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1 Valley-symmetry-preserved transport in ballistic graphene with gate-defined carrier guiding Minsoo Kim 1, Ji-Hae Choi 1, Sang-Hoon Lee 1, Kenji Watanabe 2, Takashi Taniguchi 2, Seung-Hoon Jhi 1, and Hu-Jong Lee 1,* 1 Department of Physics, Pohang University of Science and Technology, Pohang , Republic of Korea 2 Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba, , Japan * Correspondence and requests for materials should be addressed to H.-J.L. ( hjlee@postech.ac.kr). NATURE PHYSICS 1

2 1. Device geometry Figure S1. SEM image of devices. SEM image of a, GNR device and b, AB interferometer device from a vertical view. The scale bar is 2 m. Graphene is highlighted in false color. 2 NATURE PHYSICS

3 SUPPLEMENTARY INFORMATION 2. Basic transport properties of graphene Figure S2. Basic transport properties of hbn-encapsulated graphene. a, Schematic of a Hall-bar device without top gates. b, Lateral resistance Rxx as a function of back-gate voltage Vbg showing a sharp full width at half maximum (FHWM) of ~2 V. c, A log-log plot of the lateral conductance Gxx = 1/Rxx of the hole side as a function of carrier density n. Conductance saturates for n below ~ cm -2. d, Resistance measured in a van der Pauw configuration as a function of Vbg. Except for the top-gate structure, the devices shown in Figs. 1a and 4a in the main text were in Hall-bar-type measurement configurations similar to the schematic configuration given in Fig. S2a. Figures S2a,b,c show the basic conduction properties of the device in Fig. 4a obtained by operating the back gate only with the top gate kept being grounded. As shown in Fig. S2b, a sharp lateral resistance (Rxx) peak appears at the back-gate voltage of VCNP = 0.5 V with the full width at half maximum (FWHM) of ~2 V. Rxx was measured by injecting a bias current I between the leads A and D while measuring the potential difference between the leads B and C. From the sheet conductance, we derived that le 1 m for Vbg - VCNP > 5 V. Figure S2c is the lateral conductance Gxx = 1/Rxx of the hole side as a function of the carrier NATURE PHYSICS 3

4 density n. The conductance is saturated for n below ~ cm -2, which is the residual carrier density of device due to electron-hole puddles 1. The negative resistance measured with a van der Pauw configuration shown in Fig. S2d reveals the ballistic transport nature of the device. For the measurement, a bias current was injected between the leads A and B and the potential difference V was monitored between the leads F and D. Resistance becomes negative for T below ~160 K, and then it saturates below 20 K. Saturation of negative resistance implies that the mean free path of graphene becomes larger than the size of the device Reproducibility of quantised conductance We obtained the reproducible data in different devices used in the study, which are plotted in Fig. S3. Conductance plotted in Fig. S3a (b) was obtained in the hole side of the device No. 2 (3). Devices No. 2 and No. 3 had similar geometry with the same channel width W of ~120 nm but with different quasi-1d channel length of L=0.7 and 1.0 m for the devices No. 2 and 3, respectively. We could not observe the effect of different values of L on the conductance quantisation. Figure S3c (d) was obtained from the hole (electron) side of the device No. 4. L and W of this device were ~0.7 m and ~150 nm, respectively. 4 NATURE PHYSICS

5 SUPPLEMENTARY INFORMATION Figure S3. Reproducibility of quantised conductance. Conductance G of graphene quasi- 1D channels as a function of Fermi wave vector kf and the corresponding transconductance dg/dkf, a, in the hole side of GNR device No. 2, b, in the hole side of GNR device No. 3, c, in the hole side of GNR device No. 4, d, in the electron side of GNR device No. 4. NATURE PHYSICS 5

6 4. Energy dependence of quantised conductance Figure S4. Energy dependence of quantised conductance. a, Development of the transconductance dg/dkf at different temperatures T = 0.15 K (navy), 8 K (blue), 20 K (sky blue). dg/dkf is almost flat at T = 60 K (cyan). b, Finite bias spectroscopy of the device. G is plotted as a function of dc bias Vdc with different Vbg, applied between 8 and 13 V in steps of 0.1 V. Plateaus at 10, 14 e 2 /h are clearly visible at low bias, the feature of which disappears when Vdc is applied beyond 5 mv. Figure S4a shows the temperature (T) dependence of transconductance dg/dkf for T = 0.15, 8, 20, and 60 K, which shows that the conductance quantisation survives up to 20 K. Until T reaches 8 K from below, the transconductance amplitude varies little from the value at T = 0.15 K. For a 65 nm-wide constriction, the spacing between two energy subbands of 1D channels E = ħvf /W * should be ~30 mev (ħ= h/2 and vf is the Fermi velocity). But, as 6 NATURE PHYSICS

