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Twin extra-high photoluminescence in resonant double-period quantum wells C. H. Chang, Y. H. Cheng, and W. J. Hsueh* Photonics Group, Department of Engineering Science and Ocean Engineering, National Taiwan University, 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan *Corresponding author: [email protected] Received August 13, 2014; revised October 9, 2014; accepted October 17, 2014; posted October 20, 2014 (Doc. ID 220862); published November 17, 2014 Twin extra high photoluminescence (PL) in resonant quasi-periodic double-period quantum wells (DPQWs) for higher-generation orders is demonstrated. In the DPQW, the number of maxima in the maximum values of the PL intensity is two, which is different from other quasi-periodic quantum wells (QWs) and traditional periodic QWs. The maximum PL intensity in a DPQW is also stronger than that in a periodic QW under the anti-Bragg condition and that in a Fibonacci QW. Although the peaks of the squared electric field for the twin PL are both located near the QWs, their field profiles are distinct. © 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.39.006581
Resonant photonic crystals have received a great deal of interest because of their optical properties and their potential application in various optical devices [1–5]. Because of their advantages in terms of fabrication and cost, one-dimensional resonant photonic crystals based on multiple quantum wells (QWs) with a periodic arrangement, namely resonant periodic quantum wells (PQWs), have attracted much attention with regard to their optical properties during the past decades [6–8]. There are two specific conditions, the Bragg and antiBragg conditions, which determine the completely distinctive optical properties in PQWs. Under the Bragg condition, d λω0 ∕2 at the exciton resonance frequency, ω0 , there is a broad bandgap and a large reflectance in the reflection spectra. These unique characteristics of the condition have been used in various studies [9–12]. However, a great increase in absorbance and photoluminescence (PL) occurs under the anti-Bragg condition, where d λω0 ∕4. Therefore, a PQW under the anti-Bragg condition is much more suitable for application in light emitters than a PQW under the Bragg condition [13–15]. Since quasi-crystals were first proposed, quasi-crystalline materials have attracted a lot of attention and interest in various fields of research due to their unique physical properties [16–21]. Recently, multiple QW structures with quasi-periodic arrangements have been studied, both theoretically and experimentally. In previous studies, the optical properties of quasi-periodic QWs have been investigated and have been compared with those of periodic ones [22–25]. In particular, in terms of light emission, the results have shown that the PL intensity in a Fibonacci quantum well (FQW) is significantly stronger than that in a PQW, not only under the Bragg condition, but also under the anti-Bragg condition [23,26]. It has also been found that the properties of PL intensity in a Thue–Morse quantum well (TMQW) are similar to those in a PQW [5]. Therefore, the PL properties of a TMQW and a FQW are quite different, even though they are both quasi-periodic QWs. In addition to the Fibonacci and Thue–Morse sequence, the doubleperiod sequence is one of the most representative 0146-9592/14/236581-04$15.00/0
structures of quasi-periodic systems [19]. It is of interest to investigate whether the PL characteristics in resonant double-period quantum wells (DPQWs) are different from those in other quasi-periodic QWs or those in PQWs. However, few studies have so far verified the luminescence properties of a DPQW. If the PL emission from the DPQW is stronger than that from a PQW under the anti-Bragg condition or even that from a FQW, this structure has great potential to serve as an excellent light emitter. A DPQW with two different spacings, A and B, and which obeys a double-period sequence: A → AB and B → AA, is studied [19]. For example, the generation orders, v 1, 2, 3, and 4, indicate the different structures, A, AB, ABAA, and ABAAABAB. The number of QWs in the DPQW is equal to N 2v−1 because each of the spacings contains a QW. The DPQWs are placed between a finite forward barrier and a semi-infinite rear barrier. The cladding thickness, dC , between the vacuum interface and the center of the first QW is equal to λω0 ∕2. It is assumed that there is no dielectric contrast between the QWs and the barriers. It is also assumed that the barrier thicknesses are sufficient to prevent interaction between excitons in the QWs, so the excitons only couple with the electromagnetic field. Each QW is also assumed to be identical. The intensity of the luminescence emitted from the DPQW depends on the incident angle, the contribution of the exciton to the light polarization, and the generation order and the spacing thickness of the structure. This is calculated based on the introduced resonant source term, the radiation boundary condition, and the following two assumptions: one, that there is no correlation between the source functions in the QWs and two, that the distribution of the direction of the noncoherent exciton polarization is isotropic in the plane of the structure [5,7,23]. This study only considers the PL spectra radiated to the left side of the DPQWs and along the growth direction, which are described as the summation of radiation created by each source in P a noncoherent 2 j manner, expressed as Iω 4ΞjqS∕φ1 j2 N j1 jℑ ωj . The spectral density Ξ is relevant to the Fourier transformed source correlation, q is the light wave vector, © 2014 Optical Society of America
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Fig. 1. Maximum values of the PL intensity, defined as PLmax , in the DPQWs versus the filling factor, F, for increasing the generation order, v. A and B mark the positions of the anti-Bragg and Bragg conditions, respectively. The gray squares and green diamonds indicate the maximum positions. The maximum positions in the DPQWs where v 6, 7, and 8 are specially marked by C, D, E, F, G, and H. I and J mark the hollow positions in the DPQWs where v 7 and 8, respectively. The parameter values for the system are as follows: ℏω0 1.523 eV, ℏΓ0 25 μeV, ℏΓ 180 μeV, and nb 3.59.
