11institutetext:  Fluids and Flows group, Department of Applied Physics and J.
M. Burgers Centre for Fluid Dynamics, Eindhoven University of Technology, P.O.
Box 513, 5600 MB Eindhoven, The Netherlands, EU

Thermal convection Rotating and swirling flows Velocity measurements

# Velocimetry in rapidly rotating convection: spatial correlations, flow
structures and length scales

Matteo Madonia 11 Andrés J. Aguirre Guzmán 11 Herman J. H. Clercx 11 Rudie P.
J. Kunnen E-mail: 11<EMAIL_ADDRESS>11

###### Abstract

Rotating Rayleigh–Bénard convection is an oft-employed model system to
evaluate the interplay of buoyant forcing and Coriolis forces due to rotation,
an eminently relevant interaction of dynamical effects found in many
geophysical and astrophysical flows. These flows display extreme values of the
governing parameters: large Rayleigh numbers $Ra$, quantifying the strength of
thermal forcing, and small Ekman numbers $E$, a parameter inversely
proportional to the rotation rate. This leads to the dominant geostrophic
balance of forces in the flow between pressure gradient and Coriolis force.
The so-called geostrophic regime of rotating convection is difficult to study
with laboratory experiments and numerical simulations given the requirements
to attain simultaneously large $Ra$ values and small values of $E$. Here, we
use flow measurements using stereoscopic particle image velocimetry in a
large-scale rotating convection apparatus in a horizontal plane at mid-height
to study the rich flow phenomenology of the geostrophic regime of rotating
convection. We quantify the horizontal length scales of the flow using spatial
correlations of vertical velocity and vertical vorticity, reproducing features
of the convective Taylor columns and plumes flow states both part of the
geostrophic regime. Additionally, we find in this horizontal plane an
organisation into a quadrupolar vortex at higher Rayleigh numbers starting
from the plumes state.

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##  1 Introduction

Rayleigh–Bénard convection, the flow in a fluid layer between two parallel
horizontal plates where the bottom plate is at a higher temperature than the
top, is a canonical model for buoyancy-driven flows. The addition of rotation
is a popular and interesting extension, as the combination of buoyant forcing
and rotation captures two of the principal constituents of many flows in
geophysics and astrophysics. It is a mathematically well-defined problem that
lends itself well to both numerical and experimental investigation (with
appropriate lateral confinement) and stands out as a turbulent flow problem
where numerical and experimental results can be compared one-to-one to great
success [1].

Three principal parameters are required to describe rotating Rayleigh–Bénard
convection (RRBC). Here we shall use the Rayleigh number $Ra=g\alpha\Delta
TH^{3}/(\nu\kappa)$, quantifying the strength of thermal forcing relative to
dissipation, the Ekman number $E=\nu/(2\Omega H^{2})$, the ratio of viscous
forces to Coriolis forces, and the Prandtl number $Pr=\nu/\kappa$, describing
the diffusive properties of the fluid. Here $g$ is the gravitational
acceleration, $\Delta T$ the temperature difference between the plates and $H$
their vertical separation, $\Omega$ represents the rotation rate, and
$\alpha$, $\nu$ and $\kappa$, respectively, are the coefficient of thermal
expansion, kinematic viscosity and thermal diffusivity of the fluid. Another
popular parameter is the convective Rossby number $Ro_{c}=E\sqrt{Ra/Pr}$ that
combines the three previous parameters to directly compare the strength of
buoyancy to Coriolis forces. The most popular geometry for experiments is an
upright cylinder, where the diameter-to-height aspect ratio $\Gamma=D/H$
characterises its shape.

As a result of their vast proportions, geophysical and astrophysical flows are
governed by extreme values of the dimensionless parameters defined before,
with $Ra\gtrsim 10^{15}$ and $E\lesssim 10^{-10}$ [2, 3, 4, 5]. At the same
time, the critical Rayleigh number $Ra_{C}$ for onset of convective motion is
significantly higher when rotation is applied: for $Pr>0.68$ linear stability
theory gives the asymptotic result $Ra_{C}=8.6956E^{-4/3}$ in the limit of
small $E$ [6]. So, despite the huge $Ra$ values most of these natural flows
are still dominated by rotation, implying that the supercriticality
$Ra/Ra_{C}$ retains more modest values. This rotational constraint leads to
the dominant force balance being between pressure gradient and Coriolis force,
the so-called geostrophic balance [7]. In recent years it was found that this
geostrophic state of rotating convection displays a diverse set of flow
structures (cells, convective Taylor columns (CTCs), plumes and geostrophic
turbulence (GT)) [8, 9]. Each of these flow states is expected to display
characteristic scaling behaviour for flow statistics, like the convective heat
transfer and intensity of velocity fluctuations. To enter this flow state one
needs to resort to specialised tools: asymptotically reduced models (e.g.
Refs. [10, 8]), large-scale numerical simulations on fine meshes (e.g. Refs.
[11, 12, 13]) or large-scale experiments [14]. Next to heat transfer
measurements, there is a great demand for experimental flow analysis of
geostrophic convection; here we present the first results in that direction.

In this work we employ our large-scale convection apparatus TROCONVEX [14, 15]
to obtain experimental flow data with extensive coverage of the geostrophic
regime, achieving unprecedented values of the governing parameters. The
measurement method is stereoscopic particle image velocimetry (SPIV) [16], a
technique that resolves full three-component flow velocity vectors in a planar
cross-section. We explore the various flow features that make up the
geostrophic regime and quantify their characteristic horizontal length scales
using spatial correlations of vertical velocity and vertical vorticity,
testing recent scaling arguments [17, 18] for the horizontal length scale of
rapidly rotating convection. This correlation procedure has been employed to
interpret results of numerical simulations [19]. We have successfully applied
this method to experimental velocimetry results from a smaller experiment
[20]. We distinguish the different flow structures from these spatial
autocorrelations and the related length scales. At the same time, we search
for so-called large-scale vortices (LSVs): simulations have shown that the GT
state can develop a striking upscale energy transfer leading to the growth of
domain-filling LSVs [8, 21, 22, 12, 13]. LSVs are so far only observed in
numerical simulations of RRBC without lateral confinement: depending on the
fluid and the operating parameters, a single LSV or a large dipolar vortex
could be observed [23, 13]. It is thus not clear whether LSVs can manifest in
RRBC experiments, too. Finally, with these flow measurements we explore the
regime of rotation-influenced turbulence (RIT) that we inferred from our
earlier heat flux and temperature measurements [15], with largely unknown flow
properties.

##  2 Experimental arrangement

TROCONVEX is a large-scale rotating convection apparatus using water as the
working fluid. Here we provide a short description; for details we refer to
our previous works [14, 15]. The convection cell is an upright cylinder of
inner diameter $D=0.39\un{m}$. Its modular structure allows heights $H$
between $0.8$ and $4\un{m}$; here we use $H=2\un{m}$ exclusively for an aspect
ratio $\Gamma\approx 1/5$. The top and bottom plates are made of copper. The
bottom plate is electrically heated and regulated to a constant temperature
$T_{b}$. The top plate is equipped with a double spiral groove through which
cooling liquid is circulated; a thermostated bath and chiller combined
regulate the temperature to a constant value $T_{t}$. This convection cell is
placed on a rotating table.

