- Email: [email protected]
Exciton-polariton confinement in Fibonacci quasiperiodic superlattice
Surface Science 600 (2006) 4337–4341 www.elsevier.com/locate/susc
Exciton-polariton confinement in Fibonacci quasiperiodic superlattice F.F. de Medeiros a, E.L. Albuquerque
a,*
, M.S. Vasconcelos b, G.A. Farias
c
a
Departamento de Fı´sica, Universidade Federal do Rio Grande do Norte, Campus Universitario, 59072-970 Natal-RN, Brazil Departamento de Cieˆncias Exatas, Centro Federal de Educac¸a˜o Tecnolo´gica do Maranha˜o, 65025-001 Sa˜o Luı´s-MA, Brazil Departamento de Fı´sica, Universidade Federal do Ceara´, Caixa Postal 6030 Campus do Pici, 60455-760 Fortaleza-CE, Brazil
b c
Available online 22 May 2006
Abstract We investigate the propagation of exciton-polaritons (bulk and surface modes) in quasiperiodic superlattices of Fibonacci type, truncated at z = 0, where z is defined as the growth axis. For our purposes, the Fibonacci structure can be realized experimentally by juxtaposing two basic building blocks A and B, following a Fibonacci sequential rule. Here A is the spatially dispersive medium modelled by a semiconductor from the nitride family (GaN, for instance), which alternates with a common dielectric medium B (sapphire). Our main aim is the investigation of the modified optical properties of the exciton polariton modes for a better understanding of the dynamics of the excitation in confined systems. The dispersion relation shows a bottleneck profile for the superlattice modes, whose behavior is similar to those found in the bulk crystal. The Cantor-like energy spectrum is also investigated, and scaling properties are shown for the Fibonacci quasiperiodic structure. 2006 Elsevier B.V. All rights reserved. Keywords: Computer simulation; Molecular beam epitaxy; Surface waves-Polaritons; Gallium nitride; Quantum wells; Semiconductor–insulator interfaces
1
vðk; xÞ ¼ SðDk 2 X2 Þ ;
1. Introduction
2
Polaritons, coupled-mode excitations made up from dipole-active elementary excitations (such as phonons, plasmons, magnons, etc.) interacting with photons, have attracted considerable attention during the past two decades (for a review, see Ref. [1]). Exciton polaritons are coupled-mode excitations made up from dipole-active excitons interacting with photons. They show the effect of spatial dispersion, that is, an extra dependence on wavevector k of the dielectric function, i.e, ðk; xÞ ¼ 1 þ vðk; xÞ;
ð1Þ
where 1 is the background dielectric constant, and v(k,x) is the susceptibility given by
*
Corresponding author. Tel.: +55 842153793; fax: +55 842153791. E-mail address: [email protected] (E.L. Albuquerque).
0039-6028/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2006.02.076
2
X ¼ x þ ixC
x20 :
with
ð2Þ ð3Þ
Here x0 is the frequency of the uncoupled exciton (the band-gap frequency less the binding frequency), D = h x 0/ M, where M = me + mh is the exciton mass, C is a phenomenological damping parameter, and S ¼ 4pa0 x20 is the oscillator strength at x = 0 and k = 0. The exciton’s susceptibility differs from the corresponding phonon form by the spatial-dispersion term Dk2. Exciton polaritons in direct band-gap semiconductors were extensively studied in the seventies and eighties (for a review, see [2]). However, only recently have there been investigations on their optical properties, and nowadays they have become a widespread research field. For instance, propagation of short light pulses in semiconductor slabs containing excitons has been performed using the scattering-state technique and the steepest-descent method, in order to calculate numerically and to describe
4338
F.F. de Medeiros et al. / Surface Science 600 (2006) 4337–4341
analytically the temporary behavior of the dielectric polarization inside the films [3]. Polarized photoluminescence in semiconductor microcavities, controlled by changing the exciton-cavity detuning, were recently theoretical and experimentally studied [4,5], as well as the observation of a phase transition from a classical thermal mixed state to a quantum-mechanical pure state of exciton polaritons in a GaAs multiple quantum-well microcavity [6]. Furthermore, polarization properties of the exciton-polariton modes in supported (with sapphire as the substrate) and unsupported films made up of wide-band-gap semiconductor-like gallium nitride (GaN), have been recently investigated both theoretically [7] and experimentally [8]. In fact, when a nitride semiconductor is placed in a substrate, dramatic changes in the intensity of the optical response, the energy of the optical