S is related to the exciton susceptibility, φ1 is the projection of the exciton wave function on the even solution, u1 , of d2 Ez∕dz2 q2 Ez 0, and ℑj ω is a Green’s function depicting the field radiated by each source [5,7]. The transmission, reflection, and absorption spectra of the structures are obtained by the transfer matrix method [7,12,23]. Initially, the overall effect of the thickness filling factor of the system, defined by F dA ∕D, where D dA dB λω0 , on the maximum values of the PL intensity in the DPQWs is considered for increasing generation orders, as shown in Fig. 1. The maximum values of the PL intensity are defined as PLmax . Only the curves of PLmax within this range are shown, due to the symmetry points at F 0.25, 0.5, and 0.75. It is seen that the values of PLmax at F 0.25 and 0.5 under the anti-Bragg and Bragg conditions become, respectively, greater and smaller when the number of QWs increases [14,23]. Because of the anti-Bragg and Bragg conditions, the values of PLmax at F 0.25 and 0.5 in the DPQW are equal to those in a TMQW with the same number of QWs [5]. As the filling factor increases from 0.25 to about 0.4, the values of PLmax in the DPQWs with v 4 and 5 decrease slightly and then increase. Subsequently, the values of PLmax increase sharply until they reach their own minimum values. Therefore, there is only one maximum, around F 0.4, in the values of PLmax for lower generation orders. However, there are two maxima, near F 0.4 and 0.45, for higher-generation orders. When the generation order is greater than 6, the first maximum occurs around F 0.4, and a second maximum emerges near F 0.45. These two maxima both increase as the generation order increases. In comparison, the profiles of PLmax in the DPQW are different from those in the FQW, the TMQW, and the PQW [5,23]. Strictly speaking, the profiles of PLmax in the DPQW are more similar to those in the FQW, since the PL intensity is significantly enhanced in both of these structures. The main difference between them is that there are two maxima in the values of PLmax in the DPQW and one in the FQW for higher-generation orders. In order to give a clear comparison between the PL characteristics of the different QW structures, the maximum PLmax and the filling factors that correspond to the maximum PLmax , defined as F max , in the DPQWs, the
Fig. 2. (a) The maximum PLmax and (b) F max , defined by the filling factors corresponding to the maximum PLmax , in the QW structures versus the number of QWs, N. The values for the DPQWs in (a) and (b) correspond to the gray squares and green diamonds in Fig. 1.
FQWs, the PQWs, and the TMQWs, are plotted as a function of the number of QWs in Fig. 2. It is clearly seen that the PL properties of the DPQW are more similar to those of the FQW than to those of the PQW and the TMQW. Figure 2(a) shows that the maximum PLmax in the DPQW and the FQW sharply increases as the number of QWs increases. However, there is only a gradual increase in the maximum PLmax in the PQW and the TMQW [5,23]. Figure 2(b) shows that there is a variation in the value of F max in the DPQW and the FQW when the number of QWs increases. In contrast, the value of F max in the PQW and the TMQW remains at 0.25, the position of the antiBragg condition, regardless of any increase in the number of QWs. However, there are some major differences in the PL characteristics of the DPQW and the FQW, even though they are similar. Two values for the maximum PLmax occur near F 0.4 and 0.45 in the DPQW when the number of QWs exceeds 32, as shown in Figs. 2(a) and 2(b). The second value of the maximum PLmax increases more rapidly than the first. The former eventually exceeds the latter when the number of QWs reaches 128. However, there is only one value for the maximum PLmax in the FQW. The values of F max in the FQW are between those in the DPQW when the number of QWs is greater than 32. It is also seen that the two values of the maximum PLmax in the DPQW with N 128 are both greater than that in the FQW with N 144. Therefore, the PL intensity in the DPQW is stronger than that in the FQW for a relatively smaller number of QWs. The effect of the filling factor on the PL, transmission, and reflection spectra in the DPQW is illustrated in Fig. 3. The evolution of these spectra with the filling factor is clearly shown. Figure 3(a) shows that the PL profiles