Abstract
Quantum point contacts are cornerstones of mesoscopic physics and central building blocks for quantum electronics. Although the Fermi wavelength in highquality 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 sizeconfined 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 × 10^{6} 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.
Introduction
The observation of unique transport phenomena in graphene, such as Klein tunnelling^{1}, evanescent wave transport^{2}, or the halfinteger^{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 confinementinduced 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 disorderdominated^{12,13,14,15} transport behaviour to a quasiballistic 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 zigzag edges^{18}, an unusual Josephson effect^{19}, unconventional edge states^{20}, magnetic edgestate excitons^{21} or topologically protected quantum spin Hall states^{22}.
In this work we report on the observation of size quantization and localized trap states in ballistic transport through graphene constrictions approximating quantum point contacts. Away from the Dirac point, the current features evenly spaced, reproducible kinks superposed on a linear background, in agreement with transport simulations. Scattering at the rough constriction edges reduces quantization steps to kinks in both experiment and theory. The kink spacing, and their evolution with magnetic field, allows us to unambiguously identify them as signatures of size quantization. Close to the Dirac point, deviations from ballistic behaviour allow for probing the density of localized trap states.
Results
Ballistic transport
We prepared fourprobe devices based on highmobility 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 (Fig. 1a–c). The graphene leads are sidecontacted^{8} by 80nmthick chrome/gold electrodes. A backgate voltage is applied on the highly doped Si substrate to tune the carrier density in the graphene layer, , where α is the socalled lever arm and is the gate voltage of the minimum conductance, that is, 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 and Supplementary Fig. 1). From this data we extract a carrier mobility in the range of around 150.000 cm^{2} V^{−1} s^{−1} (Supplementary Note 1), resulting in a mean free path exceeding 1 μm at around ΔV_{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.
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 (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 Wk_{F} reaches a multiple of π
where 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 equation (1) yields
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 the higher Fourier components 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 zerothorder term in the expansion (equation (2)) survives. This mean conductance G^{(0)} of a constriction of width W thus features a linear dependenc on k_{F}, or, equivalently, a squareroot dependence as a function of backgate voltage assuming an energyindependent transmission c_{0} of all modes, in accord with Fig. 1d.
By measuring the carrierdensitydependent quantum Hall effect at high magnetic fields^{4,24}, we can independently determine the gate coupling α for each device (Supplementary Fig. 2, Supplementary Table 1 and 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 with its width W (Fig. 1f) determined from scanning electron microscopy (SEM) images (see, for example, Fig. 1b). The estimates for c_{0}W extracted from G^{(0)} lie only slightly 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
For small k_{F}<50 × 10^{6} m^{−1} (that is, low carrier concentrations) the measured conductances systematically deviate from the expected linear behaviour (Fig. 1e). This deviation from the squareroot relation between G and n (that is, ΔV_{g}) becomes more apparent when focusing on G around the charge neutrality point (CNP). The conductance as function of n for two different cooldowns of the same graphene constriction (W≈230 nm, Fig. 2a), shows marked cooldowndependent lowcarrierdensity regions with substantial deviations from . Far away from the CNP, the conductance as function of n for both cooldowns shows (i) an identical behaviour 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 (Supplementary Fig. 8). These observations suggest that the squareroot relation between the Fermi wave vector k_{F} and the gate voltage V_{g}, that is, n needs to be modified. While the quantum capacitance of ideal graphene can be neglected^{25,26,27}, a small additional contribution n_{T}(ΔV_{g}) from, for example, localized trap states modifies the relation between n and k_{F} to
Far away from the Dirac point (), we recover the expected squareroot relation. Close to the Dirac point, however, αΔV_{g} will be strongly modified by deviations n_{T} from the linear density of states of ideal Dirac fermions and approaches n_{T}(ΔV_{g}) near the CNP. The trap states do not contribute to transport, yet they contribute to the charging characteristics^{28}. Such trap states can for instance be found at the rough edges of patterned graphene devices, which feature a significant number of localized states. A tightbinding simulation of the local density of states of the experimental geometry yields a strong clustering of localized states at the device edges (Fig. 2c), which energetically lie close to the CNP (Fig. 2e). The deviation of G from the scaling also opens up the opportunity to extract n_{T} from experimental conductance data (for example, Fig. 2d), and thus a new pathway for device characterization. Inspired by the tightbinding simulation, we approximate the distribution of trap states 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} (Fig. 2b, Supplementary Fig. 3 and Supplementary Note 3). We find good qualitative agreement between simulation and experiment (compare Fig. 2d,e). Quantitative correspondence would require a detailed, microscopic model for the trap state density n_{T}. Note that the only difference between different traces in Fig. 2a,b,d is the exposition of the device to air for several days leading to a wider carrier density region of substantial deviations (green trace). The number of trap states (that is, 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 cooldown dependence we observe in our measurements. While this interpretation seems plausible and is consistent with our data, alternative explanations such as electron–hole puddles^{29} or charged impurities^{13} 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^{30} (see blue trace in Fig. 2b): we simulate the fourprobe constriction geometry taken from a SEM image, scaled down by a factor of four to obtain a numerically feasible problem size^{31}. To account for the etched edges in the devices, we include an edge roughness amplitude of ΔW=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 behaviour of G(k_{F}), we find reproducible modulations (kinks) in the conductance (Fig. 3a–c and Supplementary Fig. 4). The kinks are well reproduced for several cooldowns (see arrows in Fig. 2a, Supplementary Figs 5 and 6 and Supplementary Note 4), as well as for different devices (Supplementary Fig. 7), generally showing a spacing ΔG varying in the range of (2−4)e^{2}/h (see arrows in Fig. 3b,c). The ‘step height’ and its sharpness depend on the carrier density (that is, 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 ΔG 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 Supplementary Fig. 4b)
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 (Fig. 3a), instead of sharp size quantization steps^{32}. The replacement of sharp quantization steps by kinks reflects the strong scattering at the rough edges of the device^{33,34}, resulting in the accumulation of random phases in the Fourier components of G (equation (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 edgedisorderinduced scattering while neglecting other scattering channels such as electron–electron 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, for example, semiconducting GaAs heterostructures show very clear, pronounced quantization plateaus^{35}. In these heterostructures, the electron wavelength near the Γ point is very long, and cannot resolve edge disorder on the nanometre 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 zerothorder Fourier component∝k_{F} (or ), the superimposed modulations of the conductance δG(k_{F})=G−G^{(0)} provide direct information on the quantized conductance through the constriction (equation (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 Fig. 3d,e) is consistent with the constriction width W derived from the SEM image. 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 tightbinding calculation.
