THE TRANSFER MATRIX METHOD AND THE SYLVESTER THEOREM. INTERACTING MODES AND THRESHOLD EFFECTS IN 2DEG

Transcription

1 Available at: 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 Física Teórica y Materia Condensada, UAM-Azcapotzalco, Av. S. Pablo 180, C.P , México D.F., México and P. Pereyra Física Teórica y Materia Condensada, UAM-Azcapotzalco, Av. S. Pablo 180, C.P , 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.

2 I. INTRODUCTION The transfer matrix method is being used with success to study different types of problems 1,2, in particular problems regarding transport and optoelectronic properties of quasi-1d disordered and 1-D periodic systems 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 proposed 20. Recently, analytic methods were developed and applied to study multichannel finite periodic systems 12,15,16. 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 polynomials 12,15 p N,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 variables 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

3 (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, E y, 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 T ij and the conductance g = T rtt, in units of e 2 /π h. II. SCATTERING THEORY IN LOCALLY PERIODIC SYSTEMS In this Section we shall recall some well-known results and procedures to formally obtain 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 simplicity, the potential V (x, y, z) will be considered as a stepwise function of z, with discontinuities at z = z mr, (where r = 0,..., I, z m0 = z m, and z mi = z m+1 ), and infinite outside the strip {0 x w x, 0 y w y }. The coordinates z m (with m = 0, 1,..., n) denote the end points of the cells. We shall take V (x, y, z < z 0 ) = 0 and V (x, y, z > z n ) = 0. Let us now consider a basis of eigenfunctions {ϕ m i j (x, y)} satisfying the boundary conditions ϕ m i j as (0, y) = 0, ϕ m i j (w x, y) = 0, ϕ m i j (x, 0) = 0, ϕ m i j (x, w y ) = 0, to expand the 3-D wave function ψ m i (x, y, z) = φ m i j (z)ϕ m i j (x, y), (1) j=1 There are two natural choices for the eigenfunctions {ϕ m i 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 w x w y. In the first choice, i.e. considering ϕ m i j (y) = 2 w y sin( πjy w y ), we obtain the coupled equations h 2 2m d 2 dz 2 φm i j (z) k V m i j,k φm i k (z) + (E E T j) φ m i j (z) = 0, (2) 3

4 FIG. 1: A periodic potential, laterally bounded by infinite hard walls, for fixed x. where E T j = h 2 π 2 j 2 /2m w 2 y, and V m i j,k V m i j,k = 2 w y w y 0 are the coupling matrix elements dy V m i (y)ϕ m i j Here V m i (y) is the transversal potential for m i z m i+1. (y)ϕ m i k (y), i, j, k = 1, 2,... (3) 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 ϕ m i j (y) satisfies the eigenvalue problem with ɛ m i j ( h2 2m ( d2 dy 2 + V m i (y) ) ϕ m i j the exact transversal eigenvalues. In this case, we have h 2 d 2 ( 2m dz 2 φm i j (z) + (y) = ɛ m i j ϕ m i j (y), for z mi z z mi+1, (4) E ɛ m i j ) φ m i 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 V j,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 f j = φ j and f j+n = φ j. Hence, the set of coupled equations can be written as f (z) = U r f(z), z [z mr, z mr+1 ], (6) with U r = 0 I N 2m h 2 (V r EI N ) + K 2 T 0, (7) 4

5 a 2N 2N matrix. Here K T = diag(k T 1, k T 2,..., k T N ) and (V r ) j,i = V j,i (z) for z [z mr, z mr+1 ]. Since V r is symmetric and real, U r corresponds to an infinitesimal symplectic transformation, i.e. it belongs to the non-compact Lie algebra sp(2n,r) and satisfies the relation Ur T Σ y + Σ y U r = 0. (8) Where Σ y = σ y I N. Equation (5) has the solution f(z) = W (r) (z z mr ) f(z mr )., with z mr < z z mr+1, (9) and W (r) (z z mr ) = exp{(z z mr )U r }. (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 If we define the symmetric matrix such that (W (r) ) T Σ y W (r) = Σ y. (11) u 2 r = 2m h 2 (V r EI N ) + K 2 T (12) U r = 0 u 2 r 0 I N, (13) and expand W (r) in power series, we obtain as an alternative representation the following: W (r) (z) = cosh zu r u 1 r sinh zu r. (14) u r sinh zu r cosh zu r In the next Section, we will show that the matrix functions cosh(zu r ) and u ±1 r sinh(zu r ) can be written as polynomials of degree N 1 in the matrix variable u r, and we will apply this representation in Section IV to study a specific example. It is worth noticing here that at each interface z mi, we must impose the continuity of the wave function j=1 φ m i j (z mi )ϕ m i j (y), = φ m i+ j (z mi )ϕ m i+ j (y). (15) j=1 5

