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Linear polarization study of microwave-radiation-induced magnetoresistance oscillations: Comparison of power dependence to theory

We present an experimental study of the microwave power and the linear polarization angle dependence of the microwave-induced magnetoresistance oscillations in the high-mobility GaAs/AlGaAs two-dimensional electron system. Experimental results show the sinusoidal dependence of the oscillatory magnetoresistance extrema as a function of the polarization angle. Yet, as the microwave power increases, the angular dependence includes additional harmonic content, and it begins to resemble the absolute value of the cosine function. We present a theory to explain such peculiar behavior.


INTRODUCTION
Microwave radiation-induced zero-resistance states [1,2] are an interesting phenomenon in two dimensional electron systems (2DES) because, for example, coincidence of microwaveinduced zero-resistance states with quantum Hall zero-resistance states leads to the extinction of the associated quantum Hall plateaus. [3] The microwave radiation-induced magnetoresistance oscillations that lead into the zero-resistance states include characteristic traits such as periodicity in B −1 ,[1, 2] a 1/4-cycle shift, [4], distinctive sensitivity to temperature and microwave power, [5,6] along with other specific features. . The linear-polarizationsensitivity of these oscillations has been a topic of intensive recent experimental study [13,17,23,26,28,34,35]; these studies have shown that the amplitude of microwave radiation-induced magnetoresistance oscillations changes periodically with the linear microwave polarization angle.
Looking at the linear polarization characteristics in greater detail, at fixed temperature and polarization angle the amplitude of microwave radiation-induced magnetoresistance oscillations increases with the microwave power. It follows, approximately, [5,6] where A is the amplitude of microwave radiation-induced magnetoresistance oscillations, A 0 and α ≈ 1/2 are constants and P is microwave power. At fixed temperature and microwave power, amplitude of microwave radiation-induced magnetoresistance oscillations changes sinusoidally with the linear polarization angle. The experimental results have shown that the longitudinal resistance R xx vs linear polarization angle θ follows a cosine square function [17], i.e., R xx (θ) = A ± Ccos 2 (θ − θ 0 ) (A and C are constants, θ 0 is the phase shift, which depend on microwave frequency [27]), at low microwave power. Note tha this angular dependence can also be rewritten as R xx (θ) = A±(C/2)(1+cos[2(θ−θ 0 )]. Deviation from this functional form was noted for higher microwave intensities.
From the theoretical point of view [39-44, 46-52, 54-59], there are many approaches to understand the physics of the microwave radiation-induced magnetoresistance oscillations.
These include the radiation-assisted indirect inter-Landau-level scattering by phonons and impurities (the displacement model) [39,43], the periodic motion of the electron orbit centers under irradiation (the radiation driven electron orbit model) [44], and a radiation-induced steady state non-equilibrium distribution (the inelastic model) [46]. Among these approaches, the radiation driven electron orbit model intensively considered temperature [36,45], mi-crowave power [6] and microwave polarization direction [35] as factors that could change the amplitude of microwave radiation-induced magnetoresistance oscillations. Here, we report an experimental study of microwave power and linear polarization angle dependence of the radiation-induced magnetoresistance oscillations and compare the results with the predictions of the radiation driven electron orbit model. The comparison provides new understanding of the experimental results at high microwave powers.

