Fabry–Perot type resonant modes of exciton luminescence in Cu₂O nanowires

Semiconductor optical micro- cavities have received considerable attention due to the need to develop new sources of enhanced light emission for the field of photonics [1–3]. Recently, much interest has been focused on light–matter interaction in nanostructures [4] as an example that offers fascinating optical phenomena to be explored. Semiconductor nanostructures have the potential to create varying types of optical micro-cavities having many different geometries and a range of refractive properties that are desirable to produce strong field confinement from photons created by the annihilation of excitons in a crystalline lattice [5, 6]. Importantly, semiconductor nanowires offer numerous advantages such as a matrix for both creating spontaneous and stimulated emission [7–11]. They subsequently act as a medium for guiding and amplifying the light produced for delivery with minimum losses. An increasing number of studies have shown that effective light emission within semiconductor nanowires can derive from photoluminescence produced by a process of continuous generation and annihilation of excitons. Excitons in various semiconductors present an array of different properties, such as large effective Bohr radius, varying binding energy to lattice atoms or impurities, variable lifetimes before recombination, differing degrees of coupling to lattice phonons, etc Among these excitons in the direct-gap semiconductor Cu₂O have some of these attractive benefits such as large binding energy (150 meV [12]), sizeable Bohr radius, and the long lifetime up to 10 μs [12]. Moreover, the existence of multiple Rydberg energies in Cu₂O, their susceptibility to size variation by external magnetic fields, and the ability to select excitons having different principal quantum numbers offers the opportunity to produce multiple wavelengths in the photoluminescence series [13]. Currently, Fabry–Perot (F-P) type resonance modes in Cu₂O nanowire micro-cavities have not received the attention while that commercially important semi-conductors, such as Si, and Ga(In)As materials, have received. In this work, exciton luminescences arising from the F-P type resonant modes of Cu₂O nanowires were detected experimentally using micro-photoluminescence (μ-PL) spectroscopy at room temperature. We carried out Finite-Difference Time Domain (FDTD) simulations on modeled Cu₂O nanowires using COMSOL Multiphysics software. These revealed the F-P type resonance exciton eigenmodes that were remarkably similar to the experimentally observed frequencies. We feel this work provides a path toward the development of potential excitonic light sources and helps to advance the understanding of the role that the micro-cavity plays in determining nanowire optics.

Typical Cu₂O NWs were fabricated by a previously reported method [24]. Figure 1 shows a top view of the scanning electron microscopy (SEM) images obtained from a typical preparation of NWs. As shown in figure 1(a), the diameters of characteristic NWs fall in the range of 90–285 nm. A magnified SEM image (figure 1(b)) shows an isolated single NW having nearly uniform average diameter of 110 and 153 nm at the upper end and lower end labeled with blue arrows with reasonably straight edges and longitudinal terminations, the single NW has the length of 4.06 μm, as shown in figure S1(d) is available online at stacks.iop.org/JPCO/3/085006/mmedia.

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In figure 2(a), a schematic experimental geometry for performing PL characterization is present. The PL spectrum of a representative multiple Cu₂O NWs on a quartz substrate was obtained at room temperature, as shown in figure 2(b). The PL measurements were obtained using the EDINBURGH Instrument FLS980 Spectrometer. Contributions arising from luminescence of quartz substrate are also shown in figure 2(b). As seen in figure 2(b), the substrate without Cu₂O nanowires shows no luminescent peak. The PL spectrum from Cu₂O NWs, on the other hand, exhibits a maximum emission peak at 525 nm (i.e. 2.36 eV) as well as multiple ripples comprising subsidiary peak intensities out to beyond 600 nm. We note that this series of weaker emission peaks on the low-energy side of the maximum emission line falls within an energy range from 2.32 to 2.14 eV. It is generally accepted that the band gap energy of bulk Cu₂O lies in a range of 2.1–2.2 eV (590–563 nm) depending on surface bonds and impurities [14, 15]. Thus, the series of multiple emission lines on the low-energy side of the maximum emission peak most likely arises from different resonance cavity modes of the NWs [16]. Additionally, there is a weak emission peak at the wavelength of ~405 nm (3.06 eV). This result is consistent with a previous report [17] that attributes the peak to a high energy exciton process in Cu₂O.

