SReferences for Supplementary Material

# Coulomb Drag between a Carbon Nanotube and Monolayer Graphene

Laurel E. Anderson  Department of Physics, Harvard University, Cambridge,
Massachusetts 02138, USA  Austin Cheng  Department of Applied Physics, Harvard
University, Cambridge, Massachusetts 02138, USA  Takashi Taniguchi
International Center for Materials Nanoarchitectonics, National Institute for
Materials Science, Tsukuba 305-0044, Japan  Kenji Watanabe  Research Center
for Functional Materials, National Institute for Materials Science, Tsukuba
305-0044, Japan  Philip Kim  Department of Physics, Harvard University,
Cambridge, Massachusetts 02138, USA  Department of Applied Physics, Harvard
University, Cambridge, Massachusetts 02138, USA<EMAIL_ADDRESS>
###### Abstract

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.

††preprint: APS/123-QED

##

Coulomb interactions generate a plethora of novel emergent phenomena in
condensed-matter systems, particularly when electronic confinement to fewer
than three spatial dimensions increases the relative strength of potential to
kinetic energy [1]. One experimental tool for studying interaction-driven
effects in low-dimensional systems is Coulomb drag [2, 3, 4, 5]. When a
current is driven in a conductor that is near but electrically isolated from
another, Coulomb interactions between the charge carriers in the two
conductors generate a voltage drop in the “passive” conductor. The drag
resistance is thus a direct probe of interlayer charge carrier interaction.
Most of the past theoretical and experimental efforts have focused on drag
between 2-dimensional (2D) conductors, such as electrons confined in
semiconductor heterointerfaces [3, 4] and graphene [5, 6, 7], revealing
several new emergent phenomena including exciton condensation under strong
magnetic fields [5]. Drag experiments have also been performed between
1-dimensional (1D) conductors [8, 9, 10], showing signatures of Wigner crystal
and Luttinger liquid behavior.

Coulomb drag experiments between mixed-dimensional systems, e.g. 1D-2D
conductors, have also been conceived [11, 12] to investigate the effects of
dimensionality on electron-electron interactions. Such a system was recently
probed experimentally [13] using an InAs nanowire as 1D conductor and graphene
as 2D conducting layer. This recent 1D-2D drag experiment shows an anomalous
temperature and density dependent drag response that might be related to
energy drag [14, 15, 16] due to the large mismatch in thermal conductivities
between InAs and graphene. However, the breakdown of layer (Onsager)
reciprocity and subsequent thermopower measurements in these devices [17]
suggest local heating-induced thermoelectric effects may also play a
substantial role in the reported drag results.

In this letter, we report Coulomb drag in a new 1D-2D conducting system, a
metallic single-walled carbon nanotube (SWNT) and monolayer graphene separated
by an atomically thin (2-4 nm) insulating barrier of hexagonal boron nitride
(hBN). Since SWNTs and graphene have similar linear dispersion relations with
comparable Fermi energies and work functions [18, 19], the interaction-driven
momentum and energy transfer between carriers in separate layers are enhanced,
amplifying the drag signal. Due to the small ($\sim$2 nm) diameter of the
SWNT, driving current in the nanotube provides an extremely localized 1D drag
source in the graphene channel. We measure density, temperature, and distance
dependence of the drag effect to probe the carrier interactions between these
1D and 2D conductors. This work is an early step toward experimentally
addressing the specific impact of increased confinement on interaction
effects, and may also be a new test system for hydrodynamics in graphene.

A scanning electron microscope image of an example SWNT-graphene drag device
is shown in Figure 1(a). Details of the fabrication are given in the
Supplemental Material (SM) [20]. In brief, monolayer graphene is encapsulated
in hBN and then transferred on top of a metallic SWNT. The hBN flake
separating the SWNT and graphene is 2-4 nm thick, so that the two conductors
are sufficiently close together for interlayer Coulomb interactions, but they
remain electrically isolated, without a significant tunneling current. The
graphene and SWNT have individual electrical contacts, allowing them to be
characterized separately. We can use graphene or nanotube as either drive or
drag layer. While we focus on one SWNT in the following discussion, several
SWNT/graphene devices were measured, and similar results were obtained (see SM
[20]).

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) $V_{drag}$ versus
$I_{drive}$ for reciprocal layer configurations: nanotube-drive, graphene-drag
(orange, filled symbols) and graphene-drive, nanotube-drag (blue, open
symbols). Data were taken at $T=300\;\textrm{K}$ and $V_{g}=21\;\textrm{V}$,
with averaging gate voltage window $\Delta V=\pm 1$ V to enhance the signal-
to-noise ratio. Dashed curves are lines of best fit. Additional details can be
found in the SM [20]. (c) SWNT conductance as a function of gate voltage. The
dip is a local conductance minimum, not the charge neutrality point (see SM
[20] for discussion). (d) Graphene resistance as a function of gate voltage.
(e) $V_{drag}$ in graphene versus SWNT $I_{drive}$ and $V_{g}$. (f) $R_{drag}$
versus $V_{g}$. Dashed line marks zero drag signal.

Measurements of the drag resistance were performed by applying DC current
$I_{drive}$ through the drive layer (SWNT or graphene) while the voltage
$V_{drag}$ was measured in the drag layer (graphene or SWNT). Example data for
both configurations are presented in Figure 1(b). When using the SWNT as drive
layer, $V_{drag}$ in graphene is measured with the voltage probes nearest the
SWNT, at a distance $x=800$ nm away (closed circles). At temperature
$T=300\;\textrm{K}$, there is a linear relationship between $I_{drive}$ and
$V_{drag}$, whether the SWNT or the Gr is the drive layer. Using the graphene
as the drive layer and measuring $V_{drag}$ across the SWNT (open circles)
results in a noisier signal than the reciprocal drag scheme, due to the higher
resistance of the SWNTs (see SM [20] for discussion). Even so, both biasing
configurations yield the same current-voltage relationship. This Onsager
reciprocity when drive and drag layers are exchanged demonstrates that the
system is in the linear response regime, allowing the extraction of the drag
resistance from the slope: $R_{drag}=\Delta V_{drag}/\Delta I_{drive}$.

In our devices, the SWNTs are beneath the graphene (Fig. 1(a) inset), enabling
the carrier densities in both SWNT and graphene to be tuned by a voltage
$V_{g}$ applied to a back gate. Figure 1(c) and (d) show the conductance
$G_{NT}$ of SWNT and resistance $R_{Gr}$ of graphene, respectively, as a
function of $V_{g}$ measured at $T=300\;\textrm{K}$. The gradual decrease of
$G_{NT}$ as $V_{g}$ increases indicates the SWNT is hole-doped. In the
graphene, $R_{Gr}$ exhibits a peak corresponding to the charge neutrality
point (CNP) around gate voltage $V_{0}=13\;\textrm{V}$. We also measure the
drag response as a function of $V_{g}$, as shown in Figure 1(e). We extract
$R_{drag}$ in the linear response regime in as a function of $V_{g}$, as
described above. For $V_{g}<V_{0}$, $V_{drag}$ and $I_{drive}$ have opposite
sign, while for $V_{g}>V_{0}$, they have the same sign. As shown in Figure
1(f), $R_{drag}$ thus changes sign at $V_{g}=V_{0}$ where the dominant carrier
type in graphene switches from electrons to holes. This behavior is
qualitatively similar to previous measurements of momentum-transfer Coulomb
drag in double-layer graphene systems [6, 7]. The higher magnitude of e-h
compared to h-h drag can be attributed to the higher density of holes in the
SWNT at more negative gate voltages. Due to heavy SWNT doping, the $k_{F}$’s
of the SWNT and graphene do not overlap within our experimental gate window
(further discussion in SM [20]), preventing us from investigating the double
neutrality point, where the chiral nature of the 1D-2D Dirac system can be
explored [43].

As T decreases, the relationship between drive current and drag voltage
becomes increasingly nonlinear (Fig. 2(a)). To quantitatively address this
change, we fit $V_{drag}$ with a 3rd-order polynomial in $I_{drive}$:
$V_{drag}=I_{drive}R_{drag}+\gamma I_{drive}^{2}+\eta I_{drive}^{3}$, where
$\gamma$ and $\eta$ are fitting coefficients. The nonlinear effect sensitively
varies with $V_{g}$; Figure 2(b-c) shows the $V_{g}$ dependence of $\gamma$
and $\eta$ at several fixed temperatures. We find that $\gamma>0$ and $\eta<0$
for all gate voltage ranges we probe, and both quantities have larger
magnitude nearer the CNP of graphene ($V_{g}\approx V_{0}$) and at lower
temperatures. This increasingly nonlinear effect also breaks Onsager layer
reciprocity at low temperatures. As shown in Figure 2(d) and (e), the drag
resistance from SWNT-drive and graphene-drive configurations show
progressively worse correspondence at lower temperatures, as the nonlinear
part of the relation between $I_{drive}$ and $V_{drag}$ becomes appreciable.

Figure 2:  (a) $V_{drag}$ versus $I_{drive}$ at $V_{g}=22$ 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
$I_{drive}$. (d) $R_{drag}$ from linear fit versus $V_{g}$ (subtracting the
graphene CNP voltage) for reciprocal configurations at $T=180\;\textrm{K}$
(same colors and symbols as Fig. 1(b)). (e) Same measurement at
$T=117\;\textrm{K}$.

Higher-order dependence of $V_{drag}$ on $I_{drive}$ is best explained by
development of a temperature gradient in the SWNT due to the Peltier effect or
Joule heating, which can be efficiently transferred to the nearby graphene
[14, 15, 16]. Such a temperature gradient in the graphene generates a
thermoelectric voltage and causes a temperature-dependent change in
$R_{drag}$. Both can give rise to nonlinear terms in $V_{drag}$ (see SM [20]
for further analysis).

To avoid the nonlinear drag phenomena discussed above, we focus on linear drag
resistance measured at small drive current at relatively high temperature
($T>100\;\textrm{K}$). Figure 3(a) shows $R_{drag}$ as a function of gate
voltage $V_{g}$, referenced to the CNP of graphene $V_{0}$, at different fixed
temperatures in this regime. In general, $R_{drag}$ changes sign at the
graphene CNP, and that $|R_{drag}|$ grows linearly, peaks, then rapidly
decreases as the graphene carrier density, $n_{Gr}\propto V_{g}-V_{0}$,
increases. Fig. 3(b) shows that $R_{drag}\sim(V_{g}-V_{0})^{-\beta}$, where
$1<\beta<2$ at different temperatures. This behavior resembles 2D-2D graphene
drag, where $1<\beta<2$ has also been observed [6, 7].

