Momentum relaxation due to polar optical phonons in AlGaN / GaN heterostructures

Using the dielectric continuum (DC) model, momentum relaxation rates are calculated for electrons confined in quasi-two-dimensional (quasi-2D) channels of AlGaN/GaN heterostructures. Particular attention is paid to the effects of half-space and interface modes on the momentum relaxation. The total momentum relaxation rates are compared with those evaluated by the three-dimensional phonon (3DP) model, and also with the Callen results for bulk GaN. In heterostructures with a wide channel (effective channel width >100 Å), the DC and 3DP models yield very close momentum relaxation rates. Only for narrow-channel heterostructures do interface phonons become important in momentum relaxation processes, and an abrupt threshold occurs for emission of interface as well as half-space phonons. For a 30-Å GaN channel, for instance, the 3DP model is found to underestimate rates just below the bulk phonon energy by 70% and overestimate rates just above the bulk phonon energy by 40% compared to the DC model. Owing to the rapid decrease in the electron-phonon interaction with the phonon wave vector, negative momentum relaxation rates are predicted for interface phonon absorption in usual GaN channels. The total rates remain positive due to the dominant half-space phonon scattering. The quasi-2D rates can have substantially higher peak values than the three-dimensional rates near the phonon emission threshold. Analytical expressions for momentum relaxation rates are obtained in the extreme quantum limits (i.e., the threshold emission and the near subband-bottom absorption). All the results are well explained in terms of electron and phonon densities of states.


I. INTRODUCTION
GaN-based wide-band-gap semiconductor heterostructures such as AlGaN/GaN have attracted intense research interest due to their envisaged applications in novel high-power and high-mobility devices such as heterostructure field-effect transistors (HFETs).At the interface of such a heterostructure, a two-dimensional (2D) electron gas forms due to internal spontaneous polarization plus possible strain-induced polarization by the piezoelectric effect.As the 2D electron gas (2DEG) arises in the absence of doping, impurity scattering can be minimized and high electron mobility can occur.Therefore, the electron transport in AlGaN/GaN heterostructures is determined by phonon scattering processes, which at room temperature are dominated by polar optical phonon scattering.
It is well known that the momentum relaxation rate 1 (MRR) or, equally, its inverse, the momentum relaxation time, is a key parameter for describing the dynamics of the electron transport in semiconductors under an external electric field.While it is governed by the microscopic electron-phonon scattering processes, the MRR is closely related to dynamic quantities of interest such as the electron mobility and the electron drift velocity.Hence, knowing the MRR, or more specifically its dependence on the electron energy for different AlGaN/GaN heterostructures, is fundamental to the optimization of the HFET device performance.
There have been studies of electron momentum relaxation in GaAs-and GaN-based quantum wells.For example, in an early study for GaAs quantum wells, 2 both electron scattering rates and MRRs were numerically calculated and compared with analytical results 3 obtained from the momentum conservation approximation to examine the validity of the approximation.For GaN quantum wells, Anderson et al. 4 studied the momentum relaxation and low-field electron transport in degenerate 2DEGs by solving a linearized Boltzmann equation.In most calculations, the polar optical phonons of the quasi-two-dimensional (quasi-2D) systems are simply taken as the longitudinal-optical (LO) phonons of the bulk material.This is referred to as the three-dimensional phonon (3DP) approximation.According to the well-established dielectric continuum (DC) model, the polar modes of a single heterostructure include half-space LO modes and interface modes, and all these modes interact with the electrons in the quasi-2D channel.Indeed, using the DC model, MRRs and mobility of electrons due to phonon absorption were calculated for AlGaN/GaN heterostructures. 5Mori and Ando showed that the sum of the form factors associated with half-space and interface modes was equal to the form factor for bulk phonons.This implied that the total rates calculated with the 3DP approximation would be close to those obtained from the DC model. 1,6However, this is only partly true, and caution should be taken in studying momentum relaxation with the 3DP approximation.The argument is as follows.First, evaluation of the MRR depends on not only the form factor, but also the electron-phonon interaction strength and detailed electron and phonon densities of states.Second, since the potential of interface modes decreases exponentially from the interface according to e −q|z| (Sec.II below), scattering with interface modes is weak in wide wells making the 3DP approximation a reasonable model.Indeed, for GaAs wells with widths greater than 100 Å, the 3DP model suffices for the evaluation of scattering rates 7,8 and energy loss rates. 9For narrow wells, however, the interface phonons are important scattering partners, and the two phonon models yield quite different rates. 8,9However, as far as we know, there has been no comparison of MRRs based on the two phonon models.Third, the dependence of the MRR on the electron energy is of great interest.Previous scattering rate calculations 8 have shown that, when the electron energy is near to the phonon energy, the 3DP estimation becomes poorer.It is predicted that this will be true for MRRs as well.Fourth, the points above are made assuming no hot-phonon effects.In the hot-phonon regime, energy loss rates have been found to depend on the phonon models used. 6,9MRRs are thought to do so as well.
