Elastic scattering by hot electrons and apparent lifetime of longitudinal optical phonons in gallium nitride

III-Nitride based high electron mobility transistors (HEMTs) can be microwave devices of choice when both high frequency and high power operation are required.^1–3 At high fields both current density and cut-off frequency of HEMTs are determined to a large degree by the saturation velocity of electrons $vsat$⁠. GaN has a high breakdown field and a parabolic conduction band (CB) with satellite valleys far removed from the CB edge. Furthermore, the energy of longitudinal optical (LO) phonons is very high, $ℏωLO=90 meV.$ The combination of these fortuitous characteristics causes one to expect the saturation velocity in GaN to be on the scale of $vsat≈ℏωLO/mc=2.8×107 cm/s$⁠. In reality $vsat$ is less than that, especially when the electron density $ne$ is high, i.e., in the regime where the power-amplifying HEMT is expected to operate.^1–3 Our most recent experiments⁴ have shown that $vsat$ steadily falls from about $1.9$ to less than $0.9×107 cm/s$ as $ne$ is increased from $1018 cm−2$ to $1019 cm−2$ as shown in Fig. 1. Here $ne$ is the “average” 3D density, estimated from the 2D density $n2D$ also shown in the figure and the full width at half maximum of the wavefunction using Schrodinger-Poisson simulation. The source of this reduction has been discussed extensively, and, while details may differ, there is a broad consensus that the reduction in saturation is caused by the hot phonons accumulating in the channel.^5–7 The accumulation is caused by the fact that the lifetime of LO phonons in GaN, as measured in Raman experiments, can be as long as $τLO=2−3 ps$ (Refs. 8 and 9) at low $ne$⁠. The lifetime is so long because $ℏωLO$ in GaN is high, and an LO phonon cannot split into two acoustic (TA or LA) ones.¹⁰ The only channel available for LO decay is the slow Ridley channel,¹¹ in which an LO phonon splits into a TO and a TA (LA) phonon. Since the group velocity of LO phonons is very small, they accumulate in the HEMT channel, causing an increase in the electron temperature $Te$⁠. In addition, the momentum relaxation rate increases and the velocity saturates.^5,6 Naturally, the higher $ne$⁠, the higher the density of LO phonons, which in turn reduces the $vsat$⁠.

This simple explanation assumes that $τLO$ is constant and does not depend on electron concentration and temperature. It is understood that electrons are not directly involved in the decay of LO phonons. This commonsense assumption of constant $τLO$ has been challenged in the last decade. The time resolved Raman measurements¹² in which decay times of the Anti-Stokes line, caused by hot phonons and thus associated with $τLO$⁠, has been shown to decrease from 2.5 ps to less than 300 fs as $ne$ increased from 10¹⁷ to 10¹⁹cm⁻³. Apparent decrease of $τLO$ was also inferred from the measurements of $Te$⁠,via hot-electron noise at microwave frequencies.^13–16 Since hot electron and hot phonons are at thermal equilibrium with each other, the rate of hot electrons cooling had been associated with LO phonon lifetime. This measured lifetime had also shown a decrease with increased $ne$ and $Te$⁠.

The mechanism of the decrease in $τLO$ with increase of $ne$ is not obvious,¹⁷ since at first look only the anharmonicity of lattice vibrations defines $τLO$⁠. In 2006 Dyson and Ridley¹⁸ came up with an ingenious explanation based on coupling between LO phonons and plasmons. The resonant frequency of plasmons near the Brillion zone (BZ) center is $ωp=(e2ne/εmc)1/2$⁠, where $mc=0.2m0$ is the CB effective mass, and $ε=8.9$ is a low frequency dielectric constant. When $ne$ approaches 10¹⁹cm⁻³, $ωp$ becomes comparable with $ωLO$⁠, and LO phonon and plasmon couple forming two hybridized phonon-plasmon (HPP) longitudinal modes with frequencies $ω+$ and $ω−$⁠. Higher energy HPP $ω+$ has large longitudinal electric field and thus efficiently scatters electrons. Unlike LO phonons, the HPP mode is dispersive and has a group velocity on the scale of 10⁶cm/s. There are two ostensive reasons why the HPP's decay faster than “bare” LO phonons.^18–20 First, the rate of Ridley process of decay, LO → TO + LA is proportional to LA frequency, $ωA1=ωLO−ωTO$⁠, and to the LA phonon density of states, which according to Debye model is $ρA1∼ωA12$⁠. Then, the LO phonon lifetime is $τLO∼(ωLO−ωTO)−3$⁠, and when the LO phonon hybridizes with a plasmon, the lifetime of the HPP $τ+∼(ω+−ωTO)−3$ decreases dramatically in comparison with that of a bare LO phonon simply because $ω+>ωLO$⁠. The second reason for apparent decrease of LO phonon lifetime is the migration of coupled plasmon-phonon quasiparticles out of the HEMT channel. Indeed, with the group velocity of 10⁶cm/s it would take only 300 fs for the hot quasiparticles to move out of the channel.