To determine the carrier densities (and thus Fermi energies) of each conductor
as a function of $V_{g}$, we employ a finite element analysis of the graphene
channel and SWNT together with the hBN separation layers and silicon back gate
(Fig. 1(a) inset). Since the SWNT locally screens the graphene channel from
the back gate, the local carrier density in graphene is reduced in the
graphene channel directly above the SWNT and maximized away from the SWNT. To
estimate the carrier density (and thus chemical potential) of the SWNT, we
also need to consider device geometry and quantum capacitance (detailed
discussion in [20]). Figure 3(b) summarizes the estimated upper and lower
bounds of the Fermi energies of graphene $E_{F}^{Gr}$ and SWNT $E_{F}^{NT}$.
While the SWNT remains a heavily p-doped degenerate 1D conductor in the
experimental gate voltage range, our analysis suggests that $E_{F}^{Gr}$ is
comparable to or even smaller than $k_{B}T$ in the temperature range
$T>100\;\textrm{K}$, where $k_{B}$ is the Boltzmann constant, for all $V_{g}$
where the drag signal is measurable.

Near the CNP of the graphene channel, disorder becomes more relevant, creating
charge puddles [44]. The $n_{Gr}$-dependent conductance of the graphene
channel is accordingly expected to saturate at low temperatures for
$|n_{Gr}|<\delta n$, where $\delta n$ is the residual density due to charge
puddles, which can be estimated from the temperature-dependent conductance $G$
of the graphene [45]. Figure 3(d) shows $G(n_{Gr})$ measured in the graphene
channel of our device for $T\lesssim 150\;\textrm{K}$. From the saturation of
$G(n_{Gr})$ near the CNP, we estimate $\delta n\approx 1.1\times
10^{10}\;\textrm{cm}^{-2}$. For $|n_{Gr}|<\delta n$, the electron and hole
contributions of Coulomb drag cancel, resulting in linearly vanishing
$R_{drag}$ with $n_{Gr}$ as observed in the experiment (shaded region in Fig.
3(a)). We also estimate the puddle energy scale $k_{B}T_{d}=\hbar
v_{F}\sqrt{\pi\delta n}$ where $\hbar$ and $v_{F}=10^{6}\;\textrm{m/s}$ are
the reduced Plank constant and Fermi velocity of graphene, respectively. From
$\delta n$ experimentally obtained above, we find the disorder temperature
scale $T_{d}\approx 140\;\textrm{K}$, which separates the low temperature
regime where disorder effects are dominant and the high temperature regime
where thermal broadening is appreciable.

Figure 3: (a) Drag resistance as a function of $V_{g}$ at various
temperatures: 265 K (top), 235 K, 200 K, 160 K, 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
SM [20] for details of calculation. (c) Log-log plot of $R_{drag}$ versus
$V_{g}-V_{0}$ at selected temperatures, with
$R_{drag}\propto(V_{g}-V_{0})^{-1}$ (dot-dashed) and
$R_{drag}\propto(V_{g}-V_{0})^{-1.5}$ (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
$T=118\;\textrm{K}$).

We find the drag near the graphene CNP depends sensitively on temperature.
Figure 4(a) shows $T$-dependent $R_{drag}$ at fixed density (reported as
$n_{Gr}^{bulk}$, the upper bound of the estimated graphene carrier density).
For $n_{Gr}^{bulk}=\pm 1.3\times 10^{10}\;\textrm{cm}^{-2}$, close to the peak
value of $|R_{drag}(n_{Gr}^{bulk})|$, $R_{drag}$ increases linearly in
$T-T_{d}$ in the high temperature regime ($T>T_{d}$). In the low temperature
regime ($T<T_{d}$), however, the linear response $R_{drag}$ is difficult to
determine, due to the nonlinear drag effects and broken Onsager reciprocity
discussed above. At larger density (e.g. $n_{Gr}^{bulk}=\pm 8.4\times
10^{10}\;\textrm{cm}^{-2}$, far from the CNP), we observe a similar trend,
although $|R_{drag}|$ is reduced. A broader range of the density and
temperature dependent $R_{drag}(n_{Gr}^{bulk},T)$ is shown in Figure 4(b),
where the magnitude of the drag resistance appears to increase approximately
linearly at all densities. For $T>T_{d}$, the density dependence of
$\frac{dR_{drag}}{dT}$ behaves similarly to $R_{drag}(n_{Gr}^{bulk})$ [Fig.
4(c)].

Figure 4:  (a) Drag resistance as a function of temperature for various
graphene charge carrier densities (stated value is the upper bound). For
$T<140\;\textrm{K}$ (the shaded region in the graph), nonlinear effects
increasingly dominate the signal.(b) Drag resistance versus temperature and
graphene carrier density. (c) Average $\frac{dR_{drag}}{dT}$ for $T>T_{d}$
versus temperature at a range of graphene carrier densities.

The temperature dependent drag behavior discussed above is distinctly
different from 2D-2D drag in graphene, where a crossover between
$R_{drag}(T)\sim$ constant and $R_{drag}(T)\sim T^{-2}$ is expected [46] in
the parameter range of our experimental regime, $E_{F}^{Gr}\lesssim k_{B}T\sim
E_{F}^{NT}$. For 1D-2D drag between SWNT and graphene, Badalyan and Jauho
calculated the Coulomb drag effect in the Fermi liquid regime of both
conductors ($k_{B}T\ll E_{F}^{NT},E_{F}^{Gr}$) [47], predicting
$R_{drag}(T)\sim T^{\alpha}$, where $\alpha\approx 3.7$ at low temperatures. A
more general theory of 1D-2D drag [12] predicts a transition to $1<\alpha<2$
at higher temperatures ($T>T_{d}$). While a more extensive model extending to
the nondegenerate Dirac fluid limit in the presence of disorder is required
for further quantitative comparison, our experiments show qualitatively
similar behavior in the high temperature limit.

Finally, we studied the relationship between the drag signal strength and the
distance of the graphene voltage probes from the SWNT. Previous experiments
have demonstrated signatures of hydrodynamic electron flow from current
injection into a rectangular graphene channel [48, 49, 50], with discernable
effects even at room temperature [49, 50]. Viscosity of the electron fluid
causes the injected current to draw neighboring regions along with it,
resulting in a negative potential near the injection contacts and creating
current whirlpools in certain confined geometries [48, 49, 51, 52]. Our SWNT-
graphene Coulomb drag device geometry provides a unique experimental probe of
hydrodynamic flow of graphene charge carriers, as the current flowing in the
SWNT generates a direct dragging force on the graphene carriers without
injecting current in graphene. This approach should have the benefit of
eliminating diffusive “spray” from the contacts that could mask hydrodynamic
transport signatures.

Figure 5(a) shows $R_{drag}$ measured at pairs of voltage probes in the
graphene channel laterally displaced by distance $x$ away from the SWNT.
$R_{drag}$ decreases as $x$ increases, becoming almost unmeasurable for
$x>2\;\mu$m. In Ohmic transport, such a diminishing drag signal can be
understood with a diffusive model, where the escaping current density in the
graphene just above the SWNT is expected to decay as $J_{esc}(x)\sim e^{-\pi
x/W}$, where $W$ is the channel width [53]. In the diffusive transport regime,
we therefore expect driving current in the SWNT to cause a drag voltage in the
probes at distance $x$ away following $R_{drag}(x)\sim e^{-\pi x/W}$. The
inset of Figure 5(a) shows that the measured $R_{drag}(x)$ follows such an
exponential decay. We obtain the effective channel width $W_{\textrm{eff}}$ by
fitting this functional dependence. Figure 5(b) shows $W_{\textrm{eff}}$ as a
function of $V_{g}$. Interestingly, we find $W_{\textrm{eff}}$ is larger than
the physical channel width $W=1\;\mu$m in our device when the graphene is in
the Dirac fluid regime, enhanced by about a factor of 2 at the CNP. Based on
previous observations that the electron fluid of graphene is highly viscous in
this temperature range near the CNP [48, 54], the increase in
$W_{\textrm{eff}}$ may hint at a hydrodynamic contribution to the transport
behavior.

Figure 5:  (a) $R_{drag}$ versus $V_{g}$ at $T=300\;\textrm{K}$ for pairs of
voltage probes at increasing distance $x$ from the SWNT: 800 nm (circles), 1.2
$\mu$m (squares), 1.4 $\mu$m (triangles), and 2 $\mu$m (stars). Inset:
$R_{drag}$ at $V_{g}=24\;\textrm{V}$ for increasing distances. Dashed curve is
an exponential fit (see main text). (b) Effective channel width
$W_{\textrm{eff}}$ versus $V_{g}$, extracted from fit for $R_{drag}$ at
$T=300\;\textrm{K}$. Dashed line marks actual device width ($W=1\;\mu$m).

In summary, we present an experimental study of mixed-dimensional Coulomb drag
between a SWNT and graphene. Our drag measurements in a SWNT-graphene
heterostructure are qualitatively consistent with momentum transfer between
the drive and drag layers, although we also observe an onset of nonlinearity
due to local energy transfer combined with temperature dependent drag effects
at lower temperatures. Within the linear response regime, the dependences on
temperature, carrier density, and distance have subtleties that suggest an
interplay of different mechanisms at work in this novel hybrid system. Further
measurements with higher spatial resolution, such as current imaging [55, 50],
would be necessary to gain a deeper understanding of the current flow
patterns, and samples with less disorder should amplify hydrodynamic transport
signatures in the graphene [51].

###### Acknowledgements.

We acknowledge Leonid Levitov, Patrick Ledwith, and Artem Talanov for useful
discussions. Sample preparation and device fabrication was supported by ONR
MURI (N00014-16-1-2921). P.K. and A.C. acknowledge support from the ARO
(W911NF-17-1-0574) for a part of measurement and analysis. L.E.A. acknowledges
support from the ARO through the NDSEG Fellowship Program. K.W. and T.T.
acknowledge support from the Elemental Strategy Initiative conducted by the
MEXT, Japan (grant number JPMXP0112101001) and JSPS KAKENHI (grant numbers
19H05790 and JP20H00354). This work was performed, in part, at the Center for
Nanoscale Systems (CNS), a member of the National Nanotechnology
Infrastructure Network, which is supported by the NSF under Grant No.
ECS-0335765. CNS is part of Harvard University.