In this paper, we study the momentum relaxation of electrons in the channels of AlGaN/GaN heterostructures.As the 3DP treatment is convenient for practical usage, one of course wants to know the discrepancy between MRRs estimated by this model and those calculated with the DC model.This comparison can be made with regard to only the total rates.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 momentum relaxation rate can be singled out.Thus, it is also of great interest to find and understand behaviors of the half-space modes and interface modes in the momentum relaxation process.These are the purposes of this study.Therefore, using the two phonon models, the momentum relaxation rate is evaluated as a function of the electron kinetic energy and the effective channel width for the respective phonon emission and absorption processes.The MRRs from the scattering with half-space phonons and interface phonons are compared and examined, focusing on the energy dependences, the peak rates variation, and their energy shift as the heterostructure parameter (e.g., the effective channel width) is varied.We found that negative MRRs occur in all interface phonon absorption processes, and this is due to the stronger forward scattering from the long-wavelength interface modes involved.Particular attention is also paid to a comparison of the total MRRs calculated with the DC model and the 3DP approximation for a number of effective channel widths to examine the latter phonon model in the estimation of MRRs.These results are also compared with the bulk situation to find how both electron and phonon confinement and their densities of states affect the momentum relaxation.
This paper is organized as follows.In Sec.II, the dielectric continuum model for single heterostructures is briefly described where the phonon modes and associated electron-phonon interactions are given.Then, a formulation of the momentum relaxation rate in such heterostructures is presented in Sec.III.In Sec.IV, first we present results of the momentum relaxation rate in a typical AlGaN/GaN heterostructure by comparing the respective contributions to the momentum relaxation from the half-space and interface modes.Then, we show the momentum relaxation results for heterostructures with different well widths, and also compare them with the 3D bulk case.This is to investigate how quantum confinement in both electronic states and phonon modes affect the momentum relaxation.In order to examine the usual 3D phonon approximation in the evaluation of MRRs, we further compare the momentum relaxation rates obtained from the 3D phonon and DC models for a variety of AlGaN/GaN heterostructures.The phonon confinement effects on the momentum relaxation are discussed in detail.Finally, Sec.V summarizes the main results obtained.In Appendix A, a derivation of the final expressions for the MRRs is outlined for both the half-space and interface phonons.An effective numerical technique in calculating these rates is described in terms of handling the singularities of the integrals involved.It is of great interest to obtain, if at all possible, an analytical formula for the MRR in a quasi-2D system such as the heterostructure.To do this, two special cases are considered in Appendix B: (i) when the kinetic energy of the electron is sufficiently close to a phonon energy that a half-space or interface phonon can be emitted, and (ii) when the electron kinetic energy is close to zero, a phonon is absorbed.In the former case, we obtain an analytical formula for the momentum relaxation rate and, further, this rate is equal to the usual electron-phonon scattering rate.For the latter case, in contrast, we find from the obtained analytical formulas that the momentum relaxation rate is always smaller than the usual scattering rate; further, the MRR for interface phonons can have a negative value, whereas the MRR for half-space modes is always positive.These analytical expressions are used to check and interpret our numerical results.

II. OPTICAL PHONON MODES AND ELECTRON-PHONON INTERACTIONS IN THE DIELECTRIC CONTINUUM MODEL
To study the momentum relaxation in heterostructures, we need electron states, lattice vibration modes, and electronphonon interactions.Let the interface of the heterostructure be at z = 0, with the barrier in the space −L 1 < z < 0 and the electron-containing active region in the space 0 < z < L 2 .Let ρ = (x,y) be the position vector in the plane parallel to the interface.Due to confinement in the growth direction z, the motion of an electron can be described by the wave function ψ nk (r) = 1 √ A φ n (z)e ik•ρ , and the corresponding electron energy E nk = ε n + E k , where A is the sample area, n indexes the subband, and φ n (z) is the confinement envelope function corresponding to energy ε n .k is the electron wave vector parallel to the heterostructure interface, and E k is the electron kinetic energy E k = h2 k 2 /2m * , with m * being the electron effective mass.
