February 1, 2014 / Vol. 39, No. 3 / OPTICS LETTERS
489
Exciton photoluminescence in resonant quasi-periodic Thue–Morse quantum wells W. J. Hsueh,* C. H. Chang, and C. T. Lin Photonics Group, Department of Engineering Science and Ocean Engineering, National Taiwan University, 1, Sec. 4, Roosevelt Road, Taipei 10660, Taiwan *Corresponding author: [email protected] Received November 15, 2013; accepted December 15, 2013; posted December 20, 2013 (Doc. ID 201464); published January 21, 2014 This Letter investigates exciton photoluminescence (PL) in resonant quasi-periodic Thue–Morse quantum wells (QWs). The results show that the PL properties of quasi-periodic Thue–Morse QWs are quite different from those of resonant Fibonacci QWs. The maximum and minimum PL intensities occur under the anti-Bragg and Bragg conditions, respectively. The maxima of the PL intensity gradually decline when the filling factor increases from 0.25 to 0.5. Accordingly, the squared electric field at the QWs decreases as the Thue–Morse QW deviates from the antiBragg condition. © 2014 Optical Society of America OCIS codes: (230.5590) Quantum-well, -wire and -dot devices; (250.5230) Photoluminescence; (160.5293) Photonic bandgap materials. http://dx.doi.org/10.1364/OL.39.000489
Since quasi-crystals were first proposed, quasi-crystalline materials have attracted a great deal of interest because of their distinctive physical properties and their applications in various fields of research [1–8]. In recent years, quasi-crystals have been used in one-dimensional resonant photonic crystals based on multiple quantum wells (QWs). Numerous studies have focused on the properties of light waves propagating in multiple quantum well structures with different quasi-periodic arrangements [9,10]. Prior studies have indicated that the optical properties of quasi-periodic structures are different from those of periodic structures. For resonant periodic QWs, there is a wide bandgap and large reflectance under the Bragg condition, where the period, d, is equal to the half-wavelength at the exciton resonance frequency, ω0 . These structures provide an advantage in the design of various optical devices and slow light applications [11–15]. However, in terms of light emission, a periodic QW possesses a better light-emitting ability under the anti-Bragg condition of d λω0 ∕4 than under the Bragg condition, because of its greater absorbance and stronger photoluminescence (PL) [13,16,17]. For quasi-periodic QWs, relevant studies have shown that the wide bandgap and large reflectance around the exciton resonance frequency are replaced by a striking structural dip in the reflection spectra [10]. This unique characteristic results in absorbance and PL emission near the exciton resonance in a quasi-periodic QW that are very different from those in a periodic QW under the Bragg condition [18]. The difference between the properties of PL intensity in periodic QWs and those in resonant Fibonacci quantum wells (FQWs) has been recently studied [19,20]. The results show that the PL intensity in a FQW is significantly larger than that in a periodic QW under the anti-Bragg condition in certain cases because of the enhanced field at the QWs. For quasi-crystalline systems, the Thue–Morse sequence is another representative structure besides the Fibonacci one [21]. Whether the PL characteristics of resonant quasi-periodic Thue–Morse quantum wells (TMQWs) differ from those of a FQW is of interest. However, 0146-9592/14/030489-04$15.00/0
few studies have determined the properties of PL intensity in a TMQW. If the luminescence characteristics of a TMQW can be clarified, this structure will be useful in the design of light-emitting devices. This Letter considers a TMQW that consists of two different intervals, A and B, which obeys the recurrence rule: A → AB and B → BA [8,21]. The generation order of the system, v, indicates the different structures, such as A, AB, ABBA, and ABBABAAB, for v 1, 2, 3, and 4, respectively. There is a QW in each of the intervals, so the number of QWs in the TMQW is (N 2v−1 ). In this study, the barrier thicknesses of the structures are assumed to be thick enough to avoid coupling between excitons in the QWs, so the excitons only interact with the electromagnetic field. The dielectric contrast between QWs and barriers also is neglected. The TMQWs are placed between a finite forward barrier and a semiinfinite rear barrier. The cap thickness between the vacuum interface and the center of the first QW is set to dC λω0 ∕2. The interaction between the electromagnetic wave and the excitons in the QW structures is described by Maxwell’s equation: 2 ω ⇀ ∇×∇×E D; c ⇀
⇀
⇀
⇀
(1)
