[table]capposition=top

# Development of a cylindrical mirror analyzer electron spectrometer and
associated data acquisition system to study inner shell electron emission
following ion-atom collision

Rohit Tyagi  Department of Physics, Indian Institute of Technology Kanpur,
Kanpur - 208016, India  A. H. Kelkar<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Kanpur, Kanpur - 208016, India

###### Abstract

In this paper we report on the development and performance of a cylindrical
mirror analyser electron spectrometer for ion atom collision experiments. A
low cost data acquisition system using Arduino microcontroller has also been
developed and tested. We have measured the Auger emission spectra for various
gaseous targets in collision with 1 MeV proton beams. Relative total Auger
emission cross sections have also been measured for N2 molecular target as a
function of proton energy.

††preprint: APS

##  I Introduction

Electron Spectroscopy involves the study of the energy and angular emission
pattern of secondary electrons emitted from atoms, molecules, clusters and
solid surfaces upon excitation via charged particles (electrons or heavy ions)
and photons. In ion-atom collision, the nature of Coulomb interactions
manifests in various ionization processes such as soft electron emission,
binary encounter and Auger electron emission etc. Study of these process is
useful in atomic and molecular physics, material science, space research, and
plasma diagnostics etc. In the last couple of decades, there has been an
increase in interest in probing state-selective atomic and molecular phenomena
such as Inter-atomic/Intra-molecular Coulombic decay (ICD) and molecular ion
dissociation dynamics Murphy1988  using kinematically complete ion-electron
yield measurements JUllrich_1997 . To study these processes, an electron
energy analyzer is usually combined with a recoil ion momentum
spectrometerJUllrich_1997 ; Albert2005i ; Saha2013 .Various designs for an
electron spectrometer, for example, hemispherical analyzer, parallel plate
analyzer, cylindrical analyzer, magnetic bottle electron spectrometer etc are
routinely employed in electron spectroscopy investigations.

Here we present the development of a new electron spectrometer, based on a
cylindrical mirror analyzer, for measuring secondary electron emission from
gaseous targets induced by fast ion impact. A cylindrical mirror electron
analyzer (CMA) has a simple design, excellent sensitivity, and low signal-to-
noise ratio. The set-up consists of an ultra high vacuum compatible
cylindrical mirror analyzer (CMA), designed and assembled at the 1.7 MV
Tandetron accelerator laboratory at IIT Kanpur, to measure electron yields as
a function of electron energy for fast ions interacting with atoms/molecules.
In the following sections we will discuss the design parameters, simulation
and performance of the CMA in detail. In addition to the CMA, we will also
dicuss the implementation of an automated data acquisition and control system,
based on Arduino Uno microcontroller.

##  II Experimental set-up and CMA design

Performance characterization of the CMA was carried out by studying proton
collision induced Auger electron emission for various atomic and molecular
targets. The experiments were conducted at the 20o beam line of 1.7 MV
tandetron accelerator facility HVEE  at IIT Kanpur. Intitally a beam of H-
ions, produced using a duoplasmatron source, is accelerated to a potential of
10 kV - 15 kV. this beam is mass analyzed using a 90o magnet before entering
the tandem accelerator. In the tandem accelerator the H- beam gets accelerated
to the terminal potential ($TV=200$ kV $-1.7$MV), where it collides with high
pressure nitrogen gas. The collision process strips the H- ions of the
electrons, leading to creation of H+ beam which is further accelerated by the
same potential in the outgoing channel. In general, for the tandem
accelerator, the final energy of the ion beam is given as $(q+1)TV$, where $q$
is the ion charge state and $TV$ is the terminal voltage. The positive ion
(proton) beam is further focused using an electrostatic quadrupole triplet and
directed toward the 20o beam line using a switching magnet. The 20o beam line
is equipped with a set of steering plates for alignment of the ion beam. In
addition, the exit end of the beam line is slightly off centered from the beam
line and a set of deflector plates are used to deflect the ion beam, thereby
filtering out the neutrals in the main ion beam created during its passage
though the beam line. The filtered projectile ion beam is collimated to a size
of 2$\times$2 mm2 using a pair of four jaw slits before entering the main
scattering chamber. A differential pumping chamber is connected between the
four jaw slits to maintain high background vacuum ($<5\times 10^{-8}$ mbar) in
the main scattering chamber. The CMA is mounted in the scattering chamber with
its symmetry axis orthogonal to both, the ion beam and target gas jet as shown
in fig 1.

###  II.1 Cylindrical mirror analyser

Fig. 1: Schematic of experimental setup. In Left Schematic of the the CMA
assembly. OE (outer electrode) cylinder diameter is 84 mm and length 96 mm. IE
(Inner electrode) diameter is 40 mm. slit to slit distance is 77 mm. Total
length of CMA is 118 mm. In middle Assembled CMA mounted on CF-150 flange is
shown. In Right, CAD drawing of scattering chamber is shown.

The Cylindrical mirror analyzer (CMA) (see fig 1) consists of two coaxial
cylindrical shells made of non-magnetic stainless steel. A conical electron
stopper is also connected to the inner cylinder to improve the energy
resolution (discussed later). CMAs are widely used as electron energy
analyzers and their ideal design parameters are well known. Electrons emitted
from the interaction zone, located on the axis of the CMA enter through a slit
on the inner cylinder electrode, traverse the electric field region between
the inner and outer cylinder and exit through another slit on the inner
cylinder. The strength of electric field and distance between the two slits
determine the pass energy of the electrons. The optimum location of the
electron source (interaction region) on the axis is such that it subtends an
entrance angle of 42.3o with respect to the axis of the cylinder Sar‐El1971 .
However, due to geometrical constraints, the interaction zone in the present
setup is located 30.5 mm away from first slit of the inner electrode. The
entrance angle of the electrons, therefore, is 35.5o in our setup. The design
parameters of the CMA have been chosen such that it can be coupled later with
a recoil ion momentum spectrometer setup Duley2022  for performing combined
recoil ion - electron spectroscopy following molecular fragmentation. Using
the geometrical parameters of the CMA and classical trajectory calculation for
motion of a charged particle in cylindrical capacitor field Moore2009 , we can
obtain the value of electrostatic potential on the outer electrode for
transmission of electrons with energy $E_{e}$ (in eV) as:

$\mathrm{V_{outer}=\frac{E_{e}\ln\frac{R_{2}}{R_{1}}\sin^{2}\theta}{\ln(\frac{z}{4R_{1}}\tan\theta+1)}}$
(1)

or,

$\mathrm{V_{outer}=0.48E_{e}}$ (2)

Here, $R_{1}$ and $R_{2}$ are the radii of inner and outer cylinders,
respectively, $\mathrm{z}$ is the distance between the two slits and
$\mathrm{\theta}$ is the entrance angle.

In order to keep the interaction zone free of any electric field, the inner
electrode was kept at ground potential. the outer surfaces of the inner
electrode and electron stopper as well as the inner surface of the outer
electrode were coated with carbon soot from an ethylene welding torch. the
carbon soot coating helps in suppressing the emission of secondary electrons
within the CMA due to collision of high energy electrons with the walls of the
electrodes. A channel electron multiplier (CEM) detector is placed 24 mm away
from the exit slit of inner cylinder (on the axis of CMA) to detect the
electrons.

Fig. 2:  Electron trajectories In (a) with stopper and (c) without stopper, in
(b) and (d) potential contour lines are shown .

###  II.2 SIMION Simulation

Fig. 3:  Outer Electrode potential(negative) vs pass energy of electrons. Data
is generated using SIMION sofware.

The design of the CMA was optimized by performing particle trajectory
simulations using SIMION 8.0 simion . For electron trajectory simulation, the
inner electrode was kept at ground potential and a negative potential was
applied to the outer electrode. The interaction zone, kept 30.5 mm away from
the entrance slit, was simulated as a spherical region of 2 mm radius.
Thousands of electrons were generated in this region with a kinetic energy
distribution $\mathrm{E_{e}\pm 10\%}$. The particle velocities (direction)
were constrained within an emission cone with half angle $\mathrm{\theta\pm
1^{o}}$, where $\theta$ is the geometric acceptance angle of the CMA. This
approach aimed to streamline the simulation process and reduce computation
time.

