Abstract
Twodimensional electrons in a magnetic field can form new states of matter characterized by topological properties and strong electronic correlations as displayed in the integer and fractional quantum Hall states. In these states, the electron liquid displays several spectacular characteristics, which manifest themselves in transport experiments with the quantization of the Hall resistance and a vanishing longitudinal conductivity or in thermodynamic equilibrium when the electron fluid becomes incompressible. Several experiments have reported that dissipationless transport can be achieved even at weak, nonquantizing magnetic fields when the electrons absorb photons at specific energies related to their cyclotron frequency. Here we perform compressibility measurements on electrons on liquid helium demonstrating the formation of an incompressible electronic state under these resonant excitation conditions. This new state provides a striking example of irradiationinduced selforganization in a quantum system.
Introduction
The discovery of the integer and fractional quantum Hall effects^{1,2,3,4,5,6} revealed the existence of new states of matter characterized by topological properties and strong electronic correlations triggering an intense theoretical and experimental research activity. These efforts lead to a detailed microscopic understanding of the main experimental phenomena and to some of the most beautiful conceptual breakthroughs in condensed matter physics^{7}. The observation of a new dissipationless transport regime at low magnetic fields under microwave irradiation^{8,9} raised a new challenge regarding our understanding of twodimensional electron systems. Microwaveinduced zeroresistance states (ZRSs) appear at highmicrowave excitation powers when the ratio between the photon energy ħω and Landau level spacing ħω_{c} is close to a value within the fraction series: J=ω/ω_{c}=1+1/4,2+1/4,... In theory, however, the motion of electrons in a magnetic field in ultraclean samples is well described as a harmonic oscillator where the selection rules only allow transitions between nearby oscillator states corresponding to J=1. The theoretical explanations proposed, so far, have attempted to resolve this contradiction by considering the role of sharp inhomogeneities due to a shortrange disorder potential^{10,11,12}, edges^{13,14} and contacts^{15}. The dominant microscopic picture for ZRS is currently an ensemble of domains with vanishing local conductivity^{12} but the formation of a collective state with longrange order has also been suggested^{16,17}. So far, the experimental evidence does not provide a definite proof in support of one of the available models and despite intense experimental efforts^{18,19,20,21,22,23,24} the microscopic nature of ZRS remains a puzzle. One of the difficulties is that while the manifestations of ZRS in transport phenomena are spectacular, such as vanishing longitudinal resistance, other indications of these novel electronic states remain elusive.
The observation of ZRS for surface electrons on helium under intersubband excitation^{25,26} has opened a new research direction in this field since the strong Coulomb interactions in this system allow collective effects, such as, for example, Wigner crystallization, to be observed more readily^{27,28}. Previously, we reported a strong redistribution of the electron density under irradiation that coincided with the appearance of ZRS^{29}, although the underlying mechanism was not elucidated. Since the formation of ZRS in a Hall system coincides with vanishing conductivity, the observed redistribution may simply be a consequence of the expected long charge relaxation rates in this regime. In the experiments presented here, we systematically study the behaviour of the electronic density under irradiation and demonstrate a regime in which electrons stabilize at a fixed steadystate density independent of their initial density profile and the electrostatic confinement potential. Since in this regime the electron density is not changed by an increase of the holding electrostatic forces, which tend to compress the electron cloud, we describe this new phase of the electron gas as an incompressible state.
Results
Description of the system
A ensemble of electrons was trapped on a liquid helium surface forming a nondegenerate twodimensional electron gas. The energy levels accessible to the surface electrons in a quantizing perpendicular magnetic field are shown in Fig. 1. They are formed by Landau levels separated by energy ħω_{c} and intersubband excitations of energy ħω perpendicular to the helium surface^{27,28}. Our experiments are performed at a temperature of T=300 mK much smaller than the Landau level spacing k_{B}T≪ħω_{c} so that in equilibrium the electrons mainly fill the lowest Landau level, whereas under resonant irradiation at energy ħω, they can be excited into another subband manifold^{25,26}. Note that the twolevel system formed by the two subbands has been proposed as a candidate system for quantum computing^{30}. Spatially, the electrons are distributed between the two regions on the helium surface (see Fig. 2), a central region above the discshaped electrode at potential V_{d} and a surrounding guard region above the ring electrode held at potential V_{g}. We denote n_{e} and n_{g} as the mean electron densities in the central and guard regions, respectively. As in a field effect transistor, n_{e} and n_{g} can be controlled by changing the potentials V_{d} and V_{g}. The key difference here is that for surface electrons the total number of electrons N_{e} in the cloud is fixed as long as the positive potentials V_{d} and V_{g} are sufficiently strong to balance the electron–electron Coulomb repulsion. Examples of simulated electron density profiles for different values of V_{g} are shown in Fig. 2, the simulations were performed within an electrostatic model as described in refs 31, 32, 33.
