The exciton bound to a pair of nitrogen atoms situated at nearby lattice sites in dilute GaAs:N provides an energetically uniform electronic system, spectrally distinct from pairs with larger or smaller separations, and can even be grown with a uniform pair orientation in the crystal. We use photoluminescence excitation spectroscopy on an ensemble of N pairs to study the narrow continuous energy distribution within two of the individual exchange- and symmetry-split exciton states. Inhomogeneous linewidths of 50–60 *μ*eV vary across the crystal on a mesoscopic scale and can be 30 *μ*eV at microscopic locations indicating that the homogeneous linewidth inferred from previous time-domain measurements is still considerably broadened. While excitation and emission linewidths are similar, results show a small energy shift between them indicative of exciton transfer via phonon-assisted tunneling between spatially separated N pairs. We numerically simulate the tunneling in a spatial network of randomly distributed pairs having a normal distribution of bound exciton energies. Comparing the ensemble excitation-emission energy shift with the measured results shows that the transfer probability is higher than expected from the dilute pair concentration and what is known of the exciton wavefunction spatial extent. Both the broadening and the exciton transfer have implications for the goal of pair-bound excitons as a single- or multi-qubit system.

## I. INTRODUCTION

In GaAs:N, the linewidth of the exciton bound to a N pair has taken on renewed importance with the interest in using the exciton as a quantum computing qubit^{1} and the demonstration of coherent control over a single exciton.^{2} The ensemble linewidth measured in photoluminescence (PL) reflects the ability of various growth technologies to create N pairs with a repeatable controlled electron-trapping potential, which is an expected requirement for advanced schemes of multi-qubit systems. Unlike the similar system of strain-driven quantum dots, the potential energy well surrounding the N pair can be very uniform such that even in ensemble measurements the exchange and symmetry driven splittings of the exciton on the order of 100 *μ*eV can be resolved.^{3,4} A further advantage of the randomly occurring pair system is that the pair probability itself is a function of nitrogen concentration and thus pair densities can be grown several orders of magnitude below the atomic densities that can be controlled as dopants. At the lower limit of this attainable pair density range, a single pair-bound exciton can be optically resolved;^{5} at the other limit, above 0.12% N, the exciton wavefunction spatially overlaps those at nearby pairs, forming an extended band of states.^{6} Exciton tunneling between the low-density pairs, which would impact multi-qubit operation, has been assumed to be minimal. For example, spectroscopic observation of tunneling in the similar system of GaP:N could only be accomplished by introducing a disorder from alloying the host with GaAs.^{7–9} Without this intentional disorder, the energy distribution of isolated N states in GaP was considered to have vanishing width.^{10}

In the case of ultradilute pairs in GaAs:N, there is importance even in the small amount of broadening that remains in the absence of alloying. Experimentally, the linewidth and the degree of exciton transfer are not well understood because published linewidth values are close to the resolution of grating spectrometers. This includes the PL linewidth of single pairs^{11} even though their time-domain measurements^{2} indicate a much smaller homogeneous linewidth of $\u210f\tau lifetime=6\mu eV$. In this paper, we use photoluminescence excitation spectroscopy (PLE) to measure the ensemble absorption linewidth of the 1.508 eV N pairs. The PLE technique has the advantage of a spectral resolution limited only by the laser linewidth, and it measures the relevant excitation spectrum which can differ from the emission spectrum in cases of charge-carrier transfer. We find that growth homogeneity and energy diffusion between spatially separated pairs remain significant challenges to the multi-qubit goal and to precise control of the pair system in general.

## II. EXPERIMENT

Two ultradilute 1-*μ*m thick GaAsN samples were used: a metal-organic chemical vapor deposition (MOCVD)-grown sample previously characterized^{3} as having sharp linewidths and superior brightness, and a molecular beam epitaxy (MBE) sample grown to have nitrogen pairs preferentially oriented in the growth plane and along the $[11\xaf0]$ crystal axis.^{12} For both samples, nitrogen atomic densities of 1 × 10^{18} cm^{−3} were determined by growth parameters that were calibrated using secondary ion mass spectroscopy (SIMS). That determination is also supported by Ref. 18 where absorption and SIMS were measured at different N concentrations. It was found that strong broadening and redshift of the 1.508 pair occurred for concentrations above a threshold that is between 1 and 3 × 10^{18} cm^{−3}, establishing the latter as an upper limit for any samples that exhibit the exchange and symmetry-split transitions seen here. From this limit, we obtain a density of 5 × 10^{14} cm^{−3} for the randomly occurring pairs with 4 × 10^{2} total pairs in the ensemble. This upper limit was used for our discussions and calculations because the results below suggest a concentration higher than random. Such an aggregation was also suggested in Ref. 1.

