Coulomb Drag between a Carbon Nanotube and Monolayer Graphene

We have measured Coulomb drag between an individual single-walled carbon nanotube (SWNT) as a one-dimensional (1D) conductor and the two-dimensional (2D) conductor monolayer graphene, separated by a few-atom-thick boron nitride layer. The graphene carrier density is tuned across the charge neutrality point (CNP) by a gate, while the SWNT remains degenerate. At high temperatures, the drag resistance changes sign across the CNP, as expected for momentum transfer from drive to drag layer, and exhibits layer exchange Onsager reciprocity. We find that layer reciprocity is broken near the graphene CNP at low temperatures due to nonlinear drag response associated with temperature dependent drag and thermoelectric effects. The drag resistance shows power-law dependences on temperature and carrier density characteristic of 1D Fermi liquid-2D Dirac fluid drag. The 2D drag signal at high temperatures decays with distance from the 1D source slower than expected for a diffusive current distribution, suggesting additional interaction effects in the graphene in the hydrodynamic transport regime.

Figure 1

(a) False color scanning electron microscope image of a typical SWNT-graphene drag device. Graphene (green) is encapsulated in hBN (dark blue) and transferred on top of a metallic SWNT (dashed line in center of blue charged region). Electrical contacts (gold) are made to the graphene and SWNT. Inset: cross-section schematic of the device. (b) $V_{\mathrm{drag}}$ versus $I_{\text{drive}}$ for reciprocal layer configurations: nanotube-drive, graphene-drag (orange, filled symbols) and graphene-drive, nanotube-drag (blue, open symbols). Data were taken at $T=300\mathrm{K}$ and $V_g=21\mathrm{V}$, with averaging gate voltage window $\mathrm{\Delta }V=\pm 1\mathrm{V}$ to enhance the signal-to-noise ratio. Dashed curves are lines of best fit. Additional details can be found in the Supplemental Material [20]. (c) SWNT conductance as a function of gate voltage. The dip is a local conductance minimum, not the charge neutrality point (see Supplemental Material [20] for discussion). (d) Graphene resistance as a function of gate voltage. (e) $V_{\mathrm{drag}}$ in graphene versus SWNT $I_{\text{drive}}$ and $V_g$. (f) $R_{\mathrm{drag}}$ versus $V_g$. Dashed line marks zero drag signal.

Figure 2

(a) $V_{\mathrm{drag}}$ versus $I_{\text{drive}}$ at $V_g=22\mathrm{V}$ for varying temperatures. Dashed lines are 3rd order polynomial fits. (b) 2nd order coefficient versus $V_g$. (c) 3rd order coefficient versus $V_g$. Regions of noisy data in (b) and (c) are due to SWNT local conductance dip limiting $I_{\text{drive}}$. (d) $R_{\mathrm{drag}}$ from linear fit versus $V_g$ (subtracting the graphene CNP voltage) for reciprocal configurations at $T=180\mathrm{K}$ [same colors and symbols as Fig. 1]. (e) Same measurement at $T=117\mathrm{K}$.

Figure 3

(a) Drag resistance as a function of $V_g$ at various temperatures: 265 (top), 235, 200, 160, and 130 K (bottom). Disorder-dominated range is indicated by gray shading. (b) Estimated range of Fermi energies for SWNT (purple) and graphene (blue) in a range of $V_g$ near the graphene CNP. Dot-dashed line is approximate center of SWNT $E_F$ range. See Supplemental Material [20] for details of calculation. (c) Log-log plot of $R_{\mathrm{drag}}$ versus $V_g-V_0$ at selected temperatures, with $R_{\mathrm{drag}}\propto (V_g-V_0{)}^{-1}$ (dot-dashed), and $R_{\mathrm{drag}}\propto (V_g-V_0{)}^{-1.5}$ (dotted) for comparison. (d) Graphene conductance versus charge carrier density for temperatures in (a). The residual carrier density $\delta n$ is estimated by the intersection of the line at minimum conductivity (black) and a linear fit to log(G) away from charge neutrality (dark red dashed line shows example fit for $T=118\mathrm{K}$).

Figure 4

(a) Drag resistance as a function of temperature for various graphene charge carrier densities (stated value is the upper bound). For $T<140\mathrm{K}$ (the shaded region in the graph), nonlinear effects increasingly dominate the signal. (b) Drag resistance versus temperature and graphene carrier density. (c) Average $(d{R}_{\mathrm{drag}}/dT)$ for $T>T_d$ versus temperature at a range of graphene carrier densities.

Figure 5

(a) $R_{\mathrm{drag}}$ versus $V_g$ at $T=300\mathrm{K}$ for pairs of voltage probes at increasing distance $x$ from the SWNT: 800 nm (circles), $1.2\mu \mathrm{m}$ (squares), $1.4\mu \mathrm{m}$ (triangles), and $2\mu \mathrm{m}$ (stars). Inset: $R_{\mathrm{drag}}$ at $V_g=24\mathrm{V}$ for increasing distances. Dashed curve is an exponential fit (see main text). (b) Effective channel width $W_{\mathrm{eff}}$ versus $V_g$, extracted from fit for $R_{\mathrm{drag}}$ at $T=300\mathrm{K}$. Dashed line marks actual device width ($W=1\mu \mathrm{m}$).