# Empirical Tight-Binding Parameters for Wurtzite group III-V(non-Nitride) and
IV Materials

Joseph Sink<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Iowa  Craig Pryor  Department of Physics and Astronomy, and
Optical Science and Technology Center, University of Iowa  craig-
<EMAIL_ADDRESS>
###### Abstract

Many commonly used nearest neighbor tight binding models for cubic
semiconductors often result in inaccurate band structures when transferred to
hexagonal polytypes. The resulting bandgaps are systematically too small, and
in some cases calculations may erroneously predict a metal for the wurtzite
polytype. We have calculated a set of $\rm spds^{*}$ tight binding parameters
for the hexagonal phases of non-nitride III-V and group V semiconductors fit
to experimental data and empirical pseudopotential calculations. Our fitting
procedure constrains the parameters to be as close as possible to those for
the cubic polytype so as to make the the parameters maximally transferable.
Our parameters, when combined with the existing cubic parameters, provide a
model suitable for modeling the electronic structure of polytypic
heterostructures containing both the wurtzite and zincblende crystal phases.

##  I Introduction

Non-nitride III-V and group V semiconductor materials are well known to have
cubic symmetry (3C-zincblende or diamond) when grown in bulk, but may exhibit
exotic crystal phases (2H-wurtzite or lonsdaleite, 4H, etc) when grown in high
aspect ratio morphologies such as nanowires. By carefully tuning growth
conditions[1, 2, 3, 4, 5], it is possible to deliberately alternate between
the cubic and hexagonal phases to form polytypic heterostructures in which the
chemical composition is constant throughout the structure. While chemically
identical, the differences in the bonding geometry between polytypes gives
rise to differences in their electronic properties. There has been a great
deal of interest in this new and novel degree of freedom. [6, 7, 8, 9, 1, 2,
3, 10, 4, 5, 11, 12, 13, 14, 15, 16]

Extracting bulk electronic properties from wuzrtite samples is complicated by
their small dimensions, resulting in a lack of experimental data available to
constrain empirical models. Photoluminescence, high-field magneto-
photoluminescence spectroscopy[9] and polarized photocurrent spectroscopy[11]
can be used to measure the bandgap, band symmetry ordering and exciton
effective mass as a function of position along the nanowire. However, many
other properties currently rely on theoretical calculations based on highly
transferable models such as density functional theory[17] and empirical
pseudopotentials[18, 19]. For small changes in the local electronic
environment, pseudopotentials that are empirically fit to the band structure
of the cubic phase may be transferred to the positions of the atoms in
hexagonal phase without changing them. The band structures resulting from this
method are in good agreement with existing experimental results and provide
additional band structure information[18, 19].

While pseudopotentials have been used to directly compute the electronic
structure of semiconductor nanostructures[20, 21], it has been more common to
use the tight binding approach, in which the single electron wave function in
a material is expressed as a sum of localized orbitals centered on the atoms.
In the Empirical Tight-Binding (ETB) approach [24, 25, 26, 27] the Hamiltonian
matrix elements between localized orbitals are treated as parameters that are
fit to experimental or ab initio results for bulk materials. For inhomogeneous
systems the Hamiltonian matrix elements between localized orbitals are assumed
to be the same as in the bulk, depending only on the atoms begin coupled.
Additional approximations are usually made for computational efficiency: 1)
only nearest neighbor (NN) orbitals are coupled 2) Hamiltonian matrix elements
involving orbitals on two different atoms and the potential from a third atom
(three-center integrals) are ignored 3) the orbitals on different atoms are
assumed to be orthogonal. The last of these may be ensured by Loẅdin
orthogonalization[28] but the first two are true approximations.

Tight binding parameters are said to be transferable when the Hamiltonian
matrix elements between orbitals can be transferred between similar materials.
This would mean, for example, that the matrix elements between orbitals in
adjacent Ga and As atoms would be the same in GaAs and in an AlGaAs alloy.
This relationship is only approximate since the material surrounding the Ga-As
pair will effect the local electronic environment, altering the local atomic
wavefunctions and thereby the strength of the hopping interactions. The
effects of this perturbation are generally small between like materials, and
manifest as chemical trends such as those observed in Ref. 29.

The cubic and hexagonal phases of III-V (zincblende and wurtzite) and group IV
(diamond and lonsdaleite) semiconductors would appear to be ideal candidates
for polytype transferability. Not only are the materials chemically identical,
but the four nearest neighbors and 9 out of 12 second nearest neighbors are at
the same positions, with very small deviations that can be treated with strain
modifications. The similarities of the polytypes are confirmed by the small
differences in experimental bandgaps between polytypes. The agreement between
pseudopotential calculations and experiment suggests that the crystal
potentials of the two polytypes are very similar, with a rearrangement of the
atomic positions leading to subtle changes in the band structure.

However, transferring parameters from the widely used cubic $spds^{*}$ model
of Ref. 29 to hexagonal crystals results in bulk band structures in
significant disagreement with experiment[30, 11, 31, 32] and theory[18, 19].
This indicates that the hexagonal polytypes require different parameters which
must be fit to reliable target data. It has been observed for SiC using an
$sps^{*}$ model that excellent transferability may be expected among the
hexagonal 2H, 4H, 6H and 8H polytypes [33]. We have carried out such a
procedure, fitting $spds^{*}$ tight binding parameters to experimental data
when it is available and to empirical pseudopotential band structures when
experimental data is not available. We impose a penalty function in the
fitting procedure to keep the parameters for the hexagonal phase as close to
the cubic parameters as possible. Our result is a set of tight binding
parameters for the non-nitride III-V and group IV semiconductors in the
hexagonal phase that is still relatively close to the parameters for the cubic
phase, thus maintaining approximate transferability.

Section II reviews the relevant formalism and notation we use. Section II.2
discusses the ETB, the approximations typically made, and their implications
for transferability. In section III we describe our fitting procedure in some
detail, and present the results. Section IV concludes with a discussion of
possible applications of our new parameters and what the lack of
transferability implies for the accuracy of tight binding models in general.

##  II Background

###  II.1 Geometry: WZ and ZB

The WZ crystal is defined by four interpenetrating hexagonal lattices, with
conventional lattice vectors

$\displaystyle\vec{a}_{1}=\\{\frac{1}{2},\frac{\sqrt{3}}{2},\\}a$
$\displaystyle\vec{a}_{2}=\\{\frac{1}{2},-\frac{\sqrt{3}}{2},0\\}a$
$\displaystyle\vec{a}_{3}=\\{0,0,c\\}$ (1)

and basis vectors

$\displaystyle\vec{t}_{1}=\\{0,0,0\\}$ $\displaystyle\vec{t}_{2}=\\{0,0,uc\\}$
$\displaystyle\vec{t}_{3}=\\{0,\frac{\sqrt{3}a}{3},\frac{c}{2}\\}$
$\displaystyle\vec{t}_{4}=\\{0,\frac{\sqrt{3}a}{3},(\frac{1}{2}+u)c\\}$ (2)

where $a$, $c$ and $u$ are material dependent lattice constants. In the ideal
WZ configuration, all nearest neighbor bond lengths
($|\vec{t}_{1}-\vec{t}_{2}|=|\vec{t}_{2}-\vec{t}_{3}|=|\vec{t}_{3}-\vec{t}_{4}|$)
are equal providing constraints for $u(=3/8)$ and $c/a(=\sqrt{8/3})$. The
$c/a$ ratio for many WZ phase materials has shown to be close to ideal via
X-ray diffraction (XRD), in excellent agreement with predictions from Density
Functional Theory (DFT) lattice relaxation calculations. Experimental
measurements for the internal structure parameter, $u$, requires measurement
of not only the location of the Bragg peaks but also their amplitudes [32] and
has thus proven difficult to extract experimentally. Estimates for $u$ in this
work are taken from DFT lattice relaxation calculations (See Table 1 for
values).

| ZB | WZ
---|---|---
Mat. | $a_{zb}(\text{\r{A}})^{a}$ | $\frac{a_{wz}}{a_{zb}}-\frac{1}{\sqrt{2}}$ | $\frac{c_{wz}}{a_{wz}}-\sqrt{\frac{8}{3}}$ | $u-\frac{3}{8}$
AlP | 5.4635 | -0.0018b | 0.0089 | -0.0001
AlAs | 5.6600 |  0.0019b | 0.0077 | -0.0003
AlSb | 6.1355 |  0.0031b | 0.0059 | -0.0001
GaP | 5.4509 | -0.0021b | 0.0092 | -0.0004
GaAs | 5.6532 | -0.0021b | 0.0127 | -0.0004
GaSb | 6.0959 | -0.0028b | 0.0139 | -0.0007
InP | 5.8687 | -0.0013b | 0.0078 | -0.0004
InAs | 6.0583 | -0.0014b | 0.0089 | -0.0005
InSb | 6.4794 | -0.0014b | 0.0094 | -0.0006
C | 3.5668 | -0.0048c | 0.0273 | -
Si | 5.4300 | -0.0073d | 0.0196 | -
Ge | 5.6563 | -0.0021e | 0.0164 | -0.0007

aReference 29
---
bReference 34
cReference 35
dReference 36
eReference 37

Table 1: ZB and WZ lattice parameters. For WZ, the differences from ideal
ratios are shown. Lattice constants for the WZ III-V’s are taken from ab
initio DFT. The LDL group-IV lattice constants are from XRD. The ideal value
of $u$=3/8 was used for C and Si.

