Extracting net current from an upstream neutral mode in the fractional quantum Hall regime

Abstract

Upstream neutral modes, counter propagating to charge modes and carrying energy without net charge, had been predicted to exist in some of the fractional quantum Hall states and were recently observed via noise measurements. Understanding such modes will assist in identifying the wavefunction of these states, as well as shedding light on the role of Coulomb interactions within edge modes. Here, operating mainly in the ν=2/3 state, we place a quantum dot a few micrometres upstream of an ohmic contact, which serves as a 'neutral modes source'. We show that the neutral modes heat the input of the dot, causing a net thermo-electric current to flow through it. Heating of the electrons leads to a decay of the neutral mode, manifested in the vanishing of the thermo-electric current at T>110 mK. This set-up provides a straightforward method to investigate upstream neutral modes without turning to the more cumbersome noise measurements.

ARTICLE

Received 28 May 2012 | Accepted 9 Nov 2012 | Published 18 Dec 2012

Upstream neutral modes, counter propagating to charge modes and carrying energy without net charge, had been predicted to exist in some of the fractional quantum Hall states and were recently observed via noise measurements. Understanding such modes will assist in identifying the wavefunction of these states, as well as shedding light on the role of Coulomb interactions within edge modes. Here, operating mainly in the n 2/3 state, we place a

quantum dot a few micrometres upstream of an ohmic contact, which serves as a neutral modes source. We show that the neutral modes heat the input of the dot, causing a net thermo-electric current to ow through it. Heating of the electrons leads to a decay of the neutral mode, manifested in the vanishing of the thermo-electric current at T4110 mK. This set-up provides a straightforward method to investigate upstream neutral modes without turning to the more cumbersome noise measurements.

DOI: 10.1038/ncomms2305

Extracting net current from an upstream neutral mode in the fractional quantum Hall regime

I. Gurman1,*, R. Sabo1,*, M. Heiblum1, V. Umansky1 & D. Mahalu1

1 Braun Center for Submicron Research, Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel. * These authors contributed equally to this work. Correspondence and requests for materials should be addressed to M.H. (email: mailto:[email protected]

Web End [email protected] ).

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Thirty years after the discovery of the celebrated fractional quantum Hall effect1, there are still open questions regarding electronic characteristics in this regime; one of

them is the formation and nature of upstream chiral neutral modes. These modes, which carry energy without net charge210, are at the focus of recent experimental and theoretical studies, and are encouraged by the advent of the search for Majorana quasiparticles11,12. Certain fractional states, dubbed as hole-conjugate states13,14, with the most prominent one v 2/3,

support reconstructed chiral edge modes, which due to interaction and disorder, give birth to counter propagating neutral modes24. Among two possible models of the upstream neutral modes: (i) coherent dipole-like excitations, which carry energy; and (ii) classical-like excitations, namely, classical heat waves15; recent results favour the latter one.

Being chargeless, the observation of neutral modes presented a challenge. Bid et al.5 allowed upstream neutral modes, emanating from an ohmic contact, to scatter off a partially pinched quantum point contact (QPC). This has resulted with excess noise but, as expected, with zero net current. In a recent work by Gross et al.7, a few new features of these modes were reported: rst, that a standard (non-ideal) ohmic contact can detect neutral modes, very much like a QPC; second, injecting charge onto a QPC also excites upstream neutral modes; and third, a simple model of classical heating was found to explain much of the observed data. Moreover, quantum dots (QDs) were used as thermometers to detect temperature increase due to neutral modes8,9. The results, shown in these precursory works, support energy propagation via upstream chiral neutral modes, and exclude transport of lattice phonons.

In this work, we follow a recent proposal, which suggested employing a QD to convert the neutral modes energy into a thermo-electric current10. We impinged an upstream neutral mode, emanating from the back side of a biased ohmic contact, on the input QPC of a Coulomb blockaded (CB) metallic QD16. Conductance peaks exhibited substantial broadening when

increasing the bias on the ohmic contact. Furthermore, a net thermo-electric current was generated, due to the temperature gradient across the QD. The results agree with a toy model of a metallic QD having two leads at different temperatures, and the electrons obeying Fermi-Dirac distribution (see more in Supplementary Note S1 and ref. 7). Most of the measurements were carried at v 2/3 fractional state, though other lling

fractions were also tested.

ResultsMeasurement set-up. The sample was fabricated in a GaAs-AlGaAs heterostructure, embedding a two-dimensional electron gas (2DEG) 130 nm below the surface. Schematics of the QD and the measurement set-up are depicted in Fig. 1a; accompanied by an SEM (scanning electron microscopy) micrograph in Fig. 1b. Because of the relatively short decay length of the neutral modes5, the dot, with internal size B2 2 mm2, was placed 10 mm

upstream of contact N, which served as source for the neutral modes. The current traversing through the dot was measured at contact D, and its root mean square (RMS) value was extracted using a spectrum analyser, thus loosing phase (polarity) information. The sample was cooled to 30 mK in a dilution refrigerator.

