The competition of electron-electron interband scattering (ee) and longitudinal optical phonon emission (e-ph) as electron capture mechanisms is theoretically investigated in III-nitride quantum wells. The non-trivial separation of their scattering probabilities is discussed, and compact expressions for capture time are obtained in the framework of the quantum many-body formalism. At the typical operating conditions of light emitting diodes (LEDs), the model predicts an increasing importance of ee scattering as a capture mechanism with increasing carrier density. Verifications against recent experiments are presented to support this finding and confirm the need for population-dependent capture time expressions including both ee and e-ph mechanisms for an accurate description of LED carrier dynamics and efficiency.

Intense investigations are currently ongoing about the nature of the internal quantum efficiency (IQE) droop in GaN-based light-emitting diodes (LEDs) at high current and temperature.1–5 Numerical simulations of quantum well (QW) LEDs are largely based on drift-diffusion simulators, whose inadequacies related to several nitride-specific issues have been generally addressed with ad hoc models (e.g., for ballistic overshoot, carrier overflow, tunneling assisted by phonons, and crystal defects), often based on poorly known or ill-defined physical parameters.6–8 

In semiclassical models of LED transport dynamics, the carrier capture and escape times between bulk unbound states and QW bound states are among the most critical quantities for a correct estimate of the IQE droop.5,9 In spite of their importance, capture/escape times are usually treated as fitting parameters, and their dependence on the excitation conditions is customarily neglected.

Electron capture in QWs may take place through electron-electron (ee) scattering, relaxation on defects, longitudinal optical (LO) phonon emission (e-ph), multi-phonon emission, tunneling, etc. (see, e.g., Refs. 10 and 11 and references therein). Among these processes, capture via LO-phonon emission τeph, besides concurring to the determination of the total QW capture time to a considerable extent,11–13 is interesting because it is determined by the LO-phonon-electron self-energy Σeph, a quantity that also affects the phonon-assisted Auger recombination lifetime.14–18 At high carrier densities, capture via ee scattering may also become very effective, and the relative importance of ee and e-ph mechanisms should be assessed. (In the present work, only electrons will be considered, but the formalism also applies to hole capture.)

In a previous work,19 we presented a quantum model which is able to reproduce the experimental capture time in InGaN/GaN QWs as a function of bulk and QW carrier densities N and NQW, considering, however, only the e-ph process, without addressing ee scattering. The importance of the latter as a capture mechanism in III-nitride QWs is controversial: ultrafast pump-probe experiments12,20–23 indicate a probable contribution from both mechanisms, in one case12 reporting also numerical estimates of the overall capture time τ as a function of N. In recent experiments, David et al.9 obtained indications of a τ with two components, one compatible with LO-phonon emission and a second one possibly related to ee scattering. For the III-nitride materials system, no theoretical, first-principles works allow to clearly distinguish among individual contributions to the overall capture rate. In conventional III-V-based QWs, the relative importance of ee and e-ph scattering rates was calculated in Ref. 24 for AlGaAs/GaAs QWs within the static screening and second-order Born approximations, and ee scattering was found to be quite effective. Also, employing simple static screening, the two capture mechanisms were found to be of comparable magnitude in Ref. 10, at least for high carrier densities. Valuable theoretical calculations25–27 employing frequency dependent screening in the Random Phase Approximation (RPA) formalism28,29 revealed a significant decrease in the capture time with N; however, these fully numerical approaches do not lead to formulations ready to be employed in device-level simulators.

In Sec. II, following the quantum many-body formalism and employing the RPA dielectric function in the dynamical form (frequency and momentum dependent) of the Single Plasmon Pole (SPP) approximation,28–30 we present explicit, analytic expressions of capture time for the ee scattering mechanism, as functions of carrier density and InGaN/GaN QW parameters. Electron eigenfunctions are evaluated here within the flat band and effective mass approximations, with parameters according to Ref. 31, but the proposed approach can be applied with no modifications also when potential and carrier density profiles are determined by solving the Poisson and Schrödinger equations selfconsistently,32 also taking into account the interface polarization charges.11,33 The results obtained with the present method are compared in Sec. III with the experimental data from Ref. 9 (for ee scattering) and from Ref. 12 (for e-ph and ee processes).

