1.
Introduction

Theoretical calculations^[1–3] and experimental measurements^[4–6] of the carrier transport characteristics depict that wurtzite phase gallium nitride (Wz-GaN) is a potential candidate for high power^[4], high frequency^[7] and optoelectronic^[8] semiconductor devices. Wide band-gap and other suitable material parameters of Wz-GaN (E_g = 3.40 eV at 300 K) make it a promising material to be the base semiconductors of ultraviolet photodetectors^[8], blue light emitting diodes^[9], high electron mobility transistors^[10], impact avalanche transit time (IMPATT) diodes^[7], etc. Monte Carlo (MC) simulations^[1–3] and experimental measurements^[4–6] of electron drift velocity versus field characteristics carried out by several researchers revealed that the peak electron drift velocity in Wz-GaN^[1–6] lies 2.0 × 10⁵ – 3.0 × 10⁵ m/s at the electric fields of 1.80 × 10⁷ – 2.25 × 10⁷ V/m. However, in high frequency devices like IMPATT diodes, the electric field reaches much higher values (~10⁸ V/m) within the active layer as compared to the field corresponding to the peak electron drift velocity, especially at millimeter-wave (30–300 GHz) and terahertz (0.3–5.0 THz) frequencies^{[7, 11]}. Thus, electron drift velocity is bound to reach the saturation value within the active layers of millimeter-wave and terahertz IMPATT diodes based on Wz-GaN^{[7, 11]}, which operate under the avalanche breakdown condition. But, so far as the author’s knowledge in concerned, no theoretical or experimental report is available in published literature predicting or measuring the saturation value of electron drift velocity in Wz-GaN under the condition of impact ionization.



In this paper, the author has developed an analytical model of hot-electron transport in Wz-GaN by considering all possible scattering mechanisms such as deformation potential, piezoelectric acoustic phonon scattering, optical phonon scattering, electron-electron scattering and ionizing scattering. The scattering limited saturation drift velocity of electrons in Wz-GaN at high electric fields (> 10⁷ V/m) have been calculated by using the said model and its dependences on temperature and doping density have been investigated. The analytical expression developed in this paper will be very useful for the modelling and simulation of some important semiconductor devices that operate under avalanche breakdown conditions such as IMPATT diodes, avalanche photodiodes, etc. The said analytical expression will be specifically useful for the simulation of the above-mentioned devices via the drift-diffusion or quantum-corrected drift–diffusion models reported earlier by the author^[11–18]. The details of the theoretical model developed in the present work have been discussed in the next section. The numerical results and comparison of those with results obtained from MC simulations and experimental measurements carried out by other researchers are presented in Section 3. Finally, the paper is concluded in Section 4.

2.
The theoretical model

At high electric field, under the avalanche breakdown condition, the mobile electrons in Wz-GaN undergo deformation potential acoustic phonon scattering, piezoelectric acoustic phonon scattering, optical phonon scattering, electron-electron scattering, in addition to the impact ionizing scattering^{[19, 20]}. The threshold energy and mean free path associated with the above-mentioned collision events are assumed to be $E_{left( {
m {dfp}}
ight)}^{left( {
m e}
ight)}$, $E_{left( {
m {pz}}
ight)}^{left( {
m e}
ight)}$, $E_{left( {
m {op}}
ight)}^{left( {
m e}
ight)}$, $E_{left({
m {ee}}
ight)}^{left( {
m e}
ight)}$, $E_{left( {
m i}
ight)}^{left( {
m e}
ight)}$ and $l_{left( {
m {dfp}}
ight)}^{left( {
m e}
ight)}$, $l_{left( {
m {pz}}
ight)}^{left( {
m e}
ight)}$, $l_{left({
m {op}}
ight)}^{left( {
m e}
ight)}$, $l_{left( {
m {ee}}
ight)}^{left( {
m e}
ight)}$, $l_{left({
m i}
ight)}^{left( {
m e}
ight)}$ respectively; here the subscripts dfp, pz, op, ee and i stand for potential acoustic phonon collision, piezoelectric acoustic phonon collision, optical phonon collision, electron– electron collision and ionizing collision respectively, and the superscript e stands for the electrons (e.g. $E_{left( i
ight)}^{left({
m e}
ight)}$ and $l_{left( i
ight)}^{left( {
m e}
ight)}$ stand for the threshold energy and mean free path associated with the ionizing collisions). The inter-valley scattering is another significant scattering mechanism in group III–V semiconductors at low electric fields; however, at the field range of impact ionization under consideration, all the conducting electrons must lie in the higher energy satellite valley of the conduction band which lead to the probability of inter-valley scattering vanishing.



