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Nonlinear acousto-electric transport in a two-dimensional electron system

We study both theoretically and experimentally the nonlinear interaction between an intense surface acoustic wave and a two-dimensional electron plasma in semiconductor-piezocrystal hybrid structures. The experiments on hybrid systems exhibit strongly nonlinear acousto-electric effects. The plasma turns into moving electron stripes, the acousto-electric current reaches its maximum, and the sound absorption strongly decreases. To describe the nonlinear phenomena, we develop a coupled-amplitude method for a two-dimensional system in the strongly nonlinear regime of interaction. At low electron densities the absorption coefficient decreases with increasing sound intensity, whereas at high electron density the absorption coefficient is not a monotonous function of the sound intensity. High-harmonic generation coefficients as a function of the sound intensity have a nontrivial behavior. Theory and experiment are found to be in a good agreement.


Introduction
The interaction between surface acoustic waves (SAW's) and mobile carriers in quantum wells is an important method to study the dynamic properties of twodimensional (2D) systems. The SAW can trap carriers and induce acoustic charge transport as has been investigated in a number of systems in view of possible device applications [1]. Also, the SAW-method was applied to study the quantum Hall effects [2,3,4], electron transport through a quantum-point contact [5], lateral nanostructures [6], and commensurability effects in a 2D system [7]. However, all those experiments have been done in the regime of small signals and linear interaction. A recent Letter by Rotter et al. [8] reports strongly nonlinear acousto-electric effects in a 2D electron gas (2DEG), which become possible in hybrid structures based on A 3 B 5 semiconductors and LiNbO 3 [9][10][11]. In these experiments an intense SAW breaks a 2DEG into moving electron stripes and all characteristics of the acousto-electric interaction are strongly modified as compared to the linear case.
In modern hybrid structures [11] the SAW-induced potential amplitude can become comparable with the band-gap of a semiconductor. The previous Letter [8] on nonlinear effects in the hybrid structures with a 2DEG includes a brief qualitative analysis. Here we present a detailed theoretical study of nonlinear acousto-electric effects in a 2D electron system and develop a coupled-amplitude method for intense SAW's interacting with a 2DEG. Using our theoretical results we can explain main experimental observations. Including the effect of electron diffusion we find a good quantitative agreement between theory and experiment for the case of the SAW absorption coefficient in the nonlinear regime.
Nonlinear acoustic waves in bulk piezocrystals with free carriers were discussed in a number of theoretical papers [12][13][14][15][16]. The application of a dc voltage to the crystal can result in the current amplification of sound [12,14,17] and in the formation of stationary nonlinear waves [12,14,16]. Analytic results in the theory of nonlinear acoustic waves in bulk piezocrystals with free carriers were mostly obtained in the limit of small amplitudes or for the case of very intense acoustic waves [12,13,16].
For nonlinear SAW's in crystals with a 3D electron gas, a theory was developed in the limit of very high amplitudes, when the SAW bunches electrons near the crystal surface [18]. Another theoretical aspect related to the generation of the second harmonic of a SAW was studied in Ref. [19] by using the coupled-amplitude method and perturbation theory. A theory of acousto-electric interactions in a 2D electron system was developed mostly for the linear regime of interaction [20,21].
Here, we study both theoretically and experimentally the transition from the linear regime of the acousto-electric interaction to the limit of strongly nonlinear effects in a 2DEG. Our theoretical results are applied for a description of experimental data on hybrid structures [8,11]. We pay attention to density-dependences of the absorption coefficient and the SAW-velocity shift because the electron density is a tunable parameter in experiments on 2D systems. Such dependences were not discussed in detail in the context of 3D systems [12][13][14][15][16]. It turns out that for low densities, the absorption coefficient is a decreasing function of the sound intensity caused by the trapping of electrons in the SAW piezoelectric potential. At sufficiently high electron density, however, the absorption coefficient is a non-monotonous function of the sound intensity. This behavior can be understood in terms of a dynamical screening effect. Also, our quantitative analysis shows that the absorption coefficient at room temperature is strongly reduced due to electron diffusion. A non-monotonous behavior with increasing sound intensity was also found for the intensities of higher harmonics in a short device.
