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PHYSICAL REVIEW LETTERS

PRL 111, 266802 (2013)

Effect of Coulomb Interaction on Microwave-Induced Magnetoconductivity Oscillations of Surface Electrons on Liquid Helium 1

2

Denis Konstantinov,1,* Yuriy Monarkha,2,† and Kimitoshi Kono3,4

Okinawa Institute of Science and Technology, Tancha 1919-1, Okinawa 904-0412, Japan Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, Kharkov 61103, Ukraine 3 Low Temperature Physics Laboratory, RIKEN, Hirosawa 2-1, Wako 351-0198, Japan 4 RIKEN CEMS, Hirosawa 2-1, Wako 351-0198, Japan (Received 21 July 2013; published 26 December 2013)

The experimental observation of the strong Coulombic effect on magneto-oscillations of the photoconductivity of surface electrons in liquid helium is reported. The observed broadening of the oscillations and shifts in positions of conductivity extrema with increasing electron density are in good agreement with the linear transport theory, which takes into account an internal electric field of fluctuational origin. These results provide important evidence for identification of the mechanism of the oscillations and zero-resistance states developed in their minima. DOI: 10.1103/PhysRevLett.111.266802

PACS numbers: 73.20.-r, 73.21.-b, 78.20.Ls, 78.56.-a

Recent studies of electron transport in two-dimensional (2D) electron systems subjected to a perpendicular magnetic field and to microwave (MW) radiation reveal spectacular phenomena. For electrons in GaAs=AlGaAs heterostructures, the resistivity exhibits magneto-oscillations controlled by the ratio of the MW frequency ω to the cyclotron frequency ωc, and at high radiation power the minima of the oscillations evolve into zero-resistance states (ZRS) [1,2]. Positions of the resistance minima reported [1] obey a universal law: ω=ωc ¼ integer þ 1=4. Oscillations of the magnetoconductivity σ xx and ZRS were also observed in the nondegenerate 2D electron system formed on the free surface of liquid helium [3,4] when the system was tuned to the resonance for MW intersubband absorption. The energy spectrum of surface electron (SE) subbands in liquid helium resembles the energy spectrum of a hydrogen atom (Δl ∝ l−2 , where l ¼ 1; 2; …) [5,6]. The corresponding Rydberg energy is rather small: ℏω2;1 ≡ Δ2 − Δ1 is about 3.2 K for liquid 3 He. SEs form a highly correlated electron system, which undergoes a Wigner solid transition [7] when the ratio of the average interaction potential energy to the average kinetic energy pffiffiffiffiffiffiffiffi P ≡ e2 πne =T e ≃ 131, where ne and T e are the SE density and temperature, respectively. Even for the lowest ne used in experiments [3,4], the plasma parameter P ≫ 1. It is well established that the Coulomb interaction strongly affects σ xx of SEs at high electron densities [8,9]. Therefore, the Coulomb interaction should be very important for magneto-oscillations of σ xx as well. A variety of different theoretical mechanisms [10–13] was proposed to explain magneto-oscillations and ZRS observed in semiconductor electron systems. Since these theories are not critical to the condition ω ¼ ω2;1 used in experiments on liquid helium [3,4], they can hardly be applied to the SE system. 0031-9007=13=111(26)=266802(5)

A theory [14] of magneto-oscillations in the electron system formed on liquid helium is based on a nonequilibrium population of the second surface subband induced by the MW resonance: ρ¯ 2 > ρ¯ 1 expð−ℏω2;1 =T e Þ, where ρ¯ l are fractional occupancies of surface subbands. The important point is that energy conservation for quasielastic intersubband scattering from l ¼ 2 to l0 ¼ 1 is essentially the same as energy conservation for photon-assisted intrasubband scattering [15]: ℏω2;1 stands instead of ℏω. Therefore, similar to Ref. [15], displacement of the electron orbit center X0 − X in the direction of the driving electric field E∥ changes its sign near the level-alignment points Bm defined by the condition ω2;1 =ωc ¼ n0 − n ≡ m > 0 [here n is the Landau level (LL) number]. In spite of obvious differences in description of intersubband scattering and photon-assisted scattering the basic reason for negative conductivity effects is the same in both cases: scattering (displacement of an orbit center) against the driving field caused by energy conservation. At equilibrium, reverse scattering from l ¼ 1 to l0 ¼ 2 leads to the opposite displacement, and the sign-changing terms of the momentum relaxation rate compensate each other. Therefore, the sign-changing intersubband momentum relaxation rate does not appear under conditions of experiments on usual magnetointersubband oscillations [16], or on interference oscillations [17] with an arbitrary MW frequency (ω ≠ ω2;1 ). Thus, an excess electron population of the second subband caused by the MW resonance is necessary to trigger the intersubband displacement mechanism of magneto-oscillations and negative conductivity. As shown earlier [18], a system with σ xx < 0 is unstable, which leads to ZRS. The single-electron theory [14] was extended to include effects of strong Coulomb interaction between SEs [19], and important predictions were made about the broadening

