Fractional Coulomb blockade for quasi-particle tunneling between edge channels

Magnetic field periodic conductance of a quantum dot in fractional quantum Hall regime indicates fractional charge tunneling.

Abstract

In the fractional quantum Hall effect, the elementary excitations are quasi-particles with fractional charges as predicted by theory and demonstrated by noise and interference experiments. We observe Coulomb blockade of fractional charges in the measured magneto-conductance of a 1.4-micron-wide quantum dot. Interaction-driven edge reconstruction separates the dot into concentric compressible regions with fractionally charged excitations and incompressible regions acting as tunnel barriers for quasi-particles. Our data show the formation of incompressible regions of filling factors 2/3 and 1/3. Comparing data at fractional filling factors to filling factor 2, we extract the fractional quasi-particle charge e^*/e = 0.32 ± 0.03 and 0.35 ± 0.05. Our investigations extend and complement quantum Hall Fabry-Pérot interference experiments investigating the nature of anyonic fractional quasi-particles.

INTRODUCTION

Large quantum dots (QDs) can be used to study physical processes in the quantum Hall regime. Nevertheless, no effects of quasi-particle tunneling have been observed for Coulomb blockaded QDs in the fractional quantum Hall regime so far (1). The reason is that tunneling barriers connecting a QD in the Coulomb blockade to source and drain regions are strongly backscattering and therefore only allow for electron tunneling (2). Operating barriers in the regime of weak backscattering allows for quasi-particle tunneling. Evidence for fractionally charged quasi-particles has previously been observed in experiments on shot noise of a quantum point contact (3–7), capacitively probed localized states (8), anti-dots (9), photo-assisted shot noise (10, 11), or quantum Hall Fabry-Pérot interferometers (12). In general, quantum Hall Fabry-Pérot interferometer experiments (13–17) in the fractional quantum Hall regime offer the opportunity to study anyonic statistics of fractional quasi-particles (12, 18–27). Anyonic fractional statistics were recently detected in two seminal experiments by Bartolomei et al. (28) and by Nakamura et al. (29). Mach-Zehnder interferometers in the quantum Hall regime (30) were proposed as an alternative probe to study quasi-particle statistics (31, 32). The close relation between QDs and Fabry-Pérot interferometers in the quantum Hall regime (33–37) promises complementary experimental observations on fractional quasi-particles in QDs.

Here, we study the magneto-transport through a large QD containing roughly 900 electrons in the fractional quantum Hall regime for filling factors ν < 1. The QD forms concentric compressible regions separated by incompressible regions (38). In the integer quantum Hall regime, this has been established by experiments (39–42) and theory (43). By reconstructing the charge carrier distribution in the QD at zero magnetic field, we can show that in two specific regions of magnetic field, the incompressible region corresponds to a fractional filling factor ν_in = 1/3 or 2/3, respectively. In our experiments, the QD is weakly tunnel-coupled to its leads and occupied by an integer number of electrons N. In this regime, conductance peaks arise each time the chemical potential of the Nth electron state and the leads are degenerate, as usual in Coulomb blockade experiments. While only an integer number of electrons can tunnel between the QD and the leads (2), we observe fractional quasi-particle tunneling between the compressible regions inside the QD with a fractional charge corresponding to e^* = e/3, as predicted theoretically. In the properly tuned regime, the tunnel coupling across the incompressible region is strong enough to enable tunneling of fractionally charged quasi-particles and weak enough to lead to a detectable Coulomb blockade signal via rearrangement of (fractional) charges. As each compressible region forms a QD, the quasi-particle tunneling can be treated in a capacitive single particle model as the fractional Coulomb blockade between two nested QDs. We construct a phase diagram of stable charge, which was previously proposed theoretically (44) and measured for integer Landau levels (37, 45–50).