In addition to controlling the density profile of the electron cloud ρ(r), the potentials V_{d} and V_{g} also change the perpendicular holding field in the cell E_{z}. To avoid this unwanted effect in our experiments, we fixed the value of V_{d} and changed the potential V_{tg} simultaneously with V_{g} keeping the difference V_{tg}–V_{g} between top and bottom guard electrode voltages equal to V_{d}. This choice ensures a uniform value of E_{z} across the cell. Since lowering the potential V_{g} compresses the electron cloud towards the centre of the helium cell, we define the compressibility of the electron system as χ=−dn_{e}/dV_{g}. In this definition, the electrostatic potential plays the role of the chemical potential in quantum Hall systems. This difference is due to the nondegenerate statistics for electrons on helium for which the Fermi energy is much smaller than the thermal energy.
Compressibility in equilibrium
We developed the following method to measure the compressibility of the electron cloud. An a.c. voltage excitation with amplitude V_{a.c.}=25 mV was applied on the top and bottom guard electrodes (see Fig. 2) at a low frequency f_{a.c.}≃2 Hz, for which the electron density quasistatically follows the driving potential. The induced modulation of n_{e} was measured by recording the a.c. current i_{a.c.} created by the motion of image charges on the top central electrode with radius R_{i}=0.7 cm. The correspondence between the variation of n_{e} and i_{a.c.} was established using plane capacitance electrostatics, for which an electron trapped at the middle of the cell induces half an image charge of e/2 on the top electrode. The use of a simplified electrostatic model is justified here, since the gradients of ρ(r) are located away from the electrode on which i_{a.c.} is measured. This leads to the following expression for the compressibility χ=−dn_{e}/dV_{g}:
The presented measurement technique has a strong similarity to that used in compressibility experiments in the quantum Hall regime^{34}, providing an additional justification for our definition of compressibility. In Supplementary Note 1, we provide a more thorough discussion on our definition of compressibility and give a detailed derivation of Equation 1, which is explained in Supplementary Note 2.
Using Equation 1, the dependence n_{e}(V_{g}) can be reconstructed by integrating χ with respect to V_{g} starting from the high V_{g} limit where n_{e}=0. The obtained results are illustrated in Fig. 3, which shows i_{a.c.} and the corresponding density n_{e} as a function of V_{g} for several N_{e} values. The experimental curves for n_{e}(V_{g}) are compared with the results of electrostatic simulations^{31,32,33,35} that use the total electron number N_{e} as the single fitting parameter for each of the obtained curves. The simulations exhibit an extremely good agreement with the experimental results. The dependence n_{e}(V_{g}) can be understood as follows: a large positive potential V_{g} attracts the electrons towards the guard electrodes, whereas at low V_{g} the electrons are repelled from the guard region and concentrate at the centre of the cell. The potential from the guard electrodes is then almost completely screened, leading to a value of n_{e} that is almost independent of V_{g}. At intermediate V_{g}, electrons occupy both the central and guard regions. In this case, n_{e} decreases linearly with increasing V_{g} until the central region becomes completely depleted. At this point, n_{e}=0 regardless of the value of V_{g}. In the intermediate regime, the compressibility χ=−(dn_{e})/(dV_{g}) depends only weakly on the total number of trapped electrons N_{e} and the values of the confining potentials. This observation can be understood by considering a simplified electrostatic model in which the two electron reservoirs in the disc and guard are treated as plane capacitors. This model, which is presented in more detail in Supplementary Note 3, leads to a value for χ that depends only on the geometrical cell parameters (provided the disc/guard reservoirs are not empty):
Here, h=2.6 mm is the cell height, and are the surface areas of the bottom disc/guard electrodes (R_{g} being the outer radius of the guard electrodes) and is the vacuum permittivity. We experimentally find χ_{0}≃2.9 × 10^{6} cm^{−2} V^{−1} in good agreement with the estimation obtained using the geometrical cell parameters R_{d}=1 cm and R_{g}=1.3 cm. This reference value will be used to normalize the compressibility in our following experiments.
