Doping-enhanced radiative efficiency enables lasing in unpassivated GaAs nanowires

Nanolasers hold promise for applications including integrated photonics, on-chip optical interconnects and optical sensing. Key to the realization of current cavity designs is the use of nanomaterials combining high gain with high radiative efficiency. Until now, efforts to enhance the performance of semiconductor nanomaterials have focused on reducing the rate of non-radiative recombination through improvements to material quality and complex passivation schemes. Here we employ controlled impurity doping to increase the rate of radiative recombination. This unique approach enables us to improve the radiative efficiency of unpassivated GaAs nanowires by a factor of several hundred times while also increasing differential gain and reducing the transparency carrier density. In this way, we demonstrate lasing from a nanomaterial that combines high radiative efficiency with a picosecond carrier lifetime ready for high speed applications.

somewhat longer than the doped NW in (b). Approximating their geometry as truncated cones, the undoped NW has a tip diameter of 340 nm, a base diameter of 390 nm and a length of 2.06 m to give a volume of 0.22 m 3 . The doped NW has a tip diameter of 299 nm, a base diameter of 350 nm and a length of 1.52 m to give a volume of 0.13 m 3 , which is some 40% less than that of the undoped NW. operating at 850 nm (TT Electronics OPV302) was used as the reference source. The output of this reference laser was firstly measured by an optical power meter (Thor PM100D) before its spectra was collected by our system using the same parameters as later power-dependant photoluminescence experiments. Being of small spatial size and low divergence, the entire reference beam was collected by our system enabling the ratio between photons collected and CCD counts to be measured. When calculating the absolute external efficiency of our NWs, we assumed light emission to be isotropic. (b) N A = 8.9x10 17 cm -3 , Injection= 1.0x10 9 cm -3 s -1 (c) N A = 2.8x10 19 cm -3 , Injection= 1.0x10 9 cm -3 s -1 (d) N A = 2.8x10 16 cm -3 , Injection= 1.0x10 24 cm -3 s -1 (e) N A = 8.9x10 17 cm -3 , Injection= 1.0x10 24 cm -3 s -1 (f) N A = 2.8x10 19 cm -3 , Injection= 1.0x10 24 cm -3 s -1 (g) N A = 2.8x10 16 cm -3 , Injection= 1.0x10 29 cm -3 s -1 (h) N A = 8.9x10 17 cm -3 , Injection= 1.0x10 29 cm -3 s -1 (i) N A = 2.8x10 19 cm -3 , Injection= 1.0x10 29 cm -3 s -1 (j) N A = 2.8x10 16 cm -3 , Injection= 1.0x10 34 cm -3 s -1 (k) N A = 8.9x10 17 cm -3 , Injection= 1.0x10 34 cm -3 s -1 (l) N A = 2.8x10 19 cm -3 , Injection= 1.0x10 34 cm -3 s -1 . Note that band bending is reduced by both increased doping and increased rates of carrier generation.   700 nm and appears more pronounced as approaches zero at shorter wavelengths. By considering the response at several particular wavelengths close to the band gap (as plotted in Supplementary   Fig. 3c), we estimate the minority carrier lifetime to be approximately 1 ps. This is in good agreement with the up-conversion measurement presented in Fig. 2 of the main manuscript.
Beyond carrier dynamics, TRS spectroscopy also provides insight into band structure as carriers relax towards their equilibrium distributions following excitation. 8

Supplementary Note 3 | Low temperature PL
In all cases the lifetime of PL emission at low temperature was found to be less than the 80 ps system response of our time correlated single photon counting (TCSPC) setup (see Supplementary   Fig. 4b). Such short lifetimes are consistent with previous measurements [9][10][11][12][13] of GaAs nanostructures at low temperature and indicate the continued dominance of a non-radiative recombination pathway. The surface recombination velocity of GaAs has been found 12-14 to remain relatively high (>1x10 5 cm s -1 ) at cryogenic temperatures and the lifetime of GaAs nanostructures has previously been linked to surface recombination at these temperatures. 12,[14][15][16] In the case of InGaAs/InP wires, surface recombination velocity was found to vary as the square root of absolute temperature between 4 and 77 K. This suggests a capture cross section that is independent of temperature. Such behaviour can be expected for traps with a large capture cross section such as those present at the surface of unpassivated GaAs. 17,18

