Generalized eigenvalue problem criteria for multiband

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Generalized eigenvalue problem criteria for multiband-coupled systems: hole mixing phenomenon study

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Phys. Scr. 84 (2011) 055702 (10pp)

doi:10.1088/0031-8949/84/05/055702

Generalized eigenvalue problem criteria for multiband-coupled systems: hole mixing phenomenon study A Mendoza-Álvarez1 , J J Flores-Godoy1 , G Fernández-Anaya1 and L Diago-Cisneros1,2 1 2

Departamento de Física y Matemáticas, Universidad Iberoamericana, México DF, CP 01219, México Facultad de Física, Universidad de La Habana, CP 10400, Cuba

E-mail: [email protected]

Received 3 August 2011 Accepted for publication 5 September 2011 Published 31 October 2011 Online at stacks.iop.org/PhysScr/84/055702 Abstract Non-linear eigenvalue equations straightforwardly determine fundamental physical quantities of a wide variety of areas. We retrieve a root-locus-like procedure, as a new technique for directly analyzing specific physical phenomena involving multiband-multichannel charge–carrier coupled modes. A new explicit necessary and sufficient condition is presented for a generalized eigenvalue problem, associated with an N -coupled components matrix boundary equation. Within our approach, the uncoupled-system case is nicely recovered. We tested the present scheme by applying it to heavy and light holes, described via the Kohn–Lüttinger model, and found good agreement for our proposition even at medium-intensity band mixing. We simulated the multiband-hole band-mixing-phenomenon by monitoring the root-locus for the quadratic eigenvalue problem, and by plotting the metamorphosis of the effective band offset profile, for bulk and layered heterostructures, respectively. Several new features have been observed; for example, for light holes solely, an appealing interchange of quantum-well-like versus barrier-like roles has been detected for several III–V semiconductor binary compounds. PACS numbers: 73.23.−b, 02.60.Lj (Some figures in this article are in colour only in the electronic version.)

quantum transport studies of holes [5] via QEP calculations of ortonormalized eigen-spinors, provided they have been taken as the basis to construct the system’s propagating wave modes. Different from that earlier effort of a QEP-technique collateral use [5], we present here a suitable new approach to directly analyze specific physics phenomena involving multiband-multichannel charge–carrier coupled modes. Indeed, to our knowledge, neither pure theoretical nor numerical applications of the location for roots for the QEP eigenvalues have been reported previously, to explicitly describe several standard III–V semiconductor binary-compound bulk systems. While reviewing standard tools on classical control design analysis [6], we are inspired by a plot of the complex plane with the evolution of the eigenvalues of a system when a parameter is varied. This

1. Preliminary remarks The quadratic eigenvalue problem (QEP) is currently receiving much attention because of its wide variety of applications in areas such as the dynamic analysis of structural mechanics, acoustic systems, electrical circuit simulation, fluid mechanics, linear algebra problems, signal processing, micro(nano)-electronic modeling of mechanical systems [1–3], electronic properties of quantum dots and quantum rings [4]. We emphasize the feasibility of the QEP technique as a new modus operandi to deal with problems of electronic and transport properties, which demand the solution of a multiband-coupled differential system of equations. Recently, in the area of charge–carrier quantum transport, outstanding results were achieved in 0031-8949/11/055702+10$33.00

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Phys. Scr. 84 (2011) 055702

A Mendoza-Álvarez et al

evolution is known in the literature as the root locus [6]. Although we are not invoking any of the rules mentioned in [6], we do recognize, on the grounds of some experience in classical control theory, the remarkable graphical resemblance of Evans’ approach [6] to our root-locus-like technique. We will refer to this last terminology throughout the paper whenever we proceed to graph the evolution of the eigenvalues for the QEP for different values of the band mixing parameter on the complex plane. It is worthwhile to stress the importance of the eigenvalue-equation roots as well as the orthonormalization of the correlated basis for the well-founded physical problem solution. Looking for tunneling characteristic quantities for systems that involve N -component spinors, some authors run into difficulties by preserving classical rules such as the charge conservation and unitarity of the scattering matrix [7–10]. This subject leads them to introduce auxiliary normalization coefficients in transmission magnitudes, to mimic general principles such as the flux conservation, neglecting the relevance of the so-called complete basis orthonormalization (for an overview, see [5] and references therein). Despite the usefulness of the nonlinear eigenvalue equations, they have not so far received wide direct application in N -component effective mass theory in the framework of the envelope function approximation (EFA) [11, 12]. There have been a few earlier theoretical attempts within EFA (N > 2) models [5, 13], and they seem very promising. The critical consideration by these works consists in a correct manipulation of the basis that spans the coupled spinors needed for flux conservation. In this sense, it might suffice to consider some widely accepted N -band approaches of second-order linear differential systems, namely [14]: (i) the Dimmock model (N = 2), to study diffusion through heterostructures of II–VI compounds [15, 16]; (ii) the Bogolioubov model (N = 2), for superconducting elementary excitations [17]; (iii) the full phenomenological model for long-wave polar optical modes (N = 3) [18]; (iv) the Kohn–Lüttinger (KL) model (N = 4), to study the dynamics of heavy (hh) and light holes (lh) [19]; (v) the extended KL model (N = 6) [19] and the Kane model (N = 8) [20], for coupling the conduction and valence bands with spin–orbit interaction and widely used to describe transport processes by manipulating the spin. These last two models are excellent condensed-matter physics theoretical schemes, for the rapidly emerging area of spintronics [21]. The aim of this paper is not only to introduce a QEP-based root-locus-like procedure as a novel methodology for directly analyzing specific physical phenomena for multiband-coupled charge carriers in low-dimensional systems, but also to prove that there exist sufficient and necessary conditions for a simultaneous triangularization procedure of the generalized eigenvalue problem (GEP), derived from a coupled-components boundary equation, and to study the band mixing effects on these conditions. A previous report that focused just on finding circumstantial solutions to a GEP within the EFA for multiband-uncoupled systems [13] is being improved here not merely as an extension, but also as a complementary and deeper insight into related topics. In that sense, it is worthwhile to stress