Two types of polar optical modes occur in the wurtzite structure owing to anisotropy in uniaxial crystals.The anisotropy, however, causes only a small difference in the two LO-like phonon frequencies. 10Thus, in this study, the polar modes are simply taken to be cubiclike LO modes.In the dielectric continuum model, the polar vibration modes of a single heterostructure consist of half-space modes and interface modes. 11The half-space modes have the frequencies of the polar modes associated with the two constituent materials with their optical vibrations, electric fields, and scalar potentials occurring in the respective constituent regions.Different from half-space modes, the interface modes have different frequencies from the polar modes of both constituent materials, and an interface mode has lattice vibrations and electric fields in both constituent regions.The barrier is a binary alloy in which the polar optical modes consist of both GaNand AlN-like vibrations.When treating phonon modes of Al x Ga 1-x N/GaN heterostructures, we neglect all GaN-like vibrations in the alloy Al x Ga 1-x N. 12 This is based on two considerations: (i) the half-space modes associated with the alloy have very weak interactions with electrons in the GaN channel; (ii) interface modes and their momentum relaxation rates can be dealt with in a simple way.
Let L represent the dimension of a constituent half-space in the growth direction L = Na, with a being the lattice constant.That is, L = L 1 (L = L 2 ) for the half-space of Al x Ga 1-x N (GaN).The half-space modes then can be simply indexed by (q z ,q), all having the LO frequency of the material ω LO .Here, q is the in-plane phonon wave vector, and q z = nπ/L (n = 1,2, . . .,N − 1).The Hamiltonian of an electron interacting with these half-space modes can be written as where a q z (q) and a + q z (q) are the annihilation and creation operators for the half-space mode of (q z ,q).γ LO is a constant given by γ , where 0 and ∞ are the static and high-frequency dielectric constants of the constituent material, and e is the electron charge.V is the volume of the constituent region (V = AL).We use cgs units throughout the paper.
Let (ω) be the lattice dielectric function in the active region , where ∞ , ω LO , ω TO are the high-frequency dielectric constant, the LO and transverse optical (TO) phonon frequencies of the active region.When GaN-like vibrations are neglected, the dielectric function in the barrier is then given by ¯ , where ¯ ∞ , ωLO , ωTO are the high-frequency dielectric constant, the LO and TO phonon frequencies of AlN.The frequencies of the interface modes are determined by (ω) + ¯ (ω) = 0, which yields two solutions ω ν (ν = 1,2; let ω 1 < ω 2 ).This shows that the interface phonons have no dispersion.The interface modes can be simply indexed by (ν,q).The electron-interface-phonon interaction Hamiltonian can be written as where γ ν is given by γ . a ν (q) and a + ν (q) are the annihilation and creation operators for the interface mode (ν,q).

III. MOMENTUM RELAXATION RATES IN A SINGLE HETEROSTRUCTURE
By knowing the interaction Hamiltonians, the usual electron-phonon scattering rates for half-space and interface modes can be given by Fermi's golden rule.The electron momentum relaxation rate describes the response of electrons to electric fields, and can be approached by weighing the scattering rate by the appropriate decrease in momentum. 1et k be the wave vector of an electron before it is scattered by a phonon with wave vector q, and let θ be the angle between the two 2D wave vectors.Due to momentum conservation, the emission (absorption) of the phonon causes a fractional increase of momentum of −q cos θ/k (q cos θ/k) in the direction of k.
Thus, the momentum relaxation rate for an electron at state (n,k) due to emission (upper signs) or absorption (lower signs) of the half-space phonons can be written as where M q z ,q (n ,k ; n,k) is the electron-phonon interaction matrix element for the electronic transition from state (n,k) to state (n ,k ) due to the scattering with a half-space mode phonon of (q z ,q) (frequency ω LO ), i.e., M q z ,q (n ,k with T L being the lattice temperature.
Similarly, the momentum relaxation rate associated with the interface modes can be given by where M ν,q (n ,k ; n,k) are the interaction matrix elements associated with the interface modes (ν,q), M ν,q (n ,k ; n,k) = n ,k |H i |n,k .Still, the upper (lower) signs are for phonon emission (absorption).For half-space phonon scattering [Eq.( 3)], each matrix element carries an integral over z, and only the matrix elements depend on q z .Thus, summation over q z can be simply performed for |M q z ,q (n ,k ; n,k)| 2 .Replacing q z by L π ∞ 0 dq z and expressing the matrix element in terms of the electron and phonon envelope functions, we convert this summation q z |M q z ,q (n ,k ; n,k)| 2 to a triple integral over q z ,z,z .The integration over q z can be performed analytically, reducing the triple integral to a double integral (over z, z ) that yields a form factor F dependent on q, n, n .Therefore, we obtain with the form factor given by where z, z run through a half-space.
