PACS: 71.70.Ej; 72.25.Dc; 73.21.Hb; 73.23.Ad

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 mixing^{1}, *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 tunneling^{2} and had pointed up its relevance
for experimental and technological applications^{1} . 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 regions^{3}. 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 potential^{4}
. A first-principles study on valence-band (VB)-mixing, established a non-linear
response for a pseudo-potential in series of the atomic distribution function^{5} . 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 dots^{11}. 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
case^{14}, 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 problem^{12} and
somehow to the unstrained multiband-hole problem^{13} . 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 negligible^{14} . 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*_{eff}^{15} 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
photodetectors^{17}. 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 designs^{17} . 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*_{eff}^{13}, 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 problem^{14}:

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 *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]; *a*_{0} [Bohr
radius]; *E* [Energy of incident and uncoupled propagating modes];

Bearing direct association to the original matrix dynamic equation, we exclusively focus to
the case when *k*_{z} are all different and real
(symmetric) or arise in conjugated pairs **O**_{N}/**I**_{N},
stand for (*N* × *N*) null/identity matrix. The QEP’s
solutions result in the eigenvalues **Γ**_{j} . As
* Q*(

*k*) is regular, eight finite-real or complex-conjugated pairs of eigenvalues are expected. In the framework of the QEP method

_{z}^{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 *q*_{0}=det ^{19}. In the
specific case of the KL model Hamiltonian^{16}, *q*_{i}
contain the values of *γ*_{i} and the
components of the in-plane quasi-wave vector

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-mixing^{5}
and in order to achieve the target of the present study, an effective potential
operator ^{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

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 *V*_{eff} under the envisioned scenarios. We
have taken W_{11(22)} = *m*_{0} as the bare electron mass. For
*κ*_{T} ≈ 0 the VB-mixing vanishes, thereby *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 media^{21}, 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 ^{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 formulation^{22} , 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 effect^{18} .
Hence, the normal-to-plane displacement can be cast as

which remains connected to in-plane deformation *ε*_{1}
*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 reads^{17}

while for the hexagonal ones we have^{17}

To quote the parameter *c*_{s} from the
substrate wafer^{23} , 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 *j* independently and
finally *j* plays one or the other role in the heterostructure.
Worthwhile to remark the clear parabolic dependence of *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 ^{24} . By an unitary canonical transformation

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 model^{20}. Preserving the same conception
framework posted above, the potential-energy operator *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 ^{24} . However, worthwhile comment here that: Δ_{1} represents the energy splitting produced by the anisotropy of the hexagonal symmetry; *z*-direction (perpendicular to it), due to the spin-orbit (so) interaction.

On the ground of symmetry relations (18) and (20), the quoting of *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

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.

**3.1. Simulation of V**

_{eff}

**profile evolution**

On general grounds, for *κ*_{T} ≈ 0 the
*V*_{eff} is constant^{14}^{,}^{16}, while by letting grow
*κ*_{T}, the band mixing effects arise and
*V*_{eff} changes^{12}^{,}^{14}^{,}^{16}. We are focused here on evaluating first the
stress-free systems, and then the effect of a pseudomorphic strain on
*V*_{eff}.

**3.1.1. Unstressed V**

_{eff}

*metamorphosis*Pursuing a deeper understanding of the rather cumbersome *κ*_{T} impact over the *V*_{eff}, we simulate its profile
evolution for pure (*κ*_{T} ≈ 0) and mixed (*κ*_{T} ≠ 0) holes. To get this problem solved, one has to figure out
(9) for several III-V semiconducting heterostructures, having taken external
layers length as 25 Å and the embedded stripe thickness as 50 Å.

