# Analysis of Dirac exceptional points and their isospectral Hermitian
counterparts

Jose H. D. Rivero  Department of Materials Science and Engineering, University
of Pennsylvania, Philadelphia, PA 19104, USA  Liang Feng  Department of
Materials Science and Engineering, University of Pennsylvania, Philadelphia,
PA 19104, USA  Li Ge<EMAIL_ADDRESS>Department of Physics and Astronomy,
College of Staten Island, CUNY, Staten Island, NY 10314, USA  The Graduate
Center, CUNY, New York, NY 10016, USA

###### Abstract

Recently, a Dirac exceptional point (EP) was reported in a non-Hermitian
system. Unlike a Dirac point in Hermitian systems, this Dirac EP has coalesced
eigenstates in addition to the degenerate energy. Also different from a
typical EP, the two energy levels connected at this Dirac EP remain real in
its vicinity and display a linear instead of square root dispersion, forming a
tilted Dirac cone in the hybrid space consisting of a momentum dimension and a
synthetic dimension for the strength of non-Hermiticity. In this report, we
first present simple three-band and two-band matrix models with a Dirac EP,
where the linear dispersion of the tilted Dirac cone can be expressed
analytically. Importantly, our analysis also reveals that there exist
Hermitian and non-Hermitian systems that have the same (real-valued) energy
spectrum in their entire parameter space, with the exception that one or more
degeneracies in the former are replaced by Dirac EPs in the later. Finally, we
show the existence of an imaginary Dirac cone with an EP at its center.

##  I Introduction

Dirac points, the degeneracies at the center of Dirac cones, give graphene [1,
2] and other Dirac matters [3] their unusual electronic properties. Their
signature, i.e., the linear dispersion around a Dirac point in momentum space,
signals a massless fermion that differs significantly from a classical object
with the quadratic kinetic energy relation. Similar to all other degeneracies
in Hermitian systems, a Dirac point corresponds to two different quantum
eigenstates at the same energy level and hence is an example of a diabolic
point [4].

An exceptional point (EP) [5, 6], on the other hand, features coalesced
eigenstates at a degenerate energy level. Its existence is a unique feature of
non-Hermitian systems, which arise when a physical system is represented by a
Hamiltonian with partial degrees of freedom or by another operator that
describes the exchange of energy or particles with the environment (such as
the scattering matrix [7, 10, 8, 9, 11]). This approach has been adopted in
studies ranging from nuclear decay [12] to photon lifetime in optical
microcavities [13], yielding insightful results as well as providing new
directions of research [13, 14, 15, 16]. For example, the motion of
eigenfrequencies or resonances in the complex plane can be quite unusual in
the vicinity of an EP, which results in intriguing behaviors such as gain-
suppressed lasing [17, 18, 19, 20] and loss-induced lasing [21].

While a degeneracy in a Hermitian system cannot be an exceptional point, a
degeneracy in a non-Hermitian system is not necessarily an EP. Such a non-EP
degeneracy can occur accidentally or by symmetry, similar to the mechanisms in
a Hermitian system. Some examples include the zero modes in a non-Hermitian
flatband [22] and Dirac points constructed in non-Hermitian lattices [23, 24].
For the degeneracies that are indeed EPs, traditionally one associates their
perturbative dependence on a system parameter with a fractional exponent
[e.g., a square root for an EP of order 2 (EP2)], which implies a stronger
response and hence potentially enhanced sensitivity [25, 26, 27]. Furthermore,
if an EP has its roots in a non-Hermitian symmetry, such as parity-time ($\cal
PT$) [16, 28, 29, 30], anti-$\cal PT$ [31, 32, 33, 34, 35], or particle-hole
[36, 37, 38, 39] symmetries, the non-Hermitian system experiences a
corresponding spontaneous symmetry breaking or restoration across the EP. This
phenomenon is usually accompanied by dramatic changes in the energy spectrum:
It may transition from real to complex or even imaginary and vice-versa, which
is used as one experimental signature of the EP itself [32, 40, 41, 42].

Therefore, the recent, accidental finding of a Dirac EP [43] in a one-
dimensional periodic $\cal PT$-symmetric system came as a surprise: Two energy
bands connected by this Dirac EP display a linear and conical “dispersion” in
a two-dimensional hybrid space, as a function of both momentum and the non-
Hermitian parameter given by the optical gain and loss strength. In addition,
the two energy bands remain real in the vicinity of the Dirac EP, without
undergoing a spontaneous symmetry breaking. While a three-band matrix model
was introduced in Ref. [43] to capture the latter, the linear and conical
dispersion at this Dirac EP remains to be elucidated. The differences between
Dirac points, Dirac EPs, and conventional EPs are summarized in Table. 1. In
the last row, state flip is marked “possible” instead of “yes” when a
conventional EP2 is encircled, due to the non-adiabatic transition from the
low-gain/high-loss state to the high-gain/low-loss state [6]. Such non-
adiabatic behaviors caused by the different modal gain (or loss) are absent in
the vicinity of a Dirac EP thanks to its real spectrum.

Table 1: Comparison of three types of degeneracies. |  Dirac Points |  Dirac EPs |  Conventional EP2s
---|---|---|---
Algebraic multiplicity |  2 |  2 |  2
Geometric multiplicity |  2 |  1 |  1
Coalescence of wave functions |  No |  Yes |  Yes
Locally real spectrum |  Yes |  Yes |  No
Branch cuts at degeneracy |  No |  No |  Yes
Node type in energy (real part) |  Point |  Point |  Line or surface
Energy dispersion |  Conical |  Conical |  Square roots
State flip when encircled |  No |  No |  Possible

In this Report, we first introduce a revised three-band model and show
explicitly the linear dispersion of the (tilted) Dirac cone centered at the
Dirac EP, via a perturbative treatment. Our approach is different from the
standard procedure of using the alternating Puiseux series for a conventional
EP [44, 45], which is inapplicable at a Dirac EP as we will show. Because the
Dirac EP connects two bands instead of three, we further reduce this three-
band model to a two-band non-Hermitian Hamiltonian where the Dirac EP can be
found. Using a slightly different approach, we find, interestingly, that while
the same linear and conical dispersion is produced in the hybrid dimensions,
the Dirac point becomes a diabolic point instead, with two distinct
eigenstates. We show that this dilemma can be resolved by revisiting the
three-band model. More importantly, this comparison has a far-reaching
implication: There exist Hermitian and non-Hermitian systems that have the
same (real-valued) energy spectrum in their entire parameter space, with the
exception that one or more degeneracies in the former are replaced by Dirac
EPs in the later. Finally, we show the existence of an imaginary Dirac cone
with an EP at its center.

##  II Three-band model

The systems exhibiting a Dirac EP we study originate from the following
Schrödinger equation

$i\frac{d}{dt}\psi(x,t)=[-\partial_{x}^{2}+V(x)]\psi(x,t),$ (1)

where time, position, and potential are in their dimensionless forms and
$\hbar=1$. $V(x)=V_{0}(\cos\,x+i\tau\sin\,x)\,(\tau\geq 0)$ is a complex
potential with the spatial period $a=2\pi$. It is $\cal PT$-symmetric and
satisfies $V(x)=V^{*}(-x)$ [13]. The asterisk denotes complex conjugation and
represents time reversal, and the imaginary part of the potential represents
optical gain and loss with strength $\tau$ [13].

The Dirac EP in this system is found at the point contact between the second
and third bands in the first Brillouin zone, where $\tau=1$ and $k=0$. To gain
analytical understanding of this Dirac EP, Ref. [43] first expanded the Bloch
wave function in the plane-wave basis, i.e.,

$\psi_{nk}(x,t)=e^{ikx-i\omega t}\sum_{m\in\mathbb{Z}}a_{m}e^{imx},$ (2)

and derived the Bloch Hamiltonian

$\displaystyle H_{k}=\sum_{m\in\mathbb{Z}}(m+k)^{2}|m\rangle\langle m|\;$
$\displaystyle+\;t_{-}|m\rangle\langle m+1|$
$\displaystyle+\;t_{+}|m\rangle\langle m-1|$ (3)

that satisfies

$H_{k}\Psi_{nk}(m)=\omega_{nk}\Psi_{nk}(m).$ (4)

Here $n=1,2,\ldots$ is the band index, $\omega_{nk}$ is energy of the $n$th
band when the momentum is $k$,
$\Psi_{nk}(m)=[\ldots,a_{-1},a_{0},a_{1},\ldots]^{T}$is the wave function in
the plane-wave basis, and $t_{\pm}=V_{0}(1\pm\tau)/2\in\mathbb{R}$. Then by
truncating $H_{k}$ given by Eq. (3) and keeping only the $m=-1$, $0$ and 1
block, Ref. [43] showed analytically that indeed the second and third bands
remain real-valued when we increase $\tau$ across $1$, i.e., the system does
not experience a spontaneous $\cal PT$ breaking by going through the Dirac EP.
While this analysis was only performed for $k=0$ and the truncation turns out
to be a crude approximation, it shone light on how a Dirac EP could be
constructed.

Below we first use the insight from this truncation to introduce a revised
three-band model, where a Dirac EP exists and where the linear dispersion of
its (tilted) Dirac cone can be expressed analytically. In this model, we allow
$k$ to be a (small) free parameter in addition to the gain and loss strength
$\tau$, and the three-band non-Hermitian Hamiltonian is given by

$H^{(3)}=\begin{pmatrix}1-2k&t_{-}&0\\\ t_{+}&0&t_{-}\\\
0&t_{+}&1+2k\end{pmatrix}$ (5)

with the asymmetric couplings $t_{\pm}$ introduced in Eq. (3). The eigenvalues
$\omega_{i}$ of $H^{(3)}$ are the solutions of the characteristic polynomial

$\omega_{i}(1-\omega_{i})^{2}+2t^{2}(1-\omega_{i})-4k^{2}\omega_{i}=0,$ (6)

where $t^{2}\equiv t_{-}t_{+}=V_{0}^{2}(1-\tau^{2})/4$. This cubic equation
can be solved analytically, but the resulting expressions for $\omega_{i}$’s
are rather complicated (e.g., with a square root inside a cubic root) and do
not help us understand the properties of the Dirac EP. We could perform a
Taylor expansion of these expressions for $\omega_{i}$’s, but a much simpler
approach is to expand Eq. (6) directly, which gives the same results. We do
note that being a cubic equation with real coefficients, Eq. (6) indicates
that the three energy bands are either all real or one real plus a complex
conjugate pair. Therefore, it does not exclude the possibility of a
spontaneous $\cal PT$ breaking, which nevertheless does not take place at the
Dirac EP.