In fig 2 we have shown representative electron trajectories though the CMA. In
the first design iteration (see fig 2 (a) and (b)) it was observed that for an
extended interaction zone, electrons of kinetic energies different from the
pass energy of the CMA are also able to transmit though the CMA. this results
in very poor energy resolution of the CMA. In the laboratory experiments, the
interaction zone is defined as the overlap between the projectile ion beam and
the effusive target gas jet, which may extend beyond 2 mm. The background gas
density in the scattering chamber would also contribute as extended
interaction zone along the path of the projectile ion beam. This poses a major
challenge in achieving desired energy resolution of the analyzer. To alleviate
this, we modified the design of the CMA by introducing a conical electron
stopper (see fig 2 (c and d)). the electron stopper was kept at the same
potential as the inner electrode. As evident from the simulated trajectories
(see fig 2 c) the electron stopper creates a narrow annular region for
electrons to enter the CMA, thereby restricting the acceptance of electrons
emitted in a limited interaction volume on the axis of the spectrometer.
Introduction of the electron stopper results in improved energy resolution of
the CMA and from the simulated trajectories, we obtained the energy resolution
of the CMA $\mathrm{\sim 3\%}$.

In fig 3 we have plotted the voltage on the outer electrode vs the electron
kinetic energy. The voltage varies linearly with the electron kinetic energy,
as expected from equation 1 above. A linear fit to the simulated data in fig
3, gives a slope of 0.49, which agrees well with equation 2 for the present
CMA geometry.

###  II.3 Data acquisition system

Fig. 4: Schematic of Data Acquisition system. Fig. 5: (a) Schematic of the
circuit for Arduino PWM to Analog signal converter. (b) PWM output of Arduino
with 50% duty cycle (Yellow square wave signal) and analog dc output of the
sallen-key filter (Red solid line). Fig. 6: Sallen-Key filter output Vs power
supply output. Black rectangular points show the measured values. Red solid
line is a linear fit to the data.

A schematic diagram of the data acquisition system is shown in figure 4.
During the experiment, the outer electrode of the CMA is floated at the
desired voltage using an externally controlled ortec power supply ().
Electrons passing through both slits of the CMA are detected by the CEM
detector, which generates a pulse of $\sim$ 10 mV ($\sim$ 20 ns) for each
incident electron. The CEM is operated in saturation mode for maximum
detection efficiency. The output signal of the CEM is amplified using a fast
amplifier (Ortec ) and fed to a constant fraction discriminator (CFD). The CFD
generates a corresponding TTL output which is stored in the PC after
digitization. The projectile ion beam current falling on the Faraday cup is
measured using a charge integrator module (ortec). The charge integrator also
generates TTL pulses which are recorded in the PC after digitization. In order
to record the electron spectrum over a given energy range, automatically, the
data acquisition system is required to handle three signals. i) Generate an
analog control signal for the dc power supply connected to the outer electrode
of the CMA, ii) digitize and record the TTL signal corresponding to the CEM
detector output and iii) record the TTL output from the charge integrator
module. A low cost data acquisition system was designed in-house, to automate
the process using off-the-shelf Arduino microcontroller Arduino  and Raspberry
Pi RaspberryPi  personal computer.

The Arduino microcontroller is a versatile open-source platform designed for
electronics projects. It features analog and digital input/output pins,
enabling it to interface with sensors, collect data, and communicate with
other devices. The programming interface for Arduino Uno is highly user-
friendly extensive community support makes it a popular choice for creating
customized data acquisition setups. Arduino Uno microcontroller has a digital
i/o pins which can be used to record the TTL signals. It also has a 12 bit
digital-to-analog converter and analog output pins which generate a
corresponding PWM output. The analog remote signal for controlling the outer
electrode power supply was generated using the PWM output of the Arduino Uno
(Pin 12). The PWM (pulse width modulated) output is a 5 V square wave signal
with varying duty cycle, corresponding to each analog value. We designed a
second order Sallen-Key low-pass filter Sallen1955  convert the PWM output
into a true analog dc signal. This signal is then used to control the ortec
power supply connected to the outer electrode of the CMA. The Schematic
circuit of the Sallen-Key filter is shown in figure 5a. This circuit
incorporate a basic IC 741. The second order nature of the filter ensures a
more gradual roll-off of high-frequency components, resulting in a stable and
continuous output. The PWM output (50$\%$ duty cycle) from Arduino and
filtered dc output are shown in figure 5b. The measurements were performed
with a 1 GHz Oscilloscope (Yokogawa model no. ). The transfer function H(s)
and cutoff frequency $\omega_{0}$ of the filter is given as,

$H(s)=\frac{\omega_{o}^{2}}{s^{2}+2\alpha s+\omega_{0}^{2}}$ (3)
$\omega_{0}=\frac{1}{C_{1}C_{2}R_{1}R_{2}}$ (4)

where $s=j\omega$ and $\alpha$ is attenuation constant.

To further verify the dc characteristics of the Sallen-Key output, we measured
the output of the outer electrode power supply for a range of control signal
amplitudes. The plot in figure 6 shows a perfectly linear response of the
power supply controlled using the frequency filtered PWM output of the Arduino
microcontroller.

As discussed above, digital i/o pins of Arduino microcontroller were employed
to record the TTL pulse outputs from the CFD and Charge integrator. Typical
count rate from the CEM detector (and CFD) during the experiments was kept
below 1 kHz. The projectile beam current was also $\sim$ 50 nA. This results
in TTL output frequency of $\sim$ 500 Hz from the charge integrator. The clock
frequency of Arduino Uno microcontroller is 16 MHz, which is high enough for
faithful recording of these data at the digital i/o pins. Nevertheless, an
experimental validation is useful for assigning confidence in the measured
electron energy spectrum. A characteristic feature of ion-atom collision
process is the probabilistic nature of electron emission. Therefore, the
measured electron count, N, at a fixed energy (CMA voltage) follows Poisson
distribution and over multiple measurements the number N has an uncertainty of
$\sqrt{N}$, under otherwise identical experimental conditions. We have
measured this distribution for various pass energies of the CMA. For the
measurement, the outer electrode potential was kept at a fixed voltage, and
electron count data (the CFD output) was recorded multiple times for a fixed
number of pulses received from the charge integrator module. In figure 7, we
have shown one such measured histogram for electrode voltage = -50 V. The
histogram is fitted well with a Gaussian distribution and the standard
deviation ($\sigma$) is comparable to $\sqrt{N}$, where N is the peak of the
Gaussian distribution.

Fig. 7:  Electron counts statistics while keeping the outer electrode at -50
V. Measurement has been done 911 times.

The process of control signal generation as well as simultaneous recording and
storage of the CEM detector and charge integrator outputs was automated using
a custom written Python python  script. To facilitate real-time data
communication between the experimental setup and the PC, the pyserial 3.5
pyserial  library is employed. The Arduino board was programmed to send the
necessary information through the serial port, accessible to the PC through
the Python script. The Arduino script continuously sends the relevant data,
such as electron detection events from the CFD and the beam current from the
charge integrator, through the serial port. The Python script utilizes
Matplotlib module in Python to create online electron energy spectrum,
therefore enabling visualization of the experimental data in real-time.

Fig. 8: Electron count distribution passing from both slit from an electron
gun with extraction voltage kept at (a) 100 V (b) and 145 V. Electron gun
filament was at 3.2 V and 1.7 A.

###  II.4 CMA energy resolution

For estimating the resolution of the CMA, we used a diverging beam electron
gun in a separate experiment. The electron gun was placed along the axis of
the CMA and electron spectra were measured for several energies of the emitted
electrons. The emitted electron energy had a spread of $\mathrm{\sim 10\%}$.
The energy resolution of a CMA spectrometer is given as:

$\mathrm{\frac{\Delta E}{E}=\frac{w}{L}}$ (5)

where $\mathrm{w}$ is the slit width and $\mathrm{L}$ is the distance between
the slits.

The measured electron energy spectra are sown in figure 8 for two electron
energies, 100 eV and 145 eV. Using a Gaussian fit to the measured electron
spectrum, we have estimated the energy resolution of the CMA $\mathrm{\sim
3-4\%}$. A similar resolution was estimated from SIMION simulation.

##  III Calibration and measurements

Fig. 9: Measured Auger electron energy peak of Ar, Oxygen in $CO_{2}$, Carbon
in $CH_{4}$ and nitrogen in N2 on Y axis and peak from reference on x axis by
1 MeV proton bombardment are shown. Straight line fit slope is 1.02 with
standard error 0.026. Error bar of 3% are shown Fig. 10: Measured electron
energy spectrum of N2 by 1.2 MeV proton bombardment (at Top). And $CO_{2}$,
$CH_{4}$ and Ar ( from Top to bottom) by 1 MeV proton bombardment. Fig. 11:
Total cross section after integrating DDCS for different projectile energy.

