Hot electron energy relaxation in lattice-matched InAlN/AlN/GaN heterostructures: The sum rules for electron-phonon interactions and hot-phonon effect

Nitride compound semiconductors such as GaN and AlN which have a wide energy gap withstand high breakdown electric fields and support excellent thermal stability.¹ The heterostructure field-effect transistor (HFET) using a GaN-based heterostructure is very promising for high-power radio frequency and high-mobility operations, owing to its high electron density, high electron drift velocity as well as its high drain breakdown voltage. Particularly interesting are lattice-matched InAlN/AlN/GaN heterostructures,^2–5 in which a two-dimensional (2D) electron gas forms in the undoped GaN layer near the interface of the GaN layer and the one nanometre thick AlN spacer, due to internal spontaneous polarization alone, i.e., with no strain-induced polarization by the piezoelectric effect. As such an electron gas arises in the absence of doping and lattice strain, high electron density (2 × 10¹³/cm²) as well as high drift velocity (3 × 10⁷cm/s) can occur in the lattice-matched heterostructures. The electrons in the quasi-two-dimensional (quasi-2D) GaN channel are heated up due to high electric power, with the electron temperature T_e elevated above the lattice temperature T₀ (i.e., room temperature). Electron temperatures up to 2500 K have been measured using a microwave noise technique for a lattice-matched InAlN/AlN/GaN heterostructure with an areal electron density of 1.2 × 10¹³/cm² in the GaN channel.² Heat dissipation in the GaN channel is a complicated process which includes energy relaxation (ER) of the hot electrons mainly by emission of polar-optical phonons, decay of the polar-optical phonons into acoustic phonons via anharmonic interactions and diffusion of the excess acoustic phonons into the remote heat sink. The optical phonon lifetime has been measured for GaN-based heterostructures, which, in general, falls in the range from 0.1 to 1.7 ps^2,5,6—except for the case of the electron density 8 × 10¹¹/cm², and depends on both the electron density⁶ and electron temperature.^2,6 As the optical phonon lifetime is much longer than the Fröhlich scattering time (∼10 fs for GaN⁷), the emission of polar optical phonons is very fast compared to their decay into acoustic phonons, and a large population of non-equilibrium (hot) phonons are accumulated leading to a slowdown in ER [termed hot-phonon effect (HPE)]. Hot phonons also result in an increase of electrical resistance and impose limitations on electron drift velocities.^5,8

The average power dissipation (PD) and ER time τ_E are two key parameters for describing electron ER in semiconductors under an external electric field. Apparently, both quantities depend on the electron temperature, and knowing how they depend on T_e is fundamental to the optimization of the HFET devices.¹ In experimental studies of hot-electron ER in GaN-based heterostructures,^2,9 the dependence of the electron temperature T_e on the supplied power, which is equal to the power dissipated to the lattice by the hot electrons under steady-state conditions, was directly measured using the microwave noise technique, and then the electron ER time as a function of the electron temperature was deduced by $τE(Te)=kBdTe/dPs$⁠, where P_s is the average power supplied to each electron. For the lattice-matched heterostructures, the relaxation time was found to fall sharply at the low electron temperatures and decrease very slowly at the high electron temperatures (>1200 K).² 

Microscopically, ER of the hot electrons is governed by scattering with polar-optical phonons (scattering with acoustic phonons is weak and neglected¹⁰). In bulk GaN, the polar modes are treated as the longitudinal optical (LO) modes of a single frequency ω_LO. In a simplification, therefore, the polar-optical phonons of the heterostructures are usually taken to be the LO phonons of the bulk material such as GaN. This is referred to as the three-dimensional phonon (3DP) model. According to the dielectric continuum (DC) model, however, the eigenmodes of the polar-optical vibrations in the heterostructure include half-space (HS) modes and interface (IF) modes, and all these modes interact with the electrons in the quasi-2D channel. Both phonon models have been used to study ER and momentum relaxation for GaAs-based quantum wells.^11,12 For GaN-based quasi-2D systems, the 3DP model was used for an early study on the momentum relaxation and low-field electron transport in GaN quantum wells.¹³ In principle, the phonon eigenmodes of quasi-2D GaN heterostructures should be considered for electron transport studies. Indeed, the DC model has been employed in recent years to study electron-phonon (e-p) scattering associated with the various phonon modes,¹⁴ as well as the electron momentum relaxation¹⁵ and ER¹⁶ in GaN heterostructures. Mori and Ando derived a sum rule¹⁷ from the DC model which showed that the sum of the form factors associated with HS and IF modes was equal to the form factor for bulk phonons. Later, using a microscopic model Register derived a similar sum rule¹⁸ to prove that the total e-p scattering rate in the heterostructure was independent of the phonon basis sets, as long as each set was orthonormal and complete. These sum rules seem to imply that the DC and 3DP phonon models would yield equivalent results for the ER in GaN heterostructures. However, this claim is only partly true, and caution should be taken in applying the sum rules to the ER in GaN heterostructures. First, in Register's scattering rate sum rule, polar scattering with electrons is assumed to be made by phonons of a uniform frequency ω_LO¹⁸ (i.e., all the phonon modes are degenerate). There is no scattering rate sum rule as the normal modes of the heterostructure including the IF and HS modes are not degenerate. In fact, the IF phonons differ significantly from the HS phonons in both phonon frequency and e-p coupling strength. Second, as the scalar potential of the IF modes decreases exponentially from the interface according to $e−q|z|$⁠, scattering with IF modes is weak in wide wells making the 3DP evaluation accurate. Indeed, for quantum wells with widths greater than 100 Å, the 3DP model suffices for the evaluation of scattering rates,^12,19,20 energy loss rates,²¹ and momentum relaxation rates.¹⁵ For narrow wells, however, scattering with IF phonons becomes increasingly important, and as such, the two phonon models yield quite different rates for GaAs quantum wells^20,21 as well as GaN heterostructures.¹⁵ For GaN heterostructures with a 30 Å-wide channel, the 3DP model underestimates momentum relaxation rates just below the bulk LO phonon energy by 70%, and overestimates rates immediately above the LO phonon energy by 40% compared to the DC model.¹⁵ However, as far as we know, there has been no comparison of ER rates for GaN heterostructures based on the two phonon models. Third, in GaN heterostructures, screening from the mobile quasi-2D electrons is strong due to the high electron density. The scattering rate sum rule becomes invalid causing quite different DC and 3DP rates when screening is accounted for. Fourth, the scattering rate sum rule is valid with an equilibrium phonon distribution being taken as an important prerequisite.²² In the GaN HFET where the hot-phonon effect must be taken into account, phonon modes are clearly in non-equilibrium, with the consequence that different modes make different contributions to the ER process. In this circumstance, it is necessary to use the correct normal modes of the GaN heterostructure to calculate the ER. Previous calculations for GaAs quantum wells have shown that the ER rates in the hot-phonon regime depend on the phonon models used.^21,23

