# Tailoring the Emission Wavelength of Color Centers in Hexagonal Boron
Nitride for Quantum Applications

Chanaprom Cholsuk<EMAIL_ADDRESS>Abbe Center of Photonics,
Institute of Applied Physics, Friedrich Schiller University Jena, 07745 Jena,
Germany  Sujin Suwanna  Optical and Quantum Physics Laboratory, Department of
Physics, Mahidol University, Bangkok 10400, Thailand  Tobias Vogl
<EMAIL_ADDRESS>Abbe Center of Photonics, Institute of Applied
Physics, Friedrich Schiller University Jena, 07745 Jena, Germany  Fraunhofer-
Institute for Applied Optics and Precision Engineering IOF, 07745 Jena,
Germany  Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK

###### Abstract

Optical quantum technologies promise to revolutionize today’s information
processing and sensors. Crucial to many quantum applications are efficient
sources of pure single photons. For a quantum emitter to be used in such
application, or for different quantum systems to be coupled to each other, the
optical emission wavelength of the quantum emitter needs to be tailored. Here,
we use density functional theory to calculate and manipulate the transition
energy of fluorescent defects in the two-dimensional material hexagonal boron
nitride. Our calculations feature the HSE06 functional which allows us to
accurately predict the electronic band structures of 267 different defects.
Moreover, using strain-tuning we can tailor the optical transition energy of
suitable quantum emitters to match precisely that of quantum technology
applications. We therefore not only provide a guide to make emitters for a
specific application, but also have a promising pathway of tailoring quantum
emitters that can couple to other solid-state qubit systems such as color
centers in diamond.

single photons; defects; quantum emitters; density functional theory; strain
tuning; charge states

##  I Introduction

Optical quantum technologies promise groundbreaking applications with quantum
communication [1], quantum computing [2], and quantum metrology [3]. Using
single photons as flying qubits for quantum information processing has several
advantages: photons travel at the speed of light, resulting in a fast
transmission in communication systems, and they only interact weakly with the
environment, which translates in robust qubit states [4, 5, 6]. Moreover,
single quantum states can be easily manipulated with linear optics to process
the quantum information. These advantages, however, come at the expense of
difficult multi-qubit operations, as the photons do not interact directly with
each other. The controlled not (cnot) gate required for universal quantum
computing is thus not realizable with linear optics [7] and needs a nonlinear
medium as a mediator (e.g. through cross-phase modulation [8]). This necessity
can be relaxed with one-way or measurement-based quantum computing, which uses
entangled qubits as a resource and then manages to perform universal quantum
computations with single qubit gates realized with linear optics [9, 10]. The
generation of entangled states from single photons, however, requires any
single photon source (SPS) to be of high quality.
Many well-known single photon emitting systems have been investigated,
including semiconductor quantum dots [11, 12], color centers in solid-state
crystals such as diamond [13], silicon carbide [14], or indium selenide [15],
as well as trapped ions [16]. While single photons emitted from quantum dots
have near-ideal photon purity and indistinguishability [17], they require low
temperatures for operation. The widely investigated NV- center in diamond
suffers from a low Debye-Waller (DW) factor below 4% at room temperature [18]
which limits its optical coherence due to the strong phonon sideband. Other
color centers in diamond such as SiV- have a much higher DW factor of up to
70%, but only a relatively low quantum efficiency of 3.5% [19]. The SiV-
centers in SiC host crystals also have high quantum efficiencies around 70%
but medium DW factors around 33% [14]. This fuels the search for novel quantum
emitters with both high quantum efficiencies and DW factors at room
temperature.
The recently discovered color centers hosted by two-dimensional (2D) hexagonal
boron nitride (hBN) [20] have demonstrated DW factors as high as 82.4% [21]
and quantum efficiencies of 87% at room temperature [22], alongside bright and
pure single photon emission. Due to the 2D geometry of the host crystals, it
is easily possible to integrate the emitters with waveguides and fiber
networks [23, 24]. In addition, the robustness of the emitters, their long-
term stability, and a fast radiative decay lifetime allowing high repetition
rates [21, 25], has led to the general understanding that these emitters can
be used in practical quantum information processing applications. In fact,
single photons emitted from hBN have been used for quantum random number
generation [26, 27] and single photon interferometry [28]. Their linewidth at
room temperature, however, still needs substantial improvement in order to be
used for optical quantum computing [29].
Around the nature of the emitters, there has been a large debate among the
community, due to the large distribution of zero-phonon lines (ZPLs) [30, 31].
Recent works have correlated experimentally observed emission spectra with
theoretically calculated defect complexes [32]. For instance, the emitter with
a zero-phonon line around 2 eV was assigned to the (2)3B1 to (1)3B1 transition
of the V${}_{\text{N}}$C${}_{\text{B}}$ defect [33]. Likewise, the
C${}_{\text{B}}$C${}_{\text{N}}$ complex was shown to be responsible for the 4
eV emission via calculations of the coupling to the vibrational degrees of
freedom [34] and via delta self-consisted field ($\Delta$SCF) calculations
[35, 36]. The influence of carbon in the emitter formation was also shown
experimentally [37]. Besides carbon-based defects, oxygen complexes such as
O${}_{\text{B}}$V${}_{\text{N}}$O${}_{\text{B}}$ [38] and the negatively
charged boron vacancy V${}_{\text{B}}^{-}$ [39] are believed to be responsible
for quantum emission in the visible spectrum. Most theoretical studies take
the approach of matching the calculated spectra to experimentally observed
data, e.g. the photoluminescence spectrum [38, 40, 32, 31, 41, 42].
In this work, we calculate a large number of defect complexes in order to
uncover their radiative transitions using spin-polarized density functional
theory (DFT) with the Heyd-Scuseria-Ernzerhof (HSE06) functional. In contrast
to previous works, we do not aim to match computational results with
experimental data, but rather to discover new defects which have radiative
transitions at important wavelengths for quantum technologies. This should
facilitate one to fabricate defects with the desired wavelength for specific
applications. Frequency conversion schemes with non-ideal efficiency are thus
not necessary. Our comprehensive study includes the electronic structure of
267 defect complexes with substitutional atoms from periodic main groups III
through VI, transition metals, and multi-defect complexes. For a selected
subset of defects, we have also investigated the effects of their charge
states (neutral and $\pm 1$). Ultimately, we develop criteria to identify
promising defects based on the transition type (radiative/non-radiative),
localization (deep/shallow), and transition energy. Based on these criteria,
we select emitters whose wavelengths are close to the important ones for
quantum technologies and apply strain-tuning to tailor their emission
wavelengths. As exemplary wavelengths, we chose here the resonance wavelengths
of solid-state quantum emitters and qubits in diamond and silicon carbide,
typical quantum memory wavelengths in rare-earth ion doped crystals and alkali
vapors, as well as low-loss telecom wavelengths for long-distance quantum
communication. We therefore provide a comprehensive list of promising quantum
emitters in hBN and a mechanism to tailor them to enable potential
applications in quantum technologies.

