ARTICLES
Electronic transport in extended systems: Application to carbon nanotubes
Marco Buongiorno Nardelli
Department of Physics, North Carolina State University, Raleigh, North Carolina 27695-8202
共Received 30 March 1999兲
We present an efficient approach to describe the electronic transport properties of extended systems. The
method is based on the surface Green’s function matching formalism and combines the iterative calculation of
transfer matrices with the Landauer formula for the coherent conductance. The scheme is applicable to any
general Hamiltonian that can be described within a localized orbital basis. As illustrative examples, we calcu-
late transport properties for various ideal and mechanically deformed carbon nanotubes using realistic orthogo-
nal and nonorthogonal tight-binding models. In particular, we observe that bent carbon nanotubes maintain
their basic electrical properties even in the presence of large mechanical deformations.
关S0163-1829共99兲10935-4兴
I. INTRODUCTION
Recent years have witnessed a great amount of research in
the field of quantum conductance in nanostructures.
1
These
have become the systems of choice for investigations of elec-
trical conduction on a mesoscopic scale. The improvements
in nanostructured material production have stimulated devel-
opments in both experiment and theory. In particular, the
formal relation between conduction and transmission, the
Landauer formula,
2
has enhanced the understanding of elec-
tronic transport in extended systems and has proven to be
very useful in interpreting experiments involving the conduc-
tance of nanostructures. Among all the possible nanostruc-
ture materials, carbon nanotubes have attracted much atten-
tion since their discovery in 1991,
3
because of their special
geometrical and electronic properties. Their electronic and
transmission properties have been studied both
experimentally
4–8
and theoretically.
9–13
In particular, from
the theoretical point of view, the sensitivity of their elec-
tronic properties to their geometry makes them truly unique
in offering the possibility of studying quantum transport in a
very tunable environment.
The problem of calculating quantum conductance in car-
bon nanotubes has been addressed with a variety of tech-
niques that reflect the various approaches in the theory of
quantum transport in ballistic systems. Tian and Datta
9
pre-
dicted an Aharonov-Bohm-type effect in graphitic tubules in
an axial magnetic field combining the Landauer formula with
a semiclassical treatment of the transmission probability.
Saito et al.
10
studied the tunneling conductance of connected
carbon nanotubes via the direct calculation of the current
density. Chico et al.
11
addressed the problem of quantum
conductance in carbon nanotubes with defects, efficiently
combining a surface Green’s-function approach
14
to describe
the interface between different tubes with a scattering
matrix-based calculation of the transmission function, thus
obtaining the conductance via a multichannel generalization
of the Landauer formula. The variation of the conductance
with the diameter of the carbon nanotubes has been studied
by Tamura and Tsukada.
12
They combined the Landauer for-
mula with the explicit calculation of the scattering matrix
and an effective-mass approximation to clarify the physical
origin of this scaling law. Most recently, the conductance of
carbon nanotube wires in the presence of disorder has been
addressed by Anantram and Govindan.
13
They have devel-
oped an efficient numerical procedure to compute the elec-
tronic transmission using a Green’s function formalism.
All the previous calculations derive the electronic struc-
ture of the carbon nanotube from a simple
-orbital tight-
binding Hamiltonian that describes the bands of the graphitic
network of the carbon nanotube via a single nearest-neighbor
hopping parameter. Since the electronic properties of carbon
nanotubes are basically determined by the sp
2
orbitals, the
model gives a reasonably good qualitative description of
their behavior and, given its simplicity, it has become the
model of choice in a number of theoretical investigations.
However, although qualitatively useful to interpret experi-
mental results, this simple Hamiltonian lacks the accuracy
that more sophisticated tight-binding 共TB兲 models or ab ini-
tio methods are able to provide. In the present paper we
present an efficient scheme that is particularly suitable for
realistic calculations of electronic transport properties in ex-
tended systems. The approach we have designed is inspired
by the one outlined in Ref. 11, differing from the latter in the
use of the generalized Landauer formula for the transmission
function proposed by Meir and Wingreen.
