# The positively charged carbon vacancy defect as a near-infrared emitter in
4H-SiC

Meysam Mohseni  Wigner Research Centre for Physics, PO. Box 49, H-1525
Budapest, Hungary  Loránd Eötvös University, Pázmány Péter Sétány 1/A, H-1117
Budapest, Hungary  Péter Udvarhelyi  Wigner Research Centre for Physics, PO.
Box 49, H-1525 Budapest, Hungary  Budapest University of Technology and
Economics, Műegyetem rkp. 3., 1111 Budapest, Hungary  Gergő Thiering  Wigner
Research Centre for Physics, PO. Box 49, H-1525 Budapest, Hungary  Adam Gali
Wigner Research Centre for Physics, PO. Box 49, H-1525 Budapest, Hungary
Budapest University of Technology and Economics, Műegyetem rkp. 3., 1111
Budapest, Hungary

###### Abstract

Certain intrinsic point defects in silicon carbide are promising quantum
systems with efficient spin-photon interface. Despite carbon vacancy in
silicon carbide is an elementary and relatively abundant intrinsic defect, no
optical signal has been reported associated with it. Here, we revisit the
positively charged carbon vacancy defects in the 4H polytype of silicon
carbide (4H-SiC) by means of ab initio calculations. We find that the excited
state is optically active for the so-called h-site configuration of carbon
vacancy in 4H-SiC, with zero-phonon line at $0.65~{}\mathrm{eV}$. We propose
this defect as an exotic paramagnetic near-infrared emitter in the IR-B
region.

##  I Introduction

Silicon carbide (SiC) is one of the most promising material platforms for the
integration of quantum defects. Wafer scale single crystals and isotope
engineered samples are readily available. Scalable arrays of integrated
quantum emitters were already demonstrated in this platform [1, 2]. Advanced
micro-fabrication techniques and recent advances in the integrated photonic
devices [3, 4, 5, 6] enable for improved magneto-optical properties for the
hosted quantum emitters. Various point defects in the 4H polytype of SiC have
been already utilized for quantum communication [7], quantum computing [8] and
nanoscale sensing [9, 10, 11, 12] applications. Most of the promising defects
are vacancy-related, created by irradiation techniques [13, 14] or laser
writing [15]. Isolated silicon vacancy ($\text{V}_{\text{Si}}$) is a well
known fundamental defect in silicon carbide associated with single photon
emitters with coherently controllable electron spin even at room temperature
[16, 17, 18, 19, 20, 21, 22, 23]. Divacancy
($\text{V}_{\text{Si}}\text{V}_{\text{C}}$) is a defect complex of adjacent
silicon and carbon vacancies [24] created by annealing. Its neutral charge
state is a color center [25, 26, 27] with a triplet ground state for which
coherent manipulation was firstly demonstrated among the quantum defects in
SiC [28, 29]. For the other constituent of the divacancy, the single carbon
vacancy defect, no associated emission was reported in experiments, which is a
critical information for the initialization and readout of the spin state. As
the electron irradiation creation of this fundamental defect is even more
efficient than that of the silicon vacancy [30], we investigate its possible
application as an abundant quantum defect in SiC.

The carbon vacancy ($\text{V}_{\text{C}}$) defect in 4H-SiC was investigated
by various experimental techniques before. Deep level transient spectroscopy
(DLTS) [31] and electron paramagnetic resonance (EPR) [32] measurements
assigned two paramagnetic centers, EI5 and EI6, to the
$\text{V}_{\text{C}}^{+}$ charge state corresponding to the quasi-cubic (k)
and hexagonal (h) defect sites, respectively. The four Si neighbours of the
defect showed considerable hyperfine constants. The EI5 center was reported to
be Jahn-Teller (JT) distorted to $\text{C}_{1\text{h}}$ symmetry [33].
Temperature activated averaging to $\text{C}_{3\text{v}}$ symmetry was also
reported above T $\approx 50$ K, with an activation energy of
$0.014~{}\mathrm{eV}$ [34, 33]. From photo-EPR measurements, Son et al. [35]
revealed that the EI5 center is a stable deep donor, with $(+/0)$ charge
transition level at 1.47 eV above the valence band maximum [31]. However, no
optical signal has been associated with these basic defects in 4H-SiC which is
surprising as photoluminescence is a very sensitive technique when compared to
DLTS or EPR methods.

