Consistent Parametrization of Multiband Hamiltonians: Mathematical Foundations and Data-Driven Applications in Nanoscience

Abstract

Bandstructure methods occupy a central place in the physics of nanostructures, and the multiband $\mathit{k}·\mathit{p}$ theory of Luttinger, Kohn, and Kane has served as one of the most widely used computational frameworks for modelling electronic states and energies in low-dimensional semiconductor systems for several decades. Despite its broad success, the theory harbours a fundamental mathematical difficulty that has been largely overlooked: the multiband Luttinger–Kohn Hamiltonians are non-elliptic partial differential operators for the overwhelming majority of common III–V and III-nitride crystalline materials, a fact that violates the axiomatic requirements of quantum mechanics and is the root cause of the long-standing problem of spurious solutions. In this paper, we derive ellipticity conditions rigorously for the $6\times 6$, $8\times 8$, and $14\times 14$ zinc-blende Hamiltonians, demonstrating that non-ellipticity affects a substantially larger class of materials than previously reported. We develop and justify a systematic parameter rescaling procedure for the $8\times 8$ Kane Hamiltonian and obtain admissible parameter sets for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaN, AlN, and InN. The inversion-asymmetry parameter B is shown to play an essential and previously unrecognized role in maintaining ellipticity, and it is used to optimize the bandstructure fit of the rescaled parameter sets. Analysis of several known $14\times 14$ models reveals structural sources of non-ellipticity, pointing to the need for a revision of perturbative assumptions regarding out-of-basis band contributions. The consistent parametrization framework developed here provides the rigorous mathematical foundation required by inverse design methodologies, AI-enhanced electronic structure calculations, and data-driven multifidelity approaches in nanoscience and nanotechnology.

1. Introduction

The collection of methods known as effective mass theory is one of the fundamental topics in the physics of nanostructures. The theory has been used to describe a wide variety of physical phenomena ranging from the formation of electronic bands in periodic solids to realistic field–matter interactions in modern semiconductor materials. Furthermore, the theory establishes a robust computational framework for simulating observable quantum-mechanical states and the corresponding energies in low-dimensional nanoscale systems, including quantum wells, wires, and dots.

In the original Luttinger–Kohn work [1], the authors applied perturbation theory to the Schrödinger equation with a smooth potential and constructed a representation for the valence band Hamiltonian near the high-symmetry point $\mathsf{\Gamma }$ of the first Brillouin zone in bulk zinc-blende (ZB) crystals with a large fundamental bandgap. Soon after that, Kane showed how to extend the model to narrow-gap materials such as InSb and Ge, where one can also account for the influence of the conduction bands [2]. One of the advantages of the $\mathit{k}·\mathit{p}$ theory is its universality and flexibility when it comes to the simulation of electronic transport phenomena in the presence of electromagnetic and/or thermoelastic fields [3,4]. Indeed, the theory has also been extended to cover wurtzite (WZ)-type crystals, materials with inclusions, heterostructure materials, and superlattices [5,6,7]. Another advantage of the effective mass theory is its flexibility, as one can easily adjust the models to include additional effects such as strain [8], piezoelectricity, magnetic field, and the respective nonlinear effects. These inbuilt multiscale effects are crucial for such applications as light-emitting diodes, lasers, high-precision sensors, photo-galvanic elements, hybrid bio-nanodevices, and many others [9].

For a wide range of applications the Luttinger–Kohn models have provided good, computationally feasible, and efficient approximations that agree well with experimental results [10,11]. However, for some types of crystal materials, band structure calculations based on such multiband models lead to solutions with unphysical properties [12,13], or the so-called spurious solutions [14,15,16,17,18,19].

As a result, there have been various attempts to explain the origin of the spurious solutions and to develop reliable procedures to avoid them [16,19,20,21]. These approaches rely on three main ideas: (a) to modify the original Hamiltonian and remove the terms responsible for the spurious solutions [15,22], (b) to change bandstructure parameters [14,20,23], and (c) to identify and exclude from simulations the physically inadequate observable states [11,24] or to change the numerical scheme to avoid such states altogether [21,25]. All of the mentioned approaches suffer from a common weakness—the lack of clear justification of the underlying theoretical procedure and thus the limitations in their applicability [16,17]. In this work we show that spurious solutions are a consequence of a more fundamental problem in applications of the effective mass theory: the non-ellipticity of the multiband Hamiltonian in the position representation.

The systematic study of connections between the structure of $6\times 6$, $8\times 8$, and $14\times 14$ Hamiltonians, their ellipticity in the position representation, and the material parameters for ZB crystals allows us to conclude that the widely adopted $\mathit{k}·\mathit{p}$ Hamiltonians turn out to be non-elliptic (hyperbolic) for a broad class of known material parameters. The solution space of the corresponding Schrödinger equation is wider than the space of norm-bounded observable states. The obtained stationary models are therefore susceptible to unphysical solutions even in the bulk case. Furthermore, due to non-ellipticity the time-dependent Schrödinger equation loses the fundamental property of probability current conservation [26].

These facts lead to an important assertion. Since any qualitative multiband approximation of the Schrödinger Hamiltonian must preserve its core physical properties—such as ellipticity and, as a consequence, semi-boundedness of the set of energy states—the lack of ellipticity for certain materials implies that the use of a multiband Hamiltonian for such materials is fundamentally incorrect. This results in substantial ramifications for the applications of effective-mass theory to bulk solids and heterostructures.

The whole procedure of obtaining the material parameters from experiment and their incorporation into mathematical models of effective mass theory needs to be revisited, taking into account the general ellipticity constraints derived in the present work. Before this is carried out, we propose here sets of elliptic Hamiltonian parameters for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaN, AlN, and InN, optimized in terms of the bandstructure fit. We also supply a parameter rescaling procedure used to obtain these sets from the available non-elliptic parameters.

The rest of the paper is organized as follows. Section 2 revises the basic properties of the Schrödinger equation and its $\mathit{k}·\mathit{p}$ approximations, placing the multiband effective-mass framework in its broader multiscale and multiphysics context. Section 3 develops the envelope-function and effective-mass formalism in bounded domains, discusses the role of boundary conditions, and establishes the connection between ellipticity of the governing operator and mathematical well-posedness. Section 4 examines the direct relevance of ellipticity to inverse design, data-driven, and AI-enhanced methodologies in nanoscience and nanotechnology, motivating the consistent parametrization framework developed in the subsequent sections. Section 5 presents a detailed ellipticity analysis of the classical $6\times 6$ Luttinger–Kohn Hamiltonian for ZB crystals; the ellipticity constraints take the form of explicit linear inequalities on the Luttinger parameters, and their evaluation across all gathered parameter sets from major material-data sources [27,28,29,30,31] establishes that only carbon possesses admissible parameters among all analyzed materials. Section 6 is devoted to the $8\times 8$ Kane Hamiltonian: it first carries out the ellipticity analysis in the absence of the inversion-asymmetry parameter B [32], then extends the ellipticity conditions to the case of non-zero B, and develops and justifies the systematic parameter rescaling procedure, yielding admissible parameter sets for all twelve studied III–V and III-nitride materials, together with a quantitative assessment of the bandstructure fit. Section 7 examines the ellipticity of two existing $14\times 14$ ZB Hamiltonian models [33,34], demonstrating that their non-ellipticity has structural origins and identifying the need for a revision of perturbative assumptions regarding out-of-basis band contributions. Section 8 places all results in a broader scientific landscape, discussing connections to boundary conditions and open quantum systems, nanoscale multiphysics, inverse problems, quantum control, and the AI-assisted inverse design of nanostructures and arguing that mathematically consistent forward models are an indispensable foundation for any reliable data-driven pipeline at the nanoscale. The conclusions are given in Section 9.

2. Overview of Luttinger–Kohn Bandstructure Theory and Hamiltonian Parametrization

The material properties (such as fundamental bandgaps and spin–orbit splitting energies) obtained experimentally represent real quantum phenomena, whereas models based on multiband Hamiltonians are meant to approximate them. As such, these models are derived from the stationary Schrödinger equation that represents averaged charge carrier interactions in the crystalline structure [35,36]. The derivation scheme involves the application of the Bloch wave representation and the projection of the original Hamiltonian onto the orthogonal subspace of the reduced phase space [1,37]. The projective part of the Hamiltonian is then adjusted with the help of perturbation theory [1,37,38] to account for the influence of outer bands. However, this last step lacks a rigorous theoretical foundation as it does not guarantee the convergence of the perturbative expansion [39,40]. The result is that the derived Hamiltonian, although directly based on the experimental parameters (Table 1, Table 2 and Table 5), represents a totally different mathematical object compared to its origin. The physical evidence to support this claim has already been reported for $\mathrm{GaAs}$ [41] and for $\mathrm{Si}$ [42]. While the practical utility of Luttinger–Kohn approximations remains well-established for many systems, their mathematical consistency is fundamentally compromised in certain crystalline classes where the underlying operator fails to satisfy ellipticity criteria, directly precipitating the observed unphysical solutions.

We start with the Schrödinger equation

$$H_0\psi \left(x\right)\equiv \left({\displaystyle \frac{{\mathbf{p}}^2}{2m_0}}+V\left(x\right)+H_{SO}\right)\psi \left(x\right)=E\psi \left(x\right),$$

(1)

where $\mathbf{p}=-i\hslash \nabla$ is the momentum operator of a charge carrier with mass $m_0$, $V\left(x\right)$ is the effective potential, and $x\in \mathsf{\Omega }\subset {\mathbb{R}}^3$. The unknown E stands for the eigenenergy of the system and the function $\psi \left(x\right)$ is the corresponding eigenstate. The Hamiltonian $H_{SO}$ accounts for relativistic effects of the spin.

In the finite domain $\mathsf{\Omega }$ we supplement (1) by the boundary conditions

$$\psi \left(x\right)=f\left(x\right),x\in \partial \mathsf{\Omega },$$

(2)

assuming that the combination of a given $\mathsf{\Omega }$ and $f\left(x\right)$ endows operator $H_0$ with all necessary properties postulated by the standard axiomatic approach to quantum mechanics [43]. The operator $H_0$ is an elliptic partial differential operator (PDO). It is symmetric over its domain of definition $D\left(H_0\right)\subset {\mathcal{H}}^2\left(\mathsf{\Omega }\right)$. Furthermore, we require that the boundary $\partial \mathsf{\Omega }$ is sufficiently smooth, so that a self-adjoint extension of $H_0$ exists and possesses the property of probability current conservation [43,44]. All mentioned assumptions can be satisfied in the bulk case [45], which will be our main focus throughout the work. The approach we develop in this paper is not limited to boundary conditions (2), and other general boundary formulations can be handled, both in the context of general mathematical treatments with other types of boundary conditions [46] and in the context of multiband/envelope-function models for nanostructures which may include Robin boundary conditions [47].

If $V\left(x\right)$ is a gently varying function over the unit cell [1], the original operator $H_0$ can be approximated by another operator H (using Bloch theorem), determined by the projection P of $H_0$ onto the considered eigenspace and Löwdin perturbation theory [1,37]. The last step in this approximation procedure accounts for the influence of the elements from the space (so-called class of states B) complement to the chosen eigenspace (so-called class of states A) by the formula

$$H=P{H}_0+{\displaystyle \sum _{i=1}^r}{\delta }^i{H}^{\left(i\right)}$$

(3)

up to the order r. Setting $\delta =1$ leads to the final approximation, under the assumption that the series (3) is convergent for such $\delta$. Despite the wide applicability of such approximations, the intrinsic ellipticity requirements for the realizations of H have not been explicitly verified in a systematic manner (see [1,2,20,37,48], as well as other works [14,15,18,22,25,33,34]). The only work known to us where this has been carried out, for the case of InAs, $\mathrm{GaAs}$, and ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$, is [16]. Hence, in what follows we analyze such requirements systematically for all common $6\times 6$, $8\times 8$, and $14\times 14$ ZB Hamiltonians.

3. Envelope Functions and Effective Mass Approximations in Finite Domains for Quantum Mechanical Applications

Before we proceed with our tasks, in this section, we provide the reader with a broader picture that motivates and contextualizes the work carried out in the subsequent sections.

All quantum systems, including low-dimensional nanostructures, are open in the sense that they are coupled to and interact with their environment in a complex way. Hence, formulations of boundary conditions for such systems are always approximate. In some cases, corresponding approximations can be derived in a rigorous manner if we know the surroundings of the low-dimensional nanostructures where the major part of the interactions takes place. This is the case, for example, for many self-assembled quantum dots (QDs), including InAs on GaAs, where the wetting layer is connected to their growth [49]. The properties of such quantum dots and strain distributions may be seriously affected by the wetting layer, which influences the QD states in ways that are important to account for in most applications of such low-dimensional systems, including quantum information processing and computation. In general, however, an adequate approximation of boundary conditions for such systems is not easy to deduce, given that the interactions between the system and its environment result in a complex dynamic process that cannot be accurately described solely by unitary operators. In contrast, once the (unrealistic) assumption that the system is closed is made, its evolution can be described by unitary operators acting on the system. This assumption is routinely invoked in the literature without recognition that one is dealing with an approximation. The problem with this consideration is noticeable in most schemes proposed to date for eliminating the phenomenon of unphysical spurious solutions for $\mathit{k}·\mathit{p}$ models, both those based on some forms of modification applied to the original theory and those based on modifications of $\mathit{k}·\mathit{p}$ bases. This is carried out without accounting for the approximate nature of boundary conditions and/or verifying the Hamiltonian’s ellipticity [14]. The model at hand is sensitive to both the structure of the Hamiltonian, as is well-known from early works on the subject (e.g., [50]), and boundary conditions. It is also well-known that small perturbations in the boundary conditions and their types may substantially influence the calculated solution to (1), especially in cases in which the involved operator $H_0$ is hyperbolic (non-elliptic), leading to the propagation of discontinuities inside $\mathsf{\Omega }$ from its boundary (e.g., [51] and references therein). As a result of this influence, parametrizations of the model, in particular those based on unitary transformations [14], will also be affected.

3.1. Multiscale Problem, Its Approximation with the Envelope-Function Approach, and Handling Boundary Conditions

The key to the $\mathit{k}·\mathit{p}$ theory, its modifications, and generalizations, culminating in modelling complex bandstructures of materials with quantum confinement and low-dimensional nanostructures, is a pioneering observation by Luttinger and Kohn of an intrinsic connection between the envelope-function and effective-mass (Schrödinger) models for a free particle in a magnetic field. Since then, this observation grew into the multiband $\mathit{k}·\mathit{p}$ theory that has fruitfully served scientists in different areas over several decades. All the resulting methods have their foundation in the envelope function of the wave function [52]. This is a long-range function compared to the lattice period. A physically transparent illustration of the envelope-function concept is provided by the original observations of Luttinger [38] and Luttinger and Kohn [1]: the equation of motion of a conduction-band electron under the influence of a magnetic field, when projected onto the slowly varying envelope function $F_c\left(\mathbf{r}\right)$, is identical in form to the effective-mass Schrödinger equation for a free particle of mass $m^{*}$ in the same field. This equivalence—that the envelope function satisfies the same equation as a particle in vacuum with the band-edge effective mass substituted for the free electron mass—is the foundational result from which the entire $\mathit{k}·\mathit{p}$ perturbation theory develops. It is also the reason why the envelope function is frequently identified with the wave function of the electron in the applied-field problem. The mathematical well-posedness of this identification, however, requires that the multiband effective-mass operator approximating the full Schrödinger Hamiltonian preserve its essential operator-theoretic properties, most critically its ellipticity, which is the subject of the present paper.

Therefore, the original problem is multiscale from the outset, because the envelope function and the material’s lattice periodic function vary on different length scales. This envelope-function approximation of the complete wave function has many convenient consequences; for example, quantum mechanical quantities expressed by matrix elements are calculated via integrals over the envelope function alone and not over the complete electronic wave function. Among other things, the developed $\mathit{k}·\mathit{p}$ methods may differ by the sets of basis functions chosen for constructing the model, but all of them use a perturbation approach as their starting point. Another convenient feature of the multiband $\mathit{k}·\mathit{p}$ theory is that other fields, such as mechanical, piezoelectric, and thermal, can be incorporated within the coupled Hamiltonian equations of the model, leading to a multiphysics problem. At the same time, the theory has well-known difficulties, particularly when dealing with semiconductor structures containing different chemical compositions, e.g., heterostructures. One difficulty is related to the formulation of correct boundary conditions for multiband Hamiltonian problems and the other to the violation of the underlying model’s ellipticity conditions, which is still largely overlooked. One of the main consequences of these difficulties, known for a long time, reveals itself in spurious non-physical solutions of the underlying mathematical models. In addressing this, attempts were made to formulate first-principles-based envelope-function approximations (sometimes referred to as “exact envelope-function theories”). One of the most common ways to obtain them is via envelope-function expansions, as was carried out in earlier works by M. Burt and others, eventually resulting in the Burt–Foreman (BF) theory that can be applied to heterostructures, allowing their modelling with multiband approximations. One of the distinctive features of this theory was its attempt to handle the above-mentioned difficulty of the $\mathit{k}·\mathit{p}$ Luttinger–Kohn (LK) models related to boundary conditions, which is clearly pronounced when dealing with heterostructures in the multiband case. Earlier, other approaches to address this issue were also reported [24]. A series of systematic comparisons were made based on the results obtained with LK and BF models (e.g., [53] and references therein). Different geometries were possible to handle by using spherical and cylindrical representations of Hamiltonians obtainable with appropriately chosen bases (e.g., the Vahala–Sercel basis [54]). Other formalisms of the $\mathit{k}·\mathit{p}$ theory, including those based on generalized plane-wave considerations, have also been discussed in the context of higher numbers of bands compared to the standard eight-band model [55]. For several decades, the multiband $\mathit{k}·\mathit{p}$ theory has been a crucial tool for successfully modelling a wide range of quantum systems, incorporating spin-splitting and strain, handling various nonperiodic problems, and accounting for excitonic, electromagnetic, and electromechanical effects. Moreover, it has also allowed the treatment of important nonlinear effects [56,57,58]. Advances in developing effective-mass models for obtaining electronic states and energies have been largely driven by the goal of achieving a reasonable balance between accuracy and computational cost, with the latter scaling favourably compared to ab initio atomistic-based methodologies. In recent years, the BF theory has further been extended to handle several important classes of nanostructures with various compositions, growth directions, and dopings (e.g., [59]).

3.2. Multiband Effective Mass Approximations, Hamiltonian-Based PDE Operators, and Ellipticity Conditions

An important observation regarding the coupled systems of PDEs that arise from the application of the multiband $\mathit{k}·\mathit{p}$ theory is that we have to deal with elliptic operators, and hence ellipticity conditions must be satisfied. While mathematically this is apparent, from a practical perspective, it was first noticed in the context of, and linked to, spurious solution problems appearing in the application of the $\mathit{k}·\mathit{p}$ envelope function theory (e.g., [16,19] and references therein). It was shown in [16] that the BF operator ordering (see Section 3) with experimental $\mathit{k}·\mathit{p}$ input parameters mildly violates ellipticity requirements, while symmetrization procedures would violate them strongly in the case of four-band and six-band Hamiltonians. The eight-band Hamiltonian was also analyzed and the authors concluded that the parametrization of multiband $\mathit{k}·\mathit{p}$ models must be reviewed; this was carried out systematically for the first time, for most common cases, in [19]. Some of the main sources of violation of ellipticity conditions, and hence of inadequate multiband Hamiltonian mathematical models with non-elliptic operators, were related to unsuitable renormalizations of the effective mass parameters and/or incorrect operator orderings. These initial works stimulated a deeper interest in the problem of spurious solutions from the point of view of numerical discretizations and software developments (e.g., [18,60,61,62]). The ellipticity requirements, from theoretical and computational perspectives, have been discussed in [17,63]. At the same time, even after these findings, many works have continued appearing that followed the ideas of [20], other operator ordering techniques, and their combinations with other methodologies that did not address the fundamental ellipticity issue (e.g., [64,65]). The latter category includes also works developed in the context of data-driven approaches. For example, Ref. [66] reported an interesting approach where subpixel smoothing replaces discontinuous parameters of the original models based on the eight-band effective-mass Luttinger–Kohn and Burt–Foreman Hamiltonians. Such smoothing is practically achieved either by eliminating the first-order perturbation terms in energy or by applying the Hellmann–Feynman theorem. Yet, ellipticity requirements for the resulting models have not been analyzed with the rigour they deserve. It should be noted that a number of earlier works addressed this problem with different levels of success (e.g., [16,18,19] and references therein), with extensions summarized in [67], largely driven by practical applications involving the appearance of spurious solutions. The important point to be made is that resorting to more bands in the $\mathit{k}·\mathit{p}$ theory cannot lead to reliable results due to a lack of mathematical foundation at the root of the problem; the analysis of the $14\times 14$ models in Section 7 provides concrete evidence for this conclusion.

3.3. Mesoscopic Description with Empirical Methods, Data Integration, and Atomistic-to-Continuum Approaches

From a broader perspective, the $\mathit{k}·\mathit{p}$ theory is part of a larger group of empirical/semi-empirical methods, together with the tight-binding and pseudopotential approaches. Each of these methods is based on different formalisms, uses many assumptions and approximations, and obtains some parameters from empirical data. Their development since the early days of computers was stimulated by the fact that empirical calculations are substantially faster than their ab initio atomistic counterparts. On the other hand, their results can potentially be wrong if “non-admissible” parameters are used to parametrize the method. By non-admissible parameters we understand those that may violate some of the fundamental properties, such as spectral properties, of the operators of the underlying mathematical models. In the case of the $\mathit{k}·\mathit{p}$ theory, such operators are based on elliptic-type PDE operators for Luttinger–Kohn and Kane models, where formally defined bandstructure parameters are closely related to the effective masses of the charges, describing coupling between the states.

In the empirical tight-binding approach, the parameters of the Hamiltonian are treated phenomenologically, with a phenomenological dependence fitted to reproduce the (bulk) band structure obtained from experiments or more refined atomistic calculations. Another approach from this group is the pseudopotential method, which is frequently based on plane-wave semi-empirical approximations (e.g., [68] and references therein). As has recently been pointed out, there is a connection between these two approaches once a fully discrete approximation is used (e.g., [69], with further discussion on discrete and continuous bases found in [70]).

The main advantage of all approaches from this group of empirical/semi-empirical methods is that they are simple and computationally efficient. The main challenge lies in their parametrization. The reason behind this challenge is that all methods from this group can provide only a mesoscopic level of description of the system, and therefore the quality of parametrization defines the quality of the results obtained with such methods. As system complexity increases and data from higher levels of computation, such as atomistic calculations, become more readily available, the integration of data must be carried out with due care, without violating the fundamental structure of the underlying mathematical models. For the $\mathit{k}·\mathit{p}$ theory, this means, among other things, that the ellipticity requirements must be preserved after the parametrization procedure is completed.

The above parametrization procedure must be consistent, given that the integration of data from the microscopic level within mesoscopic models can formally be viewed as an atomistic-to-continuum multiscale coupling approach [71,72,73] that combines the accuracy of atomistic models (and/or experiments) with the computational efficiency of the $\mathit{k}·\mathit{p}$ theory. As noted earlier, one of the major advantages of this approach is the ability to integrate other coupled fields within the continuum-like description based on multiband Hamiltonian models of this theory, including mechanical (strain), electromagnetic, electromechanical (piezoelectric), and thermal fields. For the representative examples systematically analyzed in Section 5, Section 6 and Section 7, we consider six-, eight-, and fourteen-band Hamiltonians with an underlying procedure that can be applied to any number of bands. While more bands have been employed in the literature (e.g., [74,75,76] and references therein), none of these works offered a consistent parametrization procedure. Increasing the number of bands without recognizing the limitations of the model used and ensuring its mathematical validity is a path that needs to be avoided. Hence, following [67], the present paper is the first publication in which the analysis of the most common Hamiltonians and materials has been carried out systematically with a fully consistent parametrization procedure.

Fundamentally different from the empirical and semi-empirical theories, such as $\mathit{k}·\mathit{p}$, are atomistic methodologies based on density functional theory (DFT) calculations or other ab initio (first-principles) techniques, as well as multiscale multifidelity approaches. The DFT methodology is an ab initio approach, since no parameter characterizing the Hamiltonian of the system is tuned to empirical data and simulation results are thus free from empirical bias. At a practical level, the term ab initio refers to the use of first-principles calculations in software based on DFT, such as Quantum ESPRESSO, Gaussian, or VASP, among others.

Such DFT-based approaches provided an important path to the development of multiscale modelling and subsequently to multifidelity methodologies. Already in the late 1990s, Nieminen demonstrated that DFT offers a robust and practical framework for parameter-free computation of ground-state electronic structures, bonding properties, and related quantities in condensed matter systems, while also pointing to its natural role as a microscopic foundation for multiscale coupling to coarser-grained descriptions [77]. This trajectory was further developed in subsequent work [78], which outlined hierarchical modelling strategies linking quantum-mechanical DFT calculations to atomistic and molecular dynamics simulations, and eventually to continuum-level finite-element descriptions—precisely the kind of cross-scale framework that motivates the parametrization approach presented here.