in the DPQW where F 0.25 and 0.5 are very distinct from each other because they pertain to the anti-Bragg and Bragg conditions, respectively [13,15,23]. There are two peaks that are symmetrical, with respect to the exciton resonance frequency in the PL spectra for F 0.25. In contrast, the PL intensity for F 0.5 is very weak throughout the frequency range, although the PL profile is also symmetrical. For F 0.5, the approximate expression for the PL intensity can be expressed as Iω ≈ NΞq2 u22 jS∕φ1 j2 jtj2 , where u2 is the odd solution and t is the transmission coefficient [5,7]. This expression shows that the PL of the Bragg QW structure is restrained by the low transmittance. However, when the DPQW deviates from the anti-Bragg and the Bragg conditions, the PL spectra become quite sharp and asymmetrical, again with respect to the exciton resonance frequency. On the left side of the exciton resonance frequency, the two
December 1, 2014 / Vol. 39, No. 23 / OPTICS LETTERS
Fig. 3. (a) PL, (b) transmission, and (c) reflection spectra in a DPQW with v 8 and N 128 for different filling factors. A partial close-up of (a) is shown in the inset in (a). The normalized frequency is defined as Ω ω − ω0 D∕2πc.
great peaks, respectively, correspond to F 0.415 and 0.463, which are the values of F max denoted by G and H in Fig. 1. It is clearly seen that the peak value for F 0.463 is larger than that for F 0.415. There are also two small peaks on the left side of the frequency for F 0.434, at the hollow position marked by J in Fig. 1. These two small peaks for F 0.434 are higher than those for F 0.25, even though the larger value corresponds to the hollow position shown in Fig. 1. In addition, on the right side of the frequency, there is a gradual decrease in the peak values as the filling factor increases from 0.415 to 0.463. Because of the close relationship between the absorbance and the PL intensity, the transmission and reflection spectra are related to the formation of the PL spectra [5,7]. Figures 3(b) and 3(c) show that the frequency range for low transmittance and the linewidths of the reflection spectra for F 0.5 are much greater than those for the other filling factors because of the Bragg condition. It is also observed that the variation near the exciton resonance frequency in the reflection spectra is much more complicated than that in the transmission spectra. The transmittance in the middle of the spectra remains very small as the filling factor increases. In contrast, some significant dips and peaks appear in the reflection spectra, which lead to the peaks and dips in the PL spectra. In particular, the dip on the left side of the exciton resonance frequency in the reflection spectra for F 0.463 is deeper than that for F 0.415, which means that the maximum PL intensity of the former is stronger than that of the latter. Although the dip for F 0.425 is also deeper than that for F 0.463, it is not reflected in the PL intensity because there is greater transmittance. The PL, transmission, and reflection spectra corresponding to the two maxima in the values of PLmax in the DPQWs for higher-generation orders are of interest. As shown in Fig. 4(a), in the PL spectra for the first maxima, there are two peaks that are sharp and asymmetrical with respect to each other. The peak value and the corresponding frequency become, respectively, greater and farther from the exciton resonance frequency as the generation order increases. Figure 4(b) clearly shows that the PL spectra are primarily dependent on the reflection spectra. It is seen that the frequency range for low transmittance becomes larger in the transmission spectra as the generation order increases, but the transmittance near the exciton resonance frequency remains very small. In contrast, the peaks located at almost the exciton resonance frequency
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Fig. 4. (a) PL and (b) transmission and reflection spectra in DPQWs with v 6, 7, and 8 for the respective values of F max , marked by C, E, and G, the first maxima in Fig. 1. (c) PL and (d) transmission and reflection spectra in the DPQWs with v 6, 7, and 8 for the respective values of F max , marked by D, F, and H, the second maxima in Fig. 1. The numbers of QWs in the DPQWs are N 32, 64, and 128. The insets in (a) and (c) show a partial close-up of (a) and (c), respectively.