Compressibility under irradiation
We next present our compressibility measurements in the presence of microwave irradiation, focusing on the ZRS fraction J=ω/ω_{c}=6.25, with ω=2π × 139 GHz and a magnetic field of B=0.79 T. Outside ZRS regions the compressibility is not changed by microwave irradiation since it is independent of the conductivity of the system σ_{xx} provided it remains finite. The holding field, identical in the central and guard regions (V_{d}=V_{g}–V_{tg}=4.24 V), was chosen to tune the photon energy in resonance with the intersubband transition.
The compressibilities measured in the dark χ_{d} and under irradiation χ_{M} are compared in Fig. 4. Two singular regions where χ_{M} and χ_{d} differ are present: region (I) at high V_{g} and region (II) at low V_{g}. Hereafter on we use different notations for the average densities in the dark n_{eD} and n_{gD} and under microwave irradiation n_{eM} and n_{gM} to avoid ambiguity. In region (I), the dark density in the guard is higher than at the centre n_{gD}>n_{eD}, whereas in region (II) we have n_{gD}<n_{eD}. In both regions, a strong suppression of the compressibility is observed and χ_{M} strikingly vanishes in most of region (I). In contrast, at V_{g}=V_{d}=4.24 V where the electrons are distributed evenly between centre and the guard, the compressibility remains almost unchanged under irradiation: χ_{M}≃χ_{d}≃χ_{0}.
To clarify the physical origin of the anomalous regions (I) and (II), we convert the experimentally controlled variables V_{g} and N_{e} to physically more relevant densities n_{eD} and n_{gD}. We obtained n_{eD} and N_{e} from the compressibility measurements in the dark (as in Fig. 3). The quantity n_{gD} could not be measured directly in a reliable manner owing to the unavoidable effect of the density gradients in the guard region. We, thus, calculated n_{gD} by averaging the simulated density profiles ρ(r) over the guard region. This procedure is justified by the excellent agreement between the compressibility measurements and our numerical simulations that we demonstrated in absence of irradiation.
Using this method, we summarize on the n_{eD},n_{gD} plane the changes in compressibility δχ=χ_{M}–χ_{d} under irradiation measured at different values of V_{g} and N_{e} while fixing all the other parameters (magnetic field, microwave frequency and power, temperature and perpendicular electric field; the dependence on microwave power and magnetic field is shown in Supplementary Figs 1–5). The results are shown in the colourscale panels in Fig. 4. The anomalous regions (I) and (II) are upper bounded by lines of constant density in the centre n_{eD}=n_{c} and in the guard n_{gD}=n_{c} where we introduced n_{c}≃3 × 10^{6} cm^{−2}. Indeed, for n_{gD}>n_{c} and n_{eD}>n_{c} the change in compressibility δχ≪χ_{0} is negligible and the electron density is still described by the electrostatics of the gas phase. Similarly, at low densities: n_{eD} and n_{gD}<1.5 × 10^{6} cm^{−2}, we also find no deviations from the equilibrium χ values. The incompressible regions (I) and (II) occupy the space n_{eD}<n_{c}<n_{gD} and a fraction of the space n_{gD}<n_{c}<n_{eD}, they are characterized by δχ/χ_{0}≃−1 in Fig. 4. Finally, in the remaining area on the n_{eD},n_{gD} plane, χ becomes negative under irradiation, for example, at n_{eD}=n_{gD}=2 × 10^{6} cm^{−2}. To highlight the robustness of our results, Fig. 4 also shows similar data obtained at two other ZRS fractions: J=5.25 and J=10.25. Incompressible regions appear for these cases as well, we note that for J=10.25 the position of the boundary n_{c} is displaced towards significantly lower densities n_{c}≃1.3 × 10^{6} cm^{−2}.
