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Strong photoluminescence (PL) emission from a resonant Fibonacci quantum well (FQW) is demonstrated. The maximum PL intensity in the FQW is significantly stronger than that in a periodic QW under the Bragg or anti-Bragg conditions. Moreover, the peaks of the squared electric field in the FQW are located very near each of the QWs. The optimal PL spectrum in the FQW has an asymmetrical form rather than the symmetrical one in the periodic case. The maximum PL intensity and the corresponding thickness filling factor in the FQW become greater with increasing generation order.
©2013 Optical Society of America
The propagation of light waves in resonant photonic crystals has attracted a lot of interest and has been extensively investigated during the past few years [1–3]. Some kinds of resonant photonic crystals are based on quantum dots, quantum wires or quantum wells. Among these materials, periodic quantum wells (PQWs) are very interesting in regard to the aspect of fabrication. Numerous studies have concentrated on the optical properties of PQWs [4–7]. For PQWs, two conditions govern the greatly different optical properties. First, a great change in the reflection spectra takes place under the Bragg condition d = λ(ω0)/2 at the exciton resonance frequency ω0. Since the Bragg condition was first discussed, numerous theoretical and experimental works have been studied [8–12]. However, relevant studies have indicated that the PL intensity is very weak under the Bragg condition [5,13,14]. Secondly, another specific condition is the anti-Bragg condition d = λ(ω0)/4, which has also been studied experimentally and theoretically [15–17]. For this condition, the linewidth and magnitude of the reflection spectrum are significantly smaller than those under the Bragg condition. This results in a situation where the magnitude of the absorption and PL for the former is strikingly larger than that for the latter according to the relation between the absorption and PL spectra [13,16,17].
Since the discovery of quasicrystals, photonic crystals with quasiperiodic sequences have received a great deal of interest [18–22]. Up to now, these structures have not possessed the ability to emit light because of their optically inactive properties. Recently, quasicrystals have been applied to multiple quantum wells in several theoretical and experimental studies [14,23]. For quasiperiodic systems, one of the famous structures is the Fibonacci sequence. Research has shown that the PL emission from Fibonacci quantum wells (FQWs) can be stronger than that from PQWs under the Bragg condition [14]. However, serving as an ideal light emitter, the anti-Bragg condition is more appropriate than the Bragg condition due to its much more efficient PL emission. Thus, it is worthwhile to investigate whether the PL enhancement ability in the FQW can surpass that in the PQW under the anti-Bragg condition due to the potential impact of such research on light-emitting devices.
Here, we make a detailed comparison between the PL and absorption spectra in FQWs and PQWs. The PL intensities in the structures are simulated via a phenomenological approach to describe luminescence. The result shows that the maxima of the PL intensity in the FQWs with higher generation orders are significantly stronger than those in the PQWs under the Bragg or anti-Bragg conditions. The maximum PL intensity and the corresponding thickness filling factor in the FQW become greater with increasing generation order. Moreover, we find that the optimal PL spectrum and the squared electric field in the FQW are much different from those in the PQW.