Resonantly excited micro-PL was measured in an arrangement similar to Ref. 13 using confocal detection for a spatial resolution of 1 *μ*m. Excitation with power density of 1 W/cm^{2} was by a single-frequency (5 MHz = 20 neV) laser controlled with a wavemeter having 20 MHz repeatability. Excitation spectra were collected by a single spectrometer and CCD having 50 *μ*eV resolution while tuning the laser cavity through mode hop-free scans. Light detection was in the polarization orthogonal to the laser and required remounting the MBE sample between measurements. All measurements were at 5 K.

## III. RESULTS

### A. Linewidth measurement

Figures 1(b), inset, and 1(c) show spatial and spectral PL plots of the 1.508 eV N pair-bound excitons in the MOCVD sample under off-resonant (1.96 eV) excitation. The 15-*μ*m spatial map shows PL intensity variations that originate in the sample. The spectrum in Fig. 1(c) is similar to Ref. 3, being dominated by the four transitions corresponding to in-plane N pairs. Due to the higher temperature here, exciton occupation of the four levels is more equal. The spectral weighting of the four PL peaks is also affected by their dipole moments and transfer dynamics, the net effect being the larger magnitude of the higher-energy peaks X_{2} and Y_{2}. We use the notation of Ref. 11 for C_{2v} symmetry pairs, i.e., first- or fourth-nearest neighbors aligned along the ⟨110⟩ direction. X, Z, and Y refer to the symmetry-split exciton oriented, respectively, along the pair direction, along its C_{2} direction, or perpendicular to both, and the subscripts 1 and 2 refer to the lower and upper exchange-split set.

In general, the PLE technique is useful when there is a small spectral shift between absorption and emission allowing the latter's intensity to be recorded free from laser scatter. Scanning the laser then measures the absorption spectrum under the assumption of a constant transfer rate between absorbing and emitting states. Within a single N pair, this occurs by exciton transfer between states like X_{2} and Y_{2} which we term intrapair transfer. The complementary interpair transfer between different pairs has been previously observed only at higher densities and accompanied by strong broadening. The intrapair transfer here allows PL from one state to monitor the exciton population at the other.^{2} Because the energy splitting between X_{2} and Y_{2} is small compared to k*T*, the transfer can occur in either direction. Figure 1(a) shows this in the energy relaxation mode which measures the excitation linewidth of X_{2} by detecting PL at Y_{2}. CCD spectra (vertical profiles) are mapped as a horizontal function of the laser energy as it sweeps through X_{2}. The brightest band is from laser scatter superposed on the resonant X_{2} PL, and at lower energy is a weaker signal from excitons transferred to Y_{2}. The magnified section shows the X_{1} PL peak partially resolved from the Y_{2} tail. In order to separate these overlapping signals, peak fitting was used to extract their linecenters and the signal strengths. The latter provides the X_{2} PLE spectrum in Fig. 1(b). It peaks close to the spectrometer-measured PL curve in Fig. 1(c) but is narrower with a width of 51 *μ*eV compared to 130 *μ*eV.

The energies of the laser and detection can be switched allowing the absorption of Y_{2} to be scanned. Figure 2 shows the excitation spectra of X_{2} and Y_{2} lines for both samples. Measurements were done at sample locations chosen for better X_{2}-Y_{2} resolution and therefore are more homogeneous than the sample average. Peak energies and the irregular line shapes varied on a length scale similar to the intensity variations shown in the inset of Fig. 1(b). This is particularly evident in the MBE sample which was remounted between measurements. The growth inhomogeneities evident in the inset of Fig. 1(b) may include N density variations or a spatial correlation of pair energies that would explain these results. Small broad components to the line shape are also seen in Fig. 2, suggesting more than one broadening mechanism to the local potential fluctuations. These complexities and the linewidths shown illustrate the benefit of the high-resolution PLE technique over the 50-*μ*eV resolution PL spectrum in Fig. 1(c). The absorption linewidth for our ensemble of pairs is approximately 50 *μ*eV and in select locations of the MBE sample approaches 29 *μ*eV. This still represents significant site-to-site fluctuation imposed by the growth process upon the single-pair 6-*μ*eV linewidth that is inferred from the temporal measurements of Ref. 2.

