††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Direct laser-written optomechanical membranes in fiber Fabry-Perot cavities

Lukas Tenbrake  Institute of Applied Physics, University of Bonn, Germany
Alexander Faßbender ID Institute of Physics, University of Bonn, Germany
Sebastian Hofferberth ID Institute of Applied Physics, University of Bonn,
Germany  Stefan Linden ID Institute of Physics, University of Bonn, Germany
Hannes Pfeifer ID<EMAIL_ADDRESS>Institute of Applied
Physics, University of Bonn, Germany

(August 27, 2024)

###### Abstract

Integrated micro and nanophotonic optomechanical experiments enable the
manipulation of mechanical resonators on the single phonon level. Interfacing
these structures requires elaborate techniques limited in tunability,
flexibility, and scaling towards multi-mode systems. Here, we demonstrate a
cavity optomechanical experiment using 3D-laser-written polymer membranes
inside fiber Fabry-Perot cavities. Vacuum coupling strengths of
$\sim$30\text{\,}\mathrm{kHz}$$ to the fundamental megahertz mechanical mode
are reached. We observe optomechanical spring tuning of the mechanical
resonator by tens of kHz exceeding its linewidth at cryogenic temperatures.
The extreme flexibility of the laser writing process allows for a direct
integration of the membrane into the microscopic cavity. The direct fiber
coupling, its scaling capabilities to coupled resonator systems, and the
potential implementation of dissipation dilution structures and integration of
electrodes make it a promising platform for fiber-tip integrated
accelerometers, optomechanically tunable multi-mode mechanical systems, or
directly fiber-coupled systems for microwave to optics conversion.

##  I Introduction

Cavity optomechanical experiments have been implemented on a multitude of
different platforms [1] ranging from the canonical movable end mirror of a
Fabry-Perot cavity [2], over membranes in cavities [3, 4], toroidal[5] and
optomechanical crystal resonators [6], down to the vibrational modes of
molecules in a plasmonic picocavity [7]. This platform diversification and
constant improvement advanced the field during the past years leading to key
achievements like ground-state cooling of mechanical resonators [8, 9],
optomechanical state teleportation experiments [10], efficient microwave to
optical conversion [11, 12], or sensing of the mechanical resonator below the
standard quantum limit [13, 14, 15]. Among the current challenges in the field
are the realization and addressing of multi-mode optomechanical systems and
the integration of optomechanical elements for different sensing applications.

Optomechanical devices with high optical field concentration and
correspondingly large optomechanical coupling are usually realized with on-
chip platforms [6, 5, 16]. A different approach is taken by miniaturized
Fabry-Perot cavities with concave mirror structures fabricated on optical
fiber tips [17, 18]. These fiber Fabry-Perot cavities (FFPCs) have been
established as a platform for light-matter interaction during the past years,
including realizations of optomechanical experiments [19, 20, 21, 22, 23, 24,
25]. They feature a direct fiber-coupled optical access, small cavity lengths,
high optical finesse, and an open resonator volume. This allows to introduce
both conventional membrane-type resonators as well as more unconventional
resonators like standing waves in liquid Helium.

With its first demonstration in the 1990s [26], 3D direct laser writing (DLW)
enabled the emergence of free-formed three- dimensional polymeric structures
with sub-micrometer resolution. It led to the miniaturization of on-chip
optical components [27], waveguides [28], and mechanical structures [29]
including 3D acoustic metamaterials [30], and has been used for mask
applications [18]. Apart from large planar substrates, writing on fiber-ends
has been tackled for applications in endoscopy [31] or sensing [32, 33].

In this article, we demonstrate the integration of mechanical polymer membrane
resonators into highly miniaturized FFPCs. We realize a miniaturized, fiber-
coupled membrane-in-the-middle (MIM) experiment [3, 4] with superior scaling
capabilities due to the 3D DLW fabrication process both integrated into highly
stable monolithic FFPCs [34] and on flat distributed Bragg reflector (DBR)
substrates. We analyze the achievable optomechanical coupling strength, and
demonstrate a dispersive optomechanical spring effect tuning of the mechanical
resonace in the presence of a thermal optical nonlinearity exceeding the
mechanical linewidth at cryogenic temperatures. Our results pave the way for
high-sensitivity fiber cavity integrated accelerometry and DLW-enabled
mechanical multi-mode structures interfaced using FFPCs including multiple
membranes inside a miniaturized Fabry-Perot cavity [35] and extended
mechanical metamaterials on DBR substrates.

##  II Results

### Polymer membrane in a fiber cavity

Figure 1: Overview of the system. In a a schematic overview of the
experimental structure is shown. The polymer drum resonator is fabricated on a
highly reflective DBR substrate. The fiber mirror is positioned above the
polymer drum and realizes a fiber Fabry-Perot cavity with the DBR beneath
enclosing the drum membrane. The inset shows a microscope picture of an
approaching fiber mirror (diameter: $125\text{\,}\mathrm{\SIUnitSymbolMicro
m}$) to a polymer drum array. In b the magnitude of the displacement field
$\mathbf{u}$ of the fundamental drum mode is shown as retrieved from finite
element simulations. The height $h_{\text{drum}}$ of the polymer drum supports
and the drum membrane thickness $t_{\text{drum}}$ determine the positions of
the optical cavity mode intensity maxima and minima with respect to the
membrane interfaces. In c an SEM micrograph of a fabricated rectangular
polymer drum structure is shown.

The mechanical resonator in our experiments is a drum-like polymer membrane of
$1-$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ thickness. It is fabricated
using 3D direct laser writing (see methods) on top of a highly reflective DBR
mirror ($10\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ transmission111High
reflectivity range: $750\text{\,}\mathrm{nm}$ to $805\text{\,}\mathrm{nm}$)
located either on an extended substrate or on an optical fiber-tip. A typical
rectangular membrane used here spans
$\sim$45\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\times$60\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and is placed on support
bars that suspend the membrane $5-$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
above the mirror surface (see Fig. 1 a and c). The supports reduce the free
membrane surface by $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ on each side.
More elaborate geometries like multi-membrane structures or suspensions like
soft-clamping [36] can be directly realized in the 3D DLW fabrication. Finite-
element simulations of the fundamental mechanical mode, see Fig. 1 b and
methods section, yield an expected mechanical resonance frequency of
$2.1\text{\,}\mathrm{MHz}$ and an internal quality factor of $\sim 20$ due to
the comparably large elastic loss tangent of the 3D DLW polymer resist
material at ambient conditions. As no specific dissipation dilution is
implemented in this simple design, the mechanical quality factor reflects the
intrinsic $Q$ value of the material [37].

The membrane is integrated into an optical FFPC by approaching a fiber mirror
– a fiber-tip with a concave-shaped facet and high-reflection coated surface
[17, 18]. The cavity length is adjusted to
$\sim$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, but can be scanned using a
piezo-electric element attached to the fiber mirror. The optical fiber leading
to the approached fiber mirror serves as both input and output of the cavity.
The transmission of $2000\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$1 is chosen
to retrieve a single-sided cavity geometry, with an approximate balance of the
cavity-to-input-fiber coupling rate and the internal cavity losses. The
internal cavity losses are dominated by scattering from the polymer surface
that exhibits a roughness of $<$5\text{\,}\mathrm{nm}$$222RMS surface profile
variation in AFM measurement after post-fabrication polishing using an oxygen
plasma ashing process. The optical cavity mode field extends over the membrane
and the open cavity domains on both sides of the membrane due to its low
reflectivity. The optical cavity mode spectrum is characterized by scanning
the cavity resonance over a probe laser tone with modulated sidebands (see
methods, and [38, 34]). We measure optical linewidths of
$\kappa/2\pi\sim$\text{\,}\mathrm{GHz}$$ corresponding to finesse values of
$\approx 1000$. The variations are caused by different optical intensities on
the membrane surfaces depending on the individual membrane geometry.

