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Numerical simulations of nonlinear interaction of space charge waves in microwave and millimeter wave range in n-InN films have been carried out. A micro- and millimeter-waves frequency conversion using the negative differential conductivity phenomenon is carried out when the harmonics of the input signal are generated. An increment in the amplification is observed in n-InN films at essentially at high-frequencies f < 450 GHz, when compared with n-GaAs films f < 44 GHz. This work provides a way to achieve a frequency conversion and amplification of micro- and millimeter-waves.

Semiconducting group-III nitrides have attracted a lot of attention in the recent decade because of, mainly, the large gap (0.7 - 6.2 eV) that can be covered by the nitrides and their alloys. Their electrical and optical proper- ties are highly suitable for novel high frequency electronics, optoelectronics and photonics applications. Among those, in particular, Indium Nitride (InN) has become the focus of intense research after recent reports of an un- expectedly low band gap of 0.7 eV at room temperature [

Nonlinear interaction of space charge waves in microwave and millimeter wave range in n-InN films using the negative differential conductivity will be one of the most relevant topics in microelectronics and communica- tions in the coming years, due to the potential it represents in terms of amplification of micro- and millimeter- waves. Therefore, we present two-dimensional numerical simulations of propagation and amplification of space charge waves in InN films and microwave frequency conversion under negative differential conductivity phe- nomenon. We use a high quality, single crystal wurtzita InN film placed onto substrate without a acoustic con- tact. It is assumed that the electron gas is localized in the center of film. The thickness of the n-InN film is 2 h ≤ 1 μm, see

The coordinate system is chosen as follows: X-axis is directed perpendicularly to the film, the electric field E_{0}_{}

The structure of the n-InN traveling-wave amplifier with an epitaxial layer

is applied along Z-axis, exciting and receiving antennas are parallel to Y-axis. 2D model of electron gas in the n-InN film is used. Taking a constant surface sheet density of 2D n_{0} = 2.5 × 10^{13} cm^{−2} for a 1 µm thick film. There has been some evidence to suggest the presence of electron accumulation at the surface of InN, including measurements of the sheet carrier density as a function of InN film thickness and capacitance-voltage profiling [_{0} = v(E_{0}), E_{0} = U_{0}/L_{z}, are considered, where U_{0} is bias voltage, L_{z} is the length of the film. Generally, a non-local dependence of drift velocity v_{d} of electrons on the electric field takes place. In simulations, an approximation of two-dimen- sional electron gas is used. The set of balance equations for concentration, drift velocity, and the averaged energy to describe the dynamics of space charge waves within the n-InN film takes a form, like in GaAs film [

where v_{d} is drift velocity, φ is the of potential, n = n_{0} + ñ where n_{0} is constant electron concentration, ñ is the varying part, w is the electron energy, D is the diffusion coefficient, and ε_{0} is the lattice dielectric permittivity of n-InN, m(w) is averaged effective mass, q is the electron charge, τ_{p}_{,w}(w) are relaxation times, E_{ext} is a small mi- crowave electric signal applied to the input antenna and E_{0} is the bias electric field. It is assumed that a condition of occurring negative differential conductivity is realized. Because the signal frequencies are in microwave or millimeter wave range, it is possible to separate diffusion and drift motions. For the sake of simplicity, instead of relaxation times, the parameter E_{s} is introduced [

In such a representation, the mean energy and mean effective mass of electron are denoted by w and m(w), the equilibrium value of w is w_{0}; A direct correspondence between local field dependence and non-local effects is well seen. Because a dependence E_{s} = E_{s}(w) is unique, it is possible to express the parameters w and v_{s} through the value of E_{s}. The dependencies of drift velocity, averaged electron energy, and effective mass versus electric field in InN films were calculated by our Monte Carlo procedure, which are pretty similar as experimental re- sults [_{iv} = ħω_{LO}), mass density (ρ = 6.81 g/cm^{3}), and static and high frequency dielectric constants (ε_{s} = 15.3 and ε_{∞} = 8.4) are adopted from [_{s} = 11.0 and ε_{∞} = 6.7) recently proposed in [

Electron drift velocity (a), average electron energy (b), averaged mass (c) versus electric field used in simulations

field applied in InN. A large negative differential conductance appears when increasing electric field more than the threshold fielded; the electrons have not enough energy to make the inter-valley scattering. Beyond this val- ue, the optical scattering mechanisms play a drastic role rather than acoustic and ionized impurity scattering.

