Phase Diagram of the Electron System Without Applying a Periodic Potential
First, we checked the performance of the fabricated device by applying the same potential to both electrodes Ch1 and Ch2 of the central microchannel (\(V_\text {ch}=V_\text {ch1}=V_\text {ch2}\)) and measuring the current in the device I while applying a peak-to-peak AC voltage \(V_\text {ac}=5\) mV to the device (Fig. 2) for various values of \(V_\text {ch}\) and bias \(V_\text {sg}\) applied to the split-gate electrode of the central microchannel. To understand this diagram, it is convenient to use a simplified capacitance model to find the relationship between the density of electrons in the central microchannel and voltages applied to different electrodes of the device [23]. First, we define the total capacitance of the liquid surface in the central microchannel \(C_\varSigma =C_\text {ch}+C_\text {sg}\), where \(C_\text {ch}\) and \(C_\text {sg}\) are capacitances between the liquid surface and the channel’s bottom and split-gate electrodes, respectively. It is also convenient to introduce the dimensionless coupling constants \(\alpha =C_\text {ch}/C_\varSigma \) and \(\beta =C_\text {sg}/C_\varSigma \), which satisfy the obvious relation \(\alpha +\beta =1\). Then, the potential at the uncharged liquid surface can be written as \(V_\text {b}=\alpha V_\text {ch}+\beta V_\text {sg}\). When the device is charged with electrons, the potential of the charged liquid surface \(V_\text {e}\) must be the same everywhere, owing to high mobility of the surface electrons on liquid helium. The value of \(V_\text {e}\) is determined by voltages applied to the reservoir’s bottom and guard electrodes and the number of electrons in the reservoir, and is assumed to be fixed once the device is charged. Note that occasionally, loss of electrons from the device is observed, which is reflected in discontinuous jumps of the measured current I. Such data are not considered here. Then, by the definition of capacitance, we can define the total charge Q of electrons in the channel \(Q=C_\varSigma (V_\text {e}-V_\text {b})\). A further simplification can be made by assuming a uniform density distribution of electrons in the channel, that is, \(Q=-en_\mathrm{s}S\), where \(n_\mathrm{s}\) is the areal density of surface electrons, \(e>0\) is the electron charge and S is the channel area. Such a parallel-plate capacitance approximation is partially justified by a large aspect ratio (\(\sim 10\)) of the wide, shallow microchannel used in our device. Using \(C_\text {ch}=\epsilon \epsilon _0 S/d\), where \(d=550\) nm is the height of the microchannel in our device, \(\epsilon =1.056\) is the dielectric constant of liquid helium and \(\epsilon _0=8.85\times 10^{12}\) F/m is the permittivity of free space, we obtain the relation
$$\begin{aligned} n_\mathrm{s}=\frac{\epsilon \epsilon _0}{\alpha e d} \left( \alpha V_\text {ch} + \beta V_\text {sg} - V_\text {e} \right) . \end{aligned}$$
(1)
The above equation is useful to characterize the device and to estimate various quantities. For example, the maximum density of electrons corresponds to the condition \(V_\text {e}=V_\text {sg}\), in which electrons cease to be confined across the microchannel by the split-gate potential, from which we find \(n_s^{(\text {max})}=\epsilon \epsilon _0(V_\text {ch}-V_\text {sg})/(ed)\). In contrast, the zero density of electrons in the central microchannel corresponds to the condition \(\alpha V_\text {ch} + \beta V_\text {sg}=V_\text {e}\), which determines the threshold value of the channel voltage for given values of \(V_\text {sg}\) and \(V_\text {e}\)
$$\begin{aligned} V_\text {ch}^{(\text {th})}=\frac{1}{\alpha }V_\text {e}- \frac{1-\alpha }{\alpha } V_\text {sg}. \end{aligned}$$
(2)
Below this threshold value, the potential at the uncharged surface in the central microchannel \(V_\text {b}\) is lower than \(V_\text {e}\); therefore, the central microchannel is completely depleted of electrons and the current I in the device is zero. The experimental values of \(V_\text {ch}^{(\text {th})}\) are plotted in Fig. 2 with a dashed (white) line. By fitting this line using Eq. (2), we obtain \(V_\mathbf e =0.92\) V and \(\alpha =0.77\) (therefore \(\beta =0.23\)).
