Excitonâpolaritons in nanostructured nitride ... - Semantic Scholar
Microelectronics Journal 36 (2005) 1006–1010 www.elsevier.com/locate/mejo
Exciton–polaritons in nanostructured nitride superlattices F.F. de Medeirosa, E.L. Albuquerquea,*, M.S. Vasconcelosb b
a Universidade Federal do Rio Grande do Norte, Departamento de Fı´sica, 59072-970 Natal, RN, Brazil Centro Federal de Educac¸a˜o Tecnolo´gica do Maranha˜o, Departamento de Cieˆncias Exatas, 65025-001 Sa˜o Luı´s, MA, Brazil
Available online 2 June 2005
Abstract In this work we study the propagation of exciton–polaritons (bulk and surface modes) in a binary superlattice ABAB., truncated at zZ0, where z is defined as the growth axis. Here A is the spatially dispersive medium, which alternates with a common dielectric medium B (sapphire-Al2O3). The excitonic medium (A) is modelled by a semiconductor from the nitride’s family (III–V semiconductor) that has, as a main characteristic, a wide-direct gap. The exciton-polariton spectrum is determined, in both s and p-polarization, by using standard electromagnetic boundary conditions, together with an additional boundary condition (ABC) for the Wannier–Mott excitons, employing a transfer-matrix formalism to simplify the algebra. The dispersion relation shows a bottleneck profile for the superlattice modes, whose behavior is similar to those found in the bulk crystal. Furthermore, interesting properties are revealed from the ABC as well as from different ratios of the thickness of the two superlattices alternating materials. q 2005 Elsevier Ltd. All rights reserved. Keywords: Nanostructures; Semiconductors; Optical properties; Exciton–polaritons
1. Introduction Exciton–polaritons [1–5] are mixed modes resulting from the interaction between the exciton (the electron-hole pair) and the photon in the band-gap frequency region between the valence and the conduction band. They show the effect of spatial dispersion and, as a consequence, the dielectric function has an extra dependence with the wavevector k, i.e. 3ðk; uÞ Z 3N C S= u20 C Dk2 K u2 K iuG ; (1) where 3N is the background dielectric constant and SZ 4pa0u20 is the oscillator strength at uZ0 and kZ0. Here, u0 is the frequency of the uncoupled exciton (the band-gap frequency less the binding frequency) and DZ-u0/M, where MZmeCmh is the exciton mass, u is the incident light frequency with uyu0 and G is a phenomenological damping parameter. The k-dependence of the dielectric function gives rise to a rich phenomenology when compared to the corresponding phonon-polaritons, due to the spatialdispersion term Dk2. * Corresponding author. Tel.: C55 842153793; fax: C55 842153791. E-mail address: [email protected] (E.L. Albuquerque).
0026-2692/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2005.04.018
Exciton–polaritons have been studied extensively in direct band-gap semiconductors such as GaAs [6,7], CdS [8], ZnO [9] and CuCl [10]. Particular attention has been given to their optical properties (photoluminescence, absorption spectra, reflectance spectra, etc.) since optical characterization is one of the best experimental tools to investigate their band structures and electronic properties. In contrast with Si and conventional III–V semiconductors that do not have band gaps sufficiently large, III–V nitride semiconductors are suitable for designing and fabricating optoelectronic devices in the violet, blue and green region of the spectrum required for full color display. They display noticeable crystalline robustness, relative indifference to chemical aggressions, remarkable thermal stability and direct band gap at the zone center. Moreover, semiconductor components which have the spectrum range extended to the blue wavelengths, can then emit and detect the three primary colours of the visible spectrum, producing a major impact on imaging and graphics applications [11–13]. In the last decade, much effort was expended to grow and characterize GaN, AlN and InN and their alloys (for a review see Ref. [14]). As a result, overall performance of the III–V nitride devices improved very fast. It is the aim of this work to study the propagation of exciton–polaritons in nanostructured nitride binary superlattices. In particular we are interested here in the GaN properties with the sapphire as substrate, which has been
F.F. de Medeiros et al. / Microelectronics Journal 36 (2005) 1006–1010
extensively employed as a substrate for the growth of the III–V nitride semiconductors, in part due to its good quality available at low cost. It is transparent, stable at high temperature and the technology of growth of the nitrides on sapphire is well established [15]. The plan of this work is as follows: in Section 2 we present a general theory to characterize the excitonpolariton s- and p-polarized, respectively. Numerical results are then discussed in Section 3, stressing the excitonpolariton spectra. The conclusions of this work are presented and Section 5.