### References

  * Lucas and Fong [2018] A. Lucas and K. C. Fong, Journal of Physics: Condensed Matter 30, 053001 (2018), 1710.08425 .
  * Narozhny and Levchenko [2016] B. N. Narozhny and A. Levchenko, Reviews of Modern Physics 88, 025003 (2016), 1505.07468 .
  * Gramila  _et al._ [1991] T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer, and K. W. West, Physical Review Letters 66, 1216 (1991).
  * Hill  _et al._ [1996] N. P. R. Hill, J. T. Nicholls, E. H. Linfield, M. Pepper, D. A. Ritchie, A. R. Hamilton, and G. A. C. Jones, Journal of Physics: Condensed Matter 8, L557 (1996).
  * Liu  _et al._ [2017] X. Liu, L. Wang, K. C. Fong, Y. Gao, P. Maher, K. Watanabe, T. Taniguchi, J. Hone, C. Dean, and P. Kim, Physical Review Letters 119, 056802 (2017), 1612.08308 .
  * Kim  _et al._ [2011] S. Kim, I. Jo, J. Nah, Z. Yao, S. K. Banerjee, and E. Tutuc, Physical Review B 83, 161401(R) (2011), 1010.2113 .
  * Gorbachev  _et al._ [2012] R. V. Gorbachev, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. Tudorovskiy, I. V. Grigorieva, A. H. MacDonald, S. V. Morozov, K. Watanabe, T. Taniguchi, and L. A. Ponomarenko, Nature Physics 8, 896 (2012), 1206.6626 .
  * Yamamoto  _et al._ [2006] M. Yamamoto, M. Stopa, Y. Tokura, Y. Hirayama, and S. Tarucha, Science 313, 204 (2006).
  * Laroche  _et al._ [2011] D. Laroche, G. Gervais, M. P. Lilly, and J. L. Reno, Nature Nanotechnology 6, 793 (2011), arXiv:1008.5155v3 .
  * Laroche  _et al._ [2014] D. Laroche, G. Gervais, M. P. Lilly, and J. L. Reno, Science 343, 631 (2014), 1312.4950 .
  * Sirenko and Vasilopoulos [1992] Y. M. Sirenko and P. Vasilopoulos, Physical Review B 46, 1611 (1992).
  * Lyo [2003] S. K. Lyo, Physical Review B 68, 045310 (2003).
  * Mitra  _et al._ [2020a] R. Mitra, M. R. Sahu, K. Watanabe, T. Taniguchi, H. Shtrikman, A. K. Sood, and A. Das, Physical Review Letters 124, 116803 (2020a), 2002.09874 .
  * Song and Levitov [2012] J. C. W. Song and L. S. Levitov, Physical Review Letters 109, 236602 (2012), 1205.5257 .
  * Song  _et al._ [2013] J. C. W. Song, D. A. Abanin, and L. S. Levitov, Nano Letters 13, 3631 (2013), 1304.1450 .
  * Song and Levitov [2013] J. C. W. Song and L. S. Levitov, Physical Review Letters 111, 126601 (2013), 1303.3529 .
  * Mitra  _et al._ [2020b] R. Mitra, M. R. Sahu, A. Sood, T. Taniguchi, K. Watanabe, H. Shtrikman, S. Mukerjee, A. K. Sood, and A. Das, (2020b), arXiv:2009.08882 .
  * Laird  _et al._ [2015] E. A. Laird, F. Kuemmeth, G. A. Steele, K. Grove-Rasmussen, J. Nygård, K. Flensberg, and L. P. Kouwenhoven, Reviews of Modern Physics 87, 703 (2015), 1403.6113 .
  * Castro Neto  _et al._ [2009] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Reviews of Modern Physics 81, 109 (2009), 0709.1163 .
  * [20] See Supplemental Material for detailed fabrication and measurement methods, discussion of the effect of the graphene chemical potential on the observed drag, estimation of graphene and SWNT charge carrier densities, and additional experimental data. It also contains Refs. [21-42].
  * Cheng  _et al._ [2019] A. Cheng, T. Taniguchi, K. Watanabe, P. Kim, and J.-D. Pillet, Physical Review Letters 123, 216804 (2019), arXiv:1910.13307 .
  * Sfeir  _et al._ [2006] M. Y. Sfeir, T. Beetz, F. Wang, L. Huang, X. M. H. Huang, M. Huang, J. Hone, S. O’Brien, J. A. Misewich, T. F. Heinz, L. Wu, Y. Zhu, and L. E. Brus, Science 312, 554 (2006).
  * Huang  _et al._ [2005] X. M. H. Huang, R. Caldwell, L. Huang, S. C. Jun, M. Huang, M. Y. Sfeir, S. P. O’Brien, and J. Hone, Nano Letters 5, 1515 (2005).
  * Cao  _et al._ [2015] Q. Cao, S.-J. Han, J. Tersoff, A. D. Franklin, Y. Zhu, Z. Zhang, G. S. Tulevski, J. Tang, and W. Haensch, Science 350, 68 (2015).
  * Huang  _et al._ [2015] J.-W. Huang, C. Pan, S. Tran, B. Cheng, K. Watanabe, T. Taniguchi, C. N. Lau, and M. Bockrath, Nano Letters 15, 6836 (2015).
  * Kim  _et al._ [2005] W. Kim, A. Javey, R. Tu, J. Cao, Q. Wang, and H. Dai, Applied Physics Letters 87, 173101 (2005).
  * Pitner  _et al._ [2019] G. Pitner, G. Hills, J. P. Llinas, K.-M. Persson, R. Park, J. Bokor, S. Mitra, and H.-S. P. Wong, Nano Letters 19, 1083 (2019).
  * Dean  _et al._ [2010] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, Nature Nanotechnology 5, 722 (2010), arXiv:1005.4917 .
  * Wang  _et al._ [2013] L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
  * Li  _et al._ [2016] J. I. A. Li, T. Taniguchi, K. Watanabe, J. Hone, A. Levchenko, and C. R. Dean, Physical Review Letters 117, 046802 (2016), arXiv:1602.01039 .
  * Peres  _et al._ [2011] N. M. R. Peres, J. M. B. Lopes dos Santos, and A. H. Castro Neto, EPL (Europhysics Letters) 95, 18001 (2011).
  * Ilani  _et al._ [2006] S. Ilani, L. A. K. Donev, M. Kindermann, and P. L. McEuen, Nature Physics 2, 687 (2006).
  * Wong and Akinwande [2010] H.-S. P. Wong and D. Akinwande,  _Carbon Nanotube and Graphene Device Physics_ (Cambridge University Press, Cambridge, 2010) pp. 139–148.
  * Ng [1995] K. K. Ng,  _Complete Guide to Semiconductor Devices_ , 1st ed. (McGraw-Hill, New York, NY, 1995).
  * Zhang  _et al._ [2017] K. Zhang, Y. Feng, F. Wang, Z. Yang, and J. Wang, Journal of Materials Chemistry C 5, 11992 (2017).
  * Laturia  _et al._ [2018] A. Laturia, M. L. Van de Put, and W. G. Vandenberghe, npj 2D Materials and Applications 2, 6 (2018).
  * Seol  _et al._ [2010] J. H. Seol, I. Jo, A. L. Moore, L. Lindsay, Z. H. Aitken, M. T. Pettes, X. Li, Z. Yao, R. Huang, D. Broido, N. Mingo, R. S. Ruoff, and L. Shi, Science 328, 213 (2010).
  * Elias  _et al._ [2011] D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Morozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V. Grigorieva, K. S. Novoselov, F. Guinea, and A. K. Geim, Nature Physics 7, 701 (2011), arXiv:1104.1396 .
  * Nicklow  _et al._ [1972] R. Nicklow, N. Wakabayashi, and H. G. Smith, Physical Review B 5, 4951 (1972).
  * Lu  _et al._ [2007] W. Lu, D. Wang, and L. Chen, Nano Letters 7, 2729 (2007).
  * Kozinsky and Marzari [2006] B. Kozinsky and N. Marzari, Physical Review Letters 96, 166801 (2006), arXiv:0602599 [cond-mat] .
  * Blundell and Blundell [2009] S. J. Blundell and K. M. Blundell,  _Concepts in Thermal Physics_, 2nd ed. (Oxford University Press, Oxford, 2009) pp. 413–415.
  * Hwang  _et al._ [2011] E. H. Hwang, R. Sensarma, and S. Das Sarma, Physical Review B 84, 245441 (2011).
  * Martin  _et al._ [2008] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing, and A. Yacoby, Nature Physics 4, 144 (2008), 0705.2180 .
  * Couto  _et al._ [2014] N. J. G. Couto, D. Costanzo, S. Engels, D. K. Ki, K. Watanabe, T. Taniguchi, C. Stampfer, F. Guinea, and A. F. Morpurgo, Physical Review X 4, 041019 (2014), 1401.5356 .
  * Narozhny  _et al._ [2012] B. N. Narozhny, M. Titov, I. V. Gornyi, and P. M. Ostrovsky, Physical Review B 85, 195421 (2012), 1110.6359 .
  * Badalyan and Jauho [2020] S. M. Badalyan and A. P. Jauho, Physical Review Research 2, 013086 (2020), 1906.05517 .
  * Bandurin  _et al._ [2016] D. A. Bandurin, I. Torre, R. K. Kumar, M. Ben Shalom, A. Tomadin, A. Principi, G. H. Auton, E. Khestanova, K. S. Novoselov, I. V. Grigorieva, L. A. Ponomarenko, A. K. Geim, and M. Polini, Science 351, 1055 (2016), 1509.04165 .
  * Berdyugin  _et al._ [2019] A. I. Berdyugin, S. G. Xu, F. M. D. Pellegrino, R. Krishna Kumar, A. Principi, I. Torre, M. Ben Shalom, T. Taniguchi, K. Watanabe, I. V. Grigorieva, M. Polini, A. K. Geim, and D. A. Bandurin, Science 364, 162 (2019), 1806.01606 .
  * Ku  _et al._ [2020] M. J. H. Ku, T. X. Zhou, Q. Li, Y. J. Shin, J. K. Shi, C. Burch, L. E. Anderson, A. T. Pierce, Y. Xie, A. Hamo, U. Vool, H. Zhang, F. Casola, T. Taniguchi, K. Watanabe, M. M. Fogler, P. Kim, A. Yacoby, and R. L. Walsworth, Nature 583, 537 (2020).
  * Levitov and Falkovich [2016] L. Levitov and G. Falkovich, Nature Physics 12, 672 (2016), 1508.00836 .
  * Pellegrino  _et al._ [2017] F. M. D. Pellegrino, I. Torre, and M. Polini, Physical Review B 96, 195401 (2017), 1706.08363 .
  * Abanin  _et al._ [2011] D. A. Abanin, S. V. Morozov, L. A. Ponomarenko, R. V. Gorbachev, A. S. Mayorov, M. I. Katsnelson, K. Watanabe, T. Taniguchi, K. S. Novoselov, L. S. Levitov, and A. K. Geim, Science 332, 328 (2011).
  * Bandurin  _et al._ [2018] D. A. Bandurin, A. V. Shytov, L. S. Levitov, R. K. Kumar, A. I. Berdyugin, M. Ben Shalom, I. V. Grigorieva, A. K. Geim, and G. Falkovich, Nature Communications 9, 4533 (2018), 1806.03231 .
  * Sulpizio  _et al._ [2019] J. A. Sulpizio, L. Ella, A. Rozen, J. Birkbeck, D. J. Perello, D. Dutta, M. Ben-Shalom, T. Taniguchi, K. Watanabe, T. Holder, R. Queiroz, A. Principi, A. Stern, T. Scaffidi, A. K. Geim, and S. Ilani, Nature 576, 75 (2019), 1905.11662 .