and the dips on the left side of the frequency in the reflection spectra result in the corresponding dips and the peaks that occur in the PL spectra. Compared to the PL spectra for the first maxima, there is a greater difference between the values of the two peaks in the PL spectra for the second maxima, as shown in Fig. 4(c). The variation becomes more irregular in the PL profiles for the second maximum as the generation order increases. As in Fig. 4(b), it is also observed in Fig. 4(d) that the PL profiles are mainly influenced by the reflection profiles. The peaks located close to the exciton resonance frequency in the reflection spectra for the second maximum are higher than those for the first maximum. This results in the smaller dip values that are almost at the frequency in the PL spectra for the former, as seen in Figs. 4(a) and 4(c). Finally, a comparison of the squared electric fields in the DPQWs for different filling factors and the PQW under the anti-Bragg condition for the same number of QWs is shown in Fig. 5. This study refers to the frequency where the PL intensity in the structure is maximal as ωPL max . Figures 5(a) and 5(b) show that the field profiles in the DPQWs for the two maxima, indicated by E and F in Fig. 1, are very distinct from each other. It is seen that there are two points near z∕λω0 6.5 and 21, K and L, that divide the field profile for F 0.408 into three groups in which the respective average
Fig. 5. Squared electric fields in the DPQWs for (a) F 0.408, (b) F 0.461, (c) F 0.428 for v 7 and N 64, and (d) the PQW with F 0.25 for N 64. The numerical calculations are performed by ωPL max . The black line separates the vacuum region from the structure. The blue lines denote the QW positions. The incident, reflected, and transmitted waves are, respectively, denoted by the horizontal lines, I I , I R , and I T . The squared electric fields are equal to jEωPL max j2 for z > 0, multiplied by the background refractive index for clarity. The field values at the vacuum interface and the QWs are marked by red circles.
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field values are very different. However, there is a great difference in the field profile for F 0.461, which is more similar to that in the FQW, since many of the peaks in the field are situated very near the QWs [23]. In comparison, the field profile for F 0.428, which is the hollow position marked by I in Fig. 1, is similar to that for F 0.408, as seen in Fig. 5(c). It is observed that the field profile for F 0.428 is also divided into three groups by the two points near z∕λω0 4.5 and 20, M and N, although there is a decrease in both the peaks of the field and the field values at the QWs. Although the field profiles in the DPQWs are distinct, their PL intensities are all stronger than that in the PQW under the anti-Bragg condition, where the peaks of the field are situated at the QWs, but these peak values are not great, as shown in Fig. 5(d). This can be attributed to the enhanced lightmatter coupling because of the larger field at the QWs in the DPQWs [5,23,26]. This Letter demonstrates twin extra-high PL in resonant quasi-periodic DPQWs. Two maxima occur in the values of PLmax for higher-generation orders. It is found that the second maximum begins to emerge near F 0.45 next to the first maximum, which is around F 0.4, when the generation order exceeds 6. There is a dramatic rise in the second maximum, which eventually becomes greater than the first maximum, when the generation order reaches 8. The reflection and transmission spectra also show that the PL spectra for the second maxima are more asymmetrical than those for the first maxima, because the PL profiles are significantly dependent on the reflection profiles. These results for DPQWs are more similar to those for FQWs than to those for TMQWs and PQWs. The maximum PL intensity in a DPQW is even stronger than that in the PQW under the anti-Bragg condition and that in a FQW. The field profiles for the two maxima are also very different, although the field values at the QWs for both are greater than those for the PQW under the Bragg or anti-Bragg conditions. The authors acknowledge the support in the form of partial funding by the National Science Council of Taiwan under grant nos. MOST 103-3113-E-002-001 and MOST 103-2221-E-002-118. References 1. D. Goldberg, L. I. Deych, A. A. Lisyansky, Z. Shi, V. M. Menon, V. Tokranov, M. Yakimov, and S. Oktyabrsky, Nat. Photonics 3, 662 (2009).
