Peculiar Magneto-Transport Features of Ultra-Narrow Graphene Nanoribbons Under High Magnetic Field

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1 1 Supporting Information Peculiar Magneto-Transport Features of Ultra-Narrow Graphene Nanoribbons Under High Magnetic Field Haoliang Shen, Alessandro Cresti, Walter Escoffier, Yi Shi, Xinran Wang, and Bertrand Raquet 1. Device fabrication The GNRs were synthesized by unzipping MWCNTs in solution as reported previously [1]. The 300nm SiO 2 /p ++ silicon substrate with pre-patterned metal markers was soaked in the solution for two hours to deposit a good density of GNRs. The substrate was then annealed in Ar atmosphere at 400 C to remove the poly(m-phenylenevinylene-co-2,5-dioctoxy-p-phenylenevinylene) surfactant. AFM was used to locate and characterize individual bilayer GNRs. E-beam lithography was then applied to pattern the source/drain, followed by evaporation of 30nm Pd and liftoff to form metal leads. The samples were finally annealed in Ar atmosphere at 200 C for 15min to improve the contact quality. 2. AFM characterization of the bilayer GNRs The as-deposited GNRs were characterized by AFM. The typical height and the average width of the bilayer GNRs are 1.5±0.2 nm and ~20 nm, respectively, in line with what reported in literature [1]. Figure S1. AFM characterization of the bilayer GNRs. Insets: height profile at the section, indicated by black dashed line. The structural parameters and the capacitive simulation results of our device samples are indicated as below.

2 2 Number # Width Channel Length Capacitive Coupling Carrier Density Efficiency Number of subbands (nm) (nm) (µf/cm 2 ) (/m 2 /V) at V g= -30 V S S S S S S S Table S1. Structural parameters of measured GNR device samples. We note that the carrier density efficiency is approximately three times as large as the one from a simple parallel capacitor model α / (m 2 V). This deviation comes from the fact that the ribbon width is much smaller than the gate thickness. The estimation of the number of subbands in Table S1 has used three approximations: 1) to obtain the Fermi energy E(Vg) we assumed linear dispersion like in 2D graphene. 2) The band structure is assumed to have 1D subbands with equal energy separation δe. 3) The spin degeneracy is preserved in these subbands while the valley degeneracy (g v =2) is lifted since our ribbon edge geometry is most probably mixed. Since our ribbon consists of two layers, the number of total subband includes a layer degeneracy (g L =2). In such a scenario, the number of spin-degenerate subbands is estimated by N = Int [E(V g )/(δe(w)/ g v )] g L. We note that this approach cannot provide a very precise value because of the complex band structures of GNRs. The estimate, however, does provide a qualitative assessment of the sample quality. 3. Thermal robust conductance modulations at low temperature In the doped region, we observed some reproducible conductance modulations to appear below 80K, which are robust to thermal cycling. These structures are likely related to the electronic structure of the GNR. Figure S2. G-V g characteristics of device S2 zoomed in the p-doped region, from 60 K to 4 K.

3 3 4. Reproducibility of the quantized resistance plateau and the plateau perturbation In all of the measured bilayer GNR devices, we systemically observe a ~h/(4e 2 ) resistance plateau (see fig. 2 and fig. S3), resulting from the onset of chiral edge channels under high magnetic field. In some samples, the plateau is accompanied by some clearly visible irregularities (see devices S3 in fig. 2 (b), and S5 in fig. S4). As discussed and explained in the paper, such a perturbation, in the form of a resistance peak inside the plateau, shifts to higher field when the doping level is increased. Figure S3. Magneto-resistance measurement of device S1, S4, S6 for selected back-gate voltages, showing a quantized plateau. All data were measured at 4.2K. Figure S4. Magneto-resistance of device S5 at 4.2 K and under selected back-gate voltages between -25 V and -42 V. Inset: resistance plot as a function of B/Vg to indicate detailed plateau features. 5. Primitive cell geometry and tight-binding model We define the ribbon from a 2D bilayer system with given twisting angle between top and bottom layers. To this aim, we start with a simple AA stacking and then turn the top layer by an angle θ. In order to keep the commensurability between the lattices of the two layers, we only consider discrete angles corresponding to the rotation of the vector mt 1 +nt 2 onto the vector nt 1 +mt 2, where t 1 =[3,- 3]a/2 and t 2 =[3, 3]a/2, with a the in-plane C-C distance, are the usual graphene translation vectors, and m and n are integer numbers. Indeed, θ is determined by the ratio s n/m as θ = arctan [ 3 (1-s 2 )/(1+4s+s 2 )].

