Magnetic field dependent equilibration of fractional quantum Hall edge modes

Tanmay Maiti,1 Pooja Agarwal,1 Suvankar Purkait,1 G J Sreejith,2

Sourin Das,3 Giorgio Biasiol,4 Lucia Sorba,5 and Biswajit Karmakar1, ∗

1Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata 700 064, India2Indian Institute of Science Education and Research, Pune 411008, India

3Department of Physical Sciences, IISER Kolkata, Mohanpur, West Bengal 741246, India4Istituto Officina dei Materiali CNR, Laboratorio TASC, 34149 Trieste, Italy

5NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza San Silvestro 12, I-56127 Pisa, Italy

Fractional conductance is measured by partitioning ν = 1 edge state using gate-tunable fractionalquantum Hall (FQH) liquids of filling 1/3 or 2/3 for current injection and detection. We observetwo sets of FQH plateaus 1/9, 2/9, 4/9 and 1/6, 1/3, 2/3 at low and high magnetic field ends of theν = 1 plateau respectively. The findings are explained by magnetic field dependent equilibration ofthree FQH edge modes with conductance e2/3h arising from edge reconstruction. The results revealremarkable enhancement of the equilibration lengths of the FQH edge modes with increasing field.

Topological phases of matter such as the quantumHall states possess gapless protected surface states thatcarry the current, while the bulk remains insulating. Atthe smooth boundary Coulomb interaction leads to re-construction of the edge states in integer quantum Hall(IQH) systems [1–3], fractional quantum Hall (FQH) sys-tems [4–8], quantum spin Hall insulators [9, 10] andgraphene nanoribbons [11] etc. Edge reconstruction [12–15] can result in additional integer and fractional edgemodes as well as neutral modes with short equilibra-tion lengths [9–11, 16] due to quasi-particle scatteringbetween the modes. Transient FQH edge modes withshort equilibration lengths of a few micrometers havebeen demonstrated in IQH [17] and FQH [18] systems.Robust reconstructed FQH edge modes with long equili-bration lengths [19, 20] can have non-trivial implicationson investigating braiding statistics [20–24], quantum in-terferometry [25–27] and design of hybrid quantum Hallsystems [28]. A suitable system to explore robust FQHedge modes is ν = 1 IQH state, where three edge modeswith conductance e2/3h [1] are formed by edge recon-struction. Robustness of the FQH edge modes is investi-gated here for the first time by varying magnetic field.

In this letter we show enhancement of equilibrationlength of a FQH edge mode of conductance e2/3h uponincreasing magnetic field within the ν = 1 IQH plateau.The equilibration length of the mode is estimated to beas high as 777±40 µm at the high field end of the plateau.Enhancement of the equilibration length with increasingfield is corroborated by investigating equilibration lengthof FQH edge modes at bulk filling ν = 2/3. Our resultsreveal the way to find the robust FQH edge modes.

The experiments are carried out on a modulationdoped heterostructure, in which the 2DEG resides ina GaAs/AlGaAs heterointerface located 100 nm belowthe top surface. Figure 1 (inset a) shows a topologicallyequivalent schematic device structure [29], where the twosides of IQH region of filling ν is bounded by regions offilling fractions ν1 and ν2 that are tunable using top-gatesg1 and g2 as in Ref. 30. For transport measurements, four

standard Ohmic contacts (S1,S2,D1,D2) are deployed onthe device. The sample is mounted in a dilution refrig-erator equipped with 14 T superconducting magnet atbase temperature 7 mK, where the lowest electron tem-perature achieved is about 30 mK. All measurementsare carried out at 30 mK unless stated otherwise. Thecarriers are injected by light illumination at 3 K with aGaAs LED and the injected carriers are persistent at lowtemperature [31]. After illumination, the sample carrierdensity and mobility become n ∼ 2.27 × 1011 cm−2 andµ ∼ 4 × 106 cm2/Vs respectively. At S2, a customizedpreamplifier SR555 (the RC filter is removed from thebias input) is deployed to facilitate simultaneous mea-surement of the output current and application of ACexcitation voltage. Source contacts S1 and S2 are excitedby 25.8 µV at frequency 17 Hz and 26 Hz respectively[29], such that the excitation injects a system current of1 nA corresponding to a quantized conductance of e2/h.Output currents at D1 and D2 are measured by lock-intechnique using suitable current to voltage preamplifiers.

To set the filling fraction ν = 1 of the 2DEG, two ter-minal magneto-resistance (2TMR) is measured betweencontacts S1 and D2 (Fig. 1) disconnecting all other con-tacts and setting gate bias V g1 = V g2 = 0 V [29]. Aν = 1 IQH plateau is formed in the magnetic field Brange of 8 to 11 T. Figure 1 (inset b) shows the trans-mitted conductance between S1 to S2 (blue curve) andthe reflected conductance between S1 to D2 (red curve)at B = 9.95 T keeping the top gate g2 in pinch-off condi-tion. This g1 gate transmission characteristic shows in-compressible FQH plateaus at filling fractions ν1 = 1/3and 2/3 similar to Ref. 32. The transmitted conductancedoes not reach 1 at V g1 = 0 V , because of insufficientelectron density beneath the top gates. Total conduc-tance stays fixed (magenta line) at unity as expected fromcurrent conservation. The characteristics of both g1 andg2 gates are identical and the positions of 2/3 and 1/3plateaus shift to higher gate voltage bias with increasingB field [32] to achieve the same incompressible state.