It is straightforward to verify that this EP has energy
$\omega=1\equiv\omega_{0}$ (again $\hbar=1$) and exists at $\tau=1,k=0$. Its
coalesced eigenstate is given by
$[a_{-1},a_{0},a_{1}]^{T}=[0,0,1]^{T}\equiv\Psi_{0}$. This EP is the point
contact of the second and third bands that form a tilted Dirac cone (see Fig.
1), similar to the original Hamiltonian $H_{k}$ shown in Ref. [43]. To derive
the dispersion relation near this Dirac EP, we write
$\omega_{i}\equiv\omega_{0}+\Delta\omega_{i}\,(|\Delta\omega_{i}|\ll\omega_{0})$
and study how $\Delta\omega_{i}$ depends on the two small parameters
$\Delta\tau\equiv\tau-1$ and $k$.

Figure 1: Tilted Dirac cone of $H^{(3)}$ centered at a Dirac EP. $k$ is
momentum and $\Delta\tau$ is the change of the non-Hermitian strength from its
value (i.e., 1) at the Dirac EP. $V_{0}=1$ is used in the couplings $t_{\pm}$.
The intersection of the Dirac cone and two parallel planes given by
$\omega=1-s\Delta\tau\pm d$ are also shown, where $\omega_{0}=1$ is the energy
at the Dirac EP, $s=\sqrt{2}-1$, and $d=1/40$. The arrow shows the diagonal
direction used in Fig. 2(b).

Here we take the advantage of knowing that the dispersion is linear and
conical, which implies that $\Delta\omega_{i}$ is of the same order as $k$ and
$\Delta\tau$. Consequently, Eq. (6) becomes

$\Delta\omega_{i}^{2}-2t^{2}\Delta\omega_{i}-4k^{2}=0,$ (7)

to the leading order of $\Delta\omega_{i}$ (i.e., $\Delta\omega_{i}^{2}$),
where $t^{2}\approx-(V_{0}^{2}/2)\Delta\tau$ is also of the same order as
$\Delta\tau$ (with $V_{0}$ chosen to be of order unity). We then find

$\Delta\omega_{i}=t^{2}\pm\sqrt{t^{4}+4k^{2}}$ (8)

or

$\Delta\omega_{i}\approx(-\alpha\pm\sqrt{4+\alpha^{2}})k$ (9)

in any arbitrary direction
$\Delta\tau=(2\alpha/V_{0}^{2})k\,(\alpha\in\mathbb{R})$ from the origin of
the $\Delta\tau$-$k$ plane.

The square root in Eq. (9) directly captures the linear and conical dispersion
of the Dirac cone shown in Fig.1, with the linear term in front of it
explaining the tilt of this Dirac cone. We also note that this expression is
exact when $\Delta\tau=0$ (i.e., along the $k$ direction from the Dirac EP),
which can be checked easily by setting the second term in Eq. (6) to be zero.
In Fig. 2, we show the comparison of Eq. (9) and the actual band energies
along two other directions, i.e., the $\Delta\tau$ axis and the one with
$\alpha=2.5$ (diagonal direction in Fig. 1). We observe that the band closer
to $\omega=1$ is better approximated by Eq. (9), which will be explained using
the equivalent two-band model below where the analysis becomes easier.

We also note that our approach is different from the perturbative expansion
used at a conventional EP, which utilizes alternating Puiseux series [45] with
fractional powers of a small parameter. In fact, this standard treatment is
inapplicable at a Dirac EP as we show below. This approach calculates the
perturbative corrections to the eigenstates and their energies when the
Hamiltonian changes from $H_{0}$ to $H=H_{0}+\epsilon H_{1}\,(\epsilon\ll 1)$,
and when an EP is the result of two coalescing eigenstates (i.e., of
multiplicity 2) [44], the alternating Puiseux series assume integer and half-
integer powers of $\epsilon$, i.e.,

$\displaystyle\omega_{\pm}$
$\displaystyle=\omega_{0}\pm\epsilon^{1/2}\omega_{1}+\epsilon\omega_{2}\pm\epsilon^{3/2}\omega_{3}+\ldots$
(10) $\displaystyle\Psi_{\pm}$
$\displaystyle=\Psi_{0}\pm\epsilon^{1/2}\Psi_{1}+\epsilon\Psi_{2}\pm\epsilon^{3/2}\Psi_{3}+\ldots$
(11)

Here $\omega_{\pm}$ are the energies of the two eigenstates $\Psi_{\pm}$ that
become coalesced at the EP, with energy $\omega_{0}$ and wave function
$\Psi_{0}$. Even if we ignore the $\epsilon^{1/2}$ term, Eq. (10) indicates
clearly that $\omega_{\pm}$ have the same linear dependence on $\epsilon$. In
other words, they stay the same to the linear order and hence cannot form a
Dirac cone in this perturbative expansion.

Figure 2: Linear dispersion of the tilted Dirac cone along (a) $\Delta\tau$
and (b) the diagonal direction given by $\Delta\tau=5k$ in Fig. 1. Solid and
dashed lines show the actual and approximated band energies, which cannot be
distinguished by eye for the band closer to $\omega=1$.

##  III Two-band model

Since the Dirac EP only connects two bands of $H^{(3)}$ [i.e., the second and
third bands; represented by the two signs in Eq. (8)], it should be possible
to reduce $H^{(3)}$ to a two-band Hamiltonian, which not only simplifies the
understanding of the Dirac EP but also provides another instance where it
exists. To this end, we first write down the time-dependent Schrödinger
equation corresponding to $H^{(3)}$:

$\displaystyle i\frac{d}{dt}a_{-1}$ $\displaystyle=(1-2k)a_{-1}+t_{-}a_{0}$
(12) $\displaystyle i\frac{d}{dt}a_{0}$ $\displaystyle=t_{+}a_{-1}+t_{-}a_{1}$
(13) $\displaystyle i\frac{d}{dt}a_{1}$ $\displaystyle=(1+2k)a_{1}+t_{+}a_{0}$
(14)

Using $a_{0}\propto e^{-i\omega_{i}t}$ in an eigenstate with energy
$\omega_{i}$, we obtain

$a_{0}=(t_{+}a_{-1}+t_{-}a_{1})/\omega_{i}$ (15)

from Eq. (13) and use this expression to eliminate $a_{0}$ in Eqs. (12) and
(14). The result is a two-band Hamiltonian

$H_{a}=(1+t^{2}/\omega_{i})\bm{1}+\begin{pmatrix}-2k&t_{-}^{2}/\omega_{i}\\\
t_{+}^{2}/\omega_{i}&2k\end{pmatrix},$ (16)

where $\bm{1}$ is the identity matrix. Because no approximations have been
used in deriving $H_{a}$, it has the same eigenvalues $\omega_{i}$’s as
$H^{(3)}$. Furthermore, we note that the eigenvalue $\omega_{i}$ appears in
the Hamiltonian $H_{a}$ itself, and hence this problem can be treated as a
nonlinear eigenvalue problem [46]. Nevertheless, if we replace $\omega_{i}$ by
$\omega_{0}=1$ in $H_{a}$ itself near the expected EP, i.e.,

$H_{a}\rightarrow
H_{a}^{\prime}=(1+t^{2})\bm{1}+\begin{pmatrix}-2k&t_{-}^{2}\\\
t_{+}^{2}&2k\end{pmatrix},$ (17)

we immediately find that $H_{a}^{\prime}$ takes the Jordan normal form when
$k=0$ and $\Delta\tau=0$, at which $t_{-}$ vanishes as well that leads to an
EP with energy $\omega_{0}=1$. It is easy to check that the two eigenvalues of
$H_{a}^{\prime}$ are the same as those given by Eq. (8), and hence it gives
the same linear and conical dispersion relation (9).

If we have not replaced $\omega_{i}$ by $\omega_{0}=1$ in Eq. (16), we can
also express the nonlinear eigenvalues of $H_{a}$ in a self-consistent way,
i.e.,

$\omega_{i}=(1+t^{2}/\omega_{i})\pm\sqrt{4k^{2}+t^{4}/\omega_{i}^{2}},$ (18)

which is equivalent to Eq. (6). It is then clear that the errors introduced by
taking $\omega_{i}\approx\omega_{0}$ in Eq. (17) only originate from the
$t^{2}$, $t^{4}$ terms in Eq. (18). They are partially cancelled (increased)
in the solution with the “$-$” (“$+$”) sign in Eq. (18), which is also closer
to (further from) the energy at the Dirac EP. This is exactly what we have
observed in Fig. 2.