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In order to test the hypotheses that the series of PL emission lines arise from exciton resonant modes, we examined a representative sample of different NWs to determine the effects of different lengths and diameters on the PL peak distributions. The energy of the excitation from the He-Cd laser operating at 325 nm (3.82 eV) was generally much greater than the exciton resonances we observed (2.17–2.30 eV at room temperature). Figure 2(c) summarizes the micro-PL spectra obtained from single Cu₂O NWs using different samples (S_a, S_b, and S_c) having measurably dissimilar diameters and lengths. The corresponding optical images of the three samples are shown in figures S1(a)–(c) of the supporting information (SI). As seen in figure 2(c), all samples exhibit emission peaks in the range between 590–528 nm (2.1–2.35 eV). We attribute their presence to various exciton resonant modes in the different NW cavities. PL intensities of the sample S_c are slightly greater than that of S_b, and the intensity of the sample S_a is the lowest of all these samples. To determine the subsidiary peaks in these PL spectra, we focus on the specific peaks from each sample and label them with tags from a–m in figures 2(d)–(f). Every label points to a peak which likely represents one of distinct resonant modes in the NWs [8]. As shown in figure 2(d), the emission peaks of the sample S_a show a peak spacing between neighboring modes of 21.7 meV. The mode spacing can be calculated according to the following equation: Δλ = λ²/2 L[n–λ(dn/dλ)] [18], where n is the refractive index at the wavelength λ, and L is the measured length of the NW. Here n is assumed to be 2.7, and dn/dλ is about −0.52 μm⁻¹ [19]. We derive a mode spacing of 18.1 nm at the peak resonance of 2.26 eV (i.e. ∼549 nm) for a cavity of measured length L = 3.6 μm. The mode spacing gradually increases to 20.6 nm at 2.10 eV (i.e. 590 nm). As shown in figures 2(e) and (f), the samples S_b and S_c exhibit similar features to S_a at the region of the exciton peak. The exciton PL at the peak at 549 nm, obtained from sample S_b, consists of a series of identical ripples, labeled from a to j in figure 2(e). Their mode spacing is approximately 7.42 nm at the PL peak. Compared with the mode spacing of the sample S_a, the sample S_b presents the smaller mode spacing at the same PL wavelength (∼550 nm). In figure 2(f), the exciton emission peaks from the sample S_c at the 546 nm peak consists of ten identical lines labeled a to j. The measured mode spacing is about 11.35 nm at the peak. The mode spacing, calculated to be 9.7 nm for a measured cavity length L = 4.8 μm, is nearly consistent with the experimental spacing for sample S_c.

In general, the average mode spacing for experimental lines was found to scale inversely with the resonator cavity lengths [19, 20]. Based on optical images of the three samples (shown in figure S1, supplementary information) and the N.A. of the lens (N.A. = 0.66), the optical resolution of the lens is obtained by the equation of d = 0.61*λ/N.A., the value of d is about 0.3 μm. Thus, the resonator lengths could be estimated to be 3.6 ± 0.6, 5.8 ± 0.6 and 4.8 ± 0.6 μm for Sa, Sb, and S_c, respectively. We found for all investigated NWs that the experimental mode spacing closely tracks a linear relationship to the inverse cavity length, as shown in figure S2, supplementary information. This mode spacing was found to be independent of NW diameter (as characterized by the SEM images) over the range between 90 and 285 nm. The mode spacing of the interference peaks is inversely proportional to nanowire length, demonstrating that the nanowire functions as an optical Fabry-Perot (F-P) resonator along its length [21]. Hence, we attribute the multiple subsidiary peaks in the PL spectra to a series of standing-wave Fabry–Perot (FP) resonant modes. For the PL spectra modulated by FP-type resonances, mode energy splitting of the FP modes can be calculated by ΔE = hc/2nL [15], where n is energy dependent refractive index, and L is the cavity length of the sample. For the sample S_a, the calculated mode energy spacing ΔE is determined to be 13.05 meV, a value somewhat smaller than the experimental result above (21.7 meV). Meanwhile, the energy spacing ΔE of S_b is 5.37 meV, and that of the sample S_c is 8.305 meV, confirming that the energy spacing scales with the inverse length of the NWs.

To understand the mechanism by which F-P modes alter the transmission of light down a cylindrical fiber we performed Finite Difference Time Domain (FDTD) simulations of the photon field intensity in Cu₂O NWs possessing a step index change to air, as described in the Methods section. For schematic purposes we assumed a generic right circularly symmetric and cylindrical NW having length L, a diameter d, and ask which modes propagate as a function of wavelength (figure 3(a)). In figure 3(b), we calculate the two-dimensional projections of the cross-sectional electric-field distributions along the axial direction for a NW of specific length L = 3 μm, diameter d = 160 nm, and an emission wavelengths λ = 549 nm. These values are similar to the parameters that describe sample Sa. Figure 3(b) shows that the calculated electric-field mode pattern exhibits a periodic enhancement of the electric-field intensity with a fixed axial spacing. In this figure, the deeper the red color that appears, the more intense is the electric field. Figures 3(c)–(e) shows the radial electric-field profiles for the E_x, E_y, E_z components when the effective mode index (n_eff) of the NW is set constant at 1.6236. Cross-sectional plots of the normalized electric field (Norm.E) are shown in figures 3(f) and (g) for two different effective mode indices. In this case we attribute each distribution to the same HE_11a,b mode and consider changes in field distributions produced only by altering the effective FP mode indices [20, 22]. The reasons to consider this possibility are as follows: first, the fundamental HE₁₁ mode is the better confined mode for sustaining a standing wave pattern along the entire wire length compared with the higher order modes; second, the FP resonant modes can be seen to provide the greater confinement of the field distributions within the NWs and maintain internal polarization as shown in figures 3(f) and (g). As both experimental and simulated measurements appear to give similar results for the NW of sample Sa, we conclude that FP resonant cavity modes (standing waves) exist in this sample.

