# Novel topological phases of a semi-Dirac Chern insulator in presence of
extended range hopping

Sayan Mondal  Department of Physics, Indian Institute of Technology Guwahati,
Guwahati 781039, Assam, India  Saurabh Basu  Department of Physics, Indian
Institute of Technology Guwahati, Guwahati 781039, Assam, India

###### Abstract

We study topological properties and the topological phase transitions therein
for a semi-Dirac Haldane model on a honeycomb lattice in presence of an
extended range (third neighbour) hopping. While in the absence of a third
neighbour hopping, $t_{3}$, the system exhibits gapless electronic spectrum,
its presence creates an energy gap in the dispersion. However, the nature of
the spectral gap, that is, whether it is trivial or topological needs to be
ascertained. We find that the answer depends on the value of $t_{3}$, and its
interplay with the value of the onsite potential that breaks the sublattice
symmetry, namely, Semenoff mass ($\Delta$). To elucidate our findings on the
topological phases, we demonstrate two kinds of phase diagrams using the
available parameter space, one in which the phases are shown in the
$\Delta$-$t_{3}$ plane, and the other one in a more familiar $\Delta$-$\phi$
plane ($\phi$ being the Haldane flux). The phase diagrams depict presence of
Chern insulating lobes comprising of Chern numbers $\pm 2$ and $\pm 1$ for
certain values of $t_{3}$, along with trivial insulating regions (zero Chern
number). Thus there are phase transitions from one topological regime to
another which are characterized by abrupt changes in the values of the Chern
number. To support the existence of the topological phases, we compute the
counter propagating chiral edge modes in a ribbon geometry. Finally, the
anomalous Hall conductivity shows plateaus either at $e^{2}/h$ or $2e^{2}/h$
corresponding to these topological phases.

##  I Introduction

In condensed matter systems, the information whether a material possesses a
topological phase, has been an immense interest since the discovery of quantum
Hall effect (QHE) [1]. The latter demonstrates that in presence of a strong
magnetic field, the Hall conductivity of a two-dimensional electron gas
acquires a series of plateaus quantized in unit of $e^{2}/h$. This
quantization was due to the presence of the discrete magnetic Bloch bands [2,
3, 4, 5, 6, 7] or the Landau levels [8, 10, 9, 12, 11] owing to the presence
of a magnetic flux. The topological invariant that defines the quantization of
the Hall conductivity, from a general perspective, is known as the Thouless-
Kohmoto-Nightingale-Nijs (TKNN) invariant [2].

An external magnetic field, initially appeared to be necessary to achieve QHE,
however, Haldane had proposed that even in absence of an external magnetic
flux, QHE still can be observed [13]. He introduced a direction dependent
complex next nearest neighbour hopping in a honeycomb lattice, such as
graphene, which breaks the time reversal symmetry (TRS). This broken TRS is
the only necessary criterion to observe QHE. The model proposed by Haldane is
a two band system, with the bands being characterized by a topological
invariant known as the Chern number, and these quantized (integer) values of
the Chern number yield a plateau in the Hall conductivity when the Fermi
energy lies in the bulk gap. Further, a non-zero value of the Chern number and
hence the quantized value of the Hall conductivity can be seen for a finite
value of the phase, $\phi$ (we shall call it Haldane flux) of the complex next
nearest neighbour hopping. A non-zero value of the Semenoff mass, $\Delta$,
that breaks the sublattice symmetry in graphene, opens or closes a gap in the
bandstructure [14]. The variation of the Semenoff mass with the Haldane flux
presents a phase diagram that encodes opening and closing of the band gap
alternately at the two Dirac points, which are usually called as the
$\mathbf{K}$ and the $\mathbf{K}^{\prime}$ points [13, 15].

In recent years, exploration of the topological properties associated with the
Haldane model have progressed rapidly in quantum many body systems [16, 17,
18, 19] both from the experimental and the theoretical perspectives. The study
in the two dimensional Dirac systems have also been explored, such as, Fe
based ferromagnetic insulators, $\mathrm{XFe_{2}(PO_{4})_{2}}$, where X may be
Cs, K La etc. [20], in Dice lattice [21] etc. which host isotropic low energy
Dirac like dispersions. However, there exists certain other materials that
display anisotropic dispersions at low energies, for example, say quadratic
along one direction in the Brillouin zone (BZ), and linear along the other one
[22, 23, 24], which are known as the semi-Dirac systems. The semi-Dirac
dispersions have been found in a variety of materials, such as, phosphorene
under pressure and doping [25, 26], electric fields [27, 28], in multilayered
structures of $\mathrm{TiO}_{2}/\mathrm{VO}_{2}$ [29, 30],
$\mathrm{BEDT}$-$\mathrm{TTF_{2}I_{3}}$ organic salts under pressure [31, 32],
oxidized silicene layer [33], deformed graphene [34] etc. Experimentally, the
semi-Dirac dispersion has been observed in layers of black phosphorene
obtained via in situ doping of potassium atoms [35].

A natural question arises: whether the semi-Dirac system will show similar
topological phases as their Dirac counter part. To have an insight on the
answer to this, we wish to explore whether and how the lowering of symmetry
induced by anisotropic dispersion modifies the topological properties of the
system. However, to achieve a topological phase, we need to break the TRS of
the system, either by including a perpendicular magnetic field or via adding
the complex second neighbour hopping (the Haldane term). Here, we choose the
latter option. However, unlike the Dirac case, addition of the complex second
neighbour hopping in the semi-Dirac system does not open a gap in the
electronic spectrum, and hence the system still remains a semi-metal, with the
conduction and the valence bands touching each other at a point intermediate
to the $\mathbf{K}$ and the $\mathbf{K}^{\prime}$ points in the BZ (the
$\mathbf{M}$ point). A little introspection reveals that, in such a scenario
we can add a real third neighbour hopping to open up a gap in the energy
spectrum, and hence look for the existence of the topological phases. The
semi-Dirac system with the Haldane term has been discussed in literature [36].
However to the best of our knowledge, the semi-Dirac system with the Haldane
term and a real third neighbour hopping have never been discussed and hence is
new to the scientific community. An important dividend of such an exercise
will be accessing regions in the phase diagram with large values of the Chern
number, which also facilitates studying the topological phase transitions
between phases with different Chern numbers.

Motivated by the above scenario, here we discuss the topological properties of
the semi-Dirac system in presence of a third neighbour (between different
sublattices) hopping. We shall show that inclusion of the third neighbour
hopping shifts the band minima from the boundary towards the interior of the
BZ and makes the system a Chern insulator, with Chern numbers $\pm 2$ for
certain values of the hopping amplitude. Addition of the Semenoff mass to the
problem changes Chern number from $\pm 2$ to $\mp 1$. Consequently, we obtain
the plateaus of the Hall conductivity that are quantized as $Ce^{2}/h$, with
$C$ being its Chern number and acquires values $\pm 1$ and $\pm 2$.

The paper is organized as follows, in section II we show the semi-Dirac
Hamiltonian in presence of a Haldane term and a real third neighbour hopping
on a honeycomb lattice. In section III, we investigate the topological
properties by computing the Chern number for various values of the amplitude
of the third neighbour hopping, and obtain the phase diagrams that demonstrate
the existence (or absence) of the non-trivial topological phases. In section
IV, we study the structure of the edge modes in a nanoribbon for various
relevant values of the second and the third neighbour hopping amplitudes.
Hence, we compute the anomalous Hall conductivities in section V that exhibit
plateaus quantized in units of $e^{2}/h$ and finally conclude with a brief
summary of our results in section VI.

##  II Model Hamiltonian

We consider a tight binding Hamiltonian on a honeycomb lattice with hopping
between the various neighbours that can be written as,

$\displaystyle H=$
$\displaystyle-\sum_{\left<i,j\right>}t_{ij}c_{i}^{\dagger}c_{j}+t_{2}\sum_{\left<\left<i,j\right>\right>}e^{i\phi_{ij}}c_{i}^{\dagger}c_{j}+$
$\displaystyle+t_{3}\sum_{\left<\left<\left<i,j\right>\right>\right>}c_{i}^{\dagger}c_{j}+\sum_{i}\Delta_{i}c_{i}^{\dagger}c_{i}+{\rm{h.c.}}$
(1)

The first term is the nearest neighbour (N1) hopping. The N1 hopping strengths
along the $\boldsymbol{\delta}_{2}$ and the $\boldsymbol{\delta}_{3}$
directions are $t$, while in the third direction, that is along
$\boldsymbol{\delta}_{1}$, the strength is $t_{1}$ as shown in Fig. 1. The N1
vectors are given by $\boldsymbol{\delta}_{1}=a(0,1)$,
$\boldsymbol{\delta}_{2}=a(\sqrt{3}/2,-1/2)$ and
$\boldsymbol{\delta}_{3}=a(-\sqrt{3}/2,-1/2)$. In the first term,
$t_{ij}=t_{1}$ or $t$ when $j$ connects the neighbours
$\boldsymbol{\delta}_{1}$ or $\boldsymbol{\delta}_{2,3}$ that belongs to the
other sublattice respectively. We have assumed two different values of
$t_{1}$, such as, $t_{1}=t$ and $t_{1}=2t$. The value $t_{1}=t$ represents the
well known isotropic Dirac case, such as graphene, while $t_{1}=2t$ denotes
the semi-Dirac case and is the focus for this work. The second term is the
Haldane term comprising of a complex second neighbour (N2) hopping with an
amplitude, $t_{2}$ and a complex phase denoted by $\phi_{ij}$, where $\phi$
assumes positive (negative) values if the electron hops in the clockwise
(anti-clockwise) direction. The third term represents the third neighbour (N3)
hopping between different sublattices and the fourth term represents the
onsite energy (Semenoff mass), that assumes values $+\Delta$ and $-\Delta$ for
sublattices A and B respectively. Performing a Fourier transform of Eq. II,
the Hamiltonian in the momentum space can be written as,

Figure 1: A honeycomb lattice is shown where the red and the blue circles
represent the sublattices A and B respectively. In the
$\boldsymbol{\delta}_{2}$ and $\boldsymbol{\delta}_{3}$ directions the N2
hopping strengths are same ($t$), while in the $\boldsymbol{\delta}_{1}$
direction it is $t_{1}$. The N3 hopping is shown by the yellow arrow.