More recently, the multiscale perspective has evolved into what is now broadly referred to as multifidelity modelling, in which models of varying computational cost and accuracy are systematically combined to balance efficiency with predictive fidelity. In this context, uncertainty quantification becomes a necessary ingredient, since propagating information across scales and fidelity levels inevitably introduces approximations whose effect on the final predictions must be controlled. A promising and increasingly prominent bridge between these fidelity levels is provided by machine learning (ML): as demonstrated by Batra and Sankaranarayanan [79], ML algorithms and data-driven methods can effectively link high-fidelity ab initio molecular dynamics with more computationally tractable classical and coarse-grained simulations, opening new avenues for materials discovery and design. This connection between multifidelity scale-bridging and emerging artificial intelligence (AI) tools further underscores the relevance of consistent, well-parametrized mesoscopic models such as those developed in the present work, since they can serve as reliable intermediate-fidelity representations within such broader computational frameworks.

4. Ellipticity as a Foundation: Multiband Hamiltonians for Inverse Design and Data-Driven Applications

The mathematical well-posedness of multiband Hamiltonian models carries direct consequences that extend well beyond the immediate context of band structure calculations. As data-driven methodologies and inverse design frameworks become increasingly prevalent in materials science and nanostructure engineering, the consistency and ellipticity of the underlying forward models become essential prerequisites—not merely desirable properties. This section outlines the connection between the mathematical framework developed in this work and these broader computational paradigms.

One of the motivations for the present work has been the problem of inverse design of semiconductor heterostructures, where one seeks to determine structural parameters—compositions, layer thicknesses, and potential profiles—that yield prescribed electronic properties. Early contributions to this problem were framed in the language of inverse quantum scattering [80,81,82], where the goal is to reconstruct the potential from spectral data. This line of inquiry naturally connects to a broader mathematical theory of inverse problems [83,84,85,86,87], which provides the rigorous foundation for understanding existence, uniqueness, and stability of solutions. In quantum mechanical settings specifically, the role of boundary conditions is non-trivial [88,89] and the structure of the Hamiltonian fundamentally constrains what can be reliably recovered from data.

Inverse problems arising in quantum statistical physics have been discussed in the context of various model hierarchies, including quantum hydrodynamic models and related continuum approximations [90,91]. As the modelling level shifts from continuum descriptions to systems of coupled Schrödinger equations at the discrete level, the role of inverse problems becomes more pronounced, and the sensitivity to the mathematical properties of the operator—in particular, its ellipticity—increases accordingly. In a quite general mathematical framework, inverse problems also arise in the context of Hamilton–Jacobi equations [92], though the approximate nature of the Hamiltonian in such formulations is often left unexamined. Related deterministic and stochastic dynamics governed by hyperbolic Hamilton–Jacobi–Bellman-type equations have been studied in [93], where a sequence of PDE approximations based on the Steklov–Poincaré operator technique was proposed, and more recently in the context of quantum control problems [94]. The present work directly addresses this gap by ensuring that the multiband Hamiltonians considered here satisfy the necessary ellipticity conditions before they are deployed in any forward or inverse simulation context.

The key role of inverse problems in this context can be understood as follows. In quantum mechanics one seeks a relationship between observables and the potential [88,89], yet neither of the two is known precisely. The conventional formulation of the Schrödinger equation is therefore conditional on the quality of the corresponding approximations. As a representative example, it suffices to note that the potential can be influenced by classical effects—for instance, by the piezoelectric effect produced by coupled electromechanical fields, a situation directly relevant to the nanostructures considered in this paper. On the other hand, an appropriate framework for observables in quantum mechanics may be developed via non-self-adjoint operators [95], which introduces serious additional challenges. In the Schrödinger formulation, quantum mechanical operators are conventionally taken to be Hermitian, the principal consequence of which is that the eigenvalues are always real. The discussion of this assumption has a long history in the scientific community [96], and extensions beyond it have been proposed, with quantum effects observed even in classical systems [97]. Lyantse’s framework for non-self-adjoint singular differential operators [98] is especially relevant in this context, as it handles the spectral behaviour of open or dissipative systems where the standard self-adjoint spectral theorems fail. Ensuring ellipticity conditions in such settings allows one to guarantee invertibility of the operator’s principal symbol, thereby preventing the numerical instabilities common in non-rigorous $\mathit{k}·\mathit{p}$ expansions. More recent works have extended this analysis to Schrödinger operators with complex potentials [99] and to non-self-adjoint elliptic problems arising in dissipative and directed dynamics in biological and nanoscale networks [100]. Balancing the quality of approximations in this landscape represents a fundamental challenge, and data-driven modelling frameworks constitute one of the key elements of further progress. This is also reflected in the application of probabilistic approaches such as the Bayesian framework to inverse problems [101].

The methodologies to solve inverse problems have traditionally relied on the exploitation of additional information about the problem structure. Much of recent research is oriented towards combining specific knowledge of physics-based models with data, leading to various data-driven modelling frameworks in which machine learning and other artificial intelligence algorithms play an increasing role [102]. A particularly important development in this direction is the introduction of rigorous alternatives to purely neural-network-based approaches: the ODIL framework [103], for instance, replaces standard neural networks with grid-based discretizations, achieving significantly faster and more accurate solutions for inverse problems while maintaining a mathematically transparent structure. Complementing this, a Bayesian reformulation providing a precise notion of information density in inverse problems [104] offers a principled way to quantify which parts of an inverse solution are genuinely determined by data versus effectively guessed—a distinction of direct relevance when the forward model, such as a multiband $\mathit{k}·\mathit{p}$ Hamiltonian, carries its own structural approximations. Among many other applications, data-driven approaches are becoming increasingly important in bandgap engineering [105] and other problems in nanoscience and nanotechnology. In this context, the correct definition of physics-based models satisfying a mathematically coherent foundation becomes critical, as the subsequent sections of this paper demonstrate.

Mathematically, inverse design constitutes an inverse problem: given target properties, find the structure. The considerable recent interest in this formulation within nanophotonics [106,107,108,109,110] and more broadly across photonic and metasurface design [111] reflects the growing recognition that forward simulation tools alone are insufficient for high-performance device engineering. Inverse design has been applied to a wide range of physical systems, including quantum spin Hall-based phononic topological insulators [112], quantum nanophotonics [107], and nanoparticle design [113]. The computational methodology of inverse design has also been reviewed from a general materials perspective [114,115], and the connection to crystal structure generation via invertible representations has been explored [116]. Across these applications, a common thread is the need for reliable forward models: an inverse design procedure is only as trustworthy as the simulator it inverts.

In the specific context of semiconductor heterostructures, inverse scattering provides the natural mathematical framework for recovering potential profiles from spectral or scattering data [80,81,82]. Mathematical algorithms and procedures for the solution of such problems have been developed in a series of contributions [117,118,119], covering both the reconstruction of quantum dot potentials and the multiband $\mathit{k}·\mathit{p}$ Riccati equation for heterostructure transport. Beyond electron tunnelling applications, these approaches extend naturally to spin-dependent tunnelling in spintronics and to the emerging field of multichannel inverse scattering, where desired transmission or reflection properties serve as input for designing heterostructure potential profiles. Theoretically, inverse scattering problems have been extensively investigated [85,86], and spectral approaches have recently been extended to surface-localized transmission eigenmodes [120]—of direct relevance to nanostructure boundaries where $\mathit{k}·\mathit{p}$ theory is most sensitive to symmetry breaking. Further extensions cover non-self-adjoint Dirac operators with periodic potentials [121], which model narrow-gap semiconductors and topological insulators through a Riemann–Hilbert formulation capable of recovering potentials even when the spectrum acquires non-real components. The importance of boundary conditions in this setting cannot be overstated [122]: consistent boundary conditions for multiband envelope functions in semiconductor nanostructures were derived in [49], providing a rigorous foundation for the forward models upon which inverse design procedures must rest. Challenges associated with spurious solutions and the multiband effective mass theory applied to low-dimensional nanostructures were systematically analyzed in [18], further motivating the consistent parametrization approach developed here.

The advent of machine learning and deep learning has substantially transformed the inverse design landscape. In two-dimensional Dirac materials such as graphene, physics-constrained neural networks have been deployed to solve the inverse design problem for quantum dot structures, replacing the direct solution of the Dirac equation with a machine-learning surrogate and demonstrating that scattering efficiency can be designed to vary over two orders of magnitude through appropriate gate potential combinations [123]. Deep neural networks have been applied to the prediction of optical properties and free-form inverse design of metamaterials and metasurfaces [108], to inverse design in quantum nanophotonics via local density of states as a bridge between structure and quantum functional characteristics [107], and to multi-target inverse design of nanoparticles [113]. Inverse design combined with optimal and robust quantum control [94,124] further extends this landscape to dynamical problems, where the goal is to derive control fields producing desired quantum state transformations that are robust with respect to experimental imperfections. A comprehensive review of machine learning methods for inverse design in materials and composite structures, including a systematic classification of problem types—interpolation, extrapolation, multifidelity, and small-data regimes—is provided in [125], and the connection to generative models and deep learning for materials discovery is reviewed in [115].

This classification is directly relevant here, since multiband $\mathit{k}·\mathit{p}$ simulations naturally produce structured datasets of varying fidelity depending on the number of bands and the level of parametrization consistency.

Strain engineering represents a further dimension of the inverse design problem for nanostructures. Strain and lattice deformation in two-dimensional materials such as graphene produce a rich variety of novel electronic effects [126], and the interplay between mechanical, electronic, and electrochemical degrees of freedom—including in graphene electrode systems [127]—makes the forward modelling problem inherently multiphysical. Machine learning has also been applied to shape memory graphene nanoribbons with relevance to biomedical engineering [128], illustrating the breadth of nanoscience problems in which data-driven approaches are now being deployed. In all such cases, the quality of the inverse design output depends critically on the mathematical consistency of the underlying physical model.

The Bayesian approach to inverse problems—in which the goal is to characterise the posterior distribution of model parameters conditioned on measured data—has been intensively studied theoretically [101,129] and applied to inverse scattering with topological priors [130]. An additional challenge that arises in practice is that experimental observations of complex systems are often limited to a small fraction of the system under study; the resulting subsampling bias can lead to substantial errors when inferring collective properties, and mathematical tools to overcome it remain an active area of research [131]. In the nanoscience context, this challenge is compounded by the fact that the forward model itself may introduce approximation errors—for instance, through the use of a non-elliptic Hamiltonian—that are difficult to distinguish from measurement noise within a purely statistical framework.

The ellipticity conditions studied in this paper are also of relevance beyond the immediate scope of materials science. In the context of gauge-invariant quantization, ellipticity requirements have been identified as a fundamental constraint on the well-posedness of quantum field theories [132,133]: the fundamental laws of physics can be derived from the requirement of invariance under suitable classes of transformations alongside the need for a well-posed mathematical theory, and boundary operators in quantum field theory are subject to the same elliptic constraints that govern the multiband Hamiltonians studied here. While in materials science and elasticity [84,134] ellipticity conditions are well recognized as being of fundamental importance, their role in the quantum mechanical setting—and in particular in the $\mathit{k}·\mathit{p}$ theory of semiconductor nanostructures—has been largely overlooked. The results derived in the subsequent sections of this paper thus carry consequences that extend to the broader theoretical framework of low-dimensional quantum systems.

Having established this range of motivations—spanning inverse design, data-driven modelling, non-self-adjoint operator theory, Bayesian inference, strain-coupled multiphysics, and the mathematical foundations of quantum field theory—we are now in a position to turn our attention to the specific examples that demonstrate how to handle the parametrization of multiband Hamiltonians consistently, closing the gap that exists in the literature on this topic. Further applications are discussed in Section 8.

5. Six-Band Hamiltonian Analysis

This section is devoted to the ellipticity analysis of the classical $6\times 6$ Hamiltonian for $\mathrm{ZB}$ [1]-type crystals, demonstrating our approach in detail. In this work we use the Luttinger parameter notation, which is common in recent works on the subject. When necessary, the parameters will be converted from other parameter notations [36].

The Luttinger–Kohn (LK) Hamiltonian is defined as follows [37,38]:

$$H^{LK}=\left(\begin{array}{cccccc}P+Q& S& R& 0& -{\displaystyle \frac{1}{\sqrt{2}}}S& -\sqrt{2}R\\ S^{★}& P-Q& 0& R& \sqrt{2}Q& \sqrt{{\displaystyle \frac{3}{2}}}S\\ R^{★}& 0& P-Q& -S& \sqrt{{\displaystyle \frac{3}{2}}}{S}^{★}& -\sqrt{2}Q\\ 0& R^{★}& -S^{★}& P+Q& \sqrt{2}{R}^{★}& -{\displaystyle \frac{1}{\sqrt{2}}}{S}^{★}\\ -{\displaystyle \frac{1}{\sqrt{2}}}{S}^{★}& \sqrt{2}Q& \sqrt{{\displaystyle \frac{3}{2}}}S& \sqrt{2}R& P-{\Delta }_{SO}& 0\\ -\sqrt{2}{R}^{★}& \sqrt{{\displaystyle \frac{3}{2}}}{S}^{★}& \sqrt{2}Q& -{\displaystyle \frac{1}{\sqrt{2}}}S& 0& P-{\Delta }_{SO}\end{array}\right)$$

with

$$\begin{array}{c}P=-{\displaystyle \frac{{\hslash }^2}{2m_0}}{\gamma }_1{\mathbf{k}}^2,\hfill \\ Q=-{\displaystyle \frac{{\hslash }^2}{2m_0}}{\gamma }_2(k_x^2+k_y^2-k_z^2),\hfill \\ R=-{\displaystyle \frac{{\hslash }^2}{2m_0}}{\displaystyle \frac{-\sqrt{3}}{2}}\left[({\gamma }_2+{\gamma }_3)k_{-}^2+({\gamma }_2-{\gamma }_3)k_{+}^2\right],\hfill \\ S=-{\displaystyle \frac{{\hslash }^2}{2m_0}}(-2\sqrt{3}){\gamma }_3{k}_{-}{k}_z,\hfill \end{array}$$

where ${\mathbf{k}}^2=k_x^2+k_y^2+k_z^2$, $k_{\pm }=k_x\pm i{k}_y$. Each of P, Q, R, and S is a second-order differential operator in the position representation or, equivalently, a second-order polynomial in the momentum representation [1].

Our aim is to check the type (elliptic, hyperbolic, or essentially hyperbolic) of $H^{LK}$ as a partial differential operator (PDO), keeping in mind that the Schrödinger operator from Equation (1) is elliptic. Only the second-order derivative terms play the dominant role in the following analysis, because contributions from the terms linear in the components of $\mathbf{k}$, as well as from the potential, are bounded in the domain $D\left(H^{LK}\right)$ [135]. It follows that the results for more complicated physical models with potential contributions from additional fields (e.g., strain, magnetic field, etc.) remain the same as for the original $H^{LK}$ analyzed here. The fact that the Hamiltonian is a linear operator guarantees that this is also true for any other representation of $H^{LK}$ obtained by linear (basis) transformations.

In a more general sense, for any m-dimensional matrix PDO $H={\left\{h_{ij}\right\}}_{i,j=1}^m$, where each element $h_{ij}$ is a second-order PDO [136,137],

$$h_{ij}={\displaystyle \sum _{k,l=0}^n}{h}_{ij}^{kl}{\displaystyle \frac{{\partial }^2}{\partial x_k\partial x_l}},$$

(4)

the associated quadratic form (also known in the mathematical literature as a principal symbol) is defined by

$$G({\xi }_1,\dots ,{\xi }_{nm})=vM{v}^T,v=\left({\xi }_1,\dots ,{\xi }_{nm}\right).$$

(5)

Here M is an $mn\times mn$ matrix composed from the elements $h_{ij}^{kl}$. The $\mathit{k}·\mathit{p}$ multiband Hamiltonians are a special case of such matrix PDOs defined by their elements (4). They are symmetric as matrix PDOs, so the associated quadratic form G will have M with only real eigenvalues ${\lambda }_i$ (e.g., [16]).

Using these notations, the procedure of obtaining the ellipticity condition for H is reduced to the question about the sign of ${\lambda }_i$ for the associated M. More precisely, the matrix differential operator H will be elliptic if and only if all eigenvalues of the corresponding Hermitian matrix M have the same sign [135,136].

In general, it is a challenging task to calculate the eigenvalues of M explicitly, even for Hamiltonians with dimension as small as $3\times 3$; however, this has proved to be possible [40] for highly symmetric and sparse band-structure Hamiltonians such as $H^{LK}$ and several others considered here.

Taking into account the fact that the sequence of eigenenergies of $H_0$ is semi-bounded from below, for an approximation $H^{LK}$, we obtain

$${\lambda }_i<0,i=0,1,\dots ,nm.$$

(6)

Constraints (6) guarantee the ellipticity (in the strong sense [137]) of Hamiltonian H. The operator H possesses a self-adjoint extension in $D\left(H\right)\subset {\mathcal{H}}^{n+2}\left(\mathsf{\Omega }\right)$, $n>0$, provided that the boundary $\partial \mathsf{\Omega }$ is sufficiently smooth, as we have assumed in the previous section. Then it can be extended to a Hermitian operator by a closure in the norm ([136], p. 113), or via the Lax–Milgram procedure [135]. From the physical point of view, the smoothness characteristics of $D\left(H\right)$ fulfil the natural assumption of quantum theory that the state of the system must be a continuous function of spatial variables even when some coefficients of H have finite jumps, as is the case for heterostructures consisting of different materials [10,48].

The direct calculation via (5) for $H=H^{LK}$ ($n=3$, $m=6$) leads us to the $18\times 18$ matrix $M=M^{LK}$ with the following distinct eigenvalues:

$$\begin{array}{cc}{\lambda }_1=-E({\gamma }_1+4{\gamma }_2+6{\gamma }_3),& {\lambda }_2=E(3{\gamma }_3-{\gamma }_1-4{\gamma }_2),\\ {\lambda }_3=E(2{\gamma }_2-{\gamma }_1+3{\gamma }_3),& {\lambda }_4=E(2{\gamma }_2-{\gamma }_1-3{\gamma }_3),\end{array}$$

(7)

having multiplicity 2, 4, 6, and 6, respectively. The quantities ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ are the Luttinger material parameters mentioned above, and $E={\displaystyle \frac{{\hslash }^2}{2m_0}}$. By substituting Equation (7) into Constraints (6), we obtain the system of linear inequalities with respect to ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$. They describe the feasibility region ${\Lambda }_{-}$ in the space of ordered triplets ${\gamma }_1,{\gamma }_2,{\gamma }_3$. In this work we call a triplet of numbers a, b, c feasible if $(a,b,c)\in {\Lambda }_{-}$. More generally, we call a set of material parameters admissible if the Hamiltonian based on this set is an elliptic partial differential operator. The region ${\Lambda }_{-}$ comprises an unbounded pyramid in ${\mathbb{R}}^3$ (cf. Figure 1) with the following rays as its edges:

$$\begin{array}{cc}{l}_1=(8t,t,-2t),\hfill & l_2=(2t,t,0),\hfill \\ l_3=(3t,0,t),\hfill & l_4=(4t,-t,0),\hfill \end{array}$$

where $t\in [0,\infty )$, and the vertex is situated at the origin ${\gamma }_1={\gamma }_2={\gamma }_3=0$. The boundary of ${\Lambda }_{-}$ and the edges $l_1$, $l_2$, $l_3$, $l_4$ are illustrated in Figure 1.

When $({\gamma }_1,{\gamma }_2,{\gamma }_3)\in {\Lambda }_{-}$, the Hamiltonian $H^{LK}$ is an elliptic PDO with a semi-bounded sequence of eigenvalues. One can use similar reasoning to obtain the corresponding inequalities for other common representations of $H^{LK}$. The LK Hamiltonian in the A, B, C-parameter notation [1] was considered in [19].

To determine the ellipticity of $H^{LK}$, we gathered in Table 1 the material parameters ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaN, AlN, InN, and C, and we evaluated ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, ${\lambda }_4$ for the gathered triplets (Full accuracy data are available at https://doi.org/10.5281/zenodo.20435094). As it turns out, the eigenvalues ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_4$ are negative for all analyzed parameter sets. In that case, ellipticity is determined solely by the value of ${\lambda }_3$. We provide the values of ${\lambda }_3/E$ along with two other parameter-dependent quantities that are important for the ellipticity analysis of the present work. The first is the signed distance d from the parameter point $({\gamma }_1,{\gamma }_2,{\gamma }_3)$ to the boundary $\partial {\Lambda }_{-}$ of ${\Lambda }_{-}$: it is positive for points lying outside ${\Lambda }_{-}$ and negative for points inside. The second is the absolute ratio $\rho$ between the positive and negative values among ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, ${\lambda }_4$. While ellipticity remains a strict binary concept mathematically, the distance d provides a practical indicator of a $\mathit{k}·\mathit{p}$ model’s viability or the necessity of including more bands. Additionally, d can be utilized as a robust, unbiased constraint within optimization schemes when fitting Hamiltonian parameters to ab initio data [138,139], offering a tool similar to other measures of non-ellipticity [17,139].

Table 1. Material parameters for ZB-type crystals; d denotes the distance from the point $({\gamma }_1,{\gamma }_2,{\gamma }_3)$ to the feasibility region ${\Lambda }_{-}$.

#	El	${\mathit{\gamma }}_1$	${\mathit{\gamma }}_2$	${\mathit{\gamma }}_3$	${\mathit{\lambda }}_3/\mathit{E}$	d	$\mathit{\rho }$	#	El	${\mathit{\gamma }}_1$	${\mathit{\gamma }}_2$	${\mathit{\gamma }}_3$	${\mathit{\lambda }}_3/\mathit{E}$	d	$\mathit{\rho }$
1	GaAs [29]	6.980	2.060	2.930	5.930	1.585	0.117	26	InP eg	5.040	1.560	1.730	3.270	0.874	0.094
2	GaAs a	7.100	2.020	2.910	5.670	1.515	0.111	27	InP [30]	6.280	2.085	2.755	6.156	1.645	0.129
3	GaAs b	7.800	2.460	3.300	7.020	1.876	0.121	28	GaSb [29]	13.400	4.700	6.000	14.000	3.742	0.134
4	GaAs c	6.950	2.250	2.860	6.130	1.638	0.119	29	GaSb af	11.800	4.030	5.260	12.040	3.218	0.132
5	GaAs d	6.850	2.100	2.900	6.050	1.617	0.120	30	GaSb b	13.100	4.500	6.000	13.900	3.715	0.136
6	GaAs e	6.800	2.400	1.000	1.000	0.267	0.025	31	GaSb [28]	13.300	4.400	5.700	121.600	3.367	0.125
7	GaAs f	7.200	2.500	1.100	1.100	0.294	0.025	32	GaSb [30]	11.000	3.000	4.368	8.105	2.166	0.105
8	GaAs [30]	7.150	2.030	2.959	5.788	1.547	0.113	33	AlSb [29]	5.180	1.190	1.970	3.110	0.831	0.090
9	AlAs [29]	3.760	0.820	1.420	2.140	0.572	0.087	34	AlSb [28]	4.150	1.010	1.750	3.120	0.834	0.108
10	AlAs [28]	3.760	0.900	1.420	2.300	0.615	0.091	35	AlSb [30]	4.120	1.045	1.715	3.115	0.832	0.108
11	AlAs [30]	4.030	1.045	1.697	3.150	0.842	0.110	36	InSb [29]	34.800	15.500	16.500	45.700	12.214	0.154
12	InAs [29]	20.000	8.500	9.200	24.600	6.575	0.148	37	InSb a	36.130	16.240	17.340	48.370	12.927	0.156
13	InAs [28]	20.400	8.300	9.100	23.500	6.281	0.142	38	InSb b	36.410	15.940	16.990	46.440	12.412	0.151
14	InAs [28]	19.670	8.370	9.290	24.940	6.665	0.151	39	InSb c	35.080	15.640	16.910	46.930	12.543	0.156
15	InAs [30]	19.700	8.400	9.280	24.939	6.665	0.151	40	InSb [30]	35.000	15.700	16.821	46.864	12.525	0.156
16	GaP [29]	4.050	0.490	1.250	0.680	0.182	0.030	41	GaN [29]	2.670	0.750	1.100	2.130	0.569	0.111
17	GaP afh	4.200	0.980	1.660	2.740	0.732	0.096	43	GaN [28]	3.080	0.860	1.260	2.420	0.647	0.110
20	AlP [29]	3.350	0.710	1.230	1.760	0.470	0.081	44	GaN [140]	5.050	0.600	1.787	1.511	0.404	0.051
21	AlP afh	3.470	0.060	1.150	0.100	0.027	0.006	45	AlN [29]	1.920	0.470	0.850	1.570	0.420	0.115
23	InP [29]	5.080	1.600	2.100	4.420	1.181	0.118	47	InN i	3.720	1.260	1.630	3.690	0.986	0.129
24	InP a	5.150	0.940	1.620	1.590	0.425	0.052	50	C [28]	2.540	−0.100	0.606	−0.922	<0.000 0.00	0
25	InP bf	6.280	2.080	2.780	6.220	1.662	0.130	51	C [28]	3.610	0.090	1.101	−0.127	<0.000 0.00	0

[a] Set 1 from [28]; [b] Set 2 from [28]; [c] Set 3 from [28]; [d] Set 4 from [28]; [e] Set 5 from [28]; [f] obtained by extrapolation from 5-level model; [g] measured at $T=300\mathrm{K}$; [h] set from [30]; [i] the sets from [28,29].

Table 1. Material parameters for ZB-type crystals; d denotes the distance from the point $({\gamma }_1,{\gamma }_2,{\gamma }_3)$ to the feasibility region ${\Lambda }_{-}$.

