# Monte Carlo Calculations of Amplification Spectrum for GaN THz Transit-time Resonance Maser

E. Starikov, P. Shiktorov, V. Gruinskis, L. Reggiani, L. Varani, J.-C.

Vaissière, Jian Zhao

► To cite this version:

E. Starikov, P. Shiktorov, V. Gruinskis, L. Reggiani, L. Varani, et al., Monte Carlo Calculations of Amplification Spectrum for GaN THz Transit-time Resonance Maser. VLSI Design, 2001, Computational Electronics; Papers Presented at The Seventh International Workshop on Computational Electronics, 13 (1-4), pp.37-43. 10.1155/2001/65961. hal-02398104

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# Monte Carlo Calculations of Amplification Spectrum for GaN THz Transit-time Resonance Maser

E. STARIKOV<sup>a,c</sup>, P. SHIKTOROV<sup>a</sup>, V. GRUŽINSKIS<sup>a</sup>, L. REGGIANI<sup>b,*</sup>, L. VARANI<sup>c</sup>, J. C. VAISSIÈRE<sup>c</sup> and JIAN H. ZHAO<sup>d</sup>

We report Monte Carlo calculations of the amplification spectrum of microwave generation in bulk GaN and its dependence on applied electric fields, doping level, lattice temperature, *etc*. The amplification is shown to occur in a wide frequency range of 0.05 to 3 THz with an optimal generation efficiency of about 1~2%.

Keywords: THz generation; GaN; Monte Carlo simulation

## INTRODUCTION

During the last decades significant efforts have been devoted to realize a tunable THz semiconductor radiation source in view of its broad range of applications. In bulk semiconductors, a physical mechanism appropriate to this purpose is the so called optical phonon transit-time resonance (OPTTR) [1]. It consists of the periodic motion of carriers inside the optical phonon sphere of momentum space, resulting from the combined action of carrier quasi-ballistic acceleration by an applied electric field up to the optical phonon energy and the subsequent emission of an optical

phonon, which pushes the carrier back near the sphere center. As a consequence, the motion takes an oscillatory character and a dynamic negative differential mobility (DNDM) can appear at frequencies near the inverse of the transit-time and its harmonics [1]. In standard III-V semiconductors, such as GaAs and InP, the maximum generation frequency was found to be limited in the range 300 to 400 GHz [2]. By contrast, for the wide-gap materials, such as GaN, InN, SiC, *etc.*, one can expect a considerable increase of the maximum generation frequency and a general improvement of the conditions for the DNDM to occur, as a result of a higher value of the optical

<footnote><sup>a</sup>Semiconductor Physics Institute, A. Goštauto 11, 2600 Vilnius, Lithuania;</footnote>

<footnote><sup>b</sup>Dipartimento di Ingegneria dell'Innovazione, Istituto Nazionale di Fisica della Materia, Università di Lecce, Via Arnesano s/n, 73100 Lecce, Italy; <sup>c</sup>Centre d'Electronique et de Micro-optoélectronique de Montpellier, (CNRS UMR 5507) Université Montpellier II, 34095 Montpellier Cedex 5, France; <sup>d</sup>SiCLAB, Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08855, USA</footnote>

<footnote>*Corresponding author. Tel.: (3702) 614920, Fax: (3702) 627123, e-mail: jane@pav.pfi.lt</footnote>
phonon energy and a stronger interaction of electrons with optical phonons. The aim of this work is to confirm the above expectation by calculating the amplification band and the maximum gain for an OPTTR maser based on bulk zincblende and wurtzite GaN.

## COMPUTATIONAL PROCEDURES

In general, the amplification spectrum (or the gain, $\alpha(f)$) of microwaves (MW) which can propagate in some active medium is described by the frequency dependence of the real part of the carrier MW mobility, $Re[\mu(f)]$, as:

$$ \alpha(f) = -Re[\mu(f)]n \frac{e}{c\varepsilon_0\sqrt{\varepsilon}} \quad (1) $$

where $n$ is the carrier concentration, $c$ the light velocity in vacuum, $\varepsilon_0$ the permittivity of vacuum, and $\varepsilon$ the static dielectric constant of the material. Under linear conditions, $\alpha(f)$ determines the frequency region of amplification, gives a threshold value of the net losses for generation to appear, allows one to choose the optimal doping level $N_D$ of a sample, etc. Under nonlinear conditions, when $\alpha(f)$ depends on the MW field amplitude, it allows one to determine the energy and power characteristics of generation. Under multi-signal operation it allows one to determine the characteristics of each radiation mode, the spectral behavior of the amplification band under single-mode generation, etc.