The ZB primitive unit cell is constructed from two interpenetrating fcc
lattices, with conventional lattice

$\displaystyle\vec{a}_{1}=\frac{a}{2}\\{1,1,0\\}$
$\displaystyle\vec{a}_{2}=\frac{a}{2}\\{1,0,1\\}$
$\displaystyle\vec{a}_{3}=\frac{a}{2}\\{0,1,1\\}$ (3)

and basis vectors

$\displaystyle\vec{t}_{1}=\\{0,0,0\\}$
$\displaystyle\quad\vec{t}_{2}=\frac{a}{4}\\{1,1,1\\}$ (4)

where $a$ is the ZB lattice constant (See Table 1 for values).

The most obvious difference between the WZ and ZB crystals is the number of
atoms in the primitive unit cell. This results in the reciprocal-space volume
($V=(2\pi)^{3}/|\vec{a}_{1}\cdot\vec{a}_{2}\times\vec{a}_{3}|$) of ideal-WZ
($V_{WZ}$) being half that of ZB ($V_{ZB}$). Consequently, band-folding must
occur in order to accommodate the same number of states states in half the
(reciprocal-space) volume.

It is useful to consider the non-primitive ZB unit cell which has been doubled
along the $[001]$ direction such that $V_{WZ}=V_{ZB}$. This four atom unit
cell, $ZB_{4}$, has lattice and basis vectors

$\displaystyle\vec{a}^{\prime}_{1}=\vec{a}_{1}$
$\displaystyle\vec{a}^{\prime}_{2}=\vec{a}_{2}$
$\displaystyle\vec{a}^{\prime}_{3}=2\vec{a}_{3}$ (5)
$\displaystyle\vec{t}^{\prime}_{1}=\vec{t}_{1}$
$\displaystyle\vec{t}^{\prime}_{2}=\vec{t}_{2}$ (6)
$\displaystyle\vec{t}^{\prime}_{3}=\vec{t}_{1}+\vec{a}_{3}$
$\displaystyle\vec{t}^{\prime}_{4}=\vec{t}_{2}+\vec{a}_{3}.$

Here, the non-primitive (primed) vectors for $ZB_{4}$ are expressed in terms
of the primitive ZB (un-primed) vectors. Doubling the ZB unit cell along
$[001]$ results in a zone folding of the primitive $L_{z}$ point back onto
$\Gamma$, allowing half the WZ $\Gamma$-point state to be identified with WZ
$\Gamma$-point states, and the other half with zone-folded states. This
correspondence of states between the WZ and ZB can be used to understand the
band ordering of the WZ $\Gamma_{8c}$ and $\Gamma_{7c}$ states (Fig. 1).

Figure 1: Illustrative band diagram for ZB4 and WZ near the $\Gamma$-point.
The band folded $L$ point conduction band valley is shown. The ordering of the
red and purple bands are material dependent. The ordering shown corresponds to
GaAs, InP, InAs and InSb, while AlP, AsAs, AlSb, GaP and GaSb have reversed
purple ($L_{6c}/\Gamma_{8c}$) and red bands ($\Gamma_{6c}/\Gamma_{7c}$).

At 1NN distances the crystal field environment for the two polytypes are
identical, with both materials being tetrahedrally coordinated and having the
same bond lengths. At 2NN distances the crystals share a majority of their
atomic arrangement with 9 of 12 2NN atoms in the same configuration and the
remaining 3 of 12 merely rotated by $\pi/3$ (Fig.2). This rotation demotes the
crystal symmetry from $T_{d}$ in ZB ($O_{h}$ in Diamond) to $C_{6v}$ in WZ
($D_{6h}$ in Lonsdaleite). The identical 1NN configuration and extremely
similar 2NN configuration serves as the basis to expect a high degree of
polytypic transferability of the electronic parameters between the WZ and ZB
crystal-phases.

(a) ZB4

(b) WZ

(c) ZB4

(d) WZ

Figure 2: Crystal Structure for WZ and ZB. a) ZB viewed along $[111]$, b) WZ
viewed along $[0001]$ showing first and second nearest neighbors about a
central atom. The white spheres in a) and b) are the central atoms, while the
blue and pink spheres are first and second nearest neighbors, respectively.
The opaque shells indicate first and second nearest neighbor distances. c)
non-primitive four atom ZB unit cell and nearest neighbors, d) primitive WZ
unit cell and nearest neighbors. Comparisons of a) c) with b) d) demonstrate
the key difference between the WZ and ZB crystals is the three atoms at 2NN
distances which are rotated by $\pi/3$.

###  II.2 Tight Binding

ETB is a short range atomistic theory that can be used to yield sparse real-
space matrices that can be solved efficiently for systems with many atoms
[38]. Many approximations, some of which are discussed below, are required in
order to reduce the number of empirical parameters to a manageable set.

The TB Hamiltonian within the one-electron approximation is expressed as a sum
over effective atomic potentials, $V_{i}(\vec{r})$, centered on atomic sites
in the crystal

$\displaystyle\hat{H}_{1e}(\vec{r})=\hat{T}+\sum_{j,i}\hat{V}_{i}(\vec{r}-\vec{R}_{j}-\vec{t}_{i})+\hat{H}_{so}$
(7)

where $\hat{T}$ is the kinetic energy operator ($\frac{\hat{p}^{2}}{2m_{e}}$),
$\vec{R}_{j}$ is the location of the $j^{th}$ unit cell in the crystal,
$\vec{t}_{i}$ the position of the $i^{th}$ atom in the crystal basis, $V_{i}$
an effective atomic potential for an atomic species occupying the $i^{th}$
position in the crystal basis ($\vec{t}_{i}$) and $\hat{H}_{so}$ the spin-
orbit interaction.

To emphasize the bonding nature of the electronic hopping, the ETB basis is
typically taken to be constructed from atomic(-like) orbitals centered on the
atoms in the unit cell [39, 29, 40]

$\displaystyle\ket{\psi_{\eta}}=\ket{\phi_{\beta},\vec{R}_{j}+\vec{t}_{i}}\ket{\chi}$
(8)

where $\eta$ is shorthand for the indices ${\\{i\text{(basis)},\ j\text{(unit
cell)},\ \beta\text{(orbital)},\ \chi\text{(spinor)}\\}}$.

The atomic(-like) orbitals may be taken to be orthogonal between different
lattice sites
($\bra{\psi_{\eta}}\ket{\psi_{\eta^{\prime}}}=\delta_{\eta,\eta^{\prime}}$)
via an orthogonalization procedure such as Löwdin or by using Wannier
functions. In practice, the orbital wavefunctions are rarely calculated as
only knowledge of how they transform is required for most applications. Non-
orthogonal ETB models have been shown to increase transferability in some ETB
models[41] at the expense of computational efficiency[42] as the basis
requires orthogonalization to resolve ambiguities in various operators (ie
position operator for charge localization).

Expanding the one-electron Hamiltonian in Eq. 7 in an orthogonal (Löwdin)
basis and collecting like terms yields yields

$\displaystyle H_{\eta,\eta^{\prime}}(\vec{r})=$
$\displaystyle\left<\psi_{\eta}\left|\hat{T}+\hat{V}_{\eta}(\vec{r})\right|\psi_{\eta^{\prime}}\right>\delta_{\eta,\eta^{\prime}}$
(9)
$\displaystyle+\left<\psi_{\eta}\left|\hat{V}_{\eta}(\vec{r})\right|\psi_{\eta^{\prime}}\right>(1-\delta_{\eta,\eta^{\prime}})$
$\displaystyle+\sum_{\eta^{\prime\prime}}\left<\psi_{\eta}\left|\hat{V}_{\eta^{\prime\prime}}(\vec{r})\right|\psi_{\eta^{\prime}}\right>+\left<\psi_{\eta}\left|\hat{H}_{so}\right|\psi_{\eta^{\prime}}\right>$
$\displaystyle H_{\eta,\eta^{\prime}}(\vec{r})=$
$\displaystyle\mathcal{E}_{\eta}\delta_{\eta,\eta^{\prime}}+t_{\eta,\eta^{\prime}}+\sum_{\eta^{\prime\prime}}\gamma^{\eta^{\prime\prime}}_{\eta,\eta^{\prime}}+\Delta^{so}_{\eta,\eta^{\prime}}$
(10)

where $\hat{V}_{\eta}(\vec{r})=\hat{V}_{i}(\vec{r}-\vec{R}_{j}-\vec{t}_{i})$
and $\eta$,$\eta^{\prime}$ and $\eta^{\prime\prime}$ in the 3CI sum correspond
to different atomic sites in the crystal. The terms in Eq. 9 are the onsite-
energy $\mathcal{E}_{\eta}$, two-center integrals (2CI)
$t_{\eta,\eta^{\prime}}$, three-center integrals (3CI)
$\gamma^{lm}_{\eta,\eta^{\prime}}$ and spin-orbit coupling
$\Delta^{so}_{\eta,\eta^{\prime}}$, respectively.