#	El	${\mathit{\gamma }}_1$	${\mathit{\gamma }}_2$	${\mathit{\gamma }}_3$	${\mathit{\lambda }}_3/\mathit{E}$	d	$\mathit{\rho }$	#	El	${\mathit{\gamma }}_1$	${\mathit{\gamma }}_2$	${\mathit{\gamma }}_3$	${\mathit{\lambda }}_3/\mathit{E}$	d	$\mathit{\rho }$
1	GaAs [29]	6.980	2.060	2.930	5.930	1.585	0.117	26	InP eg	5.040	1.560	1.730	3.270	0.874	0.094
2	GaAs a	7.100	2.020	2.910	5.670	1.515	0.111	27	InP [30]	6.280	2.085	2.755	6.156	1.645	0.129
3	GaAs b	7.800	2.460	3.300	7.020	1.876	0.121	28	GaSb [29]	13.400	4.700	6.000	14.000	3.742	0.134
4	GaAs c	6.950	2.250	2.860	6.130	1.638	0.119	29	GaSb af	11.800	4.030	5.260	12.040	3.218	0.132
5	GaAs d	6.850	2.100	2.900	6.050	1.617	0.120	30	GaSb b	13.100	4.500	6.000	13.900	3.715	0.136
6	GaAs e	6.800	2.400	1.000	1.000	0.267	0.025	31	GaSb [28]	13.300	4.400	5.700	121.600	3.367	0.125
7	GaAs f	7.200	2.500	1.100	1.100	0.294	0.025	32	GaSb [30]	11.000	3.000	4.368	8.105	2.166	0.105
8	GaAs [30]	7.150	2.030	2.959	5.788	1.547	0.113	33	AlSb [29]	5.180	1.190	1.970	3.110	0.831	0.090
9	AlAs [29]	3.760	0.820	1.420	2.140	0.572	0.087	34	AlSb [28]	4.150	1.010	1.750	3.120	0.834	0.108
10	AlAs [28]	3.760	0.900	1.420	2.300	0.615	0.091	35	AlSb [30]	4.120	1.045	1.715	3.115	0.832	0.108
11	AlAs [30]	4.030	1.045	1.697	3.150	0.842	0.110	36	InSb [29]	34.800	15.500	16.500	45.700	12.214	0.154
12	InAs [29]	20.000	8.500	9.200	24.600	6.575	0.148	37	InSb a	36.130	16.240	17.340	48.370	12.927	0.156
13	InAs [28]	20.400	8.300	9.100	23.500	6.281	0.142	38	InSb b	36.410	15.940	16.990	46.440	12.412	0.151
14	InAs [28]	19.670	8.370	9.290	24.940	6.665	0.151	39	InSb c	35.080	15.640	16.910	46.930	12.543	0.156
15	InAs [30]	19.700	8.400	9.280	24.939	6.665	0.151	40	InSb [30]	35.000	15.700	16.821	46.864	12.525	0.156
16	GaP [29]	4.050	0.490	1.250	0.680	0.182	0.030	41	GaN [29]	2.670	0.750	1.100	2.130	0.569	0.111
17	GaP afh	4.200	0.980	1.660	2.740	0.732	0.096	43	GaN [28]	3.080	0.860	1.260	2.420	0.647	0.110
20	AlP [29]	3.350	0.710	1.230	1.760	0.470	0.081	44	GaN [140]	5.050	0.600	1.787	1.511	0.404	0.051
21	AlP afh	3.470	0.060	1.150	0.100	0.027	0.006	45	AlN [29]	1.920	0.470	0.850	1.570	0.420	0.115
23	InP [29]	5.080	1.600	2.100	4.420	1.181	0.118	47	InN i	3.720	1.260	1.630	3.690	0.986	0.129
24	InP a	5.150	0.940	1.620	1.590	0.425	0.052	50	C [28]	2.540	−0.100	0.606	−0.922	<0.000 0.00	0
25	InP bf	6.280	2.080	2.780	6.220	1.662	0.130	51	C [28]	3.610	0.090	1.101	−0.127	<0.000 0.00	0

[a] Set 1 from [28]; [b] Set 2 from [28]; [c] Set 3 from [28]; [d] Set 4 from [28]; [e] Set 5 from [28]; [f] obtained by extrapolation from 5-level model; [g] measured at $T=300\mathrm{K}$; [h] set from [30]; [i] the sets from [28,29].

From Table 1 one can observe that among all analyzed materials only carbon possesses admissible sets of parameters (the last two sets in Table 1, indicated by 0 in the $\rho$ column). All other gathered parameter sets yield ${\lambda }_3>0$. That is why the Hamiltonian $H^{LK}$ is not elliptic for those materials. It may even have no symmetric domain $D\left(H^{LK}\right)$, as opposed to the original partial differential operator $H_0$. Moreover, instead of the inclusion $D\left(H^{LK}\right)=D\left(H\right)\subset D\left(H_0\right)\subset {\mathcal{H}}^2\left(\mathsf{\Omega }\right)$, we have only

$$D\left(H^{LK}\right)\subset {\mathcal{H}}^1\left(\mathsf{\Omega }\right).$$

(8)

Then, discontinuous solutions of Equation (1) become theoretically possible. They will occur in models with jump-discontinuous coefficients [141], which is precisely the case for heterostructure materials. Additionally, the double degeneracy of ${\lambda }_3>0$ from Equation (7) means that for certain $\mathsf{\Omega }$ there exists a two-dimensional manifold within $D\left(H^{LK}\right)$ containing non-physical solutions to Equation (1). Consequently, the momentum operator from Equation (1) will be ill-defined for such eigenstates of $H^{LK}$ (by the embedding theorems ([136], p. 119)). All the above arguments allow us to conclude that $H^{LK}$ does not provide a sufficiently good approximation to $H_0$, preserving the type of the PDO, for the majority of available parameter sets.

Let us return to the admissible parameters from Table 1. For carbon, the parameter values were analyzed earlier [142], where it was noted that they do not agree well with Hall-effect experimental measurements. In our earlier work [19] we showed that experimentally consistent parameter sets for C are not admissible in terms of ellipticity. More generally, ensuring that the material parameters remain within the admissible domain is a fundamental structural necessity; violating ellipticity can trigger numerical instabilities and spurious oscillations when solving the governing equations on a coarse discretization mesh.

Concerning the remaining materials in Table 1, we observe a clear correlation between the average distance to ${\Lambda }_{-}$ per material and the size of the fundamental bandgap. Specifically, the parameter sets for the large-bandgap materials AlP, AlAs, GaP, GaN, and InP are noticeably close ($d<1$) to ${\Lambda }_{-}$. The closest set in terms of distance—set number 19 for AlP—can be made elliptic by direct adjustment. Other materials have a smaller bandgap and are consequently further from ${\Lambda }_{-}$. The average distance to ${\Lambda }_{-}$ for GaAs is around 1.7; for InAs, it exceeds 6. Indium antimonide (InSb) represents an extreme case, with a distance exceeding 12. This material has the smallest bandgap and the highest curvature of the light-hole bands. It is well established from experiment [2,143,144] that the valence-band-only LK model is insufficient for InSb-like materials, and the present analysis provides theoretical support for this fact. The ellipticity of higher-band $\mathit{k}·\mathit{p}$ models is considered in the following sections.

6. Eight-Band Hamiltonians

This section is devoted to the analysis of the Kane model [32,35]. The basis set of the $8\times 8$ Kane Hamiltonian [32] contains two more elements, $|S\uparrow \rangle$ and $|S\downarrow \rangle$, in addition to the basis set of $H^{LK}$. These new basis elements represent the influence of the innermost conduction band. Recall that the influence of the out-of-basis states is again treated perturbatively up to the second order by using the Löwdin perturbation theory. In this section we follow the exposition of [32], because it presents the most general description of the $8\times 8$ Kane Hamiltonian for zinc-blende crystals. Naturally, the results presented here remain valid for other versions [36,62,145] of the same Hamiltonian. Since our main focus is to check the ellipticity conditions, we shall drop the spin–orbit interaction part, labelled as $H_{\mathrm{s}.\mathrm{o}.}+H_{\mathrm{s}.\mathrm{o}.}^{\prime }$ in Equation (13) of [32]. This part of the Hamiltonian is linear in $\mathbf{k}$ and therefore does not affect the form of G (as noted earlier, only second-order terms in $\mathbf{k}$ are essential for the ellipticity analysis). Following Kane [35], we rewrite the resulting operator in the block-diagonal form

$$H^K=\left(\begin{array}{cc}{H}_{\uparrow }^K& 0\\ 0& H_{\downarrow }^K\end{array}\right),$$

where $H_{\uparrow }^K$ is the Kane $4\times 4$ interaction matrix [2], given by Equation (9) in the basis $|S\uparrow \rangle$, $|X\uparrow \rangle$, $|Y\uparrow \rangle$, $|Z\uparrow \rangle$ [32]. The matrix $H_{\downarrow }^K$, also defined by Equation (9), acts upon the spin-down part of the basis $|S\downarrow \rangle$, $|X\downarrow \rangle$, $|Y\downarrow \rangle$, $|Z\downarrow \rangle$:

$$H_{\uparrow }^K=\left(\begin{array}{cccc}{E}_c+\left(E+A^{\prime }\right){\mathbf{k}}^2& i{P}_0{k}_x+B{k}_y{k}_z& i{P}_0{k}_y+B{k}_x{k}_z& i{P}_0{k}_z+B{k}_x{k}_y\\ -i{P}_0{k}_x+B{k}_y{k}_z& E_v+M^{\prime }(k_y^2+k_z^2)+L^{\prime }{k}_x^2+E{\mathbf{k}}^2& N^{\prime }{k}_x{k}_y& N^{\prime }{k}_x{k}_z\\ -i{P}_0{k}_y+B{k}_x{k}_z& N^{\prime }{k}_x{k}_y& E_v+M^{\prime }(k_y^2+k_z^2)+L^{\prime }{k}_x^2+E{\mathbf{k}}^2& N^{\prime }{k}_y{k}_z\\ -i{P}_0{k}_z+B{k}_x{k}_y& N^{\prime }{k}_x{k}_z& N^{\prime }{k}_y{k}_z& E_v+M^{\prime }(k_y^2+k_z^2)+L^{\prime }{k}_x^2+E{\mathbf{k}}^2\end{array}\right).$$

(9)

Parameters $A^{\prime }$, B, $P_0$, $M^{\prime }$, $N^{\prime }$, $L^{\prime }$ are known as Kane parameters [35]; their definitions are provided in Table 4.2 of [36]. The quantities $E_c$ and $E_v$ are the conduction- and valence-band energies, respectively, and $E={\displaystyle \frac{{\hslash }^2}{2m_0}}$, as before. The parameter $A^{\prime }$ represents the influence of the higher bands on the conduction band included in the basis. The parameter $P_0$ accounts for the mixing of conduction and valence band states away from $\mathbf{k}=\mathbf{0}$. The quantity B is the so-called inversion-asymmetry parameter; it equals zero in materials with a centrosymmetric crystal structure, such as diamond [32]. By setting $B=0$ in Equation (9) we obtain a simplified version of Equation (9), known as the Bir–Pikus $4\times 4$ Hamiltonian. The general case of $H^K$ with $B\ne 0$ was studied by T. Bahder (Equation (15) in [32]). In practice, the aforementioned parameters are fitted to experimental data. It is frequently assumed in the literature that the simplified version of $H^K$ provides a sufficiently good description of the physical phenomena in ZB crystals with a face-centred lattice. As we demonstrate below, however, the Hamiltonian of such a simplified model is non-elliptic for all studied material parameter sets and is therefore prone to the appearance of spurious solutions. In particular, the parameter B cannot be set to zero for materials where $E+A^{\prime }<0$.

Similarly to the $6\times 6$ case, it is common to rewrite the Hamiltonian $H^K$ in the basis where its spin–orbit interaction part becomes diagonal. One additionally pre-multiplies the original basis functions to make interband matrix elements and other physically relevant quantities real-valued.

Direct calculation of eigenvalues for the quadratic form associated with $H_{\uparrow }^K$, described in detail for the Luttinger–Kohn case in the previous section, gives five distinct eigenvalues:

$$\begin{array}{cc}\hfill {\lambda }_1^{\prime }& =E+L^{\prime }+N^{\prime },\hfill \\ \hfill {\lambda }_2^{\prime }& =E+L^{\prime }-{\displaystyle \frac{1}{2}}{N}^{\prime },\hfill \\ \hfill {\lambda }_3^{\prime }& =E+M^{\prime }-{\displaystyle \frac{1}{2}}{N}^{\prime },\hfill \\ \hfill {\lambda }_4^{\prime }& =E+{\displaystyle \frac{2A^{\prime }+2M^{\prime }+N^{\prime }}{4}}-\sqrt{{\displaystyle \frac{{(2A^{\prime }-2M^{\prime }-N^{\prime })}^2}{16}}+{\displaystyle \frac{B^2}{2}}},\hfill \\ \hfill {\lambda }_5^{\prime }& =E+{\displaystyle \frac{2A^{\prime }+2M^{\prime }+N^{\prime }}{4}}+\sqrt{{\displaystyle \frac{{(2A^{\prime }-2M^{\prime }-N^{\prime })}^2}{16}}+{\displaystyle \frac{B^2}{2}}}.\hfill \end{array}$$

(10)

The presence of the second-order conduction-valence band mixing, characterized by the parameter B of the Kane Hamiltonian (9), is reflected in Equation (10) by the pair of eigenvalues ${\lambda }_4^{\prime }$, ${\lambda }_5^{\prime }$, which are both determined by the full set of principal Hamiltonian parameters N, M, L, $A^{\prime }$, B. Note that if one removes the mixing by setting $B=0$, this property disappears and the eigenvalues ${\lambda }_4^{\prime }$, ${\lambda }_5^{\prime }$ reduce to

$${\lambda }_{04}^{\prime }=E+M^{\prime }+{\displaystyle \frac{1}{2}}{N}^{\prime },{\lambda }_{05}^{\prime }=E+A^{\prime }.$$

6.1. Ellipticity Analysis in the Absence of Inversion Asymmetry

We analyze the set ${\lambda }_1^{\prime }$, ${\lambda }_2^{\prime }$, ${\lambda }_3^{\prime }$, ${\lambda }_{04}^{\prime }$, ${\lambda }_{05}^{\prime }$ associated with $B=0$ in $H^K$ first. The fifth eigenvalue ${\lambda }_{05}^{\prime }$ in Equation (10) is related to the conduction band of Equation (9), because its corresponding three-dimensional eigenspace (${\lambda }_{05}^{\prime }$ is triple degenerate) has only its first three coordinates as non-zero. Hence this eigenspace is orthogonal to the space associated with the valence bands, which are characterized by the eigenvalues ${\lambda }_1^{\prime }$, ${\lambda }_2^{\prime }$, ${\lambda }_3^{\prime }$, ${\lambda }_{04}^{\prime }$ with degeneracy 1, 2, 3, 3, respectively. The following system of inequalities ensures ellipticity of the $8\times 8$ ZB Hamiltonian [32] with $B=0$:

$$\left\{\begin{array}{cc}\hfill E+L^{\prime }+N^{\prime }& <0\\ \hfill E+L^{\prime }-{\displaystyle \frac{1}{2}}{N}^{\prime }& <0\\ \hfill E+M^{\prime }-{\displaystyle \frac{1}{2}}{N}^{\prime }& <0\\ \hfill E+M^{\prime }+{\displaystyle \frac{1}{2}}{N}^{\prime }& <0\\ \hfill E+A^{\prime }& >0.\end{array}\right.$$

(11)

As noted above, the eigenvalues ${\lambda }_1^{\prime }$, ${\lambda }_2^{\prime }$, ${\lambda }_3^{\prime }$, ${\lambda }_{04}^{\prime }$ are related to the valence band; hence the sign of the first four inequalities in System (11) is the same as in constraints (6). The opposite sign of the fifth inequality reflects its correspondence to the conduction band. Due to the electron–hole duality, the conduction-band eigenenergies must be semi-bounded from below. The presence of the summand E in System (11) is connected with the differences in the definition of the Dresselhaus parameters [146] and $L^{\prime }$, $M^{\prime }$ [32].

To compare the results for the $8\times 8$ ZB Hamiltonian with those previously obtained for the $6\times 6$ Hamiltonian, we define the dimensionless parameters ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, ${\gamma }_3^{\prime }$ analogously to the Luttinger triplet [36,62,143]:

$$\begin{array}{ccc}\hfill {\gamma }_1^{\prime }& =& -{\displaystyle \frac{1}{3}}(L^{\prime }+2M^{\prime }){\displaystyle \frac{2m_0}{{\hslash }^2}}-1,\hfill \\ \hfill {\gamma }_2^{\prime }& =& -{\displaystyle \frac{1}{6}}(L^{\prime }-M^{\prime }){\displaystyle \frac{2m_0}{{\hslash }^2}},\hfill \\ \hfill {\gamma }_3^{\prime }& =& -{\displaystyle \frac{1}{6}}{N}^{\prime }{\displaystyle \frac{2m_0}{{\hslash }^2}}.\hfill \end{array}$$

With these definitions, System (11) transforms to

$$\left\{\begin{array}{cc}\hfill -{\gamma }_1^{\prime }-4{\gamma }_2^{\prime }-6{\gamma }_3^{\prime }& <0\\ \hfill -{\gamma }_1^{\prime }-4{\gamma }_2^{\prime }+3{\gamma }_3^{\prime }& <0\\ \hfill -{\gamma }_1^{\prime }+2{\gamma }_2^{\prime }+3{\gamma }_3^{\prime }& <0\\ \hfill -{\gamma }_1^{\prime }+2{\gamma }_2^{\prime }-3{\gamma }_3^{\prime }& <0\\ \hfill 1+A& >0,\end{array}\right.$$

(12)

with $A=A^{\prime }/E$.

The modified and the original Luttinger parameters ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ are connected by the following relations [143]:

$${\gamma }_1^{\prime }={\gamma }_1-{\displaystyle \frac{E_p}{3E_g}},{\gamma }_2^{\prime }={\gamma }_2-{\displaystyle \frac{E_p}{6E_g}},{\gamma }_3^{\prime }={\gamma }_3-{\displaystyle \frac{E_p}{6E_g}},$$

(13)

where $E_p=P_0^2/E$ and $E_g=E_c-E_v$ is the fundamental bandgap energy, with $P_0$ being the Kane parameter from Equation (9).

As expected, four out of the five inequalities in System (12), which represent the ellipticity constraints for the valence-band part of $H^K$, have the same structure as those for the LK Hamiltonian (7). Hence, the feasibility region of the valence-band part of $H^K$ in the space of parameters $\left({\gamma }_1^{\prime },{\gamma }_2^{\prime },{\gamma }_3^{\prime }\right)$ coincides with the feasibility region ${\Lambda }_{-}$ of $H^{LK}$, depicted in Figure 1. It follows that if $\left({\gamma }_1^{\prime },{\gamma }_2^{\prime },{\gamma }_3^{\prime }\right)\in {\Lambda }_{-}$, the valence-band part of $H^K$ in the position representation is an elliptic partial differential operator. The transformation given by Equation (13) can be geometrically interpreted as a shift in the parameter space proportional to the vector ${\mathbf{v}}^{\prime }=(-2,-1,-1)$. This shift reduces the value of ${\lambda }_3^{\prime }$ and, as we shall soon see, brings the majority of the non-elliptic parameter triplets $({\gamma }_1,{\gamma }_2,{\gamma }_3)$ closer to the feasibility region.

The dimensionless parameter A appearing in the fifth inequality of System (12) is responsible for the coupling between the conduction band and other states. It is commonly assumed that the in-basis valence bands are the major contributors to A. The value of A is determined by matching it to the experimentally measured effective mass $m_c$ of the conduction band via the formula

$$A={\displaystyle \frac{m_0}{m_c}}-1-E_p{\displaystyle \frac{E_g+\frac{2}{3}\Delta }{E_g(E_g+\Delta )}}.$$

(14)

The magnitude of this parameter is clearly affected by the size of the bandgap $E_g$ and the spin–orbit splitting $\Delta$. The experimental nature of $m_c$ does not exclude other possible contributions to A. For this reason, we extended the collection of parameter sets from Table 1 by those stemming from the same sets of Luttinger parameters but with different values of bandgap energy $E_g$ (measured in different experimental setups). We also added a parameter set obtained by fitting the band structure of the $8\times 8$ Hamiltonian to the band structure calculated by ab initio methods [145]. All the data pertaining to the ellipticity analysis of the $8\times 8$ Hamiltonian are shown in Table 2. In each case, the modified Luttinger parameters ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, ${\gamma }_3^{\prime }$ were calculated using Equation (13) and the values of $P_0^2$, $E_g$ provided in the respective dataset source. For those sources in Table 2 for which $P_0^2$ is unavailable, we use the values compiled by Vurgaftman, Meyer, and Ram-Mohan [29].

The ellipticity conditions of $H^K$ are still violated for all materials listed in Table 1. The situation is, however, more complex than for the $6\times 6$ Hamiltonian. To illustrate this, we report in Table 2 the values of ${\lambda }_1^{\prime },\dots ,{\lambda }_{05}^{\prime }$ (divided by E), the distance d to the feasibility region from $\left({\gamma }_1^{\prime },{\gamma }_2^{\prime },{\gamma }_3^{\prime }\right)$, and the measure of non-ellipticity $\rho$, defined in the same way as for the $6\times 6$ Hamiltonian.

Overall, we confirm a reduction of the average distance to the feasibility region for all materials, especially for InAs and InSb. Furthermore, for several materials there exist parameter sets that are close to satisfying the full set of ellipticity constraints in System (12). These are the narrow-gap semiconductor InSb (sets #37 and #38 from Table 2) and, perhaps more surprisingly, the larger-bandgap materials InP, AlAs, and AlSb (sets #25, #10–#11, and #33, respectively). For these materials the corresponding parameter sets can be made elliptic by direct adjustment of ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, ${\gamma }_3^{\prime }$, A.

Certain parameter sets for AlP, AlSb, and InAs satisfy the ellipticity conditions for the valence-band part of the Hamiltonian but do not satisfy the conduction-band constraint (inequality 5 from System (12)). Among these, the sets #20 and #33 for AlP and AlSb reported in [29] differ sharply in the value of ${\gamma }_2$ from the other sets for these materials shown in Table 2. For AlP, this can be explained by the fact that, in the absence of direct experimental data, most of the material parameters were extrapolated from measurements on ternary alloys and from ab initio calculations, which introduces considerable uncertainty. The authors of [147] performed a readjustment of the Luttinger parameters to better match the experimental photoluminescence results on AlP/GaP heterostructures [147]. The set #33 for AlSb is based on available theoretical calculations from various sources and on the simultaneous fitting of ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ to the experimentally determined hole effective masses along the $\left[001\right]$, $\left[110\right]$, and $\left[111\right]$ directions [29].

For InAs, the size of ${\lambda }_1^{\prime },\dots ,{\lambda }_{04}^{\prime }$ indicates that the triplets $\left({\gamma }_1^{\prime },{\gamma }_2^{\prime },{\gamma }_3^{\prime }\right)$ of its parameter sets lie very close to the face of ${\Lambda }_{-}$ described by ${\lambda }_2^{\prime }=0$: two are inside (sets #12 and #13) and two are slightly outside (sets #14 and #15). The values of ${\lambda }_{05}^{\prime }$ for all four parameter sets are grouped near ${\lambda }_{05}^{\prime }=-4.8$, and thus the conduction-band part of the Hamiltonian is, again, far from being elliptic.

All material parameter sets for GaAs violate two out of four ellipticity conditions for the valence-band part, although parameter set #3 from Table 2 lies close to ${\Lambda }_{-}$ ($d=0.34$). However, it violates the ellipticity condition for the conduction-band part by the same margin of approximately $-2.9$ as do the other GaAs parameter sets; for set #8 the margin is slightly lower, ${\lambda }_{05}^{\prime }\approx -2.34$. This reduction of the margin should be attributed to the optimization procedure [145] used to obtain set #8. As far as ellipticity is concerned, this optimization procedure is no more effective than other parameter acquisition methods.

Table 2. Material data for the $8\times 8$ ZB Hamiltonian with $B=0$; d denotes the distance from the point $({\gamma }_1,{\gamma }_2,{\gamma }_3)$ to the feasibility region ${\Lambda }_{-}$. Positive values of ${\lambda }_1^{\prime }/E,\dots ,{\lambda }_{05}^{\prime }/E$ are typeset in bold.

