High-Performance Multiwavelength GaNAs Single Nanowire Lasers

In this study, we report a significant enhancement in the performance of GaNAs-based single nanowire lasers through optimization of growth conditions, leading to a lower lasing threshold and higher operation temperatures. Our analysis reveals that these improvements in the laser performance can be attributed to a decrease in the density of localized states within the material. Furthermore, we demonstrate that owing to their excellent nonlinear optical properties, these nanowires support self-frequency conversion of the stimulated emission through second harmonic generation (SHG) and sum-frequency generation (SFG), providing coherent light emission in the cyan-green range. Mode-specific differences in the self-conversion efficiency are revealed and explained by differences in the light extraction efficiency of the converted light caused by the electric field distribution of the fundamental modes. Our work, therefore, facilitates the design and development of multiwavelength coherent light generation and higher-temperature operation of GaNAs nanowire lasers, which will be useful in the fields of optical communications, sensing, and nanophotonics.


S2. Threshold gain calculations
The lasing threshold in a NW Fabry-Perot cavity is determined by the following relationship where Γ is the mode confinement factor, describing the mode overlap with the gain medium, αi is the propagation loss, L is the NW length and R is the reflectivity of the end facets.The mode confinement factor can be written as where the numerator integral is taken over the gain medium cross section, while the denominator integral is taken over all space.Here, c is the speed of light,  0 is the vacuum permittivity, nr is the refractive index of the material, and E and H are the electric and magnetic fields, respectively.The parameters Γ, αi and R can be found from finite-difference time-domain (FDTD) simulations (see Figure S3).

S3. Mode analysis
It is important to single out the dominant lasing mode in the NWs.Using the FDTD results of section S2, we simulate the lasing threshold gain of the dominant waveguided modes in the NWs (Figure S4a).For wavelengths longer than 950 nm, the HE11a and HE11b modes have the lowest threshold gain values.Since the geometry of the reference and SAE-grown NWs studied in this work is similar, the HE11a and HE11b modes are expected to dominate the lasing in both samples.This can be confirmed by looking at the mode-spacing between the lasing peaks in the PL spectra (Δλ), which for Fabry-Perot cavities is given by where λ is the light wavelength, L is the cavity length (in this case, the NW length), and ng is the group index of refraction, determined from the FDTD calculations.Figure S4b   This implies that rates of non-radiative recombination, which causes quenching of the PL intensity with increasing temperature, are the same for the two growth methods.The two samples are growth using different cooling mechanisms of the shroud, which affects the background pressure during the growth.Though the use of the higher background pressure may affect incorporation of background impurities, it does not affect the PL intensity of our NWs.As demonstrated in Figure S7, no significant difference of the PL intensity is observed for the NWs grown with a water-(black) and LN2 -cooled (blue) shroud.Thus, in our case the cooling system of the MBE chamber did not affect the optical quality of the nanowires.In combination with the results of Figures S5 and S6, this confirms that the background pressure during the NW growth does not influence the formation of non-radiative defects and, therefore, the lasing performance of the studied NWs.  Figure S9 shows the lasing threshold vs NW length for SAE-grown (squares) and reference (circles) NWs, where no correlation between these parameters can be observed.From Figs S8 and S9, it is apparent that the observed improvement in the lasing performance of the SAE-grown NWs is not primarily related to their length.

S6. Localization Energy
In highly mismatched materials, such as GaNAs, the exciton localization due to alloy fluctuations can be very large.In such materials, the exciton density of states contains a low-energy tail comprising localized exciton states, located below the free-exciton energy.For energies below the band gap energy, the density of the localized states (DLS) within such a band tail decays exponentially at lower energies as: where E is the exciton energy and E0 is the characteristic localization energy, which is a measure of the degree of localization in the material.The localization energy of an individual NW can, therefore, be extracted from the acquired PL spectra at low excitation densities, as exemplified in Figure S10.

S7. Rate equation analysis
To model the recombination dynamics of the system, we use a set of rate equations (adapted from [3] ), governing the concentration of free excitons (N), localized excitons (NLE) and photons in the lasing photon mode (S): Here, ηp is the absorption efficiency of laser power by the NW, taken to be 1% [3] , P is the excitation power, V is the NW volume calculated for a 6 µm-long NW having a 400 nm diameter, τsp,FE and τsp,LE are the spontaneous recombination lifetimes of the free and localized excitons, respectively, both taken to be approximately 3 ns, [3] C is the Auger coefficient, assumed to be 10 -30 cm 6 /s. [2]vg and Γ are the group velocity and mode confinement factor of the mode, deduced from the FDTD calculationssee section S3. gth is the threshold gain which is obtained to assure the best fit to the data of Figure 2c of the main text, τrlx is the relaxation time of free excitons into the localized states, [3] DLS is the density of localized states and β is the spontaneous emission coupling factor.gmax(N) is the peak gain, which can be calculated from where the values of g0, Ns and Ntr are taken from [3] .Table S1 displays the values for the parameters used in the simulations displayed in the main paper, where DLS and β are used as variables.