6 Here φ m i± j (z) and ϕ m i± 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 ϕ m i± 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, z m to z m+1, i.e. W (z m+1, z m ) = W (I) ( z mi z mi 1 )...W (2) (z m2 z m1 )W (1) (z m1 z m ). (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 z 0 to z n, is given by W (z n, z 0 ) = W (z n, z n 1 )...W (z 2, z 1 )W (z 1, z 0 ). (17) For reasons of simplicity, we shall denote a single cell transfer matrix W (z + l c, z) just as W, and we will represent it in general as W W (z + l c, z) = ϑ µ, (18) ν χ where ϑ, µ, ν and χ are N N real matrices. Similarly, the whole n-cell transfer matrix will be written as W (z n, z 0 ) = W n = W n = ϑ µ ν χ n = ϑ n ν n µ n χ n. (19) For some purposes, in particular for the calculation of transmission amplitudes, it is convenient 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 iκ 1/2 iκ 1/2 1 W κ 1/2 κ 1/2. (20) iκ 1/2 iκ 1/2 6

7 with κ = diag(k 1, k 2,..., k N ), being the scattering amplitudes given by t = 2κ 1/2 (ϑ T + κχ T κ 1 i(κµ T ν T κ 1 )) 1 κ 1/2 = (t ) T, (21) and r = 1 2 t κ 1/2 (ϑ κ 1 χκ + i(µκ + κ 1 ν))κ 1/2, (22) r = 1 2 κ1/2 (ϑ κ 1 χκ i(µκ + κ 1 ν))κ 1/2 t. (23) Analogous relations hold between the n-cell scattering amplitudes (r n, t n, r n, t n ) and the transfer matrix blocks (ϑ n, µ n, ν n, χ n ), as well as between the n-cell transfer-matrix M n and the scattering amplitudes 4 : M n = α n γ n β n δ n = ( ) 1 t n r n (t n) 1. (24) (t n ) 1 r n (t n ) 1 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 W n 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 nonsingular 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 g 0 W 2N g 1 W 2N ( 1) 2N 1 g 2N 1 W + ( 1) 2N g 2N = 0, (25) is nullified by W. Here, the scalar coefficients g m are the elementary homogeneous symmetric functions of the roots {λ i }, i.e. g 0 = 1, g 1 = λ i, g 2 = i<j λ iλ j, etc.. It is clear that for matrices with N > 4, the roots λ i 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 W n = W W n 1, (26) the initial conditions are W 0 = I and W 1 = W. In the Appendix B, we show that any power of the transfer matrix W can be written as W n = 2N i=1 2N 1 1 π (λ i ) k=0 2N 1 k W k m=0 7 ( 1) m g m λi 2N 1 k m λ n i, (27)

8 where π(x) = g 1 2N 2N l=0 ( 1) l g l x l = 2N i=1 (x x i ). (28) Therefore, for any analytic function of matrix variable we have the spectral decomposition 23 f(w ) = 2N i=1 2N 1 1 π (λ i ) k=0 2N 1 k W k m=0 which can also be written in the more compact form where f(w ) = ρ i (W ) = 1 2N 1 π (λ i ) k=0 2N i=1 2N 1 k W k ( 1) m g m λi 2N 1 k m f(λ i ). (29) ρ i (W )f(λ i ), (30) m=0 ( 1) m g m λi 2N 1 k m. (31) Equation (30) is precisely the statement of Sylvester Theorem where the ρ i (W ) polynomials, also known as Frobenius covariants, are defined by ρ i (W ) = These polynomials are idempotent and orthogonal to each other, i.e. Moreover, π(w ) (W λ i I 2N )π (λ i ). (32) ρ i (W )ρ j (W ) = ρ i (W )δ i,j. (33) ρ i (W ) = I 2N. i 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