2. General theory The bulk exciton-polariton dispersion relation is found by using Maxwell’s equations with the dielectric function given by (1), appropriated to an excitonic medium. The result is: 3ðk; uÞ½k K 3ðk; uÞu =c Z 0; 2
2
2
(2)
where c is the light speed in the vacuum. The solution of this equation gives a longitudinal mode if 3(k,u)Z0, i.e.: Dk2 Z u2 K S=3N:
(3)
The transverse mode is obtained when the second factor in (2) vanishes, i.e.: Dk4 K ðu2 C Du2 3N=c2 Þk2 C ðu2 3N K SÞu2 =c2 Z 0:
1007
[18,19]: ð Z 0: mPð C n vP=vz
(5)
Here m and n are integer numbers (we call it ABC3). Consider now the propagation of an exciton-polariton, p-polarized mode. The electric field is in the xz-plane, while the magnetic field points in the y-direction. The GaN/ sapphire layers are parallel to the Cartesian xy-plane. Therefore, in each layer, the p-polarized electromagnetic wave is characterized by the fields ð r ; tÞ Z ðEx ; 0; Ez Þ exp ðikx x K iutÞ; Eðð
(6a)
ð ððr ; tÞ Z ð0; Hy ; 0Þ exp ðikx x K iutÞ: H
(6b)
Inside each layer, the above fields must satisfy Maxwell’s equations ð r ; tÞ Z KvBðð ð r ; tÞ=vt; V ! Eðð
(7a)
ð ððr ; tÞ Z 30 3j vEðð ð r ; tÞ=vt; V !H
(7b)
yielding: ð r ; tÞ=v2 t; ð r ; tÞ Z K 3j =c2 v2 Eðð V !V ! Eðð
(8)
where the label j denotes A or B. The solution of Maxwell’s equation in the nth unit cell for medium A (excitonic medium), is given by Eð Tj Z 1;0;ikx =kjz Anj expðKkjz zÞC K1;0;ikx =kjz Bnj expðkjz zÞ;
(4)
(9a)
Eqs. (3) and (4) provide the exciton-polariton dispersion relations for bulk crystal. There is no stop band, with at least one propagating mode for every frequency u. Consider now the superlattice case, characterized by a binary periodic structure .ABABABA., where medium A, with thickness a, is fulfilled by GaN and is characterized by an excitonic dielectric function 3A(u). Medium B, with thickness b, is fulfilled by sapphire, characterized by a dielectric constant 3B. The unit cell of the superlattice has a length LZaCb, with its axis parallel to the z-direction. In order to study the polariton propagation in this structure, we will apply Maxwell’s boundary conditions at the interfaces between the layers. However, Maxwell’s equations provide only two boundary conditions, so an additional boundary condition is required. The need of an ABC was first recognized by Pekar [1,16], and has been since then a recurrent problem. In its simpler form it is considered that the polarization field vanish at the boundary of the dispersive medium, i.e. Pð Z 0 at zZ0 (we call it ABC1). Another simple form, proposed by other authors such as Ting et al. [17], suggest that the normal derivative of the polarization field vanish at the boundary of ð the dispersive medium, i.e. vP=vzZ 0 (we call it ABC2). A third proposal, which generalize the previous one, yields
Eð L Z 1; 0;KikL =kx AnL expðkz zÞC 1;0; ikL =kx BnL expðKkz zÞ; (9b) ðkjz Þ2 Z kx2 K kj2 ;
jZ 1; 2 with k1 and k2 being the where solutions of (4) for the two transverse modes. Also ðkLz Þ2 Z kx2 K kL2 , where kL is the solution of (3) for the longitudinal mode. The term exp(ikxxKiut) was omitted for simplicity. Analogously, the solution for the electric field in the dielectric medium B is: Eð Zð1;0;ikx =aB ÞE1n exp ðKaB zÞCð1;0;Kikx =aB ÞE2n exp ðaB zÞ (10) where aBZ[k2x -3Bu2/c2]1/2 must be real in order to ensure the localization of the superlattice’s mode. The constitutive relation between the excitonic polarizð ation field Pand the electric field Eð is given by: X Pð Z ½cL Eð L C cJ Eð T : (11) JZ1;2
Next, we apply the standard electromagnetic boundary conditions (continuity of Ex and Dz) together with (5) at the interfaces of the nth unity cell, i.e. at zZnl, zZnlCa and zZ(nC1)L. The result, with the help of a transfer-matrix
