Here we introduce a uniaxial dielectric continuum model with temperature-dependent phonon mode frequencies to study temperature- and orientation-dependent polar-optical-phonon limited electron mobility and saturation velocity in uniaxial semiconductors. The formalism for calculating electron scattering rates, momentum relaxation rates, and rate of energy change as a function of the electron kinetic energy and incident electron angle with respect to the *c*-axis are presented and evaluated numerically. Electron–longitudinal-optical-phonon interactions are shown to depend weakly on the electron incident angle, whereas the electron–transverse-optical-phonon interactions around the emission threshold energy are observed to depend strongest on the electron incident angle when varied from *π*/4 to *π*/2 (with respect to the c-axis). We provide electron mobility and saturation velocity limits in different GaN crystal orientations as a function of temperature and electron concentration. At room temperature and for an electron density of 5 × 10^{18} cm^{−3}, electron mobility limit of ∼3200 cm^{2}/V s and electron saturation velocity limit of 3.15 × 10^{7} cm/s are calculated. Both GaN electron mobility and saturation velocity are observed to be governed by the longitudinal-optical-phonon interaction, and their directional anisotropy is shown to vary less than 5% as the electron incident angle with respect to the c-axis is varied from 0 to *π*/2. Overall, we develop a theoretical formalism for calculating anisotropic properties of uniaxial wurtzite semiconductors.

## I. INTRODUCTION

Gallium nitride (GaN), with high breakdown field (3.3 MV/cm), high electron saturation velocity (2.5 × 10^{7} cm/s), and high electron mobility (2000 cm^{2}/V s), is an ideal material for electronics.^{1} Under many device configurations such as in AlGaN/GaN high electron mobility transistors, the temperature of the GaN layer (where electrons are transported through/across) can easily reach above >100 °C.^{2} Among scattering mechanisms limiting the electron mobility (ionized impurity scattering, alloy disorder scattering, interface- or surface-roughness scattering, and acoustic- and optical-phonon scattering^{3–6}), the most prominent one under such elevated temperatures is reported to be the polar-optical-phonon scattering.^{7,8} For this reason, electron–polar-optical-phonon interactions and/or electron mobility and saturation velocity in GaN-based devices have been under intense investigation by theoretical means; these investigations either solely focus on the optical-phonon limited electron transport or take into account other scattering mechanisms as well.^{3,4,7,9,10} The majority of these investigations (with a few exceptions^{11,12}) are done in the framework of the dielectric continuum model of *cubic* crystals, which treats the polar-optical-phonons in wurtzite crystals as isotropic, assuming that the anisotropy emerging from the wurtzite model is small enough to be ignored.^{13} However, such models fundamentally ignore the inherent uniaxial properties of wurtzite GaN. Thus, a theory providing an understanding of the theoretical limits of the uniaxial electronic properties of GaN materials is essential for enabling advanced devices such as vertical GaN devices.^{14} Another futuristic device where this theory portends utility is the polariton lasers in which the anisotropy of the transverse-optical-phonon modes play a larger role.^{15}

In this work, we develop the anisotropic dielectric continuum model for uniaxial crystals and investigate the electron–polar-optical-phonon interactions in bulk wurtzite GaN considering its angular variance with respect to the *c*-axis. Although we assume bulk optical-phonons in our calculations, we do not restrict our discussions to bulk materials. In two-dimensional systems, based on the dielectric continuum model, there originally exist two types of phonon modes: the half space modes and interface modes. Electrons traversing in the two-dimensional channel formed at the heterointerface of the AlGaN/GaN experience scattering processes with both types of phonons. However, it has been shown that the sum of the two form factors associated with these modes is equal to the form factor for bulk optical-phonons.^{16} Hence, the results obtained in this work may be applied to electron–phonon interactions in two-dimensional channels as well.