Performing such a Fourier analysis for several devices (Supplementary Fig. 9 and 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 equation (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^{11,36}. For example, by analysing finite bias measurements from our smallest constriction device we extract a subband energy spacing of ΔE=13.5±2 meV near the CNP (Fig. 4a,b, Supplementary Figs 10–12 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}=2WΔE/h=(1.5±0.2) × 10^{6} m s^{−1}. This is a clear signature of a substantially renormalized Fermi velocity in nanostructured graphene, possibly enhanced by electron–electron interaction^{37}. Moreover, the extracted energy scales are consistent with the weak temperature dependence of the quantized conductance (Fig. 4c, Supplementary Figs 13 and 14 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^{38} with magnetic field B. The transition from size quantization at zero Bfield 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 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 Fig. 15 and Supplementary Note 8 for similar data from a 280nm 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 (Fig. 5b,c). At higher temperatures (T=6 K) the evolution of quantized subbands to Landau levels is observed even for higher conductance plateaus (Fig. 5d,e). 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 Fig. 5e,f) 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, a signature of size quantization.
Discussion
We have shown ballistic conductance of confined Dirac fermions in highmobility graphene nanoconstrictions sandwiched by hBN. Away from the Dirac point, we observe a linear increase in conductance as function of Fermi wave vector 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). Tightbinding 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.
Methods
Experimental methods and details
The hBN–graphene–hBN sandwich structures^{8} have been etched by reactive ion etching in an SF_{6} atmosphere, prior deposition of a ∼10nmthick Cr etching mask. Residues of Cr oxide are removed by immersing the samples in a tetramethylammonium hydroxide solution for about 30–35 s. All transport measurements are performed in a fourprobe configuration using standard lockin techniques. Since the distances between the contacted currentcarrying electrodes and the voltage probes are small compared with the other length scales of the system, we have an effective twoprobe configuration. Importantly, this way we exclude the onedimensional contact resistances.
Electrostatic simulations and transport calculations
We simulate the experimental device geometry using a thirdnearest neighbour tightbinding ansatz. We rescale our device by a factor of four compared with experiment, to arrive at a numerically feasible geometry. We determine the Green’s function using the modular recursive Green’s function method^{30,39}. 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.
Additional information
How to cite this article: Terrés, B. et al. Size quantization of Dirac fermions in graphene constrictions. Nat. Commun. 7:11528 doi: 10.1038/ncomms11528 (2016).
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Acknowledgements
We acknowledge stimulating discussions with F. Hassler, F. Haupt and B.J. van Wees. Support by the HNF, the DFG (SPP1459), the ERC (GANr. 280140), the EU project Graphene Flagship (Contract No. NECTICT604391) and Spinograph, and the Austrian Science Fund (SFB041 VICOM, SFB049 NextLite and DKW1243 Solids4Fun) is gratefully acknowledged. Calculations were performed on the Vienna Scientific Clusters.
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B.T. and C.S. conceived the project; B.T. and J.P. fabricated the samples, performed the experiments and interpreted the data; S.E. assisted during measurements; B.T., D.J. and J.P. analysed the data; L.A.C. and F.L. performed the numerical calculations and theoretical analysis; A.G. and F.L. developed the numerical code; T.T. and K.W. synthesized the hBN crystals; J.B., S.V.R. and C.S. advised on theory and experiments; B.T., L.A.C., F.L., J.B. and C.S. prepared the manuscript; all authors contributed in discussions and writing of the manuscript.
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Supplementary Figures 115, Supplementary Table 1, Supplementary Notes 19 and Supplementary References (PDF 24534 kb)
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Terrés, B., Chizhova, L., Libisch, F. et al. Size quantization of Dirac fermions in graphene constrictions. Nat Commun 7, 11528 (2016). https://doi.org/10.1038/ncomms11528
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DOI: https://doi.org/10.1038/ncomms11528
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