The e-ph and the ee elementary interactions are described, respectively, by the unscreened Frölich and Coulomb potentials Vph and Vee.29 At the lowest perturbative order, two electrons can interact exchanging a virtual LO-phonon or photon, as shown in Fig. 1(a). At higher perturbative orders, the two considered scattering processes become deeply connected, as shown, e.g., by the 2nd order diagrams in Fig. 1(b). At each perturbative order, several diagrams contain both ee and e-ph elementary interactions, represented in the figure by a wavy or dashed line, respectively, and in the transition amplitude by a factor Vee or Vph, according to the Feynman's rules.29,30 Their expressions Vee=Vee/ϵ=4πe2/(ϵ0ϵ|q|2) and Vph=M|q|2D0(ωm) are presented and discussed in Ref. 29: M|q|2=(1/2)KϵωLOVee is the square of the unscreened electron-phonon matrix element, Kϵ=ϵ1ϵs1,ϵ and ϵs are the dynamic and static dielectric constants, D0(ωm)=2ωLO/(ωm2ωLO2) is the unscreened phonon propagator,19,29ωLO is the polar LO-phonon frequency, q is the virtual phonon or photon wavevector, ωm its (bosonic) frequency, and ϵ0 and e are the vacuum dielectric permittivity and the elementary charge. Thus, considering diagrams with a single polarization bubble, there are four possible diagrams (Fig. 1(b)), and diagrams with n bubbles have 2n+1 possible arrangements. The effective ee plus e-ph interaction Veff in RPA can be found summing up to infinite perturbative order all possible n-bubble diagrams (Fig. 1(c)) according to Dyson's equation

Veff=(Vee+Vph)/[1(Vee+Vph)P],

where P=(1ϵ)/Vee is the polarization of the electron gas and ϵ is the dielectric function. An important and non-trivial step is the separation of the effective dressed (dynamically screened) RPA interaction Veff into its ee and e-ph contributions Vee,s and Vph,s

Vee,s=Veeϵ,Vph,s=Vphϵ2(1M|q|2D0P/ϵ),
(1)

where the RPA-SPP dielectric function ϵ1=1+Ωpl2/(ωm2ωq2) contains the N- and |q|-dependent effective plasmon and plasma frequencies ωq and Ωpl.29,34 Despite being formally distinct, a mutual interplay between the two screened Frölich and Coulomb interactions and related self-energies does exist at a very fundamental level: a detailed derivation of Vph,s (see Ref. 29, Sec. VI C) shows that the obtained screened e-ph interaction includes the ee effects properly, and a straightforward derivation of the screened e-ph interaction from Dyson's equation without the Coulomb interaction would not include the extra factor 1/ϵ in the M|q|2D0P/ϵ term contained in the expression of Vph, s in Eq. (1).

FIG. 1.

(a) First order Feynman's diagrams for electron scattering exchanging a phonon (dashed line) or a photon (wavy line). (b) The four 2nd order Feynman's diagrams with one polarization bubble. Summing as geometrical series n-bubble contributions up to infinite order, the effective dressed RPA interaction (double wavy line) is obtained (c). The RPA self-energy (d), represented by a free particle propagator emitting and reabsorbing a quantum of dressed RPA interactions, contains both Frölich and Coulomb terms.

FIG. 1.

(a) First order Feynman's diagrams for electron scattering exchanging a phonon (dashed line) or a photon (wavy line). (b) The four 2nd order Feynman's diagrams with one polarization bubble. Summing as geometrical series n-bubble contributions up to infinite order, the effective dressed RPA interaction (double wavy line) is obtained (c). The RPA self-energy (d), represented by a free particle propagator emitting and reabsorbing a quantum of dressed RPA interactions, contains both Frölich and Coulomb terms.