##  II Computational details

###  II.1 Density Functional Theory

All spin-polarized DFT calculations have been performed using QuantumATK
(version S-2021.06) [43], which utilizes numerical linear combination of
atomic orbitals (LCAO) double $\zeta$ polarized basis sets [38]. The used
pseudopotentials and basis sets (which influence the accuracy of our DFT
calculations) are internally constructed by our DFT calculator from a
frequently benchmarked and reviewed database for all elements to provide the
most accurate results. Point-like defects were created in the center of a
$7\times 7\times 1$ supercell (see Figure 1), which was proven to be
sufficiently large to exclude any defect-defect interaction of neighboring
cells. For the lattice structural optimization, only internal coordinates were
allowed to relax using a $5\times 5\times 1$ Monkhorst–Pack reciprocal space
grid [44] until all forces were below 0.01 eV$\cdot$Å-1 and the total energy
convergence reached $10^{-4}$ eV. It is noteworthy that all supercell
calculations were initially optimized from the relaxed unit cell to save
computational time. A vacuum layer of 15 Å was added to minimize the van der
Waals interaction between layers. Since this work mainly focuses on the
electronic band structure and optical transitions, we employed the HSE06
functional [45] to avoid underestimating the band gap. We note that even using
the HSE06 functional, DFT has only a finite accuracy. The predicted band
structures are nevertheless quantitatively correct and agree well with
experiments, as a recent study comparing different DFT functionals to
calculate quantum emitters in hBN has pointed out [41, 46]. Moreover, HSE06
has demonstrated very good agreement with experiments (and in particular
better than functionals in the generalized gradient approximation) [46]. To
capture all possible states, a dense $11\times 11\times 1$ $k$-point centered
at the $\Gamma$ point was implemented.

