Two-dimensional 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, non-quantizing 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 irradiation-induced self-organization in a quantum system.
The discovery of the integer and fractional quantum Hall effects1,2,3,4,5,6 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 physics7. The observation of a new dissipationless transport regime at low magnetic fields under microwave irradiation8,9 raised a new challenge regarding our understanding of two-dimensional electron systems. Microwave-induced zero-resistance states (ZRSs) appear at high-microwave excitation powers when the ratio between the photon energy ħω and Landau level spacing ħωc is close to a value within the fraction series: J=ω/ωc=1+1/4,2+1/4,... In theory, however, the motion of electrons in a magnetic field in ultra-clean 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 short-range disorder potential10,11,12, edges13,14 and contacts15. The dominant microscopic picture for ZRS is currently an ensemble of domains with vanishing local conductivity12 but the formation of a collective state with long-range order has also been suggested16,17. So far, the experimental evidence does not provide a definite proof in support of one of the available models and despite intense experimental efforts18,19,20,21,22,23,24 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 excitation25,26 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 readily27,28. Previously, we reported a strong redistribution of the electron density under irradiation that coincided with the appearance of ZRS29, 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 steady-state 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.
Description of the system
A ensemble of electrons was trapped on a liquid helium surface forming a non-degenerate two-dimensional 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 ħωc and intersubband excitations of energy ħω perpendicular to the helium surface27,28. Our experiments are performed at a temperature of T=300 mK much smaller than the Landau level spacing kBT≪ħωc 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 manifold25,26. Note that the two-level system formed by the two subbands has been proposed as a candidate system for quantum computing30. Spatially, the electrons are distributed between the two regions on the helium surface (see Fig. 2), a central region above the disc-shaped electrode at potential Vd and a surrounding guard region above the ring electrode held at potential Vg. We denote ne and ng as the mean electron densities in the central and guard regions, respectively. As in a field effect transistor, ne and ng can be controlled by changing the potentials Vd and Vg. The key difference here is that for surface electrons the total number of electrons Ne in the cloud is fixed as long as the positive potentials Vd and Vg are sufficiently strong to balance the electron–electron Coulomb repulsion. Examples of simulated electron density profiles for different values of Vg 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 Vd and Vg also change the perpendicular holding field in the cell Ez. To avoid this unwanted effect in our experiments, we fixed the value of Vd and changed the potential Vtg simultaneously with Vg keeping the difference Vtg–Vg between top and bottom guard electrode voltages equal to Vd. This choice ensures a uniform value of Ez across the cell. Since lowering the potential Vg compresses the electron cloud towards the centre of the helium cell, we define the compressibility of the electron system as χ=−dne/dVg. In this definition, the electrostatic potential plays the role of the chemical potential in quantum Hall systems. This difference is due to the non-degenerate 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 Va.c.=25 mV was applied on the top and bottom guard electrodes (see Fig. 2) at a low frequency fa.c.≃2 Hz, for which the electron density quasi-statically follows the driving potential. The induced modulation of ne was measured by recording the a.c. current ia.c. created by the motion of image charges on the top central electrode with radius Ri=0.7 cm. The correspondence between the variation of ne and ia.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 ia.c. is measured. This leads to the following expression for the compressibility χ=−dne/dVg:
The presented measurement technique has a strong similarity to that used in compressibility experiments in the quantum Hall regime34, 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 ne(Vg) can be reconstructed by integrating χ with respect to Vg starting from the high Vg limit where ne=0. The obtained results are illustrated in Fig. 3, which shows ia.c. and the corresponding density ne as a function of Vg for several Ne values. The experimental curves for ne(Vg) are compared with the results of electrostatic simulations31,32,33,35 that use the total electron number Ne as the single fitting parameter for each of the obtained curves. The simulations exhibit an extremely good agreement with the experimental results. The dependence ne(Vg) can be understood as follows: a large positive potential Vg attracts the electrons towards the guard electrodes, whereas at low Vg 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 ne that is almost independent of Vg. At intermediate Vg, electrons occupy both the central and guard regions. In this case, ne decreases linearly with increasing Vg until the central region becomes completely depleted. At this point, ne=0 regardless of the value of Vg. In the intermediate regime, the compressibility χ=−(dne)/(dVg) depends only weakly on the total number of trapped electrons Ne 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 (Rg being the outer radius of the guard electrodes) and is the vacuum permittivity. We experimentally find χ0≃2.9 × 106 cm−2 V−1 in good agreement with the estimation obtained using the geometrical cell parameters Rd=1 cm and Rg=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/ωc=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 σxx provided it remains finite. The holding field, identical in the central and guard regions (Vd=Vg–Vtg=4.24 V), was chosen to tune the photon energy in resonance with the intersubband transition.
The compressibilities measured in the dark χd and under irradiation χM are compared in Fig. 4. Two singular regions where χM and χd differ are present: region (I) at high Vg and region (II) at low Vg. Hereafter on we use different notations for the average densities in the dark neD and ngD and under microwave irradiation neM and ngM to avoid ambiguity. In region (I), the dark density in the guard is higher than at the centre ngD>neD, whereas in region (II) we have ngD<neD. In both regions, a strong suppression of the compressibility is observed and χM strikingly vanishes in most of region (I). In contrast, at Vg=Vd=4.24 V where the electrons are distributed evenly between centre and the guard, the compressibility remains almost unchanged under irradiation: χM≃χd≃χ0.