transitions, the radiative lifetimes, and the emission patterns are produced [9,10]. Group III nitride semiconductor compounds, like GaN, have undergone a recent and worldwide rediscovery. These compounds display noticeable crystalline robustness, relative indifference to chemical aggressions, remarkable thermal stability and direct band-gap at the zone center. Recent achievements in the production of several advanced optoelectronic devices (green and blue light emitting diodes, for example) [11] and the realization of a promising breakthrough in the area of UV detectors and heterojunctions bipolar transitions, among others [12], have made these structures excellent options for technological applications and attractive objects of research. In this communication, we investigate the spectrum of the more energetic exciton coupled with a photon (p-polarized), the A coupled exciton mode, in multilayer structures composed of GaN/sapphire (Al2O3) layers arranged in a quasiperiodical fashion, which follows the Fibonacci (FB) mathematical sequence. This quasiperiodic structures can be generated by their inflation rules (in what follows, A and B are building blocks modelling the nitride GaN and the sapphire, respectively, whose thicknesses are a and b): A ! AB, B ! A. Although several theoretical techniques have been used to study the propagation of collective excitations in these structures, in the present work, we make use of the transfer matrix approach to analyze the superlattice modes, simplifying the algebra which would be otherwise quite involved (for a review, see Ref. [13]). Our main aim is the investigation of the modified optical properties of the exciton polariton bulk modes for a better understanding of the dynamics of the excitation in confined systems. The outline of this paper is as follows: in Section 2, we present the main equations related with the theory of exciton polaritons in quasiperiodic structures. The numerical results and the conclusions of the paper are presented in Section 3. 2. General theory The bulk exciton polariton dispersion relation is found by solving Maxwell’s equation with the use of the dielectric function. The result is well-known, i.e.,
ðk; xÞ½k 2 ðk; xÞx2 =c2 ¼ 0;
ð4Þ
where c is the light speed in the vacuum. The solution of this equation gives a longitudinal mode if (k,x) = 0, i.e., Dk 2 ¼ X2 S=1 :
ð5Þ
For the transverse modes we have: Dk 4 ðX2 þ Dx2 1 =c2 Þk 2 þ ðX2 1 SÞx2 =c2 ¼ 0:
ð6Þ
Eqs. (5) and (6) provide the exciton polariton dispersion relations for the bulk crystal. In this case, there is no forbidden band, with at least one propagating mode for every frequency x. Before considering the propagation of the exciton-polariton modes in a multilayer structure arranged in a quasiperiodical fashion, which follows the Fibonacci mathematical sequence, let us first consider an infinite periodic structure, namely ABABAB [14]. Suppose that a p-polarized light is incident from the vacuum on the surface (xy-plane) of the multilayer structure. Two transverse modes of wavevectors ~ k 1 and ~ k 2 and one longitudinal mode of wavevector ~ k L can propagate in medium A. In order to study the polariton propagation, we will apply Maxwell’s boundary conditions at the interfaces. However, Maxwell’s equations provide only two boundary conditions, so an additional boundary condition (ABC) is required. The need of an ABC was first recognized by Pekar [15], which considered that the polarization field vanishes at the boundary of the dispersive region, i.e., ~ P ¼ 0 at z = 0 (we will assume it here). The solution of Maxwell’s equation in the nth unit cell for medium A (excitonic medium), is given by z z ~ ETj ¼ ð1; 0; ik x =k zj ÞAnj ekj z þ ð1; 0; ik x =k zj ÞBnj ekj z ;
ð7Þ
z z ~ EL ¼ ð1; 0; ik L =k x ÞAnL ekL z þ ð1; 0; ik L =k x ÞBnL ekL z ;
ð8Þ
2
where ðk zj Þ ¼ k 2x k 2j , j = 1,2, with k1 and k2 being the solutions of (6) for the two transverse modes. Also 2 ðk zL Þ ¼ k 2x k 2L , where kL is the solution of (5) for the longitudinal mode. The term exp(ikxx ixt) was omitted for simplicity. The solution for the electric field in the dielectric medium B is: ~ E ¼ ð1; 0; ik x =aB ÞEn1 eaB z þ ð1; 0; ik x =aB ÞEn2 eaB z ; ½k 2x
2
ð9Þ
2
where aB ¼ B x =c must be real in order to ensure the localization of the superlattice’s mode. The constitutive relation between the excitonic polarization field ~ P and the electric field ~ E is given by X ~ P ¼ vL ~ EL þ ET : vJ ~ ð10Þ J ¼1;2
