# Joined optical and thermal characterization of a III-nitride semiconductor
membrane by micro-photoluminescence spectroscopy and Raman thermometry

Mahmoud Elhajhasan  Wilken Seemann  Katharina Dudde  Daniel Vaske  Gordon
Callsen<EMAIL_ADDRESS>Institut für Festkörperphysik, Universität
Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany  Ian Rousseau  Thomas F. K.
Weatherley  Jean-François Carlin  Raphaël Butté  Nicolas Grandjean  Institute
of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland  Nakib H. Protik  Institut für Physik und IRIS Adlershof,
Humboldt-Universität zu Berlin, 12489 Berlin, Germany  Guiseppe Romano  MIT-
IBM Watson AI Lab, IBM Research, Cambridge, MA, USA

###### Abstract

We present the simultaneous optical and thermal analysis of a freestanding
photonic semiconductor membrane made from wurtzite III-nitride material. By
linking micro-photoluminescence spectroscopy with Raman thermometry and other
spectroscopic techniques, we demonstrate how a robust value for the thermal
conductivity $\kappa$ can be obtained using only optical, non-invasive means.
For this, we consider the balance of different contributions to thermal
transport given by, e.g., excitons, charge carriers, and heat carrying
phonons. In principal, all these contributions can be of relevance in a
photonic membrane on different lengthscales. Further complication is given by
the fact that this membrane is made from direct bandgap semiconductors,
designed to emit light based on an InxGa1-xN ($x=0.15$) quantum well embedded
in GaN. Thus, III-nitride membranes similar to the one in the focus of this
study, have already successfully been used for laser diode structures facing
thermal limitations. To meet these intricate challenges, we designed a novel
experimental setup that enables the necessary optical and thermal
characterizations in parallel. After the optical characterization by micro-
photoluminescence, we follow a careful step-by-step approach to quantify the
thermal properties of our photonic membrane. Therefore, we perform micro-Raman
thermometry, either based on a heating laser that also acts as a probe laser
(1-laser Raman thermometry), or based on two lasers, providing the heating and
the temperature probe separately (2-laser Raman thermometry). For the latter
technique, we can obtain temperature maps over several tens of micrometers
with a spatial resolution less than $1\,\mu\text{m}$. As a result, the
temperature probe volume using the 2-laser Raman thermometry technique can be
increased by a factor exceeding $100$ compared to the conventional 1-laser
Raman thermometry technique, which impacts the derivation of the thermal
conductivity $\kappa$. Only based on our largest temperature probe volume we
derive $\kappa\,=\,95^{+11}_{-7}\,$$\frac{\text{W}}{\text{m}\cdot\text{K}}$
for the c-plane of our $\approx\,250\text{-nm}$-thick photonic membrane at
around room temperature, which compares well to our ab initio calculations,
applied to a simplified structure, yielding $\kappa\,=\,136\,$
$\frac{\text{W}}{\text{m}\cdot\text{K}}$ . Based on these calculations, we
explain the particular relevance of the temperature probe volume, as quasi-
ballistic transport of heat-carrying phonons with high relevance for $\kappa$
occurs on length scales beyond the penetration depths of the heating laser and
even its focus spot radius. The 1-laser Raman thermometry technique,
therefore, fails to determine realistic $\kappa$ values, while with the
2-laser Raman thermometry technique one can probe temperatures in sufficiently
large volumes. The present work represents a significant step towards non-
invasive, highly spatially resolved, and still quantitative thermometry
performed on a photonic membrane made of a direct bandgap semiconductor, which
is of particular relevance for photonic applications.

