2.2. PHOTOLUMINESCENCE LINE WIDTH ANALYSIS IN InGaN MULTIPLE QUANTUM WELLS USING MONTE CARLO
2.2.3. Double-scaled potential in InGaN multipple quantum wells
The near-band-edge PL spectra measured at different temperatures are presented in Fig. 2.2.1. The spectra contain a single band with two humps on the low-energy slope separated by approximately 90 meV. We attribute these features to direct and longitudinal-optical-phonon replicated recombination of localized excitons, respectively. The PL intensity maintains an almost stable value below 50 K and decreases at elevated temperatures, most probably, due to enhanced influence of nonradiative recombination.
2.65 2.70 2.75 2.80 2.85 101
102
310 K
PL Intensity (arb. units)
Photon Energy (eV) 10 K
Fig. 2.2.1. Evolution of the PL spectra with temperature in InGaN/GaN MQWs. The
temperature is incremented by 15 K starting with 10 K (the uppermost spectrum) and ending with 310 K (the lowermost spectrum). The band peak positions are indicated by dots.
The PL band peak position is highlighted by dots in Fig. 2.2.1. The peak exhibits a well-established S-shaped temperature behavior [7–9]: it slightly redhifts in the range from 10 K to 50 K, then blueshifs up to 150 K, and redshifts again afterwards. Points in Fig. 2 show a W-shaped temperature dependence of the full width at half maximum (FWHM) of the PL band
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with a characteristic kink at about 150 K. The S-shaped temperature behavior of the PL band peak position and the W-shaped temperature behavior of the line width are known to be a signature of exciton hopping over randomly dispersed localized states with a crossover from a nonthermalized to a thermalized distribution function of the excitons [17]. Under appropriate selection of the electronic potential profile, exciton hopping is able to quantitatively account for the PL line width and the Stokes shift of the PL band in respect of the average exciton energy in group-III nitride alloys as revealed by our recent three-dimensional (3D) Monte Carlo simulation in AlInGaN bulk-like layers [17]. Here we applied a Monte Carlo simulation routine for exciton hopping in 2D regime and demonstrated that the potential profile can be described based only on the experimental temperature dependence of the PL line width.
Simultaneously, we deduced the Stokes shift of the PL band and reconstructed the temperature dependence of the average exciton energy in our MQW structure using the measured PL peak positions.
The observed temperature behavior of the PL line width was simulated by using a 2D Monte Carlo procedure with the Miller-Abrahams rate for phonon-assisted exciton tunneling between the initial and final states i and j with the energies Ei and Ej, respectively,
0 Here r is the distance between the states, ij α is the decay length of the exciton wave function, and ν0 is the attempt-to-escape frequency. Hopping was simulated over a randomly generated set of localized states with the sheet density N and localization energies dispersed in accordance with a Gaussian distribution with the central position at exciton energy and the dispersion parameter (the energy scale of the band potential profile fluctuations) σ . For each excited exciton, the hopping process terminates by recombination with the probability τ0−1 and the energy of the localized state where the recombination has taken place from is scored to the emission spectrum S h0( )ν .
Fitting of the temperature dependence of the width of the simulated PL band to the experimental results reveals the important peculiarities as follows. In the initial 10 K to 150 K region, the variation of the line width is due to thermal enhancement of exciton hopping within nonthermalized energy distribution and the shape of the dependence is basically a function of the spatial (the product Nα2) and temporal (the product ν τ0 0) parameters of the hopping process. The kink in the temperature dependence of the line width at about 150 K represents a crossover from a nonthermalized to thermalized energy distribution of excitons and the crossover temperature is related mainly with the energy scale of the band potential profile fluctuations σ. Finally, an almost constant line width right above the crossover temperature (150–180 K) indicates an occurrence of a thermalized exciton energy distribution and is determined by the energy scale σ and additional inhomogeneous broadening Γ that results in a modified emission spectrum
( ) 0( ) ( , )
S hν =
∫
S hν′ G Γ hν −hν′ d hν′, (2.2.2) where G( ,Γ hν) is a standard Gaussian function with dispersion Γ. The further increase of the line width above 180 K is attributed to the influence of delocalized excitons that are not taken into account in the model used.27
(a) Temperature dependence of the full width at half maximum of the photoluminescence band in 2-nm InGaN/GaN MQWs. Points depict experimental data; dashed, dotted and solid lines show results of Monte Carlo simulation of exciton hopping for different scales of random potential fluctuations σ (indicated) with the inhomogeneous broadening Γ of 29 meV taken into account; dashed-dotted line
represents results for σ = 31 meV and Γ = 0.
(b) Temperature dependence of the Stokes shift of the PL band peak position in respect of the average exciton energy corresponding to the best-fitted line width dependence.
Solid line in Fig. 2.2.2a represents the result of the best fit obtained for the following values of the parameters: Nα2 = , 1 ν τ0 0 = ×3 105, 31 meVσ = , and Γ =29 meV. Dashed and dotted lines demonstrate sensitivity of the simulation results in respect of the hopping energy scale σ and a dashed-dotted line shows the simulated dependence without inhomogeneous broadening. position (open dots) and the
reconstructed average exciton energy (solid dots) as a function of
temperature. Solid line represents the best fit by using the Bose-Einstein-like formula.
Our simulation reveals a double-scaled band potential profile in InGaN quantum wells similar to that deduced in AlInGaN bulk-like layers [17]. Such a band potential profile implies
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hopping of excitons within isolated In-rich regions with the localized-state energy dispersion σ, whereas the average exciton energy is different in each In-rich region and is dispersed over the scale Γ. We attribute the dispersion Γ to different average indium content within the In-rich regions. Moreover in quantum wells, both the energy scale σ and the inhomogeneous broadening Γ can be increased by well-width fluctuations that modulate quantum confinement energy.
Figure 2.2.2b depicts the temperature behavior of the Stokes shift deduced from the peak position of the simulated spectra. Based on this data and the measured peak positions of the PL band (open points in Fig. 2.2.3), the temperature dependence of the average exciton energy was reconstructed (filled points in Fig. 2.2.3). The dependence is fairly well described by the Bose-Einstein-like expression
( )
( ) (0)
exp 1
E T E
T λ
= − θ
− (2.2.3)
with the best-fit parameters λ = 0.154 eV, θ = 379 K, and E(0) = 2.8452 eV. Although obtained in a rather indirect way, this result provides supplementary data on characterization of thin quantum wells with relatively high In molar fraction, where the exciton energy is difficult to precisely measure (e.g. using absorption, reflection, or photoluminescence excitation spectroscopy) because of low optical density and considerable broadening of the spectra.
2.2.4. Potential profile as a function of indium content in InGaN multipple quantum