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Abstract
The article presents analysis of the formation of charge carrier domains generated by a localized optical beam and the phenomenon of their oscillations. The research was carried out for bipolar photorefractive semiconductors characterized by nonlinear transport of electrons. The analysis allowed us to determine a set of basic quantitative parameters that have an impact on the appearance of carrier oscillations and their character.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Although it is not always noticeable, nonlinear phenomena are strongly inscribed in the nature of physical systems of the world around us. As a result, they play an important role in almost all branches of modern science. The measurable result of research on this class of phenomena are, among others, nonlinear effects occurring in semiconductor media. In this article, we would like to draw attention to the phenomena occurring in semiconductors exhibiting photorefractive nonlinearity. In particular, we will focus our attention on materials such as photorefractive GaAs [1–3] or InP [4–6] crystals and on structures of reduced dimensionality such as semi-insulating multiple quantum wells based on the GaAs/AlGaAs material system [7,8]. The materials selected for this study are well-suited for research on nonlinear phenomena and at the same time provide a good platform for their applications. The practical aspect includes the possibility of innovative solutions in integrated optics systems, with particular emphasis on applications in optical telecommunications. Considering the discussed class of media from a fundamental point of view, worth noting is the fact that they have a unique combination of two types of nonlinearity: optical nonlinearity [9] and nonlinearity of electronic transport [10,11].
Photorefractive semiconductors research is usually carried out in the context of multiwave mixing [12–14] or self-trapping of light [15,16]. Nonlinear transport of electrons, incorporated into the photorefractive mechanism, is considered in this case only as a factor having an impact on the photorefractive response, e.g., during the build-up of a photorefractive grating [17–19], or photorefractive amplification in nonlinear mixing of waves [20,21]. However, it is common knowledge that nonlinear electron transport has a wide use in electronics, being the cause for the formation of high electric field domains (the Gunn domains) [22], and a basis for the construction of high-frequency electronic oscillators. Considering the scope of this paper, it is worth mentioning that as early as at the initial phase of research on the Gunn effect, it was discovered that electric field domains could be triggered by means of light [23,24]. However, the concept of combining the generation of high electric field domains with the photorefractive mechanism was not proposed until two decades later. First, the excitation of low-frequency current oscillations, distinctly different than high-frequency Gunn oscillations was observed. The oscillations were triggered both by a single laser beam [25] and as a response to an interference pattern [26]. Next, the possibility of excitation of high-field Gunn domains was investigated in the context of amplification of a moving photorefractive grating [27]. Synchronised with the moving interference pattern, electric field domains are able to amplify optically-induced changes of the refractive index. A numerical analysis of the phenomenon, referred to as the photorefractive Gunn effect, shows that it can lead to both periodic and chaotic solutions [28]. Within the area of research on the photorefractive Gunn effect, noteworthy are also studies showing the possibility of the external control of high-frequency oscillations by varying parameters of a transient grating [29,30].
It is worth noting that research indicates several, different sources of domain instability and current oscillations. While high-frequency Gunn oscillations are caused by an intervalley scattering mechanism [10,11], low-frequency oscillations result from the mechanism of electric-field-enhanced trapping of carriers at defect levels [31]. It is commonly accepted that both mechanisms lead to the phenomenon of the negative differential resistance. Different, not related to negative differential resistance, mechanism of instability is presented by Korneev et al. [32]. In this case instability originates from a special type of photoconductivity temporal response that is related with dynamics of filling shallow level traps.
In this paper, the authors describe a new, to their best knowledge, phenomenon, which has its source in the mechanism of intervalley scattering of electrons. The described effect consists in the formation of charge carrier domains induced by a localized optical beam. We are not dealing here with a separate, traveling Gunn domain, but we observe local oscillations of electron and hole domains, which are coupled with oscillations of the electric field. The numerical analysis allowed us to isolate the basic parameters that affect the appearance of oscillations and their nature.
