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Abstract
The CsPbBr3 microwires with unique isosceles right triangle cross-sections are commonly observed via chemical vapor deposition method. In this work, we study the correlations between measured multi-mode lasing behaviors and the simulation of the mode patterns inside the triangular-rod microcavity. We confirm that lasing action with higher-order transverse modes can well sustain, even when these modes experience large optical loss due to the isosceles triangle cross-section. By comparing the experimental and simulation results, the higher-order transverse modes tend to show up prior to the fundamental transverse modes for wider microwires. We attribute this behavior to the nonuniform field distribution caused by the high absorption efficiency of CsPbBr3. We also elaborate on the difficulties to sustain the whispering gallery mode in the CsPbBr3 triangular-rod microcavity, which implies that the lateral dimension and geometry of the cavity should be considered carefully for the future design of low threshold wire-based laser devices.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
All-inorganic cesium lead halide perovskite materials (CsPbX3, X = Cl, Br, I) have received enormous attention owing to their prominent material properties, such as long diffusion length or long carrier lifetime [1], high material gain [2–4], high conversion efficiency (PCE) in solar cells [5,6], high photoluminescence quantum yield (PLQY) in quantum dot structures [7]. Several literatures also present that CsPbX3 perovskite possess better resistance to moisture and oxygen than organic–inorganic hybrid perovskite [8], which suggest reliable engineering for daily applications. The fabrication of CsPbX3 perovskite is also simple and convenient which can be achieved by solution process [9,10] or Chemical Vapor Deposition (CVD) method [11]. CsPbX3 perovskite synthesized by the CVD method exhibits high crystal quality than the solution process and can be exploited in mass production [12]. In addition, one important advantage of CsPbX3 perovskite is the tunability of bandgap by varying the mole fraction of halide ions, which strongly affects the emission (or absorption) wavelength in the entire visible regime [13,14]. Therefore, a variety of coherent light sources based on CsPbX3 perovskite with different morphologies (in micro and sub-micro scales) had been reported, including tetragons [15], hemispheres [16], spheres [17,18], plates [19], and wires [20]. In the engineering aspect, CsPbBr3 wires are generally observed by CVD method on Si or mica substrates [21–24]. The wire structures are also easily synthesized for large-area processing. Hence, wire structures are commonly adopted in the field of advanced perovskite-based devices, such as wavelength-tunable polaritonic devices [21,25], laser array devices [26]. Many reports focused on related subjects such as alignment techniques [25,27,28], polarization sensing [29], Vernier effect based single-mode selection [30], etc. Beside utilizing CsPbBr3 wires as the laser device, one of the most important and practical issues is to lower the lasing threshold value. Therefore, the analysis of the higher-order transverse modes is a direct way to unveil the leaky channels as cavity photons propagate inside the microwire cavity. The higher-order transverse modes correspond to larger transverse propagation vectors inside a microcavity, thus causing higher optical loss as cavity photons bump into lateral surfaces of the microcavity. However, most of reports only consider the Fabry–Pérot lasing action associated with the fundamental transverse mode pattern [21,23]. Whether or not the lasing action with higher-order transverse modes can sustain in perovskite wire structures is still an open question. In the case of small diameter wires such as nanowires, the nanowire structures are supposed to possess Fabry–Pérot lasing action with the fundamental transverse mode pattern due to the several tens of nanometer size in transverse directions. However, nanowires with a sub-micro meter scale in width may allow higher-order transverse modes to participate in lasing action. In addition, nano-laser also inevitably loses its quality factors due to poor lateral confinement of cavity photons. Thus, the quality factor in micro-laser structures is generally much higher than that in nano-laser structures. Hence, to find out the proper lateral size of microwire structures in which only the fundamental transverse mode can sustain in the lasing action should help optimize the threshold value and quality factor. Furthermore, microwire structures that possess whispering gallery modes (WGMs) due to total internal reflection theoretically have low threshold value and high-quality factor. However, it is not well understood that how come Fabry–Pérot like lasing action is commonly observed in CsPbBr3 triangular-rod microcavities [31]. On the contrary, the triangular, hexagonal, and square microplates on the substrate are easier to achieve lasing of the WGM modes [32]. In this work, we analyze the higher-order transverse modes of CsPbBr3 microwires with mostly observed isosceles right triangle cross-section. By comparing the simulation and experimental results, we confirm that higher-order transverse modes can sustain in CsPbBr3 microwires with a base length of a > 2.6 µm, where “a” denotes the side of isosceles right triangle cross-section that is contact with the substrate. Unlike many other reports that only consider lasing action associated with the fundamental transverse mode in wire structures, this work provides more insights about correlations between measured multi-mode lasing behaviors and transverse mode patterns of isosceles right triangular microcavity. We further observe an unusual sequence of lasing action that appears in photoluminescence (PL) spectra while excitation is increased. We ascribe the phenomena to the nonuniform field distribution in the transverse plane caused by the high absorption efficiency of CsPbBr3. Finally, by comparing with other reports of WGMs lasing in triangular, hexagonal, and square plate structures, we also elaborate on the difficulties of WGMs to sustain in microwire with isosceles right triangle cross section. The difficulties arise from the normal incidence of transverse propagating photons toward two isosceles sides and the leaky channel caused by the small refractive-index difference at the substrate interface.