Supplementary Note 4 | Quantifying radiative efficiency
We quantified radiative efficiency by calibrating our system (see Supplementary

Supplementary Note 5 | Description of the emission from wurtzite nanowires
The emission spectrum of a wurtzite (WZ) GaAs NW as presented in Fig. 1e of the main text is seen to peak at approximately 1.435 eV (865 nm) with a full width at half-maximum (FWHM) of 44 meV.
Some confusion remains in the literature regarding the expected peak position of WZ GaAs, 19 and the value found here sits between those currently published, being 10 meV greater than 20 one report and 10-25 meV less than 21-24 several others. We note that unlike the majority of samples characterised elsewhere, 20,21,23 our NWs are unpassivated and therefore required relatively high pump powers. Considering the power dependant emission of a WZ GaAs nanowire as presented in Fig. 3b of the main text, a higher energy peak appears at 1.531 eV (810 nm) for photoinjected carrier densities greater than 3x10 17 cm -3 . This second peak may be identified as the conduction band to light-hole transition 22 and its appearance is a result of photoinduced band-filling as suggested by the broadening of emission towards higher energy at increased pump powers. Both peaks are seen to slightly redshift with increasing excitation.

Supplementary Note 6 | Role of NW areal density.
The areal density of NWs was observed to strongly affect both their structure and morphology.
Supplementary Fig. 8a-c presents a progressive dilution of a 30 nm colloid used to seed growth, with Supplementary Fig. 8b showing a 25 times dilution relative to Supplementary Fig. 8a and Supplementary Fig. 8c a 250 times dilution. As areal density is reduced, NW height increases and the sidewall morphology is seen to change. Examining the TEM images in Supplementary Fig. 9 the changing appearance of the NW sidewalls can be linked to the introduction of a high density of planar defects. Both the increase in NW height and introduction of planar defects may be considered a product of increased group III supply stemming from the increase in collection area with reduced NW areal density. Where growth is group III limited, an increase in supply will increase the growth rate but also likely the supersaturation and effective V/III ratio, factors both known to affect crystal structure. 25,26 One interesting feature of the NWs grown at lower area densities Supplementary Fig. 8b,c,d is the appearance of a tapered segment near the tip of the NWs. This is likely related to a gradual depletion of Ga from the Au seed particle during the cooling phase which was conducted under arsine. 27 In NWs grown at lower areal densities, a variation in crystal structure was often evident along the length of the NW. Planar defects tended to have a higher density towards the base of the NWs which in some instances gave way to periodic twinning towards the tip of the NWs (see Supplementary Fig. 10). This variation can be related to changes in group III supply as the NW grows in height and its collection area shifts from the substrate to the NW sidewalls. 28 Interestingly, periodic twinning in these NWs grown at low density often showed overgrowth from {111} to {110} type sidewalls.