For interface modes, we find that the matrix elements are given by where f n n (q) is the form factor The delta function in Eq. ( 5) [Eq.( 7)] reflects momentum conservation in the scattering of an electron with a half-space (interface) phonon.This reduces the double summation over the electron and phonon wave vectors k , q in Eqs. ( 3) and ( 4) to a single summation over either wave vector (e.g., q), which can be transformed to an integral over the wave vector as usual.For general electron energies, however, such integrals can not be evaluated analytically, and the relaxation rates (3) and ( 4)] need to be evaluated numerically.The numerical techniques are described in Appendix A. We note that dynamic screening or antiscreening may occur, depending on electron densities (antiscreening dominates at small q, while screening is present at large q). 13,14Following a long established approximation in the literature, here we have assumed that screening and antiscreening cancel out, and ignored screening entirely.
Actual AlGaN/GaN heterostructures reported in most experiments are strained or pseudomorphic heterostructures grown on sapphire substrates, where the 2DEG is induced by polarization charges.Self-consistent calculations 15,16 have shown that, for Al x Ga 1-x N/GaN heterostructures with various compositions x, the 2DEG densities in the lowest subband alone fit well with experimental data.The single subband model will suffice for the momentum relaxation study, 5 and in what follows, the subband index 1 will be omitted for clarity.We assume that the electrons are completely confined in a channel parallel to the interface.We model the confinement envelope function for the lowest subband by the Fang-Howard wave function 17,18 where b is a variational parameter that is determined by minimizing the total energy of the 2DEG system.b is related to the areal electron density N s in the GaN channel via Therefore, the form factor associated with the half-space modes is given by and for the interface modes the form factor is We note that, for undoped Al x Ga 1-x N/GaN heterostructures, 15,16 the electron density N s varies with the alloy composition x to achieve a balance with the polarization and surface charges across the entire structure. 19In the wave-function model, an effective channel width d can be defined as twice the average penetration depth of the charge in the active GaN region; 20,21 d is related to the Fang-Howard b parameter via d = 6/b.In this study, the material parameters are taken from Refs.22-24.The LO and TO phonon frequencies used for GaN are ω LO = 91.13meV, 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 * = 0.22m 0 (m 0 is the free electron mass).The temperature is fixed at room temperature 300 K, and hot-phonon effects are not considered.Calculations are performed for a variety of AlGaN/GaN heterostructures by varying the value of the electron density.For numerical integration in q space (see Appendix A), 155 Gauss-Legendre quadrature points are used and excellent convergence is achieved.

IV. RESULTS AND DISCUSSIONS
Scattering rates and momentum relaxation rates are usually given for convenience in units of a basic rate W 0 (Ref.25)  relating to bulk parameters ). W 0 is 142 THz for bulk GaN, which is approximately 17 times higher than the W 0 value for bulk GaAs (the latter semiconductor has smaller electron effective mass and optical phonon frequencies).We first look at momentum relaxation rates calculated using the dielectric model.The optical phonons contributing for the momentum relaxation are the GaN half-space modes (hω LO = 91.13meV), and the lowerand higher-energy interface modes (hω 1 = 69.70 and hω 2 = 102.09meV, respectively).We first choose a heterostructure with an effective well width of 50 Å (corresponding to an electron density of N s = 10 13 /cm 2 ), and shall discuss later what happens to the momentum relaxation when the well width varies.We separate half-space modes from interface modes and compare their respective contributions to the momentum relaxation rate.Figure 1  (dashed line) and the interface modes (solid line, against the right y axis).For clarity, the dimensionless electron energies are introduced with reference to the bulk GaN LO phonon energy hω LO .Abrupt threshold emission occurs at hω LO for half-space modes and at hω 1 , hω 2 for interface modes.In particular, since the interface modes have two frequencies, the momentum relaxation rate displays two clear discontinuities at the two energies hω 1 , hω 2 as expected.The half-space modes dominate the momentum relaxation.For instance, the peak rate at hω LO is approximately six times that at the higher interface phonon energy hω 2 .It is interesting to note that, at the threshold for half-space or interface mode emission, analytical expressions exist for the momentum relaxation rate, and further this rate is equal to the usual electron-phonon scattering rate (a proof of this is presented in Appendix B).This is illustrated in Fig. 1, where the electron scattering rate due to the half-space modes is also shown (dotted line).In fact, the analytical solution provides a good criterion for a check of our numerical result.By substituting the material parameters into Eq.