the information we obtain by plotting the root locus for the QEP eigenvalues. On the other hand, previous examinations of the topic reported just sufficient conditions that rely on pure numerical methods [22–26]. In addition, some of them do not perform simultaneous diagonalization [22] or do not straightforwardly apply to concrete physical systems [23–25]. A previous classical review on a related matter is not quite general, because the GEP’s matrices can be demonstrated to depart from commutative matrices within the McCoy theorem sense [27]. As a consequence, further studies of generic necessary and sufficient conditions for the simultaneous triangularization of a matrix pair—followed by applications to real and specific multiband-coupled physical systems—are required. To circumvent the several drawbacks mentioned above, we focus on an explicit physical problem and its output to semiconductor-based electronic and optoelectronic heterostructures, rather than focusing on highly mathematical procedures alone, as was done earlier [23–25]. This is the main result we aim at, and hopefully the new method we present here may become a complementary tool for studying the electronic and transport properties in multiband-coupled systems. As a bonus, it could be to some extent valid to any EFA model with minor changes, if any. It is outside the scope of the present study to do an exhaustive analysis of possible physical interpretations over all focused criteria. However, a physical realization that applies properly to the hole mixing effects is discussed. The remainder of this paper is organized as follows. Section 2 is devoted to the general basic background and some definitions. Proofs and numerical results are presented in section 3. In that section, we test our proposition within the KL model Hamiltonian [19], we recover the uncoupled-hole regime case reported earlier [13] and present the root-locus-like strategy together with a valence-band mixing-event simulation. Finally, in section 4 we present the concluding remarks.

2. General background and definitions A wide class of condensed-matter physical problems—briefly presented in the above section—are described by an N-coupled system of ordinary second-order and linear differential equations, known as the Sturm–Liouville matrix generalized boundary problem [28, 29]: dF (z) dF (z) d B(z) + P (z)F (z) + Y (z) + W (z)F (z) dz dz dz = O N×1

(1) ,

where B(z) and W (z) are, in general, (N × N ) Hermitian matrices and Y (z) = −P † (z). By z we denote the coordinate along the quantization direction, which is perpendicular to the heterostructure planes of the interfaces, considered to be perfect. Along z the quantum heterogeneity is explicitly revealed and the z-component quasi-lineal momentum k z is confined. In general, F (z) is called the ‘field’, but in the present study it represents the (N × 1) envelope function, whose N amplitudes are under study [14]. Formally, F and matrix coefficients are not only a 1D function on z; they are also dependent on a 2D in-plane wavevector 2

Phys. Scr. 84 (2011) 055702


κET = k x eˆ x + k y eˆ y . In the [x, y] plane, the system preserves translational invariance. Following general considerations made for an N = 1 vibrating system [30], 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 non-zero elements are diagonal and include the quasi-bidimensional potential of the heterostructure and energy. Hereafter O N /I N stand for the (N × N ) null/identity matrix. By proposing a standard plane-wave solution to (1) in the form F j (z) ∼ ei (k z ) j z , it is straightforward to derive a nonlinear algebraic problem associated with (1): Q(k z )ϕ = k z2 M + k z C + K ϕ = O N×1 , (2)

equal to 1, or equivalently, a matrix X over a field is unipotent if and only if its characteristic polynomial is (x − 1)n , i.e. the matrix X − I n is nilpotent3 .

3. Procedures and results 3.1. Procedures This section is devoted to presenting an unpractised scheme to face the problem posed in section 1, whose pattern definitions were presented in section 2. Although the standard eigenvalue problem (SEP) derived from a corresponding GEP was plentifully proved for consistency in the hh–lh quantum simultaneous-transport phenomenon [5], some flux conservation failures arise when the hole-mixing strength is sensible [10] and/or the superlattice length is remarkably enhanced [10]. To quote the physical observables associated with (3), one needs first an explicit simultaneous diagonalization of matrices A and B in equation (3). The most striking outcome so far is that there is no trivial, if any, general closed solutions to (3). As a matter of fact, the number of practical purposes is persistently growing as the modus operandi for solving QEP improves.

i.e. a QEP. The linear momentum k z takes discrete values, while the quasi-momentum κT , and so its components, is a continuous parameter. In (2), Q(k z ) is a second-degree matrix polynomial on k z . Here the (N × N ) matrices M, C and K are, in general, functions of z. We focus our attention on the case when M, C and K are constant-by-layer, Hermitians; and M is non-singular; therefore k z is real or arises in conjugated pairs (k z , k z∗ ) [3]. These properties justify k · p approximation-scheme Hamiltonians [7, 15, 17, 20], [31–34] within the EFA [11, 12], whose eigen-systems are a potential subject of application of the scheme presented in this paper. Let A − k z B be a certain (2N × 2N ) linear matrix in k z , known as a first-step linearization of Q(λ). It has been shown that the substitution µ = k z ϕ [2, 3] is the simplest way to construct the linear form of Q(k z ), with identical eigenvalues. Then, equation (2) can be rewritten, leading to an associated GEP [3] ON N ϕ N ON ϕ − kz −K −C µ ON M µ ϕ = {A − k z B} = O 2N×1 , (3) µ