Close modal

A QW capture process via ee scattering consists of an electron belonging to a barrier state Ψk1 with energy E and wavevector k1 (the zero-point energy is the QW ground state) that interacts with a second electron with wavevector k2 ending in the nth QW state ϕn,k1q, with the exchange of a virtual photon with wavevector q and frequency ωm; the second electron can belong either to a barrier state Ψk2 or to the jth QW subband ϕj,k2. Considering only the first case, the ee scattering RPA self-energy Σk1,k2ee(E) reads

Σk1,k2ee(E)=1βq,k2,ωmΨk2ϕn,k1q|Vee,s|Ψk1Ψk2G0.
(2)

Here, β is the inverse temperature in energy units, G0=1/[iωm+iω|k1q|2/(2m*)EF/] is Matsubara's single-particle propagator19,28,29 where ω=E/, is the reduced Planck's constant, m* is the electron effective mass, and EF is the Fermi energy.

Σk1,k2ee was evaluated with the following procedure:19 (a) the k2 summation was performed by exploiting a Dirac-δ factor stemming from the calculation of Ψk2ϕn,k1q|Vee,s|Ψk1Ψk2, representing the momentum conservation at each vertex; (b) the summation over the frequency ωm was done following the Matsubara formalism (the Fermi nF and Bose nB occupation factors appear during the summation thanks to the bosonic character of ωm);29 (c) the q-summation was converted into an integral by exploiting the QW-plane translational invariance: considering for q and k1 their orthogonal and in-plane components (qz,q) and (k1,z,k1), the integration could be done analytically using residue theorems, after having extended the integration to a complex domain, first in qz, then in q, without any truncation unlike in some numerical approaches.25,26,35 In the end, we obtained

Σk1,k2ee(E)=παm*cϵ02πdθIee(θ)n,k1,k2×Ωpl2ωq(1+nBnF)Θ(Eres)2m*Eres2k12sin2(θ),
(3)

where indicates the imaginary part, α is the fine structure constant, c is the light velocity, k1,2=|k1,2|,q=|q|, θ is the angle between q and k1, Θ is the Heaviside step function, Eres=EEFΘ(EF)ωq, and the form factor Iee(θ) comes from the eigenfunctions overlap integral. The QW Fermi energy EF (NQW) was evaluated as a function of the equivalent two-dimensional (2D) QW carrier density, estimated as NQWLw, where Lw is the QW width. All implicitly q-dependent quantities in Eq. (3), like, Eres, ωq, or the Fermi distribution

nF=11+exp[β(2(k1q)22m*EF)],
(4)

for the QW states, were evaluated in the q-pole of the Green's function qp=k1cos(θ)+2m*Eres(qp)/2k12sin2(θ), expressing the momentum-energy conservation. Regarding nB, in the present work, it was always set to zero for simplicity. The corresponding quantum capture time is defined as 1/τee=(2/)Σee/(1+nBnF): since in rate equations it is customary to write the occupation factor (1 + nB − nF) explicitly, this position avoids to erroneously include it twice.

Regarding the capture process via phonon emission, it can be described as an electronic transition from an initial barrier state Ψk1 to a final QW state ϕn,k1q through the emission of a phonon of wavevector q and frequency ωm. In RPA, the self-energy related to this process is

Σk1eph(E)=1βq,ωmϕn,k1q|Vph,s|Ψk1G0,
(5)

and following a procedure still based on complex integration and described in Ref. 19, we obtained

Σk1eph,±(E)=2πKϵωLO2αm*c02πdθIeph(θ)n,k1×F±(1+nBnF)Θ(Eres±)2m*Eres±2k12sin2(θ),
(6)

where Eres±=EEFΘ(EF)ω±,Ieph(θ) is a form factor ensuing from the wavefunction overlap integral, F±=(ω±2+Ωpl2ωq2)2/[2ω±(ω±2ωq2)(ω±2ω2)], and the upper or lower signs in Σk1eph,± refer to the emission of a phonon-plasmon mode of frequency ω±, where ω±2=(ωq2+ωLO2)/2±(ωq2ωLO2)2+4KϵωLO2Ωpl2/2.

Since the two e-ph self-energies Σk1eph,± correspond to the emission of two possible and distinct LO-phonon-plasmons, two distinct quantum capture times 1/τeph±=(2/)Σeph,±/(1+nBnF) are possible. Therefore, the sum of RPA ee and e-ph self-energies represented in Fig. 1(d) provides the overall capture time τ=1/(1/τee+1/τeph++1/τeph) and depends on both N and NQW.

