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.

Quantum point contacts (QPCs) 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 nanometers, the observation of quantum confinement of Dirac electrons in nanostructured graphene systems has proven surprisingly challenging.
Here we show ballistic transport and quantized conductance of size-confined Dirac fermions in lithographically-defined graphene constrictions.At high charge 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 (v F ≈ 1.5 × 10 6 m/s) in our graphene 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 observation of unique transport phenomena in graphene, such as Klein tunneling 1 , evanescent wave transport 2 , or the half-integer 3,4 and fractional 5,6 quantum Hall effect are directly related to the material quality as well as the relativistic dispersion of the charge carriers.As the quality of bulk graphene has been impressively improved in the last years 7,8 , the understanding of the role and limitations of edges on transport properties of graphene is becoming increasingly important.This is particularly true for nanoscale graphene systems where edges can dominate device properties.Indeed, the rough edges of graphene nanodevices are most probably responsible for the difficulties in observing clear confinement induced quantization effects, such as quantized conductance 9 and shell filling 10 .So far signatures of quantized conductance have only been observed in suspended graphene, however with limited control and information on geometry and constriction width 11 .More generally, with further progress in fabrication technology, graphene nanoribbons and constrictions are expected to evolve from a disorder dominated [12][13][14][15] transport behavior to a quasi-ballistic regime where boundary effects, crystal alignment, and edge defects 16,17 govern the transport characteristics.This will open the door to investigate interesting phenomena arising from edge states, including magnetic order at zig-zag edges 18 , an unusual Josephson effect, unconventional edge states 20 , magnetic edge-state excitons 21 or topologically protected quantum spin Hall states 22 .
In this work we report on the observation of quantum confinement and edge states in ballistic transport through graphene constrictions approximating quantum point contacts.We prepared 4-probe devices based on high-mobility graphene-hexagonal boron nitride (hBN) sandwiches on SiO 2 /Si substrates and use reactive ion etching to pattern narrow constrictions (see Methods) with widths ranging from W ≈ 230 to 850 nm, connecting wide leads (Figs.1a-1c).The graphene leads are side-contacted 8 by chrome/gold electrodes.A back gate voltage is applied on the highly doped Si substrate to tune the carrier density in the graphene layer, n = α(V g − V 0 g ) = α∆V g , where α is the so-called lever arm and V 0 g is the gate voltage of the minimum conductance, i.e. the charge neutrality point.To demonstrate the high electronic quality of our graphene-hBN sandwich structures we show the gate characteristic of a reference Hall bar device (Fig. 1d).From this data we extract a carrier mobility in the range of around 150.000 cm 2 /Vs (see Supplementary Note 1), resulting in a mean free path exceeding 1 µm at around ∆V g = 4.6 V. Thus, the mean free path is expected to clearly exceed all relevant length scales in our constriction devices giving rise to ballistic transport.