Different from the previous heat transfer measurements [15], we now employ a
custom-made transparent acrylic cylinder section. A sketch of the SPIV
arrangement is shown in figure 1. A custom-made water-filled prism around the
cylinder enables optical access from outside without too much diffraction on
the cylinder surface. A laser light sheet $\approx 3.5\un{mm}$ thick crosses
the tank horizontally at mid-height, pulsing at frequencies of $7.5$ or
$15\un{Hz}$; chosen according to the typical flow speeds. The water is seeded
with polyamid seeding particles of nominal size $5\un{\upmu m}$. The
illuminated particle images are recorded with two cameras (Jai SP-500M-CXP2;
$5\un{Mpixel}$) placed on opposite sides of the cylinder at an angle of
$\approx 45^{\circ}$ with the vertical. Scheimpflug adapters [16] rotate lens
and image plane so that the full intersection area is imaged in focus. This
stereoscopic arrangement allows for SPIV evaluation [16] of all three velocity
components in the light sheet plane; the resulting velocity field fits $122$
vectors in the diameter at a vector separation of $3.2\un{mm}$ in both
horizontal directions. Here we analyse between $3000$ and $9000$ vector fields
per experiment, a duration of $200-600\un{s}$.

Figure 1: Sketch of the SPIV arrangement. A water-filled acrylic prism
surrounds the cylinder. The laser light sheet (green line) crosses the
cylinder at mid-height. The $45^{\circ}$ oblique planes of the prism
facilitate imaging of the intersection area of the laser sheet with the
cylinder using the cameras.

We have selected seven operating conditions that capture various flow
subregimes. These settings are indicated in a phase diagram in figure 2 with
red crosses. The mean temperature $T_{m}=(T_{b}+T_{t})/2$ is kept at
$31\un{{}^{\circ}C}$, i.e. $Pr=5.2$. A constant rotation rate
$\Omega=1.9\un{rad/s}$ is applied, i.e. $E=5\times 10^{-8}$. A major concern
in experiments of this kind is the influence of centrifugal buoyancy.
Centrifugal acceleration tilts the local vector of gravitational acceleration
radially outward, away from the vertical, leading to possibly significant
deviations from the intended flow configuration [24, 25]. The importance of
centrifual acceleration is typically quantified as the Froude number
$Fr=\Omega^{2}D/(2g)$. For the experiments discussed here $Fr=0.07$; at this
value we could not observe significant up/down asymmetry in our sidewall
temperature measurements [15] pointing at negligibly small impact of
centrifugal buoyancy.

We set bottom and top temperatures such that
$Ra\in\\{0.11,0.22,0.43,0.65,1.1,2.2,4.3\\}\times 10^{12}$, or,
correspondingly, $Ra/Ra_{C}\in\\{2.3,4.7,9.1,14,23,47,91\\}$. Note that it is
hard in practice to get to smaller $Ra/Ra_{C}$ values in large setups like
TROCONVEX, given that the small temperature differences required ($\Delta
T<0.5\un{{}^{\circ}C}$) cannot be controlled accurately enough at these scales
[14, 15]. That is why we cannot enter the cellular state and only partially
the CTC state. In addition, we consider one nonrotating case at $Ra=6.5\times
10^{11}$ for comparison.

Figure 2: Phase diagram for rotating convection in water. The expected flow
structures and transitions between them (based on heat-flux and temperature
measurements from TROCONVEX [15]) are labeled in the diagram. The experiments
discussed in this work are indicated with red crosses.

##  3 Spatial autocorrelation

We can quantify and evaluate the spatial structure of the flow features by
using spatial autocorrelations. This procedure has been applied to geostrophic
convection results from simulations [19], where the authors have shown that
the autocorrelation nicely recovers the typical radial structure of flow
features like cells and CTCs, and later also to experimental data based on PIV
and 3D-PTV (three-dimensional particle tracking velocimetry) [20]. Here we
define the spatial autocorrelation of a scalar variable $f$ as

$R_{f}(r)=\frac{\left<f(\vect{x})f(\vect{x}+\vect{r})\right>_{A,t}}{\left<f^{2}(\vect{x})\right>_{A,t}}\,,$
(1)

where $\vect{r}$ is the separation vector with length $r=|\vect{r}|$. The
angular brackets $\left<\ldots\right>_{A,t}$ denote a spatial averaging over
all available positions $\vect{x}$ within the cylinder cross-section $A$ and
all possible $\vect{r}$ emanating from there, and we apply averaging in time
$t$ as well. The effective area $A$ for the autocorrelation is deliberately
reduced to exclude the wall mode that is found near the sidewall [26, 27, 28,
29, 30, 31]; the thickness measure from [26] is employed. Within this reduced
area we assume horizontal homogeneity; $\vect{r}$ can be the vector connecting
any two points within that area. We use zero padding and 2D fast Fourier
transforms to calculate $R_{f}$, where the resulting (axisymmetric) 2D
correlation graphs are converted to 1D using binning with circular shells of
width $3.7\un{mm}$.

|  |
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Figure 3: (a–c) Flow velocity snapshots at mid-height for (a) $Ra/Ra_{C}=2.3$,
(b) $Ra/Ra_{C}=14$, (c) $Ra/Ra_{C}=91$. The background colour indicates
vertical velocity in $\un{mm/s}$; arrows represent the in-plane velocity
components. Only one ninth of the total number of vectors is plotted for
clarity of the images. (d–f) Snapshots of vorticity $\zeta$ in $\un{1/s}$ at
mid-height for (d) $Ra/Ra_{C}=2.3$, (e) $Ra/Ra_{C}=14$, (f) $Ra/Ra_{C}=91$.

Once the spatial autocorrelation $R_{f}(r)$ is known, we can compute the
characteristic correlation length (or integral length scale [32]) as

$L_{f}=\int_{0}^{\infty}R_{f}(r)\,\upd r\,.$ (2)

Nieves et al. [19] considered primarily correlations of temperature
fluctuations $\theta$, but also of vertical velocity $w$ and vertical
vorticity $\zeta=(\vect{\nabla}\vect{\times}\vect{u})_{z}$, where
$\vect{u}=(u,v,w)$ is the velocity vector. Rajaei et al. [20] could only
consider correlation of $\zeta$ from their regular PIV measurements. Here we
shall consider both $w$ and $\zeta$. We expect from theoretical models
describing the cellular and CTC states as single-wavenumber modes [33, 34]
that $\theta$, $w$ and $\zeta$ lead to very similar correlation graphs. For
plumes and GT (and possibly RIT) the correlations for $w$ and $\zeta$ may
develop significant differences in correlation length [19].

