# Spin-to-Charge conversion with electrode confinement in diamond

Liam Hanlon  Laser Physics Centre, Research School of Physics, Australian
National University, Canberra, ACT, Australia, 2601  Lachlan Oberg  Laser
Physics Centre, Research School of Physics, Australian National University,
Canberra, ACT, Australia, 2601  Yun Heng Chen  Laser Physics Centre, Research
School of Physics, Australian National University, Canberra, ACT, Australia,
2601  Marcus W. Doherty<EMAIL_ADDRESS>Laser Physics Centre, Research
School of Physics, Australian National University, Canberra, ACT, Australia,
2601

###### Abstract

The nitrogen-vacancy (NV) center in diamond has a wide range of potential
applications in quantum metrology, communications and computation. The key to
its use lies in how large the optical spin contrast is and the associated
fidelity of spin state readout. In this paper we propose a new mechanism for
improving contrast with a spin-to-charge protocol that relies on the use of an
external electrode and cryogenic temperatures to discretise the diamond
conduction band for spin-selective resonant photoionization. We use effective
mass theory to calculate the discrete eigenenergies in this new system and use
them to formulate a new spin-to-charge protocol that involves resonant
photoionization out the NV ground state into the diamond conduction band. The
major sources of broadening are also addressed which guide the design of the
experiment. With this mechanism we theorise an optical spin contrast that and
an associated spin readout fidelity of 85%. This significant improvement can
be applied to a number of cryogenic quantum technologies.

diamond, nitrogen-vacancy, spin-to-charge, electrode, NV, fidelity

††preprint: APS/123-QED

The nitrogen vacancy (NV) center in diamond is a unique and well-studied
defect which has been used in a variety of quantum metrology and imaging
experiments [1, 2, 3, 4] as well as quantum computing and networking
experiments [5, 6, 7]. The defect possesses the longest electron coherence
time of any solid-state defect [8], permits optical initialisation and readout
of spin-states under ambient and cryogenic conditions [9], and can be
engineered within a myriad of nanoscale geometries [10, 11, 12].

The key mechanism involved in both NV quantum sensing as well as computing is
the ability to initialise and readout the NV electron spin state. The
electronic structure that allows for spin selective readout is only available
when the NV center is in the negatively charged state (NV-), other charge
states do not exhibit spin selective properties. This presents a significant
problem in NV physics as the center can ionize into the neutral NV0 state
under optical illumination where the electron is excited into the diamond
conduction band. During this process the spin information is lost and the NV
center isn’t usable until an electron repopulates the defect, converting it
back into NV-. The recombination process is most often achieved by optically
exciting electrons into the NV center from the diamond valence band[13]. This
ionization process limits NV charge state control and is an important issue in
NV physics.

One key area of study for NV charge state control involves purposeful
photoionization via the spin-to-charge conversion (SCC) technique. With SCC,
the NV spin information is read out by optically ionizing the defect when it
is in a particular spin state. Thus, the spin of the electron is mapped to the
charge state of the NV and spin information is obtained by measuring the
charge state. The charge state can be measured optically with a laser
maximally resonant to NV-[14] or by measuring the photoelectric current
induced by ionization [15, 16]. This technique has been shown to have a larger
optical spin contrast compared to conventional intrinsic photo-luminescence
cycling techniques [15, 14, 17], which in turn increases the readout fidelity
of measuring a spin state. Whilst SCC methods improve contrast, the
improvement is not very significant. For example, Jaskula et al reported an
SCC readout contrast of 36% compared to the conventional methods which have a
readout contrast of 25% [14]. The primary reason for the limitation is that
SCC protocols still require shelving the NV electron into the singlet state
manifold via an inter-system crossing (ISC). The branching rate of the ISC is
not 100% from the $m_{s}=\pm 1$ state nor is it 0% from the $m_{s}=0$ state
[18], this lowers the spin-selectivity and increases the probability of a
false readout. Current methods also do not alter the probability of
photoionization itself, the rate of photoionization is set by the intrinsic
nature of the NV center in diamond. Being able to read out an electron spin
state consistently without introducing noise from erroneous photoionization is
essential in producing high fidelity spin measurements for quantum protocols.

We introduce a new mechanism of charge state control with the application of
an external electrode to the surface of the diamond over a near surface NV
center. The electrode creates a potential well within the diamond which has
the effect of spectrally confining the density of low lying conduction band
states in the diamond. This discretized conduction band has a two-fold effect.
Firstly, it increases the photoionization probability at frequencies resonant
to a discrete transition whilst reducing the probability of photoionization at
other frequencies. Secondly, the electrode creates energy level separation in
the conduction band which is much larger than the separation of levels in the
NV ground state triplet. These two factors allow for an SCC protocol where the
NV electron is resonantly ionized out of the ground state triplet into a
discrete conduction band state with higher probability compared to
conventional photoionization. The wide separation of the conduction band
states means that the individual triplet transitions can be addressed. These
factors create a highly selective spin to charge protocol with very high
optical spin contrast. The technique promises to vastly improve the fidelity
of spin readout which has applications for NV based quantum sensing,
communications and computation. The design also creates a discrete three level
system for stimulated Raman adiabatic passage (STIRAP) experiments [19]. The
key to this process is ensuring that the spectral lines are narrow enough to
be resolved. To achieve this the major sources of broadening need to be
addressed.

In this paper we provide a proof-of-principle demonstration of the electrode
for NV readout using extensive theoretical modelling. We introduce the
electrode and briefly describe how its design improves performance.
Simulations of the electric potential well being generated by the electrode
are performed and effective mass theory is then applied to calculate the
discrete eigenenergies and wavefunctions being produced in the well. We then
use these energies to calculate the density of states and subsequent
photoionization probability. Next, we consider the major sources of broadening
which describe the constraints on the design of the system and help outline
problems to be overcome to fully realise this technology. Finally, we describe
the spin to charge conversion protocol, outline the requirements of the system
for the protocol to be successful, and calculate the contrast and readout
fidelity this protocol can produce.

Figure 1: a) Image of the electrode setup. The grey diamond has an electrode
fabricated onto it made of 5 nm SiOx and 95 nm ITO. The cylindrical section of
the electrode sits over an NV center and a wire connects the electrode to a
voltage source. b) Simulation of the potential well generated by the
electrode. The white section is an electrode which sits on a diamond surface
and carries a 10 mV potential. c) Plot of the simulated wavefunction for the
ground state wavefunction viewed from the XZ (left) and XY plane (right
image). d) Plot of the same simulation from c) but for the first excited state
wavefunction. The $\Delta E$ term is the difference between the first and
second eignenergies.

In our design, the cylindrical electrode is placed over an NV with a thin wire
connecting to a voltage source. The electrode and wire have a thin, insulating
silicon oxide (SiOx) layer to prevent charges moving from the diamond into the
electrode. On top of the SiOx is a transparent indium-tin oxide (ITO)
conductive layer which carries the electric potential and allows for optical
illumination through the electrode. The dimensions of the electrode were
chosen to maximise the energy splitting in the diamond conduction band (see
figure 1). Smaller confinement volumes (from smaller electrodes) have larger
energy splitting, making them easier to individually address. The limiting
factor on size comes from the resolution of the nanofabrication process itself
as the wire width must be smaller than the electrode to prevent wavefunctions
occupying the space under the wire (see supplementary for more details).

The potential well generated by the electrode can be solved for using
Maxwell’s equations. The solution can then be used in a Schrodinger equation
which solves for the required eigenenergies and associated wavefunctions:

$\Big{(}-\frac{\hbar^{2}}{2}\vec{\nabla}\cdot\overleftrightarrow{\frac{1}{m}}\cdot\vec{\nabla}+V(\vec{r})\Big{)}\ket{F_{n}(\vec{r})}=E_{n}\ket{F_{n}(\vec{r})},$
(1)

where $E_{n}$ is the eigenenergy for a given energy level $n$, $V(\vec{r})$ is
the electric potential from the electrode, $\overleftrightarrow{m}$ is the
effective mass tensor of the electron and $F_{n}$ is an envelope function. The
envelope function is related to the electron wavefunction by the following:

$\psi_{n}=F_{n}(\vec{r})u_{\vec{k}}(\vec{r})e^{i\vec{k}\cdot\vec{r}},$ (2)

where $u_{\vec{k}}(\vec{r})$ is the Bloch function for the state of the
conduction band minimum in the bulk diamond unit cell and $\vec{k}$ is its
associated wavevector. A full derivation of equation 1 can be found in the
supplementary section.

These equations are applicable due to three key assumptions that we apply in
effective mass theory. The first is that for photoionization we only need to
consider states that are energetically close to the conduction band minimum
(CBM). This is due to the fact that we are only ionizing from the NV to the
lowest lying states in the conduction band. This assumption allows us to
approximate the electron energy as a free electron with an effective mass in a
confining potential. The second assumption is that the envelope function
varies on a distance much greater than the Bloch function, allowing it to be
considered constant when calculating values over the Bloch function space.
Finally, we assume that the electrode potential isn’t strong enough or varying
enough on the scale of the unit cell such that the Bloch function depends on
the potential.

Stronger electrode potentials also reduce the depth of the confining
wavefunction, requiring the placement of NVs closer to the diamond surface in
order to be effected by the electron confinement. This is undesirable as NVs
close to the diamond surface exhibit charge instability[20]. The potential
chosen was designed to create a wavefunction whose center is at the position
of the NV which is approximately 50 nm from the diamond surface. This choice
was made to maximise the eigenenergy splitting whilst maintaining NV
stability.

