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Circular-polarization-dependent study of the microwave-induced magneto-resistance oscillations in the 2D electron system
ARTICLE IN PRESS
Physica E 35 (2006) 315–319 www.elsevier.com/locate/physe
Circular-polarization-dependent study of the microwave-induced magneto-resistance oscillations in the 2D electron system J.H. Smeta,, B. Gorshunova,d, C. Jianga, L. Pfeifferb, K. Westb, V. Umanskyc, M. Dresseld, R. Meiselse, F. Kuchare, K. von Klitzinga a
Max-Planck-Institut fu¨r Festko¨rperforschung, HeisenbergstraX e 1, D-70569 Stuttgart, Germany b Lucent Technologies, Bell Labs, Murray Hill, NJ 07974, USA c Braun Center for Submicron Research, Weizmann Institute of Science, Rehovot 76100, Israel d I. Physikalisches Institut, Universita¨t Stuttgart, Stuttgart, Germany e Department of Physics, University of Leoben, A-8700 Leoben, Austria Available online 1 November 2006
Abstract We have investigated the influence of the polarization of the incident radiation on the recently discovered microwave-induced resistance oscillations in state-of-the-art highest purity 2D electron systems. A quasi-optical setup allows us to tune in situ between different circular as well as linear polarization states. We find that the microwave-induced zero resistance and the resistance oscillations are notably immune to changes in the polarization. This observation is discrepant with a number of proposed theories. Deviations for different polarizations only occur for a bolometric contribution to the resistance associated with the resonant heating at the cyclotron resonance itself. r 2006 Published by Elsevier B.V. PACS: 73.21.b; 73.43.f Keywords: Zero resistance; Microwave photoconductivity; 2D electron system
1. Introduction Zero resistance is a common Hall mark of such fundamental phenomena as superconductivity and the quantum Hall effects. Recently, zero resistance was, however, discovered in a surprising context. When a 2D electron system is exposed to quasi-monochromatic microwave radiation and a minute perpendicular magnetic field, transport along the direction of current flow may proceed in a dissipationless fashion [1,2]. An important prerequisite is that the sample is of exceptional purity. A dramatic example is illustrated in Fig. 1. The longitudinal DCresistivity oscillates as a function of the ratio o/oc, where o is the microwave frequency and oc the cyclotron frequency. As the radiation intensity is enhanced, the amplitude of the oscillations increases and the resistance values at the Corresponding author.
E-mail address: [email protected] (J.H. Smet). 1386-9477/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.physe.2006.08.017
minima saturate as they approach zero. Original photoconductivity experiments on much lower quality samples only revealed the intuitively expected resistance increase due to resonant heating at the cyclotron resonance [3]. A temperature-dependent study of the deepest resistance minima suggests activated transport [1,2], which is normally associated with the existence of a gap at the chemical potential. One might imagine that this thermal activation behavior and the vanishing of the resistance indicate the formation of a new strongly correlated manybody quantum Hall state, but in fact the Hall resistivity behaves classically, as expected at low magnetic fields, and remains largely unaffected by the microwave radiation [1,2,4,5]. These experimental observations triggered a large body of theoretical work. The majority of theoretical accounts subdivide the argumentation to explain the zero resistance into two main points. First, some mechanism produces an oscillatory photoconductivity contribution that may turn
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Fig. 2. Predicted oscillations in the magneto-conductivity (or resistivity) for both circular polarization directions based on equations contained in Ref. [16] for modest microwave power levels. The model takes into account a non-equilibrium distribution of the charge carriers. The photoconductivity sph is normalized to the dark conductivity sdc. According to theory [21,22], an instability develops when sph drops below zero and the formation of current domains prevents the resistance from crossing the sph ¼ 0-line [22–26].
Fig. 1. DC-magneto-resistance of a state-of-the-art 2D electron system in the absence (top panel) and presence of 73 GHz microwave radiation (bottom panel). The data are taken at a temperature below 100 mK.