In order to probe the mechanical mode, we lock the cavity to the probe laser
using a feedback loop on a Pound-Drever-Hall (PDH) error signal [39, 40]. The
thermal excitation of the membrane is transduced to cavity frequency noise
visible in the noise spectrum of the calibrated PDH error signal, which is
recorded using an electrical spectrum analyzer. The measured mechanical
frequencies and linewidths are in good agreement with the finite element
simulation with mechanical resonance frequencies between 1 and
$4\text{\,}\mathrm{MHz}$ depending on the particular membrane geometry. The
calibration of the PDH error signal slope at the lock point together with the
temperature of the environment furthermore allows a quantification of the
vacuum optomechanical coupling rate of the membrane modes (see methods, and
[41, 34]).

### Characterization of the optomechanical coupling

A linear, dispersive optomechanical coupling manifests as a shift of the
optical cavity frequency $\Delta\omega_{\text{cav}}$ upon a displacement
$\Delta z$ of the membrane [3, 4]. In turn, the cavity field photons act on
the pliable membrane through their radiation pressure, closing the interaction
loop. The magnitude of this effect strongly depends on the geometry of the MIM
system, in particular on the optical intensity on both sides of the membrane.
A difference in intensity will cause a non-vanishing net radiation pressure on
the membrane resulting in an optomechanical interaction. The interaction
strength is given by the vacuum optomechanical coupling rate
$g_{0}=\frac{\partial\omega_{\text{cav}}}{\partial
z}z_{\text{zpf}}\coloneqq-G^{(1)}z_{\text{zpf}}$ with $z_{\text{zpf}}$ the
mechanical resonator’s zero-point motion amplitude. As we fix our total cavity
length to $\sim$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, the relevant
parameters determining the achievable coupling are the polymer drum support
height $h_{\text{drum}}$ and the membrane thickness $t_{\text{drum}}$ of the
drum (see Fig. 1 b). Since the latter is comparable to the probe laser
wavelength $\lambda$, it has a strong effect on the local cavity intensity at
the membrane surfaces.

We compute the expected frequency-pull factor
$G^{(1)}(t_{\text{drum}},h_{\text{drum}})$ in our experiment using two
different methods. After finding the resonance condition of the optical cavity
field (see methods), the first method considers a small shift $\Delta z$ of
the membrane position. Its effect on the cavity resonance frequency as
expected from Maxwell’s equations is numerically extracted to determine the
frequency-pull. For the second method, we employ perturbation theory for
Maxwell’s equations with moving material boundaries [42] (for details, see
methods). For this, we use the cavity resonance condition to find the explicit
form of the cavity electrical field. The resulting pull-factor (see Eq. 6)
scales linearly with the intensity difference on the two membrane-air
interfaces and the polymer refractive index $n_{\text{poly}}$. Both methods
yield identical results. The resulting
$G^{(1)}(t_{\text{drum}},h_{\text{drum}})$ is shown in Fig. 2 a with some
highlighted field configurations in b. Maximum frequency-pull factors of up to
$11\text{\,}\mathrm{GHz}\text{\,}{\mathrm{nm}}^{-1}$ are predicted. The map
features a periodic pattern of well-defined minima and maxima with a
periodicity of half the (material) wavelength. The asymmetry of positive vs.
negative coupling emerges from the two air domains of the cavity not having
equal lengths.

Figure 2:  In a the calculated $G^{(1)}(t_{\text{drum}},h_{\text{drum}})$ are
shown as a color-map, with {$t_{\text{drum}},h_{\text{drum}}$} normalized to
the respective material wavelength. Measured $g_{0}$ for two exemplary polymer
drums are highlighted in green (drum $1$) and purple (drum $2$) and plotted
along the third plot-axis. By scanning the probe-wavelength, the measurements
of the two drums trace out cuts in the map. Four cavity geometries
corresponding to special coupling scenarios are highlighted:  and  correspond
to $G^{(1)}=0$4,  and  maximize $|G^{(1)}|$. In b (bottom) a sketch of the
cavity intensity $I_{\text{cav}}$ of an exemplary cavity geometry is shown.
The cavity intensity at the two polymer-air/vacuum interfaces is shown for the
four scenarios from a ( , , , ). In c the measured $g_{0}$ from image a are
plotted against the scanned probe laser wavelength $\lambda$ for both drum 1
(green) and drum 2 (purple).

We test our model by measuring $g_{0}$ for different probe wavelengths on two
exemplary polymer geometries: $\\{t_{\text{drum
1}}\sim$1.65\text{\,}\mathrm{\SIUnitSymbolMicro m}$,\,h_{\text{drum
1}}\sim$9\text{\,}\mathrm{\SIUnitSymbolMicro m}$\\}$ and $\\{t_{\text{drum
2}}\sim$1.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$,\,h_{\text{drum
2}}\sim$8.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$\\}$. The calibrated optical
cavity frequency noise spectrum $\mathcal{S}_{\nu\nu}(f)$ is measured (see
methods) and the fundamental mechanical resonance is used to extract $g_{0}$
at ambient conditions from its expected effect on the cavity frequency noise
via [41]:

$\mathcal{S}_{\nu\nu}(f)=\frac{2g_{0}^{2}}{4\pi^{2}}\cdot\frac{2\Omega_{m}}{\hbar}\cdot\frac{2\Gamma
k_{B}T}{\left(\Omega^{2}-\Omega_{m}^{2}\right)^{2}+\Gamma^{2}\Omega^{2}}.$

Here, $f=\Omega/2\pi$ denotes the noise frequency, $\Omega_{m}$ the mechanical
resonance frequency, $\Gamma/2\pi$ the mechanical linewidth, $k_{B}$
Boltzmann’s constant and $T$ the temperature of the polymer membrane. The
results are shown in Fig. 2 a and c. As the probe laser wavelength is scanned,
the coupling rate changes through the shift of the cavity intensity on the
polymer drum surfaces. This corresponds to the diagonal cuts in Fig. 2 a
across the coupling landscape for the two representative drum geometries
(green – drum $1$, purple – drum $2$). Here, tuning the probe laser wavelength
from $760\text{\,}\mathrm{nm}$ to $785\text{\,}\mathrm{nm}$ is equivalent to
adjusting the geometry by a few $100\text{\,}\mathrm{nm}$333$h_{\text{drum}}$
is chosen to be $\gtrsim$8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ for these
geometries, such that tuning over the complete probe laser wavelength-range
changes the cavity field node/anti-node positions by at least half of the
(material) wavelength.. We find a maximum coupling rate of
$|g_{0}|/2\pi=$33\pm 7\text{\,}\mathrm{kHz}$$ at
$\lambda=$772\text{\,}\mathrm{nm}$$ (see Fig. 2 c for drum 2).

Four special cases of the intensity distribution are highlighted in Fig. 2 a.
Their corresponding cavity intensity distribution at the polymer membrane
surfaces is shown in Fig. 2 b. Vanishing optomechanical coupling is observed
for equal intensities on both sides of the membrane, in the simplest case by
$t_{\text{drum}}$ being a multiple of half the material wavelength. Optimal
coupling is achieved, if a cavity field node (anti-node) is located at one
side of the polymer membrane in combination with a corresponding anti-node
(node) at the other444Note that the quadratic frequency-pull factor is
maximized for equal intensities on either side along the connection line of
the linear maxima of same $t_{\text{drum}}$. This corresponds to  in Fig. 2 b.
It also vanishes at $t_{\text{drum}}$ being equal to a multiple of half the
material wavelength ( ). The maximum expected quadratic coupling is
$G^{(2)}\gtrsim$200\text{\,}\mathrm{GHz}\text{\,}{\mathrm{nm}}^{-2}$$..