RESULTS AND DISCUSSION

Experimental setup and characterization

The QD sample is fabricated on an AlGaAs/GaAs heterostructure etched into a Hall bar structure. It hosts a two-dimensional electron gas (2DEG) 130 nm below the surface, which is contacted by annealed AuGeNi ohmic contacts. We measure a bulk electron density n_bulk = 1.44 × 10¹¹cm⁻² and electron mobility μ = 5.6 × 10⁶cm²/Vs at temperature T = 30 mK. The bulk electron density can be altered by applying a voltage to the prepatterned, overgrown back gate extending underneath the Hall bar 1 μm below the 2DEG (51). For all measurements in this paper, the back gate was grounded. However, the presence of back gates or additional gates can influence the confinement potential, which was exploited in previous experiments (12, 27, 52, 53).

The detailed gate design of the inner structure of the QD sample is shown in Fig. 1A. The QD with a width of 1.4 μm and a lithographic area of ≈ 2 μm² is formed by four metallic gates [labeled center barrier (CB), left and right barriers (LB and RB), and plunger gate (PG); yellow] that are lithographically patterned on the surface of the AlGaAs/GaAs heterostructure (dark blue). We form the QD by applying negative voltages to the gates, thereby depleting the electron gas underneath. Depletion of the electron gas below the gate occurs at −0.35 V. By applying a small voltage V_SD between the source and drain contacts and measuring the resulting current I_SD, we study the two-terminal linear conductance G_dot = V_SD/I_SD of the QD. First, we tune the QD system into the Coulomb blockade regime. The CB gate and the LB and RB gates tune the transmission of the right and left barriers, respectively. The transmitted conductance through both barriers is set to ≪ e²/h such that the QD is only weakly coupled to its leads. We fix the voltage of the CB gate V_CB = −1.2 V, while the voltage on LB and RB is changed to retune the barrier coupling of the QD for different measurements. The voltage on the PG is varied around V_PG ≈ −0.4 V and used to tune the discrete energy levels of the QD.

Fig. 1. Sample schematic and density characterization.

(A) atomic force microscopy image of the QD device. Top gates (labeled LB, PG, RB, and CB) appear in yellow, while the uncovered semiconductor is dark blue. A magnetic field is applied perpendicular to the sample surface. The overlaid schematics show the chiral compressible regions separated by an incompressible ν_in = 1 filling factor region for a bulk filling factor ν_b ≈ 2. We apply a source-drain bias V_SD and measure the current I_SD. (B) Normalized conductance G_dot/G_max through the QD as a function of the PG voltage V_PG and the magnetic field B, where G_max is the local maximum of the conductance over five adjacent PG voltage traces. Seven measurements with different barrier gate voltages V_LB and V_RB are combined by shifting in PG voltage such that the peaks match at the boundaries of the individual measurements. The boundaries between measurements are marked with white arrows. The Coulomb peaks show a 1/B-periodic behavior that can be related to integer filling factors ν_dot = 2,3,4,5,6,8,10,12, and 14 in the QD (dashed lines). Fitting these features results in an electron density n_dot = (1.11 ± 0.04) × 10¹¹cm⁻² in the QD. A corresponding filling factor (ν_dot) axis is indicated on the right.

The measurements were conducted in a dilution refrigerator at the base temperature T = 30 mK. All measurements presented within this paper were performed on the same sample during one cooldown. The presented modulation of the Coulomb peaks was reproduced in a second cooldown with similar gate voltages. In addition, similar measurements were reproduced with another sample using a different heterostructure and gate design.

