Exciton-polariton soliton wavetrains in molecular crystals with dispersive long-range intermolecular interactions
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Exciton-polariton soliton wavetrains in molecular crystals withdispersive long-range intermolecular interactions
E. Nji Nde Aboringong and Alain M. Dikand´e
Laboratory of Research on Advanced Materials and Nonlinear Sciences (LaRAMaNS), Department of Physics,Faculty of Science, University of Buea P.O. Box 63 Buea, CameroonReceived: date / Revised version: date
Abstract.
The peculiar crystal structure of one-dimensional molecular solids originates from packing ofan array of molecules in which intermolecular interactions are dominantly dispersive, including hydrogen-bond, van der Waals and London-type forces. These forces are usually relatively weaker than covalent andionic bondings such that long-range intermolecular interactions should play important role in dispersionproperties of molecular crystals such as polymers and biomolecular chain structures. In this work theeffects of long but finite-range intermolecular interactions on single-exciton dispersion energy, and henceon characteristic parameters of periodic soliton trains associated with bound exciton-polariton states inone-dimensional molecular crystals interacting with an electromagnetic field, are investigated. Long-rangeinteractions are shown to quantitatively modify the exciton-polariton soliton amplitudes, width and velocityas a result of shrinkage of the single-exciton energy spectrum. The soliton structures of interest are nonlinearwavetrains, consisting of periodically ordered single-pulse (i.e. bright) or single-kink (i.e. dark) solitons withequal separation between the constituent single-soliton modes. Periodic soliton structures are relevant andbest suited for finite-size chain systems, where periodic boundary conditions rule the generation of nonlinearwave profiles. Generally they are of weaker nonlinearity compared to their single-soliton constituents aswell established within the framework of their generation via the process of modulational instability.
PACS.
Molecular crystals are discrete chains of molecules inter-acting via van der Waals, hydrogen-bond and London dis-persion forces [1,2,3,4]. These molecular solids are char-acterized by narrow bands with transport properties atthe interface between hopping and band transports me-diated by excitons [3,4,5,6]. Excitons in molecular solidsare elementary excitations formed from bound electron-hole states that can transport energy without transport-ing net electric charge, and are known to play importantrole in free carrier photoexcitations. In fact the trans-port of energy in form of excitons in molecular crystalsis one of the fundamental physical processes characteriz-ing these systems, several related interesting and peculiarproperties have indeed been observed in organic molecularchains, biopolymers and protein chains [5,6,7,8,9,10,11,12,13,14] as for instance α -helix amid chains [15].The possible formation of solitons in one-dimensional(1 D ) molecular crystals with multi-exciton scatterings [16,17,18] has been discussed in the past, taking into accountthe Pauli character of exciton operators [19]. It has beenshown that the inclusion of exciton-exciton dimer term inthe exciton Hamiltonian with intramolecular vibrations and nearest-neighbour interactions, leads to two possiblenonlinear structures namely bright and dark exciton soli-tons. As emphasized in ref. [19], the existence and stabil-ity of any of this particular type of soliton is determinedby specific conditions on values of the system parametersincluding soliton velocity, and to this last point it was es-tablished that neither kind of soliton (i.e. bright or dark)can exist if the short-range intermolecular interaction isrepulsive.The interaction of excitons with light has also beenextensively discussed in the past [20,21], considering theprocess as a mean to regulate certain functions of molec-ular crystals as well as to enhance excitonic transport inthese materials. In biological systems [22,23,24] in partic-ular frequency selective effects of light on protein activa-tion, involving energies of the same order and nature asthe electromagnetic radiation of light, have helped revealthat protein interactions are based on resonant electro-magnetic energy transfer within the range of infra-red andvisible light [23]. Formally one can describe the interactionof excitons with light by considering the model Hamilto-nian studied in ref. [19], with an extra term accounting forthe exciton-photon coupling and the Maxwell equation de-scribing the propagation of light. This results in the model Aboringong et al.: Exciton-polariton soliton wavetrains with long-range interactions studied recently [25,26], in which the authors establishedthe existence of both bright and dark solitons associatedwith large-amplitude exciton and polariton excitations inthe system.However, the bright and dark soliton solutions pro-posed in refs. [25,26] are localized structures with a vanish-ing shape at the boundaries of an infinitely long molecularchain. This is a shortcoming when considering their appli-cation in contexts where the molecular chain is finite, re-quiring periodic boundary conditions. Also, in real molec-ular crystals, dispersive interactions between molecularunits are not limited only to nearest-neighbour molecules,the account of long-range interactions is relevant for aproper estimate not only of dispersion properties of ex-citations, but also characteristic parameters of bright anddark soliton structures.In the present work we refomulate the generation andpropagation of large-amplitude excitations in the model ofref. [25,26], taking into consideration the influence of long-range dispersive interactions between molecules along themolecular crystal. We focus on periodic soliton trains, in-stead of localized solitary-wave structures, for the firstones are more appropriate for molecular crystals with fi-nite lengths. Proceeding with we consider the model Hamil-tonian studied in ref. [26] which describes a longitudi-nal chain of interacting excitons coupled to electromag-netic radiations, but now taking into account the contri-bution of long-range intermolecular interactions on exci-ton dispersions. With this improved Hamiltonain, we ob-tain large-amplitude solutions to the coupled nonlinearexciton-polariton equations in terms of periodic short soli-ton wavetrains.
