In the format provided by the authors and unedited. DOI: 1.138/NPHYS491 Edge conduction in monolayer WTe 2 Contents SI-1. Characterizations of monolayer WTe2 devices SI-2. Magnetoresistance and temperature dependence of thick WTe2 on SiO2 SI-3. Measurements demonstrating that the edges are strongly coupled to the contacts SI-4. Length dependence of the monolayer 2D bulk conduction SI-5. Temperature dependence of the monolayer 2D bulk conduction SI-6. Temperature dependence of GG eeeeeeee at different gate voltages SI-7. Bulk contribution to the non-local signal SI-8. Linear response measurements SI-1. Characterizations of monolayer WTe2 devices Device label upper hbn (nm) lower hbn (nm) top graphite (nm) MW1 5.8 18 3 MW2 9.2 17.5 4.1 MW3 11.4 14 3.4 BW1 5.4 3 3.3 TW1 5.5 22.6 4 Table S1 Thickness of the upper hbn, lower hbn, and the top graphite used for device MW1, MW2, MW3, BW1(bilayer) and TW1(trilayer). All thicknesses were obtained from the AFM image. Figure S1 Monolayer WTe 2 device. a, Optical microscope image of device MW2 without top gate. b, Cartoon of a typical monolayer WTe 2 device. The top graphite and hbn are show separated for clarity. c, Atomic force microscope image of the area highlighted in (a). The red dashed line outlines the monolayer flake. d, Line cut along the black dashed line in (c) matches the monolayer thickness, ~.7 nm. Scale bars: 5 μm. NATURE PHYSICS www.nature.com/naturephysics 1
3 12.6 cm -1 165.5 cm -1 Raman Intensity (a.u.) 1 218.4 cm -1 85.1 cm -1 135.9 cm -1 5 1 15 25 Raman Shift (cm -1 ) Figure S2 Raman spectra of an encapsulated monolayer WTe 2. Five Raman peaks are clearly observed in the range 5 to 25 cm -1, consistent with previous reports 1, 2 on monolayer WTe2, with no signature of breathing or shear modes above 1 cm -1. Raman spectra was performed at 5 K with a 532 nm excitation laser. SI-2. Magnetoresistance and temperature dependence of thick WTe2 on SiO2 As is well established, monolayer WTe2 degrades with time in air. However, for many-layer flakes only the top few layers appear to degrade, enabling electrical conductance measurements even without hbn encapsulation. Fig. S3a shows the four-terminal magnetoresistance (MR) as a function of magnetic field parallel to the c-axis measured on a 1 nm thick WTe2 device with current along the a-axis. The large MR, 8,%, at 7 T and 7 K is comparable to the previous report 3. Fig. S3b shows a clear metal to insulator transition as layer thickness decreases for nonencapsulated flakes, consistent with the previous report 2. As the temperature of the trilayer device decreases the resistivity actually increases, different from the encapsulated trilayer device reported in the main text (Fig. 1e). We also found non-encapsulated monolayer and bilayer flakes were not stable over time. In contrast, the encapsulated monolayer WTe2 devices we measured in the main text are very stable, yielding indistinguishable results even after several months. NATURE PHYSICS www.nature.com/naturephysics 2
Figure S3 Bulk and few-layer WTe 2 on SiO 2 substrate. a, Field dependence of MR in a 1 nm thick WTe 2 with the current along the a-axis and the applied field along c-axis at 7 K. The inset is the positive data on a log-log scale. b, Temperature dependence of sheet resistivity (per square) in non-encapsulated WTe 2 devices of different thickness, from 1 nm thick down to a trilayer. SI-3. Measurements demonstrating that the edges are strongly coupled to the contacts In studying the edges in the main text (Figure 3 and 4), we focused on two-terminal measurements. Four-terminal measurements were not presented because they yield the same results, implying the contacts to the edge were perfect in the sense that the current can only pass between adjacent edges via the metal of the contact in between. To illustrate this, Fig. S4a shows the parallel field dependence of the conductance at Vg = in device MW2, in which the black and red curves were measured with two- and four-terminal configurations respectively as defined in the inset (GG 23 = II 23 /VV 23, GG 14,23 = II 14 /VV 23 ). Even as the magnetic field suppresses the edge conductance, both two and four-terminal measurements give almost identical conductance values. Figure S4 Comparison of two-terminal and multi-terminal measurements in device MW2. a, Perpendicular magnetic field dependence of the conductance of a particular edge at 6.5 K, Vg = ; black and red curves are two- and four-terminal measurements respectively as labeled, with 2 and 3 the voltage contacts in both cases. b, Gate dependence of direct two-terminal conductance GG 36 (black) and series conductance (GG 1 32 + GG 1 26 ) 1 (red), at 1.6 K and BB =. NATURE PHYSICS www.nature.com/naturephysics 3
Figure S4b compares the gate dependence of GG 36 with (GG 1 32 + GG 1 26 ) 1 in device MW2 with contact 4 grounded, here GG 36, GG 32, GG 26 are two-terminal conductances. On the plateau region, there is minimal difference between the two, even for the mesoscopic fluctuations. This implies a strong contact coupling and, again, that the conductance is determined entirely by the edge in this regime, since bulk current flow would violate this equivalence. SI-4. Length dependence of the monolayer 2D bulk conduction Above 1 K, two-terminal conduction is dominated by the 2D bulk. Fig. S5 shows the twoterminal resistance as a function of aspect ratio L/W for device MW1 at Vg =. If the edge contribution is small, the two-terminal resistance is given roughly by RR = ρρ ss LL WW + 2 RR cc. From the linear fit (for large aspect ratio a deviation from the linear fit is expected due to current spreading) we extract the sheet resistivity ρρ ss, which increases from 2 kω at room temperature to 125 kω at 155 K, consistent with the insulating behavior for zero gate voltage. The extracted contact resistance RR cc is approximately 2 kω per contact. 8 6 V ac = 1 V B = = 155 K R (k ) 4 2 K 3 K..2.4.6 L/W Figure S5 Length dependence of two-terminal resistance in device MW1. Resistance as a function of aspect ratio at Vg = for T = 3 K, K, and 155 K. NATURE PHYSICS www.nature.com/naturephysics 4