To find the bulk polariton dispersion relation, we apply the standard electromagnetic boundary conditions together with the Pekar’s ABC at the interfaces of the nth cell. The dispersion relation can be obtained with the help of the transfer-matrix method (for a better description of the
F.F. de Medeiros et al. / Surface Science 600 (2006) 4337–4341
method see Ref. [16]) to simplify the algebra which can be otherwise quite heavy. The transfer matrix T relates the coefficients of the electric field in one cell to those in the preceding cell. Taking into account also the translational symmetry of the system through the application of Bloch’s theorem, the dispersion relation for the bulk polariton modes follows, i.e., cos(QL) = (1/2)Tr(T), where Tr(T) means the trace of the matrix T, Q is the Bloch wavevector and L = a + b is the length of the unit cell of the superlattice. On the other hand, to set up the dispersion relation of the surface polariton modes, we truncate the superlattice by introducing an external surface at z = 0, where z is defined as the superlattice’s growth axis. The region z < 0 is filled by a transparent medium C (vacuum), whose frequency-independent dielectric constant is C. This truncated superlattice does not possess full translational symmetry in the z-direction, and, therefore, we may no longer assume Bloch’s ansatz as in the previous case. However, the bulk polariton equation still holds provide we replace Q by ib, i.e., cosh(b L) = (1/2)Tr(T), where b is an attenuation factor, with Re(b) > 0 as the condition for a localized mode. Considering the external layer (in contact with the vacuum) of the superlattice be medium A, one can find the surface polariton dispersion relation T11 + T12k = T22 + T21k1, where Tij (i,j = 1,2) are matrix elements of T, and k is a complex expression related with the superlattice’s physical parameters. Consider now that the multilayer structures are composed of GaN/sapphire(Al2O3) layers arranged in a quasiperiodical fashion, which follows the Fibonacci mathematical sequence. The Fibonacci generations are obtained using the following recursive rule Sn1Sn2, for n P 2, whose initial conditions are S0 = B and S1 = A. The subsequent Fibonacci generations are S2 = [AB], S3 = [ABA], S4 = [ABAAB], etc., The number of building blocks increases in accordance with the Fibonacci number Fn = Fn1 + Fn2, with F0 = F1 = 1. Also the ratio between the number of the building B in the pffiffiffi sequence tends to the golden mean number s ¼ ð1 þ 5Þ=2 for large n. The transfer matrices for the Fibonacci generations are T S nþ2 ¼ T S n T S nþ1 for any higher generation (n P 1), with 1 T S 0 ¼ N 1 B M B and T S 1 ¼ N A M A . The matrices NJ and MJ (J = A,B) are given elsewhere [14]. Therefore, the transfer matrix for any generation of the Fibonacci structure can be obtained by a simple equation relating the transfer matrices of two previous generations.
4339
filled by sapphire-Al2O3, we take the dielectric constant B = 10. The damping energy C is neglected. Fig. 1 depicts the p-polarized exciton polariton spectrum in GaN/sapphire quasiperiodic structure for the ninth Fibonacci generation. Instead of using the frequency x, we prefer to replace it by the reduced frequency x/x0 as a function of the dimensionless wavevector kxa, kx being the in-plane wavevector and a the thickness of the excitonic GaN layer. We have considered the thicknesses a = b = 50 nm. As we can infer, the bulk-mode continuum is now replaced by several undamped normal modes that have mixed surface and guided character in the spectrum (full lines in the figure), whose energies correspond to the discretized values of the exciton polariton wavevector along the z-axis. The bulk modes now occur as bands, shown shaded in Fig. 1, which are limited by the equations
3. Numerical results and conclusions We now present some numerical results to characterize the p-polarized exciton polariton spectra in the GaN/sapphire quasiperiodic Fibonacci-like structure. The physical parameters used for GaN (excitonic medium A) obtained from the A exciton in bulk GaN are [12,17]: ⁄x0 = 3487 meV; 1 = 8.75; 4pa0 = 15 · 103 and M = 1.3m0, where m0 is the rest mass of the electron. In medium B,
Fig. 1. Exciton polariton spectrum for the reduced frequency x/x0 versus kxa for the ninth generation of the quasiperiodic Fibonacci sequence. In (a), the full lines represent the surface and bulk modes, while the two transverse and one longitudinal polariton’s bulk modes are indicated by the chain-dotted lines. The almost vertical dashed lines mean the light lines. In (b), we show the modes in the region near the resonance x/ x0 = 1.