#	El	${\mathit{E}}_{\mathit{p}}$	${\mathit{E}}_{\mathit{g}}$	${\mathbf{\Delta }}_{\mathit{SO}}$	${\mathit{A}}^{\prime }$	${\mathit{\gamma }}_1^{\prime }$	${\mathit{\gamma }}_2^{\prime }$	${\mathit{\gamma }}_3^{\prime }$	${\mathit{\lambda }}_1^{\prime }/\mathit{E}$	${\mathit{\lambda }}_2^{\prime }/\mathit{E}$	${\mathit{\lambda }}_3^{\prime }/\mathit{E}$	${\mathit{\lambda }}_{04}^{\prime }/\mathit{E}$	${\mathit{\lambda }}_{05}^{\prime }/\mathit{E}$	d	$\mathit{\rho }$	${\mathbf{\Delta }}_{05}^{\mathbf{min}}$	${\mathbf{\Delta }}_{05}^{\mathbf{max}}$
1	GaAs [29]	28.80	1.519	0.341	−3.88	0.66	−1.10	−0.23	5.12	3.05	−3.55	−2.17	−2.88	0.70	1.43	5.73	6.67
2	GaAs a	28.80	1.519	0.341	−3.88	0.78	−1.14	−0.25	5.28	3.03	−3.81	−2.31	−2.88	0.73	1.36	5.69	7.15
3	GaAs b	28.80	1.519	0.341	−3.88	1.48	−0.70	0.14	0.48	1.74	−2.46	−3.30	−2.88	0.34	0.39	3.27	4.62
4	GaAs c	28.80	1.519	0.341	−3.88	0.63	−0.91	−0.30	4.81	2.11	−3.35	−1.55	−2.88	0.66	1.41	3.96	6.29
5	GaAs d	28.80	1.519	0.341	−3.88	0.48	−0.76	−2.16	15.52	−3.92	−8.48	4.48	−2.88	2.13	1.61	8.41	15.92
6	GaAs e	28.80	1.519	0.341	−3.88	0.88	−0.66	−2.06	14.12	−4.42	−8.38	3.98	−2.88	1.94	1.41	7.47	15.74
7	GaAs [30]	28.80	1.519	0.346	−3.86	0.83	−1.13	−0.20	4.89	3.09	−3.69	−2.49	−2.86	0.67	1.29	5.79	6.93
8	GaAs [145]	25.47	1.519	0.341	−3.34	1.28	−0.73	0.03	1.46	1.73	−2.65	−2.83	−2.34	0.34	0.58	3.25	4.98
9	AlAs [29]	21.10	3.099	0.280	−0.95	1.49	−0.31	0.29	−1.94	0.62	−1.26	−2.98	0.05	0.12	0.10	1.21	2.46
10	AlAs [28]	21.10	3.099	0.280	−0.95	1.49	−0.23	0.29	−2.26	0.30	−1.10	−2.82	0.05	0.06	0.05	0.59	2.15
11	AlAs [30]	21.10	3.140	0.275	−0.87	1.79	−0.07	0.58	−4.95	0.24	−0.21	−3.67	0.13	0.05	0.03	0.47	0.41
12	InAs [29]	21.50	0.417	0.390	−5.79	2.81	−0.09	0.61	−6.08	−0.62	−1.18	−4.82	−4.79	−0.12	0.00	4.79	1.98
13	InAs [28]	21.50	0.417	0.390	−5.79	3.21	−0.29	0.51	−5.08	−0.52	−2.28	−5.32	−4.79	−0.10	0.00	4.79	3.82
14	InAs be	21.50	0.417	0.390	−5.79	2.48	−0.22	0.70	−5.77	0.50	−0.84	−5.02	−4.79	0.10	0.04	4.79	1.41
15	InAs [30]	21.50	0.418	0.380	−5.81	2.55	−0.17	0.71	−6.11	0.26	−0.78	−5.02	−4.81	0.05	0.02	4.81	1.31
16	GaP [29]	31.40	2.886	0.080	−4.09	0.42	−1.32	−0.56	8.25	3.18	−4.76	−1.38	−3.09	1.13	1.86	6.30	9.43
17	GaP ae	22.20	2.880	0.080	−0.95	1.63	−0.30	0.38	−2.66	0.71	−1.11	−3.37	0.05	0.14	0.10	1.42	2.21
18	GaP b	22.20	2.880	0.080	−0.95	1.48	−0.79	−0.03	1.91	1.59	−3.17	−2.97	0.05	0.31	0.57	3.16	6.29
19	GaP [30]	31.40	2.895	0.080	−4.06	0.58	−0.82	−0.15	3.63	2.24	−2.69	−1.77	−3.06	0.50	1.32	4.45	5.34
20	AlP [29]	17.70	3.630	0.070	−1.30	1.72	−0.10	0.42	−3.82	−0.06	−0.68	−3.18	−0.30	−0.01	0.00	0.30	1.35
21	AlP ae	17.70	3.630	0.070	−1.30	1.84	−0.75	0.34	−0.86	2.18	−2.34	−4.36	−0.30	0.43	0.29	4.33	4.65
22	AlP [30]	17.70	3.630	0.070	−1.30	1.84	−0.75	0.33	−0.85	2.14	−2.34	−4.34	−0.30	0.42	0.28	4.26	4.66
23	InP [29]	20.70	1.424	0.108	−2.62	0.23	−0.82	−0.32	5.00	2.09	−2.85	−0.91	−1.62	0.69	1.89	4.08	5.57
24	InP a	16.70	1.454	0.108	0.36	1.32	−0.97	−0.29	4.34	1.69	−4.15	−2.39	1.36	0.60	0.92	3.31	8.11
25	InP be	20.40	1.559	0.108	−1.22	1.92	−0.10	0.60	−5.11	0.28	−0.32	−3.92	−0.22	0.06	0.03	0.55	0.63
26	InP df	17.50	1.465	0.108	−0.09	1.06	−0.43	−0.26	2.23	−0.12	−2.70	−1.14	0.91	0.31	0.56	1.09	5.28
27	InP [30]	20.70	1.344	0.108	−3.44	1.15	−0.48	0.19	−0.35	1.35	−1.54	−2.68	−2.44	0.26	0.29	2.63	3.01
28	GaSb [29]	27.00	0.812	0.760	−3.25	2.32	−0.84	0.46	−1.70	2.43	−2.63	−5.37	−2.25	0.48	0.25	4.07	4.40
29	GaSb ae	22.40	0.812	0.725	1.39	2.60	−0.57	0.66	−4.31	1.65	−1.75	−5.73	2.39	0.32	0.14	2.79	2.95
30	GaSb b	26.10	0.812	0.725	−2.45	2.39	−0.86	0.64	−2.81	2.97	−2.17	−6.03	−1.45	0.58	0.27	5.01	3.66
31	GaSb [28]	25.00	0.812	0.725	−1.31	3.04	−0.73	0.57	−3.52	1.59	−2.79	−6.21	−0.31	0.31	0.14	2.69	4.71
32	GaSb [30]	27.00	0.750	0.756	−5.34	−1.00	−3.00	−1.63	22.79	8.11	−9.89	−0.11	−4.34	3.13	3.09	13.50	16.48
33	AlSb [29]	18.70	2.386	0.676	−1.12	2.57	−0.12	0.66	−6.09	−0.11	−0.81	−4.79	−0.12	−0.02	0.00	0.12	1.50
34	AlSb [28]	18.70	2.386	0.680	−1.12	1.55	−0.30	0.44	−3.02	0.98	−0.80	−3.46	−0.12	0.19	0.13	1.81	1.48
35	AlSb [30]	18.70	2.300	0.673	−1.37	1.41	−0.31	0.36	−2.33	0.91	−0.95	−3.11	−0.37	0.18	0.14	1.68	1.76
36	InSb [29]	23.30	0.235	0.810	−0.46	1.75	−1.02	−0.02	2.50	2.27	−3.87	−3.73	0.54	0.45	0.63	3.37	5.75
37	InSb a	23.20	0.235	0.803	−0.13	3.25	−0.20	0.90	−7.85	0.25	−0.95	−6.35	0.87	0.05	0.02	0.37	1.41
38	InSb b	23.42	0.237	0.803	−0.37	3.44	−0.54	0.51	−4.31	0.25	−3.01	−6.05	0.63	0.05	0.02	0.37	4.47
39	InSb c	23.10	0.235	0.803	0.12	2.31	−0.74	0.53	−2.50	2.24	−2.22	−5.38	1.12	0.44	0.22	3.32	3.29
40	InSb [30]	23.30	0.180	0.810	−21.07	−8.15	−5.87	−4.75	60.16	17.39	−17.86	10.66	−20.07	8.26	4.94	25.29	25.98
41	GaN [29]	25.00	3.299	0.017	−1.90	0.14	−0.51	−0.16	2.89	1.42	−1.66	−0.68	−0.90	0.40	1.84	2.83	3.31
42	GaN [148]	25.00	3.299	0.017	−1.90	0.17	−0.50	−0.15	2.76	1.38	−1.64	−0.72	−0.90	0.38	1.75	2.75	3.27
43	GaN [28]	25.00	3.299	0.017	−1.90	0.55	−0.40	0.00	1.08	1.05	−1.37	−1.35	−0.90	0.21	0.78	2.09	2.73
44	GaN [140]	25.00	3.440	0.017	−1.59	2.63	−0.61	0.58	−3.64	1.54	−2.12	−5.58	−0.59	0.30	0.14	3.08	4.24
45	GaN [31]	16.86	3.070	0.017	−1.30	0.68	−0.28	0.06	0.07	0.63	−1.05	−1.42	−0.30	0.12	0.28	1.25	2.09
46	AlN [29]	27.10	6.000	0.019	−1.51	0.41	−0.28	0.10	0.13	1.01	−0.69	−1.27	−0.51	0.20	0.58	2.01	1.38
47	AlN [148]	27.10	5.400	0.019	−2.01	0.25	−0.37	0.01	1.14	1.26	−0.94	−1.02	−1.01	0.25	1.22	2.52	1.88
48	AlN [31]	23.84	5.630	0.019	−2.07	0.04	−0.36	−0.13	2.15	1.01	−1.13	−0.37	−1.07	0.30	2.10	2.01	2.26
49	InN [29]	25.00	1.940	0.006	−5.54	−0.58	−0.89	−0.52	7.23	2.57	−2.75	0.35	−4.54	0.99	3.69	5.14	5.50
50	InN [148]	17.20	0.780	0.005	−8.72	−3.63	−2.42	−2.05	25.56	7.16	−7.34	4.94	−7.72	3.51	5.13	14.28	14.64
51	InN [31]	11.37	0.530	0.005	−3.87	−0.34	−0.77	−0.46	6.13	2.04	−2.56	0.17	−2.87	0.84	3.25	4.06	5.11

[a] Set 1 from [28]; [b] Set 2 from [28]; [c] Set 3 from [28]; [d] Set 5 from [28]; [e] obtained by extrapolation from a 5-level model; [f] measured at $T=300\mathrm{K}$.

Table 2. Material data for the $8\times 8$ ZB Hamiltonian with $B=0$; d denotes the distance from the point $({\gamma }_1,{\gamma }_2,{\gamma }_3)$ to the feasibility region ${\Lambda }_{-}$. Positive values of ${\lambda }_1^{\prime }/E,\dots ,{\lambda }_{05}^{\prime }/E$ are typeset in bold.

#	El	${\mathit{E}}_{\mathit{p}}$	${\mathit{E}}_{\mathit{g}}$	${\mathbf{\Delta }}_{\mathit{SO}}$	${\mathit{A}}^{\prime }$	${\mathit{\gamma }}_1^{\prime }$	${\mathit{\gamma }}_2^{\prime }$	${\mathit{\gamma }}_3^{\prime }$	${\mathit{\lambda }}_1^{\prime }/\mathit{E}$	${\mathit{\lambda }}_2^{\prime }/\mathit{E}$	${\mathit{\lambda }}_3^{\prime }/\mathit{E}$	${\mathit{\lambda }}_{04}^{\prime }/\mathit{E}$	${\mathit{\lambda }}_{05}^{\prime }/\mathit{E}$	d	$\mathit{\rho }$	${\mathbf{\Delta }}_{05}^{\mathbf{min}}$	${\mathbf{\Delta }}_{05}^{\mathbf{max}}$
1	GaAs [29]	28.80	1.519	0.341	−3.88	0.66	−1.10	−0.23	5.12	3.05	−3.55	−2.17	−2.88	0.70	1.43	5.73	6.67
2	GaAs a	28.80	1.519	0.341	−3.88	0.78	−1.14	−0.25	5.28	3.03	−3.81	−2.31	−2.88	0.73	1.36	5.69	7.15
3	GaAs b	28.80	1.519	0.341	−3.88	1.48	−0.70	0.14	0.48	1.74	−2.46	−3.30	−2.88	0.34	0.39	3.27	4.62
4	GaAs c	28.80	1.519	0.341	−3.88	0.63	−0.91	−0.30	4.81	2.11	−3.35	−1.55	−2.88	0.66	1.41	3.96	6.29
5	GaAs d	28.80	1.519	0.341	−3.88	0.48	−0.76	−2.16	15.52	−3.92	−8.48	4.48	−2.88	2.13	1.61	8.41	15.92
6	GaAs e	28.80	1.519	0.341	−3.88	0.88	−0.66	−2.06	14.12	−4.42	−8.38	3.98	−2.88	1.94	1.41	7.47	15.74
7	GaAs [30]	28.80	1.519	0.346	−3.86	0.83	−1.13	−0.20	4.89	3.09	−3.69	−2.49	−2.86	0.67	1.29	5.79	6.93
8	GaAs [145]	25.47	1.519	0.341	−3.34	1.28	−0.73	0.03	1.46	1.73	−2.65	−2.83	−2.34	0.34	0.58	3.25	4.98
9	AlAs [29]	21.10	3.099	0.280	−0.95	1.49	−0.31	0.29	−1.94	0.62	−1.26	−2.98	0.05	0.12	0.10	1.21	2.46
10	AlAs [28]	21.10	3.099	0.280	−0.95	1.49	−0.23	0.29	−2.26	0.30	−1.10	−2.82	0.05	0.06	0.05	0.59	2.15
11	AlAs [30]	21.10	3.140	0.275	−0.87	1.79	−0.07	0.58	−4.95	0.24	−0.21	−3.67	0.13	0.05	0.03	0.47	0.41
12	InAs [29]	21.50	0.417	0.390	−5.79	2.81	−0.09	0.61	−6.08	−0.62	−1.18	−4.82	−4.79	−0.12	0.00	4.79	1.98
13	InAs [28]	21.50	0.417	0.390	−5.79	3.21	−0.29	0.51	−5.08	−0.52	−2.28	−5.32	−4.79	−0.10	0.00	4.79	3.82
14	InAs be	21.50	0.417	0.390	−5.79	2.48	−0.22	0.70	−5.77	0.50	−0.84	−5.02	−4.79	0.10	0.04	4.79	1.41
15	InAs [30]	21.50	0.418	0.380	−5.81	2.55	−0.17	0.71	−6.11	0.26	−0.78	−5.02	−4.81	0.05	0.02	4.81	1.31
16	GaP [29]	31.40	2.886	0.080	−4.09	0.42	−1.32	−0.56	8.25	3.18	−4.76	−1.38	−3.09	1.13	1.86	6.30	9.43
17	GaP ae	22.20	2.880	0.080	−0.95	1.63	−0.30	0.38	−2.66	0.71	−1.11	−3.37	0.05	0.14	0.10	1.42	2.21
18	GaP b	22.20	2.880	0.080	−0.95	1.48	−0.79	−0.03	1.91	1.59	−3.17	−2.97	0.05	0.31	0.57	3.16	6.29
19	GaP [30]	31.40	2.895	0.080	−4.06	0.58	−0.82	−0.15	3.63	2.24	−2.69	−1.77	−3.06	0.50	1.32	4.45	5.34
20	AlP [29]	17.70	3.630	0.070	−1.30	1.72	−0.10	0.42	−3.82	−0.06	−0.68	−3.18	−0.30	−0.01	0.00	0.30	1.35
21	AlP ae	17.70	3.630	0.070	−1.30	1.84	−0.75	0.34	−0.86	2.18	−2.34	−4.36	−0.30	0.43	0.29	4.33	4.65
22	AlP [30]	17.70	3.630	0.070	−1.30	1.84	−0.75	0.33	−0.85	2.14	−2.34	−4.34	−0.30	0.42	0.28	4.26	4.66
23	InP [29]	20.70	1.424	0.108	−2.62	0.23	−0.82	−0.32	5.00	2.09	−2.85	−0.91	−1.62	0.69	1.89	4.08	5.57
24	InP a	16.70	1.454	0.108	0.36	1.32	−0.97	−0.29	4.34	1.69	−4.15	−2.39	1.36	0.60	0.92	3.31	8.11
25	InP be	20.40	1.559	0.108	−1.22	1.92	−0.10	0.60	−5.11	0.28	−0.32	−3.92	−0.22	0.06	0.03	0.55	0.63
26	InP df	17.50	1.465	0.108	−0.09	1.06	−0.43	−0.26	2.23	−0.12	−2.70	−1.14	0.91	0.31	0.56	1.09	5.28
27	InP [30]	20.70	1.344	0.108	−3.44	1.15	−0.48	0.19	−0.35	1.35	−1.54	−2.68	−2.44	0.26	0.29	2.63	3.01
28	GaSb [29]	27.00	0.812	0.760	−3.25	2.32	−0.84	0.46	−1.70	2.43	−2.63	−5.37	−2.25	0.48	0.25	4.07	4.40
29	GaSb ae	22.40	0.812	0.725	1.39	2.60	−0.57	0.66	−4.31	1.65	−1.75	−5.73	2.39	0.32	0.14	2.79	2.95
30	GaSb b	26.10	0.812	0.725	−2.45	2.39	−0.86	0.64	−2.81	2.97	−2.17	−6.03	−1.45	0.58	0.27	5.01	3.66
31	GaSb [28]	25.00	0.812	0.725	−1.31	3.04	−0.73	0.57	−3.52	1.59	−2.79	−6.21	−0.31	0.31	0.14	2.69	4.71
32	GaSb [30]	27.00	0.750	0.756	−5.34	−1.00	−3.00	−1.63	22.79	8.11	−9.89	−0.11	−4.34	3.13	3.09	13.50	16.48
33	AlSb [29]	18.70	2.386	0.676	−1.12	2.57	−0.12	0.66	−6.09	−0.11	−0.81	−4.79	−0.12	−0.02	0.00	0.12	1.50
34	AlSb [28]	18.70	2.386	0.680	−1.12	1.55	−0.30	0.44	−3.02	0.98	−0.80	−3.46	−0.12	0.19	0.13	1.81	1.48
35	AlSb [30]	18.70	2.300	0.673	−1.37	1.41	−0.31	0.36	−2.33	0.91	−0.95	−3.11	−0.37	0.18	0.14	1.68	1.76
36	InSb [29]	23.30	0.235	0.810	−0.46	1.75	−1.02	−0.02	2.50	2.27	−3.87	−3.73	0.54	0.45	0.63	3.37	5.75
37	InSb a	23.20	0.235	0.803	−0.13	3.25	−0.20	0.90	−7.85	0.25	−0.95	−6.35	0.87	0.05	0.02	0.37	1.41
38	InSb b	23.42	0.237	0.803	−0.37	3.44	−0.54	0.51	−4.31	0.25	−3.01	−6.05	0.63	0.05	0.02	0.37	4.47
39	InSb c	23.10	0.235	0.803	0.12	2.31	−0.74	0.53	−2.50	2.24	−2.22	−5.38	1.12	0.44	0.22	3.32	3.29
40	InSb [30]	23.30	0.180	0.810	−21.07	−8.15	−5.87	−4.75	60.16	17.39	−17.86	10.66	−20.07	8.26	4.94	25.29	25.98
41	GaN [29]	25.00	3.299	0.017	−1.90	0.14	−0.51	−0.16	2.89	1.42	−1.66	−0.68	−0.90	0.40	1.84	2.83	3.31
42	GaN [148]	25.00	3.299	0.017	−1.90	0.17	−0.50	−0.15	2.76	1.38	−1.64	−0.72	−0.90	0.38	1.75	2.75	3.27
43	GaN [28]	25.00	3.299	0.017	−1.90	0.55	−0.40	0.00	1.08	1.05	−1.37	−1.35	−0.90	0.21	0.78	2.09	2.73
44	GaN [140]	25.00	3.440	0.017	−1.59	2.63	−0.61	0.58	−3.64	1.54	−2.12	−5.58	−0.59	0.30	0.14	3.08	4.24
45	GaN [31]	16.86	3.070	0.017	−1.30	0.68	−0.28	0.06	0.07	0.63	−1.05	−1.42	−0.30	0.12	0.28	1.25	2.09
46	AlN [29]	27.10	6.000	0.019	−1.51	0.41	−0.28	0.10	0.13	1.01	−0.69	−1.27	−0.51	0.20	0.58	2.01	1.38
47	AlN [148]	27.10	5.400	0.019	−2.01	0.25	−0.37	0.01	1.14	1.26	−0.94	−1.02	−1.01	0.25	1.22	2.52	1.88
48	AlN [31]	23.84	5.630	0.019	−2.07	0.04	−0.36	−0.13	2.15	1.01	−1.13	−0.37	−1.07	0.30	2.10	2.01	2.26
49	InN [29]	25.00	1.940	0.006	−5.54	−0.58	−0.89	−0.52	7.23	2.57	−2.75	0.35	−4.54	0.99	3.69	5.14	5.50
50	InN [148]	17.20	0.780	0.005	−8.72	−3.63	−2.42	−2.05	25.56	7.16	−7.34	4.94	−7.72	3.51	5.13	14.28	14.64
51	InN [31]	11.37	0.530	0.005	−3.87	−0.34	−0.77	−0.46	6.13	2.04	−2.56	0.17	−2.87	0.84	3.25	4.06	5.11

[a] Set 1 from [28]; [b] Set 2 from [28]; [c] Set 3 from [28]; [d] Set 5 from [28]; [e] obtained by extrapolation from a 5-level model; [f] measured at $T=300\mathrm{K}$.

The sets for GaN and AlN fail the first two valence-band constraints from System (12), as do most other material parameter sets. One exception is set #44 for GaN [140], in which spherical symmetry of the heavy-hole and light-hole bands is assumed. This assumption leads to larger values of ${\gamma }_1$, ${\gamma }_3$ and a smaller ${\gamma }_2$, and, as a consequence, to a more than fivefold reduction in the ratio $\rho$ between positive and negative eigenvalues.

An even more severe situation is observed for InN. The conduction-band eigenvalue ${\lambda }_{05}^{\prime }$ is noticeably below zero for all three available datasets (${\lambda }_{05}^{\prime }\approx -4.54$, $-7.72$, $-2.87$ for sets #49, #50, #51, respectively). In addition, three out of four valence-band conditions are violated. It is important to note that the recently obtained set #51 features roughly twice as large values of ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$ and noticeably smaller $E_p$ and $E_g$ than two other sets #49 and #50 reported earlier [29,148]. As demonstrated in [31], set #51 recovers the band structure better than the other two sets for InN. In terms of ellipticity, set #51 also results in a distance to ${\Lambda }_{-}$ ($d\approx 0.84$) that is smaller than others and a smaller band eigenvalue ${\lambda }_{04}^{\prime }\approx 1.17$.

For GaP and GaSb the data are inconclusive, as the sign and magnitude of the eigenvalues in Equation (10) depend on the choice of material parameter dataset. Sets #17 and #29 from Landolt–Börnstein [28], based on the earlier data of Lawaetz [27], are the most favourable in terms of ellipticity: $d\approx 0.14$, ${\lambda }_{05}^{\prime }\approx 0.05$ for GaP; $d\approx 0.32$, ${\lambda }_{05}^{\prime }\approx 2.39$ for GaSb. As a summary of the above analysis, we visualize in Figure 2 the values of ${\lambda }_{05}^{\prime }$ and d for the selected parameter sets with the material-wise minimal distance to ${\Lambda }_{-}$. In this figure, the ellipticity of the Hamiltonian corresponds to the region (shaded in grey) where ${\lambda }_{05}^{\prime }>0$ and $d<0$ simultaneously.

It is worth noting that roughly $76\%$ of parameter sets for the analyzed materials fail the conduction-band constraint ${\lambda }_{05}^{\prime }>0$. This group includes all datasets for GaAs, InAs, AlP, AlSb, GaN, AlN, and InN, all of which are important for applications. The positive (negative) sign of ${\lambda }_{05}^{\prime }$ is responsible for a positive (negative) gain in energy as we proceed from one conduction-band eigenvalue of the Hamiltonian to the next in the position representation. In the momentum representation, the sign and magnitude of the eigenvalue are responsible for the upward (downward) curvature of the conduction band. In addition to the issues caused by the non-ellipticity of the valence-band part discussed in Section 5, the condition ${\lambda }_{05}^{\prime }>0$ being violated entails the existence of conduction-band-related eigenstates of $H^K$ with energies in the bandgap or in the regions associated with the valence bands [14,16,17,62]. This clearly poses a serious problem in applications.