2. Experiment
We synthesized high quality rod-like CsPbBr3 microcavities with isosceles right triangle cross-section on mica substrates by chemical vapor deposition (CVD). CsBr and PbBr2 precursors with a molar ratio of 1:1 were placed inside a crucible placed at the center of a quartz tube in a furnace. A cleaved mica substrate was cleaned and placed at a distance of 15 cm from the crucible on the downstream side. The quartz tube was evacuated first, and then filled with high purity Ar (99.999%) gas under fixed pressure about 5∼7 torr. The furnace temperature was controlled with a ramp-up rate of 30 °C/min from room temperature (RT) to 390 °C, maintained at 390 °C for 20 mins, and then let to cool down to RT naturally.
Figure 1(a), (b) shows the morphologies of the CsPbBr3 triangular microwires under optical microscopy (OM) and scanning electron microscopy (SEM), respectively. The CsPbBr3 triangular microwires grow along the [001] directions and have (100) facets at both ends. This particular morphology is commonly observed when silicon or mica is used as the substrate [22,33]. There is an incommensurate heteroepitaxial lattice mismatch between the CsPbBr3 and mica crystal structures. The growth mechanism is attributed to the asymmetric lattice mismatch [33]. The lateral cross-section of the triangular rod is an isosceles right triangle, so we define the lateral width “a” as the base length of the isosceles triangle (the side in contact the mica substrate) and “L” as the length of the microrod. The typical dimensions of our samples are: L ≈ 2∼30 µm and a ≈ 1∼5 µm.
Fig. 1. (a) Optical image of the CsPbBr3 triangular microwires. (b) The tiled SEM image of single CsPbBr3 wire.
We use CW He-Cd laser (λ = 325 nm) as the pumping source to measure the steady-state PL of CsPbBr3 microwires. To study the lasing characteristics, we measured the power-dependent spectra using the same PL system with Nd-YAG pulse laser (λ = 355 nm, duration = 350 ps, repetition rate = 1 kHz) as the pumping source. The pumping spot size was carefully controlled to cover the whole triangular microwire such that the excitation is uniform on each sample. Both the excitation and detection light went through the same objective (10X, NA 0.25) in a backscattering configuration. The emitted light from the sample was collected and sent into a spectrometer (Horiba iHR320) mounted with a liquid nitrogen cooling CCD system.
3. Results and discussions
An illustration of triangular microwire under µ-PL excitation is shown in Fig. 2(a). Typical steady-state optical absorbtance (red line) and PL spectrum (black line) are shown in Fig. 2(b), which correspond to the band-edge optical transition of bulk CsPbBr3. The blueshift of the absorption peak relative to PL peak is attributed to the Stokes shift [34,35]. The full width at half maximum (FWHM) of the typical PL spectrum is about 17 nm that is comparable with other reports [18,19,23]. The clear absorption peak (∼512 nm) is attributed to the stable exciton binding energy ∼40 meV in RT [14,19]. The threshold (Pth) of our wires is several hundred µJ/cm2. Figure 2(c) shows the power dependence of the emission spectrum of a single wire with L=7.3 µm and a=1.8 µm above threshold to demonstrate multimode lasing action. Figure 2(d) shows the measured quality factor (Q= λ/Δλ) of this sample is ∼ 5360 at threshold. The L-L curve (dark line) and FWHM (blue line) confirm the existence of lasing action as shown in Fig. 2(e). Figure 2(f) illustrates the physical dimension of this single wire, and the dark image shows a clear interference pattern that reveals the lasing action occurred inside the microcavity.
