1. Introduction

GaN based nitrides, in particular, InGaN/GaN quantum wells (QWs), play the central role in the rapid emerging of new generation of semiconductor lighting industry [1, 2]. The transport and recombination of carriers in InGaN/GaN QWs essentially determine the underlying physical mechanisms and hence the performance of GaN-based light-emitting devices. There have been more and more evidence showing that the randomly distributed In-rich clusters with typical tens of nanometers exist in InGaN layers instead of random In composition fluctuation [3, 4]. These In-rich nanoclusters lead to the potential minima where the carriers can be quantum mechanically confined. Such potential minima strongly alter the electronic structures and thus affect the transport and recombination dynamics of carriers. Despite the fact that GaN-based light-emitting devices including laser diodes have been commercialized, the luminescence mechanism of the devices is still unveiled incompletely. In particular, future effort is required to link the carrier dynamics to the intricate structure of InGaN layers.

In general, for a perfect semiconductor system, standard exciton picture is valid. In such an ideal system, only radiative recombination process is a path of exciton “death” and time evolution of its photoluminescence (PL) intensity obeys a single exponential decay. Shorter decay time of the PL intensity means higher luminescence efficiency just for this ideal system. For real semiconductors, however, imperfections such as defects/impurities, dislocations, composition fluctuation and so forth inevitably exist inside the lattice matrix of materials. For example, for InGaN epilayers to be investigated in the present work, nano-sized In-rich clusters naturally form, giving rise to the localized electronic states. A natural extension of exciton concept in such disordered semiconductors is the localization of excitons or localized excitons. However, the exciton picture may be questioned when people try to explain certain emission characters of InGaN/GaN QWs using the extended exciton model. For instance, a recent study about the coupling between electron-hole pairs and longitudinal optical (LO) phonons in InGaN/GaN QWs and quantum boxes (QBs) structures shows that Huang-Rhys factor S characterizing the coupling strength between electrons and LO phonons increases significantly as the vertical size of the QBs or the thickness of QWs increases [5]. This behavior is suggested to be interpreted using electron and holes localized independently on separate sites rather than the picture of excitons. In the case of ideal excitons the time decaying would be exponential with a decay rate given by the lifetime of excitons. In fact, it was found that the decay in disordered InGaN/GaN quantum structures exhibits non-exponential character [6], which is difficult to bring into agreement with an exciton picture. The non-exponential or multi-exponential PL decay is quite often observed in the disordered systems such as GaN/AlGaN QWs [7], GaInP/GaAs heterostructures [8], and nanostructures [9]. These studies indicate that the transport and decaying of carriers in strongly disordered semiconductors like InGaN/GaN QWs are needed to be investigated carefully.

It is reasonable to believe in that electrons and holes are not spatially correlated in the form of exciton, but localized in uncorrelated randomly distributed fluctuation minima independently, if the amplitude of the disorder potential is higher than the binding energy of excitons. In the systems with a broad localized state distribution, photogenerated charge carriers, electrons and holes, face two choices: radiatively recombine with other partner having a real spatial distance to generate light or non-radiatively transfer to other localized states. These two processes are competitive, determining decay behavior of the PL intensity. As a result, the decay curve of the PL intensity could be non-exponential or multi-exponential. In this case, shorter time does not certainly mean higher luminescence efficiency. In fact, such simple fitting even using bi- or multi-exponential decay functions might be problematic and could not reveal the precise physical nature of the complicated systems. Morel et al. [6] proposed a model for radiative recombination of electron-hole pairs in GaInN/GaN quantum objects, in which a electron and a hole were considered as independently localized at sharp potential fluctuations, forming a two dimensional pseudo-donor-acceptor pair (DAP) [10]. However, in their model, only case of low temperature, at which the radiative recombination process is dominant, was considered. The dynamic problem of photogenerated carriers in real disordered systems is much complicated than previously handled. For example, the nonradiative recombination process becomes considerable with increase of temperature and can not be neglected any more.

In this paper, we attempt to simulate the decay curves of the photogenerated carriers in disordered InGaN/GaN QWs by means of a latest theoretical model in which electron and holes are treated not in the form of excition, but rather as independent species, which was recently developed by Rubel et al. [12]. It has been shown that the competing between the radiative recombination and phonon-assisted hopping transition between different localized states of localized carriers is responsible for the non-exponential decay of the carriers in a broad temperature range from low temperature to room temperature. We also found that in the InGaN/GaN QWs with strong disorder and deep localization states the carriers exhibit anomalously an almost ideal exponential decay, which is successfully interpreted using the model.

