Dissipationless zero energy epigraphene edge state for nanoelectronics

: The graphene edge state is essential for graphene electronics and fundamental in graphene theory, however it is not observed in deposited graphene. Here we report the discovery of the epigraphene edge state (EGES) in conventionally patterned epigraphene using plasma-based lithography that stabilizes and passivates the edges probably by fusing the graphene edges to the non-polar silicon carbide substrate, as expected. Transport involves a single, essentially dissipationless conductance channel at zero energy up to room temperature. The Fermi level is pinned at zero energy. The EGES does not generate a Hall voltage and the usual quantum Hall effect is observed only after subtraction of the EGES current. EGES transport is highly protected and apparently mediated by an unconventional zero-energy fermion that is half electron and half hole. Interconnected networks involving only the EGES can be patterned, opening the door to a new graphene nanoelectronics paradigm that is relevant for quantum computing.


3
The predicted properties of deposited quasi-freestanding graphene were so spectacular that it was expected to imminently trigger a graphene nanoelectronics revolution 1 . However the edges of lithographically patterned deposited graphene are invariably insulating due to uncontrollable chemical and structural disorder [2][3][4][5][6] . Since the edge state dominates transport in all neutral graphene nanostructures, [7][8][9][10][11][12][13][14][15] few, if any of the predictions could be tested experimentally. More importantly, the inability to stabilize patterned graphene edges 4-6 derailed the ambitious graphene nanoelectronics effort replacing it with much more modest goals. 16,17 Evidence of the protected graphene edge state 7-9,12-14 (not to be confused with quantum Hall edge states 14,[18][19][20][21][22][23][24] ) was first found in 40 nm wide self-assembled graphene ribbons that form 25 on the high temperature annealed sidewalls of steps etched on the polar faces of electronics grade hexagonal SiC. 26,27 Electron microscopy revealed that the sidewall ribbons terminated in the SiC, thereby stabilizing and passivating the edges. [28][29][30] Sidewall ribbons exhibit dissipationless (ballistic) transport over tens of microns, even at room temperature. Surprisingly, the conductance was found to be 1 G0=1 e 2 /h rather than the predicted 2 G0 7,8,[13][14][15] where e is the charge and h is Planck's constant. Unfortunately, the sidewall topology is unsuited for nanoelectronics. It is also incompatible with conventional magneto-transport measurement methods. Hence essentially none of the numerous edge state theories could be tested, which, as we show here, in fact fundamentally diverge from long-standing predictions. 13,14,24,31 We subsequently produced SiC wafers that were cut along crystal faces corresponding to the annealed sidewall facets. The epigraphene is found to be charge neutral as required for nanoelectronics so that the Fermi energy EF=0 (in contrast graphene monolayers grown on the commercial polar SiC faces have a charge density n≈10 13 cm -2 , corresponding to a EF of about 0.3 eV [32][33][34][35][36] ). The conventionally patterned interconnected nanostructures exhibit 1 G0 ballistic conduction at zero magnetic field, up to room temperature, and independent of edge chirality.
The mean free path (mfp) of the EGES is on the order of 10 µm up to room temperature, whereas the mfp of the bulk of the sample is on the order of 10 nm. This demonstrates an unexpected high degree of protection and stands in stark contrast with conventional patterned deposited graphene, where the opposite is true: micron scale bulk mfp's have been observed in boron nitride supported deposited graphene, however the edges are invariably insulating. 5 Strikingly, the quantized epigraphene edge state currents do not generate a Hall voltage, contrary to conventional theory 24 . This causes an anomalous quantum Hall effect, and signals that the EGES transport involves fundamentally different physics than generally assumed for the graphene edge state. 13,14,24 We use the simple tight-binding description of the edge state (Fig.1) as a convenient starting point for discussion, [7][8][9]14,15 which shows that the Fermi level should be pinned at EF=0 at the edge, as experimentally confirmed here. A comprehensive analysis reveals quantized transport involving an unconventional fermion, that is neither an electron nor a hole, since, for a given current, these would produce equal and opposite Hall voltages. Hence, we propose that the novel fermion that mediates this zero-energy mode is most likely half electron and half hole. While this description applies to Majorana fermions 37 alternatives may exist.