However, both of these explications appear questionable when subjected to in-depth examination. First of all, when one considers real phonon dispersion in GaN,²¹ at frequencies near and above $ωA1=ωLO−ωT0=240 cm−1$⁠, the acoustic phonon density no longer follows a square law, and, in fact, decreases with frequency as evidenced in Fig. 2(a). Therefore, when hybridization takes place, the density $ρA2$ of the acoustic modes with frequency $ωA2=ω+−ωT0>240 cm−1$ into which HPP decays is actually reduced compared with $ρA1$⁠; therefore, no decrease in the lifetime can take place. When it comes to the “migration” explanation, it is important to note that HPP frequency $ω+(x)$ is coordinate dependent and differs from $ωLO$ only where $ne$ is large, i.e., the inside of the channel as shown in Figs. 1(b) and 1(c). Therefore, the HPP moving ballistically towards the edge of the channel simply gets reflected back into the channel, as shown in Fig. 1(c), as propagating further would clearly violate energy conservation. Alternatively, one can say, that an HPP mode whose dispersion is parabolic (i.e., electron-like) simply forms a confined state inside the channel, described by the wave function $ψ+(x)$ and similar in all respects to the confined electron state. The HPP's in this state cannot possibly exit the channel of HEMT in the vertical direction. Furthermore, if $τLO$ goes down as drastically as predicted by the “plasmon hybridization” model, then $vsat$ should not change with $ne$ changing from 10¹⁸ to 10¹⁹cm⁻³.¹⁸ Yet our data in Fig. 1 (Ref. 4) does show significant decrease of $vsat$ in that density range confirming earlier results,²² albeit in disagreement with results in Refs. 23 and 24 where the electron density was determined in a different way. Finally, apparent decrease of optical phonon lifetime with $ne$ had also been measured in silicon,²⁵ and germanium,²⁶ non-polar materials where plasmons and phonons do not couple at all.

To reconcile these facts, it is essential to recognize that characteristics measured in Raman and noise experiments are different from actual phonon lifetimes. In Raman experiments, only the dynamics of phonons with wavevector q ∼ 0 is observed, while noise temperature measurements determine only the maximum LO phonon population near $q0=2mcωLO/ℏ≈0.68 nm−1$⁠, where most of LO phonons are produced. But electron-phonon interaction occurs throughout a much wider span of wave vectors in the BZ. Hence, if there existed an $ne$-dependent mechanism instigating migration or “diffusion” of the LO phonons throughout the BZ, this diffusion would quickly deplete the LO phonon population in the region where they are produced (⁠$q≈0$ in Raman experiments and $q≈q0$ in electron noise measurements). Measured, or apparent, LO phonon lifetime $τapp$ then would be much lower than $τLO$⁠. With the total number of hot LO phonons in the BZ steadily growing as $ne$ increases, $vsat$ would continue its steady decrease, albeit the rate of this decrease would experience some slow down, as measured in our experiments.

In this work we establish that this mechanism causing the decrease of $τapp$ with $ne$ does exist and is nothing but the scattering of LO phonons on hot electrons, a second-order process that can be remarkably efficient due to large density of LO phonon states. The “classical” diagram of the process is shown in Fig. 3(a) where an LO phonon with quasi-momentum $q1$ and hot electron $k1$ “collide” and the new states carry momenta $q2=q1+Δq$ and $k2=k1−Δq$⁠. Since the energy of LO phonon barely changes, the process is nearly elastic, in fact similar to Rayleigh scattering of photons in the atmosphere. The quantum nature of the process is shown in Fig. 3(b), where the distribution of hot electrons in the CB is shown as a tinted sphere, with the initial $k1$ and final $k2$ states located on the same equal energy surface. According to 2-nd order perturbation theory, scattering involves absorption of the initial phonon $q1$ and electron making a transition into the virtual state $k′=k1+q1$ followed by emission of phonon $q2$ and electron going into the final state $k2$⁠, or the emission of $q2$ occurs first as electron makes transition to a different virtual state, $k″=k1–q2$⁠. According to Fermi golden rule²⁷ and diagram in Fig. 3(b), the total rate of scattering between two LO phonon states $q1$ and $q2$ is