The coupled-amplitude method with the introduction of fast and slow variables was used before for a description of bulk systems [14,15]. However, the formulas from the bulk theory can not be directly applied to surface waves because of the complex character of the lattice vibrations in a SAW. To develop the coupled-amplitude method in a 2D system, we introduce a local velocity shift and a local absorption coefficient in an integral form through an electron current, a SAW potential, and an electro-mechanical coupling coefficient of a microstructure. The resulting formulas can then be applied to any type of SAW's interacting with a 2DEG. Moreover, using our approach, we can find solutions for higher harmonics for arbitrary SAW inten-sity. This is in contrast with perturbation methods developed earlier for 3D systems and valid at small SAW intensities [19]. At very high intensities, we find analytical asymptotic dependences for the high-harmonic intensities in a 2D system.
As has been shown before, the linear approximation holds when δn ≪ N s , where N s is the equilibrium 2DEG density and δn is a density perturbation due to a SAW.
In the linear theory, the absorption coefficient Γ 0 and the SAW-velocity shift due to the 2DEG δv 0 s are given by the well-known relation [20,21] − δv 0 where q is the SAW wave vector, v 0 s is the sound velocity in the absence of a 2DEG, σ 0 and D e are the 2DEG conductivity and the diffusion coefficient, respectively.
where ǫ ef f (q) is the effective dielectric constant in a 2D system. In most piezoelectric crystals, the coupling K 2 ef f ≪ 1. In GaAs, K 2 ef f = 0.00064, whereas in the hybrid structures as studied here, it is two orders of magnitude larger, in the range of 0.01 − 0.05 [10], but still much less then unity. The goal of this paper is to describe the acousto-electric interaction for the case of large amplitude SAW's, when δn ∼ N s and the perturbation theory is not longer valid. At the same time, the coupling K 2 ef f will be assumed to be much less than unity. Below, we will generalize the results following from Eq. 1 for the strongly nonlinear case, when δn ∼ N s . The paper is organized in the following way. In the first section, we will give the general equations for SAW's on a piezoelectric crystal. The second section is devoted to a coupled-amplitude method developed for the case of intense SAW's. The third and fourth sections are about phenomena related to large-amplitude SAW's in a 2DEG. Then, we will discuss experimental data on the hybrid structures and apply our theoretical results for the interpretation of experiments.

Model and general equations
In usual GaAs-based microstructures it is very difficult to realize SAW's with highamplitude potentials because the electro-mechanical coupling in GaAs is relatively weak. A strong piezoelectric interaction can be achieved in hybrid structures [8,9] ( Fig. 1). Those consist of a semiconductor layer being bonded to a piezoelectric host crystal, in our case, LiNbO 3 [8,9]. The semiconductor layer contains an InGaAs − AlGaAs quantum well (QW) with a high-quality 2DEG, to which Ohmic contacts are formed. The distance between the QW and the piezocrystal is only 32 nm, whereas the distance between the QW and the top transport gate is d = 450 nm. In our model x and y are the in-plane coordinates and z is the normal one. The QW plane corresponds to z = d, and the SAW travels in the x-direction (see Fig. 1). By changing the transport-gate voltage V t one can tune the electron density N s (V t ) in a QW. In this structure, traveling SAW's can induce very strong piezoelectric fields in the semiconductor layer due to the strong piezoelectricity of the host LiNbO 3 crystal. A SAW is induced and detected by the metallic interdigital transducers, IDT 1 and IDT 2, respectively, at room temperature [8]. The acousto-electric current is measured between two Ohmic contacts labeled 1 and 2 in Fig. 1.
SAW's in a piezoelectric crystal with a 2D plasma are described by the system of equations [22] where e = −|e| is the electron charge, ρ is the mass density, and ǫ is the dielectric constant. Further, c iklm is the elastic tensor, p lik is the piezoelectric tensor, u k (x, z, t) is the lattice displacement, and φ(x, z, t) and n(x, t) are the electrostatic potential and the 2D electron density, respectively. In Eqs. 2,3, we have used the notations ∂ l f = ∂f /∂x l andḟ = ∂f /∂t, and the sum convention for repeated indexes. t is the time, and x 3 = z, x 2 = y, and x 1 = x are the coordinates.
In our geometry, the SAW propagating in the x-direction is a purely Rayleigh wave, in which only two components of the displacement, u x and u z , are nonzero.
