**5. References**

	- [13] Deuflhard, P. (2004). *Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms.* Series Computational Mathematics, **35**, Springer.
	- [14] Dmitriev, V. I., Berezina, N. I. (1986). *Numerical Methods of Solution of the Synthesis Problems for the Radiating Systems*, (In Russian). Moscow, MGU.
	- [15] Feld, Ya. N. (1992). On the Quadratic Lemma in Electrodynamics. *Sov. Phys. Dokl.*, 37, pp. 235-236.
	- [16] Fourikis, N. (2000). *Advanced Array Systems. Applications and RF Technologies. Phased Array Systems*. Ascot park, Academic Press.
	- [17] Gelfand, I. M., Fomin, S. V. (2000). *Calculus of Variations*. Dover Publ.
	- [18] Kaklamani, D. I., Anastassiu, H. T. (2002). Aspects of the Method of Auxiliary Sources (MAS) in Computational Electromagnetics. *IEEE Antennas & Propagation Magazine*, vol. 44, No 6, pp. 48-64.
	- [19] Katsenelenbaum, B., Marcader del Rio, L., Pereyaslavets, M., at al. (1998). *Theory of Nonuniform Waveguides*, London, IEE Series.
	- [20] Katsenelenbaum, B. Z. (2003). *Electromagnetic Fields Restriction and Approximation*. Weinheim, Wiley-VCH.
	- [21] Katsenelenbaum, B. Z. (2006). *High-Frequency Electrodynamics*. Weinheim, Wiley-VCH.
	- [22] Khzmalyan, A. D., Kondratyev, A. S. (2003). The Phase-Only Shaping and Adaptive Nulling of an Amplitude Pattern. *IEEE Trans. Antennas and Propag.*, vol. AP-51,No 2, pp. 264-272.
	- [23] Krasnoselsky, M. A., Zabreiko, P. P. (1975). *Geometric Methods of Nonlinear Analysis*, (in Russian). Moscow, Nauka.
	- [24] Kumar, A., Hristov, H. D. (1989). *Microwave Cavity Antennas*. Boston– London, Artech House.
	- [25] Landau, L., Lifshitz, E. (1984). *Electrodynamics of continuous media*. London, Pergamon Press.
	- [26] Markov, G. T., Petrov, B. M., Grudinskaya, G. P. (1979). *Electrodynamics and Propagation of Waves*, (in Russian). Moscow, Sov. Radio.
	- [27] Milligan, T. A. (2005). *Modern Antenna Design*. Hoboken, NJ, John Wiley & Sons.
	- [28] Polak, E. (1971). *Computational Methods in Application. A Unified Approach*. New York, Academic Press.
	- [29] Savenko, P. O. (2002). *Nonlinear Synthesis Problems of Radiating Systems*. Lviv, Ukraine, IAPMM.
	- [30] Smirnov, V. I. (1958). *Higher Mathematics*, (in Russian). Vol. 4, Moscow, GIFML.
	- [31] Styeskal, H. l, Herd, J. S. (1990). Mutual Coupling Compensation in Small Array Antennas. *IEEE Trans. on AP*, Vol. 38, No. 12, pp. 1971-1975.
	- [32] Tikhonov, A. N., Arsenin, V. Y. (1977). *Solutions of Ill-Posed Problems*. New York, Wiley.
	- [33] Vapnyarskii, I. B. (2001). Lagrange Multipliers, in M. Hazewinkel, *Encyclopedia of Mathematics*, Springer.
	- [34] Veinstein, L. A. (1998). *The Electromagnetic Waves*, (in Russian). Moscow, Radio i Svyaz.
	- [35] Vendik, O. G., Parnes, M. D. (2002). *Antennas with Electrical Scanning*, (in Russian). Saint Petersburg, Science Press.
	- [36] Vladimirov et al. (1985). *Electromagnetic Compatibility of Radioelectronic Devices and Systems*, (in Russian). Moscow, Radio i Svyaz.
	- [37] Voitovich, N. N., Katsenelenbaum, B. Z. Korshunova, E. N. , at al. (1989). *Electrodynamics of Antennas with Semitransparent Surfaces*, (in Russian). Moscow, Nauka.
	- [38] Weinberg, M. M., Trenogin, V. A. (1969). *Theory of the Branching of Solutions of Nonlinear Equations*, (in Russian). Moscow, Nauka.

© 2012 García-Barrientos et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**A Numerical Study of Amplification** 

Abel García-Barrientos, Francisco R. Trejo-Macotela, Liz del Carmen Cruz-Netro and Volodymyr Grimalsky

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/47764

**1. Introduction** 

**of Space Charge Waves in n-InP Films** 

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 *et al.* & Barybin *et al.*]. The first monolithic device using space charge waves was a two-port amplifier developed in the beginning of 1970s in the United States. This device contained an *n-*GaAs film on a dielectric substrate, and a couple of source and drain ohmic contacts. A microwave signal applied to the input electrode modulates the electron density under this electrode. These modulations are drifted to the drain and amplified due to the negative

and reproduction in any medium, provided the original work is properly cited.