The electron density in the QD region is reduced compared to the bulk density of the 2DEG by the applied confining gate voltages. To estimate the effective electron density n_dot inside the QD, we analyze the conductance G_dot through the QD in the integer quantum Hall regime. Figure 1B shows the normalized conductance G_dot/G_max as a function of magnetic field B applied perpendicular to the sample surface and the PG voltage V_PG. For improved visibility of all resonances, each conductance trace (PG voltage varied, magnetic field fixed) is normalized by the maximal local conductance G_max of the five adjacent traces. In addition, the transmission of the barriers of the QD changes slowly with magnetic field. Therefore, the barrier gate voltages V_LB and V_RB needed to be retuned at some magnetic fields to ensure that the conductance through the barriers stays in the desired range of weak coupling. In Fig. 1B, we therefore combine seven measurements taken over finite magnetic field ranges where different barrier gate voltage settings were applied (transitions marked by arrows). We combine the seven measurements into one figure by shifting them in PG voltage such that the peaks match at the boundary of the individual measurements. We observe Coulomb blockade resonances as a function of the PG voltage. Their position in gate voltage shows a 1/B-periodic modulation as indicated by dashed lines in Fig. 1B. This 1/B-periodic oscillation is directly related to the changing quantum capacitance of the QD that follows the 1/B-dependent density of states at the Fermi energy. Identifying these features with integer filling factors ν_dot = 2,3,4,5,6,8,10,12, and 14 in the QD allows us to extract the electron density n_dot = (1.11 ± 0.04) × 10¹¹cm⁻² in the QD by fitting the relation ν = n_doth/(eB). All of the 1/B-periodic modulations used for determining the density n_dot in the QD are within the magnetic field range of one single measurement with one specific gate voltage setting. We will see later that this density corresponds to the maximum local density in the QD center.

Periodic modulation of Coulomb resonances for fractional filling

We now study the conductance of the weakly coupled QD in the fractional quantum Hall regime for filling factor ν_dot ≳ 2/3 and compare it to the integer quantum Hall regime at ν_dot ≈ 2. First, we look at the conductance around filling factor ν_dot ≈ 2 as a function of PG voltage and magnetic field shown in Fig. 2A. The Coulomb resonances show a distinct periodic pattern in magnetic field that has been studied in previous works (36, 37, 47, 49, 50). The pattern originates from an interplay of the two compressible regions emerging from the two filled Landau levels at filling factor 2 as schematically depicted by the light blue regions in Fig. 1A. Regions of stable charge (N₁ and N₂) (separated by white lines in Fig. 2A) can be described by a capacitance model (33, 35, 37, 44) where N₁ and N₂ correspond to the number of electrons on the outer and inner compressible region, respectively.

Fig. 2. Conductance G_dot as a function of the PG voltage V_PG and the magnetic field B for different dot filling factors.

(A) ν_dot ≈ 2, (B) ν_dot ≳ 2/3, and (C) ν_dot ≳ 1/3. The exact dot filling factor ν_dot at the middle of the magnetic field range is indicated with experimental uncertainty in parentheses. Regions of constant charge (N₁, N₂) are indicated in the charge stability diagram by white lines. The charge on the outer and inner compressible regions is denoted by eN₁ and eN₂, respectively. The situation for and 1/3 in (B) and (C) allows for fractional charging of e^* = e/3.

Changing to the fractional quantum Hall regime, the conductance depending on the magnetic field and the PG voltage is shown in Fig. 2B around filling factor ν_dot ≳ 2/3. The Coulomb resonances of the QD show a periodic modulation in the amplitude and the position in PG voltage as a function of the magnetic field. The modulations are clearly visible while being less pronounced and extended compared to filling factor 2. The visible conductance resonances are continuously connected, in contrast to the clearly separated resonances around filling factor 2. The observation of modulated Coulomb resonances suggests the existence of a nontrivial fractional quantum Hall state inside the QD. Such a periodic pattern has previously not been observed for QDs in the fractional quantum Hall regime for filling factor ν_dot < 1 to the best of our knowledge.

To further study the modulated Coulomb oscillations at filling factor ν_dot ≳ 2/3 and get quantitative predictions, we extend the model for the integer quantum Hall effect discussed in our previous work (37) to fractional filling factors. We assume the existence of two compressible regions for the fractional filling factor ν_dot ≳ 2/3 separated by an incompressible ν_in = 2/3 region as schematically depicted in Fig. 1A and similar to the situation at ν_dot = 2 where the incompressible region assumes filling factor ν_in = 1. In thermodynamic equilibrium, the charge distribution in the QD minimizes the electrostatic energy. Changing the magnetic field by δB or the PG voltage by δV_PG charge imbalances δQ_i (i = 1,2) arise between the outer (i = 1) and inner (i = 2) compressible region