## II. ELECTRON–PHONON INTERACTION IN BULK WURTZITE CRYSTALS

The dielectric continuum model of uniaxial wurtzite crystals provides a formalism that can be used to calculate properties that have a dependency on the angle between the phonon wave vector **q** and the *c*-axis, denoted here as *θ* (Fig. 1 inset). This dependency stems from the solution of Loudon's model^{17} for frequencies of extraordinary phonons, which is expressed as^{13}

where $\omega L$ and $\omega T$ are longitudinal-optical (LO) and transverse-optical (TO) phonon frequencies with the subscripts *z* and $\u22a5$, indicating directions parallel and perpendicular to the *c*-axis, respectively. These phonon modes $\omega LO$ and $\omega TO$ are purely LO and TO modes, respectively, only when the angle is $\theta =0$ or *π*/2. For intermediate angles between *θ* = 0 and *π*/2, the modes are mixed, and for this reason, we refer to $\omega LO$ as longitudinal-optical-like (LO-like) phonon mode frequency and $\omega TO$ as transverse-optical-like (TO-like) phonon mode frequency. In the case of cubic crystals, the *θ* dependence is completely absent because of the condition $\omega zL=\omega \u22a5L$ and $\omega zT=\omega \u22a5T$, which simplifies Eqs. (1) and (2) to $\omega LO=\omega zL$ and $\omega TO=\omega zT$, respectively. Hence, cubic crystals are referred to as isotropic materials, whereas wurtzite crystals are referred to as uniaxial materials since their properties must be specified considering the angle with respect to the *c*-axis. In addition, due to this mixing of the two phonon modes in wurtzite crystals, the interaction between electrons and the TO-like phonon modes cannot be ignored as in the cubic crystals. Pure TO phonon modes only produce displacements of oppositely charged planes such that their normal distance between each other is fixed; that is, the charged planes only *slide by* each other, resulting in negligible contributions to the electron–polar-optical-phonon interaction. In contrast, pure LO phonon modes produce displacements of oppositely charged planes such that the normal distance between the planes is varied, making them the dominant source of the electron–polar-optical-phonon interaction. As the TO-like phonon mode in wurtzite crystals is only purely TO mode when *θ* = 0 and *π*/2, the effect of phonon modes with intermediate angles on the electron-phonon interaction must be considered.

To fully understand the electron–polar-optical-phonon interactions and evaluate electronic properties for not only room temperature but also higher temperatures, we implement the effect of temperature on phonon mode frequencies in our calculations by using temperature-dependent phonon mode frequencies.^{18} The general expression is given by

where $\omega 0$ is the extrapolated phonon frequency at 0 K, *T* is the temperature, and *α* and *β* are the first- and second-order temperature coefficients, respectively. GaN material constants including the phonon mode frequencies at temperatures of 300 and 800 K are listed in Table I. In this work, we limit the maximum temperature to 800 K as GaN devices are reported to show degraded characteristics when the operation temperature exceeds this limit (∼500 °C).^{19} The temperature-dependent phonon frequencies are calculated using Eq. (3) with the experimentally extracted fitting parameters.^{18} For the optical dielectric constant $\epsilon \u221e$, the temperature dependency is derived based on the temperature-dependent refractive index *n* measurement.^{20} Using the relationship between the two constants $\epsilon \u221e=n2$, we obtain

where *γ* is a fitting parameter set as $\gamma =4.334$ to match the room temperature value $\epsilon \u221e=5.35$ (Ref. 21) and assume that the optical dielectric constant is isotropic,^{17} i.e., $\epsilon \u221e=\epsilon z\u221e=\epsilon \u22a5\u221e$. The static dielectric constants are calculated so that they satisfy the Lyddane-Sachs-Teller relation: $\epsilon z0=\epsilon z\u221e\omega zL2/\omega zT2$ and $\epsilon \u22a50=\epsilon \u22a5\u221e\omega \u22a5L2/\omega \u22a5T2$.

