# Evidence for Phonon-Assisted Intertube Electronic Transport in an Armchair
Carbon Nanotube Film

Davoud Adinehloo  Department of Electrical Engineering, University at Buffalo,
Buffalo, NY 14228, USA  Weilu Gao  Department of Electrical and Computer
Engineering, University of Utah, Salt Lake City, UT 84112, USA  Ali Mojibpour
Department of Electrical and Computer Engineering, Rice University, Houston,
TX 77005, USA  Junichiro Kono  Department of Electrical and Computer
Engineering, Rice University, Houston, TX 77005, USA  Department of Physics
and Astronomy, Rice University, Houston, TX 77005, USA  Department of
Materials Science and NanoEngineering, Rice University, Houston, TX 77005, USA
The Smalley-Curl Institute, Rice University, Houston, TX 77005, USA  Vasili
Perebeinos  Department of Electrical Engineering, University at Buffalo,
Buffalo, NY 14228, USA<EMAIL_ADDRESS>
###### Abstract

The electrical conductivity of a macroscopic assembly of nanomaterials is
determined through a complex interplay of electronic transport within and
between constituent nano-objects. Phonons play dual roles in this situation:
their increased populations tend to reduce the conductivity via electron
scattering while they can boost the conductivity by assisting electrons to
propagate through the potential-energy landscape. Here, we identify a phonon-
assisted coherent electron transport process between neighboring nanotubes in
temperature-dependent conductivity measurements on a macroscopic film of
armchair single-wall carbon nanotubes. Through atomistic modeling of
electronic states and calculations of both electronic and phonon-assisted
junction conductances, we conclude that phonon-assisted conductance is the
dominant mechanism for the observed high-temperature transport. The
unambiguous manifestation of coherent intertube dynamics proves a single-
chirality armchair nanotube film to be a unique macroscopic solid-state
ensemble of nano-objects promising for the development of room-temperature
coherent electronic devices.

Toward large-scale applications of nanomaterials in electronics Jariwala  _et
al._ (2013); Okimoto  _et al._ (2010); Rao  _et al._ (2018); Gao  _et al._
(2020), understanding and controlling electron transport processes, not only
within each nano-object but also between them, is crucial Zorn and Zaumseil
(2021). The overall electrical conductivity of a macroscopic assembly of
nanomaterials is determined by an array of interdependent quantities – e.g.,
defect density, doping level, Fermi energy, material size, the density of
nano-objects, purity, homogeneity, morphology, and temperature. The role of
phonons in this highly complex situation is subtle since they tend to scatter
electrons to reduce the conductivity while simultaneously assisting electrons
to go through the potential-energy landscape to increase the conductivity.
This dual role of phonons in electronic transport across macro-objects has not
been elucidated. Particularly, phonon-assisted processes, including phonon-
assisted coherent electron transfer between nano-objects, have not been
identified in a macroscopic sample.

