REVISTA CUBANA DE FÍSICA 						                                               Vol. 21, No. 1, 2004

TRANSFER MATRICES AND GREEN FUNCTIONS 
FOR THE STUDY OF ELEMENTARY EXCITATIONS 
IN MULTILAYERED HETEROSTRUCTURES
R. Pérez Álvarez1 and F. García Molliner
Universidad “Jaume I”, Castellón de la Plana, Spain

ABSTRACT
This article is concern with a mathematical tool, the Associated Transfer Matriz T, which proves useful in the study of a wide class of physical problems involving multilayer heterostructures. General properties of linear, second order differential matrix Sturn Liouville operators are discussed as a basis for establishing general properties of T, which is also generally related to the Green function G. Some identities satisfied by T are derived, which prove useful in practice to monitor the numerical quality of computational processes.

Key words: Transfer Matrix, Green functions, heterostructure.

PACS: 02.90.+p; 62.30.+d; 68.35.Ja; 68.65.Ac; 73.21.Ac; 73.21.-b.

RESUMEN
Este artículo está concebido como una herramienta matemática. El Traslado Asociado Matriz T demuestra la utilidad del estudio de una clase ancha de problemas físicos que involucran heterostructuras del multilayer. Propiedades generales de lineal, segunda matriz de diferencial del orden, se derivan de Sturn que se discuten operadores de Liouville como una base para establecer las propiedades generales 
de T, que también se relacionan, generalmente, con la Función Verde G. Identidades satisfechas por T demuestran la práctica de supervisar la calidad numérica de procesos computacionales.

Palabras clave: Traslado de Matriz, Función Verde, heteroestructura.

I. INTRODUCTION
	Problems concerned with multilayer systems arise very often in different fields of physics. One has 
to integrate differential equations across some sequence of different domains, which raises the question of repeated matching at all interfaces involved. In particular, the development of efficient techniques of epitaxial crystal growth originated a great deal of work of this nature in solid state physics, notably -but not only- in the field of semiconductor quantum heterostructures1,2.

	A great deal of activity has also been concerned with Quasiregular Heterostructures3 which follow some non periodic self-replicative sequence, of which the best known is the Fibonacci sequence4. In these cases the number of interfaces can be very large.

					
E-mail:	1rpa@fisica.uh.cu
Permanent address: Dpto. Física Teórica, Facultad de Física, Universidad de La Habana.

	It is therefore of practical value to have mathematical tools available which can be useful in the study of such systems. Among these, the Surface Green Function Matching1 has been extended to an arbitrarily large number of interfaces5. Green functions were generally related to a transfer matrix6 which transfers amplitudes and normal derivatives 
-of which more will be said presently but it was found later7,8 that it is physically more appealing to introduce a different kind of transfer matrix, here denoted T and termed the Associated Thansfer Matrix which is even simpler to operate with and
is directly related to the physically interesting magnitudes of the problem under study.

	The purpose of this article is to expound on the general relationship between T and the Green function G, to discuss some basic formal properties of T and to derive from this some general identities satisfied by T which can prove useful in practice. We shall deal explicity with the case of planar geometry, although the analysis can also be adapted to other tractable geometries –e.g. spherical or cylindrical.

	Irrespective of the specific physical nature of the states, modes, waves or quasiparticles we may be concerned with, the general setup is the following. We start from 3D linear, second order differential system and Fourier transform in the 2D (x,y) plane of the interfaces, thus introducing a 2D wavevector k. We then have a system of N coupled, k-dependent ordinary differential equations in the variable z, in the direction perpendicular to the interfaces. We stress that in practice N can be sufficiently large that an analytical study is impracticable. For instance, in electronic band structure calculations one can find multiband envelope function calculations of electronic band structure with N = 89 and N = 1410 and pseudo-potential calculations with up to N = 6511. Thus, it is interesting to study in some detail the formal properties of an object, T7, which is a useful tool for numerical computation. 