S8 Band gap energy estimation
Figure S11 shows the squared photoluminescence excitation (PLE) intensity acquired from the reference (the orange line) and SAE-grown (the purple line) NWs, respectively.From the PLE curves, the band gap energies may be estimated as the intersection of their tangents with the x-axis.By this process (see the black lines in the figure) we find the band gap energies of 1.21 and 1.20 eV for the SAE-grown and reference samples, respectively.Using these band gap energies, the nitrogen content can be estimated from the band-anticrossing model, where the relationship between the band gap energy and nitrogen content is given by: where Eg GaAs is the band gap energy of the nitrogen-free GaAs, x is the nitrogen content, E N is the energy level of the single nitrogen impurity and the V-parameter determines the strength of the interaction between the host and nitrogen states. [4]The last two parameters have been previously estimated as 1.65 eV and 2.5 eV, respectively. [4,5]From the equation, we find that the nitrogen concentrations in the SAE-grown and reference samples are 2.2% and 2.3%, respectively.

S9 Statistical analysis t-test
The t-test tests the null-hypothesis that the population means of two populations is equal: The test statistic, T, is given by where ̅ and  ̅ are the sample means,   and   are the pooled standard deviations and n and m are the sample sizes.From T, the p-value may be calculated as the probability of finding the given T if the null hypothesis is true.For a given significance level, α, the null hypothesis may be rejected if p<α.

Linear regression test
As a measure of the linear dependence of two variables A and B in a sample, the Pearson correlation coefficient may be used: Here, cov(A,B) is the covariance of the two variables, and σA and σB are the sample standard deviations.From ρ we can calculate the p-value, which is used to test the null hypothesis that the ρ of the population is 0: To determine how well the linear model can explain the variance in the data, the coefficient of determination, R 2 (=ρ 2 ), can be used.

S10 Influence of the β-factor on the lasing threshold
The β factor affects the photon density in the lasing mode, which relates to the rate of stimulated emission.However, the main influence of the lasing threshold stems from the concentration of free excitons, N, which is not directly related to β but depends on non-radiative recombination (which is the same in both structures) and also the density of localized states.By carefully evaluating the simulation results, we note that with the simulation parameters used in this manuscript (see Table S1 of section S7), changes of β within the range of 0.01-0.05do not significantly affect the lasing threshold.This is shown in Figure S12, which shows the power dependence of the lasing mode photon density, S, simulated using the parameters of Table S1 of section S7, a localized state density of 7•10 18 cm -3 , and β varying between 0.01 and 0.05.In all simulations, the lasing threshold is estimated to be ~170 mW.We also note that similar results showing a lasing threshold independent of β have been previously reported. [5]gure S12.Simulated photon density, S, as a function of excitation power density obtained assuming a fixed density of the localized states (7•10 18 cm -3 ) for several parameters β relevant to this work.

Figures
Figures S1 shows representative scanning electron microscopy (SEM) images of several nanowires (NW) grown by selective area epitaxy (SAE) (a-c) and reference (d-f) structures, transferred to gold substrates.Figure S2 similarly shows top-view SEM images of as-grown NWs grown by the SAE and reference growth methods.From the top-view SEM images, undergrowth on the substrate can be clearly observed in the reference sample (Figure S2b), while the substrate of the SAE-grown NWs does not show evidence of such undergrowth deposition.

Figure S1 :
Figure S1: SEM micrograph of NWs grown using the SAE (a-c) and reference (d-f) growth methods, transferred to gold substrates.

Figure S2 :
Figure S2: Top-view SEM micrograph of as-grown NWs grown using the SAE (a) and reference (b) growth methods.

Figure S3 :
Figure S3: Simulated wavelength dependence of Γ, αi and R for a 400 nm diameter NW.
shows the measured mode spacings for the reference (the open circles) and SAE-grown (the solid squares) NWs and the simulated dependencies (the solid lines) for the main modes using equation (S1).The majority of the NWs in both structures exhibit lasing in the HE11a or HE11b modes, though lasing in the HE21b mode is also observed.Polarization-resolved measurements of the lasing emission support the mode assignment, exemplified by Figure S4c-e (the symbols) for the HE11a, HE11b, and HE21b modes, respectively, along with the anticipated polarization dependence based on the FDTD calculations (the solid lines).

Figure S4 :Figure S5 :
Figure S4: (a) Simulated spectral dependence of threshold gain values for the five main cavity modes of a 400 nm-thick NW at the wavelength of 960 nm.(b) Mode spacing calculated for the five main modes (the dashed lines) and measured from the reference (the open symbols) and SAE-grown (the closed symbols) structures.(c)-(e) Measured (the symbols) and calculated (the lines) polar plots of the HE11a, HE11b, and HE21b modes, respectively.The gray bars indicate the NW orientation.

Figure S6 :
Figure S6: (a) Average integrated PL intensity acquired from 10 reference (the orange circles) and SAE-grown (the purple squares) NWs.The error bars indicate the standard deviation.(b) Integrated PL intensity acquired from reference (the orange circles) and SAE-grown (the purple squares) NW arrays.The measurements were performed using a cw-laser emitting at 800 nm wavelength.

Figure S7 :
Figure S7: The average PL intensity among NW samples grown using water cooling (the black dot) and LN2 cooling (the blue square).The error bars indicate the standard deviation.

Figure S8 :
Figure S8: SEM micrographs of two SAE-grown (a, c) and two reference (b, d) NWs.(e, f) Power dependence of the PL intensity of the same NWs as shown in (a-d).

Figure S10 :
Figure S10: Exponential fit of the low-energy tail of the PL spectra from an SAE-grown (a) and reference (b) NW to extract the localization energy, E0.

Figure S11 :
Figure S11: PLE spectra acquired from arrays of SAE-grown (the purple line) and reference (the orange line) NWs at 4 K.The black lines are the tangents of the PLE curves, which are used to estimate the band gap energy by their intersection with the x-axis.