Electronic Properties of Carbon Nanotubes Calculated from Density
  Functional Theory and the Empirical pi-Bond Model
Electronic Properties of Carbon Nanotubes Calculated from
Density Functional Theory and the Empirical π-Bond Model
Deep Shah, Nicolas A. Bruque,∗ Khairul Alam, Roger K. Lake,† and Rajeev R. Pandey
Department of Electrical Engineering,
University of California, Riverside, CA 92521-0204
J. Computational Electronics (accepted, 20 February 2007)
Abstract
The validity of the DFT models implemented by FIREBALL for CNT electronic device modeling
is assessed. The effective masses, band gaps, and transmission coefficients of semi-conducting,
zigzag, (n, 0) carbon nanotubes (CNTs) resulting from the ab-initio tight-binding density
functional theory (DFT) code FIREBALL and the empirical, nearest-neighbor π-bond model are
compared for all semiconducting n values 5 ≤ n ≤ 35. The DFT values for the effective masses
differ from the π-bond values by ±9% over the range of n values, 17 ≤ n ≤ 29, most important
for electronic device applications. Over the range 13 ≤ n ≤ 35, the DFT bandgaps are less than
the empirical bandgaps by 20-180 meV depending on the functional and the n value. The π-bond
model gives results that differ significantly from the DFT results when the CNT diameter goes
below 1 nm due to the large curvature of the CNT. The π-bond model quickly becomes inaccurate
away from the bandedges for a (10, 0) CNT, and it is completely inaccurate for n ≤ 8.
Keywords: FIREBALL, CNT, DFT, NEGF, Bandstructure.
∗Electronic address: nbruque@ee.ucr.edu
†Electronic address: rlake@ee.ucr.edu
mailto:nbruque@ee.ucr.edu
mailto:rlake@ee.ucr.edu
I. INTRODUCTION
Carbon nanotube (CNT) systems are of high research interest for use in sensing and
nanoelectronics. Electronic device and circuit architecture concepts for bio-assembled CNTs
have been described and demonstrated [1, 2, 3, 4, 5]. One system that we have found
particularly interesting is a molecule joining two semiconducting CNTs. Such a system can
display the electrical response of a resonant tunnel diode (RTD) [6, 7, 8, 9], and we refer to
it as a CNT-Mol-RTD.
The current-voltage response of a CNT-Mol-RTD depends on the alignment of the CNT
bandedges with molecular states of the organic group [6, 7, 8]. Therefore, quantitative
simulations of such a structure require models which are suitable for calculating the electronic
states of both semiconductors, molecules, and the chemical bonds between the molecule
and semiconductor. The models should be general enough to treat local distortions in the
semiconductor lattice and distortion of the molecule. A CNT-Mol-RTD structure tends
to be large compared to the benzene dithiol type of molecules that have been so heavily
studied for molecular electronics [10, 11, 12, 13, 14, 15, 16]. Therefore, the model and its
implementation must have the ability to handle large systems.
One model widely used for computational molecular electronics is density functional
theory (DFT). DFT is a general, flexible theory with a proven record of calculating the
electronic states of both semiconductors and molecules. The well known weakness of DFT
implemented with widely used density functionals is its underestimation of the bandgap of
semiconductors [17]. For electronic device modeling, this is not a trivial issue. The most
famous example is that of Ge which is predicted within the local density approximation
(LDA) or generalized gradient approximation (GGA) to be a metal. Another example is
Zener tunneling which is exponentially dependent on the bandgap. It is this process that
limits the maximum on-off current ratio in CNT field effect transistors. Linear errors in
the bandgap result in exponentially large errors in the simulated minimum off-current [18].
The final example is the electron current of a CNT-Mol-RTD. It depends on the alignment
of a molecular state with the conduction band edge of the CNT. If the conduction band
edge is too low, the predicted current - voltage response will be inaccurate. While hybrid
functionals that include exact exchange greatly improve the situation, they also significantly
increase the computational burden [19].
Another option for the simulation of large structures is to use empirical models [20, 21].
Such models can reproduce the bandgaps and energy levels of semiconductors with high
accuracy. However, simulations of heterogeneous CNT, metal, organic, biological systems
are difficult with empirical models due to issues of transferability of parameters. Conversely,
electronic device simulations of semiconductors are difficult with DFT models due to the
underestimation of the bandgap. One is forced to make a choice, and we have chosen the
DFT model as implemented by the code FIREBALL [22, 23] since it has demonstrated the
ability to model large biological molecules [24].