The incompressible regions with vanishing χ correspond to an unexpected regime where the density becomes independent of the compressing confinement potential V_{g}. In the integer quantum Hall effect, incompressible phases appear owing to the finite energy required to add electrons to a system in which the Landau levels available at the Fermi energy are all fully occupied. This explanation, however, is not applicable to electrons on helium since they form a nondegenerate electron gas. Experiments on the quantum Hall effect have also shown that a vanishing longitudinal conductivity σ_{xx}, can freeze a nonequilibrium electron density distribution since the charge relaxation time scales can become exponentially large^{36,37}. An explanation based on the vanishing conductivity σ_{xx} seems natural owning to the coincidence between the onset of charge redistribution and ZRS. We can determine experimentally whether the incompressible behaviour can be explained only on the basis of σ_{xx}=0. Indeed, a state with σ_{xx}=0 is expected to freeze the existing density distribution due to the very long charge relaxation rates; thus, the final state should depend in a nontrivial way on the equilibrium density profile and on the kinetics of the transition to ZRS.
We developed the following approach to determine the central density under irradiation n_{eM} for different initial densities n_{eD}. We performed compressibility measurements without irradiation as described in Fig. 3 to obtain n_{eD} as function of V_{g} at a fixed N_{e}. Then, fixing V_{g}, we irradiated the electron system with on/off pulses of millimetre waves creating a periodic displacement of the electron density δn_{e}=n_{eM}–n_{eD}. This displacement induces a transient current of image charges on the measuring electrode i_{pv}(t), which within a plane capacitor approximation is related to the change in the electron density δn_{e} through the relation:
The integral in this equation is evaluated over the time interval where the irradiation is switched off (a derivation of this relation is provided in Supplementary Note 4). Combining δn_{e} with the known values for n_{eD}, we reconstructed the dependence of n_{eM} on the guard potential V_{g}: the results are shown oi Fig. 5 for several N_{e} values.
Deviations from n_{eD} mainly appear in two voltage regions, which are in good correspondence with the regions (I) and (II) outlined in Fig. 4. In region (I), the density under irradiation exhibits a striking plateau as a function of V_{g} with a plateau density independent of N_{e}. We emphasize that this plateau appears because of the cancellation between the decrease of n_{eD} at higher V_{g} and the increase of the area below i_{pv}(t) curves (see Fig. 5c and Equation 3), it is thus a highly nontrivial experimental result. These observations confirm the existence of an incompressible phase for electrons on helium and appear to exclude an explanation based only on σ_{xx}=0 since the final state density does not depend on the density distribution in equilibrium for a wide range of parameters. Instead, our experiments imply the existence of a dynamical mechanism that stabilize the electron density to a fixed value.
We next comment on the sign of δn_{e}. In region (I), the electrons migrate from the guard, where the densities are higher, to the centre of the electron cloud, whereas in region (II), the trend is opposite and electrons flow from the centre towards the edges of the electron cloud. In the latter region, narrow density plateaux are also observed, however, the plateau density value depends on N_{e} in contrast to region (I). An approximate calculation of n_{gM} shown in Fig. 5 suggests that in region (II), the transition to an incompressible state occurs owning to the pinning of the density inside the guard region. Thus, in the incompressible phase, the electron cloud transfers electrons from a highdensity reservoir region, increasing the density in the lowdensity regions up to a plateau value.