Supplemental Material for “Coulomb Drag Between a Carbon Nanotube and
Monolayer Graphene”

Contents

I: Fabrication of SWNT-graphene devices I A: SWNT synthesis, characterization,
and transfer I.1 B: Graphene-hBN heterostructure fabrication and transfer I.2
C: Additional nanofabrication I.3 II: SWNT conductance II III: DC measurement
technique III IV: Additional Onsager data IV V: Graphene chemical potential
and its effect on Coulomb drag V VI: Estimation of SWNT and graphene carrier
densities VI VII: Derivation of nonlinear contributions to the drag voltage
VII VIII: Comparison of individual layer transport and drag resistance in
different measurements VIII References References

###  I I: Fabrication of SWNT-graphene devices

More complete details of some device fabrication processes are described
elsewhere, particularly in [1].

####  I.1 A: SWNT synthesis, characterization, and transfer

Carbon nanotubes are grown in a chemical vapor deposition (CVD) furnace using
the method described in [2]. The growth substrate is a 5 mm $\times$ 5 mm
silicon chip with a slit in the center [Fig. S1(a)], oriented perpendicular to
the gas flow direction. A cobalt-molybdenum-based catalyst is applied to the
chip on the side of the slit nearer to the gas inlet, so that nanotubes grow
suspended across the slit. Suspended nanotubes were located and characterized
using Rayleigh scattering spectroscopy [2] and imaging [Fig. S1(d)]. By
matching peaks in Rayleigh scattering intensity with nanotube optical
transition energies, the chiral indices (and thus diameter and
metallic/semiconducting nature) of the nanotube can be determined [Fig.
S1(c)]. The scattered light is also routed to a camera, providing an image of
the nanotube spanning the slit [Fig. S1(b)]. All the SWNTs used in devices
described in this paper were metallic.

Electron beam (e-beam) lithography was used to define a $\sim 50\times
50\;\mu$m resist-free window on a SiO2/Si chip coated with $\sim$100 nm of 495
K Polymethyl methacrylate (PMMA) A4 resist. The resist layer helps the SWNTs
transfer to the chip S_Huang2005. When a single SWNT with the desired
characteristics has been located, it is aligned with the heterostructure so
that it crosses the center of the PMMA window [Fig. S2(a-b)]. The growth chip
and PMMA-coated sample are pressed together until mechanical contact is
evident [Fig. S2(c)], then heated to 180 °C for 5 minutes to soften the
resist. The chips are then cooled to 90 °C and slowly separated [Fig. S2(d)].
Successful SWNT transfer is confirmed by scanning electron microscope (SEM) or
atomic force microscope (AFM) imaging. The same PMMA window can be reused
several times in the event of unsuccessful transfer, or to transfer multiple
SWNTs for parallel device fabrication. Finally, the PMMA is removed by high-
temperature annealing in vacuum.

To anchor the SWNT to the substrate and confirm its suitability to be
incorporated into a device, electrical contacts were made to either end of a
30-50 $\mu$m section of the SWNT, inside the window region [Fig. S2(e)]. The
mask for the contacts was defined by e-beam lithography, and metal was
deposited by e-beam evaporation (10 nm Cr/60 nm Au). Many other recipes for
high-quality SWNT contacts have been reported in literature, including some
measurements with quantum-limited contact resistance
S_Cao2015,S_Huang2015,S_Kim2005,S_Pitner2019. However, this recipe was found
to remain the most reliable through the additional fabrication steps.

####  I.2 B: Graphene-hBN heterostructure fabrication and transfer

Boron nitride-encapsulated graphene heterostructures were prepared using
standard techniques S_Dean2010,S_Wang2013. The top hBN flake (20-40 nm) is
picked up with a polypropylene carbonate (PPC) film on a Polydimethylsiloxane
(PDMS) stamp, and then used to pick up graphene and bottom hBN (2-5 nm)
flakes. It was critical to ensure that the bottom hBN flake was both thin and
large enough to completely cover the graphene (at least in the region intended
to be near the SWNT), in order to allow interaction between the graphene and
SWNT without shorting. Once assembled on a stamp, the stack was transferred on
top of a contacted SWNT [Fig. S2(f-g)]. The PPC and stack were detached from
the stamp by heating to 150° C to melt the PPC, which was then removed by
high-temperature vacuum annealing.

####  I.3 C: Additional nanofabrication

Following stack transfer, the heterostructure was shaped into a bar (with
additional extended regions above the SWNT to avoid etching it) by reactive
ion etching the heterostructure with CHF3 through a resist mask defined by
e-beam lithography. A second lithography step defined the graphene contact
electrodes [Fig. S2(h)], which were made by reactive ion etching to expose a
clean graphene edge, and then depositing a metallic trilayer (2 nm Cr/8 nm
Pd/50 nm Au) using thermal evaporation (as described in S_Wang2013). A final
device is shown in [Fig. S2(i)].

We briefly note that it is possible to invert the order of the layers in the
SWNT-graphene heterostructure, so that the hBN-encapsulated graphene is placed
on the substrate first, and then a CNT is transferred on top. This improves
the quality of the graphene by enabling the use of a thicker hBN layer between
the graphene and the SiO2. However, the extremely low friction between SWNTs
and clean hBN can result in the SWNT bending and shifting from the intended
transfer position, sometimes shorting to exposed parts of the graphene or
breaking due to stress in fabrication. In contrast, transferring the SWNT to
SiO2 pins the SWNT to the rougher substrate, and allows for screening out
poorly-conducting SWNTs before transferring a stack. It was thus found to be a
more reliable fabrication method, although the yield of working devices after
SWNT transfer was still low due to poor electrical contact to the SWNT, SWNTs
breaking during stack transfer, and the thin, bottom hBN flake shifting or
cracking during stack transfer and allowing the graphene to short the SWNT.

###  II II: SWNT conductance

This section is a brief note on the transport behavior of the SWNTs in our
devices. In Figure 1(c), the conductance of the SWNT decreases with increasing
gate voltage $V_{g}$, characteristic of a hole-doped nanotube. There is also a
notable dip in the conductance around $V_{g}=40$ V. While at first this may
appear to be the charge neutrality point of the metallic SWNT, it was found to
shift its position depending on the direction of the gate voltage sweep, as
shown in Figure S3(a). However, the overall SWNT conductance away from the dip
generally decreased with increasing $V_{g}$, regardless of the gate sweep
direction. Furthermore, the drag resistance measured during the same gate
voltage sweeps does not change polarity when the SWNT conductance dip is on
the left side of the graphene CNP as opposed to on the right (Fig. S3(b)). We
therefore conclude that the SWNT appears to remain hole-doped at all
accessible $V_{g}$ values. The most likely origin of this conductance dip
shift is doping from charges on the substrate, which are reconfigured as the
silicon gate is kept at a particular voltage (as before the start of a gate
sweep) and then swept. The uncertainty in the nanotube CNP position is
incorporated into estimates of the SWNT Fermi energy plotted in the main text
(Fig. 3(b) of main text). More details of the calculation are in Section VI
below.

Due to the high hole density in the SWNT, the Fermi wavevectors of the SWNT
and graphene are significantly mismatched when the drag signal is maximized
near the graphene CNP (see Section V for further discussion of drag
mechanisms). Although the increased electron-electron scattering phase space
when $k_{F}^{Gr}=k_{F}^{NT}$ may generically lead to an enhancement
$R_{drag}$, we expect that the high $k_{F}$ where this would occur would mean
the Coulomb interaction is already quite suppressed, reducing the drag signal.

In some devices, we do observe a change in the polarity of the drag resistance
on either side of a dip in SWNT conductance. We provide an example in a
double-gated bilayer graphene (BLG)-SWNT drag device, also used to estimate
capacitance ratios in Supplemental Material section VI. Figure S4(a) shows the
SWNT conductance tuned by the back gate (the top gate has almost no effect on
the SWNT, as it is generally screened by the graphene) and (b) is a color plot
of drag resistance as a function of back gate and top gate voltages. There is
a clear sign reversal with both the graphene CNP (tuned by both top and back
gates and thus diagonal on this plot) and SWNT conductance dip (tuned only by
the back gate). This double sign reversal allows us to say with some
confidence that the SWNP conductance dip is in fact the CNP.

###  III III: DC measurement technique

Low-frequency AC measurements are a standard technique for electrical
transport experiments, including many Coulomb drag and other double-layer
measurements (e.g. Refs. [10, 11]). However, drag measurements are sensitive
to parasitic effects that can obscure the behavior that truly arises from
interlayer charge carrier interactions [12, 13]. For example, when a bias is
applied to the drive layer to initiate current flow, the layer acquires a non-
zero potential due to contact and layer resistances of the drive layer, which
may cause an asymmetric gating effect on the drag layer. This is particularly
problematic for AC measurements, since the alternating potential on the drive
layer can capacitively couple to the drag layer and generate an alternating
current, which can cause a spurious alternating voltage signal in the drag
layer due to its contact and layer resistances [13].

The interlayer potential can be manually adjusted to approximately zero using
resistance bridges [12, 13]. However, as the gate voltage is changed during
the measurement, the layer resistances change substantially, unbalancing the
bridge circuit. Another problem in our system is the disparity in contact
resistances between the graphene and SWNT; while graphene contacts typically
have resistances on the order of 100 $\Omega$, the lowest SWNT resistances we
achieved in these devices were $\sim 100$ k$\Omega$. This renders the SWNT-
graphene drag devices very susceptible to interlayer bias effects.

DC measurements avoid introducing spurious signals from capacitive coupling
but are more sensitive to noise and must be carefully monitored to ensure the
signal in the drag layer is not purely thermoelectric in origin, as addressed
in the main text. To circumvent these issues, we performed our measurements by
symmetrically DC biasing the drive layer, applying $\pm V_{drive}/2$ to either
side of the SWNT or to a pair of contacts in the graphene (adding a pair of
resistors in series to limit the current; the resistor values can be modified,
or an additional resistor inserted, to account for differences in contact
resistance) and measuring the resulting $I_{drive}$. The circuits are shown in
Figure S3. The value of $V_{drive}$ (and thus $I_{drive}$) is swept through a
range of values (typically
$V_{drive}=0.3\;\textnormal{V}\rightarrow-0.3\;\textnormal{V}$ on one contact,
and $-0.3\;\textnormal{V}\rightarrow 0.3\;\textnormal{V}$ on the other)
multiple times, and the detected $V_{drag}$ values at each $I_{drive}$ were
averaged to reduce noise. The error for each displayed $V_{drag}$ data point
is calculated using the standard error on the mean:
$\sigma_{\bar{x}}=\sigma/\sqrt{n}$, where $\bar{x}$ represents the average of
a population of measurements, $\sigma$ is the standard deviation, and $n$ is
the number of independent measurements.