$$\begin{array}{c}\mathrm{\delta }{Q}_1=\mathrm{\Delta }{n}_1-{\mathrm{\nu }}_{\text{in}}\mathrm{\delta }B\overline{A}/{\mathrm{\varphi }}_0-C_1\mathrm{\delta }{V}_{\text{PG}}/e\\ \mathrm{\delta }{Q}_2=\mathrm{\Delta }{n}_2+{\mathrm{\nu }}_{\text{in}}\mathrm{\delta }B\overline{A}/{\mathrm{\varphi }}_0-C_2\mathrm{\delta }{V}_{\text{PG}}/e\end{array}$$ (1)

The charge imbalances δQ_i are denoted in units of the elementary charge e. The Δn_i describes discrete changes in charge of the respective region due to quasi-particle tunneling to the other compressible region or the leads. In the fractional quantum Hall regime, this can take fractional values corresponding to a multiple of the fractional charge e^* for tunneling events between the compressible regions and is not required to be an integer number of the elementary charge e. Changing the magnetic flux through the area $\overline{A}$ enclosed by the incompressible stripe at ν_in = 2/3, a Hall current ${\mathrm{\nu }}_{\text{in}}\mathrm{\delta }\mathit{B}\overline{A}/{\mathrm{\varphi }}_0$ will flow from the outer to the inner compressible region. Consequently, the (fractional) charge ν_ine will be shifted when adding one flux quantum ϕ₀ = h/e, not necessarily corresponding to the quasi-particle charge e^*. The PG couples to the compressible regions over the effective capacitances C_i > 0. The change in total electrostatic energy can then be calculated to be

$$\mathrm{\delta }E=\frac{1}{2}{K}_1\mathrm{\delta }{Q}_1^2+\frac{1}{2}{K}_2\mathrm{\delta }{Q}_2^2+K_{12}\mathrm{\delta }{Q}_1\mathrm{\delta }{Q}_2$$ (2)

where the K_i (i = 1,2) describes the charging energies of the compressible regions and K₁₂ the cross-charging energy due to capacitive coupling between the compressible regions. This model predicts hexagonally shaped regions of stable charge as a function of the magnetic field and PG voltage as the system minimizes the energy functional (Eq. 2) by assuming suitable Δn_i.

We interpret the data in Fig. 2B at ν_dot ≳ 2/3 according to this model and draw the charge stability diagram. We will now look at charging events where a fractional charge e^* is rearranged between the two compressible regions, i.e., Δn₁ = +e^*/e, Δn₂ = −e^*/e. The corresponding magnetic field spacing ΔB coincides with the magnetic field period as indicated in Fig. 2B. The measured magnetic field period ΔB is shown in Fig. 3A as a function of magnetic field B for the regions where periodic modulations are observed. We find a stable period of ΔB = 5.6 mT (corresponding to an area $\overline{A}=0.74 \mathrm{\mu }{\mathrm{m}}^2$) at ν_dot ≈ 2 (blue dots) slowly rising toward and diverging at ν_dot ≈ 1, which is directly related to the decreasing area enclosed by the incompressible ν_in = 1 stripe as the upper spin-split branch of the lowest Landau level is depopulated (37). For ν_dot ≳ 2/3 (green squares), the period quickly increases with increasing B as well.

Fig. 3. Analysis of periodicity in magnetic field and gate voltage.