##  IV Isospectral Hermitian and non-Hermitian systems

As mentioned in the introduction, there is more than one way to reduce
$H^{(3)}$ to a two-band Hamiltonian. Because the amplitudes $a_{\pm 1}$ are
coupled indirectly through $a_{0}$ in $H^{(3)}$, one may seek eigenstates in
the forms of symmetric and antisymmetric superpositions of $a_{\pm 1}$,
weighted by the couplings $t_{\mp}$:

$a_{\pm}=t_{+}a_{-1}\pm t_{-}a_{1}.$ (19)

Similar to the first approach above, we also eliminate $a_{0}$ and express it
in terms of $a_{\pm}$. The resulting two-band Hamiltonian for $a_{\pm}$ is
then found by multiplying Eqs. (12) and (14) by $t_{+},t_{-}$ respectively and
taking the summation and difference of the results:

$H_{b}=\begin{pmatrix}1+2t^{2}/\omega_{i}&-2k\\\ -2k&1\end{pmatrix}.$ (20)

Again, by approximating $\omega_{i}$ in $H_{b}$ by $\omega_{0}=1$ at the Dirac
EP, i.e.,

$H_{b}\rightarrow H_{b}^{\prime}=\begin{pmatrix}1+2t^{2}&-2k\\\
-2k&1\end{pmatrix},$ (21)

it is straightforward to show that the two eigenvalues of $H_{b}^{\prime}$ are
given by Eq. (8), and we recover the linear and conical dispersion relation
(9). However, $H_{b}^{\prime}$ (and $H_{b}$) becomes an identity matrix at
where we expect to find the Dirac EP, i.e., $k=\Delta\tau=0$, and hence the
degeneracy $\omega_{0}=1$ is a diabolic point instead of an EP, with two
distinct eigenstates $[0,1]^{T}$ and $[1,0]^{T}$. This should not be
surprising, however, because $H_{b}^{\prime}$ is a real symmetric matrix, and
hence it is Hermitian and cannot have an EP.

This apparent contradiction can be easily resolved by realizing that one step
leading to $H_{b}$ [i.e., multiplying Eq. (14) by $t_{-}$] fails when
$t_{-}=0$ (or equivalently, $\Delta\tau=1$) where the degeneracy exists.
Therefore, $H_{b}$ is not equivalent to $H_{a}$ at $\Delta\tau=1$ and it is
allowed to differ from $H_{a}$, i.e., having a diabolic point instead of an
EP.

More importantly, this comparison reveals a far-reaching implication: There
exist Hermitian and non-Hermitian systems that have the same (real) energy
spectrum in their entire parameter space, with the exception that one or more
degeneracies in the former are replaced by Dirac EPs in the latter.

This observation holds for the pair of linear Hamiltonians
$H_{a}^{\prime},H_{b}^{\prime}$ in the entire $\tau$-$k$ parameter space. Here
$H_{b}^{\prime}$ should be treated as given, and hence the illegitimacy
mentioned above from $H^{(3)}$ to $H_{b}$ at the Dirac EP is irrelevant.
Although there is only one degeneracy (a diabolic point) replaced by one EP in
our examples here, cases with more or even all degeneracies replaced by EPs
can be trivially constructed. For example, one can generate a series of two-
band Hamiltonians similar to $H_{a}^{\prime}$ but with different energy shifts
(i.e., replacing $\omega_{0}=1$ in its diagonal elements by an increasing
series $\Delta_{m}$’s) and then stack them to form a block diagonal non-
Hermitian Hamiltonian $H_{A}^{\prime}$. By following the same process but
using $H_{b}^{\prime}$ instead, we end up with another block diagonal
Hamiltonian $H_{B}^{\prime}$ which is Hermitian. It is easy to see that they
have the same energy eigenvalues throughout the parameter space $\tau$-$k$,
with $\Delta_{m}$’s being diabolic points in $H_{B}^{\prime}$ but EPs in
$H_{A}^{\prime}$.

We note that the isospectral property between a Hermitian and a non-Hermitian
system we report here is stronger than that found in Ref. [43], where this
equivalence was only established in the $\cal PT$-symmetric regime of a linear
non-Hermitian system and away from its EP. The more general claim here is made
possible partly by the elimination of the conventional EPs of the system
studied in Ref. [43], which occur at the edge of the Brillouin zone when
$\tau=1$. Near these conventional EPs, the band energies become complex along
the $\Delta\tau$ direction and hence lose the equivalence to their Hermitian
counterparts. If we restrict our discussion to the one-dimensional parameter
space along $k$ with $\tau$ fixed at 1, one may attempt to claim that this
system also have the more general isospectral property reported here: Its
entire band structure is real valued in the first Brillouin zone and identical
to that of a Hermitian system with $V(x)=0$; its EPs, both the conventional
ones at the band edge and the unconventional ones at the center of the
Brillouin zone (including the Dirac EP), are replaced by degeneracies in the
Hermitian system. However, one quickly realizes that with $V(x)=0$, this
“crystal” is just free space with a single dispersion relation $\omega=k^{2}$.
Therefore, its degeneracies in the band analysis are artifacts of applying the
periodic boundary condition to a “unit cell” of an arbitrary length, resulting
in the folding of this single energy relation into the first Brillouin zone.

##  V linear and conical dispersion

Figure 3: Energy surfaces of $H_{a}^{\prime\prime}$ intersecting at a Dirac
exceptional line instead of a Dirac EP. The parameters are the same as in Fig.
1.

While both our three-band model $H^{(3)}$ and linearized two-band model
$H_{a}^{\prime}$ host a Dirac EP, there is a noticeable difference between
them: The small changes in the former, i.e., $H^{(3)}=H_{0}+\Delta H$ where

$H_{0}=\begin{pmatrix}1&0&0\\\ V_{0}&0&0\\\ 0&V_{0}&1\end{pmatrix},\quad\Delta
H=\begin{pmatrix}-2k&-g&0\\\ g&0&-g\\\ 0&g&2k\end{pmatrix},$ (22)

and $g\equiv V_{0}\Delta\tau/2$, are of the same order and linear in terms of
$k$ and $\Delta\tau$; there are, however, higher-order terms proportional to
$\Delta\tau^{2}$ in the two-band model $H_{a}^{\prime}=H_{0}+\Delta H+\Delta
H^{\prime}$, where

$H_{0}=\begin{pmatrix}1&0\\\ V_{0}^{2}&1\end{pmatrix},\quad\Delta
H=-\frac{V_{0}^{2}}{2}\Delta\tau\bm{1}+\begin{pmatrix}-2k&0\\\
V_{0}^{2}\Delta\tau&2k\end{pmatrix},$ (23)

and

$\Delta H^{\prime}=\frac{V_{0}^{2}}{4}\Delta\tau^{2}\begin{pmatrix}-1&1\\\
1&-1\end{pmatrix}.$ (24)

Without these higher-order terms, especially the upper right element in
$\Delta H^{\prime}$, $\omega_{0}=1$ is still an EP of the resulting
Hamiltonian $H_{a}^{\prime\prime}\equiv H_{0}+\Delta H$, and the dispersion at
this EP is still linear:

$\Delta\omega_{i}=-\frac{V_{0}^{2}}{2}\Delta\tau\pm 2k.$ (25)

However, these two energy surfaces intersect at an EP line instead of a Dirac
EP (see Fig. 3). One may refer to this line as a Dirac exceptional line (node)
following the terminology of a Dirac line node in condensed matter systems
[47, 48].

In fact, a two-band model cannot host a Dirac EP in a two-dimensional
parameter space, when the perturbation is just first-order. We note that such
two-band models are widely used to generate Dirac cones in Hermitian systems
[49], and hence this finding highlights another difference between Hermitian
and non-Hermitian systems in terms of their Dirac points. To show this
difference, we note that all two-band models with an EP can be put into the
Jordan normal form after a similar transformation. Therefore, we can take

$H_{0}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$ (26)

without loss of generality. We then express the perturbation as

$\Delta H=\Delta_{+}\sigma_{+}+\Delta_{-}\sigma_{-}+\Delta_{3}\sigma_{3},$
(27)

where we have neglected perturbations proportional to the identity matrix
because they merely cause a shift of the whole spectrum. Here
$\sigma_{\pm}=(\sigma_{1}\pm i\sigma_{2})/2$ and $\sigma_{i}\,(i=1,2,3)$ are
the three Pauli matrices. $\Delta_{\pm},\Delta_{3}\in\mathbb{C}$ are three
complex perturbation amplitudes of the same order. The two energy eigenvalues
are then given by

$\omega_{\pm}=\pm\sqrt{4\Delta_{-}(1+\Delta_{+})+\Delta_{3}^{2}}.$ (28)

Clearly, due to the leading order term $4\Delta_{-}$ in the randicand, the
dispersion of this system cannot be conical; only when $\Delta_{-}$ is second
order (i.e., $\Delta_{-}=\delta_{-}^{2}\sim\Delta_{3}^{2}$) do we recover a
conic dispersion to the leading order:

$\omega_{\pm}\approx\pm\sqrt{4\delta_{-}^{2}+\Delta_{3}^{2}}\quad(\delta_{-},\Delta_{3}\in\mathbb{R}).$
(29)

In comparison, $H_{0}$ in a two-band Hermitian model with a Dirac point would
simply vanish, and we have

$\omega_{\pm}=\pm\sqrt{4\Delta_{-}\Delta_{+}+\Delta_{3}^{2}}$ (30)

instead. A Dirac cone is then found, e.g., by letting
$\Delta_{3}\in\mathbb{R}$ together with either
$\Delta_{-}=\Delta_{+}\in\mathbb{R}$ or $\Delta_{-}=-\Delta_{+}\in
i\mathbb{R}$.

##  VI Imaginary Dirac cone

If we multiply a non-Hermitian Hamiltonian with a Dirac EP by $i$, it is clear
that the Dirac cone now exists in the imaginary part of the energy, which is
uniquely non-Hermitian. Moreover, the general analysis in the last section,
particular Eq. (29), indicates that it is unnecessary to change the
unperturbed Hamiltonian $H_{0}$ to construct an imaginary Dirac cone with an
EP at its center; we just need to change $\delta_{-},\Delta_{3}$ from real to
imaginary in this two-band model.