Material constants . | Symbols . | at 300 K . | at 800 K . |
---|---|---|---|

A_{1}(LO) phonon frequency (cm^{−1}) | $\omega zL$ | 735 | 720 |

A_{1}(TO) phonon frequency (cm^{−1}) | $\omega zT$ | 534 | 527 |

E_{1}(LO) phonon frequency (cm^{−1}) | $\omega \u22a5L$ | 743 | 727 |

E_{1}(TO) phonon frequency (cm^{−1}) | $\omega \u22a5T$ | 561 | 553 |

Static dielectric constant along the z-axis | $\epsilon z0$ | 10.12 | 10.02 |

Static dielectric constant perpendicular to z-axis | $\epsilon \u22a50$ | 9.38 | 9.30 |

Optical dielectric constant along the z-axis | $\epsilon z\u221e$ | 5.35 | 5.38 |

Optical dielectric constant perpendicular to z-axis | $\epsilon \u22a5\u221e$ | 5.35 | 5.38 |

Electron conduction effective mass | $m*$ | $0.22m0$ | |

Electron density of states (DOS) effective mass | $mDOS*$ | $1.5m0$ |

Material constants . | Symbols . | at 300 K . | at 800 K . |
---|---|---|---|

A_{1}(LO) phonon frequency (cm^{−1}) | $\omega zL$ | 735 | 720 |

A_{1}(TO) phonon frequency (cm^{−1}) | $\omega zT$ | 534 | 527 |

E_{1}(LO) phonon frequency (cm^{−1}) | $\omega \u22a5L$ | 743 | 727 |

E_{1}(TO) phonon frequency (cm^{−1}) | $\omega \u22a5T$ | 561 | 553 |

Static dielectric constant along the z-axis | $\epsilon z0$ | 10.12 | 10.02 |

Static dielectric constant perpendicular to z-axis | $\epsilon \u22a50$ | 9.38 | 9.30 |

Optical dielectric constant along the z-axis | $\epsilon z\u221e$ | 5.35 | 5.38 |

Optical dielectric constant perpendicular to z-axis | $\epsilon \u22a5\u221e$ | 5.35 | 5.38 |

Electron conduction effective mass | $m*$ | $0.22m0$ | |

Electron density of states (DOS) effective mass | $mDOS*$ | $1.5m0$ |

The LO-like phonon mode frequency $\omega LO$ and the TO-like phonon mode frequency $\omega TO$ and their corresponding phonon energies as a function of the angle *θ* for two different temperatures *T* = 300 K (solid lines) and 800 K (dashed lines) are calculated using Eqs. (1) and (2) and are shown in Fig. 1. For both temperatures, the LO-like phonon mode frequency shows a weak variation upon the angle, whereas the TO-like phonon mode frequency exhibits some anisotropy. For *T* = 300 K, the minimum LO-like phonon frequency is obtained at *θ *= 0 as $\omega LO=\omega zL=735\u2009cm\u22121$ and corresponds to the LO phonon emission threshold energy of electron, 91.1 meV. On the other hand, the minimum TO-like phonon frequency is obtained at *θ *= *π*/2 as $\omega TO=\omega \u22a5T=534\u2009cm\u22121$ and corresponds to the TO phonon emission threshold energy of electron, 66.2 meV. For *T* = 800 K, LO and TO phonon emission threshold energy decrease to 89.3 and 65.3 meV, respectively.

To understand how these angle *θ* variant optical-phonons interact with electrons, we use the Fröhlich interaction Hamiltonian to describe the electric polarization (i.e., the relative displacement of positively and negatively charged ions) produced by the LO-like and TO-like polar-optical-phonons as^{22}

where *e* is the elementary electron charge, *V* is the crystal volume, and $a\u2212q*$ and $aq$ are the creation and annihilation operators.

From this Fröhlich interaction Hamiltonian, the transition matrix element $Mq$ may be written as

where $nph$ is the phonon occupation number given by

Here, plane-wave electron states normalized in volume *V* are used to derive $Mq$.

The probability of polar-optical-phonon induced electron scattering from electron state **k** to **k ′** per unit time, $W(k,k\u2032)$, is calculated from the Fermi golden rule as

where $Ek$ and $Ek\u2032$ are the initial and final electron kinetic energy, respectively, and $\u210f\omega $ is the transferred phonon energy with the upper sign “+” (the lower sign “–”) corresponding to phonon emission (absorption). The electron kinetic energies are calculated assuming parabolic effective mass, i.e., $Ek=\u210f2k2/2m*$, where $m*=0.22m0$ is the electron conduction effective mass and $m0$ is the electron mass.

