1. Introduction
The millimeter and sub-millimeter microwave ranges are very important for applications in communications, radar, meteorology and spectroscopy. However, the structure of semiconductor devices (transistors, diodes, etc.), required for such a short wavelength, becomes very complex, which makes its fabrication difficult and expensive. One potential alternative to explore the use of such a part of the electromagnetic spectrum resides in the use of non-linear wave interaction in active media. For example, the space charge waves in thin semiconductor films, possessing negative differential conductivity (InP, GaAs, GaN at 300K and strained Si/SiGe heterostructures at 77K), propagate at frequencies that are higher than the frequencies of acoustic and spin waves in solids. This means, for example, that an elastic wave resonator operating at a given frequency is typically 100000 times smaller than an electromagnetic wave resonator at the same frequency. Thus attractively small elastic wave transmission components such as resonators, filters, and delay lines can be fabricated.
The scope of space charge waves’ applications is very large, because it can be useful to implement monolithic phase shifters, delay lines, and analog circuits for microwave signals. Space charge waves have been researched since a long time ago, which can be traced back to the 1950s [Benk]. The early experimental work on the amplification of space charge waves with a perturbation field started in the 1970s [Dean] and continues today [Kumabe
The study of microwave frequency conversion under negative differential conductivity will be one of the most relevant topics in microelectronics and communications in the coming years, due to the potential it represents in terms of amplification of micro- and millimeter-waves. However, in order to understand the behavior of non-stationary effects, a special attention must be paid to the transverse inhomogeneity, carrier-density fluctuations, in the plane of the film, because it may affect, in a negative way, the non-linear wave interaction. Thus, a creation of effective algorithms and computer programs for simulations of non-linear interaction of space charge waves in semiconductor films, where the effects of non-locality and transverse inhomogeneity should be taken into account, becomes of high importance.
2. The equations for space charge waves
Consider n-InP film placed onto substrate without an acoustic contact. It is assumed that the electron gas is localized in the center of film. The thickness of the n-InP film is
![](http://cdnintech.com/media/chapter/39226/1512345123/media/image1.png)
Figure 1.
The structure of the
In simulations, an approximation of two-dimensional 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
where
In such a representation, the mean energy and effective mass of electron are denoted by w and
A small microwave electric signal
For potential calculations, we use two dimensional Poisson’s equation
We use the Fast Fourier Transform, the sine transform along the Z axis and cosine transform along the Y axis with boundary conditions:
by replacing the equation (4) in (3) we obtain
![](http://cdnintech.com/media/chapter/39226/1512345123/media/image4.png)
Figure 2.
Electron drift velocity (a), average electron energy (b), averaged mass (c) versus electric field used in simulations; (-) MC data, (○) Fischetti, and (□) Gonzalez and experimental data: (Δ) Boers, (*) Glover, (□) Hamilton, and (○) Windhorn.
the solution of equation (7) is in the follow form:
For calculations
For
For calculations
ordering the terms to get a system from equation (13)
This difference equation is only for
And the same form for
also
for calculations
where
we use the approximation
A transverse inhomogeneity of the structure in the plane of the film along Y axis is taken into account. The following parameters are chosen: 2D concentration of electrons in the film is
3. Spatial increment of space charge waves
In this section, a description of propagation of longitudinal space charge waves in a negative differential conductance medium, using the drift - diffusion equations to find the dispersion relation, is presented; the equation of continuity (22) and Poisson’s equation (23) for electrons are
where
If one assumes that
The equation (27) is equivalent to:
If one substitutes the equation (32) into the equation (26) one obtains
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(
![](http://cdnintech.com/media/chapter/39226/1512345123/media/image5.png)
Figure 3.
Spatial increments of instability
4. Results and simulation
The propagation and amplification of space charge waves in
![](http://cdnintech.com/media/chapter/39226/1512345123/media/image6.png)
Figure 4.
Spectral components of the electric field of space charge waves. The effective excitation of harmonics is presented. The input carrier frequency is
A small microwave electric signal
![](http://cdnintech.com/media/chapter/39226/1512345123/media/image7.png)
Figure 5.
The spatial distributions of the alternative part of the electric field component
The spatial distributions of the alternate component of the electric field
5. Conclusions
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.
Acknowledgement
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 mobility program for professors.
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