where D εzE 4πP exc is the displacement vector and the growth⇀ direction of the structures is set to the coordinate, z. P exc is the contribution of the exciton to the dielectric polarization of the entire QW structure, where a resonant source term is introduced to describe the luminescence properties [14,22,23]. In this study, the s-polarized wave and the 1s excitons are considered and the local nonresonant permittivity function, εz, is a constant that is equal to the background refractive index squared, n2b . Based on the solution of Eq. (1), the transmittance, reflectance, and absorbance are obtained using the total transfer matrix through the entire structure. Radiation boundary conditions are also imposed to determine the radiation emitted from the QW structures. For © 2014 Optical Society of America
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these conditions, only outward-bound waves exist outside the structures [19,22]. Thus, the intensity of luminescence radiated to the left side of the structure is given by: I PL ω; k∥ 4
N X j1
j κS ω; k∥ 2 jI j ω; k∥ j2 ; φj 1
Ξj
(2)
where the spectral density Ξj is related to the correlation function for the Fourier-transformed source, κ2 ωnb ∕c2 − k2∥ , k∥ is the in-plane wave vector, and S j ω; k∥ is the function relevant to the exciton susceptibility. The summation indicates that the radiation emitted by each source located at the jth QW adds in a noncoherent manner. In this study, each QW is considered to be identical and the waves are emitted along the growth direction, namely k∥ 0. Therefore, Eq. (2) can be reduced to I PL ω 4ΞjqSω∕φ1 j2
N X
jI j ωj2 ;
j1
where q ωnb ∕c, I j ω −th−jM R
j1 Y mN
Mm
sj1 ; sj2
and t is the transmission coefficient of the entire structure. We first investigated the influence of higher generation orders and larger numbers of QWs on the transmission, reflection and PL spectra in the TMQWs for the same spacing thicknesses, dA 0.412λω0 and dB 0.588λω0 . The numbers of QWs in the TMQWs with v 6, 7, and 8 are N 32, 64, and 128, respectively. Figure 1(a) shows that the transmittance in the middle and at both sides of the spectra remains very small and becomes larger, respectively, as the generation order increases. In addition, the frequency range for low transmittance increases as the generation order increases. As shown in Fig. 1(b), there are notable structural dips near the exciton resonance frequency in the reflection spectra. When the generation order increases, the frequency
Fig. 1. (a) Transmission, (b) reflection, and (c) PL spectra in TMQWs with v 6, 7, and 8 for the same spacing thicknesses. The parameter values of the system are ℏω0 1.523 eV, ℏΓ0 25 μeV, ℏΓ 180 μeV, and nb 3.59. The normalized frequency, Ω, is defined as Ω ω − ω0 D∕2πc, where D dA dB λω0 .
ranges that corresponds to these dips and the linewidths and the maxima of the reflection spectra all become greater. However, there are only small changes in the minimal reflectance in the dips. The reflectance at both sides of the spectra also becomes smaller as the generation order increases. Figures 1(a) and 1(b) show there is significant absorption near the exciton resonance frequency, which corresponds to the areas of low transmittance in the transmission spectra and the dips in the reflection spectra. This results in the PL profiles shown in Fig. 1(c) because of the close relationship between the absorbance and the PL intensity [22]. Specifically, the linewidths of the PL spectra increase when there is an increase in the frequency ranges for the dips in the reflection spectra. Although the number of QWs increases significantly, the maxima of the PL intensity are still weak as the generation order increases, because the minimal reflectance in the dips remains at almost the same magnitude. Next, the influence of the filling factor of the system, defined as F dA ∕dA dB , on the transmission, reflection, and PL spectra in the TMQW is illustrated in Fig. 2. Here, there is a significant change in these spectra as the filling factor increases from 0.25 to 0.5, namely from the anti-Bragg to the Bragg conditions. In fact, the spacing thicknesses at F 0.25 are dA λω0 ∕4 and dB 3λω0 ∕4, which is equivalent to the anti-Bragg condition, because they have the same spectral characteristics [16,17,19]. As shown in Fig. 2(a), the frequency range for low transmittance becomes larger as the filling factor increases. In particular, the frequency ranges for F 0.45 and 0.5 are much broader than those for F 0.25 and 0.35. The transmittance at both sides of the spectra also becomes larger when the filling factor is greater. In contrast to the transmission spectra, the variations in the reflection spectra are more complicated, as seen in Fig. 2(b). In general, the magnitude and linewidths of the reflection spectra become greater as the filling factor increases. The reflectance at both sides of the spectra decreases as the filling factor increases. However, the reflection profiles for the respective filling factors are very different from each other. Specifically, there is only one small peak at the exciton resonance frequency in the reflection profile for F 0.25 [17]. When the filling factor
Fig. 2. (a) Transmission, (b) reflection, and (c) PL spectra in a TMQW with v 8 for different filling factors. The number of QWs in the TMQW is N 128. The other parameter values are the same as those in the caption of Fig. 1.
