# Nonlinear magnetoresistance of an irradiated two-dimensional electron system

###### Abstract

Nonlinear magnetotransport of a microwave-irradiated high mobility two-dimensional electron system under a finite direct current excitation is analyzed using a dc-controlled scheme with photon-assisted transition mechanism. The predicted amplitudes, extrema and nodes of the oscillatory differential resistance versus the magnetic field and the current density, are in excellent agreement with the recent experimental observation [Hatke et al. Phys. Rev. B 77, 201304(R) (2008)].

###### pacs:

73.50.Jt, 73.40.-c, 73.43.Qt, 71.70.DiThe predictionRyz-1970 () and detectionZud01 (); Ye () of radiation induced magnetoresistance oscillation (RIMO) in two-dimensional (2D) electron systems (ES), especially the discovery of the zero-resistance stateMani02 (); Zud03 (), have stimulated intensive experimentalDor03 (); Yang03 (); Zud04 (); Mani04 (); Willett (); Du (); Kovalev (); Mani05 (); Dor05 (); Stud (); Smet05 (); Yang06 (); Bykov05 (); Bykov06 () and theoreticalShi (); Durst (); Lei03 (); Lei04 (); Ryz03 (); Vav04 (); Dmitriev03 (); DGHO05 (); Torres05 (); Ng05 (); Ina-prl05 (); Kashuba (); Andreev03 (); Alicea05 (); Auerbach05 (); Mikhailov04 () studies on this extraordinary transport phenomenon of electrons in very high Landau levels.

Despite the fact that basic features of RIMO have been established and the understanding that it stems from impurity scattering has been reached, so far there has been no common agreement as to the accurate microscopic origin of these giant resistance oscillations. Presented in different forms, many theoretical modelsShi (); Durst (); Lei03 (); Lei04 (); Ryz03 (); Vav04 (); Torres05 (); Ina-prl05 (); Kashuba (); Ng05 (); Lei07-2 (); Auerbach07 () consider RIMO to arise from electron transitions between different Landau states due to impurity scattering accompanied by absorbing and emitting microwave photons. This origin is called the ”photon-assisted transition” or ”displacement” mechanism. A different origin, called the ”inelastic” or ”distribution function” mechanism,Dor03 (); Dmitriev03 (); DGHO05 () considers RIMO to arise from a microwave-induced nonequilibrium oscillation of the time-averaged isotropic electron distribution function in the density-of-states (DOS) modulated system. Both mechanisms exist in a real 2D semiconductor and have been shown to produce magentoresistance oscillations qualitatively having the observed period, phase and magnetic field damping. The ”displacement” mechanism predicts a well-defined photoresistivity with given impurity scattering and Landau-level broadening, while the ”inelastic” mechanism yields an additional factor proportional to the ratio of the inelastic scattering time to the impurity-induced quantum scattering time .Dmitriev03 () Since the inelastic scattering time or the thermalization time ,Lei03 () being the property of a nonequilibrium state and contributed by the direct Coulomb interactions between electrons and by all other possible impurity- and phonon-scattering mediated effective electron-electron scatterings,Lei03 () is very hard to determine theoretically or to measure experimentally, the sharp controversy whether or , i.e. which mechanism plays the dominant role in the experimental systems,Lei03 (); Dmitriev03 () has been an unsolved issue. The detailed comparison between theoretical predictions and experiments may provide a useful way to distinguish them.

Introducing additional parameters into microwave-illuminated 2DESs, such as dc excitations, can be of help to distinguish different models and mechanisms. It has been shown that a finite current alone, can also induce substantial magnetoresistance oscillation and zeroresistance without microwave radiation.Yang02 (); WZhang07 (); JZhang07 (); Bykov07 (); Lei07-1 (); Vav07 () Simultaneous application of a direct current and a microwave radiation leads to very interesting and complicated oscillatory behavior of resistance and differential resistance.WZhang07-2 (); Lei07-2 (); Auerbach07 () Recent careful measurementsWZhang08 (); Hatke08 () disclosed further details of such nonlinear magnetotransport in a high-mobility 2D semiconductor under both ac and dc exitations, allowing a careful comparison with theoretical predictions.