Taking into consideration the above-mentioned five types of scattering mechanisms (except inter-valley scattering), the energy-balance equation at the dynamic equilibrium can be formed, simply, the energy-balance equation at this condition must depict the fact that the energy acquired by an electron per unit time from the externally applied electric field should be equal to the total energy loss per unit time due to the aforementioned collision events. It can be written as^{[19, 20]}

$${v_{ m e}}leftlangle {frac{{{ m d}E}}{{{ m d}x}}} ight angle _{left( xi ight)}^{left( { m e} ight)} = sumlimits_{z in R} {,left( {{v_z}leftlangle {frac{{{ m d}E}}{{{ m d}x}}} ight angle _z^{left( { m e} ight)}} ight)} ,$$

(1)

where $R = { {
m {(dfp),(pz),(op),(ee),(i)}}} $, v_e is the average electron drift velocity, $leftlangle {{
m d}E/{
m d}x}
ight
angle _{left( xi
ight)}^{left( {
m e}
ight)}$ is the average energy acquired by an electron per unit length from the applied field (ξ), which can be simply written as

$$leftlangle {frac{{{ m d}E}}{{{ m d}x}}} ight angle _{left( xi ight)}^{left( { m e} ight)} = qxi ,$$

(2)

where q = 1.6 × 10^?19 C is the magnitude of the charge of an electron, $leftlangle {{
m d}E/{
m d}x}
ight
angle _{left( z
ight)}^{left( {
m e}
ight)}$ is the average energy loss per unit length by an electron due to any of the aforementioned five collision events denoted by a $z $-type collision event, where $z in R$, v_z is the final velocity of an electron just before undergoing the $z $-type collision event.



The author has already formulated the expressions of average energy loss per unit length by an electron due to potential acoustic phonon scattering, piezoelectric acoustic phonon scattering, optical phonon scattering, electron–electron scattering and ionizing scattering in his earlier work^[20]. Those are given by^[20]

$$leftlangle {frac{{{text{d}}E}}{{{text{d}}x}}} ight angle _{left( {{text{dfp}}} ight)}^{left( {text{e}} ight)} = left( {frac{{4{E_0}{{left( {m_{text{e}}^{text{*}}} ight)}^3}{k_{text{B}}}T}}{{pi { ho _{text{v}}}{hbar ^4}E_{left( {text{i}} ight)}^{left( {text{e}} ight)}}}} ight)left[ begin{gathered} E_{left( {text{i}} ight)}^{left( {text{e}} ight)}left{ {1 - ln left( {1 + exp left( {frac{{E_{left( {text{i}} ight)}^{left( {text{e}} ight)} - E_{text{F}}^{left( {text{e}} ight)}}}{{{k_{text{B}}}T}}} ight)} ight)} ight} + {k_{text{B}}}Tln left( {frac{{1 + exp left( {frac{{E_{text{F}}^{left( {text{e}} ight)} - E_{left( {text{i}} ight)}^{left( {text{e}} ight)}}}{{{k_{text{B}}}T}}} ight)}}{{1 + exp left( {frac{{E_{left( {text{i}} ight)}^{left( {text{e}} ight)}}}{{{k_{text{B}}}T}}} ight)}}} ight) hfill - {k_{text{B}}}Tleft{ {L{i_2}left( { - exp left( {frac{{E_{left( {text{i}} ight)}^{left( {text{e}} ight)} - E_{text{F}}^{left( {text{e}} ight)}}}{{{k_{text{B}}}T}}} ight)} ight) - L{i_2}left( { - exp left( {frac{{ - E_{text{F}}^{left( {text{e}} ight)}}}{{{k_{text{B}}}T}}} ight)} ight)} ight} hfill end{gathered} ight],$$