### B. Spectral diffusion

Also in Fig. 1(b) are the linecenters of the fitted peaks plotted across the PLE FWHM region as a function of laser energy. The laser scatter (upper points) follows the ≈26 GHz laser scan with a measured slope of 1.03 which is within experimental error of the required value of unity. However, the recombination energies of excitons transferred to the lower-energy Y_{2} and X_{1} states have much flatter slopes as indicated by their linear fit values on the graph. These slopes give information on interpair exciton transfer that is not directly observable by the spectrometer resolution. Because the linewidths (≈50 *μ*eV) are much smaller than the 200–600 *μ*eV splittings in these samples, we start with the simplest interpretation, in which the inhomogeneous broadening is due to local potential fluctuations at each pair. This is illustrated in Fig. 3 for a small set of N pairs. This pair energy distribution adds a constant energy shift to the X_{1}, Y_{1}, Y_{2}, and X_{2} spectrum leaving their internal energy separation unchanged. In the absence of interpair transfer [Fig. 3(a)], PL will be redshifted from the scanning laser by a fixed energy equal to the X_{2}-Y_{2} intrapair splitting. The PLE slope for this noninteracting system will be 1. With increasing pair transfer probability, PL will emit from lower-energy pairs and in the limiting case the PLE slopes measured in Fig. 1(b) would be zero as excitons find the lowest pairs, independent of their excitation energy. This highly interacting system is diagrammed in Fig. 3(b) and its inset. The intrapair and interpair transfers are assumed independent and can occur in either order.

The results of Fig. 1(b) indicate that interpair transfer is indeed occurring. The results were unchanged at ×2 higher power and at temperatures up to 11 K, beyond which the pair PL was quenched. Slopes varied at different sample locations but were always less than 0.5. Results were different, however, in the Y_{2} scan shown in Fig. 4: there is a difference in the positively and negatively shifted PL signals. The slope of the lower-energy light is again flatter as in Fig. 1(b), while that of the higher-energy PL is steeper, closer to the laser. These results were also found consistently across both samples.

### C. Numerical modeling

We next seek to quantify the exciton spectral diffusion. In systems of large broadening, it was analyzed from the emission–absorption shift;^{7,9} however, this is very small in the present case. We therefore use a model that treats the PLE slope seen in Figs. 1 and 4. We assume that the intrapair exciton transfer such as from X_{2} to Y_{2} can be decoupled from the interpair transfers such as between the Y_{2} states in neighboring pairs (see Fig. 3). The system can then be approximated using only the intrapair final state (Y_{2} in this example) accompanied by inhomogeneous broadening, spectral diffusion, and nonsaturating generation at an energy within the distribution. We refer to this as the excitation or laser energy for simplicity. Calculations of energetic diffusion through a discrete^{8} or continuous^{7,9,14} distribution of states have been used to analyze cw and time-resolved PL and the related phenomena of charge transport. Physically, these include a spatially and energetically random distribution of lattice sites through which charge carriers can tunnel with a probability of^{15}

where *ν*_{0} is a base tunneling rate, *a*_{0} measures the spatial extent of the bound charge wavefunction, and *i,j* label the initial and final space coordinates, *R,* and energy, *E*.

The nonreciprocity of Eq. (1) results in energy relaxation of carriers initially created at either a single energy or over the entire pair energy distribution. This has been calculated using an analytical approximation,^{7,9} Monte Carlo simulation,^{14,16} or numerical rate equations.^{8} In order to treat resonant excitation, we modify the approach of Ref. 14 to also include a random set of nitrogen pairs at spatial density *n* and radiative recombination with rate 1/*τ*. Coordinates for the pairs were randomly generated with a uniform distribution. The GaAs discrete lattice was approximated by continuous space which is valid for our nearest-pair spacing of approximately 70 nm. A Gaussian-distributed energy with standard deviation *σ* was also generated for each pair, and Eq. (1) was repeatedly applied to a single exciton originating at the centermost pair with the excitation energy. This was continued until recombination. A choice of 100 pairs ensured that for our density, the generated space was larger than 99.99% of the random walks. With small modifications, the histogram of recombination energies calculated using this method compared well with the analytical line shape^{7} calculated for isolated N atoms in GaAs_{1−x}P_{x}. Tunneling is strong in that system due to a high site density of 10^{18} cm^{−3} and k*T* is small compared to the Gaussian energy distribution width *σ*.