	The general formulation of the problem is setup in Section 2 and basic questions related to the hermicity of linear, second order matrix differential Sturm-Liouville operator are discussed in Section 3. This provides a basis to study infinite domains and regularity at infinity –Section 4- after which the general relationship between T and the Green function G is established in Section 5. Finally, general properties and identities satisfied by T in Section 6.

II. FORMULATION OF THE PROBLEM

	The N coupled differential involve N couples amplitudes corresponding to the physical model under study -e.g. the components of an electronic wavefunction in an N-band envelope function 
model or coupled elastic vibration amplitudes and electrostatic potential in piezoelectric waves. Loosely speaking, we shall refer to the 'primary field', for reasons presently explained. 

	The N amplitudes of the primary field will 
be condensed in an object, F(z), denoted for convenience as 'N-vector', which need not imply that it transforms literally as a vector.

	We now consider the vast class of physical problems for which the differential system can be cast in a compact matrix form as 

L(z)  F(x) =  W(z)  F(z) = 0,
(1)

where we have defined another N-vector, namely

                 A(z) = B(z)   P(z) · F(z).             (2)

	We denote this linear differential form, derived from F, as the secondary field. For instance, if F describes elastic strains, then A describes stresses. More precisely, normal stress components1. Further-more, we are concerned with eigenvalue problems. The eigenvalue parameter, henceforth denoted , 
is contained in W. The dependence on k and  is understood throughout.
	The z dependence of some, or all, of the (matrix) coefficients of (1) and (2) is an essential feature of heterostructures, even if piecewise homogeneous domains are involved.

	Now, F(z) is obviously continuous everywhere and, furthermore, integration across any of the interfaces of the heterostructure proves that A(z) is also continuous, which bears out the convenience of this compact formulation. Since both, the primary field and the secondary field are continuous, this suggests the convenience of defining, instead of a transfer matrix M1,6 which transfers F(z) and F'(z), a different one, here denoted as the Associated Transfer Matrix (ATM) T, which transfers the two fields that are continuous everywhere. Thus we define the 2N – vector

			      		  (3)

and then, starting from any initial point z0, we define T(z, z0) which transfers  from z0 to any z by

			  (z) = T(z, z0)  ( z0)		  (4)

	Unlike the coefficient transfer matrix, often used for systems with piecewise constant coefficients12,13, 
M(z, z0) and T(z, z0) do not depend on knowing a basis of 2N linearly independent solutions. Both, M(z, z0) and T(z, z0), pertain to initial value problems, while the Green function G(z, z') depends on the choice of boundary conditions. Thus, for a given differential system M and T are unique, while G is not. The relationship between G and M was discussed in1,6,15. Here we discuss the relationship between G and T, for which we first study the general properties of the latter, for which first some observations concerning the differential system are in order.

III. THE HERMITICITY OF THE DIFFERENTIAL 
       OPERATOR

	The differential matrix L(z) of (1) is the representa-tions of a linear differential operator which we now discuss. We recall14 that the full definition of the operator requires the specification of the manifold S of functions –here termed 'vectors' F(z)- on which 
it acts.

	Let  be an operator represented by the operating rule 

			          p = 		  (5)
	So far, is formally Hermitean14, [Chapter 3]. 
Now let represented by the operating rule (1), be an operator defined to act on a given manifold S and let F denote the generic function of this particular manifold. We write the system (1) explicitly as 

L(z)  F(z) =  +

                + Y(z)  .	               (6)

	Thus, the operator represented by this equation is

		   = -B  + iP + iY   + W(z)  	  (7)

where the dots indicate that, as a matrix, this is 
to act on vectors F. Furthermore, in most physical problems  related to heterostructures the matrix coefficients satisfy the conditions

              B = Bt,          P = - Yt, 	       W = Wt,	  (8)

	Thus, for the large class of problems we are concerned with we assume (8) to hold. Then it is easily seen that the Hermitean adjoint of t is is

               =        (9)
which so far only ensures that  is formally Hermitean.