Hence, the electric field E is also polarized in the (xy)-plane. This case corresponds to the hybrid structures studied in experiments [8,9,10], where the 128 0 -rotated Y -cut of LiNbO 3 is used. The surface of the thin GaAs film is (001). The SAW propagates in the [110]-direction of GaAs and X-direction of LiNbO 3 .
The electron 2D plasma is described by the usual hydrodynamic equations where j is the 2D electron current, σ = |e|µn s (x, t) is the 2DEG conductivity, µ is the mobility, and D e is the electron diffusion coefficient. Eq. 5 is valid in the long wavelength limit, when ql e ≪ 1, where l e is the electron mean free path.
For simplicity, we assumed above that the 2DEG is located directly at the semiconductor-piezocrystal interface.

Coupled-amplitude method
The system of nonlinear equations (2-5) can be simplified in the limit of weak electromechanical coupling, K 2 ef f ∼ p 2 /(cǫ) ≪ 1. In this limit, we can introduce two coordinates, the "slow" variable x and the "fast" variable [14,15]. The solution is of the form (see Appendixes 1 and 2), This solution is written as a sum of harmonics with wave vectors q n = nq, where q > 0 is the wave vector of the initially generated SAW near x = 0. The vec- and the quantity δq n are determined by a linear system of equations as given in the Appendix 2. The envelope functions a n (x) are slowly changing on the scale of λ = 2π/q. It is assumed that the SAW intensity related to the vector A 0 [z; q] is unity and thus the total SAW intensity is The functions u 0 (x, z) and φ 0 (x, z) describe the static spatial distributions, that can be induced by a SAW.
The functions j(x, x 1 , t) and n(x, x 1 , t) can be written in the standard form is an envelope function. Also, the electric field E can be written in the way similar to Eq. 7.
The solution (6,7) is a sum of linear-like SAW's, which slowly vary in the amplitude and in the z-profile. The z-distribution of lattice displacement U 0 is a sum of exponential functions exp [−γ j (q n + δq n (x))z], where γ j (q) are the coefficients depending also on material constants [22]. On short distances (∼ λ) the envelope functions a n (x) can be regarded as constants and we can solve the equations (2)(3)(4)(5) considering only the "fast" variable x 1 . n n (x) and φ n (x) should be considered here as the functions of the parameters a 1 , a 2 , ..., n 0 , and E 0 . Then, having n n and φ n as functions of a 1 , a 2 , ..., n 0 , and E 0 , we can find the behavior of a n (x) on long-range The electrostatic potential is written in a self-consistent way: where φ ind and φ saw are the potentials induced by a 2DEG and by piezoelectric charges of a SAW. Using Eq.3 we write φ ind and φ saw can be expressed by the harmonic amplitudes φ ind n and φ saw n . For exam- Appendix 1) and from the conservation of charge constant including the gate electrode effect, and ǫ p and ǫ s are the dielectric constants of a host piezocrystal and a semiconductor, respectively [23].
The nth-harmonic of the SAW-potential φ saw n is given only by a n (x) and by material constants and can be easily found from the Poisson equation (10). At z = d we have φ saw n (x, d) = C n a n (x), where the coefficient C n depends on the geometry and the material constants. For example, in a crystal of the type of GaAs, C n = p 4 g(q n ), where g(q n ) is a complicated function of q n . Below, we will give the necessary relations for the hybrid structures.
To find the harmonic amplitudes n n and φ n , we have to solve Eqs. 3, 4, and 5 regarding the "slow" variable x as the constant. The slowly-varying quantities n n (x) and j n (x) can be found as Fourier components of the solution n(x 1 , x) from a "fast" equation in terms of x 1 . In a self-consistent approach, the electron density .. C n a n (x)e iqnx 1 + c.c., through a non-linear equation [8]. By using Poisson's equation, the results of Appendix 1, and Eqs. 4 and 5, we obtain where b 0 is a constant, which occurs after one integration in the conservation-ofcharge equation (4). Taking into account Γ 0 λ ≪ 1, we assume that the solution of Eq. 12 is periodic in where n 0 (x) plays the role of a "local" 2D density.