In a three-band model, we find that the following Hamiltonian hosts an
imaginary Dirac cone:

$H=\begin{pmatrix}ik&g&1\\\ g&1&g\\\
0&g&-ik\end{pmatrix}\quad(k,g\in\mathbb{R}).$ (31)

The unperturbed Hamiltonian still has real eigenvalues $0$ (EP) and $1$. The
real and imaginary parts of the three energy eigenvalues are shown in Fig. 4,
where the real parts of the two coalesced eigenvalues at the EP stay the same
in the two-dimensional parameter space. In other words, these two bands are
complex conjugates, with the other band being real. These observations are
consistent with the characteristic equation

$\omega^{3}-\omega^{2}+(k^{2}-2g^{2})\omega-(k^{2}+g^{2})=0,$ (32)

which again has real coefficients. When expanding near the EP, we can drop the
higher-order cubic term and solve the remaining quadratic equation. The result
is

$\omega_{\pm}=\pm i\sqrt{k^{2}+g^{2}}$ (33)

to the leading order, showing the explicit conical and linear dispersion.

Figure 4: Imaginary Dirac cone with an EP at its center. (a) and (b) show the
real and imaginary parts of the three energies of the Hamiltonian given in Eq.
(31).

##  VII Conclusion and discussions

In summary, we studied a novel type of non-Hermitian degeneracies around which
the energy spectrum remains real, while they manifest eigenvector coalescence
in accordance with the definition of EPs. These Dirac EPs are characterized by
a conical dispersion around them, for which the standard perturbative
description using the alternating Puiseux series fails. We also identified its
imaginary counterpart, where a Dirac cone is formed in the imaginary parts of
the energies with an EP at its center.

The Dirac EPs we presented emerge in a non-Hermitian three-band model, which
can can be reduced to two-band models in several ways. This reduction may
result in either Hermitian or non-Hermitian models which are isospectral, and
both host degeneracy points. This observation led to a startling discovery:
There are Hermitian and non-Hermitian systems that have the same real-valued
energy spectrum in their entire two-dimensional parameter space, and their
degeneracy points consist of diabolic points in the former, and Dirac EPs in
the latter. While isospectral non-Hermitian and Hermitian systems have been
reported before (see, for example, Ref. [43]), one with EPs and existing in an
entire two-dimensional parameter space has not. As mentioned, we were able to
achieve this isospectral property thanks to the elimination of conventional
EPs, whose branch cuts would make the non-Hermitian spectrum complex in that
region of the two-dimensional parameter space.

Finally, we showed that for two-band models, it is impossible to generate
Dirac EPs with linear dispersion along all directions in a two-dimensional
parameter space, if only first-order perturbations are introduced to an
underlying non-Hermitian Hamiltonian at the EP; second-order terms are
necessary to produce Dirac EPs with a conical dispersion. We note that this
result holds for higher-dimensional systems as well, as our derivation based
on Eq. (27) is independent of the physical dimensions. This property is in
stark contrast to Hermitian systems, where the conical dispersion around a
Dirac point is produced with first-order perturbations. This observation
represents another intriguing difference between Dirac points in Hermitian and
non-Hermitian systems.

The results in this work broaden our understanding of non-Hermitian
degeneracies. We break the traditional link between eigenvector coalescence
and the characteristic integer root dispersion of the eigenvalue spectrum
around the exceptional point in a two-dimensional parameter space. While a
similar finding was known in the mathematical literature [50], it was based on
a single (complex) parameter turning, where the real and imaginary parts of
the perturbed Hamiltonian are collinear. When two independent perturbations
are allowed instead as we do here (i.g., either using $\Delta\tau,k$ or
$\alpha,k$), this result was found to break down in general and no traces of a
Dirac EP were found [51]: The dispersion near the EP is no longer linear
except for one particular direction [52], consistent with previous findings
[50]. Furthermore, although the enhanced sensitivity stemming from the
nonlinear dispersion of an EP is useful for applications in sensing, it
simultaneously imposes a challenge for tuning and observing exceptional
points. This fact, combined with the complex character of the non-Hermitian
spectrum, often obscure the physics of eigenvector coalescence at the EP. Our
work provides a more tolerant platform towards studying eigenstates
coalescence in a two-dimensional parameter space, which may be generalized to
higher dimensions as well.

This project is supported by NSF under Grant No. PHY-1847240 and and
ECCS-1846766.

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# Electronic and optical properties of core-shell InAlN nanorods: a
comparative study via LDA, LDA-1/2, mBJ and $G_{0}W_{0}$ methods

Ronaldo Rodrigues Pela  Supercomputing Department, Zuse Institute Berlin
(ZIB), Takustraße 7, 14195 Berlin, Germany<EMAIL_ADDRESS>Ching-
Lien Hsiao  Lars Hultman  Jens Birch  Gueorgui Kostov Gueorguiev  Thin film
Physics Division, Department of Physics, Chemistry and Biology (IFM),
Linköping University, SE 581 83 Linköping, Sweden

###### Abstract

Currently, self-induced InAlN core-shell nanorods enjoy an advanced stage of
accumulation of experimental data from their growth and characterization as
well as a comprehensive understanding of their formation mechanism by the ab
initio modeling based on Synthetic Growth Concept. However, their electronic
and optical properties, on which most of their foreseen applications are
expected to depend, have not been investigated comprehensively. $G_{0}W_{0}$
is currently regarded as a gold-standard methodology with quasi-particle
corrections to calculate electronic properties of materials in general. It is
also the starting point for higher-order methods that study excitonic effects,
such as those based on the Bethe-Salpeter equation. One major drawback of
$G_{0}W_{0}$, however, is its computational cost, much higher than density-
functional theory (DFT). Therefore, in many applications, it is highly
desirable to answer the question of how well approaches based on DFT, such as
e. g. LDA, LDA-1/2, and mBJ, can approximately reproduce $G_{0}W_{0}$ results
with respect to the electronic and optical properties. Thus, the purpose of
the present paper is to investigate how the DFT-based methodologies LDA,
LDA-1/2, and mBJ can be used as tools to approximate $G_{0}W_{0}$ in studies
of the electronic and optical properties of scaled down models of core-shell
InAlN nanorods. For these systems, we observed that band gaps, density of
states, dielectric functions, refractive indexes, absorption and reflectance
coefficients are reasonably well described by LDA-1/2 and mBJ when compared to
$G_{0}W_{0}$, however, at a much more favorable computational cost.

††preprint: AIP/123-QED

##  I Introduction

Wurtzite InAlN semiconductor alloys have a direct band gap that span a wide
spectrum range from 0.65 eV (InN) to 6.25 eV (AlN). Taniyasu, Kasu, and
Makimoto (2006); Wu  _et al._ (2002); Hsiao  _et al._ (2007) Therefore, many
optoelectronic devices can possibly be fabricated from InAlN alloys, which are
applicable in a wide wavelength range covering deep-ultraviolet (DUV) to near
infrared (NIR), such as light-emitting diodes, laser diodes, solar cells, and
photodetectors.Baten  _et al._ (2021); Alias  _et al._ (2018); Huang  _et al._
(2019); Li  _et al._ (2017); Filho  _et al._ (2023) However, InAlN thin film
often contains large number of structural defects and compositional
inhomogeneity owing to a wide-range composition immiscibility of the InxAl1-xN
($0.1<x<0.9$), low dissociation temperature of InN ($\sim 550$ °C), and
mismatches in lattice and coefficient of thermal expansion to common
substrates.Nanishi, Saito, and Yamaguchi (2003); Ferhat and Bechstedt (2002);
Palisaitis  _et al._ (2017) Alternatively, InAlN grown in the form of low-
dimensional nanostructures can provide an opportunity to overcome the effects
of lattice mismatch like threading dislocations formation and substrate-film
strain.

In the context of the InAlN low-dimensional nanostructures, self-induced core-
shell InAlN nanorods (NRs) have been successfully synthesized by reactive
magnetron sputter epitaxy (MSE) while their formation mechanism was elucidated
by modeling the relevant precursor prevalence and their corresponding
energetics using the DFT-based synthetic growth concept (SGC).Filho  _et al._
(2023) SGC is an extensive approach designed for modeling of nanostructures
with complex morphology and accounting for the role of the precursors in their
formation when employing a wide spectrum of vapor-phase deposition
techniques.Goyenola  _et al._ (2012); Gueorguiev  _et al._ (2011, 2006);
Goyenola  _et al._ (2011)

Very high-crystal-quality nanomaterials can be grown on various substrates,
including metals, metal nitrides, oxides and Si,Kamimura  _et al._ (2007);
Prabaswara  _et al._ (2020); Serban  _et al._ (2017) which opens the
possibility of integration with mature device-fabrication technology for
integrated circuit industry. Furthermore, the form of nanostructure enables to
fabricate nanodevices with high performance benefited from the reduced
geometry. For instance, InAlN nanospirals with tailored chirality have been
demonstrated to reflect circularly polarized light with corresponding
handedness, through tuning internally compositional distribution and external
geometry, which is very promising for fabricating high-performance optical
elements.Hsiao  _et al._ (2015); Kuo  _et al._ (2018) High-sensitivity
photodetectors based on InAlN nanophotonic structure is applicable from deep
DUV to NIR region.Yao  _et al._ (2017); Alias  _et al._ (2018); Li  _et al._
(2017) With controlling composition of InAlN with In-content $\sim 0.17$,
strain-less multilayer InAlN/GaN-distributed Bragg reflectors (DBRs) with high
a peak reflectivity can be grown directly onto nanodevice’s structures for
fabricating vertical-cavity surface-emitting lasers (VCSELs).Ristić  _et al._
(2005); Takeuchi  _et al._ (2018)

To aid the development of nanodevices based on core-shell InAlN NRs, it is
crucial to have a theoretical tool to test different design scenarios and to
help the interpretation of the electronic properties of as-synthesized core-
shell InAlN NRs. Reliable simulation of their optical properties provides a
strategic tool for tuning the core-shell InAlN NRs to potential electronic and
optoelectronic applications. In this sense, it is desirable a methodology that
accurately describes the excitations of such nanostructures across a wide
energy range, especially around the bandgap-energy region. The solution to
this problem is given by the $G_{0}W_{0}$ approximation within many-body
perturbation theory, which is considered the state of the art in _ab initio_
calculations of single-particle excitations.Aryasetiawan and Gunnarsson
(1998); Onida, Reining, and Rubio (2002); Bechstedt (2015); Golze, Dvorak, and
Rinke (2019); Reining (2018) It can provide accurate quasi-particle
corrections to (generalized) Kohn-Sham eigenvalues, yielding electronic
structures in excellent agreement with experiments and with higher-order
methods.Nabok, Gulans, and Draxl (2016); Bechstedt (2015); Govoni and Galli
(2015); van Setten  _et al._ (2015); Krause, Harding, and Klopper (2015)
However, a major drawback of $G_{0}W_{0}$ is its high computational cost,
which can complicate its application to complex systems with hundreds or
thousands of atoms.Pela, Marques, and Teles (2015)

For this purpose, it is interesting to find approaches based on DFT that can
reproduce $G_{0}W_{0}$ results, with reasonable accuracy, but much less
computationally involved. Among the various possibilities, in this paper, we
explore two: LDA-1/2 and the modified Becke-Johnson (mBJ) functional.

