GaN/AlxGa1−xN quantum wells were grown by molecular beam epitaxy on high quality bulk (0001) GaN substrates. The quantum well thickness was set in the 6–8 nm range to favor the photoluminescence emission of indirect excitons. Indeed, such excitons are known to be spatially indirect due to the presence of the internal electric field which spatially separates the electron and hole wave functions. The growth conditions were optimized in view of minimizing the photoluminescence peak broadening. In particular, the impact of growth temperature (up to 900 °C) on the surface morphology, structural, and photoluminescence properties was studied. The diffusion of indirect excitons on the scale of tens of micrometers was measured with a micro-photoluminescence setup equipped with a spatially resolved detection. A dedicated model and its analysis allow us to extract from these measurements the exciton diffusion constant and to conclude on the optimum growth conditions for the GaN/AlxGa1−xN quantum well structures suited for studies of quantum collective effects in indirect exciton liquids.

GaN/AlGaN quantum wells (QWs) are commonly used for a variety of devices or applications such as UV laser diodes,1–4 UV light emitting diodes,5,6 room temperature polariton lasers,7 micro-disk or micro-ring resonators,8,9 ridge waveguides,10–12 intersubband optoelectronic in the infrared and the THz ranges,13 and thermal emitters coupled to photonic crystal.14 

Most of the papers regarding GaN/AlGaN QWs were based on structures grown by heteroepitaxy on sapphire or silicon substrates using metalorganic vapor phase epitaxy (MOVPE)15,16 or molecular beam epitaxy (MBE).17,18 These structures are affected by a large density of threading dislocations typically in the 108–1010 cm−2 range. The negative impact of these defects was highlighted on different properties such as the internal quantum efficiency of GaN/AlGaN QWs.19 For example, it was shown that the room temperature photoluminescence (PL) intensity of a eight mono-layer-thick GaN/Al0.1Ga0.9N QW was 60 times larger when grown on bulk GaN substrate rather than on sapphire substrate.20 Many fundamental properties of GaN/AlGaN QWs were elucidated despite the presence of a high defect density: in particular, the major influence of the quantum confined Stark effect on the PL properties.21–33 

However, a large dislocation density is a major issue for applications requiring long excitonic radiative lifetimes (trad) of the GaN/AlGaN QWs, since the ratio giving the internal quantum efficiency IQE = (1 + trad/tnr)−1 becomes very small for a fixed short non-radiative lifetime (tnr). This situation occurs in wide GaN QWs (>5 nm). Excitons in wide QWs belong to the family of dipolar or indirect excitons (IX).34–37 IXs can be created in heterostructures where electron and hole are both confined, but maintained spatially separated either by type-II band alignment (like in specifically engineered alignment of exciton states in GaAs/AlAs QW structures)38 and in some recently discovered vertically stacked transition metal dichalcogenide (TMD) heterostructures39 or by an electric field.40–45 In GaN/AlGaN QWs, formation of IX is due to the electric field resulting from the difference of piezoelectric polarizations in the well and in the barrier, while in GaAs/AlGaAs and TMD-based structures, the electric field is applied externally. IX excitons are promising for studies of quantum collective states that these excitons can form, including Bose–Einstein condensation, that has been reported for trapped GaAs/AlGaAs IXs.43–46 However, such condensation has still not been demonstrated in the GaN/AlGaN system which holds the promise to grant access to condensates at higher temperatures and with higher exciton densities than in the GaAs/AlGaAs system.

Compared to nanostructures based on gallium arsenide (GaAs), the choice of gallium nitride is motivated by two factors. First, this material is expected to enable collective quantum effects at higher temperatures due to higher exciton binding energy (26 meV in the GaN massive material47 in comparison to 4.2 meV for GaAs48) and smaller Bohr radius (3 nm in bulk GaN and 13 nm in GaAs).49,50 Second, due to spontaneous and piezoelectric polarizations, GaN/AlGaN QWs are subject to strong build-in electric field, which pushes electrons and holes within the excitons toward opposite interfaces of the QW. Therefore, such excitons become spatially indirect even in the absence of any externally applied electric bias. This is an advantage compared to AlGaAs/GaAs QWs were application of an electric field in the growth direction is mandatory to observe IX optical emission.40–45 

Nevertheless, important challenges must be faced to achieve IX condensation in GaN/AlGaN QWs, and, in particular, IX non-radiative emission rates need to be minimized. Using high quality GaN substrates is, therefore, a prerequisite to get efficient light emission from IXs in GaN/AlGaN QWs. This was previously illustrated by demonstrating room temperature transport of IXs for a GaN/AlGaN QW structure grown on a free-standing GaN substrate, while this transport was undetectable for the same structure grown on the sapphire substrate.35 Until now, no specific study investigating the growth parameters was realized to enhance the quality of GaN/AlGaN QWs for IX transport.

The purpose of this work is then to optimize and characterize wide GaN/AlGaN QWs whose properties are adapted to the propagation of IXs. Indeed, all sources of inhomogeneities and non-radiative recombinations have to be avoided. This is all the more difficult since this should be obtained on a scale of several tens of micrometers, the typical distance for IX diffusion. This is the reason why, in this work, we exclusively used low dislocation density (<104 cm−2) GaN bulk substrates in order to increase, as far as possible, the quality of the epitaxial structures.