Here we study macroscopic assemblies of carbon nanotubes (CNTs), which provide
an ideal model system in which to address the above issues and questions.
Since their discovery in the early 1990s,Iijima (1991); Iijima and Ichihashi
(1993); Bethune  _et al._ (1993) extensive studies have revealed and
established the truly unique electronic and optical properties of these one-
dimensional nano-objects, particularly on individual nanotube levels.
Depending on the precise atomic arrangements of $sp^{2}$-bonded carbon atoms
in the honeycomb lattice, specified by a pair of integers called chirality
indices, ($n$,$m$), both metallic and semiconducting species of CNTs exist;
details of the band structure, most importantly the band gap, are determined
by ($n$,$m$).Dresselhaus  _et al._ (2001); Avouris  _et al._ (2007, 2008);
Nanot  _et al._ (2013); Weisman and Kono (2019) Intertube electronic transport
has also been studied theoreticallyNakanishi and Ando (2001); Buldum and Lu
(2001); Maarouf and Mele (2011) and experimentally.(Fuhrer  _et al._ , 2000a,
b) Based on the momentum-matching conditions on the initial and final states
of electrons in a transfer between two individual CNTs with a crossing angle
$\theta$ (see Figure 1a), the existence of special values of $\theta$, where
the intertube conductance becomes maximum and minimum has been identified
Nakanishi and Ando (2001); Buldum and Lu (2001); Maarouf and Mele (2011). We
show that after structural relaxation in a CNT- crossing geometry, the overall
junction conductance increases by an order of magnitude with a quantitatively
similar crossing angle dependence. Moreover, following Reference 19 for
twisted bilayer phonon-assisted junction conductance, we demonstrate here that
the intertube phonon-assisted junction conductance at room temperature is
comparable to the pure electronic conductance. At some crossing angles,
phonon-assisted junction conductance exceeds its electronic counterpart by an
order of magnitude. These two effects – geometry relaxation and the additional
phonon-assisted mechanism – enable us to achieve better quantitative agreement
with prior experimentsFuhrer  _et al._ (2000a) as well as the data reported
here.

In our previous work,(Gao  _et al._ , 2021) we investigated the temperature
dependence of conductivity in a set of macroscopic single-chirality CNT films
with well-defined ($n$,$m$). Distinctly ($n$,$m$)-dependent and strongly
temperature-dependent conductivity was observed, and the overall behaviors
were explained through the Mott variable range hopping (VRH) model in a wide
temperature range. However, one of the samples – an armchair (6,6) CNT film –
exhibited clear deviation from the VRH behavior, especially at elevated
temperatures, indicating that a different transport mechanism was at work.
Through detailed quantitative analysis of the temperature-dependent
conductivity, we found that the localization length was longer in (6,6) CNT
film than in the other chirality samples such that intertube transport, as
opposed to intratube transport, dominated in the (6,6) film at high
temperatures. Here, through modeling and calculations of both electronic and
phonon-assisted junction conductances, we unambiguously identify phonon-
assisted conduction as the dominant mechanism for the observed high-
temperature transport behaviour. The clear manifestation of this coherent
dynamic process makes a single-chirality armchair CNT film a unique
macroscopic object in which to study quantum transport processes at room
temperature.

The device we used in this study was based on a thin film of randomly
oriented, chirality-enriched (6,6) CNTs. The device had four electrodes, and
the channel between the two inner electrodes, shown in the bottom panel of
Figure 1a, had a length ($L$), width ($W$), and thickness ($t$) of 8 $\mu$m, 5
$\mu$m, and $\sim$12 nm, respectively. Further experimental details are fully
described in Reference 20. The experimentally measured film conductance,
$G_{\rm film}$, which is equal to the film conductivity times $Wt/L$, is
plotted against temperature in Figure 1b, showing that $G_{\rm film}$
monotonically but sublinearly increases with temperature as the temperature is
raised from 4.2 K to 300 K.

Figure 1:  (a) Schematic illustration of the device used in this study.
$\theta$ \- crossing angle, $\varphi$ \- rotating angle, $h$ \- interlayer
distance, and $S$ \- sliding shift between CNTs. (b) Experimentally measured
(red circles) and theoretically calculated (blue solid) film conductance of
the (6,6) CNT film as a function of temperature. (c) Theoretically obtained
total resistance of the film (blue solid) and the decomposed contributions:
intratube (red dashed) and a sum of the two electronic and phonon-assisted
contributions to the intertube junction resistance (green dotted dashed). The
calculations were performed at Fermi energy $E_{\text{F}}=10$ meV and applied
bias across the junction $V=1$ mV.

