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Optical properties of neodymium ions in nanoscale regions of gallium nitride

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Abstract

Wide bandgap semiconductors are increasingly important for bioimaging applications, as they can possess good biocompatibility and host a large range of fluorescent defects spanning the visible to infrared. Gallium nitride is one promising host for photostable fluorophores. In particular, neodymium (Nd)-doped gallium nitride (GaN) shows bright near-infrared fluorescence and narrow room temperature linewidth and is therefore a candidate material for fluorescent probes for bioimaging. To explore the conditions necessary to generate biomarkers based on Nd:GaN, this paper reports the room temperature photoluminescence (PL) properties of small ensembles of Nd ions implanted into the nanoscale regions of GaN epilayers. The minimum volume of Nd-implanted GaN that can be optically detected in this study is about 8×104 nm3 and the minimum detected ensemble of Nd ions is about 4×103, although not all of implanted Nd ions activate as luminescence centers. We show from the PL excitation spectra that the strongest resonant excitation appears at 619 nm, attributed to the 4I9/24G5/2 (4G7/2) transition in the 4f-shell. We measure the luminescence lifetime to be several tens of microseconds. We also identify the presence of a different excitation mechanism from the resonant excitation when excited below 510 nm (above 2.43 eV).

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Corrections

Shin-ichiro Sato, Manato Deki, Hirotaka Watanabe, Shugo Nitta, Yoshio Honda, Tomoaki Nishimura, Brant C. Gibson, Andrew D. Greentree, Hiroshi Amano, and Takeshi Ohshima, "Optical properties of neodymium ions in nanoscale regions of gallium nitride: erratum," Opt. Mater. Express 11, 524-524 (2021)
https://opg.optica.org/ome/abstract.cfm?uri=ome-11-2-524

1. Introduction

Gallium Nitride (GaN) is an important material for biological applications due to its biocompatibility [1], relatively mature fabrication and doping processes [25] and chemical functionalization [6,7]. These properties have led to reports of GaN-based biosensors [8,9], and fabrication of nanometer-sized GaN particles/wires [1012], opening the door to lifescience applications such as bioimaging and biosensing.

By exploring lanthanide (Ln)-doped GaN, it is likely possible that a suite of materials with nominally similar but tailored properties can be created. Ln-doped GaN is extremely photostable because energy levels in the 4f-shell, which are involved in the luminescence transitions, are surrounded by filled 5s and 5p orbitals [13,14]. This photostability should be contrasted with biological fluorophores, organic dyes, and quantum dots have been developed for fluorescent probes, which exhibit bleaching and blinking, limiting their effectiveness [13,15]. In addition, the narrow optical linewidth of Ln-doped GaN at room temperature (RT) enables signals to be distinguished from the broad autofluorescence of biological components [16].

In the context of bio-imaging, Nd-doped GaN is of particular interest due to its strong near infrared (NIR) luminescence at 0.9∼1.1 µm, and sharp linewidth (about 2 nm) at RT [17,18], which overlaps with NIR biological windows [19,20]. To realize the potential of Nd-doped GaN for bioimaging/sensing, small ensembles of Nd ions must be optically detectable at RT with a high signal-to-noise (S/N) ratio. Another important aspect for biological applications is to determine the resonant excitation conditions. Resonant excitation enables higher brightness from fluorescent probes with lower excitation power, providing better contrast against background luminescence such as autofluorescence of biological tissues, and also minimizing the effects of phototoxicity on biological samples [21].

Here we show the RT PL properties of small ensembles of Nd ions in a GaN epilayer. We used lithographically-defined masks to implant the Nd ions into nanoscale regions of the GaN epilayers. Isolated nanoscale Nd-implanted regions in GaN epilayers were chosen to mimic the spatial dimensions of Nd-doped GaN nanoparticles as fluorescent probes. The PL excitation (PLE) spectra reveal the efficient excitation conditions. The luminescence lifetime and the excitation wavelength dependence of PL spectrum are also determined.