Figure 2 confirms the fixed-height
*V*_{z} as a reliable input
QB-energy for *hh* (blue lines), valid even for a strong
VB-mixing [see panel (b)]. The opposite reveals panel (a) for
*lh* (red lines), considering the
*V*_{z} trend to diminish
with *κ*_{T}. This kind of evidence suffices to
demonstrate the essentialness of introducing a mutable effective
*band offset V*_{eff}, to correctly characterize
scattering processes for holes. There are further features that deserve to
be referred, indeed: the left(right) *V*_{eff} edges
move upward non-rigidly nearly 0.5 eV, meanwhile middle border stays
practically unchangeable. Worthwhile to remark that we recovered here a
phenomenology of this sort, previously obtained for the first bound states
of *lh* and *hh* in a QW ^{26}. We had found the same behavior of the
embedded layer for other materials of real-word interest
(*AlSb*, *AlP*, *AlN*).

An appealing situation arises at a specific entry of the transverse momentum. An earlier
detailed study on this subject^{12} , has predicted the existence of quantity *V*_{eff} becomes constant
along the entire layered heterostructure. In the case envisioned here, due
to the presence of *hh* and *lh*, we have

being V_{o} =
*V*_{eff}(*κ*_{T} = 0) and
A/B standing for concomitant cladding/middle layer. A direct consequence for
*V*_{eff} being flat at
*κ*_{To} is the existence of a crossover of
*V*_{eff} respect to (24).

In other words, if a QW-like profile is present for

In Fig. 3(a) the V_{eff} valence-band mixing
dependence, exhibits a neatly permutation of the
*V*_{eff} character as the one predicted for
electrons^{12} . This
permutation pattern is what we call “*keyboard*” effect, and
was detected for *lh* only in stress-free systems. This
striking interchange of roles for QB-like and QW-like layers, whenever the
in-plane kinetic energy, varies from low to large intensity, represents the
most striking contribution to the present study. For a single-band-electron
Schrödinger problem, some authors have predicted that both QW and QB may
appear in the embedded layers of a semiconductor superlattice, depending on
the transverse-component value of the wave vector^{12} . It has recently been unambiguously
demonstrated that the effective-*band offset* energy
*V*_{eff}, “*felt*” by the two
flavors of holes, as *κ*_{T} grows, is not the same.
Inspired by these earlier results, we have addressed a wider analysis of
this appealing topic, displayed in Fig.
3, pursuing a more detailed insight. We have considered a
InAs/AlSb/InAs heterostructure. Panel (a)/(b) of Fig. 3 shows explicitly the metamorphosis of
*V*_{eff}, felt by both flavors of holes
independently, respect to concomitant-material slabs. From panels (a) and
(b), it is straightforward to see that for *hh* (blue lines)
an almost constant *V*_{eff} remains, while
*κ*_{T} varies from 0 (uncoupled holes) to 0.1
Å^{-1} (strong hole band mixing), despite the respective
band-edge levels have changed. Contrary to *hh* [blue lines,
panel (b)], the *lh* exclusively [red lines, panel (a)]
exhibit the *keyboard* effect, *i.e.* they
feel an effective *band offset* exchanging from a QW-like
into a QB-like one, and *viceversa* for an InAs/AlSb/InAs
heterostructure while *κ*_{T} increases. The evident
*keyboard* effect of *V*_{eff},
resembles a former prediction for electrons . This observation means that in
the selected rank of parameters for a given binary-compound materials, a
*lh* might “*feel*” a qualitative
different *V*_{eff} (QW or QB) during its passage
through a layered system, while the degree of freedom varies in the
transverse plane. Former assertions can be readily observed in Fig. 3(c)-(d), where we have plotted the
evolution of *V*_{eff} profile [panel (c)], as well
as the progression of the *band offset* [panel (d)], with
*κ*_{T} at a fixed transverse plane of the
heterostructure. Both upper-edge and lower-edge move in opposite directions
[see panel (c)] and the zero-*band offset* point
configuration is detected in the vicinity of *κ*_{T}
≈ 0.066 Å^{-1} [see panel (d)]. The permutation holds for other
in-plane directions, as can be seen from panel (c). Although not shown here
for simplicity, the *keyboard* effect, remains robust for
other middle-layer binary compounds, namely: AlAs, AlP, and AlN.