To understand the observed temperature dependence of the film conductance, we
consider a series resistance model of the CNT conductivity and junction
resistance, which contribute to the overall resistance of the film asShim  _et
al._ (2015a):

$G_{\text{film}}^{-1}=C\left(\frac{L_{\text{cnt}}}{\sigma_{\text{intra}}}+\frac{1}{G_{\text{inter}}}\right),$
(1)

where the first and second terms originate from the intratube conductivity
$\sigma_{\text{intra}}$ and the intertube junction conductance
$G_{\text{inter}}$, respectively. A geometrical prefactor, $C=L(d+h)/(tW)$, is
determined by the average number of junctions in the film, where $d$ is the
CNT diameter, $h$ is the interlayer distance between CNTs, and
$L_{\text{cnt}}$ is the average length of CNTs. If we assume that the average
crossing angle in a randomly oriented CNT film is $\theta=45^{\circ}$, $C$
should be $0.16$, according to our geometry, while the best fit to the data
was obtained for $C=0.09$, as shown in Figure 1b.

A temperature-dependent film conductance can arise from various sources. Our
previous study on single-chirality CNT filmsGao  _et al._ (2021) has shown
that VRH is the dominant mechanism at low temperatures. Therefore, we model
$\sigma_{\text{intra}}$ by the 1D Mott VRH model(Gao  _et al._ , 2021), i.e.,
$\sigma_{\text{intra}}=\sigma_{0}\exp[-(T_{0}/T)^{1/2}]$. In Figure 1b, the
theory curve according to Equation (1) includes the electronic and phonon-
assistant contributions to the intertube (junction) conductance (discussed
below) as well as the intratube conductivity. Since the sample is randomly
oriented and the junction conductance depends on the angle $\theta$ (see
Figure 1a), we averaged the intertube junction conductance over $\theta$
between $20^{\circ}$ and $70^{\circ}$. The relative contributions of the
intratube and intertube resistances are shown in Figure 1c. At temperatures
below 50 K, the intratube resistance dominates, whereas, at temperatures above
50 K, the intertube junction resistance gives a dominant contribution to the
film resistance.

Figure 2:  (a) Electronic conductance, $G_{\text{inter, el}}$, of crossed CNTs
as a function of the crossing angle, $\theta$, between armchair CNTs of
different diameters. (b) Surface plot of $G_{\text{inter, el}}$ versus $S$ and
$\varphi$ in CNTs. $G_{\text{inter, el}}$ versus $\varphi$ in (c) and versus
$S$ in (d) at horizontal and vertical cuts depicted in (b). In (b), (c), and
(d), we used $\theta=70^{\circ}$ and (6,6) CNTs. All results were calculated
at room temperature, $E_{\text{F}}=0$ eV, and $V=1$ mV.

Below we discuss calculations of the intertube junction conductance. In the
case of an ideally transparent contact, the maximum intertube conductance
equals $4e^{2}/h\approx 155$ $\mu$S. However, an experimentally measured
conductanceFuhrer  _et al._ (2000a) is substantially smaller; therefore, we
use perturbation theory (see Methods) to calculate the intertube junction
conductance as a sum of two parallel channels: electronic conductance $G_{\rm
inter,el}$ due to the overlap of $\pi$ orbitals on neighboring tubes described
by the tight-binding model and phonon-assistant conductance $G_{\rm inter,ph}$
(see Methods). Since the distance between $\pi$-orbitals in neighboring CNTs
determines the hopping overlap, intertube junction conductance varies with the
CNT structure registry. Atomic positions in an unrelaxed structure depend on
the crossing angle ($\theta$), rotation angle ($\varphi$), and sliding shift
($S$) between adjacent CNTs; see Figure 1a. In Figure 2a we show
$G_{\text{inter, el}}$ for an unrelaxed structure versus crossing angle
$\theta$ in different armchair CNTs, including (6,6) CNTs; $G_{\rm inter,el}$
can vary by more than two orders of magnitude as $\theta$ changes. Figures
2b–2d show that $G_{\rm inter,el}$ varies with $\varphi$ and $S$ by about 30%
in (6,6) CNTs. Figure 2b is a surface map of $G_{\rm inter,el}$ versus
$\varphi$ and $S$ with $\theta=70^{\circ}$, while Figures 2c and 2d are
detailed $\varphi$ and $S$ dependences of $G_{\rm inter,el}$, corresponding to
the labeled solid lines ($\o 1-\o 4$) in Figure 2b. The variation of $G_{\rm
inter,el}$ with $\varphi$ and $S$ are periodic with periods of $60^{\circ}$
and 2.46 Å, respectively, and hence, $G_{\rm inter,el}$ in Figure 2a was
obtained as an average over $\varphi$ and $S$.

