Available at: http://www.ictp.it/~pub−off IC/2005/057
United Nations Educational Scientific and Cultural Organization
and
International Atomic Energy Agency
THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
THE TRANSFER MATRIX METHOD AND THE SYLVESTER THEOREM.
INTERACTING MODES AND THRESHOLD EFFECTS IN 2DEG
A. Anzaldo-Meneses
Fısica Teorica y Materia Condensada, UAM-Azcapotzalco,
Av. S. Pablo 180, C.P. 02200, Mexico D.F., Mexico
and
P. Pereyra†
Fısica Teorica y Materia Condensada, UAM-Azcapotzalco,
Av. S. Pablo 180, C.P. 02200, Mexico D.F., Mexico
and
The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
Abstract
Based on the Sylvester and Frobenius theorems, we drastically enhance the feasibility of the
transfer-matrix approach to deal with problems involving a large number of propagating and
interfering modes, which require the solution of coupled differential equations and the evaluation
of functions of matrix variables. We report closed formulas for the spectral decomposition of
this type of functions. We study the transmission properties of a two-dimensional multi-channel
electron gas in the presence of a channel-mixing transverse electric field, and calculate physical
quantities which have not yet been measured nor calculated for this kind of system. We observe
interesting threshold and resonant coupling effects, which we conjecture are responsible for the
appealing but not so neatly understood giant-conductance and resistance phenomena.
MIRAMARE – TRIESTE
May 2005
†Senior Associate of ICTP.
I. INTRODUCTION
The transfer matrix method is being used with success to study different types of problems1,2,
in particular problems regarding transport and optoelectronic properties of quasi-1D disordered
and 1-D periodic systems3–19. The numerical calculations using transfer matrices have been,
nevertheless, discouraged because of blowing up problems when the system’s size (related to the
number of cells n, in the growing direction) and the transfer matrix order (related to the number
of propagating modes N and the system’s dimensionality) is relatively big. As a consequence, a
number of modified transfer matrix methods have been proposed20. Recently, analytic methods
were developed and applied to study multichannel finite periodic systems12,15,16. These new
developments have shown that the annoying transfer matrix multiplication procedure can easily
be circumvented for periodic systems. In fact, simple, compact and closed formulas have been
derived for the evaluation of the whole superlattice scattering amplitudes and related physical
quantities, which depend basically on the single-cell transfer matrix M and on noncommutative
N ×N matrix polynomials12,15 pN,n. Within this approach, the analytical calculation of single
cell transfer matrices M , and the evaluation of functions of matrix variables play a crucial
role. However, such calculations can be rather involved and not such a simple problem. The
experience of deriving multichannel transfer matrices and noncommutative polynomials is quite
modest and has been limited to a small number of propagating modes and simple scattering
potentials. Therefore, further analysis to simplify the mathematical procedures in the scattering
theory and transfer matrix methods, is very much called-for. The main purpose of this paper is
to use the Sylvester’s Theorem and to apply it to obtain the spectral decomposition of analytic
functions of matrix variables21–23. To illustrate the use of this method we study a specific but
non trivial system with several interacting modes. We evaluate transmission properties of a 2D
electron gas moving by any of N propagating modes through a semiconductor heterostructure
subject to an external transverse electric field E = F/e.
In Section II, we shall introduce some well-known basic definitions of the multichannel transfer
matrices of the first and second kind (relating the wave function and its derivative (matrix W )
and state vectors (matrix M) at any two points of the scatterer system, respectively), and useful
relations of these matrices with the principal scattering amplitudes. We will present a suitable
procedure to determine single-cell transfer matrices for systems with arbitrary potential in the
transverse plane x-y but sectionally constant in the growing direction z. We want to stress
here that the present method is adequate for arbitrary transverse potential, whether or not the
corresponding transverse wave functions can be given in terms of known special functions. For
arbitrary potentials, the number of modes N (open channels) will in general limit the accuracy
of the method, even in the case where the equation (in the transversal direction) can be solved
exactly. We discuss this point in Sections II and IV to make clear that we can follow one of
two possible choices: either we use a single basis and the ensuing system of coupled equations
2
(whose dimension is cut when it equals the number of open channels N), or we use the exact
basis for each region plus the boundary conditions. In this case also, the boundary conditions
are solved only approximately by a finite series expansion, which, on the other hand, can be
cast into a system of coupled equations. In this paper we follow the first choice.