(B1), one obtains the theoretical rate for threshold ( 1 τ h ) th = 0.631 789 743W 0 .Numerical calculation of 1 τ h as given by Eq. ( A9) with E k /hω LO = 1 + 10 −10 yields a rate of 0.631 789 784W 0 , showing that an accurate calculation has been achieved.We now turn to phonon absorption.The results of the rates are shown in Fig. 2 (well width of 50 Å).Clearly, the MRRs for both half-space and interface modes vary continuously with the electron kinetic energy.There is no special electron energy where rates of momentum relaxation and scattering coincide, and the scattering rates are always faster than the MRRs for both half-space and interface phonons (compare the rates, for example, for half-space modes, namely, the dashed and dotted curves in Fig. 2).We checked numerically that this is true also for heterostructures with different well widths.These are quite different from the phonon emission case above.In fact, in the limiting k → 0 case where analytical solutions exist, it can be proved that the scattering rate is always larger than the MRR for half-space and interface modes (Appendix B).Furthermore, in the phonon absorption case, interestingly the MRR from the interface modes is negative and approaches zero as the electron energy increases (solid line in Fig. 2).This can be explained as follows.According to the expression for the fractional increase of momentum [Eq.(A1)], in a phonon absorption process, the phonon wave vector q and the phonon energy hω ν counteract each other in changing the crystal momentum: the former decreases the momentum through backward scattering, while the latter increases the momentum by forward scattering.Let k ν denote the electron wave vector for threshold emission of an interface phonon of frequency ω ν to occur, that is, k ν = √ 2m * ω ν /h.At a particular electron energy (E k = 0), the momentum relaxation occurs due to the absorption of all the interface phonons with wave vectors q ∈ [q − ν ,q + ν ] [Eq.(A6)]: As q − ν < k ν < q + ν , the interface modes of q < k ν contribute a negative MRR, whereas the modes of q > k ν make a positive contribution.For interface modes, their interaction with the electron in the lowest subband is very different from the interaction of half-space modes.Since the derivative of the form factor is f (q) = −6f (q)/(b + q) < 0, the electron-interface phonon interaction decreases faster at smaller phonon wave vectors q, as is seen from the form factor in Fig. 3 (the form factors of the half-space modes and bulk modes are also shown for comparison).Thus, when the contribution of the q < k ν modes dominates, a negative MRR occurs.It is understood that the higher-frequency modes are more likely to cause a negative MRR.We have checked and found that, indeed, the negative MRRs in Fig. 2 are mostly due to the higher-frequency interface modes.These rates, however, have a smaller magnitude than the half-space mode rates (because of the larger density of states of half-space modes), so the total rates remain to be positive.Negative total MRRs have been found to occur in one-dimensional systems 25,26 where forward scattering processes dominate.
In the special case k → 0, the MRR can be obtained analytically (Appendix B) and the rate due to absorption of the ω ν interface phonons is proportional to 1 B4) and (B6)].For a negative MRR to occur, therefore, the logarithmic derivative of f (k ν ) multiplied by the critical wave vector k ν must be smaller than −1, or simply k ν > b/5.This further proves that it is the rapid decrease in the form factor that causes the negative MRR.This analytical solution can be clearly represented using a geometric diagram as shown in Fig. 3 6 , exactly twice the f (b/5) value and making the derivative term contribution balance with that of the form-factor term (this balance is given by |AB| = |OA| in Fig. 3).That is, zero MRR occurs at k ν = b/5.When k ν > b/5, one has |AB| > |OA|, i.e., the derivative term k ν f (k ν ) is larger in magnitude than the form factor f (k ν ), then the momentum relaxation rate becomes negative.For the lower-and higher-frequency interface modes, the k ν values are 0.06 and 0.08 Å−1 , respectively, and one verifies that, in the 50-Å well, indeed k ν > b/5 = 0.024 Å−1 , and a negative rate due to the interface phonons occurs (Fig. 3).To have a positive rate, on the other hand, one needs to raise the electron density N s and, consequently, the b value according to Eq. (10); this requires that N s > 500 0 ω ν 3/2 √ 2m * h/(33πe 2 ).That is, it is required that N s >1.5, 2.7 ×10 14 /cm 2 corresponding to the lower and higher frequencies ω 1 , ω 2 .These values are unacceptably larger than the known electron density values of AlGaN/GaN heterostructures. 15,16,27Therefore, the small-k rates are negative at realistic electron densities for both the ω 1 and ω 2 phonons (as shown in Fig. 4).
