**1. Introduction**

28 Will-be-set-by-IN-TECH

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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

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

resistance effect. The amplified signal is taken from the output electrode placed near the drain, see Fig. 1a. Obviously, the output signal is maximal when all the waves reach the output electrodes with the same phase [Wang *et al.*]. The majority of devices based on space charge waves are fabricated on GaAs films, although InP is recognized to have superior characteristics compared to GaAs for power generation in the millimeter wave range [Wandinger & Dragoman *et al.*]. The threshold electric field is 3.2 kV/cm for gallium arsenide and 10 kV/cm for indium phosphide. The peak electron drift velocity is about 2.2 x 107 cm/s for gallium arsenide and 2.5 x 107 cm/s for indium phosphide. The maximum negative differential mobility is about -2400 cm2/V-s for GaAs and -2000 cm2/V-s for indium phosphide [Sze]. An electric field in excess of 10 kV/cm applied to an *n-*InP sample causes the differential electron mobility to become negative. To analyze wave phenomena in GaAs films semiconductors, a set of equations to describe the charge transport is commonly used [Kazutaka]. In the case of InP, the same set of equations can be used, because the band structure of InP is very similar to the one of GaAs. In this theory, with small initial perturbations, continuity, momentum and energy equations, and Poisson's equation are solved. The solutions show that the modulations of electron density travel along the beam in the form of waves called space charge waves; these results, for n-GaAs thin films, are in [Grimalsky, *et al.*]. In this paper, the non-linear interaction of space charge waves including the amplification in microwave and millimeter wave range in *n*-InP films, possessing the negative differential conductance phenomenon, is investigated theoretically; also the spatial increment of space charge waves in InP films is compared with the results in GaAs. The results suggest that the increment observed in the gain is due to the larger dynamic range in n-InP than in n-GaAs films, where the optical scattering mechanisms play a drastic role rather than acoustic and ionized impurity scattering.

A Numerical Study of Amplification of Space Charge Waves in n-InP Films 221

to Y-axis. 2D model of electron gas in the *n-*InP film is used. Thus, 2D electron concentration is presented only in the plane *x* = 0. The space charge waves possessing phase velocity equal to drift velocity of the electrons *v*0 = *v*(E0), E0 = U0/Lz, are considered, where U0 is bias voltage, Lz is the length of the film. Generally, a non-local dependence of drift velocity *v*d of electrons

**Figure 1.** The structure of the *n-*InP traveling-wave amplifier fabricated with an epitaxial layer (a),

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 *n-*InP film takes a form, like in GaAs film

scheme in device surface for calculating nodes *j* and *k* (b).

[Carnez, *et al.*]:

on the electric field takes place.

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 millimeterwaves. 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 nonlinear interaction of space charge waves in semiconductor films, where the effects of nonlocality 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 *2h* < 1 μm, see Fig. 1a. The coordinate system is chosen as follows: X-axis is directed perpendicularly to the film, the electric field E0 is applied along Z-axis, exciting and receiving antennas are parallel to Y-axis. 2D model of electron gas in the *n-*InP film is used. Thus, 2D electron concentration is presented only in the plane *x* = 0. The space charge waves possessing phase velocity equal to drift velocity of the electrons *v*0 = *v*(E0), E0 = U0/Lz, are considered, where U0 is bias voltage, Lz is the length of the film. Generally, a non-local dependence of drift velocity *v*d of electrons on the electric field takes place.

220 Numerical Simulation – From Theory to Industry

rather than acoustic and ionized impurity scattering.

**2. The equations for space charge waves** 

importance.