The general procedure of frequency and momentum integration in the complex-plane in order to obtain self-energies is well known in the literature. Nevertheless, it is not customary in the derivation of capture times, and numerical approaches are usual when the dynamical RPA dielectric function, adopted in this work, is considered. However, an obvious drawback is the loss of the possibility to obtain simple, compact expressions, like Eq. (3) that we obtained following a long but straightforward procedure. Instead, considering the phonon emission, the method was already outlined by one of the authors in a previous work36 and applied to III–V compounds. The formulation presented in Ref. 19 explained better several points, besides specializing the model to nitride-based QWs. In addition, it is worth emphasizing that the efforts in obtaining simple expressions like Eqs. (3) and (6) are rewarded by the possibility to achieve further simplified limiting forms (e.g., in low or high carrier density regimes, see an example in the discussion at the end of Section III), allowing to make comparisons and estimates in an easier way.

In order to validate the present model, we considered two sets of experimental data. The first (referred to as set A in the following) was obtained by Fan et al.12 employing time-resolved differential transmission spectroscopy to evaluate the dependence of the overall capture time τ from N, for a nominally undoped 2.5 nm/7.5 nm In0.08Ga0.92N/GaN well-barrier system. The second (set B in the following) was recently presented by David et al.:9 with a small-signal analysis of a 4 nm/30 nm In0.09Ga0.91 N/GaN single QW-barrier heterostructure, the authors obtained indications of a τ with two components: a fast one, around 1 ps and attributed to the e-ph process, and a second one, proportional to 1/N, much slower in the experimentally explored interval of N (1012 < N < 1014 cm−3), with a probable signature of ee scattering.

Adopting material parameter values reported in Refs. 19 and 31, we tested the present combined ee plus e-ph model against set A and set B data, separating the individual capture rate contributions and considering only the capture to the QW ground state.

A rate equation system like

(7)
dNdt=IeV(1+nBnF)Nτ+NQWτesc,
(7a)
dNQWdt=NQWτrecNQWτesc+(1+nBnF)Nτ,
(7b)
is able, in principle, to describe the time-evolution of the barrier and QW carrier density (the capture, escape, and recombination times are described, respectively, by τ, τesc, and τrec; I is the injection current and V the active region volume). Nevertheless, before proceeding, an important point to clarify is how to correctly manage the RPA approximation. In Ref. 37, chap. 14, an extensive discussion shows that the build up of the screening after a pulse excitation needs some time which is of the order of an inverse plasmon frequency in the system of excited carriers (a few femtoseconds).38 As a consequence, the static approximation of the RPA is not suitable, and the dynamic RPA formulation (adopted in the present work, see Sec. I) is considered a better one. However, the solution of Eq. (7) in the time domain would lead to wrong results even in the latter case37,38 since the RPA itself does not provide a rigorous description of the physics during a fast transient. A better approach is given, e.g., by the time-dependent Non-Equilibrium Green's Function (NEGF) formalism,37–39 which is well beyond the scope of the present contribution, focused on obtaining expressions suitable for fast modeling tools, although approximate. Therefore, we adopted a different method.

Regarding set A (the pump-probe experiment), we calculated τ(N) without making use of Eq. (7) but instead considering the definition of τ, Eqs. (3) and (6), setting NQW to a low value (1013 cm−3, realistic during the very initial part of the transient), and varying N. This choice assumes the screening in barriers as already built-up, and the obtained τ(N) may be considered representative of the capture time experienced by electrons during the initial part of the transient.

By contrast, experimental data of set B refer to a voltage-driven single-QW LED, operated in the steady-state at current density regimes spanning from low to high injection. Therefore, in the study of set B, we self-consistently calculated τ(N, NQW) and NQW itself as functions of N, ruled by Eq. (7), in which the time-derivatives were set to zero (steady-state). For this purpose, it can be noticed that only Eq. (7b) is needed, whereas Eq. (7a) may be eventually employed to connect carrier and current densities N and J.

