1. Synopsis of Fundamentals and Motivation

For many actual practical solutions and technological applications, due to the impressive development of low-dimensional electronic and optoelectronic devices, it is drastically important to include the valence-band mixing¹, i.e. the degree of freedom transverse to the main transport direction, whenever the holes are involved. Previous theoretical studies had focused this topic in resonant tunneling² and had pointed up its relevance for experimental and technological applications¹ . If the electronic transport through these systems, engage both electrons and holes, the low-dimensional device response threshold depends on the slower-heavier charge-carrier’s motion through specific potential regions³. It is unavoidable to recognize that in the specialized literature there is plenty of reports studying several physical phenomena derived from hole mixing effects and strain, via standard existing methods. Some authors had managed to determine optimal situations in a resonant tunneling of holes under internal strains, disregarding scattering effects and assuming a spatial symmetry for a constant potential⁴ . A first-principles study on valence-band (VB)-mixing, established a non-linear response for a pseudo-potential in series of the atomic distribution function⁵ . Over the past few decades VB-mixing and/or strain had been extensively studied in several nanostructures ranging from quantum wells^4,6,7,8 to quantum wires^9,10 and to quantum dots¹¹. However, just a few reports are available regarding the very evolution itself of the effective potential due to several causes, as a central topic of research. We do not focus in the present paper on the VB-mixing and strain effects problem themselves, but rather on the evolution of the effective potential when tuning both effects and on their possible influence over the scattering properties. We hope to make some progress in the understanding of the underlying physics as well as determining whether or not the VB-mixing and strain are competitor mechanisms in the evolution of the effective potential. Few direct measurements -if any- are available because they have not been usually carried out on the subject. We hope to trigger further experimental works on hole-coupling effects and strain phenomenon in the same setup, even in complementary fields whenever scattering events of mixed holes through stressed heterostructures are involved.

Earliest striking elucidations due to Milanović and Tjapkin for electrons,¹² and recalled much later by Pérez-Álvarez and García-Moliner for a fully unspecific multiband theoretical case¹⁴, are fundamental cornerstones in this concern. The metamorphosis of the effective band offset potential V_eff, “felt" by charge carriers depending on the transverse momentum value, is a very persuasive workbench to graphically mimic, the phenomenon of the in-plane dependence of the effective mass, widely known as the VB-mixing for holes. In a few words, a hole band mixing is crucial for bulk and low-dimensional confined systems possessing quantal heterogeneity, as shall be discussed in this paper, similarly to the single-band-electron problem¹² and somehow to the unstrained multiband-hole problem¹³ . Particularities of the appealing evolution features of V_eff for holes, in the presence of gradually increasing VB-mixing under strain and assuming their effects are well understood, could be of interest for condensed-matter physicists, working in the area of quantum transport through standard quantum barrier(QB)-quantum well(QW) layered systems in multiband-multichannel models.

As common assertion, the V_eff input value derives from bulk band offset lineup as long as the transverse momentum (κ_T) values are negligible¹⁴ . For finite κ_T, this assertion is no longer valid and the mixing effects appear. Such phenomenology is a typical non-linear κ_T effect, whose most striking consequence is the predicted interchange of functions between QW and QB^12,14 . Past theoretical studies have emphasized the existence of energy shifted bound states under κ_T-dependent V_eff¹⁵ and a larger reduction for V_eff as a function of κ_T for light holes (lh) in respect to that of heavy holes (hh) . These former works^{5,12,14,15,16}, were motivating enough and put us on an effort to develop a more comprehensive vision, of how V_eff evolves spatially with κ_T and strain for hh and lh. This paper is devoted to demonstrate, that V_eff profile evolution, QB-QW permutations and bandgap changes, are reliable tools for tuning the response threshold of a layered semiconductor system with spatial-dependent effective mass, and that VB-mixing and stress effects are possible concurrent mechanisms.

