Conductance quantization suppression in the quantum Hall regime

Conductance quantization is the quintessential feature of electronic transport in non-interacting mesoscopic systems. This phenomenon is observed in quasi one-dimensional conductors at zero magnetic field B, and the formation of edge states at finite magnetic fields results in wider conductance plateaus within the quantum Hall regime. Electrostatic interactions can change this picture qualitatively. At finite B, screening mechanisms in narrow, gated ballistic conductors are predicted to give rise to an increase in conductance and a suppression of quantization due to the appearance of additional conduction channels. Despite being a universal effect, this regime has proven experimentally elusive because of difficulties in realizing one-dimensional systems with sufficiently hard-walled, disorder-free confinement. Here, we experimentally demonstrate the suppression of conductance quantization within the quantum Hall regime for graphene nanoconstrictions with low edge roughness. Our findings may have profound impact on fundamental studies of quantum transport in finite-size, two-dimensional crystals with low disorder.


Supplementary Note 1. Device quality: Graphene mobility and contact resistance
Graphene strips of widths W = 1 µm (see Supplementary Fig. 1a, inset) were measured to estimate the bulk mobility  (thus, mean free path, mfp l ) of our graphene flakes and the contact resistance c R of the devices prior to the definition of the actual nanoconstrictions. We use a two parameter fitting to extract the average mobility  and c R values 1,2 at a given temperature T : Here, 2pt R is the measured two-point resistance of the device, e is the elementary charge, res n is the residual carrier concentration and n is the back-gate induced carrier concentration given by: Our devices show (Supplementary Fig.1a) a relatively low disorder density (inhomogeneities) at . This FWHM establishes an upper bound of disorder- Edge disorder plays a key role in the transport characteristics of narrow graphene devices [8][9][10][11][12] . In particular, the conductance quantization suppression (CQS) phenomenon reported in this work is highly dependent on the achievement of a low level of edge roughness in experimental devices. CQS appears in our devices etched with Ar/O 2 reactive ion etching, RIE (Sample type 1) an etching procedure with introduces much less edge disorder than O 2 plasma ashing (Sample type 2). 12 We assess the amount of edge roughness in type 1 devices using transmission electron microscopy (TEM . We note that this fact cannot be due to charge impurities in the graphene channel since the electrical bulk-flake behavior does depend on temperature ( Supplementary Fig. 1).  Lastly, we note that the periodic conductance kinks in Sample type 1 ( Supplementary Fig. 5) tend to be smaller (blue arrows) and disappear towards higher gate voltages. This behaviour agrees with theoretical calculations of coherent transport in graphene nanoribbons with moderate edge disorder 8,17 . It is explained by the fact that edge-defect scattering occurs equally in all the subbands, resulting in a stronger suppression of G  at higher gate voltages, where more subbands are available 8 . Although these theories consider non-interacting systems, our calculations generally show a similar behavior for systems with and without edge charge accumulation ( Supplementary Fig. 7). In general, the average occupation increases faster for the non-uniform potential as the gate voltage is increased, but the larger accumulation of charge at the edges also makes these systems more susceptible to edge disorder.

Supplementary Note 4. Geometrical corrections for the quantum Hall conductance in homogeneous two-terminal graphene nanoconstrictions
The presence of a nonzero longitudinal conductivity might cause deviations of the conductance from its corresponding quantized value in two-terminal configuration in spatially uniform and homogeneous 2D conductors, depending on the device geometry. 7,18,19 Here, we exclude this possibility in our nanoconstrictions which exhibit conductance quantization suppression CQS (Sample type 1).
The geometrical correspondence between homogeneous conductors of arbitrary shape and equivalent rectangles 18,19 , allows us to simplify the problem of calculating the two-terminal device conductance G in our full nanoconstriction geometry to that one of a rectangle with an aspect ratio 8 .

/W L
. This is easily done by transforming the Cartesian coordinates ) , ( y x to elliptic coordinates ) , (   as follows: where  is a real number and . Therefore, the geometry of our graphene constrictions ( Supplementary Fig. 8a In consequence, these calculations demonstrate that features introduced by geometrical corrections can explain some of the effects observed in the constrictions of type 2 (rough edges).
However, they are not consistent with the increased conductance with suppressed quantization associated with the CQS phenomenon, observed in our constrictions of type 1 (smooth edges). The CQS effect in our samples emerges due to the appearance of pairs of counter-propagating states at high magnetic field in the presence of a inhomogeneous charge density across our devices. We consider the conductance in the presence of such states 20,23 through a simple

