Elsevier

Annals of Physics

Volume 420, September 2020, 168255
Annals of Physics

Unitarity and symmetries of the multicomponent scattering matrix

https://doi.org/10.1016/j.aop.2020.168255Get rights and content

Highlights

  • Unitarity for N–component mixed flux demands: complete-orthonormalized basis.

  • Symmetries for an arbitrary basis require: structured unitarity.

  • Matrix condition number, tunes threshold-energy for available tunneling channels.

Abstract

We present a theoretical procedure, which is fundamental for unitarity preservation in multicomponent–multiband systems, for a synchronous mixed-particle quantum transport. This study focuses the problem of (N×N) interacting components (with N2), in the framework of the envelope function approximation (EFA), and the standard unitary properties of the (N=1) scattering matrix are recovered. Rather arbitrary conditions to the basis-set and/or to the output scattering coefficients, are not longer required to preserve flux conservation, if the eigen-functions are orthonormalized in both the configuration and the spinorial spaces. We predict that the present approach, is valid for different kind of multiband–multicomponent physical systems of coupled charge–spin carriers, within the EFA, with small transformations if any. We foretell the interplay for the state-vector transfer matrix eigen-values, together with the large rates of its condition number, as novel complementary tools for a more accurate definition of the threshold for tunneling channels in a scattering experiment.

Introduction

The simulation of quantum transport phenomenon, is one of the Physics’ domains that has a lengthy annals, almost as aged as the very Quantum Mechanics itself. Would be enough to retrieve the early prominent elucidation by MacColl, far ago in 1932 [1], about how time-consuming is the tunneling of spin–charge carriers, across a single-potential quantum barrier, and the large number of proposals since then, in reply to this tremendous challenge. The know-how about this subject steadily grew, but it was not all at a sudden that it has arrived at the summit, to which it has currently come, with the rising of such emergent areas as: spintronics, meta-materials and topologically-phase transitions, just to mention a few. One successful technique to settle this defiance, is the scattering matrix (SM) approach, whose attains could be assigned short later the time of MacColl’s trial [1]. Even now, there are more that remains obscure to understand about SM universal-symmetry properties, than what we see thoroughly clear; an issue to be addressed soon after, in the framework of multicomponent–multiband systems disclosed by Hermitian–Hamiltonian models. It does not really matter what name or form, the authors of different schemes, are of their wont to give to the unitarity condition, in accordance to the envisioned phenomenon and its modeling, ultimately all come to this, that we are all agree on this outstanding requirement of the Physics.

What cumbersome intrinsic details of the multicomponent–multiband quantum transport, of charge–spin carriers in the framework of the envelope function approximation (EFA), and might be in general, of scattering processes for EFA systems, can hardly be explained to those innocent of deep specialized knowledge of it, do not mean in any sense, that this significant exercise could be avoided. Nor can we here present optimum facts, to be convinced enough about the thorough one must be, when dealing with the multicomponent SM properties within the EFA, particularly with those regarding the outstanding unitarity condition (UC). It is not, that there is not a clear definition of the SM’s UC for one-component and single-band fluxes. Rather the fact is, that in order to understand the imperative standard-UC reshape, whenever dealing with streams of multicomponent–multiband charge–spin carriers, propagating jointly and simultaneously throughout layered heterostructures – described via EFA –, there are concepts and procedures that must first be comprehended. Further difficulties arise, if the physical-problem’s solution domain, allows complex-number entries, leading then to somewhat unexpected breaks of the basic SM properties.

Perhaps, the first systemic analysis – as far as we know –, devoted to this relevant issue, was addressed by Sánchez and Proetto, back in 1995 [2]. They surveyed the charge-carrier quantum transport, with the SM technique and have focused close attention on the hypothesized lack of the standard UC, for the multicomponent–multiband scattering theory (MMST). By examining the heavy- and light-hole quantum scattering, in the framework of the 2-band Kohn–Lüttinger (KL) model and using clever assumptions, these authors managed to introduce the so-called pseudo-UC, for the SM within the MMST, thus preserving a cornerstone conservation principle. Despite this remarkable theoretical achievement by Sánchez and Proetto, their methodology is not, so far, of common usage for solid-state theoretical-physicists. The last, taking into account the scarce number of citations to the reference [2] up-to-date (mostly by Diago-Cisneros et al.); in opposite to the plethora of quantum-transport studies, reported upon the workbench of the MMST. Yet, there is no doubt that the pioneer Sánchez–Proetto proposition [2], deserves a widely-held view and consequently should be more extensively applied.