$\displaystyle H(\mathbf{k})$
$\displaystyle=h_{x}(\mathbf{k})\sigma_{x}+h_{y}(\mathbf{k})\sigma_{y}+h_{z}(\mathbf{k})\sigma_{z}+h_{0}(\mathbf{k})I$
$\displaystyle=\mathbf{h(\mathbf{k})}\cdot\boldsymbol{\sigma}+h_{0}(\mathbf{k})I$
(2)

where,

$\displaystyle h_{x}(\mathbf{k})=$ $\displaystyle\left\\{t_{1}\cos
k_{y}+2t\cos\frac{k_{y}}{2}\cos\frac{\sqrt{3}k_{x}}{2}\right\\}+$
$\displaystyle t_{3}\left\\{\cos 2k_{y}+2\cos
k_{y}\cos\sqrt{3}k_{x}\right\\},$ (3) $\displaystyle h_{y}(\mathbf{k})=$
$\displaystyle\left\\{-t_{1}\sin
k_{y}+2t\sin\frac{k_{y}}{2}\cos\frac{\sqrt{3}k_{x}}{2}\right\\}+$
$\displaystyle t_{3}\left\\{\sin 2k_{y}-2\sin
k_{y}\cos\sqrt{3}k_{x}\right\\},$ (4) $\displaystyle
h_{z}(\mathbf{k})=M-2t_{2}\sin\phi\left\\{2\sin\frac{\sqrt{3}k_{x}}{2}\cos\frac{3k_{y}}{2}-\sin\sqrt{3}k_{x}\right\\}$
(5)

and,

$\displaystyle
h_{0}(\mathbf{k})=2t_{2}\cos\phi\left\\{2\cos\frac{\sqrt{3}k_{x}}{2}\cos\frac{3k_{y}}{2}+\cos\sqrt{3}k_{x}\right\\}$
(6)

where $\sigma_{i}$ ($i\in x,y,z$) denote the 2$\times$2 spin-1/2 Pauli
matrices which represent the sublattice degrees of freedom, and $I$ is the
2$\times$2 identity matrix. The energy dispersion can be obtained as,

$E(\mathbf{k})=h_{0}(\mathbf{k})+\sqrt{h_{x}(\mathbf{k})^{2}+h_{y}(\mathbf{k})^{2}+h_{z}(\mathbf{k})^{2}},$
(7)

where the $\pm$ signs refer to the upper (conduction) band and the lower
(valence) band respectively. In the absence of $t_{2}$ and $t_{3}$, the band
dispersion is linear along one direction and quadratic along its perpendicular
direction [37] about the band touching $\mathbf{M}$ point in the BZ. We refer
to this as the zero mode in our subsequent discussion.

Now, if we add a small N3 hopping, namely, $t_{3}$, then the zero modes shift
from the $\mathbf{M}$ point towards the interior of the BZ as shown in Figs.
2(b)-2(d). There are four zero modes inside the first BZ for a non-zero value
of $t_{3}$. Let us call these points where the zero modes occur as
$\boldsymbol{\Lambda}$ points. For example, one of the zero modes for a
particular value of $t_{3}$, namely, $t_{3}=t$ occurs approximately at a
particular $\boldsymbol{\Lambda}$ point, namely, $\boldsymbol{\Lambda}_{1}$ =
$\left(\frac{2.0023\pi}{9a_{0}},\frac{1.5487\pi}{3a_{0}}\right)$, while the
same for a different value of $t_{3}$, namely $t_{3}=3t$ approximately occurs
at another,
$\boldsymbol{\Lambda}_{2}=\left(\frac{1.0805\pi}{3\sqrt{3}a_{0}},\frac{1.2403\pi}{3a_{0}}\right)$.
For other values of $t_{3}$, namely, say $t_{3}>3t$, the zero modes remain
fixed at the same locations as that for $t_{3}=3t$.

Now, if we turn on the N2 hopping, $t_{2}$ then the spectral gaps open up at
these $\Lambda$ points in the BZ where the zero modes occur, and hence the
system behaves as an insulator. However, in the absence of $t_{3}$ (with
$t_{2}$ being non-zero), there is no gap at the $\mathbf{M}$ point, and the
dispersion is anisotropic linear (linear along both the directions, but with
different velocities along the $x$\- and the $y$-directions) about the
$\mathbf{M}$ point, which makes the system a semi-metal as discussed in Ref.
[36]. In Figs. 2(e) and 2(f), we have shown the bandstructures for the semi-
Dirac system in the absence, and in the presence of $t_{2}$ respectively for
non-zero values of $t_{3}$. In our calculations, we have fixed the values of
the Haldane flux, $\phi$, N1 hopping, $t_{1}$ and the Semenoff mass, $\Delta$
to be $\pi/2$, $2t$ and zero respectively. The corresponding bandstructures
for the Dirac system have been discussed in Ref. [38, 39], and we skip them
here to make our discussion concise.

It may be noted that $t_{3}$ is indeed a parameter and the values used may not
have experimental relevance. The reason being that the value of the real
second neighbour hopping is of the order of 0.1eV [43], which would mean that
$t_{3}$ is even smaller. However, the phase diagrams presented in section III
demands the value of $t_{3}$ to be of the order of $t$ or even larger in order
to access topological phases with different Chern numbers.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: The top view of the bandstructure for the semi-Dirac ($t_{1}=2t$)
system is depicted for (a) $t_{3}/t=0$, (b) $t_{3}/t=0.5$, (c) $t_{3}/t=1$ and
(d) $t_{3}/t=3$. The hexagons in each figure represent the first Brillouin
zone. In the calculations, we have fixed $t_{2}=0$, $\Delta=0$. In (e) and (f)
a three dimensional depiction of the bandstructure for the semi-Dirac system
are shown for $t_{2}=0$ and $t_{2}=0.5t$ respectively, where we have used
$\Delta=0$ and $\phi=\pi/2$.

##  III The Phase diagram

In this section we obtain the phase diagram by numerically calculating the
Chern number of the system. Since in this model, the complex N2 hopping term
breaks the TRS, non-zero values and hence non-trivial phases with finite Chern
number are expected. The inversion symmetry breaking onsite energies,
$\pm\Delta$ on different sublattices open or close energy gaps in the energy
spectrum at the $\Lambda$ points. We compute the Chern number via [40, 41],

$\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int\int_{\mathrm{BZ}}\Omega(k_{x},k_{y})\mathrm{d}k_{x}\mathrm{d}k_{y},$
(8)

where $\Omega(k)$ denotes the $z$-component of the Berry curvature which is
given by,

$\displaystyle\Omega(k_{x},k_{y})=\frac{\mathbf{h}}{2|\mathbf{h}|^{3}}\cdot\left(\frac{\partial\mathbf{h}}{\partial
k_{x}}\times\frac{\partial\mathbf{h}}{\partial k_{y}}\right)$ (9)

(a)

(b)

(c)

(d)

Figure 3: The Chern number of the lower band is depicted as a function of
$\Delta$ and $t_{3}$ for (a) $t_{1}/t=2$ and (b) $t_{1}/t=2$. The coloured
regions signify the Chern insulating regions with non-zero Chern numbers
($C=+1$ for the red region and $C=-2$ for the green one), while the white
region denotes the trivial insulating phase with $C=0$. We have shown the
variation of $C$ as a function of $\Delta$ for a particular value of $t_{3}$,
say $t_{3}=3t$, in (c) and (d) for $t_{1}/t=2$ and $t_{1}/t=1$ respectively.
In this calculation, the Haldane flux $\phi$ is kept fixed at $\pi/2$. The
topological phase transitions are implied via $C$ discontinuously changing
values between $1\rightarrow 0\rightarrow-2$.