2. Theoretical background
Semiconductors such as GaAs or InP have an interesting band structure. In the conduction band of both materials apart from the central $\mathrm{\Gamma }$ valley, characterised by high electron mobility, there are several side valleys with higher energies. The impact of side valleys on material properties, described in the framework of the so-called intervalley scattering model [10,11], relies on the assumption that ‘heated’ by the electric field electrons pass from the central $\mathrm{\Gamma }$ valley to the side L valleys. The L valleys (0.31eV away for GaAs and 0.53eV for InP), have significantly smaller curvature than the central valley and the associated higher values of the effective mass of electrons. As a consequence, after exceeding the critical value of the electric field strength, the previously increasing speed of electrons begins to decrease. This phenomenon is referred to as negative differential resistance (NDR). Devices which use this effect are known as transferred-electron-devices (TED) [33].
From a formal point of view, the modeling of nonlinear electron transport can be reduced to establishing the appropriate relationship between the speed of electron drift and the electric field. Among the various possible approaches (whose discussion goes beyond the scope of this paper), good results can be obtained based on a simple interpolation formula [11]:
where ${v_s}$ - drift saturation velocity, ${E_s}$ - saturation field, while A and $\beta $ are constants determined on the basis of the value of ${\mu _n}$ denoting electron mobility in the central valley. For bulk GaAs crystal, the values are, respectively, ${v_s} = 8.5 \times {10^4}m/sec$, ${E_s} = 1.7 \times {10^5}V/m$, $A = 0.04$, and $\beta = 4$. For quantitative description of nonlinear transport of electrons, an additional parameter ${\mu _{dn}} = d{v_n}/dE,$ describing the so-called differential electron mobility is introduced. Figure 1 shows the relations described by Eq. (1), i.e., the dependence of electron drift velocity and differential electron mobility on the electric field intensity for bulk GaAs crystal. It can be seen that a decrease in electron velocity and negative values of differential mobility are available for electric fields higher than 4 kV/cm. For an InP crystal, relationships ${v_n}(E )$ and ${\mu _{dn}}(E )$ are shaped in a similar way, save for the fact that the threshold value of the electric field is higher (the side L valley is located higher) and equals ca. 12 kV/cm [34].
Fig. 1. Dependence of electron drift velocity and differential mobility on intensity of the electric field, for bulk GaAs crystal.
The area of negative differential electron mobility is also available for GaAs/AlGaAs heterostructures. The threshold value of the electric field in this case is assumed at ca. 4 kV/cm, similar to that for bulk GaAs crystal [18,21,35].
An important factor affecting the considered oscillations is the bipolar nature of charge carrier transport. It is assumed therefore that the media under analysis are characterised by nonlinear transport of electrons and linear transport of holes. Outcomes of most of the numerical calculations shown herein have been obtained for comparable concentrations of electrons and holes (with only a slight advantage of one type of carrier). A theoretical description of the photorefractive mechanism occurring in materials with such characteristics can be based on a standard system of nonlinear differential equations, including equations of generation, transport and trapping of carriers, continuity equation, and Gauss's law. In the case under analysis, deep traps are donors with a concentration ND, which compensate the impact of shallow acceptors with a concentration NA. Such system of dopants ensures that dark conductivity is low and concentration of ionised donors is different from zero. The model under analysis is limited to physical processes occurring in one dimension. However, to make the description more clear, it can be assumed that the described processes are induced by an optical beam of intensity I, propagating towards the z axis and polarised towards the x axis. Below is a system of equations which meet the assumptions referred to above:
The issue described by Eqs. (1) and (2) can be solved numerically, by the finite differences method with an algorithm similar to that discussed in [36]. In order to adapt the numerical procedure described therein to the issue under analysis, it has been complemented with nonlinear electron transport and extended with hole transport. In the simulations carried out as part of this study, the semiconductor was illuminated by an optical beam with the Gaussian distribution and a step switching characteristic:
where L - is the width of the tested sample, w - beam radius, $\theta (t )$ - the Heaviside step function. The problem of domain instability analyzed with numerical methods is inextricably linked with the issue of boundary conditions. Contrary to the case when traveling electric domains cause current oscillations in the external circuit, the phenomena analyzed here can be seen as local, i.e. the oscillations of electric charge domains occur in the vicinity of a narrow optical beam illuminating much wider sample. For this reason, the boundary conditions used here are similar to those used in the soliton propagation analysis, i.e. the constant electrical potential values were used at the edges of the sample. It is also worth adding that the disadvantage of our numerical approach is the long computational time. Since the dynamics of the phenomena under analysis depends on light intensity, in order to shorten the calculation time, most of the simulations were performed for a light intensity relatively high for photorefractive materials (more than 500 mW/cm2). The studied sample had a width of $L$ = 2 mm, and the optical beam was positioned in the centre.3. Results
The photorefractive mechanism is based on an electro-optic phenomenon. For semiconductors, several types of this effect are available. The most obvious is to use the linear electro-optic effect that occurs, for example, in bulk GaAs crystals. The effect is available in a wide spectral range, in which it is easy to find a point with a small absorption coefficient (that allows propagation of light). However, it should be remembered that the value of the linear electro-optic coefficient for GaAs is relatively low, which may require the use of high electric fields. Another option is to use the quadratic electro-optic effect which occurs in such type of media. This effect results from the Franz-Keldysh phenomenon present around the edge of absorption of semiconductors [2,37]. In this case, the effect strength is greater, however, it is associated with a higher value of absorption. The third option is to use the phenomenon of near-resonant excitonic electro-absorption occurring near the absorption edge of quantum well structures. It is worth noting here that these structures can work in transmission geometry [7,12,14] (with the light propagating across quantum wells), or constitute the guiding layer of the waveguide [15,38].
As a point of reference in this study, we chose an analysis carried out as part of research into self-trapping of light, where the linear electro-optic effect in bulk GaAs crystal is used [16]. In spite of certain differences in the model, the phenomena presented herein can be treated as an extension of the work [16] in the field of analyzing the impact of nonlinear electron transport on the photorefractive response of gallium arsenide. It is worth adding, however, that the phenomena under analysis do not depend on the type of electro-optic effect and will occur regardless of which of the above-mentioned options we choose. All the results presented in this paper have been generated using material parameters typical of semi-insulating GaAs crystals. These parameters are listed below in Table 1.
Table 1. Material parameters used in calculations.
For non-diffractive propagation of a Gaussian beam of a radius w = 14.5 µm (FWHM ≈ 24 µm), discussed in [16], an electric field of about 34 kV/cm is required. These values determine the conditions for which the initial calculations have been made. A preliminary analyses allowed us to extract a set of basic physical quantities which have an impact on the occurrence and nature of local oscillations of charge carrier domains. The parameters whose values are controllable include: a) the degree of nonlinearity of electron transport (described, in quantitative terms, by the electric field-dependent value of negative differential mobility), b) the ratio of electron to hole concentration, which depends mainly on the concentration of dopants (or traps), and c) the optical beam intensity.
For a better understanding of the study results, let us begin with a situation in which the medium does not produce oscillations, i.e., the transport of charge carriers is dominated by holes. As an illustration of this situation, the case corresponding to the concentration of traps ${N_D} = 2.11 \times {10^{16}}c{m^{ - 3}}$, ${N_A} = 0.6 \times {10^{16}}c{m^{ - 3}}$ was chosen. For homogenous lighting with IB = 1 W/cm2 and the parameters referred to above, the ratio of electron to hole concentration n0/p0 ≈ 0.24. Figure 2 shows time evolutions of the charge carrier concentration and the electric field intensity (Visualization 1, Visualization 2, and Visualization 3). Time evolution of electron and hole concentration can be divided into two phases. The first, initial phase, begins when the lighting is switched on and finishes once quasi-stationary distributions are obtained (Visualization 1). The duration of this phase can be roughly estimated based on time constants ${\tau _n} = 1/({{\gamma_n}{N_A}} )$, ${\tau _p} = 1/{\gamma _p}({{N_D} - {N_A}} )$. In the case under analysis, quasi-stationary distributions settle after about 2 ns.
Fig. 2. Photorefractive response of GaAs, induced by a Gaussian beam of a full width at half medium of 24 µm and the maximum light intensity of I0 = IB = 1 W/cm2 for the ratio of electron to hole concentration n0/p0 ≈ 0.24 and the external electric field of 34 kV/cm. A higher electric potential was applied to the left electrode (i.e. for x = 0). The simulation results cover: a) the initial phase of the evolution of electron and hole distributions (Visualization 1), b) the second phase of the evolution of electron and hole distributions (Visualization 2), and c) the time evolution of the electric field distribution (Visualization 3).