### Small-signal Response

Under linear conditions the MW mobility is independent of the MW field amplitude and is determined by the Fourier transform of the linear response function $K_{xx}(s)$ as [2, 3]:

$$ \mu_{xx}(\omega) = e \int_{0}^{\infty} K_{xx}(s) \exp(-i\omega s) ds \quad (2) $$

where $\omega = 2\pi f$ is the circular frequency, and only the longitudinal velocity response in the direction of the constant applied electric field $E_0 = (E_0, 0, 0)$ is taken to be of interest. In turn, when the single-particle history is simulated by the Monte Carlo (MC) method, the longitudinal linear response function can be expressed in terms of velocity averaging over before- and after-scattering ensembles as [2]:

$$ K_{xx}(s) = \frac{1}{eE_0\langle\tau\rangle} [\langle v_x(s) \rangle_b - \langle v_x(s) \rangle_a] \quad (3) $$

where

$$ \langle v_x(s) \rangle_b = \frac{\langle \tau \rangle}{T-s} \int_s^T \nu[\mathbf{p}(t-s)] v_x[\mathbf{p}(t)] dt \quad (4) $$

$$ \langle v_x(s) \rangle_a = \frac{1}{N} \sum_{i=1}^{N} v_x(t_i + s) \quad (5) $$

are the probable velocities of a carrier at time $s$ under the condition that at time $s=0$ it was just before or just after a scattering event, respectively. Here $N$ is the total number of scattering events in the time interval $[0, T]$ simulated by the MC procedure, $\langle\tau\rangle = T/N$ the mean time of free flight, $\nu(\mathbf{p})$ the scattering rate for momentum $\mathbf{p}$, $t_i$ the time moments of scattering events. Then, substitution of $Re[\mu]$ into Eq. (1) gives the static gain, $\alpha_0(f)$, defined in the absence of the MW field. The advantage of this procedure is that, for a given $E_0$ and $N_D$, it allows one to obtain the whole amplification/absorption spectrum during a single MC simulation.

### Large-signal Response

Under nonlinear conditions, the carrier mobility depends on the MW field amplitude and the MW mobility is directly calculated from the velocity response. For this sake the MW electric field of the amplified mode $E_{mw}(t) = Re[E_\omega\exp(i\omega t)]$ is directly introduced into the equation of motion $\dot{p}_x(t) = eE_0 + eE_\omega \cos(\omega t)$, which describes the free
flight of a trial particle in momentum space. During the simulation of a single-particle random trajectory $\mathbf{p}_t$ the sampling time of the average drift velocity is determined by averaging over a large number $M$ of MW electric field periods with duration $T_\omega$

$$ \langle v(s) \rangle_\omega = \frac{1}{M} \sum_{i=0}^{M-1} v_x(\mathbf{p}_t=s+T_\omega i) \quad (6) $$

where $s$ belongs to the time interval $0 \le s < T_\omega$. In accordance with the Fourier analysis, the real part of the velocity response is calculated as:

$$ v_{\omega}^{r} = \frac{2}{T_{\omega}} \int_{0}^{T_{\omega}} \langle v(s) \rangle_{\omega} \cos(\omega s) ds \quad (7) $$

In this case the MW mobility is determined from the real part of the velocity response as: $Re[\mu_\omega](E_\omega) = v'_\omega/E_\omega$. The dynamical gain, $\alpha_d(E_\omega)$, which already depends on the MW field amplitude, is then calculated from $Re[\mu_\omega]$ by using Eq. (1).

