PHYSICAL REVIEW B
VOLUME 62, NUMBER 4
15 JULY 2000-II
Transfer-matrix approach for modulated structures with defects
T. Kostyrko
Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee, 37996-1200
and Institute of Physics, A. Mickiewicz University, ulica Umultowska 85, 61-614 Poznań, Poland
共Received 22 October 1999兲
We consider scattering of electrons by defects in a periodically modulated, quasi-one-dimensional structure,
within a tight-binding model. Combining a transfer matrix method and a Green function method we derive a
formula for a Landauer conductance and show its equivalence to the result of Kubo linear response theory. We
obtain explicitly unperturbed lattice Green functions from their equations of motion, using the transfer matrices. We apply the presented formalism in computations of the conductance of several multiband modulated
structures with defects: 共a兲 carbon nanotubes 共b兲 two-dimensional 共2D兲 superlattice 共c兲 modulated leads with
1D wire in the tunneling regime.
I. INTRODUCTION
The transfer matrix method is known to be an efficient
tool for studying properties of one-dimensional 共1D兲 disordered systems.1 It may be used to obtain a density of states,
a localization length of a wave function and transport properties of the linear chain. The method was also applied in
numerical studies of localization in 2D and 3D lattices,
where the properties of 2D or 3D systems may be deduced
by scaling from studies made on a quasi-1D system of finite
cross section 共a stripe or a bar兲.2
In the present paper we consider application of the
method to study electronic properties of quasi-1D modulated
structures. The system may be described as a quasi-1D chain
including l groups of atoms within each unit cell and schematically represented by a series:
•••⫺ 关 A1 •••Al 兴 m ⫺ 关 A1 •••Al 兴 m⫹1 ⫺•••,
where Ak denote consecutive groups of atoms within cells
and m is a cell index. We describe this system with the help
of a tight binding Hamiltonian
H⫽
†
†
a m,k
W k a m,k⫹1 ⫹H.c.兲
共 H k ⫹⌬ m,k 兲 a m,k ⫹ 兺 共 a m,k
兺
m,k
m,k
⫹
†
t m a m,l ⫹H.c.兲 .
兺m 共 a m⫹1,1
共1兲
†
is an N a dimensional vector made of electron
Here, a m,k
creation operators for the atoms of kth group in the mth unit
cell, k⫽1•••l.3 First term in Eq. 共1兲 is a contribution from
individual groups of atoms and Hermitian matrices H k include intragroup hopping and site energy parameters. This
term describes also an interaction with pointlike defects and
the diagonal matrices ⌬ m,k include the corresponding change
of site energy parameters. The second term in the Hamiltonian describes hopping between neighboring groups of atoms in the same cell and matrices W k include the corresponding hopping parameters. The last term in Eq. 共1兲
accounts for hopping between the neighboring cells and t m
denotes intercell hopping matrix.
0163-1829/2000/62共4兲/2458共8兲/$15.00
PRB 62
For a computational convenience we assume here that
each group Ak includes the same number of N a atoms, and
all the hopping matrices which appear in Eq. 共1兲 are square
and nonsingular matrices of dimension N a ⫻N a . Within this
approach possible constrictions 共or vacancies兲 in the system
may be considered by assuming a very large site energy at
the boundaries of the considered structures 共or at the vacancies兲. In Appendix D we discuss an alternative method for
the systems with constrictions, with the boundary conditions
strictly incorporated into the formalism.
Our Hamiltonian does not include explicitly electronelectron interactions but all the hopping and site energy parameters may be thought of as the effective ones, i.e., renormalized by the interactions by means of an effective field
共Hatree-Fock兲 theory. This mean-field renormalization,
which validates results of the present paper, does not apply
to infinite, strictly 1D systems, which are well known to
exhibit Luttinger liquid correlations. On the other hand with
the increase of the number of bands near the Fermi level the
quasi-1D system approaches gradually the Fermi liquid, as
discussed in detail in Ref. 4.
The Hamiltonian 共1兲 provides a common framework for a
description of a number of rather different physical systems.
In the simplest case it may be used to model a molecular
chain with Peierls distortion 共dimerization for l⫽2, tetramerization for l⫽4, etc.兲: in such a case all the hopping and site
energy matrices are reduced to numbers.
The Hamiltonian may describe a stripe of a width N a
⫻a (a is the interatomic distance兲, where the site energy is
modulated with a period l along the chain. Such a periodically modulated channel was studied earlier in Refs. 5,6.
The model is also suitable to study transport properties in
molecular wires or ribbons.7,8 In our previous work9 we used
Hamiltonian 共1兲 to calculate a reflection from defects and
localization in carbon nanotubes.
II. TRANSFER MATRIX METHOD
The use of Hamiltonian 共1兲 with a one-electron wave
function
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©2000 The American Physical Society
PRB 62
TRANSFER-MATRIX APPROACH FOR MODULATED . . .
⌿⫽e ⫺i ␻ t
†
a m,k
x m,k 兩 0 典
兺
m,k
共2兲
in the Schrödinger equation leads to a system of recurrence
equations for the N a -dimensional vector coefficients x m,k :
␻ x n,p ⫽ 共 H p ⫹⌬ n,p 兲 x n,p ⫹W †p⫺1 x n,p⫺1 ⫹W p x n,p⫹1
⫹ ␦ p,1t n⫺1 x n⫺1,l ⫹ ␦ p,l t †n x n⫹1,1 ,
共3兲
where W p ⫽0 for p⬍1 or p⬎l⫺1. Solution of the system of
equations determines the eigenstate 共2兲 of H corresponding
to the eigenenergy ␻ . The flux carried through the link between cells m and m⫹1 by an electron in the state ⌿ is
†
†
†
t m x m,l ⫺x m,l
tm
x m⫹1,1 .
iJ m ⫽x m⫹1,1
共4兲
It can be proved using Eq. 共3兲 and the fact that the matrices
H k are Hermitian, that the flux is independent of the cell
index m, i.e., it is conserved.