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approach (for a better description of the method see Ref. [20]), provide the dispersion relation for the bulk-polariton modes for the superlattice. The transfer matrix T relates the coefficients of the electric field in one unit cell to those in the preceding cell, i.e. " nC1 # " n# A1 A1 ZT n ; (12) nC1 B1 B1 where T can be found elsewhere [21]. Taking into account the translational symmetry of the system through the application of Bloch’s theorem, we have " nC1 # " n# A1 A1 Z expðiQLÞ n ; (13) B1 BnC1 1 where Q is the Bloch wavevector and L is the length of the superlattice unit cell. The bulk superlattice polaritons are obtained combining Eqs. (12) and (13) to yield: cosðQLÞ Z 1=2TrðTÞ:
(14)
3. Numerical results We now consider numerical calculation for the p-polarized exciton-polariton spectrum in GaN/sapphire superlattice, without the damping energy. The physical parameters used for GaN (medium A) are those suitable for the A exciton in bulk GaN [22,23]: 3NZ8.75; -u0Z 3487 meV; 4pa0Z15!10K3 and MZ1.3m0, where m0 is the rest mass of the electron. In medium B, considered to be sapphire-Al2O3, we take the dielectric constant 3BZ10. In Fig. 1, we have plotted the reduced frequency u/u0 as a function of the dimesionless wavevector kxa, where u0 is exciton’s rest frequency and a is the thickness of the GaN layer, kx being the in-plane wavevector. We have considered ABC1 with the thickness of the sapphire layer bZa/2, and aZ50 nm. The appearance of a major number of modes is the most remarkable aspect in relation to the excitonpolariton spectrum in bulk crystal. The bulk modes (shown shaded) occur as bands with edges corresponding to the lines
Now we truncate the superlattice by introducing an external surface at zZ0, with vacuum occupying the region z!0. This semi-infinite superlattice does not hold complete translational symmetry in the z-direction and, therefore, no longer admit the validity of the Bloch’s ansatz as in the previous case. On the other hand, it allows the appearence of another class of solutions: the so-called surface polaritons, whose field amplitudes decay with the distance. Instead of (13), we have now " nC1 # " n# A1 A1 Z expðKbLÞ n ; (15) B1 BnC1 1 where b is an attenuated factor, with Re(b)O0 as the condition for a localized mode. Eq. (14) still holds provide we replace Q by ib, i.e. cos hðbLÞ Z 1=2Tr ðT Þ:
(16)
The electromagnetic field in the region z!0 is given by Eð Z ð1; 0; Kkx =aC ÞEexp aC z ; (17) where aC Z ½kx2 K 3C u2 =c2 1=2 . Assuming the external layer (in contact with the vacuum) of the superlattice be medium A, one can find the following surface-polariton dispersion relation T11 C T12 l Z T22 C T21 lK1 ;
(18)
where Tij (i,jZ1,2) are matrix elements of T, and l is a complex quantity related with the superlattice’s physical parameters. Using a similar calculation, we obtain also the excitonpolariton dispersion relation for s-polarization, where the magnetic field is in the xz-plane, while electric field is parallel to the y-direction. In this case, there is no propagation of the longitudinal mode kL.
Fig. 1. p-polarized GaN/sapphire superlattice exciton-polariton dispersion relation considering ABC1. Here, aZ2b, with aZ50 nm. In (a), two transverse modes and one longitudinal mode (bulk modes) are indicated by dot lines, while dashed lines mean the light lines. The full lines represent the surface and the shaded areas the bulk superlattice modes. In (b), we show the modes in the region near the resonance u/u0Z1.