Then by plugging Eq. (6) into Eq. (8) and summing over all of the final electron states **k′**, the total electron scattering rate due to the Fröhlich interaction can be obtained as^{23}

where *m** is the electron effective mass and *ϕ′* is the angle between the initial electron wave vector **k** and the phonon wave vector **q**, hence yielding $cos\u2009\varphi \u2032=sin\u2009\theta \u2009sin\u2009\theta k\u2009cos\u2009\varphi +cos\u2009\theta \u2009cos\u2009\theta k$ such that $\theta k$ is the angle between **k** and the *c*-axis, and *ϕ* is the azimuthal angle between the wave vectors **k** and **q**. For the case of emission, *σ* is given as

and for absorption, *σ* = 1. Here, a screening factor of $E0=3\xd710\u221211\u2009eV1/2$ is included to facilitate efficient numerical integration in the case of emission when $Ek\u2009cos2\varphi \u2032\u2248\u210f\omega $. The wave vectors and their angles with respect to the *c*-axis are depicted in the inset of Fig. 1. We set the coordinate system so that the initial electron wave vector is on the *y*–*z* plane with the *z*-axis aligned to the optical axis (*c*-axis). As long as the z-axis is aligned to the c-axis, the calculation results are invariant to how we set the *x*- and *y*-axes in the uniaxial model frame.

The electron–optical-phonon scattering rates are numerically calculated using Eq. (9) and are plotted in Fig. 2 for the two cases of (a) *T* = 300 K and (b) 800 K. The scattering rates due to the emission and absorption of LO-like and TO-like phonons are calculated separately as a function of incident electron energy $Ek$ to show the contribution of each scattering process. The scattering rates for different electron incident angles $\theta k=0$ (solid line), *π*/4 (dashed line), and *π*/2 (dotted line) are also shown together. The abrupt increase in the emission scattering rates occurs around the minimum LO-like and TO-like phonon energies, 91.1 and 66.2 meV at 300 K, and 89.3 and 65.3 meV at 800 K, respectively. For both cases of *T* = 300 and 800 K, the LO-like phonon mode scattering shows a very weak $\theta k$ angle variance. The largest $\theta k$ angle variance is exhibited at the energy of onset of TO-like phonon emission (66.2 meV) as $\theta k$ varies from *π*/4 to *π*/2; though the results may differ depending on the numerical integration method or the value of the screening factor, according to our calculations, the TO-like phonon emission scattering rate of $\theta k$= *π*/4 is more than ten times larger than that of $\theta k$= *π*/2.

To make a clearer comparison of the extent of anisotropy of LO-like and TO-like mode phonon scattering rates, we plot them as a function of incident angle $\theta k$ for a fixed energy $Ek=0.1\u2009eV$ with *T* = 300 and 800 K in Fig. 3. The absorption rates for both LO-like and TO-like modes show almost no anisotropy. The largest anisotropy is between the TO-like emission of $\theta k=39\u2218$ and *π*/2 at 300 K; the rate at $\theta k=39\u2218$ is 2.5 times larger than the rate at $\theta k=\pi /2$. Furthermore, at this electron energy, we see that the effect of increased temperature is quite significant. Combined with the discussion of Fig. 2, the results of Fig. 3 demonstrate that for elevated temperatures, there is a large discrepancy between the emission threshold energy and the energy at which emission rates start to dominate over absorption rates. For example, at T = 300 K, the LO-like emission scattering rate starts to exceed the LO-like absorption scattering rate around 93 meV which is only 2 meV higher than the emission threshold energy (91.1 meV). On the other hand, at T = 800 K, this inversion occurs around 115 meV which is considerably higher than the emission threshold energy of 89.3 meV. Therefore, at higher temperatures, careful consideration must be given when assuming the onset energy of optical-phonon emission is equal to the optical-phonon energy itself.