3.2. Necessary and sufficient conditions for simultaneous triangularization Let A and B be invertible. If D = ABA−1 B−1 is a polynomial in A, then A and B are simultaneously triangularizable if and only if D is unipotent [37]. Next we prove that matrices A and B from (3) are simultaneously triangularizable, if the matrices N , M and K from (2) are invertible,

(4)

and the matrices

N being any non-singular (N × N ) matrix [3]. A linearization (3) is in fact the GEP under focus, and is not unique [1–3]. The physical problems we face here differ from those with a large-scale polynomial or rational eigensystems whose solutions demand iterative numerical tools [35]. Moreover, we deal with a pair (A, B) embedded within a linearized eigensystem GEP; thus a numerically stable reduction can be obtained by computing the generalized Schur decomposition [36], yielding the spectrum of Q(k z ). From the numerical standpoint, this scheme also works efficiently for systems whose threshold response is led by a few lowest physical modes of the full spectrum [3, 30]. However, here we address the analytical study to an alternative procedure. Although more restrictive and unmanageable than the generalized Schur decomposition, matrix simultaneous triangularization is more straightforward and comprehensible in dealing with generic necessary and sufficient criteria for GEP-solving. Following the above-presented direction and focusing on Schur’s unitary triangularization theorem [13], we define two matrices, A and B, as simultaneously triangularizable if there exists a similarity transformation U such that the matrices T A = U † A U and T B = U † B U are upper triangular matrices. A matrix X is called unipotent if all eigenvalues are

N MN −2 and KN K−1 M−1 are unipotent.

(5)

Firstly, we prove that D may always be chosen to be a polynomial in the variable A, more explicitly, like a polynomial of first degree, such that ABA−1 B−1 = T A. Note that the last identity is satisfied if N MN −2 ON KN K−1 M−1 CN −2 − CMN −2 KN K−1 M−1 T 1 T 2 ON N = T 3 T 4 −K −C or, equivalently, if the following linear system of matrices is satisfied: T 2 K = −N MN −2 , T 1 N − T 2 C = ON , T 4 K = CMN −2 − KN K−1 CN −2 , T 3 N − T 4 C = KN K−1 M−1 . 3

A matrix R is called nilpotent if there exists an integer n, such that Rn = O n .

3

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Since the matrices N , M and K are invertible, the solution of the last linear system is given by T 1 = −N MN −2 K−1 CN −1 , T 2 = −N MN −2 K−1 , T 3 = KN K−1 M−1 N −1 + CMN −2 K−1 CN −1 − KN K−1 CN −2 K−1 CN −1 , T 4 = CMN −2 K−1 − KN K−1 CN −2 K−1 . Then D = ABA−1 B−1 may always be chosen to be a polynomial of first degree T A . Secondly, we prove that the matrix ABA−1 B−1 is unipotent. Recall that N MN −2 ON −1 −1 ABA B = , KN K−1 CN −2 − CMN −2 KN K−1 M−1

less relevant—and its applications to concrete physical phenomena, which are the primary contributions. This model is often used to study the electronic and transport properties in III–V and II–VI semiconductor valence bands. Generally speaking, it considers in the k · p approximation the highest two valence bands degenerated in the 0-point of the Brillouin zone. The usual (4 × 4) KL Hamiltonian [5] for the hh–lh bands is of the form   H11 H12 H13 0  ∗ H22 0 −H13  , ˆ kl =  H12 H ∗  H13 0 H22 H12  ∗ ∗ 0 −H13 H12 H11 ∂2 H11 = A1 κt2 + V (z) − B2 2 , ∂z √ h¯ 2 3 γ2 (k 2y − k x2 ) + 2iγ3 k x k y , H12 = 2 m0 √ ∂ h¯ 2 3 γ3 (k x − ik y ) , H13 = i 2 m0 ∂z

where A−1 =

−K−1 CN −1 N −1

−K−1 ON

and B−1 =

N −1 ON

ON . M−1

Now D is unipotent if the matrices N MN −2 and KN K−1 M−1 are unipotent [38], but by the original hypothesis (5) these matrices are unipotent. Therefore, noting the former supposition, namely: let A and B be invertible and D a unipotent polynomial in A, we are able to conclude that the matrices A and B are simultaneously triangularizable. Some useful consequences derive straightforwardly from the above conclusion. The matrices A, B are simultaneously triangularizable if the matrices M and K are invertible and −1

N = M and the matrix KN K N

−1

∂2 . ∂z 2

Although not shown here for the sake of brevity of the forthcoming analysis, we have introduced typical parameters and relevant quantities (in atomic units) of the KL model [13]. In the (4 × 4) KL model, the matrix coefficients of equation (2) bear a simple relation to those in (1) [5]: M = −B,

C = 2iP

and K = W .

(9)

Then, after a little algebra, the matrix coefficients of (2) can be cast as   0 0 h 13 + iH13 0  0 0 0 −h 13 − iH13 , C= h 13 − iH13  0 0 0 0 −h 13 + iH13 0 0

is unipotent (6)

or N = M and [N , K] = O N ,

H22 = A2 κt2 + V (z) − B1

(8)

(7)

denoted hereafter by the notation [N , K] , N K − KN , for the additive commutator of the corresponding matrices. It is worthwhile to add a few remarks: note that the case when C = O N is considered in propositions (4) and (5). Besides, it is very easy to state that the hh–lh uncoupled regime [13] is fully recovered at this stage. All one needs to do is consider the consequence (7) and the mentioned case when C = O N . Note that the sufficient condition given by (7) generalizes the result presented in [13], since the condition [N , C ] = 0 is no longer needed, whose direct implication is a lack of valence-band mixing, i.e. κT ≈ 0.