The LDA-1/2 approach has proven to be an efficient alternative for obtaining
approximate quasi-particle corrections at low computational cost. Ferreira,
Marques, and Teles (2008, 2011); Pelá  _et al._ (2012); Pela  _et al._ (2016);
Pela, Gulans, and Draxl (2018); Matusalem  _et al._ (2013); Pelá  _et al._
(2011); Pela, Marques, and Teles (2015) In particular, electronic properties
of systems based on III-V semiconductors are well described by LDA-1/2.Pelá
_et al._ (2011); Pela, Marques, and Teles (2015); Filho  _et al._ (2013);
Santos  _et al._ (2012) For this class of materials, LDA-1/2 also provides
accurate one-particle energies and wavefunctions to solve the Bethe-Salpeter
equation and obtain optical properties.Matusalem  _et al._ (2020) Regarding
nanowires, LDA-1/2 calculations for Si, GaN, and GaP have been shown to
describe the band gap with an accuracy comparable to $G_{0}W_{0}$Ribeiro
(2015); Huang  _et al._ (2015) and in good agreement with experiments.Greil
_et al._ (2016) These facts make LDA-1/2 an attractive _ab initio_ framework
to study core-shell InAlN NRs. To what extent this is possible, however, has
not yet been addressed.

Another promising choice is the mBJ potential,Tran  _et al._ (2019); Tran and
Blaha (2009) a semilocal meta-GGA functional shown to be quite accurate for
band gap calculations. It is competitive with $G_{0}W_{0}$ and hybrid
functionals in terms of accuracy, at much lower computational cost.Koller,
Tran, and Blaha (2011); Kim  _et al._ (2010); Tran  _et al._ (2019); Jiang
(2013); Lee  _et al._ (2016) Interestingly, band gaps of III-V semiconductors
calculated with mBJ show good agreement with experiments.Wang  _et al._
(2013); Kim  _et al._ (2010) Apart from band gaps, optical properties of
several materials have been obtained with mBJ,Li and Singh (2017); Ibarra-
Hernández  _et al._ (2017); Rödl  _et al._ (2019); Singh (2010); Ondračka  _et
al._ (2017); Nakano and Sakai (2018) including III-V semiconductors,Rehman
_et al._ (2016); Nakano and Sakai (2018) and mBJ at least improves over PBE
when compared with experiment.Nakano and Sakai (2018) Studies employing mBJ
for nanowires have been conducted as well,Radzwan  _et al._ (2019); Xiang and
Yang (2017); Park  _et al._ (2022); Validžić  _et al._ (2014) some of which
have reported nice agreement with measurements.Park  _et al._ (2022); Validžić
_et al._ (2014) It it, thus, important to verify how mBJ performs for studying
core-shell InAlN NRs.

In this work, for the case of core-shell InAlN NRs, we conduct _ab initio_
calculations to analyze how LDA-1/2 and mBJ improve over LDA and how they can
approximate $G_{0}W_{0}$ for the following electronic and optical properties:
density of states (DOS), band gaps, dielectric function, refraction index,
extinction and absorption coefficients, and the reflectivity. Nanostructures
of similar structural and chemical complexity and their electronic and optical
properties including in relation to electronic applications have been
successfully studied previously by using both different flavors of GGA to DFT
levels of theory Gueorguiev, Stafström, and Hultman (2008), and the
$G_{0}W_{0}$ method. de Oliveira  _et al._ (2021) Here, to keep the
computational cost moderate in the $G_{0}W_{0}$ calculations, we select as
prototypes NRs with diameter of 14 Å and with In compositions of $0$, $12.5$,
and $25$% within their core.

The paper is divided as follows: in section II, we introduce the theoretical
aspects of this work; section III describes the computational methods
employed; in section IV, we present and discuss our results; and lastly, in
section V, we summarize the paper.

##  II Theoretical Framework

###  II.1 The LDA-1/2 method

LDA-1/2Ferreira, Marques, and Teles (2008, 2011); Mao  _et al._ (2022) is
inspired on Slater’s half-occupation scheme, which relates the ionization
potential $I$ of a KS eigenstate labeled with $i$ at its eigenvalue $E_{i}$:

$I=-E_{i}(f_{i}=1/2),$ (1)

where $f_{i}$ is the occupation of the KS state $i$.

In LDA-1/2, instead of dealing with half-occupations, KS equations are
modified as:

$\left[-\frac{1}{2}\nabla^{2}+V_{H}(\mathbf{r})+V_{XC}(\mathbf{r})+V_{S}(\mathbf{r})\right]\phi_{i\mathbf{k}}(\mathbf{r})=E_{i\mathbf{k}}\phi_{i\mathbf{k}}(\mathbf{r}).$
(2)

Here, we consider electrons in a solid with wavevector given by $\mathbf{k}$.
$\phi_{i\mathbf{k}}$ is the corresponding KS wavefunction. The KS potential,
$V_{KS}(\mathbf{r})=V_{H}(\mathbf{r})+V_{XC}(\mathbf{r})$, written as the sum
of the Hartree, $V_{H}(\mathbf{r})$, and the exchange-correlation (XC),
$V_{XC}(\mathbf{r})$, potentials has been adjusted to include
$V_{S}(\mathbf{r})$, the so-called self-energy potential.Ferreira, Marques,
and Teles (2008) The XC potential employed here is LDA.Kohn and Sham (1965)
For each atom in the solid, $V_{S}(\mathbf{r})$ is obtained from two
calculations with the isolated atom as

$V_{S}(\mathbf{r})=\Theta(\mathbf{r})[V_{KS,atom}(\mathbf{r})_{f_{i}=1/2}-V_{KS,atom}(\mathbf{r})_{f_{i}=1}],$
(3)

in which, we add an extra label $f_{i}$ to $V_{KS}$ to denote the occupation.
$\Theta(\mathbf{r})$ is a trimming function to avoid the divergence due to the
tail $1/(2r)$ coming from the difference of the two KS potentials in (3).
Historically, $\Theta(\mathbf{r})$ has been chosen as

$\Theta(\mathbf{r})=\left\\{\begin{array}[]{cr}\left[1-\left(\frac{r}{R_{CUT}}\right)^{8}\right]^{3},&r\leq
R_{CUT},\\\ 0,&r>R_{CUT},\end{array}\right.$ (4)

where $R_{CUT}$ is the cutoff radius, which is determined
variationallyFerreira, Marques, and Teles (2008) and has proven to be
transferable among different systems.Ferreira, Marques, and Teles (2011)

###  II.2 mBJ

The mBJ potential keeps the correlation potential the same as in LDA and
replaces the exchange potential with:Tran and Blaha (2009); Tran  _et al._
(2019)

$v_{x,\sigma}^{mBJ}(\mathrm{r})=cv_{x,\sigma}^{BR}(\mathrm{r})+(3c-2)\frac{1}{\pi}\sqrt{\frac{5}{6}}\sqrt{\frac{t_{\sigma}(\mathbf{r})}{\rho_{\sigma}(\mathbf{r})}},$
(5)

where $\rho_{\sigma}(\mathbf{r})$ is the density of electrons with spin
$\sigma$, $t_{\sigma}(\mathbf{r})$ is the corresponding kinetic-energy
density, and $v_{x,\sigma}^{BR}(\mathrm{r})$ is the Becke-Roussel
potentialBecke and Roussel (1989). The factor $c$ in (5) is evaluated asTran
and Blaha (2009)

$c=\alpha+\beta\sqrt{\frac{1}{2\Omega}\int_{\Omega}\mathrm{d}\mathbf{r}\left[\frac{|\nabla\rho_{\uparrow}(\mathbf{r})|}{\rho_{\uparrow}(\mathbf{r})}+\frac{|\nabla\rho_{\downarrow}(\mathbf{r})|}{\rho_{\downarrow}(\mathbf{r})}\right]},$
(6)

where $\alpha=-0.012$ and $\beta=1.023$ bohr1/2, and $\Omega$ is the volume of
a unit cell.