7 SUPPLEMENTARY INFORMATION seen in Fig S4a, the thermally activated energy spacing seems to be smaller than ~5 mev (= kb 60 K; kb is Boltzmann constant). Voltage bias spectroscopy was also performed to estimate the energy spacing between two subbands, by measuring G as a function of dc bias 3,4 Vdc. Results from a separate device No.3 of the similar geometry is shown Fig. S4b (no bias spectroscopy was carried out for the device discussed in the main text). In Fig. S4b, the measured traces of G exhibit nonlinear IV characteristics. Near Vdc = 0, enhanced overlap of traces of G is observed to form plateaus at 10, 14 e 2 /h, which confirms the results shown in Fig. 2. However, expected plateaus at higher biases (at Vdc 30 mv) were not observed so that accurate energy spacing could not be determined from these data. We believe that weakened guiding of carrier flow may have caused the absence of overlap of G traces in the higher energy (kbt, evdc > 5 mev). Available energy states near the CNP increase by the bias and/or thermal energy excitation and then transport channels open in graphene under the top gates. It implies that well-defined quasi-1d transport channels with effective width of W* are no longer maintained by the gate operation. NATURE PHYSICS 7

8 5. Magneto-transport in GNR Figure S5. Magneto-transport in a GNR. a, Measurement configuration of the diagonal conductance GD = I / VD. b, GD as a function of Vbg in perpendicular magnetic fields of B = 0, 0.1, 0.3, 0.5, 1, 2, 4, 6, 12 T, obtained from the device No. 4, whose zero-field conductance is shown in Fig. S3c. Figure S5a shows the measurement configuration 5 to obtain the diagonal conductance GD = I / VD = GQ N(B), where GQ = 4e 2 /h is the quantum conductance and N(B) is the number of channels between the split top gates. In zero field, we obtain the quantised conductance similar to the data shown in Fig. 2. In a perpendicular magnetic field (Fig. S5b), the conductance varies with the depopulation of transport channels while the quantised conductance in steps of 4e 2 /h is retained. We estimate that the cyclotron radius for B > 2 T becomes smaller than W * /2 for the measurement range of Vbg, which implies that two separate edge channels form along the boundaries of the split top gates without channel mixing. We noted that graphene entered into the quantum Hall regime for B > 0.8 T without the channel formation for Vtg = 0 (data not shown). Over the entire range of magnetic field in the 8 NATURE PHYSICS

9 SUPPLEMENTARY INFORMATION measurements the quantised conductance in steps of 4e 2 /h arose from the valley and spin degeneracy. In the higher fields of 6 and 12 T in Fig. S5b, the conductance quantisation due to sublevel splitting seems to emerge at GD = e 2 /h. 6. Details of tight-binding calculations Figure S6. Atomic edge structures of GNRs with various axial orientations. a-d, Edge structures of GNRs defined by the translational vectors Tmn. Red and blue dots indicate the edge atoms having less than three C-C bonds. Red (blue) dots belong to the A (B) sublattice. NATURE PHYSICS 9

10 Figure S7. Calculated band structures and quantum capacitance. Band structures of a, b, armchair-edged GNR with a size of N = 20 and 22, and c, d, zigzag-edged GNR with a size of N = 16 and 20. e-h, Corresponding quantum conductance for the systems in the upper panels. All the results are obtained from both first-principles calculations (red line) and tightbinding methods (green line). The size of the nanoribbons is defined according to the Ref NATURE PHYSICS

11 SUPPLEMENTARY INFORMATION Figure S8. Band structures and quantum conductance of the GNRs with various axial orientations. The band structure (left panel) and quantum conductance (right panel) of 60 nm-wide GNR with a, (0, 1) (zigzag-edged), b, (1, 1) (armchair-edged), c, (1, 8), d, (1, 5), e, (1, 3), f, (2, 5), g, (2, 3) and h, (4, 5) axial directions. The results for (1, 2) and (1, 4) orientations are presented in the main text. Except for the armchair-edged GNR, the quantum conductance clearly shows a step of 4e 2 /h, the same as for the zigzag-edged GNR. NATURE PHYSICS 11

12 7. Effect of smooth channel boundaries Figure S9. Quantum conductance of the 1D channels with smooth boundaries. a, Potential profiles to emulate various degrees of smoothness from the sharpest (V1(y)) to the smoothest (V4(y)) profile. In the calculation, it is assumed that the total width of GNR including the region covered by top-gates is 40 nm and the effective channel width is about 15 nm. b, c, Quantum conductance of 15 nm-width GNR along the zigzag (b) and (1, 4) directions (c) without potential modification. The orange-dashed lines are the linear fit to the conductance, which provides estimation of the effective channel width W* from the relation G = (4e 2 / πh)kfw*. d, e, Quantum conductance of the effective 1D channels along the zigzag (d) and (1, 4) directions (e) with smooth boundary set by the potential profiles in a (the color code of the conductance matches with that of the potential profiles). 12 NATURE PHYSICS