The optical properties of semiconductors are extensively studied, and are
therefore becoming increasingly well understood for the full size variation
from bulk crystals down to nanostructures. Measuring fundamental optical
phenomena like absorption, reflectivity, light scattering, and
photoluminescence (PL) on various length- and timescales has nowadays become a
widespread capability in many laboratories. The nanostructures commonly in
scope encompass the entire dimensionality range from quantum dots (0D), over
nanowires (1D), to quantum wells (2D). Consequently, a wide range of
prototypes and devices was realized, scaling from, e.g., light detectors Fan
(2020), over sensors Khan and Rao (2020), to light-emitting- Kneissl  _et al._
(2019), and laser-diodes Kneissl  _et al._ (2007). For instance, laser-diodes
often utilize nanostructures like quantum wells (QWs) as a built-in light
source Nakamura  _et al._ (1996); Kneissl  _et al._ (2003). However, the high
level of sophistication reached for the analysis of the photonic properties of
these structures Nippert  _et al._ (2016); Sheen  _et al._ (2022); Nippert
_et al._ (2018) is often contrasted by a limited understanding of the
interfering thermal phenomena Jagsch  _et al._ (2018). Commonly, either the
photonic or the thermal properties of micro- and nanostructures are studied,
while the interrelation of both phenomena is rarely reported Gesemann  _et
al._ (2010); Wen  _et al._ (2022), prohibiting any future mutual optimization.
So far, a link between photonic and phononic properties has been established
in the field of nano-optomechanics, focusing mostly on low-frequency phonons
(MHz - GHz range) Aspelmeyer  _et al._ (2014). However, thermal transport
represents an even more intricate phenomenon that can be mediated by a
weighted balance of, e.g., charge carriers Cahill  _et al._ (2002); Quan  _et
al._ (2021), excitons Wu  _et al._ (2014), and most prominently high-
frequency, thermal phonons up to the THz range Sood  _et al._ (2018).

Before any link between optical and thermal material properties can be
studied, one first has to develop a robust experimental technique that enables
such studies. Clearly, any interlinked understanding of optical and thermal
material properties could potentially pave the way towards mutual
optimizations. Established thermal characterization techniques like, e.g.,
3$\omega$-measurements Cahill (1990) are not straightforwardly compatible with
photonic micro- and nanostructures for size reasons and the required
fabrication of metal contacts. A similar observation seems valid for time-
domain thermal reflectance (TDTR) measurements Cahill  _et al._ (2002, 2014)
as another established thermal characterization approach, which most
frequently relies on metal transducers. Purely thermally induced reflectivity
changes in semiconductors are generally considered small over wide energy
ranges, if not located in close energetic vicinity to electronic and excitonic
resonances Matatagui  _et al._ (1968); Liu  _et al._ (2019). Thus, in practice
metal transducers with larger thermoreflectance coefficients in the wavelength
range of interest are commonly required for TDTR measurements Wang  _et al._
(2010); Wilson  _et al._ (2012). The processing of such metal transducers can
potentially deteriorate photonic structures. Therefore, a fully non-invasive
thermal characterization technique is required, which in the best case can
readily be combined with standard optical analyses like absorption and
reflectivity measurements, as well as (time-resolved) PL spectroscopy. In a
best case scenario, this thermal characterization technique should also
feature diffraction limited spatial resolution similar to, e.g., micro-TDTR
Maire  _et al._ (2017), together with a large applicable temperature range and
sufficient temperature resolution. Certainly, for many applications any
optical and thermal characterization at room temperature and beyond would be
sufficient. Nevertheless, most interesting thermal transport physics connected
to, e.g., the occurrence of various thermal phonon transport regimes, would
also motivate thermometry down to cryogenic temperatures Cepellotti  _et al._
(2015); Lee  _et al._ (2015); Anufriev  _et al._ (2017). In summary, a
quantifying thermal characterization technique with excellent integrability,
high spatial resolution, as well as a wide operating temperature range and
sufficient temperature resolution is desired.

