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Phys. Scr. 78 (2008) 035004 (4pp)
doi:10.1088/0031-8949/78/03/035004
A generalized eigenvalue problem solution for an uncoupled multicomponent system L Diago-Cisneros1,2 , G Fernández-Anaya1 and G Bonfanti-Escalera1 1 2
Departamento de Física y Matemáticas, Universidad Iberoamericana, CP 01219, DF México, Mexico Departamento de Física Aplicada, Facultad de Física, Universidad de La Habana, CP 10400, Cuba
E-mail:
[email protected]
Received 22 April 2008 Accepted for publication 9 July 2008 Published 22 August 2008 Online at stacks.iop.org/PhysScr/78/035004 Abstract Meaningful and well-founded physical quantities are convincingly determined by eigenvalue problem solutions emerging from a second-order N -coupled system of differential equations, known as the Sturm–Liouville matrix boundary problem. Via the generalized Schur decomposition procedure and imposing to the multicomponent system to be decoupled, which is a widely accepted remarkable physical situation, we have unambiguously demonstrated a simultaneously triangularizable scenario for (2N × 2N ) matrices content in a generalized eigenvalue equation. PACS numbers: 73.23.−b, 02.60.Lj
Regarding a more extended domain for physical systems of interest, for the sake of illustration, we consider as an initial platform an intrinsic scattering elastic problem. In the GEP we are interested in, for example, the confined ‘ith’-component of the linear momentum is taken as the eigenvalue to describe the scattering states. The other two components are quasi-free, and together with the incident energy are considered as parameters. Pursuing tunneling characteristic quantities within systems that involve N -component spinors, some authors encountered difficulties in preserving classical rules such as charge conservation and unitarity of the scattering matrix [8–10]. Recently, Diago-Cisneros et al recalled the outstanding importance of preserving discrete symmetries and general principles in controlling numerical simulation of multichannel–multicomponent quantum transport of particles or quasi-particles [11]. Keeping in mind charge carriers’ transport through N -component multi-layered systems, we shall regretfully restrict the present brief paper to the algebraic treatment of the GEP equation basically, which is fundamental to later guarantee the very physical laws invoked above. The remaining part of this paper is organized as follows: section 2 is devoted to the basic theoretical definitions remarks. Some preliminary analytic results and proofs are presented. In that section, we have exercised our proposition within the widely accepted Kohn–Lüttinger (KL) model Hamiltonian [12], in which interesting and nontrivial features
1. Introduction In the wide field of condensed matter physics, several relevant quantities are related to observables derived from nonlinear eigenvalue equations [1, 2]. The quadratic eigenvalue problem (QEP) is one of the most familiar in this area, due to its successful engineering applications [3]. Recently, outstanding results were achieved in quantum transport studies of holes [4, 5] via QEP calculations of orthonormalized eigenspinors, provided they have been taken as the basis to construct the system’s propagating wave modes. In this paper, we have focused on finding circumstantial direct solutions to a less known than QEP, (2N × 2N ) generalized eigenvalue problem (GEP). Within the envelope function approximation (EFA) framework [6, 7] we attempted to avoid those assumptions made to turn to the standard eigenvalue problem (SEP) [4, 5]. Though restricted to a particular case when holes are uncoupled, so far, we emphasize the usefulness of the outcome presented here as it is relatively ‘easier’ than QEP or SEP formulations in multicomponent systems. Besides, it is, at the same time, less demanding with respect to the standard problem, concerning the rigid mathematical assumptions needed to derive the last one after two-step linearization [3, 4]. It is in this sense that the appealing issue under investigation here becomes relevant.
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arise when applying it to a uncoupled regime of heavy (hh) and light (lh) holes. Finally, in section 3 we underline some concluding remarks.
