# Quantum capacitance governs electrolyte conductivity in carbon nanotubes

Théo Hennequin  Manoel Manghi<EMAIL_ADDRESS>Laboratoire de
Physique Théorique, Université Paul Sabatier–Toulouse III, CNRS, France
Adrien Noury  François Henn  Vincent Jourdain  John Palmeri
<EMAIL_ADDRESS>Laboratoire Charles Coulomb, Université de
Montpellier, CNRS, France

(September 3, 2024)

###### Abstract

In recent experiments, unprecedentedly large values for the conductivity of
electrolytes through carbon nanotubes (CNTs) have been measured, possibly
owing to flow slip and a high pore surface charge density whose origin is
still unknown. By accounting for the coupling between the quantum CNT and the
classical electrolyte-filled pore capacitances, we study the case where a gate
voltage is applied to the CNT. The computed surface charge and conductivity
dependence on reservoir salt concentration and gate voltage are intimately
connected to the CNT electronic density of states. This approach provides key
insight into why metallic CNTs have larger conductivities than semi-conducting
ones.

Although much experimental, theoretical, and molecular modeling effort has
been devoted over the past years to understanding water and ion (electrolyte)
transport through carbon nanotubes (CNTs) [1, 2, 3], the origin of the
electric charge localized on the surface of industrially important CNT based
nanofluidic systems still remains unclear (see Ref. [4] and references
therein). It has already been proposed that this surface charge could arise
from functional groups at the CNT entrances [5, 6] and/or the specific
adsorption of ions, such as OH- [7]. Although the above cited studies lead to
the conclusion that this surface charge plays a key role in governing ion
transport in CNTs, it is difficult to regulate it directly and one is left to
making inferences, for example by studying the variation of ionic conductance
$G$ with the pH or salt concentration, $c_{s}$, of the external bulk
reservoirs bounding the CNT. Intriguing results have been obtained, including
a power law behavior, $G\propto c_{s}^{\alpha}$, with $1/2\leq\alpha\leq 1$,
which could be interpreted as the manifestation of an underlying surface
charge regulation mechanism [8, 9, 10].

Through a simplified feasibility study we propose in this work that by biasing
a CNT incorporated in a nanofluidic system via an applied gate voltage,
$V_{g}$, and taking into account explicitly the quantum capacitance (QC) of
the quasi-1D CNT structure as well as the non-linear capacitance of the
confined electrolyte ion the pore, it should be possible to quantify the CNT
surface charge density $\sigma_{\rm Q}$ and establish a link between the
intrinsic CNT electronic properties and ion transport through the same
structure, such as the electrolyte conductance through the CNT. A major
conclusion it that these intrinsic electronic properties will depend
significantly, under certain conditions, on whether the CNT is metallic (M) or
semiconducting (SC).

The CNT quantum charge $\sigma_{\rm Q}$ arises from the low density of states
(DOS) for the charge carriers (electrons and holes) in this quasi-1D system.
Indeed, the applied $V_{g}$ creates a net charge on the CNT by perturbing the
equilibrium occupation of the allowed energy levels. Part of $V_{g}$ goes to
raising the electric potential of the CNT to $\psi_{0}$ and part goes to
shifting the chemical potential by $\Delta\mu=-eV_{\rm ch}=-e(V_{g}-\psi_{0})$
(where $e$ is the absolute value of the electron charge): $\psi_{0}$ will then
be lower than $V_{g}$ with (in the absence of other capacitances) the
difference between the two giving rise to the chemical contribution, $V_{\rm
ch}$. The amplitude of the chemical potential shift is determined by $V_{g}$
and the relative values of the capacitances with the smallest one providing
the most important contribution. We show here that for a wide range of salt
concentration, $10^{-3}\leq c_{s}\leq 4$ mol/L, the CNT QC, $C_{\rm Q}$, can
be lower than the nanopore electrolyte contribution, $C_{\rm P}$, and
therefore the quantum contribution controls the conductance. By comparing the
confined electrolyte conductivity calculated for a classical metallic nanopore
(with a very high, effectively infinite, DOS) [11] with those calculated for M
and SC CNTs, we bring to light clear signatures of each type of CNT behavior.