Figure 3: (a) Electronic (dashed blue) and phonon-assisted (solid red)
intertube junction-conductance of relaxed (6,6) CNTs as a function of $\theta$
at room temperature. (b) Electronic (dashed) and phonon-assisted (solid)
intertube junction conductance of crossed CNTs for $\theta=40^{\circ}$ (blue)
and $\theta=70^{\circ}$ (red). (c) shows phonon-assisted intertube junction
conductance as a function of Fermi energy for $\theta=40^{\circ}$ (dashed
blue) and $\theta=70^{\circ}$ (solid red). We used a bias of $V=1$ mV in all
panels and zero Fermi energy in panels (a) and (b).

The large-diameter limit of $G_{\rm inter}$ can be understood by considering
the junction conductance between twisted graphene bilayers,Maarouf and Mele
(2011) where the momentum conservation law of electrons in two Brillouin zones
twisted with respect to each other by angle $\theta$ governs the tunneling
probability. However, in smaller-diameter CNTs, finite curvature plays an
important role such that the finite contact area between the CNTs breaks the
translation symmetry and helps relax the momentum conservation requirement. As
a result, the conductance variations with angles diminish in magnitude, and a
local minimum at $90^{\circ}$ vanishes in small-diameter armchair CNTs, as
shown in Figure 2a.

To better simulate the experimental situation, we relaxed the atomic positions
at the junction using an atomistic valence force modelPerebeinos and Tersoff
(2009) for the internal distortions in each CNT and the Lennard-Jones
potential for atoms on adjacent CNTs. We set the van der Waals adhesion energy
to $E_{\text{a}}=60$ meV per atomZacharia  _et al._ (2004) and the interlayer
graphene bilayer spacing to $h=3.35$ Å. The $G_{\rm inter,el}$ of the relaxed
structure is increased by an order of magnitude compared to the unrelaxed
case; this arises from the fact that more atoms on neighboring CNTs get closer
due to CNT bending in the contact area. Figure 3a depicts the $G_{\rm
inter,el}$ and $G_{\rm inter,ph}$ of the relaxed structure versus $\theta$,
showing that overall they are comparable in magnitude at room temperature. The
$\theta$ dependence of $G_{\rm inter,ph}$ is weaker than that of $G_{\rm
inter,el}$ because phonons help relax the momentum conservation law. Note also
that the $S$ and $\varphi$ dependences of the conductance do not enter into
consideration for the relaxed structure.

The temperature dependence of phonon-assisted conductance $G_{\text{inter,
ph}}$ is nearly linear in Figure 3b, reflecting the fact that low energy
phonons are responsible for the phonon-assisted conductance, while the
electronic intertube conductance is independent of temperature. The crossover
temperature at which the two contributions become equal depends on the
crossing angle. At angles near $\theta\sim 30^{\circ}$, the phonon-assisted
contribution dominates the intertube conductance for almost all temperatures,
whereas at $\theta\sim 90^{\circ}$ the electronic contribution is larger even
at room temperature, as shown in Figures 3a and b.

Finally, Figure 3c show the calculated Fermi energy dependence of $G_{\rm
inter,ph}$. A small variation of the $G_{\rm inter,ph}$ with the Fermi energy
in Figure 3c reflects the fact that electron-phonon matrix elements have a
weak momentum dependence and densities of electronic states are constant. In
the case of electronic intertube conductance, we find that $G_{\rm inter,el}$
is almost independent of the Fermi energy (not shown).