We consider a FQW composed of two spacings, A and B, with a Fibonacci sequence: Sv = {Sv−1 Sv−2} for v ≥2, with S0 = {B} and S1 = {A} [19,22]. Each of the spacings includes a QW. The number of QWs in the FQW is N = 1 and 2 for v = 1 and 2, respectively. In addition, we define the PQW as (AB)u, where u is the number of periods. Thus, the number of QWs in a PQW with u periods equals N = 2u. The series of the structures are shown as follows: |B|A|AB|ABA|ABAAB|… for FQWs and |AB|AB|AB|AB|… while for PQWs. In this study, the FQWs and PQWs are sandwiched in between a finite fore barrier and a semi-infinite hind barrier. The cladding thickness dC between the vacuum interface and the center of the first QW equals λ(ω0)/2. We assume that the thicknesses of the barriers are sufficient to avoid interaction between excitons in the QWs, so that the excitons only couple with the electromagnetic field. Based on the classical Maxwell’s equations, in addition to introducing a resonant source function into the exciton contribution to polarization of a single QW, the emission properties of the structures under consideration can be described [11–13]. It is assumed that there is not difference between the refractive indices of QWs and barriers. At normal incidence, the electric field in a single layer, where a QW is located at z = 0 with the left and right surface of the layer at z− and z+, are governed by
where the coordinate z is the growth direction of the structure, q = ωnb/c is the light wave vector, nb is the background refractive index, and the resonant source function ∑ is related to the noncoherent emission created by the luminescence of the QWs. Here, for the 1s states of the heavy-hole excitons, the exciton susceptibility χexc(ω) has a single-pole form, and the wave function of the exciton Φ1s(z) is an even function of z.By solving Eq. (1), the relation between the electric fields at the left and right boundaries of layer j, zj− and zj+, can be described as (E+,E-)(zj+) = Mj(E+,E-)(zj-) + Vj, where Mj is the transfer matrix through layer j and Vj is a two-dimensional vector of the source term for layer j. It is assumed that each QW is identical. The transfer matrix and the two-dimensional vector can be presented as Mj = PjMw and Vj = -(2∑Sq/φ1)/2iq(sj1,sj2), where sj1 = exp(iqzj+)/2iq and sj2 = -exp(-iqzj+)/2iq. The matrices Pj and Mw describe the propagation of light through jth barrier and a single QW, respectively [12,13]. The function S(ω) has the single-pole form S(ω) = Γ0/(ω-ω0 + iΓ), where Γ0 and Γ are the exciton radiative and nonradiative damping rates, respectively, in a single QW. In this study, we only consider the waves radiated to the left side of the FQWs and PQWs and along the growth direction. Thus, the intensity of PL emitted from the structures can be calculated by
where Ξ is the spectral density relevant to the correlation function for the Fourier transformed source function and The matrix MR describes the propagation of light across the interface between the last barrier and the semi-infinite hind barrier. In Eq. (2), the transmission coefficient t of the entire structure can be calculated via the total transfer matrix MT, where MT = MR(MN…M1)PCML−1. The matrices PC and ML describe the propagation of light through the fore barrier with the thickness dC−dw/2 and across the interface between the vacuum region and the fore barrier, respectively. The absorbance A of the entire structure can also be obtained by A = 1-R-T, where R and T are the reflectance and transmittance.First of all, we consider the influence of the generation order and number of QWs on the PL spectra in FQWs and PQWs with the same interval thicknesses dA = 0.4λ(ω0) and dB = 0.6λ(ω0). We assume that D = dA + dB = λ(ω0) in this study. Figure 1 shows that there is a great difference in the PL spectra profiles of the FQWs and PQWs. For the FQWs, there are two peaks in the PL spectra, which are very sharp and asymmetrical with respect to the exciton resonance frequency. As the generation order increases, the peak value and the corresponding frequency gradually get larger and farther from the exciton resonance frequency, respectively. However, the PL spectra in the PQWs are also asymmetrical but not sharp. Specifically, the maxima of the PL intensity and the corresponding frequencies remain quite small and are still close to the exciton resonance frequency, respectively, regardless of the increase in the number of QWs. Thus, we can obviously see that the PL intensities in the FQWs are greatly larger than those in the PQWs. The higher peak values in the FQWs are about 2.5 times stronger than the maxima in the PQWs.
Fig. 1 PL spectra in the FQWs for generation orders v = 8,9,10 and in the PQWs for the corresponding numbers of QWs, including N = 34,54,88. The exciton parameters of the systems are as follows: ħω0 = 1.523 eV, ħΓ0 = 25 μeV, ħΓ = 180 μeV, and nb = 3.59. The normalized frequency Ω is defined by Ω = (ω-ω0)D/(2πc).