In our experiments, the current from the source S2

arX

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1v1

[co

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es-h

all]

20

Jun

2020

2

2g1g

S2

S1

D1

D2νν1 ν2

a

b

125μm

10

20

30

40

2 4 6 8 10 12

1

2T

MR

(k

)

2/3

1/3

2/3

B (T)

1

-0.30 -0.15

G1

(e2 /h

)

Vg1 (V)rG1

S1->D2

B = 9.95 T

tG1S1->SD3

Vg1 (V)

FIG. 1. (color online) Two terminal magneto-resistance(2TMR) trace. Inset-a: Topologically equivalent device struc-ture. Magenta line shows the current path connecting theohmic contacts. Inset-b: Characteristics of the top gate g1 at9.95 T. Blue and red curves represent transmitted (S1 → S2)and reflected (S1 → D2) conductance respectively and ma-genta line represents sum of those two conductance.

transmitted through the ν1 FQH fluid and the currentfrom the source S1 reflected by the ν1 fluid, flow along thetop mesa boundary (Fig. 1 inset a). At the detector side,the transmitted current through ν2 FQH fluid reachingD1 and the current reflected by ν2 FQH fluid reaching D2are measured. Corresponding low frequency (DC) two-terminal combined conductance (TTCC) is denoted byG ν1,ν2

S→D incorporating all the conditions of measurements,where S(D) denotes the source(detector) contact.

By setting ν = 1 and ν1 = 2/3, a 2/3 nA currentat 26 Hz transmitted from S2 source and 1/3 nA at17 Hz reflected current from source S1 are injected intothe edge states along the top mesa boundary. MeasuredTTCCs are plotted as a function of V g2 in Figs. 2(a-b) at

B = 9.95 T. We see an enhancement of G2/3,ν2S2→D1 from ex-

pected universal conductance values defined by the fillingfractions: ν1 × ν2 = 4/9 and 2/9 (Fig. 2(a), blue line)when ν2 is 2/3 and 1/3 respectively [33–35]. Also, we

see suppression of G2/3,ν2S1→D1 from the expected universal

values (1 − ν1) × ν2 = 2/9 and 1/9 (Fig. 2(b), orangeline) when ν2 is 2/3 and 1/3 respectively. Conservationof current is evident from the compensating nature ofthe measured TTCCs in Figs. 2(a and b). The TTCCscorresponding to the currents that reach the contact D1when injected from the sources S1 and S2 are plotted to-gether in Fig. 2(c). The sum of the two curves (magentaline) resembles the universal gate characteristics of g2 bycompensating deviations of the TTCCs.

Deviation of the TTCCs from the expected universallimits depends on the magnetic field strengths. Figures3(a-d) show the TTCCs similar to Fig. 2(c) maintainingν1 = 2/3 but at different magnetic fields within ν = 1

1/92/9

S1D2

b

tG12/3tG21/3

SD3->D1

G22/3

S1->D1

G12/3

G22/3

SD3->D1S1D1

G 2/

3, 2

(e2 /h

)

2/9

4/9

S2D2

a

S2D1

-0.30 -0.15

TOTAL

S2D1

c

2/3

Vg2 (V)

1/3 S1D1

FIG. 2. (color online) (a) Two-terminal combined conduc-tance (TTCC) vs V g2 at ν1 = 2/3 by exciting S2. Arrowsindicate deviation of the TTCC (blue curve) measured at D1from the expected values (dotted lines) (b) TTCC vs V g2 atν1 = 2/3 by exciting S1. Arrows indicate deviation of theTTCC (orange curve) measured at D1 from the expected val-ues (dotted lines) (c) Replotting of blue and orange curvesand plot of sum of the two (magenta). Gray strips indicateν2 = 1/3 (left) and 2/3 (right) region.

IQH plateau. At a field B = 8.5 T, the TTCCs reach theexpected universal quantized conductance values (Fig.3(a)). The TTCC values increasingly deviate from theuniversal limits (Figs. 3(b-d)) with increasing B fieldsand tend to saturate at high field (B = 10.9 T) to newTTCC limits tabulated in Table I. Similar deviation ofTTCCs from the expected universal limits with increas-ing B fields can be seen also maintaining ν1 = 1/3 (Figs.3(e-h)) and the new TTCC limits are tabulated in TableI. These new TTCC limits at strong B fields are the mainobservation of this letter.

These new TTCC limits cannot be explained by a sim-ple picture with an integer edge mode. The observa-tion of new TTCC limits suggests edge reconstruction atthe natural mesa boundary consistent with Ref. 1, wherethree downstream 1/3 charge modes arise from incom-pressible Laughlin like gaps [36] corresponding to fillingfraction 2/3 and 1/3 as the ν value reduces from thebulk value 1 to zero [37]. Of these three, the two outermost 1/3 charge modes are similar to the fractional edgemodes of 2/3 FQH state [13, 18], having charge equilibra-tion length lor of the order of few micrometers. These twoouter modes completely equilibrate with each other overpropagation length l = 125 µm along the top boundarywithin ν = 1 plateau. We assume that the innermost 1/3charge mode possess longer equilibration length lir � l athigh B field end and equilibrates with the others at lowB field end of ν = 1 plateau. With this model, the newTTCC limits at high B and universal TTCC values at

32

a

e f

b c d

g h

S1D1S2D1TOTAL

1/92/9

2/3

B = 8.5 T

4/9

G 2/

3, 2

(e2 /h

)

1/3

9.7 T

10.3 T

1/3

2/3

10.9 T

-0.30 -0.15

1/92/9

4/9S2D1S1D1TOTAL

2/3

1/3

B = 8.5 T

G 1

/3,

2 (e2 /h

)

-0.30 -0.15

Vg2 (V)