In Section III, we derive a general expression to evaluate any function of matrix variables in
terms of the eigenvalues and first powers of W . In Section IV we apply the method and relations
derived here to study the multichannel evolution of a 2D electron gas through a semiconducting
GaAs/AlGaAs/GaAs heterostructure where the AlGaAs layer is subject to an external electric
field E = (0, Ey , 0), acting transversely to the direction of motion. This is an interesting example
of interacting modes where the channel coupling (modes interaction strength) is tuned by the
electric force F . We will perform accurate calculations of the transmission coefficients Tij and
the conductance g = Trtt†, in units of e2/πh.
II. SCATTERING THEORY IN LOCALLY PERIODIC SYSTEMS
In this Section we shall recall some well-known results and procedures to formally ob-
tain transfer matrices and scattering amplitudes for systems governed by a 3-D Schrodinger
equation. The purpose is to describe transport properties through simple heterostuctures or
finite periodic systems laterally bounded by infinite hard walls (see figure 1). For simplic-
ity, the potential V (x, y, z) will be considered as a stepwise function of z, with discontinu-
ities at z = zmr , (where r = 0, ..., I, zm0= zm, and zmI
= zm+1), and infinite outside
the strip {0 ≤ x ≤ wx, 0 ≤ y ≤ wy}. The coordinates zm (with m = 0, 1, ..., n) denote
the end points of the cells. We shall take V (x, y, z < z0) = 0 and V (x, y, z > zn) = 0.
Let us now consider a basis of eigenfunctions {ϕmi
j (x, y)} satisfying the boundary conditions
ϕmi
j (0, y) = 0, ϕmi
j (wx, y) = 0, ϕmi
j (x, 0) = 0, ϕmi
j (x,wy) = 0, to expand the 3-D wave function
as
ψmi(x, y, z) =∑
j=1
φmi
j (z)ϕmi
j (x, y), (1)
There are two natural choices for the eigenfunctions {ϕmi
j (x, y)}. One in terms of trigonometric
functions satisfying the transverse boundary conditions. The other is to select the exact solutions
in each region, when available. Again, to make the discussion easier, we will consider from
now on only 2-dimensional systems, ignoring the x-direction, which is equivalent to considering
wx � wy. In the first choice, i.e. considering ϕmi
j (y) =√
2wy
sin(πjywy
), we obtain the coupled
equations
h2
2m∗
d2
dz2φmi
j (z)−∑
k
V mi
j,k φmi
k (z) + (E −ETj)φmi
j (z) = 0, (2)
3
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FIG. 1: A periodic potential, laterally bounded by infinite hard walls, for fixed x.
where ETj = h2π2j2/2m∗w2y, and V mi
j,k are the coupling matrix elements
V mi
j,k =2
wy
wy∫
0
dy V mi(y)ϕ∗mi
j (y)ϕmi
k (y), i, j, k = 1, 2, ... (3)
Here V mi(y) is the transversal potential for mi ≤ z ≤ mi+1. This set of coupled equations
is infinite, therefore impossible to solve in general. Thus, it is natural to cut at a finite fixed
number N , which we call the channels number.
For the second choice, while the potential as a function of z is constant, the 3-D Schrodinger
equation decouples and ϕmi
j (y) satisfies the eigenvalue problem
(
− h2
2m∗(d2
dy2+ V mi(y)
)
ϕmi
j (y) = εmi
j ϕmi
j (y), for zmi≤ z ≤ zmi+1
, (4)
with εmi
j the exact transversal eigenvalues. In this case, we have
h2
2m∗
d2
dz2φmi
j (z) +(
E − εmi
j
)
φmi
j (z) = 0, (5)
which solutions are trigonometric functions, however, although the solutions are decoupled, the
modes coupling remains due to the matching conditions with different transverse potential. In
fact, at the interfaces the continuity condition requirements can be cast into a system of coupled
equations which depends on the coupling matrix elements Vj,k.
After this brief digression on the two possible choices, we come back into Eq. (2). To solve
this equation we use a well-known method of the theory of differential equations. We set fj = φj
and fj+N = φ′j . Hence, the set of coupled equations can be written as
f ′(z) = Urf(z), z ∈ [zmr , zmr+1], (6)
with
Ur =
0 IN
2mh2 (Vr −EIN ) +K2
T 0
, (7)
4
a 2N × 2N matrix. Here KT = diag(kT1, kT2, ..., kTN ) and (Vr)j,i = Vj,i(z) for z ∈ [zmr , zmr+1].
