Journal of Low Temperature Physics - QFS2009 manuscript No. (will be inserted by the editor) Transport of Electrons on Liquid Helium across a Tunable Potential Barrier in a Point Contact-like Geometry David Rees Kimitoshi Kono Received: date / Accepted: date Abstract We present transport measurements of electrons bound to the surface of superfluid 4 He in a microchannel of width 10 µm. A set of electrodes 2 µm beneath the Helium surface, fabricated in a split-gate configuration using electron beam lithography, are used to control the current along the microchannel as in a point contact device. As the split-gate bias V SG is swept negative the current decreases to zero. The value of V SG at which the current is suppressed is dependent on the AC driving voltage applied to the electron system. We explain our results using a simple model in which a potential barrier created by the split-gate electrodes must be overcome in order to allow current to flow in the microchannel. The control of electron transport in such confined geometries may offer new possibilities for mesoscopic experiments with electrons on the surface of liquid Helium. Keywords surface-state electron potential barrier quantum wire point contact PACS 73.20.-r 73.23.-b 1 Introduction Surface-state electrons on liquid Helium form an ideal two-dimensional electron system[1]. As the Coulomb interaction between electrons is essentially unscreened, the system has been used to study many phenomena associated with strongly-interacting electron systems such as the transition to the Wigner solid at low temperatures[2]. Recently, it has been proposed that under appropriate conditions electrons on Helium may also be used to study quasi-one-dimensional phenomena such as melting in one dimension and reentrant melting due to structural phase transitions[3]. Recent advances in microfabrication technology have allowed the study of surfacestate electrons in confined geometries using devices such as microchannel arrays[4], David Rees Kimitoshi Kono RIKEN (The Institute of Physical and Chemical Research), Hirosawa 2-1, Wako, Saitama, 351-0198, Japan Tel.: +81-48-467-9327 Fax: +81-48-462-4652 E-mail: drees@riken.jp
2 a) b) Guard Guard Left Reservoir Right Reservoir LR Si RR 0.5 mm 10 µm c) R d) C ~ C Fig. 1 a) Photograph of the device showing the Guard electrode and the microchannel arrays which comprise the left and right reservoirs. b) Scanning electron microscope image of the small central microchannel. The exposed Silicon forms a split-gate electrode. c) Lumped-circuit model in which C is the capacitance of the electron system to the Source or Drain electrodes which are beneath the reservoirs and R is the electron resistance. d) Finite element modelling of the potential profile across the surface of the central channel for V G = 0 mv, V SG = 175 mv, V SD = 350 mv. Here the contour lines are of equipotential; darker regions signify regions of higher potential. single-electron traps[5], field-effect transistors[6] and charge-coupled devices[7]. However, a quantum wire in which the effective width of a conductive channel is less than the thermal wavelength of the electrons has not yet been demonstrated. As a first step towards this goal we have measured the transport properties of electrons in a small microchannel in which the confinement potential along the central axis of the channel may be controlled on the scale of the inter-electron separation (typically 0.5µm). In this article we report preliminary results and discuss factors governing the electron transport properties of the system.
3 2 Experimental The sample used in this experiment was fabricated in Gold using multi-layer lithography on a Boron-doped (p-type) Silicon wafer, and is shown in Figure 1a. The surface of the Silicon wafer was covered with a 500 nm insulating oxide layer. As in other similar devices[8], Source and Drain electrodes were fabricated beneath two sets of microchannels, 20 µm wide and 2 µm deep, defined by a Guard electrode, which was isolated from the Source and Drain by a layer of hard-baked photoresist. The two microchannel arrays, denoted Left Reservoir (LR) and Right Reservoir (RR), were connected by a smaller microchannel of width 10 µm and length 20 µm as shown in Figure 1b. At the base of this channel two thin electrodes of width 2 µm, separated by a gap of 200 nm, were fabricated using electron beam lithography. On either side of these thin electrodes the Silicon wafer surface was left exposed. The sample was mounted in an experimental cell which was filled with liquid Helium at 1.25 K until the bulk liquid surface was 0.5 mm below the sample allowing the microchannels to fill by capillary condensation. By applying a DC bias of V G = 0 V to the Guard and a bias of V SD = +0.35 V to the Source and Drain electrodes, the surface of the Helium in the microchannels was charged by thermionic emission from a small Tungsten filament directly above the sample. The saturated electron density is given by n max = ɛɛ 0 V SD /ed = 1.45 10 9 cm 2 where ɛ, ɛ 0, e and d are the vacuum permittivity, relative permittivity of Helium, the elementary charge and Helium depth respectively. An AC voltage V in of frequency 100 khz was also applied to the Source in order to drive the electron system; the induced current registered at the Drain electrode was measured using a lock-in amplifier. The behaviour of the system was found to be well-described by the lumped-circuit model pictured in Figure 1c. The exposed Silicon wafer formed a split-gate electrode which could be used to control the potential profile for electrons in the thin central channel by applying a voltage V SG to the substrate, as in a point contact device[9]. Figure 1d shows the results of a finite element model analysis which shows that at the center of the channel, at appropriate values for V SG, the