To determine the transmission through a barrier created
by a system of defects we follow the change of the amplitudes x m,1 and x m,l along the chain. Transfer matrices T m
relate the amplitudes at the consecutive links and provide a
formal solution for the amplitudes
冉 冊 冉 冊
x m⫹1,1
x m,l
⫽T m
x m,1
x m⫺1,l
⫽T m T m⫺1 •••T 1
冉 冊
x 1,1
x 0,l
共5兲
.
T m can be given with the help of an intracell transfer matrix
T cell,m as
T m⫽
冉
†⫺1
tm
A m,l
†⫺1 †
⫺t m
W l⫺1
I
0
⫻T cell,m
冉
I
0
冉
⫺1
⫺W k⫹1
W †k
I
0
冊
冊
共6兲
,
冉 冊
⌳
0
0
˜
⌳
,
0⬍k⬍l⫺2 共7兲
,
˜ ⫽I.
⌳⌳
共8兲
The complex eigenvalues of T perfect with modulus equal to
unity correspond to running waves. Besides, there are in general complex eigenvalues of T perfect related to decaying solutions for a given energy ␻ . Using T D and S, we write the flux
conservation law for the perfect lattice as11
† †
TD
S
冉
0
t
⫺t †
0
冊
ST D ⫽S †
冉
0
t
⫺t †
0
共10兲
where L and L⫹M are the first and last cells of the device,
respectively.
The particle incident at the barrier splits into the reflected
and transmitted parts. Both these parts can now have contributions corresponding to evanescent waves with amplitudes
exponentially decreasing with distance from the barrier. The
amplitude of the reflected part is determined by the reflection
matrix ␳ and the amplitude of the transmitted part is determined by the transmission matrix ␶ . Relations between these
two matrices can be obtained from the requirement that the
transmitted wave does not include an exponentially growing
contributions as well as any part corresponding to a wave
running towards the barrier 共a causality condition兲. We compare the current 共4兲 on the two sides of the barrier and far
from it. With the help of Eq. 共9兲 we obtain then a multichannel reflection law in the form
冊 冉
S⫽
˜␶ ⫽ 冑V ␶ 冑Ṽ,
˜␳ ⫽ 冑V ␳ 冑Ṽ,
⫺1
␳ ⫽Q 22
Q 21 ,
and T cell,m ⫽I for l⬍3. Here, I denotes a unit matrix of an
appropriate dimension, 0 is a matrix of zeros and A m,k ⫽ ␻ I
⫺H k ⫺⌬ m,k . In the case of a perfect lattice, the transfer
matrix T m ⫽T perfect can be diagonalized with a similarity
transformation S, what can be represented in the blockmatrix form10
T D ⫽S ⫺1 T perfectS⫽
Q⫽S ⫺1 T L⫹M •••T L⫹1 T L S,
共11兲
where
⫺W ⫺1
1 t m⫺1
⫺1
W k⫹1
A m,k⫹1
In the last equation we introduced a velocity matrix V 共see
Appendix A兲.
We analyze now a general case of a modulated structure
with defects. For the description of the experimental arrangement we divide the 1D system into three parts. The central
part 共a device兲, representing a disordered fragment of the
lattice, is sandwiched between perfect fragments of the
modulated structure 共leads兲, adiabatically extending into reservoirs. The device plays a role of a barrier for a particle
approaching it from ⫺⬁. It may be represented by a following product of the transfer matrices:
˜␶ †˜␶ ⫹˜␳ †˜␳ ⫽I,
W ⫺1
1 A m,1
where T cell,m ⫽T m,l⫺2 T m,l⫺3 •••T m,1 ,
T m,k ⫽
冊
2459
iV
⫺␥
␥†
⫺iV
冊
.
共9兲
VṼ⫽ṼV⫽ P f ,
共12兲
⫺1
␶ ⫽Q 11⫺Q 12Q 22
Q 21 .
Here, the matrices Q ␮ ␯ ( ␮ , ␯ ⫽1,2) are N a ⫻N a dimensional
block submatrices of matrix Q. The projection operator P f
projects out all evanescent modes and has the properties
˜ ⫽⌳
˜ P f ⫽ P f ⌳ † and P f V⫽V P f ⫽V. By the convention,
Pf⌳
⌳ includes eigenvalues of T perfect of the solutions allowed
behind the barrier.
It is natural to expect that the transmission matrix, which
determines the conductance in the Landauer theory
⌫⫽2
e2
tr共˜␶ †˜␶ 兲 ,
h
共13兲
is also directly related to the conductance of the Kubo linear
response theory. The formal proof of this was obtained for
the uniform leads long ago.12 In the next section we use the
Green function method to show that the equivalence of the
two approaches indeed remains true for a periodic potential
existing within the semi-infinite leads.
III. THE GREEN FUNCTION APPROACH
Using the Green functions formalism, the relations between the amplitude of a wave incident at a barrier ␣ m,k and
a scattered wave x m,k may be written as
2460
T. KOSTYRKO
x m,k ⫽ ␣ m,k ⫹
⫽ ␣ m,k ⫹
(0)
G mk,np
⌬ n,p x n,p
兺
n,p
G mk,np ⌬ n,p ␣ n,p ,
兺
n,p
共14兲
共15兲
are the Green’s functions and H0 describes the perfect lattice. The amplitudes of waves approaching the barrier from
⫺⬁ are given by
(␯)
␣ m,l
⫽S 21⌳ mf e ␯ ,
共16兲
where index ␯ numbers the solutions. Above we used blockmatrix representation of S, where S was partitioned into four
submatrices S i j (i, j⫽1,2) of dimension N a ⫻N a .