The $\text{V}_{\text{C}}$ defect was investigated in several theoretical
studies too. The formation energy was calculated for both k- and h-site to be
$4.07~{}\mathrm{eV}$ and $4.21~{}\mathrm{eV}$, respectively [36, 37].
Hyperfine constants calculated with LSDA method showed good agreement with
experimental data [38, 34]. The Jahn-Teller distortion in the k-site defect
was revealed to be pseudo-Jahn-Teller (pJT) effect, originating from the
electron-phonon interaction between $a$ and $e$ defect orbitals in the non-
degenerate ground state [34]. The lack of pJT effect for the h-site was
attributed to the larger crystal field around the core of the defect.

In this work, we determine the key optical and spin properties of the
positively charged carbon vacancy ($\text{V}_{\text{C}}^{+}$) defect at both
k- and h-sites in 4H-SiC by means of advanced density functional theory
calculations. We confirm its identification to the EI5 and EI6 EPR centers.
Our calculations reveal a zero-phonon line (ZPL) at the h-site defect in the
near-infrared wavelength region associated with a relatively large Debye-
Waller factor. This provides crucial information for the optical
initialization of the quantum state, a first step towards the coherent control
of its spin. Based on these findings, we propose the h-site
$\text{V}_{\text{C}}^{+}$ defect as a promising quantum emitter possessing a
paramagnetic ground state.

Figure 1:  Electronic level diagrams of the $\text{V}_{\text{C}}^{+}$ defect
in $\text{C}_{3\text{v}}$ symmetry at (a) h-site and (b) k-site and (c) in
$\text{C}_{1\text{h}}$ symmetry at k-site. Each Kohn-Sham level is labeled
according to the irreducible representations of the corresponding point group
symmetry. The conduction band (CB) and the valance band (VB) are shown in
green and blue colors, respectively.

##  II Methods

The electronic properties of $\text{V}_{\text{C}}^{+}$ were calculated using
density functional theory (DFT) [39, 40] as implemented in the Vienna Ab-
initio Simulation Package (VASP) plane wave based code [41, 42]. The hybrid
exchange functional of Heyd, Scuseria,and Ernzerhof (HSE06) was used [43]. The
defect structure is modeled in a $6\times 6\times 2$ supercell (576 atoms),
allowing for accurate $\Gamma$-point sampling in the k-space. The atomic
positions were optimized until the Hellman-Feynman forces acting on them were
less than 0.01 eV/Å. $\Delta$SCF or constraint occupation DFT [44] was
employed for the calculation of the electronic excitations. For the
calculation of phonon spectrum and normal modes, the functional of Perdew,
Burke and Ernzerhof (PBE) [45] was applied using density functional
perturbation theory (DFPT) method. Choosing the exchange correlation
functional of PBE over HSE06 in our phonon calculations provides reasonably
accurate results while significantly reducing the computational overhead.
Transition dipole matrix elements were calculated from the overlap of the
pseudo wavefunctions between the Kohn-Sham (KS) states involved in the
excitation, as implemented in the PyVaspwfc code [46]. Nonradiative transition
rates were calculated with the NONRAD code of Turiansky et al. [47, 48].

Partially self-consistent GW0 [49] and BSE [50, 51] calculations were
performed using VASP in a 128-atom supercell model, keeping the $\Gamma$-point
sampling. The orbital basis was calculated with DFT HSE06 functional, with
number of unoccupied orbitals more than 15-times the occupied ones. The energy
cutoff for the calculation of the response function was limited to
$100~{}\mathrm{eV}$. BSE was calculated beyond the Tamm-Dancoff approximation
[52], including 50 occupied and 50 unoccupied orbitals.