Regarding the other lifetimes, in the present calculation, τrec was evaluated considering only radiative and Auger recombination processes as in Ref. 19 (remarks about the uncertainties and limitations of standard recombination models in nitrides can be found, e.g., in Refs. 6, 40, and 41). For τesc, we adopted a value of 3 × 10−10 s (see Ref. 9) able to reproduce the experimental data well. It has been observed that τesc not only strongly depends on Lw and T but is also affected by a possible background doping in the active region and by the bias current.42 In addition, barrier tunneling competes with the thermionic escape with lifetimes around 0.1–1 ns,42,43 besides other effects possibly at play, like tunneling assisted by defects.5 As a consequence, in the present formulation, τesc should be regarded as a fitting parameter, more than a pure thermionic escape time, since it may include several not easily separable effects.

For both sets of experimental data, a fitting parameter a ≈ 0.13 multiplies both overlap integrals Iee and Ieph. This empirical factor scales the numerical values of the self-energies and is justified by the approximations considered for the wavefunctions and by our focus on estimating the relative importance of the involved processes rather than their absolute values.

Fig. 2 shows the calculated capture times for set A and set B for the three distinct processes:44e-ph via modes ω and ω+ and ee scattering. Regarding set A (Fig. 2(a)), the differences between theoretical and experimental slopes may originate from the limited number of experimental points, covering a narrow range of N. The agreement between our theoretical τee curve and the experimental data is much better for set B (Fig. 2(b)), probably also because the data span more than two decades in N, allowing to reduce the effects of local fluctuations.

FIG. 2.

Electron capture times τeph± and τee, calculated as functions of N, for set A (panel (a)) and set B (panel (b)). Experimental points are shown as symbols. The τeph peak at high carrier densities has been discussed in Ref. 19.

FIG. 2.

Electron capture times τeph± and τee, calculated as functions of N, for set A (panel (a)) and set B (panel (b)). Experimental points are shown as symbols. The τeph peak at high carrier densities has been discussed in Ref. 19.

Close modal

In Fig. 2, two intervals of N can be identified: in the low density regime, τ is mainly determined by LO-phonon emission as a ω phonon-plasmon mode, with characteristic time τeph. When the carrier density is increased, τee and τeph+ progressively reduce, competing with τeph when N is above ≈1017 cm−3. In the interval of N corresponding to set A experimental points (Fig. 2(a)) and also typical of LED operation, the capture time is mainly given by contributions coming from τee and τeph+ at a similar extent, and therefore, investigations about the IQE droop in LEDs should exclude neither of these two mechanisms, and the customary approximation of the overall capture time τ with a constant value (see, e.g., Ref. 45) cannot be considered realistic. It may also be noticed that the dissimilarity between the two panels of Fig. 2 depends partly on the difference in the two active region thicknesses and partly on different ways we obtained the two results, as explained above.

Fig. 3 reports the carrier density NQW versus N, calculated by Eq. (7b) in the steady-state for the parameters of set B, employing the present model for τ(N, NQW) (solid line) and a fixed τ of ≈4 ps (dashed line). The difference between the two curves is not very large, but it suggests that the capture process is not inhibited by the potential screening, at least in stationary conditions: on the contrary, screening favors capture by increasing the energies ω± and ωq, which get closer to EEF (resonance condition, see denominators in Eqs. (3) and (6)).

FIG. 3.

Carrier density NQW versus N calculated by Eq. (7b) in the steady-state for the parameters of set B, employing the present model for τ(N, NQW) (solid line) and a fixed τ of ≈4 ps (dashed line).

FIG. 3.

Carrier density NQW versus N calculated by Eq. (7b) in the steady-state for the parameters of set B, employing the present model for τ(N, NQW) (solid line) and a fixed τ of ≈4 ps (dashed line).

Close modal

As a further consistency check, we calculated, employing set B parameters, the recombination time τrec and the injected current density J as functions of N, employing Eq. (7) in the steady-state, self-consistently evaluating τ and NQW. Corresponding set B experimental points have been extracted from Fig. 3(a) of Ref. 9 at room temperature, and the comparison is shown in Fig. 4, where τrec is plotted vs. J, obtaining an overall satisfactory agreement.

FIG. 4.

Recombination time versus injected current density J, as obtained from the model (set B parameters), compared with the experimental data extracted from Fig. 3(a) of Ref. 9 at room temperature.

FIG. 4.

Recombination time versus injected current density J, as obtained from the model (set B parameters), compared with the experimental data extracted from Fig. 3(a) of Ref. 9 at room temperature.