9 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 = ± 2m h 2 (V E). Therefore U = 1, q 2 0 and (14) W (z) = cosh qz 1 q sinh qz. (34) q sinh qz cosh qz Using the relation in equation (20) we easily find M = which is a well-known result. q2 k2 cosh qz i 2qk sinh qz i q2 +k 2 2qk sinh qz i q2 +k 2 2qk sinh qz cosh qz + i q2 k 2 2qk sinh qz, 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 F y + V o for z L z z R, zero for z z L and z z R, and infinite outside the strip {0 y w y } (see Fig. 2). In the intermediate region, we introduce the characteristic length l through l 3 = h 2 /2m F and the coordinate ξ = y/l + λ with λ = 2m El 2 / h 2. It is well-known that with this change of variable, the transversal Schrödinger 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 ) φ i (y) = A i (Bi( λ i )Ai( y/l λ i ) Ai( λ i )Bi( y/l λ i ), (36) 9

10 V (y,z) y F z R V o w y z z L 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 F y + V o for z L z z R, zero for z z L and z z R, and infinite outside the strip {0 y w y }. For F V o /w y the attractive potential regions lead to bounded energy states in the continuum and to very appealing resonant behavior in the transmission coefficients. where A i is a normalization constant and λ i the eigenvalues obtained from Bi( λ i )Ai( w y /l λ i ) Ai( λ i )Bi( w y /l λ i ) = 0, i = 1, 2,... (37) The boundary condition at z L (and similarly at z R ) is N 2 φ j (z L) (sin πyj N + ) ), = φ + i w j=1 y w (z L)A i (Bi( λ i )Ai( y/l λ i ) Ai( λ i )Bi( y/l λ i ). (38) y i=1 For finite N ±, this condition cannot be satisfied for all 0 y w y. 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 h 2 2m d 2 dz 2 φ j (z L) k ( ) V k,j φ k (z L) + E E T l φ j (z L) = 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 V j,i (z) = V o F w y 2 i = j = 1, 2,..., N (40) V j,i (z) = 0 i j = 1, 2,..., N i + j even (41) V j,i (z) = 8ijF w y (i 2 j 2 ) 2 π 2 i j = 1, 2,..., N i + j odd (42) With N chosen according with the number of open channels. In this example the channels 10

11 FIG. 3: Exact transversal eigenfunctions (in this case Airy functions) φ i (y) for i = 1, 2 and 4 (solid line) together with their approximated functions φ N i of order N = 4 (dotted line). In this case the semiconductor heterostructure GaAs/AlGaAs/GaAs width, w y, is 10nm, and the electric field E F/e is such that w y /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 w y 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 φ i (y) with small index i (i = 1, 2 and 4), together with their approximated function φ N i 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 φ i (y) φ N i 2 dy, (43) in each of the three cases shown in figure 3 (i = 1, 2 and 4), are , and 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 T ij 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, E thj+1 and E thj, 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

12 ! FIG. 4: The Transmission coefficients T 11 and T 12 as functions of N for the 2DMEG moving through a GaAs/AlGaAs/GaAs heterostructure with F = 0.02eV/nm in the intermediate layer. z L to z R, according to Eq. (14), is written as W (z R, z L ) = cosh z bu u sinh z b u u 1 sinh z b u cosh z b u = ϑ µ ν χ, (44) where z b = z R z L and u 2 = 2m h 2 (V EI N) + K 2 T, (45) and K T = diag(k T 1, k T 2,..., k T N ) as defined before. Applying the matrix representations (30) and (32) to transfer matrix blocks, we obtain expressions like with λ i the eigenvalues of u 2 and ϑ = cos z b u 2 = N i=1 π(u 2 ) cosh z b λi π (λ i )(u 2 λ i I N ) (46) N π(u 2 ) = (u 2 λ j I N ), (47) j=1 N π (λ i ) = (λ i λ j ). (48) j i 12

13 ! FIG. 5: The Transmission coefficient T 22 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 e 2 /π h) and the transmission coefficients T ij. 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 w y and the layer thickness z b. We choose N = 4 and V o = 0.23eV. In the absence of channel coupling, i.e. for F = 0.0eV/nm, we have for each independent transmission coefficients T ii 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 V o F y. For F = 0.025eV/nm, w y = 10nm and V o = 0.23eV, the potential takes the form of a kind of wedged potential, which becomes 13