##  4 Flow snapshots

To get a first impression of the flow field that develops, we present
instantaneous snapshots of velocity and vorticity at three different
$Ra/Ra_{C}$ values in figure 3. Panels (a,d) are in the range where CTCs are
expected. We do observe vortices present in this flow, though the shields are
hard to distinguish. One contributing factor is that we measure at mid-height
where the columns are expected to be weaker; their vertical structure is
torsional, with positive (cyclonic) vertical vorticity at one vertical end and
negative (anticyclonic) vorticity at the other end [35, 33, 34]. Here we
observe in animations of the flow (see the supplementary material) that the
flow state is quite dynamical; no quasi-steady vortex grid is formed. Instead,
the vortices wander around and can interact with other vortices and the wall
mode, making them not as long-lived. Near the sidewall, the signature of the
wall mode [26, 27, 28, 29, 30, 31] can be seen: the top right half of the
circumference displays a prominent vertically upward flow (red colour), while
the opposite side shows downward flow (blue colour; though not as prominent in
this snapshot). A prominent dynamical feature of this wall mode is the
presence of two jets emanating from the azimuthal positions where the up- and
downward lobes meet pointing radially inward [26]. These jets, more easily
identified in animations of the flow (see supplementary material), appear to
play an important role in setting the flow structures in the bulk in motion.
Similar structures have been found in magnetoconvection simulations [36, 37].

At the higher $Ra/Ra_{C}=14$ (figure 3b,e), representative of plumes, a
qualitatively similar flow field is observed. Velocities are larger given that
the thermal forcing is stronger. It is readily observed that the vertical
velocity is partitioned into larger patches, i.e. a larger correlation length
is anticipated. At the same time, the vorticity remains confined to narrow
patches. The wall mode is still quite prominently visible, though the
fluctuations in the bulk have increased in magnitude compared to the amplitude
of the wall mode [26] so that its relative dynamical significance is
diminished.

Figures 3(c,f) display a snapshot at the highest $Ra/Ra_{C}=91$ considered
here, in the RIT range. Velocities are even larger than in the previous panel
due to stronger thermal forcing. The correlation length of vertical velocity
has become even larger. Contrarily, the vorticity field still reveals finer
scales comparable to the previously discussed cases (figure 3d,e).

An interesting new feature — that can be recognised from animations of the
flow field as a function of time (see supplementary material) — is the
organisation into a quadrupolar vortex state that fills the cross-sectional
area with two cyclonic and two anticyclonic swirls. This flow arrangement
forms a secondary large-scale flow somewhat reminiscent of LSV dynamics as
previously observed in simulations in laterally unbounded domains. The wall
mode couples to this quadrupolar vortex state (or the other way around) to
organise the entire flow field into a pattern that displays a slow azimuthal
drift [26]. This drift makes simple time-averaging ineffective to further
illustrate this structure. Instead, we employ an orientation averaging similar
to Ref. [26]. We determine the phase angle of the wall mode at each time step
by tracking the precession of the vertical velocity signal on a circular
trajectory close to the sidewall. Each snapshot is rotated by its
corresponding phase angle so that the orientation matches for each snapshot,
then the velocity field is averaged to get a clear view of the mean flow
profile. Figure 4 shows the result of such averaging, an orientation-
compensated mean vorticity field for the experiment at $Ra/Ra_{C}=47$. There
is an organisation as a quadrupolar vortex with four cores of vorticity that
fill the central part of the domain. We are currently considering this overall
flow structuring, its origin and any relation with the wall mode in more
detail.

Figure 4: Orientation-compensated mean vorticity field (in $\un{1/s}$) at
$Ra/Ra_{C}=47$.

For the nonrotating reference case (animation in the supplementary material)
the well-known large-scale circulation (LSC [38]; not to be confused with LSV)
of nonrotating confined convection can be observed: a principal partitioning
of vertical velocity into two patches (one with positive and one with negative
vertical velocity) that cover the entire cross-section of the cylinder.
Therefore, the largest correlation length is expected in this case, of the
order of the radius of the cylinder.

##  5 Results: autocorrelations

We plot spatial autocorrelation graphs for the rotating convection cases and
the nonrotating reference case in figure 5. In these plots we also indicate
units of $\ell_{C}$, the wavelength of convective instability
($\ell_{C}=4.8154E^{1/3}H$ for $Pr>0.68$ [6]) that plays an important role
throughout the geostrophic regime as a characteristic horizontal scale [10,
8]. For $R_{w}$, the autocorrelation of vertical velocity, we see that the
correlation is reduced as $r$ increases, reaches a negative minimum, then
approaches zero at large $r$. As $Ra/Ra_{C}$ increases, stronger thermal
forcing and more vigorous turbulence, the correlation length increases
considerably. This is true even more so for the nonrotating case, where the
minimum is out of the plotting range at $r/D=0.57$. In that case the
correlation length is determined by the presence of the LSC; it is of the
order of the cylinder radius. The $R_{w}$ results reported in Ref. [19]
display a similar elongation of the correlation as $Ra/Ra_{C}$ increases.
However, the regular oscillatory behaviour for the CTC flow range is not
reproduced here. We do not see the development of a true ensemble of shielded
Taylor columns. Instead, the bulk flow structures are moving around in a
secondary circulation that appears to be set up by the strong wall mode in
this flow range, with jets penetrating the bulk from laterally opposite sides
of the cylinder cross-section [26]. This prevents the formation and
preservation of a quasi-steady grid of CTCs as observed in simulations on
horizontally periodic domains, e.g. [8, 11], as well as in smaller-scale
experiments at larger $E$ and $\Gamma$[20].

Figure 5: Spatial autocorrelations of (a) vertical velocity $w$ and (b)
vertical vorticity $\zeta$. Length scales are normalised using the cell
diameter $D$. Vertical dashed lines display multiples of $\ell_{C}$ for
reference. Note the different horizontal axis ranges.

The vorticity correlations $R_{\zeta}$ (figure 5b) decay on comparatively
smaller scales than $R_{w}$, in agreement with the expectations based on the
snapshots of figure 3. The initial decrease at small $r$ is quite similar for
all the considered cases. This is in line with a gradual transition from the
structures of convection at small supercriticality (e.g. the single-mode
solutions [33, 34]), where equal scales are found for velocity and vorticity,
to a situation in line with nonrotating 3D turbulence composed of thin vortex
tubes (e.g. Ref. [39]) with the vorticity correlation length roughly equal to
the Kolmogorov length while the velocity correlation scale is the largest
length scale in the flow. Then, the cases at the smallest values of
$Ra/Ra_{C}$ display a small oscillation that was expected for the CTC state
based on earlier simulations [19] and experiments [20]. Its wavelength
corresponds quite nicely with the convective wavelength $\ell_{C}$. The
oscillation is not as pronounced here as in the other studies, presumably due
to the inability to form a quasi-steady CTC grid. For $Ra/Ra_{C}=9.1$ and
higher (excluding the nonrotating case) we observe the occurrence of a larger
length scale on which some correlation can be seen: starting from $r/D\approx
0.05$ these curves display a shallow downward slope, with a zero crossing at
$r/D\approx 0.2$. The curves each reach a shallow minimum at $r/D\approx
0.26-0.30$ then asymptote to zero at large $r$. This correlation signature is
a second indicator of the organisation into a quadrupolar vortex consisting of
two cyclonic and two anticyclonic cells.

