# Transition metal impurities in Silicon: Computational search for a
semiconductor qubit

Cheng-Wei Lee  Colorado School of Mines, Golden, CO 80401, USA  National
Renewable Energy Laboratory, Golden, CO, 80401, USA  Meenakshi Singh  Colorado
School of Mines, Golden, CO 80401, USA  Adele Tamboli  Colorado School of
Mines, Golden, CO 80401, USA  National Renewable Energy Laboratory, Golden,
CO, 80401, USA  Vladan Stevanović<EMAIL_ADDRESS>Colorado School of
Mines, Golden, CO 80401, USA  National Renewable Energy Laboratory, Golden,
CO, 80401, USA

###### Abstract

Semiconductors offer a promising platform for the physical implementation of
qubits, demonstrated by the successes in quantum sensing, computing, and
communication. The broad adoption of semiconductor qubits is presently
hindered by limited scalability and/or very low operating temperatures.
Learning from the NV- centers in diamond, whose optical properties enable high
operating temperature, our goal is to find equivalent optically active point
defect centers in crystalline silicon, which could be advantageous for their
scalability and integration with classical devices. Motivated by the fact that
transition metal impurities in silicon typically produce deep carrier trapping
centers, we apply first-principles methods to investigate electronic and
optical properties of these deep-level defects and subsequently examine their
potential for Si-based qubits. We identify nine transition metal impurities
that have optically allowed triplet-triplet transitions within the Si band
gap, which could be considered candidates for a Si-based qubit. These results
provide the first step toward Si-based qubits with higher operating
temperatures and spin-photon interfaces for quantum communication.

## Introduction

Quantum information science has drawn a lot of attention in the past few
decades due to its potential to transform how information is collected,
processed, and transmitted1; 2; 3. Recent developments focus on the physical
realization of qubits4, in which quantum information is encoded. Semiconductor
qubits are among the promising systems as demonstrated by the successes in
quantum sensing5; 6, quantum computing7; 8; 9; 10, and quantum
communication11; 12.

To realize a qubit, a quantum two-level system whose levels can be
initialized, coherently controlled and measured with high fidelity is
needed13. For semiconductors, three major qubit types are discussed13: (i)
gate-controlled nanostructures, (ii) shallow dopants in silicon, and (iii)
optically-addressable point defects such as NV- centers in diamond, a
negatively charged defect complex between substitutional nitrogen impurity and
an adjacent carbon vacancy. All these qubit types aim to isolate a charge or
spin from the semiconductor host and thus form the required quantum two-level
systems. Their performances are benchmarked against different quantum
applications and each has their unique strengths and weaknesses.13

NV- centers in diamond and neutral (and shallow) 31P impurities in crystalline
Si are exemplary systems for optically-addressable point defects and long
coherence times, respectively. The NV- centers have been demonstrated to work
even at room temperature, which is critical for quantum sensing14. Also, their
optical properties offer a natural spin-photon interface suitable for quantum
communication. The NV- centers have been shown to achieve quantum
communication at a distance over one kilometer12, but are ultimately limited
by the strong attenuation of their 637-nm light through optical fiber13.
Optically-addressable point defects in general meet the criteria for quantum
computing, but the probabilistic nature of where the defects are created
limits the scalability, which is essential for quantum computing13. 31P in Si
currently has the longest reported decoherence time for an electron spin in a
semiconductor qubit ($\approx$ few seconds)15. The even longer decoherence
time for nuclear spin15 ($>$10 s) makes such systems relevant for the storage
of quantum information, i.e. quantum memories. However, low temperatures
($<$10 K for 31P in Si) are required.

Among practical considerations, scalability and operating temperature stand
out as the major challenges for semiconductor qubits13. The mature
manufacturing techniques for Si-based devices makes Si an attractive host
material but the extremely low operating temperature is a major concern.
Besides, silicon-based implementations currently lack established spin-photon
interfaces, which is critical for application in quantum communication13; 16;
17 On the other hand, NV- centers in diamond operate at room temperature, but
scaling-up for diamond-based devices remains challenging. Considering the
advantages of both systems, one potential approach to address the challenge is
to find a NV- center like defect system in silicon. Specifically, resembling
NV- centers in diamond, we are searching for deep centers that can accommodate
optical triplet-triplet transitions within the Si band gap.