Since this process is inelastic, the electron energy curves have sensitive variation in its slope; in fact it does not increase as fast as increasing in initial fields (see

The spatial increment of space charge waves is investigated by the dispersion equation, D(ω,k) = 0, the relation between angular frequency, ω = 2πf, and wave vectors, consider these like complex,

but if we assume that ñ obeys the law ~exp(iωt-ikz), Equation (3) gives the dispersion relation:

In general, we consider the cases where ω = 2πf is real and _{0} = 2.5 × 10^{13} cm^{−2}, the bias electric field is E_{0} = 50 kV/cm. In curve 2, the electron concentration is n_{0} = 2 × 10^{13} cm^{−2} with the same bias electric field, E_{0} = 50 kV/cm. Curve 1 is the result for n-InN films where the electron concentration is n_{0} = 2.5 × 10^{13} cm^{−2} and the bias electric field is E_{0} = 45 kV/cm. The stationary values of E_{0} have been chosen in the regime of negative differential conductivity (dv/dE < 0) for all cases. One can see that an amplification of space charge waves in InN films occurs in a

(a) The spatial increment of space charge waves in an n-InN film is shown in the curve 3, where the electron con- centration is n_{0} = 2.5 × 10^{13} cm^{−2}, the bias electric field is E_{0} = 50 kV/cm. In curve 2, the electron concentration is n_{0} = 2 × 10^{13} cm^{−2} with the same bias electric field, E_{0} = 50 kV/cm. Curve 1 is the result for n-InN films where the electron concen- tration is n_{0} = 2.5 × 10^{13} cm^{−2} and the bias electric field is E_{0} = 45 kV/cm; (b) Spectral components of the electric field of space charge waves. The effective excitation of harmonics is presented. The input carrier frequency is f = 60 GHz

wide frequency range, and the maximal spatial increment is k'' = 12 × 10^{5} m^{−1} at the frequency f = 225 GHz. When compared with a case of the GaAs film [

When a small microwave electric signal

plied to the input antenna. Here z_{1} and y_{1} are the position of the input antenna; z_{0} and y_{0} are its half-width. When this signal is applied, the excitation of space charge waves in 2D electron gas takes place. These waves are sub- ject to amplification, due to negative differential conductivity.

The set of Equations (1) form a set of non-linear coupled time dependent partial differential equations. These differential equations are discretized using a finite-difference scheme and are solved numerically. A transverse inhomogeneity of the structure in the plane of the film along Y axis is taken into account. The following para- meters have been chosen: 2D electron concentration in the film is n_{0} = 2.5 × 10^{13} cm^{−2}, the initial uniform drift velocity of electrons is v_{0} ≈ 4 × 10^{7} cm/s (E_{0} = 50 - 100 kV/cm), the length of the film is L_{z} = 5 - 20 µm, the thickness of the film is 2 h = 0.1 - 1 μm. The typical output spectrum of the electromagnetic signal is given in _{m} = 25V/cm. Although the growth rate decreases as the rf frequency increases, for our case an amplification of 25 dB is obtained. One can see both the amplified signal at the first harmonic of the input signal and the har- monic generations of the input signal, which is generated due to the non-linearity of space charge waves.

The propagation and amplification of space charge waves in n-GaAs thin films with negative difference con- ductance have been studied in the last decade [_{0} = 2.5 × 10^{13} cm^{−2}. On the film surface are the cathode and anode ohmic- contacts (OCs), together with the input and output coupling elements (CEs). The CEs connect the sample struc- ture to microwave sources. A dc bias voltage (above the Gunn threshold, 50 kV/cm) was applied between the cathode and anode OCs, causing negative differential conductivity in the film. The CEs perform the conversion between electromagnetic waves and space charge waves, where the excitation of space charge waves in the 2D electron gas takes place.

The spatial distributions of the alternate component of the electric field E^{~}_{z} and E^{~}_{y} are shown in

The spatial distributions of the alternative part of the electric field component E^{~}_{z} (a) and E^{~}_{y} (b) of space charge wave; The length of the film is 20 µm. The transverse width of the film along Y axis is 40 µm

mulations have confirmed pointed below results on linear increments of space charge waves amplification. Also a possibility of non-linear frequency doubling and mixing is demonstrated. To get the effective frequency doubl- ing in the millimeter wave range, it is better to use the films with uniform doping.

A numerical simulation of nonlinear interaction, two-dimensional propagation and amplification of space charge waves in microwave and millimeter wave range in n-InN films using negative differential conductivity are pre- sented. A microwave frequency conversion using the negative differential conductivity phenomenon is carried out when the harmonics of the input signal are generated. A comparison of the calculated spatial increment of instability of space charge waves in n-GaAs and n-InN films is performed. An increment in the amplification is observed in n-InN films at essentially higher frequencies f > 44 GHz than in GaAs films [

This project has been funded by the CONACyT-Mexico grant CB-169062 and also it has been partially funded by PROMEP: Redes Temáticas de Colaboración under the project titled: Fuentes de Energías Alternas.