###  II.3 $G_{0}W_{0}$ approach

Taking KS eigenvalues and wavefunctions as reference, quasiparticle-corrected
eigenvalues $E_{i\mathbf{k}}^{QP}$ can be calculated in the $G_{0}W_{0}$
approximation as:Rangel  _et al._ (2020); Reining (2018); Golze, Dvorak, and
Rinke (2019); Onida, Reining, and Rubio (2002)

$E_{i\mathbf{k}}^{QP}=E_{i\mathbf{k}}+Z_{i\mathbf{k}}\\{\mathrm{Re}[\Sigma_{i\mathbf{k}}(E_{i\mathbf{k}})]-V_{XC,i\mathbf{k}}\\},$
(7)

where $Z_{i\mathbf{k}}$ is the quasiparticle renormalization factor, and
$\Sigma_{i\mathbf{k}}(\omega)$ and $V_{XC,i\mathbf{k}}$ are matrix elements of
the self-energy ($\Sigma(\mathbf{r},\mathbf{r}^{\prime},\omega)$) and the
exchange-correlation potential:

$\Sigma_{i\mathbf{k}}(\omega)=\int\mathrm{d}\mathbf{r}\mathrm{d}\mathbf{r}^{\prime}\phi^{*}_{i\mathbf{k}}(\mathbf{r})\Sigma(\mathbf{r},\mathbf{r}^{\prime},\omega)\phi_{i\mathbf{k}}(\mathbf{r}^{\prime}),$
(8)
$V_{XC,i\mathbf{k}}=\int\mathrm{d}\mathbf{r}V_{XC}(\mathbf{r})|\phi_{i\mathbf{k}}(\mathbf{r})|^{2}.$
(9)

Within the $G_{0}W_{0}$ approximation, the self-energy
$\Sigma(\mathbf{r},\mathbf{r}^{\prime},\omega)$ is given, in the time domain,
as a product of the imaginary number, the single particle Green’s function,
$G_{0}(\mathbf{r},\mathbf{r}^{\prime},t)$, and the screened Coulomb
interaction, $W_{0}(\mathbf{r},\mathbf{r}^{\prime},t)$, evaluated in the
random-phase approximation.Reining (2018); Freysoldt  _et al._ (2007)

###  II.4 Optical properties

Neglecting excitonic effects and considering an electric field applied along
the $\hat{\mathbf{e}}_{\alpha}$ direction, the tensorial component
$\alpha\alpha$ of the dielectric function $\varepsilon$, at a given frequency
$\omega$, has an imaginary part given byBassani and Pastori Parravicini
(1975):

$\mathrm{Im}[\varepsilon_{\alpha\alpha}(\omega)]=\frac{8\pi^{2}}{\Omega
N_{\mathbf{k}}}\sum_{cv\mathbf{k}}\frac{|\langle\phi_{c\mathbf{k}}|-\mathrm{i}\hat{\mathbf{e}}_{\alpha}\cdot\mathbf{\nabla}|\phi_{v\mathbf{k}}\rangle|^{2}}{\omega^{2}}\delta(\omega-\omega_{cv\mathbf{k}}),$
(10)

where $N_{\mathbf{k}}$ is the number of k-points, $c$ and $v$ are labels for
the conduction and valence states, respectively, and $\phi_{c\mathbf{k}}$ and
$\phi_{v\mathbf{k}}$ are the corresponding KS wavefunctions. The transition
energies, $\omega_{cv\mathbf{k}}$, are expressed in terms of the KS
eigenvalues as:

$\omega_{cv\mathbf{k}}=E_{c\mathbf{k}}-E_{v\mathbf{k}}.$ (11)

If the imaginary part is known, the real part can be obtained using the
Kramers-Kronig relationsYu and Cardona (2010):

$\mathrm{Re}[\varepsilon_{\alpha\alpha}(\omega)]=1+\frac{2}{\pi}\int_{0}^{\infty}\mathrm{d}\omega^{\prime}\frac{\omega^{\prime}\mathrm{Im}[\varepsilon_{\alpha\alpha}(\omega^{\prime})]}{\omega^{\prime
2}-\omega^{2}}.$ (12)

With $\epsilon_{\alpha\alpha}(\omega)$, it is possible to obtain other optical
properties, such as the refraction index $\tilde{n}$, the extinction
coefficient $\kappa$, the optical absorption $\mathcal{A}$ and the
reflectivity $\mathcal{R}$Dresselhaus  _et al._ (2018):

$\tilde{n}(\omega)=\sqrt{\frac{|\varepsilon(\omega)|+\mathrm{Re}[\varepsilon(\omega)]}{2}},\quad\kappa(\omega)=\sqrt{\frac{|\varepsilon(\omega)|-\mathrm{Re}[\varepsilon(\omega)]}{2}},$
(13)
$\mathcal{A}(\omega)=\frac{2\omega\kappa}{v_{light}},\quad\mathcal{R}(\omega)=\frac{(1-\tilde{n})^{2}+\kappa^{2}}{(1+\tilde{n})^{2}+\kappa^{2}},$
(14)

where $v_{light}$ is the light speed in vacuum. For simplicity, we dropped
down the double indexes $\alpha\alpha$ in Eqs. (13) and (14).

##  III Computational methods

In all DFT calculations, we employ the Quantum Espresso codeGiannozzi  _et
al._ (2009, 2017, 2020) with optimized norm-conserving Vanderbilt
pseudopotentialsHamann (2013) and a planewave cutoff of 100 Ry. For the
$G_{0}W_{0}$ calculations, we make use of BerkeleyGWDeslippe  _et al._ (2012);
Hybertsen and Louie (1986), taking LDA as the starting-point. We take
advantage of the static remainder approachDeslippe  _et al._ (2013) to speed
up convergence with respect to the unoccupied states. To reduce the
computational cost, we use the plasmon-pole approximation.Zhang  _et al._
(1989); Hybertsen and Louie (1986)

We start the study with bulk AlN and InN in the wurtzite phase. We employ the
experimental lattice parametersVurgaftman and Meyer (2003), relaxing the ions
positions with LDA. Then, the same relaxed geometry is used for all other
methods. We use a k-grid of $16\times 16\times 10$ for LDA, LDA-1/2 and mBJ.
For the $G_{0}W_{0}$ calculations, we consider an extrapolation scheme, as
described in Appendix A.1: we use k-grids of $4\times 4\times 3$ and $8\times
8\times 6$, and vary the cutoff for the dielectric function from 30 to 60 Ry
in steps of 10 Ry, and the number of KS states from 100 to 450 in steps of 50.

Then, we proceed to the core-shell InAlN NRs. We take as diameter $d=14$ Å, as
illustrated in Fig. 1. Even though much larger cells may be required to study
realistic NRs,Filho  _et al._ (2023) our goal here is to evaluate the accuracy
of LDA-1/2 and mBJ in approximating $G_{0}W_{0}$ for these systems. Keeping
the computational cost of $G_{0}W_{0}$ in mind, we selected these NRs with a
relatively small diameter as prototypes for our benchmark. We rationalize that
with these NRs, one is still able to draw meaningful conclusions. Then further
studies can then profit from our analysis and employ LDA-1/2 or mBJ to
investigate NRs with more realistic sizes.

To avoid dangling bonds which lead to spurious states at the Fermi energy, we
use H passivation, and, so, the chemical formula of the NR becomes
InnAl38-nN38H40. We study three different In concentrations: $n=0$, 2, and 4.
In all cases, we consider an unrelaxed geometry with bond lengths determined
from the AlN experimental lattice parameters.Vurgaftman and Meyer (2003) In
Fig. 1, we also show a possible split between core and shell regions, leading
to the compositions InnAl16-nN16 for the core, and Al22N22H40 for the shell.
According to this choice, the cases $n=2$ and 4 correspond to In compositions
of 12.5 and 25 % in the core. To isolate neighboring NRs, we employ a
supercell with dimensions 44 Bohr $\times$ 31.1 Bohr ($23.3$ Å$\times$ 20.2
Å).

Figure 1: Cell used to accommodate the passivated core-shell InAlN NRs
(InnAl38-nN38H40). The shown diameter, $d$, is 14 Å. We employ $L=44$ Bohr
(23.3 Å). We depict here the case of an AlN NR ($n=0$). In the case $n=2$, Al
atoms at sites labeled with 1 and 2 are replaced by In atoms. In the case
$n=4$, then all 4 labeled sites are replaced by In atoms.

For $G_{0}W_{0}$, we employ a k-grid of $1\times 1\times 16$ to obtain the
reference density with LDA, and then $1\times 1\times 6$ to generate the
reference KS wavefunctions and eigenvalues. To enable a fair comparison, the
same procedure is adopted for LDA, LDA-1/2 and mBJ. For the DOS and the
optical properties, we take for all methods 300, 320, and 400 KS states into
account, which is sufficient to cover transitions in the energy range $0-20$
eV. In the $G_{0}W_{0}$ calculations, we include 900, 920, 940 bands in the
summation used to build the dielectric function, with a cutoff of 20 Ry. To
speed up the $G_{0}W_{0}$ convergence with respect to the vacuum size, we
employ a Coulomb truncation for nanowires.Ismail-Beigi (2006)

##  IV Results

###  IV.1 Binaries: AlN and InN

The purely binary compositions, AlN and InN, can be seen as benchmarks in
relation to the ternary InAlN compounds and their properties are of relevance
to the present first-principles comparative study of the InAlN core-shell NRs
study.

####  IV.1.1 DOS and band gaps

In Table 1, the calculated band gaps of AlN and InN are compared with the
experimental ones. As usual, LDA band gaps are underestimated for both AlN and
InN. With LDA-1/2, although the band gap of InN is overestimated by 0.60 eV,
the band gap of AlN agrees with the experimental with an error of 0.04 eV.
With mBJ, in contrast, the band gap of InN deviates from experiment by 0.22
eV, while this error is 0.55 eV for AlN. Band gaps obtained with $G_{0}W_{0}$
agree with experiment with an error of 0.04 eV for AlN, and of 0.50 eV for
InN. Overall, there is a similar degree of agreement with experiment for
LDA-1/2, mBJ and $G_{0}W_{0}$.

Table 1: Calculated band gaps, compared with experimental gaps taken from Ref. Vurgaftman and Meyer, 2003. | AlN | InN
---|---|---
$G_{0}W_{0}$ | 6.29 | 0.28
LDA | 4.24 | $-0.23$
LDA-1/2 | 6.21 | 1.38
mBJ | 5.70 | 0.56
exp. | 6.25 | 0.78

Figure 2 depicts the DOS of AlN and InN.

Figure 2: DOS of AlN and InN. States belonging to valence and conduction bands
are plotted on the left and on the right, respectively. In each case, band
edges are placed at zero.