The perfect lattice Green function G (0) may be obtained
from its equation of motion using the transfer matrix. The
(0)
(⫹) m⫺n⫺1
(⫺) ˜ n⫺m
⫽g kp
⌳
⫹g kp
⌳
,
most general solution is G mk,np
(⫾)
where the matrices g
do not depend on n,m. Because the
solutions for the amplitudes x m,k far behind the barrier 共i.e.,
for m→⬁) should not include expanding solutions as well as
solutions returning towards the barrier 共the causality condition兲, the matrices g (⫺) must vanish. This leads to the following block-matrix form of G (0) for m⬎n:
冉 冊冉
(0)
G m1,np
(0)
G ml,np
⫽
S 11⌳ m⫺n⫺1 g p
S 21⌳ m⫺n g p
冊
,
where
⫺1 (0)
g p ⫽S 21
G nl,np .
共17兲
The transmission matrix ␶ ⬘ defines a relation between the
(␯)
incident and the transmitted wave far from the barrier x m,k
(␮)
⫽ 兺 ␮ ␶ ␮⬘ ␯ ␣ m,k
. By inserting Eqs. 共16兲,共17兲 into Eq. 共14兲 we
obtain
(␯)
˜ nf g p ⌬ np x n,p
␶ ␮⬘ ␯ ⫽ ␦ ␮ ␯ ⫹ 兺 e ␮† ⌳
.
np
共18兲
(␯)
from Eq. 共18兲 using
We now eliminate the amplitude x n,p
Eq. 共15兲. Next we eliminate the matrices ⌬ mk by applying
the Dyson equations for G
G⫽G (0) ⫹G (0) ⌬G⫽G (0) ⫹G⌬G (0) .
共19兲
From Eqs. 共17兲,共19兲 we obtain the following sum rules:
兺
np
⫺1
˜ nf g p ⌬ np G np,mk ⫽⌳
˜ j⫺1
˜ mf g k ,
⌳
S 11
G j1,mk ⫺⌳
f
† ⫺1
⫺1
g 1 ⌳⫽ P f 共 ⫺iV⫹ ␥ S 12
S 11兲 ⫽⫺i P f V⫽⫺iV.
P f S 11
⫺1 (0)
⫺1†
S 11
G j1,n1 S 11
→i⌳ j⫺n
Ṽ P f .
f
(0)
†
⫽ 具 0 兩 a m,k 共 ␻ ⫺H0 兲 ⫺1 a n,p
兩0典
G mk,np
⌳ f ⫽ P f ⌳,
共22兲
共23兲
Using Eq. 共23兲 we may write asymptotic expression for the
G (0)
j1,n1 for n→⫺⬁ in the form
†
兩0典,
G mk,np ⫽ 具 0 兩 a m,k 共 ␻ ⫺H兲 ⫺1 a n,p
共 e ␯ 兲␮⫽ ␦ ␮␯ ,
⫺1
˜ j⫺1
␶ ⬘ ⫽⌳
S 11
G j1,nr g r⫺1 ⌳ nf .
f
With some algebra we can prove 共see Appendix B兲 that
where
(␯)
␣ m,1
⫽S 11⌳ m⫺1
e␯ ,
f
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共20兲
⫺1
(␯)
⫺1
˜ j⫺1
˜ j⫺1
⌳
S 11
G j1,mk ⌬ mk ␣ m,k
⫽⌳
S 11
G j1,nr g r⫺1 ⌳ nf e ␯ ⫺e ␯ ,
兺
f
f
mk
共21兲
which are valid for j→⬁,n→⫺⬁. After inserting Eqs. 共20兲,
共21兲 into Eq. 共18兲 we obtain
共24兲
In the same limit we have
⫺1
⫺1†
⫺1
⫺1†
G j1,n1 S 11
→S 11
G j1,n1 S 11
Pf
S 11
共25兲
which can be easily deduced from Eq. 共24兲 if we write the
expression for G using the T matrix,13 G⫽G (0) ⫹G (0) TG (0) .
Using Eqs. 共23兲 and 共25兲 we obtain for the transmission for a
unit flux ˜␶ ⫽ 冑V ␶ ⬘ 冑Ṽ, the result
⫺1
⫺1†
˜␶ ⫽i⌳ j⫺1†
冑VS 11
冑V⌳ n⫺1
G j1,n1 S 11
.
f
f
共26兲
Finally, we obtain the following formula for the square of
norm of ˜␶ which defines the conductance in the Landauer
approach:
⫺1
⫺1†
⫺1 †
⫺1†
G j1,n1 S 11
VS 11
G j1,n1 S 11
tr˜␶˜␶ † ⫽tr共 VS 11
兲,
共27兲
where j→⬁,n→⫺⬁. The right-hand side of Eq. 共27兲, takes
a form of the Kubo formula for conductance.12,14 In this way
we showed equivalence of the Kubo and Landauer approaches for the conductance for an arbitrary periodic potential in the leads.
IV. APPLICATIONS OF THE FORMALISM
TO MODULATED SYSTEMS
In this section we discuss several simple applications of
the obtained formulas. The examples were chosen with the
intention to demonstrate versatility of the presented formalism. In all the cases the conductance was calculated using the
Mathematica program, which is available online,15 where
some other applications of the method are also presented.
A. Scattering from a single defect and resonant states
Let us consider a single point defect in a modulated structure. In this case the T matrix can be readily obtained as T
⫽⌬(I⫺G (0) ⌬) ⫺1 . Suppose we have a defect in first subcell
at the site L. Using Eqs. 共17兲,共22兲 we obtain a simple result
for the transmission matrix
†
(0)
˜␶ ⫽ P f ⫺i 冑ṼS 11
⌬ L1 共 1⫺G L1,L1
⌬ L1 兲 ⫺1 S 11冑Ṽ.