the overall dissipative conductivity negative near the minima. Proposed mechanisms include a spatial displacement of electrons against the DC-electric field with the assistance of impurity or phonon scattering [6–14], the establishment of a non-equilibrium distribution function [15–18], photon-assisted quantum tunneling [19] and nonparabolicity effects [20]. This list is not exhaustive. An example of predicted oscillations, obtained when adopting the non-equilibrium distribution scenario described in Refs. [16,17], is shown in Fig. 2. Second, it is argued that negative values of the dissipative conductivity render the initially homogeneous system unstable [21,22] and an inhomogeneous current domain pattern develops [22–26], which results in zero resistance in experiment. Some theoretical work does not invoke an instability-driven formation of current domains to explain zero resistance [27,28]. The sheer multitude of theoretical models indicates that no consensus has been reached on the origin of this nonequilibrium phenomenon. In order to assist in isolating the proper microscopic picture, a detailed polarization-dependent study was carried out [29]. In previous work, microwaves were guided to the sample with over-sized rectangular waveguides [1,2] or coaxial dipole antennas [4,30]. These approaches do not permit control over the polarization. Here, we have chosen an unconventional, all quasi-optical scheme [31] to guide the microwaves onto the sample and to produce any circular or linear polarization. Knowledge of the polarization dependence of the micro-
wave-induced oscillations may turn out an important litmus test for theoretical models. The mainstream theories based on a non-equilibrium distribution function [15–18] and on the phonon- or impurity-assisted displacement of electrons [6–14] both predict oscillations for the two senses of circular polarization, however, with substantially different amplitudes (see discussion below and Fig. 2). Most other models have assumed linear polarization and have not been analyzed for the case of circular polarization (for instance Refs. [19,27,28]). 2. Experimental setup A schematic drawing of the quasi-optical setup is depicted in Fig. 3 [29]. A state-of-the-art 2D electron system (electron mobility of 20 106 cm2/V s) is mounted in the Faraday geometry in the variable temperature insert of an optical cryostat with a split coil. The sample consists of a 4 4 mm2 van der Pauw geometry with eight contacts along its perimeter. The insert was operated at 1.8 K. The cryostat is equipped with 100 mm thick Mylar inner and outer windows. The outer windows are covered with black polyethylene foil to block visible light. Backward wave oscillators generate quasi-monochromatic radiation (bandwidth Df/fE105). Our studies focused on frequencies from 100 up to 350 GHz. The radiation passes through three dense wire grid polarizers (P1, P2, P3) as well as a socalled polarization transformer (PT). The latter consists of another fixed wire grid P4 and a mobile metallic mirror placed in parallel at a tunable distance d (see inset to the bottom panel of Fig. 3). Grid P4 reflects the component of the incident radiation with the electric field vector aligned along the wires. The remainder of the beam polarized perpendicular to the wires passes undisturbed through the
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setup with a circular microwave waveguide and a polarizer at room temperature at the entrance of the waveguide. However, with this approach the circular polarization degree was limited to 70% or less. 3. Microwave transmission experiment
Fig. 3. Top: quasi-optical setup. The various components are discussed in the text except for A1, which is a fixed attenuator, and A2, which is an absorber to block the radiation beam. The position of the sample is marked by DUT (device under test). Bottom panel: transmission data for 200 GHz linear, left and right-hand circular polarized radiation. The data are taken at 1.8 K. The inset highlights optical components controlling or affecting the polarization state (adapted from [29]).
grid. It is reflected by the mirror and hence acquires an additional phase shift proportional to d. Grids P1, P2 and P3 serve to continuously adjust the overall intensity as well as to ensure equal intensities of the radiation components with electric field vectors aligned with and perpendicular to wire grid P4, so that proper adjustment of d yields linear or any circular polarization state. The quartz lens L2 focuses the radiation onto the sample, while quartz lens L3 recollimates the beam after transmission through the cryostat. The degree of circular polarization Z is verified at various locations along the beam by recording the power with a pneumatic Golay detector (used in conjunction with the chopper) and an additional dense wire grid P5 for two orthogonal directions (J and ?) of the electric field vector: Z ¼ 1 jP? FPk j=ðP? þ Pk Þ. Typical values for the circular polarization purity Z before and after the transmission through the cryostat of the radiation beam have been included as percentages at the top of Fig. 3. After the first quartz lens, but before entering the cryostat, Z exceeds 98%. The warped Mylar windows of the cryostat and the quartz lenses deteriorate somewhat the polarization state, but the circular polarization character remains better than Z492% after transmission through the cryostat and sample holder in the absence of a sample. Preliminary experiments (not shown here) were also carried out in a