For calibration of the CMA electron spectrometer we measured the Auger spectra
for various atomic and molecular targets in collision with 1 MeV proton beam.
In figure 10 we have shown the electron spectra corresponding to K-LL Auger
emission from CH4, N2, CO2 molecules and L-MM Auger emission from Ar atoms. In
each spectrum, the electron counts have been normalized with respect to the
projectile beam intensity and target gas pressure. The spectrum represents
relative differential cross section for electron emission. Geometry of CMA
restricts collection of electrons in a narrow range of angle
$\mathrm{\theta=35.5^{o}}$ with respect to the projectile ion direction,
however, there is near complete ($\mathrm{2\pi}$) collection in the azimuthal
plane (normal to the spectrometer axis). Therefore, the relative differential
cross section is given as:

$\frac{d^{2}\sigma}{dE_{e}d\theta}=\frac{N_{e}-N_{nkg}}{N_{p}P}$ (6)

where, $N_{e}$ is the number of target electrons (with gas) detected at energy
$E_{e}$, $N_{bkg}$ is the number of background electrons (without gas),
$E_{e}$ is the emitted electron energy, $N_{P}$ is the number of projectile
ions and P is the target gas pressure. An ionization gauge was used to measure
the gas pressure and hence absolute number density of target gas
molecules/atoms could not be measured in the present experiments.

The energy spectra were measured in a limited energy range around the Auger
emission energy peak region in all cases. The spectra (see inset in figure 10)
show the characteristic features of secondary electron emission following
collision with energetic ions MISRA2009 . The electron emission cross section
decreases monotonically as a function of electron kinetic energy. This is
attributed to the binary collision process. A broad peak corresponding to the
Auger process is visible riding over this continuum background. In order to
extract the contribution from Auger peaks, one can approximate the continuum
background in the peak region by a polynomial function and subtract it from
the measured energy spectrum. The interpolated continuum background is shown
as solid curve in the inset of figure 10 and the Auger peaks after background
subtraction are shown in the main panel. The Auger energy distribution is a
broad peak and consists of several Auger lines corresponding to emission from
different L and M sub-shells. More than one peak in the Auger spectrum can be
identified. However, the resolution of CMA is not high enough to clearly
resolve the sub-shell contribution and we have used multiple Gaussian fits to
reveal identifiable Auger lines. Additionally, in energetic ion-atom collision
the Auger process is also accompanied by multiple ionization of the valance
electrons, thereby broadening the Auger region. In table 1, we have reported
the measured Auger energy values (using the calibration factor from SIMION
simulation) are in excellent agreement with the Auger values reported in
literature Kobayashi1976  validating the calibrating factor obtained using the
SIMION simulation.

We further studied the projectile energy dependence of $N_{2}$ K-LL Auger
emission yield. The proton beam energy was varied from 0.5 MeV to 2 MeV.
Several groups have reported detailed cross section measurements on K-LL auger
emission in the later part of last century Stolterfoht1975 ; Kobayashi1976 .
For low Z elements, the K-LL Auger cross sections are equivalent to the inner
shell ionization cross section due to extremely low fluorescence yield leading
to x-ray emission. The K-LL Auger emission cross sections reach a maximum in
the present projectile energy range and a variation within 10$\%$ has been
reported Kobayashi1976 . In figure 11, we have shown the integrated yield
(relative cross section) for N2 K-LL Auger emission in ionizing collisions
with 0.5 MeV to 2 MeV proton beam. The measured relative cross sections do not
vary much with in the experimental uncertainty ($\sim$ 15$\%$). This is in
agreement with the reported measurements in this projectile energy range for
H+ ions. The uncertainty in the measured cross sections in attributed mainly
to the fluctuation in the target gas pressure.

TABLE. 1: Auger peak energy for different targets. The measured values have been compared with those give in reference Kobayashi1976  proton energy (MeV) |  Target gas |  Auger $e^{-}$ energy (eV) |  Reference Kobayashi1976
---|---|---|---
1 |  $N2$ |  357,335,315 |  360,340,320
1 |  $CO_{2}$ |  502,469 (O,KLL) |  500,470
1 |  $CH_{4}$ |  247,231,259 (C, KLL) |  250,230,265
1 |  $Ar$ |  195,170 (LMM) |  193,175

##  IV conclusion

We have presented the design, construction, and testing of a cylindrical
mirror analyzer electron spectrometer and associated data acquisition system.
The spectrometer is capable of measuring electron spectrum with an energy
resolution of $\sim$ 4$\%$ and nearly 2$\pi$ collection of electrons in the
azimuthal plane. The spectrometer performance was evaluated by measuring
electron spectra, following collision with MeV energy proton beam, for
$\mathrm{N_{2},CO_{2},CH_{4}andAr}$ in the K-LL (L-MM for Ar) Auger energy
range. Energy of Auger peaks measured for different targets match well with
the reported values in literature.

##  V Acknowledgments

###### Acknowledgements.

The Authors would like to thank Sahan Sykam and Sandeep Bari for smooth
operation of the tandetron accelerator facility during the experiments. AHK
would also like to acknowledge support from DST-SERB grant no.
ECR/2017/002055.

## References

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††thanks<EMAIL_ADDRESS>These two authors contributed equally††thanks: These
two authors contributed equally

# Supplementary information for “Silicon-lattice-matched boron-doped gallium
phosphide: A scalable acousto-optic platform”

Nicholas S. Yama  University of Washington, Electrical and Computer
Engineering Department, Seattle, WA, 98105, USA  I-Tung Chen  University of
Washington, Electrical and Computer Engineering Department, Seattle, WA,
98105, USA  Srivatsa Chakravarthi  University of Washington, Physics
Department, Seattle, WA, 98105, USA  Bingzhao Li  University of Washington,
Electrical and Computer Engineering Department, Seattle, WA, 98105, USA
Christian Pederson  University of Washington, Physics Department, Seattle, WA,
98105, USA  Bethany E. Matthews  Energy and Environment Directorate, Pacific
Northwest National Laboratory, Richland, Washington 99352, USA  Steven R.
Spurgeon  University of Washington, Physics Department, Seattle, WA, 98105,
USA  Energy and Environment Directorate, Pacific Northwest National
Laboratory, Richland, Washington 99352, USA  Daniel E. Perea  Earth and
Biological Sciences Directorate, Environmental Molecular Sciences Laboratory,
Pacific Northwest National Laboratory, Richland, Washington 99352, USA  Mark
G. Wirth  Earth and Biological Sciences Directorate, Environmental Molecular
Sciences Laboratory, Pacific Northwest National Laboratory, Richland,
Washington 99352, USA  Peter V. Sushko  Physical and Computational Sciences
Directorate, Pacific Northwest National Laboratory, Richland, Washington
99352, USA  Mo Li  University of Washington, Electrical and Computer
Engineering Department, Seattle, WA, 98105, USA  University of Washington,
Physics Department, Seattle, WA, 98105, USA  Kai-Mei C. Fu  University of
Washington, Electrical and Computer Engineering Department, Seattle, WA,
98105, USA  University of Washington, Physics Department, Seattle, WA, 98105,
USA  Physical Sciences Division, Pacific Northwest National Laboratory,
Richland, Washington 99352, USA

##  SI.1 Device fabrication

###  SI.1.1 Membrane transfer for hybrid devices

The commercial BGaP-on-Si wafer was covered in a protective photoresist layer
(AZ1512, MicroChemicals) and diced into 1 cm$\times$1 cm chips. For the chips
used in the hybrid devices, the silicon backside was mechanically thinned to
75 µm. The protective resist was stripped in solvent and a new layer of AZ1512
was spun on. Algined-mask photolithography (ABM) was then performed in order
to define individual membranes with vias for wet transfer. Individual
membranes are 1.5 mm$\times$1.5 mm and have rectangular vias defined at 250 µm
intervals. After developing the resist, the exposed BGaP is etched in an
Ar:Cl2:N2 inductively coupled plasma reactive ion etch (ICP-RIE) to transfer
the membrane pattern into the BGaP layer. The chip is then cleaved between the
membranes for individual use. A single-membrane chip is then flipped onto a
sapphire substrate (silicon side facing up) and the sample is exposed to a
XeF2 vapor etch (SPTS) to remove the remaining silicon. The BGaP membrane
adheres to the sapphire substrate which is then submerged in water, releasing
the membrane onto the surface (the resist is hydrophobic and orients upward).
The membrane can then be picked up on the target substrate (SiNx in this work)
and allowed to dry. After drying the adhesion to the substrate increases and
the remaining photoresist can be stripped in solvent.