(10)



a1 h 12 − iH12 K=  0 0  m1 0 M= 0 0

3.3. An example: the two-band KL model Hamiltonian (4×4) We found it useful to test the necessary and sufficient conditions deduced above, the root-locus-like modus operandi and the band-mixing-phenomenon numerical simulation, in the framework of the KL two-band model, due to its widely accepted accuracy in describing the dynamics of elementary excitations in the valence band. We think that in this way we achieve a weighted balance between the imperative mathematical description of the topic—being

0 m2 0 0

h 12 + iH12 a2 0 0 0 0 m3 0

 n1 0 N = 0 0

0 0 a2 h 12 − iH12

  0 B2 0 0 = 0 0 m4 0 0 n2 0 0

0 0 n3 0

0 B1 0 0

 0 0 . 0 n4

 0  0 , h 12 + iH12  a1 (11)  0 0 0 0 , B1 0  0 B2 (12)

(13)

For matrices given by (10)–(12) the associated QEP is regular and non-singular; therefore there are eight finite eigenvalues that are real or complex conjugate pairs. Also, 4

Phys. Scr. 84 (2011) 055702


we observed that det(k z2 M + k z C + K) is a polynomial of the eighth degree with only even power of k z and real coefficients, i.e. det(k z2 M + k z C + K) = q0 k z8 + q1 k z6 + q2 k z4 + q3 k z2 + q4 , where qi are a function of the Lüttinger semi-empirical valance band parameters and the components of κT ; q0 = det M as expected [3]. We analyzed the cases of GaAs and AlAs, and we found four different eigenvalues with algebraic multiplicity of 2. Each of these eigenvalues also has a geometric multiplicity of 2. From these lines of evidence, we strongly suspect that any III–V-like compound will have the same characteristic regarding their spectrum. For the KL model we introduce H12 = −h 12 + iH12 and H13 = −h 13 + iH13 ; then the associated GEP (3) has the matrix A given by   0 0 0 0 n1 0 0 0  0 0 0 0 0 n2 0 0     0 0 0 0 0 0 n3 0     0 0 0 0 0 0 0 n4   (14) A= ∗ −a1 H∗ 0 0 0 0 H13 0  12   ∗   H12 −a2 0 0 0 0 0 −H13   ∗  0 0 −a2 H12 H13 0 0 0  0 0 H12 −a1 0 −H13 0 0

Table 1. Numerical test for sufficient conditions (4) and (5). We have assumed k x = k y = 1 × 10−6 Å−1 , E = 0.475 eV and V (z) arbitrary. The Lüttinger parameters γ1,2 were taken from [39]. In the last column, k[N , K]k2 stands for the spectral norm. Materials AlSb AlP AlN InAs GaP GaN InP

γ2

k[N , K]k2

5.18 3.35 1.92 20.0 4.05 2.67 5.08

1.19 0.71 0.47 8.5 0.49 0.75 1.60

2.263 × 10−9 6.694 × 10−10 4.369 × 10−9 8.552 × 10−9 1.076 × 10−9 9.023 × 10−11 3.675 × 10−10

i.e. N = M = mI 4 . Taking this into account, it is clear that m 1 = m 2 = m 3 = m 4 ≡ m, and from equation (3), it is straightforward to show that I N ON B=m = mI 4 . ON I N Thus, a linearized secular equation (3) from the matrix polynomial Q(k z ) strongly simplifies to a familiar SEP, i.e. det (A − k z B) = det(A − k z mI 2N ) = O 2N . It is straightforward to prove that the commutators are equal to zero, because the matrix N is now a scalar times the identity matrix. The restriction (7) demands the hole effective masses

and then it can be proved that det{A} 6= 0. By considering (9) and (12), it is simple to find that ∗ 2 det{B} ≡ det{N } det{M} = n 1 (n)2 n 3 n 4 (m ∗hh )2 (m lh ) obviously departs from zero as det{N } 6= 0. In particular, 2 4 2 2 det{K} = a12 a22 − 2a1 H12 a2 − 2a1 a2 h 212 + H12 + 2H12 h 12 + h 412 is different from zero, in general. Therefore, we can use propositions (4) and (5) and in particular derivations (6) and (7) to achieve the necessary and sufficient conditions or sufficient conditions for a simultaneous triangularization of the GEP’s pair (A, B). Let us now briefly discuss the solutions to corollaries (6) and (7) for the KL model. Suppose now that we are interested in the requirements imposed by (6). In this case, we consider (11)–(13) and quote the eigensystem of KN K−1 N −1 , which yields eigenvalues λ1,2 =

γ1

m ∗hh =

m0 (γ1 − 2γ2 )

∗ m lh =

m0 (γ1 + 2γ2 )

and

∗ to satisfy m ∗hh ≈ m lh , where m 0 stands for the bare electron mass. The last implies the duplicated Lüttinger parameter γ2 to be disregarded with respect to γ1 . In table 1 is summarized a numerical test on several III–V binary compounds, which nicely fulfill sufficient conditions (7). As a criterion for [N , K] = C NK = O N , we have taken their two-norm or spectral norm [49], q † kC NK k2 = max νi (C NK · C NK ) , (17)

p 1 2 ((b12 + b22 )(H12 + h 212 ) ± G 12 (b2 − b1 ) d12 − 2a1 a2 b1 b2 ), (15)