Another relevant quantity, which can be calculated under large-signal operation at a given frequency, is the power generated by an active medium, $P_\omega(E_\omega) = env_\omega^r E_\omega/2$, which is negative for generation. To analyze the stable generation of an active medium placed inside an external resonant system, it is often suitable to eliminate $E_\omega$ from $P_\omega(E_\omega)$ and $\alpha_d(E_\omega)$, i.e., to express $P_\omega$ as a function of $\alpha_d$ only. Under stable generation, the dynamic gain is equal to the coefficient of the net loss in the resonant system, $\alpha_L = \Gamma + \Lambda$. It includes both the outside power extraction, as described by coefficient coupling the active medium with the external free space, $\Gamma$, and all the parasitic losses, as described by the coefficient of internal losses, $\Lambda$. Therefore, one can finally represent the power generated inside the resonator as a function of $\alpha_L$ by: $P_{gen}(\alpha_L) = -P_\omega(\alpha_d)$. In such case, the outside power extraction is given by:

$$ P_{\text{out}} = P_{\text{gen}}(\alpha_L) \frac{\Gamma}{\Gamma + \Lambda} \quad (8) $$

The generation efficiency, $\eta$, is estimated as $\eta = P_{gen}/P_0$, where $P_0 = en\langle v \rangle_0 E_0$ is the total constant power absorbed by the sample, and $\langle v \rangle_0$ the average velocity of carriers.

### Multi-signal Response

To analyze the interaction among various possible modes in a resonant system it is necessary to investigate the system response in the presence of two or more strong signals. Accordingly, to find the conditions favourable for a single-mode operation of a maser, it is often preferable to know the behavior of the small-signal amplification spectrum under the stable single-mode generation operation, i.e., in the presence of a large-amplitude MW electric field at a given frequency. In such case, MC simulations are carried out by applying several MW fields at different frequencies $\omega_j$, i.e., by taking the electric field in the form of $E(t) = E_0 + \sum_j E_{\omega_j} \cos(\omega_j t + \phi_j)$. Then, similarly to the previous case, independent velocity averagings in accordance with Eq. (6) are performed in parallel to each frequency, so that further calculations of the MW mobility and generated power for the given frequency set can be performed separately. For example, by applying one strong signal at frequency $\omega$ and scanning the amplification spectrum with another small signal at different frequencies $\omega'$, one can study the variations of the amplification spectrum under stable generation.

### NUMERICAL RESULTS

The above computational procedures are applied to calculate the amplification and generation spectra in bulk zincblende and wurtzite GaN at low lattice temperatures under OPTTR conditions. The parameters of the band models are taken from Refs. [4, 5]. For acoustic and polar optical phonon scatterings the usual expressions are used [6]. The scattering mechanism from ionized impurities has been numerically optimized to shorten the cpu
time associated with the large number of small
angle scatterings. Moreover, due to the essential
role played by impurity scattering, three models of
screening have been checked, namely: Conwell-
Weiskopf, Brooks-Herring with the lattice tem-
perature and Brooks-Herring with the electron
temperature determined by the average energy of
electrons. At low doping levels ($N_d \le 10^{15} \text{cm}^{-3}$)
all three models give the same result in a wide
range of electric fields where runway from low-
energy impurity scattering already takes place.
With the increase of doping level, the most severe
restrictions for the onset of OPTTR conditions are
found in the Conwell-Weiskopf approximation.
Therefore, below we present only numerical results
obtained under this approximation, since other
models of screening will lead only to improve
OPTTR conditions.

Linear Operation

Calculations of the differential mobility spectrum
show that the OPTTR manifests itself through a
sequence of minima corresponding to the first and
higher harmonics of the resonant value ν<sub>E</sub> = 1/τ<sub>E</sub>,
where τ<sub>E</sub> = p<sub>0</sub>/eE<sub>0</sub> is the time of an electron free
flight from the center of the optical phonon sphere
to its boundary. However, the broadening of the
minima increases significantly for higher harmon-
ics so that DNDM is usually achieved in the first
minimum only, with the exception of the low-field
region E<sub>0</sub> ≤ 2 kV/cm where DNDM is found to
take place also at the second harmonic. Figure 1
shows the field dependence of the maximum gain
α(f) for the first and second amplification bands
of a zincblende GaN sample with N<sub>D</sub> = 10<sup>15</sup> cm<sup>-3</sup>
at T<sub>0</sub> = 10 K (curves 1 and 2 in the lower part of
the figure) and the amplification range for these
bands (regions bounded by curves 4, 5 and 6, 7 for
the first and second amplification bands, respec-
tively, in the upper part of the figure). At low
doping levels (N<sub>D</sub> ≤ 10<sup>15</sup> cm<sup>-3</sup>) the gain appears
at sufficiently low electric fields E<sub>0</sub> ≈ 200 V/cm
and quickly reaches the maximum values at
E<sub>0</sub> ≈ 400 V/cm. The appearance and rapid growth