Discussion
Since the formation of a nonequilibrium density profile increases the electron–electron repulsion energy, it is important to estimate the associated energy cost. Using the simplified electrostatic model (see Supplementary Note 5 for a detailed derivation), we find that the energy cost of the redistribution per electron Δ_{e} is approximately:
From the experimental values of δn_{e}=1.5 × 10^{6} cm^{−2} and N_{e}=12.4 × 10^{6} (estimated at point V_{2} in Fig. 5), we find Δ_{e}≃0.1 eV. The presence of this large electrostatic barrier can explain why the incompressible regions occupy a narrower V_{g} range in the photocurrent data than in the compressibility measurement: for example, region (I) has a width at least 0.6 V in Fig. 4 but a width of only 0.3 V in Fig. 5. The main difference between the two techniques is that during compressibility measurements microwaves are always present, maintaining the system in a nonequilibrium state, whereas in the photocurrent measurement the microwave on/off pulses continuously reset the system back to its equilibrium state. Thus, in the photocurrent measurements, the electrons must overcome an increasingly large energy barrier to reach the incompressible state as V_{g} increases. When the barrier becomes too large, the systems remains in its equilibrium state and the photocurrent vanishes abruptly. In contrast, in the compressibility measurement, the electron system remains in the incompressible state as V_{g} changes and the electrostatic barrier does not need to be overcome directly, allowing the incompressible state to exist over a wider parameter range. (In the Supplementary Figs 6–9, we provide a detailed comparison between the two measurement techniques and show that they are fully consistent once the described hysteretic behaviour is taken into account; additional experimental data is also provided in ref. 38).
The estimated charging energy must be provided by the microwave irradiation since it is the only energy source in our system, its amplitude corresponds to a surprisingly large number of photons absorbed per electron Δ_{e}/ħω≃170, particularly in comparison with a twolevel system that cannot absorb more than one photon. The energy of the absorbed photons thus needs to be transferred efficiently to other degrees of freedom. It could be transferred by the excitation of higher Landau levels. However, the theoretical calculations performed by Y. Monarkha to explain the origin of microwaveinduced resistance oscillations for electrons on helium suggest that Landau levels higher than the photon energy are unlikely to be strongly populated^{39,40}. These calculations give an accurate prediction for the phase of the resistance oscillations allowing us to exclude strong interLandau level excitation^{41}.
The energy can also be accumulated by transitions within the same manifold of quasidegenerate Landau levels as they are bent by the confinement potential. A possible mechanism for this absorption is provided by the negative conductivity models introduced theoretically to explain ZRS in heterostructures^{12}. It has been predicted that a negative resistance state will stabilize through the formation of domains with a fixed builtin electric field E_{c}. We performed simulations of the electron density profiles for an electric field dependent conductivity model with σ(E)∝(E^{2}−E_{c}^{2}), which is believed to describe domain formation. Our simulations show that this model tends to fix the electron density gradients to generate the builtin field E_{c}. This changes the compressibility of the electron cloud depending on the number of stable domains in the system but without inducing an incompressible state. Thus, new theoretical developments are required to make the domain picture consistent with our experiments.
A recent theoretical proposal suggests that another instability can occur even before the conductivity becomes negative^{42,43}. Due to fluctuations of the microwave electromagnetic field on the wavelength scale λ≃2 mm ratchet internal currents are expected to appear under irradiation inside the electron system. The velocity of the induced electron flow will scale as where E_{ω} is the amplitude of the microwave electric field and μ_{2} a nonlinear response coefficient^{43}. Under steadystate conditions, this internal flow must be compensated by a counterflow created by internal electric fields with maximal value . The balance between the two flows sets a lower bound on the longitudinal mobility that is μ_{xx} stable under irradiation: . Mobilities below this threshold are expected to lead to the formation of electron pockets on a scale λ with a certain similarly to the stripe/puddles instability occurring in quantum Hall systems^{44}.
The above argument predicts the existence of an instability without explaining the pinning of the electron density. A mechanism selecting a particular density value is thus desirable. As a possible mechanism, we propose the following scenario. If we assume that a significant fraction of the energy carried by the absorbed photons is transferred into ripplons with a typical wave number given by the inverse magnetic length (ref. 45), then the helium surface will vibrate at frequency where γ and ρ are the helium surface tension and density, respectively^{28}. Provided frequencies are matched, this vibration can interact resonantly with electronic modes. The frequency of the expected surface vibrations is ω_{r}≃2π × 30 MHz (for B=0.79 T) and has the same order of magnitude as the low wavelength magnetoshear modes^{46}, which have frequency . Equating these frequencies ω_{r}∼Ω_{s}, we find an equation on the plateau density:
For B=0.79 T, this formula gives the right order of magnitude for the plateau density n_{e}≃5.8 × 10^{6} cm^{−2}, and the predicted dependence on B is consistent with our observations at different J=ω/ω_{c} shown in Fig. 4. Even if the described mechanism requires further theoretical studies, it correctly captures the critical dependence on the electronic density observed in the experiments. For electrons on helium, the electronic density mainly controls the strength of electron–electron interactions and the key role played by this parameter indicates that the formation of incompressible states is a collective effect involving many electrons.