Raman thermometry appears as a promising thermal characterization technique in
the light of these requirements Beechem  _et al._ (2007, 2015); Jaramillo-
Fernandez  _et al._ (2018). Raman measurements can not only be performed in a
micro-PL ($\mu$PL) setup, one can also directly map micro-Raman ($\mu$Raman)
spectra with a spatial resolution close to the diffraction limit Beechem  _et
al._ (2007) over a wide range of temperatures. However, these promising
aspects of Raman thermometry pair with a number of challenges, whenever the
Raman laser is not exclusively used as a temperature probe to map, e.g., the
Joule heating of a transistor structure under operating conditions Kuball
(2001). As soon as the Raman laser is simultaneously used for the sample
heating and the temperature probing based on Raman spectra, the situation
complicates Jaramillo-Fernandez  _et al._ (2018); Sandell  _et al._ (2020).
The applied laser heating challenges the data analysis in contrast to, e.g.,
common 3$\omega$ or TDTR measurements that apply Joule heating to a
semiconductor via metal contacts or transducers. In general, the thermal
resistance of the metal/semiconductor interface represents a manageable
challenge for 3$\omega$ and TDTR experiments and directly correlated thermal
simulations Cahill  _et al._ (2014). In contrast for Raman thermometry,
depending on the light penetration depths of the heating laser, one has to
either model the laser heating by a semiconductor surface heating or a fully
volumetric heating, which requires precise knowledge of the absorption
coefficient and its wavelength dependence Lax (1977); Jaramillo-Fernandez  _et
al._ (2018). Even more severe, in stark contrast to the Joule heating via
metallized semiconductor surfaces, the over bandgap optical excitation of
semiconductor samples by a suitable heating laser can excite charge carriers
Cahill  _et al._ (2002); Quan  _et al._ (2021) or even generate optical
excitations like excitons Wu  _et al._ (2014) and exciton-polaritons Ciers
_et al._ (2020). As a result, not only thermal phonons are generated, but all
these additional fundamental excitations, which diffuse away from a laser-
induced heatspot. This situation is then further complicated by the optical
decay of, e.g., excitons and exciton-polaritons during their propagation. Any
measured temperature rise in a semiconductor that is directly induced by laser
heating (i.e., without a metal transducer) is challenged by all these
additional energy transport mechanisms. Commonly, with thermal phonons we
refer to high frequency heat carrying phonons that predominantly contribute to
thermal transport at all temperatures in the scope of the present work. For
sake of completeness, also radiative heat transfer and related near-field
phenomena Biehs (2007) can be added to the list of these mechanisms. However,
these mechanisms can safely be neglected for the present work due to the
occurring temperature differences and distances as well as dominating
contributions from the phononic type of thermal transport.

Consequently, Raman thermometry that relies on laser heating can only lead to
a solid thermal characterization of a semiconductor, if its optical properties
are thoroughly studied in-parallel. Not only the absorption must be known
Beechem  _et al._ (2015), but also the overall quantum efficiency of the
sample is required for an estimation of how much of the absorbed light
contributes to the heating of the sample. Here, the fact that common Raman
thermometry applies the same laser for the laser heating and the temperature
probing represents a strong limitation as the electronic Weatherley  _et al._
(2021) or thermal excitations Minnich  _et al._ (2011) connect to different
characteristic mean free path lengths. Thus, such one-laser Raman thermometry
(1LRT) is best extended to two-laser Raman thermometry (2LRT), transferring
the role of the temperature probe to a second laser, which can be scanned away
from the laser heat spot, over the entire structure of interest. Based on
temperature maps extracted from Raman mapscans, it is consequently possible to
observe temperature distributions that are caused by a weighted balance of the
different mechanisms contributing to thermal transport. Thus, as the related
mean free path lengths vary, 2LRT measurements have the potential to
disentangle different contributions to thermal transport by varying the
distance between the temperature probe and the laser-induced heatspot.
Initially, Hsu et al. have applied 2LRT measurements to carbon nanotubes Hsu
_et al._ (2011), while also its applicability to silicon membranes Reparaz
_et al._ (2014) and polycrystalline nanomembranes of MoS2 Sledzinska  _et al._
(2016) was shown. However, all existing experimental 2LRT setups, as described
in detail by Reparaz et al. Reparaz  _et al._ (2014), require full optical
access from the sample front- and backside to overlap the heating and the
probe laser focus spots on the sample. While such an approach is still
feasible, e.g., for micro-structured silicon membranes Graczykowski  _et al._
(2017), most photonic micro- and nanostructures can only be accessed from one
side as a full substrate removal is often detrimental for delicate photonic
structures Vico Triviño  _et al._ (2012); Jagsch  _et al._ (2018); Triviño
_et al._ (2015).