SI-5. Temperature dependence of the monolayer 2D bulk conduction Figure S6a shows two-terminal linear-response conductance measurements in device MW2 made in a similar way to the measurements on the pincer-shaped device (MW3) in the main text (Fig. 2). Again a conductance plateau is seen at 4.2 K with mesoscopic fluctuations, indicating there is only edge conduction in this region. The conductance of the plateau is relatively low because of the combination of longer edges and ZBA. When we short out all the edge current, as shown in the insets, the conductance I/V is suppressed to an unmeasurably small level in the plateau region implying that the conductance through the bulk is negligible at this temperature. Figure S6b shows its temperature dependence at Vg =. Above ~ 1 K, the conductance rises roughly linearly with temperature. At lower temperatures, it is approximately activated with activation energy ~5 mev (red trace in the right inset). This illustrates the sense in which the bulk becomes insulating below 1 K. Figure S6 Gate and temperature dependence of bulk conductance in device MW2. a, Comparison of I/V as a function of gate voltage for the two experimental configurations shown. Analogous to Fig. 2c in the main text, the red trace is the total two-terminal resistance which contains both edge and bulk contributions, while the black trace only contains a bulk contribution. b, Temperature dependence of the bulk measurement at Vg =. Inset: Arrhenius plot, showing approximately activated behavior below ~1 K (red trace, 5 mev). No signal was detectable above the background noise at temperatures below the lowest one shown here (2 K). NATURE PHYSICS www.nature.com/naturephysics 5
SI-6. Temperature dependence of GG eeeeeeee at different gate voltages The inset of Fig. 3a shows the temperature dependence of the edge conduction at Vg = +.8 VV, which saturates below ~1 K. In Fig. S7 we plot the temperature dependence at various other gate voltages. Figure S7 Temperature dependence of GG eeeeeeee at different gate voltages for a particular edge in device MW2. a, Gate dependence of GG eeeeeeee at TT = 1.6 KK, BB =. b, TT dependence of GG eeeeeeee at selected gate voltages, each gate corresponds to a vertical line in a. c, Smoothed 2D map of GG eeeeeeee as a function of Vg and TT. NATURE PHYSICS www.nature.com/naturephysics 6
SI-7. Bulk contribution to the non-local signal Here we show a simulation of the non-local signal for the geometry approximating device MW2. Assuming the conductivity is isotropic, the electric potential satisfies the Laplace equation. In Fig. S8, the dashed rectangles represent metal contacts. All contacts except the source and drain are floating, implying constant potential and vanishing net current as boundary conditions. The simulated non-local ratio VV nnnn /VV is ~1%, in reasonable agreement with the measured value VV nnnn /VV ~ 5 % seen in Fig. 2b in the limit of high temperature or large gate voltage where we expect edge effects are negligible. The difference might come from the uncertainty of the exact monolayer shape and in-plane anisotropy in conductivity. Figure S8 Simulation of potential distribution in MW2 for the nonlocal measurement. NATURE PHYSICS www.nature.com/naturephysics 7
SI-8. Linear response measurements Fig. S9-12 are transfer characteristics of different edges from base temperature 1.6 K to near room temperature in monolayer devices MW1, MW2, MW3 and bilayer device BW1 respectively. A small enough ac bias of 1 V was used to ensure linear response. As can be seen from the temperature dependences, the strength of the ZBA varies for different devices and edges. For the edge conductance values plotted in Fig. 4e in the main text, we used the averaged value over a gate range of about 1 V (different for different edges) near Vg =, where the plateau presents. In the case of the shortest edge of at 1 K in Fig. 4e, a bulk contribution of 2 S was substracted. 3 1 3 1 L = 218 nm L = 517 nm L = 234 nm L = 772 nm L = 315 nm L = 1,5 nm T (K) 1.6 4.2 1 22.4 3 5 75 1 125 155 25 3-3 3-3 3-3 3 Figure S9 Temperature dependence of transfer characteristics of different edges (adjacent contact pairs) in monolayer device MW1, with contact separation LL ranging from 218 to 1,5 nm. 12 8 L = 1.3 m T (K) 1.6 2.3 4.2 1 2 5 1 15 25 4-4 -2 2 4 Figure S1 Temperature dependence of the transfer characteristic of a particular edge in monolayer device MW2. The contact separation is 1.4 μm. NATURE PHYSICS www.nature.com/naturephysics 8
1 L = 165 nm L = 275 nm L = 185 nm L = 47 nm L = 24 nm L = 48 nm T (K) 1.6 4.2 1.6 3 43 65 8 1 15 25 25 3 1-5 5-5 5-5 5 Figure S11 Temperature dependence of transfer characteristics of different edges in monolayer device MW3, with contact separation LL ranging from 165 to 48 nm. 6 4 6 4 L = 26 nm L = 57 nm L = 38 nm L = 8 nm L = 48 nm L = 1,49 nm T (K) 1.6 2.6 4.3 6.5 1 15 2 3 4 5 75 1 15 25 3-3 3-3 3-3 3 Figure S12 Temperature dependence of transfer characteristics of different edges in bilayer device BW1, with contact separation LL ranging from 26 to 1,49 nm. NATURE PHYSICS www.nature.com/naturephysics 9
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