The combination of essentially dissipationless single-channel transport at zero energy in conventionally nanopatterned graphene on a commercially available electronic single crystal SiC is a very large step towards realizing the envisioned epigraphene electronics paradigm shift, 38 using coherent tunneling devices and phase coherent interconnected structures that are relevant for quantum computing.

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The neutral epigraphene edge state Figure 1A shows the tight binding (TB) electronic subbands (electronic waveguide modes) in a charge neutral 700 nm wide zigzag graphene ribbon near the K point. 7,8,15,39 This band structure is generic for all graphene ribbons with chiral edges, excluding only perfect armchair ribbons, as predicted and observed. 12,40,41 The n=0 subband, i.e. the graphene edge state (GES), is special. It is composed of a flat band at E=0, that is narrowly confined to the physical edges of the ribbon.
It evolves at the K and K' points into linear dispersing electron and hole bands that are delocalized over the entire ribbon. In neutral graphene, the hole bands are occupied and the electron bands are unoccupied. The predicted flatband is half-filled in neutral graphene and gives rise to a peak in the density of states (the 0-DOS peak, Fig. 1B) at the graphene charge neutrality point (CNP, i.e. the Dirac point at E=0). The 0-DOS peak is a robust feature of "graphene molecules" 42-44 and has been experimentally confirmed in general for chiral graphene ribbons shifting CNP by DE= ℏ * $ " below the Fermi level where c*≈10 6 m/s. (The inverse happens for negative VG). However, theoretically, the 0-DOS peak pins the Fermi level at E=0 by depleting charges near the edge 45 , analogous to a Schottky barrier. 46 The resulting electric fields cause band bending of the edge-localized branch of the n=0 band. The delocalized bands on the other hand essentially rigidly shift down by DE. Transport in the n=0 subband is expected to involve both the edge-localized and 2D branches of the GES, i.e. where the Fermi level intercepts n=0 subband as indicated in Fig. 1C.
The TB approximation is an excellent starting point to describe the basic properties of graphene 7,8,13,24 but electron-electron interactions significantly modify the properties of the 6 flatband, 13,47-50 making the GES a topic of great theoretical interest 31,37,[49][50][51] . While those interactions will typically broaden the 0-DOS peak, pinning (which is an electrostatic effect) is expected to survive, so that the Dirac point properties of the GES are accessible as long as the 0-DOS peak is not saturated.
The 0-DOS saturates when ncda0=1/3 where nc is the gate induced bulk charge denstiy, d is the depletion region, which is a fraction the dielectric thickness, and a0 is the C-C bond length 45 .
Hence saturation is expected for nc>10 13 cm -2 in our geometry, i.e. above our experimental range.
Epigraphene grown on commercial SiC wafers is highly charged, [32][33][34]52 which can saturate the 0-DOS peak, so that the EGES is not seen there. The edges in BN/exfoliated graphene/BN heterostructures are heavily n-doped and distorted 5,6 and the GES is not observed.