(a)

(b)

(c)

Figure 4: The Chern numbers of the lower band is shown as a function of
$\Delta$ and $\phi$ for (a) $t_{1}/t=2$ and $t_{3}/t=1$, (b) $t_{1}/t=2$ and
$t_{3}/t=3$ and (c) $t_{1}/t=t$ and $t_{3}/t=3$. In each figure, the green and
the orange regions denote the Chern insulating phase with the Chern numbers -2
and +2 respectively, while the red and the blue regions imply the Chern
numbers +1 and -1 respectively. Further, the white region denotes trivial
topological regime with zero Chern number.

where $\mathbf{h(\mathbf{k})}$ is defined in Eq. II. It is to be noted that in
the absence of an N3 hopping, the Chern number is always zero for any
arbitrary value of $\Delta$ and $\phi$ for the semi-Dirac case ($t_{1}=2t$),
even though the time reversal symmetry remains broken. However in presence of
the non-zero N3 hopping, we may obtain non-zero values for the Chern number.
In Fig. 3(a) we have depicted the Chern number corresponding to the lower band
as a function of $\Delta$ and $t_{3}$ for Haldane flux, $\phi=\pi/2$. As can
be seen from Fig. 3(a), there are two regions denoted by the red and the green
colours. The region in red indicates the value of the Chern number, $C=1$,
while the green region indicates $C=-2$. In addition, there is also a finite
region denoted by the white colour, which corresponds to a trivial region with
$C=0$. In absence of $t_{3}$ or at small values of $t_{3}$, namely
$t_{3}<0.68t$, for all values of $\Delta$, the trivial region prevails. We
observe the topological phase with the Chern number $C=-2$ beyond a certain
value of the N3 hopping, $t_{3}$, namely $t_{3}\gtrsim 0.68t$, for a zero
Semenoff mass ($\Delta=0$). If we increase the value of $\Delta$, then we
observe the $C=-2$ phase for a range of values of $t_{3}$, such as,
$0.68t\lesssim t_{3}\lesssim 1.9t$. However, for $t_{3}\gtrsim 1.9t$, there
are two topological phases with Chern numbers $C=1$ and $C=-2$, which depend
on the value of $\Delta$. For example, for $t_{3}=3t$, there is phase
transition occurring from a $C=0$ to a $C=1$ phase at
$\Delta\simeq-5.04t_{2}$. $C$ again drops to zero at $\Delta\simeq-3.11t_{2}$.
Beyond $\Delta\simeq-2.56t_{2}$ the Chern number becomes $-2$ and stays at
$-2$ until $\Delta\simeq 2.56t_{2}$, when the Chern number vanishes again. The
Chern number becomes 1 at $\Delta\simeq 3.11t_{2}$, and finally vanishes again
for $\Delta\gtrsim 5.04t_{2}$. Thus there are a series of phase transitions
occurring at $t_{3}=3t$. There is always a trivial region (with $C=0$) in
between the two Chern insulating regions having two different Chern numbers
(the white region between the red and the green regions). Further, as one
increases the value of $t_{3}$, the vanishing Chern numbers are obtained for
lesser values of $\Delta$. As a result, the width of the Chern insulating
region with $C=-2$ (the green region) shrinks with the increase of $t_{3}$, or
equivalently, we can say, the width of the $C=1$ region increases with
increase in $t_{3}$. The trivial region (shown in white in Fig. 3(a)) gets
narrower as one increases the value of $t_{3}$.

This phenomenon is somewhat different in the Dirac case (see Fig. 3(b)), where
we can see a non-zero Chern number (namely, $C=1$) even in absence of the N3
hopping, that is, the Haldane model. The phase persists for very small values
of $t_{3}$. However in the presence of N3 hopping, we obtain a phase with
Chern number $C=-2$ or $C=1$ depending on the value of $\Delta$. Further,
unlike the semi-Dirac case, there is no trivial regime in between the two
different Chern insulating regimes, that is, the red and the green regions. If
we fix the value of $t_{3}$, say, $t_{3}=3t$, and calculate the Chern numbers
for increasing values of $\Delta$, then we observe the Chern number to jump
from $C=-2$ to $C=1$ at $\Delta\simeq 2.55t_{2}$. Finally, the Chern number
drops to zero from a value $C=1$ at $\Delta=3\sqrt{3}t_{2}$. The values of
$\Delta$, at which the Chern number changes from a value $C=-2$ to $C=1$,
depend on the value of the N3 hopping $t_{3}$ (see the shoulder like region in
Fig. 3(b)). However, the values of $\Delta$ at which the Chern number vanishes
from a value $C=1$ does not depend upon $t_{3}$. It should be noted that the
calculations are done for a Haldane flux, $\phi=\pi/2$. If we change $\phi$ to
$-\pi/2$ then the phase diagram will remain identical, except that the Chern
numbers will undergo a sign change.

In Figs. 3(c) and 3(d) we have shown the variation of Chern numbers as a
function of $\Delta$ for the semi-Dirac ($t_{1}=2t$) and the Dirac systems
($t_{1}=t$) respectively for a particular value of $t_{3}$, say, $t_{3}=3t$.
As can be seen for the semi-Dirac case (see fig. 3(c)), there are phase
transitions occurring from $C=0$ to $C=1$ and then again to $C=0$ as one
increases $\Delta$. With further increase of $\Delta$, $C$ drops to $-2$. To
quote some numerical values, the plateau at $C=-2$ exists for a range of
$\Delta$, that is, $-2.56t_{2}\lesssim\Delta\lesssim 2.56t_{2}$. With further
increase in the value of $\Delta$, $C$ drops to zero and then again rises to 1
and finally vanishes. The plateaus at $C=1$ persist for some values of
$\Delta$, such that, $-5.04t_{2}\lesssim\Delta\lesssim-3.11t_{2}$ and
$3.11t_{2}\lesssim\Delta\lesssim 5.04t_{2}$. A similar phase transitions are
observed for the Dirac case (see Fig. 3(d)), except that there is direct phase
transitions from $C=1$ to $C=-2$ or vice versa. The $C=-2$ plateau occurs for
$-2.55t_{2}<\Delta<2.55t_{2}$, while the plateaus at $C=1$ occur for
$-3\sqrt{3}t_{2}\leq\Delta\lesssim-2.55t_{2}$ and $2.55t_{2}\lesssim\Delta\leq
3\sqrt{3}t_{2}$.

Fig. 4 shows the phase diagram in the $\Delta$-$\phi$ plane corresponding to
the lower band for both the semi-Dirac and the Dirac cases. For both of them,
the values of the Chern number depend on the value of N3 hopping amplitude
(see Fig. 3). In Fig. 4(a), we have shown the phase diagram for the semi-Dirac
case for $t_{3}=t$. As can be seen, there are two Chern insulating regions
with Chern numbers $C=-2$ (green region) and $C=+2$ (yellow region). The phase
diagram is similar to that of the Haldane model, except that the values for
the Chern number are different in this case. Further, the widths of the Chern
insulating lobes are smaller than those in the Haldane model. Now, if we
increase the value of $t_{3}$ (say, $t_{3}=3t$), we shall see additional Chern
insulating regions emerge, with the Chern numbers given by $C=+1$ (red region)
and $C=-1$ (blue region) as depicted in Fig. 4(b). There exists a trivial
insulating phase with $C=0$ in between the two Chern insulating regions, that
is, between the green and the red regions, or between the yellow and the blue
regions. These types of phase diagrams are in complete contrast with the Dirac
case, where in the latter, the trivial insulating phase is absent as shown in
Fig. 4(c). The width of the Chern insulating region with $C=-2$ is greater in
the Dirac case compared to that of the semi-Dirac case (see Fig. 4(c)), as is
evident from the $\Delta$-$t_{3}$ phase diagram (Fig. 3(b)). For $t_{3}$ to be
vanishingly small, we note that the phase diagram becomes similar to that of
the Haldane model.

The phase diagrams presented in Figs. 3 and 4 aid us in identifying specific
values of $t_{3}$ and $\Delta$ to explore he nature of the topological phases.
We achieve that via the numerical computation of the edge states and the
anomalous Hall conductivity as discussed below. These quantities are
investigated for $t_{3}=0.5t$, $t$ and $3t$, where we have considered
$\Delta=0$ corresponding to $t_{3}=0.5t$ and $t$, while the $t_{3}=3t$ case
has been studied for $\Delta=0$ and $\Delta=4t_{2}$, which, correspond to
$C=-2$ and $C=1$ respectively.

Figure 5: A schematic diagram of the ribbon is shown. The edge current flows
along the zigzag edges ($x$ axis) of the ribbon.

(a) (b) (c) (d) (e) (f) (g) (h)

Figure 6: The energy spectra of the ribbon as a function of the dimensionless
momentum $k$ (here $k$ denotes $\sqrt{3}a_{0}k_{x}$) for the semi-Dirac system
($t_{1}/t=2$) are shown in (a) $t_{3}/t=2$, $\Delta/t_{2}=0$ (b) $t_{3}/t=3$,
$\Delta/t_{2}=4$ and (c) $t_{3}/t=0.5$, $\Delta/t_{2}=0$, while for the Dirac
system ($t_{1}/t=1$) it is shown in (d) $t_{3}/t=0.1$, $\Delta=0$, (e)
$t_{3}/t=1$, $\Delta=0$ and (f) $t_{3}/t=3$, $\Delta/t_{2}=4$. The green dots
in each figure signify the intersection of the edge states with the Fermi
energy $E_{F}$ (shown via the red dashed line). A schematic diagram of a part
of the ribbon is shown in (g) and (h). The arrows in (g) represent the edge
currents corresponding to the points of figure (a) and (e), while in (b) the
edge currents are shown corresponding to the points of figure (b), (d) and
(f).