### Optomechanical spring effect

We now consider dynamical effects of the intracavity photon number on the
mechanical modes of the polymer membrane. As we are working in the fast-
cavity/Doppler-regime, where $\kappa\gg\Omega_{m}$, the optomechanical spring
effect shifts the fundamental flexural mode frequency $\Omega_{m}$ as

$\Delta\Omega_{m}(\Delta)={g_{0}}^{2}n_{\text{cav}}\cdot\frac{2\Delta}{\kappa^{2}/4+\Delta^{2}}$
(1)

with cavity detuning $\Delta$, cavity photon number $n_{\text{cav}}$ and
vacuum optomechanical coupling rate $g_{0}$ [1]. Effects on the mechanical
linewidth are neglected.

To measure $\Delta\Omega_{m}$, we make use of a two-tone measurement scheme
that utilizes a probe laser beam fixed at $780\text{\,}\mathrm{nm}$ to which
the cavity is locked in a PDH feedback loop (for details, see methods).
$\Omega_{m}$ is measured from $\mathcal{S}_{\nu\nu}(f)$ as extracted from the
calibrated error signal of the feedback loop. To impose a frequency shift of
the mechanical resonator, a tunable pump-laser
($$760\text{\,}\mathrm{nm}$-$785\text{\,}\mathrm{nm}$$ ) is additionally
coupled into the input-port fiber addressing another optical cavity resonance
separated by one free spectral range ($\sim$5\text{\,}\mathrm{THz}$$) from the
probe. The probe path is decoupled from the pump-laser radiation with a narrow
pass-band interference filter. The pump-laser is scanned over the optical
resonance modulating the detuning and the intracavity photon number
$n_{\text{cav}}$. During this process, $\mathcal{S}_{\nu\nu}(f)$ is recorded
to extract $\Delta\Omega_{m}$. The measurements are repeated for a sweep of
different input powers $P_{\text{in}}$.

Fig. 3 d shows a typical measurement of $\mathcal{S}_{\nu\nu}(f)$ of the
mechanical polymer drum resonance against the cavity detuning $\Delta$ for a
comparably large pump input power of
$P_{\text{in}}=$6.8\text{\,}\mathrm{mW}$$. At such a power level an additional
photothermal nonlinearity of the optical cavity resonance shifts the resonance
position during the detuning scan [43] leading to a discontinuous behavior
(black dashed cross) of the signal. The photothermal nonlinearity of the
optical mode is also observed in the reflection signal of the pump-laser shown
in Fig. 3 c. An optomechanical bistability as the origin of this behavior can
be excluded as our coupling rate $g_{0}$ and $n_{\text{cav}}$ are too small
for this effect [1]. The nonlinearity is likely related to absorption on the
polymer surfaces as stronger nonlinearites are observed in geometries with
intensity maxima at the polymer-air interface.

Figure 3: a Normalized pump-reflection signal ${P_{\text{PD}}}_{\text{pump}}$
and mechanical resonance frequency shift $\Delta\Omega_{m}$ induced by the
optomechanical spring against cavity detuning $\Delta$ for input power
$P_{\text{in}}=$6.8\text{\,}\mathrm{mW}$$. The dashed lines represent the
frequency shift corresponding to unstable or not scanned solutions of Eq. 2.
In c the pump-reflection signal ${P_{\text{PD}}}_{\text{pump}}$ is plotted
against the cavity detuning $\Delta$ for different input powers
$P_{\text{in}}$. In b the corresponding mechanical resonance frequency shift
$\Delta\Omega_{m}$ induced by the optomechanical spring is shown. The legend
in the lower right indicates the corresponding pump powers. d shows the
normalized frequency noise spectra $S_{\nu\nu}/S_{\nu\nu\max}$ against cavity
detuning $\Delta$ for $P_{\text{in}}=$6.8\text{\,}\mathrm{mW}$$. The dashed
black lines highlight the effect of the optomechanical spring on the
mechanical resonance frequency $\Omega_{m}$.

As the photothermal absorption is slow compared to the optomechanically
induced dynamics, we include the shift and bistability of the optical
resonance in our analysis by treating it as an additional static detuning of
the cavity. The cavity detuning $\Delta$ is therefore modified to
$\Delta^{\prime}=\Delta-\alpha n_{\text{cav}}$ with $\alpha$ being the
photothermal frequency-pull factor. To find $n_{\text{cav}}$ for a given
detuning, we insert $\Delta^{\prime}$ in the steady-state solution of the
equation-of-motion of the averaged complex light amplitude
$\langle\hat{a}\rangle$ as obtained from input-output formalism [44]. This
leads to a third order polynomial equation in the cavity photon number
$n_{\text{cav}}$:

$0=\alpha^{2}n_{\text{cav }}^{3}-2\Delta\cdot\alpha
n_{\operatorname{cav}}{}^{2}-\left(\Delta^{2}-\kappa^{2}/4\right)n_{\text{cav
}}-\kappa_{\text{ex}}n_{\text{in }}$ (2)

with input photon number rate $n_{\text{in}}$ and the cavity input coupling
$\kappa_{\text{ex}}$. The two stable solutions of $n_{\text{in}}$ correspond
to the photon number expected from the two directions of the detuning scan,
whilst the third, unstable solution is not reached [43]. As shown in Fig. 3 a
(red curve) – c, we start the scan from $\Delta<0$. As more and more photons
are coupled into the cavity, the resonance shifts away from the approaching
pump-laser following one of the stable branches for the cavity reflection
signal. When the pump-scan catches up with the drifting resonance at the
maximum amount of intracavity photons, any further detuning abruptly reduces
the thermal frequency-drift of the resonance resulting in a sudden jump to the
reflection signal baseline. Using the solutions of Eq. 2, the extracted cavity
photon number $n_{\text{cav}}(\Delta)$ can be inserted in Eq. 1 with $\Delta$
being replaced by $\Delta^{\prime}$. This model is then used to fit the
measured data displayed in Fig. 3 a (blue curve) & b. The unstable solution
and the not-scanned branch of the reflection signal are included as dashed
lines tracing a loop shape. Towards low pump-powers, the transition to a
normal dispersive lineshape of the optomechanical spring effect can be
observed.

The maximum optomechanical frequency shift measured here was
$\Delta\Omega_{m}/2\pi=$31\text{\,}\mathrm{kHz}$$ for the maximum pump-power
that we reached of $P_{\text{in}}=$6.8\text{\,}\mathrm{mW}$$. The cavity
finesse in this case was $\mathcal{F}=1400\pm 300$ and the photothermal
frequency-pull factor $\alpha/2\pi=$8\pm 2\text{\,}\mathrm{kHz}$$. Improved
cavity finesse by stronger surface polishing, and higher pump-powers will
allow us to increase this shift. Currently it already surpasses the mechanical
linewidth at cryogenic temperatures. As the shift is on the order of the
current frequency disorder of printed polymer drums it could be used to
dynamically tune single drums in a multi-mode system into and out of
collective resonances.

### Mechanical resonance linewidth

Under ambient atmospheric conditions, the mechanical linewidth of the polymer
membrane oscillators is limited to $\sim 20$ by about equal parts through
damping of the membrane motion by surrounding gas and internal losses of the
polymer. In comparison, radiation of mechanical energy into the substrate is
negligible due to the impedance mismatch of sound in the polymer and the glass
material below. Internal losses of the polymer are characterized by the loss
tangent – the tangent of the phase between imaginary and real part of the
dynamic modulus – of the material and vary strongly between different DLW
resists [45]. In addition, the mechanical properties of polymers can exhibit a
rich temperature dependence [46] with possible secondary glass transitions due
to conformation changes of polymer chains.