νi 6=0

which must fulfil λ1,2 = 1. Here d12 = 2b1 b2 [H 212 + h 212 − a1 a2 ] and 2 2 G 12 = H12 + h 212 H12 + h 212 (b1 + b2 )2 − 4a1 a2 b1 b2 .

of a matrix C NK with complex entries, where νi ranges over the spectrum of the matrix (C NK · C †NK ). Here C †NK stands for the conjugate transpose of C NK . Formula (17) guarantees a reliable measure of closeness to the expected matrix element values by means of the implicit matrix difference in (16). On general grounds, all the norms in Rn are equivalent. Thus (17) equals the Euclidean norm considered in table 1 of [13], up to the precision of an arbitrary constant. Note that for the selected materials, the duplicated Lüttinger parameter γ2 can be disregarded with respect to γ1 , leading the spectral norm of (16) to be sufficiently close to zero. With the above analysis of case (7), we readily recover the situation described in an earlier report [13]. Furthermore, the criteria we have implemented here improve former considerations [13], because they are less demanding and focus on deeper insight into related topics as

For the additional case (7), the requirement [N , K] = O N implies for [N , K ] =   ∗ 0 H12 (n 1 − n 2 ) 0 0 H12 (n 2 − n 1 )  0 0 0   ∗  0 0 0 H12 (n 3 − n 4 ) ∗ 0 0 H12 (n 4 − n 3 ) 0 (16) that n 1 = n 2 and n 3 = n 4 . Moreover, as N = M we have m 1 = m 4 and m 2 = m 3 . Then n 1 = n 2 = n 3 = n 4 , 5

Phys. Scr. 84 (2011) 055702


well. Therefore, more III–V semiconductor heterojunctions satisfy (7), allowing the pencil (A, B) to be simultaneously triangularizable. 3.4. QEP spectral distribution Before discussing several consequences of the spectral distribution in (15), we would like to highlight some problems about hole mixing effects and procedures to treat them. For many actual practical solutions and technological applications—involving holes as charge carriers—it is very important to include band mixing, i.e. the degree of freedom transverses to the main transport direction [5, 14, 40]. The consideration of heterostructures such as semiconductor superlattices, resonant tunneling diodes and quantum well photodetectors needs the explicit inclusion of finite off-zone-center transverse momentum components to explain the hole quantum transport mechanisms [5, 40]. It is unavoidable to recognize that in the specialized literature there are a large number of reports studying several physical phenomena derived from hole mixing effects, via standard existing methods. However, just a few reports are available concerning the band mixing itself as a central topic of investigation [14, 41]. As a matter of fact, what we are really proposing here is neither superiority of the present approach nor possible improvements in accuracy or computational effort, but rather a remarkably different view to manage a problem not usually addressed in most theoretical attempts, to deal with multiband-coupled quantum phenomena. It is in this sense that a comparison with existing standard methods in the area just mentioned above is both unfair and inaccurate. In other words, hole mixing is crucial for bulk and low-dimensional confined systems possessing quantal heterogeneity, a question soon to be considered partially in this paper within quite an unusual framework, namely the evolution of the root locus for the QEP eigenvalues. It is then tempting, as well as useful, to have a look for the QEP spectral distribution at functional-zeroed hole band mixing, regarding its ‘ideal’ sense, on the grounds of fulfilment of the constraint (15), and accordingly the contrast to a medium-intensity hole band mixing case. Precisely, the aim of figures 1 and 2 is to give some insight into the spectrum (15) at zero and finite hole band mixing, respectively. The ultimate goal is to determine whether the requirement λ1,2 = 1 depends or not on hole band mixing, as this is remarkable when we study most of the physical phenomena at the valence band [5, 40]. In figure 1, we illustrate the contours of both kRe(λ1 )k and kIm(λ1 )k by means of a density-map format. We have selected k x ≡ k y ranging in the interval [10−12 ; 10−5 ] Å−1 , i.e. κT ≈ 0 (low-intensity band mixing), and the system is said to yield an uncoupled hole regime. Assuming a logarithmic distribution of points, it is straightforward to show that log10 [kRe(λ1 )k] = [ − 10−15 ; −5 × (−10)−16 ] ⇒ kRe(λ1 )k ≡ 1 within machine accuracy, and log10 [kIm(λ1 )k] = [ − 7.3; −7.24] ⇒ kIm(λ1 )k ≈ 0. The plot shows that a relatively low band mixing perturbation of the order of [10−12 ; 10−5 ]Å −1 is unable to move the eigenvalues λ1,2 from the established requirement. Thus extremely good agreement with the imposed condition (λ1,2 = 1) is found for uncoupled hh and

Figure 1. Spectral distribution for the fulfilment of the necessary and sufficient conditions (15) of GEP simultaneous triangularization for uncoupled holes. The top panel shows the density-map contours for Re(λ1 ) as a function of the in-plane quasi-momentum components k x and k y . For the target quantity Re(λ1 ), we have assumed log10 [kRe(λ1 )k], pursuing a better resolution. The bottom panel plots the same for the imaginary part by means of log10 [kIm(λ1 )k]. We take the parameters of GaAs from [39] with κT ∈ [10−12 , 10−5 ] Å−1 . The side bar in each panel shows the corresponding scaled mapping of colors.