###  II.2 Strain-tuning

In order to being able to manipulate the transition energy, bi-axial strain
was applied to the crystal structures by elongating and compressing them in
both $a$ and $b$ directions isotropically (for definition of the axial
directions see Figure 1). To create strain in the crystal lattice, the built-
in pressure function in QuantumATK has been used. The pressure is applied by a
$3\times 3$ stress tensor, where we define identical values for the $xx$ and
$yy$ components (and all other components set to zero) to simulate bi-axial
strain. The geometry is relaxed again and the applied strain $s$ can be
calculated using the initial length of the unstrained lattice parameter
$L_{0}$ and the length difference of the strained lattice parameter $\Delta
L$, where the strain is given by $s=\frac{\Delta L}{L_{0}}$.

Figure 1: Examples of $7\times 7\times 1$ supercells with defect complexes
consisting of boron atoms (green), nitrogen (gray), aluminium (red), and
titanium (blue). The $a$ and $b$ directions are in-plane. (a)
Al${}_{\text{N}}$V${}_{\text{B}}$ defect, where a substitutional Al atom
replaces a nitrogen atom next to a boron vacancy. (b) Ti${}_{\text{BN}}$ where
a Ti atom sits in the center of a bi-vacancy V${}_{\text{BN}}$.

###  II.3 Criteria for promising high quality defects

In principle, the photoluminescence properties of any quantum emitter depend
on the Huang-Rhys factor, Debye-Waller factor, its quantum efficiency (QE),
and the excited-state lifetime. These excited-state properties depend on the
electronic transition, which can be classified in general into two types:
radiative and non-radiative transitions. Both transition types are disparate
in the sense of the electron-phonon coupling. While the radiative transition
can happen without coupling to any phonon (i.e. into the zero-phonon line),
the non-radiative transitions dissipate the excitation mostly through phonons.
Ideally, any quantum emitter should feature a low Huang-Rhys factor, a high DW
factor, high QE, and short excited-state lifetime.
To select suitable defects for quantum technologies among the large number of
investigated complexes, we define the following criteria for promising quantum
emitters:

  1. 1.

Electronic transition type: the defect should form a two-level system, where
the transition between the highest occupied and the lowest unoccupied defect
state is radiative. We determine this by considering the imaginary part of the
dielectric function or the optical absorption spectrum. Non-radiative
transitions do not exhibit any characteristic peaks.

  2. 2.

Transition energy: the emitter should have an optical emission wavelength
useful for quantum technology applications. We estimate this from the energy
difference between the ground (highest occupied) and excited (lowest
unoccupied) states, which correspond to the characteristic peak observed in
the imaginary part of dielectric function.

  3. 3.

Localization in the band structure: defect states can naturally occur in
either shallow or deep regions in the band gap. For a high quantum efficiency
at room temperature, the states should be well isolated from the band edges or
other defect states (i.e. deep defects). Those deep-lying states should also
exhibit a flat line in both the density of states and the electronic band
structure, as this implies they have the same energy in every high symmetry
point. In other words, such deep-level defects are well isolated from
interaction with neighboring atoms.

It is important to point out that this work considers only first-order
transitions between the highest occupied state and the lowest unoccupied
state. While there might be optically allowed transitions to higher unoccupied
states possible, such excitation would have multiple decay channels which
reduces the quantum efficiency of any given transition. Moreover, based on the
first-order electronic transition, half of all defects are classified as being
non-radiative. In another ongoing, work we employ the $\Delta$SCF method to
uncover the allowed optical transition pathways of those defects from the
zero-phonon line.

##  III Results and discussion

Figure 2: Distribution of optical transition energies (neutral charge states)
of identified defect complexes in the optical spectrum. The defects have been
classified according to the position in the periodic table of the involved
impurities. The inset shows a histogram (with a bin width of 0.25 eV) of all
205 investigated (neutral) defects classified in their transition type
(radiative, non-radiative, and degenerate). Note that in this article V
denotes a vacancy, while for the chemical element vanadium the symbol Va is
used.