Since Vr is symmetric and real, Ur corresponds to an infinitesimal symplectic transformation,
i.e. it belongs to the non-compact Lie algebra sp(2N,R) and satisfies the relation
UTr Σy + ΣyUr = 0. (8)
Where Σy = σy ⊗ IN . Equation (5) has the solution
f(z) = W (r) (z − zmr) f(zmr)., with zmr < z ≤ zmr+1, (9)
and
W (r)(z − zmr) = exp{(z − zmr )Ur}. (10)
This expression suggests, in principle, an infinite power series to determine the matrix W . We
shall see, however, that the number of terms that has to be taken into account for this kind of
functions, using the Sylvester theorem, is finite and depends on the matrix dimension.
Using the flux conservation requirement, it is possible to show that the real matrices W (r)
belong to the non-compact real symplectic Lie group Sp(2N,R) satisfying
(W (r))T ΣyW(r) = Σy. (11)
If we define the symmetric matrix
u2r =
2m
h2 (Vr −EIN ) +K2T (12)
such that
Ur =
0 IN
u2r 0
, (13)
and expand W (r) in power series, we obtain as an alternative representation the following:
W (r)(z) =
cosh zur u−1r sinh zur
ur sinh zur cosh zur
. (14)
In the next Section, we will show that the matrix functions cosh(zur) and u±1r sinh(zur) can
be written as polynomials of degree N − 1 in the matrix variable ur, and we will apply this
representation in Section IV to study a specific example.
It is worth noticing here that at each interface zmi, we must impose the continuity of the
wave function
∑
j=1
φmi−j (zmi
)ϕmi−j (y),=
∑
j=1
φmi+j (zmi
)ϕmi+j (y). (15)
5
Here φmi±j (z) and ϕmi±
j (y) are the eigenfunctions at the left (−) and right hand side (+) of
the interface. Similar equations must be imposed for their derivatives. Evidently, if the wave
functions ϕmi±j (y) are distinct at each side of the interface these conditions can not be satisfied
for finite number of channels. Thus, as mentioned before, the matching conditions must be seen
as finite series expansions in terms of a particular basis of eigenfunctions. Since we are not
primarily interested on the wave functions of a particular region, a natural choice is to use a
single basis of transverse eigenfunctions for all regions. In the semi-infinite leads the transverse
eigenfunctions are the trigonometric functions, hence this basis seems to be the most adequate
one. In the example we discuss further on this point. From here on we will take the trigonometric
basis.
Using the matching conditions, the well known multiplicative property of the transfer matrices
follows. We can obtain the transfer matrix from, say, zm to zm+1, i.e.
W (zm+1, zm) = W (I) (zmI− zmI−1
)
...W (2)(zm2− zm1
)W (1) (zm1− zm) . (16)
It is worth noticing that using different bases in different regions, this fundamental multiplicative
property can not be satisfied for finite channels number. As a consequence the treatment that
follows would not be possible.
In a similar way, i.e. using the multiplicative property, the global transfer matrix for the
whole n-cell finite periodic system, extending from z0 to zn, is given by
W (zn, z0) = W (zn, zn−1) ...W (z2, z1)W (z1, z0) . (17)
For reasons of simplicity, we shall denote a single cell transfer matrix W (z + lc, z) just as W ,
and we will represent it in general as
W ≡W (z + lc, z) =
ϑ µ
ν χ
, (18)
where ϑ, µ, ν and χ are N ×N real matrices. Similarly, the whole n-cell transfer matrix will be
written as
W (zn, z0) = Wn = W n =
ϑ µ
ν χ
n
=
ϑn µn
νn χn
. (19)
For some purposes, in particular for the calculation of transmission amplitudes, it is conve-
nient to deal with the transfer matrix M (relating state vectors) and the scattering matrix S.
These matrices can be obtained from W by a similarity transformation (see Appendix A).
