Complexity of the dipolar exciton Mott transition in GaN/(AlGa)N nanostructures

The Mott transition from a dipolar excitonic to an electron-hole plasma state is demonstrated in a wide GaN/(Al,Ga)N quantum well at $T=7$ K by means of spatially resolved magnetophotoluminescence spectroscopy. Increasing optical excitation density, we drive the system from the excitonic state, characterized by a diamagnetic behavior and thus a quadratic energy dependence on the magnetic field, to the unbound electron-hole state, characterized by a linear shift of the emission energy with the magnetic field. The complexity of the system requires taking into account both the density dependence of the exciton binding energy and the exciton-exciton interaction and correlation energy that are of the same order of magnitude. We estimate the carrier density at Mott transition as $n_{\mathrm{Mott}}\approx 2\times {10}^{11} {\mathrm{cm}}^{-2}$ and address the role played by excitonic correlations in this process. Our results strongly rely on the spatial resolution of the photoluminescence and the assessment of the carrier transport. We show that in contrast to GaAs/(Al,Ga)As systems, where transport of dipolar magnetoexcitons is strongly quenched by the magnetic field due to exciton mass enhancement, in GaN/(Al,Ga)N the band parameters are such that the transport is preserved up to 9 T.

Figure 1

(a) IX phase diagram. Four phases theoretically expected are shown by different colors. The green region delimited by dotted line represents the phase space area explored in this paper. The critical density corresponding to the transition toward plasma states is determined in this paper. (b) Sketch of the the GaN/(Al,Ga)N QW bands in the absence (red dashed line) and in the presence of optical excitation (black solid line). Corresponding electron and hole wave functions and energy levels are shown with the same color code. The band-to-band transitions are indicated by arrows. (c) Artist view of the two ways to study IX emission: IX in a trap (upper sketch)—pointlike excitation followed by spatially separated PL analysis in the trap (used in Ref. [32]) and free space (lower sketch)—broad excitation followed by detection in the center of the excitation spot (used in this paper).

Figure 2

PL intensities (color encoded in log scale) measured at $T=7$ K and two different excitation power densities: $P=90$ mW/${\mathrm{cm}}^2$ (a), (b); $P=2.8$ mW/${\mathrm{cm}}^2$ (c), (d); and two magnetic field values: $B=0$ (a), (c); $B=9$ T (b), (d). The measured laser intensity profile (normalized to unity) is shown by dashed lines. Dotted lines are Gaussian-shaped guides for the eye showing the decrease of the IX emission energy when the distance from the excitation center increases.

Figure 3

PL intensities (color encoded in log scale) measured at $T=14$ K and two different excitation power densities: $P=90$ mW/${\mathrm{cm}}^2$ (a), (b); $P=2.8$ mW/${\mathrm{cm}}^2$ (c), (d); and two magnetic-field values: $B=0$ (a), (c); $B=9$ T (b), (d). The measured laser intensity profile (normalized to unity) is shown by dashed lines. Dotted lines are Gaussian-shaped guides for the eye showing the decrease of the IX emission energy when the distance from the excitation center increases.

Figure 4

(a)–(c) IX emission energy shift with respect to the $B=0$ emission energy as a function of the magnetic field at $T=7$ K (a), $T=10$ K (b), $T=14$ K (c) for various excitation power densities from $P=2.8$ mW/${\mathrm{cm}}^2$ (black symbols) to $P=90$ mW/${\mathrm{cm}}^2$ (magenta symbols). Solid lines are parabolic fits to the diamagnetic shift model. The model does not fit the highest power data in (a) and the linear fit is shown by the long-dashed line. Short-dashed line in (a) shows the cyclotron shift expected for unbound electron-hole pairs. (d)–(f) show the emission energy at $B=0$ for each power density and three different temperatures. The horizontal dashed line shows zero-density limit $E_{\mathrm{IX}0}$ of the exciton emission energy. The vertical dashed arrow illustrates the density-induced blueshift of the emission line at $P=90$ mW/${\mathrm{cm}}^2$. (g)–(i) Values of Bohr radius extracted from the quadratic fit (a)–(c) as a function of the zero-field density-induced energy shift. Excitation power density is shown with the same color code in (a)–(c), (d)–(f), and (g)–(i).

Figure 5

PL spectra collected from $25\text{−}\mu \mathrm{m}$-diameter area in the center of the excitation spot at different power densities. Two temperatures ($T=7$ K, $T=14$ K) and two magnetic field values ($B=0$ and $B=9$ T) are shown for each power. We identify the two emission lines as IXs and unbound electron-hole pairs (EH).

Figure 6

Exciton Bohr radius (solid lines) as a function of the IX density calculated within exponential model assuming $n_{\mathrm{Mott}}=2\times {10}^{11} {\mathrm{cm}}^{-2}$ (dashed lines) and ${\text{Ry}}_0=25$ meV. These assumptions are identical for the three different temperatures (a)–(c). Symbols show the Bohr radii extracted from the diamagnetic shift at different powers (same as in Fig. 4, same color code) as a function of the carrier density calculated from the corresponding emission energy at $B=0$. Open (filled) symbols show the calculation within Model C (Model NC). At 7 K, crosses on top of the circles indicate the density points where magnetic field induced energy shift is linear, so the interpretation in terms of the diamagnetic shift is not satisfactory (the emission is expected to be dominated by the electron-hole plasma).

Figure 7

Experimental setup.

Figure 8

Calculated spatial distribution of the emission energy shift (a), (b) and the intensity (c), (d) for three different values $M\left(B\right)/M$ and two power densities. Dashed lines in (c), (d) show measured IX intensity at $T=14$ K (there is no difference in the $B=0$ and $B=9$ T profiles).

Figure 9

Energy shift (a), (b) and intensity increase (c) in the center of the excitation spot calculated as a function of $M\left(B\right)/M$ for two different values of power density. Dashed lines point to the value $M\left(B\right)/M=1.3$, the estimated limit for our system.

Figure 10

A typical IX photoluminescence spectrum (blue) and the result of the fitting procedure (red dashed).

Figure 11

Effect of the photoinjected carriers on the conduction (a) and valence (b) band diagram and wave function. The band profiles and wave functions are calculated for an empty QW (black lines) and for a pair density of $n=2\times {10}^{11} {\mathrm{cm}}^{-2}$ (red dashed lines).

Figure 12

Pair correlation function calculated for various values of the particle density at $T=7$ K and dipole length $d=7.8$ nm. Thick lines indicate the calculations in the low density limit ($n={10}^7 {\mathrm{cm}}^{-2}$, dark blue), at Mott density estimated in this paper ($n=2\times {10}^{11} {\mathrm{cm}}^{-2}$, cyan) and at $n=7\times {10}^{11} {\mathrm{cm}}^{-2}$, corresponding to the saturation of $g\left(r\right)$. Dashed line points to $r_0$, the mean-field exclusion radius given by Eq. (A17).

Figure 13

(a) Interaction energy (or, equivalently, density-induced energy shift) as a function of particle density for $T=$ 7 K. Model NC (self-consistent solution of the coupled Schrödinger and Poisson equations, Appendix pp1-s5a), black line; model C solved with (blue line) and without adaptive procedure, see Appendix pp1-s5b. Dashed lines indicate the estimated Mott density and the corresponding interaction-induced energy shift.