2. Experimental

The samples used in this study were device-grade undoped GaN epilayers with thickness of 5 µm grown by metalorganic vapor phase epitaxy (MOVPE) on n-type GaN (n-GaN) substrates. The GaN epilayer was highly resistive and the donor concentration was below 1×1013 cm−3 according to capacitance-voltage characteristics. The detail of GaN epilayers fabricated under similar conditions has been reported elsewhere [22]. Commercially available 6 µm-thick undoped GaN epilayers on Sapphire substrates (POWDEC K.K.) was also used. Using these GaN epilayers, three types of samples were fabricated as listed in Table 1. Resist films were coated on the GaN epilayers and periodic dot patterns were formed using either photolithography or electron beam (EB) lithography techniques to perform Nd implantation into nanoscale regions (circles and squares) of GaN epi-layers. The minimum area of implantation regions was 1×1 µm2 squares for the photolithographic pattern and 60×60 nm2 squares for the EB lithographic pattern. The samples were then implanted with 100 keV or 150 keV Nd ions at the fluence of 1.0×1014 cm−2 at RT at Takasaki Advanced Radiation Research Institute, National Institutes for Quantum and Radiological Science and Technology (QST). The implanted-Nd peak depth, the thickness (half-width) of implanted layer, and the concentration were estimated to be 24 nm, 21 nm, and 4.5×1019 cm−3 for 100 keV, and 32 nm, 29 nm, and 3.3×1019 cm−3 for 150 keV, respectively, according to Monte Carlo simulation code, SRIM (see Fig. 1) [23]. Since all Nd ions were implanted within 80 nm from the surface, luminescence properties of implanted-Nd ions were unaffected by the substrate. The resist films were removed after implantation and a 50 nm thick SiN cap layer was formed on the surface using a magnetron sputtering method to suppress the deterioration of crystallinity by preferential evaporation of nitrogen during post thermal annealing [24]. The Nd-implanted samples were then annealed at either 1200 °C for 1 min or 1250 °C for 2 min under N2 atmosphere (1 atm) using an infrared furnace to remove radiation induced defects and to activate implanted Nd ions as luminescence centers. The furnace temperature rose to the set temperature in 1 min, then kept at the temperature for the set time, and was then naturally cooled down to RT for about 15 min. The SiN cap layer of the samples were subsequently removed by hydrofluoric acid treatment (HF:H2O = 1:5, 20 min). No significant diffusion of Nd after thermal annealing was found in this study.

 figure: Fig. 1.

Fig. 1. Distribution of 100 keV and 150 keV Nd ions implanted with GaN calculated by SRIM. The ion fluence and the mass density of GaN were 1×1014 cm−2 and 6.1 g/cm3, respectively.

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Tables Icon

Table 1. Fabrication conditions of the samples used in this study.

Luminescence properties of Nd-implanted regions in the samples were characterized by a home-built laser scanning confocal microscopy (CFM). A Supercontinuum wavelength-tunable pulsed laser (6 ps pulse width, 0.1∼80 MHz repetition rate) was used for excitation. Here, the center wavelength and the bandwidth of excitation laser are represented by 600/20 nm, for instance (600 nm center wavelength and 20 nm bandwidth). Photon emission from the Nd-implanted regions was collected with an objective lens (numerical aperture, NA = 0.85) and detected by a Si avalanche photo-diode (APD). The PL spectra were investigated by an imaging spectrometer installed in the CFM. A 900 nm long pass filter was placed in front of the APD and the spectrometer to selectively collect photons emitted from the implanted-Nd ions. The PLE spectra ranging from 420 nm to 850 nm and the luminescence lifetime were also measured using the same measurement setup. All measurements were performed at RT.

3. Results and discussion

Representative CFM images of the Nd-doped GaN sample B are shown in Figs. 2(a)-(e). The implanted-Nd ions were excited with 620/12 nm light for the resonant excitation, which will be discussed in more detail in the next paragraph. The obtained CFM images reproduced the designed EB lithography pattern as shown in Figs. 2(a)-(b), indicating the EB lithography pattern was successfully transferred to the Nd implantation pattern. In Figs. 2(c)-(e), array structures of 200×200 nm2, 100×100 nm2, and 60×60 nm2 square dots were studied. In these cases, the observed luminescence spots were larger than the actual sizes of implanted regions and the images reflected the shape of the CFM point spread function (PSF), since the lateral resolution (445 nm according to the Rayleigh limit) of CFM was larger than the implanted regions [25].

 figure: Fig. 2.