**
3.1.2. Keyboard effect versus pseudomorphic
strain
**

Turning now to built-in elastic stressed layered heterostructures, we are interested in
answering a simple question: whether or not the existence of a pseudomorphic
strain becomes a weak or a strong competitor mechanism able to diminish the
*keyboard* effect on *V*_{eff}, or
even make it rises/vanishes occasionally. Thereby, we need to account the
accumulated strain energy resulting from the tensile or compressive stress
acting on the crystal slabs. The last requires to solve ([for:Weff-s]),
assuming the heterostructure sandwiched into a pseudomorphically strained
QW/QB/QW-sequence.

Figure 4 is devoted to demonstrate that
the *keyboard* pattern for *lh* remains robust
in a
*InSb*:*InSb*/*AlN*/*InSb*
pseudomorphically strained layered heterostructure [see panel (b)], respect
to that of the stress-free system [see panel (a)]. In this case, we conclude
that maximized *U*
_{s} ([for:Strain]) do not represent any antagonist mechanism
regarding to valence-band mixing influence on
*V*_{eff}. Identical middle-layer nitride
material (*AlN*) may not follow a same evolution of
VB-offset, if the concomitant cladding layers change. This in shown in panel
(c), whose displayed behavior is the opposite to that in panel (a),
*i.e.* we found no evidences of the
*keyboard* effect. In this case, none zero-*band
offset* point configuration were found even at strong VB mixing,
though a remarkable anisotropy was detected, despite a common trend of
*V*_{eff} is neatly preserved. On the contrary, a
robust isotropic character in the VB-offset progression for the same
middle-layer binary compound embedded in the strained heterostructure, have
been detected and shown in panel (d).

Figure 5(b) exhibits an unexpected
*keyboard* effect for *lh* in a
InP:GaP/AlSb/GaP layered heterostructure under a pseudomorphic strain, in
comparison with a stress-free heterostructure [see panel (a)]. This evidence
encourages to suggest *U*_{s}
([for:Strain]) as a trigger mechanism in the presence of valence-band mixing
to influence *V*_{eff}, forcing a qualitative
distinctive phenomenology to arise in the effective VB-offset profile
(*keyboard* effect). Besides, it is worth to underline
the inversion of the valence-band line-up, which is also remarkable
considering the appealing interplay from QW-like to QB-like behavior (and
*viceversa*) when comparing both stress-free [see panel
(a)] and strained [see panel (b)] heterostructures. It is worth noticing the
difference in the phenomenology of several stressed antimonide-based
systems, respect to that discussed for nitride ones in Fig. 4(d). As can be seen in Fig. 5(c), an anisotropic pattern characterizes the
*lh* case in a InP:GaP/AlSb/GaP layered heterostructure
under a pseudomorphic strain. Importantly, the *keyboard*
effect on *V*_{eff} stays steady along [10] and [01]
in-plane directions, while vanishes in the [11]. Thus a topological tuning
of this striking effect reveals possible.

Figure 6 shows another striking
performance of *V*_{eff}, when unexpectedly a
*keyboard* effect has arisen for *hh* in a
InSb:GaP/AlAs/GaP strained layered heterostructure [see panel (b)]. Indeed,
if we consider the stress-free case of Fig.
6(a), we can see that the standard rectangular distribution for
*V*_{eff} remains consistent in a wide range of
*κ*_{T}. Furthermore, a valence-band line-up
inversion takes place, which is yet another remarkable performance for
QW-like and QB-like slabs. For the sake of continue the qualitative insight
into the influence of the concomitant cladding/middle layers as presented in
the discussion of Fig. 4(a,c), we alter
here just the cladding ones. At the opposite to the mentioned previous case
for *lh*, the same kind of *V*_{eff}
evolution for *hh* remains [see panels (a) and (c)], however
a difference of tendency along [10] and [01] in-plane directions, respect to
that of the [11], was observed. We underline in panel (d), the possibility
for topological tuning of the *keyboard* effect on
*V*_{eff}, as for *hh* now the
[11] in-plane direction solely shows a zero-*band offset*
point at the vicinity of *κ*_{T} ≈ 0.075
Å^{-1}.

