the transfer matrix method and the sylvester

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  • Available at:−off IC/2005/057

    United Nations Educational Scientific and Cultural Organization


    International Atomic Energy Agency




    A. Anzaldo-Meneses

    F́ısica Teórica y Materia Condensada, UAM-Azcapotzalco,

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


    P. Pereyra†

    F́ısica Teórica y Materia Condensada, UAM-Azcapotzalco,

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


    The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.


    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.


    May 2005

    † Senior Associate of ICTP.


    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


  • (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.


    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


    ψmi(x, y, z) = ∑


    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



    2m∗ d2

    dz2 φmij (z)−


    V mij,k φ mi k (z) + (E −ETj)φ

    mi j (z) = 0, (2)


  • ��


    � ������

    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 ∫


    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


    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

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