##  IV Edge states

In order to understand whether the nature of the band gaps are topological or
trivial, we look for the existence (or absence) of the edge states. To achieve
this, we have considered the system to have semi-infinite ribbon geometry.
Such a scenario breaks the periodicity along a particular direction, while the
translational symmetry is preserved along the perpendicular direction. We take
the semi-infinite ribbon [42, 43, 44] to be finite along the $y$-direction,
and infinite along the $x$-direction. We further label the sites along the
$y$-direction as A1, B1, A2, B2, …. AN, BN etc as shown in Fig. 5. Since the
translational invariance is preserved along the $x$-direction, we can fourier
transform the operators along the $x$ direction only, that is, use
$c_{x,y}^{\dagger}=\sum_{k}e^{ikx}c_{k,y}^{\dagger}$. This yields two sets of
coupled eigenvalue equations for the wave functions which can be written as,

$\displaystyle E_{k}a_{k,n}=$
$\displaystyle-\left[t\left\\{1+e^{(-1)^{n}ik}\right\\}b_{k,n}+t_{1}b_{k,n-1}\right]$
(10)
$\displaystyle-2t_{2}\left[\cos(k+\phi)a_{k,n}+e^{(-1)^{n}\frac{ik}{2}}\times\right.$
$\displaystyle\left.\cos\left(\frac{k}{2}-\phi\right)\\{a_{k,n-1}+a_{k,n+1}\\}\right]$
$\displaystyle+t_{3}\left[b_{k,n+3}+2b_{k,n-1}\cos k\right]$ $\displaystyle
E_{k}b_{k,n}=$
$\displaystyle-\left[t\left\\{1+e^{(-1)^{n+1}ik}\right\\}a_{k,n}+t_{1}a_{k,n+1}\right]$
(11)
$\displaystyle-2t_{2}\left[\cos(k-\phi)b_{k,n}+e^{(-1)^{n+1}\frac{ik}{2}}\times\right.$
$\displaystyle\left.\cos\left(\frac{k}{2}+\phi\right)\\{a_{k,n-1}+a_{k,n+1}\\}\right]$
$\displaystyle+t_{3}\left[a_{k,n-3}+2a_{k,n+1}\cos k\right]$

where $n$ denotes the site index. $n$ assumes integer values in the range
$[1:N]$ with $N$ being the total number of unit cells along the $y$-direction.
In Eqs. 10 and 11, $a_{k,n}$ and $b_{k,n}$ are the coefficients of the wave
functions corresponding to the $n$-th A and B sublattices respectively. Here
$k$ is the momentum along the periodic $x$-direction, which is rendered
dimensionless by defining, $k=\sqrt{3}a_{0}k_{x}$. The width $D$ of the ribbon
along the $y$-direction is related to $N$ via
$D(N)=a_{0}\left(\frac{3N}{2}-1\right)$. In our work, we have used $N=256$ and
hence the ribbon has a width of 383$a_{0}$. By solving Eqs. 10 and 11 one can
get the bandstructure of the nanoribbon as shown in Fig. 6 for a fixed value
of the Haldane flux, namely $\phi=\pi/2$. As can be seen, one of the edge
modes from the lower band crosses over to the upper band as a function of
$k_{x}$, and another one crosses over in the opposite direction. These edge
modes are responsible for a finite value of the Hall conductivity, provided
the Fermi energy lies in the bulk gap. In Fig. 6(a), we show the edge states
for the semi-Dirac ($t_{1}=2t$) system corresponding to a particular value of
the N3 hopping, for example, $t_{3}=t$. The red dashed line represents the
Fermi energy, $E_{F}$ and the points where the edge modes intersect the Fermi
energy are shown by green dots. The edge currents corresponding to the points
‘$p$’ and ‘$r$’ flow along one edge of the ribbon and the edge currents
corresponding to the points ‘$p$’ and ‘$s$’ travel along the other edge (see
Fig. 6(g)). However their flow are in the opposite directions, since the
velocity of the electron is proportional to $\partial E/\partial k$ which
alters sign at $(q,s)$ compared to those at $(p,r)$.

Owing to the presence of a pair of edge states, there will be finite Hall
conductivity with a plateau occurring at a value $2e^{2}/h$, with the factor
‘$2$’ denoting the number of edge modes [45]. This result is consistent with
the Chern number phase diagram (see Fig. 3(a)), where the Chern number is
found to have a value -2 for $t_{3}=t$ and $\Delta=0$. In contrast, we get a
single edge mode, along either edge of the ribbon for $t_{3}=3t$ and
$\Delta=4t_{2}$ as shown in Fig. 6(f). In this case, we show the edge currents
corresponding to the points ‘$q$’ and ‘$r$’ in Fig. 6(h). This result is also
consistent with the phase diagram (Fig. 3(b)), where we find $C=1$. For
$t_{2}=0.5$ and $\Delta=0$, the edge modes are shown in Fig. 6(c). It is
clearly visible that the edge modes have split from the bulk. Thus, one can
say that the edge modes do not contribute to the edge current and hence the
system possesses zero Hall conductivity.

The edge states for the Dirac system ($t_{1}=t$) are shown for comparison in
Figs. 6(d), 6(e) and 6(f). The plots show the presence of single edge mode at
the points ‘$p$’ and ‘$s$’ for $t_{3}=0.1t$ and $\Delta=0$ (Fig. 6(d)), which
propagates at two opposite edges of the ribbon corresponding to the points
‘$p$’ and ‘$s$’. Such a situation yields a plateau in the Hall conductivity at
$e^{2}/h$. This result is similar to that of the Haldane model. Now, if we
increase the value of the N3 hopping, $t_{3}$ say, $t_{3}=t$ (Fig. 6(e)), a
pair of edge modes appear, and they propagate along two different edges of the
ribbon, however these two pairs are counter propagating at the opposite edges
(see Fig. 6(h)). In this case, the Hall plateau will be quantized in unit of
$2e^{2}/h$. The pair of edge currents will be there as long as the Semenoff
mass, $\Delta$ remains at a zero value. However, as we introduce a finite
value of $\Delta$, there is possibility that there will be single edge mode at
each edge as depicted in Fig. 6(f). Here we see that the edge modes are along
either edge of the ribbon corresponding to the points ‘$q$’ and ‘$r$’ as shown
in Fig. 6(h). The number of edge currents along either edge of the ribbon is
consistent with their values for the Chern numbers, namely, $C=1$ and $C=-2$
(see Fig. 3(b)).

##  V Anomalous Hall conductivity

(a)

(b)

Figure 7: The variation of anomalous Hall conductivity, $\sigma_{xy}$ is shown
as a function of the Fermi energy $E_{F}$ for (a) $t_{1}/t=2$ and (b)
$t_{1}/t=1$. Here $\sigma_{0}=e^{2}/h$ is the unit of the Hall conductivity.
In this calculation we have fixed the N2 hopping $t_{2}$ at $0.5t$ and the
Haldane flux $\phi$ at $\pi/2$.

The anomalous Hall conductivity requires a non-zero local Berry curvature. In
order to calculate the Hall conductivity, we first obtain the Berry curvature
of the system using Eq. 9 and then use the following formula [46, 47, 48],

$\sigma_{xy}=\frac{\sigma_{0}}{2\pi}\sum_{\lambda}\int\frac{\mathrm{d}k_{x}\mathrm{d}k_{y}}{(2\pi)^{2}}f\left(E^{\lambda}_{k_{x},k_{y}}\right)\Omega(k_{x},k_{y})$
(12)

where $E^{\lambda}(k_{x},k_{y})$ denotes the electronic energies and
$\lambda=$ $+1$ and $-1$ represent the upper and the lower bands respectively.
$\sigma_{0}(=e^{2}/h)$ sets the scale for $\sigma_{xy}$.
$f(E)=\left[1+e^{(E-E_{F})/K_{B}T}\right]^{-1}$ is the Fermi-Dirac
distribution function with $E_{F}$ being the Fermi energy and $T$ is the
absolute temperature. Using Eqs. 9 and 12, the Hall conductivity is calculated
numerically at zero temperature ($T=0$) as a function of $E_{F}$ and are shown
in Fig. 7(a) corresponding to the semi-Dirac system ($t_{1}=2t$). We see that
as long the Fermi energy lies in the gapped region, the Hall conductivity
shows a plateau quantized in unit of $2e^{2}/h$ for $t_{3}=t$ and $\Delta=0$.
Since the integral is performed over the occupied states for a given value of
$E_{F}$, the Hall conductivity decreases as $E_{F}$ moves away from the gapped
region, that is, towards the bulk. If we consider the Semenoff mass, $\Delta$
to be zero, we see that the plateaus occur at $2e^{2}/h$ as shown by the green
and the red curves in Fig. 7(a). However, in presence of a finite value of
$\Delta$, there is a possibility of getting a plateau at $e^{2}/h$ occurring
in $\sigma_{xy}$ as shown by the blue curve in Fig. 7(b). These results are
supported by the respective values for the Chern numbers. Thus, corresponding
to $C=-2$, we get the Hall plateau quantized at a value $2e^{2}/h$, and for
$C=1$, we get it is quantized at $e^{2}/h$.

To compare with the Dirac case, that is, for $t_{1}=t$, we have shown the
anomalous Hall conductivity in Fig. 7(b). As can be seen, with a small N3
hopping (say, $t_{3}=0.1t$), the Hall plateau is quantized in unit of
$e^{2}/h$ (the red curve in Fig. 7(b)). If we increase the value of the N3
hopping, the quantized Hall conductivity is seen at $2e^{2}/h$. Now, if we add
the Semenoff mass term, the quantized Hall conductivity acquires a plateau at
$e^{2}/h$. The value of $\Delta$ till which the $e^{2}/h$ plateau is retained
depends on the value of $t_{3}$. In Fig. 7(b), we show the Hall conductivity
for a non-zero $\Delta$ by the blue curve corresponding to $t_{3}=3t$. The
existence of the $e^{2}/h$ Hall plateau is noted for a certain range of
$\Delta$, that is, $2.5t_{2}\lesssim\Delta\lesssim 3\sqrt{3}t_{2}$
corresponding to a fixed value of N3 hopping, namely, $t_{3}=3t$. Similar to
the case of the semi-Dirac system, the quantized Hall conductivity of the
Dirac system is fully consistent with the corresponding Chern number phase
diagrams (see Fig. 3(b)).