4 4 The primitive cell of the bilayer contains N=4(m 2 +n 2 +mn) atoms and its translation vectors are R 1 = nt 1 +mt 2 and R 2 = -mt 1 +(m+n)t 2. We can now define the ribbon translation vector (along its axis) as T=αR 1 +βr 2, where α and β are integer numbers. The cell is then repeated with period T along the ribbon axis, and along the transverse direction with translation vector V=-(α+2β)R 1 + (2α+β)R 2 as many times as necessary to cover the ribbon width W. The ribbon is then cut to have a final width W and any remaining dangling edge atom is removed. The relative angles of bottom and top layers with respect to the armchair direction are given by φ bottom = arcos{( 3/2)(1+s+sσ)/[ (1+s+s 2 ) (1+σ+σ 2 )]} and φ top = arcos{( 3/2)(1+s+σ)/[ (1+s+s 2 ) (1+σ+σ 2 )]}, where σ β/α. Of course, θ= φ top - φ bottom. The two parameters s and σ, together with the ribbon width W, fully characterize the bilayer ribbon. Figure S5. Schematic of the bilayer GNR structure building procedure. In this illustration, we choose (m,n)=(3,1), (α,β)=(1,1), corresponding to φ bottom =46.1, φ top =13.9, and twisting angle θ=32.2. As a consequence, R 1 = 3t 1 +t 2 and R 2 = -t 1 +4t 2, while T=2t 1 +5t 2 and V=-12t 1 +9t 2. The basic primitive cell exploited to build the ribbon is indicated by a red box. We adopt the single-orbital tight-binding model proposed by [2,3] and based on ab initio calculations. The coupling energy between two atoms at positions r i and r j is given by t ij =n 2 V ppσ (r ij )+(1-n 2 )V ppπ (r ij ), where r ij = r j - r i, r ij = r ij and n=z ij /r ij is the direction cosine of r ij along the z-axis. The Slater-Koster parameters are V ppπ (r ij ) = -γ 0 exp[q π (1-r ij /a)]/[1+exp[(r ij -r c )/l c ] and V ppσ (r ij ) =γ 1 exp[q σ (1-r ij /a 1 )]/[1+exp[(r ij -r c )/l c ], with a=0.148nm the in-plane C-C distance, a 1 =0.3349nm the interlayer distance, γ 0 =2.7eV and γ 1 =0.48eV the coupling parameter, q σ = and q π = the decaying rate, r c =0.614nm and l c =0.0265nm the cut-off distance and decaying rate appearing in the additional smooth cut-off function at the denominator. The magnetic field is included in the description by means of the Peierls phase factor [4], which modifies the coupling energies as t ij t ij exp[ie/(ħc) ʃ dl A(r)], where A(r) is the vector potential and the line integral is performed along the path from r i to r j. The ribbon band structure depends very sensitively on both the twisting angle θ and the edge orientation, i.e. φ top and φ bottom. In particular, the twisting angle is known to renormalize the Fermi velocity [5, 6] and to shift the Dirac points [7], thus inducing van Hove singularities at higher energies.

5 5 This entails different spacing and sequences for low-energy Landau levels (LLs) [8], and the presence of additional LL series [9-11]. The lateral confinement in narrow ribbons further complicates this scenery by introducing quantized transverse wave vectors and thus fragmenting the band structure at low magnetic fields and inducing chiral edge states at high fields. The specific edge geometry, when containing zigzag sections, may entail the presence of states strongly confined at the edges [12], which are present also in the absence of magnetic field [13] and have the same nature as the edge states observed in zigzag monolayer ribbons [14]. Importantly, these states are very sensitive to disorder and especially edge disorder. As shown in the main text and in section 8, they are thus expected to easily localize and not to significantly contribute to electron transport. 6. Band structure of ribbons with different geometry Let us consider some typical 20nm-wide ribbons with different twisting angles and edge directions, see Fig. S6, and show the corresponding band structures at B=0 and B=50T in Fig. S7 and S8. For the sake of completeness, we also consider an armchair ribbon with Bernal stacking in Fig. S9. Figure S6. Top and bottom edge geometries of a bilayer ribbon with W 20nm and (a) θ 32.2 (m=3, n=1), φ bottom =-16.1,φ top =16.1 (α=1, β=0); (b) θ 32.2 (m=3, n=1), φ bottom =46.1,φ top =13.9 (α=1, β=1); θ 6 (m=6, n=5), φ bottom =-3,φ top =3 (α=1, β=0) ;(d) θ 6 (m=6, n=5), φ bottom =27,φ top =33 (α=1, β=1). The red and blue lines indicate the armchair direction of the bottom and top layers, respectively. Figure S7. Band structure of a bilayer ribbon with W 20nm, θ 32.2 and (a) φ bottom =-16.1,φ top =16.1 and B=0T; (b) φ bottom =46.1,φ top =13.9 and B=0T; (c) same as (a) for B=50T; (d) same as (b) for B=50T.