The second phase of evolution of charge carriers lasts longer (Visualization 2). In the case under analysis, the time window in which this phase takes place extends to 1 ms. Changes in the distributions of charge carriers are correlated with the establishment of a stationary distribution of the electric field (Visualization 3). A characteristic feature of the photorefractive response shown in Fig. 2 is its nonlocality. The induced electron and hole concentrations as well as the space charge field are characterized by asymmetry in distribution. Asymmetry of the electric field causes asymmetry of the refractive index distribution, and in consequence, the curvature of the trajectory of the self-trapping optical beam. The effect is discussed in [16]. For better clarity, all the distributions shown in the media have been rescaled by dividing the obtained values by the boundary values. As the boundary values of the carrier concentrations, the values generated under homogeneous illumination were used: ${n_0} = 2.89 \times {10^{14}}c{m^{ - 3}}$, ${p_0} = 1.18 \times {10^{15}}c{m^{ - 3}}$.
Let us see how the photorefractive response changes in the situation when electron transport have a stronger impact on it than in the case discussed above. An indicator of such a situation may be a sufficiently high ratio of electron to hole concentration. As an example, let us analyse the response obtained for the concentration of traps: ${N_D} = 5 \times {10^{16}}c{m^{ - 3}}$, ${N_A} = 0.6 \times {10^{16}}c{m^{ - 3}}$. For such parameters, the ratio of electron to hole concentration (for a homogenous lighting IB = 1 W/cm2) is n0/p0 ≈ 2.07. The results are shown in Fig. 3, Visualization 4, Visualization 5 and Visualization 6. A presentation of the first, initial phase of evolution of charge carrier distributions has been skipped here, as its course is similar to that shown in Visualization 1.
Fig. 3. Photorefractive response of GaAs, induced by a Gaussian beam of a full width at half medium of 24 µm and the maximum intensity of I0 = IB = 1W/cm2, for the ratio of electron to hole concentration n0/p0 ≈ 2.07 and the external electric field of 34 kV/cm. A higher electric potential was applied to the left electrode. The following boundary values were adopted: ${n_0} = 8.43 \times {10^{14}}c{m^{ - 3}}$, ${p_0} = 4.06 \times {10^{14}}c{m^{ - 3}}$, $N_D^ + \; = 6 \times {10^{21}}c{m^{ - 3}}$. The simulation results cover: a) the evolution of electron and hole distributions (Visualization 4), b) the time evolution of the donor concentration (Visualization 5), and c) the time evolution of the electric field distribution (Visualization 6).
As illustrated by the movie Visualization 4, during the initial phase of evolution of electrons and holes, the charge carrier domain separates and travels (electrons and holes in the same direction) towards the electrode of a higher potential. The detachment of the domain occurs at the border of the area illuminated by the optical beam. Having travelled a certain distance (300 µm in the case under analysis, i.e. about 12.5 beam widths), the domain stops and begins to oscillate. In order to quantify the dynamics of the phenomenon under consideration, it is worth referring the characteristic times to the dielectric relaxation constant. For assumed material parameters and illumination of 1 W/cm2, the dielectric relaxation time has a value of about τd = 1.6 µs. The formation time of the domain is about 70 times longer, while the time it takes for the domain to travel (until it stops and oscillates) is about 1100 τd. An estimated domain speed is around 15 cm/s. The oscillations of the carrier domains are coupled with oscillations of the space charge and the oscillating distribution of the electric field, as shown in Visualization 5 and Visualization 6. Careful observation of the oscillating distributions of carrier concentrations shows that once the domains of electrons and holes reach point x = 680 µm, they begin to disappear, entailing a change in the space charge and the electric field distribution in this area. At the same time, several dozen micrometers behind (at point x = 730 µm), new local maxima are formed. The whole process repeats itself many times with a period of about 0.55 ms. It is worth noting here that the spatial charge generated in the described process takes the form of a dipole domain, one part of which breaks off, moves a certain distance and begins to oscillate. Although in this article we have focused on the oscillations of the free-charge domains, it should be remembered that they are coupled to oscillations of a much larger space charge. This fact may turn out to be interesting in the context of optical phenomena, the analysis of which, however, is beyond the scope of this article. Since the films presented herein cover only several periods of oscillation, Fig. 4 shows the results of a simulation carried out in a time window of 8 ms. The results shown in this figure lead to a finding that the oscillation amplitude within the time window under analysis does not change (the oscillations are non-damped).