The sum involved in calculating the 3CI’s generates a large number of
parameters that are typically ignored
($\gamma^{\eta^{\prime\prime}}_{\eta,\eta^{\prime}}=0$). This truncation is
justified post-fact by arguing their influence must be negligible compared to
first-nearest neighbor 2CI terms, and reasonable agreement between the two-
center model and experimental data remains possible. It has been shown via
DFT-TB [43, 44, 45, 46] that the inclusion of 3CI and 2CI possibly out to
third-nearest neighbor interaction are necessary for polytypic transferability
of ETB parameters. The quasi-cubic approximation

$\displaystyle\Delta_{xtal},\Delta_{SO}=\frac{1}{2}((\Delta_{AB}+\Delta_{AC})$
$\displaystyle\pm\sqrt{(\Delta_{AB}+\Delta_{AC})^{2}-16u\Delta_{AB}\Delta_{AC}})$
(11)

can be used to estimate the crystal ($\Delta_{xtal}$) and spin-orbit
($\Delta_{SO}$) splittings in WZ materials in terms of the $\Gamma_{9v}$ and
$\Gamma_{7v}$ band splittings ($\Delta_{AB}=-(\Gamma_{9v}-\Gamma_{7v}),\
\Delta_{AC}=-(\Gamma_{9v}-\Gamma_{7v^{\prime}})$). The deviation from the
cubic approximation observed in InP, seen in the differing spin-orbit
splittings in the ZB and WZ [31] crystal phases, implies non-negligible
contributions from $N^{th}-NN$ and $3CI$ terms.

The Slater-Koster (SK) [39] formalism allows for the decomposition of the two-
center integrals in terms of products of radial and angular components. The
angular dependence is known from the positions of atoms in the structure,
while the radial-overlap terms are empirically determined. For small
deviations about an ideal geometry, it is common to approximate the unknown
scaling of the radial terms with a power scaling law
($\rho\rightarrow\rho(d/d_{0})^{\beta}$) [29] fit to deformation potential
data.

The ZB crystal field splits the 5-fold degenerate d-band into a doublet
($\Gamma_{15}=\\{d_{x^{2}-y^{2}},d_{3z^{2}-r^{2}}\\}$) and triplet
($\Gamma_{12}=\\{d_{xy},d_{yz},d_{zx}\\}$). In the WZ crystal field the p-band
splits into a singlet ($\Gamma_{1}=\\{p_{z}\\}$) and doublet
($\Gamma_{6}=\\{p_{x},p_{y}\\}$), and the d-band into a singlet
($\Gamma_{1}=\\{d_{3z^{2}-r^{2}}\\}$) and two doublets
($\Gamma_{5}=\\{d_{xy},d_{x^{2}-y^{2}}\\}$ and
$\Gamma_{6}=\\{d_{yz},d_{zx}\\}$). Here, as in Ref. 47, we ignore the p-band
splitting in WZ and represent the d-band onsite energies in terms of the ZB
onsite energies as $E_{d_{3z^{2}-r^{2}}}=E_{d_{15}}$,
$E_{d_{xz}}=E_{d_{yz}}=(E_{d_{12}}+2E_{d_{15}})/3$ and
$E_{d_{x^{2}-y^{2}}}=E_{d_{xy}}=(2E_{d_{12}}+E_{d_{15}})/3$.

A perturbing field that reduces the crystal point symmetry from the ideal
configuration will mix the diagonal onsite energies as

$\displaystyle E^{\prime}_{i}=C_{ij}E_{j}$ (12)

where $E^{\prime}_{i}$ is the strained onsite energy, $E_{i}$ is the ideal
onsite energy and the $C_{ij}$ is a symmetric matrix whose coefficients are
functions of the local strain ($C_{ij}=C_{ij}(\epsilon)$).

For the ETB model, these coefficients are determined by comparison with
deformation potentials. In the presence of shear strain, the d-bands in our
model respond via

$\displaystyle E^{\prime}_{xy}$
$\displaystyle=E_{xy}\left(1+2b_{d}(\epsilon_{zz}-\epsilon_{xx})\right)$ (13)
$\displaystyle E^{\prime}_{yz}$
$\displaystyle=E_{yz}\left(1-b_{d}(\epsilon_{zz}-\epsilon_{xx})\right)$
$\displaystyle E^{\prime}_{zx}$
$\displaystyle=E_{zx}\left(1-b_{d}(\epsilon_{zz}-\epsilon_{xx})\right)$
$\displaystyle E^{\prime}_{x^{2}-y^{2}}$
$\displaystyle=E_{x^{2}-y^{2}}\qquad\qquad
E^{\prime}_{3z^{2}-r^{2}}=E_{3z^{2}-r^{2}}$

where $b_{d}$ is the shear parameter for the d-bands and $E^{\prime}$ is the
strained onsite energy in terms of the unstrained value for the ideal crystal.

While only the macroscopic strain appears in Eq. 13, an accurate description
of both the macroscopic ($\epsilon$) and internal ($\Lambda$) strain tensors
are important for describing the valence band states. In AlN, the deviation
from ideal $u$ ($u^{\prime}-u\approx 6.975\ 10^{-3}$) is responsible for the
inversion of the $\Gamma_{9v}$ and $\Gamma_{7v}$ band ordering[47], although
no such band rearrangement occurs among the non-Nitride III-V’s. The strain
tensors in terms of the real and ideal lattice constants are given by

$\displaystyle\vec{a}^{\prime}_{i}$
$\displaystyle=(\mathbf{1}+\mathbf{\epsilon})\cdot\vec{a}_{i}$ (14)
$\displaystyle\vec{t}^{\prime}_{j}$
$\displaystyle=(\mathbf{1}+\mathbf{\Lambda_{j}})\cdot(\mathbf{1}+\mathbf{\epsilon})\cdot\vec{t}_{j}$
(15)

The non-zero strain components of $\epsilon$ and $\Lambda_{i}$ from ideal to
the real crystal lattice are
$\epsilon_{xx}=\epsilon_{yy}=\frac{a^{\prime}}{a}-1$ and
$\epsilon_{zz}=\frac{c^{\prime}}{c}-1$ and
$\Lambda_{2,zz}=\frac{u^{\prime}}{u}-1$ and
$\Lambda_{4,zz}=\frac{1/2+u^{\prime}}{1/2+u}-1$, respectively.

###  II.3  Nearest Neighbor and 2-Center Approximations

As noted in Section II.1, WZ and ZB are remarkably similar in their structure
out to second nearest neighbors. Importantly, both ideal-WZ and unstrained-ZB
are tetrahedrally coordinated, with the deviation of the real WZ crystal from
ideal being small ($<1\%$) for most III-V and IV materials. Assuming the
first-nearest-neighbor approximation remains valid for the hexagonal phase, WZ
and ZB crystals should both equally well described by the same set of ETB
parameters.

The 1NN 2CA orthogonal spds∗[29] model is the simplest model that can describe
well simultaneously the valence band and lowest conduction band states,
including the location of the conduction band minima. As the first nearest
neighbor bond lengths in ideal-WZ and unstrained-ZB are by definition the
same, the radial overlap parameters transfer from ZB to WZ intact.

Some results for the ‘naive transferability’ attempt are shown in Figs. 3. The
largest observed error and general trend in these plots are the systematic
underestimation of the WZ band structure as a result of the top of the valence
band ($\Gamma_{9v}$) migrating upward in energy. For the extreme case of InSb,
this underestimation leads to the ‘naïve’-model predicting a transition to a
metal in the WZ phase.

(a) WZ InSb Figure 3:  Electronic band structures for InSb calculated using
Slater-Koster parameters fit to ZB taken from Ref. 29 are shown in blue
(dashed-dotted). Red (dashed) curves are EPP band structures taken from Ref.
19. Assuming polytypic transferability between WZ and ZB within the ETB model
results in WZ-InSb to become metallic. Band structures calculated with EPP are
sufficiently transferable such that only modest changes to the WZ-InSb band
gap are observed, as expected.

The failure of the model parameters to be directly transferable between the ZB
and WZ phases is not all-together unexpected, and is likely due to the effects
of second nearest and/or three-center interactions inadvertently being
compensated for by the remaining 1NN hopping elements during optimization.
This compensation is necessary to describe the ZB band structure in the
truncated basis, but consequently causes the model to fail to transfer to the
WZ configuration.

While strict transferability of the ETB parameters generates obviously
incorrect results, only moderate corrections to the parameters (reflecting the
subtle changes in long range interactions) are required to correct many of
these discrepancies. We propose a new set of ETB parameters empirically fit to
available WZ experimental and theoretical empirical pseudo-potential (EPP)
data [18, 19], subject to the constraint that they remain close to the ZB
reference model in Ref. 29. Polytype transferability is accomplished by
starting with the ZB parameterization and allowing the electronic parameters
to deviate slightly to yield better agreement with experimental and
theoretical data.