##  VI Conclusion

We have shown that in the semi-Dirac system, adding a third neighbour hopping
causes the zero modes to move inward into the BZ from its boundary. Addition
of the Haldane term creates spectral gaps at those points. We have obtained
two different phase diagrams, namely, in the parameter spaces defined by
$\Delta$-$\phi$ and $\Delta$-$t_{3}$ by computing the Chern numbers. The
$\Delta$-$t_{3}$ phase diagram is an addition to the existing literature.
Further, the $\Delta$-$t_{3}$ phase diagram for the semi-Dirac case shows
different scenario than the Dirac case in the following sense. There is always
a trivial regime in between the two Chern insulating regimes ($C=-2$ and
$C=1$), which is absent for the Dirac case. The $\Delta$-$\phi$ phase diagram
for the semi-Dirac case shows that one may have Chern insulating regions
either with $|C|=2$ and $|C|=1$, or only $|C|=2$, depending on the value of
the N3 hopping, $t_{3}$. The computation of the edge states show additional
crossing of the edge modes corresponding to $|C|=2$. Finally, the anomalous
Hall conductivities, for several values of $t_{3}$, demonstrate the existence
of Hall plateaus quantized either at $2e^{2}/h$ or at $e^{2}/h$ depending on
the values of their Chern number.

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# Describing Globally Distributed Software Architectures for Tax Compliance

Michael Dorner1, Oliver Treidler2, Tom-Eric Kunz2, Ehsan Zabardast1,
Daniel Mendez13, Darja Šmite1, Maximilian Capraro2, Krzysztof Wnuk1
1Department of Software Engineering, Blekinge Institute of Technology, Sweden
2Kolabri, Germany  3fortiss, Germany

###### Abstract

The company-internal reuse of software components owned by organizational
units in different countries is taxable. To comply with international taxation
standards, multinational enterprises need to consider a geographical
perspective on their software architecture. However, there is no viewpoint
that frames the concerns of tax authorities as stakeholders towards a globally
distributed software architecture.

In this article, we introduce the reader to the concerns of tax authorities as
stakeholders and we investigate how software companies can describe their
globally distributed software architectures to tax authorities.

In an _in-virtuo_ experiment, we (1) develop a viewpoint that frames the
concerns of tax authorities, (2) create a view of a large-scale, globally
distributed microservice architecture from a multinational enterprise, and (3)
evaluate the resulting software architecture description with a panel of four
tax experts.

The panel of tax experts found that our proposed architectural viewpoint
properly and sufficiently frames the concerns of taxation stakeholders.
However, the resulting view falls short: Although the architecture description
reveals that almost 70% of all reuse relationships between the 2560
microservices in our case are cross-border and, therefore, taxable, unclear
jurisdictions of owners and a potentially insufficient definition of ownership
introduce significant noise to the view that limits its usefulness and
explanatory power of our software architecture description.

Although our software architecture description provides a solid foundation for
the subsequent step, namely valuing software components, we stumbled over
several theoretical and practical problems when identifying and defining
ownership in distributed teams, which requires further interdisciplinary
research.

###### Index Terms:

architecture description, architecture viewpoint, architecture view, software
component, microservice, taxation

##  I Introduction

Code is not tied to a physical, geographical location due to its intangible
nature. The code owner of a software component, however, is. Code ownership
refers to the concept of assigning responsibility and accountability for a
particular piece of code or software component to an individual, a team, or an
organizational unit within an organization.

The geographical location of those software component owners has a legal
implication that is too often underestimated: Using software components with
owners in different countries is taxable as it constitutes an often implicit
form of licensing [1]. To comply with the international standard of taxation
[2], multinational enterprises must be able to report a geographical
perspective on their software architecture. However, there is no architectural
viewpoint that frames the concerns of tax authorities as stakeholders towards
a globally distributed software architectures which could serve as foundation
for a software architecture description.

In this article, we investigate the concerns of tax authorities as
stakeholders of globally distributed software architectures and seek an answer
to the research question

How can multinational enterprises describe their globally distributed software
architectures to tax authorities?

We address this research question in an _in-virtuo_ experiment with a
multinational enterprise aiming to report an architecture description of its
globally distributed software architecture. Instead of all involved tax
authorities, we build a panel of four tax experts that evaluates the
usefulness and completeness of the two core elements of the software
architecture description, namely the architecture viewpoint and the
architecture view. This experimental design enables us to focus on evaluating
emerging technical, procedural, and legal issues in practice while reporting
software architecture descriptions to tax authorities without any potential
legal implications and consequences—which is crucial in this sensitive
context.

The main contributions of this study are the following:

  * •

An introduction to the concerns of tax authorities as stakeholders of globally
distributed software architectures

  * •

A novel software architecture viewpoint specifying the conventions for
constructing and using a view for the concerns of tax authorities towards
geographically distributed software architectures

  * •

An empirical view of an industrial globally distributed microservice
architecture using our novel viewpoint

  * •

An evaluation of the software architecture description for tax compliance by a
panel of tax experts

Throughout this paper, we use the terminology and definitions of ISO/IEC/IEEE
42010:2022 [3]. In particular, we distinguish between architecture
description, architecture view, and architecture viewpoint: An _architecture
description_ includes one or more architecture views. An _architecture view_
(or simply, _view_) addresses one or more of the concerns held by the system’s
stakeholders. An architecture view expresses the architecture of the system of
interest in accordance with an _architecture viewpoint_ (or simply,
_viewpoint_). There are two aspects to a viewpoint: the concerns it frames for
stakeholders and the conventions it establishes on views. Viewpoint
conventions can include languages, notations, model kinds, design rules,
and/or modelling methods, analysis techniques, and other operations on views.

The remainder of the article is structured as follows: After establishing the
background and related work in Section II and discussing our study design as
an experimental simulation in Section III, we define the viewpoint (Section
IV-B) and how we use it to generate a view for the concern of tax authorities
(Section IV-C). In Section V, we present and discuss the evaluation results
for our novel viewpoint (Section V-A and the view (Section IV-C) in detail. We
conclude the paper with a summary of our findings and the implications for
research and practice prior to outlining future work. Figure 1 provides an
overview of our study.

Section V Evaluating with expert panel Section IVCreating software
architecture description for tax compliance Section IV-A Define viewpoint
Section IV-B Construct view Section IV-C Evaluate viewpoint Section V-A
Evaluate view Section V-BConcernViewViewpoint Software architecture Figure 1:
An overview of the structure of our study as _in-virtuo_ experiment and its
two components: (1) creating a software architecture description with a
viewpoint derived from the concerns of tax authorities and a view of an
empirical software architecture using the viewpoint and (2) evaluating the
software architecture description with a panel of tax experts.

##  II Related work

This article connects taxation with software architecture distributed across
multiple countries. Therefore, we provide related work on globally distributed
software development, ownership in the context of software engineering and
taxation, and the emerging field of taxation in software engineering in the
following subsections.

###  II-A Globally distributed software development

Globally distributed software development, especially in large enterprises, is
a common practice. Tough competition and a lack of resources motivated many
software companies to look for ways to optimize their costs by establishing
their own subsidiaries in lower-cost premises locally and nationally or even
in other countries or by teaming up with allies that can help in delivering
products more cost-effectively. As a result, distributed working across
national boundaries has become widespread [4]. However, cross-border
collaborations are far from being challenge-free. They are recognized to be
considerably more complex than the most complicated one-site development
endeavors due to the significant communication and coordination overhead. The
solutions to the problems of global projects have evolved through the years.
They could be characterized by a long journey from initial problematic
experiences of collaborative work through strictly independent work in
distributed sites to the evolved ways of working across borders.

Architectural modularization was one of the most prominent early solutions to
the coordination and communication challenges in global software engineering.
In 1968, Conway published his observation that software architecture cannot be
separated from the structure of the organization that develops the software,
known as Conway’s law [5]. Inspired by this observation, architectural
modularization has emerged as a way to minimize dependencies, structure and
allocate tasks, and encapsulate the work in a single geographic development
site as much as possible so that communication, coordination, and
synchronization needs are minimized [6, 7, 8]. Although the modularity
approach to coordination has proven very useful, it has many limitations [9,
10, 11]. While some researchers advocate that strict modularization of work is
the possibly best way to alleviate the challenges of distributed work, others
argue that work in isolation has many disadvantages [9], such as redundant
work, cheap and dirty architectural decisions, misplaced functionality [12,
10] and integration problems [13, 14]. Software components are never truly
independent and even the best designs are never error-free, while changes are
never completely predictable [14].

Further, strict code ownership is also found to introduce communication
overhead and delays [11]. For these reasons, and due to the increased use of
agile methods, there has been an increasing interest in collaboratively
developing software products. This includes a shift from modularization based
on geographically distributed locations towards componentization through
microservices [15] on a team level, as well as a trend towards establishing
collective ownership of source code and other documents to allow everyone in
the project to work on any model or artifact [11]. Collaborative work includes
keeping track of which team owns and does what (e.g., through Scrum of Scrum
fora or organizing cross-unit demos), submitting contributions to other’s
components and reviewing each other’s code [1, 16], and fostering reuse of
artifacts to reduce redundant work.

###  II-B Ownership in software engineering and taxation

Ownership plays a crucial role in taxation in general. However, as we learned
in the course of this study (see Section V), the definition of ownership in
the context of software engineering and the definition of ownership in
taxation overlap but are not congruent due to the different context and
perspective towards ownership.