Fig. 4. Time evolution of concentration distributions of: a) electrons and b) holes, determined for a time window of 8 ms. The calculations have been performed taking into account the parameters of the simulation shown in Fig. 3.
An important parameter characterising oscillations of carrier domains is their frequency. In the Gunn effect, domain-induced oscillations of the external current usually have frequencies in the gigahertz range. The phenomenon described in this article has lower frequencies. In the numerical experiments carried out as part of the study, estimated frequencies of the oscillating domains equalled less than ten kilohertz. What is important, both the frequency and the amplitude of oscillations are influenced by the parameters referred to above, such as: the value of the external electric field, which determines the degree of nonlinearity of electron transport, the ratio of electron to hole concentration, and the intensity of the optical beam. The impact of these quantities on the observed oscillations is briefly discussed below.
3.1 Impact of light intensity
The value of light intensity is a parameter on which the concentration of optically excited carriers depends. Since the concentration of carriers is an important parameter in the phenomenon under analysis, we have conducted a series of numerical experiments consisting in the observation of oscillations frequency and amplitude for various light intensities. All the experiments were carried out at the same values of the external electric field and the ratio of electron to hole concentration. Figure 5 shows how the frequency f and amplitude of oscillations A depend on the maximum optical beam intensity I0 whose value is varying between 0.5 W/cm2 and 3 W/cm2. The measurement points marked in the chart are determined for the external electric field of a value of 34 kV/cm and n0/p0 ≈ 2.07. The background illumination in each experiment had an intensity equal to the maximum beam intensity IB = I0.
Fig. 5. Dependence of frequency a) and amplitude b) of charge carrier domain oscillations on light intensity. The simulations have been performed for E0 = 34 kV/cm, n0/p0 ≈ 2.07, and IB = I0.
As can be seen in the analyzed range of light intensity, the frequency of observed oscillations increases linearly. However, the amplitude curve has a different shape. It remains almost unchanged within the range above 1 W/cm2. Below this threshold, the value of amplitude increases as the light intensity decreases. As mentioned earlier, in the initial phase of evolution of carrier distribution, the formed domain travels towards the electrode of a higher potential. The observations show that velocity of the detached domain falls in line with a decline in intensity of the optical beam. As a consequence, the time after which oscillations are observed increases. For example, for a light intensity of 3 W/cm2, oscillations are observed about 640 µs after the domains parts detaches, while for a light intensity of 500 mW/cm2, oscillations occur after 4 ms. In both cases, the domains oscillate after traveling a distance of about 300 µm from the optical beam.
Detachment of the charge carrier domain is not observed when the light intensity is too low. As an example, Fig. 6 shows time evolution of electron concentration distribution determined for an optical beam intensity of 10 mW/cm2 and 100 mW/cm2 . During the first simulation, which lasted 13.5 ms and was carried out for an optical beam of 10 mW/cm2, the creation of a domain was not observed. During the simulation for an optical beam of 100 mW/cm2, a domain was formed; however, the time window was too short to observe oscillations in it. Both simulations were carried out for an electric field of E0 = 34 kV/cm and the same ratio of electron to hole concentration n0/p0 ≈ 2.07. A characteristic feature that differentiates both cases is the value of electron concentration, which in the first case is lower than in the second by as much as an order of magnitude.
Fig. 6. Time evolution of the electron concentration distribution for a light intensity of a) I0 = 10 mW/cm2 and b) I0 = 100 mW/cm2, within a time window of 13.5 ms.