The samples were grown by molecular beam epitaxy on bulk (0001) ammonothermal GaN substrates (1 × 1 cm−2 or 1 in. diameter) from Ammono with an offcut of 0.5° toward the m-plane.51,52 The dislocation density of these substrates is estimated to be below 5 × 104 cm−2. Two sample sets were grown using different substrate preparations before the molecular beam epitaxy. For the first one, the substrates are de-oxidized using a HF-based solution. For the second one, the GaN substrates, after in situ de-oxidation, are overgrown with a 1.1 μm GaN layer grown by metal organic vapor phase epitaxy at 1000 °C. This second approach leads to smoother surfaces as will be shown in the following. The samples are mounted onto indium-free 3-in. molyblock holders adapted to the shapes of the substrates. NH3 is used as the nitrogen source. Ga and Al atoms are evaporated from a double filament effusion cell and a cold-lip effusion cell, respectively. The growth temperature is deduced from the sublimation rate of GaN under vacuum measured on calibration samples.53 

The samples were initially heated at 780 °C under an ammonia flow of 100 standard centimeter cubes per minute (SCCM) for 10 min in order to remove residual surface contaminants such as C and O. A 600 nm GaN buffer layer was grown before a 100 nm-thick AlxGa1−xN layer, a GaN QW, and a 50 nm-thick AlxGa1−xN top layer. For the first series of samples, the growth temperature was set at 743 °C (sample A), 773 °C (sample B), 796 °C (sample C), and 818 °C (sample D). For the second series of samples (with a MOVPE GaN buffer layer), the growth temperature was set at 780 °C for sample A′ and 900 °C for sample B′. For this last sample, the surface was annealed under ammonia at 900 °C for 20 min before the growth of the first AlxGa1−xN layer. All the data regarding the samples can be found in Table I.

TABLE I.

Growth conditions of the GaN/AlGaN layers, the Al composition (xAl), the quantum well thickness, and the root mean square roughness of 10 × 10 μm2 atomic force microscopy images for the first set (A–D) and the second set (A′ and B′) of samples.

SampleSubstrate preparation before MBEGrowth temperature (°C)NH3 flow (SCCM)xAlQW thickness (nm)10 × 10 μm2 rms roughness (nm)
Chemical 743 100 0.082 7.8 1.2 
Chemical 773 100 0.081 7.8 0.6 
Chemical 796 100 0.084 7.8 2.3 
Chemical 818 100 0.078 7.8 6.4 
A′ MOVPE GaN 780 100 0.096 6.4 0.25 
B′ MOVPE GaN 900 500 0.108 6.4 0.17 
SampleSubstrate preparation before MBEGrowth temperature (°C)NH3 flow (SCCM)xAlQW thickness (nm)10 × 10 μm2 rms roughness (nm)
Chemical 743 100 0.082 7.8 1.2 
Chemical 773 100 0.081 7.8 0.6 
Chemical 796 100 0.084 7.8 2.3 
Chemical 818 100 0.078 7.8 6.4 
A′ MOVPE GaN 780 100 0.096 6.4 0.25 
B′ MOVPE GaN 900 500 0.108 6.4 0.17 

Atomic force microscopy in the tapping mode was used to characterize the surface of the samples.

The Al content, quantum well, and barrier thicknesses were deduced from ω/2θ high-resolution x-ray diffraction and summarized in Table I.

The high-resolution cathodoluminescence was performed with a MonoCL4 GATAN system equipped with a high-sensitivity photomultiplier mounted on a field emission gun scanning electron microscope (FEG-SEM; JEOL JSM700F). The electron beam current ranges typically from 1 to 4 nA and the voltage beam is fixed at 5 keV. Dispersed light is collected through a paraboloidal mirror and quantified as a whole using a CCD detector with a sensitivity range from 250 to 1050 nm. Acquisition time is fixed to 350 μs/pixel. All observations are done at room temperature.

Morphological and structural analyses of the considered samples were performed using a ThermoFisher Titan SPECTRA 200 transmission electron microscope (TEM) operating at 200 kV equipped with a cold FEG and a Cs aberration probe corrector. All the analyses were performed using a probe convergent semi-angle of 29.4 mrad and a collection angle between 109 and 200 mrad allowing STEM-HAADF Z-contrast imaging.

Low temperature PL at 10 K was performed using the 244 nm line of a frequency-doubled Ar laser with a spot diameter of 130 μm as the excitation source.

Spatially resolved micro(μ)-PL measurements were conducted at 65 K using a continuous wave laser emitting at 355 nm, focused onto a0 ≈ 1.5 μm-radius spot on the sample surface. The PL images were collected by a microscope objective that allows for ten times magnification and filtered through a vertical slit, which selects a 250 × 1 μm2 area on the sample surface. These spatial areas were analyzed by a spectrometer equipped with a 1200 lines/mm grating and a 2048 × 512 pixels CCD camera. The PL images that can be obtained with this set up are characterized by spatial and spectral resolution of ≈ 1 μm and ≈ 1 meV, respectively.

The surfaces of GaN (or AlGaN with a low Al composition typically below 0.2) grown by molecular beam epitaxy using ammonia as the nitrogen source under N-rich conditions are generally constituted by hillocks (or mounds)54 which are not related to a spiral growth but rather to the existence of a step-edge barrier for the diffusion of adatoms (Ehrlich–Schwoebel barrier).55,56 This growth regime is unstable and induces a coarsening of the mounds and an increase of the surface roughness when the epitaxial layer thickness increases.54 In order to improve the surface morphology, a step-flow growth regime would be preferable and one solution to reach this regime would be to increase the growth temperature.56 This is the purpose of the study described below.