Three separate solutions from equation 1 were obtained to account for the
three effective masses in the separate Cartesian directions along the diamond.
In each solution, 6 eigenenergies/wavefunctions were obtained and added
together to give a total of 18 conduction band energy levels. The
eigenenergies calculated were then be used to calculate the transition rate
from the NV to a particular conduction band state by using Fermi’s golden
rule:

$\Lambda_{n}(E)=\frac{2\pi}{\hbar}\sum_{n}|\mu_{b}C(E)\sqrt{A_{e}}F_{n}(r)|_{r=NV}\mathcal{E}(E)|^{2}\mathcal{L}(E-E_{n}),$
(3)

where $\mathcal{L}(E-E_{n})$ is a Lorentzian function whose peaks are at the
energy levels of the conduction band states, $E_{n}$, $\mu_{b}$ is the
transition dipole moment in bulk diamond, $\mathcal{E}(E)$ is the electric
field of the interrogating laser and $C(E)$ is the dimensionless Franck-Condon
factor which describes the vibrational overlap of states under a Born-
Oppenheimer approximation. The electron confinement envelope function is
represented by $F_{n}(r)$ which is defined at the position of the NV and is
normalised to the volume of the diamond with the constant $A_{e}$.

Figure 2: Plot of the transition cross section in both bulk diamond (blue) and
in the confined region created by the electrode (orange). The cross section is
proportional to $\sqrt{E}$ in bulk diamond whereas the confined electron has
Lorentzian peaks at each eigenenergy calculated using equation 1. The
linewidth: $\phi=1$ GHz, is chosen based on the error modelling and shows how
thin lines are easily distinguishable. Taking the ratio of the first peak with
the equivalent bulk value gives a transition rate that is 2.4 times more
likely for a confined electron.

Figure 2 shows the transition cross section for photoionization in a confined
electrode (orange) and bulk diamond (blue) for the first 12 energy levels. The
plots are normalised to the dipole moment, electric field and Franck-Condon
factor which is common to both transition cross sections. The equations have
been put in dimensionless units such that the ratio of photoionization between
the two curves can be assessed (see supplementary for details). Whilst the
bulk diamond transition rate is a smooth function proportional to $\sqrt{E}$,
the transition rate in the confined system shows clear Lorentzian peaks at
each eigenenergy calculated using equation 1. When the electron is confined,
the probability of resonant ionization to the first conduction band level is
2.4 times higher compared to bulk diamond, increasing charge state control.

Distinguishing between adjacent transitions is a key aspect of the SCC
protocol. Broadening of the transition lines will increase the probability of
an unwanted transition and will prevent accurate readout of the NV spin state.
From figure 2, the transition linewidths are approximated to be about 1 GHz
and the transitions are qualitatively distinguishable, however this needs to
be considered in more detail. The three main sources of linewidth broadening
are identified as: fluctuations in the confining potential from the electrode,
fluctuations in the confining potential due to surface charge traps and
fluctuations in the conduction band energy levels from electron-phonon
broadening.

Inconsistencies in the electrode potential will alter the potential at the NV
and the resultant diamond conduction band states. This source of broadening
was assessed using the following equation:

$\Gamma_{electrode}=\sigma_{V}\Big{(}\frac{\partial E_{n}}{\partial
V}-\frac{1}{h}\frac{\partial E_{NV}}{\partial V}\Big{)},$ (4)

where $\sigma_{V}$ is the RMS uncertainty of the electrode which was assumed
to be $\pm 0.001mV$, $\frac{\partial E_{n}}{\partial V}$ is the change in
conduction band energy as a function of potential (GHz/mV) and $\frac{\partial
E_{NV}}{\partial V}$ is the change in NV energy level as a function of
potential. The change in conduction band energy can be calculated by solving
for the eigenenergies with an offset to the confining potential by $\pm
0.001mV$ and calculating the slope (see supplementary for more details and
figures). The change in NV energy is given by $-eV$ where $e$ is the electron
charge. Putting all these factors together gives a linewidth of 0.257 GHz.

Electron-phonon (e-p) scattering is the process by which the energy of an
electron is altered slightly by a phonon interaction. In bulk diamond this can
be an issue even at low temperatures (4 K). We can model this broadening with
Fermi’s golden rule:

$\begin{split}\Gamma_{ep}=\alpha\sum_{n}\int|\bra{n}\Psi_{k}(\vec{r})\ket{1}|^{2}\omega_{k}^{3}n_{B}&(\omega_{k})\delta(\omega_{n}-\omega_{k})d^{3}k\\\
\alpha=\frac{\Theta^{2}}{2(2\pi)^{2}\hbar\rho c_{l}^{4}},\end{split}$ (5)

where $\Theta$ is the acoustic deformation potential, $\rho$ is the density of
diamond and $c_{l}$ is the longitudinal speed of sound in diamond. The
equation is a sum of all the interactions of the phonon modes
$\Psi_{k}(\vec{r})$ with the discretised energy levels of the conduction band
minimum from 1 to n, integrated over the phonon k-space. The phonons have a
wavelength $\omega_{k}$ and a temperature dependent distribution given by a
Bose-Einstein distribution $n_{B}$ [19]. The full derivation of this equation
is given in the supplementary material. For simplicity of calculation, the
confining potential is modelled as a square well and then solved for a range
of volumes (plots available in the supplementary). For a square well that has
roughly the same dimensions as the electrode confining potential volume, the
e-p broadening does not exceed 0.5 GHz.

Imperfections in the diamond structure as well as its surface termination
allow for surface charge traps that can hold electrons [21]. Surface charge
traps can be occupied briefly by electrons both from the NV as well as from
defects in the bulk diamond. Charges from these surface traps can then ionize
back into the bulk diamond or can hop from trap to adjacent trap. The result
is a continuous fluctuation of a local electric field from the constant change
in the position of the charges relative to the NV. The noise from the electric
field fluctuation can affect both the NV and conduction band energy similar to
noise from the electrode. To calculate the broadening we assume that the
surface charge traps are uniformly distributed on the diamond and that the
electrons occupying the traps can only move from trap to adjacent trap. In
this picture, the charge motion can be modelled using Redfield theory for a
two level fluctuator [22] where the linewidth is given by:

$\Gamma_{sc}=\frac{2\pi}{\hbar^{2}}\overline{|\delta E|^{2}}S(0),$ (6)

where $S(\omega)$ is the Lorentzian noise spectrum and $\overline{|\delta
E|^{2}}$ is the variance of the change in energy which is given by:

$\overline{|\delta E|^{2}}=\eta
e^{2}\int_{\Omega}|\bra{F_{0}}V(\vec{r},\vec{q})\ket{F_{0}}-V(\vec{r}_{NV},\vec{q})|^{2}d^{2}q,$
(7)

where $\eta$ is the surface density of acceptors and $V(\vec{r},\vec{q})$ is
the potential generated by an electron at position $\vec{q}$ on the surface of
the diamond away from the NV at position $\vec{r}$. A full derivation of this
broadening can be found in the supplementary material. By treating the ground
state conduction band envelope function, $F_{0}$, as a Gaussian then the
effect of a single surface trap on the Gaussian distribution can be modelled
as a multipole expansion. The solution of a single expansion can then be
integrated over the number of surface traps to obtain the overall electric
field noise and its associated broadening. When solving for the overall
linewidth using this model the broadening is actually significant. Even when
considering extremely fast charge motion on the surface (THz in order) the
broadening can be as high as $10^{15}$ Hz for a typical trap density
($10^{18}$ m-2)[23, 24], as much as 6 orders of magnitude broader than the 1
GHz broadening shown in figure 2.

The ideal way to mitigate the surface trap issue is to reduce the density of
charges on the surface of the diamond. Whilst physically reducing the surface
trap density is difficult (although not impossible), an alternate solution is
to fill the traps with a nitrogen donor layer in the diamond which is
considered in more detail by Oberg et. al.[23]. The donor layer nitrogen’s
will pass their electron to the surface traps, resulting in a complete filling
of the traps. The filling of the traps will mitigate any opportunity for
electrons to hop within the traps, thereby reducing the AC electric field they
might produce. The amount of trap filling that occurs depends on the
concentration of donors relative to the trap concentration as well as the
distance the donors are from the traps. Calculating the occupation of the
traps can be achieved by defining the trap energy and density in a Fermi-Dirac
distribution (see supplementary for full derivation). Full surface trap
occupation can be achieved by adding a donor nitrogen layer 100 nm from the
surface of the diamond and reducing the trap density to $\approx 10^{15}$ m-2
with an equal concentration of donors. Fully occupied traps allow no change in
the surface electric field from charge motion as all the traps are filled,
thereby reducing the broadening from surface traps to effectively zero.

Figure 3: a) Diagram of the spin to charge readout protocol. With the
electrode in place, the low lying conduction band states are discretized. This
allows for resonant ionization into the upper energy states which represent
the ionized NV0 2E state with an electron in one of the first two conduction
band levels $c_{0}$ and $c_{i}$ respectively. Thus a laser (light blue) can be
applied to ionize out of the ground state $m_{s}=0$ and a microwave/laser
(yellow/dark blue respectively) combination can be used to ionize out of the
$m_{s}=\pm 1$. b) As long as the linewidth $\Gamma$ is small and the energy
difference between the two transitions is large then the two transitions can
be distinguished. In the error equation, $p_{1,0}$ is the probability of
ionizing from the wrong spin state when intending ionization from the other,
$p_{0,0}$ is the probability of ionizing correctly from the correct spin state
and $p_{a}$ is the probability of absorbing instead of ionizing. Thus, the
error is the ratio of erroneous photoionization compared to the total amount
probability of other processes (see supplementary for more details).

With the electrode and diamond design characteristics outlined, the spin-to-
charge protocol is relatively simple. When the NV electron is in the ground
triplet, an ionizing laser excites the electron from the ground state to the
upper energy states which represent the ionized NV0 2E state with an electron
in one of the first two conduction band levels $c_{0}$ and $c_{i}$
respectively (figure 3). The NV0 charge state can then be read out either
optically or through measuring the photoelectric current [14, 15]. The
ionization will present as a drop in fluorescence from the NV- or by an
increase in current as the NV is ionized into the neutral charge state. This
process can occur when ionizing from $m_{s}=0$ to the conduction band or
$m_{s}=\pm 1$ to the conduction band. Fine structure of the first two
conduction band states is negligible due to the fact that there is no spin-
orbit effects in the conduction band minimum [19].