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© 2013 American Physical Society

PRL 111, 266802 (2013)

PHYSICAL REVIEW LETTERS

of oscillations and locations of σ xx extrema as functions of the electron density ne and the level-alignment number m ¼ roundðω21 =ωc Þ. Experimental observation of these Coulombic effects would provide crucial evidence for determining the origin of oscillations and ZRS observed for SEs on liquid helium. In this Letter, we report our investigation of Coulombic effects on MW-induced oscillations of σ xx in the 2D electron system formed on liquid 3 He. We also present a comparison of our observations with the many-electron theory [19], which indicates a substantial heating of the electron system near Bm. The free surface of liquid helium is contained in a flat cylindrical cell attached to the mixing chamber of a dilution refrigerator. Electrons are emitted from a heated tungsten filament, which is located above the surface, and attracted toward the liquid phase by application of a positive potential to a circular metal plate (back gate) located in the liquid, parallel to the surface. Due to the large repulsive barrier at the interface, a circular pool of electrons is trapped on the free surface. The conductivity σ xx of the electron layer is measured by a capacitive coupling method, using a pair of circular Corbino electrodes placed above and parallel to the liquid surface [20]. Numerical values of the conductivity are extracted from experimental data using an analysis described elsewhere [21]. Excitation of the second subband is produced by millimeter-wavelength MWs (ω=2π ≈ 79 GHz) transmitted through a waveguide coupled to the experimental cell. The energy difference between the ground subband and the first excited subband is tuned to match the energy of MW photons (ω21 ≅ ω) by applying the positive bias at the back gate, and shifting subband energy levels due to the Stark effect. After adjusting the voltage, the equilibrium distribution of electrons becomes different from that obtained after deposition of electrons on the surface. Therefore, the electron density ne is determined from numerical calculations of the distribution of the electrostatic potential caused by the biased electrodes and the electron layer [21]. It is instructive to compare the behavior of SE conductivity with and without radiation. The σ xx measured is shown in Fig. 1. The theoretical lines (solid) represent three major approximations for equilibrium magnetotransport: the classical Drude equation, the self-consistent Born approximation (SCBA) [22], and the many-electron (me) theory [19]. The many-electron theory is based on the idea [23] that internal electric fields of fluctuational origin acting on each electron can be considered as quasiuniform fields because the magnetic length LB ¼ ðℏc=eBÞ1=2 is much smaller than the average electron spacing. The typical value of the fluctuational field depends strongly pffiffiffiffiffi on electron temð0Þ perature and density [24]: Ef ≃ 3 T e n3=4 e . Self-energy effects (collision broadening of LLs) can be combined with the Coulombic effect by considering the dynamic structure

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FIG. 1 (color online). σ xx versus B measured for SEs at T ¼ 0.2 K and ne ¼ 1.3 × 107 cm−2 without MW radiation (dash-dot-dotted line, blue) and with it ω=2π ¼ 79 GHz) for two values of input power: −5 dBm (dash-dotted line, olive) and 0 dBm (dashed line, red). Solid lines represent theoretical approximations, as described in the text.

factor (DSF) of an ensemble of electrons moving fast in the fluctuational field Ef [6,25]. The Coulombic effect reduces σ xx in the low magnetic field range. As a result, the manyelectron curve transforms steadily from the SCBA result (high fields) to the Drude result (low fields). An onset magnetic field B0 , above which the Landau quantization affects magnetotransport of SEs and conductivity deviates from ð0Þ the Drude result, is proportional to Ef [8]. The onset field obtained for SEs without radiation (dashdotted line in Fig. 1) is about 0.14 T, which roughly agrees with the calculations (solid lines). For irradiated electrons, with the same areal density the Drude regime extends to significantly higher fields for large radiation intensities (cf. the red dashed line). This indicates heating of SEs by radiation, which increases the fluctuational electric field. The electron temperature under irradiation can be roughly estimated from the root square dependence of B0 on T e . For irradiated electrons, B0 determines the onset of oscillations. For the red dashed line, we have B0 ≈ 0.34 T, which gives an estimation T e ≈1 K. Note that the calculated line for the Drude regime with T e ¼ T falls slightly lower than the one expected from extrapolation of experimental data (dash-dot-dotted line). This suggests a measured zero-field scattering rate for cold electrons ν0 ¼ 2.6 × 107 s−1 , which is a bit higher than that obtained from calculations. The evolution of the shape of σ xx oscillations with an increase in electron density is shown in Fig. 2. The oscillations are the most pronounced for the lowest electron density. A noticeable influence of the Coulomb interaction on σ xx oscillations, which magnifies with m , is a strong suppression of the amplitude and an increase in the broadening of conductivity extrema. Internal forces cause also nontrivial changes in the location of conductivity extrema. These