Recently, using the DC model, the authors calculated hot-electron ER in a typical lattice-matched InAlN/AlN/GaN heterostructure.¹⁶ We found that the experimentally observed dramatic fall at low T_e² was caused chiefly by the fast decreased HPE and electron screening, while the very slow decrease at the high temperatures was due to the fast optical-phonon decay. In this paper, we study ER of the hot electrons in the heterostructure with both DC and 3DP models. As the 3DP approach is relatively simple and thus convenient for practical calculation, one of course wants to know how ER results estimated by this model differ from those calculated with the DC model, and further what causes the discrepancies. We pay particular attention to a quantitative comparison of the T_e-dependencies of the PD and ER time at high temperatures calculated with the two models, as the high T_e relaxation process is of great interest in terms of HFET devices. This comparison can be made with regard to only the total power loss and relaxation time. On the other hand, there is an advantage of the use of the DC model, in that the contributions from the quasi-2D phonon modes to the ER can be singled out. It is therefore of great interest to find and understand the behaviours of the HS and IF modes, in particular, those of the IF modes, in the ER process in such narrow channel GaN heterostructures. The comparison with the 3DP calculations also provides a simple means to examine how the IF phonons contribute. Equally important from the electron gas side is screening. Clearly, dynamic effects of screening from the electrons need to be included owing to the high frequencies of the polar modes. For these purposes, a comprehensive study needs to be carried out for the ER in the GaN heterostructures, in which an emphasis is put on how ER results from the two phonon models differ when both hot phonons and screening are taken into account. Therefore, using the two phonon models, the power loss and ER time are calculated as functions of the electron temperature, for a number of phonon scattering processes with electron screening included or excluded. The ER results from scattering with the HS and IF phonons are compared and examined, and then the total power loss and relaxation time are further compared with those obtained from the 3DP approximation. The sum rules for the e-p interactions are closely checked for the spontaneous phonon emission process. For the stimulated phonon emission process where the hot-phonon re-absorption is accounted for, special attention is paid to the difference in the DC and 3DP calculations to examine the 3DP model in the evaluation of ER in particular at high electron temperatures. We found that the two models yield very close power loss values and ER times, the discrepancies being caused by how much the high energy IF phonons contribute to the ER. With no hot phonons or screening, the power loss calculated from the 3DP model is 5% smaller than the DC PD, whereas slightly larger 3DP power loss (by less than 4% with a phonon lifetime from 0.1 to 1 ps) is obtained throughout the electron temperature range after including both the HPE and screening.

This paper is organized as follows. In Sec. II, following a brief description of the DC and 3DP models for lattice-matched InAlN/AlN/GaN heterostructures, a formulation of the PD and ER time is presented, taking into account non-equilibrium polar optical phonons, electron degeneracy, and screening. In Sec. III, first we show results of the non-equilibrium phonon occupation numbers for both HS and IF modes in a typical lattice-matched InAlN/AlN/GaN heterostructure. These results are used to analyze how hot phonons slow down the quasi-2D electron ER in the high-temperature region. Then, by choosing two GaN heterostructures with different channel widths, we compare PD results from the DC and 3DP models for the simple case with no screening. This is to check the sum rules as well as investigate phonon confinement effects and roles the HS and IF modes play in the respective spontaneous and stimulated phonon emission processes. In order to examine the usual 3DP approximation in the evaluation of ER, we further compare the DC and 3DP results of power loss and ER time in the lattice-matched heterostructure for a number of detailed phonon scattering processes with or without electron screening. Comparisons are also made with the experimental data as well as the bulk GaN situation, and the hot-phonon and screening effects are discussed in great detail. Finally, Sec. IV summarizes the main results obtained.