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We also modeled the cross-sectional electric-fields for the E_x, E_y, E_z components for a PL wavelength of 549 nm, with NW length and diameter equal to those found for samples S_b and S_c. In figure 4 we show the transverse fields for a NW of length L = 5 μm, and diameter d = 200 nm (sample S_b). Similarly, in figure 5 we show the same transverse fields for a NW of length L = 3 μm, and diameter d = 220 nm (sample S_c). Both results were obtained by solving Maxwell's equations for a right circular cylinder of appropriate dimensions. As seen from the simulations, the electric-field intensity projections along the NW axis exhibits periodical dependence on longitudinal position. When the NW diameter exceeds 200 nm, we find two guided modes are supported: both HE₁₁ and TE₀₁ modes become allowed. The cross-sectional plots of the normalized electric fields for these two radial modes are shown in figures S3 and S4 of the supplementary information for several values of the effective mode index.

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To verify the physical origin of the FP modes, we calculated the effective mode indices at 549 nm for the nanowires with various diameters using COMSOL Multiphysics (version 5.3). The results shown in figure S6(a) indicate that effective mode indices drop monotonically as the NW diameter decreases. The effective mode index approaches ∼2.4 for a diameter of ∼320 nm. As the diameter decreases to approximately 140 nm, there are four calculated effective mode indices with n_eff = 1.0743, 1.18, 1.3635, and 1.4991, which could all support the fundamental HE₁₁ mode. However, the mode indices of 1.0743 and 1.18 do not totally confine the optical field to the nanowire, as shown in figure S5 of supplementary information. As the NW diameter is further decreased, light becomes weakly confined inside the nanowire. More optical field is leaked into the surroundings and the substrate. In figure S6(b), the reflectivity of optical power is calculated using the formula R = (N_eff − n₀)²/(N_eff + n₀)² [19], where n₀ is the refractive index of the surroundings. Here, the reflectivity gradually decreases with a decrease in the diameter and then decreases more quickly as the diameter drops below 200 nm. Finally, the reflectivity approaches zero as the diameter approaches ∼143 nm. It exhibits almost zero FP interference peaks as little of the optical field remains inside the NW. Our results for Cu₂O nanowires are found to be generally consistent with the properties of Fabry-Pérot micro-cavity modes discovered in single InGaAs/GaAs and CdS nanowires [19, 23].

In summary, we report observations of PL eigenmodes of band edge excitons in Cu₂O NW micro- cavities. The origin of the eigenmodes is attributed to Fabry–Perot (FP) type resonant modes inside the optical cavity of each NW. The number of modes in the band-edge emission region were observed for three investigational samples that showed a varying number of longitudinal modes (16, 12, and 14) with different splittings. The mode energy spacing depended on the inverse of the NW length. Experimental results and computer modeling of radiation fields demonstrated that FP resonant modes are set up inside the NWs, with modal density favoring the HE11 mode particularly for NW diameters used in this study. In addition, both HE11 and TE01 radial modes (superposed on the FP longitudinal modes) appeared inside the NW as its diameter exceeded 200 nm. The results presented here advance the understanding of exciton PL in the micro-cavity of Cu₂O NWs. They show approaches to improving the weak optical processes, such as spontaneous emission, in these devices and suggest potential applications in exciton-related optoelectronic devices where light emission may be enhanced by optical micro-cavities.

The authors thank Senior Engineering Yu Wang, A/Prof. Qing Su and Zhenghua Ju for assistance in PL measurements. This work was supported by the National Natural Science Foundation of China (Grants 61204106, 61405077), the Fundamental Research Funds for the Central Universities (Grants lzujbky-2016-119), the Natural Science Foundation of Gansu Province, China (Grants 2016GS08252), and the Science and technology development project of Chengguan District, Lanzhou, China (Grants 2016 CGKJ280).