##  III Results

The carbon vacancy model in 4H-SiC exhibits $\text{C}_{3\text{v}}$ symmetry
without considering the electron-phonon interaction (by restricting the
symmetry). The vacancy dangling bonds introduce two defect levels, $a$ and
$e$, in the band gap, labeled by the irreducible representations of the
symmetry group. In the positively charged ground state of the defect, a single
electron occupies the $a$ level, while the double degenerate $e$ level is
empty, resulting in a spin doublet state. After lifting the symmetry
constraints, we obtain a pJT distorted relaxed structure at k-site with
$\text{C}_{1\text{h}}$ symmetry with a pJT relaxation energy of
$82~{}\mathrm{meV}$. However, the same type of calculations at the h-site
results in a negligible pJT energy, in line with previous findings [34]. The
defect-level-diagram for both site defects is shown in Fig. 1. In Fig. 2, we
visualize the geometric structure and the ground state spin density for both
h- (a) and k-sites (b). For the $\text{C}_{3\text{v}}$ symmetric h-site
defect, the distance between the three top-side silicon atoms
($\text{Si}_{2}$, $\text{Si}_{3}$ and $\text{Si}_{4}$) and their distance to
$\text{Si}_{1}$ are $3.08~{}\text{\AA}$ and $3.22~{}\text{\AA}$, respectively.
For $\text{C}_{1\text{h}}$ symmetry at k-site, the distance between Si1-Si2
and Si3-Si4 are $3.15~{}\text{\AA}$ and $3.02~{}\text{\AA}$, respectively. The
Si3,4-Si2 and Si3,4-Si1 distances are $3.10~{}\text{\AA}$ and
$3.26~{}\text{\AA}$, respectively.

Figure 2:  Spin density of the V${}_{\text{C}}^{+}$ defect at (a) h-site with
$\text{C}_{3\text{v}}$ symmetry and (b) k-site with $\text{C}_{1\text{h}}$
symmetry. The hyperfine parameters for the atoms labeled here are given in
Table 1.

###  III.1 Hyperfine parameters

Table 1:  HSE06 calculated and experimental [53] hyperfine principal values ($A_{xx}$, $A_{yy}$ and $A_{zz}$) for the nearest neighbor silicon atoms (see Fig. 2) in the $\text{V}_{\text{C}}^{+}$ defect. The core contributions is included in the results. $\theta$ is the polar angle of the $A_{zz}$ principal axis, measured from the c-axis of the 4H-SiC crystal.  |  |  | HSE06 calculation |  |  |  |  |  | Experimental |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---
| Atoms | $A_{xx}~{}(\mathrm{MHz})$ | $A_{yy}~{}(\mathrm{MHz})$ | $A_{zz}~{}(\mathrm{MHz})$ | $\theta~{}(^{\circ})$ |  |  | $A_{xx}~{}(\mathrm{MHz})$ | $A_{yy}~{}(\mathrm{MHz})$ | $A_{zz}~{}(\mathrm{MHz})$ | $\theta~{}(^{\circ})$ |
h-site | Si1 | -323.3 | -323.3 | -492.4 | 0 |  |  | 297.29 | 297.29 | 433.75 | 0 |
| Si2-Si3-Si4 | -17.1 | -15.0 | -30.5 | 90.33 |  |  | 39.23 | 39.23 | 59.12 | 98 |
k-site | Si1 | -130.7 | -127.2 | -198.4 | 8.7 |  |  | 124.4 | 94.14 | 181.01 | 7.7 |
| Si2 | -67.1 | -64.7 | -105.5 | 125.79 |  |  | 91.06 | 89.38 | 132.81 | 121.5 |
| Si3-Si4 | -99.45 | -96.3 | -148.6 | 102.23 |  |  | 107.88 | 106.76 | 154.67 | 103.2 |

For the EPR-active doublet ground state of the V${}_{\text{C}}^{+}$ defect, we
calculate the hyperfine parameters using the HSE06 hybrid functional. The
hyperfine tensor describing the interaction between the nuclear spin at
$R_{I}$ and the electron spin density $\rho_{s}$ of the defect is given by