We turn our attention to the influence of the thickness filling factor F of the system, defined by F = dA/(dA + dB), on the properties of PL intensity and absorbance in the FQWs and PQWs. Let us consider a FQW with v = 10 and a PQW with u = 44. The number of QWs in the FQW and PQW are N = 89 and 88, respectively. Figure 2 shows that the PL intensities and absorbance are similar near the exciton resonance frequency in both the FQWs and PQWs. Moreover, particularly at F = 0.25 and 0.5, the profiles of these spectra in the FQWs are nearly identical to those in the PQWs. Specifically, the interval thicknesses at F = 0.25 are dA = λ(ω0)/4 and dB = 3λ(ω0)/4. This is called the effective anti-Bragg condition. We find that the PL and absorption spectra under this condition are identical to those under the anti-Bragg condition, in which dA = dB = λ(ω0)/4 [16]. Hence, for the purposes of this study, the effective anti-Bragg condition can be regarded as the anti-Bragg condition. In contrast, the interval thicknesses at F = 0.5 are dA = dB = λ(ω0)/2, which satisfies the Bragg condition. Therefore, Figs. 2(a) and 2(b) show that the PL and absorption spectra in the FQW and PQW with F = 0.25 both have a splitting form composed of two peaks symmetrical with respect to the exciton resonance frequency. The magnitude of these spectra is quite large near the exciton resonance frequency. However, the PL intensities in the structures with F = 0.5 are very weak throughout the frequency range [5,13]. The magnitude of the absorbance is also quite small near the exciton resonance frequency.
Fig. 2 (a) PL and (b) absorption spectra in the FQW with v = 10 and the PQW with u = 44. The interval thicknesses of the structures vary with the thickness filling factors F = 0.25, 0.35, 0.45, and 0.5. The other parameters are consistent with those mentioned in the caption in Fig. 1.
On the other hand, the two peaks in the PL and absorption spectra in the FQWs with F = 0.35 and 0.45 become very sharp and asymmetrical with respect to the exciton resonance frequency. When the thickness filling factor increases from 0.25 to 0.45, the higher peak values of the PL intensity and absorbance in the FQWs still become stronger. Moreover, there is one dip that gradually gets deeper at the exciton resonance frequency in both the PL and absorption spectra. In contrast, the magnitude and linewidths of the PL and absorption spectra in the PQWs drastically decline near the exciton resonance frequency as the thickness filling factor increases from 0.25 to 0.5. Accordingly, at the same thickness filling factors of F = 0.35 and 0.45, it can be seen that the magnitude of the PL and absorbance near the exciton resonance frequency in the FQWs is significantly greater than that in the PQWs. Furthermore, the higher peak values of the PL intensity and absorbance in the FQWs with F = 0.35 and F = 0.45 are stronger than the peak values in the FQW and PQW with F = 0.25.
It is interesting to see the overall effect of the thickness filling factor F on the maxima of the PL intensity in the FQWs and PQWs. Here, we define the maxima of the PL intensity as PLmax. Figure 3 presents the curves of PLmax in the FQWs and PQWs for changes in thickness filling factor, the generation order of the FQW, and the number of QWs in the PQW corresponding to the generation order. Owing to the symmetry points at F = 0.25, 0.5, and 0.75, we only present the profiles of PLmax for F = 0.25 to F = 0.5. Specifically, the profiles for F = 0.0 to F = 0.25 and the ones for F = 0.25 to F = 0.5 are symmetrical at F = 0.25, and so on. As seen in Fig. 3, under the anti-Bragg and Bragg conditions, the values of PLmax at F = 0.25 and 0.5 in the FQWs for the respective generation orders are almost equal to those in the PQWs with the corresponding number of QWs [16,17]. Only a slight difference exists between these values because of a slight difference between their numbers of QWs. Moreover, the values of PLmax at F = 0.25 and 0.5 become larger and smaller, respectively, with the increase in the number of QWs.
Fig. 3 The maxima of the PL intensity, defined as PLmax, in the FQWs and PQWs versus the thickness filling factor F. The solid and dashed lines represent the numerical results of the FQWs for the generation orders v = 6 to 10 and the PQWs where the numbers of QWs are N = 12 to 88, respectively.