Figure 1 shows the AFM images of the surface of the first series of samples (A–D). The growth temperature has a strong impact on the surface morphology: the root mean square (rms) roughness of these 10 × 10 μm2 AFM images is 1.2, 0.6, 2.3, and 6.4 nm for growth temperatures of 743, 773, 796, and 818 °C, respectively. At 743 °C, the surface is composed of a mix of mounds and step meandering [Fig. 1(a)] with a characteristic size of ∼0.8 μm. At 773 °C [Fig. 1(b)], the mounds tend to disappear and the surface is nearly exclusively composed of step meandering (the meanders or valleys are perpendicular to the steps) with a mean distance between valleys of ∼1.2 μm. The valleys are approximately oriented along the [ 11 2 ¯ 0] axis while the steps are oriented parallel to the [ 1 1 ¯ 00] axis. For temperatures larger than 800 °C [Figs. 1(c) and 1(d)], the surface becomes strongly pitted because of the large desorption rate of N and NHx species57 and an insufficient ammonia flow to stabilize the surface. The size and the density of the pits increase with the temperature. The mean equivalent diameter of the pits is 54, 86, and 146 nm, and the pit density is 6.7 × 10−7, 3.0 × 10−8, and 5.5 × 10−8 cm−2 for growth temperatures of 777, 796, and 818 °C, respectively. The formation of pitted GaN surfaces grown by MBE at relatively high temperature was already observed.55,58 This issue can be solved by increasing the ammonia flow to compensate the N desorption. With such strategy, the growth of GaN by ammonia-MBE in a quasi-step-flow regime at 920 °C was demonstrated showing smooth surfaces with a rms roughness of 1.1 nm for 5 × 5 μm2 AFM image.56 

FIG. 1.

10 × 10 μm2 atomic force microscopy images of the AlGaN surface of samples A–D [(a)–(d)]. The growth temperature increases from 743 °C (left) to 818 °C (right). The scale bar corresponds to 2 μm.

FIG. 1.

10 × 10 μm2 atomic force microscopy images of the AlGaN surface of samples A–D [(a)–(d)]. The growth temperature increases from 743 °C (left) to 818 °C (right). The scale bar corresponds to 2 μm.

Close modal

In the second set of samples, the GaN substrates were systematically overgrown by a 1.1 μm-thick GaN layer by MOVPE. We observed that this kind of substrate preparation before MBE generally leads to smoother GaN/AlGaN surfaces in a more reproducible way than with the chemical surface preparation we used for the first set of samples. Sample A′ is grown at a temperature close to the one of sample B, and, therefore, the impact of the surface preparation can be evaluated. Sample B′ is grown at 900 °C with an ammonia flow which is five times larger than for the first set of samples.

Figure 2(a) shows a 5 × 5 μm2 AFM image of the surface of a 600 nm-thick GaN layer grown with the same conditions as for sample B′. The surface is constituted by parallel steps indicating a step-flow epitaxial growth. The rms roughness is 0.1 nm which favorably compares with previous works about GaN homoepitaxy by ammonia-MBE.56,58 The rms roughness of the top AlGaN surface for sample B′ is 0.16 nm [Fig. 2(b)]. This larger rms roughness and the appearance of meanders (with a small peak-valley amplitude ∼0.5 nm) is related to the lower surface diffusion of Al and also to the possible increase of the step-edge barrier for Al.

FIG. 2.

5 × 5 μm2 atomic force images of (a) a bare GaN surface grown at 900 °C with an ammonia flow of 500 SCCM and of (b) sample B′ grown under the same conditions but with the GaN/AlGaN quantum well. The scale bar is 1 μm.

FIG. 2.

5 × 5 μm2 atomic force images of (a) a bare GaN surface grown at 900 °C with an ammonia flow of 500 SCCM and of (b) sample B′ grown under the same conditions but with the GaN/AlGaN quantum well. The scale bar is 1 μm.

Close modal

The surface properties of sample A′ and B′ are compared in Fig. 3. The AFM image of sample A′ [Fig. 3(a)] is characteristic of a growth in the step meandering regime while that of sample B′ [Fig. 3(b)] approaches what is expected for a growth in the step-flow regime. The rms roughness is 0.25 and 0.17 nm for samples A′ and B′, respectively. The corresponding height profiles are shown in Fig. 3(c). The peak-valley amplitude is about 1.5 nm for sample A′ and 0.5 nm for sample B′. The height variation is quasi-periodic with a period of 1.2 μm for sample A′ and 2 μm for sample B′. This height modulation is characteristic from the step meandering growth instability on vicinal surfaces (also known as the Bales–Zangwill instability), the period depends on growth parameters such as the growth rate or the temperature.56,59 The period of the height modulation is the same for samples B and A′ as expected from the growth temperatures of 773 and 780 °C which are very close. The substrate preparation for sample A′ gives a smoother surface (rms roughness of 0.25 nm compared to 0.6 nm for sample B) without any pits detected on a 10 × 10 μm2 AFM image.

FIG. 3.

Surface characteristics of samples A′ and B′. 10 × 10 μm2 atomic force microscopy images of sample A′ (a) and B′ (b). The scale bar is 2 μm. Height profile extracted along the line drawn in blue for sample A′ and in green for sample B′ (c).

FIG. 3.

Surface characteristics of samples A′ and B′. 10 × 10 μm2 atomic force microscopy images of sample A′ (a) and B′ (b). The scale bar is 2 μm. Height profile extracted along the line drawn in blue for sample A′ and in green for sample B′ (c).

Close modal

We will see in Sec. IV that there are no clear correlations between surface roughness and exciton transport: despite larger peak-valley amplitude of the surface fluctuations on the micrometer scale, IX transport in sample A′ is more efficient than in sample B′.

In order to determine whether the high temperature growth process can degrade the crystalline quality of the nitride layers, we have evaluated the dislocation density using panchromatic cathodoluminescence at room temperature. This technique is accurate to evaluate the number of dislocations that are effectively non-radiative centers, an underestimation of the total dislocation density up to factor 2 has been reported.60 No dark spots related to the presence of threading dislocations are visible on the CL panchromatic image of sample B′ shown in Fig. 4(a), while a dark spot density of 1.7 × 106 cm−2 is found for a sample with a similar structure grown on a free-standing GaN substrate [Fig. 4(b)]. According to the size of the image (100 × 100 μm2), this represents a threading dislocation density lower than 104 cm−2, in line with the Ammono GaN substrate specifications.

FIG. 4.

Cathodoluminescence images in the panchromatic mode at room temperature of sample B′ (a) and for a sample with a similar structure grown on a free-standing GaN substrate (halide vapor phase epitaxy) (b).

