# Success of the small mass ratio approximation during the final orbits of
binary black hole simulations

Sergi Navarro Albalat  Center for Gravitational Physics, University of Texas
at Austin, Austin, Texas 78712, USA  Aaron Zimmerman  Center for Gravitational
Physics, University of Texas at Austin, Austin, Texas 78712, USA  Matthew
Giesler  Cornell Center for Astrophysics and Planetary Science, Cornell
University, Ithaca, New York 14853, USA  Mark A. Scheel  Theoretical
Astrophysics, Walter Burke Institute for Theoretical Physics, California
Institute of Technology, Pasadena, California 91125, USA

###### Abstract

Recent studies have shown the surprising effectiveness of the small mass-ratio
approximation (SMR) in modeling the relativistic two-body problem even at
comparable masses. Up to now this effectiveness has been demonstrated only
during inspiral, before the binary transitions into plunge and merger. Here we
examine the binding energy of nonspinning binary black hole simulations with
mass ratios from 20:1 to equal mass. We show for the first time that the
binaries undergo a transition to plunge as predicted by analytic theory, and
estimate the size of the transition region, which is $\sim 10$ gravitational
wave cycles for equal mass binaries. By including transition, the SMR
expansion of the binding energy is accurate until the last cycle of
gravitational wave emission. This is true even for comparable mass binaries
such as those observed by current gravitational wave detectors, where the
transition often makes up much of the observed signal. Our work provides
further evidence that the SMR approximation can be directly applied to current
gravitational wave observations.

## Introduction –

The direct detection of gravitational waves (GWs) [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14] has provided a new view of the universe, revealing
populations of binaries composed of black holes and neutron stars [15, 16, 17]
and enabling tests of relativity in the dynamical and strong-field regime,
e.g. [18, 19, 20, 21, 22, 23, 24]. Accurate modeling of the relativistic two-
body problem is crucial for carrying out sensitive GW searches and measuring
the parameters of detected binaries. For example effective one body models
(e.g. [25, 26, 27, 28, 29, 30, 31, 32, 33]) and phenomenological models (e.g.
[34, 35, 36, 37, 38, 39, 40]) predict the GWs emitted during the binary
inspiral, merger, and post-merger phases.

The construction of such models requires input from a number of methods, such
as post-Newtonian (PN), gravitational self-force (GSF) and numerical
relativity (NR) techniques, each with different limitations. For example the
PN approximation is valid at large separations and slow velocities. It worsens
late in the inspiral, and can be a poor approximation for systems with small
mass ratios (SMRs) $q^{-1}$, where $q=m_{1}/m_{2}\geq 1$, which spend many
cycles at close separations. In contrast, for SMR binaries, GSF methods based
on an expansion of the metric about a black hole background are more accurate.
GSF is well-suited to describe extreme-mass-ratio inspirals (EMRIs) with
$q^{-1}\sim 10^{-4}$–$10^{-6}$, which are promising targets for the LISA
mission [41]. It has recently been pushed to second-order (2GSF) for
nonspinning binaries [42, 43, 44, 45]. Meanwhile, numerical relativity (NR)
provides two-body solutions exact up to numerical errors, and can be used to
assess the validity of the PN and SMR approximations in the regime of
comparable masses and small separations accessible to NR [46]. NR is also used
to calibrate full inspiral-merger-postmerger models, or to build surrogates
which can interpolate waveform predictions between simulations, e.g. [47, 48,
49, 50, 51, 52, 53].

Surprisingly, comparisons between NR and SMR approximations of binary black
hole systems have shown that the latter is effective at describing even
comparable mass ratio systems, e.g. [54, 55, 56, 57, 58, 44, 45, 59]. These
include the binding energy, GW fluxes and GW phasing, and is achieved by re-
expanding the SMR series in the symmetric mass ratio $\nu\equiv
m_{1}m_{2}/(m_{1}+m_{2})^{2}$ rather than $q^{-1}$ [56, 57, 45]. This
promising result indicates that GSF methods could model GWs from binaries
detectable by current [60, 61, 62] and future ground-based detectors [63, 64,
65], and may be key for modeling intermediate-mass-ratio inspirals (IMRIs),
$q^{-1}\sim 10^{-2}$–$10^{-4}$, a regime which remains challenging for NR [66,
67, 68, 69]. It is especially exciting when considering recent detections of
systems with $q^{-1}\sim 10^{-1}$, such at GW190814 [70, 3, 4] and GW191219
[5]. These lie in a challenging regime where current models are not well-
calibrated to NR simulations.

In this study we tackle an important limitation of previous analyses. In all
cases, the agreement between NR and SMR predictions breaks down as the binary
approaches the innermost stable circular orbit (ISCO). This is expected: first
because higher-order SMR coefficients may grow at high frequencies, but also
because previous SMR predictions expand around adiabatic inspiral, and must
eventually fail as the binary transitions into plunge and merger.
Distinguishing between these two effects is crucial for modeling the binary
near merger and is and particularly relevant for IMRIs and comparable-mass
systems, such as those currently observed by ground-based GW detectors, where
the transition region is large.

The transition can be understood as a singular boundary layer in between the
slow inspiral through a sequence of circular orbits and a direct plunge with
timescale $T\sim M$, $M$ the total mass. In the SMR approximation, this
transition occurs over a region of characteristic size $\sim M\nu^{2/5}$
around the location of the ISCO $r_{*}=6M$, and with a dynamical timescale
$T\sim M\nu^{-1/5}$. Following the initial description of the transition
dynamics [71, 26], a number of studies have refined and generalized the
analytic approximations to the dynamics [72, 73, 74, 75, 76, 77, 78]. Recently
GSF corrections have been incorporated into a generic expansion of transition
equations and their solutions [79, 80], which we use here.

Here we investigate the binding energy $E$ of nonspinning, quasicircular
binary black holes at comparable masses and show that $E$ follows well-behaved
SMR inspiral and transition expansions even through ISCO. During the inspiral
we recover the geodesic and post-geodesic coefficients as functions of in
invariant radius $r_{\Omega}\equiv M^{1/3}\Omega^{-2/3}$. During the
transition, we find that $E$ follows the expected fractional power expansion
in $\nu$ [80], with the coefficients functions of the rescaled transition
radius $R_{\Omega}\equiv\nu^{-2/5}(r_{\Omega}-r_{*})$. This expansion breaks
down near merger and towards inspiral as expected, and we estimate the region
of validity to be $-2\lesssim R_{\Omega}/M\lesssim 7$, corresponding to
$r_{\Omega}\sim r_{*}-2M\nu^{2/5}$. The leading $\mathcal{O}(\nu^{4/5})$
coefficient from the transition fits is in good agreement with predictions. We
extract higher-order coefficients up to $\mathcal{O}(\nu^{9/5})$, where
unknown 2GSF contributions first appear, and we find them to be negligible
within the uncertainty of our analysis. Thus, our results indicate that an SMR
expansion can provide accurate predictions for gravitational waves for
comparable mass systems up to the final GW cycle before merger, consistent
with recent success of EMRI surrogate models [51, 52] and 2GSF-accurate
inspiral waveforms [45]. Our analysis also shows that the transition
constitutes a large portion of many signals observed by current ground-based
detectors, indicating that an SMR approximation scheme augmented by transition
dynamics, may have direct application to GW astronomy in the near future.

From here we set $G=c=M=1$ and use ′ for derivatives with respect to
$r_{\Omega}$ or $R_{\Omega}$, depending on the context. Quantities evaluated
at ISCO are indicated by $*$.

## NR simulations –

We select a set of high-resolution, nonspinning and quasicircular binary black
hole simulations produced with the Spectral Einstein Code (SpEC) [81, 82] with
mass ratios ranging from $q=1$ to $q=20$. These simulations have low initial
eccentricity $e\lesssim 10^{-4}$, a relatively large number of orbital cycles
$N_{\rm cycles}\sim 20$–$45$, and in most cases two resolution levels, which
allows us to assess numerical uncertainties.