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QL = 0 and QL = p. The surface modes are found among the bands formed by the bulk modes. The dispersion relation exhibit bottleneck-like behavior, whose aspect has some resemblance to those found in the bulk crystal. The chain-dotted lines describe the propagation of the bulk transverse polariton modes, with wavevector kj(j = 1,2) obtained from (6) and the longitudinal mode, with wavevector kL given by (5). Observe that the modes in the quasiperiodic sequence have their origin from the light line, represented by the almost vertical dashed line. The energy band width bellow the bulk longitudinal polariton mode are broader than the bulk bands above this value. Due to the extra wavevector dependence in the dielectric response function ð~ k; xÞ, leading to the so called spatial-dispersion effects in the optical region of exciton resonance, surface polaritons in thin excitonic films, as well as in quasiperiodic superlattices, have the property of coexisting with bulk modes in energy between transverse and longitudinal modes, and consequently, the energy transfer between these modes are possible. Moreover, the polariton spectrum is highly directional, since it comes from non-thermalized excitons. This means that, taking into account real thin films, the energy differences between excitons localized in different quantum wells (due to the well-width fluctuations) acts as a dephasing mechanism and, as a consequence, tend to disappear mainly inside the center of the bottleneck spectrum, as shown in Fig. 1. We do not expect to see this effect experimentally, unless high-quality materials are been used, but unfortunately it is not easy to fabricate them [18]. In Fig. 2(a) we show the distribution of the allowed and forbidden energy bandwidths of the bulk exciton polaritons, plotted against the Fibonacci generation index n, for a fixed value of the in-plane wavevector, namely kxa = 2.875. Notice that, as n increases, the allowed band regions get narrower and narrower, as an indication of greater localization of the modes, characterizing a Cantor-like spectrum. In fact, the total width D of the allowed energy regions (the Lebesgue measure of the energy spectrum) decreases with n as the power law D F d n . Here Fn is the Fibonacci number and the exponent d (the diffusion constant of the spectra) is a function of the common in-plane wavevector kxa. This exponent can indicate the degree of localization of the excitation [19]. To investigate this bandwidth scaling behavior, we show in Fig. 2(b) a log-log plot of these power laws for five different values of kxa, namely 2.875, 2.9, 2.925, 2.95, and 2.975 (from top to bottom). As it can be seen, there is a dependence of the d exponent with the dimensionless wavevector kxa for the Fibonacci structure. This is a completely different behavior from the one found for magnetostatic modes propagating in these structures [20], where the linear coefficient is virtually the same. The inset of this figure shows an interesting linear behavior of the scale exponent d against the reduced wavevector kxa. In summary, we have described the spectra, scaling, and localization of exciton polaritons modes which can propa-
Fig. 2. (a) The distribution of the bandwidths for the exciton polariton as a function of the Fibonacci generation number n. (b) Log–log plot for the total allowed bandwitdth D against the Fibonacci number Fn. The inset shows the behavior of the scale exponent d against the reduced wavevector kxa.
gate in quasiperiodic structures following a Fibonacci sequence. As the defining rules of this sequence impose long range correlations on it, it is plausible to search for global (universal) consequences of these correlations. The global aspects of the sequence were found in their band widths structures, which are fractal objects presenting a Cantor-like profile, as exemplified in Fig. 2(a). Much more interesting, however, are the power laws which govern the scale of the spectra depicted in Fig. 2(b), where we found that the scaling index d depends linearly on the dimensionless wavevector kxa. The most important experimental technique to investigate the optical properties of exciton-polaritons is that of resonance Brillouin scattering (RBS for short) proposed by Brenig et al. [21] in 1972. Using this spectroscopic
F.F. de Medeiros et al. / Surface Science 600 (2006) 4337–4341
method it is possible to determine the basic exciton and photon parameters, as well as information about the dynamics of the exciton–phonon coupling which produces the scattering interaction. Furthermore, with RBS it is possible to probe the exciton-polariton dispersion curve at wavevectors which are a significant fraction of Brillouin zone values. RBS can sometimes be observed in poorer quality samples near the exciton resonance, but usually only weakly. Surely our model can be realized experimentally using this technique. Acknowledgements We would like to thank partial financial support from CNPq, CNPq-CTEnerg, and FINEP-CTInfra (Brazilian Research Agencies). References [1] E.L. Albuquerque, M.G. Cottam, Polaritons in Periodic and Quasiperiodic Structures, Elsevier, Amsterdam, 2004. [2] J.L. Birman, in: E.I. Rashba, M.D. Sturge (Eds.), Excitons, North Holland, Amsterdam, 1982. [3] G. Malpuech, A. Kavokin, G. Panzarini, Phys. Rev. B 60 (1999) 16788. [4] M.D. Martin, G. Aichmayr, L. Vin˜a, R. Andre´, Phys. Rev. Lett. 89 (2002) 077402.