The bottom panel of Fig. 3 displays the outcome of a transmission experiment with the quasi-optical setup for both circular polarization directions and linear polarization of the incident radiation. Transmission experiments for unpolarized radiation were reported previously in Ref. [32]. Data are shown for both positive and negative values of B. Active cyclotron resonance absorption (CRA) should only occur for the proper sense of circular polarization with respect to the B-field orientation, so that the radiation field accelerates the electron along its cyclotron motion. Indeed, the transmitted power drops nearly to zero for CRA in the active sense of the polarization, whereas reversing the B-field orientation while maintaining the circular polarization direction turns the cyclotron resonance mode inactive (CRI). For linear polarization, the transmitted power does not drop below 50% as it should. These transmission data confirm the quality of the various polarization states. Noteworthy is also the absence of absorption features at the harmonics of oc. This contrasts with the magneto-resistance, which exhibits microwaveinduced oscillations up to the 10th harmonic of the cyclotron resonance (Figs. 1 and 4). 4. Polarization-dependent photoresistivity Fig. 4A illustrates the influence of the circular polarization sense on the microwave- induced resistance oscillations for different frequencies: 200, 183 and 100 GHz [29]. The absorption signals for 200 GHz have been included at the top to easily compare the position of features with the cyclotron resonance. At fields below the CRA (the regions on the left of the dotted lines which mark the o ¼ oc condition) where the microwave-induced oscillations occur, the magnetoresistivity traces for both senses of circular polarization are nearly indistinguishable. This is true for the entire frequency range (100–350 GHz) covered in our experiments. The same conclusion holds for linearly polarized radiation irrespective of the orientation of the electric field vector (an example is shown in Fig. 4B). Hence, the experiments here only support theoretical mechanisms in which the polarization state of the microwaves is not relevant. The experiments in a circular waveguide system with 70% polarization led to the same conclusion. Only at fields near the cyclotron resonance, where significant absorption takes place, deviations between the CRA and CRI curves do arise, likely because resonant heating at the cyclotron resonance produces a second contribution to the photoresistivity. In microwave photoconductivity experiments prior to the discovery of the zero
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resistance phenomenon, the CRA was already detected as a small resistance increase [3]. In the context of the recently discovered microwave-induced resistance oscillations, signatures of the CRA in the magnetoresistivity remained unidentified, presumably because they were masked by the much larger microwave-induced resistance oscillations. The insensitivity of the latter on the polarization state (as proven above) and the ability to alter the circular polarization direction in situ make it straightforward to reveal the bolometric cyclotron resonance contribution. A striking example, at the highest microwave power we have available, is illustrated in Fig. 4B for 243 GHz [29]. The resistance enhancement at the cyclotron resonance can be surprisingly large and develops fine structure. Its close connection to the cyclotron resonance is established by comparing with transmission data and by comparing its amplitude for the different polarization states. We conclude that we can distinguish two microwave-induced contributions to the resistance: the bolometric and polarization-dependent resistance enhancement at the cyclotron resonance due to resonant heating of the electron system and the polarization insensitive contribution related to the oscillatory photoresistivity below and near oc.
(ooc)2 larger for the CRA polarization sense [Eqs. (16) and (17) in Ref. [16] after adaptation to circular polarization; see also Eq. (6.11) in Ref. [14]]. For the maxima (or minima) near o ¼ 2oc and o ¼ 3oc, this amounts to a factor of about 9 and 4, respectively. We refer to Fig. 2 where theoretical predictions for both senses of circular polarization are compared. For large microwave powers, the microwave-induced correction to the dark DC conductivity no longer obeys a linear law but rather a sub-
5. Comparison with theory The complete immunity of the microwave-induced resistance oscillations to the polarization state may be regarded as a crucial test for theories. For instance, this cannot be reconciled with the two most frequently cited theories: the non-equilibrium distribution function scenario [15–18] and the picture based on impurity- and phononassisted electron displacement against the DC-electric field [6–14]. For both the mechanisms, a dependence on the sense of circular polarization enters through the AC Drude conductivity sD ac. Excluding the narrow vicinity of the cyclotron resonance, the AC Drude conductivity is proportional to (ooc)2 if the microwave field accelerates electrons (CRA) and (o+oc)2 if the microwaves decelerate electrons (CRI). For small DC-fields and not to strong a microwave field, the correction to the dark DC dissipative conductivity is linearly proportional to this Drude conductivity and hence is a factor (o+oc)2/ Fig. 4. (A) Comparison of the DC-magnetoresistivity Rxx under microwave radiation for both senses of circular polarization at three frequencies and a temperature of 1.8 K. In the white regions where the microwaveinduced oscillations occur, the magneto-resistance is independent of the microwave polarization. The transmitted power Ptrans (arb. units) at 200 GHz is plotted at the top for comparison with the 200 GHz transport data. The black dotted curve in the bottom panel displays Rxx in the absence of microwaves. (B) Rxx for 243 GHz radiation for various polarizations at the highest available power. Curves are offset for clarity (CRA blue dashed line, CRI red line and linear polarization black line). The dashed line at the top represents the transmitted microwave power (no ordinate shown). Resonant heating at the cyclotron resonance is held responsible for the sharp resistance peak. This peak, as opposed to the oscillatory photo-resistance, is very sensitive to the polarization state [adapted from [29]].
