Robust quantum point contact operation of narrow graphene constrictions patterned by AFM cleavage lithography

Detecting conductance quantization in graphene nanostructures turned out more challenging than expected. The observation of well-defined conductance plateaus through graphene nanoconstrictions so far has only been accessible in the highest quality suspended or h-BN encapsulated devices. However, reaching low conductance quanta in zero magnetic field, is a delicate task even with such ultra-high mobility devices. Here, we demonstrate a simple AFM-based nanopatterning technique for defining graphene constrictions with high precision (down to 10 nm width) and reduced edge-roughness (+/- 1 nm). The patterning process is based on the in-plane mechanical cleavage of graphene by the AFM tip, along its high symmetry crystallographic directions. As-defined, narrow graphene constrictions with improved edge quality enable an unprecedentedly robust QPC operation, allowing the observation of conductance quantization even on standard $SiO_2/Si$ substrates, down to low conductance quanta. Conductance plateaus, were observed at $ne^2/h$, evenly spaced by $2e^2/h$ (corresponding to n = 3, 5, 7, 9, 11) in the absence of an external magnetic field, while spaced by $e^2/h$ (n = 1, 2, 3, 4, 5, 6) in 8T magnetic field.

cutting) provides a viable solution to overcome this problem, and to realize confinement. In this case, the main factors defining the electronic transport characteristics are the charge carrier mobility in graphene and the structural quality of the constriction edges. A major bottleneck is the extended charge localization area, implying that the detrimental effects of edge disorder are felt even relatively far from the constriction 17  Although, e-beam lithography is capable of defining constrictions down to a few tens of nanometers width, the size of the investigated constrictions is typically in the hundred nanometers range. This is required to mitigate the effect of edge disorder 20 that precludes obtaining well-defined QPC characteristics in narrower constrictions. However, to reach lower conductance quanta, in such wide constrictions, the Fermi wavelength of graphene has to be of comparable scale to the constriction width. This imposes highly demanding requirements on the graphene quality for enabling QPC operation. Such requirements can only be met by suspending graphene or encapsulating it between hexagonal boron nitride layers. Only in the best ultra-high mobility devices can the signatures of size quantization be observed, in the form of more or less evenly spaced modulations (kinks) superimposed on the linear conductance 21,22,23,24 . However, in the absence of magnetic field, these plateaus only become well-defined further away from the Dirac point, corresponding to high conductance quanta (typically, σ > 10 e 2 /h). Reducing the constriction width is important as it can ensure a more robust QPC operation, allowing the observation of plateaus at lower conductance quanta, and enabling QPC characteristics to persist up to higher temperatures, by increasing the energy separation between transversal modes. However, reducing the constriction width cannot be efficient without improving the edge quality. These are two of the next important steps towards improving the feasibility and reliability of graphene QPCs.
Besides e-beam lithography, scanning probe microscopy-based techniques can also be employed for patterning graphene. For instance, AFM-based local anodic oxidation lithography has been applied on graphene to form nanoribbons, quantum dots or nanorings, by etching, down to 30-nm-wide, insulating trenches 25 , 26 , 27 . The precise direct mechanical cutting of graphene by AFM turned out quite difficult, as graphene tends to tear and fold along various directions during the scratching 28 . An underlying thin layer of polymethyl methacrylate (PMMA) 29 has been employed, to provide stronger adhesion to the substrate, and improve the cutting precision. On conducting substrates scanning tunneling microscope lithography can be employed, for defining graphene nanoribbons with nanometre precision 30,31 . However, the conductive substrates required by STM do not allow the direct integration of the patterned graphene structures into electronic devices.
Here, we have developed an improved AFM lithographic method enabling a much more precise direct mechanical patterning (in-plane cleavage) of graphene sheets. This technique enabled us to define graphene nanoconstrictions down to 10 nm width, with edges of improved structural quality (lower roughness), by cleaving the graphene sheet along its high symmetry crystallographic directions (armchair, zigzag). The as-defined graphene nanoconstrictions allow the observation of conductance quantization down to a few conductance quanta, even on standard SiO2/Si substrates, at temperatures up to 40 K.