(A) Magnetic field period ΔB indicated in Fig. 2 (A to C) as a function of magnetic field B. A periodic modulation is only observed for the shaded magnetic field regions. (B) Zero magnetic field density n_{0, dot} of the QD as a function of the radius r calculated from the magnetic field periodicities in (A) according to Eqs. 3 and 4, mirrored around r = 0. We assume a magnetic field periodicity ΔB corresponding to a flux quantum ϕ₀ for 2 > ν_dot > 1 [(C), blue dots] and ν_dot ≳ 1/3 [(E), orange triangles] and ϕ₀/2 for ν_dot ≳ 2/3 [(D), green squares]. Assuming a flux quantum periodicity ϕ₀ and an incompressible stripe at ν_in = 1/3 for ν_dot ≳ 2/3 instead (light green empty diamonds), the calculated charge distribution does not agree with the data of other filling factors. The blue line shows a fit according to Eq. 5. (C to E) Ratio V₁₂/V₁ of the PG voltages indicated in Fig. 2 (A and B) depending on B for dot filling factors around (C) ν_dot ≈ 2, (D) ν_dot ≳ 2/3, and (E) 1/2 > ν_dot ≳ 1/3. Quasi-particle charge ratios e^*/e calculated by Eq. 8 are indicated.

Periodic modulations of the conductance are also observed for filling factors 1/2 > ν_dot > 1/3 as shown in Fig. 2C as a function of PG voltage and magnetic field (see also Fig. 4, G and H). The charge stability diagram is indicated and very similar to ν_dot ≳ 2/3 in Fig. 2B. For the region where ν_dot ≳ 1/3, the QD exhibits two compressible regions separated by an incompressible region at ν_in = 1/3, and we can apply the same model as described above for ν_dot ≳ 2/3 where ν_in = 2/3. The corresponding magnetic period for 1/2 > ν_dot > 1/3 is displayed in Fig. 3A (orange triangles) and shows a slow increase similar to the behavior close to ν_dot ≈ 2. For all three regimes (ν_in = 1, 2/3, or 1/3) in Fig. 3A, a periodic modulation is only observed for ν_dot ≳ ν_in within the experimental uncertainty that stems from the uncertainty in the calculated dot density n_dot.

Fig. 4. Conductance G_dot as a function of the PG voltage V_PG and the magnetic field B for decreasing dot filling factors 2>ν_dot >1/3.

The filling factor ν_dot is indicated with the corresponding experimental uncertainty in parentheses. Periodic modulation of the Coulomb peaks are observed in (A), (D), (E), (G), and (H) corresponding to filling factors in the regions marked in Fig. 3A. No periodic modulations are observed in (B), (C), and (F).

Extracting the dot density distribution

The magnetic field spacing between two rearrangements resulting from the model is $\mathrm{\Delta }B=(e^{*}/e){\mathrm{\varphi }}_0/({\mathrm{\nu }}_{\text{in}}\overline{A})$. Assuming a circular charge distribution of the QD, we can calculate the radius of the incompressible region according to

$$r(B)=\sqrt{\frac{e^{*}/e}{{\mathrm{\nu }}_{\text{in}}}\frac{{\mathrm{\varphi }}_0}{\mathrm{\Delta }B\mathrm{\pi }}}$$ (3)

with a density in the incompressible region

$$n(r,B)=\frac{\mathit{eB}}{h}{\mathrm{\nu }}_{\text{in}}$$ (4)

Using these two equations enables us to reconstruct the zero magnetic field charge density distribution n_{0, dot} in the QD from the measured magnetic period ΔB(B) in Fig. 3A. We assume fractional charge e^* = e/3 for ν_dot ≳ 1/3 and 2/3 and charge e for ν_dot ≈ 2 while having an incompressible region at ν_in = 1/3, 2/3, and 1, respectively. This results in the dot density n_0,dot shown in Fig. 3B with a flux quantum periodicity ϕ₀ for ν_dot ≈ 2 (blue dots) and ν_dot ≳ 1/3 (orange triangles) and a half-flux quantum periodicity ϕ₀/2 for ν_dot ≳ 2/3 (green squares). All three different regimes line up to form a smooth radial density dependence. The slightly higher radii for ν_dot ≳ 1/3 probably reflect the lower voltages applied to the barrier gates in this regime. For ν_dot ≳ 2/3, we can exclude periodic modulations spaced by a full flux quantum ϕ₀ that would originate from an incompressible region of ν_in = 1/3. They would lead to the light green, empty diamonds in Fig. 3B that do not line up with the points of the other regimes.