(3)

$$leftlangle {frac{{{text{d}}E}}{{dx}}} ight angle _{left( {{text{pz}}} ight)}^{left( {text{e}} ight)} = left( {frac{{{k_{text{B}}}T}}{{pi {varepsilon _{text{r}}},E_{left( {text{i}} ight)}^{left( {text{e}} ight)}}}} ight){left( {frac{{q{k_{text{m}}}m_{text{e}}^{text{*}}{u_0}}}{hbar }} ight)^2}left[ begin{gathered} E_{left( {text{i}} ight)}^{left( {text{e}} ight)}left{ {1 - ln left( {1 + exp left( {frac{{E_{left( {text{i}} ight)}^{left( {text{e}} ight)} - E_{text{F}}^{left( {text{e}} ight)}}}{{{k_{text{B}}}T}}} ight)} ight)} ight} + {k_{text{B}}}Tln left( {frac{{1 + exp left( {frac{{E_{text{F}}^{left( {text{e}} ight)} - E_{left( {text{i}} ight)}^{left( {text{e}} ight)}}}{{{k_{text{B}}}T}}} ight)}}{{1 + exp left( {frac{{E_{text{F}}^{left( {text{e}} ight)}}}{{{k_{text{B}}}T}}} ight)}}} ight) hfill - ,, {k_{text{B}}}Tleft{ {L{i_2}left( { - exp left( {frac{{E_{left( {text{i}} ight)}^{left( {text{e}} ight)} - E_{text{F}}^{left( {text{e}} ight)}}}{{{k_{text{B}}}T}}} ight)} ight) - L{i_2}left( { - exp left( {frac{{ - E_{text{F}}^{left( {text{e}} ight)}}}{{{k_{text{B}}}T}}} ight)} ight)} ight} hfill end{gathered} ight].$$

(4)

$$leftlangle {frac{{{ m d}E}}{{ m dx}}} ight angle _{left( { m {op}} ight)}^{left( { m e} ight)} = left( {frac{{E_{left( { m {op}} ight)}^{left( { m e} ight)}}}{{l_{left( { m {op}} ight)}^{left( { m e} ight)}}}} ight),$$

(5)

$$leftlangle {frac{{{ m d}E}}{{{ m d}x}}} ight angle _{left( { m {ee}} ight)}^{left( { m e} ight)} = left( {frac{{4pi ,{q^4},n}}{{varepsilon _0^2varepsilon _r^2E_{left({ m i} ight)}^{left( { m e} ight)}}}} ight)ln left( {frac{{E_{left( { m i} ight)}^{left( { m e} ight)}}}{{E_{left( { m {op}} ight)}^{left( { m e} ight)}}}} ight),$$

(6)

$$leftlangle {frac{{{ m d}E}}{{{ m d}x}}} ight angle _{left( { m i} ight)}^{left( { m e} ight)} = left( {frac{{E_{left( { m i} ight)}^{left( { m e} ight)}}}{{l_{left( { m i} ight)}^{left( { m e} ight)}}}} ight),$$

(7)

where $m_{
m e}^*$ is the electron effective mass, ${u_0}$ is the average acoustic velocity, k_B = 1.38 × 10^?23 J/K is the Boltzmann constant, T is temperature in Kelvin (K), E₀ is the effective deformation potential constant, ${
ho _v}$ is the density of the material, $hbar = h/2pi $ is the normalized Planck’s constant (h = 6.62 × 10^?34 J·s), $E_{
m F}^{left( {
m e}
ight)}$ is the quasi Fermi energy level, k_m is the piezoelectric coupling constant, ${varepsilon _{
m r}}$ is the static dielectric constant of the material, ${varepsilon _0}$ = 8.85 × 10^?12 F/m is the permittivity of the vacuum and n is the electron concentration. In Eqs. (3) and (4) $L{i_2}left( x
ight) = intlimits_x^0 {left[ {ln left( {1 - {x'}}
ight)/{x'}}
ight],{
m d}{x'}} $ is the dilogarithm of the variable $z $. It can also be written in summation form as $L{i_2}left( x
ight) = sumlimits_{s = 1}^infty {left( {{x^s}/{s^2}}
ight)} $.