Finally, Fig. 7 confirms the existence of a strong
competitor mechanism between strain and *keyboard* effect. We
can see in Fig. 7(b) how the
*keyboard* pattern vanishes in a AlAs:InAs/AlN/InAs
layered heterostructure under a pseudomorphic strain. The envisioned effect
was apparently robust for the stress-free analogous systems shown in panel
(a), within the selected range of *κ*_{T}. In panel
(d), a resembling phenomenon was found for the strained InSb:InSb/AlAs/InSb
layered heterostructure, with an appealing bonus of the interplay from
QW-like to QB-like behavior (and *viceversa*) of the slabs.
The last inversion of the valence-band line-up was not obtained for the
stress-free correspondent system, exhibited in Fig. 7(c).

Although not depicted here, we have found qualitative patterns alike *keyboard* effect under pseudomorphic strain, for several *III* - *V* binary compounds (see Table I). As a bonus, a numerical evaluation of the polynomial interpolation (16) is presented in the last column of the Table I, thus complementing the characterization of the selected systems.

**3.2 Influence of the pseudomorphic strain on
k**

_{z}

**-spectrum**

The QEP *k*_{z}-spectrum is a meaningful, and well-founded physical
quantity that can be obtained *via* the
*root-locus-like* procedure^{19} by unfolding back in the complex plane the
dispersion-curve values for bulk materials, determined by stress-induced effects
on the stress-free heterostructure. Thus, we take advantage of the
*root-locus-like* know-how, to promptly identify evanescent
modes, keeping in mind that complex (or pure imaginary) solutions are forbidden
for some layers and represent unstable solutions underlying the lack of
hospitality of these slabs for oscillating modes. The opposite examination is
straightforward and also suitable for propagating modes, which become equated
with stable solutions for given layers.

To obtain the QEP *k*_{z}-spectrum in a periodic pseudomorphically
strained heterostructures of QB-acting/QW-acting/QB-acting materials, we first
use ([for:Weff-s]) and substitute it in ^{13}. Next, we solve again the
characteristic problem (9), whose eigenvalues allow us to obtain the new
expression for the QEP-matrix *k*_{z}. Once we have quoted the eigenvalues
*k*_{z} of (7), it is then straightforward to
generate a plot in the complex plane, symbolizing the locations of
*k*_{z} values that rise as band mixing parameter
*κ*_{T} changes. Keeping in mind that complex (or
pure imaginary)/real solutions of (7) represent forbidden/allowed modes, we take
advantage of the *root-locus-like* map to identify
evanescent/propagating modes for a given layer. Thus, we are able “to stamp” on
a 2*D*-map language, a frequency-domain analysis of the
envisioned heterostructure under a quantum-transport problem. This way, we are
presenting an unfamiliar methodology in the context of quantum solid state
physics, to deal with low-dimensional physical phenomenology.