Ballistic transport.
We measure the conductance as function of gate voltage for a number of constrictions with different widths W (Fig. 1d; see labels in Fig. 1e).The observed square root dependence G ∝ ∆V g ∝ √ n (see dashed lines in Fig. 1d) is a first indication of highly ballistic transport in our devices.Indeed, according to the Landauer theory for ballistic transport, the conductance through a perfect constriction increases by an additional conductance quantum e 2 /h whenever W k F reaches a multiple of π, where k F = √ πn is the Fermi wave number, the factor four accounts for the valley and spin degeneracies, θ is the step function, and we have neglected minor phase contributions due to details of the graphene edge 23 for simplicity.Fourier expansion of Eq. (1) yields (2) For an ideal constriction c 0 = 1, φ j = 0, and c j = 1/(jπ), j > 0. In the presence of edge roughness, c 0 is reduced to a value below 1 due to limited average transmission, and higher Fourier components c j are expected to decay in magnitude and acquire random scattering phases φ j = 0. Consequently, the sharp quantization steps turn into periodic modulations as will be shown below.Averaged over these modulations only the zeroth order term in the expansion [Eq.(2)] survives.This mean conductance G (0) of a constriction of width W thus features a linear dependence on k F , or, equivalently, a square-root dependence as a function of back-gate voltage assuming an energy-independent transmission c 0 of all modes, in accord with Fig. 1d.By measuring the carrier density dependent quantum Hall effect at high magnetic fields 24,25 , we can independently determine the gate coupling α for each device (see Supplementary Note 2).We can thus unfold the dependence on V g and study both the electron and hole conductance as function of k F (Fig. 1e).From the linear slopes of G(k F ), the product c 0 W can be extracted for each device and compared to its width W (Fig. 1f) determined from scanning electron microscopy (SEM) images (see, e.g., Fig. 1b).The estimates for c 0 W extracted from G (0) lie just little below the width W , where c 0 decreases for decreasing width.This suggests that for the narrower devices reflections, most likely due to device geometry and edge roughness, are playing a more important role.From the data in Fig. 1f we can extract c 0 ≈ 0.56 for our smallest constriction.Below we will show that, indeed, reflections at the rough edges of the constriction and not a reduction in active channel width is responsible for the deviation of the experimentally extracted c 0 W from the SEM width W .
Localized states at the edges.
For small k F < 50 × 10 6 m −1 (i.e.low carrier concentrations) the measured conductances systematically deviate from the expected linear behavior (see Fig. 1e).
This deviation from the square-root relation between G and n (i.e.∆V g ) becomes more apparent when focusing on G around the charge neutrality point (CNP).The conductance as function of n for two different cool-downs of the same graphene constriction (W ≈ 230 nm, Fig. 2a), shows marked cool-down dependent low carrier density regions with substantial deviations from G ∝ √ n.Far away from the CNP, the conductance as function of n for both cool-downs shows (i) an identical √ n behavior leading to the very same c 0 W and (ii) almost identical, regularly spaced kink structures (see arrows in Fig. 2a), which are, however, slightly shifted relative to another on the carrier density axis n.These observations suggest that the square-root relation between the Fermi wave vector k F and the gate voltage V g , i.e. n needs to be modified.While the quantum capacitance of ideal graphene can be neglected [26][27][28] , a small additional contribution n T (∆V g ) from, e.g., localized trap states modifies the relation between n and k F to The size of the first peak is substantially reduced for both experiment and theory due to the presence of localized states that lead to additional scattering.f, Comparison of width WF extracted from the Fourier transform of the conductance traces (as in panels e, f) to geometric constriction width W from four different devices (extracted from SEM images).
Far away from the Dirac point (k 2 F πn T ), we recover the expected square root relation.Close to the Dirac point, however, α∆V g will be strongly modified by deviations n T from the linear density of states of ideal Dirac fermions and approaches n T (∆V g ) near the CNP.
The trap states do not contribute to transport, yet they contribute to the charging characteristics 30 .It is important to note that electron-hole puddles 29 or charged impurities would only smear out the density of states but would not add additional trap-state density n T .This is in contrast to graphene edges, in particular rough graphene edges, which feature a significant number of trap states.For example, a tight-binding simulation of the local density of states of the experimental geometry yields a strong clustering of localized states at the device edges (see Fig. 2c), which energetically lie close to the CNP (Fig. 2e).The deviation of G from the √ n scaling also opens up the opportunity to extract n T from experimental conductance data (e.g.Fig. 2d), and thus a new pathway for device characterization.Inspired by the tight-binding simulation, we approximate the trap state density n T as function of Fermi wave vector by a Gaussian distribution.We fit the position, height and width of the Gaussian by minimizing the difference between the measured G(k F ) and the corresponding linear extrapolation to very low values of k F (see Fig. 2b and Supplementary Note 3).We find good qualitative agreement between simulation and experiment (compare Figs. 2d and 2e).Quantitative correspondence would require a detailed, microscopic model for the trap state density n T .Note that the only difference between different traces in Figs.2a, 2b and 2d is the exposition of The dense regions correspond to plateaus in conductance.b, Transconductance ∂g/∂Vg in units of e 2 /hV (see color-scale) as a function of bias and back gate voltage for a different cooldown of the same device (see also Supplementary Note 6).At V b = 0, the transitions between conductance plateaus appear as red spots.At finite bias voltage, we observe a diamond like shape, which provides an energy scale for the subband energy spacing ∆E ≈ 13.5 ± 2 meV (see dashed black lines and white arrow), which is also in good agreement with the energy scale observed in panel a (see also Supplementary Note 6).c, Conductance traces as a function of temperature and back gate voltage.We observe features with different temperature dependencies.Above around 10 K only kinks related to quantized conductance survive (see arrows).
the device to air for several days leading to a wider carrier density region of substantial deviations (green trace).The number of trap states (i.e., the deviations around the CNP) is significantly enhanced (compare also green and black trace in Fig. 2d).As the active graphene layer is completely sandwiched in hBN only the graphene edges are exposed to air and, very likely, experience chemical modifications.In line with our numerical results, we thus conjecture that localized states at the edges substantially contribute to n T , leading to the strong cool-down dependence we observe in our measurements.While this interpretation seems plausible and is consistent with our data, alternative explanations cannot be ruled out.
Away from the CNP our data agrees remarkably well with ballistic transport simulations through the device geometry using a modular Green's function approach 8 (see blue trace in Fig. 2b): we simulate the 4-probe constriction geometry taken from a SEM image, scaled down by a factor of four to obtain a numerically feasible problem size 7 .To account for the etched edges in the devices, we include an edge roughness amplitude of ∆W = 0.2W for the constriction.This comparatively large edge roughness (which is consistent with the systematic reduction of transmission through the constriction when using the average conductance) is probably due to microcracks at the edges of the device.