Figure 4: Temperature dependence of the mechanical resonator properties. The
optomechanical cavity geometry for the experimental setup inside a liquid
Helium flow cryostat uses an FFPC with two fiber mirrors (FM1 and FM2) inside
a slotted glass ferrule that is tunable by an attached piezo [34]. The polymer
drum is directly fabricated on one of the fiber mirrors as shown in a. Upon
cool-down the polymer stiffens causing the resonance frequency to increase as
shown in b. In c, the mechanical linewidth at different sensor temperatures is
shown. For some exemplary measurements the quality factors and measured
spectra are included.

To investigate the temperature dependence of the mechanical properties, we
place a ferrule-based, passively stable FFPC configuration (for details, see
[34]) in a liquid Helium continuous flow cryostat. The polymer membrane
structure is fabricated on a highly reflective fiber mirror (see Fig. 4 a).
The sample holder, on which the FFPC is mounted, sits in an evacuated chamber
and can be cooled from room temperature down to $\sim$4\text{\,}\mathrm{K}$$.
The sensor temperature is, however, only a lower bound to the local
temperature of the membrane. When cooling down the membrane resonator, the
polymer material stiffness and pre-strain increase, which leads to an increase
of the mechanical resonance frequency from about $3\text{\,}\mathrm{MHz}$ to
$\sim$3.5\text{\,}\mathrm{MHz}$$ (see Fig. 4 b). At the same time the
mechanical linewidth drops from about $150\text{\,}\mathrm{kHz}$ down to
$\sim$6\text{\,}\mathrm{kHz}$$ corresponding to a mechanical quality factor of
$\sim 600$ as shown in Fig. 4 c. Kinks in the temperature dependence of the
mechanical linewidth may indicate possible secondary glass transitions [46].

The remaining loss, decoherence or linewidth broading of the polymer membrane
resonators at cryogenic temperature can be caused by various mechanisms. A
possible candidate is a mechanical coupling to the conformation changes of
polymer molecule chains. Also scattering of high energy phonons can appear,
which would be suppressed at even lower temperatures. Due to the comparably
large thickness of the membrane resonators, thermo-elastic dissipation [47,
48] is expected to play another major role. Engineering the mechanical mode to
reduce strain gradients and further polishing using oxygen plasma ashing to
thin down the membrane will help to reduce this loss channel and can be
combined with isolation and soft-clamping techniques to further increase the
mechanical quality factor [36, 37]. The exploration of the material properties
of other 3D DLW resists under cryogenic conditions will help to further
identify low-loss materials. Moreover, 3D DLW structures can be used as blanks
for subsequent material deposition, where the resist blank is later removed by
oxygen plasma ashing or suit as stamp frames (e.g. from PDMS) in combination
with other membranes or 2D materials like transition metal dichalcogenides.

##  III Discussion

We have demonstrated an FFPC-integrated optomechanical membrane-in-the-middle
experiment with 3D direct laser-written membrane structures. The larger
membrane thickness compared to other MIM-experiments [3, 19, 20, 24, 25] of
several $\lambda/2$ allows us to realize large optomechanical couplings of
$g_{0}\sim$30\text{\,}\mathrm{kHz}$$ due to the maximized intensity
differences on the two membrane-air interfaces despite its relatively low
reflectivity.

The unconventional polymer material stemming from the 3D DLW fabrication
process results in moderate intrinsic mechanical quality factors at ambient
conditions. However, a significant gain in the mechanical Q-factor is realized
at cryogenic temperatures. Furthermore, the large range of specialized 3D DLW
materials [45], the possible implementation of dissipation dilution techniques
[37], the use of 3D DLW structures as blanks for material growth, and their
capabilities as stamping frames will allow a fast advancement of the presented
platform. In turn, optomechanical low-temperature experiments enable highly
sensitive material characterization of DLW materials, which will contribute to
the current rapid development of specialized resists.

The large flexibility of the 3D DLW fabrication allows the combination of this
highly integrated platform with additional structures like electrodes [18] for
electromechanical coupling, or emitters, and makes it scalable towards multi-
membrane experiments. These can be both realized as membrane-stacks [35] in a
single FFPC e.g., on a fiber mirror or as a planar 2D mechanical metamaterial
on a DBR substrate. Such metamaterials would benefit from the large
optomechanical spring effects allowing for tunable mechanics and optical
reconfiguration of mechanical multi-mode circuits with much less elaborate
tuning techniques than required in other optomechanical platforms [49]. This
will enable vibration-routing in 2D metamaterials, distributed sensing in
multi-mode mechanical structures, and fiber-tip-integrated sensing of motion
and force.

## Methods

### Sample fabrication

The polymer drum structures were fabricated using a commercial 3D lithography
system (Nanoscribe Photonic Professional GT+, Nanoscribe GmbH & Co. KG,
Germany) that uses two-photon-polymerization. A $63\times$ objective was used
in combination with the photoresist IP-S in dip-in configuration to print on
the end-facet of the single-mode fiber or DBR substrates using the system’s
piezo-mode. The system features a femtosecond laser centered at 780 nm that
patterns and polymerizes the resist at the objective’s focus. The hatching
distance for the feet and frame were chosen comparably coarse, while the
membrane was patterned fine ($65\text{\,}\mathrm{nm}$) using a laserpower of
$22\,\%$ (corresponding to $11\text{\,}\mathrm{mW}$) and a scanspeed of
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{s}$. The
unpolymerized resist part was removed via immersion in propylene glycol methyl
ether acetate (PGMEA) ($30\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$) and a
subsequent bath in isopropyl alcohol
($30\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$). In a post-development step, the
structure was flood exposed with a $6\text{\,}\mathrm{W}$ UV-lamp while
sitting in isopropyl alcohol ($10\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$).
Further post-processing was performed using oxygen plasma-ashing for surface
polishing of the membrane to reduce scattering losses from the interface.
Using this process, several different geometries were realized. To reduce the
complexity of the analysis, the comparably simple rectangular membrane design
is presented in this article.

The FFPC designs used in this work are based on either a hemi-cavity design
with a fiber mirror that can scan structures on a flat, highly reflective
substrate inspired by [50], or a passively stable, monolithic FFPC realization
with two fiber mirrors inside a glass ferrule as demonstrated in [34]. The
polymer structures are fabricated on the higher reflective DBR substrate or
fiber mirror ($10\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ transmission).

### Experimental setup

Figure 5: Overview of the experiment setup. The green highlighted pump laser
is only used within the optomechanical spring measurements. The optical cavity
with the mechanical membrane resonator (gray box) is either located in ambient
conditions (single fiber mirror and DBR substrate) or situated on the sample
holder of a liquid Helium flow cryostat (two fiber mirrors in a glass
ferrule). The optical cavity length can either be scanned or locked to the
probe laser (highlighted in red) using either a Pound-Drever-Hall-type
(highlighted in blue) or a side-of-fringe lock. For this purpose and as a
frequency-meter in case of a scanned cavity, an electro-optic modulator is
used to create sidebands to the main probe laser tone at adjustable RF-
frequency up to $12\text{\,}\mathrm{GHz}$. The photodiode reflection signals
and the electrical noise spectrum of the lock can be measured using
conventional oscilloscopes and an electric spectrum analyzer.