Ownership in software engineering, on the one hand, refers to the assignment
of responsibility and accountability for specific portions of a software to
individuals or teams within a development organization. There are different
perspectives on ownership and autonomy in software development organizations.
Software development organizations usually adopt the weak ownership principle
[17]. This principle entails assigning ownership of specific components (or
microservices) to individual teams. In an ideal scenario, the designated team
assumes responsibility as the primary (or exclusive) contributor to the
associated component. Nevertheless, empirical analysis of real-world instances
often reveals a departure from this idealized model [18, 19, 20, 21]. In
practice, a given component may offer support to multiple business streams,
undergo development by distinct teams, and necessitate the involvement of
specialized teams to address non-functional requirements, such as security and
reliability. Consequently, while a specific team retains responsibility for
the component, the extent of its contribution to the component can exhibit
considerable variability [22].

Tax professionals, on the other hand, begin from a civil law perspective and
understand ownership in terms of the legal or economic owner regarding the
right of exploitation [2]. While the identification of the legal or economic
owner is relevant for balance sheet allocations, in the context of transfer
pricing, the legal owner is considered (only) as the starting point for a
functional and risk analysis that aims for a profit allocation commensurate
with the value-adding activities performed by the legal entities within the
group, the so-called _functional ownership_ [2].

Describing and discussing the intersection as well as the difference of
definitions is beyond the scope of this work and requires future
interdisciplinary research.

###  II-C Taxation

Taxation in software industry has been debated for many decades [23]. The
problem with taxing the final result of software engineering, the software
product or service, for example, has shown to be challenging to tackle and is
still subject to ongoing and broad discussion [24]. This debate has also
reached software engineering, the construction of the software product; at
first in the context of inner source, use of open source software development
practices and the establishment of open source-like communities within an
organization [25]. To address the taxation challenges in inner source software
development, Buchner and Riehle proposed a cost-based approach [26]. However,
Treidler et al. showed that a cost-based approach is not applicable since
inner source does not reflect the function and risk profile, the starting
point for any transfer pricing discussion, for a cost-based approach [27].

In a previous article [1], we identified the use of software components owned
by teams or individuals who represent separate geographically distributed
subsidiaries of the same enterprise as type of licensing—a key intangibles in
software engineering that are taxable. For an in-depth discussion on the
concerns of tax authorities on software architecture, we refer the reader to
Section IV-A in which we discuss a simplified example of a multinational
enterprise developing software and how tax authorities become a stakeholder to
a globally distributed software architecture.

##  III Study design

We have designed our study as an _in-virtuo_ experiment (or experimental
simulation [28]). In such a setup, the behaviors of actors that a researcher
aims to observe and measure are natural and rooted in their professional
experience and expertise, but the setting in which these behaviors occur has
been specifically created for the purposes of the study. That is, without the
study, the setting would not exist. Thus, the term “simulation” refers in this
context to the artificially created setting that aims to recreate a concrete
type of setting in which the “experiment” (i.e., observed behavior) takes
place [28].

In our study, we simulate how software companies can describe their globally
distributed software architectures for tax compliance. The simulation consists
of two parts: (1) creating a view using our viewpoint at a multinational
enterprise and (2) the evaluation by a panel of tax experts in which we
observe and measure the usefulness and completeness of an architecture
description for tax compliance.

On the one hand, the behaviors of actors we aim to observe and measure are all
natural: (1) the multinational enterprise fulfilling its reporting obligations
regarding their globally distributed software architecture to comply with
international taxation standards, (2) the tax advisors representing and
consulting companies and creating a line of reasoning, and (3) tax auditors
from tax authorities scrutinizing the line of reasoning. We instructed all
participants to imagine it to be in the context of a tax audit.

On the other hand, the setting is specifically created for the purpose of this
study, also as a means to ensure the anonymity. Obviously, the evaluation of
our proposed architecture description in a real tax audit (i.e., a natural,
real setting) is not possible: Tax audits are highly sensitive and, if not
brought to court, confidential. Our setting must therefore provide a safe
space for all participants without any legal implications for the company or
the tax authorities.

However, our setting is not plucked out of thin air but emulates a tax audit
as it involves all relevant parties: The panel of four tax experts consists of
two tax auditors and two tax advisors representing both opposing perspectives,
the company perspective as the provider and the tax authority perspective as
the recipient of such documentation. Their expertise and experiences will give
us a first evaluation of our preparatory steps and the resulting software
architecture description of globally distributed software architectures for
tax compliance .

Although this evaluation is a key pillar of our study, our study goes beyond a
traditional judgment study as it also describes the practical issues for
creating and using a view based on our novel viewpoint in an empirical case.
We discuss our approach to the limitations in Section VII.

Last but not least, we would like to emphasize that an in-depth legal
discussion and evaluation of our work is beyond our reach.

###  III-A Data collection

All involved parties want to remain anonymous including our case company used
as a source for creating our view. Therefore, we are not able to describe the
case and the data collection in detail. However, we may well argue that our
case company is exemplary for a multinational enterprise developing software.

In brief, for creating the view, we extracted the software architecture and
the team information from an internal system that extracts the component
structure of the system from the code repositories on a daily basis.

For the evaluation of the usefulness and completeness of our software
architecture description for its intended purpose, we conducted semi-
structured interviews with four tax experts. The main goal of the evaluation
is to seek generalizability over the responses rather than generalizability to
a population of actors [28]. We ensured privacy and confidentiality for all
actors (i.e., the involved companies, tax authorities, and tax advisors) by
conducting the interviews separately and removing all company-specific
information. We also altered the geographical distribution of the subsidiaries
to make our company not identifiable in any sense on the one hand but also to
give the reader a better impression of globally distributed software
architecture on the other hand.

The interview guidelines are publicly
available111https://zenodo.org/records/10214942. The group of four tax experts
represent the two opposing parties in a tax audit: internal or external tax
advisors representing the company and tax auditors representing the tax
authorities.

We conducted two pilot interviews to address didactical challenges since the
interviews were conducted with tax experts with no software engineering
background. A pair of a co-author with a taxation background and a co-author
with a software engineering background conducted the interviews with the tax
experts virtually between 2023-10-24 and 2023-11-03. Table I provides an
overview of our interviewees and their background. We recorded all interview
sessions.

TABLE I: An overview of the interviewees and their backgrounds. ID | Role |  Background
---|---|---
A | Tax auditor |  Former tax auditor, majoring in international taxation, OECD advisor for transfer pricing from 2016 to 2018, currently not involved in transfer pricing
K | Tax advisor |  Transfer pricing advisor, previously transfer pricing expert at one of the Big Four accounting firms for 20 years
M | Tax auditor |  Tax auditor and key expert on transfer pricing for other tax auditors for more than eight years
R | Tax advisor |  Managing director at a transfer price consulting firm; tax inspector before, 35 years of experience in international taxation

###  III-B Data analysis

As our interview guidelines show, large parts of the interviews were dedicated
to providing contextual software engineering understanding for the tax
experts. Therefore, we translated and paraphrased only the answers from the
interviewees. While one researcher of the interview team analyzed the recorded
interviews using thematic analysis [29], the second one ensured the quality of
the findings and mappings. After the coding, we again reflected on the
findings.

##  IV A software architecture description for tax compliance

In the following subsections, we provide a gentle introduction to
international taxation for software architects and specify the set of
conventions for the creation, interpretation, and use of an architecture view
to frame the concern of tax authorities.

###  IV-A International taxation standards as concern and tax authorities as
stakeholder

Consider the fictional _devnullsoft Group_ , a multinational enterprise that
develops and sells a software product. The software is split up into 18
components. Each component has an owning team that is responsible and
accountable for its components. The teams are assigned to one of the legal
entities of the enterprise in three countries depending on the location of the
team members: _devnullsoft GmbH_ in Germany, _devnullsoft AB_ in Sweden, and
_devnullsoft Ltd._ in the UK. In Figure 2, we provide a schematic overview of
a simplified globally distributed software architecture with owning teams
assigned to three different subsidiaries in Germany, Sweden, and the UK. Using
code components across different jurisdictions is taxable.

Section V Evaluating with expert panel Section IVCreating software
architecture description for tax compliancedevnullsoft ABdevnullsoft
GmbHdevnullsoft Ltd.[width=0.5cm, framewidth=0mm]SE[width=0.5cm,
framewidth=0mm]DE[width=0.5cm, framewidth=0mm]GBSoftware
productComponentTaxable dependencyNon-taxable dependencydevnullsoft Group
Figure 2: A schematic overview of a globally distributed software architecture
developed by the fictional _devnullsoft Group_ , a multinational software
enterprise with three subsidiaries developing software. Using code components
across different jurisdictions is taxable.

Only the German _devnullsoft GmbH_ sells this resulting product to customers.
Without any further consideration, solely the German _devnullsoft GmbH_
generates profits, which are then fully taxed in Germany according to German
law. The Swedish and English tax authorities are left out in the cold because
_devnullsoft AB_ and _devnullsoft Ltd._ have no share of the profit that could
be taxed in Sweden or in the UK, although both subsidiaries maintain, test,
and develop, or simply own components that are crucial for the software
product that made the success of the software possible.