Fig. 2. (a) The schematic picture of CsPbBr3 triangular microwire under µ-PL excitation. (b) Typical PL and absorbtance spectrum under excitation of 325 nm CW laser. (c) Power dependent spectrum above Pth in log scale. (d) Measured quality factor (e) L-L curve (black) and FWHM (blue) demonstrate lasing action occurs. (f) The dark image showing a clear interference pattern above Pth.
As mentioned above, various morphologies of CsPbBr3 microcavities can be synthesized by controlling specific conditions such as the molar ratio of precursors, growing temperature, and pressure. To further identify the relationships between field patterns and the corresponding lasing modes, we systematically measured the power-dependent PL spectra of the triangular microwires with different sizes in RT. The wavelength region that lasing occurs is between 530 nm to 545 nm for our samples. We simulated the cavity modes in this unique triangular-rod cavity with isosceles right triangle cross section by COMSOL software. Because we do not have the exact information of material gain (or loss) for different excitation power. The simulation is done by assuming a cold cavity, which does not consider the effect of active region with optical gain or absorption. Therefore, the simulated mode intensity cannot be directly related to the intensity of emission peaks observed experimentally. For the simulation setting, the cavity transverse field pattern and spectrum are obtained based on 2D COMSOL EM frequency-domain stationary solver with dipole source. Resonant modes are found by parametric sweeps in which the longitudinal propagation constants are specified as kz ∼ mπ/L (L: cavity length; m: integer). The computational window boarder is defined as scattering boundary condition to avoid reflected waves. Finally, the wavelength dependent refractive index of CsPbBr3 is incorporated based on the previous report [36], and the index of mica substrate is given as 1.6 [37].
Figure 3(a) and (c) show the multi-mode lasing spectrum of a single triangular microrod (named sample A) with L=8.3 µm and a=1.3 µm. As shown in Fig. 3(d), the wavelengths of resonance modes of simulation are close to the peak positions in measured spectra. Regarding the mode patterns of electrical fields, the field concentrates in the central region, which implies the resonance modes are Fabry-Pérot modes (in the longitudinal direction) associated with the fundamental transverse mode patterns.
Fig. 3. (a) The power dependent spectra in linear scale of single CsPbBr3 triangular microrod. Threshold of sample A is 182.2 µJ/cm2. (b) The illustration of sample dimensions. (c) Selected power dependent spectra of lasing modes in log scale. (d) Simulated intensity and mode patterns (norm of electrical fields) of cavity modes.
We further measured another PL measurement from a sample B with L=29.6 µm and a=4.3 µm. As shown in Fig. 4(a) and (c), the lasing mode spacing (∼0.7 nm) is much smaller than sample A (∼2.5 nm) in Fig. 3, which results from the increase of the whole cavity size (both L and a). As shown in Fig. 4(d), the simulation shows that higher-order transverse modes corresponding to shorter wavelengths (higher energy) start to appear first inside the triangular microcavity. At first sight, the result seems to go against intuition since higher-order transverse modes appear prior to fundamental transverse modes. Especially when the lateral cross-section is close to an isosceles right triangle, the optical loss in higher order transverse modes is significantly larger. In contrast to previous reports [21,23], which only considered fundamental transverse modes propagating back and forth inside the triangular microcavity and dominating all lasing characteristics. Our simulation and experimental results are evidenced to match the real situation in CsPbBr3 microwires with isosceles right triangle. The assumption of lasing action dominated by Fabry−Pérot modes associated with the fundamental transverse mode pattern is reasonable only when the lateral width approximates to sub-micro or nanometer scale. Here, our triangular microwire cavities have larger lateral dimension and can sustain lasing action with higher-order transverse modes, which is attributed to the high optical gain of CsPbBr3 [38,39] that can compensate for the lateral optical loss.
Fig. 4. (a) The power dependent spectra in linear scale of single CsPbBr3 triangular microrod. Threshold of sample B is 454.2 µJ/cm2. (b) The illustration of sample scales. (c) Selected power dependent spectra of lasing modes in log scale. (d) Simulated spectrum and selected cavity mode patterns (norm of electrical fields).