From each simulation we take the GW strain $h$ extrapolated to infinity [83,
81] and corrected for the binary center of mass motion [84, 85]. From the
strain we define the invariant radius $r_{\Omega}$ using an orbital frequency
$\Omega$ inferred from the $\ell=2$, $m=2$ mode of the gravitational waves,
$h_{22}$. Although the quantity of interest for our analysis is $E$, only the
energy flux $\dot{E}$ is directly accessible from the strain. Thus, we analyze
the gradients $E^{\prime}(r_{\Omega})=\dot{E}/\dot{r}_{\Omega}$ during
inspiral and $E^{\prime}(R_{\Omega})=\dot{E}/\dot{R_{\Omega}}$ during the
transition as an indirect measure of $E$.

We find that $h_{22}$ exhibits small modulations beyond those expected from
quasicircular inspiral, which become particularly noticeable in
$\dot{\Omega}$. While the origin of these modulations is uncertain, during
early inspiral they are dominated by residual junk radiation and at later
times appear to be due to modulations of the center of mass, see e.g. [59]. To
mitigate them, we apply a low-pass filter to $\dot{\Omega}$ during the early
inspiral, with a cutoff frequency chosen conservatively high so that the
overall chirping of $\dot{\Omega}$ is not biased. Towards the transition
regime the dynamics are fast enough that the filtering can still potentially
bias the result. For $r<9.5$ we smooth the modulations with a rolling fit of
$E^{\prime}(R_{\Omega})$ to a quadratic over a fiducial window size of $\Delta
R_{\Omega}=\pm 2$. Further details of these procedures are in the
Supplementary Materials.

## Inspiral expansion –

During the adiabatic inspiral, post-geodesic corrections to $E$ can be
calculated from 1GSF corrections to the redshift factor $z$, an invariant
quantity constructed from the conservative piece of the metric perturbation
[86]. The connection between $E$ and $z$ is a consequence of the first law of
binary mechanics (FLBM) [87, 88], which assumes a helical symmetry with
killing vector field $K=\partial_{t}+\Omega\partial_{\phi}$. While this
symmetry does not hold for dynamical binaries, the FLBM has been found to be
surprisingly accurate when comparing analytic predictions to NR, e.g. [55,
89]. In our comparison we require the $O(\nu)$ corrections to the derivative
of the binding energy, $E^{\prime}(r_{\Omega})$. For that, we use $E$ and $z$
expressions given in Ref. [55] and translate these directly into predictions
for $E^{\prime}(r_{\Omega})$ as detailed in the Supplementary Materials.
Without assuming the relationships derived from the FLBM, 2GSF information is
required to compute $\mathcal{O}(\nu)$ contributions to $E$ [42]. We compare
our NR result to both of these predictions.

## Inspiral results –

Figure 1: Top panel: Difference between the leading coefficient
$E_{0}^{\prime}$ from a free fit to our simulations and the geodesic limit,
showing the recovery of the energy gradient $E^{\prime}(r_{\Omega})$ from the
data during inspiral. Middle panel: Inspiral results when using the geodesic
limit as a baseline for our fit. We plot the subleading coefficients
$E^{\prime}_{1}$ (solid red) and $E^{\prime}_{2}$ (solid green) of the
inspiral expansion of $E^{\prime}$. Also plotted is the corresponding
$E^{\prime\rm{4PN}}_{1}$ prediction [57] and two SMR results: one based on the
FLBM [55] and one a post-adiabatic expansion including 2GSF corrections [42].
Bottom panel: Difference between the NR and FLBM results for $E^{\prime}_{1}$.

To compare NR and SMR approximations during the inspiral, we perform a least-
square fit of $E^{\prime}(r_{\Omega})$ to an expansion in integer powers of
$\nu$,

$\displaystyle
E^{\prime}(r_{\Omega})=\sum_{i=0}E^{\prime}_{i}(r_{\Omega})\nu^{i},$ (1)

Following the approach of [58, 59], we first fit the NR data at fixed $r$
values to both a first and second degree polynomial in $\nu$, without
reference to the SMR prediction. From this we extract values for the
coefficients as a function of $r_{\Omega}$, and we recover the geodesic
prediction $E^{\prime}_{0}(r_{\Omega})$ from the NR data alone. This is shown
in the top panel of Fig. 1, where we plot the difference of $E^{\prime}_{0}$
and $E_{\rm geo}$, finding remarkable agreement.

Having confirmed the test particle limit, we fit the remainder
$\nu^{-1}[E^{\prime}(r_{\Omega})-E^{\prime}_{\rm geo}(r_{\Omega})]$ which
allows us to extract the $\mathcal{O}(\nu)$ and $\mathcal{O}(\nu^{2})$
coefficients more accurately. Figure 1 shows the fitted coefficients
$E^{\prime}_{1}$ and $E^{\prime}_{2}$. The result is in good agreement with
the first-order prediction from the FLBM, with systematic deviations starting
$r_{\Omega}\lesssim 10$. The NR data agrees better with the FLBM than with the
post-adiabatic result [42] for $E$ based on a 2GSF calculation. This shows the
importance of understanding subtle differences in the definitions of energy
and orbital frequency that are used when comparing NR and GSF methods, see
[42]. We also find evidence of a small but non-zero $\mathcal{O}(\nu^{2})$
term during the inspiral.

We have repeated the inspiral analysis using the orbital angular momentum
$L^{\prime}$ derived from the angular momentum flux. As with the binding
energy, a fit to $\mathcal{O}(\nu^{2})$ provides excellent agreement
throughout the inspiral regime, with a breakdown for $r_{\Omega}\lesssim 10$.
We also find that the adiabatic condition for circular orbits is satisfied for
each of our extracted coefficients during inspiral, with
$E^{\prime}_{i}/L^{\prime}_{i}=\Omega$ to within the uncertainty in our fits,
including the $\mathcal{O}(\nu^{2})$ coefficients. This agreement provides
further evidence for the accuracy of the FLBM during inspiral.

## Transition expansion –

Using an SMR expansion around the Schwarzschild metric, the binding energy and
radius of the orbit during the transition take the form [80]

$\displaystyle E$
$\displaystyle=E_{*}+\Omega_{*}[\nu^{4/5}\xi(\nu,s)+\nu^{6/5}Y(\nu,s)]\,,$ (2)
$\displaystyle r$ $\displaystyle=r_{*}+\nu^{2/5}R(\nu,s)\,.$ (3)

where $s\equiv\nu^{1/5}(\tau-\tau_{*})$ is the transition time parameter. The
transition variables can be expanded in fractional powers of $\nu$:
$\xi=\sum\xi_{i}\nu^{i/5}$, $Y=\sum Y_{i}\nu^{i/5}$, $R=\sum R_{i}\nu^{i/5}$.
The transition equations provide a method for iteratively solving for each of
$\xi_{i}$, $R_{i}$ and $Y_{i}$, with their boundary conditions fixed by
matching to inspiral at early times $s\rightarrow-\infty$. They also take as
input the self-force $F_{\mu}$ in the neighborhood of $r_{*}$. For example,
angular momentum conservation reveals
$\xi=F^{1}_{\phi*}s+\mathcal{O}(\nu^{2/5})$ [80]. For the self-force
$F^{1}_{\phi*}$, we use first-order flux data at the ISCO, taken from [75,
90]. Note that our gauge-invariant $R_{\Omega}$ differs from $R$ at
$\mathcal{O}(\nu^{2/5})$, and we have re-expanded the small parameter $q^{-1}$
in terms of $\nu$ which alters the usual transition expansion at
$\mathcal{O}(\nu^{9/5})$, where 2GSF corrections appear. This means that our
final fitted transition functions differ from those of [79, 80] beyond the
leading order. For our analysis we numerically solve for the leading order
terms $R_{0}(s)$ and $\xi_{0}(s)$. The leading transition equations we use are
in the Supplementary Materials.