To calculate the overall optical readout contrast the following equation is
applied:

$C=\frac{1-\mathcal{L}(\Delta E)}{1+\mathcal{L}(\Delta E)+g/f},$ (8)

in the above equation, $g$ is a constant factor that describes the ratio of
photoionization to absorption at the expected photoionization energy (2.6 eV
[13]) which is taken from Razinkovas et al [25]. The $f$ term is a constant
that describes the increase in photoionization cross section due to the
discretization of the conduction band minimum taken from figure 2. Finally,
$\mathcal{L}(\Delta E)$ is the Lorentzian function that describes the
photoionization spin selectivity. The splitting of the NV ground state triplet
($m_{s}=0$ and $m_{s}=\pm 1$ respectively) is known to be: $\Delta D\approx
2.87$ GHz [26, 9] and the splitting of the first two conduction band levels
under the electrode is calculated to be: $\Delta C\approx 9.945$ GHz using
equation 1. The Lorentzian is then a function of the difference in energy
separation between the conduction band energy levels and the ground state NV
triplet energy levels ($\Delta E=\Delta C-\Delta D=7.075$ GHz) as well as the
total linewidth broadening which is found by adding the broadening sources
considered in this paper ($0.757$ GHz). Equation 8 is derived in more detail
in the supplementary section and gives an optical spin contrast of 85%. The
associated spin readout fidelity of the readout can be expressed with the
following equation:

$F=1-\Big{(}\frac{\mathcal{L}(\Delta E)+g/f}{1+\mathcal{L}(\Delta
E)+g/f}\Big{)},$ (9)

where a full derivation can be found in the supplementary section. From this
equation the readout fidelity can be calculated to also be 85%.

Readout fidelity is one of the main limiting factors in NV performance for a
variety of quantum technologies. Typically in the NV, spin readout occurs when
the electron is pumped into the singlet levels from the triplet manifold via
an ISC. The branching ratio in the ISC reduces the probability of electrons
transitioning to the singlet from the $m_{s}=\pm 1$, reducing overall contrast
[18]. This SCC protocol avoids this issue by performing a single transition
photoionization that avoids the ISC altogether. It is only achievable with the
electrode as the two spectral lines need to be distinguishable and this is
only achieved with discrete, wide-gap conduction band energy levels. Achieving
this high readout fidelity SCC using an external electrode is possible as long
as the electrode remains stable, the diamond is at cryogenic temperatures and
is engineered with a nitrogen donor layer with a surface trap density of
$10^{15}$ m-2. Future work would involve engineering the electrode, measuring
confinement and then fully realising the design parameters for the SCC readout
protocol. All these requirements are achievable, offering great potential for
the future of NV technology.

###### Acknowledgements.

The authors acknowledge the support from the Australian Research Council (DP
170103098). We would also like to thank Audrius Alkauskas and Lukas Razinkovas
for sharing absorption and photoionization cross-section data for our
calculations. Finally we wish to thank Sophie Stearn for her assistance with
proof-reading.

##  Appendix A Electrode fabrication

The electrode is fabricated using a sputtering process. The diamond is cleaned
using a three-acid boil (sulfuric, nitric and perchloric acid), after which
approximately 100nm of polymethyl methacrylate (PMMA) is spin coated onto the
diamond to be used as a resist layer. The electrode shape is then etched into
the PMMA along with the electrode wire using electron beam lithography (EBL).
The resist is then exposed to methyl iso-butyl ketone (MIBK) for approximately
1 minute to etch away the PMMA exposed by the EBL beam and is further exposed
using a low dose oxygen plasma etch in a barrel etcher. This creates the hole
in the resist layer which can be layered with 5 nm of silicon oxide (SiOx) and
95nm of indium-tin oxide (ITO) using sputter deposition. The remaining PMMA
layer is removed using a lift-off technique with acetone where only the SiOx
and ITO layers remain. The wire connects to the end of the diamond which can
be layered coarsely with a conductive material (for example: silver paste)
that can be more easily connected to a power supply for generating a potential
using a wire bonding process.

The electrode was designed to carry a potential which can create the confining
well within the diamond substrate around the NV. Numerical calculations
revealed that smaller confining regions produced larger conduction band
splitting which is necessary for the high contrast spin-to-charge readout
mechanism mentioned in the main paper. The length and width of the electrode
controls the length and width of the potential well whereas the depth of the
well is largely controlled by the magnitude of the potential at the electrode
rather than its physical height. A larger positive potential creates a smaller
confining well closer to the surface of the diamond, creating the surface
instability mentioned in the main paper. The SiOx was applied to create an
insulating layer between the electrode and the diamond surface and the ITO was
chosen in for its capacity to carry a potential as well as its optical
transparency so laser illumination can occur over the diamond surface.

##  Appendix B Effective mass theory

In order to calculate the conduction band wavefunctions and eigenenergies
which are confined by the electrode potential we apply effective mass theory.
We begin by solving the Schrodinger equation in the absence of an external
potential:

$\Big{(}\frac{-\hbar^{2}}{2}\vec{\nabla}\cdot\overleftrightarrow{\frac{1}{m}}\cdot\vec{\nabla}+V_{c}(\vec{k})\Big{)}\ket{F_{n}(\vec{r})}=E_{n}^{c}\ket{F_{n}(\vec{r})},$
(10)

where $E_{n}^{c}$ is the eigenenergy of the crystal system for a given energy
level $n$, $V_{c}(\vec{k})$ is the crystal potential, $m$ is the effective
mass tensor of the electron and $F_{n}$ is an envelope function which is
related to the electron wavefunction by the following:

$\psi_{n}=F_{n}(\vec{r})u_{\vec{k}}(\vec{r})e^{i\vec{k}\cdot\vec{r}},$ (11)

where $u_{\vec{k}}(\vec{r})$ is the Bloch function for the state of the
conduction band minimum in the bulk diamond unit cell and $\vec{k}$ is its
associated wavevector. The exponent describes the phase difference when going
between unit cells and we are considering $n$ conduction band minima as we
expect new minima in different vector directions. Expanding out equation 10 to
include the Bloch function gives:

$\Big{(}T(\vec{r})+V_{c}(\vec{k})\Big{)}F_{n}(\vec{r})\mu_{\vec{k}}(\vec{r})e^{i\vec{k}_{n}\cdot\vec{r}}=E_{n}^{c}F_{n}(\vec{r})\mu_{\vec{k}}(\vec{r})e^{i\vec{k}_{n}\cdot\vec{r}},$
(12)

where
$T(\vec{r})=\frac{-\hbar^{2}}{2}\vec{\nabla}\cdot\overleftrightarrow{\frac{1}{m}}\cdot\vec{\nabla}$,
is the kinetic energy for a free electron with an effective mass $m$. For
simplicity the Bloch function can be simplified to:
$\mu_{\vec{k}}(\vec{r})e^{i\vec{k}_{n}\cdot\vec{r}}=\ket{\phi(\vec{r})}$.
Expanding equation 12 as a product rule whilst multiplying both sides of the
equation by the complex conjugate $\bra{\phi(\vec{r})}$ gives the following:

$\begin{split}\innerproduct{\phi(\vec{r})}{\phi(\vec{r})}T(\vec{r})F_{n}(\vec{r})+F_{n}(\vec{r})\bra{\phi}T(\vec{r})\ket{\phi(\vec{r})}+\\\
F_{n}(\vec{r})\bra{\phi}V_{c}(\vec{k})\ket{\phi(\vec{r})}=E_{n}^{c}F_{n}(\vec{r})\innerproduct{\phi}{\phi},\end{split}$
(13)

note that the envelope function is not acting on the eigenstates. The envelope
function varies on distances much larger than Bloch function between diamond
unit cells. Thus when considering small length scales of the diamond unit
cell, one approximation being made is that $F_{n}$ is effectively constant and
can be moved out of the inner product. Taking the inner product:

$T(\vec{r})F_{n}(\vec{r})+E_{b}F(\vec{r})=E_{n}^{c}F_{n}(\vec{r}),$ (14)

where $E_{b}$ is the energy of the Bloch function for the conduction band
minimum. Another other key approximation being made is that an electrode
confining potential isn’t strong enough or varying enough on the scale of the
unit cell such that the Bloch function depends on the potential. This means
that the electrode potential can be added in a new Schrodinger equation as the
Bloch function energy is independent of the electrode potential. Adding in the
electrode potential gives equation 1 from the main paper:

$\big{(}T(\vec{r})+V(\vec{r})\big{)}F_{n}(\vec{r})=E_{n}F_{n}(\vec{r}),$ (15)

where $E_{n}=E_{n}^{c}-E_{b}$ so that $E_{n}$ becomes the energy of the total
wavefunction relative to the Bloch function.

Three separate solutions from equation 1 were obtained to account for the
three effective masses in the separate Cartesian directions along the diamond.
In each solution 6 eigenenergies/wavefunctions were obtained and added
together to give a total of 18 conduction band energy levels. The
eigenenergies can then be used to calculate the transition rate from the NV to
a particular conduction band state by using Fermi’s golden rule:

$\Lambda_{n}(E)=\frac{2\pi}{\hbar}\sum_{n}|\mu_{b}C(E)\sqrt{A_{e}}F_{n}(r)|_{r=NV}\mathcal{E}(E)|^{2}\mathcal{L}(E-E_{n}),$
(16)

where $\mathcal{L}(E-E_{n})$ is a Lorentzian function whose peaks are at the
energy levels of the conduction band states $E_{n}$, $\mu_{b}$ is the
transition dipole moment in bulk diamond, $\mathcal{E}(E)$ is the electric
field of the interrogating laser and $C(E)$ is the dimensionless Franck-Condon
factor which describes the vibrational overlap of states under a Born-
Oppenheimer approximation. The electron confinement envelope function is
represented by $F_{n}(r)|_{r=NV}$ which is defined at the position of the NV
and is normalised to the volume of the diamond with the constant $A_{e}$ using
the following equation:

$\frac{A_{e}}{V_{c}}\int_{V}F_{n}^{*}F_{n}dV=1,$ (17)

where $V_{c}$ is the volume of the diamond unit cell. Equation 17 is solved
for $A_{e}$ by using the results of equation 15 and numerically integrating
the envelope function over the diamond volume directly.