In a study of an anti-dot embedded into a ν = 2/3 fractional quantum Hall (FQH) state (9), a flux period of ϕ₀ was found. This experiment was analyzed with the help of an electrostatic model which assumed that the edge of the ν = 2/3 quantum Hall state consists of a downstream propagating integer channel and an upstream propagating fractional 1/3 channel (54). This edge structure was proposed by MacDonald (55). Depending on the strength of the Coulomb repulsion between these channels, the flux periodicity was found to be ϕ₀ for weak coupling and ϕ₀/2 for strong coupling. In the latter case, the two channels can be described as a single compressible region, as in our model. In the present experiment, we cannot distinguish any additional ϕ₀ period. We conclude that in our experiment the 2/3 edge does not exhibit signatures of an additional neutral mode. The absence of other incompressible stripes with fractional filling factors <2/3 might be due to the increasing steepness of the confining potential closer to the edge. Edge reconstruction will only occur when the fractional gap exceeds the potential gradient times the magnetic length.

We can fit the dot density n_0,dot(r) in Fig. 3B using a model proposed by Lier and Gerhardts (56) for the position of incompressible stripes at the gated edge of quantum Hall systems. The presence of a charged gate leads to a reduction of the bulk density n_bulk by a factor s. To calculate the density distribution, we assume two gates placed symmetrically around the center at r = 0 resulting in

$$\begin{array}{c}{n}_{0,\text{dot}}(r)=n_{\text{bulk}}s(r,r_0,d)s(r,-r_0,-d) \text{with}\\ s(r,r_0,d)=\sqrt{(r_0-r)/(r_0+d-r)}\end{array}$$ (5)

where r₀ is the radius where the density drops to 0 and d is the depletion length around the gate. The fit (blue line) to our data at ν_dot ≈ 2 is shown in Fig. 3B with r₀ = (517 ± 1)nm and d = (131 ± 4) nm. This agrees well with the lithographic dot size r_lith ≈ 0.7 μm ≈ r₀ + d.

From electrostatic simulations using COMSOL, we calculate a magnetic field period ΔB = 6.3 mT at a magnetic field of B = 10T for filling factor ν_dot ≳ 1/3 and gate voltages comparable to the experimentally applied values, which is in good agreement with the experimentally determined value. Similarly, we get ΔB = 11.6 mT at a magnetic field of B = 7T for filling factor ν_dot ≳ 2/3, again in good agreement with the experiment. The calculated total charge on the QD corresponds roughly to $N_{\text{tot}}^{(\text{sim})}=870$ electrons. This is comparable to the experimentally derived value of $N_{\text{tot}}^{(\text{exp})}=820$ from experimental values for the density n_dot and the area $\overline{A}$. The results of the electrostatic simulations agree well with the experimental observations.

Returning to the charge stability diagram in Fig. 2B, we calculate within our model the slope of an internal charging line between the two compressible regions

$${\frac{\mathrm{\delta }B}{\mathrm{\delta }{V}_{\text{PG}}}|}_{\text{rearr}.}=-\frac{1}{{\mathrm{\nu }}_{\text{in}}}\frac{{\mathrm{\varphi }}_{0}e({\mathrm{\alpha }}_1-{\mathrm{\alpha }}_2)}{\overline{A}[(K_1-K_{12})+(K_2-K_{12})]}<0$$ (6)

where α₁ = (K₁C₁ + K₁₂C₂)/e² and α₂ = (K₁₂C₁ + K₂C₂)/e² denote the lever arms of the PG on the respective compressible region. This slope is negative as generally α₁ > α₂, which limits the ways we can draw the recharging lines connecting the visible Coulomb oscillations. In addition, we calculate the slopes corresponding to a constant charge on either of the two compressible regions

$$\begin{array}{c}{\frac{\mathrm{\delta }B}{\mathrm{\delta }{V}_{\text{PG}}}|}_{\mathrm{\Delta }{n}_1=0}=-\frac{1}{{\mathrm{\nu }}_{\text{in}}}{C}_1{\mathrm{\varphi }}_0/(e\overline{A}) <0,\\ {\frac{\mathrm{\delta }B}{\mathrm{\delta }{V}_{\text{PG}}}|}_{\mathrm{\Delta }{n}_2=0}=\frac{1}{{\mathrm{\nu }}_{\text{in}}}{C}_2{\mathrm{\varphi }}_0/(e\overline{A}) >0\end{array}$$ (7)