9.7 T

-0.30 -0.15

10.3 T

-0.30 -0.15

1/6

1/3

10.9 T

2g1g

1/3

1/3

D2S1

S2ν =1

D1S2

S1

D1

D2

ν =11/3

2/3

1g 2g2g1g

S2

2/3

S1 D2

D11/

3ν =1

2g1g

S2

S1

D1

2/3

2/3

D2ν =1

i j k l

a

e f

b c d

g h

2 g1 g

1/3

1/3

D2S1

S2ν =1

D1S2

S1

D1

D2ν =11/

3

2/3

1 g 2 g2 g1 gS2

2/3

S1 D2

D11/3ν =1

2 g1 gS2

S1

D1

2/3

2/3

D2ν =1

i j k l

FIG. 3. (color online) (a-d) Two-terminal conductance (TTC) vs V g2 at different B fields when ν1 = 2/3. (e-h) TTC vs V g2at different B fields for ν1 = 1/3. Dotted lines in (a) and (e) show expected universal values of TTC at 8.5 T. Dashed lines in(d) and (h) show new limits of TTC at 10.9 T. Gray strips indicate ν2 = 1/3 (left) and 2/3 (right) region. (i-l) Paths of three1/3 modes at different ν1 and ν2 values inferred from the measurements. Non current carrying modes are not shown.

2g1g

1/3

1/3

D2S1

S2

ν = 2/3

D1

c

a b

8.8 9.6 10.4

200

400

600

800

B (T)

l i

r

(m

)

-0.30 -0.15

1/2

S1→D1

TOTAL

S2→D1

G 1

/3,

2 (e

2/h

)

Vg2 (V)

1/3

1/6

0.1 0.2 0.3

l i

r (B)

l i

r (T)

T (K)

2 g1 g

1/3

1/3

D2S1

S2ν = 2/3

D1c

a b

FIG. 4. (color online) (a) Equilbration length of the inner-most mode vs B field at T = 30 mK (green curve) and vstemperature at B = 10.55 T (red curve). (b) Two-terminalconductance (TTC) vs V g2 at ν = 2/3 for ν1 = 1/3. Graystrip indicates ν2 = 1/3 region. Dotted line shows expecteduniversal conductance limit for ν1 = ν2 = 1/3. (c) Schematicquantum Hall circuits for ν = 2/3 and ν1 = ν2 = 1/3 basedon two downstream 1/3 charge modes (blue lines).

FIG. 3. (color online) (a-d) Two-terminal combined conductance (TTCC) vs V g2 at different B fields when ν1 = 2/3. (e-h)TTCC vs V g2 at different B fields for ν1 = 1/3. Dotted lines in (a) and (e) show universal values of TTCC at 8.5 T. Dashedlines in (d) and (h) show new limits of TTCC at 10.9 T. Gray strips indicate ν2 = 1/3 (left) and 2/3 (right) region. (i-l) Pathsof three 1/3 modes at different ν1 and ν2 values inferred from the measurements. Non current carrying modes are not shown.

low B can be explained using the schematic Figs. 3(i-l).In Fig. 3(i) both the top-gated regions are at 2/3 fill-

ing, where the two 1/3 charge modes from S2 reach D1without equilibrating with the innermost mode. As a re-

sult the TTCC reaches the new limit G2/3,2/3S2→D1 = 2/3 as

observed in Fig. 3(d). In Fig. 3(j), out of two modesfrom S2, outermost 1/3 charge mode reaches D1 and theother is reflected to D2, as a consequence the new TTCC

limitG2/3,1/3S2→D1 = 1/3 is observed in Fig. 3(d). None of the

modes from S1 reach D1 in the above two cases, resulting

in the new TTCC limits of G2/3,2/3S1→D1 = G

2/3,1/3S1→D1 → 0 as

observed in Fig. 3(d). At lower magnetic field all threemodes completely equilibrate leading to equipartition ofthe injected current and hence the TTCCs reach the uni-versal values (Table I) as observed in Fig. 3(a).

When ν1 = 1/3 (Figs. 3(k-l)), the outermost 1/3charge mode from S2 completely equilibrates with themiddle 1/3 mode along the top boundary. One (Fig.3(l)) or both (Fig. 3(k)) of these two modes reach D1depending on ν2 values 1/3 or 2/3 respectively. As a

consequence TTCCs reach to the new limits G1/3,1/3S2→D1

= 1/6 and G1/3,2/3S2→D1 = 1/3 (blue line, Fig. 3(h)). At

ν1 = 1/3, the current in the middle 1/3 charge mode in-jected from S1 similarly equilibrates with the outermostmode. Again, one (Fig. 3(l)) or both (Fig. 3(k)) of thesetwo modes reach to D1 depending on ν2 values. Result-

ing new TTCC limits can be estimated to be G1/3,1/3S1→D1 =

Two-terminal combinedconductance (TTCC)

Universallimit at low B

New limit athigh B

G2/3,2/3

S2→D1 4/9 2/3

G2/3,1/3

S2→D1 2/9 1/3

G2/3,2/3

S1→D1 2/9 0

G2/3,1/3

S1→D1 1/9 0

G1/3,2/3

S2→D1 2/9 1/3

G1/3,1/3

S2→D1 1/9 1/6

G1/3,2/3

S1→D1 4/9 1/3

G1/3,1/3

S1→D1 2/9 1/6

TABLE I. Low and high field limits of conductances (e2/h).

1/6 and G1/3,2/3S1→D1 = 1/3 (orange line, Fig. 3(h)) when ν2

is 1/3 and 2/3 respectively. At low B fields (Fig. 3(e))the TTCCs reach the universal conductance values (Ta-ble I). Using the model (Figs. 3(i-l)), all the TTCC canbe represented in terms of the transmission probabilities[34] of the fractional edge modes [29].