In this paper, we compare predictions of the properties relevant to electronic device
modeling of semiconducting (n,0) CNTs calculated from both DFT theory and an empirical,
tight-binding, π-bond model. Two important band-edge properties are the bandgap and the
effective masses. For larger diameter CNTs, the empirical, tight-binding model is expected to
predict the bandgap and band-edge effective masses with good accuracy since the empirical
parameters have been chosen by fitting to those quantities. Therefore, for the larger diameter
CNTs, we use the values of the bandgap and effective masses obtained from the π-bond model
to assess the values obtained from the DFT model. For smaller diameter CNTs where the
curvature becomes significant, the π-bond model breaks down.
Bandedge quantities alone are not sufficient for electronic device modeling. Since elec-
tronic devices are operated at biases on the order of a volt, accurate modeling of the higher
energy states away from the bandedges is necessary for device modeling. The quantity that
best characterizes the higher energy spectra for device simulations is the transmission coef-
ficient. Therefore, we compare transmission coefficients calculated from the DFT and the
empirical, tight-binding models.
Below, we calculate and compare the bandgaps, effective masses, and transmission spectra
of semiconducting zigzag CNTs, ranging from (5, 0) to (35, 0) corresponding to diameters
ranging from 0.39 nm to 2.8 nm. The bandgaps and effective masses are calculated, plotted,
and compared for every non-metallic (n, 0) CNT with 5 ≤ n ≤ 35. Selected transmission
coefficients are plotted and compared for n = 10, 20, 31, and 35.
II. METHOD OF CALCULATION
The FIREBALL calculations are performed using the local density approximation (LDA)
(the Ceperley-Alder [25] form as parameterized by Perdew and Zunger [26]) and the BLYP
exchange-correlation functional [27, 28]. A self-consistent calculation is performed using a
generalization of the Harris-Foulkes [29, 30] energy functional referred to as DOGS after
the authors of the original paper [31, 32]. A separable non-local pseudopotential [33] and
a minimal sp3 FIREBALL basis are used. The localized pseudoatomic orbitals are slightly
excited due to hard wall boundary conditions imposed at certain cutoffs, r2sc and r
c [32]. An
excitation energy of approximately 2.0 eV is used to preserve the chemical bonding trends
of carbon which results in r2sc = 4.0 Å and r
c = 4.5 Å. Further details are given in [23].
After the FIREBALL DFT calculation finishes, the device Hamiltonian matrix elements,
the overlap matrix, and the device-to-contact coupling matrices are extracted. The spatial
extent of the non-zero matrix elements (the sparsity of the matrices) is determined by the
pseudopotential cutoff limits and the FIREBALL orbital radii. Each CNT system, pictured
in Fig. 1, consists of one CNT unit cell composed of 4 atomic layers periodically repeated
in the axial direction. Non-zero matrix elements of a given atomic layer extend to the left
and right 4 atomic layers, or one unit cell of the zigzag CNT. In terms of the 4-atomic layer
unit cells, there is only nearest neighbor unit-cell coupling.
Transmission is calculated using the non-equilibrium Green function formalism (NEGF).
The CNT is partitioned into a ‘device’ consisting of one unit cell and a left and right
‘contact.’ The left and right ‘contacts’ are taken into account exactly by self-energies Σ`
and Σr, respectively, as illustrated in Fig. 1. Details of the NEGF algorithm are described
in [6, 7, 8].
For the empirical π-bond model, we use a nearest-neighbor model with matrix element
Vppπ = −2.77 eV and �p = 0.0 eV [34]. The NEGF algorithm is the same as that used
with the FIREBALL matrix elements. The effective mass for both models is calculated from
the 1-D dispersion using 1/m∗ = 1~2∂
2E/∂k2. The energy band gap is determined from an
E − k calculation by reading the difference between the highest occupied band energy and
the lowest unoccupied band energy at the Γ point.
III. RESULT AND DISCUSSION
Figure 2(a) compares the calculated band gaps for (n, 0) CNTs with 5 ≤ n ≤ 35 leaving
out the metallic CNTs with n values that are integer multiples of 3. The n values are
shown on the bottom horizontal axis and the corresponding CNT diameters are shown on
the top horizontal axis. On the plot itself, the data points indicate the n values for which
calculations were performed. All other n values in that range correspond to metallic CNTs.
