle anharmonicity in periodic one-dimensional lattices can lead to the existence of intrinsic localized modes (ILMs) [1-4]. These novel modes with frequencies outside the harmonic phonon bands have highly localized displacement patterns and can occur at any lattice site. Their movement [5-7], interaction with impurities [8, 9], stability [7, 10, 11], and interrelations with extended lattice modes [10-12] are just a few properties which have been investigated. Although most of the earlier work is restricted to the dynamics of u n d r i v e n nonlinear lattices, the experimental verification of ILMs requires an external probe. We consider here the case of an applied harmonic electric field, e.g. infrared radiation, and investigate how the presence of this spatially extended driving force affects ILM existence and properties. Recent related papers [13, 14] are restricted to weak anharmonicity and/or weak driving forces, whereas these restrictions are absent in our work. We find an interesting variety of driven ILMs in periodic lattices with realistic anharmonic interparticle potentials. These are obtained within a rotating wave approximation (RWA) outlined below and are verified by molecular dynamics simulations. Here we sketch only a few results; numerous additional aspects are detailed in Refs. [15, 16]. We focus on one-dimensional monatomic or diatomic lattices of N particles with masses mn at positions r,,(t) (n -1. . . . . N ), subject to periodic boundary conditions. The par* Corresponding author. 0921-4526/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSD1 0921-4526(95)00753-9
ticles interact via full realistic two-body central potentials V ( r ) , as well as their Taylor series expansions about the equilibrium configuration. Alternating charges qn = ( - 1)"q are assumed, and the lattice is externally driven by an applied harmonic electric field ~ cos(o)t). The equation of motion for particle n is m,?',~ = - ~ - ~ [ V ' ( r , - r, l
t) - V ' ( r , + l - r,~)]
I
+ ( - 1 )"qg cos(~t),
( 1)
where V'(r,, - r , - f ) = [~ V(r)/~r]r=r,_,. ~ and the summation is over the neighbors included in the interaction. For the steady state, we make an RWA by assuming that the particles oscillate harmonically at the driving frequency rn(t) = c, cos(~ot) + b, + na.
(2)
The b,'s describe the static distortions accompanying ILMs for asymmetric interparticle potentials, and a is the nearestneighbor distance [17]. Proceeding as in the undriven case [11], we insert Eq. (2) into Eq. (1), multiply the result by cos(o~t) or unity, and average over a period of the driving force. This leads to two sets of coupled nonlinear equations for the coefficients {c,, bn}, which are solved numerically for all particles in the lattice (N = 96 in this paper). Since a// particles are externally driven, it is important to allow for arbitrary nonzero displacements at every site. Recall that in the harmonic limit the only response to extended optical driving has the displacement pattern of the
388
72 ROssler. J.B. Page/ Physica B 219&220 (1996) 387 389
optically active mode in the undriven system. For a diatomic lattice this is the k = 0 optic mode, while for a monatomic lattice of alternating charges it is the zone boundary mode (ZBM). As expected, the corresponding driven anharmonic lattices can also respond with these displacement patterns, with the particles moving either in-phase or n out-of-phase with the external force. It is perhaps more surprising that localized response is also found in these anharmonic lattices, for full realistic potentials or their (k2, k3, k4) series expansions, at frequencies consistent with these systems' undriven ILMs. To illustrate, we consider a diatomic lattice with full Born-Mayer plus Coulomb (BMC) nearest-neighbor interactions. Undriven ILMs in the harmonic gap were reported previously for similar systems [9, 18]. Here we take Na! masses and BMC potential parameters obtained from shell-model harmonic force constants. The top panel of Fig. 1 shows the dynamic and static displacements near the center of a driven oddparity gap ILM. Despite the fact that all particles are driven, the response appears virtually identical with this system's
0.12 -
1.
¸ . ~
s~
~
I
0"010 A/a 0.2
0.00-
E
tttttttttttttttttttt
5.10 "4•" o
x170
0.00 -5"10 4 35
50
65
particle Fig. 1. Driven odd-parity gap ILM in a 96-particle diatomic nearest-neighbor BMC lattice with NaI parameters, for a driving frequency of ~o = 0.85OJmax, where (.Omax is the maximum harmonic frequency of the undriven lattice. The magnitude of the applied force is qg = 0.001 eV/]~, and the mode is centered on a light mass. Top panel: dynamic displacements ({cn}, filled circles) and static displacements ({b~}, filled squares). The insert gives the driving frequency versus central particle amplitude. Middle panel: applied force pattern. Lower panel: magnification of the dynamic displacements (filled circles), revealing the extended lattice mode response. For comparison, the open circles show the corresponding undriven dynamic displacements.