15
This generaliza-
tion makes the present method extremely flexible and appli-
cable to any system described by a Hamiltonian with a lo-
calized orbital basis. The present formulation allows us to
also fully consider the complete microscopic structure of the
semi-infinite leads 共in a lead-conductor-lead geometry兲 with
a very limited computational cost. Moreover, the only quan-
tities that enter into the present formulation are the matrix
elements of the Hamiltonian operator, with no need for the
explicit knowledge of the electron wave functions for the
multichannel expansion. The last fact makes the numerical
PHYSICAL REVIEW B 15 SEPTEMBER 1999-IVOLUME 60, NUMBER 11
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calculations particularly efficient also for systems described
by multiorbital localized-basis Hamiltonians.
The paper is structured as follows: in Sec. II we introduce
the relation between conductance and Green’s functions; in
Sec. III we introduce the Green’s function formalism to com-
pute bulk conductance 共III A兲 and the transmission through
an interface 共III B兲. Section IV is devoted to the discussion
of the first results obtained with the present method. In par-
ticular, we computed the electronic transport in a bent carbon
nanotube using a full sp
3
tight-binding Hamiltonian.
16
Our
calculations predict that bending does not drastically change
the conductivity of the system, and that carbon nanotubes
maintain their electric properties even under severe deforma-
tions in the absence of topological defects. As an application
to the nonorthogonal orbital basis case, we compute the elec-
tronic and transport properties of various nanotubes using a
nonorthogonal tight-binding scheme
17
based on ab initio
density-functional theory. We conclude with some final re-
marks in Sec. V. The appendixes are devoted to the exten-
sion of the present method to the general case of a system
described by a nonorthogonal Hamiltonian model 共Appendix
A兲 and to a truly three-dimensional system 共Appendix B兲.
II. ELECTRON TRANSMISSION
AND GREEN’S FUNCTIONS
Let us consider a system composed of a conductor C con-
nected to two semi-infinite leads, R and L, as in Fig. 1. A
fundamental result in the theory of electronic transport is that
the conductance through a region of interacting electrons
共the C region in Fig. 1兲 is related to the scattering properties
of the region itself via the Landauer formula:
2
C⫽
2e
2
h
T,
where T is the transmission function and C is the conduc-
tance. The former represents the probability that an electron
injected at one end of the conductor will transmit to the other
end. The transmission function can be expressed in terms of
the Green’s functions of the conductors and the coupling of
the conductor to the leads:
18,19,15
T⫽ Tr
共
⌫
L
G
C
r
⌫
R
G
C
a
兲
,
where G
C
兵
r,a
其
are the retarded and advanced Green’s func-
tions of the conductor, and ⌫
兵
L,R
其
are functions that describe
the coupling of the conductor to the leads. To compute the
Green’s function of the conductor we start from the equation
for the Green’s function of the whole system:
共
⑀
⫺ H
兲
G⫽ I, 共1兲
where
⑀
⫽ E⫹ i
with
arbitrarily small and I is the identity
matrix. In the hypothesis that the Hamiltonian of the system
can be expressed in a discrete real-space matrix representa-
tion, the previous equation corresponds to the inversion of an
infinite matrix for the open system, consisting of the conduc-
tor and the semi-infinite leads. The above Green’s function
can be partitioned into submatrices that correspond to the
individual subsystems,
冉
G
L
G
LC
G
LCR
G
CL
G
C
G
CR
G
LRC
G
RC
G
R
冊
⫽
冉
共
⑀
⫺ H
L
兲
h
LC
0
h
LC
†
共
⑀
⫺ H
C
兲
h
CR
0 h
CR
†
共
⑀
⫺ H
R
兲
冊
⫺ 1
, 共2兲
where the matrix (
⑀
⫺ H
C
) represents the finite isolated con-
ductor, (
⑀
⫺ H
兵
R,L
其
) represent the infinite leads, and h
CR
and
h
LC
are the coupling matrices that will be nonzero only for
adjacent points in the conductor and the leads, respectively.