Within the QH edge modes, energy is equilibrated byheat transfer [38–40] and charge is equilibrated by quasi-particle scattering [17, 18]. In this experiment chargeequilibration length lir of the innermost mode can be es-timated from the current measured between S1 and D1

4

2

a

e f

b c d

g h

2g1g

1/3

1/3

D2S1

S2

ν =1

D1S2

S1

D1

D2

ν =11/3

2/3

1g 2g2g1g

S2

2/3

S1 D2

D1

1/3ν =1

2g1g

S2

S1

D1

2/3

2/3

D2

ν =1

i j k l

S1→D1

S2→D1

TOTAL

1/9

2/9

2/3

B = 8.5 T

4/9

G 2

/3,

2 (

e2/h

)

1/3

9.7 T

10.3 T

1/3

2/3

10.9 T

-0.30 -0.15

1/9

2/9

4/9

S2→D1

S1→D1

TOTAL

2/3

1/3

B = 8.5 T

G 1

/3,

2 (e

2/h

)

-0.30 -0.15

Vg2 (V)

9.7 T

-0.30 -0.15

10.3 T

-0.30 -0.15

1/6

1/3

10.9 T

a

e f

b c d

g h

2 g1 g

1/3

1/3

D2S1

S2ν =1

D1S2

S1

D1

D2ν =11/

3

2/3

1 g 2 g2 g1 gS2

2/3

S1 D2

D1

1/3ν =1

2 g1 gS2

S1

D1

2/3

2/3

D2ν =1

i j k l

FIG. 3. (color online) (a-d) Two-terminal conductance (TTC) vs V g2 at different B fields when ν1 = 2/3. (e-h) TTC vs V g2at different B fields for ν1 = 1/3. Dotted lines in (a) and (e) show expected universal values of TTC at 8.5 T. Dashed lines in(d) and (h) show new limits of TTC at 10.9 T. Gray strips indicate ν2 = 1/3 (left) and 2/3 (right) region. (i-l) Paths of three1/3 modes at different ν1 and ν2 values inferred from the measurements. Non current carrying modes are not shown.

2g1g

1/3

1/3

D2S1

S2

ν = 2/3

D1

c

a b

8.8 9.6 10.4

200

400

600

800

B (T)

l i

r

(m

)

-0.30 -0.15

1/2

S1→D1

TOTAL

S2→D1

G 1

/3,

2 (e

2/h

)

Vg2 (V)

1/3

1/6

0.1 0.2 0.3

l i

r (B)

l i

r (T)

T (K)

2 g1 g

1/3

1/3

D2S1

S2ν = 2/3

D1c

a b

FIG. 4. (color online) (a) Equilbration length of the inner-most mode vs B field at T = 30 mK (green curve) and vstemperature at B = 10.55 T (red curve). (b) Two-terminalconductance (TTC) vs V g2 at ν = 2/3 for ν1 = 1/3. Graystrip indicates ν2 = 1/3 region. Dotted line shows expecteduniversal conductance limit for ν1 = ν2 = 1/3. (c) Schematicquantum Hall circuits for ν = 2/3 and ν1 = ν2 = 1/3 basedon two downstream 1/3 charge modes (blue lines).

FIG. 4. (color online) (a) Equilibration length of the in-nermost mode vs B at T = 30 mK (green curve) and vs Tat B = 10.55T (red curve). (b) Two-terminal combined con-ductance (TTCC) vs V g2 at ν = 2/3 for ν1 = 1/3. Graystrip indicates ν2 = 1/3 region. Dotted line shows expecteduniversal conductance limit for ν1 = ν2 = 1/3. (c) Schematicquantum Hall circuits for ν = 2/3 and ν1 = ν2 = 1/3 basedon two downstream 1/3 charge modes (blue lines).

when ν1 = ν2 = 2/3 (Fig. 3(i)). The current injectedfrom S1 in the innermost mode is transferred by quasi-particle scattering into two outer modes over the prop-agation length l [29] that reach D1 and correspondingTTCC can be derive [29, 41–43] as:

G2/3,2/3S1→D1 =

2

9− 2

9e−3l/2l

ir , (1)

where the pre-factors are fixed by the boundary condi-tions - no scattering into outer modes at l = 0 and fullequilibration at l � lir. Figure 4(a) (green line) showsthe B field dependence of the equilibration length esti-

mated from variation of G2/3,2/3S1→D1 obtained from measure-

ments similar to Figs. 3(a-d). Estimated equilibrationlength of the innermost mode increases from about zeroto 777± 40 µm with increasing B field. Qualitatively Bfield dependent equilibration length can be understoodfrom decreased equilibration rate due to increase in sep-aration between the modes resulting from shrinking ofthe ν = 1 region (Figs. 3(i-l)) with increasing B field.For a fixed magnetic field lir generaly increases with low-ering temperature and saturates to quantum scatteringlimit [29] as shown in Fig. 4(a) for B = 10.55 T.

Increased equilibration length of the innermost modewith B field (Fig. 4(a)), suggests the possibility of higherequilibration length lor of the two outermost modes athigh B field beyond the ν = 1 plateau. At higher mag-netic field, inner most 1/3 charge mode disappears asthe bulk filling fraction reaches ν = 2/3 and the outermost two 1/3 charge modes remain in the FQH system[1, 13, 18]. To examine the equilibration properties of the

two 1/3 charge modes at ν = 2/3, we measure the TTCCfrom S1 to D1 by setting ν1 = ν2 = 1/3 as shown in Fig.4(c). If fully equilibrated, half of the current form S1should reach D1 and the TTCC limit would be 1/6. Atthe largest field B = 13.98 T accessible; we find TTCCvalue of 0.155± 0.003 instead of 1/6 (Fig. 4(b)) and thisTTCC can be written as [29, 41–43]

G1/3,1/3S1→D1 (ν = 2/3) =

1

6− 1

6e−2l/l

or . (2)

The equilibration length of the 1/3 edge modes at ν =2/3 is increased to lor = 104± 4 µm at B = 13.98 T. Thelor must be increasing within ν = 1 plateau in micrometerrange but much smaller lor results in full equilibration ofthe outer modes.

