Resonant Photovoltaic Effect in Surface-State Electrons on Liquid Helium

We observed an ultra-strong photovoltaic effect induced by resonant intersubband absorption of microwaves in a two-dimensional electrons system on the surface of liquid helium. The effect emerges in the regime of microwave-induced vanishing of dissipative conductance reported previously [D. Konstantinov and K. Kono: Phys. Rev. Lett. 105 (2010) 226801)] and is characterized by a nonequilibrium spatial distribution of electrons in the confining electrostatic potential. The electrostatic energy acquired by an electron exceeds other relevant energies by several orders of magnitude.

We observed an electromotive-force effect induced by the resonant intersubband absorption of microwaves in surface-state electrons on liquid helium subjected to a perpendicular magnetic field. The effect emerges at the minima of radiation-induced conductance oscillations reported previously [D. Konstantinov  Under electromagnetic irradiation, a net flow of charged particles may occur even in the absence of an average macroscopic force acting on them. As a result, the separation of positive and negative carriers causes an electrical potential difference, known as an electromotive force (EMF), to develop across the system. In semiconductors, such a photovoltaic effect can result, for example, from the diffusion of nonequilibrium charged carries produced by light or their separation at p-n junction [1]. The latter mechanism makes it possible to convert the energy of radiation into electricity and has important applications in photodiodes and photovoltaic cells. Recently, significant effort has been made to increase the efficiency of solar cells by employing new concepts based on, for example, nanostructured materials [2], hot-carrier transfer [3], and surface plasmon resonance [4]. In a number of experiments [5][6][7][8], photovoltaic effects were reported in conjunction with microwave-induced oscillations and zero-resistance states (ZRS) in a two-dimensional electron system (2DES) in high-mobility GaAs/AlGaAs heterostructures [9,10].
In this Letter, we report the first observation of an EMF effect induced in surface-state electrons (SSE) on liquid helium by the interaction of electrons with a resonant microwave radiation field. In such a system, surface state subbands with energies ǫ n (n = 1, 2, ..) are formed owing to the attractive image force, the repulsive surface barrier, and an electric field E ⊥ applied perpendicular to the surface [11]. Below 1 K, almost all electrons are frozen into the lowest subband. Recent work has provided evidence that an effect similar to the microwaveinduced ZRS of a quantum degenerate 2DES occurs in a nondegenerate system of SSE on helium [12]. In the experiment in Ref. 12, the intersubband n = 1 → 2 transition was excited using microwaves with angular frequency ω such thathω = ǫ 2 −ǫ 1 . Under such conditions, the longitudinal conductivity σ xx , measured using capacitively coupled Corbino electrodes, underwent oscillations upon varying the perpendicular magnetic field B, exhibiting a sequence of minima shifted to lower values with respect to B satisfying the relation ω/ω c = l, where ω c = eB/m is the cyclotron frequency and l = 4, 5, ... The vanishing conductivity σ xx → 0 at the minima implied vanishing resistivity ρ xx in accordance with the standard tensor relation. Remarkably, the vanishing conductivity exhibited a hysteretic response, indicating the intrinsic instability of the electronic states. Here, we demonstrate that at the minima of the conductivity oscillations the radiation causes the separation of electrons from the neutralizing background provided by the positively charged metallic electrodes, resulting in a large EMF on the order 1 V across the electron layer. Our finding suggests a novel photovoltaic effect occurring in a 2DES subjected to a perpendicular magnetic field. Figure 1 shows a schematic diagram of the experimental apparatus. Liquid 3 He is condensed into a cell (a flat cylindrical cavity), which contains a sintered silver heat exchanger to ensure sufficient heat contact between the liquid and the cell body. The liquid level is set midway between two parallel plates, each having a diameter of 26 mm, separated by d = 2.6 mm. The bottom plate is divided into electrodes B and G 1 by a gap at a distance of 10 mm from the center of the plate, while the top plate is divided into electrodes C 1 , C 2 and G 2 by two gaps at distances of 7 and 10 mm from the center. Each of the three gaps has a width of 0.2 mm. Electrodes C 1 and C 2 form a Corbino disk (dc-grounded), from which σ xx for the SSE can be determined as described previously [12]. Electrons are produced by briefly heating a tungsten filament F, while a positive bias V B = 0.4 V is applied to B with all other electrodes grounded. The amount of charge, which forms a circular pool on the liquid surface above electrode B, is determined by assuming the condition of complete screening of the electric field above the liquid and corresponds to about 5 × 10 6 electrons.