###  SI.1.2 Photonic device fabrication

Starting with either a BGaP-on-Si or transferred BGaP-on-substrate chip, the
sample is cleaned in a solvent rinse and allowed to dehydrate on a hotplate at
150∘C. Electron-beam lithography (EBL) resist, hydrogen silsesquioxane (HSQ),
is spun on to around 100 nm thick and soft baked at 80∘C. The photonic
structures are then patterned in EBL and the HSQ is developed in a 25%
tetramethylammonium hydroxide (TMAH) bath. The device pattern is transfered to
the BGaP layer by ICP-RIE either completely etching through the membrane (as
in the hybrid devices) or partially etching the membrane. If the devices need
to be suspended then AZ1512 photoresist is spun on and etch vias are patterned
using an aligned direct write (Heidelberg DWL66 plus). The resist is developed
and the etch vias are defined in ICP-RIE. At this stage XeF2 vapor etch is
used to undercut the silicon substrate for the suspended devices.

###  SI.1.3 Suspended AO device fabrication

The BGaP photonic waveguides and grating couplers are patterned with EBL using
PMMA resist. The pattern is transferred into BGaP by partial etching via ICP-
RIE using chlorine-based chemistry. The 290 nm thick ZnO film is deposited
using an RF magnetron sputtering system and lifted-off in a sonicated acetone
bath. The IDT pattern is written using the EBL and lifted-off after depositing
220 nm aluminum film using an electron-beam evaporator. The releasing vias are
patterned with photolithography and etched in ICP. Finally, XeF2 etching of
silicon is used for suspending the BGaP layer.

Figure SI.1: Fabrication process Schematic representations and end results of
the fabrication processes for (a) hybrid devices, (b) suspended photonics, and
(c) supsended AO devices.

Figure SI.2: Membrane transfer optical images. Optical images of the hybrid
BGaP (a, b) and the un-doped MBE GaP (c, d) devices. (a) Transferred BGaP
membrane for the devices used in the paper (approximately 1.5 mm on both
edges, vias are spaced at 250 µm). Several bubbles formed on the bottom-left
side of the membrane. The large particle on the bottom is suspected to be some
of the photoresist which did not immediately dissolve. The remaining top-right
corner is well-adhered and used to pattern devices. (b) One of the final BGaP
5-µm disk devices. (c) The transferred un-doped GaP membrane after device
patterning (prior to etching). The membrane is partial as it is from the edge
of the wafer. (d) Final devices after etching. The highlighted devices are the
high-$Q$ optical devices. In both cases, the design of the 5-µm disks are
identical.

Figure SI.3: Suspended devices. (a) Real-color optical microscope image of the
measured device set after fabrication. Encircled device sets are (i) 40-µm
ring telecom devices, (ii) 20-µm ring telecom devices, and (iii) 10-µm ring
visible devices. The light blue color is real and corresponds to the undercut
area. (b) Profilometer measurement (Bruker DektakXT) of the devices prior to
undercutting the substrate with XeF2. The height is zeroed to partially etched
GaP height giving the partial etch depth to be 168 nm. Likewise, the etch via
depth is measured to be around 110 nm which is consistent with the
approximately 268 nm thick membrane and some slight etching of the silicon
substrate by the ICP-RIE. Optical microscope images of (c) visible devices,
and (d, e) 40-µm ring telecom devices using cross-polarized bright-field
imaging. The membrane appears transparent and the undercut silicon surface can
be seen. SEM images of the (f) 10-µm ring visible devices, and (g,h) 40-µm
ring telecom devices.

##  SI.2 Device measurement

###  SI.2.1 Photonic device transmission measurements

Samples were loaded into custom-built confocal microscope systems and
transmission measurements were performed using a supercontinuum laser
(Fianium), scanning diode laser (Sacher), and/or scanning telecom laser (HP)
sent through an erbium-doped fiber amplifier (Thorlabs). The transmission
signal was filtered through a combination of spatial filtering and cross-
polarized optics. The resulting light could be collected on a spectrometer
(Princeton instruments) or avalanche-photodiodes (Excelitas, ThorLabs).

##  SI.3 Material characterization

###  SI.3.1 Atomic-force microscopy

The surface roughness of the material was measured using a Bruker Icon atomic
force microscope in tapping mode under ambient conditions.

###  SI.3.2 Variable-angle spectroscopic ellipsometry

Variable angle spectroscopic ellipsometry (J. A. Woollam RC2) is performed
over GaP transparency window (500 nm–2500 nm) on an unused BGaP-on-Si chip to
determine the refractive index and (measurable) optical absorption. Assuming
layer thicknesses of 41 nm and 37 nm for the top and bottom GaP layers, and
using known GaP refractive index data, the BGaP layer thickness is fitted
against a Gaussian oscilaltor model. Importantly, the backside of the silicon
substrate (750 µm) is polished and must be accounted for in the model. The
fitting is performed in the CompleteEASE software (J. A. Wollam) for
measurements at 55∘, 65∘, and 75∘ and wavelength range between 500–2500 nm.

Figure SI.4: VASE fit. Data and fit for the VASE ellipsometry data obtained
from CompleteEASE.

###  SI.3.3 Scanning Transmission Electron Microscopy

Cross-sectional TEM samples were prepared using a FEI Helios NanoLab DualBeam
Focused Ion Beam (FIB) microscope and a standard lift out procedure. Images of
the as-grown and post-irradiated samples were collected using a probe-
corrected JEOL GrandARM-300F STEM operating at 300 kV with a semi-convergence
angle of 27.5 mrad and a collection angle of 82–186 mrad (HAADF) and 48–150
(MAADF). Additional drift-corrected high-resolution HAADF images were
collected using a probe-corrected Thermo Fisher Scientific Themis Z microscope
operating at 300 kV, with a convergence semiangle of 25.2 mrad and an
approximate collection angle range of 65 to 200 mrad.

##  SI.4 Temporal coupled-mode theory model for the add-drop resonator

We model the disk/ring resonator devices using temporal coupled-mode theory
(CMT). This section derives the device transfer functions from the coupled-
mode equations as used in the main text.

We assume throughout that the resonance linewidths are significantly smaller
than the free spectral range. Additionally we consider a coupling geometry
corresponding to the four-port add-drop resonator configuration as shown in
Fig. SI.5. The cavity field terms $a_{j}(t)$ give the amplitude and
(temporally varying) phase of the $j$-th propagating cavity resonance,
normalized such that $|a_{j}|^{2}$ gives the energy in the cavity. Likewise,
the waveguide field terms $s_{k\pm}$ give the incoming/outgoing waveguide
field amplitudes, normalized such that $|s_{k\pm}|^{2}$ gives the power in the
corresponding mode. It is assumed that the only incoming mode is at the input
port $s_{i+}$ and that there is no reflection near the coupling region.

Figure SI.5: Coupled-mode theory model. Schematic representations of the
coupling between the waveguide and cavity mode(s) for (a) ideal, (b)
symmetrically split, and (c) asymmetrically split cases.

###  SI.4.1 Ideal case

We first consider the idealized case shown in Fig. SI.5a in which a single
counter-clockwise propagating mode $a(t)$ at frequency $\omega_{0}$ with
intrinsic decay rate $\gamma$ is coupled to each waveguide with a rate
$\kappa$. Time evolution of the mode under driving $s_{i+}$ is described by
the differential equation

$\derivative{a}{t}=\left(-i\omega_{0}+\frac{\Gamma}{2}\right)a+\sqrt{\kappa}e^{i\theta}s_{1+},$
(SI.1)

where $\Gamma=\gamma+2\kappa$ is the total energy decay rate. Note the factor
of $2\kappa$ corresponds to the fact that there are two symmetrical waveguides
for coupling.

The outgoing fields at the drop ($s_{d-}$) and pass ($s_{p-}$) ports are
related to the add port ($s_{a+}$), input field ($s_{i+}$), and cavity field
$a$ through the linear system of equations [1]

$\displaystyle s_{p-}$
$\displaystyle=e^{i\phi}(s_{i+}-\sqrt{\kappa}e^{-i\theta}a),$ (SI.2)
$\displaystyle s_{d-}$
$\displaystyle=e^{i\phi}(s_{a+}-\sqrt{\kappa}e^{-i\theta}a),$ (SI.3)

where $\phi$ corresponds to the phase winding incurred between the port
reference planes. Here and throughout we assume that $s_{a+}=0$. These
equations in combination with equation (SI.1) constitute the coupled-mode
equations for a single-mode resonator in the add-drop configuration.