We can further quantify and compare these autocorrelation results by deriving
characteristic length scales from them. We consider the integral scales
$L_{w}$ and $L_{\zeta}$ as defined before. Additionally, we define length
scales based on the correlation magnitude: $\ell_{0.5,w}$ and
$\ell_{0.5,\zeta}$ for the $r$ where the corresponding autocorrelation has the
value $0.5$, and $\ell_{0,w}$ where $R_{w}$ crosses zero for the first time.
The zero crossing is not as informative for $R_{\zeta}$ given the longer
positive correlation that is observed. These length scales are plotted as a
function of $Ra/Ra_{C}$ in figure 6. It is clear that the vorticity-based
scales are always smaller than their velocity-based counterparts. While
$L_{\zeta}$ shows some variation with $Ra/Ra_{C}$, the smaller
$\ell_{0.5,\zeta}$ remains more or less constant throughout. Indeed, the
initial decay of $R_{\zeta}$ is quite similar in all cases. At the two
smallest $Ra/Ra_{C}$ values considered here the velocity-based scales $L_{w}$
and $\ell_{0.5,w}$ are of comparable size to their vorticity-based
counterparts, as expected based on prior results for the CTC state [33, 34,
19, 20]. For larger $Ra/Ra_{C}\gtrsim 9$ the velocity-based scales become
increasingly larger. Based on this observation we expect that the CTC-to-
plumes transition takes place between $Ra/Ra_{C}=4.7$ and $9.1$. This is in
agreement with the reported transition $RaE^{4/3}=55$ (or $Ra/Ra_{C}=6.3$) for
the asymptotic simulations and fully in line with conclusions based on our
earlier heat-flux and temperature measurements in the same setup [15]. Beyond
that transition, in the plumes state, the velocity correlation widens, though
for the highest two $Ra/Ra_{C}$ values some saturation can be observed. The
correlation graphs for these cases, in the RIT range, are in line with the
plumes cases in terms of shape. The saturation of $\ell_{0,w}$ at such length
is in line with the organisation into a quadrupolar structure, where
correlation up to about one fourth of the diameter is expected. Note the
significant difference with the nonrotating case, where correlation continues
up to about half the diameter due to presence of the LSC with the cross-
sectional area divided into one half upward and one half downward flow.

Two recent works have considered the horizontal length scale of convection in
the geostrophic regime. Guervilly et al. [17] combine results of various
numerical models to find an effective scaling $\ell\sim
Ro_{U}^{1/2}\sim(RaE^{2}/Pr)^{1/2}$ with the Rossby number $Ro_{U}$ based on a
measured velocity scale $U$. In our notation this amounts to $\ell\sim
Ro_{c}$. They only find this scaling at very small $E\lesssim 10^{-9}$. Aurnou
et al. [18] provide theoretical scaling arguments based on the so-called CIA
(Coriolis–Inertial–Archimedean) force balance that also predict $\ell\sim
Ro_{c}$. In our experiments at constant $E$ with variation of $Ra$ this
translates to $\ell\sim Ra^{1/2}$. This scaling slope is included in figure 6
with the solid black line; a trend clearly steeper than our data. A power law
fit to our data for $4.7\leq Ra/Ra_{C}\leq 47$ (dashed black line) renders a
scaling $\ell\sim Ra^{0.38}$. Looking at figure 4(b) of Guervilly et al. [17],
our shallower scaling corresponds nicely to the shallower trend of their data
for $10^{-9}\lesssim E\lesssim 10^{-7}$, which indeed encloses our $E$ value.
While the scaling of the length scale is similar, comparison of the magnitude
is not possible due to differences in domain (sphere vs. cylinder) and $Pr$
value (0.01 vs. 5.2).

Figure 6: Correlation length scales as a function of $Ra/Ra_{C}$. The line
segments with open symbols are the corresponding results for the nonrotating
reference case (same symbols and colours; not to scale in terms of
$Ra/Ra_{C}$). Error intervals (not plotted) are equal to the symbol size or
smaller. The horizontal dashed line indicates the convective wavelength
$\ell_{C}$. Vertical dotted lines indicate the regime transitions of figure 2.
The black solid line indicates the scaling $\ell\sim Ra^{1/2}$; the black
dashed line is a fit $\ell\sim Ra^{0.38}$ to the data.

##  6 Conclusion

We have performed stereoscopic particle image velocimetry measurements in
rotating thermal convection in the geostrophic regime at small Ekman number
$E=5\times 10^{-8}$ where the effects of rotation are prominent. The flow
phenomenology has been quantitatively analysed using spatial correlations of
vertical velocity and vertical vorticity. The correlation length scales based
on vertical vorticity remain reasonably constant over the considered range of
supercriticality values $2.3\leq Ra/Ra_{C}\leq 91$, in line with observations
that the critical wavelength $\ell_{C}$ for onset of convection is an
important horizontal length scale throughout the geostrophic regime.
Correlation length scales of vertical velocity grow with increasing
$Ra/Ra_{C}$ and can be used to identify different flow states: the state of
convective Taylor columns (CTC) for $Ra/Ra_{C}\lesssim 6$, the plumes state
for $Ra/Ra_{C}\gtrsim 6$ changing gradually into the previously
uncharacterised state of rotation-influenced turbulence (RIT), in good
agreement with earlier results from reduced numerical models in the same $E$
range [17]. The plumes/GT and RIT ranges display an interesting new
organisation into a quadrupolar vortex (recognised from orientation-
compensated mean vorticity fields) which is presumably preferred over a single
LSV or a dipole [23, 13] in this confined domain.

Throughout these measurements we can identify the wall mode, a coherent
vertical flow structure near the sidewall consisting of one lobe with upward
transport on one half of the circumference and one downward lobe on the other
half. The prominence of the wall mode relative to the fluctuations in the
interior diminishes at higher $Ra/Ra_{C}$. At low $Ra/Ra_{C}$ the jets
emanating from the wall mode set the vortical structures in horizontal motion;
contrary to the results from horizontally periodic simulations no quasi-steady
CTC grid forms. The nature and origin of the coupling of the wall mode and the
interior flow, at both high and low limits of $Ra/Ra_{C}$, is an open question
that we want to address later.

The study of rotating convection in the geostrophic regime poses challenges to
experimentalists and numericists alike. Nonetheless, recent numerical works
have elucidated the rich flow phenomenology that develops in this convection
setting. Here, we contribute experimental results at more extreme values of
the governing parameters that expand our understanding of this intriguing and
geo-/astrophysically relevant flow problem, despite the influence of
unexpected yet intriguing features like the persistent wall mode and its
interaction with the bulk flow.

###### Acknowledgements.

M.M., A.J.A.G. and R.P.J.K. received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (Grant agreement No. 678634).