For our case of N pairs in GaAs, there is a lower density, by a factor of 10^{3}, and a shorter radiative lifetime, both of which contribute to fewer tunneling events per exciton. This reduces energy relaxation that would lower the PL energy. In addition, the substantially narrower linewidth, with *σ* << k*T* changes the relaxation behavior as tunneling events to higher energies are nearly as probable as ones to lower energies. Figure 5(a) shows recombination (PL) spectra for several temperatures with excitation at a single energy chosen to be *E*_{0} + 1.5 *σ* for illustration.

The experimental conditions of a 60 *μ*eV linewidth found in Fig. 2 and *T* = 5 K correspond to the k*T/σ *= 7 solid curve. Tunneling in this case distributes excitons almost uniformly through the inhomogeneous energy distribution resulting in a recombination spectrum just slightly down-shifted from the distribution (dashed curve). At low temperatures, tunneling to lower-energy pairs becomes the more favorable path, and this is seen in the more pronounced energy relaxation of the peak and in the threshold at the excitation energy. Examination of the individual tunneling events reveals that for high temperature, excitons can cycle through a small number of sites repeatedly until escaping to more distant sites. In the absence of low-energy trapping, spatial diffusion is slightly larger as seen in the higher temperature curves of Fig. 5(c).

Total PL intensity is fixed due to the absence of any nonradiative recombination in the simulation; however, excitons which recombine at the exact excitation energy are not displayed in the recombination curves of Figs. 5(a) and 5(c). This includes zero-tunneling recombinations as well as cyclical tunneling paths which return to the exact initial energy and location. For the lowest temperature, k*T*/*σ* = 0.056, the unplotted recombinations at the generation energy comprises 73% of the total PL and occur mostly due to the zero-tunneling events. This agrees with the nearest-neighbor approximation of Ref. 17, which was derived for low temperature. At high temperature, the unrelaxed recombination increases to 80% due to greater contribution from the tunneling cycles. The detected PL energy is an average of the unrelaxed excitons and the histograms of Fig. 5(a), giving an energy redshifted from the laser. This calculation is shown in the symbols of Fig. 5(b) as a function of the laser energy. The slopes of 0.46–0.89 are comparable to the experimental results of Figs. 1 and 4. Empirically, the slope is found to decrease with increase of either of the three independent factors entering the calculation: *ν*_{0}*τ*, $a03n$, or *σ/*k*T*. The first two affect the balance of unrelaxed and thermalized excitons discussed above, whereas *σ/*k*T* allows the thermalized excitons to cool below the linecenter.

## IV. DISCUSSION

Overall, the experimental Stokes shift indicates a greater spectral diffusion than predicted by the available information on these parameters. In Fig. 5, we have set *ν*_{0} from the upper range of values measured^{17} in GaP. The radiative lifetime *τ*, set here equal to 250 ps, was shown in Ref. 11 to be strongly masked by exciton capture and interlevel transfer rates that dominate the longer nonresonant time-resolved PL lifetimes. *τ* for the bright states was in the range 200–670 ps from Ref. 11. The pair density, *n*, used here derives from the nitrogen atomic density of 3 × 10^{18} cm^{−3} measured in Ref. 18 where the pair-bound excitons were found to shift and broaden above this concentration. *a*_{0} is a much stronger factor in the tunneling probability. The rough agreement between Fig. 5(b) and, e.g., the energy-relaxed experimental result of Fig. 4 does not allow the modeled value of 225 Å to be varied by more than ±20%. Comparison with other measurements is difficult because the quantity in Eq. (1) is intended to approximate the long-range dependence of the isoelectronic trap-bound exciton wavefunction, which should differ in form from that of the free exciton. Using the free exciton model, Ref. 4 analyzed diamagnetic shifts of excitons in deeper GaAs:N traps to indicate wavefunction root-mean-square values of approximately 40 Å. Similar values of the diamagnetic shift were found for the 1.508 eV pair studied here.^{18,19} Within the theoretical framework used here, our experimental results of Figs. 1 and 4, therefore, can be understood only under the assumption of a slower wavefunction decay or a spatial correlation in the pair probability. Clumping of the pairs into regions of higher density, suggested by the inhomogeneous PL map of the inset in Fig. 1(b), would increase the spectral diffusion beyond that calculated for the average density.