Characterizing of the QD. The QD was tuned to the CB regime, having a maximal conductance Gmax 23 e2h 0:35 on resonance.

The charging energy was deduced via the ubiquitous non-linear differential conductance (the Coulomb diamond in Fig. 1c)17, UcE70 meV and the plungers levering factor aP Du/(eDVP)

0.005, where Du is the dots potential. Being relatively large, the QD was metallic, namely dooTooUc, with d the level spacing and T the electron temperature. Therefore, the dots excitation spectrum was practically continuous (Fig. 1c), and the width of

a b

VN

VP

Downstream

charge mode

P

Upstream

neutral mode

N

N

2 [afii9839]m

G1 G2

C D

VC

0

50

110

2.85 2.84 2.83 2.82 2.81

100 0.3

50 0.2

0.1

0

c

V C (V)

VP (V)

Figure 1 | Quantum dot scheme. (a) A sketch of the device. Either a charge current injected from contact C (blue) or an upstream neutral current injected from contact N (red) are impinged on the input of the QD. The current at the output of the QD (dashed blue) is measured at contact D through an LC (f0 995 kHz) circuit and a homemade cold amplier. A constant ow of charged mode xed at ground potential (contact G2) is impinged

on the output side of the dot (blue dotted line). The charge mode emitted from contact N ows directly into the ground (G1, blue) and inuences neither the QD nor the experiment. (b) SEM image showing the QD and the ohmic contact used to inject the neutral mode. The upstream neutral current travels a distance of 10 mm along the edge before impinging at the QD (red arrow). All the other ohmic contacts (shown only in panel a) are tens of micrometres away from the dot. The QD itself is made out of two QPCs (opening of 650 nm) and a plunger gate (300 nm wide). The lithographic area of the QD is

B2 2 mm2. (c) Coulomb diamond characterization of the QD in v 2/3, obtained by measuring the QDs conductance, @I@V , while scanning VP and

changing VC in small DC increments. The scale bar is in units of the quantum conductance at v 2/3 G (2e2/3h).

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each conductance peak was expected to be temperature limited16,18.

Heating of the QDs Input. Heating of the QD was monitored by observing the broadening of the conductance peaks in the presence of an upstream neutral mode (red arrow in Fig. 1a). As VN increased, the conductance peaks broadened (colour scale in Fig. 2a) while their height hardly changed (see Supplementary Note S2 and Supplementary Figs S1 and S2). Two cuts, one at VN 0

and another at VN 200 mV (dashed lines in Fig. 2a), are shown in

Fig. 2b. Employing a two-temperature toy model for the QD, with an input temperature Tin and an output temperature of 30 mK, led to Tin 125 mK for VN 200 mV. A more complete description of

Tin (VN) is given in Fig. 2c; it is rather symmetric with respect to VN 0, with a change in slope around VN 150 mV. Such

dependence was also observed in previous works58.

The above measurements cannot determine whether the dot was heated homogenously or if a temperature gradient was created. In the most nave approach, current ows in the dot due to the difference between the non-equilibrium energy distributions in both leads19. Therefore, when an external bias is applied, electrons will ow from high to low potential with resultant CB peaks (Fig. 3a). However, the thermo-electric current Ite, driven solely by a temperature gradient across the dot, will strongly depend on the position of the dots relevant energy level (ED) with respect to the Fermi energy (eF)20,21. When EDoeF electrons will ow from the cold to the hot lead, whereas when ED4eF

they will run in the opposite direction (Fig. 3b). In Fig. 3c, the blue curve is the measured linear differential conductance, whereas the red curve is the RMS value of thermo-electric current, when VN is AC modulated (amplitude of 10 mV). More information about the sign inversion of Ite is presented in Supplementary Fig. S3 and Supplementary Note S3. This result

a b

c

VN=0V

VN=200V

0.25

0.2

0.15

0.1

0.05

0

0.2

200

100

2 /3 h )

0.1

V N (V)

G(2e2 /3 h )

T in (mK)G(2e

0

3.03

3.02

3.01

VP (V)

VN (V)

0

130

100

80

200

3.04 3.03

3.02

3.01

30 200

100

0

100 200

VP (V)

Figure 2 | Broadening of CB peaks due to neutral mode. (a) 2D plot showing the broadening of three consecutive CB peaks and serving as a proof for the heating caused by the neutral mode. (b) Two cuts of the 2D plot showing CB peaks at VN 0 (blue) and VN 200 mV (red). The dashed lines in

panel a show the lines along which the cuts were made. (c) The hot leads temperature (Tin) as function of VN. The values were deduced from the width of the CB peaks using a toy model of a metallic QD with different temperatures at its leads.