## Transition results –

Figure 2: Top panel: Leading order coefficient $E^{\prime}_{4/5}(R_{\Omega})$
of the transition expansion of $E^{\prime}(R_{\Omega})$ obtained from the NR
fit (blue) compared to the leading-order SMR prediction (red). The recovery of
the analytic prediction gives direct evidence for transition dynamics in
comparable-mass NR simulations. Bottom panel: Difference between the SMR
transition prediction and the NR result. Figure 3: Subleading coefficients of
the transition expansion for $E^{\prime}(R_{\Omega})$ from a fit to the NR
data, fixing the first two coefficients. The solid lines correspond to the
result of the fit with terms up to $\mathcal{O}(\nu^{8/5})$, and include our
estimated uncertainties. The dashed color lines represent the result of the
fit including a $\mathcal{O}(\nu^{9/5})$ term where 2GSF effects first enter.
This term (bottom panel) is consistent with zero within our uncertainties.

For the transition analysis we follow the same method as for the inspiral but
we fit the NR data at fixed $R_{\Omega}$ to the fractional power expansion

$\displaystyle E^{\prime}(R_{\Omega})$ $\displaystyle=\sum_{i=4}^{i_{\rm
max}}E^{\prime}_{i/5}(R_{\Omega})\nu^{i/5}.$ (4)

Following the expectation from Eq. (2), we set $E^{\prime}_{5/5}=0$ in our
first fit and let $i_{\rm{max}}=8$, which includes all transition orders where
only 1GSF effects are present. We find this number of terms is required to
guarantee the stability of the fit and leave no structure in the residuals.
Figure 2 shows the results of the transition coefficients $E^{\prime}_{4/5}$
from this fit. The leading SMR result is in excellent agreement for
$-2\lesssim R_{\Omega}\lesssim 7$, confirming the predicted transition
dynamics even at comparable masses.

Having recovered the leading prediction $\nu^{4/5}\Omega_{*}d\xi_{0}/dR_{0}$
purely from the NR data, we next fit the residual between $E^{\prime}$ and it.
In principle, the term we subtract is not accurate through $\mathcal{O}(\nu)$
and can introduce a term at $E^{\prime}_{5/5}$. As such, we first fit the NR
data including this coefficient, and find the result is fully consistent with
$E^{\prime}_{5/5}=0$ 111 This result is potentially surprising, since it seems
to imply that the leading radial self-force effect, $f^{r}_{[0]}$ in [79, 80],
vanishes so that the $\mathcal{O}(\nu^{1/5})$ term in $R$, $R_{1}$, can be set
to zero along with $Y_{1}$. We speculate this may be because we work with the
gauge-invariant $R_{\Omega}$ and the energy directly, while the piece
$f^{r}_{[0]}$ sourcing $R_{1}$ is instead gauge-dependent. . We then set
$E^{\prime}_{5/5}=0$ and fit the scaled residual
$[E^{\prime}-\nu^{4/5}E^{\rm{SMR}}_{4/5}]\nu^{-2/5}$. This allows us to
extract accurate coefficients using either $i_{\rm max}=8$ or $i_{\rm max}=9$,
with the latter providing an estimate for the $\mathcal{O}(\nu^{9/5})$ term.

Figure 3 shows the resulting higher-order coefficients. Generally the
coefficients are comparable to $E_{4/5}(R_{\Omega})$ in a region around ISCO
but grow at larger $R_{\Omega}$, consistent with a breakdown of the transition
expansion towards inspiral. The coefficient $E^{\prime}_{7/5}$ is consistently
larger than $E^{\prime}_{6/5}$ and similar to $E^{\prime}_{8/5}$, which is why
we require terms up to $E^{\prime}_{8/5}$ to recover the leading-order result.
The leading result alone is never accurate at these mass ratios. Including
$E^{\prime}_{9/5}$ results in clear overfitting of the residuals, and itself
is consistent with zero, which further demonstrates that truncating the series
at $i_{\rm max}=8$ is appropriate when describing $E^{\prime}$ using a
transition expansion. Our expansion fails for $R_{\Omega}\gtrsim 7$, as is
clear from the failure to recover the leading-order result, and from the blow-
up of the subleading coefficients. Similar results for the angular momentum
and the variable $Y$ which describes the departure from circularity are given
in the Supplementary Materials. The latter analysis demonstrates that the
combination of higher order transition terms
$E^{\prime}_{6/5}-\Omega_{*}L^{\prime}_{6/5}$ are also in agreement with
analytic predictions.

## Conclusions –

Figure 4: Depiction of the domains of applicability of each expansion for a
sample of mass ratios $q=1,$ $5$, $20$. We plot $dE/dr_{\Omega}$ from the NR
data after removing the oscillations (blue), the raw NR data (light gray), the
inspiral prediction from the FLBM (dashed red lines), and the transition
expansion resulting from our fit, excluding $\mathcal{O}(\nu^{9/5})$ and
higher) (black dashed lines). The markers indicate the number of GW cycles
left before merger: 1 (circle), 2 (down-triangle), 4 (diamond), 6 (up-
triangle) and 10 (square). A combination of the analytic inspiral
approximation and transition dynamics including 1GSF information models the
binding energy accurately up until the final $1$-$2$ GW cycles.

We have extracted for the first time the SMR limit from nonspinning,
quasicircular NR simulations in the transition region around the ISCO. Our
work extends previous analyses of the validity of the SMR approximation at
comparable masses, which were restricted to the inspiral region (but see also
[52]). We find that an adiabatic SMR expansion, together with the FLBM, is in
good agreement with our simulations for $r_{\Omega}\gtrsim 10$. The failure of
the FLBM result at smaller radii can be explained by the onset of transition
dynamics. Using a transition expansion for the binding energy and angular
momentum as functions of
$R_{\Omega}\equiv\nu^{-2/5}(r_{\Omega}-r_{\rm{isco}})$, we can recover the
leading-order SMR result [71, 26] in a region of width $-2\lesssim
R_{\Omega}\lesssim 7$ around the ISCO. We find that terms up to
$\mathcal{O}(\nu^{8/5})$ are necessary to recover this result. We also give a
prediction for the value of the higher-order coefficients and show that the
$\mathcal{O}(\nu^{9/5})$ contribution is zero to within our uncertainties,
suggesting a small 2GSF contribution to $E$.

Our results are summarized in Fig. 4, which shows the NR data for
$dE/dr_{\Omega}$, for three binaries with $q=1$, $q=5$, and $q=20$. We compare
our raw NR data with the smoothed and filtered data that we fit, along with
the $\mathcal{O}(\nu)$-accurate FLBM inspiral prediction and the results of
our transition fit up to $\mathcal{O}(\nu^{8/5})$. This illustrates the
failure of the inspiral treatment near ISCO for higher $q$, the narrowing of
the transition region with increasing $q$, and the fact that a combination of
the two treatments describes the energy accurately until the last cycle before
merger in all cases using only 1GSF information.

The next step would be to explore transition contributions to the GW phasing,
extending the results of [58] to merger. SMR predictions in this regime would
be enabled by combining transition modeling [79] with 2GSF-accurate fluxes
[44] and waveforms [45]. It then is critical to include spins using an SMR
expansion around Kerr. Another direction would be to examine eccentric
binaries. These areas represent the frontier of 2GSF calculations. If
achieved, GSF could provide a complete, first-principles model for the two-
body problem, applicable from EMRIs to equal masses.

## Acknowledgements –

We would like to thank the authors of the SpEC simulations used in this
analysis: Serguei Ossokine, Joohean Yoo, Vijay Varma and Jonathan Blackman. We
also thank Adam Pound, Niels Warburton, Barry Wardell and Leanne Durkan for
discussions about the transition expansion and filtering method and for
generously sharing the 2GSF flux data from [44]. For the simulations used in
this work, computations were performed on the Wheeler cluster at Caltech,
which is supported by the Sherman Fairchild Foundation and by Caltech; and on
Frontera at the Texas Advanced Computing Center [92]. We also thank the
developers of Scri [84, 85], which was used to calculate the energy and
angular momentum fluxes. This work makes use of the Black Hole Perturbation
Toolkit [90]. S.N.A. and A.Z. are supported by NSF Grant Numbers PHY-1912578
and PHY-2207594. M.G. is supported by NSF Grant Number PHY-1912081 at Cornell.
M.A.S. is supported in part by the Sherman Fairchild Foundation and by
National Science Foundation (NSF) Grant Nos. PHY-2011961, PHY-2011968, and
OAC-1931266 at Caltech.