Nevertheless implemented solely to the valence band, worthwhile mentioning several works, that somehow consider the ideas and/or methods of the pseudo-UC [2]. Indeed, in 1997 T. Kumar et al. [3] developed a SM approach to survey the coherent-hole quantum transport, trespassing an emitter-base junction of a bipolar transistor. Shortly after, K. Roenker et al. [4] analyzed hole anti-resonances, by scheming their scattering coefficients across a multiple quantum-well system, via the SM formalism. Besides, in 1999, these authors have stressed the prominence of the spin–orbit split-of valence subband on the scattering properties of holes in GaAs-based layered heterostructures [5]. We remark that, concurrently with the pseudo-UC modus operandi [2], in the period elapsed, approximately, from 1989 up to 2001, raised other scenarios [3], [6], [7], [8], [9], [10]. Even though, it becomes difficult to say, what possible indispensable advantages might be hoped, in distinguishing these last modeling from the pioneer scheme of the pseudo-UC. Despite of being less fundamental from both the pure physical (including experimental) and mathematical point of views, they have became much popular. This last, probably due to an easier numerical development, together with a consequent lower computational effort. For these reasons we do believe, it is not feasible to step forward following the path envisioned by Sánchez and Proetto [2], which is in our opinion a better-grounded way of proceeding, before its essentials are sufficiently clearly seen, and thereby generalized to the MMST case, although this time, without somewhat arbitrary physical assumptions and/or numerical simplifications, that substantially depart from real multicomponent–multiband systems. This is the main goal of the present Solid-State-Physics theoretical approach and its related mathematical tools.

In spite of the pseudo-UC [2], seems to be specially appropriate and successful, for a particular case of the (2 × 2) KL Hamiltonian with pure holes, is yet inconvenient for describing a wider class of multicomponent–multiband fluxes. Indeed, we are talking about the leading distinction from pseudo-UC treatment [2], if kinetic-coefficient amplitudes and/or the interplay for involved particles (or elemental-excitation quantals) moving towards a scatterer, can naturally be increased or decreased without limit, but still undergoing a containable shape, since the SM’s UC is directed to an unavoidable conservation purpose. In short, one must think of some primary physical quantities as being constantly the same magnitude – during the experiment (real or gedanken) –, while secondary ones change through all increasing–decreasing stages, at will and simultaneously. Motivated by the challenge of the latter question, in 2005 Diago-Cisneros et al. [11] revisited this topic and have briefly demonstrated, though not exhaustively, that the standard UC, no matter how subtle it is, can be fulfilled in the more general context of mixed holes.

There are several studies, that focus free- and interacting-field theory, identifying two classes of transformations, one of which leads the SM to be shifted in energy [12]. Unlike the free-field formulation, annihilation–creation processes do not allow certain commutations relations, so other than Bogoliubov transformations are then required [12]. Instead of finding new symmetries to match these non-Bogoliubov permutations, the suggestion is to identify those ones, that do not alter neither the SM, nor its symmetry identities [12]. Typically, the interacting-field models, exhibit a factorization of the SM and its eigenvalues can be solved exactly [12]. More recently, deviations from parity-time (PT) invariance and non-Hermitian (NH) Hamiltonians, have attracted an intense awareness [13], [14], [15], [16], [17]. In that sense, some authors have derived certain generalized unitarity identities, fulfilled by scattering amplitudes for both real and complex PT-symmetric potentials [13], [15]. While others have discussed that, the momentum and total probability, are not preserved in NH scattering events [14], [17]. Worthy to attention, is the existence of asymmetrical scattering for NH potentials, with direct implications to the SM elements [16]. The usefulness of the symmetries for NH Hamiltonians is out of question, but their mathematical operationalization as well as basic notions required to be tailored [16]. There are theoretical proofs for the unitarity breakdown of the SM, corresponding to NH-Hamiltonian systems, but it still obeys a generalized UC of the form [17] SˆS=SSˆ=IN,which bounces back to the usual UC, if the single-component Hamiltonian recovers Hermiticity,1 being IN the (N×N) identity matrix.

