Commun. Theor. Phys. (Beijing, China) 35 (2001) pp. 609 614 c International Academic Publishers Vol. 35, No. 5, May 15, 2001 Nonlinear Gap Modes in a 1D Alternating Bond Monatomic Lattice with Anharmonicity PAN Liu-Xian, 1,2 ZHOU Guang-Hui, 1 XIA Qing-Lin 3 and YAN Jia-Ren 1 1 Department of Physics and Institute of Nonlinear Physics, Hunan Normal University, Changsha 410081, China 2 Department of Physics, Yiyang Teacher s College, Yiyang 413049, Hunan Province, China 3 Department of Applied Physics, China Center South University, Changsha 410083, China (Received October 3, 2000; Revised November 27, 2000) Abstract We analytically study the nonlinear localized gap modes in a one-dimensional atomic chain with uniform atomic mass but two periodically alternating force constants between the nearest neighbors by means of a quasi-continuum approximation. This model simulates a row of atoms in the 111 direction of a diamond-structure type of crystals or molecular crystals with alternating double and single bonds. For this lattice system, we find that the harmonic plus quartic anharmonic terms of inter-site potential produce a new type of nonlinear localized gap modes with a slightly asymmetry distribution of atomic displacements. These localized gap modes are somewhat different from widely studied localized gap modes with a symmetry atomic displacement distribution in diatomic ion lattices. PACS numbers: 63.50.+x Key words: diamond-structure lattice, nonlinear gap modes, quasi-continuum approximation 1 Introduction Since some important properties of solids are mainly or partly determined by the crystal anharmonicity, nonlinear localized modes (or discrete breathers) in a perfect lattice have been extensively studied for this decade. [1] The pioneer work of Sievers and Takeno [2] on one-dimensional monatomic lattice with quartic inter-site anharmonicity by the lattice Green-function technique found the existence of localized modes which involve only a few lattice site vibration with the frequencies above the maximum harmonic frequency in the case of sufficient large atomic vibrational amplitude. Later, Yoshimura et al. [3] proved that this highly localized vibrational mode is the extreme case of an envelope lattice soliton by the multiple-scale method. Since then, many works have emphasized on connections to the soliton-like envelope localized modes for both monatomic and diatomic lattices, but particular attention has been paid to the diatomic lattice which may produce nonlinear localized gap modes [4 13] with symmetry pattern of atomic displacement distributions and with frequencies between the acoustic and optical branches instead of above the plane wave spectrum for monatomic lattice. [2,3] The nonlinear localized gap modes can be produced with smaller atomic vibrational amplitude and lower frequency in comparison with localized modes in monatomic lattice. Moreover, the property of gap modes without interaction with linear waves (e.g., phonons) leads to its considerable stability. [7] Therefore, the importance of gap modes is that it may be a good candidate for experimental detection [14] of this nonlinear-induced localized modes and for telecommunication application. Franchini et al. [15] numerically studied an atomic chain with uniform atomic mass but two periodically alternating force constants between the nearest neighbors. This model simulates a row of atoms in the 111 direction of a diamond-structure crystal (such as important semiconductors of silicon and germanium), or a molecular chain with periodically alternating single and double bonds. In this lattice system, they observed [15] the nonlinear localized gap modes with symmetry atomic displacement distribution, which are similar to those in diatomic ion lattice. [4 13] Because of the physical importance of this lattice system and few works on localized modes available, recently we analytically studied [16] this lattice system by means of a multiple-scale expansion. And pulse solitonlike gap modes were obtained. In the present paper we restudy this lattice model by a different analytical method of quasi-continuum approximation. We find that the nonlinear localized gap modes in this lattice system have a slight asymmetry pattern of atomic displacement distribution, which are somewhat different from the results of Refs [15] and [16]. Multiple-scale method has been extensively used to study localized modes in anharmonic lattices. It reduces the lattice equation of motion into a nonlinear Schrödinger equation, so that the produced localized modes [8,9,16] are The project supported by the Science Foundation of Hunan Education Commission (Grant No. 980506) and partially by National Natural Science Foundation of China (Grant No. 19775013)