##  Appendix A Supplementary materials

## Details of numerical simulations –

Table 1 gives further details on the numerical simulations used in this study,
which includes $q=14$ and $q=15$ simulations associated with the surrogate
described in [53]. The initial data types correspond to Superposed Kerr Schild
(SKS) [93] and Superposed Harmonic Kerr (SHK) [94]. The initial data makes use
of improved prescriptions to minimize the initial center of mass motion [95],
and the initial eccentricity is reduced using an iterative method [96, 97,
98]. The resolution level is indicated by the Lev argument, and corresponds to
internal tolerances of the adaptive mesh refinement algorithm used in SpEC
[99, 100]. Due to the nature of adaptive mesh refinement, and the variable
history of refinement for each simulation, strict spectral convergence is not
expected for derived quantities such as the gravitational wave strain $h$
[81]. For this reason, we assess numerical uncertainties by including
simulations with different resolutions where available in our error estimates.
From each simulation we take the extrapolated gravitational wave strain
$h=h_{+}-ih_{\times}$ at future null infinity [83, 81], expanded in $(\ell,m)$
modes of spin-weighted spherical harmonics. We use an $N=4$ extrapolation
setting (fifth-order in $r^{-1}$) as appropriate for inspiral [83, 81] for all
our simulations. The strain $h$ is further corrected by applying a translation
and boost that minimizes the effect of the center of mass motion present in
the simulations [84, 85].

The orbital frequency is estimated using the $(2,2)$ mode of the gravitational
wave strain,

$\displaystyle\phi_{22}$ $\displaystyle=\arg h_{22}\,,$ $\displaystyle\Omega$
$\displaystyle=\dot{\phi}_{22}/2\,,$ (5)

and the flux is computed using the standard formula

$\displaystyle\dot{E}$
$\displaystyle=\lim_{r\rightarrow\infty}\frac{r^{2}}{16\pi}\sum^{\ell,m}|\dot{h}_{\ell
m}|^{2},$ (6)

summing over all modes with available to us from our simulations, $\ell\leq
8$. In principle one can integrate the flux to find the binding energy,
choosing the integration constant by matching either to the mass of the final
black hole or to PN theory early in the inspiral. We find that this procedure
introduces undesired errors in the analysis, which is why we focus on the
gradient of the binding energy in our analysis.

## Details of the data analysis and error estimates –

During inspiral, we use a second-order forward and backward Butterworth filter
$\mathcal{B}$ to filter $\dot{\Omega}$. To mitigate the impact on the
frequency sweep, we find that find that applying the filter after subtracting
a good estimate of the data improves its performance by reducing the overall
variation of the data. For this we use the 2GSF prediction for $\dot{\Omega}$
from [45]. Further, we apply the filter to $\dot{\Omega}$ as a function of its
index. Since the time step in our simulation is adaptive, the strain data $h$
is not uniformly sampled, but we find that resampling to uniform time steps
makes finding an appropriate cutoff frequency more challenging than for the
non-uniform sampling. The reason is that with the denser sampling rate at late
times effectively brings the late-time chirp to lower frequencies, so that a
uniform cutoff frequency better targets the actual noisy behavior at all
times. The cutoff frequencies are chosen as $f_{c}=a+b(\nu-\nu_{q=20})$, with
$a\in\\{6,8,10\\}\times 10^{-4}$ and $b=1.95\times 10^{-3}$. The filtered data
is then

$\displaystyle\dot{\Omega}_{{\rm filtered},i}=[\mathcal{B}*(\dot{\Omega}_{{\rm
raw},i}-\dot{\Omega}_{{\rm SMR},i})]+\dot{\Omega}_{{\rm SMR},i}\,.$ (7)

Since the filtering process biases the data late in the simulation, and
because we have no SMR data beyond ISCO, we choose a cutoff radius
$r_{\Omega,c}=9.5$ beyond which we simply switch to a rolling fit of
$E^{\prime}(\Omega)$ (or $L^{\prime}(R_{\Omega})$ as discussed below) to a
quadratic, over a window $\Delta R_{\Omega}\in\pm\\{1,2,3\\}$ around the
fitted point. For our fiducial analysis, we select the one corresponding to
$\Delta R=2$ and $a=8\times 10^{-4}$.

To create our uncertainty bands for our fitted quantities, we vary all of the
parameters involved in the filtering and smoothing of our data within the
stated ranges, as well as repeating our analysis with simulations at a lower
resolution. The error bands are created by taking the envelope of the
variation in our fitted parameters, over the different resolutions, and the
$1$-$\sigma$ errors of our least-squares fits at each frequency point.

$q$ | Type | $M\Omega_{0}$ | $N_{\text{cycles}}$ | $e_{0}$ | Levs | SXS ID
---|---|---|---|---|---|---
1 | SKS | 0.01233 | 27.96 | 1.355e-4 | 5,6 | 2513
1.5 | SKS | 0.01250 | 28.98 | 5.77e-5 | 2,3 | 2331
2 | SHK | 0.01554 | 20.70 | 2.408e-4 | 2,3 | 2497
2.5 | SKS | 0.01512 | 22.49 | 7.580e-4 | 2,3 | 0191
3 | SHK | 0.01707 | 20.44 | 9.64e-5 | 2,3 | 2498
3.5 | SKS | 0.01477 | 27.76 | 2.665e-4 | 4,5 | 2483
4 | SKS | 0.01600 | 25.67 | 8.702e-4 | 4,5 | 2485
4.5 | SKS | 0.01616 | 27.37 | 8.289e-4 | 4,5 | 2484
5 | SKS | 0.01589 | 29.13 | 2.236e-4 | 4,5 | 2487
5.5 | SKS | 0.01592 | 30.81 | 4.442e-4 | 4,5 | 2486
6 | SKS | 0.01588 | 32.62 | 5.864e-4 | 4,5 | 2489
6.5 | SKS | 0.01599 | 34.43 | 7.263e-4 | 4,5 | 2488
7 | SKS | 0.01577 | 36.16 | 3.612e-4 | 4,5 | 2491
7.5 | SKS | 0.01597 | 37.89 | 3.694e-05 | 4,5 | 2490
8 | SKS | 0.01584 | 39.53 | 6.688e-4 | 5 | 2493
8.5 | SKS | 0.01594 | 41.31 | 8.578e-4 | 5 | 2492
9 | SKS | 0.01583 | 43.16 | 2.010e-4 | 4,5 | 2495
9.5 | SKS | 0.01585 | 44.93 | 1.584e-4 | 4 | 2494
14 | SHK | 0.02292 | 27.70 | 3.814e-4 | 2,3 | 2480
15 | SHK | 0.02317 | 27.94 | 3.692e-4 | 2,3 | 2477
20 | SKS | 0.02321 | 34.38 | 2.506e-4 | 3,4 | 2516
Table 1: Details of the nonspinning quasicircular SpEC simulations used in
this analysis. The subscript zero denotes the reference time (time at which
junk radiation has sufficiently decayed).