To avoid this scenario and to provide a common ground for international
taxation, reducing uncertainty for multinational enterprises, and preventing
tax avoidance through profit shifting, nearly all countries in the world
agreed on and implemented the so-called _arm’s length principle_ as defined in
the _OECD Transfer Pricing Guidelines for Multinational Enterprises and Tax
Administrations_ [2]. This has become the guiding principle and the de-facto
standard for the taxation of multinational enterprises that requires
associated enterprises to operate as if not associated and regular
participants in the market from a taxation perspective. To comply with the
arm’s length principle, multinational enterprises must pay a so-called
_transfer price_. Transfer prices are the prices at which an enterprise
transfers physical goods or—for software much more relevant—intangibles or
provides services to associated enterprises. Those transfer prices are
established on a market value basis and aim to avoid profit shifts from high
to low tax regions. Although other types of intangibles, like cross-border
code contributions, code reviews, bug reports, etc., are also relevant from a
taxation point of view, we focus in this article on another highly relevant
type of intangible in software engineering: licenses.

Although maybe not even explicit, the company-internal use and reuse of
software components is, in fact, an instance of licensing. A software license
is a legal instrument governing the use or redistribution of software.
Software licenses and their obligations are widely discussed in the context of
open-source software. But in contrast to open source, components and their use
are protected by the confines of a company. Third parties are excluded from
any use or redistribution right of the software components. Without further
considerations—i.e., transfer prices—this exclusion contradicts the arm’s
length principle and makes multinational enterprises like our _devnullsoft
Group_ not compliant.

In this way, tax authorities become a stakeholder in globally distributed
software architectures as they have the concern of identifying cross-border
reuse of software components as an instance of taxable licensing. Determining
the actual transfer pricing for the cross-border reuse of software components
is a vast and separate challenge on its own and not part of this work. Our
work focuses on the architectural description comprising all necessary
information for reporting the software architecture to tax authorities.

To conclude, the concern of tax authorities on software architecture is the
(potentially implicit) license structure of the software-intensive product
within the multinational enterprise. From this concern, we can derive three
main questions:

  * •

How is the software-intensive product structured?

  * •

What legal entities (implicitly) license those substructures of the software?

  * •

Where are those legal entities geographically located?

All three questions must be answered in the context of software architecture.

###  IV-B Architecture viewpoint

To address the concern of tax authorities and to answer the three derived
questions, our viewpoint requires four elements: Defining the component and
component owner, decomposing the component structure of the software product,
identifying component owners, and identifying the jurisdictions of the
component owners.

####  IV-B1 Defining component and component owner

A software component is a self-contained, reusable piece of software that
encapsulates the internal construction and exposes its functionality through a
well-defined interface so other components can use the functionality. That
enables software components to be used, developed, tested, and maintained
individually by different teams and separately from the rest of the software
product. Software components can take many forms, including libraries,
modules, classes, functions, or even entire applications.

Component ownership refers to the concept of assigning responsibility and
accountability for a particular software component to an individual or an
organizational unit within an organization. Although there are different
notions of code ownership (for example, strong, weak, or collective code
ownership [17]), component ownership is more strictly defined in this context:
We requires only one organizational unit is ultimately responsible and
accountable for a software component.

The definition of a component or a component owner is company-specific and may
vary. Thus, an in-depth rationale for the definitions in the context of the
reporting company is required.

####  IV-B2 Decomposing the component structure of the software product

To reveal the component structure and the dependencies of software components,
the software product must be decomposed into the component structure. This
results in a dependency graph reflecting the dependencies between the software
components. The type of dependency depends on the definition of a software
component. Therefore, a rationale is required.

####  IV-B3 Identifying component owners

Each component must be assigned to one owner. This usually results in a table
of components and their owners. As discussed previously, the mapping of owner
to component is a one-to-many mapping.

####  IV-B4 Identifying jurisdictions of component owners

Each organizational unit (or individual) owning a software component must be
assigned to one jurisdiction. In most cases, a jurisdiction is the country
where the local subsidiary is located. This usually results in a table of
component owners and countries. As discussed previously, mapping an owner to a
country is one-to-one.

###  IV-C Architecture view

In this section, we discuss the practical view construction using the
presented viewpoint at a large multinational enterprise.

In detail, we use microservices as the most suitable abstraction layer for
software components for the specific software architecture in our case. A
microservice represents a discrete piece of a larger application’s
functionality and operates as a self-contained unit. Microservices are
typically designed to be lightweight, modular, and independently scalable,
allowing for greater flexibility and agility in software development [15].
Therefore, microservices meet the definition of a software component.

The software architecture of the software product consists of 2560 different
microservices. We consider only microservices in production; experimental or
other non-production microservices are excluded. Microservices in production
imply that those microservices face the customer directly or indirectly.

In our case company, a component owner is responsible and accountable for its
software component. That implies ownership includes functional and non-
functional requirements, such as security, reliability, and quality, and
extends throughout the software lifecycle, including development, testing, and
maintenance. It also involves all aspects of operating that code in
production, such as deployment, monitoring, troubleshooting, and remediation
of operational and security incidents.

In our case company, each microservice is usually owned by a single team. We
excluded 42 microservices (1.64%), each owned by one individual because the
locations of individuals were not accessible for this research due to privacy
concerns. Those remaining 2518 microservices have 16533 dependencies between
them and are owned by 336 different teams.

The geographical locations of the teams are self-assigned: Teams report their
location in a central system. We find this self-reporting most realistic in
the world of distributed and remote teams. However, we were not able to
retrieve or confirm the actual geographical location of 140 teams (41.16%)
with this self-reporting approach. This is mostly due to remote and
distributed teams that do not have a single location that we could assign to a
single jurisdiction. This affected 8097 use relationships, i.e., 48.94% of all
use relationships.

Due to the large amount of data, we present the architecture view in two
aggregated visualizations instead of a plain list, as graph and as table.

The directed graph as in Figure 3 depicts jurisdictions (countries) as
vertices and dependencies as directed edges. An edge goes from the component
owner to the component user. We aggregate the number of dependencies between
countries as edge weights.

Section V Evaluating with expert panel Section IVCreating software
architecture description for tax compliancedevnullsoft ABdevnullsoft
GmbHdevnullsoft Ltd.[width=0.5cm, framewidth=0mm]SE[width=0.5cm,
framewidth=0mm]DE[width=0.5cm,
framewidth=0mm]GBGermanyUSAFranceNetherlandsUK113026140691648850431116410824241415326252115192
Figure 3: The geographical distribution of the microservice architecture of
our case company using our architectural viewpoint. The arrows indicate the
direction of the dependency from microservice user to the microservice owner.
The countries are partially substituted by equivalent countries to ensure
confidentiality for the company.

This graph, however, lacks of the information of unclear locations. Therefore,
we provide also a tabular form of the graph including the information of
missing or unclear location. Table II depicts this information.

TABLE II: An equivalent tabular representation of Figure 3 but including unknown dependencies. |  | Component owner
---|---|---
|  | DEU | GBR | NLD | FRA | USA | N/A
Component user | DEU | 2 | 2 | 0 | 0 | 0 | 4
GBR | 15 | 164 | 2 | 261 | 43 | 141
NLD | 3 | 6 | 19 | 11 | 5 | 8
FRA | 24 | 108 | 21 | 4069 | 850 | 1767
USA | 14 | 24 | 15 | 1130 | 1648 | 642
N/A | 27 | 70 | 14 | 2283 | 970 | 2171

##  V Evaluation results

In this section, we report the evaluation results from the panel of tax
experts (see Section III for more details) evaluating the software
architecture description with its viewpoint that framed the concerns of tax
authorities and the view of software architecture from our industry partner.

###  V-A Viewpoint

Our panel confirmed the correctness of our viewpoint and agreed with the
distilled three questions raised from the concern of international taxation.
As interviewee A stated: “I think it is possible to reveal the license
structure [by your viewpoint], at least by 90%, there will always be
fuzziness.”

However, the panel also agreed that this viewpoint can only be the beginning.
“You are approaching it from a taxation perspective from a very detailed
level”, as interviewee R said. “[Your viewpoint] is an important first step
towards understanding who is actually contributing to the software product.
But to answer the follow-up question, which is the valuation of the software
component, you need to add another layer. [Components] are not additive.” In
the words of interviewee K: “Value is not counting components.” The
classification of software components potentially reflects how easily the
components can be replaced or how differentiating the components from
competitors are.

In this context, all participants emphasized the importance of ownership from
a taxation perspective. Interviewee M and interviewee K highlighted
accountability as a key aspect. They raised the question of who is actually
accountable for the software product or the components from a legal
perspective.

Our panel emphasized that although aggregating over jurisdictions can be
acceptable in our case since there is only one subsidiary per jurisdiction, it
does not generalize: If there is more than one subsidiary in a jurisdiction,
aggregating over countries is not applicable. Interviewee K also mentioned
that assigning legal entities to jurisdiction can become complicated referring
to so-called _permanent establishments_ , where, in complex situations,
subsidiaries may have international branches and sites fall into jurisdiction
X but belong to jurisdiction Y. “Assigning jurisdiction is complex. There are
gray areas.” However, those special cases are likely not relevant for those
considerations.

Interviewee M wondered how often software architectures may change and
commented that the viewpoint does not consider such changes over time but
represents a snapshot of software architecture. As a tax auditor, the
interviewee would question the representativeness of the presented snapshot.

###  V-B View

When creating a view using the viewpoint, our panel identified two major
issues: (1) the significant amount of unclear links between owner and
jurisdiction and (2) the missing qualitative insights into the software
architecture.