##  III Methods

The parameters were fit by minimizing a weighted least squares objective
function constructed from the absolute fractional errors with the fitting
targets. The objective function was

$\displaystyle\chi(\vec{p})=\sum^{N}_{i=1}W_{i}\left|\frac{t_{i}-f_{i}(\vec{p})}{t_{i}}\right|^{2}$
(16)

where $\vec{p}$ is the vector of parameters, $t_{i}$ is a target value,
$f_{i}(\vec{p})$ is a function of the computed band-structure corresponding to
$t_{i}$ and $W_{i}$ are weights. $W_{i}$ allows for control of the importance
a particular target (such as the band gap energy or a valley mass) has during
optimization.

Many important electronic properties of semiconductors are governed by the
energy landscape for electrons and holes at or near the band-gap.
Consequently, properties such as valley states, band-curvatures and energy
splittings receive a great deal of attention in the description of electronic
states of semiconducting materials. In general, states close to the gap
receive greater emphasis (weighted higher) compared to states more remote in
energy since transport and optical properties are determined largely by the
energy landscape at the bandgap. Particular attention is given to direct and
indirect bandgaps, energy splittings (ie $\Delta_{ab},\Delta_{ac}$), and
effective masses.

To enforce the ‘closeness’ constraint of the ETB-parameters to the ZB
reference model for maximal transferability, a quadratic penalty function was
used

$\displaystyle\mathcal{P}(\vec{p})=W_{0}\sum_{i}\left|\frac{(p_{i}-p_{i,0})}{p_{i,0}}-\delta\right|^{2}$
(17)

where $\delta$ and $W_{0}$ control the effective width of the confining well
centered on the vector $\vec{p}_{0}$. Confinement in this way proved effective
in preventing ‘parameter wonder’ during fitting. The inherent non-uniqueness
of ETB models is a consequence of the large number of degrees of freedom
relative to the number of constraints. In particular, parameters involving the
s∗ band are nearly unconstrained. They are kept as they improve the
transferability of the s bands. A confining weight and width of $W=.1$ and
$\delta=.05$ were used during optimization.

Since we expect small differences between the WZ and ZB parameters,
$\vec{p}_{0}$ for each WZ material was taken to be the corresponding ZB
parameterization from Ref. 29. We initialize the d and s∗ onsite energies with
parameters from Ref. 29, in which a single value was used

$\displaystyle E_{d_{12},a}=E_{d_{15},a}$
$\displaystyle=E_{d_{12},c}=E_{d_{15},c}$ (18) $\displaystyle E_{s^{*},a}$
$\displaystyle=E_{s^{*},c}.$

The parameters in Eq.18 were then allowed to split during optimization, as is
done for the WZ III-Nitrides in Ref. 47. The ETB-parameters were then
optimized to be in general agreement with available experimental and
theoretical data [18, 19] while maintaining maximal closeness with the ZB
polytype phase enforced via Eq.17. The WZ model values and ZB deviations can
be found in Tables 2, 3 and the corresponding comparisons with target values
in Tables 9, 9, 9.

Limiting the fitting to only small corrections, it was found to be only
possible to reproduce the direct gap and lowest remote valley. The WZ K-point
proved difficult to reproduce while simultaneously maintaining the desired
level of transferability with the ZB parameters. Accurate fitting of the WZ-K
conduction band valley minima strongly perturbs the ZB-L point conduction band
minima as well as in general causing significant deviation of the parameters
from the ZB values. As the WZ-K valley is generally remote in energy and in
k-space, a general preference of WZ-ZB transferability and direct gap over
enforcement of remote targets was adopted.

##  IV Results and Discussion

All models are now in close agreement with the expected bandgap energies and
locations of the conduction band minima. Parameters and comparisons for the WZ
group III-V and IV materials can be found in Tables 2, 3 and Tables 9, 5,
respectively, and corresponding band structures in Figs. 4.

Notably, InSb is no longer metallic and InAs , as was the case in with the
naïve transferability. All materials now have the expected band symmetry at
the gap and location of the conduction band minima. Most materials are direct
band-gap materials in the WZ phase with the exception of C and Si which have
indirect minima at K and M, respectively.

| C | Si | Ge
---|---|---|---
$\lambda$ |  0.0030 | ( 0.00) |  0.0193 | ( 0.01) |  0.1301 | ( 0.02)
$E_{s}$ | -0.6252 | ( 0.01) | -2.5797 | (-0.02) | -3.8222 | (-0.01)
$E_{p}$ | 7.0591 | ( 0.06) | 4.2663 | (-0.05) | 4.0307 | ( 0.03)
$E_{d_{12}}$ | 28.8484 | (-0.02) | 14.3263 | (-0.05) | 12.0352 | ( 0.04)
$E_{d_{15}}$ | 23.4616 | ( 0.17) | 11.995 | ( 0.12) | 13.7182 | (-0.09)
$E_{s^{*}}$ | 38.8847 | (-0.01) | 19.2604 | (-0.01) | 18.8331 | (-0.01)
$ss\sigma$ | -4.3234 | ( 0.01) | -1.8439 | ( 0.05) | -1.5651 | (-0.04)
$sp\sigma$ |  6.5614 | (-0.19) |  2.8499 | (-0.02) |  2.7295 | ( 0.02)
$sd\sigma$ | -2.4101 | ( 0.13) | -2.6879 | ( 0.04) | -2.881 | (-0.03)
$ss^{*}\sigma$ | -2.4022 | (-0.01) | -1.7396 | (-0.02) | -1.8272 | ( 0.05)
$pp\sigma$ |  8.4125 | (-0.11) |  4.1471 | (-0.01) |  4.5227 | (-0.06)
$pp\pi$ | -2.9551 | (-0.12) | -1.5099 | ( 0.05) | -1.6515 | ( 0.00)
$pd\sigma$ | -2.3075 | (-0.07) | -2.1762 | (-0.03) | -2.2941 | (-0.04)
$pd\pi$ |  4.2872 | (-0.09) |  1.9477 | ( 0.03) |  1.6439 | ( 0.13)
$s^{*}p\sigma$ |  5.0475 | ( 0.02) |  2.7509 | ( 0.03) |  2.8755 | (-0.02)
$dd\sigma$ | -3.2526 | ( 0.22) | -1.2144 | ( 0.01) | -1.2208 | ( 0.00)
$dd\pi$ |  5.9774 | (-0.20) |  2.6039 | (-0.04) |  2.4763 | ( 0.01)
$dd\delta$ | -3.9583 | ( 0.01) | -2.4427 | ( 0.01) | -1.9936 | ( 0.07)
$s^{*}d\sigma$ | -2.3285 | (-0.01) | -0.7057 | (-0.01) | -0.6200 | ( 0.00)
$s^{*}s^{*}\sigma$ | -2.6714 | ( 0.00) | -3.2861 | ( 0.01) | -3.6254 | (-0.01)
$E_{n.i.}$ | -0.0757 | - | -0.0790 | - | -0.0089 | -
Table 2: LDL group-IV SK-parameters in eV. Values in parenthesis are the
fractional difference between our LDL parameters and the DMD parameters from
Ref. 29. The onsite energies have been shifted to make the top of the valence
band have zero energy ($E(\Gamma_{9v})=0$) for the ideal WZ crystal (Table 1).
$\Delta E_{n.i.}=E^{\prime}(\Gamma_{9v})-E(\Gamma_{9v})$ is the shift in the
valence band maximum when going from ideal to non-ideal WZ. To obtain the
parameters for the non-ideal structure, subtract $\Delta E_{n.i}$ from the
ideal onsite energies.

While there is insufficient deformation data available to re-fit the radial
scaling coefficients for WZ, the similarity between the ZB and WZ ETB-
parameters (Table 2, 3) as well as the character of states near the gap
improves the likely hood that the the radial scaling coefficients will be
transferable between the ZB and WZ phase. In Table 10 we compare the pressure
coefficients for the ZB phase and the WZ phase for similar states near the
band gap. The ZB pressure coefficients were calculated using the ZB optimized
Slater-Koster and radial-scaling coefficients from Ref. 29. The WZ pressure
coefficients were calculated using the WZ optimized parameters from the
present work and the radial-scaling coefficients from Ref. 29. Many of the
pressure coefficients are similar for the WZ and ZB phase, in qualitative
agreement with results from Ref. 48 where it was found that only small
variations between the ZB and WZ deformation potentials were present for the
III-V nitrides.