Quantized conductance.
Superimposed on the overall linear behavior of G(k F ), we find reproducible modulations ("kinks") in the conductance (see Figs. 3a-3c and Fig. S4b).The kinks are well reproduced for several cool downs (see arrows in Fig. 2a and Supplementary Note 4) as well as for different devices, generally showing a spacing ∆G varying in the range of (2 − 4)e 2 /h (see arrows in Figs.3b and  3c).The "step height" and its sharpness depend on the carrier density (i.e.k F ) as well as on the constriction width and is strongly influenced by the overall transmission c 0 (Fig. 1f).Remarkably, we observe a spacing ∆G of the steps close to 4e 2 /h for one of our wide samples (W ≈ 310 nm) at elevated conductance values on both the electron and hole sides (see arrows and horizontal lines in Fig. 3c and Fig. S4b) Our assignment of the conductance "kinks" as signatures of quantized flow through the constriction is supported by our theoretical results.Theory and experimental data from the smallest constriction show similar smoothed, irregular modulations (see Fig. 3a), instead of sharp size quantization steps. 33The replacement of sharp quantization steps by kinks reflects the strong scattering at the rough edges of the device 34,35 , resulting in the accumulation of random phases in the Fourier components of G [Eq. ( 2)].We note that calculations with smaller edge disorder show a larger average conductance, yet very similar "kink" structures.As the present calculation includes only edge-disorder induced scattering while neglecting other scattering channels such as electronelectron or electron-phonon scattering, the good agreement with the data suggests edge scattering to be the dominant contribution to the formation of the "kinks".By contrast, both experimental and theoretical investigations of, e.g., semiconducting GaAs heterostructures show very clear, pronounced quantization plateaus 36 .In these heterostructures, the electron wave length near the Γ point is very long, and cannot resolve edge disorder on the nanometer scale.By contrast, K-K scattering in graphene allows conduction electrons to probe disorder on a much shorter length scale.Consequently, edge roughness substantially impacts transport.The comparison between experimental and theoretical data (Fig. 3a) unambiguously establishes the observed modulations to be consistent with the smoothed size quantization effects predicted by theory.By subtracting the zeroth-order Fourier component ∝ k F (or √ n), the superimposed modulations of the conductance δG(k F ) = G − G (0) provide direct information on the quantized conductance through the constriction [Eq.( 2)].One key observation is that the Fourier transform of δG(k F ) offers an alternative route towards the determination of the constriction width complementary to that from the mean conductance G (0) .For example, the pronounced peak of the first harmonic at 230 nm (red arrows in Figs.3d and 3e) is consistent with the constriction width W derived from the SEM image.Interestingly, our simulation also correctly reproduces the experimental observation that the peak in the Fourier spectrum of δG(k F ) is more pronounced on the electron side (Fig. 3d) than on the hole side.This results from the slightly asymmetric energy distribution of the trap states relative to the CNP, which is accounted for in our tight-binding calculation.
Performing such a Fourier analysis for several devices (Supplementary Note 5) yields much closer agreement with the geometric width W (Fig. 3f and horizontal axis of Fig. 1f) than an estimate based only on the zerothorder Fourier component c 0 W [first term in Eq. ( 2), see vertical axis of Fig. 1f].Fourier spectroscopy of conductance modulations thus allows to disentangle reduced transmission due to scattering at the edges (c 0 W ) from the effective width of the constriction, and proves the relation between the observed Fourier periodicity and the device geometry.
Bias voltage spectroscopy measurements yield an estimate for the energy scale of the size quantization steps 11,37 .For example, by analyzing finite bias measurements from our smallest constriction device we extract a subband energy spacing of ∆E = 13.5 ± 2 meV near the CNP (Figs. 4a, 4b and Supplementary Note 6).With the geometric width of 230 nm also confirmed by the Fourier spectroscopy (Fig. 3c) we can estimate the Fermi velocity near the CNP as v F = 2W ∆E/h = (1.5 ± 0.2) × 10 6 m/s.This is a clear signature of a substantially renormalized Fermi velocity in nanostructured graphene, possibly enhanced by electron-electron interaction 38 .Moreover, the extracted energy scales are consistent with the weak temperature dependence of the quantized conductance (Fig. 4c and Supplementary Note 7).