(A)
(B)
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linear dependence on the microwave power and these factors are reduced. However, the dramatic difference with experiment here remains. 6. Conclusions We conclude that a polarization-dependent study of the microwave photoconductivity has revealed two contributions: a bolometric and strong polarization-dependent component associated with resonant heating at the cyclotron resonance and the polarization insensitive microwave-induced oscillations. This immunity of the microwave-induced oscillations to changes of the polarization direction of the microwave radiation contradicts the mainstream theories. This puzzling discrepancy will likely reinvigorate the controversy on their origin and there is clearly a strong need to analyze the polarization dependence of other proposed theoretical mechanisms. Acknowledgment We gratefully acknowledge financial support from the German Science Foundation (DFG), the German Ministry of Science and Education (BMBF), the German Israeli Foundation (GIF), and the Transnational Access program of the EU (RITA-CT-2003-506095 WISSMC). References [1] R.G. Mani, J.H. Smet, K. von Klitzing, V. Narayanamurti, W.B. Johnson, V. Umansky, Nature 420 (2002) 646. [2] M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 90 (2003) 046807. [3] E. Vasiliadou, G. Mu¨ller, D. Heitmann, D. Weiss, K. von Klitzing, Phys. Rev. B 48 (1993) 17145. [4] S.A. Studenikin, M. Potemski, P.T. Coleridge, A.S. Sachrajda, Z.R. Wasilewski, Issue Series Title: Solid State Commun. 129 (2004) 341. [5] R.G. Mani, V. Narayanamurti, K. von Klitzing, J.H. Smet, W.B. Johnson, V. Umansky, Phys. Rev. B 69 (2004) 161306.
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[6] V.I. Ryzhii, Sov. Phys. Solid State 11 (1970) 2078. [7] V.I. Ryzhii, R.A. Suris, B.S. Shchamkhalova, Sov. Phys. Semicond. 20 (1986) 1299. [8] A.C. Durst, S. Sachdev, N. Read, S.M. Girvin, Phys. Rev. Lett. 91 (2003) 086803. [9] V. Ryzhii, R. Suris, J. Phys.: Condens. Matter 15 (2003) 6855. [10] V. Ryzhii, V. Vyurkov, Phys. Rev. B 68 (2003) 165406. [11] V. Ryzhii, Phys. Rev. B 68 (2003) 193402. [12] X.L. Lei, S.Y. Liu, Phys. Rev. Lett. 91 (2003) 226805. [13] V.B. Shikin, J. Exp. Theor. Phys. Lett. 77 (2003) 236. [14] M.G. Vavilov, I.L. Aleiner, Phys. Rev. B 69 (2004) 035303. [15] S.I. Dorozhkin, Pis’ma ZhETF 77 (2003) 681; S.I. Dorozhkin, JETP Lett 77 (2003) 577. [16] I.A. Dmitriev, M.G. Vavilov, I.L. Aleiner, A.D. Mirlin, D.G. Polyakov, Phys. Rev. B 71 (2005) 115316; I.A. Dmitriev, M.G. Vavilov, I.L. Aleiner, A.D. Mirlin, D.G. Polyakov, Physica E 25 (2004) 205. [17] I.A. Dmitriev, A.D. Mirlin, D.G. Polyakov, Phys. Rev. Lett. 91 (2003) 226802. [18] J. Dietel, L.I. Glazman, F.W.J. Hekking, F. von Oppen, Phys. Rev. B 71 (2005) 045329. [19] J. Shi, X.C. Xie, Phys. Rev. Lett. 91 (2003) 086801. [20] A.A. Koulakov, M.E. Raikh, Phys. Rev. B 68 (2003) 115324. [21] A.L. Zakharov, Sov. Phys. JETP 11 (1960) 478. [22] A.V. Andreev, I.L. Aleiner, A.J. Millis, Phys. Rev. Lett. 91 (2003) 056803. [23] F.S. Bergeret, B. Huckestein, A.F. Volkov, Phys. Rev. B 67 (2003) 241303. [24] V. Ryzhii, A. Satou, J. Phys. Soc. Japan 72 (2003) 2718. [25] A.F. Volkov, V.V. Pavlovskii, Phys. Rev. B 69 (2004) 125305. [26] A. Auerbach, I. Finkler, B.I. Halperin, A. Yacoby, Phys. Rev. Lett. 94 (2005) 196801. [27] P.H. Rivera, P.A. Schulz, Phys. Rev. B 70 (2004) 75314. [28] J. Inarrea, G. Platero, Phys. Rev. Lett. 94 (2005) 16806. [29] J.H. Smet, B. Gorshunov, C. Jiang, L. Pfeiffer, K. West, V. Umansky, M. Dressel, R. Meisels, F. Kuchar, K. von Klitzing, Phys. Rev. Lett. 95 (2005) 116804. [30] R.L. Willett, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 93 (2004) 026804. [31] G. Kozlov, A. Volkov, Top. Appl. Phys. 74 (1998) 51. [32] S.A. Studenikin, M. Potemski, A.S. Sachrajda, M. Hilke, L.N. Pfeiffer, K.W. West, Phys. Rev. B 71 (2005) 245313.