In conclusion, we have shown that electrons trapped on a helium surface exhibit incompressible behaviour under resonant irradiation conditions corresponding to the formation of a microwaveinduced ZRS. Their density becomes pinned to a fixed value independent of the applied electrostatic force and on the initial electron distribution profile for a wide range of parameters. The transition to the incompressible state is achieved by overcoming an impressively high electrostatic energy barrier of up to 0.1 eV per electron. We described the possible energy conversion processes within the electron system that can transform the energy of the absorbed photons into charging energy, and we proposed several competing mechanisms that can render the equilibrium density profile unstable. Since the incompressible behaviour emerging from our experiments is very elegant, we believe that its understanding will stimulate the emergence of new concepts for selforganization in quantum systems.
Methods
Our experiments were performed in a Leiden dilution refrigerator with base temperature around 25 mK, the magnetic field was provided by a homemade superconducting magnet with maximal field of 1 T. The experimental cell was filled with liquid helium until half filling by monitoring the cell capacitance during helium condensation. Electrons were deposited on the helium surface by thermionic emission from a heated tungsten filament. Control of the total electron number N_{e} in the cloud was achieved by trapping an initially high concentration of electrons and then lowering the confinement voltages allowing excess electrons to escape. The obtained N_{e} value was determined from compressibility measurements in equilibrium. The frequency of the intersubband transition was tuned in resonance with the photon energy using the linear Starck shift induced by the electric field perpendicular to the helium surface, which was fixed during our compressibility and photocurrent experiments to keep the system at intersubband resonance. The current probe electrodes were grounded through Stanford Research (SR570) current amplifiers, while the potential of the other electrodes was set by Yokogawa DC voltage supplies, this ensure a stable d.c. voltage on all cell electrodes independently of the irradiation. Numerical simulations were performed by solving the Laplace equations for our cell geometry using a finite elements method^{47}.
Additional information
How to cite this article: Chepelianskii, A.D. et al. An incompressible state of a photoexcited electron gas. Nat. Commun. 6:7210 doi: 10.1038/ncomms8210 (2015).
References
Klitzing, K. v., Dorda, G. & Pepper, M. New method for highaccuracy determination of the finestructure constant based on quantized hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).
Tsui, D. C., Stormer, H. L. & Gossard, A. C. Twodimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).
Jain, J. Composite Fermions Cambridge Univ. Press (2007).
Novoselov, K. S. et al. Twodimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).
Zhang, Y., Tan, Y. W., Stormer, H. L. & Kim, P. Stormer and Philip Kim experimental observation of the quantum Hall effect and Berry's phase in graphene. Nature 438, 201–204 (2005).
Bolotin, K. I., Ghahari, F., Shulman, M. D., Stormer, H. L. & Kim, P. Observation of the fractional quantum Hall effect in graphene. Nature 462, 196–199 (2009).
Ezawa, Zyun Francis Quantum Hall Effect: Field Theoretical Approach and Related Topics World scientific (2008).
Mani, R. G. et al. Zeroresistance states induced by electromagneticwave excitation in GaAs/AlGaAs heterostructures. Nature 420, 646–650 (2002).
Zudov, M. A., Du, R. R., Pfeiffer, L. N. & West, K. W. Evidence for a new dissipationless effect in 2D electronic transport. Phys. Rev. Lett. 90, 046807 (2003).
Durst, A. C., Sachdev, S., Read, N. & Girvin, S. M. Radiationinduced magnetoresistance oscillations in a 2D electron gas. Phys. Rev. Lett. 91, 086803 (2003).
Dmitriev, I. A., Mirlin, A. D. & Polyakov, D. G. Cyclotronresonance harmonics in the ac response of a 2D electron gas with smooth disorder. Phys. Rev. Lett. 91, 226802 (2003).