FIG. 4.

Cathodoluminescence images in the panchromatic mode at room temperature of sample B′ (a) and for a sample with a similar structure grown on a free-standing GaN substrate (halide vapor phase epitaxy) (b).

Close modal

To obtain a better insight on the morphology and the crystalline quality of the epitaxial GaN/AlGaN QWs and on the possible presence of defects at the interface, we carried out high-resolution STEM-HAADF observation on cross sections prepared using standard polishing TEM preparation method for samples A′ and B′. Figures 5(a) and 5(c) show low magnification STEM-HAADF images of the corresponding cross section illustrating 0.2 μm long area representative of each sample. From these images, we can infer that the GaN/AlGaN QWs present perfectly flat and continuous layers with no mixing at this scale.

FIG. 5.

HAADF-STEM analysis of a cross section of samples A′ [(a) and (b)] and B′ [(c) and (d)] prepared by standard polishing TEM preparation method: (a) and (c) low magnification HAADF-STEM image of a thin area within the cross section illustrating the morphology of the GaN/AlGaN QWs and (b) and (d) HR-HAADF-STEM image representing a zoom on the GaN/AlGaN QWs interface marked by the rectangle area.

FIG. 5.

HAADF-STEM analysis of a cross section of samples A′ [(a) and (b)] and B′ [(c) and (d)] prepared by standard polishing TEM preparation method: (a) and (c) low magnification HAADF-STEM image of a thin area within the cross section illustrating the morphology of the GaN/AlGaN QWs and (b) and (d) HR-HAADF-STEM image representing a zoom on the GaN/AlGaN QWs interface marked by the rectangle area.

Close modal

A closer analysis of the GaN/AlGaN QWs for both samples at higher magnification represented in Figs. 5(b) and 5(d) allowed us to observe the defect free crystallinity of each layer individually and the abrupt GaN/AlGaN interfaces. Note that the dark spot in the upper part of the GaN quantum well [Fig. 5(b)] is a beam damage effect.

The PL properties were only evaluated for the samples A′ and B′ which possess better surface morphologies compared to the first set of samples. Two different PL setups were used for this evaluation, see Sec. II. The first one consists of a classic PL setup (macro-PL) at low temperature (10 K) with a large diameter spot (130 μm) of the excitation laser. The results of these experiments can be easily compared with literature data. The second setup is more specific as it is a micro-PL setup at 65 K equipped with a CCD camera allowing it to detect the diffusion of excitons within tens of micrometers range.

Note that sample areas studied at 10 and 65 K are not exactly the same and correspond to different distances from the wafer center so that the Al composition slightly differs in these two sets of experiments. From the measured low power IX emission energies, we estimate this difference to be less than 1%.

Figure 6(a) shows the 10 K PL spectra of samples A′ and B′ at an excitation power density of 0.7 W/cm2. The PL spectra are dominated by the QW emission. The corresponding electronic state is an IX, the transition energy is situated close to 3.2 eV and accompanied by the longitudinal-optical phonon replica at 92.5 meV lower energy. PL peaks attributed to the GaN buffer layer and the AlGaN barrier layer are also visible at ∼3.47 and 3.67 eV, respectively. The PL peak from the QW of sample A′ is at 3.242 eV, while it is at 3.213 eV for sample B′. This difference is due to the larger Al composition of the AlGaN barrier in sample B′ which induces a larger internal electric field and, therefore, a stronger quantum confined Stark effect.

FIG. 6.

Low temperature photoluminescence (10 K) using a 244 nm laser with a spot diameter of 130 μm as the excitation source for samples A′ (growth temperature 780 °C) and B′ (growth temperature 900 °C). (a) Photoluminescence spectra in the log scale with an excitation power density of 0.7 W/cm2. (b) Full width at half maximum of the AlGaN photoluminescence peak of samples A′ and B′ compared to literature data.16,61–66 Excitation power density dependence of the photoluminescence peak energy (c) and full width at half maximum (d).

FIG. 6.

Low temperature photoluminescence (10 K) using a 244 nm laser with a spot diameter of 130 μm as the excitation source for samples A′ (growth temperature 780 °C) and B′ (growth temperature 900 °C). (a) Photoluminescence spectra in the log scale with an excitation power density of 0.7 W/cm2. (b) Full width at half maximum of the AlGaN photoluminescence peak of samples A′ and B′ compared to literature data.16,61–66 Excitation power density dependence of the photoluminescence peak energy (c) and full width at half maximum (d).

Close modal

The IX emission intensity in sample A′ is 3.2 times larger than sample B′. This could be due to a reduction of the exciton oscillator strength in sample B′, as the quantum confined Stark effect is larger for this sample. However, calculations indicate (see Sec. IV C) that the oscillator strength of the fundamental QW transition of sample A′ is only 1.6 times larger than sample B′. Therefore, it cannot account for the PL intensity difference between two samples if we assume identical non-radiative recombination. Therefore, we conclude that the non-radiative recombination rate is more important in sample B′ than in sample A′. This is not related to extended defects such as dislocations [see Fig. 4(a)] and should be attributed to the presence of a larger density of non-radiative point defects in sample B′.

The homogeneity of the AlGaN alloy can be evaluated by measuring the full width at half maximum (FWHM) of the corresponding PL peak. The FWHM is 8.6 and 8.1 meV for samples A′ and B′, respectively. These values are comparable with the best results published in the literature [Fig. 6(b)].16,61–66

Figures 6(c) and 6(d) show the excitation power dependence of the PL peak energy and the FWHM of the IX emission. IX peak shifts toward high energies due to the progressive screening of the internal electric field by the photogenerated carriers whose density increases with the excitation power.67 

The PL linewidth increases with the excitation power density due to the exciton–exciton scattering processes which becomes more probable when the exciton density increases. The PL broadening as a function of the power density is faster for sample B′ compared to A′ which can be explained by the larger carrier density for a same excitation power density. This issue will be addressed in greater detail in Sec. V. The minimum of the IX emission FWHM is 8.3 and 8.9 meV for samples A′ and B′, respectively. They are reached at an intermediate power density of 0.7 W/cm2. At lower power densities, the IX emission linewidth slightly increases. We attribute this effect to the inhomogeneous broadening induced by IX localization. Measuring FWHM of the IX emission allows us to evaluate the homogeneity of the AlGaN alloy. We estimate that rms fluctuation of the Al fraction do not exceed 0.005.