As the Indium content in the outer barrier is small (x < 0.2), the lattice matched heterostructure In_xAl_1−xN/AlN/GaN is treated as a single heterostructure AlN/GaN,^14,16 with the barrier in the space –L₁ < z < 0 and the electron-containing GaN region in the space 0 < z < L₂ (L₂ = N₂a, a being the lattice constant). In the DC model, the polar vibration modes of the heterostructure consist of half-space modes and IF modes.¹⁷ Neither the HS nor IF phonons have dispersion. The half-space modes have the frequencies of the bulk polar modes of the two constituent materials, whose scalar potentials and electric fields occur in the respective constituent regions. The HS modes are indexed by (q_z, q), where q is the in-plane phonon wavevector and q_z is determined by the fixed end boundary condition imposed on the potential function (∝ sin q_zz), for the GaN HS modes (frequency ω_LO), for instance, $qz=lπ/L2$ (l = 1,2,…,N₂ − 1). The IF modes, which are indexed by (n, q) (n = 1,2), have different frequencies from the bulk polar modes, and an IF mode has lattice vibrations and electric fields in both constituent regions. In the 3DP model, the phonon modes are simply bulk LO modes, which are indexed by the three-dimensional phonon wavevector $Q=(q,Qz)$ and normalized to the sample volume of the entire heterostructure. The electron-LO-phonon interaction is given by the Fröhlich interaction. In the growth direction, the wavevector Q_z differs from the q_z of the HS modes. The two sets of discrete wavevector values result from the two different types of boundary conditions on the respective mode potential functions.¹² 

In the GaN-based heterostructures, the narrow and shallow confinement allows us to consider only the lowest subband¹⁴ which is densely populated by the electrons (electron energy $Ek=ℏ2k2/(2m*)$⁠, k being the in-plane electron wavevector). The degenerate statistics of a high density of electrons is described by the Fermi-Dirac distribution function, $f(E)=1/(1+e(E−EF)/kBTe)$⁠, where E_F is the Fermi energy of the quasi-2D electron gas, which is determined by the areal electron density n_A and temperature T_e. The thermal equilibrium population of the phonons of frequency ω at temperature T is given by the Bose-Einstein distribution function, $N(ω,T)=1/(eℏω/kBT−1)$⁠. The hot electron energy relaxation is calculated by Fermi's golden rule. For an IF mode (n, q) of frequency ω_n, for instance, the number of phonons which are generated by the hot electrons per unit time can be expressed as $Wn(q)=[N(ωn,Te)−gn(q)]/τn(q)$ (see Refs. 7 and 24), where $gn(q)$ is the non-equilibrium IF phonon occupation number and $1/τn(q)$ is the IF phonon generation rate⁸ 

$Mn,q(k;k+q)$ is the interaction matrix element associated with electron states k and $k+q$ and phonon mode (n, q) due to the IF phonon scattering. The phonon population $gn(q)$ can be calculated at steady state of energy relaxation,^2,6,9$gn(q)=[1τn(q)N(ωn,Te)+1τpN(ωn,T0)]/(1τn(q)+1τp)$⁠, which depends on the relative magnitude of the phonon generation and decay rates $1/τn(q)$ and $1/τp$ (τ_p is the phonon lifetime, its definition being given by Eq. (5) of Ref. 25 for the standard three-phonon decay process such as the Ridley channel). Similarly, the generation rate $1/τqz(q)$ and hot-phonon population $gqz(q)$ for half-space phonons can be obtained.

Therefore, in the DC model, the average power dissipated per electron is given by

where A is the sample area. In the 3DP model, the average PD is given by

where the generation rate $1/τ(Q)$ has the same expression as $1/τn(q)$ above [Eq. (1)] except that LO phonon frequency ω_LO and the Fröhlich interaction matrix element should be used instead, and $g(Q)$ is the hot bulk LO phonon occupation number. Knowing P_d, then a hot-electron ER time τ_E can be defined in the hydrodynamic model²⁶ through $Pd(Te)=kB(Te−T0)/τE$⁠.

We now include screening. The e-p interactions are screened by the mobile electrons. The response of the electron plasma to a polar disturbance from the lattice is encapsulated by the dielectric function of the electron gas. For a high temperature electron gas as considered here, the Boltzmann distribution function is used to approach the energy distribution of the hot electrons. Then, the Lindhard dielectric function of the quasi-2D electron gas reduces to the following form:

where Z(s) is the plasma dispersion function²⁷ 

with s being complex and κ_D is the 2D Debye screening wavenumber, $κD=2πnAe2/(ϵ0kBTe)$⁠, with ϵ₀ being the static dielectric constant. In the $ϵ(q,ω)$ expression, the real arguments of Z are determined by the two dimensionless quantities y and a

In Eq. (4), F(q) is a form factor which accounts for the confinement effect on the electron-electron Coulomb interaction due to the finite heterostructure channel width¹² 

where $ϕ(z)$ is electron confinement envelope function.