Figure 1: (a) Density of states (DOS) $g(E)$ for metallic (M,left) and semi-
conducting (SC,right) CNTs computed by the tight-binding method (solid orange
lines) and the ${\bf k}\cdot{\bf p}$ approximation (blue dashed lines). (b)
Quantum surface charge density and (c) integral QCs vs. the chemical
contribution, $V_{\rm ch}$ for these M (solid curve) and SC (dotted curve)
CNTs.

The CNT QC has already been studied in setups where an electrolyte surrounds
the CNT and acts as an electrolyte liquid gate, the main goal was however to
investigate the electrical properties of the CNT itself [15, 12, 13, 14, 16,
17]. Within this context, Lemay’s group [12, 13, 14] has carried out an
interesting study, although the capacitance of the external electrolyte was
modeled using an overly simplified form (of the type used for Stern layers)
that does not capture the full complexity of the non-linear capacitance of an
electrolyte. The QC of graphene sheets and ribbons have also been studied
theoretically and experimentally using both ionic liquid and electrolyte gates
[19, 14, 18]. The gating of a generic nanopore has also been studied
theoretically for confined electrolytes in a nanofluidic setup where the gate
voltage is applied to an electrode in contact with an insulating layer making
up the finite-thickness surface of a nanopore [20, 21]. The confined
electrolyte conductance was investigated, but the important special case of a
CNT was not addressed: the system was treated as being purely classical and
therefore the capacitances considered were the geometrical one coming from the
insulating layer, together with the Stern and non-linear Graham capacitances
of the electrolyte.

In contrast to these previous studies we show here that by examining the
confined electrolyte conductivity the CNT QC can provide a direct adjustable
handle on the all important, but elusive, nanopore surface charge density. By
shedding light in a fundamental way on ion transport in this type of
nanofluidic system, it should be possible to pave the way to the tuning of
transport properties for particular practical applications.

We briefly summarize the electronic characteristics of single-walled carbon
nanotubes (SWCNTs) obtained by simplified theoretical methods and tight-
binding calculations (see Ref. [23] for more details). A SWCNT can be
considered as a graphene sheet rolled up into a cylinder and its electronic
nature, metallic (M) or semi-conducting (SC), depends on its atomic structural
characteristics fixed by two _chiral_ indices $(n,m)$ that determine its
diameter and chirality [22, 12, 24, 23]. The simplified theoretical ${\bf
k}\cdot{\bf p}$ method, which neglects curvature effects, reveals that in the
M case, $|n-m|=3q$, where $q\in\mathbb{N}$. In the SC case, $|n-m|\neq 3q$ and
the size of the band gap is inversely proportional to the SWCNT diameter $d$
111the inclusion of curvature in more sophisticated methods shows that only
armchair SWCNTs are truly metallic, the other putatively metallic SWCNTs pick
up a small band gap that can be considered for our purposes to be smeared out
at room temperature. The band structure is composed of multiple 1D subbands
sliced from the Dirac dispersion cone of graphene and, as shown by simplified
theoretical analyses [22], exhibits a certain amount of (approximate)
universality. The $i$th electron-hole subband has the following dispersion
relation

$E_{\pm}(i,k)=\pm\sqrt{(\hbar v_{F}k)^{2}+E_{i}^{2}},$ (1)

where $v_{F}\approx 8\times 10^{5}\leavevmode\nobreak\ $m/s is the Fermi
velocity at the Dirac cone, $\hbar$ the reduced Planck’s constant, $k$ the
charge carrier wave vector, and $E_{i}=2\hbar v_{F}i/(3d)$. The allowed values
of $i$ depend on whether the SWCNT is M or SC [22, 25]. The $E_{i,g}=2E_{i}$
are the individual energy gaps determined by the distance of the quantized $k$
states from the center of the Dirac cone situated where both $k$ and $E$
vanish. The DOS per unit surface (the usual DOS per unit length divided by
$\pi d$), for a SC SWCNT is given by [22, 12, 24, 23]