resistance effect. The amplified signal is taken from the output electrode placed near the drain, see Fig. 1a. Obviously, the output signal is maximal when all the waves reach the output electrodes with the same phase [Wang *et al.*]. The majority of devices based on space charge waves are fabricated on GaAs films, although InP is recognized to have superior characteristics compared to GaAs for power generation in the millimeter wave range [Wandinger & Dragoman *et al.*]. The threshold electric field is 3.2 kV/cm for gallium arsenide and 10 kV/cm for indium phosphide. The peak electron drift velocity is about 2.2 x 107 cm/s for gallium arsenide and 2.5 x 107 cm/s for indium phosphide. The maximum negative differential mobility is about -2400 cm2/V-s for GaAs and -2000 cm2/V-s for indium phosphide [Sze]. An electric field in excess of 10 kV/cm applied to an *n-*InP sample causes the differential electron mobility to become negative. To analyze wave phenomena in GaAs films semiconductors, a set of equations to describe the charge transport is commonly used [Kazutaka]. In the case of InP, the same set of equations can be used, because the band structure of InP is very similar to the one of GaAs. In this theory, with small initial perturbations, continuity, momentum and energy equations, and Poisson's equation are solved. The solutions show that the modulations of electron density travel along the beam in the form of waves called space charge waves; these results, for n-GaAs thin films, are in [Grimalsky, *et al.*]. In this paper, the non-linear interaction of space charge waves including the amplification in microwave and millimeter wave range in *n*-InP films, possessing the negative differential conductance phenomenon, is investigated theoretically; also the spatial increment of space charge waves in InP films is compared with the results in GaAs. The results suggest that the increment observed in the gain is due to the larger dynamic range in n-InP than in n-GaAs films, where the optical scattering mechanisms play a drastic role

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 millimeterwaves. 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 nonlinear interaction of space charge waves in semiconductor films, where the effects of nonlocality and transverse inhomogeneity should be taken into account, becomes of high

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 *2h* < 1 μm, see Fig. 1a. The coordinate system is chosen as follows: X-axis is directed perpendicularly to the film, the electric field E0 is applied along Z-axis, exciting and receiving antennas are parallel

**Figure 1.** The structure of the *n-*InP traveling-wave amplifier fabricated with an epitaxial layer (a), scheme in device surface for calculating nodes *j* and *k* (b).

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 *n-*InP film takes a form, like in GaAs film [Carnez, *et al.*]:

$$\begin{aligned} \frac{d\left(m(\boldsymbol{\nu})\bar{\boldsymbol{\nu}}\_{d}\right)}{dt} &= \mathbf{-}q\left(\bar{E} \cdot \frac{\bar{\boldsymbol{\nu}}\_{d}\bar{E}\_{s}}{\nu\_{s}}\right) \\ \frac{d\boldsymbol{\nu}}{dt} &= \mathbf{-}q\left(\bar{E}\bar{\boldsymbol{\nu}}\_{d} \cdot \boldsymbol{E}\_{s}\boldsymbol{\nu}\_{s}\right) \\ \frac{\partial\bar{n}}{\partial t} &+ d\boldsymbol{\nu}\left(n\bar{\boldsymbol{\nu}}\_{d} \cdot D\bar{\boldsymbol{\lambda}}\boldsymbol{n}\right) = 0 \\ D\left(\boldsymbol{\nu}\right) &= \frac{2}{3}\frac{\mathsf{T}\_{\rho}\left(\boldsymbol{\nu}\right)}{m\left(\boldsymbol{\nu}\right)}\left(\boldsymbol{\nu}\cdot\frac{1}{2}m\boldsymbol{\nu}^{\boldsymbol{\nu}}\right) \\ \bar{E} &= \bar{\boldsymbol{e}}\_{x}E\_{0} \cdot \bar{\boldsymbol{\lambda}}\boldsymbol{\rho} + \bar{\bar{\boldsymbol{e}}}\_{x}\bar{E}\_{\text{ext}} \\ \Delta\boldsymbol{\rho} &= \frac{q}{\varepsilon\_{d}\boldsymbol{\varepsilon}}(\mathbf{n}\cdot\boldsymbol{n}\_{\boldsymbol{\rho}})\boldsymbol{\delta}\left(\boldsymbol{x}\right) \end{aligned} \tag{1}$$

$$\begin{aligned} \frac{m(\boldsymbol{\nu})}{\tau\_p(\boldsymbol{\nu})} &= \frac{E\_s}{\nu\_s(E\_s)}\\ \frac{\boldsymbol{\nu} \cdot \boldsymbol{\nu}\_0}{\tau\_\boldsymbol{\nu}(\boldsymbol{\nu})} &= qE\_s \nu\_s \left(E\_s\right) \end{aligned} \tag{2}$$