###  IV IV: Additional Onsager data

We present here additional data comparing the drag response using the two
different circuit configurations: driving current in the SWNT and measuring
voltage across two probes on graphene, and driving current in the graphene and
measuring voltage across the SWNT.

Figure S6 shows a color plot of drag voltage versus drive current and gate
voltage at $T=300$ K, using the graphene as the drive layer and SWNT as the
drag layer. We have applied a narrow ($\pm 3$ V range) moving average filter
in the x-direction (gate voltage axis), essentially binning the data from
adjacent gate voltages together, to reduce noise and make overall trends in
the data more apparent. The same general behavior is apparent in this
reciprocal configuration as shown in the main text for SWNT-drive, graphene-
drag measurements (Figure 1(e) of the main text), although the higher degree
of noise in measurements using the SWNT as the drag layer complicates a
straightforward comparison based on the color plots alone.

Additional examples of drag voltage versus drive current data for reciprocal
drag circuit configurations at individual gate voltages are shown in Figure
S7. Our moving average filter is restricted to an even smaller range ($\pm 1$
V) for better quantitative comparison. The range of $I_{drive}$ is restricted
to avoid nonlinearites due to high bias, although some nonlinearity in the
SWNT-drive data (orange, filled circles) at gate voltages nearer the graphene
CNP (e.g. Fig. S7(c-e)). Although there is more noise in the measurements for
which the SWNT was the drag layer (blue, open circles), the general degree to
which the two data sets coincide and the overall linearity of both
measurements provide strong evidence that Onsager reciprocity is respected in
this regime.

The higher level of noise in the measured drag voltage when the SWNT is the
drag layer, which is consistently observed across all gate voltages and
temperature, merits some discussion. There are a few potential reasons for
increased voltage fluctuations when the SWNT is the drag layer. First of all,
the drag resistance is on the order of a few Ohms, while the channel
resistance of graphene is less than $\sim$ 1 k$\Omega$, and the resistance of
the SWNT is larger than 1 M$\Omega$. Such a large disparity between the SWNT
resistance and the drag resistance makes it challenging to detect small
resistance variations when we use the SWNT as drag layer. The higher
resistance of the SWNT means that small current fluctuations result in larger
voltage noise than would be observed if another graphene sheet were used as
the drive layer.

An additional contributing factor may be random reconfiguration of mobile
charges on the SiO2 substrate. In our device geometry, the graphene channel is
encapsulated, but the SWNT is in direct contact with the SiO2. Thus,
stochastic charge fluctuations in the charge traps on the substrate can induce
voltage fluctuations in the SWNT when it is being used as drag layer to probe
potential. For SWNT drive, we apply a relatively large bias voltage to obtain
the same amount of drive current, thus usually fluctuations in the charge
environment would not affect the driving current. The graphene drag layer is
much less disordered and thus less susceptible to charge fluctuations on the
SiO2.

Finally, the increased noise level in the graphene drive-nanotube drag
configuration may be a result of the high SWNT resistance relative to the
amplifier impedance. The input impedance of the voltage amplifier we use is
100 M$\Omega$ (SR560) while the load resistance of nanotube is $\sim$2
M$\Omega$. In order to avoid spurious capacitive coupling, we measure the DC
drag at slow speed, corresponding to 300 ms of averaging time and $\sim$1
second per data point acquisition. In this regime of low frequency measurement
with large source resistance, the noise figure (NF) of our preamp is $\sim$0.5
dB (https://www.thinksrs.com/products/sr560.htm), suggesting that the noise is
completely dominated by Johnson noise across the source resistance. The
effective rms noise is $\sim$0.5 $\mu$V, comparable to the fluctuation of the
data we observed in the graphene drive-nanotube drag measurement. Note that
for the nanotube drive-graphene drag measurement, the source resistance drops
to $\sim$1 k$\Omega$, where the fluctuation in the measured signal is
dominated by amplifier noise (NF $\sim$20). We estimate that the rms noise for
this measurement is $\sim$20 nV due to the reduced source resistance despite
the increased amplifier noise.

###  V V: Graphene chemical potential and its effect on Coulomb drag

This section describes a method to qualitatively relate the charge carrier
density dependence (more precisely, the gate voltage dependence) of $R_{drag}$
in the SWNT-graphene system by comparison to perturbation theory for graphene-
graphene drag S_Narozhny2012,S_Peres2011. Although this theory cannot account
for the mixed-dimensional nature of our devices, its applicability in a broad
range of carrier densities and temperatures make it a useful framework to
understand some of the behavior in our system. Furthermore, the discrepancies
between the 2D-2D theory and 1D-2D experiment may illuminate the ways in which
dimensionality plays a role in the Coulomb drag behavior.

In comparing our experimental data with theoretical models, it is critical to
know the charge carrier densities in the SWNT and graphene ($n_{NT}$ and
$n_{Gr}$), as well as their relationship with the chemical potential $\mu$ in
each conductor. The net charge carrier density of single-gated graphene
heterostructures is generally well approximated by a parallel-plate capacitor
formula:

$n_{Gr}=\frac{C(V_{g}-V_{0})}{e},$ (S1)

where $V_{0}$ is the gate voltage of the graphene charge neutrality point
(CNP) and $C$ is the capacitance per unit area between the gate and graphene.
However, this formula ignores the effect of the metallic SWNT on the local
electrostatic environment of the graphene. Similarly, we cannot simply apply
the analytic formula for the capacitance between a wire and a conducting plane
[16] to estimate $n_{NT}$ because the nearby graphene will substantially
screen the electric field from the back gate, even though it is not situated
between the gate and SWNT. In addition, our experiments are performed at
relatively high temperature, so the approximation $\mu\approx E_{F}$ is not
necessarily valid.

A more accurate approach to modeling the carrier densities and chemical
potentials of the two layers (neglecting momentarily the local effect of the
SWNT on the nearest region of the graphene) starts with the coupling between
the back gate and the conductors, given by:

$\displaystyle eV_{g}=\mu_{Gr}+\frac{e^{2}(n_{Gr}+n_{NT}/2\pi r_{NT})}{C_{1}}$
(S2a) $\displaystyle\mu_{Gr}=\mu_{NT}+\frac{e^{2}n_{NT}}{C_{2}}$ (S2b)

where $C_{1}$ is the capacitance per unit area between the graphene and back
gate, $C_{2}$ is the capacitance per unit length between the SWNT and
graphene, and $r_{NT}$ is the radius of the SWNT. This is analogous to the
single-gated graphene-graphene drag device considered in [14, 15, 17]. The
SWNT takes the role of the “top” (more heavily screened) layer despite its
position between the graphene and back gate because it is much closer to the
graphene than to the back gate (2 nm versus 1 $\mu$m). In contrast, the
screening effect of the SWNT on the graphene is significant but confined to
$<10$ nm on either side of the SWNT (see Fig. S5 and discussion of a finite
element method based on COMSOL simulations in the following section for more
details). Figure S4 schematically illustrates the effect of the back gate on
the graphene and SWNT bands.

To verify that $n_{Gr}$, $n_{NT}$ are linearly proportional to $V_{g}$, we can
find the density $n_{Gr}^{*}$ at which the electrical and chemical potentials
become comparable:

$\hbar v_{F}\sqrt{\pi n_{Gr}^{*}}=\frac{e^{2}n_{Gr}^{*}}{C_{1}}\rightarrow
n_{Gr}^{*}\approx 7\times 10^{6}\;\textnormal{cm}^{-2},$ (S3)

which is several orders of magnitude smaller than the residual impurity
density, $\delta n\sim 1.1\times 10^{10}\;\textnormal{cm}^{-2}$, and
corresponds to $V_{g}=0.3$ mV, smaller than the resolution of our data points.
Thus, the quantum capacitance of graphene can be neglected in our experiment,
and $V_{g}\propto n_{Gr}$ for all densities under consideration. Furthermore,
the SWNT carrier density does not seem to substantially affect the drag
behavior, apart from reducing the magnitude of $R_{drag}$ as $n_{NT}$
increases (evident in lower magnitude of the h-h drag on the left side of the
graphene CNP compared to the e-h drag on the right side of the graphene CNP,
e.g. in main text Figure 1(f)). We observe essentially similar drag behavior
regardless of the position of the SWNT conductance dip discussed in Section
II, as long as it does not overlap with the CNP of the graphene channel. Since
experimentally, the SWNT is observed to be heavily p-doped, and remains
degenerate, $V_{g}\propto n_{NT}\propto\mu_{NT}$, considering the constant
density of states of the 1D SWNT band structure [18]. It should also be noted
that, due to the chiral nature of graphene, the enhancement of electron-
electron scattering at matched Fermi wavevectors (when $q=2k_{F}$) that is
typical in 2D electronic systems is not present in graphene-graphene drag
systems [19]. Investigation of such enhancement in the SWNT-graphene system
requires low-doping nanotubes and transport measurements near the nanotube
CNP, an avenue for future study.

We can now consider 3 possible regimes of drag response, based on $\mu_{Gr}$
and $T$. When both are small ($\mu_{Gr}$, $k_{B}T$ $<\tau^{-1}$, where
$\tau(\mu_{Gr},T)$ is the scattering time), the transport is disorder-
dominated and $n_{Gr}$ becomes temperature-independent [14]:

$n_{Gr}(\mu_{Gr},k_{B}T<\tau^{-1})=\frac{\mu_{Gr}}{\hbar v_{F}^{2}\tau}.$ (S4)

The relevant expression for Coulomb drag in this regime is either

$\rho_{drag}(\mu_{Gr},\mu_{NT}\ll T)\approx
1.41\alpha^{2}\frac{\hbar}{e^{2}}\frac{\mu_{Gr}\mu_{NT}}{k_{B}^{2}T^{2}},$
(S5)

where $\alpha=e^{2}/v_{F}$ is the interaction strength, or at higher
$\mu_{NT}$ (with $\mu_{NT}>\mu_{Gr}$ in our experiment),

$\rho_{drag}(\mu_{Gr}\ll
k_{B}T\ll\mu_{NT})=5.8\alpha^{2}\frac{\hbar}{e^{2}}\frac{\mu_{Gr}}{\mu_{NT}}.$
(S6)

In either case, $\rho_{drag}\propto\mu_{Gr}$, and since we have established
that $\mu_{Gr}\propto n_{Gr}\propto V_{g}$ in this regime, the theory predicts
$\rho_{drag}\propto V_{g}$. This prediction agrees with our $R_{drag}$ data
for gate voltages close to the graphene CNP [see for example Fig. 1(d),(f) of
the main text]. Due to the high SWNT hole density in the accessible gate
voltage range in our devices, as discussed in Section II, we expect that the
regime of Equation S5 is not observed in our experiments.