F.F. de Medeiros et al. / Microelectronics Journal 36 (2005) 1006–1010
1009
QLZ0 and QLZp, while the surface modes (thin lines) appear between the bulk modes. The energy band width above the resonant frequency u0 are narrower than the bulk bands bellow this value. The dotted lines in Fig. 1(a) describe propagation of the bulk transverse polariton modes, with wavevector kj (jZ1,2) obtained from (4), and the longitudinal mode, with wavevector kL given by (3). Observe that the superlattice modes have origin from the light line, here represented by the dashed line. Because of the spatial dispersion effect, surface polaritons in thin excitonic films, as well as in superlattices, have the property of coexisting with bulk modes in energy region between the transverse and longitudinal bulk modes, and consequently, the energy transfer between these modes are possible [24]. Fig. 2 shows the exciton-polariton spectrum considering ð ABC2 (i.e. vP=vzZ 0). The main differences compared with Fig. 1 are observed more explicitly for u O uL, where uL is the longitudinal bulk mode frequency. In this region there is no propagation of the exciton. Finally, Fig. 3 shows the exciton polariton spectrum considering ABC3 (i.e. ð Pð C vP=vzZ 0, choosing mZnZ1). For all ABCs considered here, the energy bands turn narrow as thickness of the sapphire layer reduce, showing
Fig. 3. p-polarized GaN/sapphire superlattice exciton-polariton dispersion relation for ABC3, using the same parameters as in Fig. 1.
that photon–exciton interaction is favored when the sapphire layer is less thick than the GaN layer. However exciton–polaritons ‘notice’ the superlattice structure independently of the choice of the ABC employed. On the other hand, since the thicknesses a and b are bulk parameters, the surface modes are less affected by their changes. For completeness we show in Fig. 4 the exciton–polariton spectrum, s-polarized, considering aZ2b. The overall profile is very similar to the p-polarized counterpart, with a big difference related with the modes near the exciton resonance.
4. Conclusions
Fig. 2. p-polarized GaN/sapphire superlattice exciton-polariton dispersion relation for ABC2, using the same parameters as in Fig. 1.
The study of the electronic properties in semiconductor superlattices had a revival in the last two decades. Nevertheless, the comprehension of the optical properties of these artificial materials continues to require more investigation since their optical characterization provides valuable information concerning their electronic properties. The exciton–polariton curves in GaN/sapphire superlattice give a rich phenomenology. The dispersion relation has a ‘signature’ characteristic of a ‘bottleneck’ form, similar to
1010
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Acknowledgements The authors acknowledge the partial financial support of CNPq, CNPq-CTEnerg, FINEP-CTInfra, MCT-NanoSemiMat, and CAPES-Procad (Brazilian Research Agencies).
References
Fig. 4. s-polarized GaN/sapphire superlattice exciton-polariton dispersion relation for ABC1, using the same parameters as in Fig. 1.
those observed in the bulk crystal. In p-polarization, polariton curves for the three different ABCs provide distinct results as we compare the spectra shown in Figs. 1–3. The photon-exciton interaction is stronger for thinner sapphire layers. In s-polarization, the polariton curves near resonance have a different behavior, as compared to the p-polarized case, deserving further investigation. Although the ABC problem has been so far the subject of various discussions, the choice of the appropriate ABC can be found experimentally or obtained, in general, as a consequence of the Maxwell’s equations and boundary conditions. The surface polariton dispersion relation studies have been also valuable to elucidate the applicability of the ABCs [25].