Increased equilibration lengths lir and lor of the FQHmodes potentially results from high stability of incom-pressible FQH regions between the modes due to en-hancement of Coulomb energy at higher B fields. Quan-titative analysis of magnetic field dependent equilibrationof the 1/3 charge modes is left for future investigations.

Our model relies on adiabatic continuity of the edgemodes of the FQH fluids beneath the gates g1/g2 intothe FQH edge modes of the ν = 1 IQH fluid (Figs. 3(i-l)). Such adiabatic continuity is not expected if multiplemodes coupled to one [32] or a FQH edge mode partiallyreflected at this interface. Inhomogeneous carrier dis-tribution [44] or high disorder [17] in samples can alsoprevent such adiabatic continuity. These complicationsare absent in our device, allowing quantized partition ofν = 1 edge state.

In conclusion, we have demonstrated three fractionalmodes at a ν = 1 edge originating from dominated in-compressibility of FQH states at filling 1/3 and 2/3 alongthe smooth mesa boundary. Our results suggest a pos-sibility of finding more complex edge reconstruction incleaner 2DEGs. Fractionalized IQH edge presents inter-esting new possibilities to explore fractional quasiparticleinterferometry, tunneling, equilibration and statistics.Acknowledgements: We thank K Sengupta, G.

Baskaran and J K Jain for useful comments. We thankA Ratnakar for collaboration on a closely related calcula-tion. GJS acknowledges support from DST-SERB grantECR/2018/001781.

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5

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[30] B. Karmakar, D. Venturelli, L. Chirolli, F. Taddei,V. Giovannetti, R. Fazio, S. Roddaro, G. Biasiol,L. Sorba, V. Pellegrini, and F. Beltram, “Controlled cou-pling of spin-resolved quantum hall edge states,” Phys.Rev. Lett., 107, 236804 (2011).

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6

mission of heat modes across a potential barrier,” Naturecommunications, 8, 1–7 (2017).

[41] G. Muller, D. Weiss, A. V. Khaetskii, K. von Klitzing,S. Koch, H. Nickel, W. Schlapp, and R. Losch, “Equi-libration length of electrons in spin-polarized edge chan-nels,” Phys. Rev. B, 45, 3932–3935 (1992).

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and T. Fujisawa, “Charge equilibration in integer andfractional quantum hall edge channels in a generalizedhall-bar device,” Phys. Rev. B, 99, 195304 (2019).

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1

SUPPLEMENTAL MATERIAL: MAGNETIC FIELD DEPENDENT EQUILIBRATION OF FRACTIONALQUANTUM HALL EDGE MODES

In the supplemental material, we discuss the experimental techniques, present further experimental data and analy-sis. In section I, we discuss the device structure and dimensions as well as the circuit diagram used in the experiment.In section II, we discuss the current distribution under the gate g1 to address formation of a uniform fractional fluidof filling ν1 = 1/3 or 2/3 underneath the gate. We discuss charge equilibration between the co-propagating frac-tional edge modes in section III and the TTCC values are estimated as a function of equilibration length. In sectionIV, TTCC is represented by transmission probabilities of the fractional edge modes and observed TTCC limits areexplained in terms of transmission probabilities. In the last section emergent characteristics features of the chargeequilibration length lir are discussed.

I. DEVICE STRUCTURE, CIRCUIT DIAGRAM AND DETAILS OF MEASUREMENTS

125 m

66

m

D2

g2

25 m

g1

S2 D1

S1

Figure 1. Optical image of the device structure. Pink lines denote current path in the mesa under the quantum Hall regime.Separation between the two gates g1 and g2 is l = 125 µm. Length of the gate boundary beside the central mesa region islg = 25 µm.

2g1g

S2

S1

D1

D2

νν1 ν2

Special

Preamp

3I

2V

1V2I

1I

1Vg 2Vg

Customized

SR555 Preamp

Vg1 Vg2

V1 = 25.8 V, 17 Hz

V2 = 25.8 V, 26 Hz

Figure 2. Topologically equivalent device structure and measurement circuit diagram under quantized condition.

In this experiment, we use a one-sided modulation-doped AlGaAs/GaAs hetero-structure grown by molecularbeam epitaxy. The two-dimensional electron gas (2DEG) resides at the AlGaAs/GaAs hetero-interface located 100

arX

iv:2

006.

1162

1v1

[co

nd-m

at.m

es-h

all]

20

Jun

2020

2

nm below the top surface. A Si δ-doped layer placed 42 nm above the hetero-interface for modulation doping. Themesa is defined by wet chemical process and the Ohmic contacts are made by thermal deposition of metallic layers ofNi/AuGe/Ni/Au of thickness 10/100/10/100 nm after photolithography. Ohmic contacts are annealed for a durationof 90 s at 450oC using a rapid thermal annealer. The gates are made by thermal evaporation of 10 nm Ti followed by100 nm Au [1].