lh, making the pair (A, B) simultaneously triangularizable. It is worthwhile to emphasize this case, as it departs from a purely theoretical exercise. In recent years, no band-mixed holes have attracted much attention in spintronics experimental measurements, owing to their stronger Rashba effect compared to that of electrons [42–44]. We also examined the fulfilment of Re(λ1 ) = 1 when a medium-intensity hh–lh band mixing is assumed (k x ≡ k y varying in the interval [10−5 ; 10−3 ] Å−1 ). This is the purpose of figure 2. Note the stripes in panels (c) and (d), corresponding to log10 [kRe(λ1 )k] = [ − 7.6 × 10−7 ; −5.5 × (−10)−7 ] ⇒ kRe(λ1 )k ≈ 1 and log10 [kIm(λ1 )k] = [−2.81; −2.73] ⇒ kIm(λ1 )k ' 0.002, respectively. Despite an unwanted degradation in accuracy involved in this estimation, as the mixing increases (see figure 1 for reference), no remarkable violation of requirements Re(λ1 ) = 1 in the coupled hh–lh regime can be observed. Thereby, even for band mixing perturbation two orders higher than the uncoupled regime discussed in figure 1 the requirement λ1,2 = 1 is reliably stable, allowing GEP simultaneous triangularization to proceed. 3.5. Hole band mixing effects What we know about a system possessing quantal heterogeneity is mainly determined by what we know of its wave function whose components can be extracted by quoting an eigenvalue equation such as (2). Commonly, the wave function’s outstanding role in a quantum problem is dramatically diminished due to its meaningfulness. Consequently, the phase is often dropped as a precise vehicle for understanding several physical properties. This viewpoint is biased and requires corresponding reactions. Major benefits to the readers, of the numerical simulations to come next, derive from a couple of aspects. Firstly, 6

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Figure 2. Spectral distribution for the fulfilment of the necessary and sufficient condition (15) of GEP simultaneous triangularization for coupled holes. Panel (a)/(b) displays 3D perspective contours for Re(λ1 )/Im(λ1 ) as a function of the in-plane quasi-momentum components k x and k y . Panel (c)/(d) plots density-map contours for Re(λ1 )/Im(λ1 ). For Re(λ1 )/Im(λ1 ), we have assumed log10 [kRe/Im(λ1 )k] for better resolution. The parameters were taken from [39] for GaAs with κT ∈ [10−5 , 10−3 ]Å−1 .

the simulations are devoted to enhancing the feasibility of the QEP-based root-locus-like technique, as a new modus operandi to directly analyze specific physical phenomena involving multiband-multichannel charge–carrier coupled modes, via a phase-like quantity. Secondly, they are intended to make neatly understood the influence of the band mixing parameter κT on the effective-potential exchange for III–V binary-compound layered semiconductors. In assuming perfect interfaces in the [x, y] plane, we ensure that the low-dimensional confined system possesses quantal heterogeneity along the z direction. Thus, the relevant information-related elementary excitations are carried out by the z-component envelope function F (z). In our case, when k z loci are depicted, we are investigating at the same time, the behavior of the discrete-quantity component of the envelope-function’s phase. We underline that this fact, as a bonus, makes it possible not only to derive necessary and sufficient conditions (4) and (5), but also to make some related numerical authentications.

Figure 3. Root locus for the eigenvalues k z from QEP (2) as a function of k x = k y with κT ∈ [10−6 , 10−1 ] Å−1 for (a) AlAs, (b) AlSb, (c) AlP and (d) AlN binary compounds. We assumed V (z) = 0.498 eV and the longitudinal energy E = 0.475 eV. The map evolves the outward coordinate center of the imaginary axis, with different values of k z for both lh and hh.

3.5.1. Root-locus-like technique. We retrieve a procedure that relies on a robust classical control theory technique, the so-called root locus [6]. Once we have quoted the eigenvalues k z from (2), it is straightforward to generate a complex-plane plot, that symbolizes the evolution of k z allocations, while κT changes. We take advantage of the root-locus-like knowhow to promptly identify evanescent modes, keeping in mind that complex (or purely imaginary) solutions are displayed in the up (down)-half plane (see figures 3 and 4). These values are forbidden for some layers and represent unstable solutions. The opposite examination is

also suitable for propagating modes, which become patterned in the left (right)-half plane, and are equated to stable solutions for given layers. Thus we are able ‘to stamp’, using 2D-map language, the stability domain analysis of the envisioned heterostructure under a quantum-transport problem. This way, we are presenting an unfamiliar methodology to deal with solid-state low-dimensional physical phenomenology. 7

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Figure 4. Root locus for the eigenvalues k z from QEP (2) as a function of k x = k y with κT ∈ [10−6 , 10−1 ] Å−1 for (a) GaAs, (b) InAs, (c) GaP and (d) GaN binary compounds. We assumed V (z) = 0, and the longitudinal energy E = 0.475 eV. The maps evolve toward the coordinate center of the real axis. At large κT , some lh-maps evolve the outward coordinate center of the imaginary axis (see panels (a) and (b)).

Figure 5. Root locus for the eigenvalues k z from QEP (2), as a function of κT for (a) GaAs and (b) InAs binary compounds. We assumed V (z) = 0, and the longitudinal energy E = 0.475 eV.