To simplify the calculations, the plasma dispersion function [integral expression (5)] is approached by a two-pole padé approximant²⁸ 

Using the properties of the plasma dispersion function,²⁷ it is found that in the static limit $ω→0$⁠, the dielectric function [Eq. (4)] reduces to $ϵ(q,0)=1+F(q)κD/q$⁠. This is the familiar Debye screening formula which is used to evaluate screening for high-temperature non-degenerate electron gases.¹² In this study, screening from the mobile electrons is handled by dividing the scattering potential, or equally the e-p interaction matrix elements by the dielectric function of the quasi-2D electron gas. The polar disturbance is of course not static and occurs at the finite frequency of a particular phonon mode, for instance, an HS or IF mode. To account for the dynamic effect of screening the frequency ω in the dielectric function is substituted for the corresponding phonon frequency, and the dielectric function is treated as a function of wavevector q for each finite phonon frequency.

In-plane isotropy is used to simplify the ER calculations. Then, the rate expression [Eq. (1)] is reduced to a form (Ref. 24) that is proportional to the difference of two complete Fermi-Dirac integrals of order −1/2. Accurate evaluation of the integral $F−12(x)$ is important in obtaining the correct ER results. The calculation should also be efficient as the integration values are input to calculating the generation rates for all phonon modes in a large wavevector space. To calculate the Fermi-Dirac integral the integrand is transformed to $e−xeμx/(1+eμ−x)2$ such that the Gauss-Laguerre quadrature technique is used to achieve fast and excellent convergence (25 quadrature points are used). Then, these generation rates are inserted into Eqs. (2) and (3) to calculate P_d. Numerical integration over q is then carried by using the Gauss-Legendre quadrature method, 105 quadrature points being used with the cut-off of q taken to be 8k₀ (k₀ is the threshold electron wavevector for LO phonon emission, $k0=2m*ωLO/ℏ$⁠).

We model the electron envelope function $ϕ(z)$ for the triangular potential well by the Fang-Howard wave-function,^29,30 $ϕ(z)=b3/2ze−bz/2$⁠, where b is a variational parameter related to the areal electron density n_A via $b=[ 33πe2m*nA/(2ϵ0ℏ2) ]1/3$⁠. An effective channel width^31,32 d is defined as twice the average penetration depth of the charge in the active GaN region, d = 6/b.

In this study, the material parameters are taken from Refs. 33–35. The LO and TO phonon frequencies used for GaN are ω_LO = 91.13 meV, ω_TO = 66.08 meV, and, for AlN, we use ω_LO = 110.7 meV, ω_TO = 76.1 meV. The high-frequency dielectric constants are taken to be 5.29 and 4.68 for bulk GaN and AlN, respectively. The electron effective mass for GaN is m^*= 0.22m₀ (m₀ is the free electron mass), and the lattice temperature is fixed at room temperature 300 K. The optical phonon lifetime is a key parameter in the electron ER study. Thus, a range of optical phonon lifetime values from 0.1 to 2 ps are taken to examine the hot-phonon effect.

The optical phonons contributing to hot-electron ER are the GaN HS modes $(ℏωLO=91.13 meV)$⁠, and the lower- and higher-energy IF modes (⁠$ℏω1=69.70$ and $ℏω2=102.09 meV$⁠, respectively). We first look at the hot-phonon occupancy in phonon wavevector space. We choose an electron temperature of 1000 K for the electron gas of the areal density of 1.2 × 10¹³/cm² in a typical lattice-matched InAlN/AlN/GaN heterostructure. The high-T_e experimental value of optical-phonon lifetime 0.1 ps is used for all HS and IF modes.² We calculated the hot-phonon occupation numbers as functions of the in-plane phonon wavevector q for the HS modes, q_z = lπ/L₂, with mode indices 1 ≤ l ≤ 100, as most of these phonons participate in significant e-p scattering. We found that for a given wavevector q, the HS phonon occupation number increases with the mode index l and then decreases after it reaches the maximum value of a certain l_a mode. This is illustrated in a contour plot of the HS phonon occupation numbers in the mode plane q-q_z (Fig. 1). This result can be explained as follows. The hot electrons are confined in a very narrow channel (effective width 44 Å), whereas the HS phonons interacting with the electrons are present in the entire GaN region (0 < z < L₂). Thus, the e-p overlap integral strongly depends on q_z or equally the mode index l. Therefore, the q_z-dependence of the squared interaction matrix element dictates the phonon generation rate $1/τqz(q)$ and the variation of the HS phonon population with q_z. This is different from what happens in the usual square quantum wells such as GaAs/AlGaAs quantum wells, where the hot-phonon population decreases as the confined-mode index in the growth direction z increases.²¹ This is because both the electrons and phonons are confined in the same well region,¹² resulting in stronger e-p interaction with a larger overlap integral for the lower-order phonon mode than the higher-order mode. We also see from Fig. 1 that the densest phonon population is confined in a region with the modes given by q ≤ k₀ and l ≤ 23, and these phonons constitute the hottest HS phonons. The HS modes on both the low- and high-order sides have narrowed population in a small q-region, whilst the modes in the middle with approximately 10 < l < 23 have their population distributed over a broad q-region, and thus, these HS phonons make a large contribution to the electron energy dissipation. The IF phonon occupation numbers are shown in Fig. 2(a) for both the lower-frequency (ω₁) and higher-frequency (ω₂) IF modes. The lower-frequency phonons IF₁ have a smaller population, whilst the higher-frequency modes IF₂ have a higher peak and are densely populated in a broader q-region where the IF₂ phonon generation rate $1/τ2(q)≫1/τp$ because they have much greater e-p coupling strength than the IF₁ modes (the Fröhlichlike coupling constants for the IF modes¹⁷ are $αIF1=0.02$ and $αIF2=0.5$⁠). For both HS and IF phonons, the occupation number curves have a steep edge on the small-wavevector side and a slow slope on the large-q side. We found that this originates from the wavevector q-dependence of the minimum electron kinetic energy for phonon absorption [refer to Fig. 9(a) and Appendix B of Ref. 24]. Further, we found that the q-dependent non-equilibrium phonon occupation number is governed by the Fermi-Dirac integral $F−12(ξq)$ versus phonon wavevector q (refer to Appendix B of Ref. 24).