Now, the energy-balance equation in Eq. (1) can be rewritten as

$$qxi {v_{ m e}} = left( {frac{{E_{left({ m {op}} ight)}^{left( { m e} ight)}}}{{l_{left( { m {op}} ight)}^{left( { m e} ight)}}}} ight){v_r} + left{ {left( {frac{{E_{left( { m i} ight)}^{left( { m e} ight)}}}{{l_{left( { m i} ight)}^{left( { m e} ight)}}}} ight) + sumlimits_{y in R - left{ {left( {op} ight),left( i ight)} ight}} {left( {leftlangle {frac{{{ m d}E}}{{{ m d}x}}} ight angle _y^{left( { m e} ight)}} ight)} } ight}{v_{ m e}},$$

(8)

where v_r is the average random electron velocity. The expressions of v_e and v_r by considering a Maxwellian distribution with an effective electron temperature T_e, can be written as^[21]

$${v_{ m r}} = {left( {frac{{8{k_{ m B}}{T_{ m e}}}}{{text{π} ,m_{ m e}^*}}} ight)^{frac{1}{2}}},$$

(9)

$${v_{ m e}} = {left( {frac{2}{{pi ,m_{ m e}^*,{k_{ m B}}{T_{ m e}}}}} ight)^{frac{1}{2}}}left( {frac{{2,q,l_{ m {eff}}^{left( { m e} ight)},xi }}{3}} ight),$$

(10)

where

$$l_{ m {eff}}^{left( c ight)} = {left{ {sumlimits_{z in R} {{{left( {l_z^{left( { m e} ight)}} ight)}^{ - 1}}} } ight}^{ - 1}},$$

(11)

is the effective mean free path of electrons that have undergone all five possible types of scattering events under consideration. The term T_e can be eliminated from Eqs. (9) and (10) and one can obtain

$${v_{ m r}}{v_{ m e}} = left( {frac{{8,q,l_{ m {eff}}^{left( { m e} ight)},xi }}{{3pi ,m_{ m e}^*}}} ight).$$

(12)

The square of the electron drift velocity can be obtained from Eq. (8) as

$$v_{ m e}^2 = frac{{E_{left( { m {op}} ight)}^{left( { m e} ight)}{v_{ m r}}{v_{ m e}}}}{{l_{left( { m {op}} ight)}^{left( { m e} ight)}left[ {qxi - left{ {left( {frac{{E_{left( { m i} ight)}^{left( { m e} ight)}}}{{l_{left({ m i} ight)}^{left( { m e} ight)}}}} ight) + sumlimits_{y in R - left{ {left( {op} ight),left( i ight)} ight}} {left( {leftlangle {frac{{{ m d}E}}{{{ m d}x}}} ight angle _y^{left( { m e} ight)}} ight)} } ight}} ight]}}.$$

(13)

Now the factor (v_ev_r) in Eq. (13) can be substituted from Eq. (12) in order to obtain the final expression of electron drift velocity under the condition of impact ionization. It is given by

$$begin{split} {v_{ m e}} = & left( {frac{{E_{left( { m {op}} ight)}^{left( { m e} ight)}}}{{3pi ,m_{ m e}^*}}} ight)left[ {left{ {1 + left( {l_{left( { m {op}} ight)}^{left( { m e} ight)}sumlimits_{w in R - left{ {left( {op} ight)} ight}} {{{left( {l_w^{left( { m e} ight)}} ight)}^{ - 1}}} } ight)} ight}} ight. hfill & times {left. {left{ {1 - left( {frac{1}{{qxi }}} ight)left( {left( {frac{{E_{left( i ight)}^{left( { m e} ight)}}}{{l_{left( { m i} ight)}^{left( { m e} ight)}}}} ight) + sumlimits_{y in R - left{ {left( {op} ight),left( i ight)} ight}} {left( {leftlangle {frac{{{ m d}E}}{{{ m d}x}}} ight angle _y^{left( { m e} ight)}} ight)} } ight)} ight}} ight]^{ - frac{1}{2}}}. hfill end{split} $$

(14)