Supplementary Note 7 | Numerical modelling of recombination with surface bandbending
Unpassivated GaAs is known to be characterised by a high concentration of surface trap states leading to surface band-bending and depletion. [29][30][31] These effects have furthermore been reported to significantly affect emission from unpassivated GaAs NWs 9 and may be expected to vary with optical excitation. In order to assess the potential relevance of surface band-bending to our experiments we performed finite volume modelling using the commercial software package COMSOL Multiphysics. In an approach previously reported, 32 we solved a system of three coupled differential equations in 1D to give electric potential, hole concentration and electron concentration as a function of radial  Fig. 13) shows qualitative similarity to that derived from an analytical model (see Supplementary Note 8 and specifically Supplementary Fig. 16). We further note that the energy distribution and density of surface trap states on GaAs surfaces is relatively difficult to measure and will vary with orientation, 31, 33, 34 treatment 35-38 and doping 33, 39-44 and as such a full quantitative treatment should be considered beyond the scope of this paper. Our results here thus represent a qualitative assessment of how the spatial extent of surface band-bending is reduced by both doping and photoexcitation. They suggest that for the high doping densities or high photoexcitation levels that we fit in Fig. 3 of the main manuscript, surface band-bending may be safely neglected. It is also important to note that a reduced surface state density has previously been associated with p-type GaAs, 42 which would act to further reduce the effects of surface Fermi-level pinning for doped NWs.
Supplementary Fig. 11 plots modelled carrier concentrations along the radial direction for increasing acceptor concentrations from left to right and increasing excitation powers from top to bottom. At the lowest of doping densities and excitation levels ( Supplementary Fig. 11a), carrier concentration is seen to vary across the entire radial profile indicating full depletion. With increasing excitation the depletion width of the minority carriers (electrons) is firstly seen to reduce ( Supplementary Fig. 11d), before at the highest of pump powers the electron and hole concentrations are seen to be both equal and unvarying with radial position (Supplementary Fig. 11j). At a higher doping density, depletion is observed to be only partial even at low excitation ( Supplementary Fig. 11b).
Interestingly, in some specific cases of partial depletion, the superposition of carrier diffusion to the surface and surface band bending was seen to generate a minimum (maximum) in electron (hole) concentration near the NW surface ( Supplementary Fig. 11e). At the highest of doping densities assessed, surface band bending was negligible even at the lowest of excitation powers ( Supplementary Fig. 11c). Considering Supplementary Fig. 11 in its entirety, it is apparent that both heavy doping (right of figure) and high excitation (bottom of figure) act to reduce the spatial extent of band bending.
In order to visualise the spatial extent of surface band bending, we defined a threshold electric field strength (0.1 kV cm -1 ) below which material was considered to be unaffected by surface depletion.
The distance from the surface at which the electric field first reached this threshold was considered the 'depletion width'. Supplementary Fig. 12 plots depletion width as a function of excitation for various doping densities. As determined from Supplementary Fig. 11a, at the lowest of doping densities, 2.8x10 16 cm -3 , surface depletion extends to the centre of the NW which may be considered fully depleted. At a pump density of approximately 1x10 28 cm -3 s -1 this depletion width is seen to fall dramatically before approaching zero. At this point surface band-bending can be considered screened by the photoexcited carrier concentration.
While higher doping densities give smaller depletion widths at low excitation (around 5 nm for 2.8x10 19 cm -3 ), a reduction in depletion width does not occur until higher excitation levels due to the stronger electric field strengths associated with the space charge region of heavily doped materials.
The x-axis at the top of Supplementary Fig. 12 relates the photoexcited carrier densities used in modelling to equivalent powers of a 522 nm 3 m spot-size CW pump laser. In most cases depletion width is not seen to alter significantly until pump powers of approximately 1 mW suggesting that depletion will be significant across the usual power ranges accessed by CW pumping.
Supplementary Fig. 13 plots IQE, as defined by the rate of radiative recombination divided by the carrier generation rate, as a function of photoexcitation for various doping concentrations. As is similarly observed from the analytical modelling plotted in Supplementary Fig. 16, IQE is seen to be constant at low excitation before beginning to increase beyond a given threshold excitation level towards a peak value at extremely high excitation. This behaviour may be related to magnitude of the photoexcited carrier density relative to the background doping density.
Where the photoexcited carrier density is less than the background carrier density, both nonradiative and radiative recombination increase linearly with excitation density. Beyond this, as the majority carrier concentration is increased by photoexcitation, radiative recombination increases as the square of photoexcitation. The threshold photoexcitation density in this case may be approximated by the background doping density divided by the minority carrier lifetime. At the highest of photoexcitation densities, Auger recombination begins to dominate and radiative efficiency is reduced.
The greatest contrast between the results presented here and those derived from the rate equation analysis, as presented in Supplementary Fig. 16, is the magnitude of the IQE advantage enjoyed by doped NWs at low photoexcitation densities. Whereas the rate equation analysis finds heavily doped NWs (4x10 19 cm -3 ) to be up to 3 orders of magnitude more efficient than undoped NWs (1x10 16 cm -3 ) at low excitation, the modelling here finds an IQE advantage closer to 11 orders of magnitude. This large discrepancy stems from the significant carrier depletion found by the present modelling at lower doping densities (see Supplementary Fig. 11a). Carrier depletion acts to reduce IQE by reducing the rate of radiative recombination more quickly than the rate of non-radiative recombination. Lower equilibrium carrier concentrations also, however, reduce the pump power required for the photoexcited carrier concentration to exceed the equilibrium carrier concentration and produce an increase in IQE. Considering the x-axis at the top of Supplementary Fig. 13 we can thus observe that for 1 W CW excitation the difference in radiative efficiency reduces to around 5 orders of magnitude. At powers beyond this the two models converge as the depletion width approaches zero.
To better understand the conditions under which surface depletion may be neglected, we repeated the current steady state modelling for a surface trap density of zero (N t =0 cm -2 ). This scenario is effectively equivalent to simple rate equation modelling as the SRV is maintained at 2.2x10 6 cm s -1 but there is no band-bending or carrier depletion. Supplementary Fig. 14  concentrations. As all our experimental data fulfils these conditions, it may be modelled using a more straightforward rate equation approach.