Physica E 35 (2006) 315–319 www.elsevier.com/locate/physe
Circular-polarization-dependent study of the microwave-induced magneto-resistance oscillations in the 2D electron system J.H. Smeta,, B. Gorshunova,d, C. Jianga, L. Pfeifferb, K. Westb, V. Umanskyc, M. Dresseld, R. Meiselse, F. Kuchare, K. von Klitzinga a
Max-Planck-Institut fu¨r Festko¨rperforschung, HeisenbergstraX e 1, D-70569 Stuttgart, Germany b Lucent Technologies, Bell Labs, Murray Hill, NJ 07974, USA c Braun Center for Submicron Research, Weizmann Institute of Science, Rehovot 76100, Israel d I. Physikalisches Institut, Universita¨t Stuttgart, Stuttgart, Germany e Department of Physics, University of Leoben, A-8700 Leoben, Austria Available online 1 November 2006
Abstract We have investigated the influence of the polarization of the incident radiation on the recently discovered microwave-induced resistance oscillations in state-of-the-art highest purity 2D electron systems. A quasi-optical setup allows us to tune in situ between different circular as well as linear polarization states. We find that the microwave-induced zero resistance and the resistance oscillations are notably immune to changes in the polarization. This observation is discrepant with a number of proposed theories. Deviations for different polarizations only occur for a bolometric contribution to the resistance associated with the resonant heating at the cyclotron resonance itself. r 2006 Published by Elsevier B.V. PACS: 73.21.b; 73.43.f Keywords: Zero resistance; Microwave photoconductivity; 2D electron system
1. Introduction Zero resistance is a common Hall mark of such fundamental phenomena as superconductivity and the quantum Hall effects. Recently, zero resistance was, however, discovered in a surprising context. When a 2D electron system is exposed to quasi-monochromatic microwave radiation and a minute perpendicular magnetic field, transport along the direction of current flow may proceed in a dissipationless fashion [1,2]. An important prerequisite is that the sample is of exceptional purity. A dramatic example is illustrated in Fig. 1. The longitudinal DCresistivity oscillates as a function of the ratio o/oc, where o is the microwave frequency and oc the cyclotron frequency. As the radiation intensity is enhanced, the amplitude of the oscillations increases and the resistance values at the Corresponding author.
E-mail address: [email protected] (J.H. Smet). 1386-9477/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.physe.2006.08.017
minima saturate as they approach zero. Original photoconductivity experiments on much lower quality samples only revealed the intuitively expected resistance increase due to resonant heating at the cyclotron resonance [3]. A temperature-dependent study of the deepest resistance minima suggests activated transport [1,2], which is normally associated with the existence of a gap at the chemical potential. One might imagine that this thermal activation behavior and the vanishing of the resistance indicate the formation of a new strongly correlated manybody quantum Hall state, but in fact the Hall resistivity behaves classically, as expected at low magnetic fields, and remains largely unaffected by the microwave radiation [1,2,4,5]. These experimental observations triggered a large body of theoretical work. The majority of theoretical accounts subdivide the argumentation to explain the zero resistance into two main points. First, some mechanism produces an oscillatory photoconductivity contribution that may turn
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Fig. 2. Predicted oscillations in the magneto-conductivity (or resistivity) for both circular polarization directions based on equations contained in Ref. [16] for modest microwave power levels. The model takes into account a non-equilibrium distribution of the charge carriers. The photoconductivity sph is normalized to the dark conductivity sdc. According to theory [21,22], an instability develops when sph drops below zero and the formation of current domains prevents the resistance from crossing the sph ¼ 0-line [22–26].
Fig. 1. DC-magneto-resistance of a state-of-the-art 2D electron system in the absence (top panel) and presence of 73 GHz microwave radiation (bottom panel). The data are taken at a temperature below 100 mK.