Overall, the optical properties of these samples are very close, except the larger QW PL intensity of sample A′.

Figure 7 shows PL spectra of sample A′ [(a)–(d)] and B′ [(e)–(h)] color-encoded in the logarithmic scale. The spectra are taken at different excitation power densities, and the excitation spot is positioned at x = 0. The QW exciton emission and its longitudinal-optical (LO) phonon replica (IX + 1LO) can be readily identified. Indeed, their energies decrease with the distance from the excitation spot, reaching a constant value, E0, on the scale of several tens of micrometers, depending on excitation power. This behavior is consistent with previous observations on similar samples and is due to density-dependent screening of the built-in electric field: at highest power and at x = 0, where maximum exciton density is reached, the highest emission energy, Emax, is observed. Dipole–dipole interaction between excitons pushes them away from the excitation spot, leading to the expansion of the exciton cloud. Thus, due to radial dilution via diffusion and recombination, IX emission energy decreases with increasing distance from the excitation spot. IX emission energy observed in the limit of large distances (x > 30 μm), where it does not change any more with the distance, can be interpreted as the “zero-density” exciton energy, E0. The “zero-density” energy in sample A′ is higher than in sample B′ due to the larger Al composition of the AlGaN barrier in sample B′ which induces a larger internal electric field F, see Table II. This confirms the conclusions drawn from macro-PL experiments presented in Sec. IV A.

FIG. 7.

Spatially resolved μPL spectra (color-encoded in the log scale) at 65 K for sample A′ [(a)–(d)] and B′ [(e)–(h)] at different excitation power densities (indicated on top of each column). Dotted lines indicate ‘zero-density' energies, E0. Solid lines are the results of the modeling by Eq. (4). Arrows indicate the extension of the IX cloud that is power dependent, but systematically larger for sample A′ then for sample B′.

FIG. 7.

Spatially resolved μPL spectra (color-encoded in the log scale) at 65 K for sample A′ [(a)–(d)] and B′ [(e)–(h)] at different excitation power densities (indicated on top of each column). Dotted lines indicate ‘zero-density' energies, E0. Solid lines are the results of the modeling by Eq. (4). Arrows indicate the extension of the IX cloud that is power dependent, but systematically larger for sample A′ then for sample B′.

Close modal
TABLE II.

Parameters obtained from the self-consistent solution of Schrödinger and Poisson equations, from the modeling of IIX(EIX) dependence, and from the modeling of the IX transport.

ParameterDefinitionUnitsSample A′Sample B′Origin
E0 “Zero-density” energy eV 3.254 3.230 Fit IIX(EIX
Mean free path Nm 12.5 8.5 Fit IIX(x), EIX(x) 
Diffusion constant cm2/s 4.1 2.8 From L, D=Lvth 
Ω0 Squared overlap of the electron/hole wavefunctions × 10−4 2.5 1.6 Schrödinger–Poisson 
t rad 0 Radiative recombination time at n= μ0.42 0.65 Fit IIX(x), EIX(x), and t rad , B 0 = t rad , A 0 × Ω 0 , A / Ω 0 , B  
tnr Non-radiative recombination time μ0.07 0.03 Fit IIX(x), EIX(x) 
α Number of IX per incident photon 0.064 0.064  68  
Φ0 Mean-field IX interaction energy meV × cm2 × 10−11 8.7 8.9 Schrödinger–Poisson 
γ Density dependence of the IX emission intensity cm−2 × 1011 5.14 5.23 Schrödinger–Poisson 
F Built-in electric field kV/cm 658 699 Schrödinger–Poisson 
ParameterDefinitionUnitsSample A′Sample B′Origin
E0 “Zero-density” energy eV 3.254 3.230 Fit IIX(EIX
Mean free path Nm 12.5 8.5 Fit IIX(x), EIX(x) 
Diffusion constant cm2/s 4.1 2.8 From L, D=Lvth 
Ω0 Squared overlap of the electron/hole wavefunctions × 10−4 2.5 1.6 Schrödinger–Poisson 
t rad 0 Radiative recombination time at n= μ0.42 0.65 Fit IIX(x), EIX(x), and t rad , B 0 = t rad , A 0 × Ω 0 , A / Ω 0 , B  
tnr Non-radiative recombination time μ0.07 0.03 Fit IIX(x), EIX(x) 
α Number of IX per incident photon 0.064 0.064  68  
Φ0 Mean-field IX interaction energy meV × cm2 × 10−11 8.7 8.9 Schrödinger–Poisson 
γ Density dependence of the IX emission intensity cm−2 × 1011 5.14 5.23 Schrödinger–Poisson 
F Built-in electric field kV/cm 658 699 Schrödinger–Poisson 

Because IX density is proportional to the emission energy blue shift with respect to E0 [see also Eq. (2)], spatial extension of the exciton cloud can be characterized by the distance at which the emission energy blue shift is reduced by a factor of ∼2 with respect to the blue shift (Emax–E0) observed at x = 0. The cloud extension length is highlighted by arrows in Fig. 7 for both samples and at each excitation power. It is known to be strongly dependent on the excitation power,35 and this is clearly observable in Fig. 7 for both samples. One can also see that at any excitation power excitonic emission persists at larger distances in sample A′ than in sample B′.