The first major issue refers to the problem that we were not able to identify
the jurisdiction of 41.16% of all owners in our view on the given software
architecture. Although the data source, the company-internal platform for
teams, is promising (“I go along with that one. […] That is, in my opinion, an
excellent data source”) since the data is not biased from a taxation
perspective (interviewee M), the amount of unclear jurisdiction of owners is
too high. The experts suggested different additions (jurisdiction of the team
manager) or complements (a cascade of different other data sources, for
example, a questionnaire sent out to all owners or geographical distribution
of the team to approximate the jurisdiction of the owners). Although “missing
some percentages is no problem”, but “half of the dependencies missing is too
much” as interviewee M explicitly said. “50% not assignable implies that 50%
of all transfer prices are not properly assigned. […] Without data, we are in
vacuo.” (interviewee A).

In this context, interviewee M and interviewee A emphasized the importance of
traceability in tax audits. Therefore, it would also be better to reveal the
process and characteristics of how teams came to their self-assignment.
Indications like regular socializing in a certain country can be such an
indicator. In this context, the panel agreed that the physical location of the
team members and developers can be secondary from a transfer pricing
perspective.

Our panel appreciated the graph representation and found it useful and
utilizable. For example, interviewee M said, “I like the [graph]
visualization; it is helpful”. Interviewee A detailed that the visualization
as a graph is “helpful for a high-level view, a good starting point for
looking into details”.

The tabular representation was seen as a useful extension of the graph.
Interviewee M preferred the presented absolute numbers over relative numbers
since absolute numbers represent the raw information and “relative numbers are
easy to calculate” anyway.

The second major issue raised by our experts is grounded in the quantitative
nature of our visualization as graph and as table. The panel of tax experts
unanimously warned that the numbers may be misinterpreted as relative value
added to the software product. As already discussed in the previous section,
considering all components equal is not an acceptable simplification. “The
edge weight does not reflect the relevance” as stated by interviewee R. It
requires a classification of the components (interviewee K).

Besides those two major issues, the panel also raised further issues.
Interviewee R wondered about the effort required to generate those insights.
The interviewee emphasized that the cost-benefit ratio must be positive.
Another important remark was related to our decision to exclude non-production
components. Interviewee M reminded us that our ultimate goal is to identify
the value of software components. Although not in non-production or completely
experimental, those components can have a significant and not negligible value
for the company. Therefore, excluding them removes explanatory power from the
software architecture description.

###  V-C Software architecture description

At the end of each interview, we asked the participants about their general
and high-level impression of our software architecture description with its
viewpoint and view in its current state.

The overall impression was positive and, according to our panel, the software
architecture description fulfilled its main purpose at its current state to a
large extent. Although “transfer pricing is not a science, it is art”,
interviewee R correctly summarized our intention as follows: “This is just
fact-driven, it is no interpretation. When you ask people, they may or may not
know who is involved. The nice thing about it that it is not a subjective
view, it is about facts, it is about technical details. This is what is used
if you like it or not. That is relevant because you have objectivized the
contributions [to the software product]. Those are the parties involved. This
allows you to have a very objective starting point to consider the potential
parties you have to consider in greater detail.” The visualization raises
awareness of parties involved in the software development. The panel also
confirmed the importance of software architecture for taxation: “I think
software architecture is the right perspective to understand how software is
composed conceptionally” (interviewee K). Interviewee R highlighted the need
for software that can manage the complexity and support the transfer pricing
processes. Interviewee M also appreciated the company’s efforts in collecting
and presenting the data in a tax audit: “A company with such detailed and
thorough documentation would be among the top 10 or top 5 percent of what a
tax auditor usually sees.” Interviewee K agrees: “I really like the approach
aiming for objectiveness. Let’s face it: Sometimes there are preposterous
debates in tax audits.”

##  VI Discussion

In this section, we critically reflect on the evaluation results and feedback
from our panel of tax experts and what it means for our software architecture
description and future work.

We found ambiguous assignments of owners to jurisdiction in about half of all
dependencies. We expected that this significant amount of unclear
jurisdictions per team create a strong headwind. Statistical estimators cannot
be applied to make any assumptions on the jurisdictions because we have a
strong bias: The owners with missing or unclear jurisdictions come mostly from
teams that have team members from different subsidiaries or are completely
virtual. Those are inherently relevant for taxation purposes and need further
investigation.

The discussion on ownership, however, cast doubt on our definition of
ownership. We learned that the term _ownership_ triggers a certain association
in the context of taxation: Tax experts think of legal ownership when they
read ownership. Although we emphasized in our viewpoint that we require an
owner to be accountable and responsible, the term ownership led to different
misunderstandings and triggered in-depth legal discussions that go beyond the
scope of this paper. We conclude that our definition of ownership can be
indeed too weak: Our understanding of accountability may fall short and does
not suffice the legal accountability. Future research needs to address the
legal aspects of code (component) ownership.

We were surprised that the quantitative nature of our description was
conceived that all components are normalized and of equal importance and
relevance for the software product. Interviewee R even suggested removing the
numbers completely and just indicating by the edges that there is a dependency
that needs further investigation to avoid this misconception. Although we
agree that those numbers can misinterpreted in such a way that all components
are equally important and relevant, we still believe that the weighted edges
provide value for prioritizing: The more dependencies, the more potential
uncertainty is in a tax audit that needs to be addressed. To avoid the
impression that dependencies are additive, we consider replacing the absolute
numbers with intervals (e.g., $(0,10)$, $(10,100)$, and $(100,\infty)$) in
future iterations of the software architecture description.

We are not aware of studies addressing at what rate or to what extent
microservices architectures or software architecture in general change over
time (a fiscal year), which would also may be not generalizable and context-
specific anyway. Since the effort to create the views once the data sources
are identified is reasonable, we suggest regularly generating snapshots of the
software architecture. Furthermore, we hypothesize that changes in software
architecture also may trigger changes in ownership, directly or indirectly.

##  VII Limitations

In this section, we critically reflect on the two components of our
simulation-based approach: the software architecture description from our
industry partner and the evaluation of the software architecture description
in the contrived setting.

###  VII-A Software architecture description

We created the view as described in the paper. We relied on historical
information extracted from an internal system that extracts the software
product’s component structure (which also comprises microservices) from the
code repositories. We believe that this system is a reliable and trustworthy
data source for extracting the data we need since the data is also used
internally for other purposes. The internal team responsible for maintaining
the system helped us to avoid misinterpretation of the data. The date of
extraction was randomly selected from the first two quarters in 2023. Although
we did not collect this data, we assume that the microservice architecture
does not change at a daily rate but is stable at least for over weeks.

###  VII-B Contrived setting

The evaluation of the software architecture description for tax compliance
highly depends on our tax panel. The contrived setting and the strict
anonymization of the case company required us to remove context and
potentially relevant information, which includes the specific business model,
the revenue stream, and organizational structure of our case company. Not
being able to provide that information limits the generalizability of our
evaluation. However, we believe that we provided enough context and
information for the panel to provide solid and high-level feedback for the
early stages of our research.

Both representatives for the tax authorities come from only one country.
Although the taxation standards are international and our panel consists of
tax experts advising taxation internationally, our study may not fully capture
the nuances in interpreting the standards. Therefore, our results only apply
to the fundamentals of describing software architectures to tax authorities.
Future research is required to address the nuances and country-specific
specialization of international taxation.

To maintain strict anonymity for the company, we decided not to include
participants from the company. The author team spoke on their behalf. In
future work addressing the subsequent questions of valuing software
components, the involvement of internals will become crucial. We believe that
in this early stage of the software architecture description, the author team
who extracted the data brought enough context to the table.

The interviews were held in English and German. To avoid translation mistakes
and misunderstandings, the second participating interviewer checked the
findings after coding and the final version of the paper was sent to all
interviewees, asking for confirmation.

##  VIII Conclusion

In this study, we proposed and evaluated a novel software architecture
description for reporting globally distributed software architecture for tax
compliance. The architecture description of our case company reveals that
almost 70% of all reuse relationships between the 2560 microservices in our
case are cross-border and, therefore, are taxable.

The panel of tax experts found that our proposed architectural viewpoint
properly and sufficiently frames the concerns of taxation stakeholders. The
panel also appreciated the evidence-based, empirical approach for visualizing
the software architectures to tax experts as software-engineering laymen.

However, we identified several challenges that must be addressed in future
work. The most critical and likely most complex issue to solve is our
insufficient definition and understanding of ownership, which needs to be
rethought for taxation purposes in this line of research. Also the question
what does or will constitute to the software product and what not deserves
further attention. Last, but not least, the subsequent step, adding value to
each component, is yet another unsolved problem in the context of taxing
software-intensive products. We believe that the software architecture
research community can play a vivid role in addressing those challenges and
paving the way for tax compliance in software engineering.

At this point, we would like to emphasize that the quality and rigor of the
data curation at our case company is extraordinary and, from our experience,
not industry standard. Not always, companies can rely on such a solid
foundation grounded in historical and accurate data on their software
architectures. This uncertainty introduces further risks and lets tax audits
stray away from an empirical-based, data-driven discussion on the value of
software. Although this research is only the beginning, we recommend companies
start continuously collecting information about their software architecture
and ownership. Accurately collecting this information is never trivial but
nearly impossible in hindsight for complex software systems.

Although “there is no black or white in tax audits but a lot of gray”, as
interviewee R summed up, we hope to shed some light on the discussions in tax
audits and lay the foundation for a data-driven approach for taxing globally
distributed software architectures.

## Acknowledgment

We would to thank our industry partner and the panel of tax experts for their
great support which made this research possible.

This work was supported by the KKS Foundation through the SERT Project
(Research Profile Grant 2018/010) at Blekinge Institute of Technology.

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