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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

F́ısica Teórica y Materia Condensada, UAM-Azcapotzalco,

Av. S. Pablo 180, C.P. 02200, México D.F., México

and

P. Pereyra†

F́ısica Teórica y Materia Condensada, UAM-Azcapotzalco,

Av. S. Pablo 180, C.P. 02200, México D.F., México

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/πh̄.

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 Schrödinger

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 {ϕmij (x, y)} satisfying the boundary conditions ϕmij (0, y) = 0, ϕ

mi j (wx, y) = 0, ϕ

mi j (x, 0) = 0, ϕ

mi j (x,wy) = 0, to expand the 3-D wave function

as

ψmi(x, y, z) = ∑

j=1

φmij (z)ϕ mi j (x, y), (1)

There are two natural choices for the eigenfunctions {ϕmij (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 ϕmij (y) = √

2 wy

sin(πjywy ), we obtain the coupled

equations

h̄2

2m∗ d2

dz2 φmij (z)−

∑

k

V mij,k φ mi k (z) + (E −ETj)φ

mi j (z) = 0, (2)

3

�

��

��

� ������

�

FIG. 1: A periodic potential, laterally bounded by infinite hard walls, for fixed x.

where ETj = h̄ 2π2j2/2m∗w2y, and V

mi j,k are the coupling matrix elements

V mij,k = 2

wy

wy ∫

0

dy V mi(y)ϕ∗mij (y)ϕ mi 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 Schrödinger

equation decouples and ϕmij (y) satisfies the eigenvalue problem

(

− h̄ 2

2m∗ ( d2

dy2 + V mi(y)

)

ϕmij (y) = � mi j ϕ

mi j (y), for zmi ≤ z ≤ zmi+1 , (4)

with �mij the exact transversal eigenvalues. In this case, we have

h̄2

2m∗ d2

dz2 φmij (z) +

(

E − �mij )

φmij (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 tran