Nowadays there are plenty of studies in underlying topics, for instance: Spintronics [18], [19], [20], [21], [22], [23], metamaterials with topologically-phase transitions [24], multi-mode quantum dots [25], multi-terminal waveguides [26] and full-transmission wave tunneling [27]; which are all intrinsically focused to tune charge–spin carriers passage across nano-structures or the transference of electromagnetic waves. This later is fundamental for current-technology data-transmission appliances and some authors have experimentally proved, that otherwise impenetrable barrier can be fully-tunneled, if the electromagnetic microwaves are biased by a PT-symmetric regime [27]. For time-domain response of multi-terminal gaussian waveguides as well as for systems with PT-symmetric microwaves tunneling, there are series of structured-symmetry identities, that the SM must satisfy for validating of solutions [26], [27]. Another major day-to-day task, is the coherent-wave propagation in nonlinear media, among them: nonlinear optics and disordered photonic lattices. These systems consider random potentials and are depicted with the aim of a non-linear Schrödinger Hamiltonian [28]. Whether the randomness of the potential is linear, or strictly nonlinear (induced by the wave itself), makes the difference at the moment to analyze the dynamic and the scattering as well [28]. Shortly, random-potential nonlinearity, induces a power-law decay for the coherent-wave diffusion [28]. Even so this peculiar behavior, together with the inherent difficulties to relate the coherent-wave spreading to the kinetic transmission–reflection coefficients, these last two scattering quantities, observe the standard statistic rule according to what, its summation is less or equal to 1 [28]. A new phenomenon have been reported and consist in the observation of a lower bound for spin quantum-diffusivity in strongly interacting regime [23]. As a consequence, complementary degrees of freedom arise, leading to new length scales to be introduced; thereby, separate conditions for unitary scattering are expected to hold [23]. The classification of topological phases, is widely given in terms of closed-system Hamiltonians [23], [24]. However, for open-system Hamiltonians [25], the SM formalism provides a more innate start up to address the UC pattern. Topologically nontrivial symmetry classes, have been proposed elsewhere [25], with regard to the scattering matrix S, for example: S=ΣySΣy,for the particle–hole symmetry class CII, while S=ΣySTΣy,holds for the time-reversal invariance (TRI) symmetry class. Hereinafter Σi=σiIN, stands for the enlarged (2N×2N) Pauli matrix σi (see Appendix A). Albeit not cited by the authors of the Ref. [25], the TRI condition (1.3) has been explicitly deduced in various previous references, using quite analog expressions of the form [29], [30], [31] Msv(z,z0)=ΣxMsv(z,z0)Σx1,for systems with spin-independent interactions; while for systems with spin-dependent interactions, this relation changes slightly and can be cast as Msv(z,z0)=KMsv(z,z0)K1,being the (2N×2N) matrix K, a time-reversal operator component (see Appendix D). Remarkably, all above posted constraints (1.2)–(1.5), chase a given structured pattern, which seems to be innate for multiple-particle beams. It is widely-known, that the state-vector transfer matrix (TM) Msv (see Appendix B), is a robust benchmark of the quantum transport, because it carries the whole input–output data of the system’s scattering process, in the same way as the SM does [11], [32]. As a matter of facts, Msv suitably connects incoming–outgoing amplitudes, throughout different available scattering channels, at a given energy [11], [32], [33]. From (1.4), (1.5), there followed clue aids in their further development, together with some of the open questions whose answers, were sought and pursued through the structured unitarity condition proved in Section 4. In perfect physical systems (without losses or gains) the particle-conservation principle, compels flux unitarity to the SM [29], [34]. Additionally, one can also comply with other requirements, enforced by the system’s symmetries. In that concern, three general classes can be established, and labeled as: orthogonal, symplectic and unitary [29], [34]. Thus if TRI collapses, the only outlasting rule is the SM unitarity [34]. Otherwise, the SM remains symmetric as long as the TRI holds [29], [34]. On the other hand, the existence of a strong spin–orbit-coupling, yields spin-rotation symmetry failing and then the SM may be restored by a quaternionic self-dual unitarity matrix [35]. For systems, far from being ideal – i.e., in the occurrence of absorption –, the SM turns sub-unitary and the UC would not be acceptable [35]. In this context, when a system undergoes energy-gain, the SM is often regarded as the reflection matrix solely, since it quantifies the portion of the incoming flux, which remains unabsorbed by the scattering system [35]. Of course, the smaller the loss-gain becomes, the closer the standard UC is approached. An alternative rewarding approximation has been introduced, in such cases whereas the loss-gain phenomenon cannot be disregarded; if so, the absorption–emission events could be devised inside the whole-energy system’s balance [35]. This last, is performed via the long-lasting Wigner–Smith time-delay (WSTD) matrix [36], [37] Q=iħSE(S),proposed almost at the quantum-theory dawn. However, it follows that much recently, the WSTD-matrix has attracted a noticeable concern once more, due to its tie with the energy’s loss-gain circumstances [35]. Ultimately, what is worthy to merit about the WSTD-matrix within the just mentioned ambience, is its capability to measure the “deficit of unitarity” of the backscattering matrix. As a bonus, it complies the temporal-durability information of the scattering phenomena [35], which undeniably is not the routine case in the framework of the TM–SM formalism. The key point of the WSTD-matrix technique, is the assumption of certain fictitious channels, carrying the absorption–emission data, tight-coupled to real ones, for describing the entire-system scattering process [35]. The Qii matrix elements, deliver scattering modes’ amplitudes regarding the injectances, while off-diagonal Qij matrix elements, bear no more and no less, than the widely-known and very useful phase-time delay [35]; whose paramount role in many applications and links with the experiment, have been vastly recognized (see Refs. [33], [35], [38], and references therein).