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Mitigation of Nitrogen Vacancy Ionization from Material Integration for Quantum Sensing

Jacob Henshaw
Pauli Kehayias
Luca Basso
Michael Jaris
Rong Cong
Michael Titze
Tzu-Ming Lu
Michael P. Lilly
Andrew M. Mounce

The nitrogen-vacancy (NV) color center in diamond has demonstrated great promise in a wide range of quantum sensing. Recently, there have been a series of proposals and experiments using NV centers to detect spin noise of quantum materials near the diamond surface. This is a rich complex area of study with novel nano-magnetism and electronic behavior, that the NV center would be ideal for sensing. However, due to the electronic properties of the NV itself and its host material, getting high quality NV centers within nanometers of such systems is challenging. Band bending caused by space charges formed at the metal-semiconductor interface force the NV center into its insensitive charge states. Here, we investigate optimizing this interface by depositing thin metal films and thin insulating layers on a series of NV ensembles at different depths to characterize the impact of metal films on different ensemble depths. We find an improvement of coherence and dephasing times we attribute to ionization of other paramagnetic defects. The insulating layer of alumina between the metal and diamond provide improved photoluminescence and higher sensitivity in all modes of sensing as compared to direct contact with the metal, providing as much as a factor of 2 increase in sensitivity, decrease of integration time by a factor of 4, for NV $T_1$ relaxometry measurements.

§ INTRODUCTION

The rapidly-developing field of 2-D materials has the opportunity to provide advances in the fields of data storage, magnetometry, and quantum information processing. However, due to their low-dimensional nature, established bulk characterization techniques, such as nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) spectroscopy, lack the sensitivity to properly probe electron dynamics responsible for phenomena such as magnetism and superconductivity. A series of measurements have been proposed[1, 2] and realized[3, 4]offering the nitrogen-vacancy (NV) center as a new probes of low dimensional electronic phases of quantum materials. Due to its long lived electronic spin state and ease of read-out and control, the NV is an excellent sensor of magnetic and electric noise with bandwidth ranging from DC to GHz provided through a variety of sensing modalities[5, 6, 7, 3, 8].

A challenge of working with NV centers is preserving the notable spin properties as the NVs form closer to the diamond surface. Without extensive oxidation treatments[7, 9, 10], the diamond surface can provide a strong upward band bending, depleting the NV of it electrons and converting the NV to its magnetically-insensitive neutral and positive charge states. Additionally, the surface provides a source of noise that worsens the NV's dephasing, decoherence, and relaxation[11, 12]. These problems can be exacerbated by the integration of metals, conductive materials, or materials with large work functions to the diamond surface.

The addition of a metal to the diamond surface forms a positive charge at the interface that is compensated by the negative charge in traps and defects in the diamond, like the aforementioned NV center. Other sources of negative charges, such as substitutional nitrogen ($\mathrm{N_s}$), may also be ionized in this process. This creates an involved competition with regards to sensing, where some ionization of $\mathrm{N_s}$ maybe beneficial in improving coherence properties, but too much ionization may result in destabilizing the $\mathrm{NV^-}$

Here, we explore how dense ensembles of NVs are affected by the integration of such materials. We deposited a thin film of metal onto the diamond surface and characterized the spin properties the NVs under the metal film, and compared to an uncoated area. We repeated this for a range of NV ensembles of different depths. We find that, depending on the depth of the NVs, spin properties such as $T_2^*$ and $T_2$ times, improve by as much as a factor of 1.4 and 1.6 respectively, while relaxation times and photo-luminescence rates are quenched due to proximity to the thin metal film. We tune this effect by adding a thin layer of alumina in between the metal and diamond, and find the photoluminescence intensity improves while preserving some degree of improvement to $T_2^*$ and $T_2$ times and providing minimal impact on relaxation times, preserving utility for $T_1$ relaxometry. We conclude by estimating the shot-noise-limited sensitivity in different sensing modalities, and find that for all forms of sensing, the addition of the insulating layer improves the sensitivity as compared to direct contact with the thin film.

§ METHODS

§.§ Sample Preparation

We start with a series of electronic-grade diamond (Element 6) with natural  abundance (1.1%). The samples are implanted with  and annealed to form NVs. The implant parameters of the samples are described in Table <ref>. We implant these samples to achieve a $\mathrm{N_s}$ concentration of 100 ppm according SRIM. We follow the annealing procedure and oxidation treatment described in our previous work[7]. After oxidation treatment, the samples have 2 nm of alumina ($\mathrm{Al_2O_3}$) deposited by atomic layer deposition (ALD) at 200C. The alumina is etched from half of the diamond and then a perpendicular half of the diamond has 50 nm of copper deposited using electron beam evaporation. This results in 4 regions: the bare diamond `Ref' region, diamond with alumina, `AlOx,' diamond with copper, `Cu,' and diamond with alumina and copper, `Cu + AlOx' (Fig. <ref>(a)). This is done so we can compare the effects of each environment in the same measurement series.

The energies, fluences, and SRIM-estimated depths for diamonds used in this work. All implants and SRIM simulations are done with an 8° tilt. Fluences are chosen so that the peak substitutional nitrogen ($\mathrm{N_s}$) concentration is 100 ppm. SRIM depth refers to the mode of the ion distribution.

Energy (keV)     Fluence (ions/cm$^2$)    SRIM depth (nm)

3     1$\times10^{13}$     5.2

4     1.2$\times10^{13}$     6.6

5.5     1.6$\times10^{13}$      8.7

7     2$\times10^{13}$      10.6

15     3.25$\times10^{13}$      20.8

25     4.5$\times10^{13}$      33.5

We excite NV centers using a 532 nm laser with an optical power of 280 mW (before the objective) and focused down to a roughly 40 $\mathrm{\mu m}$ spot [7]. The excitation provides a mechanism to spin initialize into the $\ket{0}$ spin state and readout the spin state from the spin-dependent fluorescence rate [13]. The samples rest on a sapphire substrate with a copper loop fabricated on it. The copper loop is connected to an amplified and gated microwave (mw) source to provide MW pulses for spin control. The photoluminescence (PL) is filtered using a 550 nm dichroic mirror, a 650 nm long-pass filter, and a 532 nm notch filter to suppress laser leakage and $\mathrm{NV^0}$ PL. The PL is detected by an A-CUBE-S1500-3 APD with variable gain. We perform scanning PL measurements using the stepper motors of a Thorlabs NanoMax 300.

(a) A schematic of the alumina and copper patterned on the diamond. (b) Photoluminescence(PL) image of the four quadrants of the 7 keV implanted diamond. The red dashed lines approximately indicate the lines where data was taken. (c-f) Normalized data from the four different regions for NV Polarization Time (c), Hahn Echo for $T_2$ (d), Ramsey interferometry for $T_2^*$ (e), and $T_1$ relaxation (f). The corresponding decay times of the different measurements are given as insets. For spin property measurements the pulse sequence used is shown.