In this contribution we apply 1LRT and 2LRT to a photonic membrane structure,
which has widely been described in literature Vico Triviño  _et al._ (2012);
Triviño  _et al._ (2015); Rousseau  _et al._ (2017); Jagsch  _et al._ (2018);
Rousseau  _et al._ (2018); Rousseau (2018). This membrane is mostly made from
GaN and it incorporates an InxGa1-xN ($x=0.15$) quantum well (QW) as built-in
light source. Structures very similar to this photonic membrane have already
formed the basis for publications reporting on, e.g., 1-dimensional nanobeam
lasers Triviño  _et al._ (2015); Jagsch  _et al._ (2018) and 2-dimensional
photonic crystals Vico Triviño  _et al._ (2012) with excellent optical
properties. Trivi$\tilde{\text{n}}$o et al. reported on record low threshold
blue lasing of InGaN/GaN nanobeam lasers Triviño  _et al._ (2015), which were
based on the photonic membrane also in use for the present work. As such
photonic structures can experimentally only be accessed from one side, a fully
customized 1LRT and 2LRT setup was constructed, which heats samples and probes
their temperatures from the same side by combining conventional micro-Raman
mapping and a laser scanner. To highlight the functionality of this setup, all
experimental results are introduced in a careful step-by-step approach. aiming
to achieve quantitative Raman thermometry despite the numerous challenges
posed by a photonic membrane made from direct bandgap semiconductor material.
This also includes in-parallel an analysis of the PL signal, aiming to discard
an impact of free carriers and excitons on our determination of the thermal
conductivity $\kappa$. We show that based on 1LRT or 2LRT measurements one
determines strongly different thermal conductivities, which are linked to the
temperature probe volume. 1LRT measurements are not ideally suited for a
quantitative determination of $\kappa$ in our membranes, whereas 2LRT
measurements yield
$\kappa\,=\,95^{+11}_{-7}\,$$\frac{\text{W}}{\text{m}\cdot\text{K}}$ for our
$\approx\,250\text{-nm-thick}$ photonic III-nitride membrane. We then compare
$\kappa$ derived from ab initio calculations to our experimental results,
while analyzing the dependence of the related cumulative $\kappa$ on the
phonon mean free path lengths $l^{MFP}$ provides an explanation for the
different experimental results derived from 1LRT and 2LRT measurements.
Consequently, only 2LRT measurements allow for a truly thermal quantitative
analysis of our photonic III-nitride membrane, providing a strong motivation
for any future application of this experimental technique.

The paper is structured as follows. In Sec. I, we first introduce the photonic
membrane made from III-nitride material (Sec. I.1) that we will optically
($\mu$PL) and thermally (Raman thermometry) characterize based on our
customized optical setup (Sec. I.2). Herein, we describe the mode of operation
of our optical setup in a four-step-process, which gradually raises the level
of complication for our measurements. This rigorous step-by-step approach
starts with Step A, the presentation of the $\mu$PL spectroscopy in Sec. II.
Subsequently, in Sec. III this pre-characterization of our photonic membrane
is complemented by spatially resolved, non-resonant and selected resonant
Raman measurements on GaN, which constitutes Step B in our hierarchy. Such
resonant and non-resonant Raman spectroscopy represents the basis for the
Raman thermometry described in Sec. IV, which is not yet spatially resolved
(Step C). This section is subdivided into Sec. IV.1 and Sec. IV.2, which
introduce the 1LRT and the 2LRT0 measurements without any spatial displacement
between the heat and the probe lasers. Herein, we also introduce our numerical
simulations that allow us to extract $\kappa$ from our 1LRT and 2LRT without
spatial displacement (2LRT0) measurements. Based on these experimental and
numerical foundations from Secs. I \- IV, we can turn towards the last Step D
in Sec. V that describes 2LRT measurements. Here, we show our temperature
mapscans that enable the determination of $\kappa$, which is additionally
modeled in Sec. VI based on state-of-the-art ab initio calculations. As we
experimentally find a particular scaling behavior in between the $\kappa$
values derived from 1LRT, 2LRT0, and 2LRT measurements, we discuss this
observation in Sec. VII based on our theoretical findings from Sec. VI.
Finally, we present our discussion and outlook in Sec. VIII, before
summarizing our results in Sec. IX. In addition to this paper, more technical
details and concepts are further described in the Supplementary Material (SM)
within S-Secs. I - IV.