Above the threshold line in the \(V_\text {sg}-V_\text {ch}\) plane, see Fig. 2, the current in the device is determined by the resistance R of electrons in the microchannel, which in turn depends on the phase of the electron system. For weak confinement of the electron system, which corresponds to lower values of \(V_\text {ch}\) and more positive values of \(V_\text {sg}\), the system is in the liquid phase. This corresponds to low resistance R and large current I; see Fig. 2. For stronger confinement of the electron system, which corresponds to larger values of \(V_\text {ch}\) and more negative values of \(V_\text {sg}\), the system crystallizes into a WS [24, 31]. As a result, the resistance R of electrons in the central microchannel increases due to the formation of the commensurate dimple lattice, and the measured current I drops significantly. A spectacular behavior is observed in the intermediate voltage range, where the current I exhibits a fringed pattern; see Fig. 2. This phenomenon was identified as the re-entrant melting of the WS [25, 26]. As the confining potential, therefore the width of the electron system in the microchannel, is varied by the voltage applied to the electrodes, the WS in the microchannel undergoes intermittent melting as a result of increased fluctuation of electron positions between stable configurations corresponding to different numbers of electron rows across the channel. Therefore, the fringes, which are nearly parallel to the threshold line, see Fig. 2, can be identified by the different number of electron rows in the microchannel. It is worth noting that deep in the WS phase region the threshold line slightly deviates from the fitting line. Apparently, that is because a continuous electron distribution approximation, which is used to derive Eq. (2), may not work so well for the case of a few rows of electrons in the WS state, where the granular nature of electrons has to be taken into account.
Effect of Periodic Potential
Next, we investigate the effect of a spatially periodic potential applied to electrons in the microchannel on the current. To do this, we apply potentials \(V_\text {ch1}=V_\text {ch}+\varDelta V_\text {ch}/2\) and \(V_\text {ch2}=V_\text {ch}-\varDelta V_\text {ch}/2\) to electrodes Ch1 and Ch2, where \(V_\text {ch}=1.5\) V is a fixed common bias applied to two electrodes, and \(\varDelta V_\text {ch}\) can be varied from 0 to 2 V. The absolute value of the current I in the device is plotted in Fig. 3 for various values of \(\varDelta V_\text {ch}\) and \(V_\text {sg}\). For \(\varDelta V_\text {ch}\lesssim 0.7\) V, we observe fringes of current due to the re-entrant melting of the WS, as described earlier. For the sake of illustration, the solid (red) line plots the measured current I versus \(V_\text {sg}\) for \(\varDelta V_\text {ch}=0.25\) V. For \(\varDelta V_\text {ch}\gtrsim 0.7\) V, the behavior becomes drastically different; see Fig. 3. The re-entrant melting fringes disappear and the measured current I increases significantly to a value comparable to that for electrons in the liquid phase (c.f. Fig. 2). This behavior might suggest that the application of a sufficiently strong periodic potential suppresses crystallization of the electron system into the WS phase. Under continuous electron distribution approximation, the onset of charging of electrons in the central microchannel \(\varDelta V_{\text {ch}}^{(\text {th})}\) can be expressed similarly to Eq. (2). Considering the contribution to \(\varDelta V_{\text {ch}}\) by electrodes Ch1 and Ch2, the potential threshold line can be expressed as
$$\begin{aligned} \varDelta V_\text {ch}^{(\text {th})}=\frac{2}{\gamma }\left( -V_{e}+\alpha V_{\text {ch}}+\beta V_{\text {sg}}\right) , \end{aligned}$$
(3)
where \(\gamma \) is the coupling constant maximum from the Ch2 electrode. The dashed line plotted in Fig. 3 indicates the potential threshold line determined by the experimental values of \(\varDelta V_{\text {ch}}^{(\text {th})}\) in the region of \(\varDelta V_{\text {ch}} \ge 0.7\) V. By fitting this line with Eq. (3), we obtain \(\gamma =0.02\). For low \(\varDelta V_{\text {ch}}\), where signatures of the WS phase are still prominent, again there is a deviation of the threshold line from the fitting line, suggesting that granular nature of electrons cannot be ignored in this regime.