§.§ Sample characterization

For each sample, we measure the PL intensity (Fig. <ref>(b)), NV polarization time (Fig. <ref>(c)), $T_2$ (Fig. <ref>(d)), $T_2^*$ (Fig. <ref>(e)), and $T_1$ (Fig. <ref>(f)). Representative data for a single point in each region is shown in Fig <ref>. The NV polarization time is measured by measuring two consecutive PL time traces, one with a mw $\pi$ pulse, and one without, and taking the difference between the time traces. This can be a probe of processes like Fluorescence Resonant Energy Transfer (FRET) [14], as the reduced excited-state lifetime reduces polarization efficiency. We measure $T_2$ and $T_2^*$ using Hahn echo (Fig. <ref>(d)) and Ramsey interferometry (Fig. <ref>(e)) respectively. Both measurements are done on the $\ket{0} \leftrightarrow \ket{-1}$ transition of the NV ground-state spin sublevels and are thus not immune to strain and electric field fluctuations[15, 16]. The second $\pi/2$ of both of these measurements toggles between a $\pi/2$ and $3\pi/2$ pulse and the difference between two consecutive measurements is taken to suppress noise. $T_2^*$ is the limiting timescale for ODMR linewidth and an important value for DC magnetometry with NVs. $T_2$ is the lifetime of a coherent state and is the limiting timescale for nanoscale NMR spectroscopy with NVs or sensing of low-frequency (100s kHz - 10s MHz) noise.  The $T_1$ measurement is referenced by applying a $\pi$ pulse on every other measurement and subtracting two consecutive measurements. The relaxation time, $T_1$, is sensitive to noise near the NV spin resonance frequency, providing a sensing mechanism over a wide range of frequencies tuned with a magnetic field[8, 17, 18].

In order to account for sample and microwave driving inhomogeneities, spin measurements are performed at points spaced by 25-50 $\mathrm{\mu m}$ along a pair of horizontal lines 500 $\mathrm{\mu m}$ (750 $\mathrm{\mu m}$ for $T_1$) long, from the 1Cu + AlOx' region into the 1AlOx' region and from the `Cu' to `Ref' regions (Red lines in Fig <ref>(b)). At each point we measure Rabi oscillations and NV resonance frequency along with the spin properties of the NVs. The PL rate, NV polarization time, contrast-weighted shot-noise ($C\sqrt{I_{pl}t_{ro}}$), $T_2^*$, $T_2$, $T_1$ are then averaged in each of the regions. No notable spatial dependence was observed within any given region or near the transition from one region to another for all properties except $T_1$. The $T_1$ of shallower NV ensembles did not reach a stable reference level until the probed position was a few hundred $\mathrm{\mu m}$ away from the metal edge. These measurements and processing steps are performed for all four regions across all six samples, with the exception of the 3 keV sample, where only the `Cu + AlOx' and `AlOx' areas are measured due to the signal-to-noise ratio of the `Cu' area being too small to achieve usable signal in a reasonable time. For the NV ensemble samples with depths less than 10 nm, no notable differences were observed between the `Ref' region and the 1AlOx' region. For the deepest two NV ensembles, a slight increase in PL was observed for 1AlOx' relative to `Ref.'

Due to the wide range of PL rates across the various NV ensembles, the gain of our APD needed to be adjusted from one diamond to another. Due to this, we do not quote explicit photon count rates as the APD responsivity and noise floor is not the same from diamond to diamond. To this end, all measurements or calculations that require a PL rate are expressed as ratios between regions. This still provides the critical information of relative PL rates in different regions across a single diamond.

§ RESULTS AND DISCUSSION

§.§ NV characterization

Depth dependence of NV properties critical to sensing. The ratios between different regions for Photluminescence (PL) (a), Spin Contrast (b), Polarization time (c), contrast-weighted shot-noise($C\sqrt{I_{pl}t_{ro}}$) (d), $T_2^*$ (e), and $T_2$ (f), as a function of depth. The color gradient indicates sensitivity improvement (green) or worsening (red).Error bars are the greater of the fit errors or the standard deviation of the line cut data for a region.

We perform the previously mentioned measurements across a series of diamond samples with variable depth (Table  <ref>). In Fig. <ref>, we show the ratios of NV properties critical to DC and AC sensing between different regions. We emphasize that, although for shallower NV ensembles the properties in both `Cu' and `Cu + AlOx' regions are strictly worse than the `Ref' region, the relevant comparison for sensing of nanoscale systems is `Cu + AlOx' to `Cu.' However, comparisons between the Cu coated regions and the `Ref' region do provide interesting insight into the changes in the environments in the differing regions caused by the integration of the material.

The PL rate, shown as a function of depth for the different regions, in Fig. <ref>(a), shows a gradual increase in PL relative to the `Ref' region for both `Cu' and `Cu+AlOx' regions. Importantly, we saw the `Cu + AlOx' region has consistently high PL than the `Cu' region. The level of improvement decreases for deeper NV ensembles. For deeper NV ensembles we find the PL actually increases by as much as a factor of 4. This may be due to the metal and alumina increasing collection efficiency by acting as a mirror and reflective coating or an increase in the spontaneous emission rate via plasmonic interaction [19].

We also observe the PL contrast between spin states in the `Cu + AlOx' and `Cu' regions to be much lower than that of the `Ref' region, plateauing at 10 nm. This may be due to background PL from $\mathrm{NV^0}$ for shallower NVs. Also, a slightly higher Rabi frequency was observed for the regions "Ref' and `AlOx' due to closer proximity to the mw loop. This could result in slightly lower contrast due to lower excitation bandwidth. When comparing `Cu+ AlOx' to `Cu,' we find an average 25% increase in contrast with no clear depth dependence.

We observe the polarization time increasing by as much as a factor of two for the shallowest NV ensembles (Fig. <ref>(c)), when compared to the `Ref' region. A fast polarization time for the NV is essential to reducing the overhead for of measurements. We attribute this increase to a reduced excited state lifetime due non-radiative relaxation cause by the metal through processes like FRET or Surface Energy Transfer (SET)[14]. For deeper ensembles, we find the polarization rate improves. We attribute this to the thin metal film acting as a  mirror and providing better laser excitation. The `Cu + AlOx' region is less impacted by the these processes due to an additional 2 nm stand off from the material.

An important parameter that appears in all shot-noise-limited sensitivity estimates is $C\sqrt{I_{pl}t_{ro}}$. This value is the fluorescence contrast between spin states times the shot-noise of a single measurement. When considering the comparison to the `Ref' region, we notice a dramatic drop, as low as a fifth the reference value (Fig. <ref>(d)). This is a mixture of notably reduced PL caused by band bending ionizing $\mathrm{NV^-}$, as well on non-radiative relaxation reducing photon generation from $\mathrm{NV^-}$[14] However, the comparison between `Cu + AlOx' and `Cu' regions sees a notable increase in this parameter for the shallowest NVs.

§.§ Dephasing and Decoherence

We observe an interesting phenomenon when we look at $T_2^*$, the dephasing time (Fig. <ref>(e)), and $T_2$, the decoherence time (Fig. <ref>(f)). We find the coherence properties of the NVs improve under the metal. To explore why this happens we consider what are the major causes of decoherence and dephasing at this depth and nitrogen density. At the depths of our ensembles, surface noise from dangling bonds or other surface imperfections (Fig <ref>(a)) has been seen to play a major role in decoherence[11, 12]. However, due to our very high nitrogen concentration (100 ppm), we posit paramagnetic noise from the $\mathrm{N_s}$ (Fig <ref>(a)) is the dominant decoherence and dephasing source[15].