Finally, at ≈3.298 eV, a weak but narrow emission line appears in almost the entire spatial area shown in Fig. 7. It is identical in the two samples. We assigned this emission to the second LO replica of the free exciton in the GaN buffer layer at x = 0, this light is guided in the sample plane and scattered by surface imperfections at x ≠ 0.

In this subsection, we present the modeling of the IX transport that allows us to identify and quantify the parameters that determine the observed characteristics of IX transport. The modeling procedure consists of three steps. First, each spectrum corresponding to the distances larger than 5 μm is modeled by a phenomenological fitting function introduced in Ref. 37,
I = A exp ( β l ( E E c ) ) 1 + exp ( β h ( E E c ) ) .
(1)

This allows us to determine IX peak energy EIX (slightly different from Ec), integrated intensity of the IX emission IIX, and the linewidth, that we determine as full width at half maximum (FWHM) of the spectral line. βl(βh) characterize the shape of lower (high) wing of the IX spectrum. Two spectra corresponding to the narrowest emission line for the two samples are shown in Fig. 8(a), the shaded area underlines zero-phonon IX emission. Excitation power density and distances from the excitation spot are indicated in the figure caption.

FIG. 8.

(a) μPL spectra measured at x = 17.0 μm (sample A′, blue) and x = 10.8 μm (sample B′, green) at P = 740 W/cm. Spectrally integrated intensity (b) and linewidth (c) of the IX emission in sample A′ (blue) and B′ (green) as a function of the IX energy. Different points correspond to various positions in the plane of the QW and various excitation power densities (same data as in Fig. 7). Solid lines are fitted to the data using Eq. (3). Red dashed lines indicate E0 for two samples. (d) Same data and modeling as in (b) but represented as a function of the IX density.

FIG. 8.

(a) μPL spectra measured at x = 17.0 μm (sample A′, blue) and x = 10.8 μm (sample B′, green) at P = 740 W/cm. Spectrally integrated intensity (b) and linewidth (c) of the IX emission in sample A′ (blue) and B′ (green) as a function of the IX energy. Different points correspond to various positions in the plane of the QW and various excitation power densities (same data as in Fig. 7). Solid lines are fitted to the data using Eq. (3). Red dashed lines indicate E0 for two samples. (d) Same data and modeling as in (b) but represented as a function of the IX density.

Close modal
Second, all the values of integrated intensity and FWHM measured at different excitation power densities and positions are reported as a function of the peak energy. The former dependence can be described within the model of built-in electric field that is screened by the increasing number of IXs. This leads to the increase of both energy (due to exciton–exciton interaction) and intensity (due to increasing overlap of electron and hole wave function within IX). Such behavior can be quantified by solving self-consistently Schrödinger and Poisson equations (we use NextNano software for numerical solution)69 for each sample. It appears that IX energy increases linearly with IX density, n,
E I X ( n ) = E 0 + Φ 0 n ,
(2)
where ϕ0 characterizes exciton–exciton interaction and E0 is the “zero-density” exciton energy. IX emission intensity is given by IIX(n) = nRrad, where Rrad is the radiative recombination rate. By solving self-consistently Schrödinger and Poisson equations, one can see that Rrad depends exponentially on n34 so that,
I I X ( n ) n Ω 0 e n γ ,
(3)
where Ω 0 = | d z ψ e ( z ) ψ h ( z ) | 2 is the squared electron–hole overlap integral at the zero-density excitonic transition and γ characterizes the exponential increase of this overlap with increasing exciton density. Determination of Ω 0, ϕ0, and γ for various possible energies E0 (or, equivalently, built-in electric fields) by solving self-consistently Schrödinger and Poisson equations allows us to fit Eq. (2) to the dependence IIX(EIX) shown in Fig. 8(b). From the best fit, we deduced E0, Ω 0, ϕ0, and γ for both samples. The result is shown by solid lines in Fig. 8(b), and the parameters are given in Table II. Determination of E0 and ϕ0 also provides us with the estimation of the IX density, see Eq. (2). Similar procedure has been used previously.34–37,70

The energies E0 are indicated in Fig. 8(b) by vertical dashed lines. One can see that E0A′> E0B′. This explains greater IX emission intensity in sample A′, see Fig. 8(d). At a given IX density, the emission intensity ratio IIXA′/IIXB′∼2 is almost constant and is very close to the ratio Ω 0A′/Ω 0B′∼1.6. The fact that IIXA′/IIXB′ does not depend on the IX density despite exponential density dependence of the radiative rate corroborates the result of the macro-PL analysis that pointed out the dominance of the non-radiative emission with respect to the radiative one. We will make more quantitative estimation of the corresponding times later on.

We should also comment on the origin of the excitonic emission measured at energies below E0. We interpret this emission as being due to localized excitonic states, typical for nitride QWs,22 but that cannot be accounted for within the model presented above. A non-monotonous behavior of FWHM as a function of energy corroborates this interpretation, see Fig. 8(c). Indeed, FWHM minimum is reached at energy E0. At EIX < E0, FWHM increases due to localization-induced disorder (inhomogeneous broadening). Above E0, FWHM increases due to enhanced exciton–exciton scattering.70,71 We come back to this point in Sec. V and compare the power-induced emission broadening measured in micro- and macro-PL experiments.