Color centers were rarely observed for silicon due to its small electronic
band gap. A recent review summarizes the known color centers, including 77Se+,
Er3+, and three color centers associated with particle irradiation17. Among
these color centers, Durand et al. recently demonstrated a single photon
emitter in silicon for the first time using C-doped Si16. The single photon
emitter is currently associated with the G-center, which is most likely a
defect complex made up of two substitutional C atoms and one Si
interstitial16. But currently, no color centers in silicon were found for the
same triplet-triplet optical transitions as NV- centers in diamonds. Motivated
by the established knowledge that transition metals generally form deep-level
defects in silicon, we aim to investigate their optical properties and
potential for qubits.

Figure 1:  Workflow adopted in our search for transition metal impurities in
crystalline Si. The search space consists of all 3d and select 4d and 5d
transition metals (shown at the top). Defect calculations then enable reducing
the set using various search criteria, leading to the identification of 9
candidate impurities as interstitials and substitutional defects (shown at the
bottom with specific charge state).

Specifically, we investigate the whole 3d and few selected heavier transition
metals (TM), as highlighted in Fig. 1. Heavier transition metals are chosen
based on the potential for ion implantation18; 19 and literature search on
defect levels20; 21. Fig. 1 also shows the general workflow we developed and
adopted to identify promising candidates. We utilize the present state-of-art
computational approach to evaluate defect formation energies, thermodynamic
charge transition levels (CTL), and one-particle defect-level diagrams (DLD)
for each TM as a substitutional, and an interstititial point defect in
crystalline Si, as well as a Si-vacancy/TM-substitutional defect complex.
Substitutional and interstitial point defects are the most common extrinsic
defect types and vacancy complexes are widely found as color centers in other
wide band gap host materials such as NV- centers in diamond. We also calculate
optical absorption spectra to verify that the in-gap optical transitions are
allowed. Using the selection criteria discussed later in details, we identify
five substitutional and four interstitial TM defects in silicon depicted at
the bottom of Fig. 1. While further investigations into their optical
properties and spin dynamics are required, our discovery is the first step
toward Si-based qubits with higher operating temperature and spin-photon
interface.

## Results

In search of candidate Si-host-defect system with triplet-triplet optical
transition within the Si band gap, Fig. 1 summarizes our computational
approach. For every defect, we employ the defect calculations and compute the
thermodynamic charge transition level (CTL) that can be readily extracted from
the defect formation energies (see Fig. S1 in SM for the explicit data). CTLs
represent values of the Fermi energy at which a given defect changes its
charge state, i.e. as the EF increases the electrons will be transferred from
the Fermi sea to the available empty (and localized) defect states which will
render the defect itself more negatively charged. Generally speaking, CTLs
provide a guideline how to achieve targeted defect charge states by tuning
Fermi energy. This can be done via doping or electrical gating and the latter
is preferred for semiconductor qubits in order to avoid charge or spin noises
introduced by dopants. CTLs are also useful as they determine the number of
electrons present in mid-gap defect states as a function of the Fermi energy
and are thus helpful in revealing the potential triplet-triplet optical
transitions within the Si band gap.

For a given impurity defect in Si to accommodate both a triplet electronic
ground state and a triplet excited state with the excitation energy within the
band gap of the host material, the following conditions need to be met. First,
there has to be a range of Fermi energies within which a spin-triplet
electronic configuration is stable. Consequently, there has to be at least one
CTL that will describe discharging the corresponding one-electron states (two
orbital states at minimum). Second, for the same Fermi energy range there have
to exist empty electronic states within the band gap and in the same spin
channel. These need to be separated in energy from the occupied ones so that a
triplet-triplet optical transitions with non-zero photon energy are possible.
This also means that there has to exist at least one more CTL associated with
charging of those empty states. Hence, the triplet-triplet transition
necessitates at least two CTLs within the band gap, which is the first
selection criterion we are using in our workflow.

Next, because CTLs have no direct information about the spin configuration, we
further examine the single-particle defect-level diagram of the candidates.
Aiming for triplet-triplet transitions within the Si band gap, we search for
stable defects with triplet ground state, i.e. two unpaired electrons, and at
least one empty state in the same spin channel within the band gap. One of the
unpaired electrons can be promoted to the empty state(s) and form the triplet
excited state. In addition, we use the charge density of corresponding defect
states to determine the symmetry of the electronic orbitals and confirm that
the single-particle states within the band gap are localized. Localized spin
defect states are required22 and symmetries provide the foundation to estimate
whether the optical transitions are allowed. Lastly, because the symmetry
provides no information about the actual values of the amplitudes (oscillator
strengths) of allowed optical transitions, we calculate the one-particle
absorption coefficient to verify that the specific triplet-triplet optical
transitions are indeed allowed and to estimate their brightness.