We now investigate how MRRs vary with the electron density or, equally, the well width.In a single heterostructure, the width of the triangular well is determined by the 2DEG density; 20 the higher the density is, the narrower the well is.In the Fang-Howard wave-function model we use here, changing the electron density will change the value of b [Eq.(10)] and therefore alter the effective well width d (d = 6/b).To see how MRRs vary with the well width, calculations were performed for AlGaN/GaN heterostructures at a number of electron densities.It is estimated that degeneracy will become important for electron densities in excess of 8 × 10 12 cm −2 .Compared with the nondegenerate case, it means that, at room temperature, more electrons will have the energy to emit phonons, but the effect of this will be reduced by the occupancy of the lower state inhibiting the emission.Nevertheless, it will be true that the energy range over which the interaction occurs will be generally higher, which, following the well-known property of the interaction with polar modes, will result in weakening the interaction strength.Figure 5 illustrates the results for three electron densities 10 11 , 10 13 , 5 × 10 13 cm −2 (corresponding to the effective well widths of 215, 50, and 30 Å, respectively).The MRRs shown here are the total rates due to the emission and absorption of all half-space and interface modes.The bulk result calculated from the The bulk GaN result calculated with the Callen formula is also shown (dot-dashed line).The effective well widths of these single AlGaN/GaN heterostructures are determined by the areal electron densities in the channels: the well width of 215 Å corresponds to the electron density of 10 11 /cm 2 , well width 50 Å corresponds to electron density 10 13 /cm 2 , and well width 30 Å corresponds to electron density 5 × 10 13 /cm 2 .Callen formula is also shown for comparison.We see that as the electron density increases (i.e., the well narrows), the MRRs increase; in particular, the high-rate portion just above the emission threshold increases fast, its energy region being widened from the half-space LO energy hω LO to the higherfrequency interface phonon energy hω 2 .Further, in narrow wells, interestingly, the dominant momentum relaxation rates can exceed those in bulk GaN.However, the MRRs decrease rapidly with the electron kinetic energy so we see that the rates in all the wells are smaller than the corresponding bulk values at high electron energies.To find the cause, we need to separate and check contributions to the MRRs from the interface and the half-space modes, respectively.We do so for the phonon emission case only, as the rates due to the phonon absorption are very small.In Figs.6(a) and 6(b), we show the MRRs from the interface modes and the halfspace modes, respectively, for the three electron densities above.Clearly, in the wide well (215 Å), the rates from interface modes are two orders of magnitude smaller than those from half-space modes, and thus only the abrupt feature from the half-space modes is visible in Fig. 5 (dashed line).As the well narrows, both the rates from the interface modes and those from the half-space modes increase, but the former increase rapidly.For instance, in the narrow well (30 Å), the peak rate from the higher-frequency interface modes has reached approximately half that from the half-space modes.This is again due to the different electron-phonon interactions for interface and half-space modes, as is reflected by the wave-vector dependences of their form factors: the form factor of the interface modes decreases monotonically, and the decrease becomes slower as the well narrows.This causes interface phonons to become increasingly important in narrow wells.Therefore, the sharp features from both half-space and interface modes appear on the total rate curve (solid and dotted lines, Fig. 5).At high electron energies, the quasi-2D system has smaller electronic density of states than bulk, resulting in slower 2D rates than 3D rates.We note that the 2D rates at the three emission thresholds hω 1 , hω 2 , hω LO are all finite, which is different from the situation in bulk where the zero rate occurs at the emission threshold hω LO .This has been explained in Ref. 2: In the quasi-2D system, the density of states associated with the final states of the electronic transitions is finite, while its counterpart in 3D is zero.The 3DP approximation has been widely used to study carrier-phonon interactions as well as evaluate carrier-phonon scattering rates and electron momentum relaxation rates for quasi-2D semiconductor systems. 2,3,25,28Here, we compare momentum relaxation rates in AlGaN/GaN heterostructures calculated with the DC and 3DP models.To do this, we show in Figs.7(a) and 7(b) the results of MRRs for a wide well (100 Å) and narrow well (30 Å), respectively.For the wide well, the two phonon models yield literally the same momentum relaxation rates, the two lines being virtually coincident in Figs.7(a).This is explained as follows.The potential of the interface modes decreases exponentially according to e −q|z| [Eq.( 2)].The average value of the position for electrons is z av = 50 Å.For the higher-and lower-frequency interface phonons, the characteristic wave vector k ν is 0.06 and 0.08 1/ Å, making k ν z av = 3,4, respectively.Thus, the effect of the interface modes can be neglected.In the narrow well, the MRRs calculated from the two phonon models are quite different at electron energies around the optical phonon energies: the DC model yields higher rates below hω LO , due to significant scattering from the lower-frequency interface phonons, whereas the 3DP rates are significantly higher above hω LO but become smaller beyond hω 2 than those obtained from the DC model.The 3DP model underestimates the MRRs just below hω LO by 70% and overestimates the rates just above hω LO by about 40%.At further high electron energies, the two models give similar rates.This is again because the interface phonon scattering becomes increasingly important in narrow wells.The effect above becomes more pronounced as the well width is further reduced.However, only wells wider than 10 Å have been considered since the effective mass approximation breaks down below this.We note that our results [Figs.7(a) and 7(b)] are consistent with a conservation rule for optical phonon scattering in heterostructures, 29 and this shows that the rule can be extended to momentum relaxation rates.