Built-in elastic strained layered heterostructures, has been remarkably used in the last decade, for development of light-emitting diodes, lasers, solar cells and photodetectors¹⁷. Besides, internal strain may results into a considerable modification of the electronic structure of both electrons and holes, thereby altering the response of strained systems respect to nominal behavior of strain-free designs¹⁷ . To account for strain in the present study, we additionally suppose a heterostructure sandwiched into an arbitrary configuration of pseudomorphically strained sequence of QW-acting and QB-acting binary(ternary) alloys. This assumption is motivated by actual technological interest on specific materials and configurations. On a layer-by-layer deposition, whenever the epitaxially grown layer’s lattice parameter matches that of the substrate in the in-plane direction -without collateral dislocations or vacancies- the process is referred as pseudomorphic [see Fig. 1(b)]¹⁸ . The last one is standardized and chosen for most day-to-day applications like write-read platforms, such as sound/image players and data-manipulating devices. Our motivation arises from the probable existence of a competitor mechanism able to diminish the effects of VB-mixing on V_eff¹³, or even wipe them out occasionally. Such prediction is not available yet in the specialized literature. The uncommon simultaneous treatment of the strain in the presence of hole-coupling effects remains absent or insufficiently addressed, thereof the scientific merit of the present theoretical attempt regarding earlier specialized reports^{4,6,7,8,11,13} is warranted.

The outline for this paper is the following: Sec. 2 presents briefly the theoretical framework to quote the VB V_eff for both unstressed and stressed systems. Graphical simulations on V_eff evolution, are exposed in Sec. 3. In that section, we discuss highly specialized III-V semiconductor binary (ternary)-compound cases with straight links to real-word technological applications. That section supports the main contribution of the present study and suggests possible applications. Sec. 4, contains the conclusions.

2. Calculation of the effective potential

Commonly, a wide class of solid-state physics problems, related to electronic and transport properties, demands the solution of multiband-coupled differential system of equations, widely known as Sturm-Liouville matrix generalized boundary problem¹⁴:

where B(z) and W(z) are, in general, (N × N) Hermitian matrices and is fulfilled Y(z) = -P^†(z). In the absence of external fields, standard plane-wave solutions are assumed and it is straightforward to derive a non-linear algebraic problem

called as quadratic eigenvalue problem (QEP)¹⁶, since Q(k_z) is a second-degree matrix polynomial on the z-component wavevector k_z. In the specific case of the well-known (4 × 4) Kohn-Lüttinger (KL) model Hamiltonian, the matrix coefficients of Equation (2) bear a simple relation with those in (1)¹⁶:

Then for (4 × 4) KL model, the matrix coefficients of (2) can be cast as:

Here mhh,lh* stands for the (hh,lh) effective mass, respectively. We briefly introduce some parameters and relevant quantities (in atomic units) of the KL model: γ_i , with i = 1,2,3 [Lüttinger semi-empirical VB parameters]; R [Rhydberg constant]; a₀ [Bohr radius]; E [Energy of incident and uncoupled propagating modes]; A1,2=a02Rγ1±γ2, α1,2=A1,2κT2+ V(z) - E; h12=a02R3γ2ky2-kx2; h13=-a02R3γ3kx; H13=a02R3γ3ky and H12=a02R23γ3kxky.

Bearing direct association to the original matrix dynamic equation, we exclusively focus to the case when M,C and K are constant-by-layer, hermitian, and M is non-singular; therefore k_z are all different and real (symmetric) or arise in conjugated pairs (kz,kz*). Hereafter O_N/I_N, stand for (N × N) null/identity matrix. The QEP’s solutions result in the eigenvalues kzj and the eigenvectors Γ_j . As Q(k_z) is regular, eight finite-real or complex-conjugated pairs of eigenvalues are expected. In the framework of the QEP method ^16,19, one has

is an eighth-degree polynomial with only even power of k_z and real coefficients. The coefficients q_i are functions of the system’s parameters, and q₀=det M as expected ¹⁹. In the specific case of the KL model Hamiltonian¹⁶, q_i contain the values of γ_i and the components of the in-plane quasi-wave vector kt→=kxe^x+kye^y.

Based on our, it is straightforward to know whereas k_z is oscillatory or not by dealing with (7), and thereof to retrieve the phenomenological characteristics for V_eff. To manage complex-valued eigensolutions for (7) as a function of the VB-mixing and strain, we retrieve the root-locus-like procedure ¹³, provided its robustness when pursuing a simple graphical interpretation for a non-linear eigenvalue problem (2). To our knowledge, just few pure theoretical or numerical applications of the root-locus-like algorithm, particularly for the QEP scenario, have been previously addressed to explicitly describe several standard III-V semiconductor compounds^13,19. We acknowledge the advantages of the root-locus-like technique application within the low-dimensional solid state physics^13,19 -we may be the first ones- and try to predict unknown phenomenology whenever coupled holes interplay with mutable V_eff of the stressed system. For some high specialized zinc-blenda and wurtzite systems, current knowledge of the hole quantum transport mechanism is limited. The present theoretical contribution, claims to shed light on that issue. We think readers will be interested more on how the V_eff metamorphosis under VB-mixing and strain could influence their real-world devices, rather than getting involved into the very details of the theoretical model itself. In regard to that concern, we propose a simple and comprehensive modelling procedure for V_eff to deal with, and a gedanken-like simulation for a passage of mixed holes throughout a strained-free and strained layered heterostructures. The purpose of that is hardly a hypothetic exercise, but rather pretends to show to condensed-matter theoreticians and to complementary-fields physicists, why a non-mutable V_eff under VB-mixing and strain is not acceptable and how they can re-model V_eff to improve their own works within similar conditions.