undriven gap ILM pattern at the same frequency. However, closer inspection reveals new features introduced by the driving force. Magnification of the dynamic displacements (bottom panel) shows clearly that the particles far from the mode center respond with the displacement pattern of the k = 0 optic mode, but at a very small amplitude. This "background k = 0 optic mode" does not occur in the undriven case (open circles). Furthermore, reference to the applied force pattern given in the middle panel reveals that the central particle of the driven ILM is moving n out-oJ: phase with the force, whereas the bottom panel shows that the particles in the background k = 0 optic mode move inphase with the force. At the same driving frequency, we also find a second odd-parity gap ILM, in which both the central particle and the background extended mode move in-phase with the force. Apart from this phase difference, this driven mode's dynamic and static displacements are almost identical in magnitude with those of the driven gap 1LM shown in Fig. 1. Similar results are obtained for the driven analogs of ILMs known to exist above the harmonic phonon band in undriven monatomic lattices with (k2, k3, k4) interactions [1-4]. For all of the systems studied, we thus find a wide variety of driven ILM types, corresponding to the relative phases between the driving force, the central particle, and the particles in the extended small-amplitude background. In monatomic (k2, k3, k4) lattices, the driven ILMs having different phases between the central and extended portions exhibit these phase changes abruptly, over just one or two sites. For diatomic lattices with full realistic potentials, these 'phase-domain walls' can be spread over several sites. In general, we find that the extended background and the presence of sharp phase-domain walls, when they occur, play a strong role in determining the driven ILMs' dynamical behavior as the magnitude and frequency of the driving force are varied. Detailed results are discussed elsewhere [ 15, 16]. We have verified our RWA predictions by molecular dynamics (MD) simulations, using the fifth-order Gear predictor-corrector method to integrate the exact equations of motion numerically [19]. Furthermore, we have applied a linear stability analysis following the approach introduced in Refs. [7, 11], in order to investigate the dynamical stability properties of the driven modes. The MD simulations reveal that the driven diatomic BMC odd-parity gap ILM shown in Fig. 1 and the well-localized driven even ILMs in monatomic (k2, k3,k4) lattices (not shown here) remain stationary for more than 200 oscillations, in qualitative agreement with the small instability growth rates predicted for these modes by our stability analysis. In contrast, we find that well-localized driven monatomic (k2, k3, k4) odd ILMs are stationary for at least 10 oscillations, after which they begin to move slowly from site to site, similar to the behavior of undriven odd-parity ILMs detailed in Ref. [7]. Our predicted instability growth rates for these modes are
72 Rdssler, J.R Page/ Physica B 219&220 (1996) 387 389 in quantitative agreement with rates measured in the M D simulations. This research was supported by NSF Grants D M R 9014729 and DMR-9510182.
References [I] A.S. Dolgov, Fiz. Tverd. Tela (Leningrad) 28 (1986) 1641 [Sov. Phys. Solid State 28 (1986) 907]. [2] A.J. Sievers and S. Takeno, Phys. Rev. Lett. 61 (1988) 970. [3] J.B. Page, Phys. Rev. B 41 (1990) 7835. [4] See, for instance, Section 3 of the review by A.J. Sievers and J.B. Page, in: Dynamical Properties of Solids, Vol. 7, eds. G.K. Horton and A.A. Maradudin (North Holland, Amsterdam, 1995), pp. 137-255. [5] S. Takeno and K. Hori, J. Phys. Soc. Japan 59 (1990) 3037. [6] S.R. Bickham, A.J. Sievers and S. Takeno, Phys. Rev. B 45 (1992) 10344. [7] K.W. Sandusky, J.B. Page and K.E. Schmidt, Phys. Rev. B 46 (1992) 6161, [8] Y.S. Kivshar, Phys. Lett. A 161 (1991) 80. [9] S.A. Kiselev, S.R. Bickham and A.J. Sievers, Phys. Rev. B 50 (1994) 9135.
389
[10] O.A. Chubykato, A.S. Kovalev and O.V. Usatenko, Phys. Lett. A 178 (1993) 129; A.S. Kovalev, O.V. Usatenko and O.A. Chubykalo, Phys. Solid State 35 (1993) 356. [11] K.W. Sandusky and J.B. Page, Phys. Rev. B 50 (i994) 866. [12] V.M. Burlakov, S.A. Kiselev and V.I. Rupasov, JETP Lett. 51 (1990) 544; Phys. Lett. A 147 (1990) 130; V.M. Burlakov and S.A. Kiselev, Sov. Phys. JETP 72 (1991) 854. [13] A.F. Vakakis, M.E. King and A.J. Pearlstein, Int. J. NonLinear Mech. 29 (1994) 429. [14] B.A. Malomed, Phys. Rev. B 49 (1994) 5962. [15] T. R6ssler and J.B. Page, Phys. Lett. A 204 (1995) 418. [16] T. R6ssler and J.B. Page, to be published. [17] The straightforward addition of a term s,~ sin(wt) in Eq. (2) would allow the treatment of cycle-average absorption when damping is included in Eq. (1). Interestingly, even without damping, we have found RWA solutions having nonzero s,,, in studies of the response of an integrable small anharmonic chain [T. R6ssler and J.B. Page, Phys. Rev. B 51 (1995) 11 382]. Here, we restrict our attention to zero damping and undriven ILMs having s,, - 0. [18] S.A. Kiselev, S.R. Bickham and A.J. Sievers, Phys. Rev. B 48 (1993) 13508. [19] See, for instance, M.P. Allen and D.J. Tildesley, Computer Simulations of Liquids (Clarendon, Oxford, 1987).