The role of the hole mixing effects on the electronic and quantum transport properties is widely accepted. Generally speaking, most of the existing standard techniques to do so rely on two platforms: (i) experimental observables (photoluminescence [45], outgoing current flow [40], tunneling time and time-resolved spectroscopy [46]) and (ii) non-experimental observables (energy spectrum [8], density of states and kinetic coefficients [40], conductance and phase time [5]). On the other hand, we have not yet encountered, in the literature available to us, the modus operandi presented here, which is said to yield the evolution of the root locus for the QEP eigenvalues and its straightforward outcome over the effective-potential dependence on the transverse component of the wavevector κT , except in a clearly reduced number of earlier theoretical studies [14, 41]. Regarding any eigenvalue-map scheme, their remarkable results focus on the fully unspecific theoretical case [14], or the single-band Schrödinger equation [41]. Obviously, these two cases are potentially far from the scenario we will discuss below. Nowadays the III–V semiconductor binary compound devices have attracted renewed interest considering their well-known appealing facilities for some emergent fields of technology. Low-dimensional systems based on AlAs, AlSb and AlGaN, together with quantum wells composed of GaAs, InAs, GaP, GaSb and GaN materials, are accepted as a platform for semiconductor spintronics [42–44, 47]. Furthermore, their increasing contribution to electronic and optoelectronic devices capability has been demonstrated [39]. Next, we firstly try to briefly describe several standard III–V bulk binary-compound semiconductors by mapping the root locus from QEP (2) (see figures 3–5). Secondly, we focus on providing some information to help gain a good understanding of the physical consequences from our root-locus-like analysis in bulk binary-compound semiconductors, but applied now to semiconductor layered systems. In that last concern, we plot the metamorphosis of the effective potential profile for finite values of κT (see figure 6).

Figure 6. Metamorphosis of the effective potential profile Veff for hh/lh (blue/red lines), as a function of κT and layer dimension. Panel (a)/(b) displays for GaAs/InAs-AlAs heterostructure a 3D-perspective profile as a function of the in-plane quasi-momentum components [k x = κmbT , k y = 0] Å −1 and layer thickness. Panel (c)/(d) plots, for the same heterostructures, the evolution of Veff at the interface, as κmbT grows.

Figure 3 illustrates the role of band mixing (κT ∈ [10−6 , 10−1 ] Å−1 ) on the k z spectrum from QEP (2) for several III–V alloys distinguished as barriers in layered systems. The k z eigenvalues can be imaginary, complex or real numbers, as reported earlier [7]. For low energies, as considered here, the first two cases are the expected ones and lead to evanescent modes [5]. We assume that the spatial distributions of the potential energy and the effective mass are layer-constant variables. We display the root locus evolution of the eigenvalues k z in several bulk systems, in terms of their imaginary (vertical axis) and real (horizontal axis) parts: namely, panel (a) for AlAs, one of the most recurrent material, commonly incorporated as a barrier into GaAs-based electronic and optoelectronic heterostructures; panel (b) for AlSb, which is frequently used as a barrier slab in high-mobility electronic and long-wavelength optoelectronic 8

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devices [48]; panel (c) for AlP, the largest direct-gap III–V semiconductor [48]; and finally, panel (d) for AlN, which, owing to its queer usage in technological appliances, represents some breakpoint for practical alloys [39]. In this interval of κT , the map evolves from the center of the imaginary axis, while all the eigenvalues of k z for hh (blue-cyan symbols) and lh (red-green symbols) remain purely imaginary. Fortunately, this guarantees robustness for standard properties of this layer as a potential barrier in heterostructures for most practical situations, applicable at low longitudinal energy to a wide range of hole band mixing. Figure 4 is the same as figure 3 for several bulk systems used as quantum wells in heterostructures: namely, panel (a) for GaAs, the most technologically relevant semiconductor material [39]; panel (b) for InAs, which has assumed a growing role in long-wavelength optoelectronics [48]; panel (c) for GaP, common used in visible light-emitting diodes [39]; and panel (d) for GaN, amply known in blue laser appliances and light-emitting diodes [39]. In this case, we expect real k z eigenvalues, leading to oscillatory modes at low energies. Valid for all panels and given an initial segment of κT , all eight eigenvalues are real numbers (horizontal plotting). The hh (blue-cyan symbols) and lh (red-green symbols) behave as particles in a quantum well (QW) as expected when they are allocated at these materials. However, by letting grow κT > [9.6 × 10−2 (5.5 × 10−2 )] Å−1 for GaAs (InAs), we found the four lh eigenvalues to become purely imaginary (vertical plottings in panels (a) and (b)). Previously it was shown for the Alx Ga1−x As-GaAs superlattice that both wells and barriers may interchange for each superlattice’s material, depending on the transverse (in-plane) component value of the wave vector [41]. Panels (a) and (b) of figure 4 unambiguously prove the metamorphosis of GaAs and InAs, with κT ∈ [5.5 × 102 , 10−1 ] Å−1 (i.e. with the hole band mixing). For the sake of better insight into this complicated hh–lh interference phenomena, additionally we have shown in figure 5 the evolution of the QW–barrier exchange with the valence-band mixing, for two of the most recurrent III–V alloys in practical optoelectronic devices. According to the envelope function conception, these solutions are not oscillatory allowed as quasi-bound states of the QW. In other words, the GaAs and InAs materials embedded in a layered system could become an effective potential barrier for lh, remaining as a QW for hh.