$A_{ij}=\frac{4\pi}{3}\frac{g_{N}\gamma_{N}g\gamma_{e}}{\left<\hat{S}_{z}\right>}\int\mathrm{d}^{3}r\rho_{s}(r)m_{i,j}\left(r-R_{I}\right)\text{,}$
(1)

where $g_{N}$, $\gamma_{N}$, $g$ and $\gamma_{e}$, are the $g$ factors and
gyromagnetic ratios of the nucleus and the electron, respectively. The
$m_{i,j}=[\delta_{ij}\delta(r)-\frac{1}{2}({3x_{i}x_{j}-r^{2}\delta_{i,j}}){r^{-5}}]$
is the interaction potential between electron and nuclear spins, which include
the Fermi-contact and a dipole-dipole interaction terms. Our results for the
hyperfine parameters of the first neighbor silicon atoms are listed in Table
1, at both k- and h-sites. We also compare the results to experimental data
for EI5 (k-site and C${}_{\text{1}h}$ symmetry) and EI6 (h-site and
C${}_{\text{3}v}$ symmetry) [53], showing reasonable agreement.

###  III.2 Vibronic spectrum

Next, we describe the vibronic interaction in the k-site ground state and
model the microscopic origin of thermal averaging to $\text{C}_{3\text{v}}$
symmetry, observed in the EI5 EPR center [33]. It was attributed to the
dynamic pJT effect in the ground state in Ref. 34; however, their calculated
barrier energy was considerably larger ($50~{}\mathrm{meV}$) than the thermal
activation energy of $14~{}\mathrm{meV}$. In our HSE06 calculations, we obtain
$76~{}\mathrm{meV}$ for the barrier energy, which is comparable to the pJT
energy itself ($82~{}\mathrm{meV}$). Based on these results, we cannot
consider the pJT barrier as a perturbation directly corresponding to the
dynamics of the system. Instead, we go beyond the Born-Oppenheimer
approximation and approach the problem as a dephasing process through strongly
coupled phonon excitations of the JT active modes. In this model, we identify
the thermal occupation of the first vibronic excited state of the system as
the onset of the dynamics, where the activation energy corresponds to the
polaronic gap, i.e., the energy difference of the first vibronic excited state
and the vibronic ground state.

Figure 3: Adiabatic potential energy surfaces in the function of configuration
coordinates corresponding to (a) $T$\- and (b) $E$-symmetric distortions in
the V${}_{\text{C}}^{+}$ defect at k-site. The fitted curve corresponds to the
linear Jahn-Teller solution of the $T\times(e+t_{\text{2}})$ problem.

To this end, we apply a similar model as detailed in Ref. 54. We separate the
effect of the crystal field present in the 4H-SiC crystal and describe the
defect orbitals in high symmetry $\text{T}_{\text{d}}$. In this picture, the
vacancy dangling bonds introduce a single $t_{2}$ orbital into the band gap,
occupied by a single electron. This ${}^{2}T$ electronic configuration is
Jahn-Teller unstable coupling to phonon modes of $t_{2}$ and $e$ symmetries,
called the $T\otimes(e\oplus t_{2}$) problem. The $\text{C}_{3\text{v}}$
crystal field is added as a perturbation in this model. The potential-energy
surfaces (PES) of the $T$ orbitals are formulated with pseudo-spin of three
dimensions. Therefore, the vibronic interaction can be expressed on this basis
as a $3\times 3$ matrix:

$W=\begin{bmatrix}F_{E}\left(\frac{Q_{\vartheta}}{2}-\frac{\sqrt{3}Q_{\varepsilon}}{2}\right)&-F_{T}Q_{\zeta}&-F_{T}Q_{\eta}\\\
-F_{T}Q_{\zeta}&F_{E}\left(\frac{Q_{\vartheta}}{2}+\frac{\sqrt{3}Q_{\varepsilon}}{2}\right)&-F_{T}Q_{\xi}\\\
-F_{T}Q_{\eta}&-F_{T}Q_{\xi}&-F_{E}Q_{\vartheta}\end{bmatrix},$ (2)

where the orbital degrees of freedom ($t_{2}^{(x)}$, $t_{2}^{(y)}$ and
$t_{2}^{(z)}$) are depicted by the rows and columns of the $3\times 3$ matrix
and the vibrational degrees of freedom are expressed by the $Q_{i}$
configuration coordinates. $F_{\text{T}}$ and $F_{\text{E}}$ are the linear
vibronic coupling parameters of the corresponding phonon symmetries. The
linear vibronic coupling can be expressed with the Jahn-Teller energy
($E_{\text{JT}}$) as [see Eq. (3.46, 3.48) in Ref. 55 ]