共28兲
The above result shows general features of the single defect
scattering. For a weak defect the transmission drops rapidly
close to the band edges where the velocity vanishes and the
second term in Eq. 共28兲 becomes important. For a strong
defect one may have a resonant state in the multiband
system.16 The resonant state manifests itself as an effective
enhancement of the defect strength which leads to a reduction of the transmission.
TRANSFER-MATRIX APPROACH FOR MODULATED . . .
PRB 62
We use Eq. 共28兲 to calculate the reflection due to a single
point defect in a tight-binding model9 of carbon nanotubes.
The structure of a carbon nanotube may be simply visualized
as a wrapped up graphene plane, and electronic properties of
the nanotube are determined by the vector (N,M ) which
connects equivalent atoms at this wrapping.17 We restrict
ourselves to undoped armchair (N a ,N a ) nanotubes 共with arbitrary N a ) and metallic zigzag (N a ,0) nanotubes 共with N a
⫽3,6,9, . . . ). In the undoped nanotubes, there are two conducting bands near the Fermi level ⑀ F , which is positioned at
the center of the band system, i.e., ⑀ F ⫽0.
The unit cell of armchair nanotubes includes two groups
of atoms, l⫽2, whereas the unit cell of a zigzag nanotube is
made of four subcells, l⫽4, and the total number of atoms in
the unit cell is N⫽4N a in both cases. The defect matrix ⌬ L1
takes a form (⌬ L1 ) ␮ ␯ ⫽E d ␦ ␮ , ␥ ␦ ␯ , ␥ where E d is the site energy at the defect position ␥ in the first subcell of cell L.
Because all carbon atoms in the nanotube are physically
equivalent, the transmission does not depend on the position
of the defect and the diagonal matrix element of the Green
(0)
) ␥ , ␥ is also independent of L, k, and ␥ .
function, (G Lk,Lk
Using Eq. 共28兲 and the results for the diagonalization
transformation S from Ref. 9 we obtain the transmission matrix in the subspace corresponding to the conducting channels
冉 冊
Ed /vF 1
˜␶ f ⫽I⫺
2 1⫹i P E
0 d 1
N
i
1
1
,
(0)
P 0 ⫽⫺i 共 G Lk,Lk
兲␥,␥ ,
共29兲
for the armchair nanotubes. Here v F ⫽ 冑3t 0 /2 共where t 0 is
the nearest neighbors hopping parameter兲 is the common
Fermi velocity of the bands at ␻ ⫽ ⑀ F ⫽0. The imaginary part
(0)
) ␥ , ␥ gives the density of states of the perfect nanoof (G Lk,Lk
tube, and its real part vanishes due to the symmetry of the
band structure for ␻ →⫺ ␻ . On the other hand, the density of
states in the 1D case is determined by the velocity v F . As a
result we have P 0 ⫽1/2N a v F and the conductance 共in units
of e 2 /h) is given by
1 共 E d /2N a v F 兲 2
⌫
⫽1⫺
,
nc
2 1⫹ 共 E d /2N a v F 兲 2
共30兲
where n c ⫽2 is the number of the conducting channels per
spin. Equation 共30兲 holds for the zigzag nanotubes as well,
with the only difference that in the case of zigzag nanotube
we have v F ⫽t 0 /2 for ␻ ⫽ ⑀ F ⫽0. Note that in the case of
weak E d , 兩 E d /t 0 兩 ⰆN a , we obtain a simple scaling of the
reflection coefficient R⫽n c ⫺⌫
R⫽sn c 共 E d /t 0 N a 兲 2
found numerically in Ref. 9, where s⫽1/6 for armchair
nanotubes and s⫽1/2 for the zigzag nanotubes. Result 共30兲
was also recently obtained in Ref. 18 for the case of the
armchair nanotubes, its present extension to the zigzag nanotubes is a new finding.
B. Conductance of a 2D superlattice
We consider now a problem of the influence of defects on
conductance of a 2D superlattice. The superlattice is repre-
2461
FIG. 1. The conductance of the tight binding model of the 2D
superlattice 共in units of 2e 2 /h) as a function of the Fermi energy
␻ ⫽ ⑀ F 共in units of the hopping parameter t 0 ) for the uniform stripe,
i.e., without the site energy modulation 共thin solid line兲 and with
side energy modulation 共bold solid line兲 共a兲. The reflection coefficient as a function of the Fermi energy, for the defect E d ⫽2t 0 . The
defect is positioned at the center of the area of the constant site
energy E⫽⫺1/2 共b兲, at the threshold of the site energy in the
middle of the stripe 共c兲, at the threshold of the site energy and at the
edge of the stripe 共d兲.
sented here by a stripe of a width N a ⫻a with a stepwise
modulation of a site energy along the stripe, with a periodicity of the modulation l. In terms of the parameters defined
in Eq. 共1兲 it may be written as follows:
t m ⫽W k ⫽t 0 I,
共 H k 兲 ␮ , ␯ ⫽t 0 共 ␦ ␮ , ␯ ⫹1 ⫹ ␦ ␮ ⫹1,␯ 兲
⫹
再
⫺ ␦ ␮,␯E
for
1⭐i⭐l/2,
⫹ ␦ ␮,␯E
for
l/2⬍i⭐l.
In this simple model the matrix S of the diagonalization
transformation can be readily found as described in Appendix C and can be used to compute the conductance using Eq.
共12兲. In Fig. 1共a兲 we show the dependence of the conductance on the Fermi energy for N a ⫽11, l⫽10 and E⫽1/2 共we
put t 0 ⬅1) for the perfect lattice. In Figs. 1共b兲–1共d兲 we show
the reflection 共i.e., the difference between the transmission
without defect and the transmission in the presence of the
defect兲 as a function of the Fermi level, for different positions of the single point defect of the strength E d ⫽2 at zero
temperature.