According to prior description for VB-mixing⁵ and in order to achieve the target of the present study, an effective potential operator w^eff has to be put together, which in our modelling implies to select every spatial configuration-dependent function from Hamiltonian (1)^12,16. By choosing the first-quantization axis, that of the z-coordinate direction [along the heterostructure growth seen in Fig. 1(b) ], we are required not to consider terms from (1), counting as part of their arguments the k_z -component of kt→, then

Next we follow symmetry considerations ²⁰, and solve a Schrodinger-like equation in the (2 × 2) Hilbert reduced-space of the KL model

whose eigensolutions are the expected values for V_eff. The superscript indicates the order of the corresponding matrix. The operator w^eff was previously proposed elsewhere for an unstressed system and being approximately correct, although not accurate, is reliable enough to simulate the evolution of V_eff under the envisioned scenarios. We have taken W₁₁₍₂₂₎ = A12kT2+V(z) and W12=(ℏ23/2 m0)γ2(ky2-kx2)+2iγ3kxky, with m₀ as the bare electron mass. For κ_T ≈ 0 the VB-mixing vanishes, thereby Vz=VB-VA=ΘVeff is a positive-valued constant functional step-modulated by Θ, where V_A/B stands for the potential of the concomitant materials A/B, respectively [see Fig. 1(a)]. Due the lack of a strict superlattice multiple-layered structures under study here, we have neglected the spontaneous in-layer polarization field for III-nitride constituent media²¹, thus assuming a rectangular potential profile as test-run input, rather than biased one for all III-nitride slabs of the heterostructures. As the III-nitride slabs are just a few, we mean the intrinsic-biased-profile variation to be small over the unit cell, thus negligible for the heterostructure.

Strain field may rise questions over their relative effects on the electronic structure and, in particular on the valence-band structure where shape and size of the potential profile lead to stronger hybridization of the quantum states. Lets now examine the effects of the stress in the framework of the KL model Hamiltonian. The existence of a biaxial stress applied upon the plane parallel to the heterostructure interfaces leads to the appearance of an in-plane strain. The effective potential operator w^eff in the presence of biaxial strain, can be written as ²²

where

is the accumulated strain energy resulting from the tensile or compressive stress acting on the crystal, when an epitaxial layer is grown on a different lattice-parameter substrate. Owing to strictness in formulation²² , we guess that a maximum-quota criterium (11) suffices to achieve the goal posted in Sec. 1. Being independent from κ_T, a maximized U_s was taken for granted to evaluate if there is a real challenger strain effect over VB-mixing’s influence on the metamorphosis of V_eff. In (10)-(11), the subscript s stands for strain. In (11) a_v /b represent the Pikus-Bir deformation/break potentials, describing the influence of hydrostatic/uniaxial strain. Meanwhile ε_1,3, are the in-plane, and normal-to-plane lattice displacements, respectively. For commonly used cubic and hexagonal semiconductor compounds, we assume^17,18

with a_s,l the lattice parameter of the substrate and the epitaxial layer, respectively. Though no external stress is considered along the growth direction z, the lattice parameter is forced to change due to the Poisson effect¹⁸ . Hence, the normal-to-plane displacement can be cast as

which remains connected to in-plane deformation ε₁ via the Poisson radio ν. The last is valid for zinc blende and wurtzite materials.

By changing the material and the growth plane, the value of v modifies. For cubic materials it reads¹⁷

while for the hexagonal ones we have¹⁷

To quote the parameter εc=(cs-cl)/cl, we take c_s from the substrate wafer²³ , while c_l is referred to the epitaxially-grown layer on buffer stratum.