Dmitriev, I. A., Mirlin, A. D., Polyakov, D. G. & Zudov, M. A. Nonequilibrium phenomena in high Landau levels. Rev. Mod. Phys. 84, 1709–1763 (2012).
Chepelianskii, A. D. & Shepelyansky, D. L. Microwave stabilization of edge transport and zeroresistance states. Phys. Rev. B 80, 241308 (2009).
Levin, A. D., Momtaz, Z. S., Gusev, G. M. & Bakarov, A. K. Microwaveinduced nonlocal transport in a twodimensional electron system. Phys. Rev. B 89, 161304 (2014).
Mikhailov, S. A. Theory of microwaveinduced zeroresistance states in twodimensional electron systems. Phys. Rev. B 83, 155303 (2011).
Zhirov, O. V., Chepelianskii, A. D. & Shepelyansky, D. L. Towards a synchronization theory of microwaveinduced zeroresistance states. Phys. Rev. B 88, 035410 (2013).
Chepelianskii, A. D., Pikovsky, A. S. & Shepelyansky, D. L. Synchronization, zeroresistance states and rotating Wigner crystal. Eur. Phys. J. B 60, 225–229 (2007).
Willett, R. L., Pfeiffer, L. N. & West, K. W. Evidence for currentflow anomalies in the irradiated 2D electron system at small magnetic fields. Phys. Rev. Lett. 93, 026804 (2004).
Smet, J. H. et al. Circularpolarizationdependent study of the microwave photoconductivity in a twodimensional electron system. Phys. Rev. Lett. 95, 116804 (2005).
Zudov, M. A., Du, R. R., Pfeiffer, L. N. & West, K. W. Bichromatic microwave photoresistance of a twodimensional electron system. Phys. Rev. Lett. 96, 236804 (2006).
Mani, R. G., Johnson, W. B., Umansky, V., Narayanamurti, V. & Ploog, K. Phase study of oscillatory resistances in microwaveirradiated and darkGaAs/AlGaAs devices: Indications of an unfamiliar class of the integral quantum Hall effect. Phys. Rev. B 79, 205320 (2009).
Wiedmann, S., Gusev, G. M., Raichev, O. E., Bakarov, A. K. & Portal, J. C. Microwave zeroresistance states in a bilayer electron system. Phys. Rev. Lett. 105, 026804 (2010).
Dorozhkin, S. I., Pfeiffer, L., West, K., von Klitzing, K. & Smet, J. H. Random telegraph photosignals in a microwaveexposed twodimensional electron system. Nat. Phys. 7, 336 (2011).
Mani, R. G., Ramanayaka, A. N. & Wegscheider, W. Observation of linearpolarizationsensitivity in the microwaveradiationinduced magnetoresistance oscillations. Phys. Rev. B 84, 085308 (2011).
Konstantinov, D. & Kono, K. Photoninduced vanishing of magnetoconductance in 2D electrons on liquid helium. Phys. Rev. Lett. 105, 226801 (2010).
Konstantinov, D. & Kono, K. Novel radiationinduced magnetoresistance oscillations in a nondegenerate twodimensional electron system on liquid helium. Phys. Rev. Lett. 103, 266808 (2009).
In TwoDimensional Electron Systems on Helium and Other Cryogenic Substrates ed Andrei E. Y. Kluwer Academic (1997).
Monarkha, Y. & Kono, K. TwoDimensional Coulomb Liquids and Solids Springer (2004).
Konstantinov, D., Chepelianskii, A. D. & Kono, K. Resonant photovoltaic effect in surface state electrons on liquid helium. J. Phys. Soc. Jpn. 81, 093601 (2012).
Platzman, R. M. & Dykman, M. I. Quantum computing with electrons floating on liquid helium. Science 284, 1967–1969 (1999).
Wilen, L. & Giannetta, R. Impedance methods for surface state electrons. J. Low Temp. Phys. 72, 353–369 (1988).
Kovdrya, Y. Z., Nikolayenko, V. A., Kirichek, O. I., Sokolov, S. S. & Grigor'ev, V. N. Quantum transport of surface electrons over liquid helium in magnetic field. J. Low Temp. Phys. 91, 371–389 (1993).