[1] S.I. Pekar, Zh. Eksp. Teor. Fiz. 33 (1957) 1022 [Sov. Phys. JETP 6 (1958) 785]. [2] J.J. Hopfield, Phys. Rev. 112 (1958) 1555. [3] V.M. Agranovich, A.A. Rukhadze, Sov. Phys. JETP 7 (1959) 685. [4] J.J. Hopfield, D.G. Thomas, Phys. Rev. 132 (1963) 563. [5] A.A. Maradudin, D.L. Mills, Phys. Rev. B 7 (1973) 2787. [6] T. Steiner, M.L.W. Thewalt, E.S. Coteles, J.P. Salerno, Phys. Rev. B 34 (1986) 1006. [7] J. Tignon, T. Hasche, D.S. Chemla, H.C. Schneider, F. Jahnke, S.W. Koch, Phys. Rev. Lett. 84 (2000) 3382. [8] P. Wiesner, U. Helm, Phys. Rev. B 11 (1975) 3071. [9] J. Lagois, B. Fischer, Phys. Rev. Lett. 36 (1976) 680. [10] D. Fro¨hlich, E. Mohler, P. Wiesner, Phys. Rev. Lett. 26 (1971) 554. [11] S.F. Chichibu, K. Torii, T. Deguchi, T. Sota, A. Setoguchi, H. Nakanishi, T. Azuhata, S. Nakamura, Appl. Phys. Lett. 76 (2000) 1576. [12] K. Torii, M. Ono, T. Sota, T. Azuhata, S. Chichibu, S. Nakamura, Phys. Rev. B 62 (2000) 4723. [13] S. Nakamura, T. Mukai, M. Senoh, Appl. Phys. Lett. 64 (1994) 1687. [14] J.Y. Duboz, M.A. Khan, in: B. Gil (Ed.), Physics and Applications of Nitride Compounds Semiconductors, Oxford University Press, 1997. [15] S.C. Jain, M. Willander, J. Narayan, R. Van Overstraeten, J. Appl. Phys. 87 (2000) 975. [16] S.I. Pekar, Zh. Eksp. Teor. Fiz. 34 (1958) 1176 [Sov. Phys. JETP 7 (1958) 813]. [17] C.S. Ting, M.J. Frankel, J.L. Birman, Solid State Comun. 17 (1975) 1285. [18] P. Haveli, in: R. Del Sole, A. D’Andrea, A. Lapiccirella (Eds.), Excitons in Confined Systems Springer Proceedings in Physics vol. 25, Springer Verlag, Berlin, 1988. [19] G.S. Agarwal, D.N. Pattanayk, E. Wolf, Phys. Rev. Lett. 27 (1971) 1022. [20] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 233 (1993) 67. [21] F.F. de Medeiros, MSc Dissertation, UFRN, unpublished, 2005. [22] M. Tchounkeu, O. Briot, B. Gil, J.P. Alexis, R.L. Aulombard, J. Appl. Phys. 80 (1996) 5352. [23] J.Y. Duboz, M.A. Khan, in: B. Gil (Ed.), Physics and Applications of Nitride Compounds Semiconductors, Oxford University Press, New York, 1997. [24] M.S. Vasconcelos, E.L. Albuquerque, G.A. Farias, V.N. Freire, Solid State Commun. 124 (2002) 109. [25] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 376 (2003) 225.
Exciton–polaritons in nanostructured nitride superlattices F.F. de Medeirosa, E.L. Albuquerquea,*, M.S. Vasconcelosb b
a Universidade Federal do Rio Grande do Norte, Departamento de Fı´sica, 59072-970 Natal, RN, Brazil Centro Federal de Educac¸a˜o Tecnolo´gica do Maranha˜o, Departamento de Cieˆncias Exatas, 65025-001 Sa˜o Luı´s, MA, Brazil
Available online 2 June 2005
Abstract In this work we study the propagation of exciton–polaritons (bulk and surface modes) in a binary superlattice ABAB., truncated at zZ0, where z is defined as the growth axis. Here A is the spatially dispersive medium, which alternates with a common dielectric medium B (sapphire-Al2O3). The excitonic medium (A) is modelled by a semiconductor from the nitride’s family (III–V semiconductor) that has, as a main characteristic, a wide-direct gap. The exciton-polariton spectrum is determined, in both s and p-polarization, by using standard electromagnetic boundary conditions, together with an additional boundary condition (ABC) for the Wannier–Mott excitons, employing a transfer-matrix formalism to simplify the algebra. The dispersion relation shows a bottleneck profile for the superlattice modes, whose behavior is similar to those found in the bulk crystal. Furthermore, interesting properties are revealed from the ABC as well as from different ratios of the thickness of the two superlattices alternating materials. q 2005 Elsevier Ltd. All rights reserved. Keywords: Nanostructures; Semiconductors; Optical properties; Exciton–polaritons
1. Introduction Exciton–polaritons [1–5] are mixed modes resulting from the interaction between the exciton (the electron-hole pair) and the photon in the band-gap frequency region between the valence and the conduction band. They show the effect of spatial dispersion and, as a consequence, the dielectric function has an extra dependence with the wavevector k, i.e. 3ðk; uÞ Z 3N C S= u20 C Dk2 K u2 K iuG ; (1) where 3N is the background dielectric constant and SZ 4pa0u20 is the oscillator strength at uZ0 and kZ0. Here, u0 is the frequency of the uncoupled exciton (the band-gap frequency less the binding frequency) and DZ-u0/M, where MZmeCmh is the exciton mass, u is the incident light frequency with uyu0 and G is a phenomenological damping parameter. The k-dependence of the dielectric function gives rise to a rich phenomenology when compared to the corresponding phonon-polaritons, due to the spatialdispersion term Dk2. * Corresponding author. Tel.: C55 842153793; fax: C55 842153791. E-mail address: [email protected] (E.L. Albuquerque).