$g_{\rm SC}(E)=g_{0}\sum_{i>0,\neq
3q}\frac{|E|}{\sqrt{E^{2}-E_{i}^{2}}}\Theta(|E|-E_{i}),$ (2)

where $g_{0}=4/(\pi^{2}d\hbar v_{F})$ and $\Theta$ is the Heaviside unit step
function. The SC band gap is predicted to be
$E_{g}=E_{+}(1,0)-E_{-}(1,0)=2E_{1}=0.7/d$ eV, if $d$ is in nm (between 0.2 to
0.7 eV for $1\leq d\leq 3$ nm). For purposes of illustration we choose here
$d=1.5$ nm for which $E_{g}=0.47$ eV (18 $k_{B}T$ at room temperature). The
subbands for $i=3q$ (integer $q$) are not present [see Fig. 1(a)] and
therefore the SC DOS exhibits van Hove singularities at $|E|=E_{i}$ for
$i>0,i\neq 3q$.

For true M SWCNTs, $E_{g}=0$ and the first band has a non-zero DOS
$g(0)=g_{0}$ and a half-width equal to $E_{3}$ as shown in Fig. 1(a)] [23].
The ${\bf k}\cdot{\bf p}$ method leads to a DOS given by [22]

$g_{\rm
M}(E)=g_{0}+2g_{0}\sum_{m>0}\frac{|E|}{\sqrt{E^{2}-(mE_{3})^{2}}}\Theta(|E|-mE_{3}).$
(3)

The M DOS exhibits van Hove singularities at $|E|=mE_{3}$ for $m>0$ and, aside
from the conduction band, all bands are doubly degenerate.

We show in Fig. 1(a) the approximate theoretical DOS for $d=1.5$ nm (dashed
blue curves) and the DOS computed by the group of Maruyama [26] using the
tight-binding method (solid orange curves) for a M CNT of chiral indices
$(11,11)$ ($d=1.512$ nm) and for a SC CNT of chiral indices $(19;0)$
($d=1.508$ nm). It is clear that the approximate DOS universality breaks down
for $|E|>1$ eV where the actual DOS begins to depend on the values of the
chiral indices.

At thermal equilibrium without external perturbations the chemical potential
is situated at the Dirac point taken as the reference $\mu=0$ eV. When a gate
voltage $V_{g}$ is applied to an electrode posed on a CNT, a shift in the
chemical potential, $\Delta\mu$, occurs and the occupation of the allowed
levels is modified, which leads to the creation of charges (electrons or
holes) in the various subbands labeled by $i$. The CNT DOS plays a crucial
role in determining the amplitude of this charge. The density of electrons
(number per unit surface), $n$, for a SWCNT is calculated as a function of
temperature $T$ and $\Delta\mu$ using Fermi statistics [22, 12, 24, 23],

$n(\Delta\mu)=\int_{E_{g}/2}^{\infty}\frac{g(E)}{1+\exp[\beta(E-\Delta\mu)]}dE,$
(4)

where $\beta=1/(k_{B}T)$ ($k_{B}T\approx 25$ meV at room temperature). The
density of holes, $p$, is given by Eq. (4) by replacing $-\Delta\mu$ by
$\Delta\mu$. The chemical contribution to the applied gate voltage
(electromotive force), $V_{\rm ch}=V_{g}-\psi_{0}$ engenders a charge per unit
surface,