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reaches 0.35 and 0.45, significant dips appear in the middle of the reflection spectra. The frequency ranges that correspond to the dip and the minimal reflectance in the dip for F 0.35 are wider and smaller than those for F 0.45, respectively. At F 0.5, there is the wide linewidth and large reflectance around the exciton resonance frequency in the reflection profile due to the Bragg condition [12]. Based on these transmission and reflection spectra, the PL characteristics of the TMQW for different filling factors are shown in Fig. 2(c). For F 0.25, the PL intensity is strongest because the greatest absorption occurs under the anti-Bragg condition. There also are two peaks that are symmetrical with respect to the exciton resonance frequency in the PL profile, which can be mainly attributed to the corresponding reflection spectrum, in which there is only one small peak. When the filling factor increases, the magnitude and linewidths of the PL spectra both decrease. When the filling factor is equal to 0.35 and 0.45, the dips in the corresponding reflection spectra cause greater absorption and stronger PL in the middle of the PL spectra. The dip in the PL spectrum for F 0.35 also can be attributed to the small peak in the dip of the reflection profile. However, the PL intensity for F 0.5 is very weak because of the wide linewidth and the large reflectance in the reflection spectrum [13]. As a result, Figs. 1 and 2 show that the PL spectra near the exciton resonance frequency are mainly dependent on the reflection spectra because of the low transmittance in the transmission profiles and the structural features in the reflection profiles. To compare the overall influence of the filling factor on the PL intensity in TMQWs with that in FQWs, the maximum values of PL intensity, defined as PLmax , in the TMQWs are plotted versus the filling factor in Fig. 3. The profiles of PLmax are only shown within this range because there are symmetry points at F 0.25, 0.5 and 0.75. Here, the number of QWs, N 34, in the FQW with a generation order of 8 is chosen to correspond to the number of QWs, N 32, in the TMQW with v 6 for comparison purposes. The values of PLmax at F 0.25 and 0.5 in the TMQW with v 6 are almost the same as those in the FQW with a generation order of 8 due to the anti-Bragg and Bragg conditions. There
Fig. 3. PLmax , defined by the maximum values of the PL intensity, in the TMQWs versus the filling factor, F. The numerical calculations are performed for TMQWs with v 4 to 8, which corresponds to the numbers of QWs, N 8 to 128. The A and B signs denote the positions of the anti-Bragg and Bragg conditions, respectively.
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are only slight differences between these values because the numbers of QWs are slightly different. In addition, the values of PLmax at F 0.25 and 0.5 are greater and smaller, respectively, when the generation order increases. As the TMQWs deviate from the antiBragg condition, all of the values of PLmax decline smoothly and gradually to their minima under the Bragg condition. There also is a sharper decline in the value of PLmax in the TMQW as the generation order increases. When the filling factor approaches 0.5, the decrease in all of the values of PLmax is moderated. In contrast, the values of PLmax in the TMQWs are very different from those in the FQWs when the filling factor increases from 0.25 to 0.5. There is a marked rise in the values of PLmax in the FQWs when the filling factor exceeds 0.25. After reaching their maxima, the values of PLmax drop steeply to their minima [19]. Figure 4 shows a comparison between the squared electric fields in the TMQW with v 6 and the FQW with a generation order of 8 for the same filling factors, F 0.25 and 0.412, to determine why the PL properties of these structures are different from each other. For clear comparison, the filling factor, F 0.412, was chosen to be the main comparison parameter in this study, since the maximum of PLmax in the FQW with a generation order of 8 occurs for this filling factor [19]. Here, the frequency where the PL intensity in the structure is a maximum is defined as ωPLmax . Figures 4(a) and 4(b) show the field profiles in the TMQW at F 0.25 and 0.412, respectively. At F 0.25, namely under the anti-Bragg condition, the peaks of the field are situated at the QWs although these peak values, marked by the red circles, are not very large. There also is a smooth and gradual decline in the peak values as the distance increases. However, at F 0.412, there is a decrease in the field values at the QWs although the peaks of the field increase because there is a deviation from the anti-Bragg condition. Many of the nodes of the field are even situated at the QWs. As shown in Figs. 4(c) and 4(a), the field profiles in the FQW and TMQW at F 0.25 are very similar because of
Fig. 4. Squared electric fields in a TMQW with v 6 at (a) F 0.25 and (b) F 0.412 and in a FQW with a generation order of 8 at (c) F 0.25 and (d) F 0.412. The number of QWs in the TMQW is N 32. The corresponding number of QWs in the FQW is N 34. The numerical simulations are made at ωPLmax . The horizontal lines, I I , I R , and I T , denote the incident, reflected, and transmitted waves, respectively. The vacuum region (the white area) is divided from the structure (the yellow area) by the black line. The blue lines represent the QW locations. The squared electric fields are given by jEωPLmax j2 for z > 0, multiplied by the background refractive index for clarity. The red circles denote the field values at the vacuum interface and the QWs.