The positively biased electrodes provide a neutralizing background, which counteracts the Coulomb repulsion between SSE and prevents them to spread away. Below 1 K, the average thermal kinetic energy of an electron moving along the surface does not exceed 10 −4 eV, which is much less than the typical difference of its electrical potential energy across the layer. Therefore, the effects associated with the diffusion of electrons are neglected here. In the absence of radiation, the equilibrium F y y y y y y y y y y y y y y y y spatial distribution of electrons in the confining electrostatic potential can be obtained numerically using the Green function method [13]. Examples of the equilibrium density profile calculated for 5 × 10 6 electrons and for different electrode biases are shown in the inset of Fig. 1.
Electrons are tuned for the intersubband n = 1 → 2 resonance with the applied microwaves by adjusting the bias V B , which changes the electric field E ⊥ and shifts the subband energies because of the Stark effect [14]. For ω/2π = 90.9 GHz employed in this experiment, the resonance is centered at V B = 6.4 V. In the case of 5 × 10 6 electrons and all electrodes grounded except for electrode B, this bias corresponds to the areal density of electrons n s = 2.8 × 10 6 cm −2 (see the inset of Fig. 1). Previously [12], a photoinduced change in σ xx was observed upon CW irradiation and in the presence of a driving electric field. The present work aims at detecting the photoinduced motion of the electrons in the absence of such a field. This is stimulated by the recent theoretical proposal that ZRS of 2DES in heterostructures could be explained by the stabilization of edge trajectories under microwave irradiation [15]. Indeed, such a mechanism might lead to an accumulation of charge at the edges and thus to an EMF. In this experiment, the incident microwave power is modulated using a low-frequency (0.5-2 Hz) square waveform, and the current signals I 1 and I 2 induced in electrodes C 1 and C 2 , respectively, are recorded using a digital storage oscilloscope following the current preamplifiers (100 Hz bandwidth). Alternatively, the current ∆I between electrodes C 1 and C 2 is measured using a conventional lock-in amplifier. Figure 2(a) shows an example of the transient current signals I 1 and I 2 obtained at T = 0.2 K, n s = 2.8 × 10 6 cm −2 , and B = 0.62 T. The value of B corresponds to the l = 5 conductance minimum. For reference, the modulation of the microwave power is shown in Fig. 2(b). Each trace represents the average over about 500 sweeps. The signals are comparable in magnitude and have opposite signs. From this result, we obtain the following qualitative picture. Upon switching the power on, the SSE are forced to flow along the surface, causing the depletion (accumulation) of the charge in the central (peripheral) region of the electron pool. Correspondingly, the positive (negative) current is induced in electrode C 1 (C 2 ) by the flow of the image charge. The surface charge flows until a new spatial distribution of SSE in the unchanged confining electrostatic potential is established, after which the currents I 1 and I 2 becomes zero. Correspondingly, the displacement of electrons with respect to the neutralizing background induces a nonzero electric field, and therefore EMF, to develop in the charged layer. Upon switching the power off, the displaced surface charge flows back to restore the equilibrium distribution of SSE, and a negative (positive) current consisting of the image charge is induced in C 1 (C 2 ). An important quantity, which determines the magnitude of the EMF, is the fraction of the surface charge displaced under irradiation. The latter can be estimated from the cumulative charge Q flowing, for example, through electrode C 1 . Q (in units of the elementary charge e > 0) is shown in Fig. 2(c) for three values of B corresponding to the conductance minima l = 4, 5, and 6. The total image charge induced on electrode C 1 in the dark can be determined from the following procedure. At V B = V G1 = 6.4 V, we apply a positive bias of 3.0 V to the electrode G 2 . As a result, the surface charge is pulled away from the center to form a ring beneath electrode G 2 , as confirmed numerically by calculating the charge profile (dotted line in the inset of Fig. 1). When the sign of V G2 is reversed, the charge is pushed back into the region beneath electrode C 1 , inducing a current in the electrode. Integrating this current, we find that a total image charge of 1.0 × 10 6 e is induced in electrode C 1 by the SSE. This value is in good agreement with an estimate from a simple model of a charged particle between two conducting plates, as well as precise calculations using the Green function method. The comparison of this value with Q in Fig. 2(c) shows that a significant fraction (more than 50%) of the surface charge beneath electrode C 1 can be displaced upon irradiation. For our density, a 50% displacement yields an EMF of about 0.4 V, which corresponds to the difference in the potential energy of an electron markedly exceeding other characteristic energies such as, for example, the photon energyhω ≈ 0.38 meV. Figure 3 shows a comparison between σ xx measured under CW irradiation and ∆I recorded under 100% modulation of the incident microwave power. Both sets of data are obtained upon increasing the magnetic field B at T = 0.2 K and at the same level of microwave power. A nonzero signal ∆I is observed only in the intervals of B near the conductance minima corresponding to l = 4, 5, 6, and 7. As described above, this signal is due to the currents induced in electrodes C 1 and C 2 by the displacement of the surface charge. No signal is observed when electrons are tuned away from the intersubband resonance using the microwaves by changing the electrical bias V B . Simultaneously, the detuning results in the complete disappearance of the magnetoosillations and zeroconductance states [12]. Importantly, the disappearance of ∆I in the SSE tuned away from the resonance, as well as when the surface is completely discharged by reversing the sign of V B , confirms that the current is not produced in the conducting electrodes by, for example, microwaveinduced heating, vibrations, or other effects.