The optical image of the device is presented in Fig. 1. Size of Ohmic contacts (S1, S2, D1 and D2) on the mesa isapproximately 20 × 20 µm2. The contacts S1 and D2 are directly connected to the central mesa region. The deviceis symmetric about the bottom boundary of Fig. 1 and only the top half of the device is shown for simplicity. Thetopologically equivalent circuit diagram is shown in Fig. 2. The mesa width of the device is nearly 70 µm. The pinklines show the current paths in the device at filling fraction ν = 1 in the quantum Hall (QH) regime. Ineffectivecurrent paths are not shown. From source S1, a fraction of current reaches contact S2 flowing underneath the gateg1 and rest of the current travel along the g1 gate boundary of length lg = 25 µm (Fig. 1). Similarly a part of thecurrent also travel along the g2 gate boundary of length lg = 25 µm to reach D2 contact and rest of the current reachD1 contact passing underneath the gate g2. The gate g1 controls the amount of current from S2 reaching the centralmesa region passing through the top edge underneath the gate g1, as well as the amount of current reaching S2 fromS1 passing along the bottom edge underneath the gate.

To set the central mesa region at filling ν = 1, we measure the two-terminal magneto-resistance (2TMR) betweenS1 and D2 [1]. The output current at the contact D2 is measured with a voltage excitation at the S1 contact, keepingall other contacts S2 and D1 disconnected. Therefore, no current passes through the contacts S2 and D1. As a result,total current in the central meas region is unaffected by gate voltage V g1 and V g2. In particular, at filling ν = 1of the central mesa region 2TMR must corresponds to e2/h conductance irrespective of gate voltage V g1 and V g2.During 2TMR measurement V g1 and V g2 is set to zero Volt. Inset-a of Fig. 1 of the main paper shows the 2TMRtrace as a function of the applied magnetic field. This allows us to know the range of the magnetic fields at which aν = 1 plateau is formed in the central mesa region. We have not done any correction of contact/wire resistance in the2TMR and conductance measurements because wire/contact resistance is much lower than the QH resistance 25.8kOhm. Therefore in our sample, contact resistances are low enough that quantized conductance values are obtainedwithout much error.

In Fig. 2, the measurement circuit diagram is shown with a topologically equivalent device structure under quantizedcondition. At S2 contact a customized SR555 current-to-voltage pre-amplifier is connected to facilitate simultaneousmeasurement of output current from S1 to S2 and putting voltage excitation on the contact S2. Source contactsS1 and S2 are excited with 25.8 µV at frequency 17 Hz and 26 Hz respectively. Output currents from D1 and D2at frequency 17 and 26 Hz are measured by four lock-in amplifiers using suitable current to voltage pre-amplifiers.Output current at S2 having frequency 17 Hz is measured by a lock-in amplifier. During the experiment, we measureall the above five output currents simultaneously and always monitor the quantization condition of the fractionalincompressible fluid beneath the gate g1 and IQH fluid in the central mesa region.

II. CURRENT DISTRIBUTION UNDERNEATH THE GATE g1

- 0 . 3 0 - 0 . 1 5

2 / 3

G 2/3,ν

2 (e2 /h)

1 / 3

- 0 . 3 0 - 0 . 1 5V g 2 ( V )

S 1 → S 2

S 2 → D 2

B = 1 0 . 6 T

T O T A L

B = 8 . 5 TS 2 → D 1

Figure 3. Two-terminal combined conductance (TTCC) vs V g2 at two different magnetic fields when ν1 = 2/3. The TTCCsmeasured at D1 and D2, in response to a 26 Hz voltage excitation at S2 are shown using the blue and wine curves respectively,similar to Fig. 2a of the main paper. The sum of the two TTCCs, indicated by the green curve is the total conductance of thetop edge passing underneath the gate g1. Constancy of the top edge conductance (green line) indicates conservation of currentwith V g2 sweep. Dashed line corresponds to the two-terminal transmitted conductance between S1 and S2, when S1 is excitedby 17 Hz voltage signal.

3

- 0 . 3 0 - 0 . 1 5

B = 1 0 . 6 T

2 / 9

1 / 3

V g 2 ( V )

G 1/3,ν

2 (e2 /h)

1 / 9

B = 8 . 5 T

- 0 . 3 0 - 0 . 1 5

T O T A L

S 1 → S 2

S 2 → D 2

S 2 → D 1

Figure 4. Two-terminal combined conductance (TTCC) vs V g2 similar to Fig. 3, at two different magnetic fields when ν1 = 1/3.The sum of the blue and wine curves is green line, which corresponds to the total conductance of the top edge underneath thegate g1. Dashed line corresponds to the conductance of the bottom edge underneath the gate g1.

At S2 contact a customized SR555 current to voltage pre-amplifier enables us to measure output current comingfrom S1 and passing through the bottom edge underneath the gate g1 when filling fraction ν1 is set at 1/3 or 2/3.Before taking data in Fig. 3 of the main paper, we always set the filling fraction ν1 = 1/3 or 2/3 by measuring thisconductance between S1 and S2, and accordingly V g1 is adjusted (inset b of Fig. 1 of the main paper). During dataacquisition of Figs. 3(a-h) of the main paper, we also monitor transmitted conductance S1 → S2, which is plotted(black dashed line) in Fig. 3 for the case of ν1 = 2/3 and in Fig. 4 for the case of ν1 = 1/3. The plots reveal thatthe conductance of the bottom and top edge modes underneath the gate g1 is same for any V g2, any filling fractionν1 = 1/3 or 2/3 and any magnetic field within the ν = 1 QH plateau. Quantization of these conductances to 2/3 and1/3 indicates a uniform FQH state formation under the gate g1, as any non-uniformity will lead to back scatteringbetween the top and bottom edge modes beneath the gate, resulting in a deviation from the quantization.