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[5] K.V. Kavokin, I.A. Shelykh, A.V. Kavokin, G. Malpuech, P. Bigenwald, Phys. Rev. Lett. 92 (2004) 017401. [6] H. Deng, G. Weihs, C. Santori, J. Bloch, Y. Yamamoto, Science 298 (2002) 199. [7] M.S. Vasconcelos, E.L. Albuquerque, G.A. Farias, V.N. Freire, Solid State Commun. 124 (2002) 109. [8] P.P. Paskov, T. Paskova, P.O. Holtz, B. Monemar, Phys. Status Solidi A 201 (2004) 678. [9] S.F. Chichibu, K. Torii, T. Deguchi, T. Sota, A. Setoguchi, H. Nakanishi, T. Azuhata, S. Nakamura, Appl. Phys. Lett. 76 (2000) 1576. [10] K. Torii, M. Ono, T. Sota, T. Azuhata, S. Chichibu, S. Nakamura, Phys. Rev. B 62 (2000) 4723. [11] S. Nakamura, T. Mukai, M. Senoh, Appl. Phys. Lett. 64 (1994) 1687. [12] J.Y. Duboz, M.A. Khan, in: B. Gil (Ed.), Physics and Applications of Nitride Compounds Semiconductors, Oxford University Press, 1997. [13] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 376 (2003) 225. [14] F.F. de Medeiros, E.L. Albuquerque, M.S. Vasconcelos, Microelectronics J. 36 (2005) 1006. [15] S.I. Pekar, Sov. Phys. JETP 6 (1958) 785; S.I. Pekar, Sov. Phys. JETP 7 (1958) 813. [16] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 233 (1993) 67. [17] M. Tchounkeu, O. Briot, B. Gil, J.P. Alexis, R.L. Aulombard, J. Appl. Phys. 80 (1996) 5352. [18] H. Morkoc¸, S. Strite, G.B. Gao, M.E. Lin, B. Sversodlov, M. Burns, J. Appl. Phys. 73 (1994) 1363. [19] P. Hawrylak, J.J. Quinn, Phys. Rev. Lett. 57 (1986) 380. [20] D.H.A.L. Anselmo, M.G. Cottam, E.L. Albuquerque, J. Phys. Condens. Mat. 12 (2000) 1041. [21] W. Brening, R. Zeyher, J.L. Birman, Phys. Rev. B 6 (1972) 4617.
Exciton-polariton confinement in Fibonacci quasiperiodic superlattice F.F. de Medeiros a, E.L. Albuquerque
a,*
, M.S. Vasconcelos b, G.A. Farias
c
a
Departamento de Fı´sica, Universidade Federal do Rio Grande do Norte, Campus Universitario, 59072-970 Natal-RN, Brazil Departamento de Cieˆncias Exatas, Centro Federal de Educac¸a˜o Tecnolo´gica do Maranha˜o, 65025-001 Sa˜o Luı´s-MA, Brazil Departamento de Fı´sica, Universidade Federal do Ceara´, Caixa Postal 6030 Campus do Pici, 60455-760 Fortaleza-CE, Brazil
b c
Available online 22 May 2006
Abstract We investigate the propagation of exciton-polaritons (bulk and surface modes) in quasiperiodic superlattices of Fibonacci type, truncated at z = 0, where z is defined as the growth axis. For our purposes, the Fibonacci structure can be realized experimentally by juxtaposing two basic building blocks A and B, following a Fibonacci sequential rule. Here A is the spatially dispersive medium modelled by a semiconductor from the nitride family (GaN, for instance), which alternates with a common dielectric medium B (sapphire). Our main aim is the investigation of the modified optical properties of the exciton polariton modes for a better understanding of the dynamics of the excitation in confined systems. The dispersion relation shows a bottleneck profile for the superlattice modes, whose behavior is similar to those found in the bulk crystal. The Cantor-like energy spectrum is also investigated, and scaling properties are shown for the Fibonacci quasiperiodic structure. 2006 Elsevier B.V. All rights reserved. Keywords: Computer simulation; Molecular beam epitaxy; Surface waves-Polaritons; Gallium nitride; Quantum wells; Semiconductor–insulator interfaces
1
vðk; xÞ ¼ SðDk 2 X2 Þ ;
1. Introduction
2
Polaritons, coupled-mode excitations made up from dipole-active elementary excitations (such as phonons, plasmons, magnons, etc.) interacting with photons, have attracted considerable attention during the past two decades (for a review, see Ref. [1]). Exciton polaritons are coupled-mode excitations made up from dipole-active excitons interacting with photons. They show the effect of spatial dispersion, that is, an extra dependence on wavevector k of the dielectric function, i.e, ðk; xÞ ¼ 1 þ vðk; xÞ;
ð1Þ
where 1 is the background dielectric constant, and v(k,x) is the susceptibility given by
*
Corresponding author. Tel.: +55 842153793; fax: +55 842153791. E-mail address: [email protected] (E.L. Albuquerque).