	We now, more specifically, define S as the manifold of functions defined in a given interval [a, b] 
-provisionally assumed to be finite- and satisfying given boundary conditions at its extremes. A physical example could be a quantum well with infinite barriers –vanishing amplitudes- in the external media. Let us then consider another operator , acting on another manifold S2 of F2 defined in the same interval [a, b] but otherwise satisfying so far unspecified boundary conditions.

	We define to be represented by 

L2(z)  F2(z) =  -

             - Pt(z)  .	             (10)

and then at every point z of the interval [a, b], by assumption common to S and S2, we define the residual R(z) by
R(z)  = 
      + .
	                                                                          (11)

	We then study the integral

               	(12)

which, after partial integration, is

              	(13)

	Now, since the matrix coefficients satisfy the conditions (8), the operating rule for is the same as for whence follows that the residual is

                    	(14)

and thus the residual difference appearing in (13) is

                       	(15)

	The mathematical definition of R acquires a physical meaning in terms of the primary and secondary fields, the two objects transferred by T.

	This is the central point of the analysis. The operators and are not yet fully defined, as the boundary conditions have not been specified. If we assume only that S and S2 are such that the combined effect of their boundary conditions is such that 'R(b) - 'R(a) vanishes, then

                                           (16)
and  is the Harmitean adjoint .

	If, furthermore, the boundary conditions are the same in S and S2, then  -the Hermitean adjoint of  is also equal to  and it is then that  is really Hermitean, that is to say, technically, it is totally Hermitean14 Such is the case, for instance, for electronic states with external infinite barriers, where all amplitudes vanish, or for elastic waves with external free surfaces, where all normal stresses vanish. 

	It seems reasonable to put forward this type of analysis to justify the basis of standard calculations for physical models with N > 1. Furthermore, this is the starting point to analyze a question directly relevant to the physical models employed in practice to study heterostructure.

IV. INFINITE DOMAINS: CAUSALITY 
      AND REGULARITY AT INFINITY

	This situation arises if the external media are semiinfinite or, simply, if one studies a bulk medium. So far it has been implicitly assumed that the eigenvalues are real, which is formally correct if the interval [a, b is finite. We must now analyse the situation when this is infinite, for which a brief reminder of some basic facts is at this stage convenient. In order to fix ideas it suffices to consider a simple case with N = 1.

	Firstly, we must define a sign convention for the description of stationary amplitudes F(z, t). We adopt the convention

			    F(z,t) = F0 ei(kz – wt) 	 	(17)

for a wave travelling to the right. Then, when in the time dependent picture t - , in the stationary state time independent amplitude,

			         F(z) = F0 eikz		(18)

the variable z  . In order for this to be a physically acceptable regular solution, it cannot blow up for 
z  , which requires formally (i) that the real wave-vector k, corresponding to the allowed eigenvalue of a stationary propagation eigenstate, be defined as 

			          k = 	             (19)

and (ii) that we define the limit of F(z) for z   as the regular limit.

                              	(20)

	The same analysis holds obviously for waves travelling to the left and z   and the same basic facts hold for less simple cases, e.g. for any N > 1.

	Let us now return to the standard operator  on which the analysis of the class of problems here considered is based. Since this is defined to act on the manifold S of functions which are regular at infinity, the second and fourth terms of the residual difference (15) vanish. However in order for (15) to vanish entirely, the condition on the adjoint of  is that the boundary conditions of regularity at infinity must hold for the not the F2. In fact, in order for (15) to vanish the F2 must be precisely irregular at infinity, as only then the are regular. Therefore the manifold S2 is not the same as S and we reach the non trivial conclusion that , the operator normally used in the description of infinite -or, in the event- seminfinite regular media is not totally Hermitean.