$\displaystyle F_{E}=\sqrt{2\hbar\omega_{E}E_{\text{JT}}^{E}},$ $\displaystyle
F_{T}=\sqrt{\frac{3}{2}\hbar\omega_{T}E_{\text{JT}}^{T}}\text{,}$ (3)

where $\hbar\omega$ is the phonon energy of the corresponding JT active mode
in the harmonic approximation. Our calculations results for the determination
of the above parameters can be seen in Fig. 3, where we fit a linear vibronic
coupling model for each JT modes. Thus, the adiabatic potential-energy surface
(APES)

$\displaystyle\mathbf{\varepsilon}(\mathbf{Q})=$
$\displaystyle\frac{1}{2}\hbar\omega_{\text{E}}\left(Q_{\varepsilon}^{2}+Q_{\vartheta}^{2}\right){\textbf{I}}+\frac{1}{2}\hbar\omega_{\text{T}}\left(Q_{\xi}^{2}+Q_{\zeta}^{2}+Q_{\eta}^{2}\right){\textbf{I}}$
$\displaystyle+\mathbf{W}(\mathbf{Q})-\frac{\delta}{3}\begin{pmatrix}0&1&1\\\
1&0&1\\\ 1&1&0\end{pmatrix}.$ (4)

consists of the phonon energy associated to the harmonic potential of the
electronic APES (I is the identity matrix), the vibronic interaction, and the
crystal-field splitting ($\delta$). The latter can be obtained by artificially
turning off the origin of the occupational instability, i.e., by smearing the
electron occupation on all the three $t_{2}$ defect orbitals. The residual
splitting corresponds to the crystal field parameter
$\delta=85~{}\mathrm{meV}$. The associated Hamiltonian describes three coupled
five-dimensional harmonic oscillators as

$\displaystyle\hat{H}=$
$\displaystyle\hbar\omega_{E}\left(\hat{a}_{\varepsilon}^{\dagger}\hat{a}_{\varepsilon}+\hat{a}_{\vartheta}^{\dagger}\hat{a}_{\vartheta}+1\right)$
(5)
$\displaystyle+\hbar\omega_{T}\left(\hat{a}_{\xi}^{\dagger}\hat{a}_{\xi}+\hat{a}_{\eta}^{\dagger}\hat{a}_{\eta}+\hat{a}_{\zeta}^{\dagger}\hat{a}_{\zeta}+\frac{3}{2}\right)$
$\displaystyle-
F_{E}\left(\hat{T}_{\varepsilon}\hat{Q}_{\varepsilon}+\hat{T}_{\vartheta}\hat{Q}_{\vartheta}\right)$
$\displaystyle-
F_{T}\left(\hat{T}_{\xi}\hat{Q}_{\xi}+\hat{T}_{\eta}\hat{Q}_{\eta}+\hat{T}_{\zeta}\hat{Q}_{\zeta}\right)$
$\displaystyle-\frac{\delta}{3}\left(\hat{T}_{\xi}+\hat{T}_{\eta}+\hat{T}_{\zeta}\right)\text{,}$

where $\hat{a}_{i}^{\dagger}$ is the oscillator $i$-mode creation operator,
$\hat{Q}_{i}=\frac{1}{\sqrt{2}}(\hat{a}^{\dagger}_{i}+\hat{a}_{i})$ are the
coordinate operators, and the pseudo-spin of $T$ orbitals is represented by
the orbital operators

$\displaystyle\hat{I}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},$
$\displaystyle\hat{T}_{\varepsilon}=\begin{pmatrix}\frac{\sqrt{3}}{2}&0&0\\\
0&-\frac{\sqrt{3}}{2}&0\\\ 0&0&0\end{pmatrix},$ (6)
$\displaystyle\hat{T}_{\vartheta}=\begin{pmatrix}-\frac{1}{2}&0&0\\\
0&-\frac{1}{2}&0\\\ 0&0&1\end{pmatrix},$
$\displaystyle\hat{T}_{\xi}=\begin{pmatrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix},$ $\displaystyle\hat{T}_{\eta}=\begin{pmatrix}0&0&1\\\
0&0&0\\\ 1&0&0\end{pmatrix},$
$\displaystyle\hat{T}_{\zeta}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix}.$