the overall dissipative conductivity negative near the minima. Proposed mechanisms include a spatial displacement of electrons against the DC-electric field with the assistance of impurity or phonon scattering [6–14], the establishment of a non-equilibrium distribution function [15–18], photon-assisted quantum tunneling [19] and nonparabolicity effects [20]. This list is not exhaustive. An example of predicted oscillations, obtained when adopting the non-equilibrium distribution scenario described in Refs. [16,17], is shown in Fig. 2. Second, it is argued that negative values of the dissipative conductivity render the initially homogeneous system unstable [21,22] and an inhomogeneous current domain pattern develops [22–26], which results in zero resistance in experiment. Some theoretical work does not invoke an instability-driven formation of current domains to explain zero resistance [27,28]. The sheer multitude of theoretical models indicates that no consensus has been reached on the origin of this nonequilibrium phenomenon. In order to assist in isolating the proper microscopic picture, a detailed polarization-dependent study was carried out [29]. In previous work, microwaves were guided to the sample with over-sized rectangular waveguides [1,2] or coaxial dipole antennas [4,30]. These approaches do not permit control over the polarization. Here, we have chosen an unconventional, all quasi-optical scheme [31] to guide the microwaves onto the sample and to produce any circular or linear polarization. Knowledge of the polarization dependence of the micro-
wave-induced oscillations may turn out an important litmus test for theoretical models. The mainstream theories based on a non-equilibrium distribution function [15–18] and on the phonon- or impurity-assisted displacement of electrons [6–14] both predict oscillations for the two senses of circular polarization, however, with substantially different amplitudes (see discussion below and Fig. 2). Most other models have assumed linear polarization and have not been analyzed for the case of circular polarization (for instance Refs. [19,27,28]). 2. Experimental setup A schematic drawing of the quasi-optical setup is depicted in Fig. 3 [29]. A state-of-the-art 2D electron system (electron mobility of 20 106 cm2/V s) is mounted in the Faraday geometry in the variable temperature insert of an optical cryostat with a split coil. The sample consists of a 4 4 mm2 van der Pauw geometry with eight contacts along its perimeter. The insert was operated at 1.8 K. The cryostat is equipped with 100 mm thick Mylar inner and outer windows. The outer windows are covered with black polyethylene foil to block visible light. Backward wave oscillators generate quasi-monochromatic radiation (bandwidth Df/fE105). Our studies focused on frequencies from 100 up to 350 GHz. The radiation passes through three dense wire grid polarizers (P1, P2, P3) as well as a socalled polarization transformer (PT). The latter consists of another fixed wire grid P4 and a mobile metallic mirror placed in parallel at a tunable distance d (see inset to the bottom panel of Fig. 3). Grid P4 reflects the component of the incident radiation with the electric field vector aligned along the wires. The remainder of the beam polarized perpendicular to the wires passes undisturbed through the
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setup with a circular microwave waveguide and a polarizer at room temperature at the entrance of the waveguide. However, with this approach the circular polarization degree was limited to 70% or less. 3. Microwave transmission experiment
Fig. 3. Top: quasi-optical setup. The various components are discussed in the text except for A1, which is a fixed attenuator, and A2, which is an absorber to block the radiation beam. The position of the sample is marked by DUT (device under test). Bottom panel: transmission data for 200 GHz linear, left and right-hand circular polarized radiation. The data are taken at 1.8 K. The inset highlights optical components controlling or affecting the polarization state (adapted from [29]).
grid. It is reflected by the mirror and hence acquires an additional phase shift proportional to d. Grids P1, P2 and P3 serve to continuously adjust the overall intensity as well as to ensure equal intensities of the radiation components with electric field vectors aligned with and perpendicular to wire grid P4, so that proper adjustment of d yields linear or any circular polarization state. The quartz lens L2 focuses the radiation onto the sample, while quartz lens L3 recollimates the beam after transmission through the cryostat. The degree of circular polarization Z is verified at various locations along the beam by recording the power with a pneumatic Golay detector (used in conjunction with the chopper) and an additional dense wire grid P5 for two orthogonal directions (J and ?) of the electric field vector: Z ¼ 1 jP? FPk j=ðP? þ Pk Þ. Typical values for the circular polarization purity Z before and after the transmission through the cryostat of the radiation beam have been included as percentages at the top of Fig. 3. After the first quartz lens, but before entering the cryostat, Z exceeds 98%. The warped Mylar windows of the cryostat and the quartz lenses deteriorate somewhat the polarization state, but the circular polarization character remains better than Z492% after transmission through the cryostat and sample holder in the absence of a sample. Preliminary experiments (not shown here) were also carried out in a