The influence of screening from electrons on hot-phonon population is shown in Fig. 2(b) for the IF modes. Compared to the non-screening calculation [Fig. 2(a)], static Debye screening has significantly reduced the phonon population, and, in particular, the population of the IF₁ phonons are reduced substantially as these low-frequency phonons are restricted to only a small-q region (q ≪ k₀) of wavevector space where screening from the electrons is very strong with large F(q)/q and hence large values of dielectric function ϵ(q,0). Recall that F(q) is the form factor associated with the Coulomb interaction [Eq. (7)], which increases as the wavevector q decreases. When dynamic screening is used [Fig. 2(b), dashed and dotted lines], we see that the IF phonon population becomes narrower in wavevector q-space, with the occupation numbers at small q wavevectors being increased rather than decreased, compared to the case of excluding screening in Fig. 2(a). This anti-screening arises due to the dispersion of the quasi-2D electron plasma oscillation frequency, namely, $ωpl(q)=[2πe2nAqF(q)/(ϵ0m*)]1/2$⁠, which is smaller than the phonon frequency at long wavelengths.¹² In this circumstance, the electron plasma cannot move sufficiently fast to cause screening to the polar disturbance from the lattice.¹⁰ These hot phonon population results will be used to analyze the PD calculation below.

The 3DP model has been widely used to evaluate electron energy and momentum relaxation rates for quasi-2D semiconductor systems.^12,36–38 Here, we compare hot electron PD in GaN-based heterostructures calculated with the DC and 3DP models. First, we examine the sum rules as applied for ER in GaN heterostructures. We include only the hot phonon effect and do not consider screening. To do this, we consider two heterostructures with different effective channel widths, namely, a strained Al_0.05Ga_0.95N/GaN heterostructure³⁹ with a wide well of 110 Å (corresponding to areal electron density n_A = 7 × 10¹¹/cm²) and a lattice matched In_0.18Al_0.82N/AlN/GaN heterostructure² with a narrow well of 44 Å (corresponding to n_A = 1.2 × 10¹³/cm²). Then, the average power dissipated per electron is calculated as functions of the electron temperature with hot phonons being excluded or included, for the latter case, two phonon lifetime values being used, τ_p = 0.5 and 2 ps, to investigate the HPE. Figs. 3(a) and 3(b) show the results with the electron temperatures ranging from room temperature up to 2500 K. For the strained heterostructure with a wide channel, as shown in Fig. 3(a), the two phonon models yield literally the same PD. In this case, the IF phonon scattering with potential decreasing exponentially according to $e−q|z|$ is very weak, as the average value of the position for electrons $z¯$ which is half the effective channel width¹⁷ is $z¯=55 Å$⁠, whereas the characteristic wavevector k₀ for the IF phonons is around 0.07 1/Å, making $k0z¯≈4$⁠. The form factor for the IF phonons $FI(q)$ can be neglected, and then, the PD values given by the two phonon models are equal, which is consistent with both sum rules.^17,18 For the heterostructure with a narrow 44 Å channel, in contrast, the IF phonon scattering is significantly enhanced, and the two phonon models yield different power loss values (see proof in Appendix A of Ref. 24). Therefore, a clear difference is seen between the PD curves calculated with the two phonon models [Fig. 3(b)]. In the spontaneous emission case, the DC model yields higher PD than the 3DP approximation. Taking the HPE into account, however, the 3DP power loss becomes larger. To find the cause, we need to separate and check contributions to the PD from the HS modes, the lower and higher frequency IF modes, respectively [Fig. 4(a)]. We see that the HS phonons dominate the ER process due to the large density of states of the HS modes and contribute larger PD than the IF phonons, whilst the lower-frequency IF₁ phonons make only a small contribution due to their low energy, small coupling strength, and narrow population distribution in q-space (refer to Fig. 2 and the preceding subsection III A). Without HPE, the PD due to the high-energy IF₂ phonons increases rapidly with T_e in particular above 1000 K compared to the HS phonons. For instance, the IF₂ PD at T_e = 2500 K has risen to 46% the PD due to the HS phonons. This causes a larger total PD with the DC model than the 3DP model [Fig. 3(b)]. When the hot phonons are taken into account, however, the PD due to the IF₂ phonons drops dramatically by about 85% [Fig. 4(a)], because the IF₂ phonon generation number per unit time W₂(q) is substantially reduced [compare the two curves in Fig. 4(b)], as the non-equilibrium IF₂ phonons with a broad population distribution in wavevector space (as illustrated in Fig. 2) are re-absorbed. As a result, the total DC PD becomes smaller than that evaluated with the 3DP model [Fig. 3(b)]. Therefore, the difference between the DC and 3DP results is due to the IF₂ phonons—their contribution to the power loss is enhanced at the high electron temperatures in spontaneous emission but is dramatically reduced after including the HPE. In recent studies on GaN heterostructures, we found that IF phonon absorption causes negative momentum relaxation rates,¹⁵ and also an increased interaction with the IF₂ modes leads to a reduction of phonon lifetime.¹⁴ 