To further confirm that higher-order transverse modes can sustain in isosceles right triangle microcavities, we did similar measurements on another sample C with L=10 µm and a=2.6 µm. As shown in Fig. 5, even when the lateral dimension is reduced to 2.6 µm, the emission peaks at wavelengths corresponding to higher-order transverse modes still show up. The simulation results confirm that higher-order transverse modes can well sustain in CsPbBr3 triangular microwire, even higher order transverse modes reveal that projection of wavevectors in the lateral plane are perpendicular to the two isosceles sides.
Fig. 5. (a) The power dependent spectra in linear scale of single CsPbBr3 triangular microrod. Threshold of sample C is 431.1 µJ/cm2. (b) The illustration of sample scales. (c) Selected power dependent spectra of lasing modes in log scale. (d) Simulated spectrum and selected cavity mode patterns (norm of electrical fields).
Based on the simulation and experimental results, one very intriguing phenomenon appears when we increase the pumping power for samples A, B and C. As shown in Fig. 3(a), 4(a) and 5(a). The multi-mode lasing appears toward the long wavelength (low energy) side as the excitation power increases. First, this trend is opposite to the band-filling effects in conventional semiconductor lasers, which is ascribed to the photoinduced carriers starting to populate higher-energy states. According to the PL spectrum in Fig. 2(b), conventional band filling effect in higher density of states should appear above 2.36 eV (i.e. less than 525 nm). However, we observed an opposite tendency. This special behavior was also observed from other reports but less detailed analysis [22,23]. So far, the mechanism of optical gain in CsPbBr3 system is still under debate, like non-degenerate electron hole plasma coupled with plasmon emission [40], exciton–exciton scattering at the low-temperature range and exciton-phonon scattering at higher-temperature range [17]. Another possibility might be the formation of plasmonic exciton-polaron at high pumping, which can give rise to double-peak feature in the gain spectrum with red shift tendency as pumping increases [41–45]. One might argue that self-absorption effect causes the optical gain spectrum to locate at the lower energy side [17,24]. Apparently, the self-absorption effect cannot fully explain this tendency, because the influences of self-absorption effect for short-wavelength photons should be larger than that for long-wavelength photons. That means short-wavelength modes should experience more optical loss due to self-absorption effect. However, this contradicts the observation that short-wavelength modes show up prior to long-wavelength modes. One might consider another possibility that thermal effect accounts for the sequence of lasing modes occur toward long wavelength (low energy) due to the shrinkage of band gap. However, according to literature [39,46–48], the band bap energy of CsPbBr3 perovskite typically shows a blue shift tendency at RT while operation temperature increases. Therefore, we cannot conclude that the phenomenon is due to the thermal effect, which results in a band-gap reduction as in conventional semiconductors. Besides, as shown in Fig. 3(c), 4(c), and 5(c), the peak positions of the lasing modes do not shift apparently, which implies the refractive index is almost unchanged upon different excitation power in this work. Hence, the thermal effect should also be excluded from such phenomenon.
To elaborate the sequence of lasing action that appears toward the long wavelength (low energy) side, two origins should be considered simultaneously. One is the unconventional broadening of the optical gain spectrum of CsPbBr3 with increasing pumping power. The other one is the different optical loss for each transverse modes pattern in triangular microcavity. First, we check the wavelength region that lasing modes occur. In our samples, as shown in Fig. 2(a), the spontaneous emission center is around 525 nm, and the lasing action of sample A, B, C all show up at the lower energy side of 525 nm. This result is consistent with the previous observations [39,49], that the optical gain spectra or the amplified spontaneous emission (ASE) spectra of CsPbBr3 typically take place at the lower energy side of the spontaneous emission peak. Secondly, simulation results demonstrate that transverse field patterns of each lasing mode are cavity size-dependent. Thus, in the same triangular wire, optical loss of different transverse field patterns should also be cavity size-dependent as well. Comparing the simulation results in Fig. 3(d), 4(d) and 5(d), we found that in sample A the only sustained field patterns are associated with the fundamental transverse modes due to the smaller lateral cross-section. The sample A favors the lasing action in “pure” Fabry−Pérot case, so the sequence of lasing peaks appears toward long wavelength (low energy) side that reflects the formation of unconventional optical gain spectrum broadening of CsPbBr3 as the excitation increases. In sample B and C, the increase of the size of the lateral cross-section results in the appearance of the higher-order transverse field patterns. However, in this isosceles triangle, the optical loss of higher order transverse modes should be larger than the fundamental