3.
Numerical results and discussion

The hot-electron drift velocity in Wz-GaN has been calculated as a function of electric field considering different material parameters of Wz-GaN taken from theoretical calculations and experimental measurements already reported^[22–25]. The mean free path of ionizing collisions ($l_{left( {
m i}
ight)}^{left( {
m e}
ight)}$) of electrons is taken as the inverse of the ionization rate of electrons in Wz-GaN ($alpha _{left( {
m i}
ight)}^{left({
m e}
ight)}$) which was calculated earlier by the author by using a multistage model developed by him^[20]. Since $alpha _{left( {
m i}
ight)}^{left( {
m e}
ight)}$ are functions of electric field, temperature and doping concentration, therefore $l_{left( {
m i}
ight)}^{left( {
m e}
ight)}$ also becomes the functions of those. The mean free path of optical phonon scattering ($l_{left( {
m {op}}
ight)}^{left( {
m e}
ight)}$) of electrons has been taken from published literature^[22]; it is also a function of temperature. The mean free path associated with the electron–electron collision ($l_{left({
m {ee}}
ight)}^{left( {
m e}
ight)}$) is a strong function of electron concentration; it is taken to be $l_{left({
m {ee}}
ight)}^{left( {
m e}
ight)} = {(G)^c}{n^{ - 1/3}}$; where G = 0.75 and c = 3.12 are the dimensionless fitting parameters already estimated by the author in an earlier work^[26]. All the threshold energies and mean free paths associated with deformation potential and piezoelectric acoustic phonon scattering as functions of temperature have been taken from the published literature^[22]. The hot electron drift velocity in Wz-GaN is calculated for the field range of 2.0 × 10⁷–1.0 × 10⁹ V/m, which is well above the electric field associated with the peak electron velocity in Wz-GaN reported by several researchers^[1–6]. This high electric field range under consideration ensures the hot electron transport in Wz-GaN under impact ionization condition for which the analytical expression developed in this paper is valid. All the important parameter values for the calculation of hot electron drift velocity in Wz-GaN at 300 K have been given in Table 1.

Type of parameter	Parameter	Value
Material parameter	εr	8.91
	$m_{ m {e}}^*$ (Kg) (m0 = 9.1×10?31 Kg)	0.20m0
	${E_{ m g}}$ (eV)	3.39
	$E_{left( { m {dfp}} ight)}^{left({ m {e}} ight)}$ (meV)	18.41
	$l_{left( { m {dfp}} ight)}^{left({ m {e}} ight)}$ (?)	30.90
	$E_{left( { m {pz}} ight)}^{left({ m {e}} ight)}$ (meV)	15.10
	$l_{left( { m {pz}} ight)}^{left({ m {e}} ight)}$ (?)	22.50
	$E_{left( { m {op}} ight)}^{left({ m {e}} ight)}$ (meV)	91.20
	$l_{left( { m {op}} ight)}^{left({ m {e}} ight)}$ (?)	42.00
	$E_{left( { m {i}} ight)}^{left({ m {e}} ight)}$ (eV)	3.66
	$l_{left( { m {i}} ight)}^{left({ m {e}} ight)}$ (?)	840.00
Dimensionless fitting parameters for calculating $l_{left( { m {ee}} ight)}^{left({ m {e}} ight)}$as a function of electron density	G	0.75
	c	3.12

Table1.
Values of important parameters of Wz-GaN at 300 K^[22–26].



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Type of parameter	Parameter	Value
Material parameter	εr	8.91
	$m_{ m {e}}^*$ (Kg) (m0 = 9.1×10?31 Kg)	0.20m0
	${E_{ m g}}$ (eV)	3.39
	$E_{left( { m {dfp}} ight)}^{left({ m {e}} ight)}$ (meV)	18.41
	$l_{left( { m {dfp}} ight)}^{left({ m {e}} ight)}$ (?)	30.90
	$E_{left( { m {pz}} ight)}^{left({ m {e}} ight)}$ (meV)	15.10
	$l_{left( { m {pz}} ight)}^{left({ m {e}} ight)}$ (?)	22.50
	$E_{left( { m {op}} ight)}^{left({ m {e}} ight)}$ (meV)	91.20
	$l_{left( { m {op}} ight)}^{left({ m {e}} ight)}$ (?)	42.00
	$E_{left( { m {i}} ight)}^{left({ m {e}} ight)}$ (eV)	3.66
	$l_{left( { m {i}} ight)}^{left({ m {e}} ight)}$ (?)	840.00
Dimensionless fitting parameters for calculating $l_{left( { m {ee}} ight)}^{left({ m {e}} ight)}$as a function of electron density	G	0.75
	c	3.12