We now include screening to make a comprehensive study of the electron PD calculated with the DC and 3DP models. When both screening and hot phonons are included, strictly speaking, the scattering rate sum rule is not applicable, and then one needs to find how much discrepancy the 3DP evaluation yields with respect to the DC calculation. Thus, calculations were performed using each phonon model for a number of cases, namely, (i) excluding hot phonons and screening, (ii) including only the HPE, (iii) including only static screening, (iv) including only dynamic screening, (v) including both HPE and static screening, and (vi) including both HPE and dynamic screening. The results are shown in Fig. 5 for the lattice-matched In_0.18Al_0.82N/AlN/GaN heterostructure (with a 44 -Å-wide channel), where a polar optical phonon lifetime of 1 ps is used for all the cases of including the HPE. Several points can be made by comparing the PD values in the various cases. First, the PD is substantially reduced by static screening [compare cases (i) and (iii)], whereas the reduction using dynamic screening is much smaller, which is only ∼30% the reduction caused by Debye screening [compare cases (i), (iii), and (iv)]. When both screening and hot phonons are included, the power loss values obtained with the static and dynamic screening models get closer as T_e increases; at T_e = 2500 K, the power loss is 14% smaller from Debye screening than from the dynamic screening model. Second, at the low temperatures, interestingly, both phonon models yield enhanced rather than slowed PD when dynamic screening is included. That is, anti-screening occurs when the electron temperature T_e is lower than 840 K for the DC model or T_e < 770 K for the 3DP model. This is explained as follows. Expressions (2) and (3) show that mathematically the PD is a sum of the contributions that are connected with the various in-plane phonon wavevectors q. When dynamic screening is taken into account, the bare e-p interaction is screened or anti-screened depending on wavevector.¹⁰ The large-q components in the summation contribute screening, while the small-q components which are connected to the slow motion of the electron gas contribute anti-screening,¹² due to the dispersion of the quasi-2D electron plasma frequency. At the low electron temperatures, the small-q components dominate as the degenerate distribution of the dense electrons favours the e-p scattering with small transfer wavevectors q. At the high temperatures, on the other hand, the electrons are distributed over a large k-space, and they cause screening when the large q-components dominate. We note that anti-screening for the quasi-2D electron ER was observed early in GaAs quantum wells and occurred also at low electron temperatures.⁴⁰ Third, for the three cases with no hot phonons, namely, (i), (iii), (iv), the DC model yields higher PD than the 3DP model. However, taking into account hot phonons, namely in the corresponding cases (ii), (v), and (vi), the 3DP power loss becomes larger. What causes this has become clear after our discussion above for Fig. 3(b); that is, it is due to the higher-energy IF phonons. Fourth, throughout the temperature range, the 3DP power loss is 5% smaller than the DC PD in the simplest case (i), but becomes larger (by less than 4%) after including both HPE and screening as in cases (v) and (vi). A similar deviation is obtained when reducing the phonon lifetime to 0.1 ps except for the static screening case, where only a tiny 0.2% deviation occurs. The 3DP model yields such a close result to the DC calculation, because including screening does not alter the order of the DC and 3DP power loss values in terms of their relative magnitude [that is, the DC PD is higher. Refer to cases (i), (iii), and (iv)], while accounting for the HPE does alter the order [compare cases (i), (ii)].