Based on the transfer matrices definitions, the functional dependance between W and M is
given by
M =
κ−1/2 κ−1/2
iκ1/2 −iκ1/2
−1
W
κ−1/2 κ−1/2
iκ1/2 −iκ1/2
. (20)
6
with κ = diag(k1, k2, ..., kN ), being the scattering amplitudes given by
t = 2κ1/2(ϑT + κχTκ−1 − i(κµT − νTκ−1))−1κ−1/2 = (t′)T , (21)
r = −1
2t′κ1/2(ϑ− κ−1χκ+ i(µκ+ κ−1ν))κ−1/2, (22)
and
r′ = −1
2κ1/2(ϑ− κ−1χκ− i(µκ+ κ−1ν))κ−1/2t′. (23)
Analogous relations hold between the n-cell scattering amplitudes (rn, tn, r′n, t
′n) and the
transfer matrix blocks (ϑn, µn, νn, χn), as well as between the n-cell transfer-matrix Mn and the
scattering amplitudes4:
Mn =
αn βn
γn δn
=
(
t†n
)−1r′n (t′n)−1
− (t′n)−1 rn (t′n)−1
. (24)
In the next Section, we shall introduce a powerful evaluation method that can be used in the
transfer matrix approach, we will re-derive the solutions of Wn−W n = 0, and we will show the
equivalence with the Sylvester Theorem.
III. FUNCTIONS OF MATRIX VARIABLE AND THE SYLVESTER THEOREM
Since the main interest from the mathematical point of view is to study polynomial functions
of the transfer matrix, the method we develop here has some points in common with the well
known and useful results in the theory of matrices. Some results, however, will differ. For
simplicity and mathematical clearness, we shall assume non-degenerate eigenvalues and non-
singular matrices. In principle our results can be extended to consider those cases. Let us start
with the Cayley-Hamilton theorem. This theorem states that the characteristic polynomial of a
2N × 2N (transfer) matrix W
g0W2N − g1W 2N−1 + ...+ (−1)2N−1g2N−1W + (−1)2N g2N = 0, (25)
is nullified by W . Here, the scalar coefficients gm are the elementary homogeneous symmetric
functions of the roots {λi}, i.e. g0 = 1, g1 =∑
λi, g2 =∑
i<j λiλj, etc.. It is clear that for
matrices with N > 4, the roots λi can be calculated only numerically. We shall assume that W
is non-singular, although the general case can also be considered in a similar way. Since
Wn = WWn−1, (26)
the initial conditions are W0 = I and W1 = W . In the Appendix B, we show that any power of
the transfer matrix W can be written as
W n =2N∑
i=1
1
π′(λi)
2N−1∑
k=0
Wk
2N−1−k∑
m=0
(−1)mgmλ2N−1−k−mi λn
i , (27)
7
where
π(x) = g−12N
2N∑
l=0
(−1)lglxl =
2N∏
i=1
(x− xi). (28)
Therefore, for any analytic function of matrix variable we have the spectral decomposition23
f(W ) =2N∑
i=1
1
π′(λi)
2N−1∑
k=0
Wk
2N−1−k∑
m=0
(−1)mgmλ2N−1−k−mi f(λi). (29)
which can also be written in the more compact form
f(W ) =2N∑
i=1
ρi(W )f(λi), (30)
where
ρi(W ) =1
π′(λi)
2N−1∑
k=0
Wk
2N−1−k∑
m=0
(−1)mgmλ2N−1−k−mi . (31)
Equation (30) is precisely the statement of Sylvester Theorem where the ρi(W ) polynomials,
also known as “Frobenius covariants”, are defined by
ρi(W ) =π(W )
(W − λiI2N )π′(λi). (32)
These polynomials are idempotent and orthogonal to each other, i.e.
ρi(W )ρj(W ) = ρi(W )δi,j . (33)
Moreover,
∑
i
ρi(W ) = I2N .
We have derived here the spectral decomposition of functions with matrix variable. Eqs. (27)
and (29) are of relevant interest, not only when dealing with superlattices, but also to determine
the single cell transfer matrices, as will be seen in the last Section. For a general treatment on
non-commutative polynomials see refs. [24,25]
IV. TRANSFER MATRICES AND CHANNEL MIXING EXAMPLES
To illustrate the use of the main results presented here, let us consider two simple examples.
We shall first re-derive one of the well-known quantities: the single cell transfer matrix W for
a square barrier potential. We will then consider an interesting multichannel problem: the
transport process of a charged particle with fixed energy E moving by any of the allowed
propagating modes (open channels) through a semiconductor structure subject, locally, to a
perpendicular electric field. The purpose is not only to study the channels mixing induced by
8
the electric field, but also to present a simple example of the first choice mentioned in Section
II, which means, to use, instead of the exact transverse solutions in each layer (in this case,
trigonometric and Airy functions), a single basis for all regions, leading consequently to a system
of coupled equations. We discuss also the underlying convergency problem of transmission
coefficients and exact solutions as functions of the coupled equations number N .