The third stage of the modeling consists of the quantitative analysis of the spatial IX emission patterns and determination of the IX diffusion coefficients in the two samples. We describe spatial distribution of IX density by a steady-state solution of the drift-diffusion equation including pumping and relaxation terms,36,37,72
n t = ( J d i f f + J d r i f t ) + G R n .
(4)
Here J d i f f = D n and J d r i f t = μ n ( φ 0 n ) are diffusion and drift currents, respectively. We assume that IX drift is due to dipole–dipole repulsion, and that diffusion coefficient, D, is determined by the average distance between fixed scattering centers (D = Lvth, where vth is the thermal velocity of IXs). Diffusion coefficient is related to IX mobility by Einstein relation μ=D/kBT. Two other terms account for the exciton pumping at power density P, G = α P E l exp ( x 2 / a 0 2 ) and for relaxation at rate R. The later consists of two contributions, radiative and non-radiative, R = Rrad + Rnr. El = 3.49 eV is the pumping laser energy, a0 = 1.5 μm is the laser spot radius, and α is the number of IXs created per incident photon at x = 0. From the known values of GaN absorption coefficient, α can be estimated as α = 0.065.68 The radiative recombination rate is given by R r a d = 1 t r a d 0 e x p ( n / g ), where “zero-density” radiative time t rad 0 is inversely proportional to Ω0, that we have determined previously. This imposes the relation between the radiative lifetimes in the two samples: t rad , B 0 = t rad , A 0 × Ω 0 , A / Ω 0 , B . The non-radiative rate R n r = 1 t n r is density-independent and considered as a fitting parameter. The relation between IX density and energy is given by Eq. (2) and between density and emission intensity by IIX = nRrad. More details on the IX transport modeling by drift-diffusion equation can be found in Refs. 35 and 36. Below, we present the steady-state solutions of Eq. (4) that we obtained and compare them with the experimental results.

Figure 9 shows IX emission energy [(a) and (b)] and intensity [(c) and (d)] as a function of the distance from the excitation spot measured in sample A′ [(a) and (b)] and sample B′ [(c) and (d)]. These data (solid circles) are extracted from the spectral maps like those shown in Fig. 7. Such presentation allows us to compare in the same Fig. 9 energy and intensity profiles at various power densities. Dashed lines represent the steady-state solutions of Eq. (3). The values of the fitting parameters that we obtained ( t rad , A 0, tnr, and L) as well as the deduced values ( t rad , B 0 and D) are given in Table II. Here, the recombination times and the diffusion constants are determined independently because we fit the model to both energy and intensity data. Note also, that better description of the data at low excitation powers where the model tends to overestimate IX density could be obtained by considering density dependence of the non-radiative time.

FIG. 9.

Exciton energy [(a) and (b)] and spectrally integrated intensity [(c) and (d)] as a function of the position in the QW plane (excitation spot is situated at x = 0) in samples A′ [(a) and (c)] and B′ [(b) and (d)]. Dashed lines are obtained from steady-state solutions of Eq. (4) as EIX = E0 + nϕ0, IIX = nRrad. Arrows indicate the extension of the IX cloud that is power dependent, but systematically larger for sample A′ then for sample B′.

FIG. 9.

Exciton energy [(a) and (b)] and spectrally integrated intensity [(c) and (d)] as a function of the position in the QW plane (excitation spot is situated at x = 0) in samples A′ [(a) and (c)] and B′ [(b) and (d)]. Dashed lines are obtained from steady-state solutions of Eq. (4) as EIX = E0 + nϕ0, IIX = nRrad. Arrows indicate the extension of the IX cloud that is power dependent, but systematically larger for sample A′ then for sample B′.

Close modal

The main outcome of this analysis is that the more efficient IX transport that we observe in sample A′ is due to a combination of two factors: smaller non-radiative losses, which were already evidenced by macro-PL experiments, but also larger diffusion coefficient. It seems reasonable to suppose that sample A′ presents a smaller density of point defects that leads to both longer IX mean free path L and smaller non-radiative recombination rate.

The linewidth of optical emission is an important parameter characterizing QW quality. This section is devoted to comparison of the IX emission broadening that we measure in two types of experiments in samples A′ and B′. To make this comparison meaningful, we represent the line broadening not as a function of incident power, but as a function of emission energy, which, in first approximation, is proportional to the IX density.67  Figure 10 shows the FWHM as a function of the corresponding IX emission peak energy measured in macro-PL and μPL experiments at 10 and 65 K, respectively.

FIG. 10.

FWHM of the IX emission line as a function of its peak energy extracted from μPL at 65 K (solid circles) and macro-PL at 10 K (open diamonds). Lines are linear fits of the line broadening above E0. Vertical dotted lines indicate the values of zero-density energy, E0, for each set of data. Differences between the values of E0 measured in the center (macro-PL) and in the periphery (μPL) of the wafer are highlighted by arrows.

FIG. 10.

FWHM of the IX emission line as a function of its peak energy extracted from μPL at 65 K (solid circles) and macro-PL at 10 K (open diamonds). Lines are linear fits of the line broadening above E0. Vertical dotted lines indicate the values of zero-density energy, E0, for each set of data. Differences between the values of E0 measured in the center (macro-PL) and in the periphery (μPL) of the wafer are highlighted by arrows.

Close modal

As discussed in Sec. IV C, the minimum FWHM values are reached not at the lowest IX densities, but at “zero-density” energy E0, where localized states are occupied but there are only a few free IX in the QW. This is observed in both macro-PL and μPL. These FWHM minimum values are very close in the two samples: ∼9 meV at 10 K measured in macro-PL and 17 meV at 65 K measured in μPL experiments. This indicates that surface roughness amplitude (representative of the buried interface roughness), which differs strongly in these samples (1.5 nm in sample A′ and 0.5 nm in sample B′, see Fig. 3), is not the dominant mechanism of the IX inhomogeneous broadening. Most probably, it is determined by the fluctuations of the Al content in the barriers, which are similar in our two samples. We estimate that fluctuations as small as ±0.2% could result in inhomogeneous broadening of 9 meV.

The ratio between minimum values of FWHM measured at 10 and 65 K is consistent with temperature-dependent inhomogeneous broadening if we suppose that for free excitons it is proportional to the QW area, A, explored during IX lifetime A D T. In this case, we expect that FWHM≅17 meV at 65 K would be reduced to FWHM ≅ 7 meV at 10 K, close to FWHM≅9 meV deduced from macro-PL experiments.