potential is approximately uniform in the direction along the channel whilst there is strong confinement in the lateral direction. In the reservoirs the potential is lower than in the channel. To understand the dynamics of the electron system in this region we investigated the dependence of the current through the device on both V SG and V in. 3 Results and Discussion Figure 2 shows the peak AC current I flowing through the device as V SG is varied at several values of V in. As V SG is swept negative I decreases smoothly to zero at a threshold voltage V T ; at negative values of V SG the current may be completely suppressed while at less negative bias electrons may move through the channel. Furthermore, by increasing V in not only the initial value of I but also the magnitude of V T are increased. Initially we assume that the decrease in I corresponds to an increase in R, the resistance of the electron system, which includes components from the reservoirs as well as the central channel, R R and R C. By varying V SG we change only the potential profile of the central channel; any change in resistance must be associated with R C as R R should remain constant. However, the results shown in Figure 2 and those of the
4 6 100 mv PP 5 80 mv PP 4 60 mv PP I (na) 3 2 40 mv PP 1 0 20 mv PP V T -1200-1150 -1100-1050 -1000-950 V SG (mv) Fig. 2 The current I through the central microchannel versus split-gate voltage V SG. The current is completely suppressed at a threshold voltage V T which depends on the applied AC voltage V in. finite element modelling indicate that setting V SG negative also creates a potential barrier between the two reservoirs; we see that at certain values of V SG at which the resistance is infinite we may induce current flow simply by increasing V in until electrons can pass over the barrier. As the energy of the electron system will surpass that of the barrier for only a fraction of each AC cycle, the measured current is reduced and becomes smaller, and eventually zero, as the barrier height is raised. In future work this effect will be taken into account in order to measure R R and R C explicitly. Figure 3 shows the dependence of the current on the driving amplitude at different gate voltages. Note that here we take into account the voltage dropped across each capacitance C in the lumped-circuit model and calculate the voltage across the resistance R; we denote this voltage V A. The resulting I -V A plot shows that at more positive values of V SG no potential barrier exists and the current increases linearly with increasing voltage. Here the resistance R 2 MΩ which is a typical value for such devices[10]. However, at more negative values of V SG a gap appears in the I -V A plot indicating that current may only flow once the AC voltage applied to the system is large enough for electrons to overcome the resulting potential barrier. The data in Figure 3 allows us to infer the dependence of the barrier height V B on the gate voltage. We assume that the barrier height varies linearly with gate voltage as V B = βv SG + K where β is a scaling factor and K is a constant offset. We take the value of the peak voltage at which current begins to flow as a measure of the barrier height. We find that this threshold voltage varies approximately linearly with V SG in the gap region with β = 0.23 which is in reasonable agreement with the value of 0.28 calculated using the finite element model. The value of K was found to vary from day to day, as may be seen on comparison with Figure 2, probably due to charge accumulation on the Silicon surface as a result of emission from the filament. No significant variation in K was observed between each charging procedure.
5 2.5 2.0-350 mv -360 mv -370 mv I (na) 1.5 1.0-380 mv -390 mv 0.5-400 mv 0.0 0 5 10 15 20 V A (mv PP ) Fig. 3 I versus the AC voltage V A applied to the electron system. Labels indicate the respective values of V SG. Thus the barrier height depends rather weakly on V SG, as expected for the region above the thin central electrodes where the screening of other potentials would occur, indicating that the electron system is indeed confined to the area above the thin central electrodes. At the saturated electron density this corresponds to a maximum of 500 electrons being confined in the channel as the current flows. Rousseau et al have recently reported the observation of 1-20 surface-state electrons in a microfabricated trap[11]. By controlling the electron density in the channel by changing V SG, it may be possible to study such small numbers of electrons with the current device. Klier et al have measured the DC current of electrons in a device similar to ours in which a 200 µm wide, 1 mm long channel was defined by a pair of split-gate electrodes[6]. In that experiment the electrons were confined on the surface of a thin Helium film of thickness 100 nm which led to the pinning of electrons due to the roughness of the substrate. Consequently, the current measured was much smaller than in this experiment. However, they also observed that the current decreased as the split-gate voltage was swept negative which they attributed to a change of electron density, and thus mobility, as the effective width of the channel was decreased. We anticipate that by developing a full understanding of the AC electron transport observed in our system, the mechanisms by which the current is suppressed in both devices may be compared. 4 Conclusions We measured the transport of electrons on the surface of liquid Helium through a small microchannel in which the confinement lateral to the direction of electron flow was on a scale comparable to the inter-electron spacing. The current was found to be dependent on a tunable potential barrier formed in the channel by a split-gate electrode beneath
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