###  III.1 Transition energies

We first benchmark the electronic structure of a pristine hBN monolayer with
other reports in literature. The lattice parameters, band gap, and the atomic
orbital contribution in the electronic band structure are well consistent. To
be more precise, our calculation with the HSE06 functional yields a direct
electronic band gap of $E_{\text{g}}=5.99$ eV (see Supplementary Figure S1),
which agrees well with experimental results [47] and that of calculations in
the $G_{0}W_{0}$ quasiparticle approximation [48] reporting values around 6
eV.
Then, the electronic band structures of different defect complexes were
calculated and classified according to our criteria mentioned above. Figure 2
shows the distribution of transition energies or emission wavelengths of 92
identified radiative defects. The histogram in the inset shows that most
fluorescent defects hosted by hBN have transition energies in the range of 1.6
to 2.6 eV, while a large majority is not optically active. Notably, many
transitions below 0.75 eV and above 3.25 eV mostly exhibit radiative
transitions or transitions between degenerate states, which can also be
treated as radiative. It becomes clear that the transition energy does not
correlate with the position of any impurity or dopant within one of the
periodic main groups. To be more precise, although the defects may possess the
same number of valence electrons owing to the same periodic main group, the
overlapping between the specific element and the host hBN can be distinct and
independent, conditional on how the spin polarization of the electrons is
minimized. However, we observed that impurities from group III though VI
contribute to transition energies typically ranging between 1 and 4 eV, while
transition energies below 1.5 eV originate mostly from transition metals as
dopants (the O${}_{\text{N}}$S${}_{\text{N}}$ is the exception).

####  III.1.1 Group III dopants

We now turn to a more in-depth analysis of the defect complexes, starting with
impurities from group III, namely Al, In, and Ga substituted into the
structure at boron or nitrogen lattice sites. Figure 3 (see also Supplementary
Figure S2) shows that any dopant (also outside group III) retains the large
band gap value, despite shifting the Fermi energy. This implies that intrinsic
room temperature quantum emission in general is possible, independently of the
dopant.
The Al${}_{\text{B}}$ and Ga${}_{\text{B}}$ centers do not induce any states
between valence and conduction band (VB and CB, respectively), because their
equal number of valence electrons compared to boron. In other words, there is
no additional free charge carrier left to occupy any additional states. For
the In${}_{\text{B}}$ center, however, a new deep (but unoccupied) state
appeared. For Al-based defects, only the Al${}_{\text{B}}$V${}_{\text{B}}$,
Al${}_{\text{N}}$, and Al${}_{\text{N}}$V${}_{\text{B}}$ defects have
radiative transitions around 2 eV, whereas for In-based defects, possible
transitions were found for the In${}_{\text{B}}$V${}_{\text{B}}$,
In${}_{\text{B}}$V${}_{\text{N}}$, and In${}_{\text{N}}$V${}_{\text{B}}$
center. In case of Ga-based defects, we did not find any first-order
transitions.

Figure 3:  Modified electronic band structures identifying the defect states
and the transition energy of some dopant representatives from (a) native
defects and periodic groups III, IV, and (b) periodic groups V, VI, transition
metals, and multi-defect complexes. The gray bar visualizes the states in
valence and conduction bands, the filled (unfilled) bars represent the
occupied (unoccupied) defect states, and the blue, red, black bars mean the
defect states occupied by spin up, spin down and degenerate electrons,
respectively. The transition energy between two-level systems is reported by
the value near Fermi level. The O${}_{\text{BN}}$ denotes an interstitial
oxygen placed at the center of a bi-vacancy.

