Photo- and cathodoluminescence investigations of piezoelectric GaN/AlGaN quantum wells
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THEORETICAL MODELING The exciton characteristics in GaN/AlGaN MQWs are calculated using a simple variational approach within the envelope function approximation in a two-band model. The trial function of a quasi-2D exciton is used as 2 − ρ a0 e πa 2 φ with where ψe(z) and ψh(z) are single-particle electron and hole wave functions, and a0 is a variational parameter.
ϕ ex ( ρ , z ) = f ( ρ )ψ e ( z )ψ h ( z ), f ( ρ ) =
The exciton radiative lifetime is estimated using the following equation:
(
τ 0−1 = π ab3 k0 ω LT f 2 (0) ∫ ϕ e ( z )ϕ h ( z )dz
)
2
where aB is the 3D Bohr radius, k0=nω/c and ωLT is the exciton longitudinal-transverse splitting. We used the following values of the effective masses me=0.3m0; mh=1.55m0, the conduction band offset was taken as 25% of the bandgap difference. The longitudinal-transverse exciton splitting was taken as 1 meV. We assumed homogeneous electric fields both in the GaN QW (FQW) and in AlGaN barriers (Fb.). This approach was shown to be accurate for relatively wide piezoelectric QWs [5]. The emission spectra at pulsed excitation conditions show a broad peak between 3.6 and 3.7 eV, attributed to the PL from the AlGaN barriers. This yields an independent estimate of the Al content in the barriers of 14-16%, and, according to this the value of the conduction band offset of 190 meV was used in the calculations of exciton energies. The electric field in the barrier can not be determined from the experimental data. In the calculations we kept the electric field in the barrier constant at 1x106V/cm because the results are not sensitive to the barrier field, and varied the field in the quantum wells. 0 4.5K
200
bulk GaN
400
QW field, kV/cm
PL intensity, arb. units
AlGaN
1.4 nm QW 6 nm QW
3.3
600
4 nm QW
3.4
3.5
3.6
3.7
Energy [eV] .
Figure 1 . Steady-state (bottom trace) and pulsed PL (top trace). Calculated exciton energies for varying electric field in QWs (broken ine, right scale). We were able to match the low field exciton energy in the 2 nm QW to the PL peak observed with pulsed excitation, by adjusting the QW width. The best fit was obtained for the QW thickness of
G6.12.2
1.4nm, this is about 2ML smaller than the nominal thickness and consistent with the anticipated range of thickness fluctuations. Exciton energies in wider QWs are not strongly affected by thickness fluctuations. The electric field dependence of exciton energies, is shown in Figure 1 (dashed curves). At steady-state excitation the electric field is almost the same in all the QWs of about 0.6-0.7 x 106V/cm, and possibly smaller in the narrowest QW. Under the pulsed excitation, at peak excitation intensity, the field seems to be almost completely screened. However, during the subsequent PL decay the electric field evolves around some intermediate values. This results in the PL emission peaking between 3.4 and 3.5eV. Figure 2a-c shows the exciton intrinsic lifetimes calculated for different electric fields in the barrier Fb as a function of the field in the well, FQW. A degree of correlati
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