The electron drift velocity in Wz-GaN versus electric field plots under the condition of impact ionization at room temperature (T = 300 K) for the electron concentration of 10²² m^?3 calculated by using Eq. (14) are shown in Fig. 1 and the results have been compared with the results obtained from MC simulation of Kolnik et al.^[1], Bhapkar et al.^[2] and Binash et al.^[3]. The same calculated results have been compared with the experimentally measured data of Khan et al.^[4], Wraback et al.^[5] and Barker et al.^[6]. It is clear from Figs. 1 and 2 that MC simulations^[1–3] overestimate the electron drift velocity in Wz-GaN as compared to the experimental values of the same within cold (ξ < 10 ⁷ V/m) and warm (10⁷ < ξ < 2.0 × 10 ⁷ V/m) electron transport ranges. Only the MC simulation and experimental data available within the hot (ξ > 2.0 × 10 ⁷ V/m) electron transport region are from Binash et al.^[3] and Khan et al.^[4] respectively. It is interesting to observe from Figs. 1 and 2 that the electron drift velocity values obtained from both the reports of Binash et al.^[3] and Khan et al.^[4] are in very close agreement with the calculated results in this paper within the electric field range of 7.0 × 10⁷–9.0 × 10⁷ V/m, i.e. at very high electric fields above which electron drift velocity saturates to a value of 1.44 × 10⁵ m/s at 300 K for n = 10²² m^?3. However, the values of saturated electron drift velocity obtained by Binash et al.^[3] and Khan et al.^[4] are 1.43 × 10⁵ and 1.56 × 10⁵ m/s respectively.






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Figure1.
Variations of electron drift velocity in Wz-GaN with electric field at 300 K for the electron concentration of 10²² m^?3 obtained from the MC simulations carried out by other researchers^[1–3] as well as from the analytical expression developed in this paper.






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Figure2.
Variations of electron drift velocity in Wz-GaN with electric field at 300 K for the electron concentration of 10²² m^?3 obtained from the experimental measurements carried out by other researchers^[4–6] as well as from the analytical expression developed in this paper.




The temperature dependence of electron drift velocity has been depicted in Fig. 3. Variations of hot electron drift velocity in Wz-GaN with electric field at different temperatures such as 10, 50, 100, 200, 300, 500, and 1000 K for the electron concentration of 10²² m^?3 have been calculated by using Eq. (14) and shown in Fig. 3. Fig. 3 also shows the electron drift velocity versus electric field plots in Wz-GaN obtained from the MC simulations by Bhapkar et al.^[2] (for 77, 150, 300, 500 and 1000 K) and Binash et al.^[3] (for 300, 450 and 600 K) carried out for the electron concentration of 10²² m^?3. It is evident from that figure that the MC simulation by Bhapkar et al.^[2] significantly overestimates the electron drift velocity values at both low, medium and high electric fields; this overestimation may have occurred due to the uncertainties in the values of different parameters used in their simulation. However, the simulated values of saturated drift velocity of hot electrons in Wz-GaN at 300 and 450 K reported by Binash et al.^[3] are very close in to those obtained from the analytical expression given in Eq. (14). It is noteworthy from Fig. 3 that the hot electron drift velocity is a very strong function of temperature; it decreases significantly with the increase of temperature from 10–1000 K.






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Figure3.
(Color online) Variations of electron drift velocity in Wz-GaN with electric field at different temperatures for the electron concentration of 10²² m^?3 obtained from the MC simulations carried out by other researchers^{[2, 3]} as well as from the analytical expression developed in this paper.