of the gain is connected with the onset and
dominant role played by electron runaway from
the low-energy region dominated by impurity
scattering and the concomitant formation of a
needle-like distribution in momentum space. With
a further increase of E₀, the maximum value of the
gain decreases approximately as 1/E₀ and finally
vanishes at E₀ ≈ 10 kV/cm due to the significant
penetration of carriers into the active region,
which destroys the coherence of successive free
flights. Both the first and second amplification
bands are very narrow (see the frequency regions
bounded by curves 4, 5 and 6, 7), and the optimal
amplification frequency corresponding to the
maximum gain in the band increases almost
linearly with E₀. As evidenced by curves 4 and 5
at low doping levels the amplification covers
practically both millimeter and submillimeter
ranges due to the rather wide range of electric
fields corresponding to the appearance of ampli-
fication. With the increase of the doping level and,
hence, of the impurity scattering efficiency, the
threshold field for the onset of amplification is
shifted to higher values (see curve 3 calculated
for N<sub>D</sub> = 10<sup>16</sup> cm<sup>-3</sup>). After runaway from the

<figure><img src="image_2.png" /><figcaption>FIGURE 1 Maximum values of the gain in the first and second amplification bands (curves 1 and 2) as well as minimum (4,6) and maximum (5,7) frequencies of amplification for these bands calculated for zincblende GaN as a function of the electric field E₀ at T₀ = 10 K and Nₐ = 10¹⁵ cm⁻³. The variation of the gain with the increase of the doping level is illustrated by curve 3 calculated for Nₐ = 10¹⁶ cm⁻³.</figcaption></figure>
low-energy region has taken place, the DNDM for
low and high impurity concentrations practically
coincide, since further restrictions to the occurrence
of DNDM are primarily related to carrier pen-
etration into the active region, and these are
similar in both cases. However, the gain for the
high doped sample increases significantly due to
the increase of carrier concentration.

From the experimental point of view, it is often
easier to fix the frequency (e.g., by taking a
selective detector with maximum sensitivity in a
very narrow frequency range) and investigate the
absorption (or amplification) of the MW radiation
as function of the constant electric field applied
to the sample. In this case also, the resonant
character of the DNDM manifests itself as a series
of resonant peaks and minima. This behavior is
illustrated in Figure 2, which presents the field
dependence of the amplification/absorption coeffi-
cient at a frequency of f = 1 THz calculated by the
first MC procedure at T₀ = 10 K for wurtzite GaN
with Nₐ = 10¹⁶ cm⁻³ (curve 1). Again, amplifica-
tion is achieved in a narrow range of electric field
values. We check the accuracy of calculations by

<figure><img src="image_3.png" /><figcaption>FIGURE 2 Amplification (positive) and absorption (negative) coefficients at 1 THz frequency as a function of the static electric field strength for wurtzite GaN calculated by using: (i) the averaging over before- and after-scattering ensembles (curve 1), and (ii) the velocity response to a MW electric field with an amplitude of 0.2 and 3 kV/cm (curves 2 and 3, respectively) at T₀ = 10 K and Nₐ = 10¹⁶ cm⁻³.</figcaption></figure>

using various procedures, and to illustrate the
change in amplification at large MW field ampli-
tudes curves 2 and 3 show Re[μ] calculated directly
from the velocity response to a harmonic MW field
with amplitudes E_ω = 0.2 and 3 kV/cm, respec-
tively. At small amplitudes, the differential mobi-
lities calculated by the two procedures practically
coincide (curve 2). At large amplitudes the
mobility decreases monotonously in the whole
amplification band and the amplification region
becomes more narrower (see next section for a
more detailed discussion concerning nonlinear
effects).