## Details of the inspiral prediction –

The FLBM gives [88, 55]

$\displaystyle E^{\prime}(r_{\Omega})$
$\displaystyle=E^{\prime}_{\rm{geo}}(r_{\Omega})+\nu
E^{\prime}_{\rm{FLBM}}(r_{\Omega})+\mathcal{O}(\nu^{2})\,,$ (8)

with

$\displaystyle E_{\rm{FLBM}}(x)$
$\displaystyle=\frac{1}{2}z_{\rm{1GSF}}(x)-\frac{x}{3}z^{\prime}_{\rm{1GSF}}(x)-1$
$\displaystyle+\sqrt{1-3x}+\frac{x}{6}\frac{7-24x}{(1-3x)^{3/2}}\,,$ (9)

and where $x\equiv r_{\Omega}^{-1}$. The 1GSF contribution to $z$,
$z_{\rm{1GSF}}(x)$, has been computed to high precision with multiple codes,
e.g. [101]. For the purpose of our inspiral comparison we make use of the fit
formula [88]

$\displaystyle
z_{\rm{1GSF}}(x)=\frac{2x(1-2.18522x+1.05185x^{2})}{1-2.43395x+0.400665x^{2}-5.9991x^{3}}.$
(10)

## Details of the transition formalism –

Our analysis requires that we solve the transition equations to leading order,
in order to calibrate our fits and compare with the NR results. The quantities
$R_{0}(s)$ and $Y_{0}(s)$ are solved using the leading order transition
equations [71, 26, 74, 80]

$\displaystyle\left(\frac{dR_{0}}{ds}\right)^{2}$
$\displaystyle=-\frac{2}{3}\alpha_{*}R^{3}_{0}-2\beta_{*}\kappa_{*}sR_{0}+\gamma_{*}Y_{0},$
(11) $\displaystyle\frac{d^{2}R_{0}}{ds^{2}}$
$\displaystyle=-\alpha_{*}R^{2}_{0}-\kappa_{*}\beta_{*}s,$ (12)
$\displaystyle\frac{dY_{0}}{ds}$
$\displaystyle=2\kappa_{*}\frac{\beta_{*}}{\gamma_{*}}R_{0}.$ (13)

The constants in the above equations are given by

$\displaystyle\kappa_{*}$ $\displaystyle\equiv
F_{\phi}^{1}\rvert_{r_{*}}\,,\qquad\alpha_{*}\equiv\frac{1}{4}\frac{\partial^{3}V^{\rm{geo}}}{\partial
r^{3}}\biggr{\rvert}_{\rm{isco}}\,,\qquad\gamma_{*}\equiv\frac{\partial
V^{\rm{geo}}}{\partial L}\biggr{\rvert}_{r_{*}}\,,$ (14)
$\displaystyle\beta_{*}$
$\displaystyle\equiv-\frac{1}{2}\left(\frac{\partial^{2}V^{\rm{geo}}}{\partial
r\partial L}+\Omega\frac{\partial^{2}V^{\rm{geo}}}{\partial r\partial
E}\right)\biggr{\rvert}_{r_{*}}\,,$ (15)

where $V^{\rm{geo}}$ is the effective potential of radial geodesic motion
about a Schwarzschild black hole.

## Further results: Angular momentum –

Here we present the analysis of the angular momentum in both the inspiral and
transition regimes. These results provide an independent demonstration of the
accuracy of the SMR modeling during inspiral and plunge. They are used to
confirm that the binaries are described by adiabatically evolving circular
orbits through $\mathcal{O}(\nu^{2})$ during inspiral, consistent with the
FLBM, as discussed in the main text. We also find that this condition fails at
the expected order during transition, by finding agreement with the non-
circular correction $Y_{0}$ introduced by Kesden [74].

We extract the angular momentum flux from our numerical simulations using the
standard formula

$\displaystyle\dot{L}$
$\displaystyle=\lim_{r\rightarrow\infty}\frac{r^{2}}{16\pi}\text{Im}\left(\sum_{l,m}mh^{l,m}(\dot{h}^{l,m})^{*}\right)\,.$
(16)

We use the same filtering technique applied to $\dot{\Omega}$ as done for the
analysis of $E(r_{\Omega})$ during the inspiral, and the same rolling fit to
$dL/dR_{\Omega}$ as a function of the transition radius $R$ for
$r_{\Omega}<9.5$, as done for analysis of the binding energy.

For the analytic comparisons during inspiral, we use the FLBM-derived
expansion relations [88, 55]

$\displaystyle L^{\prime}(r_{\Omega})$
$\displaystyle=L^{\prime}_{\rm{geo}}(r_{\Omega})+\nu
L^{\prime}_{\rm{FLBM}}(r_{\Omega})+\mathcal{O}(\nu^{2})\,,$ (17)
$\displaystyle L_{\rm{FLBM}}(x)$
$\displaystyle=\frac{1}{3\sqrt{x}}z^{\prime}_{\rm{1GSF}}(x)+\frac{1}{6\sqrt{x}}\frac{4-15x}{(1-3x)^{3/2}}\,,$
(18)

and the redshift factor of Eq. (10). During inspiral we use an expansion in
integer powers of $\nu$ to fit to the NR data,

$\displaystyle
L^{\prime}(r_{\Omega})=\sum_{i=0}^{2}L^{\prime}_{i}(r_{\Omega})\nu^{i}.$ (19)

As before, the fits across simulations at fixed $r_{\Omega}$ give a leading
coefficient $L^{\prime}_{0}$ in agreement with the geodesic prediction
$L^{\prime}_{\rm geo}$, and so we subtract this and fit the residual
$L^{\prime}(r_{\Omega})-L^{\prime}_{0}(r_{\Omega})$ to improve the accuracy of
the fitted $L^{\prime}_{1}(r_{\Omega})$ and $L^{\prime}_{2}(r_{\Omega})$
coefficients. The results of the inspiral analysis are depicted in Fig. 5.

Figure 5: Top panel: Recovery of the geodesic limit of the energy gradient
$L^{\prime}(r_{\Omega})$ during inspiral from the NR data. We plot the
difference between the leading coefficient $L_{0}^{\prime}$ from a free fit to
our sequence of NR simulations and the geodesic limit, together with our
estimated uncertainties. Middle panel: Inspiral results when using the
geodesic limit as a baseline for our fit. We plot the subleading coefficients
$L^{\prime}_{1}$ (solid red) and $L^{\prime}_{2}$ (solid green) of the
inspiral expansion of $L^{\prime}$. Also plotted is the corresponding
$L^{\prime\rm{4PN}}_{1}$ prediction [57] and SMR result based on the FLBM
[55]. Bottom panel: Difference between the NR and FLBM results for
$L^{\prime}_{1}$. Figure 6: Top panel: Leading order coefficient
$L^{\prime}_{4/5}(R_{\Omega})$ of the transition expansion of the binding
energy obtained from the NR fit (blue) compared to the leading-order SMR
prediction (red). Bottom panel: Difference between the SMR transition
prediction and the NR result. Figure 7: Subleading coefficients of the
transition expansion of $L^{\prime}(R)$ the NR analysis. The solid lines
correspond to the result of the fit with terms up to $\mathcal{O}(\nu^{8/5})$.
The dashed color lines represent the result of the fit including a
$\mathcal{O}(\nu^{9/5})$ term shown in the bottom panel.

For the transition analysis, the analytic approximation for $L^{\prime}(R)$
depends on the quantity $\xi$ [71, 79, 80],

$\displaystyle L$ $\displaystyle=L_{*}+\nu^{4/5}\xi(\nu,s)\,.$ (20)

This allows us to compare $L^{\prime}(R_{\Omega})$ during the transition to
analytic predictions in the same way as for $E^{\prime}(R_{\Omega})$. We carry
out the same analysis as for $E^{\prime}(R_{\Omega})$, first performing a fit
of the form

$\displaystyle L^{\prime}(R_{\Omega})$ $\displaystyle=\sum_{i=4}^{i_{\rm
max}}L^{\prime}_{i/5}(R_{\Omega})\nu^{i/5},$ (21)

with $i_{\rm max}=8$ and the only from analytic theory being that
$\xi_{1}^{\prime}(R_{\Omega})=0$. The results for the leading coefficient
shown in Fig. 6 confirm that $L_{4/5}^{\prime}$ obeys the expected transition
dynamics, in agreement with theory. Next we fix $L_{4/5}^{\prime}$ to the
analytic prediction, and verify that $L^{\prime}_{5/5}$ remains consistent
with zero. Finally fixing both $L^{\prime}_{4/5}$ and $L^{\prime}_{5/5}$, we
fit with $i_{\rm{max}}=8$ and $i_{\rm max}=9$ and extract the transition
coefficients plotted in Fig. 7. As with our analysis of the binding energy, we
see that the higher terms in the expansion are significant and well behaved
through $-2\leq R_{\Omega}\leq 7$, and that the $\mathcal{O}(\nu^{9/5})$ term
is consistent with zero to within our uncertainties. This again confirms the
small size of 2GSF contributions to the transition.