In summary, we demonstrated that the temperature dependence of CNT film
conductivity depend both on intratube and intertube (or junction) electronic
transport. The dominant conduction mechanism depends on the film preparation
conditions and the electronic structure of the CNTs.Gao  _et al._ (2021) In
this study, we showed that in films with a long localization length, such as a
chirality-enriched (6,6) CNT film, the temperature dependence of the junction
conductance can give rise to the observed overall temperature dependence of
the CNT film conductance over a wide temperature range. We evaluated the
temperature-independent electronic junction conductance and the temperature-
dependent phonon-assisted junction conductance using microscopic theory, as
shown in Figure 3. Our numerical results show that both contributions are
important at room temperature, and depending on the crossing angle, the
phonon-assisted junction conductance can be significantly larger than the
electronic intertube conductance. After averaging over the CNT crossing angle,
our model accurately reproduces the experimental data over a wide temperature
range with a single parameter fit known from the film device dimensions. We
found that structural relaxation increases the electronic conductance by an
order of magnitude, making calculated results more consistent with prior
experimental results of junction conductance at low temperatures.Fuhrer  _et
al._ (2000a) While the overall qualitative agreement between our model and the
measured film conductance is evident, the nearly perfect quantitative
agreement is likely accidental because of the oversimplified model used to
estimate the numerical value of the parameter $C$ in Equation (1). At the same
time, we underestimate by a factor of three the intertube conductance measured
on individual CNTs,Fuhrer  _et al._ (2000a) which could be due to a lack of
angle control in the experiment. However, there can be a more fundamental
reason for the observed discrepancy: the role of short-range defects in CNTs
on the electronic and phonon-assistant intertube junction conductance, which
should be a subject of further theoretical and experimental studies. Our work
on chirality-enriched films provides guidelines for reinterpretation and
analysis of other CNT network film resistances with the interplay of CNTs
intratube transport and intertube junction transport.

##  I Methods

### Conductance of 2D film:

As the thickness of the film is much smaller than the averaged length of SWCNT
($L_{\text{cnt}}$), the film can be considered to be a 2D film. Then, the
number of junctions for each percolation path can be estimated by
$N_{\text{junc}}=L/(L_{\text{cnt}}/\sqrt{2})$ assuming a saw-type 1D chain of
length $L$ made of CNT rodes with crossing angle of $45^{\circ}$. The number
of parallel percolation paths over width $W$ can be estimated as
$N_{\text{path}}=W/(L_{\text{cnt}}/\sqrt{2})$ and the number of 2D layers of
CNT monolayers in a film of thickness $t$ can be estimated as
$N_{\text{layer}}=t/(d+h)$. If each junction contributes a conductance of
$G_{\text{junc}}$, then the total conductance due to the junctions in the film
should be:

$G_{\text{film,
junc}}=N_{\text{layer}}\frac{N_{\text{path}}}{N_{\text{junc}}}G_{\text{inter}}=\frac{tW}{L}\frac{1}{(d+h)}G_{\text{inter}}$
(2)

If we consider the conductance contribution from the nanotubes themselves
(i.e., the intratube conductance), we can estimate it as follows:

$G_{\text{film,
intra}}=N_{\text{layer}}N_{\text{path}}\frac{\sigma_{\text{intra}}}{\sqrt{2}L}=\frac{tW}{L}\frac{1}{(d+h)L_{\text{cnt}}}\sigma_{\text{intra}}$
(3)

where $\sqrt{2}L$ is the average distance that electron travels in the random
film to reach from one electrode to the other and $\sigma_{\text{intra}}$ is
the intratube conductivity. The net resistance of the 2D film can be estimated
as:Shim  _et al._ (2015a)