3.2 Impact of the ratio of electron to hole concentration
The ratio of electron to hole concentration indicates how strongly nonlinear transport of electrons contributes to the photorefractive response. Technically, the dominance of electron transport over hole transport can be obtained by means of proper doping. For a high value of the electric field, such as that used in the previous simulations (34 kV/cm), the nonlinearity of electron transport is manifested as in a photorefractive response when the advantage of the concentration of electrons over those of holes is at least about twofold (like n0/p0 ≈ 2.07 in Figs. 3–5). Studies carried out for smaller concentrations of donors show that a slight decline in the value of the n0/p0 ratio causes the ceasing of the oscillation phenomenon. The threshold at which nonlinear transport of electrons is manifested in the photorefractive response is n0/p0 = 1.89. Below this threshold, the influence of hole transport on the photorefractive response is strong enough to prevent the formation of charge domains. The photorefractive response for the n0/p0 values smaller than 1.89 is similar to that shown in Fig. 2. As an example, Fig. 7(a) shows the time evolution of electron concentration distribution for n0/p0 = 1.63. For comparison, Fig. 7(b) presents electron distribution for the threshold value of n0/p0. In this case, the domain forms then it drifts towards the anode and stops after traveling a distance of about 200 µm (approximately 8 beam widths). The oscillations begin, but have a small amplitude and quickly disappear.
Fig. 7. Time evolution of electron concentration distributions for two values of the electron to hole concentration ratio: a) n0/p0 = 1.63, b) n0/p0 = 1.89. Other simulation parameters are as follows: I0 = 1 W/cm2, E0 = 34 kV/cm, IB = I0. In both cases, electron concentration distributions have been rescaled for the boundary values: a) ${n_0} = 7.47 \times {10^{14}}c{m^{ - 3}}$, b) ${n_0} = 8.05 \times {10^{14}}c{m^{ - 3}}$.
If the concentration of donors is increased, the n0/p0 ratio goes up and thus the influence of electron transport nonlinearity increases. Along with the stronger influence of electron transport, the phenomenon of domain oscillation appears. It is worth mentioning that two types of oscillation can be observed here. One is similar in character to those discussed before (Fig. 3 and Fig. 4). A domain is formed and travels towards the electrode of a higher potential, then, after passing a certain distance, starts to oscillate at a typical frequency of several kilohertz. An example of this type of oscillation is shown in Fig. 8(a), where the time evolution of electron concentration distribution has been determined for the ratio n0/p0 = 2.17. When the n0/p0 ratio goes beyond the threshold of 2.3, the oscillations change. Figure 8(b) shows oscillations of the other type, determined for n0/p0 = 2.34.
Fig. 8. Time evolution of electron concentration distributions for two values of the electron to hole concentration ratio: a) n0/p0 = 2.17, b) n0/p0 = 2.34. Other simulation parameters are as follows: I0 = 1 W/cm2, E0 = 34 kV/cm, IB = I0. In both cases, electron concentration distributions have been rescaled for the boundary values: a) ${n_0} = 8.62 \times {10^{14}}c{m^{ - 3}}$, b) ${n_0} = 8.95 \times {10^{14}}c{m^{ - 3}}$.
As can be seen, in this case the initial phase of the domain drift, preceding the oscillation phase, does not take place. The oscillating movement starts at the point where the optical beam is located. The amplitude has a significantly higher value (comparable to the length of the initial drift of the domain in the cases discussed above), and the frequency clearly falls. Further increase of the n0/p0 ratio drives up the amplitude and reduces the oscillation frequency. In consequence, the boundary phenomena occurring in the areas of electrodes grow in importance.