The values of E0 slightly differ in two types of experiments (by 12 meV in sample A′ and by 17 meV in sample B′, see arrows in Fig. 10). This is because the temperature is different, and the sample areas studied by macro-PL and μPL correspond to different areas of the wafer, which results in a small difference in Al composition in the barriers (less than 1%).

At low IX densities, corresponding to EIX < E0, IX localization leads to the increase of the linewidth. This inhomogeneous contribution to the broadening is better resolved in μPL but also visible in macro-PL data. At high IX densities, which correspond to EIX > E0, the FWHM increases linearly with the energy. As mentioned in Sec. IV C, one can expect such increase to be due to increased exciton–exciton scattering.71 At very high densities, close to Mott transition and above (for our samples we estimate the corresponding blueshifts as EIX–E0 > 30 mV), which were only reached in macro-PL experiments, the line can be additionally broadened by a step-like electron and hole joint density of states.37,73 Here, we observe that in μPL, the slope σ = (FWHM(EIX)–FWHM(E0))/(EIX–E0) ≅ 0.45 is the same for the two samples, and quite close to the results that we reported previously.70 In macro-PL, however, this slope is more than twice larger, σ ≅ 1.1. This can be easily understood. Indeed, IX energies strongly depend on their densities. Therefore, an inhomogeneous IX density profile created by a broad Gaussian-like spot on the surface will inevitably lead to a broad distribution of emission energies at different positions. In the absence of spatial resolution, such distribution appears as an additional inhomogeneous broadening, which increases with IX density. This effect must be accounted for when analyzing QW quality.

We have grown a set of GaN/AlGaN QWs to optimize IX transport in the QW plane. Surface and interface roughness, cathodoluminescence, PL, and spatially resolved μPL have been scrutinized in these samples.

By analyzing the surface morphology, we evaluated the effects of the substrate growth temperature on the growth mode (step flow vs mound formation) and dislocation densities. Further optimization has been reached by additional MOVPE growth of 1.1 μm-thick GaN layer on the substrate prior to MBE growth of the QW heterostructure. As a last step, we have comparatively studied optical emission of the two best samples obtained within the same growth procedure but keeping the substrate at different temperatures, 780 °C (sample A′) and 900 °C (sample B′).

Summarizing the results of optical experiments, we conclude that the growth protocol used for sample A′ is the most advantageous for studies of IX. In this sample, macro-PL experiments point out lower non-radiative losses, while μPL experiments confirm this observation and demonstrate IX transport over larger distances. Thus, IXs have greater diffusion constant and mobility in sample A′ than in sample B′. The corresponding mean free paths of IXs are estimated as ∼13 and ∼9 nm in samples A′ and B′, respectively. We tentatively interpret these values as an average distance between scattering centers which are also responsible for the non-radiative losses. These centers are not related to extended defects, such as dislocations since the latter are characterized by much lower densities. The corresponding diffusion constants at T = 65 K are D ∼ 4 cm2/s and D ∼ 3 cm2/s for samples A′ and B′, respectively. These values are close to those found in GaAs/AlGaAs QWs of the similar thickness by Vörös et al.,74 but about an order of magnitude smaller than those reported in Ref. 75. Importantly, the mechanisms that limit exciton diffusion are different in GaN and GaAs-based heterostructures. While in GaAs/AlGaAs QWs, the IX transport is mainly limited by the QW interface roughness, in these GaN/AlGaN QWs, the non-radiative defects seem to be the main limiting factor since the surface roughness of sample A′ is larger than sample B′ and the IX diffusion constant is larger in sample A′. The density of these defects has to be decreased to reach a quality at which the IX transport is limited by the QW interface roughness.

In contrast with IX diffusion constants, the linewidth of the IX emission is essentially the same in samples A′ and B′. We speculate that the latter is determined by Al content fluctuations in the barriers, which are much less sensitive to the growth temperature than the density of non-radiative defects.

Overall, the studied samples are promising for studies of many-body physics of dipolar bosons. They could be patterned with metallic electrodes in order to make traps able to confine spatially IX as this was demonstrated in previous work.36,70 We expect that thanks to the improved epitaxial quality of the samples and the reduced excitonic dipole length, quantum collective effects such as Bose–Einstein condensation could be demonstrated at temperatures above 1 K.

Finally, several other avenues can still be explored to improve samples quality. In particular, the AlGaN alloy fluctuations that are the main source of the broadening of the IX emission could be decreased. The straightforward way to do that would be to decrease the Al composition in the AlGaN barrier layer. Indeed, an AlGaN PL FWHM down to 5 meV was demonstrated for an Al composition as low as 0.05 [Fig. 6(b)].16 However, the limit of this approach is that the exciton confinement in the QW may decrease, which can induce the exciton escape from the QW especially for large carrier densities and for non-cryogenic temperatures. Other approaches could be considered, such as the optimization of the substrate offcut76,77 or the use of short period AlN/GaN superlattices (AlGaN digital alloy) to further minimize the impact of AlGaN alloy fluctuations.78–80 

This work was supported by the project IXTASE of the French National Research Agency (No. ANR-20-CE30-0032) and by the Occitanie region through the Quantum Technology Challenge grant. This work also benefited from the aid from the French government, managed by the National Research Agency under the project “Investissements d’Avenir” UCA JEDI (No. ANR-15-IDEX-01).

The authors have no conflicts to disclose.

B. Damilano: Writing – original draft (equal). R. Aristégui: Writing – original draft (equal). H. Teisseyre: Writing – original draft (equal). S. Vézian: Writing – original draft (supporting). V. Guigoz: Writing – original draft (supporting). A. Courville: Writing – original draft (supporting). I. Florea: Writing – original draft (supporting). P. Vennéguès: Writing – original draft (supporting). M. Bockowski: Writing – original draft (supporting). T. Guillet: Writing – original draft (supporting). M. Vladimirova: Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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