For the sake of completeness of the present theoretical framework, we have derived analytic expressions for the VB-band offset V_eff as a straightforward function of the band mixing parameters and the strain energy. In current solid-state physics studies, the later could allow a comprehensive analysis in cubic as well as in hexagonal layered-stressed semiconductor systems, whenever one manages to manipulate the accumulated pseudomorphic strain and mixing effects in a single shoot. We have considered (10)-(11) for solving (9), and thus for zinc blende (cubic) materials it may be cast

being

and

for j = A, B. In these expressions, E^j stands for the particle energy in the material j; while Usj represents the accumulated pseudomorphic strain energy due the substrate-material stress concerning each layer j independently and finally Vzj refers to QW-acting or QB-acting potentials, whenever the slab j plays one or the other role in the heterostructure. Worthwhile to remark the clear parabolic dependence of Vefflh,hh on κ_T as it readily shown in (16). The later is straightforwardly confirmed in the numerical simulation of the VB-offset depicted below.

For systems that crystalize in the wurtzite structure have been derived a (6 × 6) effective Hamiltonian H^G(k), within the envelope function approximation²⁴ . By an unitary canonical transformation U, the original Hamiltonian is re-expressed as

yielding a (6 × 6) block-diagonal Hamiltonian, quite similar in general sense to the reduced-space KL Hamiltonian we had used for the zinc blend heterostructures. The sub-scripts u(l) stand for up(low) respectively, as a resemblance of the up(down) spin-electronic states flavors and have been introduced by Broido and Sham ²⁵, who found alike transformation as (17), but for the (4 × 4) KL model. The (3 × 3) Hilbert reduced-space blocks in (17), satisfy the following symmetry relation

which is an analog to that deduced for the (2 × 2) Hilbert reduced-space of the KL model²⁰. Preserving the same conception framework posted above, the potential-energy operator W^eff6(z) for the wurtzite along the quantization axis z, is obtained by discarding all elements associated to the z-component wavevector κ_z. In doing that, we assume understood the implications of the translational invariance symmetry in the [xy] plane. Thereof

whose blocks fulfil

The up-block of (19) -in the presence of biaxial pseudomorphic strain-, has the general form

while its matrix elements are taken as

together with Fo(Go)=Δ1±Δ2; A2=1-6πz; A5=-3σ; and Δ2=(1/3)Δsoz. Owing to brevity we re-address readers to a more detailed description for matrix elements of the Hamiltonian H6^ (k) and thereof for (21), which have been accurately focused previously²⁴ . However, worthwhile comment here that: Δ₁ represents the energy splitting produced by the anisotropy of the hexagonal symmetry; A2,5 enter as some material dependent semi-empiric band parameters, and Δsoz,⊥ quote the energy splitting in the z-direction (perpendicular to it), due to the spin-orbit (so) interaction.

On the ground of symmetry relations (18) and (20), the quoting of W^eff6(z)eigen-solutions via (9), becomes a suitable task. However, for the purposes posted in the present study it suffices to figure it out the same but for W^3u, whose third-order characteristic polynomial equation reads

and has the solutions

For (22) and (23) we have taken

3. Discussion of results

Unless otherwise specified, the graphical simulations of V_eff reported here, were calculated using highly specialized III-V semiconductor binary(ternary)-compound cases for both unstressed and stressed cubic and hexagonal systems. The present numerical simulations consider different constituent media, regardless if they can be grown. In this section, we briefly present numerical exercises within the root-locus-like technique, to foretell multiband-coupled charge-carrier effects for pseudomorphically stressed III-V semiconductor layered systems.

4. Conclusions

We present an alternative graphic-based unambiguous theoretical procedure to demonstrate the VB-mixing and strain impact on V_eff metamorphosis. Have been accurately derived an analytic expression of the VB-offset for zinc blende semiconductors, while for the wurtzite its explicit eigen-values were deduced from the proper (6 × 6) Hamiltonian . The lh experience the striking keyboard effect of V_eff in stress-free and stressed systems. Nevertheless, in strained heterostructures the former behavior have been also found for hh. Evidences of this sort foretell their usefulness in experimental applications such as: VB-profile tuning, VB-offset manipulation and in theoretical analysis of hole tunneling and spectrum. Pseudomorphic strain is able to diminish the keyboard effect and also makes it emerge or even vanish eventually. We remark that the multiband-mixing and stress-induced events, are strong concurrent appliances that can not be universally neglected in layered systems. Tuning in-plane directions the keyboard effect under strain reveals anisotropic and thereof topologically tunable. The present modelling of V _eff evolution, may be a reliable workbench for testing other configurations and may be of relevance for promising heterostructure’s design guided by VB-structure modelling to enhance the hole mobility in III-V semiconducting devices provided they always lagged compared to II-IV media ⁸.