Neutral epigraphene characterization
Epigraphene is ideally suited for graphene nanoelectronics. 38,53 The quasi-freestanding graphene that self-assembles on electronics-grade silicon carbide (SiC) wafers 32-36 by thermal annealing 54,55 is of very high quality graphene and crystallographically aligned with the SiC lattice with contamination free interfaces. (for a review, see Ref. 34 ) Experiments on self-assembled 40 nm wide graphene nanoribbons, that spontaneously form on the recrystallized, thermally annealed, sloping sidewalls of trenches etched in commercial SiC wafers, 25 show them to be charge neutral 26,56,57 . At the TICNN institute, we produced wafers cut from commercial electronics grade 4H SiC stock to expose sidewall facets (i.e. 4H SiC (11 ) 0 ), n≈5 28,56 ). Neutral epigraphene (N-EG) is grown on these facets using standard confinementcontrolled sublimation methods 55 , and interrogated using a variety of surface probes (Fig. 2). In its initial stages of growth N-EG shows characteristic trapezoidal islands ( Fig. 2A) that ultimately coalesce into a continuous single film. Scanning tunneling microscopy ( Fig. 2B) reveals the graphene's hexagonal lattice structure. High-resolution angle resolved photoemission spectroscopy (ARPES) shows the iconic graphene Dirac cones (Fig. 2D)  Edge disorder in free-standing graphene is difficult to control, because the acene edge atoms (that in general produce states at E=0 in "graphene molecules") are reactive and prone to spontaneous reconstruction. 42,43 However, epigraphene is well adhered to the SiC, giving it significant mechanical, chemical and thermal stability, 58,59 while exhibiting essentially ideal graphene properties. [32][33][34][35][36] To inhibit edge distortions during fabrication, the N-EG is first coated with a 30 nm alumina film, securely embedding the graphene between alumina and silicon carbide that are both refractory materials. The sandwich is then etched through a patterned mask, 16 nm into the SiC (Fig. 3) using the inductive coupled plasma etching (ICP) technique. 60 Hence, like a nanoscale plasma welding torch, the high temperature plasma (ion temperatures can exceed 5000K) cuts through the alumina, graphene and silicon carbide. In the process C-C and Si-C bonds are formed, 61 which fuse the ribbon edges to the SiC, thereby producing stable neutral edges that terminate in the SiC, as occurs in the self-assembled ribbons that are annealed 8 at ≈1500°C. [26][27][28][29]55 The alumina coating, that is used as the top gate dielectric, greatly reduces the mobility of the graphene bulk, with mfp's <10 nm ( Supplementary Fig. S7). However it does not affect the EGES with mfp's > 20 µm, leading to the observed vivid contrast between EGES transport and bulk transport. Figure 4A (inset) shows a schematic diagram of the device. Transport measurements were performed at temperatures from T=2 K to T=300 K in magnetic fields up to |B|=9 T. Resistances are indicated by R(VG, B, T)ij,kl =Vkl/Iij, where Vkl is the voltage measured between contacts k and l and Iij is the applied current between contacts i and j. The longitudinal voltage is

Segmentation and branching of the EGES
where R0=1/G0=h/e 2 ≈25.8 kW and conductances G are defined as 1/R. A gate voltage VG induces a charge density nc according to VG=1.16×10 -7 $ " + 8.2×10 -13 nc where VG is in volts and nc in cm -2 . The first term represents the experimentally determined quantum capacitance ( Supplementary Fig. S5). The device can be decomposed in segments labeled A to H that join at junctions, i.e. at the intersections of horizontal and vertical 700 nm wide ribbons ( Fig. 4

inset).
The vertical direction in Fig, 4 inset is approximately 5° away from the zigzag direction and the horizontal direction is approximately 5° from the armchair direction. We next show that the vertical and horizontal ribbon segments are ballistic single channel conductors at the charge neutrality point. Figure 4A shows R L 40,4X (X=0, 1, 2, 3), corresponding to R L A+B+C+D; R L B+C+D; R L C+D; R L D at T=4.5 K for perpendicular magnetic fields B ranging from 0 to 9 T. Figure 4B shows RE=R L 11',10 and RE+H=R11',11' for several temperatures from Ti=2 K to 300 K, and for B=0 T and for a 9 perpendicular magnetic field B=9 T. Resistances (in units of R0=h/e 2 ) measured at CNP closely equals the number of segments, so that the resistance per segment is very close to 1 R0.