From this equation it is straightforward to obtain an explicit
expression for G
C
:
18
G
C
⫽
共
⑀
⫺ H
C
⫺ ⌺
L
⫺ ⌺
R
兲
⫺ 1
, 共3兲
where we define ⌺
L
⫽ h
LC
†
g
L
h
LC
and ⌺
R
⫽ h
RC
g
R
h
RC
†
as the
self-energy terms due to the semi-infinite leads and g
兵
L,R
其
⫽ (
⑀
⫺ H
兵
L,R
其
)
⫺ 1
are the leads’ Green’s functions. The self-
energy terms can be viewed as effective Hamiltonians that
arise from the coupling of the conductor with the leads. Once
the Green’s functions are known, the coupling functions
⌫
兵
L,R
其
can be easily obtained as
18
⌫
兵
L,R
其
⫽ i
关
⌺
兵
L,R
其
r
⫺ ⌺
兵
L,R
其
a
兴
,
where the advanced self-energy ⌺
兵
L,R
其
a
is the Hermitian con-
jugate of the retarded self-energy ⌺
兵
L,R
其
r
. The core of the
problem lies in the calculation of the Green’s functions of
the semi-infinite leads. In what follows we will present an
efficient approach to compute the self-energy terms in the
general case of an arbitrary localized-orbital Hamiltonian.
III. GREEN’S FUNCTIONS AND CONDUCTIVITY
FROM THE LAYER HAMILTONIAN
A. Transmission through a bulk system
It is well known that any solid 共or surface兲 can be viewed
as an infinite 共semi-infinite in the case of surfaces兲 stack of
FIG. 1. A conductor described by the Hamiltonian H
C
, con-
nected to leads L and R, through the coupling matrices h
LC
and
h
CR
.
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7829ELECTRONIC TRANSPORT IN EXTENDED SYSTEMS...
principal layers with nearest-neighbor interactions.
20
This
corresponds to transforming the original system into a linear
chain of principal layers. Within this approach, the matrix
elements of Eq. 共1兲 between layer orbitals will yield a set of
equations for the Green’s functions:
共
⑀
⫺ H
00
兲
G
00
⫽ I⫹ H
01
G
10
,
共
⑀
⫺ H
00
兲
G
10
⫽ H
01
†
G
00
⫹ H
01
G
20
,
共4兲
...,
共
⑀
⫺ H
00
兲
G
n0
⫽ H
01
†
G
n⫺ 1,0
⫹ H
01
G
n⫹ 1,0
,
where H
nm
and G
nm
are the matrix elements of the Hamil-
tonian and the Green’s function between the layer orbitals,
and we assume that in a bulk system H
00
⫽ H
11
⫽ ... and
H
01
⫽ H
12
⫽ ....Following Lopez-Sancho et al.,
21
this chain
can be transformed in order to express the Green’s function
of an individual layer in terms of the Green’s function of the
preceding 共or following兲 one. This is done via the introduc-
tion of the transfer matrices T and T
¯
, defined such that G
10
⫽ TG
00
and G
00
⫽ T
¯
G
10
. The transfer matrix can be easily
computed from the Hamiltonian matrix elements via an it-
erative procedure, as outlined in Ref. 21. In particular T and
T
¯
can be written as
T⫽ t
0
⫹ t
˜
0
t
1
⫹ t
˜
0
t
˜
1
t
2
⫹ ...⫹ t
˜
0
t
˜
1
t
˜
2
•••t
n
,
T
¯
⫽ t
˜
0
⫹ t
0
t
˜
1
⫹ t
0
t
1
t
˜
2
⫹ ...⫹ t
0
t
1
t
2
••• t
˜
n
,
where t
i
and t
˜
i
are defined via the recursion formulas:
t
i
⫽
共
I⫺ t
i⫺ 1
t
˜
i⫺ 1
⫺ t
˜
i⫺ 1
t
i⫺ 1
兲
⫺ 1
t
i⫺ 1
2
,
t
˜
i
⫽
共
I⫺ t
i⫺ 1
t
˜
i⫺ 1
⫺ t
˜
i⫺ 1
t
i⫺ 1
兲
⫺ 1
t
˜
i⫺ 1
2
and
t
0
⫽
共
⑀
⫺ H
00
兲
⫺ 1
H
01
†
,
t
˜
0
⫽
共
⑀
⫺ H
00
兲
⫺ 1
H
01
.
The process is repeated until t
n
,t
˜
n
⭐
␦
with
␦
arbitrarily
small.