610 PAN Liu-Xian, ZHOU Guang-Hui, XIA Qing-Lin and YAN Jia-Ren Vol. 35 always pulse-like soliton with a symmetry pattern and valid in the whole Brillouin-zone of the phonon spectrum. When the carrier wave vector of the mode is limited to the Brillouin-zone boundary, a method of decoupling ansatz plus continuum approximation can be applied to the distribution envelope functions of the modes. This so-called quasi-discrete method has also been widely used [4 6,11,14] to study nonlinear localized modes in anharmonic lattice systems. Very recently Bambi Hu et al. [17] elucidated the relation between these two analytical methods. For diatomic lattices with alternating atomic masses and uniform bond force constant or uniform mass and alternating force constants, previously used multiple-scale method [8,9,16] is suitable only for the case of wide phonon band gap and decoupling ansatz method [4 6,11,14] only for narrow band gap case. However, our results in this work indicate that the latter approach is more efficient in the view of producing localized modes with any shape of patterns for the atomic displacement distribution. The paper is arranged as follows. In Sec. 2 we formulate the lattice equation of motion for a Hamiltonian dynamical system. The qualitative analysis in the phase plane of this dynamical system is given in Sec. 3, and the asymmetry nonlinear gap modes with the vibrational frequency near the bottom and top and at the middle of the linear forbidden gap are obtained analytically in Sec. 4. Section 5 discusses the results and concludes the paper. 2 Formalism of the Model We begin with one-dimensional lattice consisting of atoms with uniform mass but two masses per primitive cell. The atoms in this lattice interact via the nearest neighbors with two alternating harmonic plus quartic anharmonic potentials. The classical equations of motion for the lattice system are given by m r n = k 2 (r n+1 r n ) k 2 (r n r n 1 ) + k 4 (r n+1 r n ) 3 k 4 (r n r n 1 ) 3, n = odd, m r n = k 2 (r n+1 r n ) k 2 (r n r n 1 ) + k 4 (r n+1 r n ) 3 k 4 (r n r n 1 ) 3, n = even, (1) where n = 1, 2,, N, m is the mass of atom and r n is the longitudinal displacement from the equilibrium position of the nth atom. The quantities k 2 and k 2 (< k 2 ) are the harmonic force constants, while k 4 and k 4 (< k 4 ) the quartic anharmonic constants. In the linear case (k 4 = k 4 ) model (1) has two branches of frequency of vibration, ω± 2 = [ k 2 + k 2 ± k2 2 + k 2 2 + 2k 2 k 2 cos(2aq) ] /m, (2) where q and a are the wavevector and the lattice spacing, respectively. The minus sign corresponds to the lowerfrequency acoustic mode and the plus sign to the upper-frequency optical mode. At Brillouin-zone boundary of q = π/2a these two modes are separated by the spectrum gap ω = ω2 2 ω1 2 = 2/m( k 2 k2 ), where ω 1 = 2 k 2 /m and ω 2 = 2k 2 /m are gap edge frequencies, and with an optical cutoff frequency ω m = ω1 2 + ω2 2 at q = 0 due to the discreteness of the system. We consider the case of narrow gap (small value of k 2 k 2 ) and seek a stationary solution for nonlinear system Eq. (1) in the form [15,16] r n = Au n cos(ωt), (3) where A is the amplitude, u n is the distribution function and ω the frequency of vibration. To make Eq. (1) linear in cos(ωt) we use the rotating-wave approximation (RWA) which was proved to be accurate enough [18] in treating (k 2, k 4 ) lattice system. Therefore, we have m ω 2 u n = k 2 (u n+1 u n ) k 2 (u n u n 1 ) + λ[ k 4 (u n+1 u n ) 3 k 4 (u n u n 1 ) 3 ], n = odd, m ω 2 u n = k 2 (u n+1 u n ) k 2 (u n u n 1 ) + λ[k 4 (u n+1 u n ) 3 k 4 (u n u n 1 ) 3 ], n = even, (4) where λ = 3A 2 /4. For the solution of wavevector near Brillouin-zone boundary, we make the following ansatz [4,5,11,19] u n = ( 1) j v 2j, n = 2j ; u n = ( 1) j w 2j+1, n = 2j + 1 ; (5)