Most of the linearizations used, in practice, are of the so-called first companion form [3], ON N N ON A= and B = , (5) −K −C ON M
2. Definitions and results
N being any non-singular (N × N ) matrix [3]. A numerically stable reduction is obtained by computing the generalized Schur decomposition [3]
2.1. Definitions There is an appealing list of technological problems leading to a second-order N-coupled system of linear differential equations [3]. These interesting problems have different origins, but are mostly dynamic under (x, y)-plane invariant translations, and can be described from a well-known Sturm–Liouville matrix generalized boundary problem [13, 14]: dF (z) d dF (z) B (z) + P (z)F (z) + Y (z) + W (z)F (z) dz dz dz = ON , (1)
W † AZ = S ,
2N X j=1
α j ei λj z ϕ j =
2N X
α j F j (z),
(6)
where W and Z are unitary matrices and S and T are upper triangular matrices. If Q(λ) is regular3 , then eigenvalues matrix Λ(Q) = {sii /tii }, with the convention that sii /tii = ∞ when tii = 0. For real pairs (A, B), there is a real generalized Schur decomposition with W and Z orthogonal and T the upper quasi-triangular matrix. Schur’s unitary triangularization theorem. Given an (N × N ) matrix A with eigenvalues λ1 , . . . , λN in any prescribed order, there is a unitary (N × N ) matrix U such that
where B(z) and W (z) are, in general, (N × N ) Hermitian matrices and are fulfilled when Y (z) = −P † (z). Following the general considerations made for N = 1 vibrating system [15], it is usual to name B(z) as the mass matrix, P (z) as the dissipation matrix and W (z) as the stiffness matrix, whose nonzero elements are diagonal and contain the Q2D potential of the heterostructure and the energy. Hereafter, ON /IN stands for the (N × N ) null/identity matrix. By z, we nominate the coordinates for quantization direction, along which the z-component of the linear momentum is confined. By proposing a solution to (1) in a form, F (z) =
W † BZ = T ,
TA = U † A U
(7)
is the upper triangular and the diagonal elements tii = λi . Furthermore, if the entries of A and its eigenvalues are all real, U may be chosen to be a real orthogonal. We define two matrices A and B as simultaneously triangularizable if there exists a similarity transformation U such that the matrices TA = U † A U and TB = U † B U are the upper triangular matrices. Let F be an algebraically closed field of characteristic 0 [21]4 and let A, B ∈ Mn (F ) (the set of (N × N ) matrices with entries in the field F ). Suppose [A, B] = f (A) for some polynomial f . In particular when [A, B] , AB − BA = O2N , then A and B can be simultaneously triangularizable, which is the simplest polynomial.
(2)
j=1
it is straightforward to derive a nonlinear algebraic problem associated to (1) Q(λ) = λ2 M + λ C + K ϕ = ON , (3)
2.2. Results In this section, we present our main results on the GEP solution for multicomponent systems. Although we are restricted here to a particular case when hh and lh are uncoupled, it is useful to remark that actual resolution in several experiments involving holes is very close to the physical situation we are considering here. It can be proved that matrices A and B from (5) are simultaneously triangularizable, if
in which the quantity λ (the eigenvalue) enters raised to a power of 2, giving rise to a QEP. Here, the (N × N ) matrices M, C and K are, in general, functions of z. We address our attention to the case when M, C and K are Hermitian, therefore λ is real or arises in conjugated pairs (λ, λ∗ ). These properties justify k · p approximation-scheme Hamiltonians [8, 16–20] within the EFA [6, 7], whose eigen systems are potentially the subject of application of the scheme presented in this paper. Let A − λ B be the certain linearization of the matrix polynomial Q(λ) in the QEP (3), i.e. Q(λ) ON = H (λ)(A − λ B)G(λ), (4) ON IN
N = M,
(8)
[C , N ] = ON ,
(9)
[N , K] = ON .
(10)
First, we prove that [A, B] = O2N . Making some algebra, we have that ON NM−N2 AB − BA = . −KN + MK −CM + MC
where H(λ) and G(λ) are (2N × 2N ) λ-matrices with constant nonzero determinants. Clearly, the eigenvalues of Q(λ) and A − λ B coincide. A linearization (4) is in fact the GEP under focus, and is not unique [1–3]. So it is important to choose one that preserves the symmetry and other structural properties of Q(λ), if possible.
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A matrix polynomial like Q(λ) is regular if its matrix coefficients fulfil to be simultaneously different from null matrix ON . 4 For example, the field complex.
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If N = M, CN = N C and N K = KN , then [A, B] =
n1 0 N = 0 0
O2N .
Now by the supposition posted above, the matrices A and B are simultaneously triangularizable.
0 n2 0 0
0 0 n3 0
0 0 . 0 n4
(14)
Considering the former matrices C , K, M and N , it is deduced that matrices A and B from (5) are simultaneous triangularizable, if n 1 = n 2 = n 3 = n 4 . We verify the sufficient conditions of the proposition 2, N = M, CN = N C and N K = KN . Then we have that CN = N C ⇔
2.2.1. An example: the (KL) model Hamiltonian [4]. We have posted above a charge carriers’ quantum transport through N -component multi-layered systems, as the target problem under study. Let us invoke traveling hole propagating modes, whose bulk plane-waves states are well described in the framework of the (4 × 4) KL Hamiltonian. We briefly introduce some parameters and relevant quantities (in atomic units) of the KL model,
0 0 n (−iH + h ) 1 13 13 0
γi , with i = 1, 2, 3 (Luttinger semi-empirical valence band (BV) parameters, typical for each semiconductor material),
0 0 = n 3 (−iH13 + h 13 ) 0
n 2 (iH13 − h 13 ) 0 0 0
R = 13.60569172 eV (Rhydberg constant),
n 3 (iH13 + h 13 ) 0 0 0
0 0 0
n 4 (iH13 − h 13 )
a0 = 0.5405 Å (Bohr radius),
⇔ n 1 = n 3 and n 2 = n 4 .
V (finite stationary barrier’s height).