Results and discussion
Graphene samples have been prepared by micro-mechanically exfoliation on the most commonly used silicon substrate with 285 nm SiO2 capping layer (see Supplementary Materials S1). For defining high-quality graphene constrictions with high precision, we have developed an improved AFM-based lithographic technique ( fig. 1a). To enable this, we have exploited the capability of AFM to determine the crystallographic orientations of the graphene lattice before patterning 32 . By imaging the surface in contact mode, the atomic potentials modulate the friction forces between tip and sample, revealing the high symmetry lattice directions of the graphene sheet ( fig. 1b, 1c). This enables us to choose the precise crystallographic orientation of the cutting direction, which -as it turns out -is of key importance for defining high quality edges. For cutting the graphene sheet, the AFM tip is lowered and pushed into the sample  Graphene constrictions can be formed by cutting two lines with one of their ends in close proximity, and the other ends reaching the edges of the flake. This, results in a narrow bridge between two large and intact graphene areas. To demonstrate the precision of our nanolithography technique in fig. 2 we show the topography of an approximately 10 nm wide constriction with the cutting lines running along an armchair direction. Note that while the two cutting lines have the same (armchair) orientation relative to the graphene lattice, they are not perfectly aligned. This slight misalignment helps avoiding strong backscattering, as discussed later. The smoothness of the edges is due to the fact, that graphene can be cut (cleaved) more easily along its high symmetry (armchair/zigzag) lattice directions 35,36 (see also Supplementary Materials S4). This is similar to cleaving bulk crystals along their high symmetry planes.
Exploiting the unique ability of AFM to image and cut graphene along these easy cleaving directions, enables us to highly improve the precision of the nanofabrication process and reduce the roughness of edges. While the fabrication of graphene constrictions with ~ 10nm width is still challenging, constrictions of a few tens of nanometers can be reliably defined by our AFM cleavage lithography technique. To quantify the characteristic edge roughness, the phase image obtained in tapping mode proved to be the most suitable, as it reveals the boundaries more clearly than topographic images. The graphene edgeindicated by the black line in fig. 2c is determined based on the deviation from the average value of the phase on the graphene sheet (44.4°). By analyzing the edge line, about +/-1 nm deviations can be measured from the average y position, indicating a ~ 2 nm edge roughness. This roughness is similar to that defined by cryo-etching of encapsulated graphene 23 , but achieved using a far simpler technique. Most importantly, our AFM based nanofabrication technique has the advantage of avoiding energetic beams and aggressive chemical etching that can induce additional disorder extending tens of nanometers inwards from the nominal edges 19 . Moreover, the quality of the edges is further preserved, since after the mechanical cleavage, constriction edges never come into contact with resist material or wet chemistry.
Residues generated by material removal during the AFM cleavage lithography, stick to the AFM tip. At the end position of the cutting, the tip often deposit debris (see Supplementary Materials S2). To make sure that the leftover debris does not short circuit the constriction additional devices have been prepared, cutting across the whole graphene flake by AFM cleavage. The two-point resistance of such devices was in MΩ range, which is about two orders of magnitude higher than corresponding to conductance quanta, measured across constrictions, proving that the conductance of the cut section can be safely neglected.  From the square root fitting of the G (Vg) characteristics, the width of the constriction can also be estimated. Using degeneracy factor of 2, and lever arm of α ≈ 7.2*10 10 cm -2 V -2 , a channel length of W ≈ 60 nm is obtained, which is in reasonably good agreement with the topographic for different devices originate from the differences of the Dirac point position. A substantial p doping of the samples originates from the charged impurities in the SiO2 substrate, as well as the annealing step employed for cleaning the samples from resist residues. Due to the relatively high p doping levels of graphene in our devices, we were not able to measure the electron branch of the characteristics. However, the significant p doping, together with the slight misalignment of constriction ends, helps us approaching the adiabatic limit, and avoiding strong backscattering, due to sharp changes of the constriction widths on the scale of the Fermi wavelength.
We also note that we could not obtain reliable conductance characteristics for the narrowest (~10 nm wide) constrictions fabricated here, as they were broken very fast during the measurements. Nevertheless, constrictions as narrow as ~30 nm could be measured ( fig. 3b) displaying quantized conductance plateaus. Most constrictions also display oscillations superposed on the step-like conductance characteristics. This is commonly observed in QPC devices, and is often attributed to transmission resonances from reflections at the two ends of the constrictions, as well as quantum interference form impurity scattering near the constrictions 8 . We have also investigated the temperature dependence of the QPC characteristics (fig 4). In accordance with expectations, by increasing the temperature, the conductance plateaus are gradually smeared out, while also acquiring a finite slope, until they are no longer resolved. In our case, conductance plateaus could be clearly observed up to 20K, and signatures of conductance quantization could be detected even at 40K. This is in accordance with the estimated energy separation of transversal modes (~5 meV for 60 nm width), expected to persist up to ~3kBT (~5 meV at 20 K) thermal energy. Moreover, the energy separation of the transverse mode in the narrowest constriction fabricated by AFM lithography (order of 100 meV) is large enough for the QPC operation to persist even up to room temperature holding the promise of opening an entirely new regime for quantum point contact devices.
In summary, we have developed a high precision nanolithography technique for the fabrication of robust graphene quantum point contact devices. We showed that defining graphene constrictions through cleaving graphene sheets by AFM along their high symmetry directions, enables us to fabricate graphene nanoconstrictions down to 10 nm width and with reduced edge roughness of +/-1 nm. Narrow graphene constrictions with high quality edges display robust QPC behavior manifesting in well-defined zero-field conductance quantization plateaus down to a few conductance quanta, even on SiO2/Si substrates, and temperatures up to 40K. In zero magnetic field, evenly spaced conductance plateaus could be detected, roughly