Transition from quantized conductance to quantum
Hall.
Additional clear fingerprints of size quantization appear in the parametric evolution of the conductance steps 39 with magnetic field, B. The transition from size quantization at zero B-field to Landau quantization at high magnetic fields occurs when the cyclotron radius l C becomes smaller than half the constriction width W .For the Landau level m the transition should occur at 2 l C = 2 √ 2m l B ≈ W with l B the magnetic length.This transition line in the B − n plane (see black dashed curve in Fig. 5a) agrees well with the onset of Landau level formation in our data (see Supplementary Note 8 for similar data from a 280 nm constriction device).The evolution of the lowest quantized steps (at B = 0 T) to the corresponding lowest Landau levels at low temperatures (T=1.7 K) can be easily tracked (see Figs. 5b  and 5c).At higher temperatures (T = 6 K) the evolution of quantized sub-bands to Landau levels is observed even for higher conductance plateaus (Fig. 5d, 5e).For a comparison, we calculate the evolution of size quantization of an infinitely long ribbon of width W as function of magnetic field.We take W ≈ 230 nm from the SEM data, which leaves no adjustable parameters.Our model ( black lines in Figs.5e and 5f) reproduces the evolution from the kinks at small fields (l B W ) to the Landau levels for large fields (l B < W ) remarkably well, further supporting the notion that they are, indeed, signature of size quantization.

DISCUSSION
We have shown ballistic conductance of confined Dirac fermions in high-mobility graphene nanoconstrictions sandwiched by hexagonal boron nitride.Away from the Dirac point, we observe a linear increase in conductance as function of Fermi wavevector with a slope proportional to constriction width.Close to the Dirac point, the charging of localized edge states distorts this linear relation.Superimposed on the linear conductance, we observe reproducible, evenly spaced modulations ("kinks").Tight-binding simulations for the device reproduce these structures related to size quantization at the constriction.We can unambiguously identify these "kinks" as size quantization signatures by both Fourier spectroscopy at zero magnetic field and their evolution with magnetic field, finding good agreement between theory and experiment.