	Although a full analysis in general terms(16, §7.5) of the conditions at infinity is outside the scope of this article, it is in order to recall that while regularity at infinity is the physical requirement in the stationary state -time independent- picture, causality is the condition required in standard physical theory, and this in turn requires w to be defined as the limit of (w + ih) for h  0. Now, for the class of problems here considered w bears a direct relation to the eigenvalue variable . For instance, for electronic states  is the energy and w is ; for most wave problems w is 1/2, etc. Then the causal formulation amounts to requiring that  is to be read everywhere as

			       =	             (21)

	Even for bound states, for which the eigenvalue is real, a small imaginary part may have to be added in practice, for instance, when calculating spectral functions numerically with a computer code. The point to stress is that (21) must be kept as such everywhere and this has a direct bearing on the foregoing analysis.

	It is correct to define the Hermitean conjugate of an operator as has been done, but in the matrices resulting in the z-representation, where  appears explicitly, it must always be kept  + i, whether  stays small but finite or it tends to zero as in (21). In other words, while the standard Hermitean conjugate of a matrix of the general form

                      m() = m1() + im2()		(22)

is

		        mt() = 		(23)

this is not the concept required for an explicit matrix analysis of the problems here considered. The concept of transconjugate matrix

		          mc() = 		(24)

was ised in6 the relationship between the standard transfer matrix M and the Green function G was established. This was further developed in1. If one used mt, instead of mc, then one would obtain a negative density of states. We now relate G to T, defined in (4), while extending sigficantly the analysis.

V. THE RELATIONSHIP BETWEEN G AND T

	We now start from

		         L(z)  L(z, z´) = IN (z – z´),		(25)

where IN is the N  N unit matrix and L(z) is the differential matrix of (1) but, instead of A(z) of (2), the differential form is now
	
     A(z – z´) = B(z)  + P(z)  G(z, z´).	(26)
	
	Defining
	
			A(z) =A(z, z´),	             (27)

a first integration of (25) yields the identity

                              A+(z) – A-(z) = -IN .		(28)

	Furthermore, a detailed formal analysis of (25) shows16 that if we define

   Z(z, z´) =  Bc(z´) + G(z, z´)  Pc(z´).	(29)

and hence

                                         Z(z) =Z(z, z´),		(30)

then there is another identity, namely

			       +Z(z) = Z(z) = - IN .		(31)

	It is now convenient to display T in the form

			      T = 		(32)

where the submatrices Ta,b (a, b = A, D) are N  N. Then, similarly to (3), we define the rectangular 
2N  N matrix

			(z, z´) =  		(33)

and thus

		         (z, z´) = T(z, z0)  (z0, z´), 	(34)

that is to say, for fixed z' (z, z ') is transferred like Y(z) and, in particular

G(z, z´) = TAA(z, z0)  G(z0, z´) + TAD(z, z0)  A(z0, z´).
 (35)

	A similar analysis can be carried out with the Green function of the Hermitean conjugate differential system, which has the same ATM. This leads to a formula analogous to (35) in which 
Tc(z´, z0), acting from the right, transfers in z´ with z fixed.

	Following this line of argument and recalling that the definition of T does not depend on the choice of z0, we arrive -details in16- at a general expression for G in terms of T of the form

G(z, z´) =
(36)

for any pair of values of (z, z´) irrespective of the position of z0. Hence

A(z, z´) =  .
 (37)

	Not all the eight (matrix) coefficients  are independent. Firstly, whether we proceed from the first or the second row of (36), we must obtain

       G(z)  G(z, z´) = CAA 	             (38)

	Then, from (28) and (37):

                               = - IN .		(39)

	Furthermore, an expression for Z(z, z') (29) can be obtained similar to (37) and then, from (31)

			   = - IN .        	             (40)

	Now, consider the configuration z  z0  z´, Then1

       G(z, z´) = G(z, z0)  G(z0) -1  G(z0, z´) .      (41)

	From this and the above equalities -together with the similar expressions for Z(z, z ')- we obtain

                            		(42)

	These relationships hold quite generally and reduce the independent coefficients. The rest of the analysis depends on the domain under study.