2. Experiment and result

Two InGaN/GaN multi-QW structures with identical layer structure and the same indium concentration but different Si-doping density, labeled by A and B, were studied in the present work. Both sample A and B were grown on c-plane sapphire substrate by metalorganic chemical vapor deposition. Each of them consists of 5 periods of InGaN/GaN QWs. The thickness of the InGaN well (GaN barrier) layer is 3 nm (12 nm). The only difference between these two samples is that the barrier layers of QWs in sample A were unintentionally-doped while the barrier layers of sample B were doped with Si to about 10¹⁸cm^-3. Structure characterization of the samples was performed using cross-sectional transmission electron microscopy (TEM) while the optical measurements of the samples were carried out with time-resolved photoluminescence (TRPL) technique. The details of structure characterization and optical measurements have been previously described elsewhere [4, 13].

 

Fig. 1. Cross-sectional TEM images of InGaN/GaN QWs: (a) for undoped sample A and (b) for doped sample B. The inset is a low-magnification image showing five periods of InGaN/GaN QWs.

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Figure 1 shows the high-resolution cross-sectional TEM images of the two samples. Five periods of alternative InGaN well and GaN barrier layers can be easily seen from the inset figures depicting the low-magnification cross-sectional TEM images of the samples. The structural evidence of the existence of the self-formed In-rich clusters, associated with strong dark contrast regions within the InGaN well layers, can be easily obtained from the high-resolution TEM images. The dark regions are due to the strong aggregation of indium atoms. A direct comparison between the high-resolution TEM images of sample A and B leads to a conclusion that the density of In-rich clusters in sample B is higher than that in sample A. This indicates that the Si doping strongly increases the disorder of InGaN layers although the doping takes place in the GaN barrier layers.

 

Fig. 2. 77 K 3D time-resolved photoluminescence spectra for InGaN/GaN QWs: (a) for undoped sample A and (b) for doped sample B.

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Fig. 3. 77 K time-resolved PL spectra for InGaN/GaN QWs at different delay times: (a) for undoped sample A and (b) for doped sample B. The solid dots represent the PL decay times as a function of photon energy and the solid lines are drawn to guide the eyes.

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Figure 2 shows the 3D time-resolved photoluminescence spectra of sample A and B measured at 77 K. It is clear that the decay time of the luminescence signal increases with increasing wavelength. Such a character could be observed in the time dependent emission spectra of DAP transitions [10]. The situation that electrons and holes were independently captured in different localized states is also similar with that of DAP. For DAP pair with reasonable large spatial separation, the energy of emission is approximately given by E(r)=E_gap -(E_a +E_d )+e ²/εr, where E_gap is the band-gap energy, E_a and E_d are the acceptor and donor binding energies, respectively, ε is the low-frequency dielectric constant and r is the donor-acceptor separation [10]. The fourth term on the right hand of the emission energy expression stems from the Coulomb attraction between the ionized donor and acceptor. Since the energy of final state (i.e., the originally neutral donor and acceptor become into ionized after the radiative recombination) is lowered by the amount of e ²/εr the emission energy is thus correspondingly increased by the amount [11]. The emission energy is inversely proportional to the distance r. It is easily understood that for the electron and hole at different localized states, the larger the distance between them, the smaller the recombination energy. As we has discussed before, the rate for a localized electron to recombination with a localized hole also depends exponentially on the distance between them. As the electron-hole separation increases the lifetimes of the states become steadily longer. In addition, a carrier that is captured at a localized center, in addition to direct recombination with a carrier of the opposite type, can transfer to a state lower in energy prior to recombination, giving rise to a lower energy photon. This also decreases the decay time of the emission at prior higher energy. Figure 3 shows the 77 K time-resolved PL spectra of the two samples at various time delays. As the time delay after the pulse excitation increases, the emission at higher energy side decays faster than the emission at lower energy side. So that, a red shift to the lower energies and narrowing of the emission band can be seen for these two samples with increasing the delay time. The pairs with a small distance will emit light with higher energy and shorter lifetime whereas the more distant pairs emit light with lower energy and longer decay time. The solid dots represent the PL decay times at different photonic energies while the solid lines are only to guide the eyes. Clearly, the emissions at higher photonic energies processes the shorter decay times. These results show that the electrons and holes in sample A and B are indeed localized in uncorrelated potential minima independently rather than in the form of exciton.

It is interesting to note that the decay curve of the emission band of sample A (unintentionally-doped) exhibits a “non-exponential” behavior, a typical character of disorder systems, whereas the emission band of sample B (intentionally Si-doped) decays at a nearly perfect exponential rate, as shown in Fig. 4. It is hard to believe that the existence of excitons in sample B whose barriers were heavily doped with Si atoms. For interpretation of this puzzling observation, we employ the theoretical model [12] for quantum structures with localized states caused by disorder to simulate the experimental curves.

 

Fig. 4. Measured PL decay profiles (empty symbols) for InGaN/GaN QWs at 77 K. The solid lines are the theoretical curves using the model described in details in the text.