Describing recombination
As the pulse duration (approximately 300 fs) in our experiments was significantly shorter than the decay process, and the interval between pulses (50 ns) significantly longer, we considered recombination as decay from an initial excited carrier concentration, , by non-radiative, radiative and Auger loss terms: where is the time dependent carrier density, is the ionised acceptor density and , and are the non-radiative, radiative and Auger recombination coefficients respectively. We included an Auger recombination term as some of our experiments employed high pump powers and/or high doping densities. Solving Supplementary Equation 1 numerically for a given photoexcited carrier concentration, , and acceptor concentration, , gives IQE through the following integration: Given the nanoscale dimensions of our emitters we further took self-absorption to be negligible and equated Supplementary Equation 2 with EQE.

Determination of the surface recombination velocity
As surface recombination is assumed to be the dominant form of carrier recombination for unpassivated GaAs NWs, 45 the minority carrier lifetime can be related to the non-radiative recombination coefficient A, the surface recombination velocity and the NW diameter in the following manner 46 : (3) Considering the up-conversion data for the 300 nm diameter doped GaAs NWs presented in Fig. 2 of the main manuscript, a minority carrier lifetime of 3.44 ps thus equates to a non-radiative recombination coefficient of 2.9x10 11 s -1 and a surface recombination velocity of 2.18 x10 6 cm s -1 .

Variation of B and C with doping
Both the radiative and Auger recombination coefficients, and , are known to vary with dopant concentration. 1, 47 In our modelling we defined these parameters as a function of dopant density on the basis of published experimental results. In the case of the radiative recombination coefficient , its variation with doping in p-type GaAs was investigated by Nelson and Sobers. 1 We fitted the following relationship to their data (see Supplementary Fig. 15): The Auger recombination coefficient is less well defined for GaAs with a range of values having been reported. [48][49][50] For p-type doping, Ahrenkiel et al. 47 found the following relationship: For doping levels of below 1.9x10 18 cm -3 , where the above relationship intersects with the value reported by Strauss et al. 48 for intrinsic GaAs (7x10 -30 cm -3 ), we took to be constant.

Trends in IQE with doping
Supplementary Fig. 16

plots IQE as modelled through Supplementary Equations 1 and 2 for GaAs
NWs with a diameter of 300 nm and a surface recombination velocity of 2.2x10 6 cm s -1 . Each curve represents a different level of doping and at low photoexcitation it is apparent that IQE increases with doping from around 0.001% for doping levels of 10 16 cm -3 to a peak of 1% for doping levels of 3.6 x10 19 cm -3 .
The increase in IQE with doping may be attributed to the second term of the differential equation governing recombination, ( ), and represents a decreasing radiative lifetime with doping.
While , IQE remains constant as the rates of non-radiative, ( ), and radiative, ( ), recombination both increase linearly with photoexcited carrier concentration. Beyond these pump powers, the rate of radiative recombination begins to increase as the square of the photoexcited carrier concentration, ( ), and IQE increases steadily with excitation.
At the highest of pump powers Auger recombination becomes significant and IQE is reduced. The threshold for this efficiency droop shifts to lower pump powers with increasing doping but is only significant at extremely high photoexcited carrier concentrations, >10 20 cm -3 ; well beyond those we achieved experimentally.
Although peak efficiency is highest (≈9%) for the lowest of doping concentrations, 10 16 cm -3 , the IQE of this NW does not exceed the IQE of a NW doped to 3.6 x10 19 cm -3 until photoexcited carrier concentrations in excess of 2x10 19 /cm 3 , a regime where thermal effects are likely to be significant.
Importantly, the increase in IQE with doping at low excitation shows a peak at around 3.6 x10 19 cm -3 beyond which Auger recombination becomes significant. The IQE of a NW doped to 1x10 20 cm -3 is seen to be less than that of the NW doped to 3.6x10 19 cm -3 for all pump powers.