Our examination is based on a current-controlled scheme of photon-assisted transport,Lei03 () which deals with a 2DES of short thermalization time having electrons in a unit area of the - plane and subject to a uniform magnetic field in the direction. When an electromagnetic wave with incident electric field irradiates perpendicularly on the plane together with a dc electric field inside, the steady transport state of this 2DES is described by the electron drift velocity and an electron temperature , satisfying the force and energy balance equationsLei03 ()

(1) | |||||

(2) |

Here, the frictional force resisting electron drift motion,

(3) |

is given in terms of the electron density correlation function , the effective impurity potential , a radiation-related coupling parameter in the Bessel function , and . The electron energy absorption from the radiation field, , and the electron energy dissipation to the lattice, , are given in Ref. Lei03, . The nonlinear longitudinal resistivity and differential resistivity in the presence of a radiation field are obtained from Eq. (1) by taking and the current density in the direction, and ,

(4) |

We have calculated the differential resistivity from above equations (taking up to three-photon processes) under different magnetic fields and bias drift velocities for a GaAs-based heterosystem with carrier density /m and low-temperature linear mobility m/V s at lattice temperature K, irradiated by a linearly -polarized microwave of frequency GHz with incident amplitude V/cm. The elastic scatterings are assumed due to a mixture of short-range and background impurities, and the Landau-level broadening is taken to be a Gaussian form with a broadening parameter .Lei03 ()

Figure 1 presents the calculated versus ( is the cyclotron frequency) at fixed bias drift velocities from ( is the Fermi velocity) to in steps of , corresponding to current densities to A/m in steps of A/m. The case exhibits typical RIMO with a sequence of resistance maxima ( and ) and negative values around the resistance minima and . With increasing to 0.16 A/m, the maxima and (minima and ) evolve into minima (maxima) having seemingly little change in the positions. Further, all the curves cross approximately at and 4.5, indicating that at this range has little effect on photoresistance there. These and other features of Fig. 1 reproduce what was exactly observed in Ref.Hatke08, .

Figure 2(a) shows the calculated versus (, is the Fermi wave vector of the 2D electron system) at fixed from to in steps of 0.0625. Traces are vertically offset in increments of for clarity. The amplitudes of - oscillations are maximized around and 3.2 and strongly suppressed at and 3.5, as shown in Fig. 2(b).note () The positions of maxima extracted are plotted in Fig. 2(c) as dots. All these are in excellent agreement with the experimental results [Fig. 2(a),(b) and (c) of Ref.Hatke08, ].

In the present current-controlled transport model the oscillations of and are referred to the behavior of function in Eq. (3). The electron density correlation function is essentially a multiplication of two energy- shifted periodically modulated DOS functions of electrons in the magnetic field.Lei03 () Its periodicity with changing frequency at low temperatures and high Landau-level occupations, determines the main periodical behavior of magnetoresistance.Lei03 () The previous examinationLei07-2 () focused on the node positions of the oscillatory peak-valley pairs of , which appear periodically roughly along the lines in the - plane, where is the control parameter of RIMOs, and is the control parameter of current-induced magnetoresistance oscillations, and , dependent on the scattering potential.Lei07-1 ()

The maxima of differential resistivity show up at lower values in the axis in comparison with the node positions of related valley-peak pairs of , and its appearance exhibits a periodicity .Lei07-1 () In the - plane, the differential resistance maxima are expected to show up roughly in the vicinity along the lines

(5) |

with , dependent on the scattering potential ( for the system on discussion). Eq. (5) qualitatively accounts for the periodical change of in a large scale in steps of .

Under strong microwave irradiation, as in the present case, the role of virtual photon process [the term in the sum of Eq. (3)] is negligible due to samll ,Lei03 () and main contributions to resistivity come from terms (single- and multiple-photon processes). Noticing that the frequency differentiate is an even function of and considering contributions from scatterings parallel and antiparallel to the drift velocity and from terms, we see that, in the case of finite bias current, the behavior is determined by the sum of two terms: (a) and (b) . Depending on the function behavior in the vicinity of , effects of these two terms can be cancelled or added, completely or partly, at different locations of . function reaches maxima (positive) at around , reaches minima (negative) at around , and passes through zero (changing sign) at around and for all integers . Thus, at or () with which all involved frequencies are located around or , contributions from (a) and (b) are almost cancelled out for modest . In the case of [], there always exists a term of frequency [(] in , and contributions from (a) and (b) are positively [negatively] additive. These clearly account for the suppression of the current effect at and , and the enhancement of it around and .