The general feature of the superlattice is the appearance
of minibands6 separated by energy gaps. Comparing the conductance of the uniform stripe 关thin solid line in the Fig.
1共a兲兴 and that of the stripe with stepwise modulation of the
potential 关bold solid line in Fig. 1共a兲兴 we note that the introduction of the modulation of the potential creates gaps in
nine out of eleven bands present at ␻ ⫽0.
The reflection of the system with the defect shows a rather
complicated pattern of the dependence on ␻ , characteristic of
the position of the defect. The reduction of the conductance
by the defect potential is very selective, which is best exemplified by the perfect transmission in the region around ␻
⫽0 in Figs. 1共b兲,1共c兲, corresponding to the defect positioned
T. KOSTYRKO
2462
at the longitudinal symmetry axis of the stripe. Evidently, the
two bands crossing at the ␻ ⫽0 point do not couple to the
defect potential at all, due to symmetry reasons. On the other
hand, moving the defect out of the symmetry axis, significantly increases the reflection at ␻ ⫽0 关Fig. 1共d兲兴.
The round maxima seen in the plot of R at certain values
of ␻ 关e.g., ␻ ⬇⫺0.4,⫹0.7 in Figs. 1共b兲,1共c兲 and ␻ ⬇0 in
Fig. 1共d兲兴 are related to the appearance of resonant 共quasibound兲 states, as was observed earlier for the uniform
stripes.16 For the energy ␻ , corresponding to the position of
the resonance, the conductance is reduced by exactly one
unit 共in the present case of the single pointlike defect兲. The
defect is especially detrimental for the conductance in very
narrow minibands, where the ratio of the defect strength to
the effective bandwidth favors appearance of the resonance
states. As a consequence the narrow spikes disappear from
the plot of conductance 共and show on the plot of the reflection兲 in the presence of the defect, independently of its position.
C. Tunneling through a semiconducting wire
Here we address a problem of the influence of the properties of measuring leads on the total conductance of the
lead-device-lead system. We demonstrate this influence using as an example a model of a 1D molecular chain connecting the leads made of superlattices of a finite width (N a
⫽11).
Let us take an infinite 1D chain with alternating site energy along the chain. It may be considered as a prototype of
a Peierls insulator where on-site charge density modulation
is accompanied by an intramolecular distortion due to a coupling of electrons to intramolecular vibrations. Such a wire
will develop a gap in the middle of its spectrum and will be
a perfect insulator at T⫽0 K for a half-filled band case with
the chemical potential at the center of the band gap. A finite
chain, on the other hand, will exhibit a finite conductance
even at zero temperature, due to the tunneling effect. In Fig.
2 we present the dependence of the conductance on a length
L of the 1D chain in the lead-wire-lead system, for the cases
of uniform leads and the modulated ones. In the latter case
we take the stepwise dependence of the site energy with l
⫽4 and E⫽1/2 共see the previous subsection兲.
For short wires, conductance depends considerably on the
parity of the number of atoms in the chain, even more than
the length of the chain itself. The conductance alternates
with addition of the consecutive atoms. This phenomenon is
due to alternation between a constructive or a destructive
interference of the particle wave multiple reflected between
the boundaries of the leads and the wire.19 The interference
is very sensitive to the dependence of the potential near the
boundary of the leads. Indeed, in the case of the modulated
leads, we observe a nonmonotonuous dependence of the conductance on L for odd values of L 共see filled circles in Fig.
2兲. The position of the maximum of the conductance first
increases, and then decreases with increasing of the modulation of the potential in the leads. The conductance at the
maximum corresponds to the perfect transmission, ⌫
⬇2e 2 /h. This effect may be interpreted as an example of a
resonance tunneling through the wire. It suggests, that it may
be possible to reach the maximum possible conductance for
PRB 62
FIG. 2. The conductance 共in units of 2e 2 /h) of the 1D wire with
site energy alternation: E n ⫽0.1⫻(⫺1) n 共given in units of t 0 ) as a
function of the wire length L 共in units of the lattice constant a), for
uniform leads 共empty circles兲, and leads with site energy modulation 共filled circles兲. An overall trend of ⌫ to decrease with L 共seen
clearly for large L), is notably modified by a tendency to alternate
between the values of ⌫ for L even and L odd, due to the interference effect 共see text兲.
quite a long molecular wire even for the Fermi energy in the
middle of the wire’s gap, by a proper choice of the amplitude
共and periodicity兲 of the potential modulation in leads.
The conductance for even L approaches smoothly the
value of conductance obtained for the uniform leads. With
increasing the wire length, the multiple reflections become
less important and points corresponding to odd and even
values of L approach a common curve. For a long enough
wire, independently of the structure of the leads, the conductance decreases exponentially with length L, with the exponent related to the value of the energy gap.
V. SUMMARY
The main objective of the present paper was to present the
transfer matrix method as a useful tool for investigation of
various modulated structures with defects. The notion of the
modulated structure we used, comprised such seemingly different systems such as carbon nanotubes, molecular wires
with lattice distortion, and superlattices.
Combining the transfer matrix and the Green function
methods we extended the proof of equivalence of the Landauer formula and the Kubo formula for the conductance, to
the systems with modulated leads. As a by-product of the
derivation we obtained some general formulas for the lattice
Green functions in modulated systems. The Green functions
were represented in terms of parameters of the transfer matrix as well as the matrix of its diagonalizing transformation.
The formulas may be useful in computations of local density
of states, solution of the Dyson equation for the problem of
local defect and calculation of the conductance in modulated
structures.