$$\frac{\partial^2 \varphi}{\partial \mathbf{y}^2} + \frac{\partial^2 \varphi}{\partial \mathbf{z}^2} = -\chi \text{fit}\,\delta(\mathbf{x})\tag{3}$$

$$\varphi(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \sum\_{j=1}^{N\_{\mathbf{z}}} \sum\_{k=0}^{N\_{\mathbf{y}}} \phi\_{jk}(\mathbf{x}) \operatorname{Stn} K\_{\mathbf{z}j} \mathbf{z} \* \operatorname{Cost} K\_{\mathbf{z}j} \mathbf{y} \tag{4}$$

$$\left. \varphi \right|\_{z=0} = \left. \varphi \right|\_{z=l\_Z} = 0; \left. \frac{\partial \varphi}{\partial \mathbf{y}} \right|\_{\mathbf{y}=0} = \left. \frac{\partial \varphi}{\partial \mathbf{y}} \right|\_{\mathbf{y}=l\_Z} = 0; K\_{\mathbf{z}j} = \frac{\pi k}{l\_{\mathbf{z}}}; K\_{\mathbf{y}k} = \frac{\pi k}{l\_{\mathbf{y}}} \tag{5}$$

$$\left.E\_{\mathcal{Y}}\right|\_{\mathbf{y}=\mathbf{0}} = \left.E\_{\mathcal{Y}}\right|\_{\mathbf{y}=l\_{\mathcal{Y}}} = \left.\mathbf{0}; \left.\boldsymbol{\upsilon}\_{\mathcal{Y}}\right|\_{\mathbf{y}=\mathbf{0}} = \left.\boldsymbol{\upsilon}\_{\mathcal{Y}}\right|\_{\mathbf{y}=l\_{\mathcal{Y}}} = \mathbf{0} \tag{6}$$

$$\frac{d^2 \phi\_{jk}(\mathbf{x})}{d\mathbf{x}^2} - \left(\left(\mathbf{K}\_{\mathbf{z}j}\,^2 + \mathbf{K}\_{\mathbf{jk}}\,^2\right)\phi\_{jk}(\mathbf{x}) = \frac{d^2 \phi\_{jk}(\mathbf{x})}{d\mathbf{x}^2} - \Theta\_{\mathbf{jk}}\,^2\phi\_{jk}(\mathbf{x}) = -\chi\tilde{\mathcal{N}}\_{jk}\delta(\mathbf{x})\tag{7}$$

$$\mathfrak{M}(\mathbf{y}, \mathbf{z}) = \sum\_{j=1}^{N\_{\mathbf{z}}} \sum\_{k=0}^{N\_{\mathbf{y}}} \mathfrak{N}\_{jk} \operatorname{Sin} K\_{\mathbf{z}j} \mathbf{z} \* \operatorname{Cos} K\_{\mathbf{z}j} \mathbf{y} \tag{8}$$

$$\phi\_{jk}(\mathbf{x}) = \frac{\chi}{2\Theta\_{\parallel k}} \mathcal{N}\_{jk} e^{-\Theta\_{\parallel k}|\mathbf{x}|} \tag{9}$$

$$\log(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \frac{\chi}{2} \sum\_{j=1}^{N\_x} \sum\_{k=0}^{N\_y} \frac{\bar{\mathbf{N}}\_{jk}}{\Theta\_{\parallel \mathbf{k}}} e^{-\Theta\_{\parallel \mathbf{k} \parallel \mathbf{x} \parallel} \mathbf{S} \text{int}K\_{\mathbf{z}j} \mathbf{z} \* \mathbf{C} \text{os}K\_{\mathbf{z}j} \mathbf{y} \tag{10}$$

$$\left. \tilde{E}\_{\mathbf{z}} \right|\_{\mathbf{x}=0} = -\frac{\partial \boldsymbol{\varrho}}{\partial \mathbf{z}} \Big|\_{\mathbf{x}=0} = -\frac{\chi}{2} \sum\_{j=1}^{N\_{\mathbf{z}}} \sum\_{k=0}^{N\_{\mathbf{y}}} \frac{\mathbb{N}\_{jk}}{\Theta\_{\parallel \mathbf{k}}} K\_{\mathbf{z}j} \mathbb{C} \text{os} \mathbf{K}\_{\mathbf{z}j} \mathbf{z} \ast \text{cos} \mathbf{K}\_{\mathbf{y}k} \mathbf{y} \tag{11}$$