The origin of our nitrogen pair linewidth, i.e., the broadening mechanism, could have several explanations including the much higher nitrogen atom concentration falling at distant lattice sites. At a similar concentration to the pairs would be typical background impurities of charged dopants. In this picture, the pairs would actually be weakly perturbed three-particle complexes. In Ref. 5, the host GaAs was designed to have a significant linewidth that imparted a few millielectron volts of energy spread on the pair energies. This was assumed to be due to lattice imperfections that would also be present here on a much smaller scale.

The simplifying assumption of independent interpair and intrapair transfer predicts one PL slope for the two excitation-detection schemes of Figs. 1 and 4. Experimentally, however, they are not equal: while all are less than unity indicating that the exciton transfers to a lower-energy pair, this effect is stronger when accompanied by a shift to lower energy from X_{2} to Y_{2}. The slopes rule out an alternate interpretation of the data: a systematic increase of the symmetry-induced splitting with increasing pair energy would have some slopes exceeding unity. Instead, the results indicate that the larger-energy internal transfer is not independent of the smaller energy transfer between pairs. In Ref. 11, the former was attributed to a phonon-assisted hyperfine interaction, proceeding indirectly through the dark and Z excitons, whereas the latter is a phonon-assisted tunneling.^{15}

It is also possible that the strength of mixing between bright and dark excitons varies among different pair sites. In quantum dots, this was modeled as a small off-diagonal element in the spin Hamiltonian and was attributed to reduced symmetry of nominally cylindrical dots.^{20} For N pairs, the interpair transfer between the nominal X_{2} state, for example, of two disparate pairs would be reduced since they no longer have identical symmetry. This is more likely in the single-impurity sample of Ref. 5 with the designed inhomogeneity of pairs. It that case, both the offset energy and the spacing of the fine structure varied significantly between pairs, and in some cases even the ordering was seen to vary.^{3} These effects should be smaller in our ensemble samples where the narrow linewidths and high degree of polarization indicate that variations in the mixing effects are small compared to the ideal fine structure splittings.

The interpair transfer described by Eq. (1) is an intrinsic rate in the quantum limit. Saturation effects applying to the ensemble may nonetheless arise from an average transition rate which depends also on occupation numbers of individual states. This was discussed in the context of the analytical approximation in Ref. 7. Saturation there was expected only in efficiently-transferring systems which would concentrate a large density of excitons into the sparse low-energy states at the tail of the distribution. In our present system, interpair transfer is a weaker process and therefore distributes the excitons nearly uniformly through the distribution. Saturation would instead be limited to the pairs at the laser energy and seen as simple absorption saturation. Experimentally, this was not observed in our measurements. In Ref. 5, the PL of a single N pair saturated at an excitation intensity approximately an order of magnitude greater than that used for our ensemble.

## V. SUMMARY

Our ability to see these temperature-driven effects while still employing the low-temperature PL technique is a rare situation found in the GaAs:N pair system. Using a single-frequency laser, we have measured an ensemble linewidth of approximately 60 *μ*eV. While narrow for a solid-state system, this is still a factor of 10 larger than the lifetime linewidth in a single N-pair exciton qubit. At the same time, we see a higher than expected transfer rate to spatially separated states within the ensemble. Being phonon-assisted, this interpair transfer is a decoherence effect and the higher rate is important for designing single-qubit studies which must consider both the planar and bulk pair densities. Our modeling could easily be extended to a quasi-2D geometry. The cyclical tunneling events should also be considered since they reduce the coherence without reducing the PLE slope.

## ACKNOWLEDGMENTS

This work was authored in part by the National Renewable Energy Laboratory, managed and operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding was provided by U.S. Department of Energy, Basic Energy Science, Division of Material Sciences. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government.