22
With this proviso, we can write the bulk Green’s function
as
G
共
E
兲
⫽
共
⑀
⫺ H
00
⫺ H
01
T⫺ H
01
†
T
¯
兲
⫺ 1
.
If we compare the previous expression with Eq. 共2兲 in the
hypothesis of leads and conductors being of the same mate-
rial 共bulk conductivity兲,
23
we can identify the present bulk
system, or rather one of its principal layers, with the conduc-
tor C, so that H
00
⬅H
C
, H
01
⬅h
CR
, and H
01
†
⬅h
LC
†
. In par-
ticular, by comparing with Eq. 共3兲, we obtain the expression
of the self-energies of the conductor-leads system:
⌺
L
⫽ H
01
†
T
¯
, ⌺
R
⫽ H
01
T.
The coupling functions are then obtained from the sole
knowledge of the transfer matrices and the coupling Hamil-
tonian matrix elements: ⌫
L
⫽⫺Im(H
01
†
T
¯
) and ⌫
R
⫽
⫺ Im(H
01
T
¯
). N.B. The knowledge of the bulk Green’s func-
tion G gives also direct informations on the electronic spec-
trum via the spectral density of bulk electronic states:
N(E)⫽⫺(1/
)Im
关
TrG(E)
兴
.
B. Transmission through an interface
The procedure outlined above can also be applied in the
case where electron transmission takes place through an in-
terface between two different media, as in the system de-
picted in Fig. 2. To study this case we make use of the
surface Green’s function matching 共SGFM兲 theory, pio-
neered by Garcia-Moliner and Velasco.
14
We have to solve Eq. 1 for H⫽ H
I
and G⫽ G
I
, where the
subscript I refers to the interface region. Using the SGFM
method, G
I
is calculated from the bulk Green’s function of
the isolated systems G
A
and G
B
, and the coupling between
the two sides of the interface, H
AB
and H
BA
. In the language
of Ref. 14, all these quantities can be expressed using 2⫻ 2
supermatrices, defined via the introduction of the appropriate
projection operators that map the subspaces of the different
materials. In particular we can define P⫽ P
A
⫹ P
B
where P is
the projector in the space of the existing orbitals, and I
I
⫽ I
A
⫹ I
B
is the projector of the interface region. The bulk
Hamiltonian H
兵
A,B
其
is meaningful only if related with the
corresponding part of P and I
I
. For example, the interface
part of the bulk Green’s function for material A is given by
G
A
⫽ I
A
G
A
I
A
⫽
冉
G
A
0
00
冊
, G
A
⫺ 1
⫽
冉
G
A
⫺ 1
0
00
冊
,
and G
A
⫺ 1
G
A
⫽ I
A
.
Let us now consider the propagation of an elementary
electronic excitation in the system. Via the calculation of the
transmitted and reflected amplitudes of an excitation that
propagates from medium A to medium B, it can be shown
that the interface Green’s function obeys the following secu-
lar equation:
14
G
I
⫺ 1
⫽
共
I
A
⑀
I
A
⫺ I
A
H
A
P
A
G
A
G
A
⫺ 1
I
A
兲
⫹
共
I
B
⑀
I
B
⫺ I
B
H
B
P
B
G
B
G
B
⫺ 1
I
B
兲
⫺
共
I
A
H
I
I
B
⫹ I
B
H
I
I
A
兲
.
In the language of layer Hamiltonians and block superma-
trices, the previous equation reads
FIG. 2. Sketch of a system containing an interface. I is the
interface region for which we need to compute the Green’s function
G
I
.
7830 PRB 60
MARCO BUONGIORNO NARDELLI
G
I
⫽
冉
G
AA
G
AB
G
BA
G
BB
冊
⫽
冉
⑀
⫺ H
00
A
⫺
共
H
01
A
兲
†
T
¯
⫺ H
AB
⫺ H
BA
⑀
⫺ H
00
B
⫺ H
01
B
T
冊
⫺ 1
. 共5兲
Once the interface Green’s function is known, we can
compute the transmission function in terms of block super-
matrices making use of the interface projection operators:
T
共
E
兲
⫽ Tr
共
⌫
L
G
I
r
⌫
R
G
I
a
兲
, ⌫
L
⫽ I
A
⌫
A
I
A
, ⌫
R
⫽ I
B
⌫
B
I
B
.