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3. Theoretical interpretation and discussion

The model [12] is based on the set of rate equations, in which electrons and holes involved in luminescence process are treated as uncorrelated carriers rather than in the form of excitons. For a semiconductor system with strong enough disorder, for example, the amplitude of potential fluctuation induced by disorder higher than the binding energy of excitons, photogenerated electrons and holes may diffuse independently at different rates. As a result, electrons and holes are most likely captured into different potential minima during their uncorrelated diffusion. These independently localized electrons and holes can either radiatively recombine or perform a phonon-assisted hopping transition to other localized states. Such dynamic behavior could be described by the rate equations which were originally proposed by Marshall [14]. The energy range where localized states are distributed is divided into a set of m energy slices with a given width. For the system, an exponential density of states (DOS) g(ε)=N ₀/ε ₀ exp(-ε/ε ₀) with a total concentration of localized states N₀ and energy scale ε₀ was adopted. For simplicity we treat the densities of localization states and localization lengths for electron and holes as equal. The rate equation for carrier density n_k in those energy slices k are formulated as follows [12]:

where Γ _r denotes the recombination rate for a localized electron to recombination with a localized hole and Γ_j→k (Γ_k→j) is the rate for a charge carrier to perform a non-radiative hopping transition from an occupied state j(k) to an empty localized state k(j) over a distance r_jk . In general, the rate for hopping transition depends exponentially on the distance involved [12].

where ε_j and ε_k are the energies of states j and k, respectively, α is the localization length and ν₀ is the attempt to escape frequency. For the transition from the slice k downward in energy, ε_k >ε_j , only the tunneling term remains and the downwards transition rate can be considered as [12]

where R_k is the typical hopping distance, determined by the concentration of unoccupied states with energy below ε_k [12]

where d_j is the concentration of localized states in the energy slice. On the other hand, the upward transition from slice j to k can be derived from the downward transition rate [12]

The rate for a localized electron to recombination with a localized hole also depends exponentially on the distance R between them [12]

where τ₀ is a time constant which depends on the particular recombination mechanism and is of the order of excitonic radiative lifetime. The most efficient recombination is the pairs of localized states in which the state for electrons is as close to the state for holes as the localization length. Correspondingly, the recombination time does not contain the exponential factor and is close to τ₀ . The concentration of such pairs is the product of the density of filled electron states n and the probability α ². The recombination rate [12] can be considered as Γ _r (R)=${\tau }_0^{-1}$ n(t)α ². The time-resolved luminescence spectrum is calculated as a convolution of carrier densities obtained by solving above equations.

As Eqs. (2) and (6) show, the transport and recombination probabilities of localized carriers depend exponentially on the distances involved. For a system with a broad localized state distribution, therefore, the recombination times of carriers will take a wide range of values. Both competitive processes jointly determine the decay behavior of the PL intensity in the disordered system. So it is not difficult to understand that the decay time of the PL intensity in a disordered system shall have a rather broad distribution and probably shows a non-exponential or multi-exponential character.

From Eqs. (2) and (6), it can be seen that the localization length α is a key parameter characterizing the disorder degree. It can be viewed as the average spreading length of the wave functions of the localized carriers in the localized states. Such a length essentially determines the dynamic behaviors of localized carriers. Using the model briefly described above, the localization length can be obtained by fitting to the experimental decay curve of the emission intensity. The solid curves in Fig. 4 are the fitting results when the values of parameters τ₀ =1 ns, ν₀ =10¹⁰ s^-1, kT=6.64 meV, ε₀ =8 meV were adopted. Note that the obtained localization length is 4.3 nm for sample A and 2.6 nm for sample B. The symbols represent the experimental decay curves of the emission intensities of sample A and B in the figure. The initial concentration of electron-hole pairs just after the excitation pulse is 4×10¹² cm^-2. Clearly, sample B (Si doped) has a shorter localization length, which indicates that the Si doping in the barrier layers of InGaN/GaN QWs can lead to further localization of carriers and can reduce the hopping rate of carriers between different localized states. In fact, the localization length is determined by the density of localized states. The lower the density of localized states, the farther distance the wave function can decay, hence the longer the localization length. This result is supported by the TEM observation (as shown in Fig. 1) which shows that sample B has the higher density of In-clusters than sample A. According to Eq. (6), the recombination time is exponentially dependent on the inverse of the localization length. The longer lifetime for the Si-doped sample, as observed in Fig. 4, is consistent with the theoretical prediction. The decay curve will tend to exhibit an exponential variation if the lifetime is long enough.

4. Conclusions

In conclusion, the non-exponential decay curves of the photogenerated carriers in disordered InGaN/GaN QWs were successfully simulated using a newly developed model considering the radiative recombination and phonon-assisted hopping transition between different localized states together. If the amplitude of the disorder potential is higher than the binding energy of excitons, electron and holes are captured in different localized states independently, similar with DAP. The electron-hole pairs with a short distance will emit light with higher energy and shorter lifetime whereas the more distant pairs emit light with lower energy and longer decay time. The anomalous near exponential decay of carriers in Si-doped InGaN/GaN QWs was found to have a longer decay time and is attributed to the shorter localization length of carriers due to the higher density of In-rich clusters and deeper localized states induced by Si doping.