Closa, F., Raphaël, E. & Chepelianskii, A. D. Transport properties of overheated electrons trapped on the a helium surface. Eur. Phys. J. B 87, 1–9 (2014).
Tessmer, S. H., Glicofridis, P. I., Ashoori, R. C., Levitov, L. S. & Melloch, M. R. Subsurface charge accumulation imaging of a quantum Hall liquid. Nature 392, 51–54 (1997).
Badrutdinov, A. O., Abdurakhimov, L. V. & Konstantinov, D. Cyclotron resonant photoresponse of a multisubband twodimensional electron system on liquid helium. Phys. Rev. B 90, 075305 (2014).
Dolgopolov, V. T., Shashkin, A. A., Zhitenev, N. B., Dorozhkin, S. I. & von Klitzing, K. Quantum Hall effect in the absence of edge currents. Phys. Rev. B 46, 12560–12567 (1992).
Jeanneret, B. et al. Observation of the integer quantum Hall effect by magnetic coupling to a Corbino ring. Phys. Rev. B 51, 9752–9756 (1995).
Watanabe, M., Nasyedkin, K., Kono, K., Konstantinov, D. & Chepelianskii, A. D. Additional experimental data is available online at https://hal.archivesouvertes.fr/hal01143216.
Monarkha, Y. P. Microwaveresonanceinduced magnetooscillations and vanishing resistance states in multisubband twodimensional electron systems. Low Temp. Phys. 37, 655–666 (2011).
Monarkha, Y. P. Coulombic effects on magnetoconductivity oscillations induced by microwave excitation in multisubband twodimensional electron systems. Low Temp. Phys. 38, 451–458 (2012).
Konstantinov, D., Monarkha, Y. & Kono, K. Effect of coulomb interaction on microwaveinduced magnetoconductivity oscillations of surface electrons on liquid helium. Phys. Rev. Lett. 111, 266802 (2013).
Entin, M. V. & Magarill, L. I. Photogalvanic current in a double quantum well. JETP Lett. 98, 38 (2013).
Entin, M. V. & Magarill, L. I. Photogalvanic current in electron gas over a liquid helium surface. JETP Lett. 98, 816–822 (2014).
Koulakov, A. A., Fogler, M. M. & Shklovskii, B. I. Charge density wave in twodimensional electron liquid in weak magnetic field. Phys. Rev. Lett. 76, 499–502 (1996).
Lea, M. J. et al. Magnetoconductivity of twodimensional electrons on liquid helium:experiments in the fluid phase. Phys. Rev. B 55, 16280–16292 (1997).
Golden, Kenneth I., Kalman, G. & Wyns, Philippe Dielectric tensor and collective modes in a twodimensional electron liquid in magnetic field. Phys. Rev. B 48, 8882–8889 (1993).
Hecht, F. New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012).
Acknowledgements
We are thankful to H. Bouchiat for fruitful discussions and acknowledge support from JSPS KAKENHI Grant Number 24000007. One of us, A.C. acknowledges support from the E. Oppenheimer fellowship and from St Catharine college in Cambridge. D.K. is supported by internal grant from Okinawa Institute of Science and Technology Graduate University.
Author information
Authors and Affiliations
Contributions
All authors contributed to all aspects of this work.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figures 19, Supplementary Notes 15 and Supplementary References (PDF 1428 kb)
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Chepelianskii, A., Watanabe, M., Nasyedkin, K. et al. An incompressible state of a photoexcited electron gas. Nat Commun 6, 7210 (2015). https://doi.org/10.1038/ncomms8210
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms8210
Further reading

Piezoacoustics for precision control of electrons floating on helium
Nature Communications (2021)

Study of narrow negative magnetoresistance effect in ultrahigh mobility GaAs/AlGaAs 2DES under microwave photoexcitation
Scientific Reports (2020)

Cyclotron resonance in the high mobility GaAs/AlGaAs 2D electron system over the microwave, mmwave, and terahertz bands
Scientific Reports (2019)

Radiationinduced magnetoresistance oscillations in monolayer and bilayer graphene
Scientific Reports (2019)

Can Warmer than Room Temperature Electrons Levitate Above a Liquid Helium Surface?
Journal of Low Temperature Physics (2019)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.