Equivalently, we can focus on the deviation from circularity during transition
[74], $Y^{\prime}(R_{\Omega})\equiv
E^{\prime}(R_{\Omega})-\Omega_{*}L^{\prime}(R_{\Omega})$. The leading
prediction for the deviation is

$\displaystyle Y^{\prime}(R)$
$\displaystyle=Y^{\prime}_{0}(R_{0})\nu^{6/5}+\mathcal{O}(\nu^{7/5}).$ (22)

We fit the NR data during transition to a fractional power series of the form

$\displaystyle Y^{\prime}(R_{\Omega})$ $\displaystyle=\sum_{i=6}^{i_{\rm
max}}Y^{\prime}_{i/5}(R_{\Omega})\nu^{i/5},$ (23)

once again setting the first correction to the leading-order behavior to zero,
$Y^{\prime}_{7/5}=0$, and letting $i_{\rm max}=8$. Figure 8 shows the fit
coefficient $Y^{\prime}_{6/5}(R_{\Omega})$ extracted from this procedure. The
leading-order prediction provides a good fit to the data over the same range
of $R_{\Omega}$ values as we find for the energy analysis, $2\lesssim
R_{\Omega}\lesssim 7$.

If we retain the $Y^{\prime}_{7/5}$ term, the near-degeneracy of the first two
terms prevents us from achieving this level of agreement with theory for
$Y^{\prime}_{6/5}$. We can justify setting $Y^{\prime}_{7/5}=0$ in two ways.
Our first comes from considering what our numerical results for
$E^{\prime}(R_{\Omega})$ imply for the transition expansion. The fact that we
find that $E^{\prime}_{5/5}=0$ after subtracting the leading transition
prediction from the NR data for $E^{\prime}(R_{\Omega})$ implies that the
first correction to $R$ vanishes, $R_{1}(R_{\Omega})=0$. This in turn implies
that first term in the transition expansion of the radial self-force,
$f^{r}_{[0]}$ of [80], vanishes. Without this term to source $Y_{1}$ in the
transition equations, $Y_{1}$ term can be set to zero in a consistent manner.
The second way is to subtract the leading-order prediction (22) from
$Y^{\prime}$ and fit the remainder,
$[Y^{\prime}(R_{\Omega})-\nu^{6/5}dY_{0}/dR_{0}]$ to the factional power
expansion in $\nu$, starting the series with $Y^{\prime}_{7/5}$. When doing so
we find the fit for $Y^{\prime}_{7/5}(R_{\Omega})$ vanishes throughout the
transition region, while the higher coefficients are nonzero, similar to what
occured for $E^{\prime}(R_{\Omega})$.

Figure 8: Top panel: Leading order coefficient $Y^{\prime}_{6/5}(R_{\Omega})$
of the transition expansion obtained from the NR fit (blue) compared to the
Kesden SMR prediction (red). Bottom panel: Difference between the SMR
transition prediction and the NR result.

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Also at ]School of Mathematics and Physics, Queen’s University Belfast,
Belfast, BT7 1NN, United Kingdom.

Also at ]Maxwell Centre, Cavendish Laboratory, University of Cambridge,
Cambridge, United Kingdom

# Influence of charged walls and defects on DC resistivity and dielectric
relaxations in Cu-Cl boracite

C. Cochard  School of Science and Engineering, University of Dundee,
Nethergate, Dundee, DD1 4HN, United Kingdom  [<EMAIL_ADDRESS>T.
Granzow  MRT Department, Luxembourg Institute of Science and Technology
(LIST), L-4362 Esch-sur-Alzette, Luxembourg  C. M. Fernandez-Posada
Department of Earth Sciences, University of Cambridge, Downing Street,
Cambridge CB2 3EQ, United Kingdom  [  M. A. Carpenter  Department of Earth
Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, United
Kingdom  R. G. P. McQuaid  School of Mathematics and Physics, Queen’s
University Belfast, Belfast, BT7 1NN, United Kingdom  J. M. Guy  School of
Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, United
Kingdom  R. W. Whatmore  Department of Materials, Faculty of Engineering,
Imperial College London, London, SW7 2AZ, United Kingdom  J. M. Gregg  School
of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN,
United Kingdom

(August 27, 2024)

###### Abstract

Charged domain walls form spontaneously in Cu-Cl boracite on cooling through
the phase transition. These walls exhibit changed conductivity compared to the
bulk and motion consistent with the existence of negative capacitance. Here,
we present the dielectric permittivity and DC resistivity of bulk Cu-Cl
boracite as a function of temperature (-140 °C to 150 °C) and frequency (1 mHz
to 10 MHz). The thermal behaviour of the two observed dielectric relaxations
and the DC resistivity is discussed. We propose that the relaxations can be
explained by the existence of point defects, most likely local complexes
created by a change of valence of Cu and accompanying oxygen vacancies. In
addition, the sudden change in resistivity seen at the phase transition
suggests that conductive domain walls contribute significantly to the
conductivity in the ferroelectric phase.

††preprint: AIP/123-QED

This article may be downloaded for personal use only. Any other use requires
prior permission of the author and AIP Publishing. This article appeared in
Appl. Phys. Lett. 119 202904 (2021) and may be found at
(http://doi.org/10.1063/5.0067846)

The boracites form a class of ferroelectrics with the general formula
M3B7O13X, where M is a metal and X a halide. As improper ferroelectrics, their
interesting crystallography and phase transitions attracted attention in the
1960’s, 70’s and 80’s with the first coupled magnetoelectric multiferroic
switching being demonstrated by Ascher et al.Ascher  _et al._ (1966) in
Ni3B7O13I and with observations indicative of potential for electro-
opticSchmid and Schwarzmüller (1976) and pyroelectric applicationsSchmid  _et
al._ (1980); Smith, Rosar, and Shaulov (1981); Whatmore, Herbert, and Ainger
(1980). However, the growth of large single crystalsSchmid (1965); Whatmore,
Brierley, and Ainger (1980) is difficult and with the development of
perovskite oxide ceramic materials possessing diverse functional properties,
interest in the boracite family subsequently waned. Recently, however, the
discoveries surrounding improper ferroelectricsCatalan  _et al._ (2012); Evans
_et al._ (2020); Feng  _et al._ (2018) and their associated charged domain
walls Guy  _et al._ (2021); McQuaid  _et al._ (2017) has rekindled interest in
the potential of boracites as functional materials. The charged domain walls
in Cu-Cl boracite are particularly remarkable: they present either enhanced
conductivity (in 90° tail-to-tail walls) or reduced conductivity (in 90° head-
to-head walls) relative to the bulkMcQuaid  _et al._ (2017). Charged walls
exist spontaneously in Cu-Cl boracite, to accommodate for the spontaneous
shear strain developing at the phase transition, but they can also be injected
and repositioned by stress and electric field, making them interesting for the
future of nanoelectronicsSeidel (2012); Whyte and Gregg (2015). Uniquely,
head-to-head charged walls have been shown to have an unconventional
electrostatic response (moving in the opposite direction to that expected
under an applied electric field), consistent with the existence of negative
capacitanceGuy  _et al._ (2021). This discovery of new properties at domain
walls prompts our understanding of the intrinsic properties of boracites to be
revisited.
The characterisation of the dielectric properties is particularly important,
as it pertains to the unusual electrostatic response of charged domain
wallsGuy  _et al._ (2021), as well as playing a role in the piezoelectric and
ferroelectric responses. Here, we focus on the characterisation of the
dielectric dispersion and resistivity, as functions of temperature, of the
same Cu-Cl boracite sample in which the conductivity of charged wallsMcQuaid
_et al._ (2017) and negative capacitance were measuredGuy  _et al._ (2021). We
observe two dielectric relaxations and a strong change of resistivity at the
phase transition (90 °C) between the high-temperature piezoelectric phase and
the low-temperature ferroelectric phaseFernandez-Posada  _et al._ (2021).
The Cu3B7O13Cl single-crystal was prepared by the phase transport
techniqueWhatmore, Brierley, and Ainger (1980). The crystal is about 5x5x1mm3
mirror polished with faces parallel. The sample is transparent with a faint
blue coloration. The transmission spectrum of a small piece of the sample was
measured on a PerkinElmer Lambda900 spectrometer between 300 nm and 900 nm.
The impedance of this crystal was measured at frequencies between 1 mHz and 10
MHz and a driving voltage of 0.5V in a large temperature range of -140 °C to
150 °C , spanning the piezoelectric-ferroelectric phase transition, with steps
of 10 or 20 °C using a Novocontrol Concept 40 dielectric spectrometer with the
unpoled sample directly contacted with parallel brass plates freshly polished
to ensure good contacts and mounted with a mechanical spring. The complex
dielectric permittivity
$\varepsilon^{*}=\varepsilon^{\prime}+i\varepsilon^{\prime\prime}$ and
resistivity $\rho^{*}=\rho^{\prime}+i\rho^{\prime\prime}$ were calculated
classically from the complex impedance.
Figure 1 presents the frequency dispersion of $\varepsilon^{\prime}$ and
$\varepsilon^{\prime\prime}$ at different temperatures. Piezoelectric
resonances are observed at frequencies above 500 kHz. With decreasing
frequency, two relaxations can be clearly observed at frequencies around 1 kHz
and 100 kHz at 150 °C. Upon further lowering of frequencies, the real and
imaginary parts of the dielelectric permittivity increase rapidly. This
increase is consistent with the existence of a resistor in series.