####  III.1.2 Group IV dopants

Table 1: Suitable defects compatible with quantum technology applications. Wavelength for | Other Systems in | Compatible hBN | Defect Transition
---|---|---|---
Quantum | Quantum | Defects | Energy (eV)/
Technology (nm) | Technology | Wavelength (nm)
552 | PbV- (diamond) | S${}_{\text{B}}$V${}_{\text{B}}$ | 2.252 / 550.7
In${}_{\text{B}}$V${}_{\text{N}}$ | 2.237 / 554.4
In${}_{\text{N}}$V${}_{\text{B}}$ | 2.236 / 554.5
589 | Na-D2 | O${}_{\text{B}}$V${}_{\text{B}}$ | 2.110 / 587.6
590 | Na-D1 | Ti${}_{\text{N}}$V${}_{\text{B}}$ | 2.100 / 590.5
V${}_{\text{N}}$V${}_{\text{B}}$Ti | 2.097 / 591.3
602 | GeV- (diamond) | Er${}_{\text{B}}$N${}_{\text{B}}$V${}_{\text{N}}$ | 2.063 / 601.1
606 | Pr3+:Y2SiO5 | * Sn${}_{\text{B}}$V${}_{\text{B}}$ | 2.037 / 608.8
620 | SnV- (diamond) | V${}_{\text{N}}$V${}_{\text{B}}$ | 2.024 / 612.7
637 | NV- (diamond) | ** Al${}_{\text{N}}$ | 1.918 / 646.6
738 | SiV- (diamond) | V${}_{\text{B}}$V${}_{\text{B}}$ | 1.678 / 738.8
780 | Rb-D2 | Er${}_{\text{B}}$V${}_{\text{B}}$ | 1.592 / 778.7
Er${}_{\text{N}}$V${}_{\text{B}}$ | 1.590 / 779.6
793 | Tm3+:Y2SiO5 | In${}_{\text{B}}$V${}_{\text{B}}$ | 1.540 / 805.2
795 | Rb-D1 | Er${}_{\text{N}}$N${}_{\text{B}}$V${}_{\text{N}}$ | 1.537 / 806.8
Al${}_{\text{B}}$V${}_{\text{B}}$ | 1.535 / 807.9
850 | Telecom-1 | Er${}_{\text{N}}$V${}_{\text{N}}$ | 1.473 / 842.0
852 | Cs-D2
862 | V${}_{\text{Si}}^{-}$ (silicon carbide) | Er${}_{\text{B}}$V${}_{\text{N}}^{-}$ | 1.427 / 869.0
894 | Cs-D1 | * Er${}_{\text{B}}$V${}_{\text{B}}^{+}$ | 1.398 / 886.9
1330 | Telecom O-band | O${}_{\text{N}}$S${}_{\text{N}}$ | 0.946 / 1310.2
1550 | Telecom C-band | * Er${}_{\text{B}}^{+}$ | 0.789 / 1572.3
* Transition between VB or degenerate (ground) state and an unoccupied non-degenerate (excited) state.
** Transition between double-occupied (degenerate) ground and unoccupied
degenerate excited state.

Out of group IV we selected C, Si, Ge, and Sn for potential defect complexes
(see Supplementary Figure S3 and S4). For C-based defects, we found a rich
variety of optically active defects, namely the
C${}_{\text{B}}$V${}_{\text{B}}$, C${}_{\text{N}}$V${}_{\text{B}}$,
C${}_{\text{N}}$V${}_{\text{N}}$, C${}_{\text{B}}$C${}_{\text{N}}$,
C${}_{\text{B}}$C${}_{\text{N}}$V${}_{\text{B}}$, and
C${}_{\text{B}}$C${}_{\text{N}}$V${}_{\text{N}}$ centers. This is consistent
with previous reports of C-based quantum emitters [49, 36, 35, 37]. The
C${}_{\text{B}}$ defect for example inherits one occupied and one unoccupied
defect states, confirming that the structure has one additional free electron
from carbon. Similarly, C${}_{\text{N}}$ also has one more free electron to
create one unoccupied defect state. The electronic structures of both defects
correspond to one spin-up occupied defect state and one spin-down unoccupied
defect state as reported by previous spin-polarized DFT calculations [49, 50,
51]. Interestingly, bi-carbon defects like C${}_{\text{B}}$C${}_{\text{N}}$
exhibit degenerate defect states, also consistent with previous work [36]. Our
transition energies at 2 and 4 eV agree well with other zero phonon line
calculations [49, 36, 35], validating our methodology. In contrast, for Si-,
Ge-, and Sn-based defects, we found first order transitions of deep-level
defects with energies in the range of 1.8 to 2.8 eV.

####  III.1.3 Group V dopants

Out of group V we selected P, As, and Sb for potential defect complexes (see
Supplementary Figure S5). Notably, doping with phosphorus yields only in case
of the P${}_{\text{B}}$V${}_{\text{B}}$ center an optical transition at 2.56
eV. Doping with arsenic in turn induces a large variety of transition energies
at 5.24 eV (As${}_{\text{N}}$), 1.72 eV (As${}_{\text{N}}$V${}_{\text{B}}$),
and 3.16 eV (As${}_{\text{N}}$N${}_{\text{B}}$V${}_{\text{N}}$). For the Sb-
based defects studied, the Sb${}_{\text{B}}$V${}_{\text{B}}$ center is the
only interesting defect for quantum emitters. Overall, this result suggests
that the first-order transitions in group V-based defects are mostly non-
radiative. We believe this is due to the structures having interacting
unpaired electrons that can minimize the energy. This flips the spin such that
the systems take a state of lowest total energy.