0039-6028/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2006.02.076
2
X ¼ x þ ixC
x20 :
with
ð2Þ ð3Þ
Here x0 is the frequency of the uncoupled exciton (the band-gap frequency less the binding frequency), D = h x 0/ M, where M = me + mh is the exciton mass, C is a phenomenological damping parameter, and S ¼ 4pa0 x20 is the oscillator strength at x = 0 and k = 0. The exciton’s susceptibility differs from the corresponding phonon form by the spatial-dispersion term Dk2. Exciton polaritons in direct band-gap semiconductors were extensively studied in the seventies and eighties (for a review, see [2]). However, only recently have there been investigations on their optical properties, and nowadays they have become a widespread research field. For instance, propagation of short light pulses in semiconductor slabs containing excitons has been performed using the scattering-state technique and the steepest-descent method, in order to calculate numerically and to describe
4338
F.F. de Medeiros et al. / Surface Science 600 (2006) 4337–4341
analytically the temporary behavior of the dielectric polarization inside the films [3]. Polarized photoluminescence in semiconductor microcavities, controlled by changing the exciton-cavity detuning, were recently theoretical and experimentally studied [4,5], as well as the observation of a phase transition from a classical thermal mixed state to a quantum-mechanical pure state of exciton polaritons in a GaAs multiple quantum-well microcavity [6]. Furthermore, polarization properties of the exciton-polariton modes in supported (with sapphire as the substrate) and unsupported films made up of wide-band-gap semiconductor-like gallium nitride (GaN), have been recently investigated both theoretically [7] and experimentally [8]. In fact, when a nitride semiconductor is placed in a substrate, dramatic changes in the intensity of the optical response, the energy of the optical transitions, the radiative lifetimes, and the emission patterns are produced [9,10]. Group III nitride semiconductor compounds, like GaN, have undergone a recent and worldwide rediscovery. These compounds display noticeable crystalline robustness, relative indifference to chemical aggressions, remarkable thermal stability and direct band-gap at the zone center. Recent achievements in the production of several advanced optoelectronic devices (green and blue light emitting diodes, for example) [11] and the realization of a promising breakthrough in the area of UV detectors and heterojunctions bipolar transitions, among others [12], have made these structures excellent options for technological applications and attractive objects of research. In this communication, we investigate the spectrum of the more energetic exciton coupled with a photon (p-polarized), the A coupled exciton mode, in multilayer structures composed of GaN/sapphire (Al2O3) layers arranged in a quasiperiodical fashion, which follows the Fibonacci (FB) mathematical sequence. This quasiperiodic structures can be generated by their inflation rules (in what follows, A and B are building blocks modelling the nitride GaN and the sapphire, respectively, whose thicknesses are a and b): A ! AB, B ! A. Although several theoretical techniques have been used to study the propagation of collective excitations in these structures, in the present work, we make use of the transfer matrix approach to analyze the superlattice modes, simplifying the algebra which would be otherwise quite involved (for a review, see Ref. [13]). Our main aim is the investigation of the modified optical properties of the exciton polariton bulk modes for a better understanding of the dynamics of the excitation in confined systems. The outline of this paper is as follows: in Section 2, we present the main equations related with the theory of exciton polaritons in quasiperiodic structures. The numerical results and the conclusions of the paper are presented in Section 3. 2. General theory The bulk exciton polariton dispersion relation is found by solving Maxwell’s equation with the use of the dielectric function. The result is well-known, i.e.,
ðk; xÞ½k 2 ðk; xÞx2 =c2 ¼ 0;
ð4Þ
where c is the light speed in the vacuum. The solution of this equation gives a longitudinal mode if (k,x) = 0, i.e., Dk 2 ¼ X2 S=1 :
ð5Þ
For the transverse modes we have: Dk 4 ðX2 þ Dx2 1 =c2 Þk 2 þ ðX2 1 SÞx2 =c2 ¼ 0:
ð6Þ
Eqs. (5) and (6) provide the exciton polariton dispersion relations for the bulk crystal. In this case, there is no forbidden band, with at least one propagating mode for every frequency x. Before considering the propagation of the exciton-polariton modes in a multilayer structure arranged in a quasiperiodical fashion, which follows the Fibonacci mathematical sequence, let us first consider an infinite periodic structure, namely ABABAB [14]. Suppose that a p-polarized light is incident from the vacuum on the surface (xy-plane) of the multilayer structure. Two transverse modes of wavevectors ~ k 1 and ~ k 2 and one longitudinal mode of wavevector ~ k L can propagate in medium A. In order to study the polariton propagation, we will apply Maxwell’s boundary conditions at the interfaces. However, Maxwell’s equations provide only two boundary conditions, so an additional boundary condition (ABC) is required. The need of an ABC was first recognized by Pekar [15], which considered that the polarization field vanishes at the boundary of the dispersive region, i.e., ~ P ¼ 0 at z = 0 (we will assume it here). The solution of Maxwell’s equation in the nth unit cell for medium A (excitonic medium), is given by z z ~ ETj ¼ ð1; 0; ik x =k zj ÞAnj ekj z þ ð1; 0; ik x =k zj ÞBnj ekj z ;