At higher $T$, the graphene charge carriers close to the CNP ($T\tau>1$,
$\mu_{Gr}\ll T$) are in the Dirac fluid regime. In this case, the carrier
density remains linear in $\mu$ but acquires temperature dependence:

$n_{Gr}(\mu_{Gr}\ll k_{B}T)=\frac{1}{\hbar^{2}\pi
v_{F}^{2}}4\textnormal{ln}(2)\mu_{Gr}k_{B}T.$ (S7)

A similar relationship between $\rho_{drag}$ and $V_{g}$ holds as in the
disorder-dominated regime, although the temperature dependence is reduced by a
factor of $T$ (since now $\mu_{Gr}\propto n_{Gr}/T$). However, since we have
relatively few data points in this regime, it is difficult to conclusively
compare the theoretical and experimental temperature dependences.

Finally, $\mu_{Gr}\gg k_{B}T$ is the Fermi liquid regime, where $n_{Gr}$ once
again loses its temperature dependence:

$n_{Gr}(\mu_{Gr}\gg k_{B}T)=\frac{1}{\hbar\pi v_{F}^{2}}\mu_{Gr}^{2}.$ (S8)

We must also consider the Fermi liquid expression for the Coulomb drag:

$\rho_{drag}(\mu_{NT}>\mu_{Gr}\gg
k_{B}T)\approx\alpha^{2}\frac{\hbar}{e^{2}}\frac{8\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\mu_{Gr}\mu_{NT}}\textnormal{ln}\left(\frac{\mu_{Gr}}{k_{B}T}\right),$
(S9)

which implies $\rho_{drag}\propto T^{2}/V_{g}^{1/2}$, neglecting any
contribution from the SWNT. While this is qualitatively similar to the
experimental behavior of $R_{drag}$ at higher $V_{g}$ (i.e. increasing with
temperature and decaying as a power law with $V_{g}$), it does not align with
a more quantitative analysis of the data (which finds $R_{drag}\propto
V_{g}^{-1}$ to $V_{g}^{-1.3}$ at $140\;\textnormal{K}<T<300\;\textnormal{K}$;
see Fig. 3(c) of the main text). This disagreement suggests a more detailed
theoretical analysis of the SWNT-graphene drag system is required to fully
understand the mechanisms at play.

###  VI VI: Estimation of SWNT and graphene carrier densities

This section expands on the estimations of SWNT and graphene carrier densities
discussed in the first part of the previous section. As stated,
straightforward application of the analytical formulae for capacitance between
parallel conducting planes (for graphene-back gate capacitance) or a wire and
a ground plane (for SWNT-back gate capacitance) would not adequately account
for the electrostatic environment of either the SWNT or graphene. As such, we
used COMSOL Multiphysics to perform a finite-element analysis of the gate,
conductors, and dielectrics and computed the induced charge on the SWNT and
graphene due to an applied gate voltage. Material parameters such as the
relative permittivity were found in Refs. [8, 18, 20, 21, 22, 23, 24, 25, 26,
27]. Figure S9 shows a color map of the electric field in this region due to
an applied $V_{g}=1$ V, along with the resulting charge carrier density in the
graphene. The simulation suggests that the local electric field (and thus the
local carrier density) is reduced by a factor of $\sim$30 in the graphene
directly above the SWNT compared to the value predicted from a parallel-plate
capacitor model. Due to the local nature of the Coulomb interaction ($\propto
r^{-3}$), we expect that this region of decreased carrier density is the part
of the graphene that most directly contributes to Coulomb drag. We use the
capacitance of this screened region as a lower bound in Figure 3(b) of the
main text, while retaining the analytic “bulk” value as an upper bound [see
Fig. S10 for equivalent capacitance circuits for both scenarios]. Since the
area of carrier depletion in the graphene is extremely narrow (the width of
this “screening well” is $\sim$7 nm, beyond which the graphene carrier density
rapidly approaches the value predicted by the analytical model), contributions
to drag from higher-$n_{Gr}$ regions could also be important.

The same simulation was used to estimate the capacitance (and thus carrier
density) of the metallic SWNT; this is its lower bound in Figure 3(b) in the
main text. Since this value may be sensitive to material parameters, such as
the dielectric permittivity of the SWNT, the precise magnitudes of which we do
not know, we need an alternative approach to validate our calculation and
estimate the possible range of the capacitance value. For this purpose, we
employ gate dependent data from SWNT/bilayer graphene (BLG) [Fig. S11(a)].
Here the SWNT acts as a local gate on BLG, which was also coupled to a gold
top gate and silicon back gate. Since the device geometry is close to the
SWNT/monolayer graphene devices in which we measure drag, electrostatic
measurements from the SWNT/BLG device allow us to infer the capacitive
coupling of the SWNT to monolayer graphene in the drag devices. Particularly,
by comparing the effects of the top and back gates on the BLG with the effect
of the SWNT gate, we can determine the degree of SWNT-BLG coupling, and
therefore how to account for the proximity of the BLG. We then rescale the
results to account for the very slightly different geometry of the devices
being considered in the main text of this paper.

For the SWNT-BLG device shown, which has a 4 nm-thick hBN flake separating the
SWNT and BLG and 1 $\mu$m of SiO2 between the heterostructure and back gate,
the BLG resistance as a function of back and top gate voltages ($V_{bg}$ and
$V_{tg}$) is shown in Figure S11(b). The slope of the line tracking the BLG
charge neutrality point as the gate voltages are changed,
$dV_{bg}/dV_{tg}=-14.66$, quantifies the strength of the capacitive coupling
to the top gate compared to the back gate ($dV_{bg}/dV_{tg}=C_{tg}/C_{bg}$).
Similarly, tracking the position of the side-peak caused by local SWNT gating
while also varying the top gate [Fig. S11(c)] gives the relative coupling of
the BLG to the SWNT and top gate, $dV_{tg}/dV_{NT}=C_{NT}/C_{tg}=-4.75$.

We can model the effective geometric capacitance of the SWNT to ground as a
series combination of back gate and BLG coupling [Fig. S10(a-b)]:

$C_{\textnormal{eff}}^{-1}=C_{NT-bg}^{-1}+C_{NT-BLG}^{-1}.$ (S10)

The first term can be calculated using the formula for wire-plane capacitance
per unit length:

$\frac{C_{NT-
bg}}{L}=\frac{2\pi\epsilon_{0}\epsilon_{r}}{\textnormal{arccosh}\left(\frac{2d_{bg}}{d_{NT}}\right)},$
(S11)

where $d_{bg}=\;1\mu$m is the SiO2 thickness, $d_{NT}$ = 2 nm is the SWNT
diameter, and $L$ is the length of the SWNT. The second term is estimated by
the experimentally-determined coupling of the BLG to the gate and SWNT:

$\frac{C_{NT-BLG}}{A}=\frac{C_{bg-BLG}}{A}\frac{dV_{bg}}{dV_{NT}}=\frac{C_{bg-
BLG}}{A}\frac{dV_{bg}}{dV_{tg}}\frac{dV_{tg}}{dV_{NT}},$ (S12)

where $C_{bg-BLG}/A=\epsilon_{0}\epsilon_{r}/d_{bg}$ is the standard parallel-
plate capacitor formula.

At this point, we must account for the difference in hBN thicknesses between
the BLG device (4 nm) and the SWNT-monolayer graphene drag device (2 nm).
Noting that the dielectric thickness enters the wire-place capacitance formula
as $1/\textnormal{arccosh}(2d_{BN}/d_{NT})$, reducing the hBN thickness from 4
to 2 nm simply requires multiplying the BLG device result by
$\textnormal{arccosh}(2\times
4\textnormal{nm}/2\textnormal{nm})/\textnormal{arccosh}(2\times
2\textnormal{nm}/2\textnormal{nm})\approx 1.567$. Multiplying our adjusted
$C_{NT-BLG}/A$ by $d_{NT}$ to convert it to capacitance per unit SWNT length,
we find $C_{\textnormal{eff}}/L=6.11\times 10^{-12}$ F/m for the geometric
capacitance.

Finally, we must also account for the contribution from the quantum
capacitance, which is particularly important for the SWNT. For a metallic
SWNT, this has the simple form [27]

$C_{Q}^{NT}=\frac{8e^{2}}{hv_{F}}\approx 310\times 10^{-12}\textnormal{F/m}.$
(S13)

This is substantially larger than geometrical capacitance estimated above but
can easily be included in series with the geometric capacitance to give the
total $C_{\textnormal{bg-NT}}/L=6.00\times 10^{-12}$ F/m.

###  VII VII: Derivation of nonlinear contributions to the drag voltage

The close connection between breakdown of Onsager reciprocity and appearance
of nonlinear drag response suggests that these behaviors may share the same
origin. We first note that driving current $I_{NT}$ in the SWNT can produce a
nonuniform temperature profile due to the thermoelectric Peltier effect
($\Delta T_{P}\propto I_{NT}$) and Joule heating ($\Delta T_{J}\propto
I_{NT}^{2}$). As the graphene and SWNT are in close proximity, this
temperature profile can be transferred from the SWNT to the graphene above,
particularly when the gate voltage is tuned near the graphene CNP [28, 29,
30]. This energy transfer gives rise to a temperature gradient across the
voltage probes in the graphene layer, $\Delta T(x)=\Delta T_{P}(x)+\Delta
T_{J}(x)$, where $x$ is the distance from SWNT to the voltage probes (fixed
for any given pair of probes, and thus omitted in the following analysis).

There are two mechanisms by which $\Delta T$ can produce a nonlinear drag
response: (i) generation of a thermoelectric voltage $V_{TE}(\Delta
T)=-S\Delta T$, where $S$ is the Seebeck coefficient of graphene and $\Delta
T\propto I_{NT}^{2}$; and (ii) temperature dependent change in the drag
resistance $R_{drag}$ producing a nonlinear drag voltage: $V_{NL}(\Delta
T)=\frac{1}{2}\frac{dR_{drag}}{dT}\Delta TI_{drive}$. The thermoelectric
voltage $V_{TE}(\Delta T)=-S\Delta T$ is a well-known phenomenon in systems
with a thermal gradient [31]. The second nonlinear contribution to the drag
voltage, $V_{NL}(\Delta T)=\frac{1}{2}\frac{dR_{drag}}{dT}\Delta TI_{drive}$,
merits further discussion.