Consider a 1 D discrete chain of molecules, in which Frenkelexcitons have formed and interact with a propagating elec-tromagnetic field. The total Hamiltonian for such systemcan be expressed [26]: H = ¯ hω N X n =1 b † n b n − N X n =1 X m = n J n − m (cid:0) b † n b m + b n b † m (cid:1) − D N X n =1 b † n b n b † n +1 b n +1 − d N X n =1 (cid:0) b † n e + n + b n e − n (cid:1) , (1)where ¯ hω is the energy associated with intramolecularvibrations, J n − m is the energy transfer integral betweenmolecular excitations on sites n and m , and D is theexciton-exciton coupling strength. The Pauli operator b † n ( b n ) creates (annihilates) a molecular exciton at site n ,and we shall consider only dimer terms formed by nearest-neighbour excitons. The quantities e + and e − in eq. (1)are the right-going and left-going components of the elec-tromagnetic field, with d the exciton-field coupling in thedipole approximation. When m = n ±
1, formula (1) reduces to the Hamilto-nian for a chain of excitons with only nearest-neighbourinteractions [26]. We are interested in the system dynamicswhen long-range interactions between molecular units arefully accounted for. In this purpose, we assume a long-range interaction whose coupling strength falls off witha power law as one moves away from the first-neighbourmolecules. One such long-range interaction is the so-calledKac-Baker potential [27,28,29,30]: J n − m = J − r r r | ℓ | , ℓ = m − n, (2)in which the real parameter r (with 0 < r ≤
1) determinesthe strength of intermolecular interactions with ℓ the dis-tance between a molecule at site n and a molecule at site m . For the long-range interaction potential eq. (2), thefirst-neighbour model considered in ref. [26] is recoveredwhen r →
0. When ℓ → ∞ (or r → ℓ = 1 or ℓ = ∞ [28,31]. In most real physical contexts of molecular crys-tals, however, long-range interactions are effectively sen-sitive only over a finite range of intermolecular distances,saturating beyond. Moreover, for molecular chains withfinite lengths as it is the case for most biological systems,the maximum value of ℓ should be finite such that theKac-Baker potential becomes unappropriate. As an alter-native we shall consider the modified Kac-Baker potential[30]: J Ln − m = J − r L − r r r | ℓ | , ℓ = m − n, (3)where L is the maximum range (i.e. the maximum of ℓ )beyond which intermolecular interactions saturate. In thenearest-neighbour limit (i.e. ℓ = L = 1), the quantity J Ln − m given in (3) reduces to J = J / r . In the same limit, the Kac-Baker potential (2)becomes J = J (1 − r ) /
2. So for the Kac-Baker potential,only when r = 0 one can recover the nearest-neighbourvalue of the intermolecular interaction strength.To derive the equation of motion for excitons from theHeisenberg formalism on the Hamiltonian formula (1), itis useful to start by remarking that being Pauli operatorsthe quantities b n and b † n obey the commutation relations: (cid:2) b n , b † m (cid:3) = (1 − P n ) δ n,m , P n = b † n b n . (4)With this, the Heisenberg formalism leads to the followingexciton equations of motion: i ¯ h ∂b n ∂t = ¯ hω b n − (1 − P n ) X m = n J Ln − m b m − Db n (cid:16) b † n +1 b n +1 + b † n − b n − (cid:17) − d (1 − P n ) e + n . (5)Let us seek for coherent structures associated with the dy-namics of single-particle states, in this goal we can readily boringong et al.: Exciton-polariton soliton wavetrains with long-range interactions 3 approximate h P n i ≈ h b n ih b † n i . Setting h b n i = α n where α n is the single-exciton occupation probability [32], eq.