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PRL 111, 266802 (2013)

PHYSICAL REVIEW LETTERS

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is determined by the collision broadening of both subbands Γ2l;l0 ¼ ðΓ2l þ Γ2l0 Þ=2, and by the Coulomb broadening xq Γ2C , which depends on the dimensionless parameter xq ¼ q2 L2 =2, and on the fluctuational field: ΓC ¼ pffiffiffi ð0Þ B 2Ef LB . Thus, the strength of the Coulomb broadening of the DSF increases with m , ne , and T e . Mutual interaction results also in an additional frequency shift of the maxima ω~ ¼ Ω − m ωc −

FIG. 2 (color online). σ xx versus ω=ωc ðBÞ obtained at T ¼ 0.2 K, input MW power W ¼ 0 dBm, and for two electron densities: ne ¼ 2 × 106 (blue), and 1.3 × 107 cm−2 (red).

Coulombic effects can be well understood from the following simple analysis. The theoretical description of magnetoconductivity oscillations [14,19] is based on an evaluation of the average probability of intersubband scattering from l to l0 accompanied by the momentum exchange ℏq due to ripplon destruction and creation ν¯ l;l0 ðqÞ, which depends on the average electron velocity V. Using the SCBA, under the conditions V y ≃−V H , and jV y j≫jV x j (here V H ¼ cE∥ =B is the absolute value of the Hall velocity), it was found that ν¯ l;l0 ðq; V H Þ ¼ 2u2l;l0 ðqÞSl;l0 ðq; ωl;l0 þ qy V H Þ;

(1)

where ul;l0 ðqÞ is a function of jqj defined by electronripplon coupling, and Sl;l0 ðq; ΩÞ is a generalization of the DSF of a nondegenerate 2D electron system taking into account that LL densities have different collision broadening Γl for different subbands (here and below the dependence of Γl on the LL number is not indicated). The qy V H represents the change of electron energy in the driving field. For interaction terms proportional to e−iq·r , the eE∥ ðX − X0 Þ ¼ eE∥ qy L2B ¼ ℏqy V H . Important relaxation rates of the multisubband electron system are found directly fromPEq. (1). For the decay rate of the excited subband νl→l0 ¼ q ν¯ l;l0 ðqÞ, the qy V H can be neglected. Since quasielastic intersubband scattering is possible only in the vicinity of the level alignment, Sl;l0 ðq; ΩÞ must have sharp maxima when Ω → m ωc . In the manyelectron theory [19] the shape of a DSF maximum reprepffiffiffi ~ ¼ expð−ω ~ 2 =γ 2 Þ= π γ, where the sents a Gaussian GðωÞ parameter γ ¼ ℏ−1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Γ2l;l0 þ xq Γ2C

(2)

Γ2l Γ2 − xq C ; 4T e ℏ 4T e ℏ

(3)

which provides the detail balance condition: νl0 →l ¼ νl→l0 expð−ℏωl;l0 =T e Þ. The momentum relaxation rate caused by intersubband scattering νinter is found by evaluation of the momentum gained by scatterers with probabilities ν¯ 2;1 ðq; V H Þ. For the two-subband model, the total momentum balance yields νinter ¼

ℏ X q ½¯ρ − ρ¯ 1 e−ℏω2;1 =T e e−ℏqy V H =T e ¯ν2;1 : (4) me V H q y 2

Here the term describing scattering from l ¼ 1 to l0 ¼ 2 was transformed using the relationship Sl0 ;l ðq;−ΩÞ¼ e−ℏΩ=T e Sl;l0 ðq;ΩÞ. Without irradiation (¯ρ2 − ρ¯ 1 e−ℏω2;1 =T e ¼ 0) it is sufficient to use ν¯ 2;1 ðq; 0Þexpanding Eq. (4) in ℏqy V H =T e . In this case, νinter has symmetric maxima in the vicinity of Bm . Under MW pumping, we have to expand ν¯ 2;1 in qy V H up to linear terms, which leads to a sign-changing contribution proportional to S02;1 ðq;ω2;1 Þ (here " 0 " means the derivative with respect to Ω). The shape ~ ¼ of these terms is asymmetric and defined by G0 ðωÞ ~ ωÞ. ~ −2γ −2 ωGð The results of the many-electron theory are plotted in Fig. 3 for T e ¼ T. Generally, the shape of σ xx oscillations and its evolution with m , illustrated by the solid blue line, agree well with data obtained for the lowest electron density (blue line of Fig. 2). Numerical differences such as a stronger broadening of conductivity extrema in the experiment can be attributed to electron heating induced by decay processes [26]. Electron heating increases the broadening parameter γ and lowers the intrasubband momentum relaxation rate νintra ∝ 1=T e . A substantial increase in electron density (red curve of Fig. 2) leads to a much stronger Coulombic effect. First, the onset of oscillatory behavior is shifted to considerably higher fields. This is also in good agreement with calculations shown in Fig. 3. Second, σ xx minima become significantly broader with an increase of ne because γ ð0Þ is determined by Ef , which increases with ne , T e , and m . A remarkable feature of the experimental data is a strong suppression of conductivity minima and ZRS with increasing density. Such a suppression is caused by the Coulomb increase in the broadening parameter γ, which determines the amplitude of νinter oscillations.