III. EQUILIBRATION ALONG THE EDGE

The experiment is conducted by selectively exciting a set of fractional modes and detecting same or another setof modes with electrostatic gating technique as described in the main paper. These populated modes co-propagatewith other unpopulated modes along the top edge of the central region during which the different modes equilibratetransferring charges into the unpopulated modes. After equilibration between the modes the current in a subset ofthe modes are then measured at D1 and D2 contacts. The resulting two-terminal combined conductances (TTCCs)depend on the the extend of equilibration between the modes. Since equilibration happens along the top edge of thecentral mesa region, it forms the key physical length scale l = 125 µm in the problem. In this section we describe thetwo-terminal combined conductance as a function of the length l.

We will first focus on equilibration processes between the two copropogating modes (labeled A and B) of identicalconductance σ via quasi-particle scattering/transfer. We follow and adapt the stochastic process of equilibrationintroduced in Ref. 2. We define a single parameter namely the equilibration length lor which corresponds to theaverage distance traveled by a quasi-particle between two inter-mode scattering event. The average number of inter-edge mode scattering events per unit length is 1/lor . Such an uncorrelated quasiparticle scattering model is suitableunder small excitation voltage and weak disorder.

We consider the case where a current is injected only into mode A at position l = 0. The current in mode Aat position l along the channel is proportional to the probability P (l) of finding each quasi-particle at a distance linjected into mode A at l = 0. Assuming incoherent scattering at the rate 1/lor per unit length, the probability P (l)evolve with l as follows

P (l + dl) = P (l)

[1− dl

lor

]+ [1− P (l)]

dl

lor, (1)

where the two terms correspond to the charge retained in A and that scattered back from B to A over a short distancedl. The solution, with the initial condition P (0) = 1, is

P (l) =1

2(1 + e−2l/l

or ). (2)

The current σV carried by the mode A at l = 0, is distributed between the modes A and B as P (l)σV and (1−P (l))σVrespectively at position l. In our experiment at bulk filling fraction ν = 2/3, two edge modes have conductance 1/3

4

each in unit of e2/h. We excite one edge mode connected to contact S1 and measure the output current at D1 (seeFig. 4c of the main paper). TTCC between S1 and D1 is linked with the current transfer between the modes andhence the TTCC is given by (Eq. 2 of the main paper)

G1/3,1/3S1→D1 (ν = 2/3) =

1

3(1− P ) =

1

6− 1

6e−2l/l

or . (3)

Based on the experimental results summarized in Figs. 3(a-h) and 4a of the main paper, we infer the existence ofthree fractional edge modes of conductance σ = 1/3 each around the bulk state of filling fraction ν = 1. The resultsalso indicate that the two outer most modes have very short equilibration lengths and completely equilibrate with eachother over the extent of ν = 1 IQH plateau. The inner mode posses finite equilibration length lir and equilibrates withthe two outer modes. We extend the model described above to the present case of three channels and assume thatequilibration of the inner mode with the outer ones happen through uncorrelated quasi-particle scattering. Duringthe measurement of the conductance between S1 and D1 (orange line in Figs. 3(a-d) of the main paper), the currentσV injected into the inner mode, partly scatters into the two outer modes at the top edge and is measured at D1.The probability Q(l) that a quasi-particle injected at l = 0 in inner mode to be found at a distance l in the samemode is governed by the condition

Q(l + dl) = Q(l)

[1− dl

lir

]+ [1−Q(l)]

αdl

lir. (4)

Here 1/lir is the rate of scattering per unit length from the inner mode to the two outer modes and α/lir is rate ofscattering per unit length from one of the outer modes to the inner mode. Solving this with boundary conditionQ(0) = 1 and assuming steady state equilibrium at large l to Q(∞) = 1/3 (current is divided equally in all modes),gives α = 1/2 and

Q(l) =1

3(1 + 2e−3l/2l

ir ). (5)

From above Eq. 5 the TTCC between S1 and D1 is given by (Eq. 1 of the main paper)

G2/3,2/3S1→D1 =

1

3(1−Q) =

2

9− 2

9e−3l/2l

ir . (6)

Moreover, the above stochastic model gives results identical to the model that is studied and verified in Ref. 3 (also seeRef. 4) with inter-channel scattering conductance per unit length. The exponential form describing the equilibrationprocess has been extensively used for inter-mode charge transfer in IQH and FQH systems[2, 5–8] and the form havebeen verified with different sample lengths [2, 7, 8].

IV. REPRESENTATION OF TTCC BY TRANSMISSION PROBABILITIES

In this section, we discuss two-terminal combined conductance (TTCC) in terms of transmission probabilitiesoriginating from equilibration between co-propagating fractional edge modes along the top boundary of the centralmesa region. TTCC between two contacts can be expressed as a combination of transmission probabilities of fractionaledge modes connecting the source and detector contacts. For the representation, the outer, middle and inner modesare labeled 1,2 and 3 respectively. Following Buttiker approach [9, 10] in fractional edge modes, the TTCC can beexpressed as

G ν1,ν2S→D =

1

3

∑

i;j

Tij , (7)

where Tij represents the probability for the transmission of a current from mode i connected to the source into modej connected to the detector. The summation i(j) is over all the modes in the top edge connected to the sourceS(D). Transmission probability (Tij) between the respective modes depend on the charge equilibration over the co-propagation length l = 125 µm and is symmetric (Tij = Tji). At high magnetic field end of the ν = 1 plateau (Fig. 1 ofthe main paper), only outer two modes completely equilibrate, but inner most mode does not equilibrate with the outermodes, hence transmission probabilities are T11 = T22 = T12 = T21 = 1/2, T33 = 1 and T13 = T23 = T31 = T32 = 0.At low magnetic field end of the plateau, all the three modes completely equilibrate with each other; hence all thetransmission probabilities become 1/3. In the intermediate values of magnetic field, the transmission probabilities Tijvaries from one limit to the other.