The inset of Fig. 3(b) shows the signal ∆I in a narrow range of B near the l = 5 conductance minimum. The signal emerges abruptly upon slowly increasing B. This feature is consistent with the abrupt drop of σ xx to zero recorded upon the upward sweep of B (solid circles). Upon the downward sweep of B, σ xx (open circles) exhibits hysteresis, which may be due to the bistability of the electronic states [12]. In contrast, the variation of ∆I with B does not show any dependence on the direction of sweeping. The absence of hysteresis of ∆I might result from the modulation of the microwave power, which is repeatedly switched on and off, thus making the observation of the metastable regime impossible. The kinetics of the radiation-induced displacement of surface charge strongly depends on different parameters such as the microwave power, the density of electrons, and the index l. The cumulative charge measured for n s = 2.8 × 10 6 cm −2 , B = 0.78 T, and at four different power levels is shown in Figure 4. For 2DES in Fig. 1, the typical time required to recover the equilibrium spatial distribution, the relaxation time τ d , can be estimated as C/σ xx , where C is the capacitance between the charged layer and the surrounding electrodes. For C ∼ 1 pF and σ xx ∼ 10 −10 S, this gives τ d ∼ 10 −2 s, which is in agreement with the fall time of Q upon switching the power off, as shown in Fig. 4. In contrast, the time required to establish the redistribution of electrons across the layer upon switching the power on, as can be seen from the rising edge of Q, can exceed 1 s at sufficiently low powers. Even at high powers, the build-up of EMF is slowed by either increasing n s or choosing the conductance minima of a high index l. Remarkably, a time delay of up to 0.1 s between the application of microwaves and the onset of the charge motion can be observed, as shown, for example, by the data labeled as 0.52 T in Fig. 2(c).
In summary, the photoresponse of SSE on liquid helium in the absence of a dc electrical driving reveals the existence of a microwave-induced current in the electron layer. The displacement of charge produces EMF, which corresponds to the difference in the electron potential energy on the order 1 eV, which exceeds, for example, its thermal kinetic energy, cyclotron energy, or intersubband energy by several orders of magnitude.
The origin of such a strong effect is presently unclear. Radiation can exert an optical force on a charged particle, as is widely used for the cooling and trapping of atoms and ions [16]. This force can be estimated as the expectation value − ∂Ĥ/∂x , whereĤ describes the interaction of the particle with the radiation field and x is the distance along the surface. Our estimations show that the optical force is too weak to directly account for the magnitude of the EMF reported here. The appearance of this effect at the minima of the conductance oscillations indicates the inherent relationship between EMF and σ xx . In addition, an important question is the relation of the effect reported here to photovoltaic effects, which accompany the microwave-induced resistance oscillations in GaAs/AlGaAs heterostructures [5][6][7][8]. In Ref. 7, the oscillations of photocurrent and photovoltage were interpreted in terms of built-in electric fields due to band bending at contacts. Clearly, such fields do not exist in our system, and here we omit such a mechanism from further consideration. The microwave field gradient in the near-contact region can generate a ponderomotive force acting on 2DES in heterostructures, which may be a possible explanation for the ZRS in this system [17]. Such an effect is unlikely to occur in our system because the SSE are not in contact with the conducting electrodes. Recently, it was proposed that microwave-induced oscillations of σ xx in SSE on helium originate from the intersubband scattering of electrons under the condition of nonequilibrium filling of the excited subband [18]. This mechanism can result in an absolute negative σ xx , causing current instability and the formation of an inhomogeneous pattern of current domains [19]. Such an effect is believed to be responsible for the EMF responses observed in Refs. 5 and 8, and may be a possible explanation for the redistribution of charge reported here. The microwave stabilization of edge trajectories, which was proposed recently to explain ZRS in heterostructures, naturally gives the correct direction of the charge displacement [15]. The creation of ballistic channels at the sample edges due to microwave irradiation combined with a pumping mechanism through the inter-subband excitation might lead to an accumulation of charge at the edges. All the possible outlined scenarios require thorough theoretical investigation.
We thank H. Bouchiat and D.L. Shepelyansky for helpful discussions. This work was supported in part by Grant-in-Aids for Scientific Research from MEXT. D.K. acknowledges an Incentive Research Grant from RIKEN.