Using Eq. 共5兲 we can write, after some matrix multiplica-
tions:
T
共
E
兲
⫽ Tr
共
⌫
A
G
AB
r
⌫
B
G
BA
a
兲
and G
BA
a
⫽ (G
AB
r
)
†
. Within the SGFM framework, the same
approach can be extended to the case of multiple interfaces,
superlattices, and the general lead-conductor-lead geometry
14
with little complication. In the previous treatment we have
assumed to have a Hamiltonian representation in terms of
orthogonal orbitals. The extension to the general case of a
nonorthogonal base is described in Appendix A. We have
also assumed a truly one-dimensional chain of principal lay-
ers, which is physical only for systems like nanotubes or
quantum wires that have a definite quasi-one-dimensional
character. The straightforward extension to a truly three-
dimensional case is described in Appendix B. N.B.Asinthe
bulk case, we can calculate the local density of states local-
ized at the interface as N
I
(E)⫽⫺(1/
)Im
关
TrG
I
(E)
兴
.
IV. EXAMPLES
As a first application of the above methodology we stud-
ied the quantum conductance of carbon nanotubes within a
nearest-neighbor
-orbital tight-binding Hamiltonian as in
Chico et al.
11
In this model, the
-orbital bands are de-
scribed via a single parameter V
pp
⫽
␥
0
⫽⫺2.75 eV. As
shown in Fig. 3, we were able to completely reproduce the
results for the conduction of a 共12,0兲/共6,6兲 matched nanotube
obtained in Ref. 11. Although this simple model can give a
reasonable qualitative description of the electronic and trans-
port properties of an ideal carbon nanotube, more sophisti-
cated models have to be used for a more general study. In
particular, geometric relaxations are ineffective in the
-orbital tight-binding model, where only the connectivity
of a given atom plays a role. In order to study the effect of
atomic relaxations on the conductance of carbon nanotubes,
we employed a full sp
3
tight-binding model already used in
the studies of electronic properties of such systems.
16
One of
the advantages of the present method to compute quantum
conductance is that it does not require periodic boundary
conditions along the direction of the principal layer expan-
sion. In quasi-one-dimensional systems such as nanotubes
and nanowires, this implies that very distorted geometries
can be analyzed with a complete convergence in the 共one-
dimensional兲 k
⬜
-point expansion 共see Appendix B兲.
In the following we present our investigations of the ef-
fect that bending has in the transport properties of a small
diameter 共4,4兲 carbon nanotube. It has recently been
observed
7
that individual carbon nanotubes deposited on a
series of electrodes behave as a chain of quantum wires con-
nected in series. The individual nanotube is broken up into a
chain of weakly coupled one-dimensional conductors sepa-
rated by local barriers. It has been argued that the local bar-
riers arise from the bending of the tube near the edge of the
electrodes, but no theoretical evidence has been produced as
yet. In the upper part of Fig. 4 we show the system that we
have studied: an initially straight tube that has been bent at
different angles
⫽ 0°,3°,6° 关Figs. 4 共a兲–4共c兲兴, where
measures the inclination of the two ends of the tubes with
respect to the unbent axis. The geometrical structure has
been optimized using an empirical many-body potential for
carbon.
24
For
⫽ 6° we observe the formation of a kink.
Since the formation of kinks in bent carbon nanotubes has
been thoroughly described both experimentally and
theoretically,
25
we do not discuss it here. In the lower part of
Fig. 4 we present our predictions for electronic conductance
and density of states of the bent tube. The presence of the
kink does not alter drastically the local density of states
共LDOS兲 at the interface nor the conductance of the system in
a substantial manner. This observation rules out the possibil-
ity that the formation of local barriers for electric transport
can be attributed solely to the microscopic deformation of
the tube wall, with no defect involved. The effect of defects
that will naturally form in the bent tube due to the strain
imposed on the system
26
has not been taken into account,
and it will be the subject of future work.
27
In Fig. 5 we show the LDOS and transmission function
for a semiconducting 共10,0兲 and metallic 共9,0兲 nanotubes cal-
culated using an orthogonal and a nonorthogonal TB model.