potential energy by considering only the linear and quadratic responses to the heterostructure perturbation [52]. To gain some insight into the rather complicated influence of the band mixing parameter κT , on the effective band offset, we display figure 6. For a single-band-electron Schrödinger problem, some authors predicted that both wells and barriers may appear in the embedded layers of a semiconductor superlattice, depending on the transversecomponent value of the wave vector [41]. Recently, it was unambiguously demonstrated that the effective-band offset energy Veff , ‘felt’ by the two flavors of holes, as κT grows, is not the same. Inspired by these earlier results, we addressed a wider analysis of this appealing topic, displayed in figure 6. We considered a width of 25 Å for the external cladding-layer GaAs(InAs), while for the AlAs layer thickness we took 50 Å. Panel (a)/(b) of figure 6 explicitly shows the metamorphosis of Veff (the difference between band-edge levels for concomitant-material slabs, i.e. the effective band offset), felt for both flavors of holes, independently. From panels (a) and (b), it is straightforward to see that for hh (blue lines), an almost constant Veff remains, while κT varies from 0 (uncoupled holes) to 0.1 Å−1 (strong hole band mixing), although the respective band-edge levels had changed. Nevertheless, for lh (red lines) in a GaAs–AlAs–GaAs heterostructure, a noticeable reduction of Veff arises, as can be seen from panel (a). Besides, panel (b) exhibits for lh (red lines), a striking exchange of roles QW–barrier for an InAs–AlAs–InAs heterostructure, such as that predicted for electrons [41]. This observation means that in the selected rank of parameters for given binary-compound materials, an lh may have ‘felt’ a qualitatively different Veff (QW or barrier) during its passage through a layered system while varying the degree of freedom in the transverse plane. The former assertions can be readily observed in panels (c) and (d) of figure 6, where we have plotted the evolution of Veff with κT at any fixed transverse plane of the heterostructure. The lh behavior, just mentioned above, is in slight compliance with the results presented in figures 4 and 5, owing to the remarkable differences separating bulk materials from layered heterostructures. These scenarios, both in bulk materials and in layered heterostructures, could attract much attention in the field of condensed-matter device designing. Whereas one can be certain of the realness of theoretical predictions only when experimental measurements appear, this unusual spectrum of k z for GaAs and InAs bulk systems and the QW versus barrier exchange in GaAs(InAs)-based superlattices foretell an effective potential barrier (or QW) selectively for light holes and could yield a filtering process, by an accurate tuning of the appropriate in-plane energy. It is common wisdom in transport studies that any kind of selection rules over propagating modes should be reflected in the relevant scattering quantities. In this sense, we hope that the appealing features presented in panels (a) and (b) of figures 4–6 could be considered in the design of optoelectronic and spintronics devices, pursuing filter-like gates for hh and lh depending on their in-plane energy and the slab they are traversing through. However, owing to the noticeable difference of the QW–barrier interchange for GaAs and InAs (see figure 5), we guess that considerable care must be taken for a vast majority of appliances if they involve hh–lh interference phenomena at a low longitudinal energy.

3.5.2. Metamorphosis of the effective band offset profile. On general grounds, for κT ≈ 0 the effective band offset is given by the difference for 3D band edge levels [5, 14]. By letting κT grow, the band mixing effects arise and the effective band offset changes. A comprehensive analysis of this subject describes a larger reduction for the piecewise constant effective barrier height Veff as a function of κT , for lh with respect to that for hh [5]. The mechanism responsible for this behavior is the increment of the term κT2 /m ∗hh,lh (z), yielding an inversion of the roles of wells and barriers [14, 41] as shown in panels (a) and (b) of figures 4 and 5. Some authors declared a shift upward in energy of the boundstates in the effective potential well as the transverse wave vector increases [51]. Recently, a valence-band mixing first-principle theory within the EFA was proposed, which approximated the superlattice 9

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4. Concluding remarks

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We presented a QEP-based root-locus-like procedure as an uncommon methodology for directly analyzing specific physical phenomena for multiband-coupled charge carriers in low-dimensional systems. If there exists a unipotent polynomial in one GEP matrix, then new explicit necessary and sufficient criteria for solving a (2N × 2N ) GEP can be derived, and one can directly decompose the GEP into triangular matrices simultaneously. By applying this method for heavy and light holes described within the (4 × 4) KL model, we do not find any unavoidable restriction to achieve necessary and sufficient conditions, in obtaining the GEP spectrum. Very good agreement with the imposed propositions (λ = 1) is found for uncoupled hh–lh in several III–V semiconductor binary compounds. As the hh–lh mixing increases by two orders, unless estimation becomes less accurate, it remains reliable enough to allow GEP simultaneous triangularization. As expected, an overall symmetric distribution of the QEP-based root locus was found to be unrelated to hole band mixing and to be in accordance with the QW-like or barrier-like intrinsic character of each material. However, the GaAs and InAs bulk materials clearly invert their role of QW, turning it into an effective potential barrier for light holes solely, as the band mixing grows. At odds with the opposite for heavy holes, the light holes exclusively feel an effective band offset exchanging from a QW-like into a barrier-like one, and vice versa for an InAs/AlAs heterostructure, while band mixing increases. These lines of evidence underline the appearance of appealing features, mediated by hh–lh coupling reinforcement [5, 50], and foretell their usefulness for experimental applications and in a theoretical analysis of hole tunneling. Some general additional cases have been deduced and the multiband-uncoupled hole regime was nicely recovered. We believe that the present scheme for GEP simultaneous solving as well as the root-locus-like modus operandi could be to some extent valid for any EFA model with minor changes, if any.

Acknowledgments This work was carried out with support from FICSAC, UIA, México. LD-C gratefully acknowledges the hospitality of Departamento de Física y Matemáticas, UIA, México.

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