Tunable transmission of quantum Hall edge channels with full degeneracy lifting in split-gated graphene devices

Charge carriers in the quantum Hall regime propagate via one-dimensional conducting channels that form along the edges of a two-dimensional electron gas. Controlling their transmission through a gate-tunable constriction, also called quantum point contact, is fundamental for many coherent transport experiments. However, in graphene, tailoring a constriction with electrostatic gates remains challenging due to the formation of p–n junctions below gate electrodes along which electron and hole edge channels co-propagate and mix, short circuiting the constriction. Here we show that this electron–hole mixing is drastically reduced in high-mobility graphene van der Waals heterostructures thanks to the full degeneracy lifting of the Landau levels, enabling quantum point contact operation with full channel pinch-off. We demonstrate gate-tunable selective transmission of integer and fractional quantum Hall edge channels through the quantum point contact. This gate control of edge channels opens the door to quantum Hall interferometry and electron quantum optics experiments in the integer and fractional quantum Hall regimes of graphene.


Supplementary Note 2: Hole doping
In the main text, we focused our analysis of the data on the electron side because we obtained more resistive contacts on the hole side, leading to not well defined Landau level fan and Hall quantization for hole doping (see Supplementary Figure 2). Such an electron-hole asymmetry usually stems from a charge transfer from the metallic contact to the graphene. In our devices, contacts induce a graphene electron-doped region in their vicinity, thus forming a p-n junction when the bulk is hole-doped. In the quantum Hall regime, in line with the mitigated equili-bration observed in the study of the QPC, the contact transmission is reduced due to these p-n junctions, resulting in a bad quantization and a noisy longitudinal resistance oscillations (See Supplementary Figure 2).

Supplementary Note 3: Negative non-local resistance
Non-local resistance R B = R 24,16 (also referred to as bend resistance), is measured by applying current between contacts 2 and 4 while voltage is measured between contacts 1 and 6 ( Fig. 1a).
This resistance as a function of backgate voltage is presented in Supplementary Figure 1b where W is the width and L the length of the device. The 7/3 state is not clearly visible in this line-cut but is revealed by pinching off the QPC (see main text). Note that the 5/3 state is not resolved in this set of data, as reported in previous works [5,6]. Supplementary Note 5: Equilibration for the QPC geometry in graphene As described in the main text, the expression for diagonal conductance of a pnp-junction with two filling factors [7,8,9,10] is no longer applicable to the QPC geometry which involves the three filling factors ν b , ν g and ν QP C . We derive in this section the two-terminal conductance for the regions I and III for such a QPC geometry -the two-terminal conductance is equal to the four-terminal diagonal conductance G D measured in our experiment-considering spin selective equilibration [11] between electron bulk edge channels and hole edge channels localized underneath the split-gates. Conductance in region II is described by Supplementary Equation (1) in the main text.

Region I -Conductance in the bipolar regime
Equilibration in region I involves two cases that depend on the sign of ν QP C . When ν QP C < 0 localized hole states underneath the split-gates extend across the QPC, whereas for ν QP C ≥ 0 electron edge channels from the bulk pass through the QPC and the hole states remain localized beneath each split-gate. We derive below the diagonal conductance for both cases.
Configuration ν QP C ≥ 0 In this configuration electron edge channels are transmitted through the QPC while hole states are localized underneath the split-gates. Supplementary Figure 5 sketches the edge channels of the bulk and those induced by the split-gate for the QPC geometry. The incoming current I in splits into two branches in the vicinity of the QPC. One part of the current is transmitted through the constriction of filling factor ν QP C ≥ 0 and the rest is backscattered along the bulk/split-gate interface.
We consider four segments of pn interfaces where electron bulk edge channels equilibriate with hole-doped edge channels. There are highlighted with grey areas in Supplementary Figure   5. Assuming that the incoming current impinging each pn interface is equally distributed among the co-propagating electron and hole channels, we can write: with r = |νg| |ν b |+|νg| and r = |ν QP C |+|νg| |ν b |+|νg| . Current conservation at the eight nodes leads to: in which we assume that no current is injected from the right lead. The outgoing current I out as a function of I in reads: The incoming current I in is given by the number of incoming edge channels ν b and the chemical potential difference µ 1 − µ 2 between the two leads: which depends solely on the three filling factors ν b , ν g and ν QP C .
Configuration ν QP C < 0 In the second configuration when ν QP C < 0, the hole states from underneath the split-gates extend across the QPC (see Supplementary Figure 6). Both hole states propagating across the QPC and those localized underneath the split-gates can equilibrate with the back-reflected electron edge channels of the bulk. As for the previous configuration equilibration occurs at the same four pn-interfaces shown in grey in Supplementary Figure 6. Similar calculations lead to : Supplementary Equations (11) and (12) are identical apart from three sign changes given by the sign change of ν QP C . They can thus be summarized into a single equation describing both configurations: In order to include spin selection, we extend Supplementary Equation (13) by summing over the spin: In (15), ν σ b , ν σ g and ν σ QP C count the number of edge channels of identical spins involved in the equilibration.
Region III -Spin-selective equilibration for V sg > 0 For V sg > 0, the charge carrier density is larger underneath the split-gates than in the bulk (ν g > ν b ). Thus all electron bulk edge channels pass underneath the split-gates along the edges of the graphene flake. The additional ν g − ν b electron edge channels are either localized underneath the split-gates or extend across the QPC (see configuration in Fig. 2i). Equilibration only has an impact on the conductance if the edge channels underneath the split-gates extend across the QPC, connecting the counter-propagating bulk edge channels and introducing backscattering.
The number of such localized edge channels connecting the counter-propagating bulk edge channels is given by ν QP C − ν b . As a result, the device becomes equivalent to an nn'n-junction.
The two-terminal conductance for spin-selective equilibration thus reads [11]: The ν b = 1 curve in Supplementary Figure 7a has a large plateau of G D = e 2 h at ν b = ν QP C = 1 ,which corresponds to the configuration where the bulk edge channel passes underneath the split-gates. The conductance remains quantized at e 2 h even for ν QP C = 2 when the second edge channel extends across the QPC (see Supplementary Figure 7c). The fact that the conductance does not decrease indicates that no equilibration occurs between the states ν b = 1 and ν QP C = 2 due to their opposite spin polarization [11] (see Landau level diagram in Supplementary Figure   7b).
The conductance of the ν b = 1 curve then decreases to 0.7 e 2 h at ν QP C = 3 indicating equilibration between the ν b = 1 state and the ν QP C = 3 state, which share the same spinup polarization (see Supplementary Figure 7d). Applying Supplementary Equation (16) with h in good agreement with the measured plateau. Note that here Equilibration with higher Landau levels The expected values of R L : computed with the above analysis of G D in the three different regions are indicated by horizontal lines in Supplementary Figure 8. The good agreement with the measured plateaux confirmed the consistency of our measurements.

Supplementary Note 7: Second QPC device
The conductance of a second QPC device is shown in Supplementary Figure 9. We find quan-