Trends in carrier lifetime with doping
As the equation governing recombination is non-linear, we defined the modelled minority carrier lifetime as the time to achieve a carrier concentration of ⁄ . Supplementary Fig. 17

Supplementary Note 9 | Diameter dependence of EQE
The minority carrier lifetime may be defined as follows: where is the non-radiative lifetime and is the radiative lifetime. In our case, surface recombination is the dominant form of recombination and ⁄ ⁄ . A similar approximation of Equation 3 from the main manuscript gives ⁄ . Substituting these approximations into Equation 1 of the main manuscript produces: Where surface recombination dominates, EQE is thus expected to be a linear function of nanowire diameter and inversely proportional to both SRV and . Supplementary Fig. 18 plots experimentally determined values of EQE for both doped and undoped NWs obtained at a photoexcitation carrier density of 9x10 16 cm -3 . A fit of Supplementary Equation 7 gives the parameters 5x10 18 cm -3 and SRV = 1.5x10 6 cm s -1 for the doped NWs and 3x10 16 cm -3 and SRV = 1.0x10 6 cm s -1 for the undoped NWs. Deviation of the doped NW dataset towards a more superlinear relationship may be related in this instance to a variation in effective doping density with varying relative shell thickness.

Supplementary Note 10 | Determining the lasing mode and estimating cavity Q factor
The threshold gain for the nanowire laser was estimated using the following expression: (8) where is the mode confinement factor, is the threshold gain, is the cavity length and is the geometric mean of the mode reflectance from each end facet. We performed finite-difference timedomain (FDTD) simulations to calculate and for the guided modes supported in a tapered nanowire lying on SiO 2 substrate. The nanowire was modelled as a truncated cone with refractive index of 3.6. The dimensions of the nanowire laser, measured from SEM images, were used for the dimensions of the truncated cone. Only modes that were supported along the entire nanowire, without being cut-off at the narrowest end, were used for these calculations. The mode confinement factor and reflectance were calculated at a fixed wavelength of 880 nm. Since mode confinement varies along the nanowire because of tapering, we calculated at the centre of the nanowire and used this value as an estimate.
The threshold gain calculated using Supplementary Equation 8 for the nanowire laser was 3390, 3450, 1140, 2570, 2480, 1050, 1700 and 1600 cm -1 for the HE11 a , HE11 b , TE01, HE21 a , HE21 b , TM01, EH11 a , and EH11 b modes, respectively. The lowest threshold gain was for the TM01 mode, which suggests that the nanowire laser in our experiments was lasing from the TM01 mode. The threshold gain above was estimated at a fixed wavelength. However the lasing spectra shown in the main manuscript has multiple peaks. We used FDTD simulations to verify that these peaks correspond to different axial modes supported in the Fabry-Perot type cavity. In these simulations, we used a dipole source orientated and positioned along the axis of the nanowire to excite TM01 guided modes. 51 As before, the nanowire was modelled as a truncated cone lying on SiO 2 substrate, with dimensions corresponding to the measured dimensions of the nanowire laser. The index of the nanowire and substrate were 3.6 and 1.5, respectively. In these simulations, the electric field was The cavity spectrum shown in Supplementary Fig. 20a enables us to estimate the group velocity, or group index of the lasing mode. The wavelength separation between resonant modes in a Fabry-Perot type cavity is given by: Where λ is the wavelength, is the cavity length and is the group index of the mode. Using Supplementary Equation 9, we estimate = 4.44 at = 883 nm for the TM01 mode. The cavity spectrum also enables us to estimate the factor for the resonant modes, using , where is the resonance frequency and is the FWHM of the spectral peak. We estimate a factor of 250-350 for the laser cavity from simulations. The factor measured from the experimental lasing spectra at threshold is 300 at λ=883 nm.