$$\left. \mathfrak{E}\_{\mathbf{y}} \right|\_{\mathbf{x}=\mathbf{0}} = -\frac{\partial \varphi}{\partial \mathbf{y}} \Big|\_{\mathbf{x}=\mathbf{0}} = \frac{\chi}{2} \sum\_{j=1}^{N\_x} \sum\_{k=0}^{N\_y} \frac{\mathfrak{N}\_{jk}}{\Theta\_{\mathbf{jk}}} K\_{jk} \mathrm{Stn} K\_{xj} \mathbf{z} \ast \mathrm{Stn} K\_{yk} \mathbf{y} \tag{12}$$

$$\begin{split} \frac{\left(\left(\boldsymbol{m}\boldsymbol{\upsilon}\_{\boldsymbol{z}}\right)\_{\boldsymbol{j},\boldsymbol{k}}^{p+1} - \left(\boldsymbol{m}\boldsymbol{\upsilon}\_{\boldsymbol{z}}\right)\_{\boldsymbol{j},\boldsymbol{k}}^{p} + \left(\boldsymbol{\upsilon}\_{\boldsymbol{z}}\right)\_{\boldsymbol{j},\boldsymbol{k}}^{p+1}} \frac{\left(\boldsymbol{m}\boldsymbol{\upsilon}\_{\boldsymbol{z}}\right)\_{\boldsymbol{j},\boldsymbol{k}}^{p+1} - \left(\boldsymbol{m}\boldsymbol{\upsilon}\_{\boldsymbol{z}}\right)\_{\boldsymbol{j}-1,\boldsymbol{k}}^{p+1}}{h\_{\boldsymbol{z}}} + \left(\boldsymbol{\upsilon}\_{\boldsymbol{z}}\right)\_{\boldsymbol{j},\boldsymbol{k}}^{p+1} \frac{\left(\boldsymbol{m}\boldsymbol{\upsilon}\_{\boldsymbol{z}}\right)\_{\boldsymbol{j},\boldsymbol{k}+1}^{p+1} - \left(\boldsymbol{m}\boldsymbol{\upsilon}\_{\boldsymbol{z}}\right)\_{\boldsymbol{j},\boldsymbol{k}-1}^{p+1}}{2h\_{\boldsymbol{y}}} \\ = \boldsymbol{\upsilon}\_{\boldsymbol{z}} \left[\left(\boldsymbol{E}\_{\boldsymbol{z}}\right)\_{\boldsymbol{j},\boldsymbol{k}}^{p+1} - \left(\frac{\boldsymbol{E}\_{\boldsymbol{s}}}{\boldsymbol{\upsilon}\_{\boldsymbol{s}}}\right)\_{\boldsymbol{j},\boldsymbol{k}}^{p+1} \left(\boldsymbol{\upsilon}\_{\boldsymbol{z}}\right)\_{\boldsymbol{j},\boldsymbol{k}}^{p+1}\right] \end{split} \tag{13}$$

$$a\_{-1}(\upsilon\_z)\_{j,k-1}^{p+1} + a\_0(\upsilon\_z)\_{j,k}^{p+1} + a\_{+1}(\upsilon\_z)\_{j,k+1}^{p+1} = f\_k \tag{14}$$

$$\left(\left(\upsilon\_{\mathbf{z}}\right)\_{j,0}^{p+1} = \frac{\left(m\upsilon\_{\mathbf{z}}\right)\_{j,0}^{p} + \chi\_{2}\tau(E\_{\mathbf{z}})\_{j,0}^{p+1}}{m\_{j,0}^{p+1} + \frac{\tau}{h\_{\mathbf{z}}}\left(\left(m\upsilon\_{\mathbf{z}}\right)\_{j,0}^{p+1} - \left(m\upsilon\_{\mathbf{z}}\right)\_{j-1,0}^{p+1}\right) + \chi\_{2}\tau\left(\frac{E\_{\mathbf{s}}}{\upsilon\_{\mathbf{s}}}\right)\_{j,0}^{p+1}}\tag{15}$$