where V is the volume, $1/ε′=1/ε∞−1/ε$⁠, and $ε∞$ is the dielectric constant in the visible/infrared region. The two terms in the parenthesis correspond to two different quantum paths, and their interference cancels divergences at $q1,2=0$⁠. Since $δ(Ek2−Ek1)=mc/(ℏ2k1Δq)δ(cos θ−Δq/2k1)$⁠, only the electron states with energies higher than $Emin(Δq)=ℏ2(Δq)2/8mc$ can scatter the LO phonons by $Δq$⁠; therefore, scattering exhibits a strong dependence on $Te$⁠. As explained earlier, for large $ne$ the LO phonon hybridizes with a plasmon and one should use the HPP mode frequency $ω+$ in place of $ωLO$⁠, which should slightly enhance scattering, but we neglect it as well as degeneracy of the electron gas at $ne$ exceeding 10¹⁹cm⁻³ in our order-of-magnitude analysis. To proceed further it is convenient to introduce normalized wave vector $qr=q/q0$⁠, normalized temperature $Tr=kBTe/ℏωLO$⁠, and normalized electron density, $Nr=neq03$⁠. Then, performing the summation in (1), which involves averaging over the angles, one obtains the estimate of the rate of the net scattering for the phonons from the state with a given value of $q$

where $nq$ is the number of LO phonons in the state $q$⁠, $Gq$ is the generation rate, which includes the net LO phonon creation by the electrons as well as their possible generation via Stokes Raman scattering, $n¯=[exp(ℏωLO/kBT)−1]−1$ is the LO equilibrium number at the lattice temperature T, and $R0=π1/2(e2q0/2πε′ℏωLO)2ωLO$ = $0.66×1015 s−1.$ In other words, if a single LO phonon had been generated, then for electron concentration of say, $1019 cm−3$ (⁠$Nr≈0.03$⁠) and $Te$ of 1000 K (⁠$Tr≈1$⁠) the apparent lifetime of it would have been only about 50 fs or less. Of course, as LO phonons are generated throughout the BZ, the scattering out of a given state q will be counter-acted by scattering into this state, so the net rate will be less and the process will acquire the features of diffusion. Indeed, if one assumes that the phonons are distributed in the BZ according to Gaussian law, $nq=n0 exp(−qr2/2σq2)$⁠, where $σq−2=−∇q2nq=0$⁠, for relatively small $∇q2nq$ the integral in (2) can be evaluated as $(3/2σq2)Tr2$ and then one obtains $dnqdt≈−Dq∇2nq−nq−n¯τLO$ where the diffusion coefficient in reciprocal space is $Dq=(3/2)R0Tr3/2Nr$⁠, or $Dq=(3/2)R0(kBT/ℏωLO)3/2neq0−1$ in absolute units of cm⁻² s⁻¹.