We solve the model Hamiltonian using the following ansatz as the basis of the
vibronic states

$\displaystyle\left|\widetilde{\Psi}\right>=\sum_{j,k,l,n,m}$
$\displaystyle\left(c_{jklnm}^{(\varepsilon)}\left|t_{2}^{(\varepsilon)}\right>+c_{jklnm}^{(\vartheta)}\left|t_{2}^{(\vartheta)}\right>\right.$
(7)
$\displaystyle+c_{jklnm}^{(\xi)}\left|t_{2}^{(\xi)}\right>+c_{jklnm}^{(\eta)}\left|t_{2}^{(\eta)}\right>$
$\displaystyle\left.+c_{jklnm}^{(\zeta)}\left|t_{2}^{(\zeta)}\right>\right)\left|j,k,l,n,m\right>\text{,}$

where $\mathcal{O}=\left(j+k+l+n+m\right)$ is the order of phonon excitations,
acting as the cutoff for the basis size. We solve for the first polaronic
excited state energy in the function of the excitation order up-to
$\mathcal{O}=10$ and extrapolate the finite order basis assuming exponential
convergence in the energy. The final result for the polaronic gap is
$14.4~{}\mathrm{meV}$, in excellent agreement with the observed activation
energy of the thermal averaging in the EI5 center.

###  III.3 Excited state calculations

In the following, we describe the electronic structure of the excited states.
The lowest lying excited state of the defect can be described as promoting a
single electron from the $a$ orbital to the $e$ degenerate orbital in the
$\text{C}_{3\text{v}}$ symmetric ground state configurations. In this vertical
excitation, the electron-hole interaction decreases the $a-e$ KS level gap. We
obtain vertical excitation energies of $0.794~{}\mathrm{eV}$ and
$0.967~{}\mathrm{eV}$ for the k- and h-site defects, respectively. The former
is in good agreement with the GW+BSE result of $0.89~{}\mathrm{eV}$ reported
in Ref. [56]. Next, we allow the relaxation of the atomic positions in the
restricted high symmetry. During the relaxation, the KS level gap is further
decreased. This results in ZPL energies of $0.656~{}\mathrm{eV}$ and
$0.839~{}\mathrm{eV}$ for the k- and h-site defects, respectively. However,
these excited states are JT unstable, further relaxing to
$\text{C}_{1\text{h}}$ symmetry. For the k-site defect, the effect of this
relaxation changes the qualitative picture of the electron promotion, as the
order of the occupied $e$ and empty $a$ level is interchanged. The resulting
electronic structure indicates that the k-site defect does not form a stable
emitter. This effect is not present at the h-site despite of the calculated JT
energy of $0.187~{}\mathrm{eV}$. We account the stability of the h-site
emitter to the larger crystal field splitting. Its final ZPL energy connecting
the JT distorted excited state to the high-symmetry ground state is
$0.652~{}\mathrm{eV}$.

We apply GW+BSE calculations in order to prove the stability of the optical
emission from the JT distorted excited state of the h-site defect. In our
partially self-consistent EVGW0 calculations in the ground state
$\text{C}_{3\text{v}}$ geometry, we obtain quasi-particle levels of 1.93 eV,
2.76 eV, and 3.53 eV energies with respect to the VBM for the a, e, CBM
levels, respectively. We note that the CBM level is obtained in the
$\Gamma$-point of the 128-atom supercell which does not fold the $M$-point
(CBM k-point). The vertical absorption in the BSE calculation is 1.029 eV. We
apply the same methods in the JT distorted excited state geometry, resulting
in the vertical emission at 0.187 eV. We obtain a 72% contribution of the
$a\rightarrow e$ transition in the excitonic wavefunction which implies that
the constructed exciton wavefunction in the $\Delta$SCF procedure is
relatively well described. The HSE06 relaxation energy between the optimised
JT distorted excited state geometry and the ground state geometry is 0.339 eV
in the ground state electronic configuration. The sum of this HSE06 relaxation
energy and the BSE vertical emission energy results in an estimate for the ZPL
energy of 0.526 eV, which is in good agreement with the ZPL energy obtained
from the $\Delta$SCF method in a larger supercell. This result confirms the
stability of the radiative transition in the h-site defect.