At first glance, the bandgaps that result from the FIREBALL calculations closely track the
bandgaps determined by the π-bond model for n ≥ 10. CNTs in this size range are the
ones that are most commonly synthesized, and they are the ones that are most important
for electronic devices [35]. Below n = 10, the π-bond model breaks down due to the large
curvature of the CNT. For the smallest CNT with n = 5, the FIREBALL calculations show
zero bandgap. Upon closer inspection of Fig. 2(a), one notices a sawtooth shape to the plot
of bandgap versus n for the DFT calculations. We observe that the bandgap resulting from
the DFT calculations corresponds closely to the bandgap resulting from the π-bond model
for (n, 0) CNTs when n = 3p+ 1 where p is an integer ≥ 3.
This is shown clearly in Fig. 2(b). in which we plot the differences between the bandgaps
calculated from the LDA and BLYP functionals and those from the π-bond model for n ≥ 10.
For n = 3p+1 ≥ 13, the LDA model underestimates the bandgap by 24 - 28 meV. For these
n values, the BLYP model underestimates the bandgap by 46 - 84 meV. For n = 3p−1 with
p an integer, the discrepancies between the bandgaps resulting from the DFT models and
the π-bond model are larger. For values of n = 3p− 1 ≥ 14, the LDA model underestimates
the bandgap by 52 - 147 meV. For these n values, the BLYP model underestimates the
bandgap by 66 - 176 meV.
Figs. 3 shows calculations and comparisons of the electron and hole effective masses.
The plots of the left column show calculations and comparisons of the electron effective
mass, and the plots of the right column show calculations and comparisons of the hole
effective mass. In each row consisting of two plots, the scale and range of values is identical
to facilitate easy comparisons between the values for electrons and holes. The calculated
electron and hole effective masses (normalized to the bare electron mass) are plotted in Figs.
3(a) and (d), respectively, versus n for all semiconducting values in the range 7 ≤ n ≤ 35.
One immediately notices that all models result in very similar values of effective masses for
both the electrons and holes. To discern the differences, we plot the difference of the mass
values calculated from the DFT models and the pi-bond model in Figs. 3(b) and (e) for
electrons and holes, respectively. In other words, Figs. 3(b) and (e) show the normalized
difference values (m∗DFT − m
π−bond)/m0. Note that on the mass difference plots, (b) and
(e), the n values range from 10 - 35 whereas on the mass plots, (a) and (d), the n values
range from 7 - 35. The range of n is reduced on the difference plots to keep the plotted
range of differences small. The difference plots show that the effective mass determined from
the BLYP model tends to be larger than the effective mass determined by the LDA model.
Over the range of n values 14 ≤ n ≤ 35, the mass determined by the BLYP model has a
maximum difference from the mass determined from the LDA model of 0.008 at n = 14
and a minimum difference of 0.002 at n = 34. Also, the sawtooth peaks in the difference
curves are out of phase with the sawtooth peaks in the mass curves above. The maximum
differences occur at the minimums of the mass curves. Overall, the differences in the mass
values predicted by the DFT models and the π-bond model are relatively small. To quantify
the differences, the percent differences, 100 ∗ (mDFT −mπ−bond)/mπ−bond, are plotted in (c)
and (f). Over a wide range of the most useful n values for device applications, 17 ≤ n ≤ 29,
the values for the effective masses from the DFT models fall to within +8% to −9% of the
values from the π-bond model.
So far, we have considered bandedge properties of bandgaps and effective masses. As
we noted above, the higher energy electronic spectra is also important for electronic device
modeling, and the best way to characterize it is to calculate the transmission coefficients.
Figure 4 shows the transmission spectra for (10, 0), (20, 0), (31, 0) and (35, 0) CNTs calcu-
lated using DFT/BLYP, DFT/LDA, and the empirical, π-bond model. In all cases, the
energy axis of the transmission curves has been shifted such that the center of the bandgap
lies at 0 eV. The energy region in Fig. 4 for which the transmission is zero is the band gap
for the CNTs.
For the 2 largest CNTs, n = 31 and 35, the transmission resulting from the DFT and
π-bond models all have similar, symmetric forms. There is some compression of the energy
scale for the transmission coefficients calculated from the DFT models compared to the
transmission coefficient calculated from the π-bond model. The energy separation between
higher modes is smaller in the DFT models then in the π-bond model.