For comparison purposes, it is important to derive the transition rates from
the NV to both the diamond conduction band in bulk as well as to the
conduction band in the confining electrode potential. To do this, the first
step is to calculate the density of states for both cases. With the electrode,
the density of states is simply the Lorentzian function with peaks at the
given energies calculated from equation 10:

$\rho_{e}(E)=\sum_{n}\mathcal{L}(E-E_{n}),$ (18)

where the Lorentzian function can be explicitly written as:

$\mathcal{L}(E-E_{n})=\frac{\Gamma/\pi}{(E-E_{n})^{2}+\Gamma^{2}},$ (19)

and $\Gamma$ is the total associated linewidth. In bulk diamond it is easier
to express the density of states when the energy is in terms of a wavevector,
$\vec{k}$:

$\rho_{b}(E)=\sum_{v}\sum_{\vec{k}}\delta(E-E_{\vec{k}}),$ (20)

where $v$ denotes a sum over the valleys in the Bloch function. The sum of
states can then be re-expressed as an integral of states over a sphere:

$\rho_{b}(E)=\frac{4\pi
V}{(2\pi)^{3}}\int_{0}^{\infty}\vec{k}^{2}\delta(E-E_{\vec{k}})d\vec{k},$ (21)

by taking an effective mass argument, the energy can be expressed as the
energy for a free particle with an effective mass, $m$, such that
$E_{\vec{k}}=\frac{\hbar^{2}}{2m}\vec{k}^{2}$. Rearranging the equation in
terms of $\vec{k}$ and substituting it into the integral gives:

$\rho_{b}(E)=\frac{V}{(2\pi)^{2}}\int_{0}^{\infty}\frac{2mE_{\vec{k}}}{\hbar^{2}}\sqrt{\frac{2mE_{\vec{k}}}{\hbar^{2}}}\frac{1}{2E_{\vec{k}}}\delta(E-E_{\vec{k}})dE_{\vec{k}},$
(22)

which when solved gives the following:

$\rho_{b}(E)=\frac{V}{2(2\pi)^{2}}\Big{(}\frac{2m}{\hbar^{2}}\Big{)}^{3/2}\sqrt{E},$
(23)

note the square root dependence on the energy, which carries into the
transition rate calculation. The transition rate in the bulk diamond is then
calculated using Fermi’s golden rule:

$\Lambda_{b}=\frac{2\pi}{\hbar}\Big{|}\mu_{b}C(E)\sqrt{A_{b}}F_{b}(r)|_{r=NV}\mathcal{E}(E)\Big{|}^{2}\sum_{\vec{k}}\delta(E-E_{\vec{k}}),$
(24)

where $\mu_{b}$ is the transition dipole moment in bulk diamond which is
constant for all wavevectors close to the conduction band minimum and the
$\sum_{\vec{k}}\delta(E-E_{\vec{k}})$ term is the density of states in bulk
diamond. The normalization constant is found by integrating the envelope
function over the bulk diamond volume:

$\frac{A_{b}}{V_{c}}\int_{V}F_{b}^{*}F_{b}dV=1,$ (25)

where $V_{c}$ is the volume of the diamond unit cell. In bulk diamond, the
envelope function encompasses all block wavefunctions so the probability of an
electron existing in the envelope function will be unity: $F_{b}^{*}F_{b}=1$,
therefore: $A_{b}=V_{c}/V$. Substituting in the normalisation and the density
of states gives the following:

$\Lambda_{b}=\frac{1}{4\pi\hbar}\Big{(}\frac{2m}{\hbar^{2}}\Big{)}^{3/2}\sqrt{E}\Big{|}\mu_{b}C(E)\sqrt{V_{c}}\mathcal{E}(E)\Big{|}^{2},$
(26)

equation 26 is plotted as the blue curve in figure 2 of the main paper. The
transition rate in the electrode confined diamond is largely the same but with
a different density of states and a different normalization:

$\Lambda_{e}=\frac{2\pi}{\hbar}\sum_{n}\mathcal{L}(E-E_{n})\Big{|}\mu_{n}\mathcal{E}(E)\Big{|}^{2},$
(27)

where the transition dipole moment in the electrode can be expressed in terms
of the bulk dipole moment mediated by the envelope wavefunction for the
confined electron: $\mu_{n}=\mu_{b}C(E)\sqrt{A_{e}}F_{e}(r)|_{r-NV}$:

$\Lambda_{e}=\frac{2\pi}{\hbar}\sum_{n}\mathcal{L}(E-E_{n})\Big{|}\mu_{b}C(E)\sqrt{A_{e}}F_{e}(r)|_{r=NV}\mathcal{E}(E)\Big{|}^{2},$
(28)

the normalization constant $A_{e}$ found using equation 17. The energy levels
($E_{n}$) are calculated using equation 1 from the main paper which can then
be input into equation 28 to create the orange peaked curve in figure 2 of the
main paper.

The two transition rates calculated cannot be directly plotted as the dipole
moment, laser electric field and Franck-Condon factor, whilst being common to
both equations, are unknown factors. However the important factor isn’t the
photoionization rates but the relative change in photoionization cross section
from the bulk diamond to the diamond in the potential well. To understand this
the transition rates in equations 26 and 28 are divided by the common factors
mentioned and multiplied by:
$1/\big{(}0.5n\sqrt{\frac{\epsilon_{0}\mu_{0}}{\hbar\omega}}\big{)}$ where $n$
is the refractive index of diamond and $\omega$ is the wavelength of the
photoionization laser. These extra factors places the dimensions of equations
26 and 28 into a photoionization cross section multiplied by the electric
charge squared, divided by the dipole moment squared, which is a dimensionless
quantity. With these changes, the two curves in figure 2 can be plotted
together, and the relative cross-sections can be compared.

##  Appendix C Broadening from the electrode

In order to create the potential well in the diamond, an electric potential
must be applied to the electrode. If however, this potential is unstable, then
it will cause a fluctuating shift in the discretized diamond conduction band
energy levels which will present as linewidth broadening. To model this
effect, the simulation can be performed with a 10 mV electrode potential and
it can also be performed with and offset based on the expected noise in the
signal generator: $\pm 0.001$ mV. The change in the potential at the NV (50 nm
from the surface of the diamond) can be measured and the change first
conduction band eigenenergy levels can also be measured.

Figure 4: Plots of the electric potential as a function of the distance from
the surface of the diamond where it meets the electrode. The 50 nm mark is
where the NV is placed. The three curves designate the solution for the 10 mV
potential as well as its offsets of $\pm 0.001$ mV. Additionally, the plot
legends show the first conduction band eigenenergy levels for each potential
solution. Overall the change generated from the potential noise very small.
Figure 5: Plots of the first conduction band eigenenergy levels as a function
of electrode potential. The change is roughly linear so a line can be drawn to
connect the three data points and a slope can be calculated to be $\approx 15$
GHz/mV which is labelled on the plot.

Figure 4 shows the electric potential as a function of distance from the
diamond surface for all three electrode potential values. The figure also has
the first conduction band eigenenergy levels labelled for each electrode
potential. By taking these energy levels and plotting them the slope of the
values can be calculated to be $\approx 15$ GHz/mV which is shown in figure 5.
The change in the transition energy (linewidth broadening) will then be the
change in the conduction band energy as a function of potential
($\frac{\partial E_{n}}{\partial V}\approx 15$ GHz/mV), minus the change in
the NV energy as a function of potential ($\frac{\partial E_{NV}}{\partial
V}$), multiplied by the RMS uncertainty of the electrode:

$\Gamma_{electrode}=\sigma_{V}\Big{(}\frac{\partial E_{c}}{\partial
V}-\frac{1}{h}\frac{\partial E_{NV}}{\partial V}\Big{)},$ (29)

where $\sigma_{V}$ is the RMS uncertainty of the electrode which was assumed
to be $\pm 0.001$ mV and the change in NV energy with potential is linear
($\frac{\partial e_{NV}}{\partial V}=-eV$) which can be calculated and
converted to $-242$ GHz/mV. Solving equation 29 gives the linewidth broadening
mentioned in the main paper of $0.257$ GHz broadening.

##  Appendix D Broadening from e-p scattering

Electron-phonon (e-p) scattering is the process by which the energy of an
electron is altered slightly by a phonon interaction. This will alter the
transition energy of photoionization as the conduction band electron states
are altered by phonon interactions, causing linewidth broadening. In bulk
diamond this can be an issue even at low temperatures (4 K). We can model this
broadening with Fermi’s golden rule:

$\begin{split}\Gamma_{ep}=\alpha\sum_{n}\int|\bra{n}&\psi_{k}(\vec{r})\ket{1}|^{2}\omega_{k}^{3}n_{B}(\omega_{k})\delta(\omega_{n}-\omega_{k})d^{3}k\\\
&\alpha=\frac{\Theta^{2}}{2(2\pi)^{2}\hbar\rho c_{l}^{4}},\end{split}$ (30)

where $\Theta$ is the acoustic deformation potential, $\rho$ is the density of
diamond and $c_{l}$ is the longitudinal speed of sound in diamond. The
equation is a sum of all the interactions of the phonon modes
$\psi_{k}(\vec{r})$ with the discretised energy levels of the conduction band
minimum from 1 to n, integrated over the phonon k-space. The phonons have a
wavelength $\omega_{k}$ and a temperature dependent distribution given by a
Bose-Einstein distribution $n_{B}$ [19]. The phonon modes can be calculated by
understanding that e-p scattering only occurs with dilational modes as they
are the only modes with a non-zero divergence. Deriving an expression for the
dilational modes as:

$\vec{u}(r)=\vec{\nabla}\psi(r),$ (31)

the scalar potential is satisfied by the wave equation:

$-\nabla^{2}\psi(r)=c_{l}^{-2}\omega^{2}\psi(r),$ (32)

which has the following solution:

$\psi(r)=A_{k}e^{i\vec{k}\cdot\vec{r}}.$ (33)