THEORETICAL MODEL
The radiation driven electron orbit model [37,44,45,53] was developed to explain the observed diagonal resistance, R xx , of an irradiated 2DES at low magnetic field, B, with B in the z-direction, perpendicular to the 2DES. Here, electrons behave as 2D quantum oscillator in the XY plane that contains the 2DES. The system is also subjected to a DC electric field in the x-direction (E DC ), the transport direction, and microwave radiation that is linearly polarized at different angles (θ) with respect to the transport direction (x-direction). The radiation electric field is given by amplitudes of the MW field and w the frequency. Thus, θ is given by tan θ = E 0y E 0x . The corresponding electronic hamiltonian can be exactly solved [38,44] obtaining a solution for the total wave function, where φ N are Fock-Darwin states, X is the center of the orbit for the electron motion, and a(t) (for the x-coordinate) and b(t) (for the y-coordinate) are the solutions for a classical driven 2D harmonic oscillator (classical uniform circular motion). The expressions for an arbitrary angle θ are given by where e is the electron charge, γ is a damping factor for the electronic interaction with acoustic phonons, w c the cyclotron frequency and E 0 is the total amplitude of the radiation electric field. These two latter equations give us the equation of an ellipse, a 2 Then, the first finding of this theoretical model is that according to the expressions for a(t) and b(t), the center of the electron orbit performs a classical elliptical trajectory in the XY plane driven by radiation (see inset of Fig. 4). This is reflected in the x and y directions as harmonic oscillatory motions with the same frequency as radiation. In this elliptical motion electrons in their orbits interact with the lattice ions being damped and emitting acoustic phonons; in the a(t) and b(t) expressions, γ represents this damping. we have phenomenologically introduced an asymmetry factor λ to deal with this important scenario affecting the obtained R xx . Since for a more intense radiation electric field E 0 , the motion in y is increasingly hindered we have introduced λ = 1 1+cE 0 where c is a constant that tends to 0 for symmetric samples and then a(t) reads: This radiation − driven behavior has a deep impact on the charged impurity scattering and in turn in the conductivity. Thus, first we calculate the impurity scattering rate W N,M between two driven Landau states Ψ N , and Ψ M [37,44]. In order to calculate the electron drift velocity, next we find the average effective distance advanced by the electron in every scattering jump [37]: ∆X M W = ∆X 0 − A * x sin wτ , where τ is the flight time that is strictly the time it takes the electron to go from the initial orbit to the final one. This time is part of the scattering time, τ S , that is normally defined as the average time between scattering events and equal to the inverse of the scattering rate. ∆X 0 is the average advanced distance without radiation. Finally the longitudinal conductivity σ xx is given by: and σ xx ≪ σ xy . Therefore,      be obtained from equation [5] and the square root between brackets. When the microwave power (electric field amplitude E 0 ) increases, the factor w 2 c w 2 1 1+cE 0 sin 2 θ gets smaller and smaller. As a result, the R xx curve begins to lose its simple sinusoidal profile. At high powers the latter factor is so small that the cos 2 factor is predominant and the square root tends to the absolute value of cos 2 : The profile observed in experiment and calculation for high powers is very similar to the one of the abs[cos θ]. This evolution can be clearly observed in the panel (c) of Figure 5.
In Figure 6, we present calculated R xx under radiation versus the microwave power for different polarization angles for peak P 1, upper panel, and for valley V 1, lower panel. For increasing angles from 0 0 to 90 0 the behavior of P 1 and V 1 are similar in the sense that the corresponding intensity of both becomes smaller and smaller. In other words, for increasing angles the height of the peak decreases and the depth of the valley decreases too. The theoretical explanation comes from the square root term as before. For increasing angles the cosine term tends to zero becoming predominant the sine term. However the latter gets smaller for increasing power due to the asymmetry factor. All curves share one important feature, the non-linearity of R xx versus P suggesting a sublinear relation. This behavior can be straightforward explained with our model in terms of: And then the R xx profile with P follows a square root dependence as experiments show.

CONCLUSION
We have presented together experimental-and theoretically-calculated-results concerning the microwave power and linear polarization angle dependence of the microwave irradiated oscillatory magnetoresistance in the GaAs/AlGaAs two-dimensional electron system. Experimental results show that, as the microwave power increases, the R xx vs. θ traces, see Intuitively, one can motivate the change in the profile of the R xx vs. θ curves with increasing power by noting that increasing the power, i.e., driving the 2DES more strongly, most likely increases the harmonic content in the R xx vs. θ lineshape, which leads to deviations from the simple sinusoidal function proposed for low power in ref. [17]. In the low power limit, this theoretical prediction of the radiation driven electron orbit model matched the experimental suggestion of R xx (θ) = A ± Ccos 2 (θ − θ 0 ) = A ± (C/2)(1 + cos[2(θ − θ 0 )] [17].