The full experiment setup is sketched in Fig. 5. The optical mode is
characterized by measuring the reflected optical power of a fixed wavelength
probe laser (probe setup red highlighted in Fig. 5, see also [38, 34]) on a
photodiode (PD${}_{\text{probe}}$), while the cavity length is scanned using a
piezo-electric element that is glued to the optical fiber. The high-voltage
drive of the piezo is generated by a $100\times$ voltage amplifier and a
subsequent low pass filter (LPF) to reject high-frequency electric noise. The
alignment of the fiber during the scan process is secured by a glass ferrule
($131\text{\,}\mathrm{\SIUnitSymbolMicro m}$ bore diameter), in which the
optical fiber can slide. The incoming and reflection signal are split on a
polarizing beam splitter (PBS). The subsequent waveplates ($\lambda/2$ and
$\lambda/4$) are used to adjust the polarization of the reflected light to be
perpendicular to the polarization of the incoming laser beam causing the PBS
to direct most of the reflection onto PD${}_{\text{probe}}$. An electro-optic
modulator (EOM) adds sidebands to the main probe laser tone. The reflection
signal measured with the oscilloscope therefore features three reflection dips
with a spacing of the RF-drive of the EOM that is thereby used as a frequency-
meter.

To characterize the optomechanical coupling strength the optical cavity is
locked to the probe laser tone using a Pound-Drever-Hall (PDH, blue
highlighted) or side-of-fringe (SoF) lock. The slope of the lock signal is
calibrated using the frequency-meter enabled by the EOM-sidebands. Using this,
the measured voltage noise of PD${}_{\text{probe}}$ can be converted to the
optical cavity frequency noise. As the bandwidth of the feedback loop is small
($\sim$1\text{\,}\mathrm{kHz}$$) compared to the mechanical resonance
frequency ($\gtrsim$1\text{\,}\mathrm{MHz}$$) a bias tee is used to split the
low from the high-frequency component. The DC-like part is used in the
feedback loop, whilst the high-frequency part is analyzed using an electric
spectrum analyzer (ESA). Mechanical mode frequency and linewidth can directly
be extracted from there. The vacuum optomechanical coupling strength is then
inferred by comparing the optical cavity frequency noise with the expected
thermal noise of a mechanical oscillator of the measured frequency and
linewidth at ambient temperature conditions [41].

In order to measure the dynamic optomechanical spring effect a second pump
laser is used (highlighted in green), while the optical cavity remains tightly
locked to the probe laser. The pump laser addresses a second optical cavity
mode separated by one free spectral range towards higher frequencies from the
probe-cavity-resonance. For variable pump laser powers, the pump frequency is
scanned. The reflection signal of the pump is recorded on a second photodiode
(PD${}_{\text{pump}}$), while the locked probe laser error signal is used for
the characterization of the mechanical mode frequency.

### Coupling calculations

To compute the expected optomechanical frequency-pull factor
$G^{(1)}(t_{\text{drum}},h_{\text{drum}})$ for a particular drum thickness and
support height, we use two methods as described in section II. For both
methods, we first numerically find the set of
$\\{l_{\text{drum}},t_{\text{drum}},h_{\text{drum}}\\}$ that fulfils the
resonance condition of the optical cavity with the probe light at
$\lambda=\frac{2\pi}{k_{0}}$. Here, $h_{\text{drum}}$ is the polymer drum
support height, $t_{\text{drum}}$ the polymer membrane thickness, and
$l_{\text{drum}}=L_{\text{cav}}-t_{\text{drum}}-h_{\text{drum}}$, where,
$L_{\text{cav}}$ denotes the separation between the two cavity mirrors (for an
overview see Fig. 1 b). For each pair $\\{t_{\text{drum}},h_{\text{drum}}\\}$,
a $l_{\text{drum}}$ at
$L_{\text{cav}}\sim$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ can be chosen
to match the resonance condition
$R(l_{\text{drum}},t_{\text{drum}},h_{\text{drum}})=0$, with (suppressing
subscripts):

$R\left(l,h,t\right)=A_{+}\left(l\right)\cdot
e^{-in_{\text{poly}}k_{0}t}-A_{-}\left(h\right)\cdot
e^{in_{\text{poly}}k_{0}t}$ (3)

and with coefficients $A_{\pm}(z)$:

$A_{\pm}(z)=\frac{1\mp in_{\text{poly}}\cdot\tan\left(k_{0}z\right)}{1\pm
in_{\text{poly}}\cdot\tan\left(k_{0}z\right)}\;.$

In the picture used here, the cavity consists of a loss-less dielectric
membrane with refractive index $n_{\text{poly}}$. It is placed between two
mirrors with perfectly conducting surfaces, enforcing field nodes at their
positions (the penetration depth of the DBR will in reality lead to a slightly
modified resonance cavity length). As the Rayleigh length $z_{R}\gg
L_{\text{cav}}$, the Gaussian-beam properties of the cavity field are
neglected and a simple standing-wave ansatz is made. There, the tangential
component of the electric field at both mirror surfaces needs to vanish and
the tangential components of both the electric- and magnetic fields need to be
continuous at the dielectric interfaces [51]. Applying these conditions to the
cavity field, we numerically find the condition that fixes the
$\\{l_{\text{drum}},t_{\text{drum}},h_{\text{drum}}\\}$ \- triplet. This is
now used to extract the optomechanical frequency-pull factor
$G^{(1)}(t_{\text{drum}},h_{\text{drum}})$:

In Method 1, we use Eq. 3 by applying a small shift to the membrane, altering
the cavity geometry to $l_{\text{drum}}\rightarrow l_{\text{drum}}+\Delta z$,
$h_{\text{drum}}\rightarrow h_{\text{drum}}-\Delta z$ and $k_{0}\rightarrow
k_{0}+\Delta k$. This simulates the effect of the linear displacement of the
membrane center region on the cavity resonance condition 555For this
mathematical treatment, $h_{\text{drum}}$ does not directly correspond to the
polymer drum support height, but more accurately to the distance between the
displaced membrane at its center and the subjacent mirror. The mode width of
the optical mode is sufficiently smaller than the effective membrane radius to
ensure that the optical mode only overlaps with an almost constant
displacement of the drum membrane .. Inserting the shifted geometry into Eq. 3
allows to numerically extract a $\Delta k$ that again fulfils Eq. 3 and
thereby reflects the shift on the resonance frequency
$\Delta\omega_{\text{cav}}$. The ratio between the frequency shift and the
small displacement allows us to numerically determine the frequency-pull
$G^{(1)}(t_{\text{drum}},h_{\text{drum}})$. Aside from the linear frequency-
pull factor, higher orders can also be numerically evaluated. E.g. for the
quadratic pull-factor values of
$\gtrsim$200\text{\,}\mathrm{GHz}\text{\,}{\mathrm{nm}}^{-2}$$ are expected.

For Method 2, we use the optical field that fulfills the resonant cavity
condition and apply perturbation theory to find the shift of the optical
resonance associated with a shift of the dielectric boundaries of our geometry
as derived by Johnson et al. in [42]. Here, Maxwell’s equations are written as
an eigenproblem of the electric cavity field $|E\rangle$ with eigenfrequency
$\omega_{\text{cav}}$666For convenience, the basis-independent representation
of the electric field as ”Bra”- and ”Ket”-vectors is utilized, with inner
product $\left\langle E\mid
E^{\prime}\right\rangle\equiv\int\mathbf{E}^{*}\cdot\mathbf{E}^{\prime}\mathop{dV}$.
given by the well-known source-free wave equation with overall cavity
permittivity $\epsilon(z)$:

$\nabla^{2}|E\rangle=\left(\frac{\omega_{\text{cav
}}}{c}\right)^{2}\epsilon(z)|E\rangle.$ (4)

Due to the vibrational motion of the drum membrane, the effective dielectric
permittivity of the cavity geometry experiences a local shift
$\mathop{\delta\epsilon}$ due to a perturbative shift in position of the drum
membrane $\mathop{\delta z}$. We expand $|E\rangle$, $\omega_{\text{cav}}$ to
first-order in $\mathop{dz}$. Plugging these expressions back into Eq. 4 and
neglecting terms of $\mathcal{O}\left(\mathop{dz}^{2}\right)$ leads to the
first-order correction of the resonator frequency in differential form [42]:

$G^{(1)}=\frac{d\omega_{\mathrm{cav}}^{(1)}}{dz}=-\frac{\omega_{\mathrm{cav}}^{(0)}}{2}\frac{\left\langle
E^{(0)}\left|\frac{d\epsilon}{dz}\right|E^{(0)}\right\rangle}{\left\langle
E^{(0)}|\epsilon|E^{(0)}\right\rangle}$ (5)

We then make use of the explicit parametrization of the cavity dielectric
function

$\epsilon(z)=\epsilon_{2}+\Delta\epsilon\big{(}\Theta(z-z_{0})-\Theta(z-(z_{0}+t_{\text{drum}}))\big{)}\;,$

with $\Delta\epsilon=\epsilon_{1}-\epsilon_{2}$ ($\epsilon_{1}$: polymer,
$\epsilon_{2}$: air/vacuum). The first drum surface $S_{\text{L}}$ is located
at $z_{0}$ and the second drum surface $S_{\text{R}}$ at
$z_{0}+t_{\text{drum}}$ following the coordinate conventions of Fig. 2 b.