The local velocity change of a SAW δv n and the local absorption coefficient Γ n can be expressed by δq n : Using the relation (see Ref. [24]) and Eqs. 11 we write Here we use the notations < f ( In this section, we have assumed that K 2 ef f ≪ 1 and neglected the terms d 2 a n /dx 2 ∝ δq n da n /dx ∝ K 4 ef f and dδq n /dx ∝ K 4 ef f . The static electric fields φ 0 (x, z) in Eq. 7, which can be induced by a SAW, will not play an important role in this paper because we will consider the case with no voltage applied to the Ohmic contacts (V 1 = V 2 ) and a relatively short device with L ≪ 1/Γ 0 . Thus, φ 0 (x, d) ≃ const and the static electric fields E 0x (x, d) can be neglected. Eqs. 14 and 15 can now be applied to various types of SAW's by introducing specific electromechanical coupling coefficients.
3 Acousto-electric transport in a two-dimensional plasma Here, we intend to consider the SAW absorption and the acoustic charge transport in the regime V 1 = V 2 . The behavior of a 2DEG in an intense SAW at small distances (∼ λ) can be assumed to be periodic and is described by nonlinear equation (12). At long distances (≫ λ) the SAW behavior is determined by complex amplitudes a n (x), that can be found by Eqs. 14 and 15. First, we define the boundary conditions at x = 0: a 1 (0) = I 1 (0) and a n (0) = 0 for n = 2, 3, .... Here, I 1 (0) denotes the SAW intensity generated by IDT1. In this case, for a relatively short sample the SAW contains mostly the fundamental SAW harmonic n = 1 with a small admixture of the higher harmonics n = 2, 3, .... So, it follows from Eqs. 14 and 15 We assume that [ The amplitudes of higher harmonics in a short sample turn out to be given by: a n (x) = ix lim an→0 a n δq n [a 1 (0), 0, 0, .., a n , ...; n 0 , 0] = ix where n = 2, 3, 4, ... . Here, we assume that [I 1 (0) − I 1 (L)] ≫ I n , which is valid for a short sample where Γ 1 L ≪ 1, and we take into account that for the strongly nonlinear case Γ 1 ∼ Γ n . The intensity of the nth-harmonic is given by I n (x) = |a n (x)| 2 and (see Eq. 17). By using Eq. 21 we find the intensities of high harmonics at x = L: I n (L) = lim an→0 |a n δq n [a 1 , 0, 0, ..., a n , ...; 0, 0]| 2 L 2 = πL 2 (K 2 ef f /ǫ ef f )q n v 0 s e 2 |n n (0)| 2 . To calculate Γ 1 and δv 1 , we numerically solve Eq. 12 for the parameters a 1 = 0 and a n = 0 when n = 2, 3, .... Then we can find the Fourier components of n(x 1 , a 1 ).
We now calculate the electron density n(x 1 , a 1 ) in the long wave-length limit qd ≪ 1.
In this limit the function given by Eq. 13 is reduced to G(x 1 −x ′ 1 ) = (4πed/ǫ s )δ(x 1 − x ′ 1 ) and Eq. 12 now writes as where the SAW-induced piezoelectric field is taken in the form E saw = F saw .
The left-hand side of the above equation is periodic and thus the right-hand side should be periodic as well. It implies that b 0 A numerical solution of Eq. 23 n(x 1 ) at T = 300K and at various averaged densities N s =< n(x 1 ) > is shown in the inset of Fig. 3. Here, we use the following parameters: λ = 33 µm, µ = 5000 cm 2 /V s, D = µ(KT /e), ǫ s = 12.5, and v 0 s = 3.8 * 10 5 cm/s. It is seen that that with decreasing N s the formerly homogenous 2DEG turns into moving electron stripes.
It follows from Eqs. 18,19: The calculated absorption coefficient Γ 1 as a function of the electron density N s for various potential amplitudes Φ saw Fig. 2. It was calculated from Eqs. 25 and the numerical solution for n(x 1 ). The potential amplitude Φ saw 1 can be easily connected with the SAW intensity I 1 by using Eqs. 17 and 22 with K 2 ef f found numerically in Ref. [9]. the maximum of the function Γ 1 (N s ) at fixed Φ saw 1 . We will show below that in the case N s < N max s Γ 1 ≃ |e|N s (v 0 s ) 2 /(I 1 µ) ∝ N s /I 1 . This asymptotic behavior follows from the Weinreich relation [25]. In the region N s ≫ N max s , the electric current σE x decreases with increasing N s because of the screening effect. Thus, the absorption coefficient decreases as well.