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the anti-Bragg condition [17,19]. However, when there is a deviation from the anti-Bragg condition, there is a great difference between the field profiles in the FQW and TMQW at F 0.412, as seen in Figs. 4(d) and 4(b). In contrast to the TMQW, both the peaks of the field and the field values at the QWs increase in the FQW. A number of the peaks of the field are situated at the QWs. To the authors’ best knowledge, if the field at the QWs is larger, the PL intensity is stronger because there is the enhanced light–matter coupling at the QWs [18]. Therefore, the PL intensity in the TMQW is weaker than that in the FQW for the same filling factor because there is the reduced light–matter coupling. The strongest PL intensity in the TMQW occurs under the anti-Bragg condition. In conclusion, we have investigated exciton PL in resonant quasi-periodic TMQWs. The transmission and reflection spectra show that the PL spectra near the exciton resonance frequency primarily depend on the reflection spectra due to the low transmittance in the transmission profiles and the structural features in the reflection profiles. We also discovered that the maximum and minimum PL intensities in the TMQW occur under the anti-Bragg and Bragg conditions, respectively. There is a smooth and gradual decrease in the values of PLmax when the filling factor increases from 0.25 to 0.5. As the generation order increases, the value of PLmax declines more sharply. These results for TMQWs are quite different from those for FQWs, in which the values of PLmax continue to rise when the filling factor exceeds 0.25. Moreover, the field values at the QWs in a TMQW are smaller than those in a FQW for the same filling factor. Therefore, the PL intensity in a TMQW is weaker than that in a FQW because there is reduced light–matter coupling. In addition, PL in the TMQWs can reach the strongest PL in resonant periodic QWs. Thus, the TMQWs can be effectively applied in optoelectronic devices. The authors acknowledge the support in part by the National Science Council of Taiwan under grant number NSC 102-221-E-002-105. References 1. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
2. W. J. Hsueh, C. T. Chen, and C. H. Chen, Phys. Rev. A 78, 013836 (2008). 3. G. Gumbs, G. S. Dubey, A. Salman, B. S. Mahmoud, and D. Huang, Phys. Rev. B 52, 210 (1995). 4. L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, Appl. Phys. Lett. 86, 261905 (2005). 5. W. J. Hsueh, C. H. Chang, Y. H. Cheng, and S. J. Wun, Opt. Express 20, 26618 (2012). 6. M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, Nature 404, 740 (2000). 7. M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008). 8. Y. H. Cheng and W. J. Hsueh, Opt. Lett. 38, 3631 (2013). 9. M. Werchner, M. Schafer, M. Kira, S. W. Koch, J. Sweet, J. D. Olitzky, J. Hendrickson, B. C. Richards, G. Khitrova, H. M. Gibbs, A. N. Poddubny, E. L. Ivchenko, M. Voronov, and M. Wegener, Opt. Express 17, 6813 (2009). 10. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, Phys. Rev. B 80, 115314 (2009). 11. J. P. Prineas, W. J. Johnston, M. Yildirim, J. Zhao, and A. L. Smirl, Appl. Phys. Lett. 89, 241106 (2006). 12. T. Ikawa and K. Cho, Phys. Rev. B 66, 085338 (2002). 13. M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, Phys. Rev. Lett. 83, 2841 (1999). 14. S. M. Sadeghi, W. Li, X. Li, and W.-P. Huang, Phys. Rev. B 74, 161304(R) (2006). 15. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, Opt. Lett. 30, 2790 (2005). 16. Y. Merle d’Aubigne, A. Wasiela, H. Mariette, and T. Dietl, Phys. Rev. B 54, 14003 (1996). 17. J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, Phys. Rev. B 61, 13863 (2000). 18. J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, A. N. Poddubny, E. L. Ivchenko, M. Wegener, and H. M. Gibbs, Opt. Express 16, 15382 (2008). 19. C. H. Chang, C. H. Chen, and W. J. Hsueh, Opt. Express 21, 14656 (2013). 20. V. Passias, N. V. Valappil, Z. Shi, L. Deych, A. A. Lisyansky, and V. M. Menon, Opt. Express 17, 6636 (2009). 21. E. L. Albuquerque and M. G. Cottam, Phys. Rep. 376, 225 (2003). 22. L. I. Deych, M. V. Erementchouk, A. A. Lisyansky, E. L. Ivchenko, and M. M. Voronov, Phys. Rev. B 76, 075350 (2007). 23. G. Malpuech, A. Kavokin, and G. Panzarini, Phys. Rev. B 60, 16788 (1999).
