Addition spectra of Wigner islands of electrons on superfluid helium

We present here an experimental study of Wigner islands formed by electrons floating over helium. Electrons are trapped electrostatically in a mesoscopic structure covered with a helium film, behaving as a quantum dot in the near-classical limit. By removing electrons one by one, we are able to find the addition spectrum, i.e., the energy required to add (or extract) one electron from the trap with occupation number $N$. Experimental addition spectra are compared with Monte Carlo simulations for the actual trap geometry, confirming the ordered state of electrons over helium in the island.

Figure 1

(Color online) RM and OM “melting” in Wigner islands sketched in the $\overline{n}$ vs $k_{B}T/E_C$ plane for small electron occupation numbers $N$—(◇) 10—(●) 11—(◻) 12—(▲) 19—(◼) 20, following Filinov et al. (Ref. 18). The OM boundaries for $N=10$, 11, and 20 are very close to the origin and are not shown. The horizontal (vertical) dash-dash lines indicate the quantum (classical) Wigner crystallization in the infinite system, for $r_s=1/n^2=37$ and ${\Gamma }_{\infty }=E_C/k_{B}T=137$, respectively.

Figure 2

(Color online) Scanning electron microscopy (SEM) picture of the complete microfabricated device (left) and magnified view of the electron trap (right).

Figure 3

(Color online) Profile of the electrostatic potential in the plane of symmetry of the trap for $V_{\text{SET}}=0.5$, 0.2, and 0 V from bottom to top. Distances along the $x$ axis are in microns from the edge of the reservoir. The difference between the top of the barrier and the minimum define the confining potential $\delta$. The top of the barrier for $V_{\text{SET}}=0.2$ and 0 V is in the gorge.

Figure 4

(Color online) Reduced charge on the SET island in terms of the potential of the reservoir electrode for several sweeps of $V_{\text{res}}$. The SET potential is $+0.6\text{V}$ while the guard electrode potential is 0. The inset shows the steps of the staircase for low occupation numbers down to zero, corrected for the baseline drift occurring between the different sweeps of $V_{\text{res}}$.

Figure 5

(Color online) Variation in the SET output signal $U$ when the reservoir voltage, $V_{\text{res}}$, is swept. The Coulomb blockade oscillation is interrupted by a change of the charge in the SET island at $\sim 0.5\text{V}$ as marked by the arrow.

Figure 6

(Color online) SET-phase variation for the values of the potential on the SET, $V_{\text{SET}}$, of 0.3, 0.5, and 0.8 V from left to right. The dots mark the escape of the last electron from the trap.

Figure 7

(Color online) Potential of the reservoir, $V_{\text{res}}$, at which the last electron leaves the trap. The intercept with the $x$ axis yields the effective contact potential between aluminum and niobium, found to be $0.206\pm 0.005\text{V}$.

Figure 8

(Color online) Top: addition spectrum for $V_{\text{SET}}=0.5\text{V}$ ($V_{\text{SET}}=0.706\text{V}$ in the simulations). Squares, (◻), lozenges, (◇), triangles, (△), and circles, (○), are experimental data for different runs and fall nearly on top of one another. Bottom: addition spectrum for $V_{\text{SET}}=0.3\text{V}$ ($V_{\text{SET}}=0.506\text{V}$ in the simulations); squares are experimental results. Dots, (●), are the results of the MC simulations, linked by straight-line segments to guide the eye. Error bars on the simulated data represent an estimate of the uncertainty.

Figure 9

(Color online) Increase in the free energy of the cluster with electron number in a parabolic trap, in volt: (▲), our results based on a discrete parabolic profile; solid line, Kong et al.’s results (Ref. 46). The relative difference in percent between the two calculations is plotted at the top, (●), referred to the vertical scale on the right.

Figure 10

(Color online) Computed addition spectra in this work and that of Bedanov and Peeters (Ref. 14), confirmed by Kong et al. (Ref. 46), (▲), at the bottom. The relative difference in percent between these calculations is plotted at the top, (●), referred to the vertical scale on the right.

Figure 11

(Color online) Wigner molecule in a parabolic trap with 31 electrons. Lines represent potential contours; the minimum of the parabola is centered on (0,0). Stars represent the electrons. The ground-state configuration is the configuration (5,11,15) as in Kong et al. (Ref. 46).

Figure 12

(Color online) Solid line: confining potential $\delta$ in terms of the potential applied on the reservoir electrode $V_{\text{res}}$. Dots are Monte Carlo simulations of the average energy per electron for one, three, and five electrons in the trap. Dash-dash lines are linear extrapolations of the Monte Carlo results as explained in the text. The crossing point between $\delta$ and this average energy gives the potential for which an electron leaves the trap.