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PHYSICAL REVIEW LETTERS

FIG. 3 (color online). σ xx versus ω=ωc ðBÞ calculated for ω ¼ ω2;1 , T e ¼ T ¼ 0.2 K, and for three electron densities ne ¼ 2.4 × 106 (solid line, blue), 5.3 × 106 (dashed line, olive), and 1.3 × 107 cm−2 (dash-dotted line, red) using the many-electron theory [19].

Another important prediction of the many-electron theory concerns positions of conductivity extrema. If we just neglect the two last terms of Eq. (3), then G0 ðω2;1 − m ωc Þ considered as a function of ω2;1 =ωc would have pffiffiffi two extrema whose shifts from an integer δþ=− ¼ γ= 2 increase their absolute values with ΓC . Assuming heating of SEs by microwaves and the corresponding increase in ΓC , this explains the power dependence of δþ=− observed previously at low MW powers [4]. For large enough γ, the overlapping of the sign-changing terms in the sum over the all m limits shifts of the extrema by a universal law δþ=− → 1=4, which coincides remarkably with that reported in Ref. [1]. The Coulomb term of Eq. (3) is independent of T e and power, and it changes locations of conductivity extrema drastically. In Fig. 3, shifts of conductivity minima δþ increase monotonically with ne and m , and eventually become substantially larger than 1=4. The behavior of shifts of conductivity maxima δ− , as a function of ne and m , is more peculiar. Since the frequency shift of Eq. (3) increases with ΓC and xq faster than the broadening parameter γ, eventually it dominates and drives maxima back to the level-alignment points. Indeed, for oscillations near m ≥9 (Fig. 3), the initial increase in jδ− j (green line) changes to a decrease (red line). Thus, δ− , as a function of density, has a minimum whose location depends on m . At a fixed electron density, δ− has a minimum as a function of m . For high densities ne ¼ 1.3 × 107 and 5.3 × 106 cm−2 , this minimum occurs at m ¼ 7 and 10, respectively (Fig. 3). It moves to larger m with lowering ne , as shown in Fig. 4 by open square symbols calculated for ne ¼ 2 × 106 cm−2. It should be noted that for semiconductor systems in the high-intensity regime, the phase of oscillations depends

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FIG. 4 (color online). Shifts of conductivity extrema δþ > 0 (circles) and δ− < 0 (squares) versus the level-alignment number m for ne ¼ 2 × 106 cm−2 : experimental data (filled symbols), and the many-electron theory (open symbols with line).

strongly on MW power due multiphoton processes [27]. For experiments with SEs on liquid helium, the resonance condition ω ¼ ω2;1 cuts off multiphoton processes. Moreover, photons are not involved directly in the intersubband displacement mechanism of oscillations as discussed in the introduction. Experimental data agree well with the above given analysis (Fig. 2). It is instructive to consider positions of conductivity extrema observed in the experiment versus m , as shown in Fig. 4. The uncertainty in the positions of oscillation extrema is mostly determined by the uncertainty in values of B. The latter was determined by in situ cyclotron resonance measurements as described earlier [4]. Note that the uncertainty in ω=ωc ∝ B−1 increases with decreasing B. For m < 6, positions of minima cannot be determined accurately because of formation of ZRS. The shift of minima δþ ¼ ω=ωc ðBþ Þ − m (red circles) increases monotonically with m , and at high m the δþ →1=2. In contrast, the shift of maxima δ− (olive squares) has a nonmonotonic dependence. After an initial increase, jδ− j reaches the maximum value of about 0.2

Effect of Coulomb interaction on microwave-induced magnetoconductivity oscillations of surface electrons on liquid helium.

The experimental observation of the strong Coulombic effect on magneto-oscillations of the photoconductivity of surface electrons in liquid helium is ...
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