14 FIG. 6: The Transmission coefficients T 11 and T 22 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 E thj. While the conductance for F = 0.025eV/nm behaves very much as that of F = 0.0/nm, the transmission coefficients exhibit interesting channel mixing effects 12,13. 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 T ii are strongly suppressed, while the crossed transmission coefficients T ij grow rapidly, starting with an infinite slope at the threshold 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 effects 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

15 FIG. 7: The Landauer conductance g = Trtt in units of e 2 /π h 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 T 11 and T 22 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 T 11 and T 12 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

16 FIG. 8: The Transmission coefficients T ii and T ij for the 2DMEG moving through a GaAs/AlGaAs/GaAs heterostructure when the external transverse electric force on the intermediate AlGaAs layer is F = 0.025eV/nm. In this four channel case w y = 10nm and z b = 20nm. The double arrows show the starting point of strong suppression of the transmission coefficients T ii 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

17 FIG. 9: The Transmission coefficients T ii and T i,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 multichannel 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

18 Σ 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 e 2 /π h VI. ACKNOWLEDGMENTS We acknowledge partial support of CONACyT Mexico (Project E). This work was done within the framework of the Associateship Scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. 18

19 APPENDIX A To establish the relation between W and the transfer matrix M, we write the wave functions φ j (z) in the propagating mode representation. It is common to use either of the following two notations φ j (z) = a j φ + j (z) + b jϕ j (z) = a j ϕ j (z) + b j ϕ j (z). In terms of these functions, the transfer matrix M is defined by c φ (z) d = M(z, z m ) a φ (z m ) φ (z) b, φ (z m ) (A1) (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 kj exp(ik j z), (A3) φ j (z) = 1 kj exp( ik j z), respectively. It is common to write the single-cell transfer matrix M(z + lc, z) in the form of M = M(z + lc, z) = On the other hand, the scattering matrix is defined by α β γ δ (A4) where ϕ (zl ) Φ in = ϕ (zr ) Φ out = SΦ in Φ out = ϕ (zl ). ϕ (zr ) (A5) (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 where the reflection and transmission amplitudes r, t and r, t, correspond to incidence from the left and right hand side, respectively. (A7) 19

20 APPENDIX B It is clear that the characteristic polynomial can also be written as the recurrence formula g 0 W 2N+l g 1 W 2N 1+l ( 1) 2N 1 g 2N 1 W l+1 + ( 1) 2N g 2N W l = 0, (B1) and we can use this relation recursively to compute all W k. The advantage of this relation against the simpler three terms recurrence formula deduced in previous papers, is the scalar characteristic of all coefficients g k. To solve (B1), we propose the generating function G(x) = W l x l, (B2) l 0 which, after replacing in the recurrence formula and rearranging terms, becomes G(x) = R(x) g 2N π(x). (B3) Here, we have defined the matrix polynomials R(x) = 2N 1 m=0 2N 1 m ( 1) m g m x l+m W l = l=0 2N 1 m=0 m x m ( 1) l g l W m l, l=0 (B4) and π(x) = g 1 2N 2N l=0 ( 1) l g l x l = 2N i=1 (x x i ), (B5) with roots x i equal to the inverses of the roots λ i of the characteristic polynomial (25). Applying the slight generalization of the Lagrange formula R(x) = 2N i=1 R(x i )π(x) π (x i )(x x i ), (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 x i 0 for all i, we find or equivalently W n = 2N i=1 R(x i ) g 2N π (x i )x n+1, (B7) i W n = W n = 2N 1 k=0 2N 1 k W k m=0 ( 1) m g m q n k m. (B8) This result is particularly interesting and coincides with the previously deduced expressions for the non-commutative polynomials p N,m of Ref. [12]. The coefficients q k are the complete homogeneous symmetric functions of the roots {λ i } i.e. q 0 = 1, q 1 = λ i, q 2 = i j λ iλ j, etc. We will show that this result is equivalent to the corresponding assertion of the Sylvester 20