To clarify the physical origin of the anomalous regions (I) and (II), we convert the experimentally controlled variables Vg and Ne to physically more relevant densities neD and ngD. We obtained neD and Ne from the compressibility measurements in the dark (as in Fig. 3). The quantity ngD 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 ngD 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 neD,ngD plane the changes in compressibility δχ=χM–χd under irradiation measured at different values of Vg and Ne 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 colour-scale panels in Fig. 4. The anomalous regions (I) and (II) are upper bounded by lines of constant density in the centre neD=nc and in the guard ngD=nc where we introduced nc≃3 × 106 cm−2. Indeed, for ngD>nc and neD>nc the change in compressibility |δχ|≪χ0 is negligible and the electron density is still described by the electrostatics of the gas phase. Similarly, at low densities: neD and ngD<1.5 × 106 cm−2, we also find no deviations from the equilibrium χ values. The incompressible regions (I) and (II) occupy the space neD<nc<ngD and a fraction of the space ngD<nc<neD, they are characterized by δχ/χ0≃−1 in Fig. 4. Finally, in the remaining area on the neD,ngD plane, χ becomes negative under irradiation, for example, at neD=ngD=2 × 106 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 nc is displaced towards significantly lower densities nc≃1.3 × 106 cm−2.
The incompressible regions with vanishing χ correspond to an unexpected regime where the density becomes independent of the compressing confinement potential Vg. 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 non-degenerate electron gas. Experiments on the quantum Hall effect have also shown that a vanishing longitudinal conductivity σxx, can freeze a non-equilibrium electron density distribution since the charge relaxation time scales can become exponentially large36,37. An explanation based on the vanishing conductivity σxx 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 σxx=0. Indeed, a state with σxx=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 non-trivial 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 neM for different initial densities neD. We performed compressibility measurements without irradiation as described in Fig. 3 to obtain neD as function of Vg at a fixed Ne. Then, fixing Vg, we irradiated the electron system with on/off pulses of millimetre waves creating a periodic displacement of the electron density δne=neM–neD. This displacement induces a transient current of image charges on the measuring electrode ipv(t), which within a plane capacitor approximation is related to the change in the electron density δne 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 δne with the known values for neD, we reconstructed the dependence of neM on the guard potential Vg: the results are shown oi Fig. 5 for several Ne values.
Deviations from neD 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 Vg with a plateau density independent of Ne. We emphasize that this plateau appears because of the cancellation between the decrease of neD at higher Vg and the increase of the area below ipv(t) curves (see Fig. 5c and Equation 3), it is thus a highly non-trivial experimental result. These observations confirm the existence of an incompressible phase for electrons on helium and appear to exclude an explanation based only on σxx=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 δne. 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 Ne in contrast to region (I). An approximate calculation of ngM 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 high-density reservoir region, increasing the density in the low-density regions up to a plateau value.
Since the formation of a non-equilibrium 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 Δe is approximately:
From the experimental values of δne=1.5 × 106 cm−2 and Ne=12.4 × 106 (estimated at point V2 in Fig. 5), we find Δe≃0.1 eV. The presence of this large electrostatic barrier can explain why the incompressible regions occupy a narrower Vg 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 non-equilibrium 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 Vg 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 Vg 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 Δe/ħω≃170, particularly in comparison with a two-level 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 microwave-induced resistance oscillations for electrons on helium suggest that Landau levels higher than the photon energy are unlikely to be strongly populated39,40. These calculations give an accurate prediction for the phase of the resistance oscillations allowing us to exclude strong inter-Landau level excitation41.
The energy can also be accumulated by transitions within the same manifold of quasi-degenerate 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 heterostructures12. It has been predicted that a negative resistance state will stabilize through the formation of domains with a fixed built-in electric field Ec. We performed simulations of the electron density profiles for an electric field dependent conductivity model with σ(E)∝(E2−Ec2), which is believed to describe domain formation. Our simulations show that this model tends to fix the electron density gradients to generate the built-in field Ec. 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 negative42,43. 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 Eω is the amplitude of the microwave electric field and μ2 a non-linear response coefficient43. Under steady-state conditions, this internal flow must be compensated by a counter-flow created by internal electric fields with maximal value . The balance between the two flows sets a lower bound on the longitudinal mobility that is μxx 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 systems44.
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, respectively28. Provided frequencies are matched, this vibration can interact resonantly with electronic modes. The frequency of the expected surface vibrations is ωr≃2π × 30 MHz (for B=0.79 T) and has the same order of magnitude as the low wavelength magneto-shear modes46, which have frequency . Equating these frequencies ωr∼Ωs, 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 ne≃5.8 × 106 cm−2, and the predicted dependence on B is consistent with our observations at different J=ω/ωc 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 microwave-induced 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 self-organization in quantum systems.
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 Ne 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 Ne 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 method47.
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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.
The authors declare no competing financial interests.
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Chepelianskii, A., Watanabe, M., Nasyedkin, K. et al. An incompressible state of a photo-excited electron gas. Nat Commun 6, 7210 (2015). https://doi.org/10.1038/ncomms8210
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