Both models reproduce the general features of the electronic
structure, in particular the opening of a pseudogap at the
Fermi energy produced by the curvature of the graphitic
walls in the 共9,0兲 nanotube.
28,29
The values of the gap calcu-
lated with the orthogonal 共0.07 eV兲 and nonorthogonal TB
共0.16 eV兲 are in very good agreement with previous
calculations,
28,29
and the nonorthogonal model reproduces
quite accurately the ab initio value.
29
The extension of the
present method to nonorthogonal Hamiltonians opens a way
to calculations of conductance using ab initio real-space
FIG. 3. LDOS and transmission function for the 共12,0兲/共6,6兲
matched tube as in Chico et al. 共Ref. 11兲. The peak in the LDOS
just below the Fermi energy 共taken as reference兲 was not shown in
Fig. 4 of Ref. 11 due to a coarser sampling of the energy 共see Ref.
31兲. Besides this, the two calculations are in complete agreement.
PRB 60
7831ELECTRONIC TRANSPORT IN EXTENDED SYSTEMS...
methods with nonorthogonal localized-orbital bases. Work in
this direction is in progress and will be the subject of a future
publication.
30
V. CONCLUSIONS
In this paper we presented an efficient approach to com-
pute the electronic transport properties of extended systems
and some applications to carbon nanotubes. The essence of
the approach relies on the iterative calculation of transfer
matrices and Green’s functions coupled with the Landauer
formula for the coherent conductance. This method is appli-
cable to any general Hamiltonian that can be described
within a localized-orbital basis and thus can be used as an
efficient and general theoretical scheme for the analysis of
the electrical properties of nanostructures. The applicability
of the method to general orthogonal and nonorthogonal tight-
binding models has been illustrated. In particular, we have
obtained a theoretical analysis of quantum conductance in
bent carbon nanotubes. Our calculations show that carbon
nanotubes maintain their basic electrical characteristics even
in the presence of large distortions and mechanical deforma-
tions.
ACKNOWLEDGMENTS
I am deeply indebted to J. Bernholc for bringing to my
attention the problem of quantum transport in carbon nano-
tubes and for his continuous support and warm encourage-
ment in the course of this work. I am very grateful to J.-L.
Fattebert for unraveling to me the subtleties of the overlap
matrices. I am also pleased to acknowledge fruitful discus-
sions with F. Cleri, L. Benedict, L. Chico-Gomez, D. Or-
likowski, and C. Roland. J. Bernholc and J.-L. Fattebert are
also thanked for a critical reading of the manuscript.
APPENDIX A
The expression for the Green’s and transmission functions
of a bulk system described by a general nonorthogonal
localized-orbital Hamiltonian follows directly from the pro-
cedure outlined in Sec. III A. All the quantities can be ob-
tained making the substitutions: (
⑀
⫺ H
00
)→(
⑀
S
00
⫺ H
00
)
and H
01
(†)
→⫺ (
⑀
S
01
(†)
⫺ H
01
(†)
). Here, we introduce the matrices
S’s that represent the overlap between the localized orbitals.
With this recipe, the equation chain 共4兲 now reads
共
⑀
S
00
⫺ H
00
兲
G
00
⫽ I⫺
共
⑀
S
01
⫺ H
01
兲
G
10
,
共
⑀
S
00
⫺ H
00
兲
G
10
⫽⫺
共
⑀
S
01
†
⫺ H
01
†
兲
G
00
⫺
共
⑀
S
01
⫺ H
01
兲
G
20
,
...,
FIG. 4. Upper panel: geometry of the bent 共4,4兲 nanotube used
in the calculations. Lower panel: LDOS and transmission function
for the different geometries: 共a兲
⫽ 0°, 共b兲 3°, 共c兲 6°. LDOS’s for
different bending angles are shifted in the picture. The Fermi en-
ergy is always taken as a reference.
FIG. 5. Upper panel: LDOS and transmission function for a
共10,0兲 carbon nanotube using the nonorthogonal tight-binding of
Porezag et al. 共Ref. 17兲 compared with the orthogonal model of
Charlier et al. 共Ref. 16兲 Lower panel: same as above for a 共9,0兲
tube. The Fermi energy is taken as reference.
7832 PRB 60
MARCO BUONGIORNO NARDELLI