Since we are interested in the steady-state solution for a periodic driving at
frequency $\omega$, we can solve the equations by Fourier transform taking
$a(t)\to A(\omega)$, and likewise $s(t)\to S(\omega)$ for the waveguide
fields. To determine the response functions for the pass $T_{p}$ and drop
$T_{d}$ ports we solve the coupled-mode equations as

$\displaystyle
T_{p}=\absolutevalue{\frac{S_{p-}}{S_{i+}}}^{2}=\frac{\Delta^{2}+(\gamma/2)^{2}}{\Delta^{2}+(\gamma/2+\kappa)^{2}},$
(SI.4) $\displaystyle
T_{d}=\absolutevalue{\frac{S_{d-}}{S_{i+}}}^{2}=\frac{\kappa^{2}}{\Delta^{2}+(\gamma/2+\kappa)^{2}},$
(SI.5)

where $\Delta\equiv\omega_{0}-\omega$ is the detuning. These correspond to a
Lorentzian dip/peak at the resonance frequency $\omega_{0}$ with full-width-
half-max (FWHM) of $\Gamma=\gamma+2\kappa$.

The total quality factor $Q$ of the resonator at this resonance is defined as
the ratio $Q\equiv\omega_{0}/\Gamma$. We can further distinguish the intrisic
quality factor $Q_{i}\equiv\omega_{0}/\gamma$ and coupling quality factor
$Q_{c}\equiv\omega_{0}/\kappa$ which distinguish between intrinsic and
coupling losses respectively. The quality factors are related by the inverse
relationship $1/Q=(1/Q_{i})+(2/Q_{c})$.

The depth of the resonance in the pass port is given by

$T_{p}(\Delta=0)=\frac{(\gamma/2)^{2}}{(\gamma/2+\kappa)^{2}}=\left(\frac{Q_{T}}{Q_{i}}\right)^{2}.$
(SI.6)

This allows for a straightforward determination of the intrinsic quality
factor of the resonator from a single measurement of the through port provided
the resonance depth and total $Q$ can be determined accurately.

###  SI.4.2 Symmetric mode splitting

We now consider the case of symmetric splitting as schematically shown in Fig.
SI.5b. The ring resonator structure supports two degenerate, counter-
propagating modes at a given frequency; one clockwise propagating (CW) and one
counter-clockwise (CCW). In the previous section we considered the CCW mode,
which, under ideal conditions, would remain independent of the CW mode.
However, imperfections in the fabricated resonator can introduce defects which
cause back-scatter thus coupling the modes at some rate $g$ and splitting the
resonance profile. In the figure, a single point defect scatterer is shown,
however the observed coupling will generally be the result of many such
defects distributed over the entire resonator.

The coupled-mode equations are modified to include the CW-propagating mode
which, by symmetry, has an identical resonance frequency $\omega_{0}$ and loss
rate $\Gamma$. Letting $a_{\mathrm{CCW}}$ ($a_{\mathrm{CW}}$) be the CCW (CW)
modes respectively, we have

$\displaystyle\derivative{a_{\mathrm{CCW}}}{t}$
$\displaystyle=\left(-i\omega_{0}+\frac{\Gamma}{2}\right)a_{\mathrm{CCW}}-iga_{\mathrm{CW}}+\sqrt{\kappa}e^{i\theta}s_{i+},$
(SI.7) $\displaystyle\derivative{a_{\mathrm{CW}}}{t}$
$\displaystyle=\left(-i\omega_{0}+\frac{\Gamma}{2}\right)a_{\mathrm{CW}}-ig^{*}a_{\mathrm{CCW}},$
(SI.8) $\displaystyle s_{p-}$
$\displaystyle=e^{i\phi}(s_{i+}-\sqrt{\kappa}e^{-i\theta}a_{\mathrm{CCW}}),$
(SI.9) $\displaystyle s_{d-}$
$\displaystyle=e^{i\phi}(s_{a+}-\sqrt{\kappa}e^{-i\theta}a_{\mathrm{CCW}}),$
(SI.10)

where the coupling is introduced through the terms containing $g$ which
corresponds to the coupling rate. This form implicitly assumes that the back-
scattering process is energy conserving which is a good approximation in most
cases [2]. The Fourier transform method can be applied again to determine

$\displaystyle T_{p}$
$\displaystyle=\absolutevalue{1-\frac{\kappa(i\Delta+\Gamma)}{(i\Delta+\Gamma)^{2}+|g|^{2}}}^{2},$
(SI.11) $\displaystyle T_{d}$
$\displaystyle=\absolutevalue{\frac{\kappa(i\Delta+\Gamma)}{(i\Delta+\Gamma)^{2}+|g|^{2}}}^{2}.$
(SI.12)

###  SI.4.3 Asymmetric mode splitting

The back-scattering process described in the previous section produces only
symmetrically split modes. However, in practice, modes are often
asymmetrically split. This is attributed to additional back-coupling
mechanisms [2] in which the resonator field is partially reflected at the
waveguide-ring coupling region due to the change in effective index. Some
amount of the reflected light couples into the counter-propagating waveguide
mode. This is shown schematically in Fig. SI.5c.

To model this the coupled-mode equations are modified to take the form

$\displaystyle\derivative{a_{\mathrm{CCW}}}{t}$
$\displaystyle=\left(-i\omega_{0}+\frac{\Gamma}{2}\right)a_{\mathrm{CCW}}-iga_{\mathrm{CW}}+\sqrt{\kappa}e^{i\theta}s_{i+},$
(SI.13) $\displaystyle\derivative{a_{\mathrm{CW}}}{t}$
$\displaystyle=\left(-i\omega_{0}+\frac{\Gamma}{2}\right)a_{\mathrm{CW}}-ig^{*}a_{\mathrm{CCW}}+f\sqrt{\kappa}e^{i\theta}s_{i+},$
(SI.14) $\displaystyle s_{p-}$
$\displaystyle=e^{i\phi}(s_{i+}-\sqrt{\kappa}e^{-i\theta}a_{\mathrm{CCW}}-f^{*}\sqrt{\kappa}e^{-i\theta}a_{\mathrm{CW}}),$
(SI.15) $\displaystyle s_{d-}$
$\displaystyle=e^{i\phi}(s_{a+}-\sqrt{\kappa}e^{-i\theta}a_{\mathrm{CCW}}-f^{*}\sqrt{\kappa}e^{-i\theta}a_{\mathrm{CW}}),$
(SI.16)

where we have introduced the back-coupling coefficient $f\in\mathbb{C}$ from
[2] with the modification of a complex conjugate in the output coupling terms.
The back-coupling coefficient parameterizes the relative amplitude and phase
of the back-coupling rate with respect to the complex forward-coupling rate
$\kappa e^{2i\theta}$. Because the back-coupling is another loss channel, the
total loss rate $\Gamma$ must be modified to include its contribution giving
$\Gamma=\gamma+2(1+|f|^{2})\kappa$.

Taking the Fourier transform as before allows for the determination of the
system response. The solutions for $A_{\mathrm{CCW}}$ and $A_{\mathrm{CW}}$
are found to be

$\displaystyle A_{\mathrm{CCW}}$
$\displaystyle=\frac{-igf\sqrt{\kappa}e^{i\theta}+(i\Delta+\Gamma)}{(i\Delta+\Gamma)^{2}+|g|^{2}},$
(SI.17) $\displaystyle A_{\mathrm{CW}}$
$\displaystyle=-\frac{|g|^{2}f\sqrt{\kappa}e^{i\theta}}{(i\Delta+\Gamma)\big{[}(i\Delta+\Gamma)^{2}+|g|^{2}\big{]}}-\frac{ig^{*}\sqrt{\kappa}e^{i\theta}}{(i\Delta+\Gamma)^{2}+|g|^{2}}+\frac{f\sqrt{\kappa}e^{i\theta}}{(i\Delta+\Gamma)}.$
(SI.18)

Substituting this into Eqs. (SI.15) and (SI.16) and simplifying terms gives
the following system response functions for the pass and drop ports

$\displaystyle T_{p}$
$\displaystyle=\absolutevalue{1+\frac{2i\kappa|g||f|\cos\beta-\kappa(1+|f|^{2})(i\Delta+\Gamma)}{(i\Delta+\Gamma)^{2}+|g|^{2}}}^{2},$
(SI.19) $\displaystyle T_{d}$
$\displaystyle=\absolutevalue{\frac{2i\kappa|g||f|\cos\beta-\kappa(1+|f|^{2})(i\Delta+\Gamma)}{(i\Delta+\Gamma)^{2}+|g|^{2}}}^{2},$
(SI.20)

where we have parameterized the splitting in terms of $(|g|,|f|,\beta)$ where
$\beta=\arg(gf)$.