In Fig. 3 we plot Γ 1 as a function of the SAW potential Φ saw 1 for a fixed 2DEG density. For electron densities less than about 10 10 cm −1 , the function Γ 1 (Φ saw 1 ) is always decreasing. At higher densities Γ 1 (Φ saw 1 ) has a maximum. We attribute this behavior to the screening effect in a 2DEG. At high density and small Φ saw 1 , the absorption is strongly suppressed because of screening. The intense SAW, however, modulates the 2DEG and, consequently, reduces screening. Thus, the absorption coefficient starts to increase with increasing Φ saw 1 , when Φ saw 1 is not so large. At larger Φ saw 1 , the plasma becomes broken into stripes and Γ 1 decreases with I 1 as 1/I 1 [8,18].
The velocity shift δv 1 numerically calculated by Eqs. 25 is shown in Fig. 4.
With increasing SAW intensity δv 1 increases and approaches to the velocity v 0 s that corresponds to the case of a totally depleted 2DEG. By analyzing Eqs. 25 and the function j(x 1 ), we can see that, in the limit I 1 → ∞, δv 1 ∝ 1/Φ saw 1 ∝ 1/ √ I 1 , like in a 3D plasma [12].
In Fig. 5 we show results of numerical calculations for the intensities of the high harmonics I n /I n,max = |n n | 2 /N 2 s , where This formula was obtained taking into account that |n n | → N s when Φ saw 1 → ∞.
The quantities I n were calculated from the Fourier components of the function n(x 1 ).
At small amplitudes Φ saw 1 , I n ∝ (Φ saw 1 ) n , being typical for weak nonlinearity. At larger Φ saw 1 , the behavior of I n is quite complex and strongly differs from the weaknonlinearity behavior. At very large Φ saw 1 , the high-harmonics intensities tends to saturate: I n → I n,max . At higher electron densities the saturation of I n occurs at larger SAW potentials (see Fig. 5).

4
Analytic results for acousto-electric effects in a 2D electron plasma In this section, we will give some analytic expressions describing various acoustoelectric effects in a system with a 2DEG. For simplicity, we will not take into account diffusion, hence assuming D e = 0 in the formula for the current (5). This assumption is well justified at large electron densities or at low temperatures. It is convenient to start with the Weinreich relation for a 2D system in the nonlinear regime [12,25].
Assuming that n(x 1 ) and j(x 1 ) are periodic functions we can rewrite Eq. 12 in the Obviously, it is valid in the limit K 2 ef f ≪ 1. Using Eq. 26 we can write < j >= |e|µ < nE x >= −µ < jE x > /v 0 s . From this equation we now get the Weinreich relation: where < jE x > is the dissipation in a SAW. To get Eq. 27, we have neglected the averaged electric fields < E x >= 0 assuming the case V 1 = V 2 .
We now consider Eq. 23 with D e = 0. At fixed Φ saw 1 , there is a critical density, N crit , for the formation of stripes in a 2DEG. When N s < N crit the plasma is split into electron stripes. If N s > N crit , the plasma is continuous but can be strongly modulated in space. For the case N s < N crit , the constant b 0 in Eq. 23 becomes zero and thus Eq. 23 has formally two solutions: The solution has to be a periodic continuous combination of these two functions.
It follows from Eq. 28 that in the limit σ 0 ≫ σ m N crit ≃ Φ saw 1 ǫ s /(4π|e|d), where σ 0 = |e|µN s . When N s < N crit the plasma turns into stripes, which means that electrons are totally trapped and the local electron velocity in a 2DEG reaches its maxi- Eqs. 23 and 24 can be used to find an asymptotic formula for Γ 1 and < j > in the large density limit when σ 0 ≫ σ m . In the limit σ 0 ≫ σ m , n ≃ N s − (ǫ s Φ saw 1 )/(4π|e|d) sin (qx 1 ) (see Eq. 28). By using the above and Eqs. 26 and 27, we have in the limit σ 0 ≫ σ m and in the region Φ saw 1 < Φ crit : where Γ max = qK 2 ef f /4 and Φ crit = N s 4π|e|d/ǫ s . This equation is valid when (Φ crit − Φ saw 1 )/Φ crit ≫ σ m /σ 0 . Eq. 29 reproduces the numerical data for Γ 1 (Φ saw 1 ) in Fig. 3 at large densities. The asymptotic formula (29) was given before in Ref. [26] without noting the condition σ 0 ≫ σ m . In the linear regime of interaction Γ 1 and δv 1 are given by the formula (1).