### Substitutional Defect Candidates

Figure 2:  Thermodynamics charge transition levels for substitutional defects
of transition metals. Semi-transparent gray areas highlight the Fermi-level
range for the final candidates with triplet-triplet optical transitions within
the Si band gap to be stable. $0$, $+$, and $-$ signs indicate the dominant
charge state within the Fermi-level range.

We first look at the substitutional defects for the transition metals in
silicon. Fig. 2 shows their thermodynamic charge transition levels within the
silicon band gap (The results of defect formation energies can be found in the
SM). Applying the the criteria of at least two CTLs leaves seven candidate
systems: ScSi, CoSi, NiSi, CuSi, ZnSi, PtSi, and AuSi.

Figure 3:  Single-particle defect level diagram and absorption spectra for
representative substitutional defects of transition metals. (a)
Co$\mathrm{{}_{Si}^{-1}}$ has C2 symmetry while (b) Zn$\mathrm{{}_{Si}^{0}}$
has no symmetry. The numbers are used to distinguish different defect states
while letters highlight their symmetries. The isosurface plots of the charge
density of corresponding defect states are shown at the density of 0.0025
$\frac{1}{a_{B}^{3}}$. Absorption coefficient ($\alpha$) are provided, with
specific peaks indicated for the corresponding transitions between labeled
defect states within the band gap. x, y, and z indicate the polarization
direction. All the other substitutional defect candidates
(Ni$\mathrm{{}_{Si}^{0}}$, Pt$\mathrm{{}_{Si}^{0}}$, and
Au$\mathrm{{}_{Si}^{+1}}$ ), which also have in-gap triplet-triplet optical
transitions, can be found in the Supplemental Materials.

Next, we search for the defect systems with stable spin triplet ground states,
i.e. having stable charge state within the band gap and spin state of $S$=1.
For this purpose, we analyze the single-particle level diagrams (see Fig. 3)
in addition to CTLs. This criterion disqualifies CuSi because it has no
triplet ground state for any of the stable charge states shown in Fig. 2.
Since triplet-triplet optical transitions are desired, we also used the
single-particle diagram to determine if there are any spin triplet excited
states within the band gap. ScSi is further disqualified since it has no spin-
triplet excited states available within the band gap.

Fig. 3 shows the electronic single-particle levels diagram for two of the
final list of candidates, Co$\mathrm{{}_{Si}^{-1}}$ and
Zn$\mathrm{{}_{Si}^{0}}$ (see Fig. S2 in SM for all the substitutional defect
candidates). Fig. 3(a) and (b) clearly show the spin triplet ground state with
either two up spins or three up spins compensated by one down spin,
respectively. The spin triplet excited states can be achieved when one of the
up or down spins was excited under spin-conservation transition for the case
of two up spins or three up spins compensated by one down spin, respectively.
Besides, we also remove candidates with defect levels that are too close to
the band edge ($\Delta E$ $<$ 0.05 eV) since electrons on these defect levels
can be easily thermally excited. The optical transitions between defect levels
and band edges also compete with transition between in-gap defect levels,
giving rise to optical signals that are too close to distinguish. We further
confirm the localization of those mid-gap defect levels by visualizing their
charge density (see Fig. 3). These charge densities also support the symmetry
determination for the defect based on atomic configurations. For the
candidates without symmetry, e.g. Fig. 3(b), we can clearly see the distortion
from mirror or rotation symmetry.

We further calculate the one-particle optical absorption coefficient for these
five candidates (see bottom subfigures in Fig. 3(a) and (b)), in order to
verify that the optical triplet-triplet transitions are not only symmetry
allowed but also bright. We focus on the photon energy range within the
silicon band gap since we are searching for in-gap optical transitions. The
first strong peaks are related to the triplet-triplet transitions for all the
candidates while the rest of the peaks are associated with transition between
band edge states and the mid-gap defect states. Also, the absorption
coefficient for all first peaks reaches $\sim$103-104 cm-1, suggesting
relatively bright transitions. The locations of these peaks that are
associated with the in-gap triplet-triplet transitions are within the photon
energy range of 0.4 - 0.6 eV ($\approx$ 3100 - 2066 nm). Unlike NV- center in
diamond ($\approx$ 637 nm 23), these optical transitions are located at the
opposite end of the optical fiber attenuation, with the signal loss dominated
by silica infrared absorption13. While the calculated transitions suggest
longer wavelength than the optimal operating bands, our results nonetheless
predict candidates that can serve as spin-photon interfaces. Optical
transitions of $\sim$1 eV is desirable for the application of quantum
communication to minimize attenuation loss in optical fiber. However, limited
by the Si band gap of $\sim$1.2 eV, in-gap states that give rise to optical
transitions close to 1 eV are inevitably close to Si band edges. Close
proximity of defect levels to Si band edges requires lower operating
temperature to maintain the fidelity of the spin state. Therefore, for design
perspective of optically addressable point defects in Si, the trade-off
between operating temperature and photon wavelength should be considered.