A rescaling procedure was introduced by Foreman [20] (see also the work of Birner [62]) and has been adopted [149,150] as a means of making ${\lambda }_{05}^{\prime }$ positive and thereby avoiding the above-described type of spurious solutions. The idea of the procedure is to adjust the momentum matrix element $E_p$ so that ${\lambda }_{05}^{\prime }$ is no longer negative. Assuming a target value ${\lambda }_{05}^{\prime }=a$, and using the definition of ${\lambda }_{05}^{\prime }$ together with Equation (14), we obtain the following [62]:

$$E_p=\left({\displaystyle \frac{m_0}{m_c}}-a\right){\displaystyle \frac{E_g(E_g+{\Delta }_{SO})}{E_g+\frac{2}{3}{\Delta }_{SO}}}.$$

(15)

Two values, $a=0$ and $a=1$, are considered in the literature as targets for rescaling. The new value of $E_p$ necessarily affects the values of the modified Luttinger parameters ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, ${\gamma }_3^{\prime }$ defined by Equation (13). To determine how this procedure impacts the ellipticity of the entire $8\times 8$ ZB Hamiltonian $H^K$, one rewrites the eigenvalues ${\lambda }_1^{\prime }$, ..., ${\lambda }_{04}^{\prime }$ as functions of $E_p$ and $E_g$:

$${\lambda }_1^{\prime }={\lambda }_1+2E{\displaystyle \frac{E_p}{E_g}},{\lambda }_{2/04}^{\prime }={\lambda }_{2/4}+{\displaystyle \frac{E}{2}}{\displaystyle \frac{E_p}{E_g}},{\lambda }_3^{\prime }={\lambda }_3-{\displaystyle \frac{E}{2}}{\displaystyle \frac{E_p}{E_g}}.$$

Combining these representations with Equation (15), we obtain a reformulation of the ellipticity constraints (12) for the valence-band part of $H^K$:

$$\begin{array}{c}\hfill {\lambda }_1+2E_m-{\displaystyle \frac{2a}{E_r}}<0,{\lambda }_2+{\displaystyle \frac{1}{2}}{E}_m-{\displaystyle \frac{a}{2E_r}}<0,\\ \hfill {\lambda }_3-{\displaystyle \frac{1}{2}}{E}_m+{\displaystyle \frac{a}{2E_r}}<0,{\lambda }_4+{\displaystyle \frac{1}{2}}{E}_m-{\displaystyle \frac{a}{2E_r}}<0,\end{array}$$

(16)

where $E_m={\displaystyle \frac{m_0}{m_c{E}_r}}$ and $E_r={\displaystyle \frac{E_g+\frac{2}{3}{\Delta }_{SO}}{E\left(E_g+{\Delta }_{SO}\right)}}$ are two material-dependent constants. Substituting $a={\lambda }_{05}^{\prime }+{\Delta }_{05}$ and solving the system of inequalities (16) with respect to ${\Delta }_{05}$, we obtain the range of values of the rescaling parameter ${\Delta }_{05}$ that renders the valence-band part of the Kane Hamiltonian elliptic:

$$E_{r}max\left\{{\displaystyle \frac{1}{2}}{\lambda }_1^{\prime },2{\lambda }_2^{\prime },2{\lambda }_{04}^{\prime }\right\}<{\Delta }_{05}<-2{\lambda }_3^{\prime }{E}_r.$$

(17)

The calculated values of the ranges from Equation (17) are provided in the last two columns of Table 2. If $-{\lambda }_{05}^{\prime }<-2{\lambda }_3^{\prime }{E}_r$, then the Hamiltonian can be made elliptic by setting ${\Delta }_{05}$ to any value within range (17) such that ${\lambda }_{05}^{\prime }+{\Delta }_{05}>0$. In practice, one also wishes to ensure that numerical inaccuracies introduced by the eigenvalue calculation procedure for $H^K$ do not overturn any of the signs of ${\lambda }_1^{\prime }$, ..., ${\lambda }_{05}^{\prime }$. To minimize that possibility while keeping ${\Delta }_{05}$ reasonably small, we propose the following selection formula:

$${\Delta }_{05}=\left\{\begin{array}{cc}2{\Delta }_m+0.1,\hfill & {\Delta }_m+{\lambda }_3^{\prime }{E}_r<-0.1,\hfill \\ {\Delta }_m-{\lambda }_3^{\prime }{E}_r,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(18)

with ${\Delta }_m=max\left\{{\displaystyle \frac{1}{4}}{\lambda }_1^{\prime }{E}_r,{\lambda }_2^{\prime }{E}_r,{\lambda }_{04}^{\prime }{E}_r,-{\displaystyle \frac{{\lambda }_{05}^{\prime }}{2E}}\right\}$.

We carried out the rescaling procedure for the material parameters from Table 2 and selected, for each material, the set with minimal ${\Delta }_{05}$. The resulting readjusted values of $E_p$ and $A^{\prime }$, together with the new values of the modified Luttinger parameters, are presented in

Table 3. Selected material parameters rescaled via Equation (18), together with the difference between the corresponding bands of the $8\times 8$ Hamiltonian [32] for the original and rescaled parameters.

El a	${\mathbf{\Delta }}_{05}$ b	${\mathit{E}}_{\mathit{p}}$	${\mathit{A}}^{\prime }$	${\mathit{\gamma }}_1^{\prime }$	${\mathit{\gamma }}_2^{\prime }$	${\mathit{\gamma }}_3^{\prime }$	${\mathit{\lambda }}_{\mathit{v}}$ c	$\mathit{err}$ d
AlN 48	2.11	11.92	0.049	0.744	−0.004	0.226	−0.050	2.779
AlP 20	0.40	16.24	−0.900	1.859	−0.036	0.484	−0.263	4.290
AlSb 33	0.22	18.14	−0.900	2.646	−0.077	0.703	−0.229	5.841
AlAs 10	0.69	18.90	−0.262	1.728	−0.116	0.404	−0.051	9.355
GaN 43	2.19	17.75	0.296	1.287	−0.037	0.363	−0.050	11.577
InP 25	0.59	19.46	−0.632	2.120	0.000	0.700	−0.020	11.691
GaP 17	1.52	17.80	0.569	2.140	−0.050	0.630	−0.050	23.338
InSb 37	0.47	23.05	0.336	3.462	−0.094	1.006	−0.067	32.664
InN 51	4.16	9.16	0.285	1.054	−0.071	0.240	−0.050	35.027
GaAs 3	3.37	23.35	−0.509	2.676	−0.102	0.738	−0.053	50.617
GaSb 29	2.87	19.63	4.263	3.740	0.000	1.230	−0.050	171.395

[a] Refer to the original dataset number from Table 2. [b] The quantity ${\Delta }_{05}$ describes the size of adjustment to $A^{\prime }$. [c] The values of ${\lambda }_v$ are calculated via ${\lambda }_v=max\{{\lambda }_1^{\prime },{\lambda }_2^{\prime },{\lambda }_3^{\prime },{\lambda }_{04}^{\prime }\}$. [d] Maximum difference in meV between the band structure for the original parameters from Table 2 and the rescaled parameters calculated using Equation (18), over the $20\%$ of the paths $\mathsf{\Gamma }L$, $\mathsf{\Gamma }K$, and $\mathsf{\Gamma }X$.

. It is also worth noting that the resulting value of $1+A$ is in our case never equal to zero or one, as was commonly assumed in earlier works [20,62]. For many materials the adjusted value of parameter A is greater than zero.

To assess the impact of rescaling on the band dispersion, in Table 3 we also report the maximum absolute difference (adjustment error) between the corresponding bands of the band structure calculated over $20\%$ of each of the three high-symmetry paths $\mathsf{\Gamma }L$, $\mathsf{\Gamma }K$, and $\mathsf{\Gamma }X$ pertaining to the first Brillouin zone (FBZ). This domain size for comparison is standard [29] and is motivated by existing evidence [145] that an accurate fit of the $\mathit{k}·\mathit{p}$ band structure to state-of-the-art ab initio calculations is achievable over this portion of the FBZ. To identify the band that contributes most to the error, we provide in Figure 3a and Figure 4 a graphical comparison of band-structure diagrams for every set from Table 3 and the original parameter sets from Table 2 on which they are based. For clarity, only bands with even indices in the representation of the $8\times 8$ Hamiltonian [32] are plotted in these figures.

As is immediately evident from the last column of Table 3, the chosen sets for AlN, AlP, AlSb, and AlAs are the least susceptible to the rescaling procedure. The differences between the band structures for the modified parameter sets (the four topmost rows of Table 3) and the original sets for this group of materials are less than $10\mathrm{meV}$. We refer to these differences as band-structure adjustment errors, or simply errors where unambiguous. For GaN and InP the errors of approximately $11\mathrm{meV}$ are also visually indistinguishable in Figure 3a and Figure 4. We therefore provide in Figure 3b their plot for GaN with an appropriate vertical scale. This plot, which is typical for all analyzed materials except InN, shows the behaviour of the band-structure adjustment error along the three main paths $\mathsf{\Gamma }L$, $\mathsf{\Gamma }K$, and $\mathsf{\Gamma }X$. For GaN, the errors along $\mathsf{\Gamma }L$ and $\mathsf{\Gamma }K$ are approximately $11\mathrm{meV}$, while the band errors along $\mathsf{\Gamma }X$ are roughly two times smaller. What makes this material noteworthy is that such small errors result from a significant change in material parameters during rescaling: the difference in $E_p$ is approximately $29\%$, and the change in ${\gamma }_1^{\prime }$ and ${\gamma }_2^{\prime }$ exceeds $100\%$.

The situation is closer to the expected for the next group of materials: GaP, InN, and GaAs. The relative differences in $E_p$ are approximately $19\%$ for all three materials; however, the error is higher for GaAs than for GaP and InN: $50.62\mathrm{meV}$ vs. $23.34\mathrm{meV}$ and $35.02\mathrm{meV}$, respectively. This can be explained by the closer proximity of p-like conduction bands in GaAs, which are treated perturbatively in the present model. For InSb, the band adjustment error of $32.66\mathrm{meV}$ (barely visible as a slightly increased curvature of the conduction and SO bands in the first panel of Figure 4) falls within the same range as for GaP, InN, and GaAs. What is unusual is that these differences in band dispersion were produced by the smallest adjustment of $E_p$ among all analyzed materials ($0.6\%$), which resulted in only approximately $10\%$ increase in ${\gamma }_1^{\prime }$ and ${\gamma }_3^{\prime }$. Such sensitivity of the error may be attributed to the very small bandgap of InSb (see Figure 4). The CB adjustment error for InSb is $10.87\mathrm{meV}$, approximately three times smaller than the valence-band adjustment error and therefore invisible in the plot. The same applies to the heavy-hole and light-hole bands.

Similar tendencies are observed for the other materials in Table 3. The adjustment of A leads to a barely noticeable change in the conduction-band dispersion. The heavy-hole (HH) and light-hole (LH) bands remain visually unaffected, even though the differences are non-zero. The rescaling also causes an increase in the curvature of the split-off (SO) band, making it the main source of the total valence-band adjustment error.

The maximum adjustment error of $171.4\mathrm{meV}$ was observed for gallium antimonide (GaSb). A detailed discussion of GaSb is deferred to the next subsection; we focus here on the following question: how can the dispersion of the CB and SO bands be corrected without compromising the ellipticity of $H^K$?

6.2. Ellipticity Analysis for the 8 × 8 ZB Hamiltonian with Inversion Asymmetry

To answer the question posed at the end of the previous subsection, we consider here the ellipticity conditions for the case of non-zero B in Equation (9). In this case the ellipticity region in the parameter space $A^{\prime },B,{\gamma }_1^{\prime },{\gamma }_2^{\prime },{\gamma }_3^{\prime }$ is described by the system of inequalities

$$\left\{\begin{array}{cc}\hfill max\left\{-4{\gamma }_2^{\prime }-6{\gamma }_3^{\prime },3{\gamma }_3^{\prime }-4{\gamma }_2^{\prime },3{\gamma }_3^{\prime }+2{\gamma }_2^{\prime }\right\}& <{\gamma }_1^{\prime }\hfill \\ \hfill E+A^{\prime }+{\lambda }_{04}^{\prime }-\sqrt{{\left(E+A^{\prime }-{\lambda }_{04}^{\prime }\right)}^2+2B^2}& <0\hfill \\ \hfill E+A^{\prime }+{\lambda }_{04}^{\prime }+\sqrt{{\left(E+A^{\prime }-{\lambda }_{04}^{\prime }\right)}^2+2B^2}& >0.\hfill \end{array}\right.$$

(19)

The first inequality is just a compact form of inequalities 1–3 from System (12), the value of ${\lambda }_{04}^{\prime }$ is equal to the one defined above, but written in a new parameter notation ${\lambda }_{04}^{\prime }=-{\gamma }_1^{\prime }+2{\gamma }_2^{\prime }-3{\gamma }_3^{\prime }$.

Despite a more complicated structure than in the situation with zero B discussed earlier, one out of the two B-dependent constraints in System (19) is always fulfilled. To be more specific: if $E+A^{\prime }\ge -{\lambda }_{04}^{\prime }$, the third inequality from System (19) is redundant; otherwise, the second one is redundant. In each case, the remaining non-redundant inequality leads to the following constraint on $B^2$:

$$B^2-2E^2(1+A)(-{\gamma }_1^{\prime }+2{\gamma }_2^{\prime }-3{\gamma }_3^{\prime })>0.$$

(20)

The combination of Equation (20) with the first inequality from System (19) yields a system of ellipticity constraints for the $8\times 8$ ZB Hamiltonian [32] with non-zero B:

$$\left\{\begin{array}{cc}\hfill max\left\{-4{\gamma }_2^{\prime }-6{\gamma }_3^{\prime },3{\gamma }_3^{\prime }-4{\gamma }_2^{\prime },3{\gamma }_3^{\prime }+2{\gamma }_2^{\prime }\right\}& <{\gamma }_1^{\prime }\hfill \\ \hfill 2E^2(1+A)(-{\gamma }_1^{\prime }+2{\gamma }_2^{\prime }-3{\gamma }_3^{\prime })& <B^2.\hfill \end{array}\right.$$

(21)

Inequality (20) is fulfilled for any B when the parameter set A, ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, ${\gamma }_3^{\prime }$ satisfies conditions 4–5 from System (12). Consequently, the admissible, in terms of System (12), material parameters with zero B remain admissible even after the value of B is set to some non-zero number. In other words, B can be treated as an additional fitting parameter to be used in the subsequent adjustment step after the rescaling procedure is performed, but the additional ellipticity-preserving readjustment of the band structure is needed. We are especially interested in correcting the improper dispersion of the CB and SO bands, since it is a major source of errors (see Table 3) for most of the rescaled material parameter sets.

Conducted numerical experiments with different values of B indicate that an increase in $\left|B\right|$ leads to an increase in the curvature of the conduction and SO bands along the $\mathsf{\Gamma }K$, $\mathsf{\Gamma }L$, and $\mathsf{\Gamma }X$ directions. For InN, this makes the CB adjustment error along those directions smaller at the expense of a larger difference in SO band dispersion between the original set and the rescaled parameter set with non-zero B. The indicated behaviour of the error and the fact that the error’s dominant contribution comes from the CB and SO bands (see Figure 3b) mean that there exists an optimal value of B that minimizes the error for the chosen energy bands. For small errors and $B>0$, the indicated behaviour is also influenced by the spin-splitting of bands away from $\mathsf{\Gamma }$. The error cannot be minimized for GaSb, because the rescaled parameters with $B=0$ already yield a visibly higher curvature of the CB than the original parameters (see the GaSb panel in Figure 4). We performed the error minimization by adjusting B for each selected parameter set from Table 3 and confirm that, for all materials except InN, an increase in $\left|B\right|$ makes the error larger.

The first, larger value of $\left|B\right|=15.006$ for InN is a result of the error minimization over two conduction bands only. The conduction-band error ${\mathrm{err}}_{\mathrm{CB}}=8.352\mathrm{meV}$ signifies that the correct dispersion of the CB can be recovered almost perfectly by selecting the appropriate $\left|B\right|$. These differences between the obtained dispersion and that of the original parameter sets are caused by the non-zero spin-splitting of the CB states for $B\ne 0$, which is also observed experimentally [151,152]. To illustrate the effects of spin-splitting and visualize the behaviour of the adjustment error, we provide in Figure 5 a band-structure plot of the direction-wise maximal absolute error between the band structure of original set #51 from Table 2 with $B=0$ and the rescaled set from Table 3 with $B=15.006$.

Notice from Figure 5b,c that the spin-splitting is even more evident for the LH and SO valence bands than for the conduction band depicted in Figure 5a. Direction-wise, the magnitude of the CB spin–orbit splitting depends on the ratio of the individual momentum components $k_x/k_y$, $k_x/k_z$, $k_y/k_z$. It is non-zero if all these ratios are different from zero, infinity, and one.

The overall eight-band adjustment error $\mathrm{err}=9.839\mathrm{meV}$ is still more than three times smaller for the calculated B than for $B=0$ (green dotted and dash-dotted lines vs. red solid line in Figure 5d). This error is now dominated by the SO band error arising from dispersion along the $\mathsf{\Gamma }K$ direction (Figure 5b). This kind of dominance is typical for the considered materials.

Starting from the same sets of parameters in Table 3, we performed another optimization procedure with the aim of verifying to what extent the overall eight-band adjustment error can be minimized with the help of B. The resulting value of $B=14.805$ and error $\mathrm{err}=9.057\mathrm{meV}$ do not differ significantly from the results of the previous optimization procedure. The corresponding band dispersion for InN is visualized in Figure 6 using the layout of the previous figure.

The obtained minimal error is approximately $10\%$ smaller than the eight-band error of the previous optimization procedure and almost six times smaller than the band-structure error of the rescaled parameters with $B=0$ (see the last column of Table 3).

For InN, the same conclusion can be drawn from Figure 6d, where the band-wise error dispersion (dotted and dash-dotted lines) is plotted together with the error dispersion of the elliptic parameter set with $B=0$. The overall error is evidently dominated by the adjustment error of the CB and SO bands (Figure 6a,b). The errors of the other bands are below $1\mathrm{meV}$.

The error plots of Figure 5 and Figure 6 are also useful to quantify the magnitude of spin-splitting in CB, LH, and SO bands for $B=15.006$ and $B=14.805$. The comparison of calculated spin-splitting parameters for GaAs and AlAs from Table 3 and the experimental values for CB along the $(1,1,0)$ direction suggests that a value of B around 70–$80\mathrm{eV}$ is needed for the $8\times 8$ model to reach the reported experimental values [153,154,155]. For such B we observed a deviation in band dispersion of around $0.35$–$0.5\mathrm{eV}$ from the dispersion for $B=0$. Thus, spin-splitting errors are more dominant than the adjustment errors for the selected material sets and possibly others (for the experimental values of B see Table 5.5 from [21] and the references therein). In order to achieve better accuracy with $B\ne 0$, one should use the band-structure diagram with realistic spin-splitting of bands as an optimization target. Having that in hand, one can possibly obtain better results by applying the implemented two-step adjustment procedure to other material data entries from Table 2 where the range $\left({\Delta }_{05}^{min},{\Delta }_{05}^{max}\right)$ for the adjustment parameter ${\Delta }_{05}$ is non-empty.

To clarify the use of $\left|B\right|$ in the above calculations, we note that, similarly to ellipticity constraints (21), the sign of B [144] has no effect on the eigenvalues of the Hamiltonian in either the momentum or position representation. It does, however, affect the eigenstates of the Hamiltonian in both representations and must therefore be taken into consideration for experiments [156,157] that make use of the eigenstates.

One can further increase the accuracy of the admissible parameter set of the $8\times 8$ Hamiltonian by fitting [145] the full set A, ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, ${\gamma }_3^{\prime }$, B and using inequalities (21) as constraints for the fitting method. Our initial results in that direction show that this is possible for a wide range of materials. In addition, the adjustment with B alone is not a universal substitute for the optimally fitted parameter set A, ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, ${\gamma }_3^{\prime }$, B, because the effect of B on the band structure disappears if two out of three momentum components are zero.

It should be noted that the non-uniqueness of parameter sets satisfying the ellipticity constraints (21) is an intrinsic feature of the $\mathit{k}·\mathit{p}$ parametrization problem rather than a limitation of the rescaling procedure [158]. The optimal admissible parameter set, in the sense of minimizing the discrepancy between the model band structure and experimental or ab initio reference data subject to the ellipticity constraints as hard inequalities, is obtained by a fully constrained optimization of the complete parameter vector A, ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, ${\gamma }_3^{\prime }$, B. The convexity of the admissible region established in this paper guarantees that such a constrained optimization problem is well-posed, and the ellipticity inequalities (21) provide the explicit, analytically derived constraints required to implement it within any standard optimization framework.

Now, let us return to the ellipticity conditions (21). So far we used B as a band-structure fitting parameter, after the ellipticity of the parameter set was established by the rescaling procedure of Equations (13), (15), and (18) with zero B. The procedure with adjustment of B can be used directly for materials with an elliptic valence-band part (e.g., sets #12, #13, #20, #33). For such materials we can overturn the negative sign of ${\lambda }_5^{\prime }$ and make the Hamiltonian fully elliptic by setting B to the appropriate non-zero value in terms of Equation (20). This allows one to bypass the rescaling procedure altogether, which may be advantageous in view of its phenomenological nature. Another, more physically motivated way to increase the accuracy of the $\mathit{k}·\mathit{p}$ Hamiltonian is to extend its basis set by adding new energy band states. The resulting Hamiltonians are analyzed in the next section.

To conclude the discussion on the ellipticity of $8\times 8$ ZB Hamiltonians, we apply the direct adjustment of B to several material parameter sets from Table 2, namely sets #12 and #13 for InAs, set #20 for AlP, and set #33 for AlSb. Only these listed parameter sets yield an elliptic valence-band part of the Hamiltonian, i.e., the corresponding distance $d=0$ in Table 2. Notably, for each of the three materials the first listed set was reported in [29]—the work that is highly regarded as a source of overall physically consistent material parameters. For InAs, the direct adjustment of B is the only option to obtain admissible parameter sets based on the data from Table 1, because the admissible range $\left({\Delta }_{05}^{min},{\Delta }_{05}^{max}\right)\ni {\Delta }_{05}$ is empty for all four of its table entries. For each of the above-mentioned parameter sets we performed a band-structure error minimization procedure over the interval $\left|B\right|\in \left({\left|B\right|}_{min},\infty \right)$ and report the resulting value of $\left|B\right|$ along with the errors in

Table 4. Results of direct band-structure error minimization procedures based on the adjustment of B.

El a	${\left|\mathit{B}\right|}_{\mathbf{min}}$	$\left|\mathit{B}\right|$ b	err c	errCB d	errHH d	errLH d	errSO d
InAs 12	25.90	25.90	0.073	0.073	0.013	0.061	0.073
InAs 13	27.21	27.21	0.081	0.077	0.013	0.064	0.081
AlP 20	5.27	5.27	0.003	0.002	0.001	0.002	0.003
AlSb 33	4.06	4.06	0.012	0.003	0.002	0.012	0.008

[a] Refer to the original dataset number from Table 2. [b] B that minimizes the band-structure error. [c] The minimal value of the error in eV calculated for the eight bands with a given B over $20\%$ of the paths $\mathsf{\Gamma }L$, $\mathsf{\Gamma }K$, and $\mathsf{\Gamma }X$. [d] The errors err_CB, err_HH, err_LH, err_SO (all in eV) of conduction bands, heavy holes, light holes, and split-off bands, respectively.

. Here ${\left|B\right|}_{min}$ is the minimal solution of Equation (20), with $0.1$ substituted in place of 0 on the right-hand side to accommodate possible numerical errors.

For brevity we do not provide band-structure or error plots based on the data from Table 4. Instead, we supply the errors of the CB, HH, LH, and SO bands as separate entries in the table. For InAs and AlP, the introduced band-structure error is dominated by the differences in the CB, SO, and LH bands. The situation is different for AlSb, for which the main contribution to the error comes from the LH bands.

The results reported in Table 4 clearly indicate that set #20 for AlP together with $B=5.272$ leads to a smaller band-structure adjustment error than the same-material set reported in Table 2 with $B=0$. We recommend this for simulations based on the $8\times 8$ ZB Hamiltonian [32] in the position representation. For InAs we suggest using set #12 with $B=25.898$. The set obtained as a result of the two-step optimization procedure reported above is the most suitable for InN. For the remaining materials analyzed in this work we recommend the sets from Table 3.

7. Ellipticity of 14 × 14 Band Models

In this section we focus our attention on the ellipticity of two $14\times 14$ ZB Hamiltonians that are frequently used in the literature [33,41,159,160,161,162,163,164,165,166,167]. These two models are based on the extended basis set: six p-like valence-band states and two s-like conduction states comprising the basis of the $8\times 8$ Hamiltonian studied in the previous section, plus six additional p-like conduction-band states. They are introduced to better describe the anisotropy of the conduction band in materials such as GaAs, InP, and InSb, where this is evidenced experimentally [33,159,161,162].

The first Hamiltonian was proposed by Zawadzki, Pfeffer, and Sigg in [160] and then extended [33,162] to account for the influence of the out-of-basis bands perturbatively. We base our analysis on this later extended version described by Equation (5) from [33]. The calculated eigenvalues ${\lambda }_1^{″}$, ..., ${\lambda }_5^{″}$ of the quadratic form associated with this $14\times 14$ ZB Hamiltonian are as follows:

$$\begin{array}{cc}\hfill {\lambda }_1^{″}& =E(-{\gamma }_1^{″}-4{\gamma }_2^{″}-6{\gamma }_3^{″})\hfill \\ \hfill {\lambda }_2^{″}& =E(-{\gamma }_1^{″}-4{\gamma }_2^{″}+3{\gamma }_3^{″})\hfill \\ \hfill {\lambda }_3^{″}& =E(-{\gamma }_1^{″}+2{\gamma }_2^{″}+3{\gamma }_3^{″})\hfill \\ \hfill {\lambda }_4^{″}& =E(-{\gamma }_1^{″}+2{\gamma }_2^{″}-3{\gamma }_3^{″})\hfill \\ \hfill {\lambda }_5^{″}& =E.\hfill \end{array}$$

(22)

The CB part of the Hamiltonian is elliptic by design, since ${\lambda }_5^{\prime \prime }>0$ independently of the material parameters. The ellipticity of the valence-band part is guaranteed when ${\lambda }_1^{″}$, ..., ${\lambda }_4^{″}$ are all negative simultaneously. Thus, we arrive at exactly the same ellipticity conditions as for the $6\times 6$ ZB Hamiltonian, albeit with different Luttinger-like parameters (compare ${\lambda }_1^{″}$, ..., ${\lambda }_4^{″}$ above with ${\lambda }_1$, ..., ${\lambda }_4$ from Equation (7)).