$\sigma_{\rm Q}(V_{\rm ch})=e(p-n),$ (5)

and is related to the chemical potential shift through $\Delta\mu=-eV_{\rm
ch}$, where $\psi_{0}$ is the electrostatic potential drop between the CNT and
the reference electrode. Note that $\sigma_{\rm Q}=0$ for $V_{\rm ch}=0$,
$\sigma_{\rm Q}>0$ for $V_{\rm ch}>0$, and $\sigma_{\rm Q}<0$ for $V_{\rm
ch}<0$ [27, 28]. The surface charge densities $\sigma_{\rm Q}(V_{\rm ch})$ for
SC and M CNTs are plotted in Fig. 1(b). For $V_{\rm ch}>1$ V, they almost
superimpose, showing some slope discontinuities associated with the van Hove
singularities of the associated DOS. As seen in the inset, however, for
$V_{\rm ch}<1$ V the two are quite different, essentially because the charge
density vanishes in the gap for the SC CNT, i.e. for $V_{\rm
ch}<E_{g}/(2e)=0.23$ V.

We distinguish between the integral QC needed here,

$C_{\rm Q}\equiv\frac{|\sigma_{\rm Q}|}{V_{\rm ch}},$ (6)

which directly determines the charge on the CNT, and the differential one,
$C^{d}_{Q}(V_{\rm ch})\equiv\partial|\sigma_{\rm Q}|/\partial V_{\rm ch}$,
which gives access directly to many features of the DOS. The integral
capacitances for SC and M CNTs are plotted in Fig. 1(c). If $|V_{\rm
ch}|<E_{3}/e$ holds, then $C_{\rm Q}\simeq C^{0}_{Q}$ for M CNTs and $C_{\rm
Q}=0$ for SC ones. For larger $|V_{\rm ch}|$, similarly to $\sigma(V_{\rm
ch})$, the two integral capacitances are quite similar.

In the simplified description we adopt here (absence of any residual
geometrical capacitance arising from the rest of the electrical circuit) the
gate voltage, $V_{g}=V_{\rm ch}+\psi_{0}$, is shared between the chemical
contribution and the electrostatic one. The equivalent electrical circuit is
shown in Fig. 2(a), the total capacitance, $C_{\rm tot}$, and amplitudes of
each contribution to $V_{g}$ depend on the individual capacitances and are
given by [12, 19, 27, 28]:

$\frac{1}{C_{\rm tot}}=\frac{1}{C_{\rm Q}}+\frac{1}{C_{\rm
P}}=\frac{V_{g}}{|\sigma_{\rm Q}|},$ (7)

where $C_{\rm P}=|\sigma_{\rm Q}|/\psi_{0}$ is the non-linear integral pore
capacitance arising from electrolyte confined to the interior of a CNT, which
leads to:

$\psi_{0}=V_{g}\frac{C_{\rm Q}(V_{\rm ch})}{C_{\rm Q}(V_{\rm ch})+C_{\rm
P}(\psi_{0})}.$ (8)

This result shows clearly that if $C_{\rm Q}\gg C_{\rm P}$ then
$\psi_{0}\lesssim V_{g}$ and $V_{\rm ch}\ll V_{g}$, indicating that quantum
effects are negligible. It corresponds to the “classical” limit. Conversely,
if $C_{\rm P}\gg C_{\rm Q}$, then $\psi_{0}\ll V_{g}$ and quantum effects are
dominant.

Figure 2: (a) Equivalent electrical circuit of the system with the quantum
capacitance $C_{\rm Q}$ and the pore one $C_{\rm P}$. The gate voltage,
$V_{g}=\psi_{0}+V_{\rm ch}$, is divided between the potential at the inner CNT
surface, $\psi_{0}$, and the chemical contribution, $V_{\rm ch}$. (b)
Dimensionless surface charge density $\sigma^{*}$ and (c) conductivity of a
KCl electrolyte vs. salt concentration $c_{s}$ (Log-Log plot) for the M (solid
lines) and SC (dashed lines) CNTs with $V_{g}=0.2$ (grey), 1 (blue), 2 (red),
and 3 V (green). The thick line is the bulk conductivity.