### Interstitial Defect Candidates

Figure 4:  Thermodynamics charge transition levels for tetrahedral
interstitial defects of transition metals.  Semi-transparent gray areas
highlight the Fermi-level range for the final candidates with triplet-triplet
optical transitions within the Si band gap to be stable. $0$, $+$, and $-$
signs indicate the dominant charge state within the Fermi-level range.

Figure 5:  Single-particle defect level diagram and absorption spectra for
representative interstitial defects of transition metals.  Both (a)
Sc$\mathrm{{}_{i}^{+1}}$ and (b) Zr$\mathrm{{}_{i}^{0}}$ have the C3v
symmetry. Numbers are used to distinguish different defect states while
letters highlight their symmetries. The isosurface plots of the charge density
of corresponding defect states are shown at the density of 0.0025
$\frac{1}{a_{B}^{3}}$ and along the principle axis of the C3 rotation, i.e.
$<$111$>$. Absorption coefficient ($\alpha$) are provided, with specific peaks
indicated for the corresponding transitions between labeled defect states
within the band gap. x, y, and z indicate the polarization direction. All the
other interstitial defect candidates (Zr$\mathrm{{}_{i}^{+2}}$ and
V$\mathrm{{}_{i}^{+1}}$), which also have in-gap triplet-triplet optical
transitions, can be found in the Supplemental Materials.

Interstitial sites are also very common for transition metal impurities in
silicon. Two kinds of interstitials exist, those that have tetrahedral or
hexagonal symmetry. Here we focus only on interstitial defects at tetrahedral
sites since, for transition metals, they generally have the lowest energies
among all the interstitial defect sites 21; 24. Applying the same selection
procedure as substitutional defects, we first look at the CTL for interstitial
defects as shown in Fig. 4. The criteria of at least two transition levels
reduces the candidate list to Sci, Tii, Vi, Cri, Zni, Zri, and Aui. After
examining for spin triplet electronic configuration within the stable charge
states, we drop Zni out of the list due to its lack of the spin triplet
configuration for Fermi energy within the band gap. A practical constraint of
having one-electron levels sufficiently away from the band edges ($\Delta E$
$<$ 0.05 eV) further removes Tii and Cri from the list. Fig. 5 shows the
single-particle defect level diagram, localized charge densities and the
optical absorption spectra for two of the final four candidates,
Sc$\mathrm{{}_{i}^{+1}}$ and Zr$\mathrm{{}_{i}^{0}}$ (see Fig. S3 for all the
interstitial defect candidates). The single-particle diagrams show that all of
them have the 3A2 triplet ground state, i.e. two unpaired electrons both on e
orbitals (see Fig. 5 and Fig. S3). Taking Sc$\mathrm{{}_{i}^{+1}}$ as example,
the spin triplet excited states, 3E, forms when one of the up spin is excited
from the e orbital to the a1 orbital. Similarly for Zr$\mathrm{{}_{i}^{0}}$,
3E state forms when the down spin is promoted from a1 to e orbitals. These
interstitial defect systems closely resemble the NV center in diamond, which
has C3v symmetry and 3A2 \- 3E transitions 25. Charge densities show the C3
symmetry along the principle axis, i.e. $<$111$>$, consistent with the
symmetry identified based on local defect configuration. The absorption
spectra show that the triplet-triplet optical transitions are allowed as shown
by the first peaks. Similar to substitutional defects, the first peaks fall on
the range between 0.4 to 0.6 eV and are far enough from other peaks.

In addition to substitutional and interstitial defects, we also investigate
the (SiV+TMSi) defect complex for all the transition metals studied in this
manuscript. We find that there are no viable candidates with optical triplet-
triplet transitions within the Si band gap, using the same work flow. They all
share a similar trend that the localized defect states are generally within
the silicon band edges, preventing practical optical applications.