V. CONCLUSIONS
In conclusion, using the dielectric continuum model, we studied momentum relaxation for quasi-2D electrons in AlGaN/GaN heterostructures.We found that half-space phonon scattering dominates the momentum relaxation processes due to the large density of states of half-space modes.For interface phonons, negative momentum relaxation rates occur in the phonon absorption processes.This stems from the peculiar electron-interface-phonon interaction that decreases rapidly as the wave vector increases, thus causing stronger forward scattering than the backward scattering.Our evaluation predicts that this negative rate will occur in normal AlGaN/GaN heterostructures with electron densities under 10 14 /cm 2 .Further, we examined the three-dimensional phonon model by comparing the total momentum relaxation rates calculated with the two phonon models.We found that the 3DP phonon model is generally a good approximation for AlGaN/GaN heterostructures with a wide channel (>100 Å).Interface phonons in the DC model, on the other hand, become important in electron momentum relaxation for only narrowchannel heterostructures, with distinctive features associated with abrupt threshold emission appearing on the rate curves.For a 30-Å GaN channel, for instance, we found that the 3DP model underestimates the MRRs just below the bulk phonon energy by 70%, and overestimates the rates just above the bulk phonon energy by 40%.We also compared these quasi-2D rates with results for bulk GaN calculated from the Callen formula.We found that, in narrow-channel heterostructures, the quasi-2D rates have very large peak values compared to the 3D rates, but become smaller at high electron kinetic energies.To interpret numerical results, we also obtained transparent expressions for momentum relaxation rates for two limiting cases, namely, the threshold emission and the near subband-bottom absorption.We discussed the results in terms of electron and phonon densities of states.
δ(E k − E k + hω LO )δ q,k .The summation over q is then carried out, followed by the summation over k , which is converted to an integral as usual.Thus, we obtain the momentum relaxation rate for threshold emission where W 0 is the basic rate . The momentum relaxation rate ( 1 τ h ) th is in fact equal to the threshold electron-phonon scattering rate. 12For interface phonons, the threshold rate can be written as where LO and ν are given by 1 We now look at case (ii).Since momentum relaxation arises from a decrease in momentum, clearly one requires that the initial momentum hk = 0. Mathematically, this is in accord with the expression for the fractional increase of momentum, which contains 1/k 2 .Recall, however, that we tackle the k → 0 phonon absorption case; that is, both the denominator and numerator of the fractional increase of momentum approach zero.This is quite different from the phonon emission case, and an analytical solution can not be sought in the direct manner above.We turn to Eq. (A9) for half-space phonons.Let the integral in this 1 τ h expression be denoted by I LO , which depends on k, i.e., I LO (k).Change the variable of integration by using q = kx + k 2 + k 2 LO , such that the new variable of integration x is dimensionless (x ∈ [−1,1]) and the k's contained in the original limits of integration q − LO , q + LO are transferred into the integrand.For simplicity, we denote the integrand by g(k,x).As k k LO , then the integrand g(k,x) can be expanded into a Maclaurin series of powers of k, i.e., g(k,x) = g(0,x) + g (0,x)k + 1 2 g (0,x)k 2 + • • • .Accordingly, the integral I LO (k) can be simply expressed as I LO (k) = 1 −1 g(0,x)dx + k 1 −1 g (0,x)dx + O(k 2 ).It turns out that g(0,x) is an odd function of x [thus making the first term in the I LO (k) expression vanish] and g (0,x) is an even function of x.Recalling that in the 1 τ h expression there is a k-dependent factor 1/k in front of I LO (k), in the limit of k → 0, therefore, only the second term of I LO (k) contributes for momentum relaxation.One finds that the final momentum relaxation rate can be expressed in a simple form in terms of the logarithmic derivative of the form factor as For the momentum relaxation due to interface phonons, one starts with Eq. (A6) (use the lower signs for phonon absorption), repeats the process as before, and obtains the MRR expression for k → 0: From Eq. (B3), clearly the sign of the ( 1 τ h ) 0 rate value is determined by the quantity in the square brackets.Using Eq. ( 11), it is found that, for any wave vector q, 1 + q F (q) F(q) = 2b 2 b + q 33b 3 + 15b 2 q + 7bq 2 + q 3 33b 4 + 54b 3 q + 44b 2 q 2 + 18bq 3 + 3q 4 > 0. (B5) Therefore, the momentum relaxation rate ( 1 τ h ) 0 due to halfspace phonons always has a positive value.For interface phonons, we find that the sign of which is determined by b − 5q.Therefore, when the critical wave vector satisfies k ν > b/5, the interface modes of frequency ω ν contribute a negative MRR.