##  V Conclusion

We have demonstrated some of the problems that arise when using ETB
parameterizations determined for bulk ZB in a WZ configuration, despite the
high degree of geometric similarity between these polytypes. The failure of
the ETB parameters to be transferable between polytypes is a direct
consequence of approximations made in the 1NN 2CA model. These approximations
result in a failure to accommodate subtle changes in the bonding environment
which are observed starting in the three-center integrals and the second
nearest neighbor two center integrals. We have produced a set of ETB
parameters for WZ non-nitride III-V’s and group IV materials that are in
significantly improved agreement with available experimental data as well as
calculations performed using empirical pseudopotentials. Special attention has
been given to the direct gap and next lowest conduction band valley states, as
well as transferability with the ZB parameters. The differences between the WZ
and ZB ETB parameters are the necessary corrections to accommodate for changes
in the crystal field. The addition of the ETB parameters presented in this
work, optimized for the hexagonal 2H (WZ) phase and transferability with
existing ETB models for cubic 3C (ZB) III-V (non-nitride) and group-IV
materials, enables the investigation heterostructures containing hexagonal and
3C polytypes.

| AlP | AlAs | AlSb | GaP | GaAs | GaSb | InP | InAs | InSb
---|---|---|---|---|---|---|---|---|---
$\lambda_{a}$ | 0.0207 | (-0.06) | 0.1715 | ( 0.00) | 0.3874 | ( 0.01) | 0.0307 | (-0.02) | 0.1813 | ( 0.01) | 0.4504 | ( 0.01) | 0.0229 | ( 0.00) | 0.175 | ( 0.01) | 0.4518 | (-0.01)
$\lambda_{c}$ | 0.0073 | ( 0.00) | 0.0072 | ( 0.00) | 0.0079 | ( 0.00) | 0.041 | ( 0.00) | 0.0408 | ( 0.00) | 0.0433 | ( 0.00) | 0.1131 | (-0.01) | 0.1248 | ( 0.00) | 0.1227 | ( 0.00)
$E_{s,a}$ | -6.6058 | (-0.19) | -6.9122 | (-0.01) | -6.0023 | (-0.07) | -6.1425 | (-0.04) | -6.0245 | (-0.02) | -5.4982 | ( 0.01) | -5.1873 | ( 0.03) | -5.7538 | ( 0.06) | -6.2602 | (-0.20)
$E_{p,a}$ | 2.8082 | ( 0.09) | 2.7705 | (-0.01) | 3.0269 | ( 0.08) | 2.441 | ( 0.10) | 3.6675 | ( 0.00) | 3.224 | ( 0.06) | 3.3852 | (-0.02) | 3.3027 | ( 0.03) | 3.7206 | ( 0.01)
$E_{d_{12},a}$ | 14.6773 | (-0.05) | 14.074 | (-0.14) | 12.0702 | (-0.11) | 15.3096 | (-0.13) | 12.6401 | ( 0.04) | 11.1386 | (-0.01) | 11.8032 | ( 0.07) | 11.5632 | ( 0.04) | 11.2501 | (-0.03)
$E_{d_{15},a}$ | 11.6797 | ( 0.16) | 10.4503 | ( 0.13) | 10.8595 | (-0.01) | 11.0311 | ( 0.17) | 13.692 | (-0.04) | 11.4367 | (-0.04) | 12.5196 | ( 0.02) | 12.4471 | (-0.03) | 10.6482 | ( 0.03)
$E_{s^{*},a}$ | 22.4105 | (-0.10) | 19.8662 | (-0.06) | 16.4681 | (-0.05) | 19.3548 | ( 0.02) | 19.131 | ( 0.02) | 16.369 | (-0.02) | 18.9953 | (-0.01) | 17.2566 | ( 0.02) | 16.8295 | (-0.06)
$E_{s,c}$ | 0.7178 | (-0.01) | 0.1144 | (-0.01) | 0.2361 | ( 0.00) | -0.9678 | ( 0.00) | -0.3144 | ( 0.00) | -1.0009 | ( 0.00) | 0.3009 | ( 0.00) | 0.1797 | ( 0.00) | 0.0426 | (-0.02)
$E_{p,c}$ | 6.5429 | (-0.07) | 5.5104 | ( 0.00) | 5.8106 | (-0.03) | 6.2643 | (-0.07) | 6.9067 | (-0.07) | 6.0132 | (-0.04) | 6.9977 | (-0.08) | 6.5234 | (-0.03) | 5.1306 | ( 0.16)
$E_{d_{12},c}$ | 11.8477 | ( 0.15) | 10.4527 | ( 0.13) | 10.6303 | ( 0.01) | 12.1366 | ( 0.10) | 12.0362 | ( 0.09) | 10.2238 | ( 0.07) | 11.865 | ( 0.07) | 10.5425 | ( 0.12) | 9.1053 | ( 0.16)
$E_{d_{15},c}$ | 16.5362 | (-0.18) | 13.7099 | (-0.12) | 11.8107 | (-0.09) | 17.4598 | (-0.28) | 13.4595 | (-0.02) | 11.978 | (-0.08) | 12.3531 | ( 0.03) | 12.1772 | (-0.01) | 12.5455 | (-0.14)
$E_{s^{*},c}$ | 20.3264 | ( 0.00) | 18.5132 | ( 0.01) | 15.456 | ( 0.01) | 18.5051 | ( 0.06) | 19.2691 | ( 0.01) | 16.0499 | ( 0.00) | 18.5897 | ( 0.01) | 17.0919 | ( 0.03) | 12.7396 | ( 0.19)
$ss\sigma$ | -1.8382 | (-0.06) | -1.8042 | (-0.04) | -1.748 | (-0.08) | -1.8177 | (-0.07) | -1.6535 | (-0.02) | -1.4552 | (-0.06) | -1.452 | (-0.04) | -1.4811 | ( 0.00) | -1.4654 | (-0.30)
$s_{a}p_{c}\sigma$ | 2.7009 | (-0.03) | 2.5436 | (-0.01) | 2.2294 | ( 0.14) | 2.6111 | ( 0.07) | 2.4913 | ( 0.00) | 2.4044 | ( 0.06) | 2.1679 | ( 0.00) | 2.1689 | ( 0.06) | 2.0664 | ( 0.19)
$s_{a}d_{c}\sigma$ | -2.2456 | ( 0.11) | -2.1428 | ( 0.16) | -2.6982 | ( 0.03) | -2.2849 | ( 0.18) | -2.6738 | ( 0.02) | -2.5768 | ( 0.01) | -2.507 | ( 0.02) | -2.5093 | ( 0.03) | -2.1788 | ( 0.15)
$s_{a}s^{*}_{c}\sigma$ | -1.4678 | (-0.03) | -1.2659 | ( 0.00) | -1.2312 | (-0.02) | -1.6108 | ( 0.02) | -1.527 | ( 0.02) | -1.6649 | ( 0.00) | -1.2652 | ( 0.02) | -1.2576 | (-0.03) | -1.4979 | ( 0.20)
$s_{c}p_{a}\sigma$ | 2.877 | (-0.03) | 2.857 | (-0.04) | 2.8108 | ( 0.04) | 3.1278 | (-0.05) | 2.9333 | ( 0.00) | 2.729 | (-0.01) | 2.549 | ( 0.04) | 2.4785 | ( 0.12) | 2.3578 | ( 0.13)
$pp\sigma$ | 4.7281 | (-0.17) | 4.4956 | (-0.06) | 4.3744 | (-0.07) | 4.5829 | (-0.09) | 4.7335 | (-0.07) | 4.5108 | (-0.01) | 4.2026 | (-0.05) | 4.1402 | (-0.01) | 3.9329 | ( 0.06)
$pp\pi$ | -1.4234 | (-0.09) | -1.5647 | (-0.17) | -1.6529 | (-0.08) | -1.6192 | (-0.13) | -1.3242 | ( 0.09) | -1.7787 | (-0.06) | -1.1877 | ( 0.07) | -1.3302 | ( 0.03) | -1.412 | ( 0.04)
$p_{a}d_{c}\sigma$ | -1.8547 | (-0.11) | -2.014 | (-0.17) | -2.240 | (-0.13) | -2.1733 | (-0.21) | -1.8308 | (-0.03) | -2.2774 | (-0.12) | -1.9619 | (-0.02) | -2.1582 | (-0.02) | -2.1305 | ( 0.01)
$p_{a}d_{c}\pi$ | 1.2727 | ( 0.32) | 1.3352 | ( 0.25) | 1.6579 | ( 0.22) | 1.3028 | ( 0.30) | 1.7763 | ( 0.00) | 1.7778 | ( 0.10) | 1.5683 | ( 0.00) | 1.5602 | (-0.01) | 1.8582 | (-0.01)
$s^{*}_{c}p_{a}\sigma$ | 2.2671 | ( 0.03) | 2.1988 | ( 0.00) | 1.8935 | ( 0.00) | 2.1737 | (-0.01) | 2.2339 | (-0.01) | 2.461 | ( 0.00) | 2.0919 | (-0.02) | 1.8675 | ( 0.02) | 2.1124 | ( 0.10)
$s_{c}d_{a}\sigma$ | -2.3231 | (-0.07) | -2.5744 | (-0.08) | -2.058 | (-0.03) | -2.6466 | (-0.14) | -2.469 | (-0.02) | -2.4401 | (-0.01) | -2.3094 | (-0.04) | -2.4926 | (-0.02) | -2.1118 | ( 0.09)
$p_{c}d_{a}\sigma$ | -1.9477 | (-0.07) | -1.9571 | (-0.11) | -2.0745 | (-0.05) | -1.9942 | (-0.10) | -1.7782 | ( 0.01) | -2.2698 | (-0.01) | -1.8735 | ( 0.01) | -1.9785 | ( 0.04) | -2.2119 | (-0.02)
$p_{c}d_{a}\pi$ | 1.599 | ( 0.27) | 1.491 | ( 0.29) | 1.3758 | ( 0.25) | 1.577 | ( 0.26) | 1.9289 | ( 0.07) | 1.6354 | ( 0.12) | 1.7121 | ( 0.04) | 1.6614 | ( 0.03) | 1.6816 | ( 0.09)
$dd\sigma$ | -1.3196 | ( 0.02) | -1.1657 | ( 0.04) | -1.1245 | ( 0.01) | -1.1846 | ( 0.03) | -1.1431 | ( 0.00) | -1.2482 | ( 0.00) | -1.2563 | (-0.01) | -1.2026 | ( 0.00) | -1.2805 | ( 0.02)
$dd\pi$ | 2.5194 | (-0.06) | 2.1358 | ( 0.02) | 2.0539 | ( 0.03) | 2.539 | (-0.12) | 2.3093 | (-0.05) | 2.2402 | (-0.02) | 2.3329 | (-0.09) | 2.2768 | (-0.04) | 2.0377 | ( 0.02)
$dd\delta$ | -1.6022 | ( 0.13) | -1.7413 | ( 0.01) | -1.5838 | ( 0.08) | -2.0501 | (-0.02) | -1.9845 | ( 0.00) | -1.6931 | ( 0.03) | -1.7761 | (-0.05) | -1.7433 | ( 0.02) | -1.3329 | ( 0.06)
$s^{*}_{c}d_{a}\sigma$ | -0.712 | ( 0.01) | -0.7354 | ( 0.01) | -0.7758 | ( 0.02) | -0.6476 | ( 0.02) | -0.6457 | ( 0.00) | -0.7981 | ( 0.00) | -0.8095 | ( 0.01) | -0.83 | ( 0.01) | -0.794 | ( 0.03)
$s_{c}s^{*}_{a}\sigma$ | -1.6871 | (-0.03) | -1.4652 | ( 0.09) | -1.4931 | ( 0.12) | -1.4465 | ( 0.10) | -2.0249 | (-0.02) | -1.8242 | ( 0.08) | -1.8984 | (-0.03) | -1.9831 | ( 0.07) | -1.8296 | ( 0.09)
$s^{*}_{a}p_{c}\sigma$ | 1.9896 | ( 0.04) | 2.1782 | (-0.03) | 2.4216 | ( 0.02) | 2.185 | ( 0.09) | 2.1337 | ( 0.02) | 3.0763 | (-0.02) | 2.5094 | ( 0.02) | 2.7347 | (-0.03) | 3.089 | (-0.13)
$s^{*}_{a}d_{c}\sigma$ | -0.7957 | (-0.02) | -0.8239 | (-0.02) | -0.7444 | (-0.02) | -0.6527 | (-0.02) | -0.6942 | (-0.01) | -0.8596 | ( 0.00) | -0.7918 | ( 0.00) | -0.8583 | (-0.01) | -0.7608 | (-0.03)
$s^{*}s^{*}\sigma$ | -3.7871 | (-0.04) | -3.6196 | ( 0.00) | -3.2994 | ( 0.00) | -3.59 | (-0.01) | -3.8005 | (-0.03) | -3.2367 | ( 0.00) | -3.7707 | (-0.02) | -4.0354 | (-0.05) | -4.0693 | (-0.26)
$\Delta E_{n.i.}$ | -0.0130 | - | 0.0130 | - | 0.0135 | - | -0.0057 | - | -0.0086 | - | -0.0025 | - | -0.0007 | - | 0.0015 | - | 0.0033 | -
Table 3: WZ III-V SK-parameters in eV. Values in parenthesis are the
fractional difference between our WZ parameters and the ZB parameters from
Ref. 29. The onsite energies have been shifted to make the top of the valence
band have zero energy ($E(\Gamma_{9v})=0$) for the ideal WZ crystal (Table 1).
$\Delta E_{n.i.}=E^{\prime}(\Gamma_{9v})-E(\Gamma_{9v})$ is the shift in the
valence band maximum when going from ideal to non-ideal WZ. To obtain the
parameters for the non-ideal structure, subtract $\Delta E_{n.i}$ from the
ideal onsite energies.