(5) becomes: i ¯ h ∂α n ∂t = ¯ hω α n − (1 − | α n | ) X m = n J Ln − m α m − Dα n (cid:0) | α n +1 | + | α n − | (cid:1) − d (1 − | α n | ) e + n . (6)As for the electromagnetic field e n , we assume the opticalresponse of the medium to the field propagation is linear.In this case the field propagation can be described by thelinear Maxwell equation with a ”source term” i.e.: (cid:18) ∂ ∂x − c ∂ ∂t (cid:19) e + ( x, t ) = 4 πdc a ∂ ∂t α ( x, t ) , (7)where c is light speed in the molecular crystal and a isthe lattice spacing. Eq. (7) suggests that the field propa-gates in a continuous medium, therefore we must carry outa continuum-limit approximation on the exciton variable α n . To this end we define the continuous position x = na ,such that α n ± ≈ α ( x, t ) ± a ∂∂x α ( x, t ) + a ∂ ∂x α ( x, t ) + ... .Treating similarly the term containing the long-range in-teraction potential J Lm − n [15], and seeking for solutions ofthe forms: α ( x, t ) = e i ( qx − ωt ) φ ( x, t ) , e + ( x, t ) = e i ( qx − ωt ) ε ( x, t ) , (8)where φ ( x, t ) and ε ( x, t ) are now purely reals, and ω and q and the exciton frequency and wavenumber respectively,eqs. (6) and (7) reduce to: i ¯ h ∂φ ( x, t ) ∂t = ( ǫ L ( q ) − ω ) φ ( x, t ) − i ¯ hv L ( q ) ∂φ ( x, t ) ∂x − D L ( q ) ∂ φ ( x, t ) ∂x − λ L ( q ) φ − d [1 − φ ( x, t ) ] ε ( x, t ) , (9)[ c ( ∂ ∂x − c ∂ ∂t ) + 2 i ( c q ∂∂x + ω ∂∂t )+ ( ω − c q )] ε ( x, t )= 4 πda [ ∂ ∂t − ω − iω ∂∂t ] φ ( x, t ) , (10)with: ǫ L ( q ) = ¯ hω − L X ℓ =1 J ℓ cos ( ℓqa ) ,λ L ( q ) = 4 D − hω − ǫ L ( q )) , (11) v L ( q ) = 1¯ h dǫ L ( q ) dq ,D L ( q ) = ¯ h dv L ( q ) dq = d ǫ L ( q ) dq . (12) The sum over ℓ in the expression of ǫ L ( q ) in eq. (11) isexact [30], yielding: ǫ L ( q ) = ¯ hω − J − r − r L cos( qa ) − r − r L K L ( q )1 − r cos( qa ) + r , (13) K L ( q ) = cos [( L + 1) qa ] − r cos( qLa ) . (14)Introducing the new coordinate z = x − vt and applyingthe slowly-varying envelope approximation [25,26], eqs.(9)-(10) now read: iP L ( q ) ∂φ∂z + M L ( q ) ∂ φ∂z − χ L ( q ) φ + A L ( q ) φ = 0 , (15) ∂ ε∂z = 1 d ( ǫ L ( q ) − ¯ hω ) ∂ φ∂z , (16)where: P L ( q ) = ¯ h ( v L ( q ) − v )+ 1 c q − ω [2( c q − ωv )( ǫ L ( q ) − ¯ hω ) − hωvΩ ] ,Ω = 4 πd ¯ ha , χ L ( q ) = ǫ L ( q ) − ¯ hω − ¯ hΩ ω c q − ω ,A L ( q ) = λ L ( q ) − ¯ hΩ ω c q − ω ,M L ( q ) = D L ( q ) + 1 c q − ω [2¯ h ( c q − ωv )( v L ( q ) − v )+ ( c − v )( ǫ L ( q ) − ¯ hω ) − ¯ hΩ v ] . (17)In general, real solutions to the nonlinear equation (15) ispossible provided P L ( q ) = 0. This condition implies thatthe solution is a nonlinear wave travelling at velocity: v = 2 c q ( ǫ L ( q ) − ¯ hω ) + ( c q − ω )¯ hv L ( q )2 ω ( ǫ L ( q ) − ¯ hω + ¯ hΩ ) + ¯ h ( c q − ω ) . (18)It turns out that because the energy ǫ L ( q ) and velocity v L ( q ) of single-exciton states depend on the intermolecularinteraction strength r and range L , the velocity v of non-linear excitations governed by eq. (15) will be sensitive tolong-range intermolecular interactions. To privide a sighton the effects of long-range intermolecular interactions onthese two relevant parameters, in figs. 1 and 2 we plotted ǫ L ( q ) and v L ( q ) respectively as a function of q , for dif-ferent values of the intermolecular interaction strength r and range L . For all the curves we used the following (arbi-trary) values for the model parameters: J = 1, ω = 0 . ω = 1 . Ω = 0 . a = 1.Fig. 1 shows that as we increase the interaction range L , the upper edge of the Brillouin zone is shifted from itsshort-range value at q = π . This shift causes a shrink-age of the Brillouin zone as L increases, and a decrease ofthe long-wavelength cut-off (i.e. low-lying single-exciton)energy ǫ L ( q = 0) as r is increased. It is also remark-able that the slope of the single-exciton dispersion energy ǫ L ( q ), near q = 0, is increased as the strength of the long-range interaction r increases. This actually indicates an Aboringong et al.: Exciton-polariton soliton wavetrains with long-range interactions