Fig. 2. (a) A wide range (200×200 µm2) CFM image of sample B at RT with 0.5 µm scan step. The excitation wavelength and power were 620/12 nm and 2.0 mW, respectively. Shape and size of implanted regions are shown as green characters. For example, “250” in the image indicates that the dot sizes of implanted regions are squares with the side of 250 nm and circles with the diameter of 250 nm. (b)-(e) Fine CFM images of 1×1 µm2, 200×200 nm2, 100×100 nm2, and 60×60 nm2 square implanted regions through excitation of 620/12 nm at 2.0 mW. The scan step was 100 nm and the interval of implanted regions was 5 µm. Note that the background counts (average counts at unimplanted regions) are unchanged for all images. (f) Photon counts from implanted regions as a function of side length of implanted regions. The estimated number of Nd ions is displayed on the top of figure. The blue straight line is the fitting curve by Eq. (1). (g) PL spectra of 200 nm, 100 nm, and 60 nm squares at RT. The ordinate is normalized by the peak intensity at 916.3 nm of 200 nm square implanted region. The spectrum at an unimplanted region is also shown in blue. The integration time was 30 s for the 200 nm square and 100 s for the other data.

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Figure 2(f) shows the photon counts from implanted regions as a function of side length of implanted regions. The estimated number of Nd ions is displayed on the top of figure. The excitation laser was focused on the center of implantation regions and the average photon counts for 30 s were recorded. The background counts, which was measured to be 1.1 kcps at an unimplanted region, was subtracted to obtain the net photon emission intensity from implanted Nd ions. The number of implanted Nd ions was derived by the product of implantation fluence (1.0×1014 cm−2) and the implantation area. The photon counts from implanted regions were stable for prolonged time and increased with increasing the number of Nd ions. The smallest region which was optically observed in this sample was the 60 nm square implanted regions. The estimated number of implanted Nd ions in the 60 nm square regions was 4×103, although not all of implanted Nd ions activated as luminescence centers after thermal annealing. Assuming that the PSF is the Gaussian function, the fraction of activated implanted-Nd ions is unchanged regardless of implantation area, and the PL intensity is proportional to the laser intensity for simplicity, the relationship between the PL intensity (I) and the side length of implanted region (d) can be expressed as:

$$ \begin{aligned} I(d) &=A \int_{-d / 2}^{d / 2} \int_{-d / 2}^{d / 2} \mathrm{d} x \mathrm{d} y \frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{x^{2}+y^{2}}{2 \sigma^{2}}\right) \\ & \propto \operatorname{erf}^{2}\left(\frac{d}{\sqrt{2} \sigma}\right) \end{aligned}$$
where σ is the standard deviation of the Gaussian function and A is the free parameter. The calculation result, shown as a solid line in Fig. 2(f), is well fitted to the experimental data and we obtain σ = 223 nm. The FWHM ($2\sqrt {2\; \textrm{ln}(2 )} \sigma $) is calculated to be 526 nm, being slightly larger than the Rayleigh limit (445 nm). This is thought to be due to misalignment of the confocal system.

PL spectra of the 200 nm, 100 nm, and 60 nm square implanted regions are shown in Fig. 2(g). The excitation wavelength was 620/12 nm. Three peaks originating from the 4F3/24I9/2 transition in Nd3+ appeared at 916.3 nm, 934.6 nm, and 944.0 nm from all the implanted regions while no peak was found in the unimplanted regions (blue line). Most of the background counts were thought to be caused by luminescence of intrinsic and surface defects. Further smaller number of Nd ions can be optically detected by the reduction of undesired defect luminescence.

 figure: Fig. 3.

Fig. 3. (a) PLE spectra of the sample A at RT. The excitation wavelength range was 595–635 nm and 760–850 nm. The emission wavelength above 900 nm was collected. The excitation laser power and bandwidth were 50 µW and below 2 nm, respectively. (b) The simplified energy diagram for the luminescence of Nd3+. The dashed blue arrows indicate non-radiative transitions.