6 6 Figure S8. Band structure of the bilayer ribbon with W 20nm, θ 6 and (a) φ bottom =-3,φ top =3 and B=0T; (b) φ bottom =27,φ top =33 and B=0T; (c) same as (a) for B=50T; (d) same as (b) for B=50T. Figure S9. Band structure of the bilayer armchair ribbon with W 20nm and Bernal stacking for (a) B=0T and (b) B=50T. At B=0, the Dirac points turn out to be placed at the Γ point (k=0) of the Brillouin zone or at intermediate positions. This depends on the geometry of the ribbon primitive cell, which determines how the hexagonal Brillouin zone of 2D graphene is projected and folded in the 1D Brillouin zone of the ribbon. As expected, Figs. S8(a,c) and S9 do not show any zero magnetic field edge states because the ribbon edges have a pure armchair geometry, see Fig. S6(c). In all the other cases, the presence of zigzag component in the edge geometry entails the presence of low-energy inter-valley edge states. Note that, even at zero magnetic field, these zigzag-related edge states are rather dispersive compared to those reported in Ref. [14]. This is due to presence of next-nearest-neighbor intra-layer coupling terms in the Hamiltonian of our model. As shown in Fig. S7(c,d) and S8(d), and also in the main text, the magnetic field does not change the non-chiral nature of these states, but simply modifies their dispersion, thus leaving them easily localizable. In contrast, the additional chiral edge states of magnetic origin, when well-developed, are rather insensitive to disorder thus constituting the main contribution to magneto-transport. At high magnetic field B=50T, the flat bands corresponding to the first LLs, together with the dispersive bands corresponding to the chiral edge states, are clearly visible in Figs. S7(c,d) and S8(c, d). The width of the plateaus is rather narrow, due to the relatively small ratio between magnetic length and ribbon width. The Bernal stacking case in Fig. S9(b) is completely analogous, even though the energy dependence of the LLs on the magnetic field is different, as expected. Curiously, the energy bands in Fig. S7(c,d) and S8(d) are not symmetric with respect to the Γ point. This is true in particular for the bands corresponding to the zigzag-like edge states, which are strongly confined at the edges. This asymmetry is the combined result of the time-reversal symmetry breaking (due to the presence of the magnetic field) and the different geometry of top and bottom edges, to which edge states are

7 7 very sensitive. The asymmetry is not observed for ribbon of Fig. S8(c), because its top and bottom edges are identical, see Fig. S6(c). Note that the adopted model contains some approximations, which we would like to briefly discuss. First of all, it does not consider the possible presence of local strain that may appear due to the twisting between the layer [15] or repulsion between the two layers in the AA stacked regions [16, 17]. These phenomena may be experimentally relevant for 2D bilayers [15, 18]. Moreover, our model does not account for possible disorder or reconstruction of the ribbon edges, which may entail specific modifications of the non-chiral zigzag-like edge states. We believe that these approximations do not significantly alter our conclusions. In fact, our simulations mainly aim at investigating the development of chiral edge states, which do not depend on the edge geometry and are only moderately affected by disorder. Indeed, the experimental observation of conductance plateaus clearly confirms that disorder is moderate. As concerns the non-chiral edge states, their specific dispersion is not important since they are anyway expected to localize and not to contribute to transport, as verified experimentally and numerically. 7. Localization of the edge states induced by zigzag sections of the ribbon due to edge roughness We consider here the ribbon investigated in the paper and sketched in Fig. S6(a), in the presence of edge roughness over a length of 100nm and under a 50T magnetic field. The random profile y(x) of the edge roughness is different for each of the four edges of the ribbon and is generated by considering the autocorrelation function C(x) = < y(x) y(x-x )> = δ 2 e - 2 x /λ, where δ determines the strength of the disorder and λ=2nm is the correlation length. The roughness profile considered in the main text is obtained with δ=0.1nm. Note that the particular choice of the correlation function is not essential to our conclusion. An example of realization is reported in Fig. S10(a). The magneto-conductance of the ribbon for different values of δ is reported in Fig. S10(b). (a) (b) Figure S10. (a) Example of edge profile for δ=0.25nm. (b) Differential conductance for different values of the edge roughness strength in a 20nm-wide and 100nm-long bilayer graphene ribbon at 50T. As already observed in the main text, we can conclude that even a moderate edge roughness (δ>0.05nm) completely suppresses the surface states transport and let the Hall plateau sequence appear. For very weak roughness (δ=0.05nm), the nonchiral edge channels do not completely localize, thus perturbing of the Hall conductance. This entails that, experimentally, only very smooth ribbons

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