13 SUPPLEMENTARY INFORMATION 8. Robust zig-zag-type edge scattering in a GNR Figure S10. Schematics of edge scattering in 1D conducting channel and the pseudospin in graphene. a,b, Edge scattering in 1D conducting channels along a, the armchair and b, (1, 4) direction. ΨI and ΨR denote the incident and reflected states, respectively, and θ is the incident angle. c,d, Pseudospin vector field (in blue or red arrows) around K and K valleys, respectively, of graphene. The crystal momentum and the pseudospin of incident (I) or reflected states (R1, R2) in the 1D conducting channels along c, the armchair and d, (1, 4) directions are denoted by the red dots and arrows, respectively, in the constant low-energy contours around K and K. The dotted lines are the axial or transverse (connecting K-K points) directions of the channel. The unique transport property in the 1D conducting channels produced by the gate constriction in graphene is understood in terms of the pseudospin conservation. Pseudospin (or NATURE PHYSICS 13

14 A-B sublattice basis) should be conserved unless the gauge field (lattice deformation) is applied 7,8. We consider the electron scattering in two contrasting cases where K and K valleys are projected onto the Γ point; 1D conducting channels along the armchair and (1, 4) directions (Fig. S10a,b). For the elastic scattering at the edge, the incident and reflected angles of electrons should be equal (i.e., the axial component is the same and the transverse component is in the opposite direction). The possible reflected states that meet this condition are R1 and R2 (Fig. S10c,d). In addition, the pseudospin should be matched. For the channel along the armchair direction, the R2 state at the K valley satisfies this condition 8-11, leading to strong intervalley scattering. For the channel along (1, 4) direction (or off the armchair direction), on the other hand, the pseudospin of R2 state is far off from that of the incident state, suppressing the intervalley scattering. The intravalley scattering (into R1 state) is dominant in this case. When K and K are projected onto different k points, the intervalley scattering is forbidden due to the crystal momentum conservation as described in the main text. 14 NATURE PHYSICS

15 SUPPLEMENTARY INFORMATION 9. Aharonov-Bohm interferometry Figure S11. Conductance of AB interferometry with dual-gate operation. G of the Aharonov-Bohm interferometry device shown in Fig. 4a, as a function of Vtg and Vbg at T = 0.15 K. Figure S11 shows the conductance plot as a function of Vtg and Vbg, where the diagonal trace was from the ring-shaped graphene 1D channel between the top gates. From the slope of the trace of conductance minimum, -12.8, the thickness of top hbn is estimated to be ~23 nm. NATURE PHYSICS 15

16 Figure S12. Fast Fourier-transform analysis on AB interference. The Fourier spectrum of the magnetoconductance shown in Fig. 4b. The vertical dashed lines indicate the expected position of the h/e peaks, as determined from the inner and outer radii of the AB ring. One can find the h/2e oscillations also. 16 NATURE PHYSICS

17 SUPPLEMENTARY INFORMATION Figure S13. Aharonov-Bohm interference and phase shift. a, Aharonov-Bohm (AB) oscillations measured at the region of high kf as a function of Vbg and B at T = 0.15 K. It shows irregular phase shift in zero field. b, Abrupt phase shift in AB oscillations. For a high kf, the scale of the Fermi wavelength F = 2 /kf becomes only about few tens of nm. In this regime there occurs an increase in probability to have difference in travel length, between the upper and lower path of Aharonov-Bohm (AB) ring, larger than F. In this case, the -phase jump of the AB oscillation is observed under the constraint of the Onsager symmetry relation. Figure S13a shows a modulation of conductance G as a function of magnetic field B and Vbg for AB oscillations with alternate -phase jumps at the regime of high kf in the range between Vbg = -30 and -25 V. Phase inversion is clearly observed with varying Vbg within much smaller variation in kf, compared to the data measured in the regime of low kf in the main text. Fig. S13b is a recast of the cyan-color NATURE PHYSICS 17

18 boxed region in Fig. S13a. A clear alternate -phase jumps are observed in the AB conductance oscillations including the B = 0 phase, for smaller values of F in the Vbg range between -30 and -25 V, as the AB path difference becomes comparable to F under the constraint of the Onsager symmetry relation (see Fig. S13a). Figure S14. Temperature dependence of AB interference. Temperature dependence of the root-mean-squared amplitude of the AB interference G for Vbg = -30 V (red) and Vbg = -5 V (blue). The solid line is an exponential fit to the data. Figure S14 shows the evolution of the root-mean-squared amplitude of the AB oscillations G measured at different temperatures. It shows the typical exponential temperature dependence of ballistic AB oscillations NATURE PHYSICS

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20 transport properties of graphene nanoribbons and presence of perfectly conducting channel. Carbon 47, (2009). 12. Hansen, A. E., Kristensen, A., Pedersen, S., Sørensen, C. B. & Lindelof, P. E. Mesoscopic decoherence in Aharonov-Bohm rings. Phys. Rev. B 64, (2001). 20 NATURE PHYSICS