ð7Þ
z z ~ EL ¼ ð1; 0; ik L =k x ÞAnL ekL z þ ð1; 0; ik L =k x ÞBnL ekL z ;
ð8Þ
2
where ðk zj Þ ¼ k 2x k 2j , j = 1,2, with k1 and k2 being the solutions of (6) for the two transverse modes. Also 2 ðk zL Þ ¼ k 2x k 2L , where kL is the solution of (5) for the longitudinal mode. The term exp(ikxx ixt) was omitted for simplicity. The solution for the electric field in the dielectric medium B is: ~ E ¼ ð1; 0; ik x =aB ÞEn1 eaB z þ ð1; 0; ik x =aB ÞEn2 eaB z ; ½k 2x
2
ð9Þ
2
where aB ¼ B x =c must be real in order to ensure the localization of the superlattice’s mode. The constitutive relation between the excitonic polarization field ~ P and the electric field ~ E is given by X ~ P ¼ vL ~ EL þ ET : vJ ~ ð10Þ J ¼1;2
To find the bulk polariton dispersion relation, we apply the standard electromagnetic boundary conditions together with the Pekar’s ABC at the interfaces of the nth cell. The dispersion relation can be obtained with the help of the transfer-matrix method (for a better description of the
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method see Ref. [16]) to simplify the algebra which can be otherwise quite heavy. The transfer matrix T relates the coefficients of the electric field in one cell to those in the preceding cell. Taking into account also the translational symmetry of the system through the application of Bloch’s theorem, the dispersion relation for the bulk polariton modes follows, i.e., cos(QL) = (1/2)Tr(T), where Tr(T) means the trace of the matrix T, Q is the Bloch wavevector and L = a + b is the length of the unit cell of the superlattice. On the other hand, to set up the dispersion relation of the surface polariton modes, we truncate the superlattice by introducing an external surface at z = 0, where z is defined as the superlattice’s growth axis. The region z < 0 is filled by a transparent medium C (vacuum), whose frequency-independent dielectric constant is C. This truncated superlattice does not possess full translational symmetry in the z-direction, and, therefore, we may no longer assume Bloch’s ansatz as in the previous case. However, the bulk polariton equation still holds provide we replace Q by ib, i.e., cosh(b L) = (1/2)Tr(T), where b is an attenuation factor, with Re(b) > 0 as the condition for a localized mode. Considering the external layer (in contact with the vacuum) of the superlattice be medium A, one can find the surface polariton dispersion relation T11 + T12k = T22 + T21k1, where Tij (i,j = 1,2) are matrix elements of T, and k is a complex expression related with the superlattice’s physical parameters. Consider now that the multilayer structures are composed of GaN/sapphire(Al2O3) layers arranged in a quasiperiodical fashion, which follows the Fibonacci mathematical sequence. The Fibonacci generations are obtained using the following recursive rule Sn1Sn2, for n P 2, whose initial conditions are S0 = B and S1 = A. The subsequent Fibonacci generations are S2 = [AB], S3 = [ABA], S4 = [ABAAB], etc., The number of building blocks increases in accordance with the Fibonacci number Fn = Fn1 + Fn2, with F0 = F1 = 1. Also the ratio between the number of the building B in the pffiffiffi sequence tends to the golden mean number s ¼ ð1 þ 5Þ=2 for large n. The transfer matrices for the Fibonacci generations are T S nþ2 ¼ T S n T S nþ1 for any higher generation (n P 1), with 1 T S 0 ¼ N 1 B M B and T S 1 ¼ N A M A . The matrices NJ and MJ (J = A,B) are given elsewhere [14]. Therefore, the transfer matrix for any generation of the Fibonacci structure can be obtained by a simple equation relating the transfer matrices of two previous generations.
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filled by sapphire-Al2O3, we take the dielectric constant B = 10. The damping energy C is neglected. Fig. 1 depicts the p-polarized exciton polariton spectrum in GaN/sapphire quasiperiodic structure for the ninth Fibonacci generation. Instead of using the frequency x, we prefer to replace it by the reduced frequency x/x0 as a function of the dimensionless wavevector kxa, kx being the in-plane wavevector and a the thickness of the excitonic GaN layer. We have considered the thicknesses a = b = 50 nm. As we can infer, the bulk-mode continuum is now replaced by several undamped normal modes that have mixed surface and guided character in the spectrum (full lines in the figure), whose energies correspond to the discretized values of the exciton polariton wavevector along the z-axis. The bulk modes now occur as bands, shown shaded in Fig. 1, which are limited by the equations
3. Numerical results and conclusions We now present some numerical results to characterize the p-polarized exciton polariton spectra in the GaN/sapphire quasiperiodic Fibonacci-like structure. The physical parameters used for GaN (excitonic medium A) obtained from the A exciton in bulk GaN are [12,17]: ⁄x0 = 3487 meV; 1 = 8.75; 4pa0 = 15 · 103 and M = 1.3m0, where m0 is the rest mass of the electron. In medium B,
Fig. 1. Exciton polariton spectrum for the reduced frequency x/x0 versus kxa for the ninth generation of the quasiperiodic Fibonacci sequence. In (a), the full lines represent the surface and bulk modes, while the two transverse and one longitudinal polariton’s bulk modes are indicated by the chain-dotted lines. The almost vertical dashed lines mean the light lines. In (b), we show the modes in the region near the resonance x/ x0 = 1.
