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Size quantization of Dirac fermions in graphene constrictions

Quantum point contacts are cornerstones of mesoscopic physics and central building blocks for quantum electronics. Although the Fermi wavelength in high-quality bulk graphene can be tuned up to hundreds of nanometres, the observation of quantum confinement of Dirac electrons in nanostructured graphene has proven surprisingly challenging. Here we show ballistic transport and quantized conductance of size-confined Dirac fermions in lithographically defined graphene constrictions. At high carrier densities, the observed conductance agrees excellently with the Landauer theory of ballistic transport without any adjustable parameter. Experimental data and simulations for the evolution of the conductance with magnetic field unambiguously confirm the identification of size quantization in the constriction. Close to the charge neutrality point, bias voltage spectroscopy reveals a renormalized Fermi velocity of ∼1.5 × 106 m s−1 in our constrictions. Moreover, at low carrier density transport measurements allow probing the density of localized states at edges, thus offering a unique handle on edge physics in graphene devices.

The data was measured at T = 1.7 K.  Figure 2).

Supplementary Note 1. Sample quality
The field-effect carrier mobility in our sandwich devices is on the order of 150.000 cm 2 V −1 s −1 .
This high sample quality is thanks to advances in sample fabrication, in particular the van-der-Waals stacking process: the graphene is fully encapsulated in hBN, resulting in significantly improved sample quality. We extract the mobility from a one µm-wide Hall bar device fabricated in the very same batch as our graphene constrictions (see Supplementary Figure 1). The dark blue trace in Figure 1d of the main manuscript is taken from this Hall bar device. As all traces from the constrictions with different widths (some of them carved out from the same hBN-graphene-hBN sandwich) lie systematically below the Hall bar trace, we exclude bulk scattering as limiting process in our devices. Independently, we have shown recently in a collaboration with A. Morpurgo, F. Guinea and coworkers 1 that in our high-quality devices the carrier mobility is not limited by charge impurity and short-range scattering but rather by nanometer-scale strain variations giving rise to long-range scattering with allowed pseudospin flips. We expect that the same limitations on the mean free path also apply to our graphene constriction devices.

Supplementary Note 2. Extraction of the gate lever arm α
Measurements of Landau levels in graphene as a function of back gate voltage V g and magnetic field B (see Supplementary Figure 2) allow for an independent determination of the gate coupling (or lever arm) α. The Landau level spectrum for massless Dirac fermions in graphene is given by where v F is the Fermi velocity and m is the quantum number of the corresponding Landau level.
Assuming a perfect linear dispersion and a constant capacitive gate coupling leads to the following relation between energy E and back gate voltage where ∆V g = V g − V 0 g , and V 0 g is the gate voltage at the charge neutrality point. As a result, the Landau levels in the B -V g plane form straight lines, i.e. B m = C m ∆V g , where the slope For a known gate coupling α, one can evaluate the measured conductance G(V g ) as a function of k F , using the standard constant capacitive coupling model k F = πα∆V g . Following the Landauer theory of conductance through a constriction of finite width W , the averaged conductance G (0) (V g ) features a square-root dependence on V g , Using Supplementary Equation 4, we obtain an implicit mapping k F (∆V g ), which depends on the functional form of n T (∆V g ) and accounts for the modified density of states in the constriction, We conjecture that the strong cool-down dependence seen in the different traces of Supplementary   Figure 3 are due to modifications in the trap state densities as the sample was exposed to air 2 .
As the graphene layer in our hBN-graphene-hBN sandwich can only interact with air at the edges, edge states presumably strongly contribute to n T . Indeed, tight-binding simulations of the constriction geometry (see main text, Figure 2c,e) yield a clustering of localized edge states close to the Dirac point. Accounting for n T by Supplementary Equation (5) should recover the 16 linear relation between Fermi wave number and conductance. We can thus determine n T from the measured conductance: we assume a Gaussian distribution of the density of trap states, and fit the width, position and height of the Gaussian distribution by minimizing deviations of the rescaled conductance G[(k F (∆V g )] from the linear conductance G (0) (k F ) of Supplementary Equation (3), see green/black traces in Supplementary Figures 3b and 3d. Note that this procedure assumes that any other sources for a deviation from a linear relation between k F and G (due to, e.g., many-body effects) are small compared to the contribution from trap states n T .