The momentum relaxation of electrons in bulk wurtzite GaN is investigated by using the approach of weighing the scattering rate by the appropriate increase or decrease in momentum.^{10,24} That is, the absorption of a phonon traveling at an angle $\varphi \u2032$ to **k** contributes to a fractional increase of momentum of $(q/k)\u2009cos\u2009\varphi \u2032$ in the direction of **k**. On the other hand, the emission of a phonon contributes to a fractional decrease of momentum of $(q/k)\u2009cos\u2009\varphi \u2032$ in the direction of **k**. Thus, the momentum relaxation rate due to emission (upper signs) and absorption (lower signs) of optical-phonons may be expressed as

Based on Eq. (4), we obtain momentum relaxation rates due to phonon emission $1/\tau me$ and absorption $1/\tau ma$:

The momentum relaxation rates due to emission and absorption of optical-phonons are numerically calculated using Eqs. (12) and (13), respectively, and their absolute values as a function of electron initial energy $Ek$ are plotted in Fig. 4(a) for temperature *T* = 300 K. The LO-like and TO-like portion of phonon emission and absorption process are calculated separately to see the contribution of each component. The incident electron angles are set to $\theta k=0$ (solid line), *π*/4 (dashed line), and *π*/2 (dotted line). The momentum relaxation rates due to LO-like modes are weakly dependent on the angle. In contrast, the rates due to TO-like modes show some anisotropy, especially near the emission threshold energy of $Ek=66.2\u2009meV$. The LO-like and TO-like momentum relaxation rates show more than two orders of magnitude difference. As mobility is proportional to the momentum relaxation time $\tau m$, the electron mobility is principally determined by interactions with LO-like phonons and the effect of the anisotropy shown in the relaxation rates with TO-like phonons is to be very small.

Through every optical-phonon scattering event an electron undergoes, the energy of the electron changes. The Dirac delta function in Eq. (8) assures that after going through a scattering event, the energy of the final electron state increases if the event was a phonon absorption process and decreases if it was a phonon emission. This energy change per unit time of electron can be directly associated with the saturation velocity as will be seen in Sec. III. To calculate the net change of energy per unit time, we subtract the power loss due to phonon emission from the power gain due to phonon absorption as

The numerical calculation results of Eq. (14) are shown in Fig. 4(b). The temperature is set to 300 K, and the incident electron angles are set to $\theta k=0$ (solid line), *π*/4 (dashed line), and *π*/2 (dotted line). The LO-like and TO-like portions of phonon emission and absorption process are calculated separately to see the contribution of each component. Similar to the case of momentum relaxation rates, we see that scattering with the LO-like phonon is the dominant electron energy relaxation process. For electrons with high kinetic energy ($Ek>91.1\u2009meV$), the power dissipated per electron through a scattering event saturates at a value of ∼16 eV/ps.

## III. ELECTRON MOBILITY AND SATURATION VELOCITY

To investigate the temperature- and orientation-dependent electron mobility and saturation velocity, we use the standard mobility–momentum relaxation time relation and the electron energy balance equation.^{25} In the literature, a two-step model,^{26,27} which treats the absorption and almost immediate emission of a polar-optical-phonon as a single elastic process, is used for calculating mobility in GaN. However, in this work, we stick to the momentum relaxation time approximation which we find reasonably accurate while at the same time satisfying our purpose of illustrating the anisotropic electron–optical-phonon interaction.

The electron mobility is given by

The averaged electron momentum relaxation time over the electron distribution $\u3008\tau m\u3009$ is given by

where $g3D(Ek)=(2mDOS*)3/22\pi 2\u210f3Ek$ is the three-dimensional density of states (DOS), $mDOS*$ is the electron density of states effective mass which is set to $mDOS*=1.5m0$, $fFD(Ek)=[expEk\u2212EFkBT+1]\u22121$ is the Fermi-Dirac distribution, $EF$ is the Fermi energy, and *N* is the spatial electron density. The momentum relaxation times for each of the four modes (LO-like emission, LO-like absorption, TO-like emission, and TO-like absorption) are calculated separately and used to derive the polar-optical-phonon limited mobility by applying Matthiessen's rule. The drift velocity is then obtained from $vd=\mu F$, where *F* is the electric field. In this work, because we only consider electron scattering with polar-optical-phonons, the electron concentration only affects the number of electrons considered over the electron distribution in the numerical calculation. Detailed description of models that account for electron-density-dependent screening factors, electron-electron scattering or nonparabolicity of the electron dispersion relation can be found elsewhere.^{28,29}