	For the finite internal domains of a heterostructure, it was shown in15 that one can define any arbitrary extended pseudomedium with a Green function satisfying arbitrary boundary conditions -except for infinite barriers- at the extremes of the said domains. In practice a considerable simplification of the algebra can be achieved in this way. The external domains are different. Sometimes they are terminated by infinite barriers, which in part of the model. The algebra is then fairly simple. A different situation arises if the external domains are assumed to extend to infinity. We now consider an infinite regular medium, which is in any case always a valid description for any domain, even if it is finite or semiinfinite.

	We take z0 = 0 and abbreviate Tab(z, z0)/Tab(z´, z0) as Tab(z)/Tab(z´). Consider the first row of (36) and take z´= 0. Then

G(z, 0) = TAD(z)  
(43)

	Regularity at - requires the vanishing of the regular limit of (43). To this effect we note that even if the coefficients of the differential system are variable, for sufficient large IzI these can be asymptotically “flattened” -i.e. replaced by their average values- at sufficiently large distances without essential loss of accuracy. Then T can be written as T(, z) and regular limits taken in the manner of (19)-(20), so we define the regular limits 

T = 

             Q = . (44)

	Now, the first factor of (43) has asymptotic oscillatory behaviour. Thus the regular vanishing of G( -, 0) requires the vanishing of the second factor in the regular limit. Following this line of argument and using the general identities just presented, we obtain the full set of parameters for the regular Green function 

CAA = T- - T+-1,

= -T-  CAA, = -T+  CAA,

= -CAA  Q+, = CAA  Q-,

          = - T-  CAA   Q+,= T+  CAA  Q-	(45)

	The regular Green function is thus fully determined from T. The latter is always an ultimate resort if the analytical solution is not viable, while G is a natural object to extract physical information directly from it. As indicated above, the set of parameters , can be considerably simplified by the convenient choice of simpler boundary condition.

VI. THE SYMPLECTIC CHARACTER 
      AND GENERAL PROPERTIES OF T

	Firstly we note that, due to the fact that the two fields transferred by T are continuous everywhere, on crossing interfaces separating different media T is obtained by simple chain multiplication of the corresponding different matrices. This is a practical useful feature for the description of heterostructures, not shared by M15. Furthermore, let us define the 2N  2N auxiliary matrix

			        J = 		(46)

	Then 

			Yc  J  Y = - Fc  A + Ac  F.  	(47)

	Now, the general definition of the flux density is

j(z) = iFc(z)  A(z) - Ac(z)  F(z) = -i Yc(z)  J  Y(z)
	 (48)

	On the other hand, the second order differential system for F is transformed in the standard way into the first order differential system for Y,

			= D(z) (z)		(49)

with

D(z) = .	(50)

	It is easily seen that

    Fc  A – Ac  F = - Yc Dc  J + J  D  Y.   (51)

	Then, since

			   Dc  J + J  D = O2N		(52)

it follows that  j(z) is constant everywhere, which is a general expression of the continuity equation.

	Now consider any arbitrary pair of points (z0, z), transfer Y from z0 to z, write T(z, zo) for brevity as T and  to j(z) to j(z0). This yields 

			  J = Tc  J  T     		(53)

	Thus T is a symplectic matrix in a general sense17. Since J has unit determinant, it follows that

			   DetT 2 = 1			(54)

	This provides a concise practical rule which can be used to a monitor the quality of the numerical computations as z grows away from z0. Furthermore, by equating the submatrices of (53) we obtain the three independent identities

		      		(55)

		     		(56)

		     .		(57)

	These can also prove useful in two ways, namely: (i) they can help simplify the algebra in the analytical formulation of a problem and (ii) other concise rules for numerical monitoring can also be extracted from them. 

	Further details and different ways to combine T and G for the study of multilayer heterostructures are given in16.