In equation 33 the normalisation constant $A_{k}$ can be found by integrating
all the dilational modes over the volume of the diamond:

$\begin{split}V=\int_{V}|\vec{u}|^{2}d^{3}r\\\
=\int_{V}|\vec{\nabla}\psi|^{2}d^{3}r\\\ =\int_{V}A_{k}^{2}k^{2}d^{3}r\\\
A_{k}=k^{-1}=c_{l}/\omega.\end{split}$ (34)

To help solve equation 30 its easier to assume that confining potential is
rectangular in shape rather than the Gaussian-like shapes they actually are as
it allows for integrals to be solved in Cartesian coordinates whose solutions
are similar to solutions to a finite square well:

$G_{n}(k)=|\bra{n}\psi_{k}(\vec{r})\ket{1}|^{2}=\Big{|}\int_{0}^{L_{x}}\int_{0}^{L_{y}}\int_{0}^{L_{z}}F_{n}\frac{c_{l}}{\omega_{k}}e^{i\vec{k}\dot{\vec{r}}}F_{1}\Big{|}^{2},$
(35)

where $L$ is the length of the confining potential in a Cartesian direction
and $F_{n}$ is the envelope wavefunction for an electron in a Cartesian box.
This assumption shouldn’t change the broadening by much as long the volumes of
the approximate and actual confining potentials are roughly the same.
Substituting equation 35 into equation 30 and writing out the integral over k
gives:

$\begin{split}\Gamma_{ep}=\alpha\sum_{n}\int_{0}^{\infty}\int_{0}^{\pi}\int_{0}^{2\pi}k^{2}\sin{(\theta_{k})}\\\
G_{n}(k)\omega_{k}^{3}\eta_{B}(\omega_{k})\delta(\omega_{n}-\omega_{k})dkd\theta_{k}d\phi_{k},\end{split}$
(36)

substituting $k=\omega/c_{l}$ and simplifying gives:

$\begin{split}\Gamma_{ep}=\frac{\alpha}{c_{l}^{3}}&\sum_{n}\int_{0}^{\infty}\omega^{5}\eta_{B}(\omega)\delta(\omega_{n}-\omega)G_{n}\big{(}\frac{\omega^{2}}{c_{l}^{2}}\big{)}\\\
&\int_{0}^{\pi}\int_{0}^{2\pi}\sin{(\theta_{k})}d\omega^{2}d\theta_{k}d\phi_{k},\end{split}$
(37)

which when solving the integral over the Dirac delta function gives:

$\begin{split}\Gamma_{ep}=\frac{\alpha}{c_{l}^{3}}&\sum_{n}\omega_{n}^{5}\eta_{B}(\omega)G_{n}\big{(}\frac{\omega^{2}}{c_{l}^{2}}\big{)}\\\
&\int_{0}^{\pi}\int_{0}^{2\pi}\sin{(\theta_{k})}d\omega^{2}d\theta_{k}d\phi_{k}.\end{split}$
(38)

The solution to equation 35 in Cartesian coordinates is effectively the
solution to a three dimensional finite square well which can be analytically
calculated to be:

$\begin{split}G_{n}(k)=\frac{8}{L_{x}L_{y}L_{z}}\frac{c_{l}}{\omega}\\\
\int_{0}^{L_{x}}\sin{\Big{(}\frac{n_{x}\pi}{L_{x}}x\Big{)}}e^{ik_{x}x}\sin{\Big{(}\frac{\pi}{L_{x}}\Big{)}}dx\\\
\int_{0}^{L_{y}}\sin{\Big{(}\frac{n_{y}\pi}{L_{y}}y\Big{)}}e^{ik_{y}y}\sin{\Big{(}\frac{\pi}{L_{y}}\Big{)}}dy\\\
\int_{0}^{L_{z}}\sin{\Big{(}\frac{n_{z}\pi}{L_{z}}z\Big{)}}e^{ik_{z}z}\sin{\Big{(}\frac{\pi}{L_{z}}\Big{)}}dz,\end{split}$
(39)

where each integral in a Cartesian direction can be calculated to be:

$\begin{split}\int_{0}^{L_{m}}\sin{\big{(}\frac{n_{m}\pi}{L_{m}}m\big{)}}e^{ik_{m}m}\sin{\big{(}\frac{\pi}{L_{m}}\big{)}}dm=\\\
\frac{2i(1+(-1)^{n_{m}}e^{ik_{m}L_{m}})k_{m}L_{m}^{2}n_{m}\pi^{2}}{k_{m}^{4}L_{m}^{4}-2k_{m}^{2}L_{m}^{2}(1+n_{m}^{2})\pi^{2}+(-1+n_{m}^{2})^{2}\pi^{4}}.\end{split}$
(40)

Using the solutions from equations 39 and 40 they can be substituted into
equation 38 and solved for a given confining volume across the number of
energy levels solved in the system (18). It is important to test the solution
across a range of energy levels to observe when the sum of broadening values
converges, 18 levels is sufficient for convergence in this case. This process
was then performed multiple times over many confining potential sizes in order
to understand how the broadening changes with volume.

Figure 6: Plot of the e-p broadening in Hz as a function of confining
potential volume. The length of the potential well is for both the x and y
coordinate whereas the depth is from the z-coordinate only.

In Figure 6 the broadening is calculated for a variety of confining potential
volumes. The x and y coordinates are changed together for the length of the
confining potential and the z coordinate is changed separately for the depth
of the potential well. For the purposes of this work, the broadening for a
potential well that is 250 nm in length and 100 nm in depth is required.
Reading off of figure 6 a confining potential of roughly this size creates
linewidth broadening which does not exceed 0.5 GHz.

##  Appendix E Broadening from surface charges

Imperfections in the diamond structure as well as its surface termination
allow for surface charge traps that can hold electrons [21]. Surface charge
traps can be occupied briefly by electrons both from the NV as well as from
defects in the bulk diamond. Charges from these surface traps can then ionize
back into the bulk diamond or can hop from trap to adjacent trap. The result
is a continuous fluctuation of a local electric field from the constant change
in the position of the charges relative to NV. This arises as noise which can
affect both the NV and conduction band energy. Calculating this noise is more
difficult compared to the other sources of broadening as the density of the
charge traps on the diamond surface and the rate of their motion from trap to
trap isn’t well known. To calculate the broadening we assume that the surface
charge traps are uniformly distributed on the diamond and that the electrons
occupying the traps can only move from trap to adjacent trap. In this picture,
the charge motion acts like a two level fluctuator which can be modelled using
Redfield theory [22] where the linewidth is given by:

$\Gamma_{sc}=\frac{2\pi}{\hbar^{2}}\overline{|\delta E|^{2}}S(0),$ (41)

where $S(\omega)$ is the noise spectrum which in a two level system is assumed
to be Lorentzian and $\overline{|\delta E|^{2}}$ is the variance of the change
in energy which is given by:

$\overline{|\delta E|^{2}}=\eta
e^{2}\int_{\Omega}|\bra{F_{0}}V(\vec{r},\vec{q})\ket{F_{0}}-V(\vec{r}_{NV},\vec{q})|^{2}d^{2}q,$
(42)

where $\eta$ is the surface density of acceptors (diamond surface trap
density) and $V(\vec{r},\vec{q})$ is the potential generated by an electron at
position $\vec{q}$ on the surface of the diamond away from the NV which is at
position $\vec{r}$. Equation 42 describes the energy variance as the change in
the conduction band energy (first term in the integral) minus the change in
the NV energy (second term in the integral). If the fluctuations are assumed
to have a zero mean, then the rate of charge motion will follow Boltzmann
statistics:

$\gamma=\gamma_{0}e^{-E_{t}/k_{b}T},$ (43)

where $E_{t}$ is the electron energy and the noise spectrum can be described
as:

$S(\omega)=\int_{-\infty}^{\infty}e^{-\gamma|\tau|}e^{-i\omega
t}=\frac{\gamma/\pi}{\omega^{2}+\gamma^{2}},$ (44)

note how equation 44 describes the spectral noise with a Lorentzian line shape
with a width given by the charge hopping rate $\gamma$.

To get an understanding of the process and to solve equation 42 for the
linewidth three separate regimes are considered. The first regime is when the
Fermi level is equal to the trap energy level and the trap occupation is
$\approx 50\%$, in this regime the electrons in the trap can hop to adjacent
traps and we can model the system like a series of 2D dipoles where the
dipoles can orient their direction as the electrons hop into adjacent traps in
any direction. The second is when the Fermi level is below the trap energy and
the trap occupation is less than $50\%$, in this regime the charges act like
monopoles and can hop anywhere in the surface via other traps or through the
diamond conduction band. In this regime the charges effectively appear and
disappear in the traps as they have a larger freedom of movement compared to
the dipole regime. The third regime is when the Fermi energy is above the trap
energy level causing the traps to be mostly occupied. Charge hopping occurs in
a similar mode to the monopole regime, but the effective trap density is
considerably lower. This mechanism doesn’t happen automatically, but it is
something that can be engineered by reducing the trap density, or, adding an
electron donor level in the diamond which preferentially donates electrons to
the unoccupied traps.