A. The one-channel square barrier transfer matrices
It is known and easy to see that for the square barrier problem, the eigenvalues of the
corresponding matrix u are q = ±√
2mh2 (V −E). Therefore
U =
0 1
q2 0
,
and (14)
W (z) =
cosh qz 1q sinh qz
q sinh qz cosh qz
. (34)
Using the relation in equation (20) we easily find
M =
cosh qz − i q2−k2
2qk sinh qz i q2+k2
2qk sinh qz
−i q2+k2
2qk sinh qz cosh qz + i q2−k2
2qk sinh qz
,
which is a well-known result.
B. Transfer matrices and channel mixing examples
We shall consider now a 2-D multichannel electron gas (2DMEG) with electrons moving
by any of N propagating modes through an electrified 2-D potential V (y, z), which is equal
to −Fy + Vo for zL ≤ z ≤ zR, zero for z ≤ zL and z ≥ zR, and infinite outside the strip
{0 ≤ y ≤ wy} (see Fig. 2). In the intermediate region, we introduce the characteristic length l
through l3 = h2/2m∗F and the coordinate ξ = y/l + λ with λ = 2m∗El2/h2. It is well-known
that with this change of variable, the transversal Schrodinger equation is transformed into the
Airy’s equation
φ′′ + ξφ = 0, (35)
which solutions are the Airy’s functions Ai(−ξ) and Bi(−ξ). Therefore, a solution satisfying the
boundary conditions is of the form
φi(y) = Ai
(
Bi(−λi)Ai(−y/l − λi)−Ai(−λi)Bi(−y/l − λi)
)
, (36)
9
z
V (y,z)
y
Fwy
zR
Vo
zL
FIG. 2: The 2-D potential V (y, z) for a three layer heterostructure, where the intermediate layer is subject
to an external electric field. The charged particles, moving by N propagating modes feel the potential
−Fy+ Vo for zL ≤ z ≤ zR, zero for z ≤ zL and z ≥ zR, and infinite outside the strip {0 ≤ y ≤ wy}. For
F ≥ Vo/wy the attractive potential regions lead to bounded energy states in the continuum and to very
appealing resonant behavior in the transmission coefficients.
where Ai is a normalization constant and λi the eigenvalues obtained from
Bi(−λi)Ai(−wy/l − λi)−Ai(−λi)Bi(−wy/l − λi) = 0, i = 1, 2, . . . (37)
The boundary condition at zL (and similarly at zR) is
N−∑
j=1
φ−j (zL)
√
2
wy(sin
πyj
wy),=
N+∑
i=1
φ+i (zL)Ai
(
Bi(−λi)Ai(−y/l−λi)−Ai(−λi)Bi(−y/l−λi)
)
. (38)
For finite N±, this condition cannot be satisfied for all 0 ≤ y ≤ wy. We are then faced with a
finite Fourier series expansion of Airy’s eigenfunctions. It is easy to see that we can cast this
condition into the system of equations
h2
2m∗
d2
dz2φ−j (zL)−
∑
k
Vk,jφ−k (zL) +
(
E −E−T l
)
φ−j (zL) = 0, (39)
which is precisely what we would have obtained from the very beginning if we had followed the
first choice of just using the same basis at each side of the interface. For the problem considered
here, the coupling matrix elements are
Vj,i(z) = Vo −Fwy
2i = j = 1, 2, ..., N (40)
Vj,i(z) = 0 i 6= j = 1, 2, ..., N i+ j even (41)
Vj,i(z) =8ijFwy
(i2 − j2)2π2i 6= j = 1, 2, ..., N i+ j odd (42)
With N chosen according with the number of open channels. In this example the channels
10
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FIG. 3: Exact transversal eigenfunctions (in this case Airy functions) φi(y) for i = 1, 2 and 4 (solid
line) together with their approximated functions φNi of order N = 4 (dotted line). In this case the
semiconductor heterostructure GaAs/AlGaAs/GaAs width, wy, is 10nm, and the electric field E ≡ F/eis such that wy/l = 5.