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Figure 3(a) shows the PLE spectrum at RT of the sample A. First the wide range scan from 420 nm to 850 nm revealed that the PL intensity was enhanced when excited through around 620 nm, 770 nm, and 840 nm due to the resonant excitation of Nd3+. The excitation wavelength was then scanned by 1 nm step and as a result at least seven peaks at 601 nm, 612 nm, 619 nm, 772 nm, 780 nm, 790 nm, and 836 nm appeared. According to the Dieke’s diagram, the three peaks appeared at 601 nm, 612 nm, and 619 nm are attributed to the transition from 4I9/2 state to 4G5/2 and/or 4G7/2 states, the three peaks appeared at 772 nm, 780 nm, and 790 nm are attributed to the transition from 4I9/2 state to 4F7/2 and/or 4S3/2 states, and the peak at 836 nm are attributed to the transition from 4I9/2 state to 4F5/2 and/or 2H9/2 states [17,18]. The energy levels between 4G5/2 and 4G7/2 states, 4F7/2 and 4S3/2 states, and 4F5/2 and 2H9/2 are undistinguishable in energy. The simplified energy diagram for the luminescence of Nd3+ is shown in Fig. 3(b). The intensity of the 4I9/24G5/2 (4G7/2) transition is higher than the other two transitions. The upward trend in the spectrum indicates the presence of at least one more peak above 850 nm, although we could not observe this in our measurement setup.

Figure 4 shows the excitation power dependence of photon emission intensity of the sample A. The ordinate is the photon counts from implanted-Nd ions with the background subtracted. Three resonant (612 nm, 770 nm, 838 nm) and two non-resonant (530 nm and 700 nm) excitation wavelengths were employed. In all cases, the photon emission intensity increased with increasing excitation power density and showed the saturation behavior. The saturation behavior of photon emission intensity can be expressed as:

$$\begin{array}{c} {I(W )\propto {{\left( {1 + \frac{{{W_0}}}{W}} \right)}^{ - 1}}} \end{array}$$
where I, W, and W0 are the photon counts from luminescence centers, the laser power density, and the saturation power density, respectively [14,25]. The calculated results, shown as solid lines in Fig. 4, are well fitted to the experimental data, and we obtain W0 = 140 kW/cm2 for 530 nm excitation, 93.7 kW/cm2 for 612 nm excitation, 294 kW/cm2 for 700 nm excitation, 289 kW/cm2 for 770 nm excitation, and 152 kW/cm2 for 838 nm excitation. It is known that W0 for non-resonant excitation is smaller as the excitation wavelength is shorter [14]. The values of W0 for 530 nm and 700 nm excitations were in agreement with this trend. In the case of resonant conditions, the value for 612 nm excitation was smaller than the values for 770 nm and 838 nm excitations, indicating the implanted-Nd ions were more effectively excited under the resonant excitation of 4I9/24G5/2 (4G7/2) transition than 4I9/24F7/2 (4S3/2) and 4I9/24F5/2 (2H9/2) transitions.

 figure: Fig. 4.

Fig. 4. Photon emission intensity of the sample A as a function of excitation power density. Excitation wavelengths (λEX) are 530 nm (black squares), 612 nm (red circles), 700 nm (green triangles), 770 nm (blue inverted triangles), 838 nm (pink diamonds). Solid lines are fitting curves by Eq. (2) and the derived saturation power densities (W0) are shown in the figure.

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Different values of the saturation power density are attributed to the different excitation cross sections and/or the different transition lifetimes. The saturation power density can be expressed as:

$$\begin{array}{c} {{W_0} = \frac{{{E_{\textrm{ex}}}}}{{{\sigma _{\textrm{eff}}} \times {\tau _1}}}} \end{array}$$
where Eex, σeff, τ1 are the excitation photon energy, the effective excitation cross section, and the luminescence lifetime of Nd3+, respectively. The luminescence lifetime of Nd3+ was characterized by the time-resolved photoluminescence (TRPL) spectroscopy on the CFM system. The pulsed laser with repetition rate of 0.1 MHz, which was the minimum repetition rate in our measurement setup, was used for excitation. Photon emission from Nd3+ ions excited by previous pulses is also included in the TRPL spectrum as the exicitation pulse interval (10 µs) is comparable to (or longer than) the expected luminescence lifetime. In this case, the decay compornent for Nd3+ ions was represented by the sum of series of exponential decay function:
$$\begin{aligned} I(t) &=A_{1} \sum_{k} \exp \left(-\frac{t+k t_{\mathrm{R}}}{\tau_{1}}\right)+A_{2} \exp \left(-\frac{t}{\tau_{2}}\right)+B \\ &=A_{1} \frac{\exp \left(-\frac{t+t_{\mathrm{R}}}{\tau_{1}}\right)}{1-\exp \left(-\frac{t_{\mathrm{R}}}{\tau_{1}}\right)}+A_{2} \exp \left(-\frac{t}{\tau_{2}}\right)+B \end{aligned}$$
where I(t) the photon counts at the time t, τ2 the residual defect luminescence lifetime (∼ns), tR the excitation pulse repetition rate (10 µs), and B the background photon count including system noise. A1 and A2 are the fitting parameters reflecting the photon emission intensities from the Nd3+ ions and the residual defects, respectively. Note that the first term becomes ${A_1}\textrm{exp}({ - t/{\tau_1}} )$ when ${t_\textrm{R}} \gg {\tau _1}$. The relevance of Eq. (4) was confirmed by characterization of transition lifetime of another color centers. Figure 5 shows a typical TRPL spectrum of the Nd-doped GaN sample A. In this measurement, the lifetime was determined to be 47.5 µs. The luminescent lifetimes on average were 45.7 µs for the sample A, and 78.3 µs for the samples B and C. No significant difference of luminescence lifetieme on excitation wavelengths was observed, since the energy transfer to Nd3+ ions (the excitation process) was much faster than luminescence lifetime [14]. In general, post thermal annealing after Nd ion implantation is peformed to recover the crystalinity of GaN host and the residual radiation damage (defects) strongly depends on the luminescence lifetime in addition to the activation ratio of Nd3+ ions. The luminescence lifetime is shortened by residual radiation damages due to the decrease in local symmetry surrounding Nd3+ ions on the Ga site [26]. The results of luminescence lifetime characterization suggest that the accumulated damage after implantation was lower in 100 keV than 150 keV and also, the better recovery from implantation induced damage was obtained by the annealing at 1250 °C than at 1200 °C. Note, however, that temperature above 1500 °C was believed to be required for the perfect recovery of GaN crystallinity [14,27] and thus the improvement of thermal annealing condition is crucial to further increase the activation of Nd3+. The effective excitaiton cross section was derived to be 4.4×10−20 cm2 for 612 nm excitation when the lifetime of 78.3 µs was used for Eq. (3).

 figure: Fig. 5.

Fig. 5. A TRPL spectrum of the sample A at RT. Closed circles are the raw data and the blue curve is the fitting by Eq. (4). The lifetime was derived to be 47.5 µs in this case.

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Figure 6 shows the PL spectra of the sample C with different exctation wavlengths at RT. Two dominant peaks appeared at the wavelengths of 910.4 nm and 916.3 nm, and two minor peaks appeared at the wavelengths of 934.6 nm and 944.0 nm. These peaks are attributed to the 4F3/24I9/2 transition in the 4f-shell and its crystal field splitting [17,18]. Focusing on the two major peaks, only the peak at 910.4 nm appeared in the case of 410 nm and 430 nm excitations, the peak at 916.3 nm appeared with the peak at 910.4 nm at 450∼510 nm excitations, and the peak at 910.4 nm dissapeared at the excitations above 530 nm. The minor two peaks unambiguously appeared at the excitations above 530 nm. It would appear that the transition of dominant peak from 910.4 nm to 916.3 nm occurs independent of the resonant excitation wavelengths shown in Fig. 3. Both peaks at 910.4 nm and 916.3 nm also appear when excited via energy transfer from free carrier (electron and hole) recombinations generated by a ultraviolet (UV) laser. This result suggests that a different mechanism from the resonant excitation is involved in the Nd3+ excitation through below 510 nm lasers (above 2.43 eV). It is known that Eu3+ ions in GaN are excited via energy transfer from carriers trapped in localized levels which are formed by Eu3+ ion itself [28,29], and the similar phenomenon would be expected in the case of Nd-doped GaN. However, unlike the case of Eu3+, this energy transfer mechanism is not efficient in Nd3+ ions, as the resonant excitation did not appear below 510 nm.

 figure: Fig. 6.

Fig. 6. PL spectra of the sample C through different excitation wavelengths at RT. Values shown in the figure represent the center excitation wavelength and the bandwidth. Dashed vertical lines are drawn to indicate observed peaks.