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Exciton photoluminescence in resonant quasi-periodic Thue–Morse quantum wells W. J. Hsueh,* C. H. Chang, and C. T. Lin Photonics Group, Department of Engineering Science and Ocean Engineering, National Taiwan University, 1, Sec. 4, Roosevelt Road, Taipei 10660, Taiwan *Corresponding author: [email protected] Received November 15, 2013; accepted December 15, 2013; posted December 20, 2013 (Doc. ID 201464); published January 21, 2014 This Letter investigates exciton photoluminescence (PL) in resonant quasi-periodic Thue–Morse quantum wells (QWs). The results show that the PL properties of quasi-periodic Thue–Morse QWs are quite different from those of resonant Fibonacci QWs. The maximum and minimum PL intensities occur under the anti-Bragg and Bragg conditions, respectively. The maxima of the PL intensity gradually decline when the filling factor increases from 0.25 to 0.5. Accordingly, the squared electric field at the QWs decreases as the Thue–Morse QW deviates from the antiBragg condition. © 2014 Optical Society of America OCIS codes: (230.5590) Quantum-well, -wire and -dot devices; (250.5230) Photoluminescence; (160.5293) Photonic bandgap materials. http://dx.doi.org/10.1364/OL.39.000489
Since quasi-crystals were first proposed, quasi-crystalline materials have attracted a great deal of interest because of their distinctive physical properties and their applications in various fields of research [1–8]. In recent years, quasi-crystals have been used in one-dimensional resonant photonic crystals based on multiple quantum wells (QWs). Numerous studies have focused on the properties of light waves propagating in multiple quantum well structures with different quasi-periodic arrangements [9,10]. Prior studies have indicated that the optical properties of quasi-periodic structures are different from those of periodic structures. For resonant periodic QWs, there is a wide bandgap and large reflectance under the Bragg condition, where the period, d, is equal to the half-wavelength at the exciton resonance frequency, ω0 . These structures provide an advantage in the design of various optical devices and slow light applications [11–15]. However, in terms of light emission, a periodic QW possesses a better light-emitting ability under the anti-Bragg condition of d λω0 ∕4 than under the Bragg condition, because of its greater absorbance and stronger photoluminescence (PL) [13,16,17]. For quasi-periodic QWs, relevant studies have shown that the wide bandgap and large reflectance around the exciton resonance frequency are replaced by a striking structural dip in the reflection spectra [10]. This unique characteristic results in absorbance and PL emission near the exciton resonance in a quasi-periodic QW that are very different from those in a periodic QW under the Bragg condition [18]. The difference between the properties of PL intensity in periodic QWs and those in resonant Fibonacci quantum wells (FQWs) has been recently studied [19,20]. The results show that the PL intensity in a FQW is significantly larger than that in a periodic QW under the anti-Bragg condition in certain cases because of the enhanced field at the QWs. For quasi-crystalline systems, the Thue–Morse sequence is another representative structure besides the Fibonacci one [21]. Whether the PL characteristics of resonant quasi-periodic Thue–Morse quantum wells (TMQWs) differ from those of a FQW is of interest. However, 0146-9592/14/030489-04$15.00/0