$\displaystyle G_{\text{film}}^{-1}$ $\displaystyle=$ $\displaystyle
G_{\text{film, intra}}^{-1}+G_{\text{film, inter}}^{-1}$ (4) $\displaystyle=$
$\displaystyle\frac{L}{tW}\,(d+h)\left(\frac{1}{G_{\text{inter}}}+\frac{L_{\text{cnt}}}{\sigma_{\text{intra}}}\right)$

which leads to an estimate of the $C=L(d+h)/(tW)$ value in Eq. (1). Note that
the $C$ value in this simple model does not depend on the CNT volume density
as opposed to more sophisticated models Komori and Makishima (1977); Shim  _et
al._ (2015b); Tripathy  _et al._ (2020). In our model, CNT volume density is
given by $n_{\rm
CNT}=N_{\text{layer}}N_{\text{path}}N_{\text{junc}}/(tWL)=2/(L^{2}_{\rm
cnt}(d+h))$ consistent with an estimate in Ref. Gao  _et al._ (2021).

### Electronic Intertube Junction Conductance:

The electronic conductance can be evaluated by Datta (2005)

$G_{\text{inter, el}}=\frac{4\pi e}{\hbar
V}\sum_{\begin{subarray}{c}\mathbf{k}ss^{\prime}\end{subarray}}M_{\mathbf{k}s}^{s^{\prime}}(f(E_{\mathbf{k}s})-f(E_{\mathbf{k}s^{\prime}}+eV)),$
(5)

where the spin degeneracy is included. Here, $\mathbf{k}$ is the 2D electron
wave vector in 2D Brillouin zone corresponding to periodically repeated
supercell containing a single junction. $s(s^{\prime})$ labels the band index,
$e$, $\hbar$, $V$, and $f$ are the elementary electronic charge, the reduced
Plank constant, the applied voltage bias between layers, and the Fermi-Dirac
distribution function, respectively. The intertube coupling
$M_{\mathbf{k}s}^{s^{\prime}}$ is given by

$M_{\mathbf{k}s}^{s^{\prime}}=|\matrixelement{\psi_{\mathbf{k}s}}{H_{\text{el}}}{\psi_{\mathbf{k}s^{\prime}}}|^{2}\delta(E_{\mathbf{k}s^{\prime}}-E_{\mathbf{k}s}),$
(6)

where
$\matrixelement{\psi_{\mathbf{k}s}}{H_{el}}{\psi_{\mathbf{k}s^{\prime}}}$ is
the tunneling matrix element for carrier scattering from state
$\psi_{\mathbf{k}s}$ to state $\psi_{\mathbf{k}s^{\prime}}$ on different CNTs.
$E_{\mathbf{k}s}$ is the carrier energy, calculated by the tight-binding
method with the first nearest-neighbor hoping $t=3.1$ eV. $H_{\text{el}}$ is
the electron intertube interaction Hamiltonian due to the hopping between
atoms on different CNTs. The hopping parameter is defined as (Perebeinos  _et
al._ , 2012)

$t_{ij}=t_{\perp}\exp(-\frac{r_{ij}-h_{0}}{\lambda_{z}})\exp[-\left(\dfrac{\xi_{ij}}{\lambda_{xy}}\right)^{\alpha}],$
(7)

where $h_{0}=3.35$ Å is the equilibrium interlayer distance, $r_{ij}$ is the
distance between atoms $i$ and $j$, $\lambda_{z}=0.6$ Å, $\lambda_{xy}=1.7$ Å,
$\alpha=2.0$, and $t_{\perp}=0.4$ eV. $\boldsymbol{\xi}_{ij}$ is the
projection of $\mathbf{r}_{ij}$ onto the layers’ bisector, and its absolute
value is given by

$\xi_{ij}=r_{ij}\sin\phi=r_{ij}\sqrt{1-\left(\dfrac{\boldsymbol{\pi}_{ij}\cdot\mathbf{r}_{ij}}{\pi_{ij}r_{ij}}\right)^{2}}\,.$
(8)