21 Theorem for powers of a matrix and for analytic functions of matrix variable. For this purpose, the symmetric functions q n, in (B8) will be rewritten as q n = 2N i=1 λ 2N+n 1 iπ (λ i ), (B9) where now π(λ) (in abuse of notation) is the monic characteristic polynomial with roots λ i. Using this representation (B8), takes the form W n = 2N i=1 2N 1 1 π (λ i ) k=0 2N 1 k W k m=0 Therefore, for any analytic function of matrix variable we have f(w ) = 2N i=1 2N 1 1 π (λ i ) k=0 2N 1 k W k m=0 ( 1) m g m λi 2N 1 k m λ n i. (B10) ( 1) m g m λi 2N 1 k m f(λ i ). (B11) 21

22 References 1 S. J. Emelett, W. D. Goodhue, A.S. Karakashian AS, K. Vaccaro J. Appl. Phys. 95, 2930(2004) 2 M. Cahay, S. Bandyopadhyay, Phys. Rev. B 69, (2004) 3 M. Pacheco and F. Claro, Phys. Stat. Sol. (b) 144, 399(1982). 4 P.A. Mello, P. Pereyra and N. Kumar, Ann. Phys. 181, 290(1988). 5 J.L. Pichard, N. Zanon, Y. Imry, et al., Jour. Phys. 51, 587(1990) 6 D.J. Griffiths and N.F. Taussing, Am. J. Phys. 60, 883(1992). 7 D.W. Sprung, H. Wu and J. Martorell, Am. J. Phys. 61, 1118(1993). 8 P.A. Mello and J.P. Pichard, J. Phys. 1, 493(1991), P. Pereyra. J. Math. Phys. 36, 1166(1995). 9 M.G. Rozman, P. Reineker, R. Tehver, Phys. Lett. A 187, 127(1994). 10 P.W. Brouwer, and K. Frahm, 53, 1490(1996). 11 C.W.J. Beenakker, Rev. Mod. Phys. 69, 731(1997). 12 P. Pereyra, J. Phys. A: Math. Gen. 31, 4521(1998); Phys. Rev. Lett 80, 2677(1998). 13 A. Anzaldo-Meneses, Ann. Phys. (Leipzig)7, 307(1998). 14 D.J. Griffihs and C.A. Steinke, Am. J. Phys. 69, 137(2001). 15 J.L. Cardoso, P. Pereyra, A. Anzaldo-Meneses, Phys. Rev. B 63, (2001), 16 P. Pereyra and E. Castillo, Phys Rev. B 65, (2002). 17 K.A. Muttalib, V.A. Gopar, Phys. Rev. B 66, (2002). 18 C. Pacher and E. Gornik, Phys. Rev. B 68, (2003), U. Merc, C. Pacher, M. Topic et al. Eur. Jour. Phys. 35, 443(2003) 19 P. Pereyra, Phys. Rev. Lett. 84, 1772(2000), F. Assaoui, P. Pereyra, J. Appl. Phys. 91, 5163(2002), L. Diago-Cisneros, P. Pereyra, H. Rodriguez-Coppola, R. Perez-Alvarez, Phys. Stat. Sol(b). 1, 125(2002), H. Simajuntak, P. Pereyra, Phys. Rev B 67, (2003), A. Kunold, P. Pereyra, J. Appl. Phys. 93, 5018(2003). 20 S.A. Rakityansky, Phys. Rev. B 70, (2004), and references therein. 21 Sylvester J.J., Phil. Mag. 16, 267 (1883), reprinted in Collected Papers 4 110; Sylvester J.J., C. R. Acad. Sci., Paris 94, 55 (1882), reprinted in Collected Papers in N. Dunford and J. T. Schwartz, Linear Operator Part I: General Theory, (Interscience Publisher Inc. NY 1958). 23 E. Merzbacher, Quantum Mechanics, (John Wiley & Sons, N.Y. 1970). 24 I.M. Gelfand, and V.S. Retakh, A Theory of 25 I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, 26 P. Pereyra, Physica E 17, 209(2003). 27 N. Giordano, E.R. Schuler, Phys. Rev. B 41, 11822(1990). 28 A.Y. Kasumov, I.I. Khodos, N.A. Kislov, et al., Phys. Rev. Letters 75, 4286 (1995). 22