0026-2692/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2005.04.018
Exciton–polaritons have been studied extensively in direct band-gap semiconductors such as GaAs [6,7], CdS [8], ZnO [9] and CuCl [10]. Particular attention has been given to their optical properties (photoluminescence, absorption spectra, reflectance spectra, etc.) since optical characterization is one of the best experimental tools to investigate their band structures and electronic properties. In contrast with Si and conventional III–V semiconductors that do not have band gaps sufficiently large, III–V nitride semiconductors are suitable for designing and fabricating optoelectronic devices in the violet, blue and green region of the spectrum required for full color display. They display noticeable crystalline robustness, relative indifference to chemical aggressions, remarkable thermal stability and direct band gap at the zone center. Moreover, semiconductor components which have the spectrum range extended to the blue wavelengths, can then emit and detect the three primary colours of the visible spectrum, producing a major impact on imaging and graphics applications [11–13]. In the last decade, much effort was expended to grow and characterize GaN, AlN and InN and their alloys (for a review see Ref. [14]). As a result, overall performance of the III–V nitride devices improved very fast. It is the aim of this work to study the propagation of exciton–polaritons in nanostructured nitride binary superlattices. In particular we are interested here in the GaN properties with the sapphire as substrate, which has been
F.F. de Medeiros et al. / Microelectronics Journal 36 (2005) 1006–1010
extensively employed as a substrate for the growth of the III–V nitride semiconductors, in part due to its good quality available at low cost. It is transparent, stable at high temperature and the technology of growth of the nitrides on sapphire is well established [15]. The plan of this work is as follows: in Section 2 we present a general theory to characterize the excitonpolariton s- and p-polarized, respectively. Numerical results are then discussed in Section 3, stressing the excitonpolariton spectra. The conclusions of this work are presented and Section 5.
2. General theory The bulk exciton-polariton dispersion relation is found by using Maxwell’s equations with the dielectric function given by (1), appropriated to an excitonic medium. The result is: 3ðk; uÞ½k K 3ðk; uÞu =c Z 0; 2
2
2
(2)
where c is the light speed in the vacuum. The solution of this equation gives a longitudinal mode if 3(k,u)Z0, i.e.: Dk2 Z u2 K S=3N:
(3)
The transverse mode is obtained when the second factor in (2) vanishes, i.e.: Dk4 K ðu2 C Du2 3N=c2 Þk2 C ðu2 3N K SÞu2 =c2 Z 0:
1007
[18,19]: ð Z 0: mPð C n vP=vz
(5)
Here m and n are integer numbers (we call it ABC3). Consider now the propagation of an exciton-polariton, p-polarized mode. The electric field is in the xz-plane, while the magnetic field points in the y-direction. The GaN/ sapphire layers are parallel to the Cartesian xy-plane. Therefore, in each layer, the p-polarized electromagnetic wave is characterized by the fields ð r ; tÞ Z ðEx ; 0; Ez Þ exp ðikx x K iutÞ; Eðð
(6a)
ð ððr ; tÞ Z ð0; Hy ; 0Þ exp ðikx x K iutÞ: H
(6b)
Inside each layer, the above fields must satisfy Maxwell’s equations ð r ; tÞ Z KvBðð ð r ; tÞ=vt; V ! Eðð
(7a)
ð ððr ; tÞ Z 30 3j vEðð ð r ; tÞ=vt; V !H
(7b)
yielding: ð r ; tÞ=v2 t; ð r ; tÞ Z K 3j =c2 v2 Eðð V !V ! Eðð
(8)
where the label j denotes A or B. The solution of Maxwell’s equation in the nth unit cell for medium A (excitonic medium), is given by Eð Tj Z 1;0;ikx =kjz Anj expðKkjz zÞC K1;0;ikx =kjz Bnj expðkjz zÞ;
(4)
(9a)
Eqs. (3) and (4) provide the exciton-polariton dispersion relations for bulk crystal. There is no stop band, with at least one propagating mode for every frequency u. Consider now the superlattice case, characterized by a binary periodic structure .ABABABA., where medium A, with thickness a, is fulfilled by GaN and is characterized by an excitonic dielectric function 3A(u). Medium B, with thickness b, is fulfilled by sapphire, characterized by a dielectric constant 3B. The unit cell of the superlattice has a length LZaCb, with its axis parallel to the z-direction. In order to study the polariton propagation in this structure, we will apply Maxwell’s boundary conditions at the interfaces between the layers. However, Maxwell’s equations provide only two boundary conditions, so an additional boundary condition is required. The need of an ABC was first recognized by Pekar [1,16], and has been since then a recurrent problem. In its simpler form it is considered that the polarization field vanish at the boundary of the dispersive medium, i.e. Pð Z 0 at zZ0 (we call it ABC1). Another simple form, proposed by other authors such as Ting et al. [17], suggest that the normal derivative of the polarization field vanish at the boundary of ð the dispersive medium, i.e. vP=vzZ 0 (we call it ABC2). A third proposal, which generalize the previous one, yields