Unless is used an alternative modeling-workbench different from the MMST, one must bring into quotations the flux conservation principle for elastic quantum-scattering events, which in terms of the SM formalism yields the preservation of the UC in its standardized formulation [29] (for free-components and single-band) or at least via the pseudo-UC [2] together with its modification in the presence of subband-interaction [11] (for mixed-components and multiple-bands). However, particularly the later case, is not commonly the main focus of current theoretical studies, concerning the multicomponent scattering phenomenologies. Hence, the present approach that deals with mathematical aspects of a technique, that can be used in the theoretical study of multicomponent–multiband quantum transport, is meaningful. We are intended to call the attention of researchers that undertake investigations in the mentioned area and/or related, that there is a way out to uphold the standard UC, which no longer requires rather unphysical premises and/or numerical lightening, to retain its full rigor. For that, becomes mandatory to build, what we have named here as complete orthonormalized basis of super-spinors. For the sake of completeness, basic properties of the SM are reviewed to form a robust theoretical procedure, which is at the same time profuse in mathematical features and physical details, while remains understandable and no difficult to use. The present package of practical resources, is an attempt to compile within a single platform, all physical guidelines that really matter to be fulfilled, when dealing with multicomponent–multiband quantum-scattering events of spin–charge streams – propagating jointly and simultaneously –, described in the framework of the MMST.

The unitarity and symmetries properties in the MMST is a subtle problem, with several difficulties to overcome. We have developed in a fairly general fashion, an analysis of the unitarity and several analytic symmetry properties of the MMST, mainly by means of the SM workbench. Though undeniably not exhaustively detailed in every mathematical entity, we thought the present theoretical modeling as a useful workbench to deal with N-component synchronous mixed-particle quantum transport. Moreover, instead of completely rigorous mathematical formalisms, we choose less abstract – as possible –, practical tools to deal with unitarity preservation and symmetries in multicomponent–multiband systems. The focus has been put in problems well described by a matrix system of second-order differential equations, with first-derivative terms (responsible for the coupled interplay) included. Provided a consistent use of the present orthonormalization procedure, no flux conservation (FC) mismatches should arise. In this study, an exercise is devoted to the quantum transport of holes in Q2D multiband–multichannel physical systems, within the framework of our theoretical procedure. The numerical simulations were based on the 2-bands KL model Hamiltonian, which only considers the two highest in energy sub-bands of the valence band (VB). It is important to stress that, most of the properties, definitions and propositions (specially the structured UC), that will be presented, are valid for any physical layered-model (with N2), as the one sketched in Fig. 1, with minor changes, if any.