These new Luttinger-like parameters ${\gamma }_1^{″}$, ${\gamma }_2^{″}$, ${\gamma }_3^{″}$ can be obtained from the conventional Luttinger parameters by subtracting from ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, ${\gamma }_3^{\prime }$ the contributions of p-like CB bands that are no longer treated perturbatively. More precisely,

$$\begin{array}{cc}\hfill {\gamma }_1^{″}& ={\gamma }_1^{\prime }-{\displaystyle \frac{Q^2}{3E{E}_0^{\prime }}}-{\displaystyle \frac{Q^2}{3E(E_0^{\prime }+{\Delta }_0^{\prime })}},\hfill \\ \hfill {\gamma }_2^{″}& ={\gamma }_2^{\prime }+{\displaystyle \frac{Q^2}{6E{E}_0^{\prime }}},{\gamma }_3^{″}={\gamma }_3^{\prime }-{\displaystyle \frac{Q^2}{6E{E}_0^{\prime }}}.\hfill \end{array}$$

(23)

Here $E_0$ is the fundamental bandgap, $E_0^{\prime }$ is the gap between the first two bottommost conduction bands; ${\Delta }_0$ and ${\Delta }_0^{\prime }$ are the spin-splitting parameters of the valence and conduction bands, respectively; and Q is the interband momentum matrix element (see Figure 1 in [33]). For a complete description of the Hamiltonian, one additionally needs to define other material-dependent parameters $P_0^{\prime }$, $\overline{\Delta }$, $\kappa$, $C_k$. These are determined by fitting the band structure to experimental data [33]. For that reason, we focus only on the sources where the full sets of fitted Hamiltonian parameters have been reported in the context of the considered $14\times 14$ model.

Besides the sets ${\gamma }_1^{″}$, ${\gamma }_2^{″}$, ${\gamma }_3^{″}$ from the original paper [33], which provides them for GaAs and InP explicitly, we used the parameter sets from Jancu et al. [168] and the Luttinger parameters recommended therein from [29] to calculate the respective values of ${\gamma }_1^{″}$, ${\gamma }_2^{″}$, ${\gamma }_3^{″}$ for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, and InSb. All calculated parameters together with the results of the ellipticity analysis are shown in Table 5.

Table 5. Material data for the $14\times 14$ ZB Hamiltonian [33]; d denotes the distance from the point $({\gamma }_1^{″},{\gamma }_2^{″},{\gamma }_3^{″})$ to the feasibility region ${\Lambda }_{-}$. Positive values of ${\lambda }_1^{″}/E$, ${\lambda }_2^{″}/E$, ${\lambda }_3^{″}/E$, ${\lambda }_4^{″}/E$ are denoted in bold.

#	El	${\mathit{\gamma }}_1^{″}$	${\mathit{\gamma }}_2^{″}$	${\mathit{\gamma }}_3^{″}$	${\mathit{\lambda }}_1^{″}/\mathit{E}$	${\mathit{\lambda }}_2^{″}/\mathit{E}$	${\mathit{\lambda }}_3^{″}/\mathit{E}$	${\mathit{\lambda }}_4^{″}/\mathit{E}$	d
1	GaAs a	0.1760	0.4210	0.1050	−2.4900	−1.5450	0.9810	0.3510	0.2622
2	GaAs b	−0.5860	−0.0210	−0.3360	2.6860	−0.3380	−0.4640	1.5520	0.4148
3	GaAs c	−1.4848	−0.0252	−0.6071	5.2281	−0.2357	−0.3868	3.2557	0.8701
4	AlAs c	−0.9087	0.3441	−0.3082	1.3816	−1.3921	0.6722	2.5214	0.6739
5	InAs c	0.7561	0.5862	0.1224	−3.8353	−2.7341	0.7834	0.0493	0.2094
6	GaP c	−1.5453	−0.1625	−0.8422	7.2485	−0.3316	−1.3063	3.7471	1.0014
7	AlP c	−1.2216	0.2206	−0.4603	3.1013	−1.0416	0.2817	3.0436	0.8134
8	InP d	0.4440	0.4580	−0.1310	−1.4900	−2.6690	0.0790	0.8650	0.2312
9	InP e	−0.4960	−0.1520	−0.6810	5.1900	−0.9390	−1.8510	2.2350	0.7129
10	InP c	−1.5396	−0.0267	−0.6633	5.6261	−0.3432	−0.5037	3.4759	0.9290
11	GaSb c	−0.3876	0.5864	−0.0283	−1.7883	−2.0429	1.4756	1.6453	0.4397
12	AlSb c	−0.9892	0.5820	−0.3248	0.6101	−2.3135	1.1788	3.1279	0.8360
13	InSb c	−2.8859	−0.9194	−1.6062	16.2004	1.7450	−3.7713	5.8656	2.2253

[a] Set 1 from [33] ($\alpha =0.065$); [b] Set 2 from [33] ($\alpha =0.085$); [c] Parameters obtained via (23) from the data in [29,168]; [d] Set 1 from [33] ($\alpha =0.12$); [e] Set 2 from [33] ($\alpha =0.2$).

Table 5. Material data for the $14\times 14$ ZB Hamiltonian [33]; d denotes the distance from the point $({\gamma }_1^{″},{\gamma }_2^{″},{\gamma }_3^{″})$ to the feasibility region ${\Lambda }_{-}$. Positive values of ${\lambda }_1^{″}/E$, ${\lambda }_2^{″}/E$, ${\lambda }_3^{″}/E$, ${\lambda }_4^{″}/E$ are denoted in bold.

#	El	${\mathit{\gamma }}_1^{″}$	${\mathit{\gamma }}_2^{″}$	${\mathit{\gamma }}_3^{″}$	${\mathit{\lambda }}_1^{″}/\mathit{E}$	${\mathit{\lambda }}_2^{″}/\mathit{E}$	${\mathit{\lambda }}_3^{″}/\mathit{E}$	${\mathit{\lambda }}_4^{″}/\mathit{E}$	d
1	GaAs a	0.1760	0.4210	0.1050	−2.4900	−1.5450	0.9810	0.3510	0.2622
2	GaAs b	−0.5860	−0.0210	−0.3360	2.6860	−0.3380	−0.4640	1.5520	0.4148
3	GaAs c	−1.4848	−0.0252	−0.6071	5.2281	−0.2357	−0.3868	3.2557	0.8701
4	AlAs c	−0.9087	0.3441	−0.3082	1.3816	−1.3921	0.6722	2.5214	0.6739
5	InAs c	0.7561	0.5862	0.1224	−3.8353	−2.7341	0.7834	0.0493	0.2094
6	GaP c	−1.5453	−0.1625	−0.8422	7.2485	−0.3316	−1.3063	3.7471	1.0014
7	AlP c	−1.2216	0.2206	−0.4603	3.1013	−1.0416	0.2817	3.0436	0.8134
8	InP d	0.4440	0.4580	−0.1310	−1.4900	−2.6690	0.0790	0.8650	0.2312
9	InP e	−0.4960	−0.1520	−0.6810	5.1900	−0.9390	−1.8510	2.2350	0.7129
10	InP c	−1.5396	−0.0267	−0.6633	5.6261	−0.3432	−0.5037	3.4759	0.9290
11	GaSb c	−0.3876	0.5864	−0.0283	−1.7883	−2.0429	1.4756	1.6453	0.4397
12	AlSb c	−0.9892	0.5820	−0.3248	0.6101	−2.3135	1.1788	3.1279	0.8360
13	InSb c	−2.8859	−0.9194	−1.6062	16.2004	1.7450	−3.7713	5.8656	2.2253

[a] Set 1 from [33] ($\alpha =0.065$); [b] Set 2 from [33] ($\alpha =0.085$); [c] Parameters obtained via (23) from the data in [29,168]; [d] Set 1 from [33] ($\alpha =0.12$); [e] Set 2 from [33] ($\alpha =0.2$).

Each of the sets considered in Table 5 fails two out of four ellipticity constraints, except the sets for AlAs, AlP, and InSb, for which three ellipticity constraints are violated. For GaAs set 1 from Table 5 we can confirm a reduction of the distance d to the feasibility region ${\Lambda }_{-}$ in the space ${\gamma }_1^{″}$, ${\gamma }_2^{″}$, ${\gamma }_3^{″}$ compared to the best GaAs sets for the $6\times 6$ and $8\times 8$ Hamiltonians. This set and set #8 are taken from the original work; both were calculated using cyclotron resonance experiments [33]. The second pair of sets #2 and #9, which are deemed more consistent experimentally [169], lies slightly outside the region ${\Lambda }_{-}$; however, the corresponding values of d are within the range of the same-material values of d from Table 3. Parameter sets for other materials are even further from ${\Lambda }_{-}$ than the unrescaled same-material sets for the $8\times 8$ Hamiltonian.

The observed increase of the distance to ${\Lambda }_{-}$ seems to be theoretically unfounded, especially in view of Equation (3). Recall that the relative norm [39] of the perturbative term from Equation (3) should decrease after eigenstates are moved from the perturbative class (class B) into the basis (class A). This can be explained as follows.

In the $8\times 8$ model, the influence of the valence bands on the CB states was represented directly by the parameters $P_0$, B and, we suppose, indirectly by the perturbative CB parameter $A^{\prime }$. The absence of $A^{\prime }$ in the CB eigenvalue from Equation (22) suggests that the $14\times 14$ model was derived under the assumption that $A^{\prime }$ depends only on the upper CB states now included in the basis. In such a situation, all cross-influence between valence and conduction bands is incorporated into P and Q through fitting to experimental data, and is then propagated to ${\gamma }_1^{″}$, ${\gamma }_2^{″}$, ${\gamma }_3^{″}$ via Equation (23). However, the terms ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, ${\gamma }_3^{\prime }$ on the right-hand side of Equation (23) were fitted to experiment under the assumption of a non-zero valence-band contribution to $A^{\prime }$. This explains why the parameter triplets ${\gamma }_1^{″}$, ${\gamma }_2^{″}$, ${\gamma }_3^{″}$ for materials with a smaller fundamental bandgap $E_0$ (InAs, GaSb, and InSb) end up having larger d.

On the other hand, the conduction-band states in materials with larger $E_0$ (AlAs, AlP, and AlSb) may in reality be influenced by higher bands not included in the basis. That influence is assumed to be zero in the model, because the CB eigenvalues are equal to E even for the newly included p-like bands. If non-negligible, this influence is accounted for by P and Q and then propagated to ${\gamma }_1^{″}$, ${\gamma }_2^{″}$, ${\gamma }_3^{″}$ by the mechanism described above. This explains the increase in d for the large-bandgap materials in Table 5 (AlAs, AlP, and AlSb).

For some parameter sets from Table 5, the ellipticity might be corrected by rescaling P and Q in a way similar to the rescaling procedure from Section 6.1. This will, of course, affect the accuracy of the band structure and therefore must involve an optimization procedure with respect to the parameters P, Q, and possibly ${\gamma }_1^{″}$, ${\gamma }_2^{″}$, ${\gamma }_3^{″}$, if our hypothesis holds true.

We now proceed to the second implementation of the $14\times 14$ ZB Hamiltonian model. The initial version of this model was derived by Rössler using the theory of invariants [170] and then extended by Mayer and Rössler [161] by adding first-order perturbative corrections to the lowest conduction and upper valence bands. The most recent version of the Hamiltonian was provided by Winkler [34]. It additionally includes the second-order conduction-valence band mixing parameters analogous to B from the Kane Hamiltonian (9).

All three mentioned versions of the $14\times 14$ ZB Hamiltonian share the common assumption that the six second-order diagonal terms related to the newly added p-like CB states are neutralized by the counter-influence of other bands; see, e.g., Table C.7 of [34]. In terms of ellipticity, such an assumption results in the presence of a zero eigenvalue among the eigenvalues of the quadratic form associated with this implementation. Our calculations confirm this. Therefore, this Hamiltonian is not elliptic by design.

It is worth noting that, unlike the first, the second implementation of the $14\times 14$ Hamiltonian [34,161,170] can be regarded as an extension of the Kane model studied in Section 6. The Hamiltonian contains the perturbative correction to s-like CB states and the conduction-valence band mixing parameters. Thus, all inter-band interaction effects embodied in the $8\times 8$ representation [32] can be properly accounted for. In our opinion, both analyzed implementations of the $14\times 14$ Hamiltonian model are less universal material-wise than the $8\times 8$ Hamiltonians, despite being more accurate at describing CB-related phenomena [166,167]. For such higher-band models, the assumptions regarding the interactions of in-basis conduction-band states require revision.

The structural complementarity between the two $14\times 14$ implementations analyzed here—the first carrying a well-defined set of valence-band ellipticity constraints but lacking a general treatment of conduction-valence mixing, and the second incorporating that mixing through Kane-type parameters but being non-elliptic by construction—suggests that a unified $14\times 14$ model that combines the perturbative correction to s-like conduction-band states from the second implementation with the elliptic operator structure of the first is both theoretically desirable and practically achievable within the parametrization framework developed in this paper. The distinction between parametrization-dependent and structure-dependent non-ellipticity established here for the $14\times 14$ case carries direct implications for higher-band models. For the first implementation, the non-ellipticity of currently available parameter sets is a consequence of the specific truncation and fitting procedure rather than of the extended basis itself, and the present parametrization framework is applicable without modification once a suitably constrained optimization of the full parameter set is undertaken. For the second implementation, the non-ellipticity is structural and independent of parameter choice, tracing directly to the modelling assumption that neutralizes the second-order diagonal terms of the newly added conduction-band states. The extension of the ellipticity analysis to 16-, 30-, and 40-band models—including the recently developed full-zone 30-band framework for III–V zinc-blende materials [171]—represents a natural continuation of the present work, for which the analytical methodology developed in Section 3, Section 4, Section 5, Section 6 and Section 7 provides the requisite conceptual and computational foundation.

8. Discussion, Impact on Other Fields, and Outlook

The results established in the preceding sections carry ramifications that extend well beyond the immediate task of correcting bandstructure parameters for a specific class of semiconductor Hamiltonians. At their core, those results address a question of mathematical well-posedness: under what conditions does a multiband effective-mass operator faithfully represent the physical Hamiltonian it is designed to approximate? The answer, grounded in the ellipticity theory of partial differential operators [135,136,137], is both definitive and consequential. When ellipticity fails, the operator loses the spectral semi-boundedness that is axiomatic in quantum mechanics [43,44], spurious solutions are admitted into the solution space [16,18], and the probability current is no longer conserved [26]. The consistent parametrization procedure developed here for the $6\times 6$, $8\times 8$, and $14\times 14$ zinc-blende Hamiltonians, together with the admissible parameter sets derived for all major III–V and III-nitride materials, therefore constitutes a necessary prerequisite for any quantitatively reliable application of the $\mathit{k}·\mathit{p}$ theory—whether that application is a conventional forward simulation or an emerging data-driven or inverse design methodology.

This section places those results in a broader scientific landscape, organized around five thematic directions:

• Section 8.1 examines the role of ellipticity conditions in neighbouring areas of nanoscience, mathematical physics, and quantum field theory, demonstrating that the issues encountered in the $\mathit{k}·\mathit{p}$ context are instances of a much more general structural problem.
• Section 8.2 addresses the mathematical foundation of multiband parametrization, examining operator ordering, the Burt–Foreman theory, ab initio fitting schemes, and quantum transport, and demonstrating that the ellipticity gap has not been systematically closed in any of these contexts prior to the present work.
• Section 8.3 develops the connection between the present parametrization framework and inverse problems, nonlocal phenomena, and size effects, all of which define the frontier of predictive modelling for low-dimensional nanostructures.
• Section 8.4 discusses the relevance of quantum control and the R-matrix scattering approach for nanostructure design within the context established by the present work.
• Finally, Section 8.5 examines the rapidly growing field of AI-assisted inverse design, arguing that mathematically consistent forward models of the type developed here are an indispensable foundation for any reliable data-driven or machine-learning pipeline operating at the nanoscale.

8.1. Ellipticity in Other Areas of Nanoscience, Nanotechnology, and Beyond

Ellipticity emerges across a remarkably broad scientific landscape—from nonlinear elasticity and strain-gradient continuum theories to quantum field theory and the Yang–Mills mass gap problem—as the universal structural prerequisite for mathematical well-posedness. This subsection demonstrates that its systematic violation in standard $\mathit{k}·\mathit{p}$ Hamiltonians, documented in the preceding sections, is not an isolated modelling deficiency but an instance of a foundational problem that recurs throughout mathematical physics, nanoscale multiphysics, and open quantum systems.

8.1.1. Ellipticity as a Unifying Structural Requirement

Ellipticity of the governing differential operator is not a technical convenience but a foundational requirement for the mathematical well-posedness of a wide class of physical models. In the theory of partial differential equations, strong ellipticity guarantees the existence, uniqueness, and regularity of solutions to boundary-value problems [135,136,137], and it is precisely this property that underpins the spectral theory of quantum-mechanical Hamiltonians [43,45]. The significance of ellipticity extends far beyond quantum mechanics: in nonlinear elasticity and strain-gradient continuum theories, strong ellipticity conditions are directly linked to infinitesimal stability and the prevention of material instabilities [134], while in probability theory and geometry, elliptic partial differential equations govern symmetry and regularity properties of fundamental importance [172]. The present work demonstrates that this structural requirement has been systematically overlooked in the parametrization of multiband $\mathit{k}·\mathit{p}$ Hamiltonians, with the result that the standard operator is non-elliptic for the overwhelming majority of common III–V semiconductor materials.

The importance of a careful treatment of Hamiltonians in nanoscience and nanotechnology applications, illustrated in the preceding sections on the example of the $\mathit{k}·\mathit{p}$ theory, becomes even more critical when coupling to additional physico-mechanical fields is required. Size-dependent effects described through strain-gradient and flexoelectric contributions [173,174], piezoelectric coupling in semiconductor rods and nanostructures [8,173], geometric phases in quantum dots subject to spin–orbit interaction [175], and electromechanical effects in graphene-based systems [127] all enrich the forward model while simultaneously tightening the mathematical requirements on the operator at its core. In each of these multiphysics settings the ellipticity of the $\mathit{k}·\mathit{p}$ backbone operator must be established before the extended model can be trusted to yield physically meaningful results. The parametrization procedure developed in the present paper provides, for the first time, a systematic and fully justified means of verifying and enforcing this requirement across the full range of materials and model dimensions considered.

The problem of spurious solutions in multiband effective-mass approximations, which motivated much of the earlier literature [15,16,17,18,20], has long been recognized as a symptom of a deeper mathematical pathology. Several remedies were proposed, including operator-ordering modifications [14,20], parameter adjustments [20,23], and numerical schemes designed to filter unphysical states [25,61,62]. What remained absent was a unifying theoretical analysis identifying non-ellipticity as the root cause and providing a constructive, material-by-material resolution. The present work supplies that analysis. It also shows that the connection between lack of ellipticity and perturbative terms describing out-of-basis band contributions is not merely an artefact of particular parameter choices but a structural feature of all standard $\mathit{k}·\mathit{p}$ Hamiltonians for materials with non-negligible conduction-valence band coupling, confirming and substantially extending the partial results reported for a restricted set of materials in [16,19]. The earlier arXiv version of the present analysis [67] first extended the ellipticity revision to all common Hamiltonians; the present paper provides its fully rigorous and comprehensive treatment. The theoretical basis for this connection—established in [16,18] and extended systematically across all standard Hamiltonians and materials in the present work—renders the explicit reproduction of individual numerical examples of spurious states superfluous: once the non-ellipticity of the operator is identified and its parameter-dependent structure is fully characterized, the occurrence of spurious solutions is an inevitable mathematical consequence, and the admissible parameter sets derived in Section 5, Section 6 and Section 7 constitute a complete and constructive remedy.

The interest in extending the $\mathit{k}·\mathit{p}$ framework to larger basis sets has grown substantially in recent years. Models with 16, 30, and even 40 bands have been developed for both zinc-blende and wurtzite material systems [74,75,76,171,176,177,178], driven by the need to describe conduction-band anisotropy and valley splitting in materials where the standard eight-band description is inadequate. The fitting of such extended models to ab initio band structures has itself become a research area [138,179,180], with fitting schemes that can assign priorities to selected bands and k-points [138]. Crucially, however, as emphasized in [74] and confirmed by the analysis of $14\times 14$ models in Section 7 of this paper, the mere increase in the number of basis states does not resolve the ellipticity problem: it may, in fact, aggravate it by introducing further perturbative corrections whose contribution to the principal symbol of the operator has not been systematically examined. The parametrization procedure developed here is applicable to Hamiltonians of arbitrary band dimension, and the present analysis of the $14\times 14$ case provides the conceptual and computational template for that extension. A fitting scheme that incorporates ellipticity constraints as hard inequality conditions—as indicated for the eight-band case in Section 6 and as illustrated for the wurtzite sixteen-band model in [138]—is the natural next step, and one that can be executed within the framework established in this paper. It must be stressed, however, that any claim to have constructed a parameter-fitting scheme for $\mathit{k}·\mathit{p}$ Hamiltonians of arbitrary complexity that is silent on ellipticity conditions is incomplete: the mathematical validity of the resulting operator cannot be guaranteed without an explicit verification of the kind performed here (see also earlier initial works [19,67]). This applies equally to ab initio-based schemes for wurtzite structures [180] and to those employing full-zone 30-band models for alloy systems such as strained ${\mathrm{Ge}}_{1-x}$${\mathrm{Sn}}_x$ [177,178].

The consecutive ellipticity analysis performed here for all standard $\mathit{k}·\mathit{p}$ Hamiltonians thus enables, for the first time, a material-wise quantification of the limits of these models’ applicability, and provides a principled basis for deciding, for a given material and a given number of bands, whether the multiband effective-mass approximation can be trusted or whether a more complete description—including additional in-basis states or a fully atomistic approach—is required.

8.1.2. Boundary Conditions, Self-Adjointness, and Open Quantum Systems

Ellipticity conditions and the correct formulation of boundary conditions are intimately related, and both are essential for the consistent application of multiband $\mathit{k}·\mathit{p}$ theory to heterostructures and low-dimensional nanostructures. The formulation of boundary conditions for multiband envelope functions in semiconductor nanostructures was placed on a rigorous footing in [49], where it was demonstrated that the self-adjointness of the multiband Hamiltonian serves as the guiding mathematical principle, and that physically consistent boundary conditions must guarantee the conservation of the probability flux density normal to the interface. Related formulations were discussed in the context of the Kane model in [24], and for the electronic structure of finite-extent embedded nanostructures in [181]. Earlier analyses by Bastard, Brum, and Ferreira [50] established that the model is sensitive to both the Hamiltonian equations and the boundary conditions, a point subsequently reinforced in [51], where small perturbations in the type of boundary conditions for elliptic operators were shown to produce profound consequences for the solution space. The approach to boundary conditions based on unitary transformations, such as that developed in [14], is itself affected by these considerations, since the parametrization of the model—including those based on such transformations—will be altered by any change in the boundary operator.

The problem of boundary conditions has been revisited in the context of newer methods for bandstructure calculation based on the multiband effective-mass approximation, including incommensurate heterostructures in momentum space [182], yet in the majority of these developments the ellipticity requirements have not been examined with the rigour they require. For open boundary conditions specifically, methods based on perfectly matched layers [183] and discrete transparent boundary conditions [184,185] have been developed for $\mathit{k}·\mathit{p}$-Schrödinger systems and represent significant technical achievements. Nevertheless, as the analysis of the present paper makes clear, these numerical advances rest on a foundation that is mathematically compromised unless the underlying Hamiltonian operator is first rendered elliptic through a consistent parametrization procedure of the type developed here. The boundary conditions derived in [49] for conical quantum dots with wetting layers, and the general boundary conditions for envelope functions derived in [186], both presuppose the self-adjointness of the Hamiltonian; the present work establishes the parameter conditions under which that presupposition is actually satisfied.

The broader issue of boundary conditions in quantum mechanics is of lasting importance [122,187]. All quantum systems, including low-dimensional nanostructures, are open in the sense that they interact with their environment in ways that cannot be fully described by unitary operators alone [188]. The dynamics of many-body entanglement under decoherence, the quantum-to-classical transition, and the control of composite quantum systems are all sensitive to the precise formulation of the boundary operator. Nonseparable states of light, in which two or more degrees of freedom are coupled in a non-separable way mathematically analogous to quantum entangled states [189], underscore the generality of the non-Hermitian and open-system framework within which boundary-condition questions arise. The framework for gauge theories on manifolds with boundary developed in [190,191] provides a rigorous setting in which boundary operators are subject to the same elliptic constraints that govern the multiband Hamiltonians studied in this paper. The non-self-adjoint setting introduces additional challenges: Lyantse’s framework for non-self-adjoint singular differential operators [98], recent results on the discrete spectrum of non-self-adjoint Schrödinger operators with complex potentials [99], and the spectral theory of non-self-adjoint elliptic problems in biological and nanoscale networks [100] all point towards settings in which ellipticity conditions remain indispensable for controlling the invertibility of the operator’s principal symbol and preventing the numerical instabilities that are endemic to non-rigorous $\mathit{k}·\mathit{p}$ expansions. The analysis of spectral enclosures and stability for non-self-adjoint discrete Schrödinger operators on the half-line [192] provides a further mathematical basis for appreciating the consequences of operator non-ellipticity in the discrete setting relevant to numerical implementations.

