**4. Results and simulation**

228 Numerical Simulation – From Theory to Industry

bias electric field, *E*0 = 20 kV/cm.

or

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If one substitutes the equation (32) into the equation (26) one obtains

��(��� � ���� � ���) � ��

�=��� � ���� � � ���

amplified instead of being damped. In general, we consider the cases where

Using the Drift-Diffusion equation, a study about of how a small, periodic disturbance may propagate in this InP film has been introduced by means the dispersion equation D(*ω*,*k*) = 0, because it determines the modes of propagation, their phase velocities and group velocities and also show if any instability can exists. For our present purpose, which is when the negative differential conductivity shows up, d*v*/d*E* < 0, space charge waves will get

and *k = k'+ik''* has real and imaginary part. The case *k'' >* 0 corresponds to spatial increment (amplification), whereas the case *k''<* 0 corresponds to the decrement (damping). In Fig. 3, the spatial increment of space charge waves in an *n-*InP film is shown in the curve 2, where the electron concentration is *n*0 = 0.8 x 1014 cm-2, the bias electric field is *E*0 = 20 kV/cm. In curve 3, the electron concentration is *n*0 = 5 x 1014 cm-2 with the same bias electric field, *E*0 = 20 kV/cm. Curve 1 is the result for *n-*GaAs films where the electron concentration is *n*0 = 5 x 1014 cm-2 and the bias electric field is E0 = 4.5 kV/cm. The stationary values of E0 have been chosen in the regime of negative differential conductivity (d*v*/d*E* <0) for all cases. One can see that an amplification of space charge waves in InP films occurs in a wide frequency range, and the maximal spatial increment is *k''* = 3x105 m-1 at the frequency *f* = 35 GHz. When compared

**Figure 3.** Spatial increments of instability *k*"(*f*) of space charge waves. Curve 1 shows results for an *n*-GaAs film [Garcia-B. *et al.*]. Curve 2 is for InP films with *E*0 = 20 kV/cm, *n*0 = 0.8x1014 cm-2 and film thickness 2*h* = 0.05 μm and curve 3 is for the electron concentration is *n*0 = 5 x 1014 cm-2 with the same

��� = �����|��� = �� ���

�� �� ��

����

��

����

(��)��� ����

(32)

= 0 (33)

 *= 2f* is real

�� � (34)

The propagation and amplification of space charge waves in *n-*GaAs thin films with negative difference conductance have been studied in the last decade [Mikhailov *et al*.], however *n-*InP films have not been addressed yet, and are subject of this work. We address the device presented in Fig. 1 by means of numerical simulations. An *n-*InP epitaxial film of thickness 0.1 - 1 μ*m* is put on an InP semi-insulating substrate. The two-dimensional electron density in the film is chosen to be *n*0 = 5 x 1014 cm-2. On the film surface are the cathode and anode ohmic contacts (OCs), together with the input and output coupling elements (CEs). Designed as a Schottky-barrier strip contacts, the CEs connect the sample structure to microwave sources. A dc bias voltage (above the Gunn threshold, 20 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. In the simulations an approximation of two-dimensional electron gas is used.

**Figure 4.** Spectral components of the electric field of space charge waves. The effective excitation of harmonics is presented. The input carrier frequency is *f* = 12 GHz.

A small microwave electric signal *Eext = Em*·sin(*ωt*)·exp(-((*z*-*z1*)*/z0*)2-((*y*-*y1*)/*y0*)2) is applied to the input antenna. Here *z1* is the position of the input antenna, *z0* is its half-width. Therefore, the parameter *2t0* determines the duration of the input electric pulse. In our simulations, this parameter is *2t0 =* 2.5 ns. The carrier frequency *f* is in the microwave range: *f* = 1 GHz – 100