VII. CONCLUSION

	The Associated Transfer Matrix T appears to be both, formally appealing and practically useful. It suggests itself in a natural way from the physics of the problem. It is unique and does not depend on having an analytical basis, so it can always be an ultimate resort, obtainable by numerical integration. It satisfied some general identities, not satisfied by other transfer matrices, which provides useful rules to monitor the numerical quality of computational processes. Finally, it is directly related to the Green function, from which direct physical information follows in a natural way.

ACKNOWLEDGEMENTS

	This work was done while one of us (R P-A) was enjoying the hospitality and finantial support of the Universitat “Jaume I”- Castellón de la Plana, Spain- which is here gratefully acknowledged. We also express our appreciation for the invaluable help of Pilar Jiménez in the preparation of the article.

REFERENCES

1GARCÍA-MOLINER, F. and V.R. VELASCO (1992): Theory of Single and Multiple Interfaces (Singapore: World Scientific.

2VINTER, B. and C. WEISBUCH (1991): Quantum Semiconductor Structures (San Diego: Academic Press).

3PÉREZ-ÁLVAREZ, R. and F. GARCIA-MOLINER (2000): “The spectrum of quasiregular heterostructures”, in Some Contemporary Problems of Condensed Matter Physics 
ed S. Vlaev and M. Gaggero-Sager (New York: Nova Science).

4MERLIN, R.; K. BAJEMA; R. CLARKE; F-Y. JUANG and P.K. BHATTACHARYA (1985): Phys. Rev. Lett. 55, 1768.

5PÉREZ-ÁLVAREZ, R.; F. GARCÍA-MOLINER And V.R. VELASCO (1995): J. Phys.: Condens. Matter 7, 2037.

6GARCIA-MOLINER, F.; R. PÉREZ-ÁLVAREZ; H. RODRIGUEZ-COPPOLA and V.R. VELASCO (1990): J. Phys. A: Math. Gen. 23, 1405..

7PÉREZ-ÁLVAREZ, R.; C. TRALLERO-HERRERO and F. GARCÍA-MOLINER (2001): Eur. J. Phys. 22, 275.

8VELASCO, V.R. (200): “Study of many interfaces and inhomogeneous systems by means of the surface Green function matching and the transfer matrix methods, in Some Contemporary Problems of Condensed Matter Physics ed S. Vlaev and M. Gaggero-Sager (New York: Nova Science).

9SZMULOWICZ, F. et al. (2003): Phys. Rev. B68, 085305.

10RÖSSLER, U. (1984):  Solid State Commun. 49, 943.

11KITCHIN, M.R. (1999): “A theoretical thesis submitted to the University of Newcastle upon Tyne for the degree of Doctor of Philosophy.

   KITCHIN, M.R.; M.J. SHAW; E. CORBIN; J.P. HAGON and M. JAROS (2002): Applied Surface Science 166, 35.
   KITCHIN, M.R.; M.J. SHAW; E. CORBIN; J.P. HAGON and M. JAROS (2002): Phys Rev. B61, 8375 (2000)
	
12ERDÖS, P. and R.C. HERNDON (1982): Advances in Physics 31, 65.

13PEREYRA, P. (1995): J. Math. Phys. 36, 1166.

14FRIEDMAN, B. (1956): Principles and techniques of applied mathematics (New York: J. Wiley).

15PÉREZ-ÁLVAREZ, R., H. RODRIGUEZ-COPPOLA;  V.R. VELASCO and F. GARCÍA-MOLINER (1988): J. Phys. C: Solid State Phys. 21, 2197.

16PÉREZ-ÁLVAREZ, R. and F. GARCÍA-MOLINER (2004): Transfer matriz, Green Function and related techniques: Tools for the study of multilayer heterostructures (Castellón de la Plana, Spain: publication of Universitar “Jaume I”).

17ARVIND DUTTA, B.; N. MUKUNDA and B. SIMON (1995): Pramana 45, 471.
   
           MEHL, C., V. MEHRMANN and H. XU (2000): Electronic J. Lin. Alg. 7, 112.