Fig. 1. (Color online) Single-exciton energy versus wavenumber q (in units of the lattice spacing a ), for different values of r and L . Fig. 2. (Color online) Single-exciton velocity versus wavenumber q (in units of the lattice spacing a ), for different values of r and L .boringong et al.: Exciton-polariton soliton wavetrains with long-range interactions 5 increase of the characteristic speed of low-lying excitons,as the range and strength of intermolecular interactionsare simultaneously increased. This last behaviour is moreevident in fig. 2, representing the velocity dispersion ofsingle-exciton modes q for different values of r and L .In the next section we derive the two possible solu-tions to the nonlinear eq. (15), laying emphasis on periodicwavetrains formed of bright and dark solitons of equal fi-nite separation (i.e. finite repetition). Shape profiles of theelectromagnetic wave coupled to the exciton, for each typeof periodic soliton solution obtained, will follow from eq.(16). The single-soliton solution to the nonlinear equation (15)depends on signs of M L ( q ) and A L ( q ). For single-pulse(i.e. bright) soliton solution propagating with velocity v given by (18), the two parameters must be of the samesign. Applying the boundary conditions φ ( z → ± ) → φ ( x, t ) = φ sech x − vtΓ , (19) Γ = s M L ( q ) χ L ( q ) , φ = s χ L ( q ) A L ( q ) . (20)In our case we are interested in a solution describing aperiodic train of pulses of the form (19), forming a pulsecrystal with a finite period. We can therefore express suchsolution formally as [33]: φ ν ( z ) = Q X i =0 φ ( z − iν ) , (21)where φ ( z ) is the single-pulse solution (19). Eq. (21) rep-resents a lattice of Q identical pulses of equal separation ν , when Q → ∞ the sum becomes exact and we find [33,34]: φ ν ( x, t ) = φ √ − κ dn x − vtΓ κ , Γ κ = p − κ Γ, (22)where dn () is a Jacobi elliptic function [33,34,35,36,37].The solution (22) can also be obtained directly by solv-ing eq. (15) with boundary conditions φ ( z ± ν ) = φ ( z ),yielding the well-known [33,34,35,36,37] periodic solitonsolution to the nonlinear Schr¨odinger equation. More ex-plicitely, formula (22) describes a periodic lattice of brightsolitons of identical amplitude φ / √ − κ , identical av-erage width Γ κ and period: ν = 2 K ( κ ) Γ κ , (23)with K ( κ ) the elliptic integral of first kind and κ (0 < κ ≤
1) the modulus of the Jacobi elliptic function. It is quiteremarkable that when κ →
1, the function dn () → sin ()and when κ → dn () → sech (). In this lastlimit the period ν → ∞ and the soliton amplitude reducesto φ , while its average width becomes Γ . When M L ( q ) and A L ( q ) are of different signs, the nonlin-ear wave equation (15) with boundary conditions φ ( z →±∞ ) = ± φ admits a single-kink soliton solution: φ ( x, t ) = φ tanh x − vtΓ , (24) Γ = s − M L ( q ) χ L ( q ) , φ = s χ L ( q ) A L ( q ) . (25)Formula (25) suggests that χ L ( q ) and A L ( q ) should bepositive, while M L ( q ) should be negative for a stable single-kink soliton solution. The wavetrain structure associatedwith the single-kink soliton (24), consisting of a periodicarrangement of identical kinks of the form (24) with equalseparation ν , is obtained either by summing φ ( z − iν )over i from zero to ∞ , or solving directly eq. (15) withperiodic boundary conditions. The two approaches resultin the following nonlinear periodic solution: φ ν ( x, t ) = r κ κ φ sn x − vtΓ ,κ , Γ ,κ = r κ Γ , (26)where sn () is another Jacobi elliptic function with a period[38,39]: ν = K ( κ ) Γ ,κ . (27)As in the case of the periodic bright soliton lattice (22),the periodic kink solution (26) tends to an harmonic wavewhen κ = 0 and reduces exactly to the single-kink solution(24) when κ = 1. Note that in this last limit the period ν → ∞ .Formula (20) and (25) suggest that in addition to theirdependence on κ , the amplitudes and widths of the single-pulse and single-kink solitons composing the bright- anddark-soliton trains eq. (22) and (26), are functions of thelong-range parameters r and L . This dependence is intro-duced essentially through the energy ǫ L ( q ) and velocity v L ( q ) of the single-exciton states, which variations withcharacteristic parameters of the long-range intermolecu-lar interaction potential were illustrated in figs. 1 and 2.In fig. 3 we sketched the bright (upper graphs) and dark(lower graphs) soliton wavetrains obtained in formula (22)and (26), for κ = 0 .