### Defect Formation Energetics

Figure 6:  Defect formation energy for candidate defects and their potential
competing defects Comparison between (a) candidate substitutional defects and
the vacancy complex counterparts and (b) candidate substitutional and
interstitial defects are shown. The number indicates the dominant charge state
at a given Fermi energy range and highlights the charge states of the
candidate substitutional and interstitial defects. Note that the intermediate
compounds are not considered here, i.e., total energies of the reference
element states are used for the chemical potentials.

An extrinsic chemical element can occupy different defect sites and their
relative abundances are important for the deterministic creation of the
defects. Candidate defects that occupy the sites of the lowest defect
formation energy are easier to create.

First, we compare the defect formation energies of the subsitutional defects
with their Si-vacancy-complex counterparts. Fig. 6(a) shows that for the
candidate subsititutional defects (indicated by their specific charge state),
their competing silicon-vacancy-complex counterparts have significantly higher
formation energies, except for Pt$\mathrm{{}_{Si}^{0}}$. This suggests that,
except for Pt$\mathrm{{}_{Si}^{0}}$, the substitutional defects would have
significantly higher defect concentration than their silicon-vacancy-complex
counterparts. But for Pt$\mathrm{{}_{Si}^{0}}$, careful tuning of Fermi level
is required to ensure that its formation energy is lower than its silicon-
vacancy-complex counterpart. Because of the focus on ion implantation as the
means to produce the defects, Fig. 6 neglects the effect of intermediate
compounds on the chemical potential term shown in eq. (1) but the relative
position of the defect formation energies and thus the discussion above
nonetheless remain the same. To be more specific, under the Si-rich condition,
considering intermediate compounds will only change the chemical potential
term for transition metals, which will change in the same way for both
substitutional defect and the complex leaving the difference in the formation
energy unchainged.

Next, we compare defect formation energies of the substitutional defect
candidates to the values of their interstitial counterparts and vice versa.
Fig. 6(b) shows the defect formation energies for candidate substitutional and
interstitial transition metals in silicon (highlighted by their stable charge
state). For the five substituional defect candidates,
Co$\mathrm{{}_{Si}^{-1}}$, Ni$\mathrm{{}_{Si}^{0}}$, Pt$\mathrm{{}_{Si}^{0}}$,
Au$\mathrm{{}_{Si}^{+1}}$, and Zn$\mathrm{{}_{Si}^{0}}$, within the Fermi
energy range at which their relevant charge states are stable, we find that
Au$\mathrm{{}_{Si}^{+1}}$ and Pt$\mathrm{{}_{Si}^{0}}$ have significantly
lower formation energies than their interstitial counterparts. This suggests
that they are the dominant defect types against the interstitial ones. On the
other hand, Co$\mathrm{{}_{Si}^{-1}}$ and Ni$\mathrm{{}_{Si}^{0}}$ have much
higher formation energies than their interstitial counterparts. This
indicates, at equilibrium, they are harder to create compared to their
interstitial counterparts, but not necessarily rule them out from the
candidate list. Unlike the rest, Zn$\mathrm{{}_{Si}^{0}}$ has similar
formation energies to its interstitial counterpart, suggesting that it
requires fine tune of Fermi level in order to be the dominant defect.
Similarly, we compare the defect formation energies of four interstitial
defect candidates (Sc$\mathrm{{}_{i}^{+1}}$, Zr$\mathrm{{}_{i}^{0}}$,
Zr$\mathrm{{}_{i}^{+2}}$, and V$\mathrm{{}_{i}^{+1}}$) to those of their
substitutional counterparts. We find that they all have lower formation
energies than their substitutional counterparts, suggesting less complexity of
creating these candidate interstitial defects.

## Discussion

### Emerging Trends

Figure 7:  Projected density of state (PDOS) for transitional metal atoms.
PDOS for transition metals in representative (a) substitutional and (b)
interstitial candidate defect systems are compared with density of state of
bulk Si (shown by semi-transparent filled curve). The magnitude of PDOS are
amplified 40 times in order to emphasize the location. Gray vertical dashed
lines highlight the position of valence band maximum and conduction band
minimum after potential alignment.