In the end of this section we consider an asymptotic behavior for the highharmonic intensities I n in the limit I 1 → ∞. The electron density n s (x 1 ) at high Φ saw 1 can be written in a parabolic approximation. Then, by calculating the Fourier components n n , we find that I n,max − I n ∝ (Φ saw 1 ) −2/3 ∝ I

Comparison with experimental data
The experiments involving SAW's were performed on the hybrid semiconductor- In 0.2 Ga 0.8 As quantum well (QW) embedded in modulation doped Al 0.2 Ga 0.8 As barriers. In these structures, the thin semiconductor layered system including a QW was tightly bound to the lithium niobate host crystal by the van der Waals forces [8,9,10]. The MBE grown quantum well structure is removed from its native GaAs substrate by etching an AlAs sacrificial layer below the active semiconductor system. The thin ELO film with a thickness of only 500nm is then transferred onto the host LiNbO 3 crystal. The parameters for this structure were described already in the beginning of Sec. 1. The geometry of the structure is shown in Fig. 1 Fig. 2). It is seen from the insert of Fig. 2 that at high SAW amplitudes the attenuation is strongly suppressed and its maximum is shifted to higher gate bias or conductivity, respectively. The experimental data for a SAW with the frequency f = 340 MHz look qualitatively similar to those for f = 114 MHz and were given earlier in Ref. [8].
The nonlinear regime of interaction is described by the theory given in Secs. 2, 3, and 4. To quantitatively compare theory and experiment, we now express the SAW potential amplitude Φ saw 1 through the input radio frequency (RF) power P .
The SAW intensity can be written as I 1 = I saw = 2(P/w)10 −IL/10 , where the width of the transducer w = 0.55 mm. The insertion losses (IL) in the transducers were measured to be 15 dB. Then, the SAW potential can be written using Eq. 22 as where K 2 ef f = 0.015 for f = 114MHz [10]. In Fig. 3 we also show the experimentally measured absorption coefficient at the gate voltage V t = −7.5 V . This voltage corresponds to the maximal attenuation for the smallest RF power, −12 dBm (see insert of Fig. 2). From the linear theory we find that the absorption coefficient is maximal at N s = 0.9 * 10 10 1/cm 2 . One can see from Fig. 3 that the experimentally measured function Γ 1 (Φ saw 1 ) for V t = −7.5 V is in a very good agreement with the calculated one for N s = 0.8 * 10 10 cm −2 . Here we did not use Thus, the shift of the SAW-velocity due to electrons decreases with increasing the SAW intensity. The experimentally observed shift is in qualitative agreement with our modeling shown in Fig. 4. However, our theory does not reproduce the character of δv 1 (V t ) in the region of small electron densities. Likely, the measurement of the SAW velocity is not so sensitive to a low-density electron system in comparison with the attenuation method.
In Fig. 6, we show the quantity Γ 1 P/I ae as a function of the RF power to verify the Weinreich relation. The acousto-electric current < j >= I ae (P ) was measured in a "short circuit geometry", where the Ohmic contacts are directly (without resistor) connected to the current measurement instrument. For more details on acousto-electric current measurements we refer to Ref. [11]. We see from Fig. 6 that the ratio Γ 1 P/I ae has a weak power-dependence. Thus, our experimental data are well described by the Weinreich relation. Slight deviations from the Weinreich relation seen in Fig. 6 can come from the density-dependence of the mobility µ(N s ), that is expected to be relatively weak at room temperature. The reason is that the main electronic scattering mechanism at high temperatures is due to acoustic phonons and is relatively insensitive to the 2D density.