For the usual electron-phonon scattering rate, however, the situation is different as the initial electron state can be exactly at the bottom of the subband k = 0.Then, the analytical solution can be obtained in a direct manner by using the approach δ(E k − E k − hω LO )δ k ,k+q = δ(E k − E k − hω LO )δ q,k , similar to that used in case (i) above.The scattering rate is W h,0 = π 2 W 0 N (ω LO )F(k LO ) for half-space phonons and W i,0 = π 2 W 0 ν ω ν ω LO LO ν N (ω ν )f (k ν ) for interface phonons.It is readily found from Eq. (B5) that 1 2 [1 + q F (q) F(q) ] − 1 < 0 and, thus, the momentum relaxation rate ( 1 τ h ) 0 is always smaller than the scattering rate W h,0 .For interface phonons, similarly, one finds from Eq. (B6) that 1 2 [1 + q f (q) f (q) ] − 1 < 0 and, therefore, ( 1 τ i ) 0 < W i,0 .

FIG. 1 .FIG. 2 .
FIG. 1. (Color online)Momentum relaxation rates from the emission of half-space modes (dashed line) and interface modes (solid line, against the right y axis) as a function of kinetic energy for an electron in the lowest subband of a single AlGaN/GaN heterostructure with an effective well width of 50 Å (see text).The dotted curve represents the electron scattering rate due to the emission of half-space phonons.The momentum relaxation rates and scattering rates are in units of W 0 (see text) and the electron kinetic energy is made dimensionless with respect to the LO phonon energy hω LO of bulk GaN.

FIG. 3 .
FIG. 3. (Color online) Form factors associated with the electronphonon interactions vs the dimensionless phonon wave vector for half-space (dashed curve), interface (solid curve), and bulk modes (dotted curve).The electron is in the lowest subband, the confinement envelope function of which is modeled by the Fang-Howard wave function.The diagram shows an analytical MRR solution due to interface phonon absorption at k → 0. Zero MRR occurs when the critical wave vector k ν is equal to b/5 (dashed vertical line).The tangent line (dotted oblique line) to the form-factor curve at the point [b/5,(5/6) 6 ] intersects the vertical axis at the point B, |AB| = |OA|.The slope of the tangent is f (b/5) = −5(5/6) 6 /b.

1 FIG. 4 .
FIG. 4. (Color online)The momentum relaxation rates at k → 0 due to absorption of lower-(ω 1 ) and higher-frequency (ω 2 ) interface phonons as functions of the common logarithm of the electron density in cm −2 .

FIG. 5 .
FIG. 5. (Color online) Total momentum relaxation rates vs electron kinetic energy obtained from the dielectric continuum model for three single AlGaN/GaN heterostructures of effective well widths 215 Å (dashed line), 50 Å (solid line), and 30 Å (dotted line).The bulk GaN result calculated with the Callen formula is also shown (dot-dashed line).The effective well widths of these single AlGaN/GaN heterostructures are determined by the areal electron densities in the channels: the well width of 215 Å corresponds to the electron density of 10 11 /cm 2 , well width 50 Å corresponds to electron density 10 13 /cm 2 , and well width 30 Å corresponds to electron density 5 × 10 13 /cm 2 .

FIG. 6 .
FIG. 6. (Color online) Momentum relaxation rates from the emission of (a) interface and (b) half-space phonons for the three single AlGaN/GaN heterostructures as specified in Fig. 5.The rate values from interface modes for the 215-Å well have been enlarged by 10 times.

FIG. 7 .
FIG. 7. (Color online) Total momentum relaxation rates vs electron kinetic energy calculated from the dielectric continuum (DC) model and the three-dimensional phonon (3DP) approximation as labeled for two single AlGaN/GaN heterostructures of effective well widths (a) 100 and (b) 30 Å.In (b), the rates for energies just below the LO phonon energy have been enlarged by 10 times for clarity.