To understand the effect of the spatially periodic potential on the electron system, it is instructive to estimate the variation in the electron density \(n_s\) in the central microchannel using the parallel-plate capacitance approximation. As described earlier, the electron density can be estimated as \(n_s=\epsilon _0\epsilon \left( V_\text {b} - V_\text {e} \right) /(\alpha e d)\), where the potential \(V_\text {b}\) at the uncharged surface of liquid helium in the central microchannel can be calculated numerically using the FEM; see Fig. 1c. We find that at the middle of the channel the density varies nearly sinusoidally with average value \(\bar{n}_s\) and amplitude \(\varDelta n_s\). In particular, for \(V_\text {e}=0.92\) V, \(V_\text {sg}=-\,0.4\) V, \(V_\text {ch}=1.5\) V and \(\varDelta V_\text {ch}=0.7\) V using the above approximation we estimate \(\bar{n}_s=3.9\times 10^{13}\) m\(^{-2}\) and \(\varDelta n_s=0.4\times 10^{13}\) m\(^{-2}\). For an infinite 2D electron system, the melting of the WS is expected to occur when the value of the plasma parameter \(\varGamma =e^2\sqrt{\pi n_s}/(4\pi \epsilon _0\epsilon k_\text {B}T)\) exceeds \(130 \,\pm \, 10\) [13, 32]. For \(T=0.86\) K, the critical density of electrons corresponds to \(n_s=1.4\times 10^{13}\) m\(^{-2}\). Therefore, a small variation in electron density due to the applied periodic potential estimated above cannot cause melting of WS for an infinite electron system. On the other hand, as was pointed out earlier, the variation in lateral confinement of the electron system in the microchannel can cause a loss of the long-range crystalline order in the quasi-1D WS due to structural transitions between two stable configurations of the electron lattice corresponding to changing the number \(N_y\) of electron rows in the channel by one [25, 26]. This is exactly the mechanism that explains the phenomenon of re-entrant melting in this system. Therefore, one can expect that a variation in \(N_y\) along the microchannel caused by the applied periodic potential can induce a similar loss of long-range positional order, which in turn strongly changes the transport of the electron system observed in the experiment. For simplicity, we assume that the smallest reciprocal lattice vector of the WS points in the x-direction. The number of electron rows can be estimated as \(N_y=\left( 4/3 \right) ^{1/4} w\sqrt{n_s}\), where w is the width of the electron system in the microchannel. For \(V_\text {e}=0.92\) V, \(V_\text {sg}=-\,0.4\) V, \(V_\text {ch}=1.5\) V and \(\varDelta V_\text {ch}=0.7\) V, we estimate that w varies from 3.63 to 3.54 \(\upmu \)m, and \(N_y\) changes from 25 to 23. Therefore, \(\varDelta N_y\approx 2\). In other words, the variation in the confining potential due to an applied periodic potential with \(\varDelta V_\text {ch}=0.7\) V is sufficient to cause a structural transition between \(N_y\)- and \((N_y+1)\)-row configurations, which increases fluctuations in the positions of electrons and suppresses nonlinear transport features usually associated with an electron system in the WS phase.
To confirm suppression of nonlinear transport features associated with the crystalline ordering of the electron system, we measured the current I as a function of the driving amplitude \(V_\text {ac}\) in the presence of periodic potential for different values of \(\varDelta V_\text {ch}\). A typical set of such IV curves is shown in Fig. 4 for four different values of \(\varDelta V_\text {ch}=0\), 0.7, 1.0 and 1.26 V. Without the periodic potential (\(\varDelta V_\text {ch}=0\)), the IV curve clearly shows two characteristic features of nonlinear transport, namely a BC plateau of current due to coherent emission of ripplons by the driven WS and a sharp rise of current due to sliding of the WS from the commensurate dimple lattice. Application of the periodic potential suppresses both features of nonlinear transport of the WS. In particular, for sufficiently large \(\varDelta V_\text {ch}\gtrsim 0.7\) V, both features essentially disappear, and the electron transport approximates that of the electron system in liquid phase. This agrees with the suppression of re-entrant melting described earlier.