In this paper, among the 3d and selected transition metals, we discover five
substitutional and four interstitial defects in silicon, which have triplet-
triplet optical transitions within the Si band gap. As highlighted in Fig. 1,
we notice a general trend that these subsitutional defect candidates are all
located at the right side of the periodic table for transitional metals while
these interstitial defect candidates are all located at the left side. To
understand these trends, we first compare the electronic projected density of
states (PDOS) on the atomic orbitals of the transition metal impurities to the
density of states of bulk Si (see Fig. 7, S6 and S7 for details). This
provides insights into the relative energies of the outermost s and d orbitals
of the transition metals to the Si band structure. We find out that, for
substitutional defects, the outermost s and d orbitals of the transition metal
are well below the valence band maximum of Si while, for interstitial defects,
the outermost s and d orbitals are dispersed around the Si band gap. Next, we
count the number of participating electrons when the defects are created and
use the relative energy positions identified previously to accommodate these
electrons. For substitutional defect candidates (see Fig. 7(a) as example),
there are sixteen electrons that need to be accommodated when a
Zn$\mathrm{{}_{Si}^{0}}$ defect is created: Zn has twelve valence electrons
and the Si dangling bonds provide four electrons. Given that outermost s and d
orbitials of Zn (black and red curves in Fig. 7(a), respectively) are deep
below the valence band maximum, twelve of the sixteen electrons fill these
states, with four remaining electrons available to occupy the defect states
within the band gap. As shown repetitively in Fig. 3 and 5, a triplet ground
state requires either two or four electrons within the band gap. For
Zn$\mathrm{{}_{Si}^{0}}$, as shown in Fig. 3(b), we indeed have four electrons
within the Si band gap, with three up spins and one down spin. Following the
same procedure, Zn$\mathrm{{}_{Si}^{+2}}$ could be a plausible candidate but
Fig. 2 shows that the charge state of +2 is not stable within the band gap.
Similarly, all the transition metals at the Zn column could potentially work
the same way if their outermost s and d orbital are deep below the valence
band maximum. For the transition metals in the columns near the Zn column, as
long as the electron count meets the requirement, i.e. two or four electrons
within the band gap, they could potentially have triplet ground states.
However, having two or four electrons within the band gap can also result in
spin singlet states, which are the cases for Cu$\mathrm{{}_{Si}^{+1}}$ and
Cu$\mathrm{{}_{Si}^{-1}}$ (see Fig. S4 for their defect-level diagrams).
Nonetheless, this procedure explains the charge state of the substitutional
defect candidates and their locations on the transition-metal periodic table.
It could facilitate a quick search as long as the outermost s and d orbitals
of transition metals are deep within the valence band maximum. Moving left-
ward, the candidate list stop at Co column for silicon because, in order to
fulfill the electron count, it requires more negative charge states, which are
not stable within the Si band gap. But if a host material can accommodate more
negative charge states and the relative position are similar, transition
metals to the left of the Co column could also be substitutional defect
candidates.

Similarly, electron counting also explains why interstitial defect candidates
are all located at the left side of the transition-metal periodic table.
Unlike substitutional defects, the outermost s and d orbitals of the
interstitial transition metals are located right near the band gap (see Fig.
7(b) and S7). Besides, unlike substitutional defects, no Si dangling bonds
pointing to the defect site exist when an interstitial defects is created.
Therefore, the total number of the outermost s and d electrons of a transition
metal is the number of electrons that occupy the defect states within the
silicon band gap. For instance, Sc has three outermost electrons, suggesting
that Sc$\mathrm{{}_{i}^{+1}}$ has two electrons within the gap and could form
a triplet ground state. Moving rightward, the candidate list ends at the V
column but it could potentially extend to Cr column with +2 charge state. We
ruled out Cr column for silicon because the triplet ground state of
Cr$\mathrm{{}_{i}^{+2}}$ are localized slightly below the valence band
maximum. Again, transition metals to the right of the V column are plausible
if a different host semiconductor can accommodate more charge states. This is
supported by the molybdenum color center observed in 4H and 6H-SiC 17, which
has significant larger band gap (Eg $\approx$ 3.0-3.2 eV)22.

Lastly, our findings are consistent with the observation that transition
metals are portable among different host materials as long as they share the
same lattice symmetry and similar bond length26. Namely, when transition-metal
color centers are identified in one host material, they are likely to be color
centers in other _similar_ host materials as defined above. We further
identify the importance of the relative energy position of the outermost s and
d orbitals with respect to the host band structure. A more comprehensive study
is still required to test the concept of portable transition metals within the
context of color centers.

### Significance and other in-gap transitions

Currently, Si-based semiconductor qubits are limited in applications of
quantum communication and sensing, due to the lack of established spin-photon
interfaces and low operating temperature13, respectively. Aiming for NV-
center-like defects in Si, our computational search finds transition metal
defects in silicon that also have optical triplet-triplet transitions within
the band gap. Intuitively, these defect systems can potentially serve as the
spin-photon interface for Si-based qubits. In addition, the defect systems
identified here also could have similar operation scheme to NV centers in
diamond, allowing higher operating temperature. But the maximum temperature is
expected to be lower than the room temperature, given that Si band gap
($\approx$ 1.2 eV) is much smaller than the diamond one ($\approx$ 5.5 eV).