(a) C

(b) Si

(c) Ge

(d) AlP

(e) AlAs

(f) AlSb

(g) GaP

(h) GaAs

(i) GaSb

(j) InP

(k) InAs

(l) InSb

Figure 4: Electronic band structures for the ideal WZ crystal phase (Table 1).
The current WZ optimized model is shown as a solid black line, and the ZB
optimized ETB parameters from Ref. 29 are shown as a blue dashed-dotted lines.
EPP band structures from Refs. 19, 18 are shown as red dashed lines.

C | Si | Ge
---|---|---
|  | TB |  |  |  | TB |  |  |  | TB |
|  EPPa | Ideal | Non-Ideal | Exp |  |  EPPa | Ideal | Non-Ideal | Exp |  |  EPPa | Ideal | Non-Ideal | Exp
$\Gamma_{7v}^{+}$ | -1.6965 | -1.2882 | -1.1583 | - | $\Gamma_{7v}^{+}$ | -0.3496 | -0.9726 | -0.9022 | - | $\Gamma_{7v}^{+}$ | -0.4896 | -0.4835 | -0.4648 | -
$\Gamma_{7v}^{+}$ | -0.003 | -0.0052 | -0.0052 | - | $\Gamma_{7v}^{+}$ | -0.0279 | -0.031 | -0.0341 | - | $\Gamma_{7v}^{+}$ | -0.1293 | -0.1321 | -0.1256 | -
$\Gamma_{9v}^{+}$ | 0.000 | 0.000 | 0.000 | - | $\Gamma_{9v}^{+}$ | 0.000 | 0.000 | 0.000 | - | $\Gamma_{9v}^{+}$ | 0.000 | 0.000 | 0.000 | -
$\Gamma_{7c}^{-}$ | 5.7365 | 5.7665 | 5.9015 | - | $\Gamma_{8c}^{-}$ | 1.4814 | 1.6703 | 1.8333 | - | $\Gamma_{8c}^{-}$ | 0.3103 | 0.3102 | 0.3666 | -
$\Gamma_{9c}^{-}$ | 5.741 | 5.7712 | 5.9062 | - | $\Gamma_{7c}^{-}$ | 2.6962 | 2.442 | 2.4875 | - | $\Gamma_{7c}^{-}$ | 0.7659 | 0.7965 | 0.7945 | -
$K_{c}$ | 4.7676 | 7.4955 | 7.5558 | - | $K_{c}$ | 1.2191 | 2.2423 | 2.3914 | - | $K_{c}$ | 1.5267 | 0.9908 | 1.0523 | -
$M_{c}$ | 5.2688 | 5.2907 | 5.288 | - | $M_{c}$ | 0.7957 | 0.7962 | 0.8309 | - | $M_{c}$ | 0.6828 | 0.682 | 0.7438 | -
$A_{7c}$ | 7.2555 | 4.7961 | 5.0482 | - | $A_{8c}$ | 2.3662 | 2.2644 | 2.2645 | - | $A_{8c}$ | 1.0951 | 3.5758 | 3.6067 | -
$H_{c}$ | 7.5599 | 7.3609 | 7.3229 | - | $H_{c}$ | 2.3229 | 1.5048 | 1.4877 | - | $H_{c}$ | 1.7489 | 1.0716 | 1.0449 | -
$L_{c}$ | 5.5528 | 6.015 | 5.9343 | - | $L_{c}$ | 1.2958 | 1.3375 | 1.3834 | - | $L_{c}$ | 0.7849 | 0.7819 | 0.8064 | -
$m_{l}(\Gamma_{9v}^{+})$ | -0.3418 | -0.5463 | -0.5639 | - | $m_{l}(\Gamma_{9v}^{+})$ | -0.5637 | -0.5861 | -0.6059 | - | $m_{l}(\Gamma_{9v}^{+})$ | -0.6035 | -0.4073 | -0.4191 | -
$m_{l}(\Gamma_{7c}^{-})$ | 0.7860 | 0.9972 | 1.0545 | - | $m_{l}(\Gamma_{8c}^{-})$ | 1.0483 | 1.5706 | 1.607 | - | $m_{l}(\Gamma_{8c}^{-})$ | 1.0563 | 1.1443 | 1.1176 | -
$m_{t}(\Gamma_{9v}^{+})$ | -0.3226 | -0.3055 | -0.3049 | - | $m_{t}(\Gamma_{9v}^{+})$ | -0.2128 | -0.1817 | -0.2179 | - | $m_{t}(\Gamma_{9v}^{+})$ | -0.0672 | -0.0559 | -0.0578 | -
$m_{t}(\Gamma_{7c}^{-})$ | 0.3785 | 0.4373 | 0.4318 | - | $m_{t}(\Gamma_{8c}^{-})$ | 0.1224 | 0.1615 | 0.163 | - | $m_{t}(\Gamma_{8c}^{-})$ | 0.0852 | 0.1086 | 0.1067 | -

aReference 19
---

Table 4: Energies ($\rm{eV}$) and effective masses ($\rm{m_{e}}$) for ideal
and non-ideal (Table 1) LDL calculated using the present ETB parameterization,
and EPP data from Ref. 19. Experimental values are shown when available and
marked with a ‘-’ otherwise. Energy values are referenced to the top of the
valence band ($\Gamma_{9v}$).