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QL = 0 and QL = p. The surface modes are found among the bands formed by the bulk modes. The dispersion relation exhibit bottleneck-like behavior, whose aspect has some resemblance to those found in the bulk crystal. The chain-dotted lines describe the propagation of the bulk transverse polariton modes, with wavevector kj(j = 1,2) obtained from (6) and the longitudinal mode, with wavevector kL given by (5). Observe that the modes in the quasiperiodic sequence have their origin from the light line, represented by the almost vertical dashed line. The energy band width bellow the bulk longitudinal polariton mode are broader than the bulk bands above this value. Due to the extra wavevector dependence in the dielectric response function ð~ k; xÞ, leading to the so called spatial-dispersion effects in the optical region of exciton resonance, surface polaritons in thin excitonic films, as well as in quasiperiodic superlattices, have the property of coexisting with bulk modes in energy between transverse and longitudinal modes, and consequently, the energy transfer between these modes are possible. Moreover, the polariton spectrum is highly directional, since it comes from non-thermalized excitons. This means that, taking into account real thin films, the energy differences between excitons localized in different quantum wells (due to the well-width fluctuations) acts as a dephasing mechanism and, as a consequence, tend to disappear mainly inside the center of the bottleneck spectrum, as shown in Fig. 1. We do not expect to see this effect experimentally, unless high-quality materials are been used, but unfortunately it is not easy to fabricate them [18]. In Fig. 2(a) we show the distribution of the allowed and forbidden energy bandwidths of the bulk exciton polaritons, plotted against the Fibonacci generation index n, for a fixed value of the in-plane wavevector, namely kxa = 2.875. Notice that, as n increases, the allowed band regions get narrower and narrower, as an indication of greater localization of the modes, characterizing a Cantor-like spectrum. In fact, the total width D of the allowed energy regions (the Lebesgue measure of the energy spectrum) decreases with n as the power law D F d n . Here Fn is the Fibonacci number and the exponent d (the diffusion constant of the spectra) is a function of the common in-plane wavevector kxa. This exponent can indicate the degree of localization of the excitation [19]. To investigate this bandwidth scaling behavior, we show in Fig. 2(b) a log-log plot of these power laws for five different values of kxa, namely 2.875, 2.9, 2.925, 2.95, and 2.975 (from top to bottom). As it can be seen, there is a dependence of the d exponent with the dimensionless wavevector kxa for the Fibonacci structure. This is a completely different behavior from the one found for magnetostatic modes propagating in these structures [20], where the linear coefficient is virtually the same. The inset of this figure shows an interesting linear behavior of the scale exponent d against the reduced wavevector kxa. In summary, we have described the spectra, scaling, and localization of exciton polaritons modes which can propa-
Fig. 2. (a) The distribution of the bandwidths for the exciton polariton as a function of the Fibonacci generation number n. (b) Log–log plot for the total allowed bandwitdth D against the Fibonacci number Fn. The inset shows the behavior of the scale exponent d against the reduced wavevector kxa.
gate in quasiperiodic structures following a Fibonacci sequence. As the defining rules of this sequence impose long range correlations on it, it is plausible to search for global (universal) consequences of these correlations. The global aspects of the sequence were found in their band widths structures, which are fractal objects presenting a Cantor-like profile, as exemplified in Fig. 2(a). Much more interesting, however, are the power laws which govern the scale of the spectra depicted in Fig. 2(b), where we found that the scaling index d depends linearly on the dimensionless wavevector kxa. The most important experimental technique to investigate the optical properties of exciton-polaritons is that of resonance Brillouin scattering (RBS for short) proposed by Brenig et al. [21] in 1972. Using this spectroscopic
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method it is possible to determine the basic exciton and photon parameters, as well as information about the dynamics of the exciton–phonon coupling which produces the scattering interaction. Furthermore, with RBS it is possible to probe the exciton-polariton dispersion curve at wavevectors which are a significant fraction of Brillouin zone values. RBS can sometimes be observed in poorer quality samples near the exciton resonance, but usually only weakly. Surely our model can be realized experimentally using this technique. Acknowledgements We would like to thank partial financial support from CNPq, CNPq-CTEnerg, and FINEP-CTInfra (Brazilian Research Agencies). References [1] E.L. Albuquerque, M.G. Cottam, Polaritons in Periodic and Quasiperiodic Structures, Elsevier, Amsterdam, 2004. [2] J.L. Birman, in: E.I. Rashba, M.D. Sturge (Eds.), Excitons, North Holland, Amsterdam, 1982. [3] G. Malpuech, A. Kavokin, G. Panzarini, Phys. Rev. B 60 (1999) 16788. [4] M.D. Martin, G. Aichmayr, L. Vin˜a, R. Andre´, Phys. Rev. Lett. 89 (2002) 077402.
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