To calculate the electrolyte capacitance, $C_{\rm P}$, we use an interpolation
formula for the electric potential at the pore wall, $\psi_{0}$, that we
previously devised for small nanopores [10]. This formula for $\psi_{0}$ as a
function of the nanopore radius $R$ ($=d/2$), electrolyte salt concentration
$c_{s}$, and nanopore surface charge density, $\sigma_{\rm P}$, becomes exact
in the good coion exclusion and homogeneous limits and is an accurate
interpolation elsewhere:

$\sinh\tilde{\psi}_{0}=\frac{\sigma_{\rm P}^{*}(1+\sigma_{\rm
P}^{*})}{\tilde{c}_{s}},$ (9)

where $\tilde{\psi}_{0}=\beta e|\psi_{0}|$ and

$\tilde{c}_{s}=\pi\ell_{B}R^{2}c_{s},\qquad\sigma_{\rm P}^{*}=\frac{\pi
R\ell_{B}}{e}|\sigma_{\rm P}|$ (10)

are respectively the dimensionless electrolyte salt concentration and surface
charge density (the electric potential at the reference electrode in the bulk
reservoir is taken to be zero). The Bjerrum length is
$\ell_{B}=e^{2}/(4\pi\epsilon k_{B}T)\simeq 0.7$ nm in bulk water at room
temperature (dielectric permittivity $\epsilon=78\epsilon_{0}$). Eq. (9) can
be rewritten by isolating $\sigma_{\rm P}^{*}$ as

$\sigma_{\rm
P}^{*}=\frac{1}{2}\left(\sqrt{1+4\tilde{c}_{s}\sinh\tilde{\psi}_{0}}-1\right).$
(11)

For a classical metal (effectively infinite DOS), formally $C_{\rm Q}/C_{\rm
P}\to\infty$ and using Eq. (8) we see that the surface charge density
$\sigma_{\rm cl}^{\mathrm{*}}$ is given by Eq. (11) with $\psi_{0}$ set to
$V_{g}$. In this case for low enough $\tilde{c}_{s}$ and $V_{g}$, $\sigma_{\rm
cl}^{*}\simeq\tilde{c}_{s}\sinh(\beta eV_{g})$ and for high enough
$\tilde{c}_{s}$ and $V_{g}\gg 25$ mV, $\sigma_{\rm
cl}^{*}\simeq\tilde{c}_{s}^{1/2}e^{\beta eV_{g}/2}$. The rapid exponential
rise in surface charge density in the latter limit appears to be compatible
with the extremely high conductivities observed in [11] for gold plated
nanopores at negative gate voltages (for which specific ion adsorption can be
neglected). Given the low DOS of CNTs we do not expect such classical
exponential behavior for them. In this case by keeping $\sigma_{\rm P}^{*}$
constant, $\tilde{\psi}_{0}$ is low when $\tilde{c}_{s}$ is high, which
increases the pore capacitance and eventually leads to $C_{\rm P}\gg C_{\rm
Q}$, i.e. quantum effects become important.

By defining $C_{\rm P}=\frac{4\epsilon}{R}\tilde{C}_{\rm P}$, the integral
pore capacitance can be found from the dimensionless one:

$\tilde{C}_{\rm P}=\frac{\sigma_{\rm
P}^{*}}{\tilde{\psi}_{0}}=\frac{\sqrt{1+4\tilde{c}_{s}\sinh\tilde{\psi}_{0}}-1}{2\tilde{\psi}_{0}}$
(12)

For $\tilde{\psi}_{0}\ll 1$, $\tilde{C}_{\rm P}\rightarrow\tilde{c}_{s}$ and
$C_{\rm P}\rightarrow\epsilon R/(2\lambda_{\rm DH}^{2})$, where $\lambda_{\rm
DH}=(8\pi\ell_{B}c_{s})^{-1/2}$ is the bulk Debye-Hückel screening length for
a monovalent electrolyte. In this limit the electrolyte filled pore acts as a
parallel plate capacitor of capacitance $\epsilon/d_{\rm eff}$ with effective
thickness $d_{\rm eff}=2\lambda_{\rm DH}^{2}/R$.