To illustrate how scattering by hot electrons determines the apparent lifetime of LO phonons, we have solved (2) for the case where the LO phonons are generated near the center of the BZ as $nq=n0 exp(−qr2/2σq2)$ where $σq=1$ (or $σq=q0$ in absolute units), $Te=600 K$⁠, $ne=5×1017 cm−3$⁠, and $τLO=2.5 ps$⁠. The first order processes of phonon creation and annihilation, contained in the term $Gq$⁠, can only maintain this distribution and not alter it, and of course they cannot create the LO phonons at large q-vectors. Therefore, in our calculations we only consider the actual decay of phonons via elastic scattering by electrons and via normal TO + LA (TA) process. In Fig. 4(a), the snapshots of non-equilibrium LO phonon distribution $nq−n¯$ from t = 0 to t = 2 ps are shown. The LO phonon density near the center of the BZ decays exponentially with $τapp=1.95 ps$⁠. In Fig. 4(b), we plot the density of per unit interval of q, i.e., $(nq−n¯)qr2$⁠, and one can clearly see that before LO phonons decay they diffuse away from the center of the BZ. Since $τapp−1=τLO−1+τdiff−1$⁠, the diffusion time is $τdiff≈9 ps$⁠, i.e., diffusion is not significant. However, when $ne$ is increased to $1019 cm−3$⁠, the apparent LO phonon decay accelerates dramatically as shown in Figs. 3(c) and 3(d). Near the BZ center $τapp$ decreases to about 0.25 ps. This is the lifetime observed in Raman experiments.¹² In the vicinity of $q0$ one can see $τapp(q0)≈0.5 ps$⁠. This is what is observed in the noise measurement.^13–15 The diffusion time near the BZ center is $τdiff≈0.29 ps$ indicating that diffusion plays the dominant role. From Figs. 3(b) and 3(d) one can also see that the total number of excess LO phonons, the area under the curve, decays with the constant lifetime $τLO$ independent of $ne$⁠.

Fig. 5 demonstrates electron-density-dependence of $τapp$ near the BZ center for different $Te$'s. More than a tenfold decrease of $τapp$ occurs between $ne=1018 cm−3$ and $ne=1019 cm−3$⁠, which is in line with the lifetime decreases reported in Refs. 12–16. The dependence of $τapp$ on $Te$ is strong at high density, when $τapp$ is dominated by diffusion, although it is somewhat weaker than $Te3/2$ dependence predicted using simplified equations earlier. Also plotted are the data from Raman measurements in Ref. 12. Agreement is good; however, one must not overlook that in addition to obvious electron concentration and temperature dependencies, the diffusion time is also strongly influenced by how the generated LO phonons are distributed in BZ, for which we have chosen a Gaussian distribution with $σq=1$⁠. Under different excitation conditions, the observed $τapp$ may vary from one experiment to another. In fact, the only way to accurately predict $τapp$ would be to obtain a self-consistent solution for dynamics of both hot LO phonons and hot electrons in each point in the BZ. Such a task is beyond the scope of this paper. Nevertheless, we can say, with confidence that the main experimentally observed trend, the order-of-magnitude reduction of $τapp$ for $ne$ exceeding 10¹⁸cm⁻³, can be faithfully replicated using realistic input parameters $σq$ and T_e. Showing that diffusion in reciprocal space can lead to tenfold decrease in $τapp$ is an important step towards a full understanding of hot phonon dynamics.

At first, the finding that LO phonon scattering by electrons occurs at rates that are only a few times slower than the rates of LO phonon generation in HEMT (typically tens of femtoseconds^5,6) can be surprising. Scattering is a 2nd-order process, while generation is a 1st order one. When dealing with, say photons, the higher order processes are usually orders-or-magnitudes weaker, but phonons have exceptionally large density of states; therefore, different order processes may have comparable strength. For instance, four–phonon scattering processes can play nearly as important a role as three phonon processes in reducing thermal conductivity.²⁷ 

Besides being a new interesting physical phenomenon, phonon scattering by hot electrons plays a key role in determining $vsat$⁠. The total number of LO phonons in the BZ is unaffected by scattering. Therefore as $ne$ increases $vsat$ steadily declines, in agreement with the experimental data.^4,22 But re-distribution of LO phonons reduces their maximum number in the vicinity of $q0$⁠, which in turn reduces $Te$ (as evidenced from noise measurements) and makes cooling more efficient. In addition, the LO phonons that spread further towards the BZ edges do not interact with electrons as strongly as the ones near $q0$⁠, so the momentum scattering rate decreases.²⁸ As a result the absolute value of the slope of $vsat$ vs. $ne$ is gradually reduced, although the slope always remains negative in full agreement with Fig. 1.

In conclusion, we have shown that LO phonon scattering by hot electrons in GaN causes diffusion of LO phonons in the BZ. This is manifested in the reduction of the measured apparent lifetime of LO phonons with an increase in electron density. Phonon diffusion strongly affects electron temperature and velocity saturation. Once it is fully incorporated into a self-consisted model of HEMTs, it will help to improve the design of the future III-Nitride HEMTs and other high speed devices, as nothing discussed here is limited to nitrides.

The authors gratefully acknowledge the support from DATE MURI (ONR N00014-11-1-0721, Dr. Paul Maki).