The bottom panel of Fig. 3 displays the outcome of a transmission experiment with the quasi-optical setup for both circular polarization directions and linear polarization of the incident radiation. Transmission experiments for unpolarized radiation were reported previously in Ref. [32]. Data are shown for both positive and negative values of B. Active cyclotron resonance absorption (CRA) should only occur for the proper sense of circular polarization with respect to the B-field orientation, so that the radiation field accelerates the electron along its cyclotron motion. Indeed, the transmitted power drops nearly to zero for CRA in the active sense of the polarization, whereas reversing the B-field orientation while maintaining the circular polarization direction turns the cyclotron resonance mode inactive (CRI). For linear polarization, the transmitted power does not drop below 50% as it should. These transmission data confirm the quality of the various polarization states. Noteworthy is also the absence of absorption features at the harmonics of oc. This contrasts with the magneto-resistance, which exhibits microwaveinduced oscillations up to the 10th harmonic of the cyclotron resonance (Figs. 1 and 4). 4. Polarization-dependent photoresistivity Fig. 4A illustrates the influence of the circular polarization sense on the microwave- induced resistance oscillations for different frequencies: 200, 183 and 100 GHz [29]. The absorption signals for 200 GHz have been included at the top to easily compare the position of features with the cyclotron resonance. At fields below the CRA (the regions on the left of the dotted lines which mark the o ¼ oc condition) where the microwave-induced oscillations occur, the magnetoresistivity traces for both senses of circular polarization are nearly indistinguishable. This is true for the entire frequency range (100–350 GHz) covered in our experiments. The same conclusion holds for linearly polarized radiation irrespective of the orientation of the electric field vector (an example is shown in Fig. 4B). Hence, the experiments here only support theoretical mechanisms in which the polarization state of the microwaves is not relevant. The experiments in a circular waveguide system with 70% polarization led to the same conclusion. Only at fields near the cyclotron resonance, where significant absorption takes place, deviations between the CRA and CRI curves do arise, likely because resonant heating at the cyclotron resonance produces a second contribution to the photoresistivity. In microwave photoconductivity experiments prior to the discovery of the zero
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resistance phenomenon, the CRA was already detected as a small resistance increase [3]. In the context of the recently discovered microwave-induced resistance oscillations, signatures of the CRA in the magnetoresistivity remained unidentified, presumably because they were masked by the much larger microwave-induced resistance oscillations. The insensitivity of the latter on the polarization state (as proven above) and the ability to alter the circular polarization direction in situ make it straightforward to reveal the bolometric cyclotron resonance contribution. A striking example, at the highest microwave power we have available, is illustrated in Fig. 4B for 243 GHz [29]. The resistance enhancement at the cyclotron resonance can be surprisingly large and develops fine structure. Its close connection to the cyclotron resonance is established by comparing with transmission data and by comparing its amplitude for the different polarization states. We conclude that we can distinguish two microwave-induced contributions to the resistance: the bolometric and polarization-dependent resistance enhancement at the cyclotron resonance due to resonant heating of the electron system and the polarization insensitive contribution related to the oscillatory photoresistivity below and near oc.
(ooc)2 larger for the CRA polarization sense [Eqs. (16) and (17) in Ref. [16] after adaptation to circular polarization; see also Eq. (6.11) in Ref. [14]]. For the maxima (or minima) near o ¼ 2oc and o ¼ 3oc, this amounts to a factor of about 9 and 4, respectively. We refer to Fig. 2 where theoretical predictions for both senses of circular polarization are compared. For large microwave powers, the microwave-induced correction to the dark DC conductivity no longer obeys a linear law but rather a sub-
5. Comparison with theory The complete immunity of the microwave-induced resistance oscillations to the polarization state may be regarded as a crucial test for theories. For instance, this cannot be reconciled with the two most frequently cited theories: the non-equilibrium distribution function scenario [15–18] and the picture based on impurity- and phononassisted electron displacement against the DC-electric field [6–14]. For both the mechanisms, a dependence on the sense of circular polarization enters through the AC Drude conductivity sD ac. Excluding the narrow vicinity of the cyclotron resonance, the AC Drude conductivity is proportional to (ooc)2 if the microwave field accelerates electrons (CRA) and (o+oc)2 if the microwaves decelerate electrons (CRI). For small DC-fields and not to strong a microwave field, the correction to the dark DC dissipative conductivity is linearly proportional to this Drude conductivity and hence is a factor (o+oc)2/ Fig. 4. (A) Comparison of the DC-magnetoresistivity Rxx under microwave radiation for both senses of circular polarization at three frequencies and a temperature of 1.8 K. In the white regions where the microwaveinduced oscillations occur, the magneto-resistance is independent of the microwave polarization. The transmitted power Ptrans (arb. units) at 200 GHz is plotted at the top for comparison with the 200 GHz transport data. The black dotted curve in the bottom panel displays Rxx in the absence of microwaves. (B) Rxx for 243 GHz radiation for various polarizations at the highest available power. Curves are offset for clarity (CRA blue dashed line, CRI red line and linear polarization black line). The dashed line at the top represents the transmitted microwave power (no ordinate shown). Resonant heating at the cyclotron resonance is held responsible for the sharp resistance peak. This peak, as opposed to the oscillatory photo-resistance, is very sensitive to the polarization state [adapted from [29]].