[afii9830]F [afii9830]F

ED ED

a b c

0.30.15 0

0.4

0.2 0

V>0

G(2e2 /3 h )

T=0

V=0

T>0

eV

I I VP

VP VP

I te (pA)

VP

1.52 1.51 1.5

VP (V)

Figure 3 | Current ow due to either voltage or temperature gradient. An illustration of a quantum dot under electrical or thermal gradient, showing how the current behaves once the QD is near resonance. The energy levels are capacitively coupled to the plunger gate (VP), and for simplicity only one energy level (ED, marked by a full line) is shown. (a) Applying a nite voltage on the left lead will result in current from left to right once the energy level is in the vicinity of the leads Fermi energy (eF, marked by dashed line); hence the single peak shown. The blue curve in panel c shows the measured conductance peaks, which t this picture. (b) Heating up the left lead will result in current from right to left as EDoeF, zero current at ED eF and

current from left to right when ED4eF. We also show the absolute value of the thermo-electric current (dotted black line) to match our measuring capabilities. The red curve in panel c shows the thermo-electric current due to voltage applied on contact N. The measurement gives a good agreement with the thermal gradient picture.

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a

a

200 180 160

8

TMC=34 mK TMC=42 mK

TMC=51 mK TMC=60 mK

TMC=69 mK TMC=103 mK

1st harmonic

2nd harmonic

6

4

140 120 100

80 60 40

45 40 35 30 25 20 15

5 0

10

I te (pA)

[p42]I te \ [p42]V P (a.u)

T in (mK)

I te (pA)

2

0

3.5

3

2.5

2

1.5

1

0.5

200 200

100

100

0

100

150 100

200 200

100

50

50 150

0

100

VN,DC (V)

VN (V)

b

b

0 200 200

Figure 4 | Dependence of the resulting thermo-electric signal on the neutral voltage. (a) Change of the thermo-electric currents rst two harmonics when applying VN VN,DC VN,AC cos (ot) (VN,ACB100 mV).

As the 1st harmonic (blue line) increases, the 2nd harmonic (green line) drops. This outcome proves that the thermo-electric current Ite is

proportional to |VN|. (b) Amplitude of the thermal transconductance as function of the DC bias on VN while keeping a constant AC modulation on the plunger gate.

VN (V)

TMC (mK)

100

0

40 120 140 160 180 200

60

80

Figure 5 | Temperature dependence of the neutral modes energy.(a) Hot leads temperature (Tin) as function of the DC bias on contact N for different base temperatures (34103 mK). As the temperature increases, the neutral modes energy decreases leading to weaker heating of the QDs input. (b) Thermo-electric current dependence on base temperature, while applying a constant AC signal on contact N. The data is tted to a quantum

dot with a temperature gradient, where DT a exp

T

T

2

. The data (blue circles) is very well described (red line) by our toy model usinga 372 and T0 66 mK.

proves a thermal gradient was formed across the dot due to the upstream neutral mode.

Thermo-electric currents dependence on neutral energy. Establishing that the heating of the QD is insensitive to the sign of VN (Fig. 2) leads to two possible scenarios: either

Ite / V2N or Ite / jVNj . This was tested by applying VN VN,DC

VN,ACcos(ot), scanning VN,DC while keeping VN,AC, constant, and measuring the 1st and 2nd harmonics in the current Ite. If Ite /

V2N V2N;DC 12 V2N;AC 2VN;DCVN;AC cos ot V

2 2 cos 2ot,

then one would expect the 1st harmonic to rise while the 2nd one stays constant while increasing VN,DC. Yet, if Itep|VN|, the 1st harmonic should increase while the 2nd decreases, to eventually disappear once VN,DC4VN,AC. Consequently, the 2nd harmonic is the best criterion to distinguish between the two options. Figure 4a shows the results of such a measurement with f o/2p 995

(497.5) kHz when measuring the 1st (2nd) harmonic, VN,AC 100

mV and VN,DC increasing in small increments. The decrease of the 2nd harmonic (green curve), until its diminishing at VN,DC 100

mV VN,AC, clearly suggests that Ite is proportional to |VN| and not

to V2N; as noted in ref. 15. In addition, the fact that Ite followed the B1 MHz excition on contact N shows that the neutral modes response time is 1 mS at most.

A similar measurement was performed by adding a small AC signal to the plunger gate and varying the DC bias on N. The

transconductance @Ite@VP was measured at resonance, (where its value

is maximal) as function of VN (this is the actual slope of Ite at resonance in Fig. 3c (red curve)). The slope in Fig. 4b is a direct representation of Ite as function of VN. The slope moderated near

VNB150 mV; attributed to a high enough temperature increase, which may allow two electrons to traverse through the QDeach contributes to an opposite polarity currentleading to the apparent saturation.