At very low temperatures (T ~ 10–20 K), both deformation potential and piezoelectric acoustic phonon scattering events dominate highly; whereas, at higher temperatures (T > 20 K), optical phonon scattering starts to dominate over the above-mentioned acoustic phonon scattering phenomena ^[20]. Fig. 4 shows the variations of average energy loss per unit length by an electron due to deformation potential acoustic phonon scattering, piezoelectric acoustic phonon scattering and optical phonon scattering with temperature; this figure depicts that the energy loss of electrons due to optical phonon scattering starts to dominate over that due to deformation potential and piezoelectric acoustic phonon scattering just over 19.3 and 16.2 K temperatures respectively. It is also noteworthy from Fig. 4 that at room temperature (300 K) and above, the influence of deformation potential and piezoelectric acoustic phonon scattering on the hot electron transport is very small as compared to that of optical phonon scattering. The saturated drift velocities of electron obtained from equation (14) are 1.91 × 10⁵, 1.84 × 10⁵, 1.73 × 10⁵, 1.59 × 10⁵, 1.44 × 10⁵, 1.14 × 10⁵ and 0.77 × 10⁵ m/s at 10, 50, 100, 200, 300, 500 and 1000 K respectively for n = 10²² m^?3; however, those are obtained as 1.43 × 10⁵, 1.17 × 10⁵ and 1.03 × 10⁵ m/s at 300, 450 and 600 K respectively from the MC simulation of Binash et al.^[3]. Thus, within the 300–500 K temperature range, the calculated values and the results reported by Binash et al.^[3] are in very close agreement.






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Figure4.
Variations of average energy loss per unit length by an electron due to deformation potential acoustic phonon scattering, piezoelectric acoustic phonon scattering and optical phonon scattering with temperature.




The dependence of hot electron drift velocity on electron concentration is depicted in Fig. 5. Variations of saturated electron drift velocity (v_e(sat)) in Wz-GaN with temperature for different electron concentrations such as 10²², 10²³, 10²⁴ and 10²⁵ m^?3 at the electric field of 5.0 × 10⁸ V/m have been calculated from Eq. (14) as shown in Fig. 5. At higher electron densities, the average energy loss of electrons due to electron–electron scattering increases sharply, which leads to a decrease in the drift velocity of electrons. Consequently,v_e(sat) decreases with the increase of doping concentration. Moreover, this effect is more pronounced at higher temperatures due to the increase in the energy loss of electrons due to electron–electron collisions as well as ionizing collisions (since$alpha _{left({
m {i}}
ight)}^{left( {
m {e}}
ight)}$ increases with temperature^[20]) at higher temperatures. The nature of the variation of v_e(at) with electron concentration agrees with MC simulation results of Bhapkar et al.^[2]; however, those MC simulated results are highly overestimated as mentioned earlier.






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Figure5.
Variations of electron saturation drift velocity in Wz-GaN with temperature for different electron concentrations at the applied electric field of 5.0 × 10⁸ V/m obtained from the analytical expression developed in this paper.

4.
Conclusion

In this paper, the author has studied the hot electron transport characteristics of Wz-GaN and the dependences of those on both temperature and doping concentration. An analytical expression of electron drift velocity under the condition of impact ionization has been developed by considering all major scattering mechanisms such as deformation potential acoustic phonon scattering, piezoelectric acoustic phonon scattering, optical phonon scattering, electron-electron scattering and ionizing scattering. Numerical calculations show that the electron drift velocity in Wz-GaN saturates at 1.44 × 10⁵ m s^?1 at room temperature for the electron concentration of 10²² m^?3. Results also show that the saturation electron drift velocity varies from 1.91 × 10⁵–0.77 × 10⁵ m/s for the change in temperature within the range of 10–1000 K, for the electron concentration of 10²² m^?3; whereas the same varies from 1.44 × 10⁵–0.91 × 10⁵ m/s at 300 K for the variation in the electron concentration within the range of 10²²–10²⁵ m^?3. The numerically calculated results have been compared with the Monte Carlo simulated results and experimental data reported earlier, and those are found to be in good agreement. The analytical expression developed in this paper will be very useful for the modelling and simulation of some important semiconductor devices operating under avalanche breakdown conditions such as IMPATT diodes, avalanche photodiodes, etc.

Acknowledgment

The Author wishes to thank Cooch Behar Government Engineering College, WB, India, for providing excellent research facilities where the present work was carried out.