transverse modes due to the non-zero components of lateral wavevectors, such that cavity photons can pump into three transverse facets which increase optical loss. Therefore, in the same triangular wire, lasing peak with higher-order transverse mode should experience larger optical loss than that of fundamental transverse modes. To elaborate on lasing action appears toward the long wavelength (low energy) side in samples B and C, we propose a model of nonuniform field distribution in the cross-sectional plane that could explain the power dependence of emission spectra. As illustrated in Fig. 6(a), the photoinduced carriers are created near the top of microcavity at lower excitation power; thus, the outer region can reach the gain status (gain > 0) first as we gradually increase the excitation power, while the inner region reaches transparency status (gain ∼ 0) at that moment. Because the fundamental transverse mode profile is centered on the core region, the net optical gain is not enough to compensate the total loss. As the excitation power increases further, as illustrated in Fig. 6(b), both the outer and inner regions of the microcavity possess significant net optical gain (gain > 0), such that the fundamental transverse mode profile can be excited. The higher-order mode has larger photon density near the top of the triangular wire, as can be seen in the simulated mode patterns of Fig. 4(d) and Fig. 5(d). Therefore, the emission process is more efficient for the higher order transverse mode at lower excitation power in wider triangular wires. We thus conclude that high absorption efficiency of CsPbBr3 accounts for the nonuniform photoinduced carrier distribution in these microcavities.
Fig. 6. (a) and (b) illustrate schematic pictures of samples under lower and higher excitations. Outer region of microcavity reaches the gain status first, and the inner core reaches population inversion only at higher pumping.
Theoretically, microwire structures which possess whispering gallery modes (WGMs) due to total internal reflection have low threshold value and high quality factor [50–52]. As a result, WGMs lasing are usually observed in triangular, hexagonal, and square plate structures [32]. However, as shown in Fig. 7(a), the geometry of the triangular plate is “equilateral triangle”, which is different from “isosceles right triangle” in our case. The optical path of WGM in an “equilateral triangle” is also an “equilateral triangle”. Thus, the incident angle to each interface is approximately 30°, and it becomes larger for square and hexagonal plate cases. The refractive index of CsPbBr3 can be approximate to 2 in 530 nm [53], so the “equilateral triangle plate” satisfies the condition for total internal reflection condition. The lateral cross sections of triangular wires synthesized on a substrate are mostly in the shape of “isosceles right triangle” [20,22,23,54]. Based on the simulation results in Fig. 4(d), 5(d), higher-order transverse mode patterns suggest normal incident to the isosceles interfaces by the transverse component of the propagation vector. We further simulated the situation when the transverse component of the propagation vector dominates inside the microcavity with the base line a = 2.6 µm (as shown in Fig. 7(b)). The norm of the electric field indicates that these cavity photons are very easy to leak out. Therefore, optical modes with higher-order transverse mode patterns experience larger optical loss. In addition, photons are easier to leak out via substrate interface due to small material refractive index difference. Hence, we did not observe WGMs lasing in these samples. The lasing action of all our samples results in interference patterns from two end facets as shown in Fig. 2(f). We believe WGMs are more difficult to sustain in CsPbBr3 microwires with “isosceles right triangle” cross section.
Fig. 7. Illustrations of optical path of cavity photons in (a) equilateral triangle plate and isosceles right triangle cross section of CsPbBr3 microwire. (b) The norm of electric field indicates a very weak confinement situation while transverse component of propagation vector dominates inside microcavity with base line a = 2.6 µm.
4. Conclusions
The CsPbBr3 triangular microwires synthesized by CVD have unique isosceles right triangle cross-section. Multi-mode lasing behaviors are generally observed in these microcavities, and the resonance modes identified by simulations which fit the PL spectra very well. We found that triangular microwires with a wider base length (a > 2.6 µm in our samples) can normally possess lasing action with higher-order transverse modes. Thus, nonuniform distribution of photoinduced carriers should be an important consideration in determining the emission spectrum for general CsPbBr3 triangular microwires. The unique isosceles right triangle cross-section is the main reason for the difficulty of observing WGMs in CsPbBr3 triangular microwires systems. The present studies directly point out that the effects of cavity size and geometry should be considered carefully for the future design of low threshold wire-based laser devices.
Funding
Ministry of Science and Technology, Taiwan (109-2112-M-001-046, 109-2112-M-006-016, 110-2112-M-006-019).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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