(A)
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linear dependence on the microwave power and these factors are reduced. However, the dramatic difference with experiment here remains. 6. Conclusions We conclude that a polarization-dependent study of the microwave photoconductivity has revealed two contributions: a bolometric and strong polarization-dependent component associated with resonant heating at the cyclotron resonance and the polarization insensitive microwave-induced oscillations. This immunity of the microwave-induced oscillations to changes of the polarization direction of the microwave radiation contradicts the mainstream theories. This puzzling discrepancy will likely reinvigorate the controversy on their origin and there is clearly a strong need to analyze the polarization dependence of other proposed theoretical mechanisms. Acknowledgment We gratefully acknowledge financial support from the German Science Foundation (DFG), the German Ministry of Science and Education (BMBF), the German Israeli Foundation (GIF), and the Transnational Access program of the EU (RITA-CT-2003-506095 WISSMC). References [1] R.G. Mani, J.H. Smet, K. von Klitzing, V. Narayanamurti, W.B. Johnson, V. Umansky, Nature 420 (2002) 646. [2] M.A. Zudov, R.R. Du, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 90 (2003) 046807. [3] E. Vasiliadou, G. Mu¨ller, D. Heitmann, D. Weiss, K. von Klitzing, Phys. Rev. B 48 (1993) 17145. [4] S.A. Studenikin, M. Potemski, P.T. Coleridge, A.S. Sachrajda, Z.R. Wasilewski, Issue Series Title: Solid State Commun. 129 (2004) 341. [5] R.G. Mani, V. Narayanamurti, K. von Klitzing, J.H. Smet, W.B. Johnson, V. Umansky, Phys. Rev. B 69 (2004) 161306.
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[6] V.I. Ryzhii, Sov. Phys. Solid State 11 (1970) 2078. [7] V.I. Ryzhii, R.A. Suris, B.S. Shchamkhalova, Sov. Phys. Semicond. 20 (1986) 1299. [8] A.C. Durst, S. Sachdev, N. Read, S.M. Girvin, Phys. Rev. Lett. 91 (2003) 086803. [9] V. Ryzhii, R. Suris, J. Phys.: Condens. Matter 15 (2003) 6855. [10] V. Ryzhii, V. Vyurkov, Phys. Rev. B 68 (2003) 165406. [11] V. Ryzhii, Phys. Rev. B 68 (2003) 193402. [12] X.L. Lei, S.Y. Liu, Phys. Rev. Lett. 91 (2003) 226805. [13] V.B. Shikin, J. Exp. Theor. Phys. Lett. 77 (2003) 236. [14] M.G. Vavilov, I.L. Aleiner, Phys. Rev. B 69 (2004) 035303. [15] S.I. Dorozhkin, Pis’ma ZhETF 77 (2003) 681; S.I. Dorozhkin, JETP Lett 77 (2003) 577. [16] I.A. Dmitriev, M.G. Vavilov, I.L. Aleiner, A.D. Mirlin, D.G. Polyakov, Phys. Rev. B 71 (2005) 115316; I.A. Dmitriev, M.G. Vavilov, I.L. Aleiner, A.D. Mirlin, D.G. Polyakov, Physica E 25 (2004) 205. [17] I.A. Dmitriev, A.D. Mirlin, D.G. Polyakov, Phys. Rev. Lett. 91 (2003) 226802. [18] J. Dietel, L.I. Glazman, F.W.J. Hekking, F. von Oppen, Phys. Rev. B 71 (2005) 045329. [19] J. Shi, X.C. Xie, Phys. Rev. Lett. 91 (2003) 086801. [20] A.A. Koulakov, M.E. Raikh, Phys. Rev. B 68 (2003) 115324. [21] A.L. Zakharov, Sov. Phys. JETP 11 (1960) 478. [22] A.V. Andreev, I.L. Aleiner, A.J. Millis, Phys. Rev. Lett. 91 (2003) 056803. [23] F.S. Bergeret, B. Huckestein, A.F. Volkov, Phys. Rev. B 67 (2003) 241303. [24] V. Ryzhii, A. Satou, J. Phys. Soc. Japan 72 (2003) 2718. [25] A.F. Volkov, V.V. Pavlovskii, Phys. Rev. B 69 (2004) 125305. [26] A. Auerbach, I. Finkler, B.I. Halperin, A. Yacoby, Phys. Rev. Lett. 94 (2005) 196801. [27] P.H. Rivera, P.A. Schulz, Phys. Rev. B 70 (2004) 75314. [28] J. Inarrea, G. Platero, Phys. Rev. Lett. 94 (2005) 16806. [29] J.H. Smet, B. Gorshunov, C. Jiang, L. Pfeiffer, K. West, V. Umansky, M. Dressel, R. Meisels, F. Kuchar, K. von Klitzing, Phys. Rev. Lett. 95 (2005) 116804. [30] R.L. Willett, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 93 (2004) 026804. [31] G. Kozlov, A. Volkov, Top. Appl. Phys. 74 (1998) 51. [32] S.A. Studenikin, M. Potemski, A.S. Sachrajda, M. Hilke, L.N. Pfeiffer, K.W. West, Phys. Rev. B 71 (2005) 245313.