The quantum transport of electrons and holes in semiconductor heterostructures, are important subjects on Solid State Physics. In comparison to electrons in the conduction band (CB), the case of VB holes has been less studied due to mathematical difficulties of the models. Nevertheless, when both charge carriers are involved, as in opto-electronic devices, the response time threshold would be determined by holes due to its bigger effective mass. Additionally, in experiments with GaAs–AlAs superlattices, when the VB is in resonance and the CB is not, the tunneling of holes occurs more rapidly than the tunneling of electrons regardless the effective masses [39]. The actual models of single-component fluxes [2], [3], [6], [7], [9] are not sufficient to describe the quantum transport of mixed multi-component fluxes, due to the lack of enough physical information about the dispersion processes. We present an alternative approach, in which all the propagating modes are taking into account collectively and simultaneously.2 Then, the multi-component and multi-channel synchronous transmission of amplitudes, can be described without arbitrary assumptions. In the present modeling, both the formalism of the TM and the N-component SM (N2) are combined, and we have called it the multi-component scattering approach (MSA) [11], [40]. Recently. a Chebyshev-polynomial view of the MSA – named after PMSA –, was developed and successfully applied in n-cell multi-channel layered heterostructures, with better results regarding several measurements, in comparison to prior theoretical reports [33]. Many physical phenomena, can be understood as scattering problems and thus, they are susceptible to be studied within the framework of the SM, which relates the incoming flux with the outgoing one. It is well known, that the SM is unitary within the single-band effective mass approximation (EMA). Nevertheless, when the problem need to be described by a matrix differential system like (2.7), within the MMST for joint and simultaneous streams, then the fulfillment of this crucial property is not a simple task. As we will see later, the properties of the basis set of expanded linear-independent solutions (LI) of the physical system, play an important role to achieve the unitarity condition on the SM. In the specialized literature for multi-band problems [3], [6], [7], [8], [9], [10], it is standard to impose the orthonormalization in the configuration space, complementing in some cases with other numerical conditions. Though successful for several practical situations, as the ones commented above in Section (1.1), that treatment is insufficient whenever the mixing and the simultaneity of spin–charge propagating carriers are involved. This for example is the case of heavy holes (hh) and light holes (lh), with different total angular momentum projection, traversing throughout a layered heterostructure [see Fig. 1] with finite in-plane energy. The first mark of this relevant problem in the framework of the MMST, was given at 1995 by Sánchez and Proetto [2], who transformed the form of the standard UC for the SM, in the particular case of the (2 × 2) KL model for pure holes. Afterwards, Diago-Cisneros et al. [11], proposed an extension to that, for the case of mixed holes.

The remainder part of this paper, is held as follows. We begin our layout, by first presenting in Section 2 a theoretical outline on the quadratic multicomponent–multiband eigenvalue problem, in the framework of the EFA, whose solutions become the cornerstone of the current procedure. Next, Section 3 turns to a crucial point, which is a multicomponent–multiband charge–spin carriers flux – propagating jointly and simultaneously –, that tunnels across a layered physical heterostructure. The last, is concerned with tunneling quantities, that must be conserved. Section 4, is deserved to the target of this paper, i.e., we derive the physical and mathematical requirements, to satisfy the structured unitarity condition (SUC) for the MMST. After the presentation of our theoretical approach, we analyze in Section 5 – in pursuit of validation –, the convergence criteria to reduce the SUC within the EFA, to the standard UC within the EMA. Then, in Section 6, we are able to apply these methods and discuss the consequences over the symmetry relations of the SM. Following that, we do the same in Section 7, but considering now the tunneling amplitudes, which are exercised for mixed hhlh. Finally, we summarize our findings in Section 8, giving as well some directions for possible applications to concrete quantum scattering problems, similar to what we have examined in the previous section. We supply several appendices, to provide complementary physical details and mathematical tools, to a better understanding of how to use the SUC methodology.