mixing depends significantly on the electric field. Before determining the transfer matrix and
calculating transmission coefficients using Eqs. (14) and (29), it might be helpful to clarify, in
terms of a specific example, the underlying approximations mentioned in Section II. If in our
problem we consider geometrical parameters in the nm scale, say wy ∼ 10nm, energies E in the
range of 1eV , and electric forces of the order of 0.05eV/nm, the number of open channels will
be around 4 or 5 (depending of course on the effective masses) and the characteristic length l
will be of the order of 2. To estimate the approximation error involved, we plot in fig 3 some
exact transversal eigenfunctions φi(y) with small index i (i = 1, 2 and 4), together with their
approximated function φNi of order N = 4 and, in figures 4 and 5, some transmission coefficients
for different approximation orders. The associated least mean square errors
∫ wy
0|φi(y)− φN
i |2dy, (43)
in each of the three cases shown in figure 3 (i = 1, 2 and 4), are 0.00008, 0.00041 and 0.0756
respectively. This shows that we have good approximations as far as N equals the number of
open channels. In figures 4 and 5 we plot the low index transmission coefficients Tij for different
values of the number of open channels N . These plots show also that the convergency is rather
good even in those regions where the transmission coefficient oscillates rapidly as a function of
the energy. We obtain a better approximation when N ≥ i, j. It is worth noticing that the
energy separation between successive open-channels thresholds, Ethj+1and Ethj
, increases with
j.
We come back now to the transfer matrix and transmission coefficients issue following the
first choice, i.e. using the same basis in both sides of the interfaces. The transfer matrix from
11
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FIG. 4: The Transmission coefficients T11 and T12 as functions of N for the 2DMEG moving through a
GaAs/AlGaAs/GaAs heterostructure with F = 0.02eV/nm in the intermediate layer.
zL to zR, according to Eq. (14), is written as
W (zR, zL) =
cosh zbu u−1 sinh zbu
u sinh zbu cosh zbu
=
ϑ µ
ν χ
, (44)
where
zb = zR − zL and u2 =2m
h2 (V −EIN ) +K2T , (45)
and KT = diag(kT1, kT2, ..., kTN ) as defined before.
Applying the matrix representations (30) and (32) to transfer matrix blocks, we obtain ex-
pressions like
ϑ = cos zb
√u2 =
N∑
i=1
π(u2) cosh zb
√λi
π′(λi)(u2 − λiIN )(46)
with λi the eigenvalues of u2 and
π(u2) =N∏
j=1
(u2 − λjIN ), (47)
π′(λi) =N∏
j 6=i
(λi − λj). (48)
12
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FIG. 5: The Transmission coefficient T22 as a function of N for the 2DMEG moving through a
GaAs/AlGaAs/GaAs heterostructure with F = 0.025eV/nm in the intermediate layer.
We can then evaluate easily the transfer matrices and physical quantities such as the Landauer
conductance g = Trtt † (in units of e2/πh) and the transmission coefficients Tij. The strength of
the couplings defines the amount of flux conveyed from one channel to another. Together with
the passage of flux between open channels, we will see resonant couplings between an open and
a closed channel. In Figs. 6−10, we plot conductance and transmission coefficients as functions
of the energy E, for different values of F , the width wy and the layer thickness zb. We choose
N = 4 and Vo = 0.23eV .
In the absence of channel coupling, i.e. for F = 0.0eV/nm, we have for each independent
transmission coefficients Tii the well-known one-mode (one-channel) behavior for a 1-D potential
barrier (see figure 6). The Landauer conductance g in figure 7, plotted for F = 0.0eV/nm (heavy
line) and F = 0.025eV (dotted line), exhibits the quantization property. The conductance steps
occur precisely at the channel’s energy thresholds.
Varying the transverse electric force F , we observe another important physical property: the
channel mixing effect. The transmission coefficients and the Landauer conductance are plotted
in figures 8−10, for F = 0.05eV/nm. In the presence of a transverse electric field the upper part
of the potential barrier behaves accordingly with Vo − Fy. For F = 0.025eV/nm, wy = 10nm
and Vo = 0.23eV , the potential takes the form of a kind of wedged potential, which becomes
13
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FIG. 6: The Transmission coefficients T11 and T22 for the 2DMEG moving through a
GaAs/AlGaAs/GaAs heterostructure when the external transverse electric force on the intermediate
AlGaAs layer is F = 0.0eV/nm.
attractive. The potential height varies from 0.23eV at y = 0 to −0.02eV at the opposite side.