Similarly, at CNP and B=9 T, RE≈1 R0, RE+H≈ 2 R0 for all temperatures studied. Hence, at CNP the resistance per segment is approximately 1 R0 at all temperatures both for nearly zigzag and armchair segments.
Conductance quantization at CNP is already obvious at zero field and is enhanced in high field.
Deviations from perfect quantization at zero field are due to weak localization at low temperatures and thermal broadening of the bulk states at high temperatures that are both overcome in a magnetic field (see below). Conductance reductions due to finite EGES mfp's are also relatively small. Away from CNP, the (low mobility) bulk increasingly participates in the transport.
The Landauer formulism treats the conductance as the sum of the contributions of the individual subbands shown in Fig. 1. The GES has the index n=0 and the bulk subbands have n≠0 indices.
Hence, the conductance G of a graphene segment of length L and width W (for B=0) can be decomposed as 62 Here a=(E-EF)/kBT, where the Fermi energy is 1 = ℏ * $ " (e.g. EF=35 meV for nc=10 11 /cm 2 ) and Qn(E) is the transmission coefficient of the n th subband. Following Ref. 48 , we assume that only the majority spin band contributes so that we neglect Θ * , ( 1 ).
Segmentation is consistent with the Landauer picture near E=0, with ballistic ribbons and isotropic scattering of the EGES occurring at the junctions. In the junction, transport is not protected so that EGES charge carriers are scattered by the random impurity potentials; 1 R0 per segment indicates Q ≈ ½ implying that forward-and back-scattering are equally probable, as for self-assembled epigraphene ribbons provided with an invasive probe. 26 Ignoring coherence effects, Gedge=G0.(1+L/ledge) -1 , where ledge is the mpf of the GES and the conductance of the n th subband is Gn=4G0.(1+L/ln) -1 , so that for the bulk, Qn(E)=(1+L/lbulk(E)) -1 for E<En where lbulk(E) =ln is the bulk mfp. 62 Four-point conductance measurements of segments B and C ( Supplementary Fig. S7) show that the bulk conductivity is s=neeµ with a bulk mobility µ≈750 cm 2 V -1 s -1 for nc>2×10 11 cm -2 , corresponding to lbulk=6.5 nm at nc=10 12 /cm 2 . For |nc|< 2×10 11 /cm 2 , the mobility increases ( Supplementary Fig. S7). The independence of the mobility on charge density for large nc is typical for graphene 13,63,64 and indicates scattering from charged impurities (of both signs) with a density |nimp|≈7×10 12 cm -2 (Ref. 63 , Eq. 1), mostly from the dielectric, however the non-conventional SiC substrate facet may also play a role.
For a 700 nm wide ribbon, Eq. 1 predicts that for T> E1/kB≈ 40 K, the thermal population of the bulk subbands increases the conductance at CNP (EF=0) with increasing temperature. This is shown in Fig. 5B where the conductance G11',11' (corresponding to segments E and H in series) at CNP is plotted for several temperatures from T=2 K to T=300 K. Using Eq.1, a good fit is found for lbulk=24 nm near CNP. A magnetic field introduces an energy gap due to Landau quantization: ELL1/kB=1300 K for B=9 T, so that the conductance increase with temperature is not observed at B=9 T even at high temperatures (Fig. 5B). The conductance increase is not seen, nor expected from Eq.1 in 40 nm self-assembled ribbons 26 up to room temperature since ELL1/kB=600 K.
For each segment X we determine the mfp lX of the EGES at CNP, and at B>2 T to overcome weak localization effects in the junctions (discussed below). Consequently, for a segment of length LX (see caption Fig. 4) GX=G0/(1+LX/lX), 62 giving lA=13 µm; lB =15 µm; lC =12 µm; lD>20 µm; lE>20 µm; lF=20 µm; lG=15 µm; lH>20 µm; lI >20 µm. Like for self-assembled ribbons 26 , lX is more than 1000 times larger than the mean free path of the bulk (Sup Mat Fig.   S7), even at room temperature. Figure 6B shows the measured conductance of Segment A at CNP, therefore that of the EGES, as a function of magnetic field for several temperatures T≤E1/kB. The conductance increases with increasing B and saturates at G≈1 G0 for B≳2 T. The minimum conductance at B=0 increases non-linearly as a function of temperature (see also Supplementary Fig. S6). Similar behavior was observed in self-assembled ribbons with graphene leads, 26,27 but not in self-assembled ribbons with metal contacts. 26 This implies that the conductance decrease at low magnetic field and low temperature involves the graphene junctions, not the segments themselves nor the metal contacts.