$$\begin{split} \frac{\left(\left(mv\_{\mathcal{V}}\right)\_{j,k}^{p+1} - \left(mv\_{\mathcal{V}}\right)\_{j,k}^{p}\right)}{\tau} &+ \left(v\_{\mathcal{V}}\right)\_{j,k}^{p+1} \frac{\left(mv\_{\mathcal{Z}}\right)\_{j,k}^{p+1} - \left(mv\_{\mathcal{Z}}\right)\_{j-1,k}^{p+1}}{h\_{\mathcal{Z}}}\\ &+ \left(v\_{\mathcal{V}}\right)\_{j,k}^{p+1} \frac{\left(mv\_{\mathcal{V}}\right)\_{j,k+1}^{p+1} - \left(mv\_{\mathcal{V}}\right)\_{j,k-1}^{p+1}}{2h\_{\mathcal{Y}}} = \chi\_{2} \left[\left(E\_{\mathcal{V}}\right)\_{j,k}^{p+1} - \left(\frac{E\_{\mathcal{S}}}{\upsilon\_{\mathcal{S}}}\right)\_{j,k}^{p+1} \left(v\_{\mathcal{V}}\right)\_{j,k}^{p+1}\right] \end{split} \tag{16}$$

$$\begin{split} -\frac{\tau}{2h\_{\text{y}}} m\_{j,k-1}^{p+1} \left( \upsilon\_{\text{y}} \right)\_{j,k}^{p+1} \left( \upsilon\_{\text{y}} \right)\_{j,k-1}^{p+1} &+ \left[ m\_{j,k}^{p+1} \left( 1 + \frac{\tau}{h\_{\text{z}}} \left( \upsilon\_{\text{z}} \right)\_{j,k}^{p+1} + \chi\_{2} \tau \left( \frac{E\_{\text{s}}}{\upsilon\_{\text{s}}} \right)\_{j,k}^{p+1} \right) \right] \left( \upsilon\_{\text{y}} \right)\_{j,k}^{p+1} \\ &+ \frac{\tau}{2h\_{\text{y}}} m\_{j,k+1}^{p+1} \left( \upsilon\_{\text{y}} \right)\_{j,k}^{p+1} \left( \upsilon\_{\text{y}} \right)\_{j,k+1}^{p+1} \\ = \left( m\upsilon\_{\text{y}} \right)\_{j,k}^{p} + \frac{\tau}{h\_{\text{z}}} \left( \upsilon\_{\text{z}} \right)\_{j,k}^{p+1} \left( m\upsilon\_{\text{y}} \right)\_{j-1,k}^{p+1} + \chi\_{2} \tau \left( E\_{\text{y}} \right)\_{j,k}^{p+1} \end{split} \tag{17}$$

$$\frac{\partial n}{\partial t} + \frac{\partial}{\partial z} \left( v\_x n - D \langle \mathbf{w} \rangle \frac{\partial n}{\partial z} \right) + \frac{\partial}{\partial y} \left( v\_y n - D \langle \mathbf{w} \rangle \frac{\partial n}{\partial y} \right) = \frac{\partial n}{\partial t} + \hat{A}\_x n + \hat{A}\_y n = 0 \tag{18}$$

$$n|\_{z=0} = n|\_{z=l\_x} = n\_0 \quad \text{and} \quad \left. \frac{\partial n}{\partial \mathbf{y}} \right|\_{\mathbf{y}=0} = \left. \frac{\partial n}{\partial \mathbf{y}} \right|\_{\mathbf{y}=l\_\mathbf{y}} = \mathbf{0} \tag{19}$$

$$\hat{A}\_z n = \frac{\partial}{\partial \mathbf{z}} \left( v\_\mathbf{z} n - D(\mathbf{w}) \frac{\partial n}{\partial \mathbf{z}} \right) \text{ and } \hat{A}\_\mathbf{y} n = \frac{\partial}{\partial \mathbf{y}} \left( v\_\mathbf{y} n - D(\mathbf{w}) \frac{\partial n}{\partial \mathbf{y}} \right) \tag{20}$$