| AlP | AlAs | AlSb
---|---|---|---
| $PP^{a}$ | TB | Exp. | $PP^{a}$ | TB | Exp. | $PP^{a}$ | TB | Exp.
|  | Ideal | Non-Ideal |  |  | Ideal | Non-Ideal |  |  | Ideal | Non-Ideal |
$E_{g}$ | 2.969 | 2.955 | 2.997 | - | 1.971 | 1.966 | 1.951 | - | 1.891 | 1.8907 | 1.8567 | -
$\Gamma_{7v^{\prime}}$ | -0.435 | -0.148 | -0.118 | - | -0.518 | -0.467 | -0.457 | - | -0.802 | -0.900 | -0.891 | -
$\Gamma_{7v}$ | -0.044 | -0.028 | -0.027 | - | -0.139 | -0.134 | -0.128 | - | -0.156 | -0.156 | -0.145 | -
$\Gamma_{9v}$ | 0.000 | 0.000 | 0.000 | - | 0.000 | 0.000 | 0.000 | - | 0.000 | 0.000 | 0.000 | -
$\Gamma_{8c}$ | 2.969 | 2.955 | 2.997 | - | 1.971 | 1.966 | 1.951 | - | 1.891 | 1.891 | 1.857 | -
$\Gamma_{7c}$ | 3.775 | 3.500 | 3.526 | - | 3.153 | 3.007 | 2.916 | - | 2.418 | 2.553 | 2.455 | -
$K_{c}$ | 3.823 | 3.523 | 3.568 | - | 2.932 | 2.605 | 2.594 | - | 2.355 | 2.414 | 2.385 | -
$M_{c}$ | 3.157 | 3.141 | 3.186 | - | 2.146 | 2.140 | 2.185 | - | 1.929 | 1.930 | 1.972 | -
$A_{c}$ | 3.684 | 5.205 | 5.205 | - | 2.745 | 4.135 | 4.162 | - | 2.478 | 3.676 | 3.722 | -
$H_{c}$ | 4.446 | 3.538 | 3.527 | - | 3.478 | 2.528 | 2.494 | - | 2.950 | 2.290 | 2.270 | -
$L_{c}$ | 3.253 | 3.275 | 3.296 | - | 2.354 | 2.333 | 2.325 | - | 2.060 | 2.108 | 2.099 | -
$m_{l}(\Gamma_{9v})$ | -0.972 | -0.599 | -0.605 | - | -0.933 | -0.508 | -0.519 | - | -0.959 | -0.523 | -0.534 | -
$m_{l}(\Gamma_{8c})$ | 1.187 | 1.397 | 1.379 | - | 1.081 | 1.198 | 1.168 | - | 1.160 | 1.210 | 1.173 | -
$m_{t}(\Gamma_{9v})$ | -0.248 | -0.217 | -0.237 | - | -0.216 | -0.174 | -0.176 | - | -0.211 | -0.155 | -0.155 | -
$m_{t}(\Gamma_{8c})$ | 0.170 | 0.216 | 0.215 | - | 0.142 | 0.1703 | 0.169 | - | 0.157 | 0.148 | 0.146 | -
| GaP | GaAs | GaSb
| $PP^{a}$ | TB | Exp. | $PP^{a}$ | TB | Exp. | $PP^{a}$ | TB | Exp.
|  | Ideal | Non-Ideal |  |  | Ideal | Non-Ideal |  |  | Ideal | Non-Ideal |
$E_{g}$ |  2.251 |  2.051 |  2.096 |  2.108a |  1.503 |  1.503 |  1.521 |  1.522b |  |  0.512 |  0.554 | -
$\Gamma_{7v^{\prime}}$ | -0.373 | -0.186 | -0.178 | - | -0.475 | -0.497 | -0.486 | - | -0.874 | -1.004 | -1.004 | -
$\Gamma_{7v}$ | -0.050 | -0.045 | -0.044 | - | -0.120 | -0.132 | -0.127 |  -0.104c | -0.142 | -0.149 | -0.149 | -
$\Gamma_{9v}$ |  0.000 |  0.000 |  0.000 | - |  0.000 |  0.000 |  0.000 | - |  0.000 |  0.000 |  0.000 | -
$\Gamma_{8c}$ |  2.251 |  2.051 |  2.096 |  2.108a |  1.503 |  2.170 |  2.216 |  1.58c |  0.509 |  0.512 |  0.554 | -
$\Gamma_{7c}$ |  2.877 |  2.866 |  2.895 | - |  1.588 |  1.503 |  1.521 |  1.522b, 1.52c |  0.851 |  0.877 |  0.904 | -
$K_{c}$ |  3.054 |  2.637 |  2.685 | - |  3.282 |  2.675 |  2.724 | - |  1.442 |  1.121 |  1.167 | -
$M_{c}$ |  2.402 |  2.376 |  2.415 | - |  2.167 |  2.144 |  2.201 | - |  0.785 |  0.788 |  0.820 | -
$A_{c}$ |  2.986 |  4.586 |  4.569 | - |  2.257 |  4.300 |  4.291 | - |  1.237 |  3.083 |  3.079 | -
$H_{c}$ |  3.653 |  2.812 |  2.787 | - |  3.318 |  2.755 |  2.725 | - |  1.890 |  1.164 |  1.134 | -
$L_{c}$ |  2.587 |  2.575 |  2.588 | - |  2.136 |  2.209 |  2.229 | - |  0.858 |  0.931 |  0.945 | -
$m_{l}(\Gamma_{9v})$ | -0.941 | -0.558 | -0.561 | - | -1.026 | -0.733 | -0.746 | - | -0.833 | -0.502 | -0.511 | -
$m_{l}(\Gamma_{8c})$ |  1.162 |  1.431 |  1.415 | - | - | - | - | - |  0.983 |  1.090 |  1.075 | -
$m_{l}(\Gamma_{7c})$ | - | - | - | - |  0.090 |  0.081 |  0.079 | - | - | - | - | -
$m_{t}(\Gamma_{9v})$ | -0.205 | -0.178 | -0.191 | - | -0.134 | -0.113 | -0.118 | - | -0.087 | -0.068 | -0.071 | -
$m_{t}(\Gamma_{8c})$ |  0.143 |  0.170 |  0.169 | - | - | - | - | - |  0.096 |  0.100 |  0.098 | -
$m_{t}(\Gamma_{7c})$ | - | - | - | - |  0.082 |  0.073 |  0.075 | - | - | - | - | -
a Reference 32
b Reference 9
c Reference 30
| InP | InAs | InSb
| $PP^{a}$ | TB | Exp. | $PP^{a}$ | TB | Exp. | $PP^{a}$ | TB | Exp.
|  | Ideal | Non-Ideal |  |  | Ideal | Non-Ideal |  |  | Ideal | Non-Ideal |
$E_{g}$ | 1.474 | 1.487 | 1.491 |  1.492d,1.508f | 0.481 | 0.480 | 0.480 |  0.482 e | 0.287 | 0.288 | 0.286 |
$\Gamma_{7v^{\prime}}$ | -0.348 | -0.338 | -0.339 | - | -0.469 | -0.490 | -0.494 |  -0.433e | -0.847 | -0.884 | -0.888 | -
$\Gamma_{7v}$ | -0.063 | -0.061 | -0.062 |  -.036d | -0.105 | -0.111 | -0.114 |  -0.068e | -0.098 | -0.101 | -0.108 | -
$\Gamma_{9v}$ | 0.000 | 0.000 | 0.000 | - | 0.000 | 0.000 | 0.000 | - | 0.000 | 0.000 | 0.000 | -
$\Gamma_{7c}$ | 1.474 | 1.487 | 1.491 | - | 0.481 | 0.480 | 0.480 |  0.482e | 0.287 | 0.288 | 0.286 | -
$\Gamma_{8c}$ | 1.712 | 2.061 | 2.080 | - | 1.222 | 1.322 | 1.339 |  1.065e | 1.116 | 0.621 | 0.634 | -
$K_{c}$ | 2.761 | 2.528 | 2.548 | - | 2.174 | 1.686 | 1.703 | - | 0.766 | 0.876 | 0.886 | -
$M_{c}$ | 2.059 | 2.054 | 2.078 | - | 1.554 | 1.544 | 1.569 | - | 0.910 | 0.911 | 0.928 | -
$A_{c}$ | 2.347 | 3.892 | 3.886 | - | 1.659 | 3.585 | 3.581 | - | 1.336 | 2.984 | 2.971 | -
$H_{c}$ | 3.186 | 2.574 | 2.553 | - | 2.742 | 2.000 | 1.977 | - | 2.049 | 1.133 | 1.113 | -
$L_{c}$ | 2.120 | 2.165 | 2.171 | - | 1.661 | 1.638 | 1.644 | - | 1.066 | 0.908 | 0.910 | -
$m_{l}(\Gamma_{9v})$ | -1.273 | -0.864 | -0.874 | - | -1.700 | -0.744 | -0.751 | - | -2.060 | -0.540 | -0.547 | -
$m_{l}(\Gamma_{7c})$ | 0.105 | 0.097 | 0.096 | - | 0.060 | 0.047 | 0.047 | - | 0.051 | 0.047 | 0.048 | -
$m_{t}(\Gamma_{9v})$ | -0.158 | -0.141 | -0.147 | - | -0.084 | -0.057 | -0.059 | - | \- 0.066 | -0.042 | -0.044 | -
$m_{t}(\Gamma_{7c})$ | 0.088 | 0.085 | 0.086 | - | 0.042 | 0.037 | 0.037 | - | 0.035 | 0.033 | 0.033 | -
d Reference 31
e Reference 11
f Reference 7

Table 5: Energies ($\rm{eV}$) and effective masses ($\rm{m_{e}}$) for ideal
and non-ideal (Table 1) WZ calculated using the present ETB parameterization,
and EPP data from Ref. 19. Experimental values are shown when available and
marked with a ‘-’ otherwise. Energy values are referenced to the top of the
valence band ($\Gamma_{9v}$)

.