Experimental methods and details
The hBN-graphene-hBN sandwich structures 8 have been etched by reactive ion etching in a SF 6 atmosphere, prior deposition of a ∼ 10 nm-thick Cr etching mask.Remaining rests of Cr oxide are removed by immersing the samples in a Tetramethylammonium hydroxide (TMAH) solution for about 30-35 s.All transport measurements are performed in a 4-probe configuration using standard lock-in techniques.Since the distances between the contacted current-carrying electrodes and the voltage probes are small, compared to the other length scales of the system, we have an effective 2-probe configuration.Importantly, this way we exclude the onedimensional contact resistances.

Electrostatic simulations and transport calculations
We simulate the experimental device geometry using a thirdnearest neighbor tight-binding ansatz.We rescale our device by a factor of four compared to experiment, to arrive at a numerically feasible geometry.We determine the Green's function using the modular recursive Green's function method 8,9 .The local density of states and transport properties can then be extracted by suitable projections on the Green's function.For more technical details see Supplementary Note 9.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 ,

Supplementary
A closer look at the traces from two different cool-downs of the narrowest device with W = 230 nm (Figs.S3a  and S3c) reveals a systematical deviation from the expected square-root dependence of G [Eq. (S3)] at low carrier concentrations, i.e for n < 0.45 × 10 12 cm −2 on the electron side and n < 0.75 × 10 12 cm −2 on the hole side (Fig. S3a).This deviation becomes more pronounced closer to the charge neutrality point (see shaded area in Figs.S3a and S3c).In the ballistic region, i.e., far from the charge neutrality point, we can use Eq. ( S3), with α extracted from the Landau level fan, and fit parameters V 0,e g for the electron (e) and V 0,h g for the hole (h) side.As expected, the conductance G evolves linearly as function of k F in the ballistic regime (see red traces in Figs.S3b and S3d), but large deviations between data and model become apparent close to the charge neutrality point.We conclude that a linear model using transport direction, where M = 0, ±1, ±2, . . . is an integer associated with the subband index (both signs emerge due to the presence of two cones), and 0 ≤ |β| < 0.5 is a Maslov index related to the boundary conditions at the edges (for simplicity we use β = 0, i.e. a zigzag ribbon).Within the energy range where the ballistic model (see red trace in Fig. S8) fits the conductance trace, the theoretical position of the subbands (marked by vertical black dashed lines in Fig. S8) for a 230 nm-wide graphene constriction (V M g = πM 2 /αW 2 , M = 1, 2, . ..) are in good agreement with the kinks in the conductance (see Fig. S8a).The agreement between model and data is also visible in the derivative of the conductance ∂G/∂V g (see Fig. S8b).Close to the charge neutrality point though, the kink signatures do not appear to follow the theoretical position of the subbands (vertical black dashed lines in Fig. S8a,b).Upon rescaling k F according to Eq. (S5) (independently determined from the average transmission), the kinks are shifted, in good agreement with the quantization model (see comparison between dashed vertical lines and the position of the kinks in Fig. S8c,d).In summary, we find that the rescaling according to Eq. (S5) will (i) realign similar, reproducible kink-structures of different cool-downs on the k F axis and (ii) shifts the kink positions to fit the simple quantization model of Eq. (S6).

Figure 1 .
Figure1.Width dependent ballistic transport in etched graphene nanoconstrictions encapsulated in hBN.a, Schematic illustration of a hBN-graphene sandwich device with the bottom-and top-layers of hBN appearing in green, the gold contacts in yellow, the SiO2 in dark blue and the Si back gate in purple.b, Scanning electron microscope (SEM) images of four investigated graphene constrictions patterned using reactive ion etching.c, False colored atomic force microscope (AFM) image of a fabricated device.Transport is measured in a four-probe configuration to eliminate any unwanted resistance of the one-dimensional contacts8 .The yellow color denotes the gold contacts, green the top layer of hBN and brown the SiO2 substrate.The white scale bar represents 500 nm.d, Low-bias back-gate characteristics of a Hall bar device (see arrow) and of five constriction devices with different widths ranging from 850 to 230 nm (color code as in panel e).The dashed grey lines are fits to the data.e, Low-bias four-terminal conductance of graphene quantum point contacts as function of kF extracted in the high carrier density limit for seven different samples.The color encodes the different samples with different constriction widths (see labels).Grey lines represent a linear fit at high values of kF , inserted as guide to the eye.Conductance deviates from the expected linear slope for small kF .Electron (hole) transport is plotted as solid (dashed) line.Data are taken at temperatures below 2 K. f, Comparison of c0W from conductance traces (panel e) with the width W (extracted from SEM images).