###  III.4 Photoluminescence spectrum

Figure 4:  Simulated PL spectrum of the h-site $\text{V}_{\text{C}}^{+}$
defect. The calculated HR and DW factors are $1.605$ and $20\%$, respectively.

The photoluminescence line-shape is calculated using the method of Alkauskas
et al. [57]. The calculated PL spectrum from h-site V${}_{\text{C}}^{+}$
defect is plotted in Fig. 4. The Huang-Rhys (HR) factor determines the
intensity of the phonon sideband. It quantifies the electronic and vibronic
states. In one-dimensional approximation, the HR factor is given by

$S=\frac{E_{\rm FC}}{\hbar\omega_{0}},$ (8)

where $\omega_{0}$ is the vibrational frequency of the effective mode and
$E_{\rm FC}$ is the Frank-Condon relaxation energy [58]. The Debye-Weller (DW)
factor is the ratio of the ZPL intensity in the total emission. It is directly
related to the HR factor by

$W_{\text{ZPL}}=e^{-S}\text{,}$ (9)

where S is the total HR factor. For our calculated $S=1.605$, the DW factor is
$20\%$.

###  III.5 Radiative lifetime

The transition dipole moment ($\mu$) between the ground and excited state is
calculated using the pseudo-wavefunctions of the defect Kohn-Sham levels $a$
and $e$ in the JT distorted excited state. The radiative transition rate of
the h-site defect is calculated as

$\frac{1}{\tau_{r}}=\frac{n\omega^{3}|\mu|^{2}}{3\pi\epsilon_{0}\hbar
c^{3}}\text{,}$ (10)

where $c$ is the speed of light, $\hbar\omega=0.652~{}\mathrm{eV}$ is the
calculated ZPL energy, $\mu=29.8~{}\mathrm{D}$ is the optical-transition
dipole moment, $n=2.647$ is the refractive index of 4H-SiC, and $\epsilon_{0}$
is the vacuum permittivity. The resulting lifetime is
$\tau_{r}=9.3~{}\mathrm{ns}$, which is comparable to the values reported for
the negatively charged nitrogen-vacancy center ($\text{NV}^{-}$) in diamond
[59]. We also note that the JT distortion has a large effect on the optical
transition dipole moment. It slightly distorts the wavefunction along with the
geometry relaxation. However, the largest effect is the greatly decreased
energy separation of the Kohn-Sham levels resulting in a large transition
dipole moment. Performing the same calculation in the restricted
$\text{C}_{3\text{v}}$ symmetric h- and k-site excited states, we obtain
optical lifetimes of $198~{}\mathrm{ns}$ and $321~{}\mathrm{ns}$,
respectively. The latter results are in line with previous GW plus Bethe-
Salpeter equation calculations reporting negligible absorption in the $a$ to
$e$ defect level transition [56].

###  III.6 Nonradiative transition

Figure 5: Calculated nonradiative lifetime of the h-site
$\text{V}_{\text{C}}^{+}$ defect in the function of temperature.

Finally, we describe the nonradiative relaxation rate from the JT distorted
excited state to the high symmetry ground state, coupled by electron-phonon
interaction for the h-site carbon vacancy defect. In this work, we consider
the interaction within the first order of electron-phonon coupling. According
this assumption the capture rate is given by Fermi’s golden rule [60];

$r=\frac{2\pi}{\hbar}g\sum_{m}\omega_{m}\sum_{n}\left|\Delta
H_{im;fn}^{e-ph}\right|^{2}\delta(E_{im}-E_{fn})\text{,}$ (11)

where $\omega_{m}$ is the thermal occupation of the vibrational state m of the
excited electronic state, $E_{im}$ and $E_{fn}$ are total energies of the
initial and the final vibronic states, $g$ is the degeneracy factor of the
final state. $\Delta H_{im;fn}^{e-ph}$ is the electron-phonon coupling matrix
element which can be expressed as