To plot the transmission coefficients in Fig. 8, and the Landauer conductance (in figure 7),
we neglect the contributions below the corresponding threshold energies Ethj. While the con-
ductance for F = 0.025eV/nm behaves very much as that of F = 0.0/nm, the transmission
coefficients exhibit interesting channel mixing effects12,13. The channels coupling, induced in
this example by the electric force, leads to very appealing channel threshold effect characterized
by discontinuities of the transmission coefficients at the energy thresholds (indicated in the figure
with arrows), where the “direct” transmission coefficients Tii are strongly suppressed, while the
“crossed” transmission coefficients Tij grow rapidly, starting with an infinite slope at the thresh-
old energy. This, so-called threshold effect, occurs as soon as the energy reaches the threshold
of the new open channel. It is characterized by a sudden passage of flux from one channel to
another. This property, closely associated with the phases of the transmission amplitudes and a
transition to a chaotic regime, results subsequently in the well known fluctuations of the physical
quantities, such as the giant conductance-resistance effects26–28.
Increasing the transverse electric field, the attractive potential regions also increase, and
consequently we have a set of bounded states, especially the bound states in the continuum. We
14
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FIG. 7: The Landauer conductance g = Trtt † in units of e2/πh for the 2DMEG moving through a
GaAs/AlGaAs/GaAs heterostructure where the AlGaAs layer is subject to an external transverse electric
force, taken here as F = 0.0eV/nm (solid line) and F = 0.025eV/nm (dotted line).
call here bound states in the continuum, those states which correspond to a transition between
a propagating mode and an evanescent mode (in the asymptotic regions), such that the energy
of the evanescent mode in the well is nearly equal to the energy of a bound state.
To plot the transmission coefficients and conductance in figures 9 and 10, we consider a larger
electric force F = 0.05eV/nm, with the remaining parameters as in figures 7 and 8. Besides
the already mentioned threshold effect, the interference phenomena leads also, in the presence
of bounded states and channel coupling interactions, to narrow resonances. In figures 9 and 10
it is evident that the resonant coupling occurs between open channels (see T11 and T22 in figure
9 above the second threshold), and also between open and closed channels. Notice that some
of these resonances involve more that two channels. The resonant couplings between, say, the
open channel 1 and the closed channel 2, can be seen below the second threshold (second arrow)
in the transmission coefficients T11 and T12 shown in figure 9. To plot and to appreciate the
conductance in figure 10, all resonances below the threshold energies have been cut off.
15
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FIG. 8: The Transmission coefficients Tii and Tij for the 2DMEG moving through aGaAs/AlGaAs/GaAs
heterostructure when the external transverse electric force on the intermediate AlGaAs layer is F =
0.025eV/nm. In this four channel case wy = 10nm and zb = 20nm. The double arrows show the starting
point of strong suppression of the transmission coefficients Tii produced by the threshold effect mentioned
in the text.
V. CONCLUSIONS
In this paper we reviewed the multichannel transfer matrix approach to study transport
properties of locally 3D periodic systems, we discussed new representations for the evaluation
of arbitrary powers of the transfer matrix W and of analytical functions of matrix variable, and
we showed their relation with the Sylvester and Frobenius Theorems.
Using the matrix generalization of the generating function for Chebyshev polynomials, we
found a simplified representation of the transfer matrix powers W k. In this representation,
arbitrary powers of the transfer matrix can be expressed in terms of its first powers.
To illustrate our results we studied the transmission of a 2D multichannel electron gas in the
presence of a transverse electric field. Interesting channels interference phenomena are described.
16
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FIG. 9: The Transmission coefficients Tii and Ti,i+1 for the same parameters as in figure 5, but F =
0.05eV/nm. In this case we see clear signatures of resonant coupling mediated by the quasi-bounded
states. The arrows show the four threshold energies.
We evaluated the conductance and the transmission coefficients for different values of the electric
field. For F = 0.0eV/nm we obtained the well known conductance quantization. Turning on
the electric field, clear threshold and resonant coupling effects are found. For electric forces
leading to attractive potential regions, the conductance quantization is distorted and bounded
states in the continuum show up. We consider that a quantitative analysis of the conductance
quantization distortion, as a function of the channel coupling strength, is an important open
problem.
The analysis and results presented in this paper provide further foundation to the multichan-
nel transfer matrix approach for finite periodic systems with interacting propagation modes.
They offer also the possibility of much easier evaluation of multichannel quantities and allow
a plenty of coherent and interfering phenomena description and the explanation of the striking
giant conductance-resistance effects.
17
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FIG. 10: Distorted conductance quantization for a system with the same parameters as in figure 6. The
resonant structure between the third and fourth threshold energy (third and fourth arrows) resembles
features of giant conductance-resistance effects. The conductance here is plotted in units of e2/πh
VI. ACKNOWLEDGMENTS
We acknowledge partial support of CONACyT Mexico (Project 29026-E). This work was done
within the framework of the Associateship Scheme of the Abdus Salam International Centre for
Theoretical Physics, Trieste, Italy.