####  III.1.4 Group VI dopants

For group-VI dopants (see Supplementary Figure S6) we found again a large
variety of allowed transition energies, in particular for oxygen-based
complexes (except for O${}_{\text{N}}$V${}_{\text{N}}$), consistent with
experimental work [21, 52]. We note that the O${}_{\text{N}}$ and
O${}_{\text{N}}$V${}_{\text{B}}$ defect states are near the band edges (i.e.
shallow defects) and are therefore not meeting our criteria for suitable
quantum emitters. We also note that previous DFT calculations have not
reported any transition for the O${}_{\text{N}}$V${}_{\text{B}}$ and
O${}_{\text{B}}$V${}_{\text{N}}$ defects [38], where we found some at 0.34 and
4.17 eV, respectively. This inconsistency can be explained by the different
functional (HSE06) we used for our calculations. It was later pointed out that
using non-hybrid functionals can underestimate the band gap [41, 46]. For S-
and Se-based we did not find any defects meeting all criteria for useful
quantum emitters.

####  III.1.5 Transition metal dopants

Transition metals being doped into the hBN lattice induce a a high density of
localized states (see Supplementary Figure S7), which is due to the high
hybridization of transition metals. We found Er-based defects fit our
selection criteria very well, as all of them yield the radiative transitions,
ranging from 1.09 to 3.43 eV. In contrast, none of vanadium-based defects have
a radiative transition. In case of Ti- and Re-based defects, there are both
radiative and non-radiative defects with a selection of different energy
differences.

####  III.1.6 Multi-defect complexes

We now expand our study to multi-defect complexes consisting of at least two
impurities from different periodic groups (e.g.
Al${}_{\text{N}}$Sb${}_{\text{B}}$, see Supplementary Figures S8-S10). We
found that these defect complexes lead to a larger range of possible
transition energies ranging from 0.58 to 5.60 eV, therefore covering almost
the full band gap of hBN. From an experimental point of view, however, it
remains a challenge to fabricate this type of defect, as different atomic
species need not only to be implanted randomly but also at adjacent lattice
sites.

Figure 4: Emission wavelength as a function of bi-axial strain. The dashed
lines mark the target wavelength and solid lines represent the emission
wavelength of the defects at (a) 552 nm (PbV- in diamond), (b) 637 nm (NV- in
diamond), (c) 780 nm (Rb-D2), 793 nm (Tm3+:Y2SiO5), and 795 nm (Rb-D1), (d)
862 nm (V${}_{\text{Si}}^{-}$ in silicon carbide), and (e) 1330 nm (telecom
O-band).

###  III.2 Defects for quantum technology applications

Our results so far show a broad coverage of the optical transition lines of
hBN defects in the visible spectrum, with a few also in the near-infrared (for
a complete list see Supplementary File S1). The question arises, if they can
be used for quantum technology applications. To demonstrate this, we selected
a number of important wavelengths for quantum technology applications. To
represent solid-state quantum emitters and qubits we chose color centers in
diamond (with the NV and group IV centers [53]) and in silicon carbide (with
the silicon vacancy [54]). Quantum memories are represented by rare-earth ion
doped crystals (such as Pr3+:Y2SiO5 and Tm3+:Y2SiO5 [55]) and alkali vapor-
based memories (e.g. sodium, rubidium, and caesium where the D1 or D2
transitions are commonly used [55]). For long-distance quantum communication,
the telecom windows are crucial with the first window at 850 nm (now commonly
used in free-space communications), and the telecom O- and C-bands at 1330 and
1550 nm, respectively. For almost all of these applications, we found a
suitable hBN quantum emitter fulfilling our selection criteria and with a
transition near the specific wavelength of the other quantum system, promising
efficient coupling to it (see Table 1). Sometimes even multiple hBN defects
are a possibility, or a defect is suitable for more than one application. We
note that for some applications we only found transitions in case of charged
defects, which are discussed in Section III.3.2. Moreover, in case of the
Sn${}_{\text{B}}$V${}_{\text{B}}$, Er${}_{\text{B}}$V${}_{\text{B}}^{+}$, and
Er${}_{\text{B}}^{+}$ defects, the transitions are between the valence band or
a degenerate (ground) state and an unoccupied (excited) state, which
technically do not meet our criteria for deep defects. As the excited state in
these cases is not degenerate, however, they can only be excited once as the
Pauli exclusion principle forbids occupying the state with two electrons with
the same spin direction. These defects therefore can, in principle, act as
single photon emitters. Unfortunately, at 637 nm (ZPL of the NV- center in
diamond) we only found the Al${}_{\text{N}}$ defect which has a nearby
transition between degenerate ground and excited states. This implies that
this defect should rather emit the two photon Fock state, which can be useful
in quantum interference experiments with many-particle states [56].
Experimentally, however, we have observed hBN quantum emitters with ZPLs
around 640 nm [57], which implies that the experimentally observed emitters
were probably not included in this theoretical study.