No. 5 Nonlinear Gap Modes in a 1D Alternating Bond Monatomic Lattice with Anharmonicity 611 with which the atomic displacements for the two sublattices are separated. Subsequently, equation (4) is transformed into [m ω 2 (k 2 + k 2 )]v n = k 2 w n k 2 w n 1 + λ[ k 4 (w n v n ) 3 k 4 (v n + w n 1 ) 3 ], [m ω 2 (k 2 + k 2 )]w n = k 2 v n k 2 v n+1 + λ[ k 4 (v n w n ) 3 k 4 (w n + v n+1 ) 3 ]. (6) Now we change the discrete argument n by the continuum coordinate x, expanding functions v and w in Taylor series. For wavevector q π/2a, it is enough to leave only the first space derivative. [4,5,11,19] Therefore, equation (6) is reduced to a set of coupled ordinary equations where the parameters are defined as dv dx = w + α v λ[β(w 3 + 3v 2 w) + β (v 3 + 3w 2 v)], dw dx = v α w + λ[β(v 3 + 3w 2 v) + β (w 3 + 3v 2 w)], (7) α = (k 2 + k 2 )/2k 2, α = (k 2 k 2 )/2k 2, β = (k 4 + k 4 )/2k 2, β = (k 4 k 4 )/2k 2, = [m ω 2 (k 2 + k 2 )]/2k 2 = m ω 2 /2k 2 α, (8) and the lattice spacing is set to unity. Equation (7) describes the dynamics of a Hamiltonian system with one degree of freedom and the following integral of motion E = 2 (w2 + v 2 ) α (wv) + λβ 4 (w4 + 6w 2 v 2 + v 4 ) + λβ (w 3 v + v 3 w), (9) and its existence allows one to integrate the system exactly. following general solutions are obtained for the system, With the help of an auxiliary function z = w/v, the ( dz ) 2 = [ (1 + z 2 ) + 2α z] 2 + 4Eλ[ β(1 + 6z 2 + z 4 ) + 4β (z + z 3 )], (10) dx v 2 = (1 + z2 ) + 2α z ± [ (1 + z 2 ) + 2α z] 2 + 4Eλ[ β(1 + 6z 2 + z 4 ) + 4β (z + z 3 )] λ[ β(1 + 6z 2 + z 4 ) + 4β (z + z 3 )] w 2 = v 2 z 2., (11) In the case of α = β = 0, all formulas in this section are reduced to those for monatomic lattice. [19] This reduction may confirm the correctness of the quasi-continuum treatment for diamond-structure lattice system. 3 Qualitative Analysis In this section we qualitatively analyze dynamical system (7) in its phase plane (w, v), where the localized solutions correspond to separatrix curves. Because parameters α, α, β and β all are positive, so the solutions are characterized only by different values of the conserved quantity E as well as parameter. If we let dw/dx = dv/dx = 0 in Eq. (7) and eliminate and α respectively, we have two coupled equations (w 2 v 2 )[λβ (w 2 + v 2 ) + 2λβ(wv) α ] = 0, (w 2 v 2 )[λβ(w 2 + v 2 ) + 2λβ (wv) + ] = 0. (12) Therefore, all possible fixed points in the phase plane are obtained by solving this set of algebra equations. Obviously, v = w = 0 is the first fixed point. The common factor in the two equations (12) leads to four fixed points v = w = ± + α 4λ(β + β ), v = w = ± α 4λ(β β ). (13) By setting the other factor of Eq. (12) to zero we have a
612 PAN Liu-Xian, ZHOU Guang-Hui, XIA Qing-Lin and YAN Jia-Ren Vol. 35 relation of v = w ± ( α )/λ(β + β ) which results in another four fixed points w = ± 1 α 2 λ(β + β ) + α λ(β β ), α v = w ± λ(β + β ) = 1 α 2 λ(β + β ) ± + α λ(β β ). (14) The phase diagram of the system depends on the sign of parameter, which changes with increasing ω. As ω increases, two subsequent bifurcations take place in the phase plane. For < α, i.e., ω < ω 1, there is only one center point v = w = 0 in the phase plane. In this case separatrix curve, and subsequently, nontrivial localized solutions are absent. However, for α < < α, i.e., ω 1 < ω < ω 2, besides a saddle point v = w = 0 there are two center points described by Eq. (13). So that the first bifurcation occurs at ω = ω 1, where the center is split into two centers plus one saddle. The phase diagram of the considered system is similar to Fig. 2 in Ref. [5] and Fig. 1 in Ref. [11] for diatomic lattice with quartic anharmonicity, except for a rotation with an axial through origin. Moreover, for > α or ω > ω 2, there are nine fixed points given by Eqs (13) and (14) plus the origin in the phase plane, and the phase diagram has the same feature in comparison with that in Refs [5] and [11]. However, for system (7) there exist only two bifurcation