Wakabayashi 39 calculated the transmission of two graphene ribbons connected by a graphene junction in the Landauer-Büttiker formulism, and predicted that the transmission of the GES at E=0 in wide junctions is Q≈½, while Q≈1 in large magnetic fields. The increase is due to the suppression of coherent back scattering, i.e. the same mechanism that causes weak localization. 62,65 The resistances on both sides of a ribbon segment are found to be identical, down to the fine structure ( Supplementary Fig. S10), which implies that the EGES involves both physical edges of the segment. This is expected for the GES (Supplementary Fig. S15) and precludes models where the edges are independent ballistic conductors.
Scattering on random impurities cause weak localization in the junction: in absence of a magnetic field at low temperatures, constructive interference increases back scattering, thereby reducing Q. In a magnetic field and/or at high temperatures, constructive interference in the junction is suppressed 62 and we find that Q increases to Q=½ (as in the case of invasive probe on ribbons 26 ).

Weak localization is suppressed when
coherence length and 2 is the coherence time. 65 Using the theoretical model of Ref. 66 for 2D, we find 2 ≈40 nm ( 2 =0.5 ps) (Fig. 6B), independent of T for T≤65 K. For comparison in Ref. 67 in 2D epigraphene on the SiC (0001)face, 2 is found to be ≈10 ps at T=4 K, and ≈1 ps at T=20 K, which extrapolates to 0.3 ps at T=65 K assuming a T -1 dependence as suggested in Ref. 67 . While the 3-parameter fits reproduce the data very well (Fig. 6B), 2D weak localization theory is not expected to be accurate for kBT<E1. This can explain why coherence times are consistent with Ref. 67 for T=65 K and not for lower temperatures.
The decoupled EGES Figure 6A shows the longitudinal conductance of segment A: G L 04,01(VG,Bi)=1/ R L 04,01. At CNP, at B=9 T and T=4.5 K the conductance is reduced by 0.22 G0 from 1 G0. Since GA=G0/(1+LA/lA) and LA=3.6 µm, therefore lA=13 µm (See Supplementary Fig. S9 for T=40 K and 65 K). As the magnetic field decreases, the conductance further reduces by DGWL due to the weak localization (WL) that is significant for B ≤2 T (Fig. 6B). Since weak localization is seen at CNP, it involves 13 the EGES current as it flows through the junction from one ribbon segment to the next. Note that the DGWL reduction of the longitudinal conductance is observed for all VG.
Important insight into the nature of the EGES is obtained by subtracting the conductance measured at CNP from all measurements. For B<2 T the resulting conductance, Fig. 6C, corresponds to Gbulk= nceµW/L with µ=750 cm 2 V -1 s -1 as expected for segment A. This effect is also observed in self-assembled ribbons (Ref. 26 , Fig. 4c-d). For B>2 T, at VG=0.6V we observe a Quantized conductance at CNP should not be confused with the disorder induced quasiquantized conductivity that was believed to occur in exfoliated graphene flakes 1,69 . The later manifests as a rounding of G(VG) at CNP and the conductance is not quantized.