Experimentally, using the microwave noise technique, the electron temperature T_e as a function of the supplied power P_s was directly measured for Si-doped bulk GaN,⁴¹ strained AlGaN/GaN,⁹ and lattice-matched In_0.18Al_0.82N/AlN/GaN² heterostructures. The total number of electrons was estimated from the measured low-field Hall mobility and channel resistance using Ohm's law. Under steady-state conditions, the supplied power is equal to the total power dissipated to the lattice by the hot electrons. Then one can obtain the experimental data of the average PD per electron versus electron temperature (see Fig. 4 of Ref. 2 for the lattice-matched In_0.18Al_0.82N/AlN/GaN heterostructure). The power loss was shown to increase with the electron temperature (from 2 nW/electron at T_e = 500 K, for instance, to 150 nW/electron at T_e = 2500 K), but the dependence is complicated by electron screening and the variation of the polar optical phonon lifetime with T_e, as was discussed in our previous study.¹⁶ In the simple approximation where neither hot phonons nor screening is included, the calculated PD is four times as large as the experimental values. Accounting for screening and HPE brings the calculation much closer to the experimental data. However, the theoretical values remain over 2.5 times higher in the low temperature region even with static screening which is generally believed to overestimate the screening effect, and the use of large phonon lifetimes there such as 20 ps (i.e., a much stronger HPE) will produce a fit with experiment. In addition, coupling of the polar optical phonons with the electron plasma is neglected and the coupled mode effect²⁶ may also play a significant role at low temperatures.

To quantify the screening effect and/or HPE, a reduction factor β is introduced, $β=Pd0/Pd$⁠, where $Pd0$ is the PD without screening or hot phonons, and P_d is the corresponding power loss when screening and/or the hot phonons are included. Fig. 6 shows the temperature-dependences of the reduction factors associated with only the HPE, only screening (Debye screening or dynamic screening), and both HPE and screening calculated with the DC and 3DP phonon models as labelled (using a polar optical phonon lifetime of 1 ps) for the lattice-matched In_0.18Al_0.82N/AlN/GaN heterostructure. When only screening is included, almost the same reduction factors are obtained from the two phonon models, with the two curves coinciding for either static or dynamic screening case. We see anti-screening again in the dynamic screening alone case for electron temperatures lower than about 800 K (below the dotted horizontal line β = 1 in Fig. 6) with reduction factor $β<$ 1, as anti-screening causes faster PD P_d than $Pd0, Pd>Pd0$⁠. In all the other cases, as $β>1$⁠, the electron ER has slowed down after including screening and/or hot phonons. We see a stronger HPE at low electron temperatures, with the reduction factors decreasing with increasing T_e. The 3DP model underestimates the HPE as expected, and as a result, the reduction factors from the 3DP calculation are smaller than the DC result even when screening is included. The reduction factor is ∼9% smaller by the 3DP approach than by the DC model in the high-T_e region. Using either of the two screening models, the reduction factor associated with both hot phonons and screening decreases as T_e increases, in both the DC and 3DP calculations, but the high-T_e reduction factor tends to be flat and the β values from the static and dynamic screening models are quite close, with β varying only from 2.5 to 3.2. That is to say, with a phonon lifetime of 1 ps, the high-temperature electron power loss is reduced approximately by a factor of 3 due to screening and the HPE.

We now turn to the ER time τ_E. Fig. 7 shows the dependences of the electron ER times on the electron temperature for the lattice-matched heterostructure, calculated with the DC and 3DP models for three cases, namely, (i) excluding hot phonons and screening, (ii) including both HPE and static screening, and (iii) including both HPE and dynamic screening. With no hot phonons or screening, both phonon models yield relaxation times around 0.05 ps but at low electron temperatures, T_e < 500 K, there is a drop in relaxation time $τE0$ (⁠$τE0$ is the ER time with no screening or HPE, $τE0=kB(Te−T0)Pd0$⁠), which is caused by the exponential rise of the generation number $ΔN(Te)=N(ωLO,Te)−N(ωLO,T0)$ with T_e. When screening and hot phonons are taken into account, the ER time can be conveniently expressed as $τE=kB(Te−T0)Pd0Pd0Pd=τE0β$⁠, where β is the reduction factor caused by screening and HPE. There is a very small difference in the relaxation times calculated from the two phonon models (upper two pair of curves with τ_p = 1 ps in Fig. 7), with the 3DP relaxation times being 4% smaller than the DC ones at high electron temperatures (Fig. 8) and the deviation staying within 5% when reducing the phonon lifetime to 0.1 ps. As the combined hot phonon and screening effect, parametrized by the reduction factor, decreases as T_e is elevated (refer to Fig. 6 above), a great fall of ER time appears at temperatures T_e < 750 K (Fig. 7). In particular, the fall is sharp when Debye screening is used as the screening wavenumber κ_D (κ_D ≈ 5.8k₀T₀/T_e) decreases fast with T_e (at T_e = 1000 K, for instance, κ_D reduces to 1.7k₀).

At high temperatures (above 1200 K), on the other hand, the relaxation time, τ_E, stays almost flat when a constant phonon lifetime of 1 ps is used throughout the temperature range. The relaxation time is 0.15 ps when static screening is used, which is slightly larger than the ∼0.12 ps value obtained with dynamic screening. This saturation in ER means that the increases in the average electron kinetic energy and PD with T_e are somewhat balanced. Experimentally, saturation in ER was observed in Si-doped bulk GaN⁴¹ and a strained AlGaN/GaN heterostructure.⁹ Experimental results^2,6 indicate that the high temperature side has phonon lifetimes τ_p one order of magnitude shorter than 1 ps. Using τ_p = 0.1 ps reduces the relaxation time τ_E to ∼0.12 ps for the static screening case (thick dashed line in Fig. 7) and to ∼0.09 ps when dynamic screening is accounted for (thick dotted line in Fig. 7), as the hot phonon occupation numbers are reduced in the phonon re-absorption processes compared to the case of the longer lifetime of 1 ps. This rapid relaxation means no bottleneck for the PD. Our calculated value ∼0.09 ps is nearly equal to the measured high-temperature relaxation time of 0.09 ps.² We note that in this case, despite it being weakened, the hot-phonon re-absorption should be included to obtain the relaxation time ∼0.09 ps, as we found that without HPE, the relaxation would be faster with τ_E ≈ 0.06 ps.