The first regime is the dipole regime, in this approach, the potential term in
equation 42 is modelled as a dipole charge:

$V(r)=\frac{e\vec{p}\cdot(\vec{r}-\vec{q})}{4\pi\epsilon_{d}|\vec{r}-\vec{q}|^{3}},$
(45)

where $\vec{p}$ is the displacement vector between neighboring traps. The idea
is that an electron can move from one trap to another, creating an
electron/hole pair that is effectively a dipole. The electron motion to a trap
is then modelled as the dipole moment flipping sign as the electron flips the
direction of the dipole. The change in conduction band energy due to a single
dipole is solved individually. This solution is then integrated over all
possible charge traps (the diamond surface) to find out the overall energy
change. Beginning with the solution due to a single dipole:

$\bra{F_{0}}V(\vec{r},\vec{q})\ket{F_{0}}=\frac{e}{4\pi\epsilon_{d}}\int
F_{0}^{2}(\vec{r})\frac{\vec{p}\cdot(\vec{r}-\vec{q})}{4\pi\epsilon_{d}|\vec{r}-\vec{q}|^{3}}d^{3}r,$
(46)

in this instance, $F_{0}$ is modelled as a Gaussian. Mathematically this
approach is actually equivalent to solving for the potential of an observer
point far from a dipole charge distribution, however, the roles are reversed,
$F_{0}$ acts like the charge distribution and the observer point is $\vec{q}$.
In this formalism the solution is obtained by using a multipole expansion. The
key assumption in the expansion is that the observer point $\vec{q}$ is
sufficiently far from the charge distribution $F_{0}$. This isn’t entirely
true for this system, so higher order terms are added in the multipole
expansion to make sure it is valid. Considering the charge distribution in
terms of the electric displacement:

$\vec{\nabla}\cdot\vec{D}=0,$ (47)

in equation 47, the solution is zero as the the charge density of a group of
dipoles will be zero. Substituting in the displacement field with the electric
field ($\vec{E}$) and polarization ($\vec{P}$) and rearranging gives:

$\vec{\nabla}\cdot\epsilon_{d}\vec{E}=-\vec{\nabla}\cdot\vec{P},$ (48)

the polarization can be described in terms of the wavefunction size and the
displacement vector between neighboring traps ($\vec{p}$). The electric field
can also be described in terms of the electric potential, giving:

$\epsilon_{d}\nabla^{2}V=-\vec{\nabla}\cdot eF_{0}^{2}\vec{p},$ (49)

electrons on the diamond surface can only form dipoles on the 2D surface
plane:

$\nabla^{2}V=\frac{-e}{\epsilon_{d}}\Big{(}p_{x}\frac{\partial}{\partial
x}F_{0}^{2}+p_{y}\frac{\partial}{\partial y}F_{0}^{2}\Big{)}.$ (50)

Using Maxwell’s law, the electric potential can be re-written in terms of the
effective charge density: $\nabla^{2}V=\rho_{eff}/\epsilon_{d}$:

$\rho_{eff}=-e\Big{(}p_{x}\frac{\partial}{\partial
x}F_{0}^{2}+p_{y}\frac{\partial}{\partial y}F_{0}^{2}\Big{)},$ (51)

with an effective charge, an equation 46 be rewritten for the change in
conduction band energy due to a surface trap with a multipole expansion:

$\begin{split}\bra{F_{0}}V(\vec{r},\vec{q})\ket{F_{0}}=\frac{e}{4\pi\epsilon_{d}}\int\frac{\rho_{eff}}{|\vec{r}-\vec{q}|}d^{3}r\\\
\approx\frac{ep_{0}}{4\pi\epsilon_{d}|\vec{r}-\vec{q}|}+\sum_{i}\frac{ep_{1i}(r_{0i}-q_{i})}{4\pi\epsilon_{d}|\vec{r_{0}}-\vec{q}|^{3}}+\\\
\frac{1}{2}\sum_{i,j}\frac{ep_{2ij}(r_{0i}-q_{i})(r_{0j}-q_{j})}{4\pi\epsilon_{d}|\vec{r_{0}}-\vec{q}|^{5}},\end{split}$
(52)

where:

$\begin{split}p_{0}=\int&\rho_{eff}d^{3}r\\\
p_{1i}=-\int\rho_{eff}&(r_{0i}-r_{i})d^{3}r\\\
p_{2ij}=\int\rho_{eff}\Big{(}3(r_{0i}-r_{i})&(r_{0j}-r_{j})-\delta_{ij}(r_{0i}-r_{i})^{2}\Big{)}d^{3}r,\end{split}$
(53)

in the above equation, $r_{0}$ is the position of the electron in the
conduction band (which is set at the origin), and the sum over i and j are for
the different dimensions of the problem (x,y,z). The first term then denotes
an asymmetric solution which is the monopole term, the second is the dipole
and the third is the octupole term. The expansion increases with higher orders
of $\vec{q}$, thus higher order terms should contribute less and less to the
overall solution. Equation 52 will be the solution for the conduction band
electron, and the solution to the NV electron can be written from equation 46
where $r=r_{0}$ as the origin is set at the centre of the confined
wavefunction where the NV is. The solutions to a single dipole charge are then
integrated over the diamond surface to account for the effects of all surface
charges. Additionally, the equations are also integrated over all the possible
orientations of the dipole, $p_{\phi}$, where $p_{x}=p\cos{(p_{\phi})}$ and
$p_{y}=p\sin{(p_{\phi})}$. The last thing to consider is the trap pair
distance which will be estimated to be 1nm. This estimation comes from the
density of traps, $p=\sqrt{1/\eta}$ where $\eta$ is the density of traps on
the diamond surface. From this derivation, equation 42 becomes:

$\begin{split}\overline{|\delta
E|^{2}}=\frac{\eta/2e^{2}}{2\pi}\int_{0}^{2\pi}\int_{\Omega}\frac{p_{dip}\cdot(\vec{r}-\vec{p})}{4\pi\epsilon_{d}|\vec{r_{0}}-\vec{q}|^{3}}-\\\
\frac{ep\cdot(\vec{r}-\vec{p})}{4\pi\epsilon_{d}|\vec{r_{0}}-\vec{q}|^{3}}+\frac{p_{hex}\cdot(\vec{r}-\vec{p})}{4\pi\epsilon_{d}|\vec{r_{0}}-\vec{q}|^{7}}d^{2}qdp_{\phi},\end{split}$
(54)

where $\eta$ has been halved as the dipoles effectively halve the number of
contributing traps. Due to the fact that the charge density is Gaussian, an
even function, equation 53 shows that the first order term in the expansion
will be an odd function and will integrate to zero. In fact, all odd terms in
the expansion will be odd functions, so only the dipole ($p_{dip}$) and
hexapole ($p_{hex}$) terms remain in the above equation which are derived from
equation 54. When adding the dipole orientations in the integral, equation 54
becomes:

$\begin{split}\overline{|\delta
E|^{2}}=\frac{\eta/2e^{2}}{4\pi}\int_{0}^{2\pi}\int_{\Omega}\frac{e\big{(}p\cos{(p_{\phi})}+p\sin{(p_{\phi})}\big{)}\cdot(\vec{r}-\vec{p})}{4\pi\epsilon_{d}|\vec{r_{0}}-\vec{q}|^{3}}\\\
-\frac{e\big{(}p\cos{(p_{\phi})}+p\sin{(p_{\phi})}\big{)}\cdot(\vec{r}-\vec{p})}{4\pi\epsilon_{d}|\vec{r_{0}}-\vec{q}|^{3}}+\frac{p_{hex}\cdot(\vec{r}-\vec{p})}{4\pi\epsilon_{d}|\vec{r_{0}}-\vec{q}|^{3}}d^{2}qdp_{\phi},\end{split}$
(55)

note how with the expansion of the dipole terms it becomes clear that the
first two terms of the integral will cancel each other out, leaving just the
hexapole term which has an analytic solution that will be referred to as
$\overline{|\delta E|^{2}}_{d}$:

$\begin{split}\overline{|\delta
E|^{2}}_{d}=\frac{15p^{2}e^{4}\eta\big{(}\alpha+\beta\big{)}}{8192\pi
q_{z}^{6}\epsilon_{d}^{2}}\\\
\alpha=3\sigma_{x}^{4}+2\sigma_{x}^{2}\sigma_{y}^{2}\\\
\beta=3\sigma_{y}^{4}-8(\sigma_{x}^{2}+\sigma_{y}^{2})\sigma_{z}^{2}+&8\sigma_{z}^{4},\end{split}$
(56)

where $\sigma_{n}$ is the standard deviation in a Cartesian direction for the
Gaussian approximation to the wavefunction and $q_{z}$ is the distance from
the diamond surface to the NV. Using the Lorentzian spectral noise density
from equation 44 the linewidth broadening due to a dipole surface of charges
will then become:

$\Gamma_{dipole}=\frac{2\pi}{\hbar^{2}}\overline{|\delta
E|^{2}}_{d}\frac{\gamma/\pi}{\gamma^{2}},$ (57)

the $\sigma$ values are chosen for a wavefunction confined to a volume tht is
that is 250 nm in length/width and 100 nm in depth, the NV is 50 nm in depth
($q_{z}$) and the density of traps is $10^{18}$ m-2. Whilst the charge hopping
rate ($\gamma$) is unknown, a range of values from kHz to GHz can be tested
see its effect on the overall linewidth. Even when considering extremely fast
charge motion on the surface (THz in order) the broadening can be as high as
$10^{15}$ Hz for a typical trap density ($10^{18}$ m-2), as much as 6 orders
of magnitude broader than the limiting requirement.