The response functions can be written in the more suggestive form of

$\displaystyle T_{p}$ $\displaystyle=|1+X|^{2},$ (SI.21) $\displaystyle T_{d}$
$\displaystyle=|X|^{2},$ (SI.22)

where

$X\equiv-\frac{\kappa}{2}\left(\frac{(|f|+e^{i\beta})(|f|+e^{-i\beta})}{i(\Delta+|g|)+\Gamma}+\frac{(|f|-e^{i\beta})(|f|-e^{-i\beta})}{i(\Delta-|g|)+\Gamma}\right).$
(SI.23)

This shows that the parameters have simple physical interpretations with
$2|g|$ defining the splitting and $(|f|,\beta)$ giving the amplitude and
direction of the asymmetry. This reproduces the form of [2] with the addition
of tracking the relative phase between the two coupling processes.

####  SI.4.3.1 Weak back-coupling approximation

Although this model can be applied to a wide range of observed resonances,
fitting can be challenging in practice. In particular, the number of
parameters and their relative co-linearity prevents high-confidence fitting
from a single measured response function (typically the pass). Instead
transmission at the pass, drop, and add ports need to be considered
simultaneously. This complicates the measurement significantly as it not only
requires measurements of the weaker drop/add signals but also necessitates
thorough characterization of optical components other than the resonator
itself (e.g. the grating couplers, waveguides, and collection optics).

Instead we make an approximation of weak back-coupling such that $|f|\ll 1$.
This approximation is intuitive and is numerically demonstrated in Section
SI.5 in which the noisy (exact model) fits closely reproduce the approximate
model results while estimating $|f|<0.01$. Enforcing this approximation allows
for the quadratic terms $|f|^{2}$ to be dropped. As a result we have

$X\approx-\frac{\kappa}{2}\left(\frac{1+2|f|\cos\beta}{i(\Delta+|g|)+\Gamma}+\frac{1-2|f|\cos\beta}{i(\Delta-|g|)+\Gamma}\right)$
(SI.24)

and $\Gamma\approx\gamma+2\kappa$ as before. This reduces the number of free
parameters by one and gives $|f|\cos\beta$ as a single real-valued parameter
describing the asymmetry of the power spectrum.

Thus we determine the transmission at the pass and drop ports to be given by

$\displaystyle T_{p}$
$\displaystyle=\left|1-\frac{\kappa}{2}\left(\frac{1+2|f|\cos\beta}{i(\Delta+|g|)+\Gamma}+\frac{1-2|f|\cos\beta}{i(\Delta-|g|)+\Gamma}\right)\right|^{2},$
(SI.25) $\displaystyle T_{d}$
$\displaystyle=\left|\frac{\kappa}{2}\left(\frac{1+2|f|\cos\beta}{i(\Delta+|g|)+\Gamma}+\frac{1-2|f|\cos\beta}{i(\Delta-|g|)+\Gamma}\right)\right|^{2}.$
(SI.26)

This form of the pass transmission is used for the fitting to determine the
parameters of the high-$Q$ telecom resonances of the main text.

##  SI.5 Fitting of optical data

The CMT analysis discussed in Section SI.4 provides analytical functions for
the resonance lineshape. This section covers specific details of the fitting
procedure relevant to reproduce the results. In all cases the data is fit to
the model as determined by the a nonlinear-least-squares optimization
algorithm (Scipy). Additional fits and data not shown in the main text are
also included for completeness.

###  SI.5.1 Drop port transmission

In the case of drop port spectra, the absolute transmission value cannot
typically be ascertained without careful (and often noisy) calibration of the
grating coupler and waveguide transmission spectra. Fortunately, for the
purposes of this work, we need only determine the linewidth of the resonance
and so the lineshape can be fitted against a generic Lorentzian

$T_{d}^{(1)}(x)=a\frac{(\Gamma/2)^{2}}{(x-x_{0})^{2}+(\Gamma/2)^{2}}+p(x-x_{0})$
(SI.27)

where $p(x)=p_{0}+p_{1}x+p_{2}x^{2}$ is a quadratic background centered at the
resonance center. The parameter $a$ corresponds to the amplitude, $\Gamma$ is
the full-width-half-max of the resonance, and $x_{0}$ corresponds to the
center position.

The above functional form assumes that the background spectra (e.g. transfer
function of the waveguide, scattered input, dark counts) varies slowly over
the resonance linewidth which is typically confirmed by the symmetrical
lineshape of most drop port resonances. Strongly asymmetrical lineshapes
correspond to interference with additional modes, such as with Fano
resonances, and can have Lorentzian-fitted linewidths which do not correspond
directly to the resonance loss. As such, these modes were excluded from this
work where possible.

For the high-$Q$ intrinsic GaP devices, the laser scaning measurements reveal
split modes. For these resonances a simplified version of Eq. (SI.23) was used
for the fitting procedure given as

$T_{d}^{(2)}=\left|\frac{a+\delta a}{i((x_{0}+\delta
x_{0})-x)+(\Gamma+\delta\Gamma)}+\frac{a-\delta a}{i((x_{0}-\delta
x_{0})-x)+(\Gamma-\delta\Gamma)}\right|^{2}+p(x-x_{0}).$ (SI.28)

where the $\delta a$, $\delta x_{0}$, and $\delta\Gamma$ terms are small
shifts to the amplitude, peak centers, and linewidths respectively. The $Q$ is
then extracted assuming that these perturbations were symmetric about the
original mode which is consistent with the CMT model.

###  SI.5.2 Pass port transmission

The pass port transmission signal is advantagoues in that the normalization
can be determined directly from the spectrum provided the noise and ambient
background is well characterized. Pass port transmission spectra for the
low-$Q$ resonances were fit with the ideal CMT model given in Eq. (SI.11) with
the modification of a multiplicative background to account for transmission
through the grating and waveguide, given as

$T_{p}^{(1)}(\omega)=\frac{(\omega_{0}-\omega)^{2}+(\gamma/2)^{2}}{(\omega_{0}-\omega)^{2}+(\gamma/2+\kappa)^{2}}\cdot
p(\omega-\omega_{0}),$ (SI.29)

where again $p(x)=p_{0}+p_{1}x+p_{2}x^{2}$ is the quadratic background
modulation. In the case of split modes, the pass transmission is fit to the
model given in Eq. (SI.25) with form

$T_{p}^{(1)}=\left|1-\frac{\kappa}{2}\left(\frac{1+2|f|\cos\beta}{i((\omega_{0}-\omega)+|g|)+(\gamma+2\kappa)}+\frac{1-2|f|\cos\beta}{i((\omega_{0}-\omega)-|g|)+(\gamma+2\kappa)}\right)\right|^{2}\cdot
p(\omega-\omega_{0})$ (SI.30)

where $\\{\omega_{0},|f|\cos\beta,\kappa,\gamma,|g|\\}$ are used as the
fitting parameters (in addition to the 3 parameters for $p(x)$).

Beyond the mode splitting, another consequence of the high $Q$ factors was
thermal optical bi-stability which results in asymmetric mode profiles. This
would prevent accurate estimation of the quality factor. To combat this we
utilized stepped sweeps at lower optical powers for which the bistability
effect was negligible. However, the lower powers yield reduced signal-to-noise
and so multiple measurements of the same resonance were employed. These
measurements could then be fitted individually to the split-mode model and
then the resulting parameters could be averaged. An example of the multiple
scans are shown in Fig. SI.6 along with the corresponding fit parameters for
one resonance. Given such a set of measurements indexed by $i$, each with fit
parameters $\mu_{i}$ and associated variance $\sigma_{i}^{2}$, the most-likely
estimator for the parameter $\mu$ is given by

$\mu\equiv\frac{1}{\sum_{i}(1/\sigma_{i}^{2})}\sum_{i}\frac{\mu_{i}}{\sigma_{i}^{2}}$
(SI.31)

and has variance

$\sigma^{2}\equiv\frac{1}{\sum_{i}(1/\sigma_{i}^{2})}.$ (SI.32)

These values are used to calculate the quality factors of the high-$Q$ telecom
resonances in the main text. The fitted parameters and associated errors as
determined by this procedure for the four high-$Q$ resonances shown in the
main text are listed below in Table SI.1.