Figure 1:  Frequency dependence of the dielectric permittivity (a) real part,
(b) imaginary part and (c) Cole-Cole plot. * highlight the two relaxations in
(b) and (c) The colour lines are separated by 20 °C except close to the phase
transition where they are separated by 10 °C .

The real and imaginary part were fitted simultaneously for frequencies below
piezoelectric resonances for all temperatures. The relaxations were modelled
as a Cole-Cole relaxation, and a term was added to describe the low-frequency
conductive behaviour, as is commonly done in ferroelectric materialsGranzow
(2017)

$\displaystyle\varepsilon^{*}=\frac{\Delta\varepsilon_{1}}{1+i(f/f_{r,1})^{a_{1}}}+\frac{\Delta\varepsilon_{2}}{1+i(f/f_{r,2})^{a_{2}}}+\frac{\sigma}{\varepsilon_{0}(i2\pi
f)^{n}}+\varepsilon_{\infty}$ (1)

where $\Delta\varepsilon_{1}$ and $\Delta\varepsilon_{2}$ are the amplitudes
of the relaxations, $f_{r,1}$ and $f_{r,2}$ the two characteristic relaxation
frequencies, $a_{1}$ and $a_{2}$ two exponents, $\sigma$ is the DC
conductivity, $n$ is an empirical constant and $\varepsilon_{\infty}$ is the
permittivity at high frequencies. Fitting the two relaxations with the
empirical Havriliak-Negami functionHavriliak and Negami (1967) improves
marginally the quality of the fit, whereas fitting with two simple Debye
relaxations decreases the quality of the fit with respect to the conduction-
dominated low-frequency part. All fits lead to similar results regarding the
changes in amplitude and frequencies of the relaxations (Supplementary Fig. S1
and S2). The results of fitting at low temperatures (below -50 °C ) was not
considered since the two relaxations have moved towards low frequencies and it
becomes difficult to differentiate them from one another and from the low
frequency conduction. The fitting parameters for the two relaxations, on the
one hand, and the DC conductivity, on the other hand, are presented in Fig. 2
and Fig. 3, respectively, and will be discussed separately.
Figure 2 presents the fitted parameters using the Cole-Cole model for the two
relaxations. Both relaxations have similar thermal evolution: the relaxation
frequencies (Fig. 2a) continuously increase with increasing temperature, the
amplitudes of the two relaxations display a jump at the phase transition(Fig.
2b) and the exponents $a_{1}$ and $a_{2}$ remain equal to 0.8 across the
temperature range investigated (Supplementary Fig. S2).
The smooth continuous increase in relaxation frequencies is a strong
indication that these relaxations are not related to Maxwell-Wagner
polarisation. Indeed, the frequency associated with this type of relaxation
are inversely proportional to the resistivity and dielectric permittivity, and
both of these quantities display a sharp increase at the phase transition.
This would lead to a sharp decrease of the relaxation frequencies at the phase
transition which is absent here.
The temperature evolution of relaxation frequencies is often modelled using
the Vogel-Fulcher law, describing an activated process of activation energy
$E_{a}$ and freezing below the temperature $T_{f}$

$\displaystyle f=f_{0}e^{-\frac{E_{a}}{k_{B}(T-T_{f})}}$ (2)

where $k_{B}$ is the Boltzmann constant and $f_{0}$ the attempt frequency. The
continuous lines in Fig. 2(a) represent the results of the fit to the measured
data. Fitting a Vogel-Fulcher law is never trivialCochard  _et al._ (2016),
but a few things can be said based on the fitted values shown in Table
LABEL:tab:VF.

Figure 2:  Fitting parameters of relaxations (a) relaxation frequency, (b)
dielectric strength. The dark green circles represent the high frequency
relaxation and the light orange squares the lower frequency one. Figure 3:
Thermal evolution of resistivity (a) as a function of temperature (b) as a
function of the inverse of temperature. Light blue squares represent the value
of resistivity measured at 1mHz. Dark blue circles depict the value obtained
from fitting. The exponent n is presented in the inset of (a).

The freezing temperatures were found to be $\sim$ 0 K (within error),
suggesting that the processes leading to the relaxations will not freeze.
Fixing the freezing temperature to 0 K, i.e. assuming an Arrhenius law, leads
to similar results with activation energies that are comparable to the one
observed for the lower frequency relaxation, but still within the error
associated with the Vogel-Fulcher fits.
A recent studyFernandez-Posada  _et al._ (2021), focusing on the elastic
properties of this sample, identified a relaxation process freezing at $\sim
40$ K, which was attributed to strain relaxation around local ferroelectric
dipoles or polarons. The low freezing temperature observed could be broadly
consistent with our observations; however, the minimum activation energy
($\sim\,0.02$ eV) associated with the strain process is more than an order of
magnitude lower than that observed in this work. The differences in activation
energies and freezing behaviour suggest different microscopic origins for the
relaxation processes reported in the present work and the one previously
observed. Specifically, the absence of freezing of the relaxation observed
here suggests that the reorientations of local dipoles are independent.
Indeed, in a process modelled by an Arrhenius law, there is no assumption of
collective response to the applied electric field.

Table 1: Fitting parameters of the Vogel-Fulcher analysis. Relaxation 1 and 2 correspond to the relaxation occurring at the highest and lowest frequency, respectively. | $T_{f}(K)$ |  $E_{a}$ (eV) |  $f_{0}$ (Hz)
---|---|---|---
Relaxation 1 | $-9\pm 13$ | $0.59\pm 0.06$ | $(8.0\pm 10.0)\cdot 10^{10}$
Relaxation 2 | $-1\pm 10$ | $0.64\pm 0.05$ | $(1.8\pm 5.3)\cdot 10^{13}$

Notably, the activations energies for both relaxations are the same within
error ($\sim\,0.6$ eV). This suggests a common origin. Domain wall motion can
largely be excluded, as both relaxation phenomena persist above $T_{c}$. While
a contribution from domains/domain walls below $T_{c}$ cannot be completely
excluded, it is certainly not the main relaxation phenomenon at play.
Perhaps the most classical origin of dielectric relaxations in dielectrics,
point defectsElissalde and Ravez (2001); Nataf  _et al._ (2016); Pramanick
_et al._ (2012) are the most likely mechanism for the observed dielectric
relaxations. Indeed, the order of magnitude of the activation energies is very
similar to that reported for defect states in perovskitesAkkopru-Akgun (2019).
UV-visible optical spectroscopy (Fig. 4) indicates the existence of states in
the band gap, strongly supporting point defects as a source of the dielectric
relaxations. The colour of the sample and the band in the transmission
spectrum at 490-540nmKim  _et al._ (1981) both suggest a change of valence of
copper Cu${}^{2+}\rightarrow\textrm{Cu}^{+}$. To conserve charge neutrality,
this change of valence is likely to be associated with positively charged
point defects, such as oxygen vacancies V${}_{O}^{\bullet\bullet}$.