Above discussions are general. The accurate behavior of resistivity inside a period scale is relevant to the detailed shape of the DOS function. Figs. 1 and 2 represent the result of a Gaussian-type DOS. The good quantitative agreement with experiment without adjusting parameters indicates that the present current-controlled scheme of photon-assisted transport captures the main physics of RIMOs in the discussed quasi-2D system.

This work was supported by the projects of the National Science Foundation of China, and the Shanghai Municipal Commission of Science and Technology.

## References

- (1) V. I. Ryzhii, Sov. Phys. Solid State 11, 2087 (1970).
- (2) M. A. Zudov, R. R. Du, J. A. Simmons, J. R. Reno, Phys. Rev. B 64, 201311(R) (2001).
- (3) P. D. Ye, L. W. Engel, D. C. Tsui, J. A. Simmons, J. R. Wendt, G. A. Vawter, and J. L. Reno, Appl. Phys. Lett. 79, 2193 (2001).
- (4) R. G. Mani, J. H. Smet, K. von Klitzing, V. Narayanamurti, W. B. Johnson, and V. Umansky, Nature 420, 646 (2002); Phys. Rev. Lett. 92, 146801 (2004).
- (5) M. A. Zudov, R. R. Du, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 90, 046807 (2003); ibid. 96, 236804 (2006); Phys. Rev. B 73, 041303(R) (2006).
- (6) S. I. Dorozhkin, JETP Lett. 77, 577 (2003).
- (7) C. L. Yang, M. A. Zudov, T. A. Knuuttila, R. R. Du, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 91, 096803 (2003).
- (8) M. A. Zudov, Phys. Rev. B 69, 041304 (2004).
- (9) R. G. Mani, V. Narayanamutri, K. von Klitzing, J. H. Smet, W. B. Johnson, and V. Umansky, Phys. Rev. B 70, 155310 (2004).
- (10) R. L. Willett, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett, 93, 026804 (2004).
- (11) R. R. Du, M. A. Zudov, C. L. Yang, Z. Q. Yang, L. N. Pfeiffer, and K. W. West, Int. J. Mod. Phys. B 18, 3462 (2004).
- (12) A. E. Kovalev, S. A. Zvyagin, C. R. Bowers, J. L. Reno, and J. A. Simmons, Solid State Commun. 130, 379 (2004).
- (13) R. G. Mani, Physica E, 25, 189 (2004); Phys. Rev. B 72, 075327 (2005); Appl. Phys. Lett. 91, 132103 (2007).
- (14) S. I. Dorozhkin, J. H. Smet, V. Umansky, K. von Klitzing, Phys. Rev. B 71, 201306(R) (2005).
- (15) S. A. Studenikin, M. Potemski, A. Sachrajda, M. Hilke, L. N. Pfeiffer, and K. W. West, IEEE Transaction on Nanotechnology, 4, 124 (2005); Phys. Rev. B 71, 245313 (2005).
- (16) J. H. Smet, B. Gorshunov, C. Jiang, L. N. Pfeiffer, K. W. West, V. Umansky, M. Dressel, R. Meisels, F. Kuchar, and K. von Klitzing, Phys. Rev. Lett. 95, 116804 (2005).
- (17) C. L. Yang, R. R. Du, J. A. Simmons, J. R. Reno,; Phys. Rev. B 74, 045315 (2006).
- (18) A. A. Bykov, J. Q. Zhang, S. Vitkalov, A. K. Kalagin, and A. K. Barakov, Phys. Rev. B 72, 245307 (2005).
- (19) A. A. Bykov, A. K. Barakov, D. R. Islamov, and A. I. Toropov, JETP Lett. 84, 391 (2006).
- (20) V. I. Ryzhii and R. Suris, J. Phys.: Condens. Matter 15, 6855 (2003); V. I. Ryzhii and V. Vyurkov, Phys. Rev. B 68, 165406 (2003); V. I. Ryzhii, Phys. Rev. B 68, 193402 (2003); Jpn. J. Appl. Phys., Part 1 44, 6600 (2005).