In the Appendices we include also some explicit results
for the parameters of the transformation diagonalizing transfer matrix of the perfect lattice for the case of modulated
PRB 62
TRANSFER-MATRIX APPROACH FOR MODULATED . . .
stripes. We applied these results to solve a problem of scattering by a local defect in the superlattice. In this work, the
single defect solution serves mainly as an example of the
approach to the problem of modulated structures with defects. On the other hand, the solution allows us to predict
some general features of scattering of electrons in the presence of many defects, if one neglects the quantum effects of
interference of electron waves scattered from different
defects.20
As another example we studied the problem of the influence of site energy modulation in leads on the conductivity
in a lead-device-lead system. We showed that the site energy
modulation in leads may significantly modify the conductance, due to change of the conditions for the constructive
interference of the electron reflected from the device-lead
boundary. This result suggests a possibility of investigation
electronic structure of molecular wires by controlled modulation of potential in leads.
Finally, we stress that the results of our work are restricted to situations where the linear response theory is valid
and a more advanced approach19 should be used to study
nonlinear effects of the bias voltage or nonlinear part of I-V
characteristics in a typical experiment. The important problem of the role of electron correlations in the device is also
out of scope of the present work. We note that an extension
of the scattering approach using Kyeldysh formalism may be
useful in this respect.21
The author is grateful to Prof. G. D. Mahan and Dr. M.
Bartkowiak for helpful discussions. Research support is acknowledged from the University of Tennessee, from Oak
Ridge National Laboratory managed by Lockheed Martin
Energy Research Corp. for the U.S. Department of Energy
under Contract No. DE-AC05-96OR22464, and from a Research Grant No. N00014-97-1-0565 from the Applied Research Projects Agency managed by the Office of Naval Research. The support from K.B.N Poland, Project Nos. 2
P03B 056 14 and 2P03B 037 17 is also acknowledged.
APPENDIX A: PROPERTIES OF THE VELOCITY MATRIX
The velocity matrix and the matrix ␥ defined in Eq. 共9兲
are given by
†
† †
tS 21⫺S 21
t S 11兲 ,
V⫽⫺i 共 S 11
˜ ⫽␥.
⌳ †␥ ⌳
†
† †
tS 21⫺S 21
t S 11兲 ⌳ m e ␯ ⫽iV ␮ ␦ ␮ ␯ .
iJm, ␮ ␯ ⫽e ␮† ⌳ m† 共 S 11
共A3兲
One can see11 that V corresponds to a usual definition of the
velocity 共a gradient of the band spectrum兲 provided that the
matrices S ␮ ␯ , determining the running waves 关see. Eq. 共16兲兴,
are properly normalized.
APPENDIX B: GREEN FUNCTIONS OF THE PERFECT
LATTICE
In this appendix we show how the perfect lattice Green
functions can be represented in terms of submatrices of the
matrix S which defines the similarity transformation for the
transfer matrix T perfect . The results may be helpful in solution of Dyson equation and determination of the local density
of states15 for modulated structures with defects.
We first rewrite the matrix T perfect for an arbitrary l in a
form which is formally similar to the transfer matrix of an
alternating lattice (l⫽2, where T cell⫽I)
冉
t ⫺1† B l
⫺t ⫺1† R l
I
0
冊冉
R ⫺1
1 B1
⫺R ⫺1
1 t
I
0
冊
.
共B1兲
The matrices B 1 , B l , R 1 , and R l which substitute A 1 , A l ,
W 1 , and W †1 respectively, in Eq. 共6兲 are defined by
B 1 ⫽A 1 ⫺Q 1 ,
⫺1
T cell,12 ,
Q 1 ⫽⫺W 1 T cell,11
B l ⫽A l ⫺Q l ,
共B2兲
†
⫺1
Q l ⫽W l⫺1
T cell,21T cell,11
,
⫺1
R 1 ⫽W 1 T cell,11
共B3兲
and as can be shown using the flux conservation Q 1 ⫽Q †1 ,
Q l ⫽Q †l , and R 1 ⫽R †l . Using Eqs. 共B2兲,共B3兲 we find relations between the perfect lattice Green functions
(0)
(0)
(0)
⫽Q 1 G n1,nl
⫹R 1 G nl,nl
,
W 1 G n2,nl
共B4兲
共A1兲
†
(0)
(0)
(0)
⫹Q l G nl,n1
.
W l⫺1
G nl⫺1,n1
⫽R l G n1,n1
共B5兲
共A2兲
The causality condition for the wave behind and ahead of a
barrier require that the following relations between the Green
functions are satisfied, respectively:
† †
†
␥ ⫽S 21
t S 12⫺S 11
tS 22 .
From the flux conservation we have
⌳ † V⌳⫽V,
eigenvalues. In the same way the matrix ␥ may be chosen to
be diagonal. Using the projection operator P f , defined in
Sec. II, we may write V⫽V P f ⫽ P f V and ␥ ⫽ ␥ (I⫺ P f )⫽(I
⫺ P f)␥.
As may be expected, the velocity matrix determines the
flux 共4兲 in the perfect lattice
T perfect⫽
ACKNOWLEDGMENTS
2463
From Eq. 共A2兲 it follows that (1⫺⌳ ␮* ⌳ ␯ )V ␮ ␯ ⫽0, i.e., V ␮ ␯
⫽0 unless ⌳ ␮* ⫽1/⌳ ␯ . Since ⌳, by assumption, excludes the
expanding solutions with 兩 ⌳ ␮ 兩 ⬎1, the last equation can be
fulfilled only for 兩 ⌳ ␯ 兩 ⫽ 兩 ⌳ ␮ 兩 ⫽1, which corresponds to running waves. As a result, V ␮ ␯ ⫽0 unless ⌳ ␮ ⫽⌳ ␯ , which is
possible 共a兲 for ␮ ⫽ ␯ , or 共b兲 when some eigenvalues of
T perfect are degenerate. In the last case we can choose the
transformation S diagonalizing T perfect , so that to have diagonal V also in the subspaces corresponding to the degenerate
(0)
⫺1 (0)
⫽S 11S 21
G nl,np ,
G n⫹11,np
(0)
⫺1 (0)
G n⫺1l,np
⫽S 22S 12
G n1,np .