For the smallest CNT, n = 10, there is a noticeable, qualitative difference between the
transmission calculated from the DFT models and the π-bond model. The transmission
resulting from the π-bond model is always symmetric around the center of the bandgap. For
the DFT models, the transmission is noticeably asymmetric. Approximately 0.5 eV above
the conduction band edge, the DFT models predict 3 bands closely spaced and 2 bands
doubly degenerate. These 7 bands, multiplied by 2 for spin, give rise to the large step of 14
in the transmission coefficient 0.5 eV above the conduction band edge in the transmission
coefficient of the (10, 0) CNT. This large increase in the transmission and density of states
0.5 eV in the conduction band is significant for device modeling. A similar large step is also
observed in a transmission calculation based on the SIESTA code for a (7, 0) CNT [36]. A
similar large increase in the transmission also occurs in the (20,0) CNT at 1.3 eV above the
conduction band edge.
The differences in the (10,0) valence band transmission resulting from the DFT and π-
bond models, while not as dramatic as those in the conduction band, are still significant
from a device modeling perspective. The 0.5 eV gap between the valence band edge and the
next lower pair of bands found from the π-bond model is reduced to approximately 0.3 eV in
the DFT models. These energies lie within the applied voltage window, (VDD), of any of the
most optimistically scaled CNT field effect transistors, and will, thus, affect the physics of
the carrier transport. While our main focus is on assessing the validity of the DFT models
for device modeling, these results also provide an assessment of the π-bond model and show
that the π-bond model should be used with care and scepticism for (10,0) CNTs.
IV. SUMMARY
The goal of this work is to assess the validity of the DFT models implemented by FIRE-
BALL for CNT electronic device modeling. Our approach is to compare the electronic
properties resulting from the DFT models with those resulting from the π-bond model since
the parameters of the π-bond model have been empirically chosen to give a good fit to the
bandgap and effective mass for CNTs with diameters that are ‘not too small.’ We have com-
pared the bandgaps, effective masses, and transmission coefficients of (n, 0) CNTs calculated
from the empirical π-bond model, DFT/LDA, and DFT/BLYP models.
For values of n in the range 17 ≤ n ≤ 29, the calculated effective masses from the
DFT models are within ±9% of those calculated from the π-bond model. For n ≥ 10,
the difference between the bandgap calculated from the π-bond model and the bandgaps
calculated from the DFT models oscillates as a function of n. The differences are smallest
for n = 3p + 1 where p is an integer, and the differences are largest for n = 3p − 1. For
n = 3p + 1 ≥ 13, the LDA model underestimates the bandgap by 24 - 58 meV and the
BLYP model underestimates the bandgap by 46 - 84 meV. For n = 3p − 1 ≥ 14, the LDA
model underestimates the bandgap by 52 - 147 meV and the BLYP model underestimates
the bandgap by 66 - 176 meV. Overall, in the important range of n values most relevant
for CNT devices, 17 ≤ n ≤ 29, the bandgaps, effective masses, and transmission coefficients
calculated from the DFT models implemented by FIREBALL are sufficiently accurate for
electronic device simulations.
These simulations also quantify what is meant by ‘not too small’ when applying the
π-bond model. For n = 10, the bandedge properties resulting from the π-bond and DFT
models agree to within 10%, however, the π-bond model quickly becomes inaccurate away
from the bandedges. The transmission from the higher energy modes resulting from the π-
bond model has differences with those resulting from the DFT models which are significant
for device modeling. For n ≤ 8, the π-bond model is completely inaccurate.
Acknowledgments
This work was supported by the Microelectronics Advanced Research Corporation Fo-
cus Center on Nano Materials (FENA), SRC/SRCEA, the NSF (ECS-0524501), and
DARPA/DMEA-CNID (H94003-04-2-0404).
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Figure Captions
FIG. 1: One unit cell (10,0) zigzag CNT with 4 atomic layers. Self-energies take into
account the semi-infinite leads for transmission calculations.
FIG. 2: (a) (n, 0) CNT band gaps as a function of n and diameter calculated from the
π-bond model and DFT with LDA and BLYP functionals. (b) Difference between the band
gap calculated from the DFT models and the π-bond model.
FIG. 3. Electron, (a) - (c), and hole, (d) - (f), effective mass comparisons. Top: Normalized
effective mass (m∗/m0), calculated from the π-bond and DFT models. Middle: Difference
between the effective mass calculated from the DFT models and the π-bond model
(mDFT −mπ−bond)/m0. Bottom: Percent difference, 100 ∗ (mDFT −mπ−bond)/mπ−bond.
FIG. 4. Transmission calculated from π-bond and DFT models for (n, 0) CNTs with n
values of (a) 10, (b) 20, (c) 31, and (d) 35.
1  2       3  4
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FIG. 4:
	INTRODUCTION
	METHOD OF CALCULATION
	RESULT AND DISCUSSION
	SUMMARY
	Acknowledgments
	References