We provide a qualitative explanation for the trend in $T_2^*$ and $T_2$ through a competition between these two noise sources, with the $\mathrm{N_s}$ being ionized by the band bending caused by the metal. The $\mathrm{N_s}$ is known to have a donor level 1.7 eV below the conduction band[20]. The $\mathrm{NV^-}$ ground state level has been found to be 2.6 eV below the conduction band[21]. We propose that for very shallow NV centers, both the $\mathrm{N_s}$ and the NV are ionized by the band bending; the Fermi level drops below the defect levels(Fig <ref>(b)). This regime provides a substantial decrease in PL, with spin properties dominated by surface noise, but also reducing the noise environment created by the $\mathrm{N_s}$. As the NVs get deeper, an ideal depth appears where there is sufficient $\mathrm{N_s}$ to charge NVs into the negatively charged state but not so much $\mathrm{N_s}$, that the NVs are still dominated by their noise. This regime is defined by the Fermi level being greater than the NV level but less than the $\mathrm{N_s}$ level. As NVs get sufficiently deep, the influence of the band bending becomes negligible, and the $\mathrm{N_s}$ keep their electron and the NVs become dominated by the nitrogen noise again, where the Fermi level approaches its bulk value determined by the nitrogen doping level. We show the explicit $T_2^*$ and $T_2$ values as a function of depth (Fig <ref>(c,d)) with relevant regions highlighted according to our qualitative description.

Proposed mechanism behind improved coherence times (a) Diagram of the diamond surface indicating major decoherence dephasing sources like surface states and $\mathrm{N_s}$. (b) Band diagram depicting the NV and ($\mathrm{N_s}$) level under the influence of the band bending caused by the metal. (c) $T_2^*$ and (d) $T_2$ as a function of depth for the various regions.

An important note to this discussion is that this is an indirect effect. The metal or material is not directly reducing the $T_2$ or $T_2^*$; it is engineering the electronic environment in such a way that has an impact on the NV magnetic noise environment. Additionally, these changes are very much nitrogen concentration dependent; lower nitrogen densities may simply not have enough electrons to compensate the surface charge induced by the metal. Any sort of relaxometry using $T_2$ or $T_2^*$ would need to account for these environmental changes. For other modes of sensing, such as DC sensing which is $T_2^*$ limited[22], or nanoscale NMR, which is $T_2$ limited[7, 22],  this change in lifetimes is a strict benefit.

§.§ Spin-Lattice Relaxation

While $T_2$ and $T_2^*$ are critical parameters for sensing of DC magnetic fields and low frequency sources on the order of MHz, many proposals for using NVs to probe nanoscale electronic states use $T_1$ relaxometry[1, 23]. $T_1$ relaxometry has already been used as a probe of conductivity in metals[24, 25]. We use the previously-established techniques to demonstrate two features: we can recover the same information from $T_1$ relaxometry measurements with and without the alumina film and, for sensing Johnson noise in conductors, the additional stand-off provides no major deficits and even improves the sensitivity of $T_1$ relaxometry in the sample dominated regime, where the induced relaxation rate from the metal is much greater than the intrinsic relaxation rate.

Measurements of thin film conductivity. (a) $T_1$ for the various regions as a function of depth. (b)Relaxation rate from external sources, extracted from data in (a) as a function of depth. This is fit to Eqn.  <ref> to estimate the conductivity of the thin metal film. The extracted conductivities agree with each other and previous measurements of thin copper films.

In order extract the conductivity from our $T_1$ data, shown in Fig <ref>(a), we follow the process laid out in Ref.[24]. We must determine the relaxation rate from external sources, $\Gamma_{ext}(d,\sigma)$. Here d is the stand-off between the NV ensemble and the metal, and $\sigma$ is the conductivity of the metal. We do this by measuring the intrinsic relaxation rate, $\Gamma_{NV,int}$, of NVs unperturbed by the metal and the relaxation rate of NVs affected by the metal, $\Gamma_{NV}(d,\sigma)$. Due to the nature of our experimental configuration, we can use the `Ref' region to determine our $\Gamma_{NV,int}$. With this we can use the following equation to determine the extrinsic relaxation rate, $\Gamma_{ext}$

\begin{equation}\label{relaxation}
\Gamma_{NV}(d,\sigma) = \Gamma_{ext}(d,\sigma) + \Gamma_{NV,int}.
\end{equation}

With $\Gamma_{ext}$, we can fit the depth dependence to determine the conductivity to the following function

\begin{equation}\label{conductivity}
\Gamma_{ext}(d,\sigma) = \gamma_{e}^2\frac{\mu_0^2k_BT\sigma}{8\pi}\frac{1}{d},
\end{equation}

where $\gamma_e$ is the electron gyromagentic ratio, $\mu_0$ is the vacuum magnetic permeability, $k_B$ is the Boltzmann constant, and $T$ is the temperature. $\sigma$ is left as a free parameter to vary for the fit. Fitting was attempted to account for the film thickness, but the thickness was consistently fit to an arbitrarily large value, indicating our range of depths is much smaller than the film thickness.

The results of the calculation of $\Gamma_{ext}$ and the estimation of the conductivity are shown Fig <ref>(b). A note, regarding the data for the `Cu + AlOx' region, the x-axis is shifted by 2 nm to account for the additional 2 nm spacing provided by the alumina. The most important feature is that the the determined conductivities agree with each quite well, and both values agree with previous examinations of thin copper films[26].

It is worth noting that $T_1$ of the `Cu + AlOx' region did not differ from the `Cu' region substantially (Fig <ref>(a)). This is because, for the case of Johnson noise, the induced relaxation rate scales as $d^{-1}$ (see Eqn. <ref>). In this case, an additional 2 nm is not a substantial change within the error of our measurement.

§.§ Influence on Sensitivity

We now discuss the idea of sensitivity and the impact that the AlOx layer has on the sensitivity of the NV ensembles for different sensing modalities. The sensitivity is the noise floor of a measurement given 1 second of integration time. This definition means a lower sensitivity provides a better sensor. We compare the sensitivity of NV ensembles in the different regions at different depths. As mentioned earlier, the contrast-weighted shot-noise appears in all sensitivity estimates with $\eta \propto \frac{1}{C\sqrt{I_{pl}t_{ro}}}$, where $\eta$ is used to denote sensitivity. Due to how dramatic the changes in the contrast-weighted shot-noise are from region to region, it plays a dominant role when comparing different regions' sensitivity. This can be seen by looking at the dependencies of the sensitivity in different modes:

\begin{equation}\label{DC_sens}
\eta_{DC} \propto \frac{1}{C\sqrt{I_{pl}t_{ro}}}\frac{1}{\sqrt{T_2^*}},
\end{equation}

\begin{equation}\label{AC_sens}
\eta_{AC}^{Mean} \propto \frac{1}{C\sqrt{I_{pl}t_{ro}}}\frac{1}{\sqrt{T_2}},
\end{equation}

\begin{equation}\label{AC_sens_var}
\eta_{AC}^{Var} \propto \frac{1}{C\sqrt{I_{pl}t_{ro}}}\frac{1}{T_2^{3/2}}.
\end{equation}

We observe improvements in the $T_2$ and $T_2^*$ by a factor of 1.5 when compared to the `Ref' region. This results in improvements of the sensitivity by a factor of roughly 1.2 ($\sqrt{1.5}$) for $\eta_{DC}$ and $\eta_{AC}^{Mean}$, the sensitivities relevant for DC magnetometry and AC magnetometry. For nanoscale NMR measurements, the sensitivity, $\eta_{AC}^{Var}$, will improve by a factor of 1.8 ($1.5^{3/2}$), due to the stronger dependence on $T_2$ for variance detection. While these are sizable improvements, the constrast weighted shot-noise is reduced by a factor of 6 as compared to the reference. In this regard, the sensitivity, as compared to the reference, is strictly worse except for the deeper ensembles which see a slight improvement.