Eð L Z 1; 0;KikL =kx AnL expðkz zÞC 1;0; ikL =kx BnL expðKkz zÞ; (9b) ðkjz Þ2 Z kx2 K kj2 ;
jZ 1; 2 with k1 and k2 being the where solutions of (4) for the two transverse modes. Also ðkLz Þ2 Z kx2 K kL2 , where kL is the solution of (3) for the longitudinal mode. The term exp(ikxxKiut) was omitted for simplicity. Analogously, the solution for the electric field in the dielectric medium B is: Eð Zð1;0;ikx =aB ÞE1n exp ðKaB zÞCð1;0;Kikx =aB ÞE2n exp ðaB zÞ (10) where aBZ[k2x -3Bu2/c2]1/2 must be real in order to ensure the localization of the superlattice’s mode. The constitutive relation between the excitonic polarizð ation field Pand the electric field Eð is given by: X Pð Z ½cL Eð L C cJ Eð T : (11) JZ1;2
Next, we apply the standard electromagnetic boundary conditions (continuity of Ex and Dz) together with (5) at the interfaces of the nth unity cell, i.e. at zZnl, zZnlCa and zZ(nC1)L. The result, with the help of a transfer-matrix
1008
F.F. de Medeiros et al. / Microelectronics Journal 36 (2005) 1006–1010
approach (for a better description of the method see Ref. [20]), provide the dispersion relation for the bulk-polariton modes for the superlattice. The transfer matrix T relates the coefficients of the electric field in one unit cell to those in the preceding cell, i.e. " nC1 # " n# A1 A1 ZT n ; (12) nC1 B1 B1 where T can be found elsewhere [21]. Taking into account the translational symmetry of the system through the application of Bloch’s theorem, we have " nC1 # " n# A1 A1 Z expðiQLÞ n ; (13) B1 BnC1 1 where Q is the Bloch wavevector and L is the length of the superlattice unit cell. The bulk superlattice polaritons are obtained combining Eqs. (12) and (13) to yield: cosðQLÞ Z 1=2TrðTÞ:
(14)
3. Numerical results We now consider numerical calculation for the p-polarized exciton-polariton spectrum in GaN/sapphire superlattice, without the damping energy. The physical parameters used for GaN (medium A) are those suitable for the A exciton in bulk GaN [22,23]: 3NZ8.75; -u0Z 3487 meV; 4pa0Z15!10K3 and MZ1.3m0, where m0 is the rest mass of the electron. In medium B, considered to be sapphire-Al2O3, we take the dielectric constant 3BZ10. In Fig. 1, we have plotted the reduced frequency u/u0 as a function of the dimesionless wavevector kxa, where u0 is exciton’s rest frequency and a is the thickness of the GaN layer, kx being the in-plane wavevector. We have considered ABC1 with the thickness of the sapphire layer bZa/2, and aZ50 nm. The appearance of a major number of modes is the most remarkable aspect in relation to the excitonpolariton spectrum in bulk crystal. The bulk modes (shown shaded) occur as bands with edges corresponding to the lines
Now we truncate the superlattice by introducing an external surface at zZ0, with vacuum occupying the region z!0. This semi-infinite superlattice does not hold complete translational symmetry in the z-direction and, therefore, no longer admit the validity of the Bloch’s ansatz as in the previous case. On the other hand, it allows the appearence of another class of solutions: the so-called surface polaritons, whose field amplitudes decay with the distance. Instead of (13), we have now " nC1 # " n# A1 A1 Z expðKbLÞ n ; (15) B1 BnC1 1 where b is an attenuated factor, with Re(b)O0 as the condition for a localized mode. Eq. (14) still holds provide we replace Q by ib, i.e. cos hðbLÞ Z 1=2Tr ðT Þ:
(16)
The electromagnetic field in the region z!0 is given by Eð Z ð1; 0; Kkx =aC ÞEexp aC z ; (17) where aC Z ½kx2 K 3C u2 =c2 1=2 . Assuming the external layer (in contact with the vacuum) of the superlattice be medium A, one can find the following surface-polariton dispersion relation T11 C T12 l Z T22 C T21 lK1 ;
(18)
where Tij (i,jZ1,2) are matrix elements of T, and l is a complex quantity related with the superlattice’s physical parameters. Using a similar calculation, we obtain also the excitonpolariton dispersion relation for s-polarization, where the magnetic field is in the xz-plane, while electric field is parallel to the y-direction. In this case, there is no propagation of the longitudinal mode kL.