When the thickness filling factor exceeds 0.25, the values of PLmax in the FQWs still become larger until reaching their own maxima. Subsequently, there is a sharp decline in the values of PLmax until they reach their own minima. These situations are particularly evident for higher generation orders. Moreover, the maximum of PLmax and the corresponding thickness filling factor become greater with increasing generation order. In comparison, all of the values of PLmax in the PQWs decline gradually as the thickness filling factor increases from 0.25 to 0.5. Thus, the values of PLmax in the FQWs are significantly stronger than those in the PQWs with the same thickness filling factors, with the exception of F = 0.25 and 0.5.
Finally, we compare the squared electric field in the FQW with that in the PQW for the same thicknesses filling factor F = 0.4 to understand the reason why the PL intensity in the FQW is enhanced. In this case, we consider the FQW with v = 8 since its number of QWs, N = 34, can exactly correspond to the PQW with u = 17. As we know, the PL intensity depends on the light-matter coupling in the QW. As the field at the QWs is greater, the PL intensity is stronger. For the FQW, the peaks of the field at the frequency where the PL intensity is maximal, namely ω(PLmax), are located very near the QWs, as shown in Figs. 4(a) and 4(d). However, Figs. 4(b) and 4(e) show that the nodes of the field at ω(PLmax) are located very near the QWs in the PQW. Thus, the FQW can result in significantly stronger PL intensity than the PQW for the same interval thicknesses, as shown in Fig. 1. Moreover, Fig. 4(c) shows the field in the PQW with F = 0.25, which is equivalent to the anti-Bragg condition. For the PQW, the strongest PL intensity occurs under the anti-Bragg condition. However, under the Bragg condition, the nodes of the field are located exactly at the QWs, thereby leading to extremely weak PL intensity [14,16]. Figures 4(c) and 4(f) show that the peaks of the field under the anti-Bragg condition are located at the QWs, but these peak values are not large. As the thickness filling factor increases from 0.25 to 0.5, the peaks of the field in the PQWs increase, but the field at the QWs decreases. This leads to a fact that the PL intensity in the PQWs becomes weaker for the increase in the thickness filling factor, as shown in Fig. 3. For the FQW, the field in the QW of the system with F = 0.25 is very similar to that in the PQW with F = 0.25. However, not only the peaks of the field but also the field at the QWs increases as the thickness filling factor increases from 0.25 to about 0.4 for the FQW with v = 8, as shown in Figs. 4(c) and 4(a). As the thickness filling factor exceeds 0.4, both the peaks of the field and the field at the QWs decrease. The above results are consistent with the change in the PL intensity of the FQWs for different thickness filling factors, as shown in Fig. 3.
Fig. 4 The squared electric field in (a) the FQW with v = 8 and F = 0.4, (b) the PQW with u = 17 and F = 0.4, and (c) the PQW with u = 17 and F = 0.25. The insets (d), (e), and (f) are a partial close-up of (a), (b), and (c), respectively. The simulation is made for a plane wave normal incidence from the left side, the vacuum region, and at ω(PLmax). The blue lines stand for the QW locations. The green line divides the vacuum region from the structure. The squared electric field equals |E[ω(PLmax)]|2 for z>0 multiplied by the background refractive index for the sake of clarity. The horizontal lines II, IR and IT indicate the incident, reflected and transmitted waves, respectively.
In summary, we have demonstrated strong PL emission from a resonant FQW. Unlike the PQW, the peaks of the squared electric field at ω(PLmax) in the FQW are located very near the QWs. Due to the enhanced light-matter coupling, we find that the values of PLmax in the FQWs for higher generation orders are significantly stronger than those in the PQWs under the Bragg or anti-Bragg conditions. The maximum of PLmax and the corresponding thickness filling factor become greater with increasing generation order. The results also show that the optimal PL spectrum in the FQW has an asymmetrical form rather than the symmetrical one in the PQW. The values of PLmax in the FQWs continue to increase until reaching their own maxima when the thickness filling factor exceeds 0.25. However, all of the values of PLmax in the PQWs decline gradually as the thickness filling factor increases.
Acknowledgments
The authors acknowledge the support in part by the National Science Council of Taiwan under grant numbers NSC 100-2221-E-002-021 and NSC 101-2221-E-002-030.
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