Sensitivity of NVs near thin metal films in different sensing modes.(a) The inverse contrast-weighted shot-noise ratios as a function of depth for the `Cu+AlOx' region and the `Cu' region. $\eta_{DC}$(b), $\eta_{AC}^{Mean}$(c), $\eta_{AC}^{Var}$(d), $\eta_{T_1}$(e), ratios as function of depth for the `Cu+AlOx' region and the `Cu' region.

Our main focus is the comparison between the two copper coated regions. We do note that the sensitivity ratios when compared to the reference region reaches 1 at around 10 nm deep and goes below 1, meaning improved sensitivity, for deeper ensembles. This change is can be appreciable, as much as 0.5 and the causes for this increase are the increased PL rate and $T_2$ or $T_2^*$. When the `Cu+AlOx' region is compared to the `Cu' region, the relevant comparison for sensing of quantum materials, it is clearly more sensitive. Although the insulating layer does decrease the degree of improvement for $T_2$ and $T_2^*$,it improves the contrast-weighted shot-noise (inverse shown in Fig  <ref>(a), by as much as a factor of 2, resulting in a reduction (improvement) in sensitivity of a factor of 2. There is also the extreme case of the 3 keV implanted sample where measurement in the `Cu' area were not feasible due to very low signal-to-noise ratio.

For all forms of quantum sensing, the `Cu+AlOx' region improves the sensitivity for all measured NV ensemble depths when compared to the `Cu' region. The impact is most prominent for the shallowest NV ensembles, but still a 20-40% improvement for the deeper NV ensembles. As mentioned, the changes in PL rate dominate resulting in very similar looking data for all modes of sensing. For DC sensing, the relative change in $T_2^*$ was effectively flat (see Fig. <ref>(e)). The DC and AC mean sensitivities scale as the square root of the $T_2^*$ and $T_2$ respectively, further flattening the small difference between the two regions resulting in trends dominated by the change in $C\sqrt{I_{pl}t_{ro}}$ (Fig. <ref>(b,c)). A slight deviation from this trend is observed in the AC variance sensitivity, which scales like $T_2^{3/2}$. In Fig<ref>(f),we saw a slight depth dependence in $T_2$, showing the $T_2$ ratio between the regions under discussion being less than 1 for shallow ensembles. The stronger $T_2$ dependence in variance sensing amplifies this dependence and a slightly weaker improvement in $\eta_{AC}^{Var}$ as a function of depth is observed.

The sensitivity of $T_1$ relaxometry needs to be treated separately due to the fact that the measurement revolves around observing changes in $T_1$. In the regime where $\Gamma_{ext} \gg \Gamma_{int}$, the sensitivity can be estimated[8, 17] by

\begin{equation}\label{T1_sens}
\eta_{T_1} \propto \frac{1}{C\sqrt{I_{pl}t_{ro}}}\frac{1}{\sqrt{\Gamma_{ext}}},
\end{equation}

This is a sensitivity to changes in $T_1$ with respect to the intrinsic $T_1$. If $\Gamma_{ext} = 1/T_1$, this means a shorter $T_1$ takes less time to sense. The sensitivity ratio between the `Cu + AlOx' and `Cu' regions is shown in Fig. <ref>(e) for different depths. As previously mentioned, the alumina provided minimal change on the $T_1$, thus this sensitivity is dominated by the PL improvement provided by the insulating layer. For the shallowest samples we see sensitivity improvements as much as a factor of 2.

§ CONCLUSIONS

This work highlights the importance of how integrating a material on the diamond surface can impact NV performance. Though we chose thin copper films, hardly a low dimensional quantum material, to demonstrate these affects, we view this as a sort of worst-case scenario. Not all process observed here will be observed to the same degree in other materials. Low dimensional materials, like magic-angle graphene, will not only provide a magnetic noise source as it approaches superconductivity[1], it will provide an acceptor for processes like FRET[14] and can reduce PL intensity. Surface charging caused by the integration of the material may cause band bending, making changes in spin properties difficult to isolate due to changes in the magnetic environment. Our approach of incorporating an insulating layer of alumina, has reduced the impact of the integration process and enabled the use of shallower NV ensembles for nanoscale quantum sensing.

We have characterized the PL rate, NV polarization time, $T_2$, $T_2^*$, and $T_1$ for a series of NV ensembles at variable depths. These measurements were performed in 4 different regions: with copper in direct contact with the diamond surface, `Cu', with copper insulated from the diamond surface by 2 nm of alumina, `Cu+ AlOx', just the alumina layer, `AlOx', and bare diamond, `Ref.' We observed a general decrease in PL rate for NV ensembles closer to the copper film, that could be tuned by with the addition of an insulating layer. We also found a relative increase of the NV polarization time as NV ensembles approached the film. We saw a non-monotonic improvement in $T_2$ and $T_2^*$ over the characterized depths. We attribute this improvement to band bending caused by the metal film, ionizing paramagnetic noise sources near the NV center inside the diamond.

We considered the impact of the alumina film in terms of sensing for different sensing modalities. For the deepest NV ensembles, the alumina and metal seemed to provide an overall improvement, in terms of sensitivity when compared to the reference region. In general, NVs at or deeper than 10 nm had sensitivities on par with the reference reference region, though the exact contributions to that sensitivity was different that that of the reference. For sensing with $T_1$ relaxometry, the `Cu + AlOx' region provides consistently superior sensitivity to the `Cu' region providing as much as a factor of 2 increase in sensitivity, decrease of integration time by a factor of 4, for $T_1$ relaxometry measurements.

There are further techniques that could be developed to suppress the influence of integrated materials. Recent work on using applied electric fields to engineer the charge environment in a more deterministic means has been demonstrated for single NVs[27]. Another approach would be to use another donor to provide charge such as phosphorus. Such co-doping has been shown to provide high NV conversion efficiency, provide, better NV properties, and would provide more charge to passivate the induced charge with minimal cost to sensor quality[28, 29, 30]

§ ACKNOWLEDGEMENTS

Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the DOE's National Nuclear Security Administration under contract DE-NA0003525. This work was funded, in part, by the Laboratory Directed Research and Development Program and performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

§ DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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