共B6兲
(0)
Using Eqs. 共B4兲–共B6兲 and equations of motion for G nk,np
with k,p⫽1,l we finally represent the lattice Green functions
in terms of the quantities that define the transfer matrix
⫺1
⫺1 ⫺1
(0)
⫺R l 共 B 1 ⫺tS 22S 12
⫽I,
兲 R 1 兴 G nl,nl
关 B l ⫺t † S 11S 21
共B7兲
2464
T. KOSTYRKO
⫺1
⫺1 ⫺1
(0)
⫺R 1 共 B l ⫺t † S 11S 21
⫽I,
兲 R l 兴 G n1,n1
关 B 1 ⫺t † S 22S 12
k⫽l⫺1
共B8兲
(0)
⫺1 ⫺1
(0)
⫽ 共 B 1 ⫺tS 22S 12
,
G n1,nl
兲 R 1 G nl,nl
共B9兲
(0)
⫺1 ⫺1
(0)
G nl,n1
⫽ 共 B l ⫺tS 11S 21
.
兲 R l G n1,n1
共B10兲
In a similar way all the relevant Green functions may be
obtained. The above equations can be simplified by using Eq.
共8兲 together with Eq. 共B1兲, to eliminate matrices B 1 ,B l ,R 1 .
†
(0)⫺1
G nl,n1
S 21⌳, which is needed to prove Eq. 共23兲, we
For S 11
use Eqs. 共B8兲, 共B10兲, and 共A1兲 to obtain
†
(0)⫺1
†
⫺1
⫺1
S 11
G nl,n1
S 21⌳⫽S 11
t 共 S 21⫺S 22S 12
S 11兲 ⫽⫺iV⫹ ␥ S 12
S 11 .
共B11兲
APPENDIX C: DIAGONALIZATION OF THE TRANSFER
MATRIX FOR THE PERFECT LATTICE
We present here explicit formulas for the matrix S in
some simple modulated structures. We can use Eqs. 共8兲,共B1兲
to find equations for the submatrices S 11 and S 12 and relations between the submatrices S ␮ ␯ of S:
˜ ⫹G †1 S 11⌳,
F 1 S 11⫽G 1 S 11⌳
共C1兲
˜,
F 1 S 12⫽G 1 S 12⌳⫹G †1 S 12⌳
共C2兲
†
˜
S 21⫽B ⫺1
l 共 t S 11⫹R l S 11⌳ 兲 ,
†
S 22⫽B ⫺1
l 共 t S 12⫹R l S 12⌳ 兲 ,
共C3兲
⫺1 †
†
F 1 ⫽B 1 ⫺tB ⫺1
l t ⫺R 1 B l R 1 ,
†
G 1 ⫽tB ⫺1
l R 1 . 共C4兲
In some cases G 1 ⫽G †1 and Eqs. 共C1兲,共C2兲 simplify to the
eigenequation for the matrix G ⫺1
1 F 1 . We may then choose
S 12⫽S 11 .
We consider a model 共described in Sec. IV B兲 of a modulated stripe of a width y⫻a where the modulating potential
does not change across the stripe 共except being infinite outside the stripe兲. In this case, assuming the hopping parameter
as our energy unit, we have
A ␮ ␯ ⫽ ␦ ␮ ⫹1,␯ ⫹ ␦ ␮ , ␯ ⫹1 ,
共C5兲
where u k is a potential parameter at kth group of atoms. All
matrices A k as well as their functions: B 1 ,B l ,R 1 ,R l are real
and commute, and G 1 ⫽G †1 . The matrix of eigenvectors of
A, given by
W k ⫽t⫽I,
A k ⫽ ␻ ⫺A⫺u k I,
共 S A 兲␮␯⫽
exp关 i ␲␯ / 共 y⫹1 兲兴
冑共 y⫹1 兲 /2
sin
冉 冊
␲␮␯
y⫹1
共C6兲
defines a unitary transformation which diagonalizes matrices
B 1 ,B l ,R 1 ,R l . As a result we have S 11⫽S 12⫽S A and
I⫹R 1,D ⌳
,
␻ ⫺u l ⫺A D ⫺Q l,D
共C7兲
where D denotes a diagonal form of the matrices. The eigenvalues of the matrices Q 1 ,Q l ,R 1 may be obtained from a
2⫻2 dimensional matrix T ␣ ( ␣ ⫽1•••y)
S 21⫽S A
˜
I⫹R 1,D ⌳
,
␻ ⫺u l ⫺A D ⫺Q l,D
S 22⫽S A
PRB 62
T ␣⫽
兿
k⫽2
冉
␻ ⫺u k ⫺A D, ␣
⫺1
1
0
冊
,
冋 册
A D, ␣ ⫽2 cos
␲␣
y⫹1
共C8兲
using the formulas, corresponding to Eq. 共B3兲
共 R 1,D 兲 ␣␣ ⫽
1
T ␣ ,11
共 Q 1,D 兲 ␣␣ ⫽⫺
,
T ␣ ,12
,
T ␣ ,11
共 Q l,D 兲 ␣␣ ⫽
T ␣ ,21
.