Figure SI.6: High-$Q$ resonator measurements. Example set of measurements for the high-$Q$ resaonce labeled as peak (vi) in the main text, Fig. 3, and corresponding fit parameters for each individual measurement.  Table SI.1: Fitted parameters for high-$Q$ telecom resonances in main text Fig. 3.  Resonance | $\omega_{0}/2\pi$ |  $\gamma/2\pi$ (THz) |  $\kappa/2\pi$ (GHz) |  $g/2\pi$ (GHz) |  $|f|\cos\beta$ (u.l.) |  $\Gamma/2\pi$ (GHz)
---|---|---|---|---|---|---
i | $194.001\pm 0.000$ | $0.862\pm 0.049$ | $0.445\pm 0.010$ | $0.596\pm 0.016$ | $-0.049\pm 0.020$ | $1.747\pm 0.064$
ii | $193.311\pm 0.000$ | $1.061\pm 0.049$ | $0.513\pm 0.010$ | $0.892\pm 0.0147$ | $-0.036\pm 0.012$ | $2.095\pm 0.063$
iii | $192.620\pm 0.000$ | $0.969\pm 0.061$ | $0.613\pm 0.013$ | $0.613\pm 0.013$ | $-0.023\pm 0.022$ | $2.196\pm 0.079$
iv | $191.929\pm 0.000$ | $1.066\pm 0.043$ | $0.530\pm 0.009$ | $1.043\pm 0.013$ | $-0.018\pm 0.009$ | $2.129\pm 0.056$

####  SI.5.2.1 Validity of the weak back-coupling approximation

In the previous section, the weak back-coupling approximation was used as the
full model over-fits individual pass spectra. To validate this approximation,
we fit using the full model given in Eq. (SI.23) without constraining the
parameters, comparisons between the constrained and unconstrained models are
shown in Fig. SI.7. In nearly all cases the parameters are remarkably similar
in both methods with almost identical values. The few cases for which the
fitting results do not converge typically correspond to the model erroneously
interpreting a background feature as a second peak which is not present in
other measurements of the same resonance. In all fits where such issues are
not present, the resulting total and intrinsic quality factors are nearly
identical, with deviations of at most a few percent. Similarly the estimated
effective back-coupling parameter $|f|\cos\beta$ similarly converges.

Figure SI.7: Unconstrained fitting. Comparison between the fitted parameters
obtained by the unconstrained (blue) and approximate (orange) asymmetric split
resonance models for two peaks correspondingly labeled in the main text Fig.
3. In nearly all cases the parameters are remarkably similar in value with and
without the inclusion of the weak back-coupling approximation.

Finally, we remark that the convergence of the (un)constrained models is not a
result of trivial over-fitting. The full model (without the weak back-coupling
approximation) over-fits the data in the very specific manner of introducing a
co-linearity between the forward- and backward-coupling, parameterized by
$\kappa$ and $|f|$ respectively. Such co-linearity, combined with the
additional degree of freedom provided by the $\cos\beta$ terms in the
numerator of Eq. (SI.23) then allows for nearly any combination of $|f|$ and
$\kappa$ to simultaneously optimize the least-square-error objective function.
As a result, the parameters $|f|$, $\beta$, and $\kappa$ typically have large
co-variances on each other and estimates of the error on these parameters
individually blow up to infinity. Fortunately, these parameters are
independent of the intrinsic loss rate $\gamma$, which is the primary
parameter of interest given its relevance to the material loss. Despite the
unbounded errors in the other parameters, estimates of $\gamma$ are determined
with high-confidence and remain nearly identical to the quoted values in the
main text in nearly all cases. Examples are shown in Fig. SI.8.

Figure SI.8: Divergence of covariance. (a,b) Raw data with fit for the 7th
measurement and associated normalized covariance matrix. (c,d) Raw data with
fit for the 10th measurement and associated normalized covariance matrix. In
(a) the fitting included the background feature on the left-hand side as part
of the split resulting in a large difference from the constrained fitting. The
resulting covariance matrix (b) also shows significantly larger covariances
between $\kappa$, $|f|$, and $\beta$ as a result of the co-linearity. The fit
in (c) shows a more reasonable splitting and minimal asymmetry, which results
in the relatively strong co-linearity of $|f|$ and $\beta$ (symmetric
splitting requires $|f|\cos\beta=0$ which has multiple solutions). (e,f,g)
Show the fitted values for $\gamma$, $\kappa$, and $|f|$ respectively. The co-
linearity results in arbitrarily large uncertainty on $\kappa$ and $|f|$,
however the estimate of $\gamma$ always retains high confidence. The
corresponding measurements (7 and 10) show respectively larger/smaller error
bars on the co-linear components.

###  SI.5.3 Quality factor versus coupling distance

In the main text Figs. 2 and 3, the quality factor as a function of coupling
distance is determined to obtain approximate asymptoting values, thus
providing an estimate of the intrinsic quality factor $Q_{i}$. To perform this
fitting we assume that the coupling rate $\kappa$ scales exponentially with
the coupling distance $d_{c}$, such that
$\kappa(d_{c})=\kappa_{0}\exp(-d_{c}/L)$ where $L$ is some characteristic
decay length for the cavity/waveguide evanescent field and $\kappa_{0}$ is the
coupling rate in the limit of $d_{c}\to 0$. Thus, we can estimate the total
quality factor $Q(d_{c})$ by the relation

$\displaystyle\frac{1}{Q(d_{c})}$
$\displaystyle=\frac{1}{Q_{i}}+\frac{2}{Q_{c}},$
$\displaystyle\approx\frac{1}{Q_{i}}+2\frac{\kappa_{0}}{\omega_{0}}\exp(-d_{c}/L).$
(SI.33)

The aggregated quality factor versus coupling distance data was fitted to this
equation to estimate the average asymptoting quality factor of $Q_{i}$.

##  SI.6 Simulation of optical resonator modes

The fundamental optical modes of the ring and disk resonators are simulated
using a finite-difference eigenmode solver (Ansys Lumerical MODE). The
simulations assume a bending radius appropriate to the specific geometry and
assume perfectly matched layer boundary conditions. A high-resolution mesh is
defined at the mode location; convergence of the results is shown in Fig.
SI.9. From these simulations we can determine the loss (per length) and group
index of the mode.

Figure SI.9: Simulation convergence. Convergence of simulation results as a
function of minimum mesh size. The increase in loss is expected to be a result
of the finer mesh size reducing the amount of numerical buffer at the material
interfaces. Despite this, the expected radiation-limited quality factor
$Q_{r}$ and real part of the group index $n_{g}$ both converges.

The simulated loss $\alpha_{[\mathrm{dB/cm}]}$ (given in terms of dB/cm;
$\alpha_{[\mathrm{dB/cm}]}\geq 0$) is related to the spatial loss rate
$\alpha$ by

$\alpha_{[\mathrm{dB/cm}]}=-10\log_{10}\big{(}e^{-\alpha\cdot(1\,\mathrm{cm})}\big{)}.$
(SI.34)

Inverting this one finds

$\alpha=-\frac{1}{(1\,\mathrm{cm})}\ln\Big(10^{-\frac{\alpha_{[\mathrm{dB/cm}]}}{10}}\Big{missing}).$
(SI.35)

The energy in the cavity follows an exponential decay in time

$\frac{U(t)}{U_{0}}=\exp(-\frac{\alpha ct}{n_{g}})$ (SI.36)

where $n_{g}$ is the group velocity (energy propagates at a speed of
$c/n_{g}$). The energy decay rate is then given as $\gamma=\alpha c/n_{g}$.

The intrinsic quality factor is defined by $Q_{i}\equiv 2\pi f_{0}/\gamma$
where $f_{0}$ is the center frequency of the cavity resonance. Expressed in
terms of the free-space wavelength $\lambda_{0}=c/f_{0}$ we have

$Q_{i}=\frac{2\pi n_{g}}{\alpha\lambda}.$ (SI.37)

This equation is used to estimate the intrinsic quality factor of the modes as
limited by radiation. The resulting radiation limited quality factors $Q_{r}$
are shown in Fig. SI.10.

Figure SI.10: Simulated radiation-limited quality factor. Simulated radiation
limited quality factors for the designated modes. For the optical ring
devices, the TM mode is the measured resonance in the main text.

##  SI.7 High-Q visible wavelength resonances

Figure SI.11: High-$Q$ visible-wavelength resonances. Scanning laser data and
fits for visible-wavelength resonances in undoped-BGaP.