few studies have determined the properties of PL intensity in a TMQW. If the luminescence characteristics of a TMQW can be clarified, this structure will be useful in the design of light-emitting devices. This Letter considers a TMQW that consists of two different intervals, A and B, which obeys the recurrence rule: A → AB and B → BA [8,21]. The generation order of the system, v, indicates the different structures, such as A, AB, ABBA, and ABBABAAB, for v 1, 2, 3, and 4, respectively. There is a QW in each of the intervals, so the number of QWs in the TMQW is (N 2v−1 ). In this study, the barrier thicknesses of the structures are assumed to be thick enough to avoid coupling between excitons in the QWs, so the excitons only interact with the electromagnetic field. The dielectric contrast between QWs and barriers also is neglected. The TMQWs are placed between a finite forward barrier and a semiinfinite rear barrier. The cap thickness between the vacuum interface and the center of the first QW is set to dC λω0 ∕2. The interaction between the electromagnetic wave and the excitons in the QW structures is described by Maxwell’s equation: 2 ω ⇀ ∇×∇×E D; c ⇀
⇀
⇀
⇀
(1)
where D εzE 4πP exc is the displacement vector and the growth⇀ direction of the structures is set to the coordinate, z. P exc is the contribution of the exciton to the dielectric polarization of the entire QW structure, where a resonant source term is introduced to describe the luminescence properties [14,22,23]. In this study, the s-polarized wave and the 1s excitons are considered and the local nonresonant permittivity function, εz, is a constant that is equal to the background refractive index squared, n2b . Based on the solution of Eq. (1), the transmittance, reflectance, and absorbance are obtained using the total transfer matrix through the entire structure. Radiation boundary conditions are also imposed to determine the radiation emitted from the QW structures. For © 2014 Optical Society of America
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these conditions, only outward-bound waves exist outside the structures [19,22]. Thus, the intensity of luminescence radiated to the left side of the structure is given by: I PL ω; k∥ 4
N X j1
j κS ω; k∥ 2 jI j ω; k∥ j2 ; φj 1
Ξj
(2)
where the spectral density Ξj is related to the correlation function for the Fourier-transformed source, κ2 ωnb ∕c2 − k2∥ , k∥ is the in-plane wave vector, and S j ω; k∥ is the function relevant to the exciton susceptibility. The summation indicates that the radiation emitted by each source located at the jth QW adds in a noncoherent manner. In this study, each QW is considered to be identical and the waves are emitted along the growth direction, namely k∥ 0. Therefore, Eq. (2) can be reduced to I PL ω 4ΞjqSω∕φ1 j2
N X
jI j ωj2 ;
j1
where q ωnb ∕c, I j ω −th−jM R
j1 Y mN
Mm
sj1 ; sj2
and t is the transmission coefficient of the entire structure. We first investigated the influence of higher generation orders and larger numbers of QWs on the transmission, reflection and PL spectra in the TMQWs for the same spacing thicknesses, dA 0.412λω0 and dB 0.588λω0 . The numbers of QWs in the TMQWs with v 6, 7, and 8 are N 32, 64, and 128, respectively. Figure 1(a) shows that the transmittance in the middle and at both sides of the spectra remains very small and becomes larger, respectively, as the generation order increases. In addition, the frequency range for low transmittance increases as the generation order increases. As shown in Fig. 1(b), there are notable structural dips near the exciton resonance frequency in the reflection spectra. When the generation order increases, the frequency
Fig. 1. (a) Transmission, (b) reflection, and (c) PL spectra in TMQWs with v 6, 7, and 8 for the same spacing thicknesses. The parameter values of the system are ℏω0 1.523 eV, ℏΓ0 25 μeV, ℏΓ 180 μeV, and nb 3.59. The normalized frequency, Ω, is defined as Ω ω − ω0 D∕2πc, where D dA dB λω0 .