2. R. A. Vlasov and A. M. Lemeza, Phys. Rev. A 84, 023828 (2011). 3. I. V. Mel’nikov and J. S. Aitchison, Appl. Phys. Lett. 87, 201111 (2005). 4. T. Iida and H. Ishihara, Phys. Rev. Lett. 90, 057403 (2003). 5. W. J. Hsueh, C. H. Chang, and C. T. Lin, Opt. Lett. 39, 489 (2014). 6. L. C. Andreani, G. Panzarini, A. V. Kavokin, and M. R. Vladimirova, Phys. Rev. B 57, 4670 (1998). 7. L. I. Deych, M. V. Erementchouk, A. A. Lisyansky, E. L. Ivchenko, and M. M. Voronov, Phys. Rev. B 76, 075350 (2007). 8. M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, Phys. Rev. Lett. 76, 4199 (1996). 9. J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, Appl. Phys. Lett. 81, 4332 (2002). 10. L. Pilozzi, A. D’Andrea, and K. Cho, Phys. Rev. B 69, 205311 (2004). 11. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, Opt. Lett. 30, 2790 (2005). 12. S. M. Sadeghi, W. Li, X. Li, and W.-P. Huang, Phys. Rev. B 74, 161304(R) (2006). 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. Y. M. d’Aubigne, A. Wasiela, H. Mariette, and T. Dietl, Phys. Rev. B 54, 14003 (1996). 15. J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, Phys. Rev. B 61, 13863 (2000). 16. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984). 17. L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and L. C. Kimerling, Appl. Phys. Lett. 86, 261905 (2005). 18. W. J. Hsueh, C. T. Chen, and C. H. Chen, Phys. Rev. A 78, 013836 (2008). 19. E. L. Albuquerque and M. G. Cottam, Phys. Rep. 376, 225 (2003). 20. Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013). 21. W. J. Hsueh, C. H. Chang, Y. H. Cheng, and S. J. Wun, Opt. Express 20, 26618 (2012). 22. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, Phys. Rev. B 80, 115314 (2009). 23. C. H. Chang, C. H. Chen, and W. J. Hsueh, Opt. Express 21, 14656 (2013). 24. A. V. Poshakinskiy, A. N. Poddubny, and S. A. Tarasenko, Phys. Rev. B 86, 205304 (2012). 25. V. Passias, N. V. Valappil, Z. Shi, L. Deych, A. A. Lisyansky, and V. M. Menon, Opt. Express 17, 6636 (2009). 26. 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).
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Twin extra-high photoluminescence in resonant double-period quantum wells C. H. Chang, Y. H. Cheng, and W. J. Hsueh* Photonics Group, Department of Engineering Science and Ocean Engineering, National Taiwan University, 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan *Corresponding author: [email protected] Received August 13, 2014; revised October 9, 2014; accepted October 17, 2014; posted October 20, 2014 (Doc. ID 220862); published November 17, 2014 Twin extra high photoluminescence (PL) in resonant quasi-periodic double-period quantum wells (DPQWs) for higher-generation orders is demonstrated. In the DPQW, the number of maxima in the maximum values of the PL intensity is two, which is different from other quasi-periodic quantum wells (QWs) and traditional periodic QWs. The maximum PL intensity in a DPQW is also stronger than that in a periodic QW under the anti-Bragg condition and that in a Fibonacci QW. Although the peaks of the squared electric field for the twin PL are both located near the QWs, their field profiles are distinct. © 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.39.006581
Resonant photonic crystals have received a great deal of interest because of their optical properties and their potential application in various optical devices [1–5]. Because of their advantages in terms of fabrication and cost, one-dimensional resonant photonic crystals based on multiple quantum wells (QWs) with a periodic arrangement, namely resonant periodic quantum wells (PQWs), have attracted much attention with regard to their optical properties during the past decades [6–8]. There are two specific conditions, the Bragg and antiBragg conditions, which determine the completely distinctive optical properties in PQWs. Under the Bragg condition, d λω0 ∕2 at the exciton resonance frequency, ω0 , there is a broad bandgap and a large reflectance in the reflection spectra. These unique characteristics of the condition have been used in various studies [9–12]. However, a great increase in absorbance and photoluminescence (PL) occurs under the anti-Bragg condition, where d λω0 ∕4. Therefore, a PQW under the anti-Bragg condition is much more suitable for application in light emitters than a PQW under the Bragg condition [13–15]. Since quasi-crystals were first proposed, quasi-crystalline materials have attracted a lot of attention and interest in various fields of research due to their unique physical properties [16–21]. Recently, multiple QW structures with quasi-periodic arrangements have been studied, both theoretically and experimentally. In previous studies, the optical properties of quasi-periodic QWs have been investigated and have been compared with those of periodic ones [22–25]. In particular, in terms of light emission, the results have shown that the PL intensity in a Fibonacci quantum well (FQW) is significantly stronger than that in a PQW, not only under the Bragg condition, but also under the anti-Bragg condition [23,26]. It has also been found that the properties of PL intensity in a Thue–Morse quantum well (TMQW) are similar to those in a PQW [5]. Therefore, the PL properties of a TMQW and a FQW are quite different, even though they are both quasi-periodic QWs. In addition to the Fibonacci and Thue–Morse sequence, the doubleperiod sequence is one of the most representative 0146-9592/14/236581-04$15.00/0