Supplementary Note 4. Reproducibility of kink signatures
We find regular kink structures in the conductance trace of our constriction devices (see, e.g., arrows in Supplementary Figure 4b). These kinks are well reproducible for different cool-downs of the same device (see Supplementary Figures 5, 6), and appear in conductance data of several different devices (see Supplementary Figure 7).
Analyzing the position of kinks as a function of back-gate voltage offers an independent check of the trap state density n T . In a first order approximation, the band structure of a graphene constriction of width W can be described as a collection of one-dimensional subbands originating from the quantization of the wave vector perpendicular to the transport direction,  Supplementary Figures 8c and 8d). In summary, we find that the rescaling according to Supplementary Equation (5)  Once the conductance is represented as a function of k F , the Fourier transform of δG(k F ) offers alternative information on the quantized conductance through the constriction. If the regular kinks we identify in our conductance data, indeed, correspond to size quantization signatures, we can extract the constriction width from the first peak of the Fourier transform. Comparison between the first peak in the Fourier transform of the measured conductance G(k F ) − G (0) (k F ) of four constriction devices (see Supplementary Figure 9 and Figure 4 in the main text) to the geometric width W of the constriction, yields good agreement (see also Figure 3f of the main text).

Supplementary Note 6. Bias spectroscopy
Using bias spectroscopy we can extract the energy scale associated with the regular kink pattern.
The differential conductance g = dI/dV ( Figure 4, Supplementary Figures 10 and 11) is measured from an AC excitation voltage V AC = 250 µV PP , using standard Lock-In techniques. We analyze six diamonds associated with kinks at the low-and high-conductance ranges (see Supplementary To check against any spurious contribution from the AC measurement technique, the bias spectroscopy measurements have been repeated in a DC configuration (Supplementary Figure 12). The conductance G = I/V b is obtained from a symmetrically applied source-drain DC bias voltage V b .
Although the resolution of the DC conductance G (Supplementary Figure 12) is not sufficient to extract the subband spacing ∆E, the conductance kinks are still visible at identical values of conductance as in the AC measurements (Supplementary Figure 11).

Supplementary Note 7. Temperature dependence
In Supplementary Figures 13 and 14 we show additional data on the temperature dependence of our transport data highlighting both (i) the high quality of our samples and (ii) the energy scale and stability of the observed kink features.

Supplementary Note 8. Evolution of size quantization with magnetic field
We provide an additional data set for the magnetic-field evolution of the size quantization signatures from the 280 nm-wide graphene constriction in Supplementary Figure 15. We find the same transition from size-quantization signatures, at low magnetic fields, to the Landau level regime, at high magnetic fields, as in the sample discussed in the main text (see Figure 5 of main manuscript).

Supplementary Note 9. Theoretical treatment
We use a third nearest neighbor tight-binding approach to simulate the constriction. We pattern the device edge using the experimental geometry determined from SEM, and a correlated random fluctuation to simulate microscopic roughness. We rescale our device by a factor of four compared to the experiment, to arrive at a numerically feasible system size. Such a rescaling by a factor of four ensures that all relevant length scales of the problem (e.g., device geometry, Fermi wavelength, magnetic length and correlation length of the edge roughness) are still much larger than the discretization length of the numerical graphene lattice, allowing to extrapolate simulation data to the experimental result 7 . We use a correlation length of 5 nm and an average disorder amplitude of 13 nm. We determine the Green's function, G(r, r ), of the device using the modular recursive Green's function method 8,9 . The local density of states, ρ(r, E), is given by To determine the evolution of subbands in a constriction of width W with magnetic field, we calculate the band structure of a perfect zigzag graphene nanoribbon of width W as a function of