Figure 4:  Transmittance of the Cu-Cl boracite sample (photo as inset)

Point defects can lead to dielectric relaxations in several ways: creating a
local dipole (or bound charge) that can be reoriented by the electric field,
providing a free charge carrier that is moved by the electric field or through
the coupling to phonons Gridnev (2002) In parenthesis, defect mobility leads
to dielectric relaxations in some cases, but have usually much higher
activation energies ($\sim$ 2 eV in boracite glassEl-Falaky, Guirguis, and Abd
El-Aal (2012)) and are rarely observed in ferroelectrics around room
temperature. The similar activation energies suggest that the two relaxations
“experience” a similar energy landscape. On the other hand, the difference in
$f_{0}$ indicates that, at infinite temperature, the energy barrier between
the minima is overcome at different frequencies. This could be explained with
different effective masses of the dipoles or charges or by the coupling to two
different phonon modes.
For example, the change of valence in copper creates a local dipole but can
also lead to polaronic hopping between sites. These two mechanisms would
probably have similar energy landscapes, but the dipole reorientation and
polaron hopping would have different effective masses. Alternatively, the
existence of anisotropy in some of the B-FO tetrahedra could create
relaxations with different effective masses. Indeed, in half of the
tetrahedra, boron is not located at the centre. This off-centring leads to the
existence of local dipoles (even in the high temperature phase) pointing
towards chlorine. The position of a V${}_{O}^{\bullet\bullet}$vacancy would
have a strong influence on this local B-O dipole and in turn change the
effective mass of the dipole created directly by V${}_{O}^{\bullet\bullet}$. A
third option would be a defect complex formed of Cu+ and
V${}_{O}^{\bullet\bullet}$; this seems particularly likely, in terms of
keeping the electroneutrality of the sample. This complex would have one
activation energy, but the Cu+\- dipoles and V${}_{O}^{\bullet\bullet}$would
again have different effective masses. In this discussion, the polarisability
of the defects was the main mechanism considered; however, defects can also
lead to dielectric relaxation through the coupling to phonon modes. We believe
that this coupling is less likely in the present case as the phase transition
does not affect the temperature dependence of the relaxation frequency,
whereas phonons would be strongly affected. At this point and by analogy with
acceptor- or donor-doped perovskiteSapper  _et al._ (2014); Zhang  _et al._
(2008) and the observed decrease in formation energy when a defect complex is
createdEichel  _et al._ (2008); Erhart  _et al._ (2007), we consider that a
charge transfer between the two defects is the most likely phenomenon.
Below the frequencies of the dielectric relaxations, a strong increase in both
$\varepsilon^{\prime}$ and $\varepsilon^{\prime\prime}$ is observed. It is
attributed to the DC conductivity of the sample, which can be determined in
two different ways: (i) as a result of fitting to the dielectric dispersion
data and (ii) considering the low-frequency (1 mHz) value of the resistivity
$\rho$ (Supplementary Figure S3a). In both cases, the temperature range was
limited to -50 °C and above. Figure 3 presents the thermal behaviour of
resistivity determined both ways. The values are in excellent agreement and
are of the order of magnitude of the resistivity previously reported in Cu-Cl
boraciteSchmid and Petermann (1977), which suggests good reliability of our
data.
At the phase transition, a clear jump in resistivity ($\Delta\rho$) of about
an order of magnitude is observed. A similar step was seen in the exponent n,
determined in the fitting procedure (see inset of Fig. 3(a)). This abrupt
change is not an artefact, since it can be observed directly on the Cole-Cole
plot (Fig. 1c): there is a clear and sudden change in the low frequency
impedance across the phase transition. Schmid and PetermannSchmid and
Petermann (1977) also reported a sudden change in resistivity at the phase
transition, although they observed that the sign of $\Delta\rho$ depended on
the sample and the poling state. This led the authors, in 1977, to conclude
that “this may possibly be due to some spurious domain orientation […] or to a
higher conduction along the walls”, even though domain wall conduction had not
been experimentally observed in any ferroelectric system at that time. We
observed a similar influence of the domain pattern in the DC resistivity
measured at room temperature for the same sample after different thermal
treatment (Supplementary Fig. S2b and S4): the highest resistivity being more
than twice as large as the smallest one. However, now that enhanced or
depressed conduction at domain wallsGuy  _et al._ (2021); McQuaid  _et al._
(2017) has been previously observed experimentally through spatially resolved
current mapping, it is more likely that the domain walls influence
predominantly the DC resistivity rather than “spurious domains”, since single
domains samples consistently show higher resistivity than multidomain
onesSchmid and Petermann (1977).
In addition to its abrupt change at the phase transition, the DC resistivity
decreases with increasing temperature, which indicates an
insulator/semiconductor behaviour. It is common to study the temperature
dependence to get insight into the physical mechanism responsible for carrier
transport. Since Cu-Cl boracite has an optical band gap of 4 eV and the
observation of defects state in the band gap, a change in resistivity due to
activated hopping between potential wells seems the most likely mechanism. In
this case, resisitivity, follows an Arrhenius law:

$\displaystyle\rho_{DC}=\rho_{0}e^{\frac{E_{a}}{k_{B}T}}$ (3)

where $\rho_{0}$ is a prefactor representing either the resistivity at at
infinite temperature, $E_{a}$ the activation energy and $k_{B}$ the Boltzmann
constant.
The activation energies were found to differ on each side of the phase
transition with $E_{a}^{\rho}=0.42\pm 0.04$ eV below and $E_{a}^{\rho}=0.72\pm
0.08$ eV above the transition, respectively. These values of activation
energies are of the same order of magnitude as the values reported in the
literature for Cu-Cl boracite, although we observe the higher activation
energy above $T_{C}$. Additionally, they are also of the same order as the one
reported for the temperature dependence of the electronic conductivity driven
by charge transfer between Cu+ and Cu2+ in a halide double perovskiteConnor
_et al._ (2021).
The thermal dependences of the resisitivity are of the same order of magnitude
as the activation energies compared to the the one observed for the two
dielectric relaxations: it would be natural to assume a similar origin. To
explain the difference on each side of the phase transition, it could be
hypothesized (see Supplementary Note 1) that, below the phase transition, the
decrease in activation energy is due to the existence of conducting domain
walls, that tend to display intrinsic semiconducting behaviourEliseev  _et
al._ (2011); Liu  _et al._ (2015); Nataf  _et al._ (2020).
In summary, the impedance spectra of Cu-Cl boracite were measured at different
temperatures. Two dielectric relaxations and an increase of permittivity at
low frequencies, consistent with the existence of a series resistor, were
found. The study of the temperature evolution of the two relaxations and the
low-frequency conductivity revealed the role of point defects. Based on the
similarity of activation energies ($\sim$ 0.6 eV) and differences in attempt
frequencies ($f_{0}=7-9\cdot 10^{12}$ Hz), we suggest that defect complexes,
related to a change of valence of Cu and oxygen vacancies, are the most
likely. Below the ferroelectric phase transition temperature, a decrease in
resistivity is observed, consistent with the appearance of the conductive
domain walls known in this system.

## Supplementary Material

See supplementary material for the difference between Debye and Havriliak-
Negami fitting, the effect of thermal treatment on the complex dielectric
permittivity and the resistivity dispersion.

###### Acknowledgements.

The assistance of C. J. Brierley at Plessey Research (Caswell) Ltd. in growing
the boracite crystals is gratefully acknowledged. The authors acknowledge
funding from the Engineering and Physical Sciences Research Council (EPSRC:
EP/P02453X/1; EP/P020194/1), the US-Ireland Research and Development
Partnership Programme (USI 120)

##  Author Declarations

### Conflict of interest

The authors have no conflicts to disclose.

## Data Availability Statement

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

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