$\phi$ is the angle between $\mathbf{r}_{ij}$ and $\boldsymbol{\pi}_{ij}$,
$\boldsymbol{\pi}_{ij}=\boldsymbol{\pi}_{i}-\boldsymbol{\pi}_{j}$, where
$\boldsymbol{\pi}_{i}$ is determined by (Perebeinos and Tersoff, 2009)

$\boldsymbol{\pi}=3\dfrac{\mathbf{r}_{ij}\times\mathbf{r}_{ik}+\mathbf{r}_{ik}\times\mathbf{r}_{il}+\mathbf{r}_{il}\times\mathbf{r}_{ij}}{r_{ij}r_{ik}+r_{ik}r_{il}+r_{il}r_{ij}},$
(9)

where $j$, $k$, and $l$ are the three nearest neighbors of $i$.
$\boldsymbol{\pi}_{i}$ is perpendicular to the surface of the CNT, and its
direction points toward the outside of CNT, and $\boldsymbol{\xi}_{ij}$ is
perpendicular to $\boldsymbol{\pi}_{ij}$.

### Phonon-Assisted Intertube Junction Conductance:

The phonon-assisted conductance is computed by (Perebeinos  _et al._ , 2012)

$G_{\text{inter, ph}}=\frac{4\pi e}{\hbar
V}\sum_{\begin{subarray}{c}\mathbf{k},\mathbf{k^{\prime}},s,s^{\prime},\mu\end{subarray}}M_{\mathbf{k^{\prime}}s^{\prime}\mu}^{\mathbf{k}s}[f(E_{\mathbf{k}s})-f(E_{\mathbf{k^{\prime}}s^{\prime}}+eV)].$
(10)

Here, $M_{\mathbf{k^{\prime}}s^{\prime}\mu}^{\mathbf{k}s}$ is given by

$\displaystyle M_{\mathbf{k^{\prime}}s^{\prime}\mu}^{\mathbf{k}s}={}$
$\displaystyle|\matrixelement{\psi_{\mathbf{k}s}}{H_{\text{e-ph}}^{\mu}}{\psi_{\mathbf{k^{\prime}}s^{\prime}}}|^{2}[n_{\mathbf{q}\mu}\delta(E_{\mathbf{k^{\prime}}s^{\prime}}-E_{\mathbf{k}s}+\hbar\omega_{\mathbf{q}\mu})$
(11)
$\displaystyle(1+n_{-\mathbf{q}\mu})\delta(E_{\mathbf{k^{\prime}}s^{\prime}}-E_{\mathbf{k}s}-\hbar\omega_{-\mathbf{q}\mu})],$

where $n$ is the Bose-Einstein function,
$\matrixelement{\psi_{\mathbf{k}s}}{H_{\text{e-ph}}^{\mu}}{\psi_{\mathbf{k^{\prime}}s^{\prime}}}$
is the electron-phonon matrix element, and $\omega_{\mathbf{q}\mu}$ is the
phonon frequency with wave vector $\mathbf{q}=\mathbf{k}-\mathbf{k^{\prime}}$
and branch number $\mu$. To obtain the phonon frequency, the structure is
relaxed using the BFGS algorithm of the quasi-Newton method.Avriel (2003) The
electron-phonon Hamiltonian are extracted by expanding Equation (7) in atomic
displacements corresponding to the phonon normal modes.

###### Acknowledgements.

We gratefully acknowledge funding from the Vice President for Research and
Economic Development (VPRED), SUNY Research Seed Grant Program, and
computational facilities at the Center for Computational Research at the
University at Buffalo (http://hdl.handle.net/10477/79221). A.M. and J.K.
acknowledge support by the Basic Energy Science (BES) program of the U.S.
Department of Energy through Grant No. DE-FG02-06ER46308 (for preparation of
carbon nanotube films) and the Robert A. Welch Foundation through Grant No.
C-1509 (for structural and electrical characterization measurements).

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