**Nonlinear Operation**

The investigation of the regime of stable genera-
tion is carried out by a direct simulation of the
carrier response to a MW electric field at a given
frequency as described above. Qualitatively, the
general situation can be described as follows. In
the presence of DNDM, an external resonant
system will lead to the onset and growth of MW
oscillations accompanied with an increase of
the amplitude of E<sub>ω</sub> inside the sample. In the
initial stage, this growth will be accompanined
with an increase of the MW power P<sub>ω</sub> which,
under small-signal conditions, is proportional to
Re[μ<sub>ω</sub>]E<sub>ω</sub><sup>2</sup>. However, a considerable increase of
E<sub>ω</sub> will lead to a decrease of the dynamical gain.
Finally, the full disappearance of the amplifica-
tion effect will take on when both the gain and
the generated power will vanish. Therefore, it
becomes mandatory the choice of the resonator
characteristics and of the output power generated
by the sample under the constraint to provide the
maximum generated power inside the sample.
Figure 3 summarizes the results at the basis of
such a choice by presenting the maximum
generated power density, P<sub>gen</sub> at four different
frequencies (and respectively, four different val-
ues of applied static fields, E<sub>0</sub> = 1.2, 2.25, 4.3,
6.35 kV/cm) as a function of the coefficient of
net losses in the resonant system, α<sub>L</sub>. For each
curve the carrier concentration was chosen to get
<figure><img src="image_4.png" /><figcaption>FIGURE 3 Power generated at the fundamental frequencies of $f=0.25, 0.5, 1$ and $1, 5$ THz (curves 1 to 4) inside a resonator as a function of the coefficient of total losses inside the resonator, $\alpha_L$, calculated in wurtzite GaN with the respective doping concentrations of $N_D=0.6, 1, 2$, and $3 \times 10^{16} \text{cm}^{-3}$ at $T_0=10$ K.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIGURE 4 Amplification spectra under steady-state conditions at $E_0=4.3$ kV/cm (solid line) and under a strong MW field of $E_\omega=3$ kV/cm at 1 THz frequency calculated for wurtzite GaN with $N_D=10^{16} \text{cm}^{-3}$ at $T_0=10$ K.</figcaption></figure>

the maximum values of $P_{gen}$. Let us recall that for a stable generation the dynamical gain must exactly compensate all the losses in the resonant system due to power extraction outside the resonator, parasitic, etc., i.e., $\alpha_d=\alpha_L$. Thus, if $\alpha_L$ exceeds the static gain given by the linear theory then any generation is absent. Therefore, with the decrease of $\alpha_L$, at each given frequency generation starts from the value of the static gain. With a subsequent decrease of $\alpha_L$, the power increases and reaches a maximum value at the optimum amplitude of the MW field. As follows from Figure 3, this optimum value is achieved when $\alpha_L \approx 0.5\alpha_d$. Here, a generation efficiency of about $1 \sim 27\%$ is achieved. With a further decrease of $\alpha_L$, the amplitude of the MW field increases so considerably that it would destroy the transit-time resonance. For this reason, the generated power decreases to finally vanish at $\alpha_L=0$.

Figure 4 compares the amplification spectra calculated when only a static field of $E_0=4.3$ kV/cm is applied (solid line), and when a strong MW field $E_\omega=3$ kV/cm at frequency 1 THz is superimposed on the static field (crosses). One can see that in the latter case the amplification spectrum is systematically shifted down at all frequencies,

which is a useful property to obtain the single-mode generation.

## CONCLUSIONS

We have presented three numerical procedures to calculate the amplification spectrum of microwave radiation under various conditions with application to an OPTTR GaN THz maser. Results confirm that GaN is a promising material for THz power generation. The MW amplification and generation caused by OPTTR occurs in the wide frequency range of 0.05 to 3 THz and persists in the THz frequency range up to liquid nitrogen temperatures and doping levels of about $5 \times 10^{16} \text{cm}^{-3}$.

## Acknowledgments

Authors acknowledge the support of the NATO collaborative-linkage grant PST.CLG.976340, the high-level grant DRB4/MDL/no 99-30 of the french *Ministère de l'Éducation nationale, de la recherche et de la technologie*, the french-lituanian bilateral cooperation n. 5380 of french CNRS, MADESS II project of the Italian Consiglio Nazionale delle Ricerche (CNR).
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[3] Price, P. J. (1983). *J. Appl. Phys.*, **54**, 3616.

[4] Foutz, B. E., Eastman, L. F., Bhapkar, U. V. and Shur, M. S. (1997). *Appl. Phys. Lett.*, **70**, 2849.

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