98 and κ = 1. The left graphs arethe single-soliton limits which, as indicated, are recov-ered when κ = 1 (i.e. when the separation between thesingle-soliton components of the two soliton wavetrains isinfinitely large). Molecular crystals offer a wealth of rich physical propertiesthat are distinct from those observed in conventional solidssuch as covalent or ionic crystals. In molecular solids, thepacking of molecules to form the molecular crystal struc-ture is determined to a large extent by relatively weakerintermolecular interactions such as hydrogen bonds and
Aboringong et al.: Exciton-polariton soliton wavetrains with long-range interactions
Fig. 3. (Color online) Sketches of the bright (upper graphs) and dark (lower graphs) soliton solutions obtained in (22) and(26) respectively, for two values of κ . Left graphs are for κ = 0 .
98, while the right graphs are the single-soliton limit (i.e. κ = 1). van der Waals. Organic molecular crystals [40] in partic-ular are interesting from both biological and biochemicalperspectives, they provide ideal candidates for unveilingthe nature of the hydrogen bond and hence to understanda variety of important biological processes in molecularsolids such as proteins and peptides.In this work we investigated the effects of long-rangeintermolecular interactions on the dispersion energy ofsingle-exciton states, and by extension on characteristicparameters of periodic wavetrains of exciton and polari-ton solitons in molecular crystals interacting with light. Byreformulating the usual Kac-Baker potential to take intoaccount the finite range of intermolecular interactions, wefound that the account of long-range interactions brings ahuge quantitative as well as qualitative changes on excitonand polariton solitons characteristic parameters. Namelywe obtained that long-range interactions will narrow theBrillouin zone of small-amplitude excitons with respectto the short-range result, while enhancing their acousticvelocity but lowering the single-exciton low-lying energystate. Large-amplitude solutions to the exciton equationwas obtained in the continuum limit in terms of periodicwavetrains of pulse and dark solitons, whose amplitudesand widths as well as periods were affected by the long-range intermolecular interactions. single and periodic soli-ton solutions to the light propagating coupled to the exci-tons, are similar to the exciton solitons for they are linkedby a simple linear relation. Molecular crystals today are used for a variety of tech-nological applications, in these applications the crystalstructure determined by the intermolecular interactionssuch as hydrogen bonds and van der Waals interactions,is key in the manipulation of the structure so as to pro-duce materials with technologically useful properties as forinstance nonlinear optical responses, short exciton pulses,robust exciton transports and so on [40]. Acknowledgments
Work supported by the World Academy of Sciences (TWAS),Trieste, Italy. A. M. Dikand´e wishes to thank the Alexan-der von Humboldt (AvH) Foundation of logistic supports.
Authors contribution statement
ENNA performed calculations, AMD proposed the themeand wrote the manuscript. Both authors agreed on curveswhich were plotted by AMD.
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