## Abstract

This article analyzes nonlinear light propagation in semiconductors with bipolar conductivity and nonlinear transport of electrons. We show how the competition between electron and hole conductivity can influence light propagation, leading to the self-bending effect of optical beam trajectory, which depending on the value of trap compensation coefficient may be stationary or transient.

© 2014 Optical Society of America

## 1. Introduction

One of the aims of research on the propagation of spatial solitons [1] is to find ways to use them for creating reconfigurable optical circuits [2]. One promising way to achieve such aim is to investigate phenomena taking place in nonlinear light propagation in photorefractive materials. Characteristic feature of photorefractive effect is that it may take place in media where free charge carriers are excited by light. Excited carriers move by diffusion and under the influence of external electric field, until the moment of their recombination with traps, which is most likely to take place in unlighted areas. If the light falling on the material has non-uniform intensity, it will cause a non-uniform spatial charge distribution, resulting in the appearance of internal electric field. This field causes the refractive index to change through the electro-optic effect, which influences the propagation of optical beam and in particular conditions can lead to photorefractive soliton formation. The space-time nonlocality [3] typical of photorefractive response offers a possibility of optical induction of waveguide channels [4,5] and a control of optical beam trajectory, which may be subject to bending [6,7].

Despite such interesting possibilities, practical implementation of the mentioned solutions is hindered by the main shortcoming of the media under consideration, that is their long response time. In this context research aimed at seeking new, fast response photorefractive materials takes on special importance. Short response time and the resultant fast switching and conversion of optical signals may be achieved in photorefractive semiconductors [8]. High mobility of free carriers in semiconductors allows to shorten a typical response time of a fraction of a second to microseconds [9]. However, semiconductors have other advantages apart from short response time. For telecommunications applications, essential features of semiconductors are their near infrared range and the fact that the technology of semiconductor growth allows to integrate optical devices with lasers and other components of integrated optics. Given the attractiveness of semiconductors for various applications, the knowledge of their nonlocal photorefractive response seems particularly valuable.

Analyses of photorefractive processes in InP [10] or CdZnTe [11] have been of much interest in the research area just mentioned. These materials demonstrate an effect of resonance amplification of space-charge field, a basis for explanation of processes observed in these materials. This article focuses on another group of semiconducting media. I have analyzed materials with two vital properties: bipolar conductivity and hot-electron effect [12,13]. Both features are essential, and their simultaneous presence combined with a selection of material parameters allows to observe self-bending of optical beams trajectories as a stationary or transient process.

## 2. Photorefractive hot-electron effect

The photorefractive mechanism in semiconductors, for instance GaAs, is particularly complex. There is an asymmetry in transport of holes and electrons, the latter being nonlinear. Gallium arsenide belongs to multi-valley semiconductors. The conductivity band of such materials consists of a central valley Γ and higher located side valleys L, characterized by large effective mass. In the state of thermodynamic equilibrium, most electrons are present in the central valley. At low values of the electric field the energy is conveyed to the lattice mainly via intra-valley scattering on acoustic phonons. As the field grows, so does the mean energy of electrons, while the energy relaxation becomes less effective. Consequently, the so called “hot electrons” are formed, which are attributed higher effective temperature than the temperature of the lattice *T _{L}* [12,13]:

*q*is an electron charge,

*k*is Boltzmann's constant,

_{B}*v*(

_{n}*E*) is a drift velocity of electrons, that depends on the electric field

*E*,

*τ*is mean time of energy relaxation ranging from 0.1 ps to 1 ps [12,14]. While the electric field further increases, a significant portion of electrons achieves an energy that allows them to pass to the side valley, where they obtain a high effective mass (inter-valley scattering on optical phonons). In descriptions of this phenomenon the mobility of electrons is usually expressed as a weighted average:where

_{r}*μ*and

_{nl}*μ*are, respectively, carrier mobilities in the central and side valleys,

_{nu}*f*(

*E*) is a distribution function describing the population of central valley, expressed by this relation:

*U*is the energy difference between the central and side minimum,

*R*is a ratio of state densities in the central and side valleys. For gallium arsenide

*R*= 94, Δ

*U*= 0.31 eV. This mechanism can be embedded in a traditional model of the photorefraction of semiconductors, describing the photogeneration, transport and trapping of the carriers. When traps are deep acceptors with a concentration

*N*compensating shallow donors with a concentration

_{A}*N*, we get this system of equations [15–19]:

_{D}*n*is a concentration of electrons,

*p*is a concentration of holes,

*β*is the coefficient of thermal generation of carriers from deep traps,

*γ*is the coefficient of the recombination of carriers with traps,

*S*is the photoionization cross section divided by photon energy,

*J*is current density,

*I*is light intensity distribution. The indices

*n*and

*p*refer to quantities characterizing, respectively, electrons and holes.