Supplementary Note 5. Further devices: Conductance at B ≠ 0 T in graphene nanoconstrictions with inhomogeneous charge density
Landauer-Büttiker analysis 24 . We consider the two-terminal device shown in Supplementary   Fig. 11. When the central part of the device is around the LL at filling factor R N , one edge of the ribbon supports R N regular edge states propagating in the same direction and the opposite edge supports an equal number of oppositely propagating states. The suppression of backscattering within the QH regime gives these states a transmission probability 1 R  T . A nonuniform electron density introduces additional conducting channels by deformation of the LLs in the previously insulating bulk (Fig. 3d, main text). This generates additional A N propagation channels 21,23,25 . By setting A T as the transmission probability of the additional A N modes, the two-terminal conductance is given as a function of A T : The conductance is quantized in the two extremal cases. For strong backscattering, 0 A  T , and This approximate expression is obtained considering that the number of additional occupied LL at a point in the constriction to be 23 : and taking into account the equality (Eq. 10, main text) is the potential variation with respect to the Fermi level and Supplementary Equation 6 indicates that the increased conductance with suppressed quantization in a given nanostructure occurs from low B until the aforementioned condition is no longer valid. The exact magnetic field at which the quantization is restored will highly depend on the sample quality. Experimentally, we can confirm the existence of the CQS phenomenon at low B . For the quality of our graphene samples, plateaus in the quantum Hall regime begin to be very clear in our large devices (width 1µm) for the LL0 at B = 3T (see Supplementary Fig. 2). At B =3-4 T, the CQS phenomenon in our narrow graphene constrictions with smooth edges is also seen for the LL0 (see Supplementary Fig. 15).

Supplementary Note 7. Conductance evolution from B = 0 T to B ≠ 0 T in narrow graphene devices: literature review
In this section, we discuss and compare magneto-transport results reported in literature in highquality and narrow graphene devices with our results. All other reported experiments show a quantized conductance at high B (see Supplementary Table 1). In general, the acute sensitivity of the conductance quantization suppression (CQS) phenomenon to the device electrostatics and disorder demonstrated in our findings explain the absence of this effect in the literature to date.
Magnetotransport measurements in different narrow, high-quality but non-ballistic graphene devices have been reported by several groups 11,[26][27][28][29][30] . These are high-quality devices in terms of bulk mobility and/or very low edge roughness, and even include pristine (natural) edges in some cases 27 . Importantly, independently of their measuring configuration in 2-or 4-terminal set-ups, none of these devices show a quantized conductance at B = 0T, but their conductance is quantized within the quantum Hall regime. The CQS effect is absent in these devices due to its high sensitivity to the presence of moderate or higher amounts of disorder, either in the bulk or at edges 20,21 .
On the other hand, high-quality and ballistic graphene constrictions have been measured in suspended devices 31   At a more general level, this result emphasizes the critical importance of quantifying the disorder in narrow, high-quality graphene devices where Coulomb interactions may need to be taken into account 23,[25][26][27]32 or not 13,28,30,31 depending on the degree of edge disorder.

Supplementary Note 10. Additional tight-binding simulations of quantization suppression features.
In this section, we address some additional features of the quantization suppression effect which emerges from our simulations, and which help us to interpret the experimental data more clearly. As discussed in the text, the appearance of a CQS peak depends on a non-uniform gating potential -however if the charge neutrality point (CNP) coincides with zero gating, then no peak will appear for the 0 th LL. Nevertheless, a peak feature is clearly visible in the experimental data. In Supplementary Fig. 14a, we address the possibility of zigzag edge states contributing to such a feature. With the 3 rd nearest neighbor tight binding model (NNTB) such states are dispersive, and lead to an electron hole asymmetry for pristine ribbons (grey curve).
However the additional hole-side channels are very quickly suppressed by even smooth edge disorder (red curve), restoring the e-h symmetry seen within the 1 st NNTB model ( Supplementary Fig. 14b).
Given their quick suppression by disorder, and the fact that large sections of zigzag edges are unlikely to occur in our samples, we do not consider such effects to be the cause of the LL0 peak in the experimental data. A more likely cause is an offset of the charge neutrality and zero gating points due to residual doping effects. This results in a non-uniform potential at the CNP, and the appearance of conductance peaks for LL0. This is shown in Supplementary Fig.14c for a simple model, where pristine zigzag nanoribbons are considered within the 1 st NNTB model.
A shift in the Fermi energy is introduced, corresponding to an additional uniform charge being added to the system, so that a finite (and thus non-uniform) gating is required to reach the CNP.
We show the case for increasing values of this shift, with the red curve corresponding to the result in the main manuscript. In each case the gate voltage is corrected, and shown relative to the charge neutrality point. The emergence of a prominent peak is clear as the Fermi energy  Fig. 2a, main text