GHz. When a small microwave signal is applied to the input antenna, the excitation of space charge waves in the 2D electron gas takes place. The space charge waves are subject to amplification, due to the negative differential conductivity. The stable implicit difference scheme is used. The following parameters have been chosen: 2D electron concentration in the film is *n*0 = 5x1014 cm-2, the initial uniform drift velocity of electrons is *v0 ≈ 2*x107 cm/s (*E0 =1*5 – 20 kV/cm), the length of the film is *Lz =* 0.1 mm, the thickness of the film is *2h* = 0.1 - 1 μm. The typical output spectrum of the electromagnetic signal is given in Fig. 4. The input carrier frequency is *f =* 12 GHz. The amplitude of the input electric microwave signal is *Em =* 25V/cm. Although the growth rate decreases as the *rf* frequency increases, for our case an amplification of 25 dB is obtained. The maximum of the input pulse occurs at *t1 =* 2.5 ns. One can see both the amplified signal at the first harmonic of the input signal and the harmonics generations of the input signal, which is generated due to the non-linearity of space charge waves.

A Numerical Study of Amplification of Space Charge Waves in n-InP Films 231

The spatial distributions of the alternate component of the electric field *E~z, E~y,* the alternate part of the electron concentration *ñ*, and the drift velocity *vz* are given in Fig. 5, at the time moment *t =* 4 ns. One can see the maximum variations are in the output antenna and the spatial distribution in direction *E~z* is bigger than in *E~y*, it is because the propagation of space charge wave. The length of the film is 0.1 mm. The transverse width of the film along Y axis is 1 mm. The duration of the input electric pulse is 2.5 ns. The spatial distributions are presented for the time moment 1.5 ns after the maximal value of the input signal. Direct numerical simulations 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 doubling in the millimeter wave range, it is

A theoretical study of two-dimensional amplification and propagation of space charge waves in n-InP films is presented. 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-InP films is performed. An increment in the amplification is observed in InP films at essentially higher frequencies *f* > 44 GHz than in GaAs films, which is due to its larger dynamic range. The maxi mum amplification (gain of 25 dB) is obtained at *f* = 35 GHz, using a distance between the input and output antennas of about 0.09 mm.

This project has been partially funded by the CONACyT- Mexico grant CB-169062 and by the ECEST-SEP (Espacio Común de Educación Superior Tecnológica) Program under the

Barybin A.A., Mikhailov A.I., Parametric interaction of space-charge waves in asymmetric

thin-film n-GaAs structures, Tech. Phys.,vol. 48, no. 6, pp. 761 – 767, 2003.

better to use the films with uniform doping.

Abel García-Barrientos and Francisco R. Trejo-Macotela

*Polytechnic University of Pachuca, Hidalgo, Mexico* 

*Polytechnic University of Altamira, Tamaulipas, Mexico* 

*Autonomous University of Morelos, Cuernavaca, Morelos, Mexico* 

**5. Conclusions** 

**Author details** 

Liz del Carmen Cruz-Netro

Volodymyr Grimalsky

**Acknowledgement** 

**6. References** 

mobility program for professors.

**Figure 5.** The spatial distributions of the alternative part of the electric field component *E*~z of space charge wave (a); the component of electric field *E*~y (b); alternative part of the electron concentration *n*~ (c); and the component of the electron drift velocity *v*z (d). The length of the film is 0.1 mm. The transverse width of the film along Y axis is 1 mm.

The spatial distributions of the alternate component of the electric field *E~z, E~y,* the alternate part of the electron concentration *ñ*, and the drift velocity *vz* are given in Fig. 5, at the time moment *t =* 4 ns. One can see the maximum variations are in the output antenna and the spatial distribution in direction *E~z* is bigger than in *E~y*, it is because the propagation of space charge wave. The length of the film is 0.1 mm. The transverse width of the film along Y axis is 1 mm. The duration of the input electric pulse is 2.5 ns. The spatial distributions are presented for the time moment 1.5 ns after the maximal value of the input signal. Direct numerical simulations 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 doubling in the millimeter wave range, it is better to use the films with uniform doping.