Data availability
The authors confirm that the data supporting the findings of this study are available within the article. Related additional data are available on reasonable request from the authors.

Robust quantum point contact operation of narrow graphene constrictions patterned by AFM cleavage lithography
Supplementary Materials S1. AFM characterization of graphene thickness is about two orders of magnitude higher than corresponding to conductance quanta measured across constrictions. Therefore, the conductance of the cut section can be safely neglected. The above indentation experiment is similar to the one presented in fig. 1d and discussed in the main text, but the lines were defined along random (not armchair, nor zigzag) orientations.

S3. Raman spectroscopy
Once graphene is cut (marked by green arrow), it develops rough edges (lower part). Then, at random points, it begins to tear along lines different from the cutting direction (AFM tip movement), in the upper part. The angles are multiples of 30 degrees in the corners, indicating that these easy tearing edges coincide with specific lattice directions (armchair, zigzag). This observation confirms the expectations that graphene tears more easily along special lattice direction (zigzag, armchair) that also implies smoother edge formation during cutting along these directions, similar to cleaving bulk crystals along their high symmetry planes. The observation of the square root dependence is another proof that the electronic transport of the charge carriers can be described within the limits of the Landauer theory in our nanoconstriction devices. According to the Landauer theory the conductance through a W wide ideal nanoscale conductor increases by an additional e 2 /h conductance quantum as WkF reaches a multiple of π (an additional channel opens):

S5. Cleaning of resist residues by sweeping in contact mode AFM
, where θ(… ) is the Heavyside function, kF is the Fermi wave number and the factor 4 is due to the four-fold degeneracy of graphene. On the one hand, kF depends on the n carrier density: = √ . The applied back gate voltage (Vg) tunes the carrier density of the graphene layer as: = ( − ), where α is the so called lever arm, and VD is the charge neutrality point, the gate voltage at the minimal conductance. If we combine these equations: On figure S4a a typical two-probe conductance measurement can be seen through graphene nanoconstriction. A good fit (R-square ~ 0.99) to square root dependence exists all the way down to G = 3 e 2 /h.
The square root dependence of the conductance further confirms our assumption that the dominant contribution to the total resistance appears to be the resistance of the constriction at the scale of our examinations. The contact resistance of our devices was estimated to be below 500 Ω; therefore, it is negligible around the Dirac point where the resistance of the constriction is more than an order of magnitude higher.
From the square root fitting, the width of the constriction can also be estimated. Using degeneracy factor of 2 instead of 4 (as explained in the main text) and lever arm of α ≈ 7.2 × 10 10 cm -2 V -2 , a channel length of W ≈ 60 nm can be calculated which is in relatively good agreement with the topographic AFM measurement (≈ 75 nm) of the constriction. Further  Figure S7. Estimation of the charge carrier mobility of graphene from conductivity measurement on intact region of the same device as displayed in the main text ( fig. 3a).

S7. Charge carrier mobility estimation
From conductance measurements we can estimate the carrier mobility 5 , as the back-gate voltage tunes the carrier density as: The estimated mobility of most of our devices has been around 3000 cm 2 V -1 s -1 , indicating a mean free path of 70 nm at a typical n ≈ 4*10 12 cm -2 . As we measured constrictions of width from 30 to 75 nm, the scale of our constrictions is comparable to the estimated length of the mean free path.