Inserting this expression back into Eq. 5, we arrive at the explicit form of
the optomechanical frequency-pull $G^{(1)}$:

$G^{(1)}=\frac{\mathop{d\omega_{\mathrm{cav}}}}{\mathop{dz}}=\frac{\omega_{\mathrm{cav}}^{(0)}}{2}\frac{\int_{S_{R}}\Delta\epsilon\left|\mathbf{E}_{\|}^{(0)}\right|^{2}\mathop{dS}-\int_{S_{L}}\Delta\epsilon\left|\mathbf{E}_{\|}^{(0)}\right|^{2}\mathop{dS}}{\int_{V}\epsilon(z)\left|\boldsymbol{E}_{\|}^{(0)}\right|^{2}\mathop{dV}}\;,$
(6)

with the total resonator volume $V$ and parallel electric field component
$\mathbf{E}_{\|}^{(0)}$ (explicit form given by Eq. 3).

As detailed in section II, Eq. 6 allows us to read-off the frequency-pull
$G^{(1)}(t_{\text{drum}},h_{\text{drum}})$ for a specific cavity geometry by
considering the cavity field distribution on both surfaces of the polymer
drum. It also provides the physical interpretation of the coupling to be
caused by differing field intensities on the membrane-air interfaces.

The coupling landscape retrieved from method 1 and method 2 agree for all
tested geometries. For our cavity geometry of interest, the results are shown
in Fig. 2 a.

### Finite element simulations

We use COMSOL Multiphysics® [52] to perform finite element simulations of the
mechanical resonator structures. The simulated geometry consists of the
polymer membrane (IP-S: Dynamic modulus
$E=$5.33+0.26\text{$\mathrm{i}$}\text{\,}\mathrm{GPa}$$, Poisson’s ratio
$\nu=0.3$, density
$\rho=$1.15\text{\,}\mathrm{kg}\text{\,}{\mathrm{dm}}^{-3}$$; [45]) with a
silica substrate below. An outer shell of the substrate is defined as a
perfectly matched layer (PML) to implement radiation losses into the substrate
(see Fig. 1 b). Whilst these radiation losses are not the dominating loss
mechanism at room temperature, they are strongly dependent on the polymer
geometry and can be reduced to limiting quality factors of $\gg 10^{6}$ for
optimized support dimensions and with included isolation cuts. By reducing the
internal polymer losses, e.g. through other resist materials, high quality
factor mechanical resonators would be feasible. Aside from the quality factor
and resonance frequency, we use the simulation to extract the effective mass
of the fundamental flexural mode (on the order of
$\sim$2.4\text{\,}\mathrm{ng}$$ depending on the specific geometry) and the
corresponding zero point motion ($\sim$3.2\text{\,}\mathrm{fm}$$) [53].

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## Acknowledgements

The authors would like to thank Prof. Dieter Meschede for supporting the first
experiments on this project. The authors acknowledge funding by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – Cluster of Excellence Matter and Light for Quantum
Computing (ML4Q) EXC 2004/1 – 390534769 as well as funding from the
Bundesministerium für Bildung und Forschung (BMBF, Federal Ministry for
Education and Research) - project FaResQ. S.H. furthermore acknowledges
funding from the European Union’s Horizon 2020 program under the ERC
consolidator grant RYD-QNLO (Grant No. 771417).

## Author contributions statement

H.P., L.T., A.F., and S.L. came up with the concept and planned the
experiments. A.F. performed the fabrication and L.T. conducted the optical
measurements. L.T., A.F., and H.P. set up the numerical simulations and
analyzed the data. All authors contributed to the writing of the manuscript.

## Data availability

The data of this study is available from the corresponding author (H.P.) upon
reasonable request.

## Competing interests

The authors declare no competing interests.


# Identification of the Nitrogen Interstitial as Origin of the 3.1 eV
Photoluminescence Band in Hexagonal Boron Nitride

###### Abstract

Nitrogen interstitials (N${}_{\textsubscript{i}}$) have the lowest formation
energy among intrinsic defects of hexagonal boron nitride (hBN) under n-type
and N-rich conditions. Using an optimized hybrid functional, which reproduces
the gap and satisfies the generalized Koopmans condition, an
N${}_{\textsubscript{i}}$ configuration is found which is lower in energy than
the ones reported so far. The (0/-) charge transition level is also much
deeper, so N${}_{\textsubscript{i}}$ acts as a very efficient compensating
center in n-type samples. Its calculated photoluminescence (PL) at 3.0 eV
agrees well with the position of an N-sensitive band measured at 3.1 eV. It
has been also found that the nitrogen vacancy (V${}_{\textsubscript{N}}$)
cannot be the origin of the three boron electron (TBC) electron paramagnetic
resonance (EPR) center and in thermal equilibrium it cannot even exist in
n-type samples.

Elham Khorasani, Thomas Frauenheim, Bálint Aradi* and Peter Deák*

Elham Khorasani
Bremen Center for Computational Materials Science, University of Bremen, P.O.
Box 330440, D-28334 Bremen, Germany.

Prof. Dr. Thomas Frauenheim
Beijing Computational Science Research Center (CSRC), 100193 Beijing, China.
Shenzhen JL Computational Science and Applied Research Institute, 518110
Shenzhen, China.
Bremen Center for Computational Materials Science, University of Bremen, P.O.
Box 330440, D-28334 Bremen, Germany.

Dr. Bálint Aradi
Bremen Center for Computational Materials Science, University of Bremen, P.O.
Box 330440, D-28334 Bremen, Germany.
Email Address<EMAIL_ADDRESS>
Prof. Dr. Peter Deák
Bremen Center for Computational Materials Science, University of Bremen, P.O.
Box 330440, D-28334 Bremen, Germany.
Email Address<EMAIL_ADDRESS>
##  1 Introduction

Materials with well-defined defect centers are highly promising for next
generation devices, with applications ranging from electronics and photonics
to quantum computing [1, 2]. Hexagonal boron nitride (hBN) is a wide-band-gap
semiconductor that exhibits a rich number of band gap levels which profoundly
impact its electronic and optical properties. These can act as recombination
centers and emit light within a specific energy range. In recent years,
numerous color centers have been reported in hBN [3, 4]. However, the physical
origin of some of these centers is not yet fully understood [5, 6]. First
principle calculations can play a crucial role in identifying the defects and
providing accurate information about their properties.