For an easier comparison of the approaches, we plot the DOS of valence and
conduction bands separately, on the left and on the right, respectively,
placing in each case the band edges at zero. It is apparent then that LDA and
$G_{0}W_{0}$ have the best agreement, confirming for AlN and InN the common
belief that $G_{0}W_{0}$ approximately shifts states rigidly. DOS obtained
with LDA-1/2 and mBJ agree well with each other and are also very close to
$G_{0}W_{0}$.

####  IV.1.2 Dielectric function

We present the $xx$ component of dielectric function in Fig. 3.

Figure 3: $xx$ component of the dielectric function: left, the real part, and
right, the imaginary part.

For AlN, the dielectric functions computed with LDA, LDA-1/2 and mBJ are red-
shifted when compared to $G_{0}W_{0}$, with LDA showing the largest deviation,
and LDA-1/2 presenting a slightly better agreement than mBJ. For InN, the
negative gap obtained with LDA causes a qualitative wrong behavior of
$\varepsilon$ for small frequencies. LDA-1/2 and mBJ show similar results,
with LDA-1/2 closer to $G_{0}W_{0}$.

###  IV.2 Core-shell InAlN NRs

####  IV.2.1 DOS

Fig. 4 displays the DOS of core-shell InAlN NRs passivated with hydrogen.

Figure 4: DOS of passivated core-shell InAlN NRs for different In
compositions. The dashed line on the top panel shows how band edges have been
evaluated to obtain band gaps shown in Table 2.

In each case, the zero energy has been defined as follows:

  1. 1.

projecting the DOS onto core atoms;

  2. 2.

identifying the valence state with the highest energy $E_{v}$;

  3. 3.

taking $E_{v}$ as reference and referring all other energies with respect to
it.

The identification of $E_{v}$ is illustrated for $G_{0}W_{0}$ DOS in the top
panel of Fig. 4 with the dashed line on the left. Similarly, we can define
$E_{c}$ by projecting the DOS onto core atoms, and taking it as the energy of
the conduction band edge. This is shown for $G_{0}W_{0}$ as the dashed line on
the right in Fig. 4 (subplot on the top).

The isolated peaks observed for energies between 0-5 eV come from states
belonging to shell atoms. The agreement between LDA, LDA-1/2, and mBJ with
$G_{0}W_{0}$ for valence states with $E<0$ is evident for the 3 NRs. For the
conduction states with $E>E_{c}$, LDA-1/2 and mBJ match $G_{0}W_{0}$ better
than LDA. It is also apparent that, the peaks in the energy range 0-5 eV are
more pronounced in LDA-1/2 than in other methods.

The definition of $E_{v}$ and $E_{c}$ allows us to compute $\Delta E$ as

$\Delta E=E_{c}-E_{v}.$ (15)

$\Delta E$ can be identified as a kind of band gap for the core region of the
NR, since it is obtained from band edges of states that belong to core atoms.

Table 2 presents $\Delta E$ for each NR. The best agreement with $G_{0}W_{0}$
is given by LDA-1/2, with $\Delta E$ approximately 1.2 eV smaller.

Table 2: $\Delta E$ of core-shell InnAl38-nN38H40, obtained from band edges of DOS projected onto atoms in the NR core. |  | $n$ |
---|---|---|---
| 0 | 2 | 4
$G_{0}W_{0}$ | 8.44 | 8.38 | 8.14
LDA | 4.69 | 4.76 | 4.75
LDA-1/2 | 7.26 | 7.19 | 7.00
mBJ | 6.66 | 6.71 | 6.55

mBJ comes next, predicting $\Delta E$ 1.6-1.8 eV smaller than $G_{0}W_{0}$.
LDA is the last one with $\Delta E$ 3.4-3.7 eV smaller than $G_{0}W_{0}$.

When compared to bulk AlN, the NRs are expected to have larger $\Delta E$ due
to quantum confinement effects. Indeed, the passivated AlN NR has $\Delta E$
larger than bulk AlN by 2.15, 0.45, 1.05 and 0.96 and eV, when calculated with
$G_{0}W_{0}$, LDA, LDA-1/2 and mBJ respectively. As estimated in Appendix B,
an enlargement of 1.9-2.5 eV is expected due to quantum confinement effects.
$G_{0}W_{0}$ best matches this expectation, followed by LDA-1/2, mBJ and LDA.

####  IV.2.2 Optical properties

Figure 5 displays the $xx$ component of the dielectric function.

Figure 5: $xx$ component of the dielectric function, with its real and
imaginary parts. The label $n$ refers to the amount of In atoms in the NR cell
according to InnAl38-nN38H40. For NRs with $n>0$, the arrows point peaks
coming from In contributions.

LDA-1/2 and mBJ agree well with each other and are red-shifted by 1.8 and 2.0
eV, respectively, in comparison with $G_{0}W_{0}$. For LDA, this amounts to
3.5 eV. By blue-shifting all Im$[\varepsilon_{xx}]$, a good agreement with
$G_{0}W_{0}$ can be observed, as shown in Appendix C. Although these shifts do
not reproduce exactly the differences in $\Delta E$, they are comparable.

Next, we consider the contribution of In atoms present in the core region to
the dielectric function of the NRs. In Fig. 5, it is evident that the presence
of In introduces peaks in $\mathrm{Im}[\varepsilon_{xx}]$, which are red-
shifted in respect to the main peak observed for AlN NRs without In. These
peaks, highlighted with arrows in Fig. 5, become evident when we
$\mathrm{Im}[\varepsilon_{xx}]$ for NRs with In and by
$\mathrm{Im}[\varepsilon_{xx}]$ for AlN NRs (not shown here). Table 3 shows
the positions of these peaks due to In.

Table 3: Peak positions due to In atoms in core-shell InnAl38-nN38H40 NRs. These peaks are highlighted in Fig. 5 with arrows.  | $n$
---|---
| 2 | 4
$G_{0}W_{0}$ | 7.70 | 6.96
LDA | 4.64 | 4.28
LDA-1/2 | 6.37 | 5.77
mBJ | 5.69 | 4.97

Peaks within $G_{0}W_{0}$ appear blue-shifted in comparison with other
methods. Peaks obtained with LDA-1/2 exhibit best agreement with $G_{0}W_{0}$,
with a difference of 1.2-1.3 eV. These numbers are 2.0 and 3.1-3.4 eV for mBJ
and LDA, respectively. Also interesting is that the red-shift of the peaks
observed by increasing $n=2$ to $n=4$ is approximately the same in
$G_{0}W_{0}$ (0.74 eV), LDA-1/2 (0.60 eV) and mBJ (0.72 eV).

In Fig. 6, we depict the refraction index $\tilde{n}$ and the extinction
coefficient $\kappa$ for the energy range of 0 to 20 eV.

Figure 6: Refraction index $\tilde{n}$ and extinction coefficient $\kappa$.

The similarity between LDA-1/2 and mBJ is apparent. Regarding the refraction
index, for the energy range 10-20 eV, LDA-1/2 and mBJ show an excellent
agreement with $G_{0}W_{0}$. Although this statement does not hold for the
extinction coefficient, there is a notable improvement over LDA: LDA-1/2 and
mBJ approximate $G_{0}W_{0}$ better than LDA.

Table 4 presents the static refractive index. The best agreement with respect
to $G_{0}W_{0}$ is observed for mBJ (difference of 12-13%), closely followed
by LDA-1/2 (14%) and, then, by LDA (26-27%).

Table 4: Refractive index $\tilde{n}$ at zero frequency of core-shell InnAl38-nN38H40 NRs.  | $n$
---|---
| 0 | 2 | 4
$G_{0}W_{0}$ | 1.59 | 1.61 | 1.63
LDA | 2.38 | 2.40 | 2.43
LDA-1/2 | 2.03 | 2.05 | 2.07
mBJ | 1.97 | 1.99 | 2.02

Figure 7 depicts the absorbance and the reflectance of the NRs. The curves for
LDA-1/2 and mBJ are very similar, and both present a better agreement with
$G_{0}W_{0}$ than LDA.

Figure 7: Absorption $\mathcal{A}$ and reflectance $\mathcal{R}$.

##  V Conclusion

We have studied electronic and optical properties of core-shell InAlN NRs with
LDA, LDA-1/2, mBJ and $G_{0}W_{0}$. For the properties, DOS, dielectric
function, refractive index, extinction coefficient, absorption coefficient,
and reflectance: results with LDA-1/2 and mBJ are similar and agree better
with $G_{0}W_{0}$ than those obtained with LDA. For band gaps and peaks in
Im$[\varepsilon]$ coming from In contributions, LDA-1/2 agrees better with
$G_{0}W_{0}$ than mBJ. Overall, LDA-1/2 and mBJ can be used as tools to
replace $G_{0}W_{0}$ with reasonable accuracy at much less computational cost.

The authors have no conflicts to disclose. The data that support the findings
of this study are available from the corresponding author upon reasonable
request.

###### Acknowledgements.

The authors gratefully acknowledge the computing time granted by the Resource
Allocation Board and provided on the supercomputer Lise and Emmy at NHR@ZIB
and NHR@Göttingen as part of the NHR infrastructure. They also acknowledge
resources provided by the National Academic Infrastructure for Supercomputing
in Sweden (NAISS) at the National Supercomputer Center (NSC) in Linköping
(NAISS 2023/5-116 and NAISS 2023/23-161) partially funded by the Swedish
Research Council through grant agreement no. 2018-05973. G. K. G., J. B., and
L. H. acknowledge support by the Swedish Government Strategic Research Area in
Materials Science on Advanced Functional Materials (AFM) at Linköping
University (Faculty Grant SFO-Mat-LiU No. 2009-00971). C.-L.H. acknowledges
support by the Swedish Research Council (Vetenskapsrådet) through grant number
2018-04198 and the Swedish Energy Agency (Energimyndigheten) through grant
number 46658-1.