This expression can be obtained by assuming a small temperature gradient
between two voltage probes separated by a distance $L$. The local drag
resistivity $\rho_{drag}$ relates the drag layer local electric field
$\varepsilon$ and drive current density $j_{drive}$ by
$\varepsilon=\rho_{drag}j_{drive}$. Now we consider a small, constant
temperature gradient $dT/dx$ between the voltage probes where $x=0,L$
correspond to the respective electrode positions. The temperature difference
between the electrodes is then $\Delta T=LdT/dx$. The drag voltage between
them can then be obtained (with $T_{0}$ the temperature of the thermal bath):

$\displaystyle
V_{drag}=\int_{0}^{L}j_{drive}\rho_{drag}(T(x))dx=j_{drive}\int_{0}^{L}\rho_{drag}(T(x))dx$
(S14)
$\displaystyle=j_{drive}L\rho_{drag}(T_{0})+\frac{1}{2}j_{drive}\frac{d\rho_{drag}}{dT}\frac{dT}{dx}L^{2}$
(S15) $\displaystyle=R_{drag}I_{drive}+\frac{1}{2}\frac{dR_{drag}}{dT}\Delta
TI_{drive}$ (S16)

The first term is the typical drag resistance $R_{drag}$ between the voltage
probes at $x=0$ and $x=L$, and the second term is the nonlinear contribution
(since $DeltaT\propto I_{drive}$ or $I_{drive}^{2}$, as discussed in the main
text) due to the temperature dependence of $R_{drag}$.

Allowing heating by both effects mentioned above, we identify terms
contributing to $V_{drag}$ that are proportional to $I_{NT}$ ($R_{drag}$ and
$V_{TE}(\Delta T_{P})$), $I_{NT}^{2}$ ($V_{TE}(\Delta T_{J})$) and
$V_{NL}(\Delta T_{P})$), and $I_{NT}^{3}$ ($V_{NL}(\Delta T_{J})$). The
$I_{NT}$-linear thermoelectric response is energy drag, which is observed to
be large in graphene systems when both layers are tuned very near the CNP [28,
29, 30, 32], but is otherwise negligible. The presence of both quadratic and
cubic nonlinearity in our experimental data, with peaks developed in both
$|\gamma|$ and $|\eta|$ at the graphene CNP (see main text Fig. 2(b) and (c)),
suggests that, at minimum, the temperature dependence of $R_{drag}$ (i.e. (ii)
above) must play a significant role. These effects are significant near the
CNP, where $|\frac{dR_{drag}}{dT}|$ exhibits large fluctuations at low
temperatures due to disorder (see main text Fig. 2(e)). The nonlinear
contribution becomes more appreciable as the linear drag signal diminishes at
lower temperature, and in SWNT devices with larger resistance (including
contact resistance; see following section for additional data), which also
supports the local heating-induced energy transfer picture.

###  VIII VIII: Comparison of individual layer transport and drag resistance
in different measurements

The data presented in the main text were primarily gathered from two separate
thermal cycles of the same device (D1), with the first set of measurements
(D1-A) occurring shortly after the completion of nanofabrication, and the
second set of measurements (D1-B) starting approximately 9 months later. In
the main text, the data in Figure 1, Figure 2(a-c), and Figure 5 are from
D1-A, and the data in Figure 2(d-e), Figure 3 and Figure 4 are from D1-B.
Comparing similar measurements for the two different data sets (for example
the drag resistance versus back gate voltage in Fig. 1(f) for D1-A and Fig.
3(d) for D1-B), it is apparent that they qualitatively follow the same
behavior, but with some quantitative discrepancies. In particular, the back
gate voltage range with an appreciable drag signal is much larger for D1-A
($V_{g}\sim$ 20 V) than for D1-B ($V_{g}\sim$ 3 V). This can be attributed to
a comparable change in the disorder in the graphene, observed as a change in
the CNP position and peak width. Figure S8 shows a direct comparison of the
SWNT conductance, graphene resistance, and drag resistance as a function of
gate voltage for D1-A [Fig. S13(a-c)] and D1-B [Fig. S13(d-f)]. In both data
sets, the drag signal width directly corresponds to the width of the graphene
CNP peak. Since all the preceding discussion about possible physical
mechanisms has relied on the interplay of various regimes (including a
disorder-dominated regime) rather than specific numerical predictions (e.g.
that the peaks occur at a specific Fermi wavevector in either graphene or
SWNT), our arguments should apply equally well in both data sets.

As an additional comparison, Figures S9 and S10 show drag resistance data from
several other devices. There are a few key differences in the geometry of the
various devices. For device pair D2, the graphene was etched into 2 bar
segments of differing widths [Fig. S14(a)]. D2-1 is 1.9 $\mu$m wide, while
D2-2 is 600 nm wide. D1 and D3 have a single bar each [Fig. S15(a)]; D1 is 1
$\mu$m wide by 7 $\mu$m long, while D3 is 1.2 $\mu$m wide by 9.7 $\mu$m long.
The hBN separating the SWNT from the graphene is significantly thicker for D2
(5 nm versus 2 nm for D1 and 3 nm for D3). Finally, the metal electrodes in D2
contact narrow, protruding sections of graphene (“noninvasive” contacts),
while in D1 and D3 they directly contact the bar (“invasive” contacts). We
also note that the SWNTs are all metallic, but the chiralities and
corresponding diameters are different for each device. The SWNT incorporated
into D1 has chiral indices (16,13) and diameter 1.97 nm, the SWNT in D2 has
chiral indices (21,5) and diameter 2.26 nm, and the SWNT in D3 has chiral
indices (21,15) and diameter 2.45 nm.

Measurements of the drag resistance as a function of the gate voltage for D2-1
and D2-2 show consistently smaller signal than D1, which is reasonable given
the larger interlayer separation. The exception is when graphene is used as
the drive layer, in which case D2-1 shows a comparatively large signal [Fig.
S14(b-c)]. Drive/drag layer reciprocity is not observed in the wider bar D2-1,
and while it is respected to a degree in the narrower bar D2-2 [Fig.
S14(e-f)], it breaks down at a higher temperature than in device D1 ($T\sim
200$ K in D2-2, compared to $T\sim 140$ K in D1). These measurements were
carried out using a small drive current (200 nA) in an attempt to remain in
the linear response regime, and the reported $R_{drag}$ in Figure
S14(b-c),(e-f) is from a linear fit of the drag voltage versus drive current
in this small range. Subsequent measurements with larger drive current [Fig.
S14(d)] show a mostly quadratic drag current-voltage relationship. It is
therefore likely that the breakdown of layer reciprocity is due to an earlier
and stronger onset of nonlinear transport effects (discussed in main text),
and even measurements with a small drive current may have a substantial
nonlinear transport contribution. Furthermore, we note that narrow,
noninvasive contacts have been predicted not to thermalize efficiently with
the electron system in graphene-graphene drag devices near charge neutrality,
leading to a breakdown of layer reciprocity even in the linear response regime
[16]. This detail of the device geometry may also contribute to the behavior
seen in the D2 devices. Since the width of the narrower device D2-2 is
comparable to the width of the noninvasive graphene contacts, the bar and
contact can thermalize more effectively, which allows some degree of layer
reciprocity to be preserved.

The geometry of device D3 is similar to D1, and it displays layer reciprocity
at comparable temperatures [Fig. S15(b)]. The drag resistance qualitatively
resembles D1, although with a smaller magnitude. This may be attributed to the
increased layer separation in device D3. The graphene quality is similar to D1
[Fig. S15(c) versus main text Fig. 1(d)], but the SWNT has substantially
higher resistance and appears quite disordered [Fig. S15(d)]. The drag signal
on the positive side of graphene CNP lacks the distinctive peak of the D1
data, likely because SWNT carrier density and current were lower than the
corresponding part of the signal in D1. The high-resistance SWNT, as well as
additional inhomogeneity appearing during and after thermal cycles, prevented
an extensive characterization of device D3. Nonetheless, the initial data we
were able to gather support the explanation of the drag behavior in the main
text.