points in the phase plane at ω 1 and ω 2, respectively. In this case the qualitative behavior of the dynamical system for diamond-structure lattice is different from that of diatomic lattice [5,11] which has three bifurcation points of ω 1, ω 2 and ω 3 (> ω 2 ), and is also different from that of monatomic lattice [19] in which only the linear cutoff frequency is the bifurcation point. Definitely, the qualitative analysis in this section supports the existence of intrinsic localized modes with frequencies above the optical branch and in the gap of phonon band. 4 Gap Modes The importance of nonlinear gap modes is that they are appeared in the linear spectrum s forbidden gap without interaction with linear waves of the system. So that gap mode may be a good candidate for experimental detection of the nonlinear-induced localized modes and for application in communication. In this section we concentrate on the nonlinear gap modes of the diamond-structure lattice system. As mentioned in the last section, when the parameters satisfy α < < α, the mode vibrational frequencies lay in the linear spectrum gap. In this case using the fact that for the spatially localized solution, i.e. such that w, v 0 for z, from Eq. (9) the invariant quantity E vanishes. Then, equation (10) is simplified as dz/dx = ±[ (1 + z 2 ) + 2α z]. (15) Its integration yields the auxiliary function in terms of x, z ± = 1 [ µ 2 3 exp µ (± µ2 2 µ ) ] / [ x 1 exp (± µ2 2 µ ) ] x, (16) where µ = α α 2 2. Principally, the amplitude envelope functions of both even- and odd-site atoms can be obtained by substituting Eq. (16) into Eq. (11). However, the resulting envelope functions may be very complicated even it is hard to analyze their qualitative behavior due to its complex dependence on z (See Eq. (11)). So that we only consider some particular cases of parameter. Near the gap bottom, the parameter α and equation (15) is integrated to z ± = (α x ± 1)/α x. (17) Substituting this simple expression into general solution (11) and after some algebraical calculations, we have both even- and odd-site atomic displacement envelope functions v 2 ± = w 2 ± = 2α 3 x 2 λ[4(β + β )(2α 4 x 4 ± 4α 3 x 3 + 3α 2 x 2 ± α x) + β], 2α (α x ± 1) 2 λ[4(β + β )(2α 4 x 4 ± 4α 3 x 3 + 3α 2 x 2 ± α x) + β], (18) where the ± signs correspond to the same signs in Eq. (17). Near the gap top, α and equation (15) is integrated to z ± = ( α x ± 1)/α x. (19)
No. 5 Nonlinear Gap Modes in a 1D Alternating Bond Monatomic Lattice with Anharmonicity 613 In the same way we have the similar displacement envelope functions for both even- and odd-sites, v 2 ± = w 2 ± = 2α 3 x 2 λ[4(β β )(2α 4 x 4 ± 4α 3 x 3 + 3α 2 x 2 ± α x) + β], 2α (α x ± 1) 2 λ[4(β β )(2α 4 x 4 ± 4α 3 x 3 + 3α 2 x 2 ± α x) + β]. (20) Comparing expression Eq. (18) with Eq. (20) we can see that the atomic displacement amplitude envelope is similar for these two cases except for a difference of coefficient β ± β in their denominators. Both gap-bottom and gaptop modes are pulse-like but with asymmetry geometry distribution. However, the pulse shape of the gap-bottom mode is sharper than that of the gap-top mode. Finally, we consider the case of = 0, i.e., ω 2 = ω 2 = (k 2 + k 2 )/m. In this case the mode vibrational frequency is in the middle of the linear spectrum gap, and the corresponding integration of Eq. (15) is z ± = e ±2α x, (21) subsequently the associated atomic displacement envelope functions are obtained from Eq. (11), v 2 ± = w 2 ± = α e 2α x λ[ β(cosh 2 2α x + 1) + 2β cosh 2α x], α e 2α x λ[ β(cosh 2 2α x + 1) + 2β cosh 2α x]. (22) These two functions are neither symmetric nor antisymmetric, but their denominators become equal and symmetric. However, the sum of two functions 2 α w + v = cosh α x λ[ β(cosh 2 2α x + 1) + 2β cosh 2α x] (23) is symmetric. Therefore, we can treat ω as the critical point of the gap, below which the pulse of mode pattern is sharper (more localized) and above which the pulse of mode pattern is wider (more delocalized). 