Now by writing $\sigma\equiv\sigma_{\rm Q}=\sigma_{\rm P}$ and combining
$\sigma_{\rm Q}(V_{g}-\psi_{0})$ from Eq. (5) and Eq. (9), we obtain
$\tilde{c}_{s}(V_{g},\psi_{0})$, which enables us to trace a parametric plot
of $\sigma$ as a function of $c_{s}$ for fixed gate voltage $V_{g}$ by varying
$\psi_{0}$ from 0 to $V_{g}$. This is shown in Fig. 2(b) for the SC and M
cases for $V_{g}=0.2,1,2$ and 3 V. For $V_{g}=0.2$ V, the surface charge
density is two orders of magnitude larger for the M CNT than for the SC one.
This is because $V_{g}$ is just below $E_{g}/2$ (gap half-width). At this
relatively low value of $V_{g}$ the surface charge density increases quickly
with $c_{s}$ at low concentrations. For higher gate voltages, $\sigma$
increases very slowly with $c_{s}$, being almost constant for $V_{g}\geq 2$ V.
For $V_{g}=1$ V (blue curves) $\sigma({\rm SC})>\sigma({\rm M})$ at low
$c_{s}$ because $\psi_{0}$ is high and $V_{\rm ch}$ is too low to reach beyond
the SC van Hove singularity occurring at $E_{g}/(2e)=0.23$ V [see Fig. 1(a)].
For large $c_{s}$, however, $\psi_{0}$ becomes lower and $V_{\rm ch}\simeq
E_{3}/e=0.7$ V higher, now reaching the large first van Hove singularity for
the M case.

The pore conductance $G$ is related to the conductivity $\kappa$ through
$G=\pi R^{2}\kappa/L$ where $L$ is the pore length. The pore conductivity,
$\kappa$, as a function of $\sigma^{*}$ and $\tilde{c}_{s}$ can be obtained
using the result obtained in Eq. 19 of Ref. [10], which takes into account the
electrical migration contribution (first two terms) and the electro-osmotic
one (third term), including flow slip at the pore surface:

$\tilde{\kappa}=(\tilde{\mu}_{+}+\tilde{\mu}_{-})\tilde{c}_{s}\sqrt{1+\left(\frac{\sigma^{*}}{\tilde{c}_{s}}\right)^{2}}-\mathrm{sgn}(\sigma)\sigma^{*}(\tilde{\mu}_{+}-\tilde{\mu}_{-})\\\
+2\sigma^{*}\left[1-\frac{\ln(1+\sigma^{*})}{\sigma^{*}}+2\tilde{b}\sigma^{*}\right],$
(13)

where $\tilde{\kappa}=(2\pi^{2}R^{2}\ell_{B}^{2}\eta\kappa/e^{2})$ is the
dimensionless conductivity ($\eta=8.94\times 10^{-4}$ Pa.s being the water
viscosity), $\tilde{\mu}_{\pm}=2\pi\eta\ell_{B}\mu_{\pm}$ the dimensionless
mobilities of ions (taken equal to the bulk values,
$\mu_{+}\approx\mu_{-}=5\times 10^{-11}$ s/kg for KCl), and $\tilde{b}=b/R$ is
the ratio between the slip length $b$ and the pore radius
[$\mathrm{sgn}(\sigma)$ denotes the sign of $\sigma$]. In Eq. (13) the
conductivity $\kappa(c_{s},V_{g})$ is a function of both $c_{s}$ and $V_{g}$
because $\sigma(c_{s},V_{g})$ is a function of both these quantities.