Figure 2 .
Figure 2. Conductance through graphene quantum point contacts a, Conductance traces of two different cool-downs (black and green curve) of the same constriction (W ≈ 230 nm) as a function of charge carrier density.For the black (green) cool-down, shaded gray (light gray) regions denote deviations from the ideal Landauer model G ∝ √ n shown in red.At higher conductance values we observe well reproduced 'kinks' with spacings on the order of 2e 2 /h (see arrows and horizontal lines).b, Experimental conductance trace as a function of kF after correction for the density of trap states (black and green curves) and theoretical simulations of graphene quantum point contact (blue curve).Theoretical results are rescaled to experimental device size as determined from panel a.Ideal transmission ∝ kF is shown in red as guide to the eye.Curves are offset horizontally for clarity.c, Local density of states of graphene quantum point contact from tight-binding simulations, at three different energies (-100 meV, -30 meV and 250 meV; see also arrows in panel e).d, Graphene density of states extracted from experiment (fit to a Gaussian) and e from simulation.Both experiment and theory find a substantial contribution from trap states around the Dirac point.

Figure 3 .
Figure 3. Size quantization signatures.a, Comparison of the low energy conductance between theory (blue) and experiment (black).b, c, Measured electron (el -black trace) and hole (ho -red trace) conductance including kink or step-like structure (see arrows) as a function of kF for two different constriction geometries (see insets).The hole conductance traces are horizontally offest for clarity.d, Fourier transform of the G − G (0) electron conductance F[δG(kF)] through the 230 nm graphene constriction, for experiment (ex -black trace) and theory (th -blue trace).The first peak of the Fourier transform clearly corresponds to the width W of the quantum point contact (marked by arrows).e, Same as d for the hole conductance.The size of the first peak is substantially reduced for both experiment and theory due to the presence of localized states that lead to additional scattering.f, Comparison of width WF extracted from the Fourier transform of the conductance traces (as in panels e, f) to geometric constriction width W from four different devices (extracted from SEM images).

Figure 4 .
Figure 4.Quantized conductance: finite bias and temperature dependence.a, Zero B field differential conductance g as a function of bias voltage V b , measured at T = 6 K, taken at fixed values of back-gate voltage Vg from −0.5 V to 3.0 V in steps of 30 mV (see lower right label).The dense regions correspond to plateaus in conductance.b, Transconductance ∂g/∂Vg in units of e 2 /hV (see color-scale) as a function of bias and back gate voltage for a different cooldown of the same device (see also Supplementary Note 6).At V b = 0, the transitions between conductance plateaus appear as red spots.At finite bias voltage, we observe a diamond like shape, which provides an energy scale for the subband energy spacing ∆E ≈ 13.5 ± 2 meV (see dashed black lines and white arrow), which is also in good agreement with the energy scale observed in panel a (see also Supplementary Note 6).c, Conductance traces as a function of temperature and back gate voltage.We observe features with different temperature dependencies.Above around 10 K only kinks related to quantized conductance survive (see arrows).