Section snippets

Theoretical preliminaries: Quadratic multicomponent–multiband eigenvalue problem

Let us consider a position-dependent mass problem, described by a system of two or more linear ordinary second-order N-coupled differential equations. The correspondent eigenvalue equation, for a multicomponent–multiband system with translational symmetry in the [x,y] plane perpendicular to axis z [see Fig. 1], can be written in the matrix form as [41] ddzB(z)dF(z)dz+P(z)F(z)+Y(z)dF(z)dz+W(z)F(z)=ON, where B(z),P(z),Y(z) and W(z) fulfill formal Hermiticity B(z)=B(z)Y(z)=P(z)P(z)=±P(z)W(z)

Flux tunneling: Conservation principle

This section focuses to a fundamental issue, which is a multicomponent–multiband charge–spin carriers flux – propagating jointly and simultaneously –, that tunnels across a layered physical heterostructure. The last, is concerned with tunneling quantities, that must be conserved. In this sense, the analysis we present next, is a premise to Section 4.

Let us turn now to the central point i.e., the flux tunneling and its conservation law within the MMST. We start from the known expression of the

Structured unitarity of the scattering matrix

This section, is devoted to the target we pursue in this paper. We develop here, the physical method and the corresponding mathematical tools, to satisfy the requirement of the structured unitarity condition, in the framework of the MMST, whenever dealing with streams of multicomponent–multiband charge–spin carriers, propagating jointly and simultaneously throughout layered heterostructures, described via EFA.

Let us begin by recalling the standard definition of the scattering matrix S [41],

Definition 4.1

Convergence from EFA to EMA: Flux and unitarity

After the main components of our theoretical approach, have been derived above in Section 4, and to the benefit of its validation, we present now the convergence criteria to reduce the SUC within the EFA, to the standard UC within the EMA. Below, we will look at how the formulations within the MMST for flux equation and the structured unitarity requirement, converge to those of the EMA representation. The clue idea for such transformation, involves mainly working with the character of the N

Symmetry relations

In this section, the goal is to apply the methodology exposed in Section 3, together with Section 4 and discuss the consequences over the symmetry relations of the scattering matrix S, with regard of its blocks. In the specialized literature these relations are usually derived from (4.38), or from the conditions imposed by means of the TRI symmetry and the spatial inversion invariance (SII) over S [30]. Owing to brevity, we drop a thorough analysis of the discrete symmetries for the MMST, since

Tunneling amplitudes

In this section, we aim to examine the procedure derived in Section 4, but considering now the tunneling amplitudes, which are exercised below for mixed hhlh. We have already commented that the MSA comprises, in a common base, two approaches of the TM formalism, potentiating the advantages of each technique. As the scattering is the central point of our approximation, now we initiate its study, and we underline that in the specialized literature, there exist different views of the SM. Next we

Concluding remarks

We hope, there is less that remains darken to comprehend about SM universal-symmetry properties for the MMST, within the EFA of Hermitian–Hamiltonian models, as result of the current approach and its straightforward implications. It is not immediately recognized, what method is to be used, to quote scattering amplitudes and verify the – sometimes cumbersome –, flux conservation principle via the SM unitarity; of absolutely all kind of physical systems. Gradually, the questions to face grow, so

CRediT authorship contribution statement

L. Diago-Cisneros: Conceptualization, Data curation, Formal analysis, Investigation, Writing - original draft, Writing - review & editing. J.J. Flores-Godoy: Data curation, Writing - original draft, Writing - review & editing. G. Fernández-Anaya: Formal analysis, Methodology. H. Rodríguez-Coppola: Investigation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

One of the authors (L.D-C) thanks for the hospitality and support of Universidad Iberoamericana (FICSAC, DINVP), Mexico.

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