The luminescence properties of hBN have been studied mainly in recent years
[5, 7, 8, 9, 10]. Museur et al. reported three photoluminescence (PL) bands at
the energies of 5.3 eV, 3.75 eV, and 3.1 eV for pyrolytic hBN (pBN) samples
[7]. pBN samples are high purity samples, free of carbon and oxygen. The
emission bands around 5.3 eV and 3.57 eV were found to be donor-acceptor pair
(DAP) type transitions. The nature of the PL band observed at 3.1 eV in pBN
samples was not elucidated. Du et al. [8] investigated the PL spectra for a
set of hBN epilayers grown by metal organic chemical vapor deposition (MOCVD)
under different ammonia ($\mathrm{NH}_{3}$) flow rates from 0.2 to 1.5
standard liters per minute (SLM). They observed four emission peaks at
energies of 4.12 eV, 3.92 eV, 3.72 eV, and 5.37 eV for hBN under the
$\mathrm{NH}_{3}$ flow rate of 0.2 SLM. The first three peaks were interpreted
as a zero phonon line at 4.12 eV with two phonon replicas. They observed a
decrease in the emission intensity for the peak at 4.12 eV upon increasing the
$\mathrm{NH}_{3}$ flow rate while the change of the peak at 5.37 eV was found
to be small. They also observed the appearance of the peak at 3.1 eV as the
flow rate of the $\mathrm{NH}_{3}$ was increased to 1.5 SLM.

Besides PL spectroscopy, electron paramagnetic resonance (EPR) is a powerful
technique often used for identifying and studying defects in semiconductors
and insulators [11]. Two types of paramagnetic centers were identified in hBN
[6, 12, 13, 14]. In one type, an unpaired electron interacts with three
equivalent boron nuclei, producing a 10-line EPR spectrum. This type is
referred to as the three boron center (TBC). In another type, an unpaired
electron interacts with a single boron nucleus, which gives rise to a 4-line
EPR spectrum. This is referred to as the one boron center (OBC). Based on
experimental observation, the carbon substitutional
($\mathrm{C}_{\mathrm{N}}$) and the nitrogen vacancy
($\mathrm{V}_{\mathrm{N}}$) were proposed [6, 14, 15] to model the TBC. In the
former, an unpaired electron was trapped at the carbon atom. This was referred
to as a carbon-associated TBC and was confirmed theoretically [15]. In the
$\mathrm{V}_{\mathrm{N}}$ model, proposed by experiments [6], the unpaired
electron is assumed to be trapped in the nitrogen vacancy. This was referred
to as an electron-irradiation produced TBC. However, its nature is still
controversial.

We have applied an optimized hybrid functional to calculate defects in hBN.
Our calculations show that the observed PL at 3.1 eV in bulk hBN is due to the
recombination of an electron with a hole trapped at a nitrogen interstitial,
$\mathrm{N}_{\mathrm{i}}$. Additionally, we have found that
$\mathrm{V}_{\mathrm{N}}$ cannot be the origin of the TBC center. In fact, in
n-type samples in equilibrium, $\mathrm{V}_{\mathrm{N}}$ turns out to be less
stable than $\mathrm{N}_{\mathrm{i}}$ even under extreme N-poor conditions,
indicating that $\mathrm{V}_{\mathrm{N}}$ can be created only by irradiation.

##  2 Computational method

It was shown that the screened hybrid functional of Heyd, Scuseria, and
Ernzerhof (HSE) [16, 17], with parameters tuned to reproduce the relative
position of the quasi-particle band edges and to satisfy the generalized
Koopman’s theorem (gKT), is capable of providing the formation energy, the
charge transition levels and the hyperfine interaction of defects very
accurately in traditional bulk semiconductors [18, 19, 20]. In our previous
work, we have shown that such a functional can be optimized for layered
compounds as well [21] and HSE($\alpha,\mu$) with a mixing parameter $\alpha$
= 0.3 and a screening parameter $\mu$ = 0.4 was found as the optimal
functional for bulk hBN.

Our calculations in the present work have been carried out with the Vienna Ab
initio Simulation Package, VASP 5.4.4, using the projector augmented wave
method [22, 23, 24]. A 420 (840) eV cutoff was applied for the expansion of
the wave functions (charge density) in hBN. The convergence condition of 10-4
eV was applied for the self-consistent electronic energy. Van der Waals
interactions between the layers were taken into account by the Tkatchenko-
Scheffler method [25], using $s_{\mathrm{R}}$ = 0.96.

The equilibrium geometry was determined for the primitive unit cell with a
$\Gamma$-centered $6\times 6\times 6$ Monkhorst-Pack (MP) $k$-point set [26],
based on constant volume relaxations and fitting to Murnaghan’s equation of
state [27]. We describe bulk hBN using an orthogonal supercell of 120 atoms
(5a1, 3a1 \+ 6a2, a3) at the calculated lattice constants (see our previous
work [21]), using the $\Gamma$-point approximation. (Here $\textbf{a}_{1}$,
$\textbf{a}_{2}$, and $\textbf{a}_{3}$ are the primitive unit vectors). The
geometries of the defects in the supercell were relaxed at fixed lattice
constants, using a force criterion of 0.01 eV/Å.

We have calculated the formation energy and the transition levels of the
intrinsic defects $\mathrm{N}_{\mathrm{i}}$ and $\mathrm{V}_{\mathrm{N}}$. The
chemical potentials $\mu_{\mathrm{N}}$ and $\mu_{\mathrm{B}}$ are related in
equilibrium, as $\mu_{\mathrm{BN}}=\mu_{\mathrm{B}}+\mu_{\mathrm{N}}$ where
$\mu_{\mathrm{BN}}$ is the energy of hBN per formula unit. For N-rich
conditions the chemical potential of nitrogen was taken as the energy of the
nitrogen atom in the molecule (at a temperature of 300 K and a pressure of 105
Pa). For the B-rich case $\mu_{\mathrm{B}}$ corresponds to the energy of the B
atom in the solid phase. For charged defects, total energies were corrected a
posteriori, using the SLABCC code [28, 29] to eliminate the artificial
interaction between the repeated charges. SLABCC is based on the scheme
proposed by Komsa and Pasquarello [30]. For the correction, the static
dielectric constant $\varepsilon_{0}$ should be used in principle if the ionic
cores are allowed to relax. However, it was shown that, due to the explicit
screening in the supercell, the high-frequency value $\varepsilon^{\infty}$
gives a better approximation [31], therefore
$\varepsilon^{\infty}_{\perp}=4.95$ and
$\varepsilon^{\infty}_{\parallel}=4.10$ [32] were used.

To explain the observed PL in bulk hBN, we have investigated electron
recombination with a hole trapped at $\mathrm{N}_{\mathrm{i}}$. Within the
accuracy of the calculation (0.1 eV), a shallow donor state cannot be
distinguished energetically from the bottom of the conduction band. Therefore,
we simulate a donor-acceptor recombination from a shallow donor to a deep
acceptor by the recombination of a bound exciton, i.e., from the conduction
band edge to the acceptor level [8]. To calculate the PL, we first relaxed the
equilibrium geometry of the bound exciton under the constraint of the orbital
occupations, i.e., with one hole at the defect level and with one electron in
the conduction band minimum (CBM). To consider the delocalization of the
electron, the total energy of the initial state was recalculated with a
(non-$\Gamma$-centered) $2\times 2\times 2$ Monkhorst-Pack set at the relaxed
geometry of the bound exciton in the $\Gamma$ approximation. The energy of the
final state after recombination was calculated at fixed geometry, using the
same $2\times 2\times 2$ $k$-point set. The PL energy is the difference
between the two total energies.

Hyperfine interaction was calculated for $\mathrm{V}_{\mathrm{N}}$. Boron has
two isotopes, ${}^{10}{\mathrm{B}}$ and ${}^{11}{\mathrm{B}}$, with the
natural abundances of $18.83\%$ and $81.17\%$, respectively. These isotopes
have different nuclear spins and magnetic moments. In this work, the hyperfine
splitting was compared to the experimental data measured in
${}^{11}{\mathrm{B}}$-enriched samples. The presented results were obtained
for the isotope ${}^{11}{\mathrm{B}}$ with $I=3/2$. The nuclear gyromagnetic
ratios of 13.66 and 3.08 were used for boron and nitrogen, respectively [33].