These constraints define the orientation of hexagons in the charge stability diagram as shown in Fig. 2B. Crossing any segment of the hexagon boundary with nonzero conductance in V_PG direction in Fig. 2B, the total charge on the QD changes by e. We see that crossing the boundary segment with negative slope implies Δn₁ = 1 and Δn₂ = 0, while crossing it along a segment with positive slope corresponds to Δn₁ = 2/3 and Δn₂ = 1/3. The charge of an electron tunneling into the QD can therefore be split among the compressible regions into fractional quasi-particle charges. This could be the reason why the Coulomb resonances are connected for ${\mathrm{\nu }}_{\text{dot}}\underset{˜}{>}2/3$ (Fig. 2B). At ν_dot ≈ 2, on the other hand, the Corbino ≳conductance across the incompressible region might be very small, such that electron tunneling into the inner compressible region is not substantial on transport time scales, giving rise to clearly separated resonances (Fig. 2A).

Fractional charge

So far, we have considered ideal values for the fractional charge to determine the flux periodicities, the density distribution, and the parameter of the charging model. Now, we extract an experimental value for the quasi-particle charge for fractional filling factors ν_dot ≳ 2/3 and 1/3 by further analyzing the charge stability diagram for the integer and fractional quantum Hall regimes in Fig. 2. To this end, we calculate from the model the voltage differences V₁ = K₁/(eα₁) and V₁₂ = (e^*/e)(K₁ − K₁₂)/(eα₁), which appear in the measurements in Fig. 2 (A to C) as separations of visible charging lines as indicated. By taking the ratio V₁₂/V₁, we eliminate the lever arm α₁. Assuming that the ratio K₁/K₁₂ is independent of the filling factor, the quasi-particle charge is then calculated by comparing the ratios for the integer and fractional filling factors

$$\frac{e^{*}}{e}=\frac{V_{12}^{(\text{fract})}/V_1^{(\text{fract})}}{V_{12}^{(\text{int})}/V_1^{(\text{int})}}$$ (8)

Figure 3 (C to E) shows the ratio V₁₂/V₁ as a function of the magnetic field for filling factors around ν_dot ≈ 2, ≳ 2/3, and ≳ 1/3. Each point corresponds to a measurement as depicted in Fig. 2 (A to C) centered around a certain magnetic field value. The extracted relevant parameters are averaged over several hexagons of such a charge stability diagram. The ratio V₁₂/V₁ is roughly constant within each filling factor regime. We extract a fractional charge e^*/e = 0.32 ± 0.03 and e^*/e = 0.35 ± 0.05, respectively, by comparison to ν_dot ≈ 2. This indicates a fractional charge e^* = e/3 for quasi-particles tunneling in the QD for both ν_dot ≳ 2/3 and ≳1/3. For ν = 1/3, fractional charge e^* = e/3 has previously been found from measurements on shot noise (3, 4), localized states (8), photo-assisted shot noise (10), or quantum Hall Fabry-Pérot interferometers (12), while at ν = 2/3, different experiments indicated different values, namely, e/3 (7, 11), 2e/3 (7, 9), and e (12).

Evolution for different filling factors

To give an overview of the evolution of the conductance measurements for decreasing filling factors 2 > ν_dot > 1/3, we show additional data in Fig. 4. As indicated in Fig. 3A, periodic modulations of the Coulomb resonances are only observed within the marked regions. The evolution from the distinct pattern at ν_dot = 2 in Fig. 4A toward filling factor 1 has been studied in detail in previous work (37) and is found here to behave in the same way. For a regime 1 > ν_dot ≳ 0.75, the Coulomb resonances show no periodic modulations as seen in Fig. 4 (B and C). In the fractional quantum Hall regime, the Coulomb resonances are modulated around filling factor ν_dot ≳ 2/3 and 1/2 > ν_dot as shown in Fig. 4 (D, E, G, and H). The two regions are interrupted by a region around filling factor ν_dot = 1/2 where no modulations are observed (see Fig. 4F).