Calculating the broadening in the monopole regime is largely the same as the
dipole regime but the potential and the subsequent expansion will be for a
monopole charge source. Following the logic from equation 52:

$\begin{split}\bra{F_{0}}V(\vec{r},\vec{q})\ket{F_{0}}=\frac{e}{4\pi\epsilon_{d}}\int\frac{F_{0}^{2}}{|\vec{r}-\vec{q}|}d^{3}r\\\
\approx\frac{ep_{0}}{4\pi\epsilon_{d}|\vec{r}-\vec{q}|}+\sum_{i}\frac{ep_{1i}(r_{0i}-q_{i})}{4\pi\epsilon_{d}|\vec{r_{0}}-\vec{q}|^{3}}+\\\
\frac{1}{2}\sum_{i,j}\frac{ep_{2ij}(r_{0i}-q_{i})(r_{0j}-q_{j})}{4\pi\epsilon_{d}|\vec{r_{0}}-\vec{q}|^{5}},\end{split}$
(58)

where:

$\begin{split}p_{0}=\int&F_{0}^{2}d^{3}r\\\ p_{1i}=-\int
F_{0}^{2}&(r_{0i}-r_{i})d^{3}r\\\ p_{2ij}=\int
F_{0}^{2}\Big{(}3(r_{0i}-r_{i})&(r_{0j}-r_{j})-\delta_{ij}(r_{0i}-r_{i})^{2}\Big{)}d^{3}r,\end{split}$
(59)

and the octupole term can be rewritten as:

$\begin{split}p_{2ij}=\frac{1}{2}\sum_{i}\frac{(2\sigma_{i}^{2}-\sum_{j\neq
i}\sigma_{j}^{2})(r_{0i}-q_{i}^{2})}{|\vec{r_{0}}-\vec{q}|^{5}}.\end{split}$
(60)

Equations 58 and 59 are almost exactly the same as equations 52 and 53 with
the exception of the substitution of $F_{0}^{2}$ for $\rho_{eff}$ to account
for the dipole nature of the charge distribution in the dipole regime. This
will change the non-zero terms in the expansion as $F_{0}^{2}$ is an even
function whereas $\rho_{eff}$ is an odd function. This means that whilst in
the dipole regime the odd terms in the expansion integrated to zero, in the
monopole regime the even terms in the expansion will integrate to zero.
Placing equations 58 and 60 into equation 42 gives:

$\begin{split}\overline{|\delta E|^{2}}=\eta
e^{2}\big{(}\frac{e}{4\pi\epsilon_{d}}\big{)}\int\Big{|}\frac{1}{|\vec{r_{0}}-\vec{q}|}-\frac{1}{|\vec{r_{NV}}-\vec{q}|}+\\\
\frac{1}{2}\sum_{i}\frac{(2\sigma_{i}^{2}-\sum_{j\neq
i}\sigma_{j}^{2})(r_{0i}-q_{i}^{2})}{|\vec{r_{0}}-\vec{q}|^{5}}\Big{|}^{2}d^{2}q,\end{split}$
(61)

equation 61 has an analytic solution which will now be referred to as
$\overline{|\delta E|^{2}}_{m}$:

$\begin{split}\overline{|\delta E|^{2}}_{m}=\big{(}\frac{\eta
e^{4}}{4\pi\epsilon_{d}}\big{)}^{2}\frac{3\pi\big{(}\alpha+\beta\big{)}}{128q_{z}^{4}}\\\
\alpha=3\sigma_{x}^{4}+2\sigma_{x}^{2}\sigma_{y}^{2}\\\
\beta=3\sigma_{y}^{4}-8(\sigma_{x}^{2}+\sigma_{y}^{2})\sigma_{z}^{2}+&8\sigma_{z}^{4},\end{split}$
(62)

substituting equation 62 into equation 41 with the Lorentzian noise spectrum
gives the linewidth broadening in the monopole regime:

$\Gamma_{monopole}=\frac{2\pi}{\hbar^{2}}\overline{|\delta
E|^{2}}_{m}\frac{\gamma/\pi}{\gamma^{2}},$ (63)

where the parameters are chosen for the same simulation as the dipole regime.
Even when considering extremely fast charge motion on the surface (THz in
order) the broadening can be as high as $10^{17}$ Hz which is 2 orders higher
than the dipole regime. This is likely due to the fact that the effective
charge density is higher in the monopole case and that charges have a much
higher freedom of movement across the diamond surface compared to the adjacent
hopping in the dipole case. In both regimes the broadening is too high for
resolving conduction band states individually. So a new approach is needed to
reduce the effective trap density. One option is to reduce the physical number
of surface traps, which is difficult to achieve, and the other is to fill the
traps with a donor layer in the diamond.

When adding a donor layer of nitrogen below the NV, the surface charges and
the donor layer act like the plates of a capacitor. The donor layer nitrogen’s
will pass their electron to the surface traps, resulting in a complete filling
of the traps. The filling of the traps will mitigate any opportunity for
electrons to hop within the traps, thereby reducing the AC electric field they
might produce. How much trap filling that occurs will depend on the donor
layer as well as the trap density. This can be described by using an effective
charge density with the following capacitor equation:

$\rho_{s}=CV=e\eta\frac{1}{e^{\frac{E_{T}-E_{N}+V(\eta,N)}{k_{b}T}+1}}.$ (64)

Where $C$ is the capacitance generated by the trap layer and the donor layer,
$E_{T}$ is the trap energy, $E_{N}$ is the donor layer energy and $V(\eta,N)$
is the potential generated in eV. The exponential function is the Fermi-Dirac
distribution which modulates the normal trap density $\eta$ based on the
energy of the system. As long as the density of donors is high enough, the
Fermi energy will be pinned to the donor layer energy and can be removed from
equation 64. Fermi pinning is a process where the surface charge density gets
high enough that the energy of the surface pins the Fermi energy at the same
point. In other words, higher dopant concentrations cause charges to go to the
surface but the surface charges in equilibrium force the charges to move back
into the bulk, pinning the energy at a point where there is only have 50%
occupation at minimum no matter how dense the dopant concentration is compared
to the trap density. Fermi pinning is in part what motivates a reduction of
the trap density instead of just increasing the dopant concentration as
pinning won’t occur if the surface trap density is low. The only unknown in
equation 64 is the potential, $V(\eta,N)$; to solve for it equation 64 is
rearranged for the trap density and the capacitance is replaced by an equation
with a capacitance per unit area for a parallel plate capacitor:
$C=\epsilon_{d}/N$ where $N$ is the distance from the diamond surface to the
nitrogen donor layer:

$\eta=\frac{eN}{\epsilon_{d}V(\eta,N)}\frac{1}{e^{\frac{E_{T}-E_{N}+V(\eta,N)}{k_{b}T}+1}},$
(65)

equation 65 can then be solved to obtain the capacitor potential $V(\eta,N)$.
The potential can be substituted into the following equation which mediates
the effective trap density by the occupation of traps created by the nitrogen
donor layer:

$\begin{split}\eta_{eff}=\eta(1-Oc(\eta,N))\\\
Oc(\eta,N)=0.5\Bigg{(}1+\frac{1}{1+e^{\frac{E_{T}-E_{N}+V(\eta,N)}{k_{b}T}}}\Bigg{)},\end{split}$
(66)

where $Oc(\eta,N)$ is the occupation of the surface traps which is a function
of the initial charge density as well as the distance the donor layer is from
the diamond surface. Equation 66 can be used in place of $\eta$ in the dipole
regime linewidth equation 57.

Figure 7: Plot of the trap occupation for a donor layer 100 nm below the
surface of the diamond. At very low surface trap densities ($\eta<\approx
4.5\times 10^{15}$ m-2) the trap occupation is 100% however when the density
is higher, then the occupation rapidly drops to 50%

Figure 7 shows shows the occupation as a function of the surface trap density
for a donor layer that is 100 nm away from the diamond surface. From equation
66 it becomes clear that as the occupation increases, the effective density
drops. If the traps are fully occupied, then the effective density is zero and
the charge hopping will also go to zero, which occurs when the trap density is
around $10^{15}$ m-2. This means that the associated AC electric field from
trap charge motion will be zero and the broadening from the surface charges
will also be zero. If however, the charge density is too high, then the
occupation will quickly drive towards 50% and the linewidth broadening will
become similar in value to the dipole regime.

To ensure that there is full trap occupation there are three options: the
first is to increase the capacitance by reducing the depth of the donor layer
($N$). This increases the ease of nitrogen donation. For example, when the
donor layer is 50 nm instead of 100 nm then the full occupation occurs at
$5.5\times 10^{15}$ m-2. There a problems with this approach, nitrogen atoms
too close to the NV center decreases coherence times and the higher
capacitance will cause the NV to donate electrons to the surface traps,
causing unwanted ionization. The second option would be to increase the Fermi
level with an external potential such as the electrode positive potential,
this will increase the Fermi level thereby increasing trap occupation for the
traps located around the electrode. This can be modelled by adding the
electrode potential to the Fermi Dirac distribution in equations 64, 65 and
66. Whilst this method will already be in effect, the potential from the
electrode is relatively small (10 mV), which has a negligible effect on the
occupation. If the electrode potential were to increase beyond 10 mV, then the
potential well would get smaller to the point where it would be difficult to
place an NV in the well volume. The third method would be to reduce the
surface trap density which whilst being technically challenging, is possible
and has the greatest overall effect on the occupation without affecting the NV
performance. This theoretical approach motivates the final result stated in
the main paper that broadening from surface traps can be completely mitigated
as long as there is a nitrogen donor layer approximately 100 nm below the
surface of the diamond (with the NV at 50 nm below the surface of the diamond)
and the surface trap density is reduced to $\approx 10^{15}$ m-2.

##  Appendix F Fidelity and contrast calculation

Readout fidelity can be understood as the probability of getting the right
answer in a process. Conversely the infidelity can be thought of as the
probability of getting the wrong answer. The probability of getting the wrong
spin state can be expressed as the sum of the probabilities for wrong
processes divided by the sum of the probabilities for all processes:

$error=\frac{p_{1}+p_{0}}{p_{0}+p_{1}+p_{a}},$ (67)

In the above equation, you are trying to get a particular spin readout
$p_{0}$, but there is a chance you could get the wrong spin state, $p_{1}$,
during photoionization or there is a chance you could get no photoionization
at all, this is because for the expected photoionization energy (2.6 eV [13]),
there is a non-zero chance of absorption which is given by $p_{a}$ where the
electron goes to the excited state in the NV instead of ionizing. The
probability of the correct readout ($p_{0}$) is given by the cross section of
photoionization at 2.6 eV in bulk diamond ($\sigma_{i,b}$), multiplied by a
further factor, $f=2.4$, which describes the increase in photoionization rate
due to the discrete energy levels (see figure 2 of the main paper). The rate
of the wrong spin readout ($p_{1}$) can be expressed as the rate of
photoionization, $\sigma_{i,b}$, multiplied by the same $f$ factor as well as
an error rate given as a Lorentzian function. The Lorentzian function
describes how the energy levels and linewidth broadening affects the
probability of getting the correct spin state in an ionization readout
process. It is a function of the difference in energy separation between the
ground state NV triplet and the conduction band energy levels as well as the
total linewidth broadening:

$\mathcal{L}(\Delta E)=\frac{\phi/\pi}{\Delta E^{2}+\phi^{2}},$ (68)

where $\phi$ is the total linewidth broadening calculated by adding all the
sources of broadening listed in this paper ($0.757$ GHz). The splitting of the
NV ground state triplet ($m_{s}=0$ and $m_{s}=\pm 1$ respectively) is known to
be: $\Delta D\approx 2.87$ GHz [26, 9] and the splitting of the first two
conduction band levels under the electrode is calculated to be: $\Delta
C\approx 9.945$ GHz using equation 1. The $\Delta E$ term in the Lorentzian
function is then the difference in energy separation between the conduction
band energy levels and the ground state NV triplet energy levels ($\Delta
E=\Delta C-\Delta D=7.075$ GHz). The final term in equation 67 is the
absorption probability $p_{a}$ which can be expressed as the photoionization
cross-section multiplied by the ratio of photoionization to absorption,
$g=0.4$, at the energy of photoionization (2.6 eV). This ratio can be found
from the photoionization/absorption cross section data by Razinkovas et al
[25]. Substituting all the relevant terms into equation 67:

$\begin{split}error=\frac{f\sigma_{i,b}\mathcal{L}(\Delta
E)+g\sigma_{i,b}}{f\sigma_{i,b}+f\sigma_{i,b}\mathcal{L}(\Delta
E)+g\sigma_{i,b}}\\\ =\frac{\mathcal{L}(\Delta E)+g/f}{1+\mathcal{L}(\Delta
E)+g/f},\end{split}$ (69)

note that equation 69 gives the error rate or the infidelity (15%). To find
the fidelity, the difference must be used (1-0.15), which is the equation used
in the main paper that gives the readout fidelity of 85%. To calculate the
optical spin contrast, the same methodology is applied, but the contrast is
given by the difference in probabilities for readout divided by the
probability for all processes:

$\begin{split}C=\frac{p_{0}-p_{1}}{p_{0}+p_{1}+p_{a}}\\\
=\frac{1-\mathcal{L}(\Delta E)}{1+\mathcal{L}(\Delta E)+g/f},\end{split}$ (70)

solving equation F4 gives an optical spin contrast of 85%. The calculation of
fidelity and contrast both dictate the requirements of the SCC protocol. The
three main factors that improve fidelity and contrast include: smaller
linewidth broadening, increasing photoionization probability and increasing
the energy splitting in the low lying conduction band states.

## References

  * Barry  _et al._ [2016] J. F. Barry, M. J. Turner, J. M. Schloss, D. R. Glenn, Y. Song, M. D. Lukin, H. Park, and R. L. Walsworth, Optical magnetic detection of single-neuron action potentials using quantum defects in diamond, Proceedings of the National Academy of Sciences 113, 14133 (2016).
  * Kucsko  _et al._ [2013] G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, Nanometre-scale thermometry in a living cell, Nature 500, 54 (2013).
  * Le Sage  _et al._ [2013] D. Le Sage, K. Arai, D. R. Glenn, S. J. Devience, L. M. Pham, L. Rahn-Lee, M. D. Lukin, A. Yacoby, A. Komeili, and R. L. Walsworth, Optical magnetic imaging of living cells, Nature 496, 486 (2013).
  * Barson  _et al._ [2021] M. S. Barson, L. M. Oberg, L. P. McGuinness, A. Denisenko, N. B. Manson, J. Wrachtrup, and M. W. Doherty, Nanoscale Vector Electric Field Imaging Using a Single Electron Spin, Nano Letters 21, 2962 (2021).
  * Maurer  _et al._ [2012] P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. Ignacio Cirac, and M. D. Lukin, Room-temperature quantum bit memory exceeding one second, Optics InfoBase Conference Papers 336, 1283 (2012).
  * Wu  _et al._ [2019] Y. Wu, Y. Wang, X. Qin, X. Rong, and J. Du, A programmable two-qubit solid-state quantum processor under ambient conditions, npj Quantum Information 5, 10.1038/s41534-019-0129-z (2019).
  * Nizovtsev  _et al._ [2005] A. P. Nizovtsev, S. Y. Kilin., F. Jelezko, T. Gaebal, I. Popa, A. Gruber, and J. Wrachtrup, A quantum computer based on NV centers in diamond: Optically detected nutations of single electron and nuclear spins, Optics and Spectroscopy (English translation of Optika i Spektroskopiya) 99, 233 (2005).
  * Balasubramanian  _et al._ [2009] G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, Ultralong spin coherence time in isotopically engineered diamond, Nature Materials 8, 383 (2009).
  * Doherty  _et al._ [2013] M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, The nitrogen-vacancy colour centre in diamond, Physics Reports 528, 1 (2013).
  * Babinec  _et al._ [2010] T. M. Babinec, B. J. Hausmann, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Lončar, A diamond nanowire single-photon source, Nature Nanotechnology 5, 195 (2010).
  * Jamali  _et al._ [2014] M. Jamali, I. Gerhardt, M. Rezai, K. Frenner, H. Fedder, and J. Wrachtrup, Microscopic diamond solid-immersion-lenses fabricated around single defect centers by focused ion beam milling, Review of Scientific Instruments 85, 10.1063/1.4902818 (2014).
  * Wan  _et al._ [2018] N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, Efficient Extraction of Light from a Nitrogen-Vacancy Center in a Diamond Parabolic Reflector, Nano Letters 18, 2787 (2018).
  * Aslam  _et al._ [2013] N. Aslam, G. Waldherr, P. Neumann, F. Jelezko, and J. Wrachtrup, Photo-induced ionization dynamics of the nitrogen vacancy defect in diamond investigated by single-shot charge state detection, New Journal of Physics 15, 0 (2013).
  * Jaskula  _et al._ [2019] J. C. Jaskula, B. J. Shields, E. Bauch, M. D. Lukin, A. S. Trifonov, and R. L. Walsworth, Improved Quantum Sensing with a Single Solid-State Spin via Spin-to-Charge Conversion, Physical Review Applied 11, 1 (2019).
  * Siyushev  _et al._ [2019] P. Siyushev, M. Nesladek, E. Bourgeois, M. Gulka, J. Hruby, T. Yamamoto, M. Trupke, T. Teraji, J. Isoya, and F. Jelezko, Photoelectrical imaging and coherent spin-state readout of single nitrogen-vacancy centers in diamond, Science 363, 728 (2019).
  * Gulka  _et al._ [2021] M. Gulka, D. Wirtitsch, V. Ivády, J. Vodnik, J. Hruby, G. Magchiels, E. Bourgeois, A. Gali, M. Trupke, and M. Nesladek, Room-temperature control and electrical readout of individual nitrogen-vacancy nuclear spins, Nature Communications 12, 1 (2021).
  * Zhang  _et al._ [2021] Q. Zhang, Y. Guo, W. Ji, M. Wang, J. Yin, F. Kong, Y. Lin, C. Yin, F. Shi, Y. Wang, and J. Du, High-fidelity single-shot readout of single electron spin in diamond with spin-to-charge conversion, Nature Communications 12, 1 (2021).
  * Goldman  _et al._ [2015] M. L. Goldman, A. Sipahigil, M. W. Doherty, N. Y. Yao, S. D. Bennett, M. Markham, D. J. Twitchen, N. B. Manson, A. Kubanek, and M. D. Lukin, Phonon-induced population dynamics and intersystem crossing in nitrogen-vacancy centers, Physical Review Letters 114, 1 (2015).
  * Oberg  _et al._ [2019] L. M. Oberg, E. Huang, P. M. Reddy, A. Alkauskas, A. D. Greentree, J. H. Cole, N. B. Manson, C. A. Meriles, and M. W. Doherty, Spin coherent quantum transport of electrons between defects in diamond, Nanophotonics 8, 1975 (2019).
  * De Oliveira  _et al._ [2017] F. F. De Oliveira, D. Antonov, Y. Wang, P. Neumann, S. A. Momenzadeh, T. Häußermann, A. Pasquarelli, A. Denisenko, and J. Wrachtrup, Tailoring spin defects in diamond by lattice charging, Nature Communications 8, 10.1038/ncomms15409 (2017).
  * Sasama  _et al._ [2020] Y. Sasama, T. Kageura, K. Komatsu, S. Moriyama, J. I. Inoue, M. Imura, K. Watanabe, T. Taniguchi, T. Uchihashi, and Y. Takahide, Charge-carrier mobility in hydrogen-terminated diamond field-effect transistors, Journal of Applied Physics 127, 10.1063/5.0001868 (2020).
  * Silchter [1996] C. P. Silchter,  _Principles of Magnetic Resonance_ , 3rd ed. (Springer, Heidelberg, 1996).
  * Oberg  _et al._ [2020] L. M. Oberg, M. O. De Vries, L. Hanlon, K. Strazdins, M. S. Barson, M. W. Doherty, and J. Wrachtrup, Solution to Electric Field Screening in Diamond Quantum Electrometers, Physical Review Applied 14, 1 (2020).
  * Stacey  _et al._ [2019] A. Stacey, N. Dontschuk, J. P. Chou, D. A. Broadway, A. K. Schenk, M. J. Sear, J. P. Tetienne, A. Hoffman, S. Prawer, C. I. Pakes, A. Tadich, N. P. de Leon, A. Gali, and L. C. Hollenberg, Evidence for Primal sp 2 Defects at the Diamond Surface: Candidates for Electron Trapping and Noise Sources, Advanced Materials Interfaces 6, 1 (2019).
  * Razinkovas  _et al._ [2021] L. Razinkovas, M. W. Doherty, N. B. Manson, C. G. Van De Walle, and A. Alkauskas, Vibrational and vibronic structure of isolated point defects: The nitrogen-vacancy center in diamond, Physical Review B 104, 1 (2021).
  * Manson  _et al._ [2018] N. B. Manson, M. Hedges, M. S. Barson, R. Ahlefeldt, M. W. Doherty, H. Abe, T. Ohshima, and M. J. Sellars, NV–N+ pair centre in 1b diamond, New Journal of Physics 20, 10.1088/1367-2630/aaec58 (2018).