Nonlinear light propagation in thus characterized medium is usually analyzed within the paraxial approximation of wave equation:

**A**is an amplitude of optical wave propagating along axis

*z*, $k=2\pi {n}_{b}/\lambda $ is a wave number, $\omega =2\pi c/\lambda $ is angular frequency, $\Delta n$ is optically induced change in the refractive index ${n}_{b}$. The system of material Eqs. (4) is commonly used in descriptions of the photorefractive mechanism in bulk, semi-insulating gallium arsenide [19] or in the multiple quantum wells structures based on this material (GaAs/AlGaAs) [14–20]. In the presented case system of Eqs. (1)–(5) is used to describe propagation in 1 + 1D configuration (for light polarized in the

*x*direction and propagating in the

*z*direction). Furthermore, I have selected an acceptor model, but note that the ratio of electron concentration to hole concentration is essential, causing the qualitative change of the obtained results. The ratio can be controlled by changing the trap compensation coefficient in acceptor and donor models alike.

## 3. Self-bending of light in hot-electron photorefractives

The effect of bending the trajectory of soliton beam can be observed thanks to the asymmetry in the distribution of the refractive index, that traps the light. Commonly, this effect is created in conditions where the beam width is comparable to the length of charge carrier diffusion. If the diffusion field created during propagation is sufficiently strong, then the soliton trajectory bends parabolically. The size of diffusion field causing asymmetry in the distribution of refractive index depends on the width of optical beam (that is why beams with relatively large widths are hardly liable to diffusion-induced self-bending). In semiconductors, an optical beam can be bent by another mechanism.

One disadvantage of materials such as GaAs is their low value of linear electro-optical coefficient (*r*_{41} = 1.43x10^{−12} m/V for *λ* = 1.06μm). In this case changes in refractive index required for the compensation of diffraction are available in two ranges: high values of the external electric field and relatively wide optical beams. Another possibility arises in an intermediate case, at a boundary region of the two experimental cases. If a beam with Gaussian distribution of intensity and a radius *w* = 14.5 μm (FWHM ≈24 μm) were to propagate in conditions in which a soliton with similar dimensions propagates, then the applied electric field should have an approximate value 35 kV/cm. Let us see which phenomena will accompany the propagation of such beam in semi-insulating gallium arsenide described by Eqs. (1)–(4) in conditions allowing for its trapping.

The presented problem of propagation can be solved numerically, by using the finite difference method discussed in the work [18]. The method is based on a self-consistent algorithm solving Eqs. (1)–(5) within two-dimensional approximation 1 + 1D. It is worth noting that used numerical algorithm has limitations related to the approximation, referred to as “time-local approximation” [21,22]. This simplified approach, however, should give a qualitatively correct insight into the phenomena occurring in the considered media and may be treated as a first approach to this complex problem. All the results yielded by the method have been generated by using material parameters for the semi-insulating gallium arsenide from the works [17,19].

I have studied two basic situations, in which I have observed different behaviour of the propagating beam. In one case the concentration of electrons is comparable to or higher than the concentration of holes. In the other case hole conductivity prevails. To illustrate the former situation, let us choose a case corresponding to the trap compensation coefficient *r* = *N _{D}*/

*N*= 0.88 (

_{A}*N*= 4.4 x 10

_{D}^{16}cm

^{−3},

*N*= 5 x 10

_{A}^{16}cm

^{−3}). For homogeneous illumination

*I*= 12 W/cm

_{B}^{2}, with the assumed parameters, the electron concentration to hole concentration ratio is

*n*

_{0}/

*p*

_{0}≈2.06.