Figure S1: Nanotube suspended growth, imaging and characterization. (a)
Schematic of nanotube growth chip with catalyst and central slit. (b) Infrared
image of suspended SWNT. (c) Optical excitation spectrum of an example
metallic SWNT. (d) Schematic of Rayleigh spectroscopy, imaging and transfer
stage setup. Figure S2: Fabrication sequence for SWNT-graphene devices. (a)
PMMA window for SWNT transfer. (b) Alignment of growth chip with CNTs to
window. (c) Mechanical contact between growth chip and target chip. (d) Growth
chip removal; SWNTs are captured by PMMA. (e) PMMA removal and
patterning/deposition of SWNT contacts. (f-g) Transfer of hBN and graphene
flakes to contacted SWNT. (h) Heterostructure shaped by etching,
patterning/deposition of graphene contacts. (i) Optical microscope image of a
final device, with SWNT position indicated by red dashed line. Gold electrode
at bottom of image is one of the SWNT contacts. Scale bar is 2 $\mu$m. Figure
S3: (a) Conductance of the SWNT in device D1 at $T=200$ K as a function of
gate voltage. Arrows indicate direction of $V_{g}$ sweep; positive to negative
(red) and negative to positive (blue). (b) $R_{drag}$ measured simultaneously
with the SWNT conductance in (a). Figure S4: (a) SWNT conductance as a
function of back gate voltage in a BLG-SWNT device (axes flipped to align with
(b)). (b) Drag resistance as a function of back gate and top gate voltages.
Figure S5: Circuit diagrams for DC measurement. (a) Schematic of symmetrically
biased SWNT (drive layer) and voltage measurement on graphene (drag layer).
Additional ammeters on SWNT and graphene can measure the SWNT conductance and
SWNT-graphene tunneling current, respectively. (b) Symmetrically biased
graphene (drive layer) and voltage measurement on graphene (to estimate
graphene resistance) and SWNT (drag layer). In this case, large ($\sim 10$
M$\Omega$) resistors are added after the voltage sources so that the graphene
is being current biased. Figure S6: Drag voltage at $T=300$ K as a function of
drive current and gate voltage, with graphene as the drive layer and SWNT as
the drag layer. Current is applied at the closest pair of voltage probes
($x=800$ nm from the SWNT). Data has been slightly smoothed in the x-direction
to reduce noise. Figure S7: Drag voltage versus drive current at various gate
voltages for reciprocal layer configurations: driving current in the SWNT
while measuring graphene voltage (orange, filled symbols) and driving current
in the graphene while measuring voltage across the SWNT (blue, open symbols)
at T = 300 K. $V_{g}$ value stated is the center of the moving average filter.
(a) $V_{g}=23$ V. (b) $V_{g}=20$ V. (c) $V_{g}=16$ V. (d) $V_{g}=-5$ V. (e)
$V_{g}=-11$ V. (f) $V_{g}=-18$ V. Figure S8: Band diagram of SWNT-graphene
heterostructure (a) at $V_{g}=0\;\textnormal{V}$ (b) at positive gate voltage.
Graphene band filling is represented in dark (light) blue and SWNT band
filling is represented in orange (yellow) for conduction (valence) band.
Adapted from [17]. Figure S9: COMSOL simulation of SWNT-graphene device. Color
map shows $|E|$ with 1 V applied to back gate for a device cross-section.
Overlay (black line) shows calculated graphene carrier density $n_{Gr}$ as a
function of distance $x$ from the SWNT. Figure S10: Equivalent capacitance
circuit of SWNT-graphene device. (a) Schematic of conductors and gate voltage.
(b) Circuit for local screening between graphene and SWNT. SWNT-graphene
capacitance is illustrated as two capacitors because each conductor screens
some of the gate electric field from the other. (c) Circuit for graphene
capacitance far from SWNT. Figure S11: (a) Optical microscopy image (upper
panel) and cross-section schematic (lower panel) of SWNT-BLG device. Scale bar
is 10 nm. (b) BLG resistance as a function of back gate and top gate voltages.
Dashed white line follows the charge neutrality point and has slope
$dV_{bg}/dV_{tg}$. Dashed red line indicate back gate voltage (0 V) and top
gate voltage range for panel (c). (c) BLG resistance as a function of top gate
voltage and SWNT local gate voltage. Dashed white line indicates secondary
Dirac peak due to local gating of the BLG by the SWNT and has slope
$dV_{tg}/dV_{NT}$. Figure S12: SWNT-BLG device in the quantum Hall regime, at
$T=1.6$ K and $V_{bg}=0$ V. (a) Hall conductance $G_{xy}$ of bulk channel as a
function of top gate voltage $V_{tg}$. SWNT local gate voltage $V_{NT}$ does
not affect the value. Inset: schematic of measurement circuit. (b) $R_{xx}$ of
bulk channel across SWNT local gate as a function of $V_{tg}andV_{NT}$.
(c)-(e) Line cuts of $G_{diag}=(R_{xx}+R_{xy})^{-1}=\nu_{local}(e^{2}/h)$ at
fixed $V_{tg}$ (indicated by dashed lines on (b)) corresponding to integer
$\nu_{bulk}$. (c) $\nu_{bulk}=16$. (d) $\nu_{bulk}=8$. (e) $\nu_{BLG}=4$.
Figure S13: Individual layer transport and drag resistance as a function of
back gate voltage for measurement D1-A. (a) SWNT conductance at
$T=200\;\textnormal{K}$. (b) Graphene resistance at $T=100\;\textnormal{K}$.
(c) Drag resistance at $T=200$ K. (d-e) Individual layer transport and drag
resistance as a function of adjusted back gate voltage for the same device
after thermal cycling and 9 months of storage (measurement D1-B). The voltage
value at the graphene charge neutrality point has been subtracted. (d) SWNT
conductance at $T=200$ K. (e) Graphene resistance at $T=200$ K. (f) Drag
resistance at $T=200$ K. Figure S14: a) Optical microscope image of device D2.
D2-1 refers to the lower, wider section and D2-2 is the upper, narrower
section. Scale bar is 4 $\mu$m. (b-c) D2-1 drag resistance versus back gate
voltage using graphene (blue) or CNT (orange) as the drive layer, at T = 182 K
(b) and 152 K (c). (d) D2-2 drag voltage in graphene versus drive current in
SWNT at $T=$ 200 K. (e-f) D2-2 drag resistance versus back gate voltage using
graphene (blue) or SWNT (orange) as the drive layer, at $T=$ 253 K (e) and 182
K (f). The CNT charge neutrality point in all measurements is at $\sim
40\;\textnormal{V}$. Figure S15: a) Optical microscope image of device D3.
Scale bar is 5 $\mu$m. (b) D3 drag resistance versus gate voltage using
graphene (blue) or SWNT (orange) as the drive layer, at $T$ = 200 K. (c) SWNT
conductance versus gate voltage at $T$ = 200 K. (d) Graphene conductance
versus gate voltage at $T$ = 200 K.

### References

  * Cheng  _et al._ [2019] A. Cheng, T. Taniguchi, K. Watanabe, P. Kim, and J.-D. Pillet, Physical Review Letters 123, 216804 (2019), arXiv:1910.13307 .
  * Sfeir  _et al._ [2006] M. Y. Sfeir, T. Beetz, F. Wang, L. Huang, X. M. H. Huang, M. Huang, J. Hone, S. O’Brien, J. A. Misewich, T. F. Heinz, L. Wu, Y. Zhu, and L. E. Brus, Science 312, 554 (2006).
  * Huang  _et al._ [2005] X. M. H. Huang, R. Caldwell, L. Huang, S. C. Jun, M. Huang, M. Y. Sfeir, S. P. O’Brien, and J. Hone, Nano Letters 5, 1515 (2005).
  * Cao  _et al._ [2015] Q. Cao, S.-J. Han, J. Tersoff, A. D. Franklin, Y. Zhu, Z. Zhang, G. S. Tulevski, J. Tang, and W. Haensch, Science 350, 68 (2015).
  * Huang  _et al._ [2015] J.-W. Huang, C. Pan, S. Tran, B. Cheng, K. Watanabe, T. Taniguchi, C. N. Lau, and M. Bockrath, Nano Letters 15, 6836 (2015).
  * Kim  _et al._ [2005] W. Kim, A. Javey, R. Tu, J. Cao, Q. Wang, and H. Dai, Applied Physics Letters 87, 173101 (2005).
  * Pitner  _et al._ [2019] G. Pitner, G. Hills, J. P. Llinas, K.-M. Persson, R. Park, J. Bokor, S. Mitra, and H.-S. P. Wong, Nano Letters 19, 1083 (2019).
  * Dean  _et al._ [2010] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, Nature Nanotechnology 5, 722 (2010), arXiv:1005.4917 .
  * Wang  _et al._ [2013] L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, 614 (2013).
  * Liu  _et al._ [2017] X. Liu, L. Wang, K. C. Fong, Y. Gao, P. Maher, K. Watanabe, T. Taniguchi, J. Hone, C. Dean, and P. Kim, Physical Review Letters 119, 056802 (2017), arXiv:1612.08308 .
  * Li  _et al._ [2016] J. I. A. Li, T. Taniguchi, K. Watanabe, J. Hone, A. Levchenko, and C. R. Dean, Physical Review Letters 117, 046802 (2016), arXiv:1602.01039 .
  * Gramila  _et al._ [1991] T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer, and K. W. West, Physical Review Letters 66, 1216 (1991).
  * Hill  _et al._ [1996] N. P. R. Hill, J. T. Nicholls, E. H. Linfield, M. Pepper, D. A. Ritchie, A. R. Hamilton, and G. A. C. Jones, Journal of Physics: Condensed Matter 8, L557 (1996).
  * Narozhny  _et al._ [2012] B. N. Narozhny, M. Titov, I. V. Gornyi, and P. M. Ostrovsky, Physical Review B 85, 195421 (2012), arXiv:1110.6359 .
  * Peres  _et al._ [2011] N. M. R. Peres, J. M. B. Lopes dos Santos, and A. H. Castro Neto, EPL (Europhysics Letters) 95, 18001 (2011).
  * Ilani  _et al._ [2006] S. Ilani, L. A. K. Donev, M. Kindermann, and P. L. McEuen, Nature Physics 2, 687 (2006).
  * Kim  _et al._ [2011] S. Kim, I. Jo, J. Nah, Z. Yao, S. K. Banerjee, and E. Tutuc, Physical Review B 83, 161401 (2011), arXiv:1010.2113 .
  * Wong and Akinwande [2010] H.-S. P. Wong and D. Akinwande,  _Carbon Nanotube and Graphene Device Physics_ (Cambridge University Press, Cambridge, 2010) pp. 139–148.
  * Hwang  _et al._ [2011] E. H. Hwang, R. Sensarma, and S. Das Sarma, Physical Review B - Condensed Matter and Materials Physics 84, 1 (2011).
  * Ng [1995] K. K. Ng,  _Complete Guide to Semiconductor Devices_ , 1st ed. (McGraw-Hill, New York, NY, 1995).
  * Zhang  _et al._ [2017] K. Zhang, Y. Feng, F. Wang, Z. Yang, and J. Wang, Journal of Materials Chemistry C 5, 11992 (2017).
  * Laturia  _et al._ [2018] A. Laturia, M. L. Van de Put, and W. G. Vandenberghe, npj 2D Materials and Applications 2, 6 (2018).
  * Seol  _et al._ [2010] J. H. Seol, I. Jo, A. L. Moore, L. Lindsay, Z. H. Aitken, M. T. Pettes, X. Li, Z. Yao, R. Huang, D. Broido, N. Mingo, R. S. Ruoff, and L. Shi, Science 328, 213 (2010).
  * Elias  _et al._ [2011] D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Morozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V. Grigorieva, K. S. Novoselov, F. Guinea, and A. K. Geim, Nature Physics 7, 701 (2011), arXiv:1104.1396 .
  * Nicklow  _et al._ [1972] R. Nicklow, N. Wakabayashi, and H. G. Smith, Physical Review B 5, 4951 (1972).
  * Lu  _et al._ [2007] W. Lu, D. Wang, and L. Chen, Nano Letters 7, 2729 (2007).
  * Kozinsky and Marzari [2006] B. Kozinsky and N. Marzari, Physical Review Letters 96, 166801 (2006), arXiv:0602599 [cond-mat] .
  * Song and Levitov [2012] J. C. W. Song and L. S. Levitov, Physical Review Letters 109, 236602 (2012), arXiv:1205.5257 .
  * Song  _et al._ [2013] J. C. W. Song, D. A. Abanin, and L. S. Levitov, Nano Letters 13, 3631 (2013), arXiv:1304.1450 .
  * Song and Levitov [2013] J. C. W. Song and L. S. Levitov, Physical Review Letters 111, 126601 (2013), arXiv:1303.3529 .
  * Blundell and Blundell [2009] S. J. Blundell and K. M. Blundell,  _Concepts in Thermal Physics_, 2nd ed. (Oxford University Press, Oxford, 2009) pp. 413–415.
  * Gorbachev  _et al._ [2012] R. V. Gorbachev, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. Tudorovskiy, I. V. Grigorieva, A. H. MacDonald, S. V. Morozov, K. Watanabe, T. Taniguchi, and L. A. Ponomarenko, Nature Physics 8, 896 (2012), arXiv:1206.6626 .