At high electron temperatures, the measured relaxation time for the investigated lattice-matched heterostructure was found to decrease slowly with the electron temperature.² Our calculation shows that the one order of magnitude shorter phonon lifetime has reduced the relaxation time τ_E by only 0.3 ps when static or dynamic screening is accounted for, and therefore, the high temperature relaxation time decreases slowly with T_e, in good agreement with experiment. These results also support the experimental finding⁶ that the polar optical phonons have a shorter lifetime at a higher electron temperature, otherwise saturation in ER would occur, similar to that in bulk GaN.^7,41

We make a comparison of the ER in bulk GaN^7,41 and the heterostructure. When hot phonons are ignored, the electron power loss is much greater in bulk GaN, which is approximately three times the PD in the heterostructure when static screening is included. This is largely because the electron density of states is much higher in bulk than in the heterostructure. However, we found that the hot phonons play an important role in determining the high-temperature ER. For Si-doped bulk GaN with a volume electron density 10¹⁸cm⁻³, the high-T_e relaxation time is around 0.2 ps [Fig. 7(a) of Ref. 7] with phonon lifetime 10 ps, which is longer than the relaxation time of ∼0.1 ps in the lattice-matched heterostructure. With a higher electron density, 10¹⁹cm⁻³, for instance, in bulk GaN, the electron ER is found to be much slower due to the combined screening and hot-phonon effect.⁷ Therefore, the rapid ER with τ_E around 0.1 ps in the heterostructure means an efficient heat transfer from the hot electron gas to the lattice, which provides the heterostructure with an advantage to use in HFET devices.

In conclusion, we have studied ER for hot electrons in the quasi-2D channel of lattice-matched InAlN/AlN/GaN heterostructures using the DC and 3DP models. The temperature of the quasi-2D electron gas in the narrow 44 -Å channel can reach above 2500 K due to high electric power, much higher than the lattice temperature (room temperature). In this study, therefore, non-equilibrium polar optical phonons as well as electron degeneracy and screening from the mobile electrons are taken into account. Particular attention is paid to the effects of the two phonon models on the hot-electron relaxation process in the GaN heterostructures. We calculated the electron temperature dependences of the electron PD and ER time using a variety of phonon lifetime values and examined the 3DP model by comparing the results calculated with the two phonon models. We found that the 3DP model yields very close results to the DC model: with no hot phonons or screening, the power loss calculated from the 3DP model is 5% smaller than the DC PD, whereas slightly larger 3DP power loss (by less than 4% with a phonon lifetime from 0.1 to 1 ps) is obtained throughout the electron temperature range after including both the HPE and screening. Very close results are obtained also for the ER time with the two phonon models (with a percent deviation smaller than 5%). As the investigated heterostructure has a channel narrower than the usual GaN-based heterostructures, therefore, the 3DP model is generally a good approximation to use for the study of ER in GaN-based heterostructures. We found that our results in the spontaneous phonon emission case are consistent with the sum rules given by Mori and Ando¹⁷ and by Register.¹⁸ The discrepancy between the DC and 3DP results is caused by how much the high energy IF phonons contribute to the ER: their contribution is enhanced in the spontaneous emission process but is dramatically reduced after including the HPE. Debye screening overestimates the high-T_e ER time by ∼0.03 ps compared to the dynamic screening model, whereas with dynamic screening included anti-screening occurs at low electron temperatures (below ∼800 K) due to the dispersion of the quasi-2D electron plasma frequency. Our calculation with both phonon models has obtained a great fall in ER time τ_E at low electron temperatures (T_e < 750 K) and slow decrease at the high temperatures with the use of decreasing phonon lifetime with T_e. The calculated temperature dependence of the relaxation time and the high-temperature relaxation time ∼0.09 ps are in good agreement with experimental results. We also compared the quasi-2D hot-electron relaxation with the electron relaxation in bulk GaN and found that the hot phonons play a key role in slowing down the high-T_e electron relaxation for bulk (τ_E ∼ 0.2 ps). For the heterostructures, in contrast, the rapid ER (τ_E ∼ 0.09 ps) and sub-picosecond phonon decay provide an advantage which benefits electron transport in the HFET devices by efficiently cooling down the extremely hot electrons to increase the electron mobility.

J.Z. acknowledges support from the Natural Science Research Funds of Jilin University, and A.D. and B.K.R. would like to thank the Office of Naval Research, U.S. for funding under Grant No. N00014-09-1-0777 sponsored by Dr. Paul Maki.