Nonlocality constitutes an additional layer of complexity that boundary conditions must eventually accommodate. Boundary conditions that involve incomplete data necessarily require regularization; the same is true for initial conditions in the presence of memory effects or non-Markovian dynamics [95]. As demonstrated in [95], nonlocal initial conditions arise naturally in quantum-mechanical models, and the framework of nonlocal partial differential equations introduces requirements on the governing operator that go beyond those of the classical elliptic setting. The present paper establishes the elliptic foundation on which such extensions must build.

8.1.3. Ellipticity in Nanoscale Multiphysics: Superlattices, Spintronics, and Straintronics

The $\mathit{k}·\mathit{p}$ theory serves as the central computational engine in a broad family of nanoscale multiphysics models that couple electronic band structure to mechanical, electromagnetic, and thermal degrees of freedom. Semiconductor superlattices provide an illuminating illustration of how this coupling enriches the phenomenology while simultaneously demanding a mathematically sound operator. The theory of spatially inhomogeneous Bloch oscillations in semiconductor superlattices [193], quantum confinement phenomena in nanowire superlattice structures [194], and quantum transport in semiconductor nanowire superlattices described by coupled quantum-mechanical and kinetic models [7] all rely, explicitly or implicitly, on the correctness of the underlying $\mathit{k}·\mathit{p}$ Hamiltonian. When the Hamiltonian is non-elliptic, the associated time-dependent Schrödinger equation loses the property of probability current conservation [26], and the dynamic simulations built upon it are susceptible to unphysical instabilities. Efficient, numerically stable multiband $\mathit{k}·\mathit{p}$ treatments of quantum transport [195] further depend on the ellipticity of the Hamiltonian to guarantee the stability of the transmitting boundary method, and the present parametrization procedure removes this vulnerability at the level of the input parameters.

In the field of spintronics, the accurate description of spin–orbit coupling, Dresselhaus and Rashba splitting, and spin-dependent tunnelling in III–V heterostructures depends critically on the eight-band Kane Hamiltonian [21,76] and on the inversion-asymmetry parameter B whose role in maintaining ellipticity was established in this work for the first time. The experimental observation of spin-splitting in InN [151,152] and the theoretical framework for Berry phase and spin precession in quantum dots without magnetic fields [175] both depend on the correct description of conduction-valence band mixing, which in turn requires the admissible parameter sets derived here. Similarly, in straintronics—the exploitation of strain to control electronic and spin properties—the incorporation of strain-induced deformation potentials within the $\mathit{k}·\mathit{p}$ framework [8,37,56,57,58] requires that the parent operator be elliptic, since strain corrections to the Hamiltonian leave the structure of the second-order principal symbol unchanged, as established in Section 5 of this paper. Nonlinear strain models in quantum dot molecules [56,58] and the coupled effects of piezoelectricity and strain in semiconductor nanostructures [8,173,174] further illustrate the breadth of applications in which the present results are directly operative.

Magnetic quantum dots doped with magnetic impurities, in which the Luttinger–Kohn $\mathit{k}·\mathit{p}$ Hamiltonian is used to account for spin–orbit interaction in the formation of magnetic polarons [196], represent yet another setting in which the admissible parameter sets derived here are of immediate practical relevance. The multiband electronic structure of such systems involves self-consistent, temperature-dependent calculations for which the mathematical validity of the underlying Hamiltonian is a prerequisite for meaningful results. Core-shell nanowires, whose band structure under n- and p-doping has been studied within the multiband $\mathit{k}·\mathit{p}$ framework [59], equally require elliptic operators for the boundary-value problems formulated at the wire–shell interface.

Novel two-dimensional materials, including graphene, silicene, and their heterostructures, introduce further ellipticity considerations. Strain-induced effects in graphene produce a rich phenomenology of novel electronic states [126], while graphene electrode systems exhibit complex interplay between mechanical, electronic, and electrochemical degrees of freedom [127]. Machine-learning approaches applied to shape-memory graphene nanoribbons [128] underscore the growing role of data-driven methods in this domain, a theme developed at length in Section 8.5. In all these cases, the forward models on which data-driven methods are trained must satisfy ellipticity conditions analogous to those derived in the present work for three-dimensional zinc-blende crystals; the systematic derivation of such conditions for non-three-dimensional and non-zinc-blende systems represents a natural and important extension of the present analysis. The generalized plane-wave formulation of the $\mathit{k}·\mathit{p}$ formalism applicable to arbitrary semiconductor nanostructures [55] provides a natural starting point for such an extension, as does the pseudopotential framework for nanoscale quantum dots [68] and the tight-binding-to-continuum connections recently developed in [69,70].

8.1.4. Ellipticity in Quantum Field Theory: The Yang–Mills Problem and the Mass Gap

The appearance of ellipticity requirements in fundamental areas of mathematical physics beyond condensed matter and nanoscience provides a striking illustration of the universality of the structural problem identified in this paper. One of the most profound open problems in mathematical physics—and one of the seven Millennium Prize Problems of the Clay Mathematics Institute—is the Yang–Mills existence and mass gap problem: the rigorous proof that, for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ${\mathbb{R}}^4$ and possesses a positive mass gap [197]. Yang–Mills theory underlies the Standard Model of particle physics, providing the mathematical foundation for quantum chromodynamics (QCD) and the electroweak theory; its predictions have been extensively tested experimentally, yet its mathematical foundation remains incomplete [198,199]. The connection between this problem and ellipticity is not superficial.

In the gauge-invariant quantization of field theories on manifolds with boundary, ellipticity conditions on the boundary operator have been identified as a fundamental constraint on mathematical well-posedness [132,133,190]. As established in [132], the fundamental laws of physics can be derived from the requirement of invariance under suitable classes of transformations combined with the requirement of a well-posed mathematical theory; the boundary operators of quantum field theory are subject to the same elliptic constraints that govern the multiband Hamiltonians studied in this paper. The perturbative quantum gauge theory on manifolds with boundary developed in [191] and the Landau-gauge Yang–Mills correlation functions studied by functional renormalization group methods [198,200] both encounter the problem of truncations—the rendering of infinite systems of functional equations finite—in ways that are mathematically analogous to the Löwdin perturbative procedure used in $\mathit{k}·\mathit{p}$ theory. In both settings, the truncation introduces approximation errors whose effect on the principal symbol of the operator, and hence on ellipticity, has not been systematically examined. The nonperturbative functional renormalization group approach to Yang–Mills theories [198,199], which has achieved quantitative agreement with lattice QCD results for gluon and ghost propagators and recently enabled the calculation of the scalar and pseudoscalar glueball spectrum from a parameter-free determination of correlation functions in Landau gauge [198], represents the state of the art in addressing the mass gap problem through functional methods; in all such approaches the ellipticity of the fundamental operator is a prerequisite for well-posedness that mirrors the requirement established here for $\mathit{k}·\mathit{p}$ Hamiltonians.

Within the eight-dimensional generalization of Yang–Mills self-duality, Bilge demonstrated that the ellipticity of the self-dual equations under the Coulomb gauge condition determines the dimensionality of the solution space and controls the global structure of solutions [201]. This result is structurally parallel to the finding of the present paper that the non-ellipticity of the $8\times 8$ zinc-blende Hamiltonian with $B=0$ admits a two-dimensional manifold of non-physical eigenstates for certain domain geometries. The connection between the lens rigidity problem for a particle in a Yang–Mills field and the recovery of the gauge potential from scattering data [202] further illustrates how ellipticity of the governing operator constrains what can be reliably reconstructed from measurements—a point of direct relevance to the inverse design framework discussed in Section 8.3.

The undecidability of the spectral gap for general quantum systems [203] places an important theoretical boundary around the Yang–Mills mass gap problem and provides context for the present results: whereas the spectral gap problem is undecidable in general, the ellipticity conditions derived here provide an explicit, computable, and material-specific criterion for spectral semi-boundedness in the $\mathit{k}·\mathit{p}$ setting. The open quantum systems perspective on Yang–Mills theory—the observation that the classical description can emerge when a quantum system is treated as open and coupled to its environment, as analyzed in the context of irreversible statistical mechanics [204,205]—connects this fundamental problem to the broader theme of open quantum systems that runs through the present work. That data-driven and machine-learning methods are beginning to penetrate this domain [199] further underscores the imperative, articulated throughout this paper, that the forward models on which such methods are trained must rest on a mathematically rigorous foundation, whether in nanoscience or in fundamental particle physics.

8.2. The Mathematical Foundation of Multiband Parametrization: Operator Ordering, Ab Initio Fitting, and the Persistent Ellipticity Gap

The analysis carried out in the preceding sections, together with the broader context established in Section 8.1, converges on a conclusion that is as simple to state as it is consequential: no parametrization procedure for multiband $\mathit{k}·\mathit{p}$ Hamiltonians can be regarded as complete unless it explicitly verifies and enforces the ellipticity conditions derived in this paper. This subsection develops that conclusion in three directions. First, it examines how the mathematical structure of the $\mathit{k}·\mathit{p}$ operator—in particular the question of operator ordering—generates or conceals ellipticity violations. Second, it surveys the state of ab initio-based fitting schemes for multiband models and demonstrates that, despite substantial recent progress, the ellipticity gap has not been systematically closed. Third, it identifies the specific modelling contexts in which the gap is most consequential and where the present parametrization procedure has the most immediate impact.

8.2.1. Operator Ordering, the Burt–Foreman Theory, and the Ellipticity Gap

The question of operator ordering in multiband envelope-function theories is inseparable from the question of ellipticity. In the standard Luttinger–Kohn formulation, the Hamiltonian is written in the momentum representation and then converted to the position representation by replacing $k_i$ with $-i\partial /\partial x_i$. When the material parameters vary spatially, as they do at heterostructure interfaces, this replacement is non-unique, and different ordering conventions yield operators with different principal symbols and hence different ellipticity properties. The Burt–Foreman (BF) theory [36,48] was developed precisely to address this non-uniqueness from first principles, deriving an envelope-function theory in which the operator ordering follows from the microscopic quantum mechanics rather than being imposed phenomenologically. As noted in [206], however, the BF operator ordering with experimental $\mathit{k}·\mathit{p}$ input parameters violates ellipticity requirements—albeit mildly—in the four-band and six-band cases, while symmetrization procedures violate them strongly. The eight-band Hamiltonian was analyzed in that work and the authors concluded that the parametrization of multiband $\mathit{k}·\mathit{p}$ models must be reviewed; the present paper carries out that review systematically and for the first time across all commonly used Hamiltonians and materials.

The significance of operator ordering extends to numerical implementations. Finite-element treatments of $\mathit{k}·\mathit{p}$ multiband envelope equations [17,61] and finite-difference schemes designed to eliminate spurious solutions through operator ordering [64,65] all depend, implicitly, on the ellipticity of the operator being discretized. When the continuous operator is non-elliptic, no discretization scheme can fully recover the correct spectral properties, and attempts to suppress spurious solutions at the numerical level [64,65] address the symptom rather than the cause. The subpixel smoothing approach developed in [66] for multiband effective-mass Hamiltonians of semiconductor nanostructures is a further example: while technically sophisticated, it proceeds without verifying whether the smoothed operator satisfies ellipticity conditions, a gap that the present work is uniquely positioned to close. The parametrization procedure developed here operates at the level of the continuous operator and its principal symbol, and it therefore provides a foundation that is independent of the subsequent choice of numerical method.

The relationship between operator ordering, boundary conditions, and ellipticity is not merely technical. As demonstrated rigorously in [51] and discussed in Section 8.1, small perturbations in the type of boundary conditions imposed on an elliptic operator can profoundly alter the solution space. When the operator is non-elliptic, this sensitivity is uncontrolled: the boundary conditions can no longer be derived from first principles, and the entire modelling framework rests on an insecure foundation. The consistent parametrization procedure developed here therefore has direct consequences for the formulation of boundary conditions in heterostructure models, complementing the first-principles derivation of envelope-function boundary conditions in [49] and the general boundary conditions for the multiband $\mathit{k}·\mathit{p}$ model derived in [186].

8.2.2. Ab Initio Fitting Schemes and the Ellipticity Requirement

The fitting of $\mathit{k}·\mathit{p}$ Hamiltonian parameters to ab initio band structures has undergone a renaissance in the past decade, driven by the availability of high-quality density-functional theory (DFT) calculations and the growing recognition that empirically determined parameters carry substantial uncertainty [74,138,179,180]. The eight-band $\mathit{k}·\mathit{p}$ formalism, which can be considered the backbone of modern semiconductor heterostructure modelling [74], has been extended to 16, 30, and 40 bands for zinc-blende and wurtzite systems [75,76,171,176,177,178], and ab initio-based methods for constructing symmetry-adapted Hamiltonians directly from DFT output have been developed [179]. A systematic computational study of bulk wurtzite III–V semiconductors using predictive ab initio methods, providing complete parameter sets for six-band $\mathit{k}·\mathit{p}$ models [180], illustrates the breadth and ambition of this programme.

Despite these advances, a critical gap persists. As the present paper demonstrates, and as the development of the sixteen-band wurtzite model in [138] acknowledges, ellipticity conditions can and should be incorporated as constraints within fitting schemes [207]. The fitting procedure in [138] uses low-discrepancy sequences to fit $\mathit{k}·\mathit{p}$ band structures beyond the eight-band scheme to hybrid-functional DFT data, and explicitly notes that ellipticity conditions can be taken into account to make the resulting parameter sets robust against spurious solutions. This is an important step, but it is a step taken for a single material system (wurtzite GaAs) and a single model dimension (sixteen bands). The present work provides the systematic ellipticity conditions—in the form of explicit, analytically derived linear inequalities on the Luttinger-like parameters—that must serve as hard constraints in any such fitting procedure, across all materials and all standard model dimensions. Any fitting scheme that does not impose these constraints as necessary conditions cannot guarantee the mathematical validity of the resulting operator, regardless of how accurately it reproduces the ab initio band dispersion.

It should be noted that a direct comparison of rescaled $\mathit{k}·\mathit{p}$ band structures with raw DFT results—while a natural first thought—is complicated by a well-known and fundamental limitation of first-principles approaches: standard LDA and GGA functionals systematically underestimate fundamental band gaps, and even hybrid-functional or GW-corrected calculations require the band gap and spin–orbit coupling to be fitted to experimental values before the resulting dispersions can serve as meaningful targets for $\mathit{k}·\mathit{p}$ parametrization [74,138,139,145,171,178,180,208,209]. The $\mathit{k}·\mathit{p}$ framework is empirical by design, with parameters calibrated to experimental observables near band extrema; its natural validation reference is therefore experimental band structure data rather than an independently approximated ab initio dispersion. The band structure fits presented in Figure 3, Figure 4, Figure 5 and Figure 6 confirm that the rescaled, elliptic parameter sets reproduce this experimental reference to the same accuracy as the original parameters, while simultaneously guaranteeing the mathematical well-posedness of the resulting operator.

The multiscale character of the problem underscores this point. The $\mathit{k}·\mathit{p}$ theory operates at the mesoscopic level, providing a computationally efficient description of electronic states that incorporates microscopic information through its parameters. This atomistic-to-continuum coupling [71,72,73] is valuable precisely because the $\mathit{k}·\mathit{p}$ framework can integrate coupled fields—mechanical, thermal, electromagnetic, piezoelectric—that are difficult to treat at the fully atomistic level. The integrity of this multiscale coupling depends on the mathematical well-posedness of the mesoscopic operator, which in turn depends on the ellipticity conditions derived in this paper. When parameters obtained from atomistic calculations or experiments are incorporated into the mesoscopic model without verifying these conditions, the integration of data across scales violates the fundamental structure of the underlying mathematical model, as has been argued in [77,78] in the context of DFT-based multiscale coupling. Machine learning methods for multifidelity scale bridging [79] face the same requirement: the intermediate-fidelity $\mathit{k}·\mathit{p}$ representations that serve as training targets or surrogate models must satisfy ellipticity conditions to be reliable components of a multifidelity pipeline.

8.2.3. Quantum Transport, Nonequilibrium Dynamics, and the Role of Ellipticity

The consequences of operator non-ellipticity are not confined to stationary eigenvalue problems. When the multiband $\mathit{k}·\mathit{p}$ Hamiltonian is used as the basis for quantum transport calculations, the non-ellipticity of the operator propagates directly into the transport formalism. Efficient and numerically stable multiband $\mathit{k}·\mathit{p}$ treatments of quantum transport in semiconductor heterostructures [195] rely on the transmitting boundary method, whose stability properties are tied to the spectral properties of the underlying operator; a non-elliptic operator undermines this stability at the most fundamental level. The quantum transmitting boundary method and its generalizations, as well as approaches based on perfectly matched layers [183] and discrete transparent boundary conditions for time-dependent $\mathit{k}·\mathit{p}$-Schrödinger systems [184,185], all presuppose that the operator being integrated is well-posed; the present parametrization procedure ensures that presupposition is actually satisfied.

The nonequilibrium dynamics of semiconductor nanostructures introduces further demands. Semiconductor superlattices and quantum wells are not isolated, closed systems: they are open and dissipative, exchanging energy and matter with their environment in ways that produce rich nonlinear spatio-temporal dynamics, including pattern formation and, under appropriate conditions, chaotic behaviour [210]. The relaxation-time approximations used in quasi-hydrodynamic models of semiconductor devices [91] and the hyperbolic Hamilton–Jacobi–Bellman equations governing deterministic and stochastic dynamics in such systems [93] are all sensitive to the spectral properties of the underlying quantum-mechanical operator. For the time-dependent Schrödinger equation based on a non-elliptic multiband Hamiltonian, the probability current is not conserved [26], the semigroup generated by the operator need not be contractive, and the distinction between physical and spurious time-evolved states becomes operationally impossible to maintain. The present parametrization procedure eliminates this problem at its source.

Open quantum systems, whose dynamics cannot be described by unitary operators alone, require special treatment of both the Hamiltonian and the boundary conditions. The open-system dynamics of entanglement [188] and the broader framework of nonseparable quantum states [189] make clear that the environment-induced decoherence processes affecting low-dimensional semiconductor nanostructures operate in a space whose structure is determined by the Hamiltonian operator. The relaxation-time and quasi-hydrodynamic models for semiconductor devices [91,93] represent an intermediate description level. These models are derived by taking moments of the Boltzmann Transport Equation, which utilize carrier velocities $\mathbf{v}\left(\mathbf{k}\right)$ and effective mass tensors defined by the first and second derivatives of the $\mathit{k}·\mathit{p}$ dispersion $E\left(\mathbf{k}\right)$, respectively. If the underlying Hamiltonian is non-elliptic, these derivatives can become ill-defined or unphysical at high momenta. A non-elliptic Hamiltonian corrupts this entire hierarchy; in macroscopic transport frameworks derived via the method of moments—including applications to the quantum Boltzmann equation—this manifests as a loss of strict hyperbolicity in the closed system of conservation laws, rendering the derived transport equations locally ill-posed and triggering severe numerical instabilities. An elliptic one, of the kind produced by the present parametrization procedure, provides a sound foundation for all levels of description.

8.3. Inverse Problems, Nonlocal Phenomena, and the Design of Low-Dimensional Nanostructures

The connection between the consistent parametrization of multiband Hamiltonians and the solution of inverse problems is not incidental. It follows directly from the structure of the forward problem: the eigenvalue problem associated with a multiband $\mathit{k}·\mathit{p}$ Hamiltonian is the mathematical model whose solution gives the electronic properties of a nanostructure; the inverse problem asks what structural parameters—compositions, layer thicknesses, potential profiles, strain distributions—yield prescribed electronic properties. An inverse problem is only as well-posed as the forward model it inverts [87,102], and the forward model is only well-posed when the underlying Hamiltonian operator is elliptic. The present work therefore provides an indispensable mathematical prerequisite for the inverse design of semiconductor nanostructures, a problem of growing practical and theoretical importance [81,82,211].

8.3.1. Size Effects, Nonlocal Models, and the Limits of Effective-Mass Approximations in Confined Quantum Systems

A fundamental manifestation of the distinction between bulk and nanoscale physics is the appearance of quantum size effects: the confinement of charge carriers to spatial dimensions comparable to or smaller than the de Broglie wavelength produces energy spectra and transport properties that depend sensitively on the dimensions and geometry of the confining structure. Low-dimensional nanostructures—quantum wells, wires, rods, and dots—derive their technological utility precisely from this sensitivity, which enables the tuning of optoelectronic properties through structural design [10,11]. Quantum dots, in particular, provide excellent materials for bioimaging, biolabeling, biosensing, and bioinspired nanoelectronics applications, including conjugation to macromolecules such as DNAs and RNAs [196]. In each of these applications, the knowledge of optoelectronic properties—which can be further influenced by mechanical, piezoelectric, and thermal fields—is essential, and it flows through the multiband $\mathit{k}·\mathit{p}$ framework whose mathematical foundation is established in this paper.

The appropriate theoretical framework for size effects is, at the deepest level, a nonlocal one. The effective-mass approximation, including its multiband $\mathit{k}·\mathit{p}$ generalization, achieves its computational efficiency by averaging over the microscopic lattice degrees of freedom, replacing them with effective parameters. This averaging is valid when the envelope function varies slowly on the scale of the lattice parameter, but breaks down when quantum confinement forces the envelope function to vary on scales comparable to the lattice period. In such cases, the local effective-mass description must be supplemented or replaced by nonlocal models that retain explicit information about the microscopic structure. Microstructural defects and inhomogeneities similarly require nonlocal treatments [212], as does the description of quantum dots doped with magnetic impurities [196], where the exchange interaction between confined carriers and localized magnetic spins produces collective effects that are intrinsically nonlocal in character.

Nonlocality can enter the governing equations, the boundary conditions, or the initial conditions [95]. In the $\mathit{k}·\mathit{p}$ context, the most immediate form of nonlocality arises through the boundary conditions at heterointerfaces, where the abrupt change in material composition forces the envelope function to satisfy matching conditions that encode information about the entire interface rather than local values alone. The consistent formulation of these conditions, as developed in [24,49,186], requires the self-adjointness of the Hamiltonian operator—which is guaranteed by the ellipticity conditions derived in the present paper. Beyond the interface problem, nonlocal initial conditions arise naturally in quantum-mechanical models when memory effects or non-Markovian dynamics are present, as demonstrated in [95], and the inverse problem of recovering such conditions from incomplete observational data represents a further frontier in the mathematical treatment of open quantum systems [101,129].

The implications of size effects for the parametrization of multiband models are direct and material. The Luttinger parameters ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ and the Kane parameters $E_p$, $A^{\prime }$, B that appear in the $8\times 8$ Hamiltonian are bulk quantities, determined from bulk measurements or bulk ab initio calculations. When these parameters are used to describe quantum-confined systems, the bulk values may require corrections that reflect the modified dielectric environment, the altered phonon spectrum, and the changed screening properties of the confined system. For ternary alloys, the parameters are calculated by linear interpolation between the constituent parameters; the ellipticity region is convex and connected in the parameter space, so the interpolated parameters will be elliptic whenever the endpoint parameters are [19]. For time-dependent parameters described by Varshni formulas, the same convexity argument applies. The admissible parameter sets derived in this paper for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaN, AlN, and InN therefore provide a reliable foundation for modelling the full range of III–V alloy nanostructures encountered in applications, provided the interpolation is performed within the elliptic region. Core-shell nanowires [59], InAs quantum dots on GaAs with wetting layers [49], and doped nanowire superlattices [194] are representative examples where this foundation is directly operative.

8.3.2. Inverse Scattering Methods and Their Renewal Through Bayesian and Machine-Learning Approaches for Nanostructure Design

The inverse problem for semiconductor heterostructures is most naturally formulated in the language of inverse quantum scattering: given spectral or scattering data characterizing the electronic states of a nanostructure, reconstruct the potential profile that gives rise to those states [81,82,85,86]. This formulation connects directly to a body of mathematical theory concerning the existence, uniqueness, and stability of solutions to inverse scattering problems [83,88,89], and to a growing set of computational algorithms for their numerical solution.

The earliest applications of inverse scattering to semiconductor heterostructure design treated the one-channel, one-dimensional problem: the reconstruction of a potential from reflection or transmission data for a single band [82]. The key computational tool in this setting is the Gelfand–Levitan–Marchenko (GLM) integral equation, whose solution gives the transformation kernel from which the potential is recovered [213,214]. The multichannel generalization, which is the natural setting for multiband $\mathit{k}·\mathit{p}$ models, is substantially more complex: the reconstruction must recover a matrix-valued potential from a matrix-valued scattering or reflection coefficient. Systematic algorithms for the multichannel inverse scattering problem in one dimension have been developed, including the derivation of the general Marchenko integral equation for N-coupled channels [215], the direct and inverse multichannel scattering theory of Lyubarskii and Marchenko [216], and the Newton–Sabatier-based method for coupled reaction channels at fixed energy [217,218]. The transmutation operator method [214] provides a further approach, reducing the inverse problem to a system of linear algebraic equations whose solution recovers the potential with remarkable numerical accuracy and stability.