structures of quasi-periodic systems [19]. It is of interest to investigate whether the PL characteristics in resonant double-period quantum wells (DPQWs) are different from those in other quasi-periodic QWs or those in PQWs. However, few studies have so far verified the luminescence properties of a DPQW. If the PL emission from the DPQW is stronger than that from a PQW under the anti-Bragg condition or even that from a FQW, this structure has great potential to serve as an excellent light emitter. A DPQW with two different spacings, A and B, and which obeys a double-period sequence: A → AB and B → AA, is studied [19]. For example, the generation orders, v 1, 2, 3, and 4, indicate the different structures, A, AB, ABAA, and ABAAABAB. The number of QWs in the DPQW is equal to N 2v−1 because each of the spacings contains a QW. The DPQWs are placed between a finite forward barrier and a semi-infinite rear barrier. The cladding thickness, dC , between the vacuum interface and the center of the first QW is equal to λω0 ∕2. It is assumed that there is no dielectric contrast between the QWs and the barriers. It is also assumed that the barrier thicknesses are sufficient to prevent interaction between excitons in the QWs, so the excitons only couple with the electromagnetic field. Each QW is also assumed to be identical. The intensity of the luminescence emitted from the DPQW depends on the incident angle, the contribution of the exciton to the light polarization, and the generation order and the spacing thickness of the structure. This is calculated based on the introduced resonant source term, the radiation boundary condition, and the following two assumptions: one, that there is no correlation between the source functions in the QWs and two, that the distribution of the direction of the noncoherent exciton polarization is isotropic in the plane of the structure [5,7,23]. This study only considers the PL spectra radiated to the left side of the DPQWs and along the growth direction, which are described as the summation of radiation created by each source in P a noncoherent 2 j manner, expressed as Iω 4ΞjqS∕φ1 j2 N j1 jℑ ωj . The spectral density Ξ is relevant to the Fourier transformed source correlation, q is the light wave vector, © 2014 Optical Society of America
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Fig. 1. Maximum values of the PL intensity, defined as PLmax , in the DPQWs versus the filling factor, F, for increasing the generation order, v. A and B mark the positions of the anti-Bragg and Bragg conditions, respectively. The gray squares and green diamonds indicate the maximum positions. The maximum positions in the DPQWs where v 6, 7, and 8 are specially marked by C, D, E, F, G, and H. I and J mark the hollow positions in the DPQWs where v 7 and 8, respectively. The parameter values for the system are as follows: ℏω0 1.523 eV, ℏΓ0 25 μeV, ℏΓ 180 μeV, and nb 3.59.
S is related to the exciton susceptibility, φ1 is the projection of the exciton wave function on the even solution, u1 , of d2 Ez∕dz2 q2 Ez 0, and ℑj ω is a Green’s function depicting the field radiated by each source [5,7]. The transmission, reflection, and absorption spectra of the structures are obtained by the transfer matrix method [7,12,23]. Initially, the overall effect of the thickness filling factor of the system, defined by F dA ∕D, where D dA dB λω0 , on the maximum values of the PL intensity in the DPQWs is considered for increasing generation orders, as shown in Fig. 1. The maximum values of the PL intensity are defined as PLmax . Only the curves of PLmax within this range are shown, due to the symmetry points at F 0.25, 0.5, and 0.75. It is seen that the values of PLmax at F 0.25 and 0.5 under the anti-Bragg and Bragg conditions become, respectively, greater and smaller when the number of QWs increases [14,23]. Because of the anti-Bragg and Bragg conditions, the values of PLmax at F 0.25 and 0.5 in the DPQW are equal to those in a TMQW with the same number of QWs [5]. As the filling factor increases from 0.25 to about 0.4, the values of PLmax in the DPQWs with v 4 and 5 decrease slightly and then increase. Subsequently, the values of PLmax increase sharply until they reach their own minimum values. Therefore, there is only one maximum, around F 0.4, in the values of PLmax for lower generation orders. However, there are two maxima, near F 0.4 and 0.45, for higher-generation orders. When the generation order is greater than 6, the first maximum occurs around F 0.4, and a second maximum emerges near F 0.45. These two maxima both increase as the generation order increases. In comparison, the profiles of PLmax in the DPQW are different from those in the FQW, the TMQW, and the PQW [5,23]. Strictly speaking, the profiles of PLmax in the DPQW are more similar to those in the FQW, since the PL intensity is significantly enhanced in both of these structures. The main difference between them is that there are two maxima in the values of PLmax in the DPQW and one in the FQW for higher-generation orders. In order to give a clear comparison between the PL characteristics of the different QW structures, the maximum PLmax and the filling factors that correspond to the maximum PLmax , defined as F max , in the DPQWs, the
Fig. 2. (a) The maximum PLmax and (b) F max , defined by the filling factors corresponding to the maximum PLmax , in the QW structures versus the number of QWs, N. The values for the DPQWs in (a) and (b) correspond to the gray squares and green diamonds in Fig. 1.