Figure 3:  (a) Electrolyte conductivity $\kappa$ vs. gate voltage $V_{g}$ in M
(solid lines) and SC (dashed lines) CNTs for reservoir salt concentration
values of (from bottom to top) $c_{s}=10^{-3},10^{-2},10^{-1},1$ mol/L. The
classical metallic CNT is shown in dotted-dashed lines. The inset is a zoom
for $c_{s}=10^{-3}$ mol/L. (b) Associated surface charge density vs. $V_{g}$
for the M CNT for the same $c_{s}$ values as in (a). The SC CNT is shown in
the inset. (c) Derivative of the electrolyte conductivity,
$\kappa^{\prime}(V_{g})$, in the M CNT for the same $c_{s}$ values as in (a).
The DOS $g(V_{\rm ch})$ is superimposed in orange. To illustrate how to read
off $\psi_{0}$ we show it and $V_{\rm ch}$ for the case $c_{s}=10^{-3}$ mol/L
and $V_{g}=1.9$ V.

One observes in Fig. 2(c), that the conductivity is enhanced compared to the
bulk case for intermediate and low salt concentrations. This is the signature
of a non-zero surface charge density. For $V_{g}>1$ V, this enhancement even
persists to high $c_{s}$, attesting to a high surface charge density. At low
salt concentration, $\kappa$ in Eq. (13) is controlled by $\sigma$ (good coion
exclusion regime) and we therefore observe in Fig. 2(c) an increase in
conductivity similar to the one observed for $\sigma(c_{s})$. We have checked
that for $V_{g}>1$ V, $\sigma^{*}>\tilde{c}_{s}$ over the whole concentration
range and $\kappa$ is therefore controlled by the first electrical migration
term in Eq. (13), which is one order of magnitude larger than the electro-
osmotic one (the second term is negligible since $\mu_{+}\approx\mu_{-}$ for
KCl). For the M case and $V_{g}<1$ V (solid grey line), $\kappa$ increases
with $c_{s}$, but much more slowly than the bulk one, with an apparent
exponent smaller than $1/2$.

By taking a slip length large enough, the electro-osmotic contribution in Eq.
(13) associated with the fluid slip at the nanopore surface,
$4\tilde{b}\sigma^{*2}$, can eventually dominate the electrical migration one.
For instance, if a reasonable value of $\tilde{b}=40$ [29] is taken, the
electro-osmotic contribution is greater by one order of magnitude over the
whole concentration range. The conductivity would then be completely
controlled by $\sigma^{*}(c_{s})$ [plotted in Fig. 2(b)], as a more complete
study (underway) will reveal.

Changing the concentration in micrometric reservoirs is not easy in practice.
One could measure more easily the electrolyte conductivity at fixed $c_{s}$ by
varying $V_{g}$. To present the conductivity in this way, we first compute
$\psi_{0}$ by solving numerically $\sigma_{\rm Q}(V_{g}-\psi_{0})=\sigma_{\rm
P}(c_{s},\psi_{0})$ using Eqs. (5) and (11) at fixed $V_{g}$ and $c_{s}$. We
then re-inject $\psi_{0}$ in $\sigma_{\rm P}$, which itself can finally be
injected into Eq. (13). In Fig. 3(a) we plot the conductivity as a function of
$V_{g}$ at fixed $c_{s}=10^{-3},10^{-2},10^{-1},1$ mol/L from bottom to top.
We confirm that $\kappa$ varies strongly at low gate voltages. For $V_{g}=0$,
$\sigma=0$ and $\kappa\propto c_{s}$ (bulk conductivity), which explains the
staggered values (separated by a decade) at low $V_{g}$. The distinction
between the three cases, semiconductor, quantum metal and “classical” metal is
clear at low $V_{g}$, where the QC suppresses the “classical” exponential
growth of conductivity with $V_{g}$ for M and SC CNTs and the SC energy gap
suppresses the conductivity with respect to that of the quantum metal. The M
CNT conductivity follows the classical one at very low $V_{g}$ within a window
the decreases in width with increasing $c_{s}$. At high $V_{g}$ all M and SC
curves are almost superimposed.