Figure 5 .
Figure 5. Magnetic field dependence of the size quantization.a, Landau level fan of the graphene quantum point contact of width W = 230 nm, measured at T = 1.7 K. Landau levels emerge at high magnetic fields.The magnetic field quantization of Landau level m dominates over size quantization as soon as 2 √ 2m lB (where the magnetic length lB ≈ 25/ B[T ] nm) is smaller than the constriction width (B field values above dashed black line).b,c, Double derivative plots of the regions delimited by thin dashed lines in panel a showing the evolution of the lowest quantization plateaus with magnetic field: we observe the full transition from quantized sub-bands (B = 0 T) to Landau levels at large B field.d, The same magnetic field evolution is visible in the conductance as a function of magnetic field and charge carrier density for a different cool-down of the same device, also measured at 1.7 K.The blue arrows highlight the expected quantum Hall conductance plateaus at 2, 6 and 10 e 2 /h.e, Double derivative plot of the conductance as a function of magnetic field and charge carrier density measured at T = 6 K.The solid black lines denote the theoretical expectations for the evolution of the size quantization with magnetic field.The thick dashed black line corresponds to the boundary of the Landau level regime, also appearing in panel a. f, Zoom-in of panel e for small magnetic fields B ≤ 1 T.
Figure S2.Landau fan and capacitive coupling.(a)-(e) Second derivative of the longitudinal conductance ∂ 2 G/∂Vg∂B as a function of magnetic field B and back-gate voltage Vg for six different devices with different widths.The red lines follow the evolution of the Landau levels.The slopes of the lines are proportional to the capacitive coupling α. (f ) The longitudinal resistivity ρ as a function of B and Vg provide an alternative way to extract α from the position of the Landau levels, marked by white lines.
Figure S7.Width dependence of the kinks in conductance.Four-terminal conductance G as a function of back gate voltage Vg for four different devices of widths 230 nm (a), 250 nm (b), 280 nm (c) and 310 nm (d).The transmission traces are shown in black (red) for electrons (holes) as a function of rescaled kF (see main text).The arrows point to kinks where the conductance jumps by about c0 × 4e 2h , with c0 as measure for the overall transmission of the device, see Eq. (S3); c0 ≈ 0.95 for the 310 nm constriction.The traces are shifted horizontally for clarity.
Supplementary Figure S8.Back-gate characteristics of the energy subbands of the 230 nm-wide graphene constriction.(a) Low-bias four-terminal conductance G as a function of back-gate voltage Vg.The theoretical position of the subbands in the Vg-axis is indicated by vertical dashed lines.Close to the Dirac point (leftmost subpanel) measurements deviate from the ideal Landau model G ∝ Vg shown in red (orange-shaded region).(b) Derivative plot ∂G/∂Vg of the conductance trace shown in panel (a).The correlation between the expected position of the subbands (vertical dashed lines) and measurements holds only at high carrier densities.(c) Same as (a) after rescaling of the charge carrier density (Eq.3).The vertical dashed lines indicating the theoretical position of the subbands matches now the positions of kinks.(d) Derivative plot ∂G/∂Vg of the conductance trace in panel (b).
S10. Bias spectroscopy of the 230 nm-wide graphene constriction.(a) Differential conductance g (upper panel) and differential transconductance ∂g/∂Vg (lower panel) as a function of back gate Vg and bias V b voltages, measured at B = 0 T and T = 6 K.The differential conductance g (top panel) is measured at V b = 0 V in the low carrier density range.The vertical black dashed lines indicate the position of the analyzed subbands.The transconductance ∂g/∂Vg (bottom color-scaled panel), of the data shown in the upper panel, is measured as a function of an applied bias voltage V b .The kinks are characterized by high values (yellow color) of transconductance.The diamond structures are highlighted by dashed gray diamonds.We extract an average subband spacing ∆E ≈ 13.5 ± 2 meV (green line).(b) Same as panel (a) measured at high carrier densities.(c) Same as panel (a) for a second cool-down of the same device.The blue trace represents the differential conductance g measured at V b = 15 mV (see blue arrow in lower colored panel).The horizontal dashed blue lines highlight the levels of conductance of the intermediate kinks, visible (blue conductance trace) for energies above the subband spacing, e.g.E ≈ 15 meV > ∆E (blue arrow in lower colored panel).