Close modal

A few additional remarks are in order. First, when N decreases below 1016 cm−3, we observed that ω ≈ ωLO and ω+ ≈ ωq. This means that ω is substantially a LO-phononic mode of frequency ωLO unless N increases enough to screen the Frölich interaction and to couple phonon to plasmon, whereas ω+ is substantially a plasmonic mode for all carrier densities. In addition, Ωpl becomes progressively much smaller than the other frequencies, allowing to obtain the limiting forms F ≈ 1/2ωLO and F+Ωpl2/(2KϵωqωLO2). Therefore, F becomes N-independent, justifying a similar behavior for τeph. Instead, F+ tends to zero for decreasing N, making τeph+(N) to loose importance when N reduces, with respect to the other two capture times. These two limiting forms of F± allow recovering known expressions of the capture time.26 

Second, the closeness of τee(N) and τeph+(N) over several decades of N (in addition, they both scale as 1/N) are worth of an explanation. Plasmons are collective excitations resulting from the quantization of carrier density oscillations, arising from a Hamiltonian for the long-range electron-electron correlations.30 Intuitively, an electron can lose part of its energy exchanging a plasmon ω+ only if the carrier density is not negligible, and the probability of this process is expected to increase with N. The same argument applies to the probability of the ee scattering; therefore, it is not unexpected that both processes exhibit the same trend with N. The argument is confirmed by the fact that, in the low-density regime, the limiting forms for self-energies become

Σeph,=απm*cKϵωLO02πdθIephΘ(Eres)2m*Eres2k12sin2(θ),
(8)
Σeph,+=απm*cΩpl2ωLO02πdθIephΘ(Eres+)2m*Eres+2k12sin2(θ),
(9)
Σee=απm*cΩpl2ϵωLO02πdθIeeΘ(Eres)2m*Eres2k12sin2(θ)
(10)

(in this regime, also the occupation factors (1 + nBnF) in the three integrands can be safely removed). It can be noticed that the prefactors of Eqs. (9) and (10) have become very similar and proportional to Ωpl2, and the frequency Ωpl, in turn, is proportional to αN.29,34 Within the approximations of the present work, all this may justify the similar behavior of τee and τeph+ with N, but the uncertainty about the overlap integrals Iee and Ieph is large enough to recommend not to speculate too much in depth about their numerical closeness. Literature simulation results46 indicate that the scenario may be more complicated: the conduction band edges of the left and right barriers of a blue-emitting InGaN QW may be separated by several hundred meVs by the effect of polarization charges, raising the capture probability by an increase in electron dwell time above the QW. Not only the polarization charges but also the incident electron kinetic energy Ek is an important parameter. Ref. 46 shows that the electron dwell time may vary by orders of magnitude with the value of polarization charges and Ek, thus a more complete transport theory (probably NEGF-based) is needed to include and describe the effects of these two parameters, although maybe preventing to obtain simple expressions for τ as in the present work.

As a final note, in the same low-density regime, from the expression of Ωpl, it follows that both Σee and Σeph,+ become proportional to α2N: since α appears raised to the power of 2, the present formulation is equivalent to the second-order Born approximation24 when N is low enough, but it provides a better description in the case of arbitrary N.

We have derived and validated a compact set of expressions for capture time taking into account ee and e-ph scattering. As suggested by Fig. 2, the authors of Ref. 9 are probably correct when attributing ee scattering to the relatively slow capture time governed by carrier population reported in the set B data. The inclusion of a realistic description of ee scattering in carrier capture models suitable for device-level simulation (see, e.g., the modeling framework described in Ref. 47) could be, therefore, crucial for LED IQE droop investigations, where high current injection regimes are considered.

In the low carrier density regime, we confirm that considering e-ph and ee processes, the emission of phonons of frequency ω (very close to ωLO) is the most effective capture mechanism. Another important remark is that the slowest elementary processes over the carrier density range considered in set B are ee scattering and the emission of phonon-plasmon ω+ modes: both contribute to a comparable extent, and they remain distinct quantum processes (but not independent, as discussed in Sec. II) with distinct probabilities.

This work was supported in part by the U.S. Army Research Laboratory through the Collaborative Research Alliance (CRA) for MultiScale multidisciplinary Modeling of Electronic materials (MSME).

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