Hall measurements of the junction of segments A and B, RHall=R04,11', exhibit a plateau near RHall≈0.25 R0 (Fig. 6D) that is observed up to T=150 K (Fig. 5A), which is unusual, since the monolayer Hall plateau RHall=½ R0 and a bilayer is ruled out (Supplementary Fig. S1). Similar behavior is observed for the other two junctions. Non-quantized pseudo-plateaus are observed for 0<VG<0.5 V, as shown in Fig. 6E at several representative VG indicated by arrows in Fig. 6D, whereas in this region R L =1/G L ≈1 R0. These anomalies are not observed in graphene Hall bars without a GES. 19,35,36,70,71 We next show that the EGES causes the anomalies.
Note that RHall=VHall/I=VHall/(Iedge+Ibulk). Applying the same procedure as used above for the longitudinal conductance, we subtract the EGES current measured at CNP (i.e. at VG=0) from the total current at any VG to determine the bulk current. Specifically, (Eq.2) Figure 6F shows that this straightforward procedure (using only measured quantities) transforms the anomalous pseudo-plateaus including the 0.25 R0 plateau, into a remarkably well defined, conventional ½ R0 monolayer graphene quantum Hall plateau that starts close to the Dirac point (E≈15 meV). Moreover, in the classical regime (low B, high nc), the expected (diffusive) 2D Hall effect is observed: RH=B/nce beyond the Hall plateaus (see also Supplementary Fig. S5).

Pinning and vanishing Hall voltage
The insensitivity of the EGES current to the gate voltage demonstrates that EGES is pinned at CNP as theoretically expected (Fig. 1C). Consequently, EF=0 along the entire edge as long as the 0-DOS peak is unfilled (i.e. for |nc|<3x10 13 /cm 2 ; |VG|<30 V). From Fig. 1C we infer that the conductance-decoupled EGES corresponds with the edge localized branch of the n=0 subband at E=0. Pinning is expected in general in for chiral edges in general since they include acene edge atoms that contribute to the 0-DOS. 42,43 Note that related Fermi level pinning at the Dirac point has previously been observed in multilayer epigraphene. 72 The increase in the conductance and of the Hall voltage with increasing VG results from the contribution from the bulk as experimentally shown above, and is consistent with Fig. 1C. The delocalized branch of the n=0 subband is probably responsible for the observed increased mobility at CNP (Supplementary Fig. S7). Moreover, Fig. 1C  We can now confidently assert that EGES properties derive from the pinned flatband at the edge, at E=0. Hence the EGES is pinned at the Dirac point singularity where the hole bands and electron bands meet. 14 Interactions within the flatband will ultimately determine its electronic structure and hence its transport properties. They will also broaden the 0-DOS peak but not significantly affect its area, so that pinning will not be significantly affected. Irrespective of the mechanism and independent of any model, the experimental transport measurements indicate that transport in the EGES is at E=0 and mediated by a spin ½ fermion, to account for both the 1 G0 transport, that is highly immune to scattering, and the absence of a Hall voltage. In a semimetal (which neutral graphene actually is 14 ) transport is mediated by both electrons and holes. Hence a "semimetallic" quasiparticle can be envisioned as a forward moving electron and a backward moving hole that each transport ½ e of charge, consistent with the properties that we observe. Theoretically an example of such a fermion (i.e. a Majorana fermion) is predicted in graphene, 37 where contact with a superconductor provides the required mixing of the electron and hole bands at the Dirac point. We speculate that in our case, the metallic contacts and/or the bonding the substrate, may play a similar role. In addition, the substrate mechanically and chemically stabilizes the edges which does not occur in deposited graphene. high resolution STM measurements (Fig. 2B inset); ZJ, YJ and TZ performed the IR spectroscopy (Fig. 2E). CB, VP, and AT performed the ARPES experiments (Fig. 2D); DW, AdC and CW performed STM and STS experiments (Fig. 2B, C). KW provided theoretical support and Supplementary Fig. S15. CB and WdH directed the Atlanta based experiments.
LM and WdH directed the TICNN efforts. WdH is primarily responsible for the analysis and interpretation.

Competing interests:
The authors declare no competing interests.