##  SI.8 The BVD model and Electromechanical conversion efficiency calculation

Figure SI.12: BVD model and fitting. The BVD model and the fitting of the
effective load impedance. (a) The effective circuit of the BVD model used to
fit the $S_{11}$ data. (b) The load impedance calculated from the measured
$S_{11}$ data, the dashed lines are the fitted BVD model. (c) The $S_{11}$
spectrum and the power ratio on the mechanical lump elements. The dash line is
the fitted BVD model. The inset shows the power ratio on the mechanical
elements $Y_{m}$.

The RF reflection spectrum is measured on a vector network analyser
(VNA)(Keysight N5230C PNA-L), on a cryogenic probe station (Lakeshore CRX-4K).
The VNA was calibrated at room-temperature and under vacuum, to the tips of
the RF probes as the reference using a calibration substrate (GGB Inc. CS-15).
To extract the IDT’s electromechanical conversion efficiency, we use the BVD
effective circuit, as shown in Fig. SI.12a, to obtain the value of the circuit
lump elements and calculate the power load on the motional elements. We can
express the load impedance $Z_{\text{L}}$ as:

$Z_{\text{L}}=R_{s}+j\omega
L_{0}+\left(\frac{1}{Z_{0}}+\sum_{N}Y^{(N)}_{m}(\omega)\right)^{-1}=\frac{1}{Y_{\text{L}}},$
(SI.38)

where

$\displaystyle Z_{0}=\frac{1}{j\omega C_{0}+1/R_{0}}$ (SI.39) $\displaystyle
Y_{m}(\omega)=\frac{1}{Z_{m}}=\frac{1}{R_{m}+j\omega L_{N}+1/j\omega C_{m}}.$
(SI.40)

$R_{s}$ accounts for the total resistance between the RF probe and the IDT and
$L_{0}$ accounts for the inductive background signal of the free space RF
signal. The shunt resistor $R_{0}$ accounts for the power dissipation/loss of
the dielectric while the shunt capacitor $C_{0}$ accounts for the static
capacitance of the IDT the electrodes. $L_{m}(\omega)$, $C_{m}(\omega)$, and
$R_{m}(\omega)$ are the mechanical elements at the target resonance frequency.
We can combine $C_{0}$ and $R_{0}$ as effective load $Z_{0}$. In our case, we
only fit one resonance, therefore $N=1$. The BVD model fit of $Z_{\text{L}}$
is shown in Fig. SI.12b and the $|S_{11}|$ fit is shown in Fig. SI.12c.

After extracting the lump element values, we can calculate the the voltage
$V_{Y_{m}}$ and current $I_{Y_{m}}$ across $Y_{m}$ using the Ohm’s law. We
find the power ratio of the total input power that is loaded on $Y_{m}$ to be

$P_{Y_{a}}=Re\left[\tilde{V}_{Y_{m}}\cdot\tilde{I}^{*}_{Y_{m}}\right].$
(SI.41)

The calculated power ratio on $Y_{m}$ is shown in the inset of Fig.
SI.12c,which shows $\sim 80\%$ of the total input power is loaded on $Y_{m}$
at 4 K.

##  SI.9 Time gating of acoustic transmission measurement

Figure SI.13: S21 signal processing. The frequency domain gated S21 signal of
the R3 mode. The inset shows the full range of the gated S21 spectrum.

The measured S21 signal includes the acoustic reflections and the free-space
RF interference, therefore, we need to perform time-gating to extract the
single-pass acoustic transmission signal. Time-gating of the S21 signal is
performed by taking an inverse Fourier transform of the measured S21,
filtering the resulting time-domain signal within a specified window and then
Fourier transforming the gated S21 signal back to the frequency domain. We
then take the intensity of the gated signal in the frequency domain as the
acoustic transmission of different delay line devices, whose transmission
signal was gated between $5\text{\,ns}<t<1300\text{\,ns}$ to remove any signal
that is much faster then the acoustic wave. The gated frequency domain signal
is shown in Fig. SI.13. We did not directly take the intensity of the time-
domain impulse signal as the time domain signal contains multiple overlapping
acoustic modes.

##  SI.10 Heterodyne measurement

Figure SI.14: Heterodyne measurement setup for the AOFS FPC: Fiber
polarization controller. AOM: Acousto-optical modulator (Brimrose model
AMF-100-1550-2FP). EDFA: Erbium-doped fiber amplifier (PriTel LNHP-PMFA-23).
PD: Photodetector (Thorlabs RXM25AF). RSA: Real-time spectrum analyser
(Tektronix RSA5100B). The input laser is split into two paths. One path goes
through the AOM, where the input laser’s frequency is up-shifted by
$\delta/2\pi=102.9$ MHz as the reference signal for the heterodyne
measurement. The other path goes through the DUT, which is driven by a VNA at
$\Omega/2\pi=6.5$ GHz. The two paths then combined and beats together to
generate the beating signal with frequency $\Omega\pm\delta$ for Stokes ($+$)
and anti-Stokes ($-$) cases.

##  SI.11 Density functional theory

The density functional calculations were performed using the periodic model
approach and plane wave basis set, as implemented in the VASP code [3]. Plane-
wave augmented pseudopotentials were used to represent the core electrons [4].
The structure optimization was performed for the 2$\times$2$\times$2 extension
of the crystallographic cell (Ge32P32) using the PBEsol exchange correlation
functional [5] in the spin-polarized mode. A $\Gamma$-centered
4$\times$4$\times$4 $k$-mesh was used for Brillouin-zone integration. The
plane wave basis set cutoff was set to 400 eV throughout. The total energy was
converged to within 10-6 eV.

Prior to density of states (DOS) calculations, the cubic cell parameters were
fixed at the value corresponding to 2% doping level to match the lattice
constant of BGaP with the Si lattice constant for consistency with
experimental studies. Then, the total energy was minimized with respect to the
internal atomic coordinates for each defect configuration using the PBEsol and
a $\Gamma$-centered 8$\times$8$\times$8 $k$-mesh.

###  SI.11.1 Determination of the lattice constant

Up to four Ga atoms in the Ga32P32 supercell were replaced with B atoms and
the energy of each configuration was minimized with respect to the atomic
coordinates and all supercell parameters. The pseudo-cubic BGaP lattice
constant, derived as the cube root of the supercell volume, decreases with
increasing B content (Fig. SI.15) and suggests that BGaP lattice match with
that of the bulk silicon is achieved by substituting $\sim$2% of Ga atoms with
B.

###  SI.11.2 Interstitial Boron defect structure

Our calculations suggest that Bi species either bind to the lattice P3- ions
or compete with the lattice Ga3+ ions for the occupancy of the cation sites,
as shown in Fig. SI.15. Specifically, a Ga3+ ions can be displaced from its
lattice site to make room for B so that both of them are coordinated by the
lattice P3- ions. As discussed in the main text, we found three such Bi
configurations that can be described as BP1, BP2, BP3. (Fig. SI.15), i.e., the
total number of P atoms coordinating the split B–Ga site remains the same but
the number of P–B and P–Ga bonds varies from one to three depending on the
configuration.

The energetically favorable boron dimers observed in the calculation can be
rationalized in terms of B3+ ions having smaller ionic radius than Ga3+ ions
(0.41 and 0.76 Å, respectively). Minimizing the supercell lattice shape and
volume in the case of the interstitial B atoms, either in ideal lattice or
bound to B${}_{\text{Ga}}$, results in the BGaP lattice constant that is
similar to or exceeding that of pure GaP as shown in Fig. SI.15.

###  SI.11.3 Calculation of the BGaP electronic density of states

The effect of the interstitial boron defects on the electronic structure are
analyzed in terms of the one electron densities of states (DOS). It is well
known that the generalized gradient approximation exchange correlation
functional, such as PBEsol used in this work, underestimates the band gap of
semiconductors by as much as a factor of 1.8. Accordingly, this deficiency
affects the energies of gap states. For consistency, we used the cubic
supercell and the lattice parameters optimized, as discussed above, for the
pure GaP (5.429 Å) and for GaP:B with 2% substitutional B defects evaluated
using the tend shown in Fig. SI.15 (5.412 Å).

Figure SI.15: Boron content dependence of lattice constant. Dependence of the
pseudo-cubic lattice parameter in GaP:B on the concentration of substitutional
B impurities and the corresponding linear fit (purple). Several BGaP
configurations were considered for the larger B concentrations. The effect of
interstitial B species in the ideal GaP lattice (Bi) and those bound to
substitutional B (B2) are shown with light green and blue, respectively.

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