We prioritize the search for optical triplet-triplet transitions within the Si
band gap as motivated by the underlying mechanism of NV centers in diamonds.
However, singlet-singlet or doublet-doublet transitions also show the
potential as spin-photon interfaces27; 28. Since different operation schemes
that utilize these two transitions are also possible, we also report all the
identified transitions in the Supplementary Materials.

### Further considerations

Lastly, we emphasize that the computational search here is only the first step
to screen candidate defect systems. In order to find an equivalent deep center
in crystalline silicon that has the same (or similar) operation scheme as NV-
centers in diamond, a few more steps are required to decide the final
candidates. Previous studies have shown the importance of having strong zero-
field splitting (ZFS), which eliminates the need of external magnetic field to
isolate the m=0 and m=$\pm 1$ states of a spin triplet system.29 Therefore,
future ZFS calculations30 can be used to further shorten the candidate list
identified in this work.

For the application of quantum communication, a sharp zero-phonon line (ZPL)
of the optical spectra is required in order to create indistinguishable
photons13. Since the rest of the spectra (phonon side bands) are not usable, a
larger Debye-Waller factor (ratio between ZPL over phonon side bands) is
desired. Future calculations of ZPL and Debye-Waller factor31 can be used for
comparisons with experimental spectra.

The way that intersystem crossing (ISC) of NV centers in diamond works enables
the initialization of the qubit at room temperatures. Based on this, another
key next step toward Si-based qubit with higher operation temperature is to
understand the ISCs in these candidates. ISC is the non-radiative relaxation
of excited electron, mediated by phonon. With the development of first-
principles phonon calculations in the past few decades32, the ISC rate can now
be quantitatively predicted and future calculations will finalize the
candidate list.

Additionally, nuclear spins and diffusion coefficient of candidate transition
metals are critical factors for physical realization of color centers in Si.
Nuclear spins of defects generally act as spin noise, reducing the coherence
time of the qubit system. However, they can be also beneficial, serving as
quantum memories. Therefore, selection criteria based on nuclear spins depends
on the actual applications. Focused ion beam techniques can readily select the
isotopes of a dopant and we notice that, among the candidate transition-metal
defects, Sc, V, Co, and Au have no naturally stable zero nuclear spin isotope.
For the perspective of reliability, a mobile color center can undermine the
performance of a Si-based qubit since all the external controls are exerted on
predefined locations. Mobility of a defect is measured by its diffusion
coefficient and the values for transition metals in Si are well documented in
the literature21; 33. Co and Ni are known to diffuse even at room
temperature21, making them less suitable if higher operating temperature is
prioritized. Au, V, and Zn are also fairly mobile33; 21; 34, with diffusion
coefficients at the order of 107 (cm2/s) at temperature of 1100 ∘C. Among
them, Zn has very low evaporation pressure, requiring encapsulation of the
device33, while V and Au stay in the bulk Si after cooling. Lastly, Sc35,
Zr36, and Pt37 have very low reported diffusion coefficient, making them
essentially immobile after cooling. Based on both factors,
Zr$\mathrm{{}_{i}^{0}}$, Zr$\mathrm{{}_{i}^{+2}}$, and
Pt$\mathrm{{}_{Si}^{0}}$ are particularly promising for future experimental
realizations.

## Methods

### Defect formation energy

We followed the standard supercell approach to calculate the defect formation
energies38; 39,

$\Delta
H_{D,q}(E_{F},\mu)=[E_{D,q}-E_{H}]+\sum_{i}n_{i}\mu_{i}+qE_{F}+E_{corr}$ (1)

where $E_{D,q}$ and $E_{H}$ are generalized DFT total energies of the defect
and host supercell, respectively. $\mu_{i}$ is the chemical potential of
element $i$ and $n_{i}$ is the number of element $i$ added ($n_{i}$ $<$ 0) or
removed ($n_{i}$ $>$ 0). $q$ is the charge state of the defect and $E_{F}$ is
the Fermi level (electron chemical potential) referenced to valence band
maximum.