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## Appendixes

##  Appendix A Convergence behavior

###  A.1 Bulk AlN and InN

Figures 8 and 9 show, for bulk AlN and InN, the convergence behavior of the
$G_{0}W_{0}$ band gap. Two parameters are varied: the number of KS states
($N_{bands}$) and the planewave cutoff ($E_{cut}$) used to build the
dielectric function.

Figure 8: AlN band gap obtained with $G_{0}W_{0}$: convergence behavior with
respect to the number of bands (left) and the planewave cutoff for the
dielectric function (right). Figure 9: Same as Fig. 8 for InN.

The convergence of $G_{0}W_{0}$ band gap is slower than that of DFT. Having a
fully converged band gap is challenging, as employing a set of fully converged
parameters comes with a very high computational cost. Therefore, we follow
here an extrapolation procedure to evaluate the $G_{0}W_{0}$ band gap,
adopting the expressionKlimeš, Kaltak, and Kresse (2014); Nabok, Gulans, and
Draxl (2016); Pela  _et al._ (2016):

$E_{g}(x)=E_{g}(\infty)+\frac{A}{x+B},$ (16)

where $E_{g}(\infty)$, $A$ and $B$ are fit coefficients, and $x$ is the
convergence parameter being tested, which can be $N_{bands}$ or $E_{cut}$.
Assuming that the extrapolation with respect to $E_{cut}$, $N_{bands}$ and the
k-grid can be carried out separately, the extrapolated band gap $E_{g}^{extr}$
can be evaluated as

$E_{g}^{extr}=E_{g}^{ref}+\delta_{kpt}+\delta_{bands}+\delta_{cut},$ (17)

where $E_{g}^{ref}=E_{g}(N_{bands}^{ref},E_{cut}^{ref},k_{grid}^{ref})$ is the
$G_{0}W_{0}$ band gap for a reference calculation employing $N_{bands}^{ref}$,
$E_{cut}^{ref}$, and $k_{grid}^{ref}$. We adopt for AlN:
$N_{bands}^{ref}=200$, $E_{cut}^{ref}=40$ Ry; and for InN:
$N_{bands}^{ref}=400$, $E_{cut}^{ref}=50$ Ry; and, in both cases,
$k_{grid}^{ref}$ as $4\times 4\times 3$.

Then $\delta_{cut}$ and $\delta_{bands}$ are obtained as

$\delta_{bands}=E_{g}(N_{bands}=\infty,E_{cut}^{ref},k_{grid}^{ref})-E_{g}^{ref},$
(18)
$\delta_{cut}=E_{g}(N_{bands}^{ref},E_{cut}=\infty,k_{grid}^{ref})-E_{g}^{ref},$
(19)

and the contribution of the k-grid to the extrapolation is approximated
asKlimeš, Kaltak, and Kresse (2014); Nabok, Gulans, and Draxl (2016); Pela
_et al._ (2016):

$\delta_{kpt}=E_{g}(N_{bands}^{ref},E_{cut}^{ref},k_{grid}^{large})-E_{g}(N_{bands}^{ref},E_{cut}^{ref},k_{grid}^{ref}),$
(20)

where $8\times 8\times 6$ is employed as $k_{grid}^{large}$.

Table 5 collects the relevant data regarding the extrapolation of the
$G_{0}W_{0}$ band gap.

Table 5: Contributions to the extrapolated $G_{0}W_{0}$ band gap. All quantities are given in eV. | $E_{g}^{ref}$ | $\delta_{bands}$ | $\delta_{cut}$ | $\delta_{kpt}$ | $E_{g}^{extr}$
---|---|---|---|---|---
AlN | 6.28 | 0.03 | 0.01 | -0.03 | 6.29
InN | 0.09 | 0.02 | 0.03 | 0.14 | 0.28

###  A.2 Nanorods

####  A.2.1 Vacuum

In Eq. (10), $\Omega$ stands for the volume of the sample. However, in most of
_ab initio_ codes, including Quantum Espresso, $\Omega$ is treated as the unit
cell volume. For systems with vacuum in the unit cell, such as the NRs studied
here, the dielectric function must be corrected by a factor $h$ equal to the
ratio between the volumes of the cell and the sample. In terms of the geometry
shown in Fig. 1, we have for the NRs

$h=\frac{A_{cell}}{A_{nanorod}}=\frac{4L^{2}}{3d^{2}},$ (21)

where $A_{cell}$ and $A_{nanorod}$ are the cross-sections of the unit cell and
the NR, respectively.

Denoting $\varepsilon^{\prime}_{xx}$ as the dielectric function without the
correction and $\varepsilon_{xx}$, the corrected one, then according to Eqs.
(10) and (12), it follows that:

$\mathrm{Im[\varepsilon_{xx}]}=h\mathrm{Im[\varepsilon^{\prime}_{xx}]},$ (22)
$\mathrm{Re[\varepsilon_{xx}]}=1+h(\mathrm{Re[\varepsilon^{\prime}_{xx}]}-1).$
(23)

Figure 10 depicts the convergence behavior of the dielectric function with
respect to cell dimension $L$.

Figure 10: Convergence behavior of the AlN NR - dielectric function obtained
with LDA.

These calculations refer to the AlN NR and have been carried out with LDA, and
a similar trend has been observed for the other cases. Fig. 10 shows that the
size $L=44$ Bohr, adopted for the NRs calculations reported in the present
work, is sufficient to guarantee satisfactory convergence within the energy
range of $0-20$ eV.

####  A.2.2 Band gap

Here, we discuss the precision level expected for our $G_{0}W_{0}$
calculations concerning the NRs. As for bulk AlN and InN, we check convergence
with respect to $N_{bands}$, $E_{cut}$ and the k-grid $1\times 1\times
N_{kpt}$. For the sake of computational cost, we restrict the analysis to the
$\Gamma\Gamma$ band gap and the dielectric function of AlN.

Figure 11 depicts on the left side the impact of $N_{kpt}$ and $N_{bands}$ on
the $\Gamma\Gamma$ band gap. Since we adopted $N_{kpt}=6$ and $N_{bands}=900$,
we estimate, by making the extrapolation procedure as described in Appendix
A.1, a correction of $\delta_{bands}=-0.06$ eV and $\delta_{kpt}=-0.08$ eV.

Figure 11: Convergence behavior of $G_{0}W_{0}$ results for the AlN NR:
$\Gamma\Gamma$ band gap (left) and dielectric function (right).

On the right side of Fig. 11, we present the imaginary part of dielectric
function. Even though the band gap has a more pronounced dependence on
$N_{kpt}$, the dielectric function exhibits good convergence already for
$N_{kpt}=6$.

Figure 12 displays the influence of $N_{bands}$ and $E_{cut}$ on the
$\Gamma\Gamma$ band gap. Following the extrapolation procedure, we evaluate a
correction of $\delta_{cut}=-0.03$ eV for the adopted $E_{cut}=20$ Ry.
Overall, by adding up $\delta_{bands}$, $\delta_{kpt}$ and $\delta_{cut}$, we
expect that $G_{0}W_{0}$ band gaps are overestimated by about 0.2 eV.

Figure 12: Impact of the number of bands on $\Gamma\Gamma$ band gap of AlN NR.

##  Appendix B Quantum confinement and band gap

The expected band gap enlargement $\Delta$ due to quantum confinement effects
can be estimated by considering a particle of mass $\mu$ confined inside a
circular quantum well of radius $r$. Following Ref. Robinett, 1996, we have,
in atomic units:

$\Delta=\frac{a^{2}_{(0,0)}}{2\mu^{*}r^{2}},$ (24)

where $a_{(0,0)}=2.40483$ is the first zero of $J_{0}(x)$, Bessel’s function
of order zero. To estimate $\Delta$ for the case of the NRs, we approximate
$\mu$ by the reduced mass of the electron and the heavy hole,
$\mu=(1/m_{e}+1/m_{hh})^{-1}$. Taking the suggested values in Ref. Vurgaftman
and Meyer, 2003: $m_{e}=\sqrt[3]{(m_{e}^{\perp})^{2}m_{e}^{\parallel}}=0.31$,
in atomic units. $m_{hh}$ is obtained from the Pikus-Bir parameters given in
Ref. Vurgaftman and Meyer, 2003 asChuang and Chang (1996)

$m_{hh}=-\sqrt[3]{(A_{2}+A_{4})^{-2}(A_{1}+A_{3})^{-1}},$ (25)

which gives $m_{hh}=1.1$, in atomic units. Therefore $\mu=0.24$.

Lastly, we consider a range for $r$ between the radius of the inscribed and
the circumscribed circles in the NRs. Taking into account the NR diameter of
$14$ Å, this implies that $r$ lies between $6.06$ and $7$ Å. Applying Eq. (24)
gives the range of $1.9$-$2.5$ eV for $\Delta$.

##  Appendix C Shift in the dielectric function

Figure 13 shows, for the NRs, $\mathrm{Im}[\varepsilon_{xx}]$ of the different
DFT approaches blue-shifted by $\delta$ to better match $G_{0}W_{0}$. To plot
these curves, we shift the vertical transitions in Eq. (10), so that the
shifted dielectric function $\varepsilon^{\prime}_{xx}$ at a given frequency
$\omega$ relates to the original $\varepsilon_{xx}$ by:

$\mathrm{Im}[\varepsilon^{\prime}_{xx}(\omega)]=\mathrm{Im}[\varepsilon_{xx}(\omega-\delta)]\frac{(\omega-\delta)^{2}}{\omega^{2}}.$
(26)

It is observed that, setting $\delta$ to 3.5, 1.8 and 2.0 eV for LDA, LDA-1/2
and mBJ, respectively, leads to an excellent agreement with
$\mathrm{Im}[\varepsilon_{xx}]$ obtained with $G_{0}W_{0}$.

Figure 13: InnAl38-nN38H40 core-shell NRs: Imaginary part of the dielectric
function blue-shifted by $\delta$. LDA, LDA-1/2 and mBJ have been shifted by
3.5, 1.8 and 2.0 eV, respectively.