The design of quantum filters with predetermined reflection and transmission properties [219] represents a direct technological application of this framework: by specifying the desired scattering matrix and applying inverse scattering theory, one obtains the potential profile—and hence the heterostructure design—that achieves those properties. The multichannel extension of this design methodology [211] encompasses the full generality of quantum systems with multiple coupled channels, including the possibility of concentrating wave functions in a chosen partial channel, engineering spatial localization, and producing resonance tunnelling with prescribed widths and energies. Toward the quantum design of multichannel systems through the inverse problem approach [211] thus represents a systematic theoretical framework for the inverse design of nanostructures whose electronic properties are described by coupled multiband equations.

The R-matrix approach, originally developed for nuclear reaction scattering [118], provides a complementary computational framework for integrating the multiband $\mathit{k}·\mathit{p}$ equation in layered semiconductor structures. The log-derivative R-matrix method based on the Jost solution to the $\mathit{k}·\mathit{p}$ equation [118] reduces the numerical instability that arises in type-II heterostructures from the simultaneous presence of propagating and evanescent states, and is directly applicable to the 14-band $\mathit{k}·\mathit{p}$ model analyzed in Section 7 of this paper. The multiband $\mathit{k}·\mathit{p}$ Riccati equation for electronic structure and transport in type-II heterostructures [80] provides yet another route to the inverse problem, connecting the scattering formalism to the reconstruction of potential profiles from transport data.

The reconstruction of potentials in quantum dots and other small symmetric structures [117] and the spectral parameter power series approach to eigenvalue problems with applications to nanostructure design [119] extend the inverse scattering framework to genuinely three-dimensional geometries, where the Radon transform and related integral transforms play the role of the GLM equation in reducing the three-dimensional problem to a sequence of one-dimensional inversions. For quantum wires, rods, and dots, where the motion of carriers is confined from two or three spatial dimensions, the development of such three-dimensional inverse techniques represents a frontier of both mathematical and applied research, one that is directly enabled by the consistent parametrization of the forward multiband model established in this paper.

A notable and consequential development in recent years has been the renewal of interest in inverse scattering methods through their integration with Bayesian statistical frameworks and machine-learning techniques. Bayesian optimization has been applied to build global potential energy surfaces for reactive molecular systems using feedback from quantum scattering calculations [220], demonstrating that accurate global surfaces can be constructed from a small number of ab initio points by an iterative process that samples the most relevant parts of configuration space. This machine-learning-assisted inverse scattering approach is directly transferable to the nanostructure design problem: the potential profile of a heterostructure can be treated as the unknown to be recovered, the target electronic properties serve as the data, and Bayesian optimization guides the iterative reconstruction. The Bayesian approach to inverse problems more generally [101] provides a rigorous statistical framework for quantifying the uncertainty in the recovered potential, distinguishing between aspects of the solution that are genuinely determined by the data and those that are effectively guessed. This distinction is of direct relevance when the forward model, such as a multiband $\mathit{k}·\mathit{p}$ Hamiltonian, carries its own structural approximations whose effect on the inverse solution must be controlled [104].

The deep-learning approach to first-principles transport simulations [221], in which machine learning maps local structural descriptors to electronic conductance properties, represents a further development in this direction. By combining first-principles transport calculations with machine-learning-based nonlinear regression, this approach demonstrates that accurate predictions of transport properties for large systems can be obtained at a fraction of the computational cost of direct first-principles calculations—but only when the local descriptors and the training data are generated by mathematically valid forward models. The consistent parametrization procedure developed in this paper ensures that the $\mathit{k}·\mathit{p}$-based forward models used in such pipelines satisfy the necessary mathematical conditions, providing a reliable foundation for the machine-learning layer that sits atop them.

The inverse design laboratory paradigm [103]—in which multi-objective Bayesian optimization is used to navigate the design space of nanostructures and identify structures with prescribed sets of properties—requires forward models that are not only accurate but mathematically well-posed. An ill-posed forward model produces a landscape of predicted properties that is corrupted by spurious features, misleading the optimization into design regions that would not, in reality, exhibit the desired properties. The ellipticity conditions established in this paper, and the admissible parameter sets derived from them, eliminate this source of corruption from the $\mathit{k}·\mathit{p}$ forward model and thereby enable reliable multi-objective Bayesian optimization for the inverse design of III–V semiconductor nanostructures.

Finally, the connection between inverse problems and nonlocal models deserves emphasis as a direction of future research that is directly enabled by the present work. The inverse problem of recovering nonlocal initial or boundary conditions from partial observations [95,131] is substantially harder than its local counterpart, because the data required to uniquely determine the nonlocal conditions may not be accessible from the limited measurements available in practice. The admissible parameter sets derived in Section 5, Section 6 and Section 7 of this paper, together with the rescaling procedure and the explicit ellipticity constraints, provide the forward model with the mathematical regularity that is a prerequisite for the stable formulation and numerical solution of any such nonlocal inverse problem: without a well-posed forward operator, the inverse problem is doubly ill-posed, and no amount of regularization can compensate for the instabilities introduced by a non-elliptic Hamiltonian. Mathematical tools to overcome the resulting subsampling bias [131] and to estimate the information content of inverse problem data [104] are active research areas whose development is directly relevant to the nanoscience context, where experimental observations of quantum systems are typically limited to ensemble averages or to quantities integrated over large spatial domains. Optimal prediction methods that compensate for the lack of resolution through the use of prior statistical information [222,223] provide a further set of tools for the partially observed nanostructure design problem. The convexity of the ellipticity region in the parameter space, established analytically in this paper, implies that any convex combination of admissible parameter sets remains admissible—a property that is directly exploitable in Bayesian and optimization-based inverse methods, where the search is naturally constrained to the feasible region. This structural feature of the elliptic parameter space thus transforms the results of Section 5, Section 6 and Section 7 from a set of material-specific parameter tables into a rigorous geometric constraint that can be embedded as a hard inequality in any data-driven or machine-learning-assisted inverse design workflow for III–V semiconductor nanostructures.

8.4. Quantum Control and the Design of Dynamical Nanostructure Responses

The discussion of inverse problems in Section 8.3 focused primarily on the recovery of static structural parameters—compositions, layer thicknesses, and potential profiles—from spectral or scattering data. A complementary and equally important class of inverse problems concerns the design of dynamical responses: given a desired quantum state transformation or time-dependent observable, determine the control fields—laser pulses, gate voltages, strain profiles—that produce it. This is the domain of quantum optimal control theory [94,224], and it connects naturally to the multiband $\mathit{k}·\mathit{p}$ framework developed in this paper.

8.4.1. Quantum Optimal Control: Mathematical Foundations and Their Connection to Ellipticity

The mathematical formulation of quantum optimal control problems [94] requires, at minimum, a well-posed forward model: the time-dependent Schrödinger equation governed by the controlled Hamiltonian must generate a unitary propagator that maps initial states to final states continuously and stably as a function of the control field. When the Hamiltonian is non-elliptic, as demonstrated here for all standard multiband $\mathit{k}·\mathit{p}$ Hamiltonians with uncorrected parameters, the semigroup generated by the operator need not be contractive, the propagator need not be unitary, and the optimal control problem may have no solution in any physically meaningful sense. The admissible parameter sets derived in this paper thus constitute a necessary condition for the mathematical well-posedness of quantum optimal control problems formulated within the $\mathit{k}·\mathit{p}$ framework.

Quantum control is concerned with active manipulation of physical and chemical processes on the atomic and molecular scale [224]. The two most critical theoretical insights in the field are the realization that ultrafast atomic and molecular dynamics can be controlled via manipulation of quantum interferences, and that optimally shaped laser pulses are the most effective means of producing desired interference patterns. These insights, combined with the concept of adaptive feedback control employing closed-loop optimization [224], have produced a body of experimental results across physics and chemistry that depends, implicitly, on the mathematical validity of the Hamiltonian used to predict and interpret the controlled dynamics. In the semiconductor nanostructure context, where the relevant Hamiltonian is a multiband $\mathit{k}·\mathit{p}$ operator, the present work establishes the parameter conditions under which that Hamiltonian is mathematically valid, and hence under which quantum control predictions can be trusted.

Optimal and robust quantum control by inverse geometric optimization [124] represents a further development, in which the control problem is formulated as a geometric optimization on the space of unitary operators. The robustness of the resulting control protocols with respect to experimental imperfections—deviations of the actual Hamiltonian from the nominal model—depends critically on the smoothness and stability of the forward model. A non-elliptic Hamiltonian introduces parameter-dependent instabilities into the forward model that cannot be characterized as experimental imperfections and cannot be corrected by robustness optimization; they can only be removed by the consistent reparametrization procedure developed in the present paper.

8.4.2. The R-Matrix Approach, Scattering-Matrix Methods, and Their Role in Nanostructure Design

The R-matrix theory, originally developed by Wigner and Eisenbud for the description of resonances in nuclear scattering, provides a powerful and numerically stable framework for computing transmission and reflection coefficients in quantum heterostructures within the multiband $\mathit{k}·\mathit{p}$ approximation [118]. The log-derivative R-matrix method based on the Jost solution to the $\mathit{k}·\mathit{p}$ equation [118] is directly applicable to the 14-band model analyzed in Section 7 of this paper, and reduces the numerical instability that arises in type-II heterostructures from the simultaneous presence of propagating and evanescent states. Its connection to the multiband $\mathit{k}·\mathit{p}$ Riccati equation for electronic structure and transport [80] provides a further route from the scattering formalism to the design of heterostructure potential profiles with prescribed transport properties.

The scattering-matrix approach occupies a central position in the connection between forward modelling and design: the scattering matrix encodes all transport information about a heterostructure in a form that is directly experimentally accessible, and the inverse problem of recovering the potential profile from the scattering matrix is precisely the design problem. For the 14-band model, the admissible parameter sets derived in this paper are a prerequisite for the numerical stability of the R-matrix computation itself: as established in Section 7, the second implementation of the $14\times 14$ Hamiltonian is non-elliptic by design owing to the vanishing of second-order diagonal terms for three upper p-like conduction bands, and no R-matrix calculation based on such an operator can be expected to yield numerically stable results for those states. The revision of the underlying assumptions and the extension of the consistent parametrization procedure to the 14-band case, outlined in Section 7, is therefore a prerequisite for the reliable application of the R-matrix and scattering-matrix methods to the full set of conduction band states in the extended-basis models.

The connection between quantum control and scattering is deepened by the inverse scattering perspective: the desired transmission or reflection matrix serves as the target of the design problem, and the potential profile (heterostructure design) is the unknown to be recovered. Optimal and robust quantum control [124] combined with the R-matrix transport formalism [118] thus constitutes a complete pipeline for the design of semiconductor nanostructures with prescribed quantum dynamical properties, one whose mathematical foundation is secured by the consistent parametrization of the underlying $\mathit{k}·\mathit{p}$ Hamiltonian established in this paper.

8.5. AI-Assisted Inverse Design of Nanostructures: The Indispensable Role of Mathematically Consistent Forward Models

With the advent of machine learning (ML) and deep learning (DL), data-driven approaches to electronic structure calculations in nanoscience and nanotechnology have received a substantial boost. The development of deep-learning methods for first-principles calculations, including deep-learning density functional theory and deep-learning quantum Monte Carlo, has extended the reach of electronic structure computations to scales and complexities previously inaccessible [225]. The generalization of deep-learning electronic structure calculations from the atomic-orbital basis to the plane-wave basis, bridging a long-standing methodological gap [226], and the machine-learning-assisted real-time feedback control of quantum dot growth [227] illustrate the pace at which AI tools are transforming the experimental and computational practice of nanoscience. Against this backdrop, the consistent parametrization of multiband Hamiltonians developed in this paper assumes a significance that extends far beyond the immediate problem of correcting bandstructure parameters: it constitutes the mathematical bedrock on which any AI-assisted design pipeline for III–V semiconductor nanostructures must rest.

8.5.1. Data-Driven Methods, Partially Observed Systems, and the Hierarchy of Models

Data-driven models for nanostructure design operate in a setting that is inherently partially observed: only a subset of the relevant degrees of freedom is accessible to measurement or computation, and the model must make reliable predictions about the full system from incomplete data. As demonstrated in [223], data-driven models can outperform physics-based models for partially observed systems precisely because they implicitly model the effects of the missing degrees of freedom through delay-coordinate embeddings and their evolution under the Koopman operator, without requiring explicit knowledge of those degrees of freedom. This empirical success, however, presupposes that the training data generated by the physics-based forward model is itself reliable: if the forward model is non-elliptic and hence admits spurious solutions, the training data will contain features that have no physical counterpart, and the data-driven model will learn to reproduce those features rather than the physical ones.

The $\mathit{k}·\mathit{p}$ theory operates at the mesoscopic level of description, above the fully atomistic but below the device level. It provides a computationally efficient bridge between the microscopic (DFT or tight-binding) and the macroscopic (device simulation) levels, and it is at this mesoscopic level that the majority of training data for ML-based nanostructure design is generated. The Wannier–Slater and Luttinger–Kohn effective-mass approximations that underpin this mesoscopic description both require careful treatment of the current density operator representation, and already at this level the conditions derived in this paper are essential: moving from the Laplacian to an effective Hamiltonian with specific current density operator representations requires that the resulting operator be elliptic, as established in Section 5, Section 6 and Section 7. When parameters obtained from atomistic calculations or experiments are incorporated into the mesoscopic model—as in the multiscale, multifidelity frameworks discussed in Section 8.1.2 [78,79]—the integration of data across scales must respect the fundamental structure of the underlying mathematical model. The consistent parametrization procedure developed in this paper is the mechanism by which that respect is operationalized.

The hierarchy of approximations implicit in the $\mathit{k}·\mathit{p}$ framework—from the full many-body problem to the single-particle Schrödinger equation, from the full Schrödinger equation to the multiband effective-mass approximation, and from the bulk Hamiltonian to the heterostructure model with interface matching conditions—introduces approximation errors at each level that must be controlled. The ellipticity conditions established in this paper control the error introduced at the level of the multiband Hamiltonian itself: they ensure that the principal symbol of the operator has the correct sign, that the spectrum is semi-bounded from below, and that the eigenstates are physically meaningful. Without this control, the errors introduced at the Hamiltonian level propagate through the entire hierarchy and contaminate every level of the data-driven model built upon it. The ODIL framework [103], which replaces standard neural networks with grid-based discretizations for solving inverse problems while maintaining mathematical transparency, is one of several approaches that require a well-posed forward model to achieve their promised efficiency and accuracy; the present parametrization procedure ensures that the $\mathit{k}·\mathit{p}$ forward model satisfies the necessary conditions.

8.5.2. Machine Learning for Bandgap Engineering, Nanostructure Design, and the Critical Role of Forward Model Validity

The application of ML to bandgap engineering [105] and to the broader problem of inverse design of materials by machine learning [114] has demonstrated that ML algorithms can efficiently navigate the high-dimensional parameter spaces of semiconductor materials to identify compositions and structures with desired electronic properties. Generative models and deep learning for materials discovery [115], invertible crystallographic representations for inverse design of inorganic crystals [116], and ML-based inverse design methods considering data characteristics and design space size [125] collectively represent a rapidly maturing methodology for data-driven materials design. Physics- constrained neural networks deployed to solve the inverse design problem for quantum dot structures in two-dimensional Dirac materials [123] demonstrate that scattering efficiency can be designed to vary over two orders of magnitude through appropriate gate potential combinations, and deep neural networks have been applied to the prediction of optical properties and free-form inverse design of metamaterials [108], to inverse design in quantum nanophotonics via local density of states [107], and to multi-target inverse design of nanoparticles [113]. Neural inverse design of nanostructures through physics-based deep learning [106] provides yet another route to automated structure optimization.

It should be noted that the ellipticity analysis performed here is specific to three-dimensional zinc-blende crystals; its extension to intrinsically non-three-dimensional materials such as graphene, silicene, and related two-dimensional systems, for which high-accuracy bandstructure data are now readily available [126,182,228,229], requires a separate derivation of the ellipticity conditions for the corresponding reduced-dimensional Hamiltonians. Such an extension would substantially broaden the scope of AI-assisted design workflows for two-dimensional nanostructures and van der Waals heterostructures.

A common thread running through all these approaches—and a thread that the present paper pulls to its logical conclusion—is the observation that the quality of the inverse design output is determined by the quality of the forward model. As emphasized in the systematic classification of ML inverse design problem types provided in [125], the reliability of interpolation, extrapolation, multifidelity, and small-data design all depend on whether the training data is generated by a mathematically valid physical model. For semiconductor nanostructures described by multiband $\mathit{k}·\mathit{p}$ theory, the present paper establishes, for the first time in full generality, the conditions under which that model is mathematically valid, and provides the explicit admissible parameter sets for all major III–V and III-nitride materials that ensure those conditions are met. Any ML-based inverse design pipeline for such nanostructures that does not incorporate these constraints—either by using non-elliptic Hamiltonians for training data generation or by failing to impose ellipticity as a hard constraint on the parameter space—operates on a compromised foundation.

The Bayesian reformulation of inverse problems [101], which provides a precise notion of information density and distinguishes between aspects of the inverse solution genuinely determined by data and those effectively guessed, is particularly instructive in this context. When the forward model carries structural approximation errors—as any non-elliptic $\mathit{k}·\mathit{p}$ Hamiltonian does—the posterior distribution of the recovered parameters will be biased in ways that cannot be corrected by more data or better algorithms. The admissible parameter sets derived in this paper remove this source of bias from the forward model, enabling Bayesian inverse design [101,129] to operate on a statistically sound foundation. The practical consequence is that the elliptic parameter space derived here, being convex and connected, is not only geometrically natural as a feasible region for optimization but is also statistically natural as a prior support for Bayesian inference: the posterior will be concentrated within the elliptic region, and the prior can be chosen to reflect this geometric structure.

8.5.3. The Broader Impact: From III–V Semiconductor Nanostructures to Biological and Biomedical Applications

The applications of low-dimensional semiconductor nanostructures extend well beyond semiconductor physics and device engineering. Quantum dots provide excellent materials for bioimaging, biolabeling, biosensing, and bioinspired nanoelectronics, including conjugation to macromolecules such as DNA and RNA. Machine learning has been applied to the automated real-time feedback-controlled growth of InAs/GaAs quantum dots [227], demonstrating that ML can tune quantum dot densities across three orders of magnitude in near-real time. However, the quality of the resulting quantum dots, as light sources, qubits, or biosensing elements, depends on the accuracy of the electronic structure model used to predict their properties. This accuracy, in turn, depends on the mathematical validity of the underlying $\mathit{k}·\mathit{p}$ Hamiltonian. Machine learning has also been applied to shape-memory graphene nanoribbons with relevance to biomedical engineering [128], as well as to synthetic biology [230] and metabolic engineering [231]. These examples further illustrate that the data-driven design paradigm is now operative across the full breadth of the life and material sciences. In each domain, the quality of the physics-based forward model determines the reliability of the data-driven output.

The ellipticity conditions and admissible parameter sets derived in this paper are also applicable to Hamiltonians with parameters that vary with temperature, such as those described by Varshni formulas for semiconductor band gaps, since the convexity of the elliptic region guarantees admissibility along any convex path in parameter space. For Hamiltonians with more complex symmetry structures where analytic calculation of the quadratic form eigenvalues is not tractable, the ellipticity of a specific parameter set can always be verified numerically, so the present framework imposes no fundamental restriction on the range of materials or model dimensions to which it can be applied.

The consistent parametrization framework developed in this paper is therefore not merely a technical correction to a set of Hamiltonian parameters. It is a foundational contribution to the mathematical infrastructure of AI-assisted nanoscience: it establishes the conditions under which the central tool of semiconductor nanostructure modelling—the multiband $\mathit{k}·\mathit{p}$ Hamiltonian—is mathematically valid, and it provides the explicit parameter sets that satisfy those conditions for all major materials. Moreover, it demonstrates that those parameter sets can be obtained from existing data by a systematic, justified, and computationally efficient rescaling procedure. The convexity of the admissible parameter space, the explicit ellipticity inequalities for 6×6, 8×8, and 14×14 Hamiltonians, and the role of the inversion-asymmetry parameter B as an additional degree of freedom for simultaneously maintaining ellipticity and improving the bandstructure fit—all of these results are directly actionable in any data-driven, optimization-based, or AI-assisted design workflow for III–V and III-nitride semiconductor nanostructures. Their impact will grow as data-driven methodologies become ever more deeply embedded in nanoscience and nanotechnology, making the mathematical consistency of the forward models on which they rely not merely desirable but indispensable.

9. Conclusions

The present paper has addressed a fundamental mathematical difficulty in one of the most widely used frameworks for modelling electronic states in semiconductor nanostructures: the non-ellipticity of multiband Luttinger–Kohn Hamiltonians in the position representation. The analysis is systematic, covering the $6\times 6$, $8\times 8$, and $14\times 14$ zinc-blende Hamiltonians, and it is exhaustive in its treatment of the materials for which these models are most commonly applied.

For the $6\times 6$ Hamiltonian, the ellipticity conditions take the form of four explicit linear inequalities on the Luttinger parameters ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$. Evaluation across all gathered parameter sets confirms that among all analyzed materials only carbon possesses admissible parameters; all other materials violate the constraint $2{\gamma }_2-{\gamma }_1+3{\gamma }_3<0$, traceable to the perturbative representation of conduction-band influence on the valence bands. This establishes that the $6\times 6$ model does not provide a mathematically valid approximation for those materials, and that the appearance of spurious solutions in numerical implementations is not an artefact of discretization but a consequence of the operator’s non-ellipticity.

The situation for the $8\times 8$ Kane Hamiltonian is more complex. The inclusion of the lowest conduction band in the basis set introduces an additional ellipticity constraint on the dimensionless parameter A, which governs the curvature of the conduction band. In the absence of the inversion-asymmetry parameter ($B=0$), no standard parameter set for any of the twelve analyzed III–V and III-nitride materials satisfies the full set of ellipticity conditions. The degree of violation in the valence-band part is, however, reduced compared to the $6\times 6$ case, and several parameter sets for InAs, AlP, and AlSb satisfy the valence-band constraints, providing direct evidence for the perturbative origin of non-ellipticity. A parameter rescaling procedure, based on adjusting the momentum matrix element $E_p$ to satisfy all ellipticity constraints simultaneously, is developed and justified. The resulting admissible parameter sets for all twelve materials are reported, together with quantitative bandstructure adjustment errors that demonstrate the quality of the bandstructure fit is preserved to within 11 meV for AlN, AlP, AlSb, AlAs, GaN, and InP and within 50 meV for GaP, InSb, and InN.

A key new result is the derivation of ellipticity conditions for the $8\times 8$ Hamiltonian with non-zero B, which shows that B cannot be set to zero for materials where $1+A^{\prime }<0$ without sacrificing ellipticity. Conversely, for the four parameter sets that already satisfy the valence-band ellipticity constraints (InAs, AlP, and AlSb), direct adjustment of B provides a route to full ellipticity that bypasses the rescaling procedure altogether. The parameter B is further demonstrated to serve as a bandstructure fitting parameter after rescaling, and its optimal value for InN is determined to reduce the total eight-band adjustment error by a factor of approximately six compared to the $B=0$ rescaled set. The elliptic parameter space in the full parameter set $\{A^{\prime },{\gamma }_1^{\prime },{\gamma }_2^{\prime },{\gamma }_3^{\prime },B\}$ is convex and connected, a property that has direct implications for optimization-based inverse design.

For the $14\times 14$ Hamiltonians, the analysis reveals that the first implementation carries ellipticity conditions structurally identical to those of the $6\times 6$ model, but written in terms of reduced Luttinger-like parameters ${\gamma }_1^{″}$, ${\gamma }_2^{″}$, ${\gamma }_3^{″}$ that account for the contribution of the in-basis p-like conduction bands. None of the available parameter sets for any analyzed material is admissible, and the source of this persistent non-ellipticity is traced to overly strict assumptions regarding the perturbative influence of out-of-basis bands on the in-basis conduction states. The second $14\times 14$ implementation is non-elliptic by design, owing to the vanishing of second-order diagonal terms for three upper p-like conduction bands. Both implementations are consequently less universal, material-wise, than the $8\times 8$ Hamiltonian, and the revision of their underlying perturbative assumptions is identified as a necessary direction for future work.

The results extend naturally to heterostructure models and ternary alloys, where the convexity of the elliptic parameter space guarantees that linear interpolation between admissible endpoint parameters yields admissible alloy parameters. The analysis covers all extensions of the considered models that preserve the structure of the second-order-in-k terms, including strain, electromagnetic, and piezoelectric contributions. The two-dimensional cases relevant to graphene, silicene, and other non-three-dimensional materials require separate analysis, as the ellipticity conditions derived here are specific to the three-dimensional zinc-blende setting.

The connection between the present results and the broader landscape of data-driven methodologies, inverse design, and AI-enhanced electronic structure calculations, developed in detail in Section 8, underscores the significance of the consistent parametrization framework established here. An inverse design procedure is only as reliable as the forward model it inverts, a data-driven surrogate is only as trustworthy as the training data it learns from, and a multifidelity modelling pipeline is only as sound as its intermediate-fidelity components.

The admissible parameter sets, the rescaling procedure, and the explicit ellipticity inequalities derived in this paper collectively constitute the rigorous mathematical foundation that all such methodologies require. This foundation’s validity extends beyond the immediate context: through the universality of ellipticity as a structural requirement for well-posed operator theory, it applies to any domain of mathematical physics and computational science in which multiband Hamiltonian models, inverse problems, and data-driven methods converge.