FQWs, the PQWs, and the TMQWs, are plotted as a function of the number of QWs in Fig. 2. It is clearly seen that the PL properties of the DPQW are more similar to those of the FQW than to those of the PQW and the TMQW. Figure 2(a) shows that the maximum PLmax in the DPQW and the FQW sharply increases as the number of QWs increases. However, there is only a gradual increase in the maximum PLmax in the PQW and the TMQW [5,23]. Figure 2(b) shows that there is a variation in the value of F max in the DPQW and the FQW when the number of QWs increases. In contrast, the value of F max in the PQW and the TMQW remains at 0.25, the position of the antiBragg condition, regardless of any increase in the number of QWs. However, there are some major differences in the PL characteristics of the DPQW and the FQW, even though they are similar. Two values for the maximum PLmax occur near F 0.4 and 0.45 in the DPQW when the number of QWs exceeds 32, as shown in Figs. 2(a) and 2(b). The second value of the maximum PLmax increases more rapidly than the first. The former eventually exceeds the latter when the number of QWs reaches 128. However, there is only one value for the maximum PLmax in the FQW. The values of F max in the FQW are between those in the DPQW when the number of QWs is greater than 32. It is also seen that the two values of the maximum PLmax in the DPQW with N 128 are both greater than that in the FQW with N 144. Therefore, the PL intensity in the DPQW is stronger than that in the FQW for a relatively smaller number of QWs. The effect of the filling factor on the PL, transmission, and reflection spectra in the DPQW is illustrated in Fig. 3. The evolution of these spectra with the filling factor is clearly shown. Figure 3(a) shows that the PL profiles in the DPQW where F 0.25 and 0.5 are very distinct from each other because they pertain to the anti-Bragg and Bragg conditions, respectively [13,15,23]. There are two peaks that are symmetrical, with respect to the exciton resonance frequency in the PL spectra for F 0.25. In contrast, the PL intensity for F 0.5 is very weak throughout the frequency range, although the PL profile is also symmetrical. For F 0.5, the approximate expression for the PL intensity can be expressed as Iω ≈ NΞq2 u22 jS∕φ1 j2 jtj2 , where u2 is the odd solution and t is the transmission coefficient [5,7]. This expression shows that the PL of the Bragg QW structure is restrained by the low transmittance. However, when the DPQW deviates from the anti-Bragg and the Bragg conditions, the PL spectra become quite sharp and asymmetrical, again with respect to the exciton resonance frequency. On the left side of the exciton resonance frequency, the two
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Fig. 3. (a) PL, (b) transmission, and (c) reflection spectra in a DPQW with v 8 and N 128 for different filling factors. A partial close-up of (a) is shown in the inset in (a). The normalized frequency is defined as Ω ω − ω0 D∕2πc.
great peaks, respectively, correspond to F 0.415 and 0.463, which are the values of F max denoted by G and H in Fig. 1. It is clearly seen that the peak value for F 0.463 is larger than that for F 0.415. There are also two small peaks on the left side of the frequency for F 0.434, at the hollow position marked by J in Fig. 1. These two small peaks for F 0.434 are higher than those for F 0.25, even though the larger value corresponds to the hollow position shown in Fig. 1. In addition, on the right side of the frequency, there is a gradual decrease in the peak values as the filling factor increases from 0.415 to 0.463. Because of the close relationship between the absorbance and the PL intensity, the transmission and reflection spectra are related to the formation of the PL spectra [5,7]. Figures 3(b) and 3(c) show that the frequency range for low transmittance and the linewidths of the reflection spectra for F 0.5 are much greater than those for the other filling factors because of the Bragg condition. It is also observed that the variation near the exciton resonance frequency in the reflection spectra is much more complicated than that in the transmission spectra. The transmittance in the middle of the spectra remains very small as the filling factor increases. In contrast, some significant dips and peaks appear in the reflection spectra, which lead to the peaks and dips in the PL spectra. In particular, the dip on the left side of the exciton resonance frequency in the reflection spectra for F 0.463 is deeper than that for F 0.415, which means that the maximum PL intensity of the former is stronger than that of the latter. Although the dip for F 0.425 is also deeper than that for F 0.463, it is not reflected in the PL intensity because there is greater transmittance. The PL, transmission, and reflection spectra corresponding to the two maxima in the values of PLmax in the DPQWs for higher-generation orders are of interest. As shown in Fig. 4(a), in the PL spectra for the first maxima, there are two peaks that are sharp and asymmetrical with respect to each other. The peak value and the corresponding frequency become, respectively, greater and farther from the exciton resonance frequency as the generation order increases. Figure 4(b) clearly shows that the PL spectra are primarily dependent on the reflection spectra. It is seen that the frequency range for low transmittance becomes larger in the transmission spectra as the generation order increases, but the transmittance near the exciton resonance frequency remains very small. In contrast, the peaks located at almost the exciton resonance frequency
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Fig. 4. (a) PL and (b) transmission and reflection spectra in DPQWs with v 6, 7, and 8 for the respective values of F max , marked by C, E, and G, the first maxima in Fig. 1. (c) PL and (d) transmission and reflection spectra in the DPQWs with v 6, 7, and 8 for the respective values of F max , marked by D, F, and H, the second maxima in Fig. 1. The numbers of QWs in the DPQWs are N 32, 64, and 128. The insets in (a) and (c) show a partial close-up of (a) and (c), respectively.