$$\frac{\frac{3}{2}n^{p+1} - 2n^p + \frac{1}{2}n^{p-1}}{\pi} + \hat{A}\_z n + \hat{A}\_y n = 0\tag{21}$$

$$\frac{\partial n}{\partial t} + \frac{\partial}{\partial z} \left[ \nu (E\_z |\_{x=0}) n - D \frac{\partial n}{\partial z} \right] = 0 \tag{22}$$

$$
\Delta \varphi = -\frac{q}{\varepsilon\_0 \varepsilon} (n - n\_0) \delta(\chi) \tag{23}
$$

$$\frac{\partial \tilde{n}}{\partial t} + \frac{\partial}{\partial z} \left[ v\_0 \tilde{n} + \frac{dv}{dE} n\_0 \tilde{E}\_s - D \frac{\partial \tilde{n}}{\partial z} \right] = 0 \tag{24}$$

$$
\Delta \tilde{\varphi} = -\frac{q}{\varepsilon\_0 \varepsilon} (\tilde{n}) \delta(\mathbf{x}) \tag{25}
$$

$$\tilde{N}\left(-i\omega + ik\nu\_0 + Dk^2 + ik\frac{d\nu}{dE}n\_0\tilde{E}\Big|\_{\mathbf{x}=0}\right) = 0\tag{26}$$

$$k\frac{d^2\phi}{dx^2} - k^2\phi = -\frac{q}{\varepsilon\_0\varepsilon}\mathfrak{N}\delta(\mathfrak{x})\tag{27}$$

$$\frac{d^2\phi}{dx^2} - k^2\phi = 0 \text{ for } \mathbf{x} \neq \mathbf{0} \tag{28}$$

$$
\left.\frac{d\phi}{d\boldsymbol{x}}\right|\_{\boldsymbol{x}=\boldsymbol{+}0} - \frac{d\phi}{d\boldsymbol{x}}\Big|\_{\boldsymbol{x}=-\boldsymbol{0}} = -\frac{q}{\varepsilon\_0 \varepsilon} \tilde{N} \tag{29}
$$

$$\phi = \begin{cases} A \exp(-kx), x > 0 \\ B \exp(kx), x < 0 \end{cases} \tag{30}$$

$$\tilde{\varphi} = \frac{q}{\varepsilon\_0 \varepsilon} \frac{\tilde{n}}{2k} \exp(-k|\mathbf{x}|) \tag{31}$$

$$\left.\tilde{E}\_{\mathbf{z}}\right|\_{\chi=0} = -ik\,\tilde{\varphi}\vert\_{\chi=0} = -i\frac{q\,\tilde{n}}{2\varepsilon\_0\varepsilon} \tag{32}$$

A Numerical Study of Amplification of Space Charge Waves in n-InP Films 229

with a case of the GaAs film, it is possible to observe an amplification of space charge waves in InP films at essentially higher frequencies *f* > 44 GHz. To obtain an amplification of 25 dB, it is necessary to use a distance between the input and output antennas of about 0.09 mm.

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

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

harmonics is presented. The input carrier frequency is *f* = 12 GHz.

**4. Results and simulation** 

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

$$
\lambda \tilde{N} (-i\omega + ik\upsilon\_0 + Dk^2) + ik \frac{d\upsilon}{dE} n\_0 \frac{(-i)q\tilde{N}}{2\varepsilon\_0 \varepsilon} = 0 \tag{33}
$$

or

$$
\omega = \upsilon\_0 k - i Dk^2 - i \frac{q n\_0}{2 \varepsilon\_0 \varepsilon} \frac{d\upsilon}{dE} k \tag{34}
$$

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 amplified instead of being damped. In general, we consider the cases where  *= 2f* is real 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 bias electric field, *E*0 = 20 kV/cm.

with a case of the GaAs film, it is possible to observe an amplification of space charge waves in InP films at essentially higher frequencies *f* > 44 GHz. To obtain an amplification of 25 dB, it is necessary to use a distance between the input and output antennas of about 0.09 mm.