DiscussionAfter establishing a clear indication of heating by the neutral mode, we investigated its dependence on the base temperature. Kane et al.3,4 predicted that the energy carried by the upstream neutral modes decays exponentially with distance, with a characteristic decay length proportional to T 2, namely, as exp

T T0

2

, with T0 a characteristic temperature. In the current set-up, raising the lattice temperature (by heating the mixing chamber of the dilution refrigerator, TMC) should

increase the temperature of the output QPC, while lowering the temperature rise at the input QPC. Consequently, the

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temperature gradient across the QD reduces rapidly, and with it the thermo-electric current should ebb.

Figure 5a shows Tin (deduced in a similar fashion to that described before, see Fig. 2c) as function of VN at different values of TMC. It is clear that as TMC increases, the temperature rise of the hot input (above TMC) becomes smaller; without further increase for TMC4100 mK. Via standard noise measurements, we veried that at this base temperature the neutral signal decreased substantially5. These results clearly point out a strong decay of the neutral mode with temperature.

Furthermore, we measured the maximal (as function of VP)

thermo-electric current Ite for different values of TMC, while keeping a constant excitation of 300 mV on VN. The data (open circles in Fig. 5b) was compared with a simulation of Ite based on

a hot input temperature Tin TMC ae

2 and a cold output (at TMC). We found that a 372 mK and T0 66 mK lead

to an excellent agreement with the data (solid red line in Fig. 5b); thus supporting the predicted temperature dependence3,4.

Similar measurements were performed at v 3/5, being also a

hole-conjugate state, with essentially the same qualitative behaviour5. However, no upstream heating had been observed at v 1 and v 2 (ref. 22) (as shown in Supplementary Fig. S4

and Supplementary Note S4); conrming that the observed heating is due to the presence of upstream neutral modes at certain fractional states.

In this work, we demonstrated a new meansexploiting a QDto transform energy carried by upstream neutral modes into net current; a method that is likely to be simpler than measuring noise. Employing a smaller QD, with a quantized excitation spectrum D4T, is expected to provide an experimental gateway to further understanding other properties of the neutral modes10. Furthermore, the measurement techniques reported in this work can be used to explore the nature of the neutral modes at the v 5/2 state5,6, which is suspected to carry with it non-Abelian

braiding statistics23.

Methods

Sample fabrication. The sample was fabricated in a GaAs-AlGaAs heterostructure, embedding a 2DEG, with areal density 8.8 1010 cm 2 and 4.2 K dark mobility

5.8 106 cm2V 1s 1, 130 nm below the surface. Schematics of the QD and the

measurement set-up are depicted in Fig. 1a; accompanied by an SEM micrograph in Fig. 1b. The dot, with internal size B2 2 mm2, was dened by two split metallic

gates (TiAu), serving as QPCs with openings 650 nm wide. A plunger gate, 300 nm wide, controlled the occupation of the dot. Because of the relatively short decay length of the neutral modes5, contact N, serving as source for the neutral modes, was placed 10 mm downstream the QD. Contacts C, D, G1 and G2 were placed tens of micrometres away from the dot. The sample was cooled to 30mK in a dilution refrigerator.

Measurement technique. Biasing contact C raised the chemical potential of the input QPC of the QD, while biasing contact N increased the QPC temperature. In both cases, a net electrical current I traversed the dot. The output drain (D) voltage VD IRH, with RH, the Hall resistance, was ltered using an LC resonant

circuit (f0 995 kHz) and amplied by homemade voltage preamplier

(cooled to 1K) followed by a room temperature amplier (NF SA-220F5).

A commercial spectrum analyser displayed the RMS signal; hence, loosing phase (polarity) information.

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Acknowledgements

We acknowledge O. Zilberberg, Y. Gross, A. Stern and G. Viola for fruitful discussions

and remarks on the manuscript. We acknowledge the partial support of the European

Research Council under the European Communitys Seventh Framework Program (FP7/

20072013)/ERC Grant agreement no. 227716, the Israeli Science Foundation, the

Minerva foundation, the German Israeli Foundation, the German Israeli Project Coop

eration and the US-Israel Bi-National Science Foundation. I.G. is grateful to the Azrieli

Foundation for the award of an Azrieli Fellowship.

Author contributions

I.G., R.S. and M.H. designed and performed the experiment and wrote the paper;

V.U. grew the 2DEG and D.M. performed the electron beam lithography.

Additional information

Supplementary Information accompanies this paper on http://www.nature.com/naturecommunications

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How to cite this article: Gurman, I. et al. Extracting net current from an upstream

neutral mode in the fractional quantum Hall regime. Nat. Commun. 3:1289 doi: 10.1038/

ncomms2305 (2012).

TMC

T0

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