##  3 Results and Discussion

###  3.1 Nitrogen interstitial

In earlier works, the nitrogen interstitial was predicted to be in a [001]
split interstitial configuration with a lattice N atom [34, 35], and was found
to be the intrinsic defect with the lowest formation energy under N-rich
conditions in n-type material. We have found, however, that the configuration
shown in Figure 1 is lower in energy by 0.70 eV. The axis of the split
interstitial tilts away from [001], so both nitrogen atoms are ideally
threefold coordinated. The $\mathrm{N}_{\mathrm{i}}$-$\mathrm{N}$ bondlength,
1.38 Å corresponds to a single bond, and $\mathrm{N}_{\mathrm{i}}$ also binds
to a boron atom of the top layer. Simple electron counting shows that the
$p$-orbital of $\mathrm{N}_{\mathrm{i}}$, which is orthogonal to the plane of
its three bonds, contains only one electron.

Figure 1: Atomic structure of the intrinsic defect $\mathrm{N}_{\mathrm{i}}$
in the lowest energy configuration. The $\mathrm{N}_{\mathrm{i}}$-$\mathrm{N}$
bondlength, 1.38 Å corresponds to a single bond. $\mathrm{N}_{\mathrm{i}}$
also binds to a boron atom of the top layer.

Figure 2 shows the formation energy calculated for the intrinsic defects
$\mathrm{N}_{\mathrm{i}}$ and $\mathrm{V}_{\mathrm{N}}$ in different charge
states, as a function of the Fermi level position between the valence band
maximum (VBM) and the conduction band minimum (CBM). The adiabatic charge
transition levels for $\mathrm{N}_{\mathrm{i}}$ are obtained at
$\mathrm{E}(+/0)=$ 0.94 eV and $\mathrm{E}(0/-)=$ 2.68 eV with respect to the
valence band edge. The latter is much deeper than the value obtained in a
previous work [34]. Weston et al. predicted $\mathrm{N}_{\mathrm{i}}$ to be
the major compensating center in n-type samples [34]. Our finding confirms
that and the lower formation energy indicates an even stronger compensating
effect.

Figure 2: Formation energies of the intrinsic point defects
$\mathrm{N}_{\mathrm{i}}$ and $\mathrm{V}_{\mathrm{N}}$ in bulk hBN as a
function of the Fermi level under (left) N-rich and (right) N-poor conditions.
Solid blue and dashed red lines represent the defects
$\mathrm{N}_{\mathrm{i}}$ and $\mathrm{V}_{\mathrm{N}}$, respectively. The
energy of the valence band maximum has been set to zero.

The PL energy was calculated for the configuration shown in Figure 1 as
described in Section 2. Assuming the donor level to be 100 meV under the CBM,
our calculated PL energy is at 3.0 eV. This can explain the origin of the
N-related observed peak at 3.1 eV in hBN in Ref. [8, 7].

###  3.2 Nitrogen Vacancy

The similarities between optical emitters in hBN and the nitrogen-vacancy
($\mathrm{NV}$) center in diamond, have brought a lot of attention to vacancy
defects in hBN. Earlier experimental studies proposed
$\mathrm{V}_{\mathrm{N}}$ as the source of the TBC [6], which was produced by
electron irradiation. $\mathrm{V}_{\mathrm{N}}$ introduces a gap level with an
unpaired electron. Our calculations show that the electron is on $p$-orbitals
orthogonal to the lattice planes, with a symmetrical charge distribution on
the three neighboring boron atoms. Figure 3 shows the spin distribution. The
calculated hyperfine coupling constants $A$ are given in Table 1. The
calculated average principle value of the hyperfine interaction tensor is
obtained at 34.23 MHz, which is significantly smaller than the value of 117.06
MHz measured for the TBC in carbon-free samples after irradiation. It has been
shown that core spin polarization may have a significant effect on the
hyperfine interaction calculated, providing accurate results in various
semiconductors [36]. Taking the core contribution into account, the calculated
magnitude of 34.23 MHz, which only involved the valence electronic spin
density, reduced to 16.09 MHz, demonstrating that the
$\mathrm{V}_{\mathrm{N}}$ can not be the origin of the three boron center.

The nitrogen instersitial defect, $\mathrm{N}_{\mathrm{i}}$ is expected to be
more stable than $\mathrm{V}_{\mathrm{N}}$ in N-rich samples. This is also
confirmed by Figure 2. However, as demonstrated by the same figure, even under
extreme N-poor conditions, $\mathrm{V}_{\mathrm{N}}$ can only exist in p-type
samples in thermal equilibrium. Actually, there is no hard experimental
evidence for the existence of $\mathrm{V}_{\mathrm{N}}$. In the earlier PL
studies [5, 8, 9, 37], the nitrogen vacancy was suggested as a possible source
of the 4.1 eV emission band in bulk hBN. Later on, however, a theoretical
study showed that the carbon dimer defect
$\mathrm{C}_{\mathrm{B}}\mathrm{C}_{\mathrm{N}}$ is responsible for the 4.1 eV
emission in hBN [38]. Jin et al. [39] used ultrahigh-resolution transmission
electron microscope imaging. They created defects by an electron beam
irradiation and found that the created defects are of the boron vacancy
($\mathrm{V}_{\mathrm{B}}$) type.

Table 1: Hyperfine coupling constants for the model defect $\mathrm{V}_{\mathrm{N}}$. The average value of $A_{\mathrm{ave}}$ was calculated as ($A_{\mathrm{xx}}+A_{\mathrm{yy}}+A_{\mathrm{zz}})/3$ in MHz. $A_{\mathrm{xx}}$ | $A_{\mathrm{yy}}$ | $A_{\mathrm{zz}}$ | $A_{\mathrm{ave}}$ |  $A_{\mathrm{ave}}$(experimental)111andrei1976point
---|---|---|---|---
21.21 | 20.81 | 60.65 | 34.23 | 117.06
Figure 3: The spin density calculated for the $\mathrm{V}_{\mathrm{N}}$ defect
as model for the TBC in bulk hBN (isovalue: 0.002 1/Å3) in (left) top view and
(right) side view. The spin density concentrated on the three neighbouring
boron atoms produces the calculated hyperfine couplings.

##  4 Conclusion

We have calculated the intrinsic defects, nitrogen interstitial and nitrogen
vacancy, using the optimal HSE (0.4, 0.3) functional for hBN as obtained in
our previous work. We have found an $\mathrm{N}_{\mathrm{i}}$ configuration
which is considerably lower in energy than the ones reported so far. The
calculated formation energies show that $\mathrm{N}_{\mathrm{i}}$ is the
intrinsic defect with the lowest formation energy under N-rich conditions. Its
acceptor level is deep and behaves as a compensating center. The calculated PL
energy of 3.0 eV for $\mathrm{N}_{\mathrm{i}}$ is in good agreement with the
experimentally observed N-related band at 3.1 eV [7, 8]. We have investigated
the nitrogen vacancy as well. Our result indicates that
$\mathrm{V}_{\mathrm{N}}$ is less stable than $\mathrm{N}_{\mathrm{i}}$ in
n-type samples even under extreme N-poor conditions and should be observable
only in irradiated samples. The hyperfine interactions have been calculated
for $\mathrm{V}_{\mathrm{N}}$, which was suspected to be a source of the three
boron centers produced under an electron-irradiation in carbon-free samples.
Our results indicate that $\mathrm{V}_{\mathrm{N}}$ cannot be the origin of
the three boron center in bulk hBN.

Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) within
the research training group (RTG) 2247.

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