C | Si | Ge
---|---|---
| Weight | Target | Result |  | Weight | Target | Result |  | Weight | Target | Result
$\Delta_{AB}[\rm{eV}]$ | 10 | 0.003 | 0.005 | $\Delta_{AB}$ | 10 | 0.028 | 0.031 | $\Delta_{AB}$ | 50 | 0.129 | 0.132
$K_{9c}-M_{c}$ | 50 | -0.501 | -0.495 | $M_{c}$ | 1000 | 0.796 | 0.796 | $\Gamma_{8c}^{-}$ | 1000 | 0.310 | 0.310
$K_{9c}-\Gamma_{c}$ | 500 | -0.969 | -0.97 | $M_{c}-\Gamma_{c}$ | 100 | -0.686 | -0.874 | $LM_{c}-M_{c}$ | 50 | -0.064 | -0.064
$K_{9c}$ | 1000 | 4.768 | 4.796 | $LM_{c}-M_{c}$ | 100 | 0.382 | 0.28 | $LM_{c}-\Gamma_{c}$ | 500 | 0.309 | 0.308
$\Gamma_{7c}^{-}$ | 10 | 5.737 | 5.766 | $M_{c}-K_{c}$ | 10 | -0.423 | -1.468 |  |  |  |
$\Gamma_{7c^{\prime}}^{-}$ | 10 | 8.825 | 6.353 | $\Gamma_{7c}^{-}-\Gamma_{8c}^{-}$ | 100 | 1.215 | 0.772 |  |  |  |
|  |  |  |  $H_{c}$-$K_{c}$ | 10 | 1.104 | -0.76 |  |  |  |
$m_{l}(\Gamma_{7c})$ | 10 | 0.786 | 0.997 | - | - | - | - | $m_{l}(\Gamma_{7c})$ | 10 | 1.056 | 1.144
$m_{l}(\Gamma_{7c^{\prime}})$ | 10 | 0.25 | 0.285 | - | - | - | - | $m_{l}(\Gamma_{8c})$ | 10 | 0.052 | 0.049

Table 6: Group-IV Target Comparison.

|  | AlP | AlAs | AlSb
---|---|---|---|---
| Weight | Target | Result | Target | Result | Target | Result
$\Delta_{AB}$ | 10 | 0.044 | 0.027 | 0.139 | 0.134 | 0.156 | 0.156
$\Gamma_{8c}$ | 1000 | 2.969 | 2.955 | 1.971 | 1.966 | 1.891 | 1.891
$M_{c}$-$\Gamma_{c}$ | 500 | 0.187 | 0.186 | 0.175 | 0.174 | 0.04 | 0.039
$LM_{c}$-$M_{c}$ | 10 | 0.039 | 0.035 | 0.161 | 0.078 | 0.048 | 0.046
$m_{l}(\Gamma_{7c})$ | 10 | 1.187 | 1.397 | 1.081 | 1.198 | 1.16 | 1.21
$m_{l}(\Gamma_{8c})$ | 10 | 0.182 | 0.182 | 0.18 | 0.159 | 0.163 | 0.152

Table 7: Al-series Target Comparison.

|  | GaP | GaAs | GaSb
---|---|---|---|---
| Weight | Target | Result | Target | Result | Target | Result
$\Delta_{AB}$ | 10 | 0.05 | 0.044 | 0.12 | 0.126 | 0.142 | 0.149
$\Gamma_{8c}$ | 1000 |  2.108a | 2.098 | - | - | 0.509 | 0.512
$\Gamma_{7c}$ | 1000 | - | - |  1.522b | 1.522 | - | -
$LM_{c}$-$\Gamma_{c}$ | 500 | - | - | 0.532 | 0.531 | -0.044 | -0.046
$M_{c}$-$\Gamma_{c}$ | 500 | 0.298 | 0.294 | - | - | - | -
$LM_{c}$-$M_{c}$ | 10 | 0.074 | 0.053 | -0.109 | -0.112 | 0.277 | 0.276
$m_{l}(\Gamma_{7c})$ | 10 | 1.162 | 1.415 | 1.05 | 1.284 | 0.983 | 1.09
$m_{l}(\Gamma_{8c})$ | 10 | 0.153 | 0.129 | 0.09 | 0.079 | 0.064 | 0.056
a Reference 32 |  |  |  |  |  |  |
b Reference 30 |  |  |  |  |  |  |

Table 8: Ga-series Target Comparison.

|  | InP | InAs | InSb
---|---|---|---|---
| Weight | Target | Result | Target | Result | Target | Result
$\Delta_{AB}$ | 10 | 0.063 | 0.061 | 0.105 | 0.113 | 0.098 | 0.101
$\Gamma_{7c}$ | 1000 |  1.492a | 1.493 |  0.481b | 0.482 | 0.287 | 0.288
$LM_{c}$-$\Gamma_{c}$ | 500 | 0.501 | 0.5 | 0.969 | 0.968 | -0.165 | -0.165
$LM_{c}$-$M_{c}$ | 10 | -0.067 | -0.068 | -0.103 | -0.104 | 0.458 | 0.457
$m_{l}(\Gamma_{7c})$ | 10 | 0.105 | 0.095 | 0.06 | 0.046 | 0.051 | 0.047
$m_{l}(\Gamma_{8c})$ | 10 | 1.094 | 1.278 | 1.276 | 1.392 | 1.781 | 1.791
a Reference 31 |  |  |  |  |  |  |
b Reference 11 |  |  |  |  |  |  |

Table 9: In-series Target Comparison.

Targets in units of $\rm{eV}$ and $\rm{m_{e}}$, weights used during fitting
and resulting fits. $LM_{c}$ refers to the valley along the L-M path. A
minimal set of targets was used in order to keep the WZ corrections to the ZB
parameters small. Carbon and Silicon required additional constraints to avoid
unphysical band structures, such as extrema with wrong symmetries. Cells
marked with a ‘-’ were not used as a targets for that material. Unless
otherwise noted, target values are EPP calculations from Refs. 18, 19.

ZB | WZ
---|---
| AlP | AlAs | AlSb |  | AlP | AlAs | AlSb
$\frac{d}{dp}E(\Gamma_{8v}-\Gamma_{6c})$ | 9.27 | 9.39 | 12.21 | $\frac{d}{dp}E(\Gamma_{9v}-\Gamma_{8c})$ | 8.66 | 8.61 | 11.14
$\frac{d}{dp}E(\Gamma_{8v}-L_{6c})$ | 4.69 | 4.67 | 6.77 | $\frac{d}{dp}E(\Gamma_{9v}-\Gamma_{7c})$ | 3.90 | 3.81 | 5.53
| GaP | GaAs | GaSb |  | GaP | GaAs | GaSb
$\frac{d}{dp}E(\Gamma_{8v}-\Gamma_{6c})$ | 10.49 | 11.62 | 14.03 | $\frac{d}{dp}E(\Gamma_{9v}-\Gamma_{8c})$ | 9.77 | 11.34 | 13.95
$\frac{d}{dp}E(\Gamma_{8v}-L_{6c})$ | 4.69 | 4.73 | 5.10 | $\frac{d}{dp}E(\Gamma_{9v}-\Gamma_{7c})$ | 3.70 | 4.73 | 4.71
| InP | InAs | InSb |  | InP | InAs | InSb
$\frac{d}{dp}E(\Gamma_{8v}-\Gamma_{6c})$ | 9.11 | 10.55 | 13.97 | $\frac{d}{dp}E(\Gamma_{9v}-\Gamma_{8c})$ | 9.04 | 9.22 | 9.92
$\frac{d}{dp}E(\Gamma_{8v}-L_{6c})$ | 3.74 | 4.40 | 4.88 | $\frac{d}{dp}E(\Gamma_{9v}-\Gamma_{7c})$ | 3.57 | 3.61 | 4.48
Table 10: Pressure coefficients in $\rm{meV/kBar}$ between the top of the
valence band ($\Gamma_{8v}$) and two lowest conduction band valley states
($\Gamma_{6c},L_{6c}$) in ZB, and the top of the valence band ($\Gamma_{8v}$)
and the lowest two conduction band states at $\Gamma$
($\Gamma_{7c},\Gamma_{8c}$) in WZ. The pressure coefficients for the ZB
crystal phase were calculated using the ETB and radial scaling parameters from
Ref. [29]. The pressure coefficients for the WZ crystal phase were calculated
using ETB parameters from the present work and radial scaling parameters from
Ref. [29]. See Fig.1 for correspondence between ZB and WZ bands.

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