N K = KN ⇔
0 n 4 (−iH13 − h 13 ) 0 0
n 1 (iH13 + h 13 ) 0 0 0
0 n 2 (−iH13 − h 13 ) 0 0
E (energy of incident and uncoupled propagating modes), a1 n 1 n 2 (−iH12 + h 12 ) 0
k x , k y (components of the transversal wavevector), A1 = a02 R(γ1 + γ2 ),
0
A2 = a02 R(γ1 − γ2 ),
a1 n 1 n (−iH12 + h 12 ) = 1 0
a1 = A1 (k x2 + k 2y ) + V − E,
0
a2 = A2 (k x2 + k 2y ) + V − E, √ h 12 = a02 R 2 3γ2 (k 2y − k x2 ), √ h 13 = −a02 R 2 3γ3 k x , √ H13 = a02 R 2 3γ3 k y , √ H12 = a02 R 2 32γ3 k x k y .
C=
0 0 h 13 − iH13 0
0 0 0 −h 13 + iH13
a1 h 12 − iH12 K= 0 0
h 13 + iH13 0 0 0
h 12 + iH12 a2 0 0
m1 0 M= 0 0
n 2 (iH12 + h 12 ) a2 n 2 0 0
0 0 a2 n 3 n 3 (−iH12 + h 12 )
0 0
n 3 (iH12 + h 12 ) a1 n 4 0 0
n 4 (iH12 + h 12 ) a1 n 4
If N = M ≡ m IN , (i) then the GEP is transformed into SEP with scaled eigenvalues, (ii) [C , N ] = ON and [N , K] = ON . Let us now prove that these two direct consequences from our main proposition (i) If m 1 = m 2 = m 3 = m 4 ≡ m, then from expression (5) it is straightforward that IN ON B=m = m I2N . ON IN
0 −h 13 − iH13 , 0 0
(11)
0 0 a2 n 3 n 4 (−iH12 + h 12 )
⇔ n 1 = n 2 and n 3 = n 4 . Therefore, if n 1 = n 2 = n 3 = n 4 by proposition 2, the matrices A and B are simultaneous triangularizable.
Now consider the case where the matrices C , K, M and N are defined as follows:
n 1 (iH12 + h 12 ) a2 n 2 0 0
0 m2 0 0
0 0 a2 h 12 − iH12
0 0 m3 0
Thus, a linearized secular equation (4) from matrix polynomial Q(λ), strongly simplifies to a familiar SEP, i.e. det(A − λ B) = det(A − λ m I2N ) = O2N . (ii) It is straightforward to prove that the commutators are equal to zero, due to matrix N being now a scalar times the identity matrix. The constriction (8) demands the hole effective masses
0 0 , h 12 + iH12 a1 (12)
0 0 , 0 m4
m ∗hh =
mo (γ1 − 2γ2 )
m ∗lh =
mo (γ1 + 2γ2 )
and (13)
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spinors, whose straightforward relation to conservation laws and symmetries in the framework of quantum transport are well known [4, 8–10].
Table 1. Numerical test for sufficient conditions (8–10). We have assumed k x = k y = 1 × 10−6 Å−1 , E = 0.475 eV and V is arbitrary. Luttinger parameters were taken after [22]. Materials Ga P Al P Al Sb
γ1
γ2
k [C , N ] k 2
k [N , K] k2
4.05 3.35 5.18
0.49 0.71 1.19
7.607 ×10−4 4.734 ×10−4 1.600 ×10−3
1.076 ×10−9 6.694 ×10−10 2.263 ×10−9
Acknowledgments This work was supported by FICSAC, UIA, México. LDC gratefully acknowledges the hospitality of Departamento de Física y Matemáticas, UIA, México.
to satisfy m ∗hh ≈ m ∗lh , where m o stands for the bare electron mass. The last implies the Luttinger parameter γ2 to be disregarded with respect to γ1 . Table 1 presents a numerical test on several heterojunctions, which nicely fulfill sufficient conditions (8)–(10). As a criterium for zero commutators in (9)–(10), we have taken their Euclidean norm [23], 1/2 N X |x j |2 , kxk2 = (15)
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j=1
because that way we guarantee a reliable measure of closeness to the expected matrix elements values in the implicit matrix difference. Notice that for the selected materials the Luttinger parameter γ2 can be disregarded with respect to γ1 and the Euclidean norm is almost zero for both commutators in each case.
3. Concluding remarks In this paper, we have focused on finding circumstantial solutions to a (2N × 2N ) GEP—within the EFA framework—in an attempt to avoid those assumptions made to turn to familiar SEP, pursuing the study of hole’s quantum transport [4]. We have exercised here with the widely accepted KL model Hamiltonian [4] under a decoupled-holes regime, which is an appealing physical situation. We have demonstrated that whenever the Luttinger parameter γ2 can be disregarded with respect to γ1 , the sufficient conditions (8)–(10) fulfil. Thus, the matrices entering the GEP can be simultaneously triangularized. As a bonus, this leads us to a SEP with scaled eigenvalues. With these results, we claim for a simplification of the rigid mathematical assumptions needed to derive the standard eigenvalue equation proposed in [4]. It is in this sense that the appealing issue under investigation becomes of certain relevance, regarding its importance for a later definition of orthonormalized
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