- (21) J. Shi and X. C. Xie, Phys. Rev. Lett. 91, 086801 (2003).
- (22) A. C. Durst, S. Sachdev, N. Read, and S. M. Girvin, Phys. Rev. Lett. 91, 086803 (2003); Physica E 25, 198 (2004).
- (23) X. L. Lei and S. Y. Liu, Phys. Rev. Lett. 91, 226805 (2003); Phys. Rev. B 72, 075345 (2005); Appl. Phys. Lett. 88, 212109 (2006); ibid. 89, 182117 (2006).
- (24) X. L. Lei, J. Phys.: Condens. Matter 16, 4045 (2004); Appl. Phys. Lett. 91, 112104 (2007).
- (25) M. G. Vavilov and I. L. Aleiner, Phys. Rev. B 69, 035303 (2004).
- (26) M. Torres and A. Kunold, Phys. Rev. B 71, 115313 (2005); J. Phys.: Condens. Matter 18, 4029 (2006).
- (27) T. K. Ng and Lixin Dai, Phys. Rev. B 72, 235333 (2005).
- (28) J. Iñarrea and G. Platero, Phys. Rev. Lett. 94, 016806 (2005); Phys. Rev. B 72, 193414 (2005).
- (29) A. Kashuba, Phys. Rev. B 73, 125340 (2006); A. Kashuba, JETP Lett. 83, 293 (2006).
- (30) I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, Phys. Rev. Lett. 91, 226802 (2003); Phys. Rev. B 70, 165305 (2004); I. A. Dmitriev, M. G. Vavilov, I. L. Aleiner, A. D. Mirlin, and D. G. Polyakov, ibid. 71, 115316 (2005); I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, ibid. 75, 245320 (2007); Phys. Rev. Lett. 99, 206805 (2007).
- (31) J. Dietel, L. I. Glazman, F. W. J. Hekking, and F. von Oppen, Phys. Rev. B 71, 045329 (2005); C. Joas, J. Dietel, and F. von Oppen, ibid. 72, 165323 (2005); J. Dietel, ibid. 73, 125350 (2006).
- (32) A. V. Andreev, I. L. Aleiner, and A. J. Mills, Phys. Rev. Lett. 91, 056803 (2003).
- (33) J. Alicea, L. Balents, M. P. A. Fisher, A. Paramekanti, and L. Radzihovsky, Phys. Rev. B 71, 235322 (2005).
- (34) A. Auerbach, I. Finkler, B. I. Halperin, and A. Yacoby Phys. Rev. Lett. 94, 196801 (2005).
- (35) S. A. Mikhailov, Phys. Rev. B 70, 165311 (2004).
- (36) X. L. Lei, Appl. Phys. Lett. 91, 112104 (2007).
- (37) A. Auerbach, and G. V. Pai, Phys. Rev. B 76, 205318 (2007).
- (38) C. L. Yang, J. Zhang, R. R. Du, J. A. Simmons, and J. L. Reno, Phys. Rev. Lett. 89, 076801 (2002).
- (39) W. Zhang, H. -S. Chiang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 75, 041304 (2007).
- (40) J. Q. Zhang, S. Vitkalov, A. A. Bykov, A. K. Kalagin, and A. K. Bakarov, Phys. Rev. B 75, 081305 (2007).
- (41) A. A. Bykov, J. Q. Zhang, S. Vitkalov, A. K. Kalagin, and A. K. Bakarov, Phys. Rev. Lett. 99, 116801 (2007).
- (42) X. L. Lei, Appl. Phys. Lett. 90, 132119 (2007).
- (43) M. G. Vavilov, I. L. Aleiner, and L. I. Glazman, Phys. Rev. B 76, 115331 (2007).
- (44) W. Zhang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 98, 106804 (2007).
- (45) W. Zhang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, Physica E, 40, 982 (2008).
- (46) A. T. Hatke, H. -S. Chiang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 77, 201304(R) (2008).
- (47) The amplitude maxima will be at and 3.25 if one considers only up to single-photon process ().