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4. Summary

In summary, we have observed photon emissions from Nd3+ ions implanted into nanoscale regions of GaN at RT. The photon emission mechanism is based on the transition among intra-4f shells in Nd3+ ions which are isolated from surrounding environments and hence is extermely stable for prolonged periods, enabling their long-term observation. The smallest volume optically detected in this study was about 8×104 nm3 (60×60×21 nm3) and was comparable with ceramic nanoparticles which have been used for fluorescence probes (less than 100 nm in diameter) [30]. Therefore, this study opens the door to lifescience and bioimaging applications of Nd-doped GaN. It was also revealed that the luminescence lifetime of Nd3+ in GaN was much longer than autofluorescence (∼ns), enabling the improvement of contrast (S/B ratio) using the time-gate method [16]. The use of NIR detectors can further enhance the detection efficiency, since 1.1 µm emissions originating from the 4F3/24I11/2 transition are also detected in addition to 916.3 nm emissions originating from 4F3/24I9/2 transition.

Funding

Australian Research Council (CE140100003, FT160100357, LE140100131); Japan Society for the Promotion of Science (JP17KK0137, JP18H01483).

Acknowledgement

A part of this work was conducted at the AIST Nano-Processing Facility, supported by “Nanotechnology Platform Program” of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, Grant Numbers JPMXP09F18AT0061 and JPMXP09F19AT0040.

Disclosures

The authors declare no conflicts of interest.

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Figures (6)

Fig. 1.
Fig. 1. Distribution of 100 keV and 150 keV Nd ions implanted with GaN calculated by SRIM. The ion fluence and the mass density of GaN were 1×1014 cm−2 and 6.1 g/cm3, respectively.
Fig. 2.
Fig. 2. (a) A wide range (200×200 µm2) CFM image of sample B at RT with 0.5 µm scan step. The excitation wavelength and power were 620/12 nm and 2.0 mW, respectively. Shape and size of implanted regions are shown as green characters. For example, “250” in the image indicates that the dot sizes of implanted regions are squares with the side of 250 nm and circles with the diameter of 250 nm. (b)-(e) Fine CFM images of 1×1 µm2, 200×200 nm2, 100×100 nm2, and 60×60 nm2 square implanted regions through excitation of 620/12 nm at 2.0 mW. The scan step was 100 nm and the interval of implanted regions was 5 µm. Note that the background counts (average counts at unimplanted regions) are unchanged for all images. (f) Photon counts from implanted regions as a function of side length of implanted regions. The estimated number of Nd ions is displayed on the top of figure. The blue straight line is the fitting curve by Eq. (1). (g) PL spectra of 200 nm, 100 nm, and 60 nm squares at RT. The ordinate is normalized by the peak intensity at 916.3 nm of 200 nm square implanted region. The spectrum at an unimplanted region is also shown in blue. The integration time was 30 s for the 200 nm square and 100 s for the other data.
Fig. 3.
Fig. 3. (a) PLE spectra of the sample A at RT. The excitation wavelength range was 595–635 nm and 760–850 nm. The emission wavelength above 900 nm was collected. The excitation laser power and bandwidth were 50 µW and below 2 nm, respectively. (b) The simplified energy diagram for the luminescence of Nd3+. The dashed blue arrows indicate non-radiative transitions.
Fig. 4.
Fig. 4. Photon emission intensity of the sample A as a function of excitation power density. Excitation wavelengths (λEX) are 530 nm (black squares), 612 nm (red circles), 700 nm (green triangles), 770 nm (blue inverted triangles), 838 nm (pink diamonds). Solid lines are fitting curves by Eq. (2) and the derived saturation power densities (W0) are shown in the figure.
Fig. 5.
Fig. 5. A TRPL spectrum of the sample A at RT. Closed circles are the raw data and the blue curve is the fitting by Eq. (4). The lifetime was derived to be 47.5 µs in this case.
Fig. 6.
Fig. 6. PL spectra of the sample C through different excitation wavelengths at RT. Values shown in the figure represent the center excitation wavelength and the bandwidth. Dashed vertical lines are drawn to indicate observed peaks.

Tables (1)

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Table 1. Fabrication conditions of the samples used in this study.

Equations (4)

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I ( d ) = A d / 2 d / 2 d / 2 d / 2 d x d y 1 2 π σ exp ( x 2 + y 2 2 σ 2 ) erf 2 ( d 2 σ )
I ( W ) ( 1 + W 0 W ) 1
W 0 = E ex σ eff × τ 1
I ( t ) = A 1 k exp ( t + k t R τ 1 ) + A 2 exp ( t τ 2 ) + B = A 1 exp ( t + t R τ 1 ) 1 exp ( t R τ 1 ) + A 2 exp ( t τ 2 ) + B
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