Conclusions
The theoretical results obtained in Sec. 2, can also be applied to study dynamics of SAW's at large distances in a long sample, where the contribution of high harmonics can be very important [28,29]. This long-distance transformation to high harmonics was studied experimentally for SAW's interacting with a 3D electron gas of a semiconductor on a piezocrystal [28,29]. In the presence of a dc voltage applied to the crystal, it is possibly to expect the appearance of nonlinear waves with a stationary profile or with a stationary energy flow [12,14,28]. In a wave with a stationary energy flow the wave shape is periodically changed in space [28]. Another scenario can relate to chaotic dynamics in an acousto-electric system [30]. Eq. 14 can be used to numerically model these phenomena in 2D electron systems at long distances.
To conclude, we have studied strongly nonlinear acousto-electric phenomena caused by the interaction between a SAW and a two-dimensional electron system.
In the experimental measurements performed on hybrid semiconductor-piezocrystal structures the SAW attenuation, the SAW velocity change, and the acousto-electric current are strongly modified in the nonlinear regime due to the formation of moving electron stripes. By using a coupled-amplitude method we have modeled the decay and transformation of SAW's in the nonlinear regime. Using our theoretical results, we could explain our experimental findings and distinguished between different regimes of the nonlinear acousto-electric interaction at large SAW intensities. n n (x) as constants and solve Poisson's equation in terms of the "fast" variable x 1 .
Again, it is valid in the limit Γ 0 /q ∼ K 2 ef f ≪ 1. As an example, we now calculate the correction δφ n in the linear regime of interaction, when n(x, x 1 , t) = N s +n 1 (x)e iqx 1 +n * From the Poisson equation we find

Appendix 2
In order to solve a system of nonlinear equations (2-5) we will use some of results from a linear-response theory [20,21]. In a linear theory the total electrostatic potential and the 2D density can be written as W [x, z, t] = W [z, k]e ikx−iωt and n(x, t) = n k e ikx−iωt , respectively. Here ω is the SAW frequency. It is convenient to introduce a quantity Π k by means of the relation is the Fourier component of the electrostatic potential at z = d. Eqs. 2,3 are now written as Above equations should be solved together with the necessary boundary conditions considered in Sec. 1. Then, we rewrite Eqs. 31,32 in the form HereL lin is a linear operator and It follows from the boundary conditions that k = q + δq lin [21], where q = ω/v 0 s and Now a solution of Eq. 33 can be written as is a vector, that can be found from the matrix given by the boundary conditions [21].
Eq. 33 contains first and second spatial derivatives and can be written asL lin A lin = Now we turn to the nonlinear theory, where the equationLA = 0 is also valid. The operatorL is determined by the equations similar to Eqs. 31-33 but with nonlinear quantity Π n (x), that is defined by Π n (x) = −en n (x)/φ n (x, d). In the nonlinear case the vector A = A 0 (x, z) + n=1,2,... a n (x)A n (x, z)e iqnx 1 + c.c..
Each term in the equationLA = 0 should be zero, and soLa n (x)A n (x, z)e iqnx 1 ≃ e iqnx 1 [a n (x)L 0 (q n )A n + a ′ n (x)L 1 (q n )A n ] = 0. To get the latter equation, we have neglected dδq n /dx ∼ K 4 ef f and d 2 a n /dx 2 ∼ K 4 ef f . We can solve the equationLA = 0 if we choose A n (x, z) = A 0 [z, q n + δq n (x)], where the vector A 0 [z; q] is defined above in the linear theory. Using Eq. 35 we get where δq n (x) is given by the equation for δq lin (Eq. 34) with corrections Π q → Π n (x) and Π 0 q → Π 0 qn = Π 0 n . Eq. 36 is used in Secs. 1,2 to describe the acousto-electric phenomena in a 2DEG. Using Eqs. 8 and 11 and the results of Appendix 1, the denominator in δq n (x) (see Eq. 34) is rewritten as 1 + Π n (x)/Π 0 n = −Π n (x)φ saw n (x)/en n (x). Then, by using Eq. 34 and the conservation-of-charge equation j n (x) = v 0 s en n (x), we obtain We use this equation in Sec. 2 (see Eq. 15). To obtain above results, we have neglected the terms like a ′′ n (x) and δq ′ n (x) assuming that K 2 ef f is a small parameter.