T ␣ ,11
共C9兲
The matrix elements of the product T ␣ , may be explicitly
obtained for an arbitrary l in a simple case of stepwise potential, studied in Sec. IV B as well as in a number of other
cases. Finally, the eigenvalues of T perfect are given by
⌳ ␣ ⫽⍀ ␣ ⫹i 冑
⍀⫽
1⫺⍀ ␣2 ,
2
⫺I
B l,D B 1,D ⫺R 1,D
2R 1,D
.
共C10兲
APPENDIX D: REFLECTION FROM A CONSTRICTION
In the transfer matrix approach as described in Sec. II, we
assumed that all hopping matrices t m , W m,k are square matrices of the same dimension. This was necessary to unambiguously relate amplitudes on consecutive pairs of groups
of atoms along the chain in Eq. 共5兲. In the case of a constriction 共or a vacancy兲 it is still possible to use a transfer matrix
approach as described in Sec. II, by assuming some very
large value of on-site potential on fictitious atoms outside the
boundary of the constriction 共or at the vacancy兲.22
An alternative solution is to include strictly the boundary
condition at the constriction into the formalism. As an example, we analyze here the scattering from a constriction of
a constant width y 1 ⫻a in a uniform stripe of a width y⫻a.
Let y-dimensional vectors x m with m⭐n(m⬎n) denote the
amplitudes of waves on the left 共right兲 side of the constriction and y-dimensional vectors x (k)
n , with 1⬍k⬍L, denote
the consecutive amplitudes of the wave at the constriction.
The vanishing of the amplitude outside the constriction
boundaries may be described with a projection operator P c
(k)
having the property that P c x (k)
n ⫽x n . As in Sec. II we consider the scattering of a particle approaching the constriction
from ⫺⬁. At the left boundary we have
冊 冉 冊冉
冉 冊冉
冉 冊 冉冊
x (1)
n
xn
⫽
u n⫹ v n
Qu n ⫹Q ⫺1 v n
x S1
x S0
⫽S ⫺1
x1
x0
vn
,
un
,
⫽
S A ⌳ n x S1
S A ⌳ ⫺n x S0
Q⫽S A ⌳S A† ,
冊
,
共D1兲
where S A is defined in Eq. 共C6兲. Using the equation x (1)
n
⫽ P c x (1)
n we find the incomplete relation between the amplitudes of incident and reflected waves
共 I⫺ P c 兲v n ⫽⫺ 共 I⫺ P c 兲 u n .
共D2兲
The amplitudes on the right side of the constriction are calculated with the help of amplitudes on the left side of the
constriction
TRANSFER-MATRIX APPROACH FOR MODULATED . . .
PRB 62
冉
P c x n⫹1
x (L)
n
冊 冉 冊
冉
T k⫽
⫽T d
x (1)
n
P cx n
,
˜␳ ⫽⫺ 冑VS A† 关 U ⫺1 W c P c ⫹U ⫺1 共 W c ⫺U c 兲
T d ⫽T L •••T 1 ,
P c 共 ␻ ⫺A⫺⌬ k 兲 P c
⫺I
I
0
冊
,
⫻ 共 I⫺ P c 兲 ⫹I⫺ P c 兴 S A 冑Ṽ,
共D3兲
where ⌬ k is a diagonal matrix describing distribution of the
potentials at atoms of a layer k. The requirement that behind
the constriction there are no solutions corresponding to the
expanding waves and the waves describing particles returning to the barrier, gives the relation
x n⫹1 ⫽Qx (L)
n .
共D4兲
calculated independently from Eqs.
Comparing x (L)
n
共D1兲,共D3兲,共D4兲 gives second relation between the amplitudes
of incident and reflected waves ahead of the barrier
P c QT d,21P c 共 u n ⫹ v n 兲 ⫹ P c QT d,22共 P c Qu n ⫹ P c Q ⫺1 v n 兲
Equation 共D2兲 combined with Eq. 共D5兲 determines unambiguously the amplitude of the reflected wave as a function
of the amplitude of the incident wave. As a result we obtain
the reflection matrix ˜␳ 关see Eq. 共12兲兴
J.M. Ziman, Models of Disorder 共Cambridge University Press,
Cambridge, 1979兲.
2
A. MacKinnon and B. Kramer, Z. Phys. B: Condens. Matter 53, 1
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3
Although it is in general necessary to give the index ␮ ⫽1••
•N a , in addition to m and k, to unambiguously identify the
position of an atom within the kth group of mth cell, the matrix
notation we use in Eq. 共1兲 allows us to suppress this extra level
of indexing.
4
D.L. Maslov and N.P. Sandler, Phys. Rev. B 55, 13 808 共1997兲.
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We restrict ourselves to the case of zero magnetic field.
11
L. Molinari, J. Phys. A 30, 983 共1997兲.
共D6兲
where
U⫽U c P c ⫹I⫺ P c ,
U c ⫽⫺ P c T d,11⫹ P c QT d,21⫺ P c T d,12P c Q⫹ P c QT d,22P c Q,
W c ⫽⫺ P c T d,11⫹ P c QT d,21⫺ P c T d,12P c Q ⫺1
⫹ P c QT d,22P c Q ⫺1 .
Note again, that the explicit formula for the submatrices of
matrix T d , describing constriction, may be obtained in a
number of interesting cases. In the case of a uniform constriction of a length L we may use the diagonal representation of T 1 ⫽•••⫽T L ,
⫽T d,11P c 共 u n ⫹ v n 兲 ⫹T d,12共 P c Qu n ⫹ P c Q ⫺1 v n 兲 . 共D5兲
1
2465
T d ⫽Z
冉
␭L
0
0
␭ ⫺L
冊
Z ⫺1 ,
共D7兲
where the matrix Z, that defines the similarity transformation
diagonalizing T 1 , may be obtained as described in
Appendix C.
D.S. Fisher and P.A. Lee, Phys. Rev. B 23, 6851 共1981兲.
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12
13