5

In case of ν1 = ν2 = 2/3 outer two modes connect S2 to D1 and inner mode connects S1 to D2 (Fig. 3i of the mainpaper). The corresponding TTCC can be expressed as:

G2/3,2/3S2→D1 =

1

3

∑

i=1,2;j=1,2

Tij =1

3(T11 + T12 + T21 + T22) and

G2/3,2/3S1→D1 =

1

3

∑

i=3;j=1,2

Tij =1

3(T31 + T32). (8)

Putting the values of Tij , we get the TTCC values as tabulated in table 1 of the main paper. Similarly for all othercombinations (Figs. 3(j-l) of the main paper), TTCC values can be estimated at the low and high magnetic field endof the ν = 1 QH plateau as tabulated in table 1 of the main paper.

Now we discuss the resemblance of the gate characteristics to the total conductance shown in Figs. 3(a-h) of themain paper. In case of ν1 = ν2 = 2/3, total conductance 2/3 of the top edge underneath the gate g1 (Fig. 3 here)can be expressed as

G2/3,2/3S2→D1 +G

2/3,2/3S2→D2 =

2

3. (9)

From the symmetry of Tij , we obtain

G2/3,2/3S2→D2 =

1

3(T13 + T23) = G

2/3,2/3S1→D1 . (10)

The above two equations suggest following TTCC relation:

G2/3,2/3S2→D1 +G

2/3,2/3S1→D1 =

2

3. (11)

The above relation (Eq. 11) indicates that the total conductance (measured by 26 Hz and 17 Hz excitation) shouldreach 2/3 for ν1 = ν2 = 2/3 for any magnetic field within the ν = 1 QH plateau. This resembling of gate characteristicis solely arising from the symmetry of transmission probabilities Tij = Tji and is experimentally verified in Figs. 3(a-d) of the main paper. This verification is the proof that the length l is the key dimension of the experiment, sincetransmission probabilities Tij depends on the co-propagation length l and equilibration lengths of the fractional edge

modes. In this experiment, the quantity G2/3,2/3S1→D1 = (T31 +T32)/3 is measured by varying magnetic field (Figs. 3(a-d)

of the main paper) and magnetic field dependent equilibration length lir of the inner most mode is estimated (Fig. 4aof the main paper).

V. CHARACTERISTICS OF lir

0.0 0.1 0.2 0.3

2

4

6

8

10

12

10.55 T 10.25 T 9.95 T

1/l i r

(m

m-1

)

Temperature (K)0.05 0.10 0.15 0.20 0.25 0.30

9.2

9.6

10.0

10.4

Temperature (K)

Mag

neti

c fi

eld

(T)

0.11

0.13

0.16

0.18

0.20

0.22

a b

Figure 5. (a) Temperature dependence of 1/lir at three different magnetic fields (b) Color plot of two terminal combinedconductance (TTCC) between (S1 → D1) at ν1 = ν2 = 2/3 as a function of magnetic field and temperature. Red shadeindicates value closer to the universal conductance of 2/9.

Temperature dependence of charge equilibration, quantified by the inverse of the equilibration length 1/lir of theinnermost mode is plotted in Fig. 5a as a function of T for three different magnetic field values. The equilibrationlengths are obtained by fitting the TTCC (between S1 and D1 when ν1 = ν2 = 2/3) to Eq. 1 of the main paper.

6

Equilibration rate decreases with lowering temperature and saturates at the lowest temperatures. The nature of thevariation suggests a quantum scattering assisted charge equilibration at low temperatures, and additional thermalscattering assisted equilibration as temperature increases. Increase of magnetic field reduces equilibration as seen inFig. 4a of the main paper.

Color plot in Fig. 5b shows the variation of the TTCC between S1 and D1 as a function of the magnetic field andtemperature for ν1 = ν2 = 2/3. Decreasing equilibration rate with increasing magnetic field and low temperatures isreflected in the deviation (green shade) of the conductance from the universal value 2/9 (red shade).

[1] B. Karmakar, D. Venturelli, L. Chirolli, F. Taddei, V. Giovannetti, R. Fazio, S. Roddaro, G. Biasiol, L. Sorba, V. Pellegrini,and F. Beltram, Phys. Rev. Lett., 107, 236804 (2011).

[2] G. Muller, D. Weiss, A. V. Khaetskii, K. von Klitzing, S. Koch, H. Nickel, W. Schlapp, and R. Losch, Phys. Rev. B, 45,3932 (1992).

[3] C. Lin, R. Eguchi, M. Hashisaka, T. Akiho, K. Muraki, and T. Fujisawa, Phys. Rev. B, 99, 195304 (2019).[4] C. Nosiglia, J. Park, B. Rosenow, and Y. Gefen, Phys. Rev. B, 98, 115408 (2018).[5] A. Wurtz, R. Wildfeuer, A. Lorke, E. V. Deviatov, and V. T. Dolgopolov, Phys. Rev. B, 65, 075303 (2002).[6] K. Ensslin, Superlattices and Microstructures, 33, 425 (2003), ISSN 0749-6036, special issue dedicated to Professor Jorg

Kotthaus on the occasion of his 60th Birthday, 29th May 2004.[7] Y. Takagaki, K. J. Friedland, J. Herfort, H. Kostial, and K. Ploog, Phys. Rev. B, 50, 4456 (1994).[8] N. Paradiso, S. Heun, S. Roddaro, D. Venturelli, F. Taddei, V. Giovannetti, R. Fazio, G. Biasiol, L. Sorba, and F. Beltram,

Phys. Rev. B, 83, 155305 (2011).[9] M. Buttiker, Phys. Rev. B, 38, 9375 (1988).

[10] C. W. J. Beenakker, Phys. Rev. Lett., 64, 216 (1990).