Even if the quantum surface charge density $\sigma$, has the same derivative
as the classical one at $V_{g}=0$, it becomes quickly linear $\sigma\simeq
C_{\rm Q}^{0}V_{g}$, with a lower slope up to $V_{g}\simeq E_{3}/e\simeq 0.7$
V, as shown in Figure 3(b). Indeed, for not too low $V_{g}$, $C_{\rm Q}\ll
C_{\rm P}$, which implies that $C_{\rm tot}\simeq C_{\rm Q}\approx C_{\rm
Q}^{0}$ and therefore $\psi_{0}\ll V_{g}$. We have checked that this is true
whenever $C_{\rm P}/C_{\rm Q}^{0}\gg 1$, which leads to
$\sqrt{\tilde{c}_{s}}>\tilde{C}_{\rm Q}^{0}$. Since $\tilde{C}_{\rm
Q}^{0}=0.03$, this last inequality is verified for $\tilde{c}_{s}>10^{-3}$
[the case for the concentrations shown in Fig. 3(a)]. For lower reservoir salt
concentrations, $\sigma$ is no longer linear in $V_{g}$. These different
behaviors are illustrated in the inset of Fig. 3(a) where we also note the
impact of the SC CNT gap, which leads to a constant $\kappa$ up to
$V_{g}=0.2\simeq E_{g}/e$ V, the edge of the gap. We conclude that the QC has
a considerable effect on both $\sigma$ and $\kappa$ as soon as
$V_{g}>k_{B}T/e\approx 25$ mV and this over the whole experimental range of
$c_{s}$.

The structure of a conductivity curve $\kappa(V_{g})$ exhibiting slope
changes, related to the DOS structure, is easier to visualize by plotting its
derivative $\kappa^{\prime}(V_{g})$ [see Fig. 3(c)] and superimposing on it
the DOS $g(V_{\rm ch})$ (in orange). Clearly the peaks in
$\kappa^{\prime}(V_{g})$ are directly connected to the van Hove singularities
of the DOS with an offset to the right which increases when $c_{s}$ decreases.
Indeed, since $\kappa$ is governed by the surface charge density $\sigma_{\rm
Q}$ which depends only on $V_{\rm ch}=V_{g}-\psi_{0}$, the higher $\psi_{0}$
is (and therefore the lower $c_{s}$ is), the more $\kappa(V_{g}-\psi_{0})$ is
shifted to the right. We can therefore directly measure the potential
$\psi_{0}$ in Fig. 3(c). For instance, for $c_{s}=10^{-3}$ mol/L and
$V_{g}=1.9$ V, we get $\psi_{0}\simeq 0.32$ V. This is a way to readily check
that for these $c_{s}$ values, $\psi_{0}\ll V_{g}$, thus confirming the
central role played by the QC on the conductivity.

As an concluding example, we consider the experimental results obtained by Liu
et al. [30], who measured, for $c_{s}=1$ mol/L, conductances of 61.0 nS for M
SWCNTs and 5.6 nS for SC ones with $0.8\leq d\leq 2$ nm and $5\leq L\leq 10$
nm. Using the DOS given in Fig. 1(a) with $d=1.5$ nm and $L=8$ nm, we can
account for these two conductance values by taking $V_{g}\simeq 0.35$ V, a
value that we interpret as an environmentally induced shift in the zero of the
gate tension.

More systematic experiments are clearly needed to ascertain to what extent the
charge density and therefore the electrolyte conductivity through SWCNTs can
be controlled by an applied gate voltage. Presumably, a more complete model
for conductivity would be needed to account for the full complexity of real
CNTs, including the influence of pH (via charge regulation), residual
geometrical capacitances, and dielectric interactions.

## References

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