18
APPENDIX A
To establish the relation between W and the transfer matrix M , we write the wave functions
φj(z) in the propagating mode representation. It is common to use either of the following two
notations
φj(z) = ajφ+j (z) + bjϕ
−j (z) = aj
−→ϕ j(z) + bj←−ϕ j(z). (A1)
In terms of these functions, the transfer matrix M is defined by
c−→φ (z)
d←−φ (z)
= M(z, zm)
a−→φ (zm)
b←−φ (zm)
, (A2)
with a, b, c, and d, diagonal N × N matrix coefficients and−→φ and
←−φ , N -dimensional vectors
whose elements are the right and left propagating functions
−→φ j(z) =
1√
kjexp(ikjz), (A3)
←−φ j(z) =
1√
kjexp(−ikjz),
respectively. It is common to write the single-cell transfer matrix M(z + lc, z) in the form of
M = M(z + lc, z) =
α β
γ δ
(A4)
On the other hand, the scattering matrix is defined by
Φout = SΦin (A5)
where
Φin =
−→ϕ (zL)
←−ϕ (zR)
Φout =
←−ϕ (zL)
−→ϕ (zR)
. (A6)
are the incoming and outgoing state vectors at the left and right hand sides of the scatterer
system. In the scattering approach to electronic transport, the S matrix for quasi-1D systems,
is written as
S =
r t′
t r′
, (A7)
where the reflection and transmission amplitudes r, t and r ′, t′, correspond to incidence from the
left and right hand side, respectively.
19
APPENDIX B
It is clear that the characteristic polynomial can also be written as the recurrence formula
g0W2N+l − g1W2N−1+l + ...+ (−1)2N−1g2N−1Wl+1 + (−1)2N g2NWl = 0, (B1)
and we can use this relation recursively to compute all Wk. The advantage of this relation
against the simpler three terms recurrence formula deduced in previous papers, is the scalar
characteristic of all coefficients gk. To solve (B1), we propose the generating function
G(x) =∑
l≥0
Wlxl, (B2)
which, after replacing in the recurrence formula and rearranging terms, becomes
G(x) =R(x)
g2Nπ(x). (B3)
Here, we have defined the matrix polynomials
R(x) =2N−1∑
m=0
(−1)mgm
2N−1−m∑
l=0
xl+mWl =2N−1∑
m=0
xmm∑
l=0
(−1)lglWm−l, (B4)
and
π(x) = g−12N
2N∑
l=0
(−1)lglxl =
2N∏
i=1
(x− xi), (B5)
with roots xi equal to the inverses of the roots λi of the characteristic polynomial (25). Applying
the slight generalization of the Lagrange formula
R(x) =2N∑
i=1
R(xi)π(x)
π′(xi)(x− xi), (B6)
where π′(x) denotes the derivative of π(x), expanding R(x)/π(x) around x = 0, and assuming,
without loss of generality, that xi 6= 0 for all i, we find
Wn = −2N∑
i=1
R(xi)
g2Nπ′(xi)xn+1i
, (B7)
or equivalently
W n = Wn =2N−1∑
k=0
Wk
2N−1−k∑
m=0
(−1)mgmqn−k−m. (B8)
This result is particularly interesting and coincides with the previously deduced expressions
for the non-commutative polynomials pN,m of Ref. [12]. The coefficients qk are the complete
homogeneous symmetric functions of the roots {λi} i.e. q0 = 1, q1 =∑
λi, q2 =∑
i≤j λiλj,
etc. We will show that this result is equivalent to the corresponding assertion of the Sylvester
20
Theorem for powers of a matrix and for analytic functions of matrix variable. For this purpose,
the symmetric functions qn, in (B8) will be rewritten as
qn =2N∑
i=1
λ2N+n−1i
π′(λi), (B9)
where now π(λ) (in abuse of notation) is the monic characteristic polynomial with roots λi.
Using this representation (B8), takes the form
W n =2N∑
i=1
1
π′(λi)
2N−1∑
k=0
Wk
2N−1−k∑
m=0
(−1)mgmλ2N−1−k−mi λn
i . (B10)
Therefore, for any analytic function of matrix variable we have
f(W ) =2N∑
i=1
1
π′(λi)
2N−1∑
k=0
Wk
2N−1−k∑
m=0
(−1)mgmλ2N−1−k−mi f(λi). (B11)
21
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