###  III.3 Tailoring the transition energy for quantum technology applications

While we have identified hBN defects compatible with common quantum technology
applications, their emission wavelength is only near the desired wavelength
for the specific application. In case of narrow transitions (e.g. the Rb-D2),
this can drastically reduce the coupling efficiency to the other quantum
system. Therefore, a fine tuning mechanism of the transition is required. This
would make it possible to select an hBN emitter close to the target wavelength
and apply this tuning method.

####  III.3.1 Strain tuning

Strain often breaks the symmetry and modifies the local structure of the
atomic orbitals, thereby shifting the energy of the system [58]. This
mechanism is particularly effective for 2D materials [33, 59, 60]. Strain-
tuning of quantum emitters in hBN has been demonstrated experimentally shortly
after their initial discovery [61]. We can apply this mechanism to tailor the
transition energy with external bi-axial strain. We note that even using the
HSE06 functional, DFT has only a finite accuracy. The exact required strain
value would have to be determined experimentally. Moreover, due to typical
residual stress in the crystal lattice, the ZPLs of hBN quantum emitters
anyway spread around their zero-stress ZPL [57]. This also requires measuring
the ZPL experimentally and in-situ tuning until the desired ZPL is achieved.
Nevertheless, our results represent qualitatively the correct experimentally
required strain.
In Figure 4 we present strain-tuning for a selection of identified compatible
defects for quantum technology applications and tailor their emission to the
precise wavelength of the corresponding quantum application (for the complete
band structures see Supplementary Figures S11-S13). For example, for the
S${}_{\text{B}}$V${}_{\text{B}}$ to be tuned into resonance with the PbV-
center in diamond at 552 nm, we need to apply 0.10% of strain. We found that
predicting the required strain is infeasible, because the local structures
with their defects react differently to the same amount of strain. Moreover,
the strain can break the C2v symmetry of the defects. Our results suggest the
lattice deformation can be classified into in-plane (e.g.
In${}_{\text{B}}$V${}_{\text{N}}$, Al${}_{\text{N}}$, and
O${}_{\text{N}}$S${}_{\text{N}}$) and out-of-plane deformations (the remaining
defects shown in Figure 4). As expected, the out-of-plane deformation results
in larger wavelength shifts in the range of 50 to 250 nm, while for in-plane
deformation this range is narrower with 30 to 40 nm. This is consistent with
the fact that out-of-plane deformation considerably alters the symmetry of the
local structures compared to in-plane.
As one might expect, many defects exhibit a linear relationship between strain
and transition energy. Some, however, also show a nonlinear behavior (see
Figure 4). This effect depends on the interplay between symmetry, initial
strain and defect configuration such as location and size [62]. Generally
speaking, we can express the strain Hamiltonian as
$H=\sum_{ij}B_{ij}\epsilon_{ij},$ where $B_{ij}$ are the electronic operators,
and $\epsilon_{ij}$ are the strain tensor components. When a quantum emitter
is embedded in hBN, the system’s strain principal axes can change due to
symmetry breaking of the defect. Depending on the direction and magnitude of
the stretching or compressing, $\epsilon_{ij}$ can be a nonlinear function of
the strain $s$. Therefore, the eigenvalues of $H$ also nonlinearly depend on
$s$. We note that even using the hybrid HSE06 functional, the DFT method still
has a finite accuracy. It is therefore possible that e.g. the trend of the
S${}_{\text{B}}$V${}_{\text{B}}$ defect in Figure 4(a) is purely linear. The
afore mentioned nonlinear effects e.g. on the
In${}_{\text{B}}$V${}_{\text{N}}$ defect in Figure 4(a), however, are larger
than this accuracy, implying that the error of our DFT calculations is not
aliasing as a nonlinear strain-wavelength relationship. Nonlinear strain
dynamics have also been reported previously [61, 58]. It was speculated that
the initial conditions of the host crystal flakes with wrinkles, cracks and
lattice mismatch from sample fabrication cause different local bond lengths
and therefore an unpredictable Stokes shift.
It is worth noting that strain can also activate quantum emitters in hBN [63].
A model to qualitatively describe this is the donor-acceptor (DAP) model.
Based on the DAP mechanism discussed in previous reports [64, 49], there is an
optimal bond length between donor and acceptor sites for strong
photoluminescence. This suggests that when the local structure is under
strain, the bond length between the active donor and acceptor is modified to
the distance where the structure is optically active, but this is not
necessarily where the structure is most stable (i.e. the zero-strain relaxed
structure). This also implies that optically inactive defects that have been
discounted due to our selection criteria could be turned active with strain.
An in-depth study of this, however, is beyond the scope of this work.