3.3 Impact of electric field
A change in intensity of the external electric field directly influences the strength of the nonlinearity of electron transport. Therefore, the electric field intensity is of fundamental importance to the phenomenon under analysis. The results discussed so far have been obtained for relatively high values of the electric field, which induce changes in the refractive index of a strength sufficient to affect a propagation of the optical beam. However, the domain oscillation effect can be also obtained at lower electric fields. A necessary condition for the oscillations to take place is the existence of an electric field within the range of negative differential mobility of electrons and the appropriate value of the electron-hole ratio. For an electric field of 34 kV/cm, differential electron mobility attains a value of ${\mu _{dn}} \approx{-} 2,2c{m^2}/s \cdot V$. At such a small value, phenomena originating in nonlinear electron transport manifest only if the n0/p0 ratio causes a greater concentration of electrons than holes. When the electric field is reduced to, e.g., 15 kV/cm, the strength of nonlinearity of electron transport increases by an order of magnitude (${\mu _{dn}} \approx{-} 52c{m^2}/s.V$). In this case, domain oscillations are observed even if the transport is not clearly dominated by electrons. Figure 9 shows how frequency and amplitude of oscillations depend on the external electric field, whose value varies between 15 kV/cm and 18.5 kV/cm. The charts have been drawn for the maximum intensity of the optical beam I0 = 1 W/cm2 and n0/p0 ≈ 1. The background illumination in each experiment has an intensity equal to the maximum beam intensity IB = I0.
Fig. 9. Dependence of frequency a) and amplitude b) of charge carrier domain oscillations on the external electric field. The simulations have been performed for I0 = 1 W/cm2, n0/p0 ≈ 1, IB = I0.
In a situation where carrier transport is not dominated by either electrons or holes (n0/p0 ≈ 1), the impact of electron nonlinearity is clearly visible for electric fields of intensity up to 19 kV/cm. For the electric field intensity close to this value, the induced oscillations of charge carrier domains have a fading character, which is shown in Fig. 10(a). Above the 19 kV/cm threshold, nonlinearity of electron transport is too weak to manifest itself in the formation of charge carrier domains. In this case, the formation of stationary, asymmetric distributions of electrons and holes, similar to those shown in Fig. 2, is observed. A characteristic feature of the photorefractive response obtained for electric fields below the 19 kV/cm threshold is an increase in the length of the initial drift of domains, which rises as the electric field decreases. In the analyzed range, it reaches about 300 µm (approximately 12.5 beam widths) for an electric field of 19 kV/cm, and about 600 µm (or 25 beam widths) for an electric field of 15.5 kV/cm (see Fig. 10(b)). At the field intensity below 15 kV/cm, the nonlinearity of electron transport is so strong that the domains reach the electrode without performing any oscillations.
Fig. 10. Time evolution of electron concentration distribution for an external electric field of a) E = 19 kV/cm and b) E = 15.5 kV/cm. Other simulation parameters are as follows: I0 = 1 W/cm2, n0/p0 ≈ 1, IB = I0. In both cases, electron concentration distributions have been rescaled for the boundary value: ${n_0} = 5.94 \times {10^{14}}c{m^{ - 3}}$.
4. Conclusions
The paper discusses the photorefractive response of a semiconductor material characterised by bipolar conductivity and nonlinear transport of electrons. The main issue to which the analysis has been limited is the phenomenon of the formation of charge carrier domains induced by a localized optical beam, and the effect of their oscillations. It follows from the analyses that bipolar nature of carrier transport has a significant impact on the occurrence of these phenomena. In highly doped n-type semiconductors oscillations of the charge carrier domains were not observed, and a nonlinear character of electron transport leads directly to the formation of high-intensity Gunn domains. Therefore, the phenomena described in this article should appear in materials in which we deal with electron-hole competition.
It has been found that light intensity, electron to hole concentration ratio, and value of the external electric field are among the basic quantitative parameters which have an impact on the character of the induced oscillations. The oscillations observed during the study had frequencies ranging from several hundred hertz to several kilohertz.
While the main aim of this article was to draw attention to the phenomenon of optically induced oscillations of the charge carrier domains, the influence of the presented process on the propagating optical beam is also interesting. Since in photorefractive materials the nonlinear beam propagation is closely related to the induced electric field distribution, we can expect that the effect described by us should be related to the induction of a wide waveguide channel on one side of the beam (depending on the direction of the electric field). It seems likely that after formation of the waveguide, the beam would shift and widen significantly. Since this is only a prediction, it can be concluded that the issue merits further investigation.
Although the phenomena considered here have been analysed using the material parameters of bulk gallium arsenide, similar effects can be expected in materials with a limited dimensionality, such as GaAs multiple quantum well structures.
Disclosures
The authors declare no conflicts of interest.
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