and the dips on the left side of the frequency in the reflection spectra result in the corresponding dips and the peaks that occur in the PL spectra. Compared to the PL spectra for the first maxima, there is a greater difference between the values of the two peaks in the PL spectra for the second maxima, as shown in Fig. 4(c). The variation becomes more irregular in the PL profiles for the second maximum as the generation order increases. As in Fig. 4(b), it is also observed in Fig. 4(d) that the PL profiles are mainly influenced by the reflection profiles. The peaks located close to the exciton resonance frequency in the reflection spectra for the second maximum are higher than those for the first maximum. This results in the smaller dip values that are almost at the frequency in the PL spectra for the former, as seen in Figs. 4(a) and 4(c). Finally, a comparison of the squared electric fields in the DPQWs for different filling factors and the PQW under the anti-Bragg condition for the same number of QWs is shown in Fig. 5. This study refers to the frequency where the PL intensity in the structure is maximal as ωPL max . Figures 5(a) and 5(b) show that the field profiles in the DPQWs for the two maxima, indicated by E and F in Fig. 1, are very distinct from each other. It is seen that there are two points near z∕λω0 6.5 and 21, K and L, that divide the field profile for F 0.408 into three groups in which the respective average
Fig. 5. Squared electric fields in the DPQWs for (a) F 0.408, (b) F 0.461, (c) F 0.428 for v 7 and N 64, and (d) the PQW with F 0.25 for N 64. The numerical calculations are performed by ωPL max . The black line separates the vacuum region from the structure. The blue lines denote the QW positions. The incident, reflected, and transmitted waves are, respectively, denoted by the horizontal lines, I I , I R , and I T . The squared electric fields are equal to jEωPL max j2 for z > 0, multiplied by the background refractive index for clarity. The field values at the vacuum interface and the QWs are marked by red circles.
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field values are very different. However, there is a great difference in the field profile for F 0.461, which is more similar to that in the FQW, since many of the peaks in the field are situated very near the QWs [23]. In comparison, the field profile for F 0.428, which is the hollow position marked by I in Fig. 1, is similar to that for F 0.408, as seen in Fig. 5(c). It is observed that the field profile for F 0.428 is also divided into three groups by the two points near z∕λω0 4.5 and 20, M and N, although there is a decrease in both the peaks of the field and the field values at the QWs. Although the field profiles in the DPQWs are distinct, their PL intensities are all stronger than that in the PQW under the anti-Bragg condition, where the peaks of the field are situated at the QWs, but these peak values are not great, as shown in Fig. 5(d). This can be attributed to the enhanced lightmatter coupling because of the larger field at the QWs in the DPQWs [5,23,26]. This Letter demonstrates twin extra-high PL in resonant quasi-periodic DPQWs. Two maxima occur in the values of PLmax for higher-generation orders. It is found that the second maximum begins to emerge near F 0.45 next to the first maximum, which is around F 0.4, when the generation order exceeds 6. There is a dramatic rise in the second maximum, which eventually becomes greater than the first maximum, when the generation order reaches 8. The reflection and transmission spectra also show that the PL spectra for the second maxima are more asymmetrical than those for the first maxima, because the PL profiles are significantly dependent on the reflection profiles. These results for DPQWs are more similar to those for FQWs than to those for TMQWs and PQWs. The maximum PL intensity in a DPQW is even stronger than that in the PQW under the anti-Bragg condition and that in a FQW. The field profiles for the two maxima are also very different, although the field values at the QWs for both are greater than those for the PQW under the Bragg or anti-Bragg conditions. The authors acknowledge the support in the form of partial funding by the National Science Council of Taiwan under grant nos. MOST 103-3113-E-002-001 and MOST 103-2221-E-002-118. References 1. D. Goldberg, L. I. Deych, A. A. Lisyansky, Z. Shi, V. M. Menon, V. Tokranov, M. Yakimov, and S. Oktyabrsky, Nat. Photonics 3, 662 (2009).
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