$E_{corr}$ is the correction term that addresses the errors intrinsic to the
supercell approach, including (1) size effect (2) potential alignment (3) band
edge problem. Specifically, we follow the correction scheme devised by Lany
and Zunger 38 and the implementation by Goyal _et al._ 39. To correct the
image charge interaction for the 216-atom cell, static dielectric constant of
11.11 is used. In this paper, the band gap problem is addressed via
generalized hybrid density functional theory, specifically HSE0640, using the
standard mixing parameter value ($\alpha=0.25$). Such choice renders
electronic indirect band gap of 1.16 eV, which is in great agreement with
experimental values of 1.17 eV 41. HSE06 also predicts the lattice constant of
5.43 $\AA$, which agrees with experimental value 42.

### Total energy calculations

All the total energies of supercells are calculated using spin-polarized
generalized hybrid density functional theory, specifically HSE0640 with the
standard mixing parameter ($\alpha=0.25$). 216-atom supercell with gamma-only
kpoint sampling is used throughout the paper unless specified. We specifically
use Vienna Ab initio Simulation Package (VASP 5.4.4 GPU version)43; 44; 45.
The wavefunction is expanded using kinetic energy cutoff of 340 eV, which is
sufficient for the projected augmented-wave (PAW) potentials46 used to
describe the electron-ion interactions (the details for the PAW can be found
in the Supplemental Materials).

### Defect concentration

Defect concentration is defined by,

$C_{D,q}=C_{0}\exp(\frac{-\Delta H_{D,q}}{k_{B}T})$ (2)

where $C_{0}$ is the concentration of available sites, $\Delta H_{D,q}$ is the
defect formation enthalpy and $k_{B}$ is the Boltzmann constant. Here we
assume negligible contribution from vibrational entropy.

### Thermodynamics charge transition state

Thermodynamics charge transition state reveals the dominant charge state at a
given Fermi level and is defined by the Fermi level that gives rise to the
same defect formation energies for two charge states of $q_{1}$ and $q_{2}$,

$(q_{1}/q_{2})=\frac{E_{D,q_{1}}-E_{D,q_{2}}}{q_{2}-q_{1}}$ (3)

where $E_{D,q_{1(2)}}$ is the total energy of defect cell with charge of
$q_{1}(q_{2})$ and the correction term for each defect cell is included.
Transition level of $(q_{1}/q_{2})$, like $E_{F}$ in Eq. 1, is referenced to
valence band maximum and it indicates that $q_{1}$ is the dominant charge
state when the Fermi level is lower than the transition level.

### Single-particle defect levels

Single-particle defect levels are the single-particle Kohn-Sham orbitals of a
defect cell and potential alignment is performed to align them with the band
structure of the host cell. We note that single-particle states within the
electronic band gap correlate with the thermodynamics charge transition levels
but by no means the same.

### Optical absorption coefficient

Absorption coefficient ($\alpha$) at a given energy difference($\epsilon_{i}$)
is calculated using the complex refractive index (n),

$\alpha(\epsilon_{i})=\frac{2\epsilon_{i}\operatorname{Im}[n(\epsilon_{i})]}{\hbar
c}$ (4)

where $\hbar$ is the reduced planck’s constant, c is the speed of light. The
complex refractive index is the square root of the frequency-dependent complex
dielectric function, in which the imaginary part can be calculated using the
linear response approach and the real part by Kramers-Kronig transformation
47. We use the VASP code (VASP 5.4.4) and the linear response calculation is
based on the generalized Kohn-Sham states of HSE06 calculation for candidate
defect systems. The complex shift of 0.009 is used in the Kramers-Kronig
transformation and the energy resolution is 0.01 eV. Around 800 empty bands
are used for the supercell with around 864 electrons, which are sufficient for
the small photon energy range of interest since the difference in absorption
intensity is negligible with increasing empty bands.

## Data availability

The data are available upon reasonable request.

###### Acknowledgements.

This work was funded by NREL’s Laboratory Directed Research and Development
program. NREL is supported by the US Department of Energy under Contract No.
DE-AC36-08GO28308 with Alliance for Sustainable Energy, LLC, the Manager and
Operator of the National Renewable Energy Laboratory. The research was
performed using computational resources sponsored by the Department of
Energy’s Office of Energy Efficiency and Renewable Energy and located at the
National Renewable Energy Laboratory.

## Author Contributions

C.-W. L. performed the defect calculations and analyzed the results. V.S. and
C.-W.L. designed the simulations. V.S. oversaw the calculations and the
analysis. V.S., M.S., and A.T. conceived the idea. All authors contributed to
the writing of the manuscript.

## Competing Interests statement

The authors declare no competing interests.

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