Supplementary Note 11 | Rate equation analysis of lasing
Rate equations were used to fit the experimental L-L curve shown in Fig. 4d of the main manuscript and thereby estimate the doping concentration, , threshold gain, , and spontaneous emission factor, . Since there were predominantly three lasing modes in the lasing spectra, multimode rate equations for three cavity modes were used. The rate equations for the carrier density in the active region, , and photon density in the th cavity mode, , are as follows: is the carrier generation rate, where , , and are the fraction of pump power absorbed, energy of pump photon, optical pump power used and volume of the nanowire, respectively. of 1% was estimated from FDTD simulations using a Gaussian source with FWHM corresponding to the laser spot size measured from experiments (see Supplementary Fig. 7). The temperature and lineshape broadening parameter ( =18.8 meV) were estimated by modelling the spontaneous emission spectrum and fitting its shape with the photoluminescence spectrum of the nanowire laser.
Lastly, and are the group velocity and mode confinement factor of the lasing mode, respectively. These parameters are dependent on the lasing mode type and mode wavelength. Here we assume that these parameters are equal for each of the three lasing modes, and use the values of and calculated for the TM01 mode at 883 nm (see Supplementary Note 10).
We also assume that parameters and are the same for each of the three lasing modes.
The rate equations in Supplementary Equation 10 were solved for the duration of one duty cycle of the pump laser with initial estimates for , and . The total photon density ( ( ) ∑ ) was then evaluated and the average photon density was calculated by integrating over time and dividing by the time span. The normalised average photon density as a function of the average pump power ( ) is shown on a log-log scale in Supplementary Fig. 21, together with the experimental data.
The four curves from the rate equation modelling are for different doping concentrations: = 0.1, 1, 2 and 5 10 19 cm -3 . The threshold gain ( = 1300 cm -1 ) and spontaneous emission factor ( = 0.015) are the same for each of these curves. The threshold gain of 1300 cm -1 is close to the threshold gain estimated for the TM01 mode (see Supplementary Note 12). The factor of 0.015 is consistent with the beta factor estimated for other nanowire lasers of similar dimensions. 3 The curve that best fits the experimental data is for = 210 19 cm -3 .
The photon density evaluated from the rate equations is converted to output power ( ) using the following equation: where ⁄ is the energy of photon, ∫ ( ) ⁄ is the time averaged photon density, where is inverse of the pump pulse frequency, ⁄ is the mode volume and is the escape rate of photons. We use the values of , and calculated from FDTD modelling (see Supplementary Note 10).
The L-L curve with absolute power units on both axes is shown in Fig. 4e of the main text. The slope of the curve above threshold is about 0.2%. Since the fraction of power absorbed in the nanowire is only about 1% of the input power (see above), the absolute slope efficiency (external efficiency) of our laser is 20%.

Supplementary Note 12 | Modelling of gain in doped GaAs
Optical gain in a direct band gap bulk semiconductor can be modelled using the following equation:  , since optical pumping methods are used in this study. We will refer to the number of injected electron-hole pairs as .
Supplementary Fig. 22a shows the peak material gain as a function of injected carrier density for doped and undoped GaAs. For the doped material, we have modelled the gain for p-type doping concentrations of 10 18 and 10 19 cm -3 . The material gain for p-type doped GaAs is much larger than for undoped GaAs. This is because p-type doping results in a downward shift in the quasi-Fermi levels, which consequently reduces the transparency carrier density required to achieve gain. 52 Therefore doping can reduce the injected carrier density, or equivalently the optical pump power, required to achieve threshold gain. In addition, p-type also increases the differential gain, because of the alignment of the quasi-Fermi levels with the band edges. 52 The differential gain ⁄ for doped and undoped GaAs is shown in Supplementary Fig. 22b. Large differential gain is required for high speed modulation applications.