Media 1in Fig. 1(a) presents time-space evolution of a Gaussian beam with maximum intensity *I _{M}* =

*I*. Although the numerical approach used allows to observe all variables in two dimensions, for clarity, in the case of space-charge field the response of the medium in one dimension is shown (Media 2 in Fig. 1(b)). In this case, the induced space-charge field and consequent changes in refractive index have an asymmetric distribution. Observed from the start of temporal evolution, the asymmetry remains in a stationary state, inducing self-bending of optical beam trajectory. The standard, approximate approach used in an analysis of transient states occurring during nonlinear propagation in photorefractive materials assumes two things: that the distribution of carrier concentration settles much faster than the distribution of space-charge field, and that this distribution is a linear function of light intensity [23]. In the presented case, carriers have distributions which are not a simple function of intensity, and their evolution extends in time much longer than time comparable to that of recombination. Media 3 and Media 4 in Figs. 2(a) and 2(b) illustrate the dynamics of carrier concentration distributions coupled with the electric field evolution. The films do not include the initial stage, whose time is comparable to the lifetime of carriers. The evolution of carriers is characterized by their different initial distributions (for

_{B}*t*= 0.1 μs). In case of electrons, a strong nonlocality relative to light intensity distribution is visible, caused by high drift length. As neither electron concentration nor hole concentration is dominant in the examined case, both types of carrier affect the propagation.

The extent of asymmetry in the refractive index distribution depends on the compensation coefficient, or precisely, on the ratio of electron concentration to hole concentration. As the hole concentration increases, the bending of trajectory becomes smaller - Fig. 3(a). When the ratio *n*_{0}/*p*_{0} is sufficiently low, the effect caused by electrons does not influence the electric field distribution, and propagation is rectilinear, as illustrated by the trajectory V in Fig. 3(a) for *r* = 0.4 (*n*_{0}/*p*_{0} ≈0.017).

To separate the effect of hot electrons, we present in Media 5 in Fig. 3(b), the propagation of optical beam with the compensation coefficient *r* = 0.88. In this case, however, the simulation has been made with an assumption that the electron transport is linear. In this situation one can observe self-bending of beam trajectory which occurs in transient states, but not in a stationary state. In this case, concentrations of carriers, despite a drift at the onset of time evolution, at a stationary state take the distributions that are approximately a linear function of light intensity distribution [17]. We can observe a similar phenomenon during the propagation of a beam in conditions of domination of holes. In such a case, however, the asymmetry of electric field, associated with initial hole drift, appears on the opposite side. This effect is shown in Media 6 and Media 7 in Fig. 4 illustrating light propagation for a compensation coefficient *r* = 0.4.

## 4. Conclusions

This work analyzes nonlinear light propagation in semiconductor materials characterized by bipolar conductivity and the hot-electron effect. Two types of optical beam self-bending are presented: stationary and transient. This analysis has been done in realistic conditions describing semi-insulating gallium arsenide. Although the discussed results refer to one value of the external electric field, it follows from the research that qualitatively similar effects can be observed for fields of other values (above the critical electric field where the nonlinear electron transport is present).

Comparison of the soliton propagation in the case of linear and nonlinear electron transport can bring out the distinctive features of the latter. The asymmetry in the distribution of the refractive index that causes self-bending, depends on the value of electric field. Size of such asymmetry increases with the increasing electric field, wherein the character of this relation is saturable. As a result, in the case of nonlinear electron transport, it is possible to bend relatively wide beams (as opposed to the asymmetry caused by the process of diffusion, which increases with a decrease of the beam width). The comparison of the time needed to reach steady-state in the case of linear and nonlinear propagation is also interesting. Contrary to the suggestions emerging in the analysis of photorefractive gratings [20], soliton formation time seems to be longer in nonlinear case.

Since the goal of this paper is to indicate that “hot electrons” can have significant impact on the light propagation in semiconductors, several issues merit further discussion. First wide and open problem is connected with higher dimensionality. In presented work, 1 + 1D model is considered. It have an applicative potential mainly in the slab waveguide configuration, which is particularly important in semiconductor optoelectronic devices. However, as 2 + 1D beams also have extensive field of applications, the influence of anisotropy associated with higher dimensionality on propagation of solitons in the regime of nonlinear transport seems to be important and valuable matter. Second interesting problem refer to the analysis of stability. Presented simulations show that bright soliton-like beams are stable, notwithstanding the bending. First look at this issue suggests that it can be considered in terms of a built-in symmetry in the nonlinear equation, as discussed in [24]. However, in considering stability as well as other important issues connected with nonlinear electron transport, valuable but difficult problem may be an attempt to obtain an analytical expression describing its influence on the propagation. From both fundamental and practical perspective, an analysis of the impact of hot electrons on the propagation of dark solitons will be also interesting. Dark solitons require lower changes of the refractive index, and may be generated by using a stronger, quadratic electro-optic effect occurring near the absorption band edge of semiconductors.

## Acknowledgments

This work has been partially supported by the National Center for Science under the project awarded by decision number DEC-2011/01/B/ST7/06234.

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