####  III.3.2 Influence of the charge-state

So far, we have only considered neutral charge states. We finally also expand
our study to charged defects (single positive/negative). We divide the defects
in Supplementary Table S1 into two groups based on whether they have a
radiative or non-radiative first-order transition. Single-charging the defect
(i.e. adding/removing one electron) naturally shifts the Fermi energy, as one
more/less state is occupied (see Supplementary Figures S14-S17 for the
complete band structures). A positive charge can depopulate the top-most
occupied state, while the negative charge can lower the bottom-most unoccupied
(which is then occupied). In many cases, the transition type is conserved,
meaning an optically active transition remains active under charging. We note
that the level structure sometimes also completely changes. Ionization and
recharging in general can lead to large shifts of the first-order transition,
an effect that is known in diamond (e.g. comparing the ZPL of the NV0 center
at 575 nm and the NV- center at 637 nm [65]). Our findings are consistent with
calculations of charged C-based defects [49].
While charging is therefore not a suitable in-situ tuning mechanism, it is
possible to generate totally new transition energies. It is worth noting that
the well-studied V${}_{\text{B}}^{-}$ was not calculated in this study, but
only the neutral boron vacancy. As we have mentioned, charging/ionization
leads to new level structures, which explains why our results for the neutral
boron vacancy differ from the literature with the negatively charged boron
vacancy. Among our charge state calculations, we have identified compatible
emitters for quantum technologies: the Er${}_{\text{B}}$V${}_{\text{N}}^{-}$
has a first-order transition near that of the V${}_{\text{Si}}^{-}$ center in
silicon carbide. The Er${}_{\text{B}}$V${}_{\text{B}}^{+}$ can couple to the
Cs-D1 transition and the Er${}_{\text{B}}^{+}$ emits in the telecom C-band.
This implies the latter can be used for long-distance quantum communication.

##  IV Conclusion

We have performed spin-polarized DFT calculations using the HSE06 functional
that allowed us to characterize the electronic band structures of 267 defects
hosted by hBN, as well as their transition type and energy. We have identified
a large distribution of color centers with their transition energy peaking in
a range from 0.25 to 5.50 eV. We have established criteria to select useful
defects that can act as single photon emitters out of our new database.
Moreover, we were able to match compatible hBN emitters for specific quantum
technology applications, including quantum communication, quantum memories,
and the coupling to other solid-state qubit systems. Furthermore, we have
theoretically demonstrated how the emission wavelengths of these hBN defects
can be tailored to exactly match the specific wavelength of the quantum
technology application. Our work therefore provides a guide for tailoring
quantum emitters with a freely choosable target wavelength, as well as an
important step towards the efficient coupling of different quantum systems in
a large-scale quantum network.

### Supplementary materials

The supporting information can be downloaded at:
https://www.mdpi.com/article/10.3390/nano12142427/s1.

### Data availability

The data set with the reported defects (sorted by type/wavelength and
classified in terms of electronic transition type, transition energy,
wavelength, and lattice deformation) can be downloaded at:
https://doi.org/10.5281/zenodo.6826694.
The raw data is available from the authors upon reasonable request.

### Funding

C.C. acknowledges a Development and Promotion of Science and Technology
Talents Project (DPST) scholarship by the Royal Thai Government. This work was
funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) - Projektnummer 445275953. The authors acknowledge support by the
German Space Agency DLR with funds provided by the Federal Ministry for
Economic Affairs and Energy BMWi under grant number 50WM2165 (QUICK3). The
computational experiments were performed on resources of the Friedrich
Schiller University Jena supported in part by DFG grants INST 275/334-1 FUGG
and INST 275/363-1 FUGG and Computing Cluster PhyClus funded by the DFG in the
framework of SFB 1375 NOA and by the state of Thuringia, project 2019 FGI 0017
Computing Cluster PhyClus. S.S. acknowledges support from NSRF via the Program
Management Unit for Human Resources & Institutional Development, Research and
Innovation (grant number B05F640051).

### Conflicts of interest

The authors declare no conflict of interest.

###### Acknowledgements.

The authors acknowledge fruitful discussions with M. Gündoğan on important
atomic transitions for quantum memories.

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