Fig. 1. p-polarized GaN/sapphire superlattice exciton-polariton dispersion relation considering ABC1. Here, aZ2b, with aZ50 nm. In (a), two transverse modes and one longitudinal mode (bulk modes) are indicated by dot lines, while dashed lines mean the light lines. The full lines represent the surface and the shaded areas the bulk superlattice modes. In (b), we show the modes in the region near the resonance u/u0Z1.
F.F. de Medeiros et al. / Microelectronics Journal 36 (2005) 1006–1010
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QLZ0 and QLZp, while the surface modes (thin lines) appear between the bulk modes. The energy band width above the resonant frequency u0 are narrower than the bulk bands bellow this value. The dotted lines in Fig. 1(a) describe propagation of the bulk transverse polariton modes, with wavevector kj (jZ1,2) obtained from (4), and the longitudinal mode, with wavevector kL given by (3). Observe that the superlattice modes have origin from the light line, here represented by the dashed line. Because of the spatial dispersion effect, surface polaritons in thin excitonic films, as well as in superlattices, have the property of coexisting with bulk modes in energy region between the transverse and longitudinal bulk modes, and consequently, the energy transfer between these modes are possible [24]. Fig. 2 shows the exciton-polariton spectrum considering ð ABC2 (i.e. vP=vzZ 0). The main differences compared with Fig. 1 are observed more explicitly for u O uL, where uL is the longitudinal bulk mode frequency. In this region there is no propagation of the exciton. Finally, Fig. 3 shows the exciton polariton spectrum considering ABC3 (i.e. ð Pð C vP=vzZ 0, choosing mZnZ1). For all ABCs considered here, the energy bands turn narrow as thickness of the sapphire layer reduce, showing
Fig. 3. p-polarized GaN/sapphire superlattice exciton-polariton dispersion relation for ABC3, using the same parameters as in Fig. 1.
that photon–exciton interaction is favored when the sapphire layer is less thick than the GaN layer. However exciton–polaritons ‘notice’ the superlattice structure independently of the choice of the ABC employed. On the other hand, since the thicknesses a and b are bulk parameters, the surface modes are less affected by their changes. For completeness we show in Fig. 4 the exciton–polariton spectrum, s-polarized, considering aZ2b. The overall profile is very similar to the p-polarized counterpart, with a big difference related with the modes near the exciton resonance.
4. Conclusions
Fig. 2. p-polarized GaN/sapphire superlattice exciton-polariton dispersion relation for ABC2, using the same parameters as in Fig. 1.
The study of the electronic properties in semiconductor superlattices had a revival in the last two decades. Nevertheless, the comprehension of the optical properties of these artificial materials continues to require more investigation since their optical characterization provides valuable information concerning their electronic properties. The exciton–polariton curves in GaN/sapphire superlattice give a rich phenomenology. The dispersion relation has a ‘signature’ characteristic of a ‘bottleneck’ form, similar to
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Acknowledgements The authors acknowledge the partial financial support of CNPq, CNPq-CTEnerg, FINEP-CTInfra, MCT-NanoSemiMat, and CAPES-Procad (Brazilian Research Agencies).
References
Fig. 4. s-polarized GaN/sapphire superlattice exciton-polariton dispersion relation for ABC1, using the same parameters as in Fig. 1.
those observed in the bulk crystal. In p-polarization, polariton curves for the three different ABCs provide distinct results as we compare the spectra shown in Figs. 1–3. The photon-exciton interaction is stronger for thinner sapphire layers. In s-polarization, the polariton curves near resonance have a different behavior, as compared to the p-polarized case, deserving further investigation. Although the ABC problem has been so far the subject of various discussions, the choice of the appropriate ABC can be found experimentally or obtained, in general, as a consequence of the Maxwell’s equations and boundary conditions. The surface polariton dispersion relation studies have been also valuable to elucidate the applicability of the ABCs [25].
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