The Fig. 8 and 9
illustrate the role of band mixing for *κ*_{T}
[10^{-6}, 10^{-1}] Å^{-1}, on the
*k*_{z} spectrum from QEP (7) for a III-V strained
alloy, clearly distinguished as QW in most layered systems of technological
interest. Importantly, by assuming two different substrates AlSb (Fig. 8) and InAs (Fig. 9), we found different patterns of the
*k*_{z} spectrum for *lh* and
*hh*. Namely for the [10] in-plane direction, the
*k*_{z}
*root-locus-like* evolution is real for *lh*, in
the range of *k*_{z} becomes pure imaginary and complex,
respectively [see Fig. 8(a), inner
green-red solid lines]. On the other hand, the
*k*_{z}
*root-locus-like* shows real values for *hh*, in
the interval *hh* and *lh* curves
are undistinguishable in this latter interval. Although not shown here, the [01]
in-plane direction exhibits the same behavior. The Fig. 8(b), displays the QEP (7 spectrum along the [11] in-plane
direction. For *lh* only, *k*_{z}
*root-locus-like* evolution starts as a real number in the range *k*_{z} spectrum for *hh*
it is always a real number in the whole selected interval *lh* and
*hh*, the *k*_{z} evolution starts as
a pure imaginary number in the range *hh* and *lh* are
indistinguishable when their *k*_{z} magnitude is the
same. Meanwhile, the panel (b) of Fig. 9
demonstrates that for the [11] in-plane direction, the
*k*_{z} values are mostly complex or pure imaginary,
except in the small interval of *k*_{z} for *hh* were found as
*κ*_{T} changes within the bounds [0.01,0.1]
Å^{-1}. The *hh* and *lh* curves are
indistinguishable in the range of *root-locus-like* map of *k*_{z} means
that the *GaP* strained-layer recovers its standard QW-behavior
for both *hh* and *lh* quasi-particles in the
stress-free configuration. On the opposite, whenever real-value map fades,
*i.e.* complex or pure imaginary magnitudes arise, none
oscillating modes can propagate through an InAs:GaP strained slabs. In this
case, the GaP might turns into an effective QB for traveling holes.

For completeness, the pseudomorphic perturbation on several stress-free hexagonal heterostructures have been considered here and we found slight modifications in the band offset compared to the unstrained case for growth planes (0001) and (1m00). However, nor permutations of V_{eff} neither *keyboard* patterns were detected in such semiconductors materials.

3.3 Bandgap and valence-band offset manipulation

It has widely been accepted that electronic properties can be tuned by elastic stress. This
assertion is clearly illustrated computationally for VB in Figs. 4, 5, 6 and 7,
which exhibit modifications of the VB-offset whenever a biaxial-pseudomorphic
strain is applied. Recently, some authors have addressed a first-principles
density functional theory calculations to a monolayer of a MoS_{2} and
indicate that both direct and indirect bandgap decrease in the presence of
biaxial strain, with transitions from direct to indirect bandgap^{27}. As uniform stress leads to
band-structure changes, we thereby expect the bandgap of exercised III-V
heterostructures displayed in Figs. 4,
5, 6 and 7, to evolve from nominal
unstrained crystal spectrum forbidden gap. It is worth noting that as shown
independently in Fig. 3(c,d), the way to
tune the V_{eff} profile as well as the VB offset is by letting grow the
*hh* - *lh* coupling. The latter represents a
complementary tool, to that proposed before^{27}^{,}^{28} in manipulating the electronic structure, but this
time for tensile strain-free systems. Importantly, VB-offset progression
depicted in Figs. 3(c,d), 5(b) and 6 for III-nitride(antimonide) heterostructure, as one of its
constituent media, demonstrates an anisotropic behavior, as a bonus to the
expected differences between *hh* and *lh*, due to
their effective masses.

In Subsecs. (3.1.2) and (3.3) we have shown the V_{eff} profile evolution and the
*keyboard* effect under the competitors hole-mixing and
strain. We hope that these appealing events would attract the attention of a
wider community of physicists beyond condensed-matter theoreticians. The present
graphic modelling of *V*_{eff} evolution, may be a
reliable workbench for testing several configurations of materials with minor
changes, if any, being useful in both experimental applications and in
theoretical analysis. The latter means that rather than considering the present
study as an end-in-itself theoretical exercise -which is not ourgoal-, readers
working in complementary and even in different fields most use (16) for zinc
blende, and (22) for wurtzite to quote *V*_{eff}
eigenvalues from (9) and this way incorporate the mutable profile of
*V*_{eff} instead of the commonly used fixed
** V**(

*z*) =

*V*-

_{B}*V*. If the studies involve electronic properties and/or scattering processes, modifications of the conductance and thereby the tunneling time, should be expected by taking realistic

_{A}*V*

_{eff}mutable quantities derived from work out the eigenvalue problem (9)-(10) rather than take a constant-guess value

*(*

**V***z*) =

*V*

_{o}. We strongly recommend to include current results to further improve own researches that should be related to the real-world experiments and day-to-day applications.

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}.