ranges that corresponds to these dips and the linewidths and the maxima of the reflection spectra all become greater. However, there are only small changes in the minimal reflectance in the dips. The reflectance at both sides of the spectra also becomes smaller as the generation order increases. Figures 1(a) and 1(b) show there is significant absorption near the exciton resonance frequency, which corresponds to the areas of low transmittance in the transmission spectra and the dips in the reflection spectra. This results in the PL profiles shown in Fig. 1(c) because of the close relationship between the absorbance and the PL intensity [22]. Specifically, the linewidths of the PL spectra increase when there is an increase in the frequency ranges for the dips in the reflection spectra. Although the number of QWs increases significantly, the maxima of the PL intensity are still weak as the generation order increases, because the minimal reflectance in the dips remains at almost the same magnitude. Next, the influence of the filling factor of the system, defined as F dA ∕dA dB , on the transmission, reflection, and PL spectra in the TMQW is illustrated in Fig. 2. Here, there is a significant change in these spectra as the filling factor increases from 0.25 to 0.5, namely from the anti-Bragg to the Bragg conditions. In fact, the spacing thicknesses at F 0.25 are dA λω0 ∕4 and dB 3λω0 ∕4, which is equivalent to the anti-Bragg condition, because they have the same spectral characteristics [16,17,19]. As shown in Fig. 2(a), the frequency range for low transmittance becomes larger as the filling factor increases. In particular, the frequency ranges for F 0.45 and 0.5 are much broader than those for F 0.25 and 0.35. The transmittance at both sides of the spectra also becomes larger when the filling factor is greater. In contrast to the transmission spectra, the variations in the reflection spectra are more complicated, as seen in Fig. 2(b). In general, the magnitude and linewidths of the reflection spectra become greater as the filling factor increases. The reflectance at both sides of the spectra decreases as the filling factor increases. However, the reflection profiles for the respective filling factors are very different from each other. Specifically, there is only one small peak at the exciton resonance frequency in the reflection profile for F 0.25 [17]. When the filling factor
Fig. 2. (a) Transmission, (b) reflection, and (c) PL spectra in a TMQW with v 8 for different filling factors. The number of QWs in the TMQW is N 128. The other parameter values are the same as those in the caption of Fig. 1.
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reaches 0.35 and 0.45, significant dips appear in the middle of the reflection spectra. The frequency ranges that correspond to the dip and the minimal reflectance in the dip for F 0.35 are wider and smaller than those for F 0.45, respectively. At F 0.5, there is the wide linewidth and large reflectance around the exciton resonance frequency in the reflection profile due to the Bragg condition [12]. Based on these transmission and reflection spectra, the PL characteristics of the TMQW for different filling factors are shown in Fig. 2(c). For F 0.25, the PL intensity is strongest because the greatest absorption occurs under the anti-Bragg condition. There also are two peaks that are symmetrical with respect to the exciton resonance frequency in the PL profile, which can be mainly attributed to the corresponding reflection spectrum, in which there is only one small peak. When the filling factor increases, the magnitude and linewidths of the PL spectra both decrease. When the filling factor is equal to 0.35 and 0.45, the dips in the corresponding reflection spectra cause greater absorption and stronger PL in the middle of the PL spectra. The dip in the PL spectrum for F 0.35 also can be attributed to the small peak in the dip of the reflection profile. However, the PL intensity for F 0.5 is very weak because of the wide linewidth and the large reflectance in the reflection spectrum [13]. As a result, Figs. 1 and 2 show that the PL spectra near the exciton resonance frequency are mainly dependent on the reflection spectra because of the low transmittance in the transmission profiles and the structural features in the reflection profiles. To compare the overall influence of the filling factor on the PL intensity in TMQWs with that in FQWs, the maximum values of PL intensity, defined as PLmax , in the TMQWs are plotted versus the filling factor in Fig. 3. The profiles of PLmax are only shown within this range because there are symmetry points at F 0.25, 0.5 and 0.75. Here, the number of QWs, N 34, in the FQW with a generation order of 8 is chosen to correspond to the number of QWs, N 32, in the TMQW with v 6 for comparison purposes. The values of PLmax at F 0.25 and 0.5 in the TMQW with v 6 are almost the same as those in the FQW with a generation order of 8 due to the anti-Bragg and Bragg conditions. There
Fig. 3. PLmax , defined by the maximum values of the PL intensity, in the TMQWs versus the filling factor, F. The numerical calculations are performed for TMQWs with v 4 to 8, which corresponds to the numbers of QWs, N 8 to 128. The A and B signs denote the positions of the anti-Bragg and Bragg conditions, respectively.