Conclusion

In conclusion, we have studied the magneto-conductance of a 1.4-μm-wide QD in the fractional quantum Hall regime for filling factors ν < 1. Around ν_dot ≳ 2/3 and 1/2 > ν_dot > 1/3, we observe periodic modulations of Coulomb resonances as a function of magnetic field. Assuming two compressible regions separated by an incompressible stripe at ν_in = 2/3 and ν_in = 1/3, respectively, we have successfully used an electrostatic model to describe the phase diagram as a function of magnetic field and PG voltage. We extract the charge density distribution of the QD at zero magnetic field. By comparing our measurements in the fractional regime with measurements at ν_dot ≈ 2, we find fractional Coulomb blockade between the compressible regions in the QD with quasi-particle tunneling of fractional charge e*/e = 0.32 ± 0.03 and e*/e = 0.35 ± 0.05 for the two fractional regimes, respectively. QDs and quantum Hall Fabry-Pérot interferometers have been shown to be closely related in the integer quantum Hall regime (33–37). We have demonstrated experimentally that this relation persists in the fractional regime. Groundbreaking recent experiments have detected anyonic phase jumps in a Fabry-Pérot interferometer at fractional filling 1/3 (29), and fractional anyonic statistics were observed in anyon collision experiments (28). While interferometry experiments in the fractional quantum Hall regime have been shown to be very intricate and sensitive (1, 12, 18, 27, 29), fully Coulomb blockaded devices may provide an alternative experimental approach. Our observations complement and extend interferometry experiments of fractional quantum Hall states.

MATERIALS AND METHODS

Sample fabrication

The sample is fabricated using standard semiconductor fabrication techniques. We use a wafer that is overgrown with a standard modulation doped single interface AlGaAs/GaAs heterostructure by the Wegscheider group. The interface where the 2DEG accumulates lies 130 nm below the surface. The Si δ-doping layer lies 60 nm below the surface with a spacer of 70 nm to the 2DEG. A patterned back gate is added before overgrowth in an established process detailed in (51) and lies roughly 1 μm below the 2DEG. The mesa is patterned using optical lithography and wet etched with diluted Piranha acid (H₂O:H₂O₂:H₂SO₄, 100:3:3) with an etching depth slightly deeper than the 2DEG. Ohmic contacts are patterned using optical lithography. Using electron beam evaporation, we deposit a eutectic mixture of Ge/Au/Ni onto the sample and anneal at 500°C for 300 s [in 200 sccm H₂/N₂ (5%) flow] after liftoff. Then, we pattern the bond pads and large gate leads with optical lithography and evaporate a Ti/Au (10 nm/80 nm) layer. The small gate structures that form the QD are patterned using electron beam lithography. The gates are then deposited by electron beam evaporation of Ti/Au (5 nm/25 nm). We check the gate structure with atomic force microscopy lithography (see Fig. 1A).

Measurement techniques

The sample is cooled down in a wet dilution refrigerator at a base temperature T = 30 mK. All lines connected to the sample are filtered with a cold lowpass RC filter (R = 10 kΩ, C = 1 nF, f_cutoff ≈ 15 kHz or R = 1 kΩ, C = 100 pF, f_cutoff ≈ 1.5 MHz) at the cold finger. The current I_SD through the sample is measured by applying a voltage bias V_SD over an current to voltage converter to the sample contact and measuring the resulting voltage over the reference resistance. The conductance is calculated as the ratio G_dot = I_SD/V_SD. The IV converter is temperature stabilized, and offset voltages are corrected for.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/7/19/eabf5547/DC1