$\Delta H_{im;fn}^{e-ph}=\sum_{k}\left<\Psi_{i}\left|\partial H/\partial
Q_{k}\right|\Psi_{f}\right>\left<\chi_{im}\left|Q_{k}-Q_{0;k}\right|\chi_{fn}\right>\text{,}$
(12)

where $H$ is Hamiltonian of the combined system of electrons and ions. The sum
runs over all phonon modes $Q_{k}$ and $Q_{0;k}$ is the projection of the
initial atomic configuration $Q_{0}$ along each of the phonon coordinates.
$W_{if}=\left<\Psi_{i}\left|\partial H/\partial Q_{k}\right|\Psi_{f}\right>$
is the electron-phonon coupling matrix element pertaining to the phonon mode
$k$. In this work, the corresponding parameters are $g=3$,
$W_{if}=0.15~{}\mathrm{eV}\mathrm{amu}^{-\frac{1}{2}}\text{\AA}^{-1}$,
$\omega_{a}=0.05~{}\mathrm{eV}$ and $\omega_{e}=0.072~{}\mathrm{eV}$. The
calculated nonradiative lifetime for the one-particle transition from $e$ to
$a$ Kohn-Sham level is $\tau_{nr}\sim 3.38~{}\mathrm{ns}$. For this
transition, the quantum efficiency (QE) of 27$\%$ was calculated as

$\text{QE}=\frac{\tau_{nr}}{\tau_{r}+\tau_{nr}}\text{,}$ (13)

where $\tau_{nr}$ is nonradiative lifetime and $\tau_{r}$ is radiative
lifetime. The calculated nonradiative lifetime at elevated temperatures is
shown in Fig. 5.

##  IV Discussion

The negatively charged carbon vacancy defect in 4H-SiC is a promising
paramagnetic defect for quantum technology applications. Owing to its non-zero
electron spin, its ground state structure have been already thoroughly
investigated by ab initio calculations. Here we report the in-depth
characterization of the excited state optical properties of the defect that
have not been detected in experiments to-date. We find that the Jahn-Teller
effect plays an important role in the stability of the excited state, enabling
a stable $\text{C}_{1\text{h}}$ configuration at the h-site. We propose the
$\text{V}_{\text{C}}^{+}$ defect as a near infrared (IR-B) emitter with the
calculated ZPL energy of $0.652~{}\mathrm{eV}$. We find the properties of this
emission very promising in the IR-B region. It shows a very short optical
lifetime of $9.3~{}\mathrm{ns}$, which is excellent compared to the values of
$11.6~{}\mathrm{ns}$, $14~{}\mathrm{ns}$ and $5.5~{}\mathrm{ns}$ values
measured for $\text{NV}^{-}$ center in diamond [59], divacancy [29] and
negatively charged silicon vacancy [61] in 4H-SiC, respectively. Moreover, the
quantum efficiency of this transition is obtained to be 27%. The optical
coherence is remarkable as well, with the calculated Debye-Waller factor of
$20\%$. We conclude that the carbon vacancy defect at the h-site in 4H-SiC can
be observed in photoluminescence and may act as a single photon emitter when
the defects are engineered to be isolated in the host material.

## Acknowledgement

AG acknowledges the Hungarian NKFIH grant No. KKP129866 of the National
Excellence Program of Quantum-coherent materials project and the support for
the Quantum Information National Laboratory from the Ministry of Innovation
and Technology of Hungary, and the EU H2020 project QuanTELCO (Grant No.
862721). The calculations were performed on the Hungarian Supercomputer Centre
at KIFÜ.

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suppSupplementary References

# An Empirical Bayes Approach to Controlling the False Discovery Exceedance

Pallavi Basu Luella Fu Indian School of Business San Francisco State
University Statistics & OR, Tel Aviv University<EMAIL_ADDRESS><EMAIL_ADDRESS>Alessio Saretto Wenguang Sun Federal Reserve Bank of
Dallas USC Marshall School of Business<EMAIL_ADDRESS><EMAIL_ADDRESS>