In order to calculate the saturation velocity, we relate the mobility to the power dissipated by optical-phonon scattering. The net electron kinetic energy increase per second is the difference between the power gained from the electric field and the power loss from the scattering. When this net energy increase per unit time becomes zero, the drift velocity saturates to the saturation velocity. Thus, we may write as

where $Fsat$ is the electric field when the net power increase is equal to zero. From Fig. 4(b), we see that the net energy loss per second is a function of electron kinetic energy $Ek$. Hence, after we obtain the saturation velocity as a function of kinetic energy, we average the saturation velocity over the electron distribution using

To predict the theoretical limit of electronic transport properties for different incident angles at elevated temperatures, we calculate the electron mobility and saturation velocity using Eqs. (15) and (18) with the momentum relaxation rate and the rate of energy change obtained from Sec. II. Figure 5 shows the (a) electron mobility *μ* and (b) saturation velocity $vsat$ as a function of temperature. The incident electron angles with respect to the c-axis are set to $\theta k=0$ (solid line), π/4 (dashed line), and π/2 (dotted line). The spatial electron concentration is fixed to *N* = 5 × 10^{18} cm^{−3} in the temperature-dependent electron mobility and saturation velocity calculations. The maximum mobility for a given temperature is obtained when the incident angle is $\theta k=\pi /2$ and mobility decreases approximately ∼5% as the angle varies to $\theta k=0$. For all incident angles, the mobility rapidly drops from ∼3100 cm^{2}/V s to ∼200 cm^{2}/V s between 300 and 800 K, falling with a power law of *T*^{−3.1}. The saturation velocity is the smallest for the angle $\theta k=\pi /2$ and largest for $\theta k=0$ with a discrepancy of ∼2%. For both mobility and saturation velocity, the angle dependence is very weak, showing that the TO-like scattering anisotropy has a little impact on the macroscopic electronic properties.

The electron mobility for higher electron concentrations is also calculated for an angle set as $\theta k=\pi /2$ and is plotted in the inset of Fig. 5(a). For all temperatures, the mobility decreases as the concentration increases. The effect of temperature increase on electron–phonon scattering rates reveals itself through the increased phonon occupation number, the decreased phonon frequencies (as shown in Fig. 2), and the dispersion of the Fermi-Dirac distribution. At a fixed concentration, as the temperature increases, more electrons with higher kinetic energy participate in the conduction; that is, electrons that are more likely to emit phonons than to absorb phonons contribute more to the average mobility. Thus, the mobility drops with the temperature increase. Apparently, the increase of phonon absorption rate associated with the increase of temperature is not enough to compensate the effect of the increase in the number of electrons with higher energy.

## IV. CONCLUSION

In conclusion, we have developed a theory of scattering rates, momentum relaxation rates, and rate of energy change due to electron interaction with polar-optical-phonons in bulk wurtzite GaN based on the dielectric continuum model and the uniaxial model. These rates are calculated numerically and are used to extract temperature and orientation dependent electron mobility and saturation velocity. The results show that the electron–optical-phonon interactions rely on the electron incident angle more in TO-like phonon modes than in LO-like phonon modes and that LO-like phonon modes are mostly independent of angle variance. The polar-optical-phonon limited mobility and the saturation velocity show a 5% variance with the orientation mostly due to the anisotropy in LO-like phonon scattering rates. We report that the macroscopic electronic properties such as the mobility and saturation velocity are predominantly determined by LO-like phonon interactions. The strong anisotropy in TO-like phonon scattering rates can be ignored in these calculations because of the two orders of magnitude difference compared to LO-like phonon scattering rates.

## ACKNOWLEDGMENTS

Part of this work, partially supported by the Air Force Office of Scientific Research (AFOSR) through Young Investigator Program Grant No. FA9550-16-1-0224, was carried out in the Micro and Nanotechnology Laboratory, University of Illinois at Urbana-Champaign, IL, USA and part of this work, partially supported by Air Force Office of Scientific Research (AFOSR) Grant No. FA9550-16-1-0227, was carried out at the University of Illinois at Chicago, IL, USA.