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are only slight differences between these values because the numbers of QWs are slightly different. In addition, the values of PLmax at F 0.25 and 0.5 are greater and smaller, respectively, when the generation order increases. As the TMQWs deviate from the antiBragg condition, all of the values of PLmax decline smoothly and gradually to their minima under the Bragg condition. There also is a sharper decline in the value of PLmax in the TMQW as the generation order increases. When the filling factor approaches 0.5, the decrease in all of the values of PLmax is moderated. In contrast, the values of PLmax in the TMQWs are very different from those in the FQWs when the filling factor increases from 0.25 to 0.5. There is a marked rise in the values of PLmax in the FQWs when the filling factor exceeds 0.25. After reaching their maxima, the values of PLmax drop steeply to their minima [19]. Figure 4 shows a comparison between the squared electric fields in the TMQW with v 6 and the FQW with a generation order of 8 for the same filling factors, F 0.25 and 0.412, to determine why the PL properties of these structures are different from each other. For clear comparison, the filling factor, F 0.412, was chosen to be the main comparison parameter in this study, since the maximum of PLmax in the FQW with a generation order of 8 occurs for this filling factor [19]. Here, the frequency where the PL intensity in the structure is a maximum is defined as ωPLmax . Figures 4(a) and 4(b) show the field profiles in the TMQW at F 0.25 and 0.412, respectively. At F 0.25, namely under the anti-Bragg condition, the peaks of the field are situated at the QWs although these peak values, marked by the red circles, are not very large. There also is a smooth and gradual decline in the peak values as the distance increases. However, at F 0.412, there is a decrease in the field values at the QWs although the peaks of the field increase because there is a deviation from the anti-Bragg condition. Many of the nodes of the field are even situated at the QWs. As shown in Figs. 4(c) and 4(a), the field profiles in the FQW and TMQW at F 0.25 are very similar because of
Fig. 4. Squared electric fields in a TMQW with v 6 at (a) F 0.25 and (b) F 0.412 and in a FQW with a generation order of 8 at (c) F 0.25 and (d) F 0.412. The number of QWs in the TMQW is N 32. The corresponding number of QWs in the FQW is N 34. The numerical simulations are made at ωPLmax . The horizontal lines, I I , I R , and I T , denote the incident, reflected, and transmitted waves, respectively. The vacuum region (the white area) is divided from the structure (the yellow area) by the black line. The blue lines represent the QW locations. The squared electric fields are given by jEωPLmax j2 for z > 0, multiplied by the background refractive index for clarity. The red circles denote the field values at the vacuum interface and the QWs.
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the anti-Bragg condition [17,19]. However, when there is a deviation from the anti-Bragg condition, there is a great difference between the field profiles in the FQW and TMQW at F 0.412, as seen in Figs. 4(d) and 4(b). In contrast to the TMQW, both the peaks of the field and the field values at the QWs increase in the FQW. A number of the peaks of the field are situated at the QWs. To the authors’ best knowledge, if the field at the QWs is larger, the PL intensity is stronger because there is the enhanced light–matter coupling at the QWs [18]. Therefore, the PL intensity in the TMQW is weaker than that in the FQW for the same filling factor because there is the reduced light–matter coupling. The strongest PL intensity in the TMQW occurs under the anti-Bragg condition. In conclusion, we have investigated exciton PL in resonant quasi-periodic TMQWs. The transmission and reflection spectra show that the PL spectra near the exciton resonance frequency primarily depend on the reflection spectra due to the low transmittance in the transmission profiles and the structural features in the reflection profiles. We also discovered that the maximum and minimum PL intensities in the TMQW occur under the anti-Bragg and Bragg conditions, respectively. There is a smooth and gradual decrease in the values of PLmax when the filling factor increases from 0.25 to 0.5. As the generation order increases, the value of PLmax declines more sharply. These results for TMQWs are quite different from those for FQWs, in which the values of PLmax continue to rise when the filling factor exceeds 0.25. Moreover, the field values at the QWs in a TMQW are smaller than those in a FQW for the same filling factor. Therefore, the PL intensity in a TMQW is weaker than that in a FQW because there is reduced light–matter coupling. In addition, PL in the TMQWs can reach the strongest PL in resonant periodic QWs. Thus, the TMQWs can be effectively applied in optoelectronic devices. The authors acknowledge the support in part by the National Science Council of Taiwan under grant number NSC 102-221-E-002-105. References 1. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
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