5 Discussion and Conclusion The structural difference between diatomic lattice and diamond-structure lattice leads to the mode pattern difference between them. Diatomic lattice with two periodically alternating masses but a single force constant has a center of symmetry on atoms, its gap mode pattern for even- (heavy or light atom) or odd-sites is either symmetric or antisymmetric. Near the gap bottom the even (or odd)-sites are at rest while the amplitude distribution for the odd (or even)-sites is a symmetric pulse [11] and both distribution functions are divergence near the gap top. On the contrary, for diamond-structure lattice with uniform mass but two periodically alternating force constants there is no center of symmetry on atoms. So that this lattice system produces asymmetric mode pattern and fruitful gap-bottom and gap-top modes. Quantitative computer drawing for expressions (18), (19) and (20) shows that the mode pattern is slightly asymmetric, i.e., is almost symmetric. The numerical work [15] did not find the gap mode with this slight asymmetric property (See Fig. 3 in Ref. [15]) due to the insufficient accuracy in the simulation. According to Ref. [17], our previous multiple-scale analysis [16] is a special case of wide phonon band gap (large value of k 2 k 2 ). So that the localized mode obtained in this paper is somewhat different from that in Ref. [16]. However, the multiplescale treatment [16] reduced the lattice equation of motion into a perturbed nonlinear Schrödinger equation, but we only considered the perturbation on the soliton parameters. Thus the first-order correction on soliton itself may add an asymmetric term and the superposition may form an asymmetric gap mode pattern. In conclusion, we have analytically studied the anharmonic localized gap modes by means of a quasi-continuum approximation in a diamond-structure lattice with equal atomic mass bonded by two different force constants that alternate from one bond to the next. The harmonic plus quartic nearest neighbor interactions (k 2, k 4 ) in lattice model is considered, and a new type of asymmetry modes inside the linear spectrum gap is found. In the gap, as the vibration frequency increases from ω 1 to ω 2 the mode pattern becomes more delocalized. However, there exists a symmetric point of frequency ω, at which the mode pattern is almost symmetric. There is a quantitative difference between our analytical result and numerical one [15] for the same lattice system, but qualitatively they are in agreement. However, the structure and the asymptotic behavior of these asymmetric gap modes are completely different from those of the widely studied symmetric gap modes for diatomic (ionic) lattice with quartic anharmonicity only and of asymmetric gap modes for the diatomic lattice with cubic anharmonicity added.
614 PAN Liu-Xian, ZHOU Guang-Hui, XIA Qing-Lin and YAN Jia-Ren Vol. 35 References [1] S. Flach and C.R. Wills, Phys. Rep. 295 (1998) 181; S. Aubry, Physica D103 (1997) 201. [2] A.J. Sievers and S. Takeno, Phys. Rev. Lett. 61 (1988) 970. [3] K. Yoshimura and S. Watanabe, J. Phys. Soc. Jpn. 60 (1991) 82. [4] Y.S. Kivashar and N. Flytzanis, Phys. Rev. A46 (1992) 7972. [5] O.A. Chubykal, A.S. Kovalev and O.V. Usatenko, Phys. Rev. B47 (1993) 3153. [6] O.A. Chubykal and Y.S. Kivashr, Phys. Rev. E48 (1993) 4128. [7] S.A. Kiselev, S.R. Bickham and A.J. Sievers, Phys. Rev. B50 (1994) 9135. [8] A. Campa, A. Giansanti, A. Tenenbaum, D. Levi and O. Ragnisco, Phys. Rev. B48 (1993) 10168. [9] HUANG Guo-Xiang, et al., Phys. Rev. B51 (1995) 12347; ibid. B57 (1998) 5746. [10] A. Franchini, V. Bortolani and R.F. Wallins, Phys. Rev. B53 (1996) 5420. [11] ZHOU Guang-Hui, DUAN Yi-Wu and YAN Jia-Ren, Phys. Rev. B53 (1996) 13977. [12] A.S. Gorshkov, O.N. Ermakova and V.F. Marchenko, Nonlinearity 10 (1997) 1007. [13] N. Flytzanis, B.A. Malomed and A. Neuper, Physica D113 (1998) 191. [14] LOU Sen-Yue, YU Jun, LIN Ji and HUANG Guo-Xiang, Chin. Phys. Lett. 12 (1995) 400. [15] A. Franchini, V. Borlolani and R.F. Wallins, Phys. Rev. B58 (1998) 8391. [16] ZHOU Guang-Hui, XIA Qing-Lin and YAN Jia-Ren, Acta Physica Sinca 49 (2000) 1741 (in Chinese). [17] Bambi Hu, Guoxiang Huang and M.G. Velarde, Phys. Rev. E62 (2000) 2827. [18] S.R. Bickham and A.J. Sievers, Phys. Rev. B43 (1991) 2339. [19] O.A. Chubykal and Y.S. Kivshar, Phys. Lett. A178 (1993) 123.