##  I Experimental details

Figure 1: (a) Layer sequence of the photonic membrane made from group-III
nitrides grown on n-type silicon (111). The origin of the layer sequence is
explained in the main text. (b) Top-view SEM image of the photonic membrane
that mainly consists of c-plane GaN. The a\- and m-plane crystal directions
are indicated. The dashed red rectangle along with the sketch on the right
indicates the freestanding part of the photonic membrane in one of its
corners, which extends up to a widths of $\approx$ 8 $\mu$m. (c) Top-view SEM
image of the photonic membrane illustrating the underetch, the roughened
silicon substrate, as also the surface and the sidewalls of the photonic GaN
membrane. (d) Top-view light microscope image of the same corner region of the
photonic membrane. Here, the color fluctuations indicate thickness
fluctuations of the photonic membrane. (e) Simplified sketch of the
experimental two-laser setup that enables micro-photoluminescence spectroscopy
(optical characterization) in parallel to one-laser- and two-laser-Raman
spectroscopy (thermal characterization).

The overall structure of the photonic membrane is introduced in Figs. 1(a) -
1(d). In addition, a simplified sketch of the experimental setup that enables
$\mu\text{PL}$ spectroscopy (optical characterization) in parallel to one-
laser- and two-laser-Raman spectroscopy (thermal characterization) is given in
Fig. 1(e). A more detailed drawing and explanation of the experimental setup
can be found in S-Sec. I.

###  I.1 The photonic membrane

The layer stack that forms the basis for the photonic membrane is made from
c-plane, wurtzite AlN, GaN, and InxGa1-xN ($x=0.15$) grown on an n-type (111)
silicon substrate as sketched in Fig. 1(a). Such material growth on silicon
(111) is key to the formation of a membrane structure, as silicon can be
selectively etched. The group-III nitride material was grown by metal-organic
vapor phase epitaxy (MOVPE) in a horizontal Aixtron® 200/4 RF-S low pressure
reactor and details regarding the growth of such epilayers on silicon (111)
can be found in Ref. Triviño (2015). First, 50 nm of c-plane AlN were
deposited on silicon (111) to avoid melt-back etching Krost and Dadgar (2002).
Afterwards, 110 nm of GaN, the 3-nm-thick InxGa1-xN ($x=0.15$) QW, and finally
another 90 nm of GaN were grown. Here, the QW is deliberately placed 10 nm
above the geometric center of the surrounding GaN, taking into account the
lower refractive index of AlN Rousseau (2018). This way, the coupling of this
built-in light source to the fundamental electromagnetic mode is enhanced. The
overall membrane thickness of $\approx\,250\,\text{nm}$ is larger than
required for single electromagnetic mode operation of this membrane, which is
of relevance as soon as, e.g., a photonic crystal is processed for the blue
spectral range Rousseau (2018). However, this represents a trade-off, aiming
to achieve the smoothest possible GaN surface without pits or cracks. The
entire stack of III-nitride material is not intentionally doped and the GaN
material exhibits a free carrier concentration of
$3\,\times\,10^{16}\,\text{cm}^{-3}$. Detailed secondary ion mass spectroscopy
of the photonic membrane can be found in Ref. Rousseau (2018), showing that
the main impurities in our photonic membrane are silicon and carbon with
concentrations of $\approx\,3\times\,10^{16}\,\text{cm}^{-3}$ and
$\approx\,5\times\,10^{16}\,\text{cm}^{-3}$.

The photonic motivation for smooth surfaces Rousseau  _et al._ (2017) is also
beneficial for the in-plane thermal transport that is studied in this work. To
quantify light scattering losses in group-III nitride photonic crystals in the
blue spectral range, Rousseau et al. Rousseau  _et al._ (2017) have conducted
detailed atomic force microscopy (AFM) on the front- and backside of precisely
the membrane that is studied in the present work. After all required
processing steps to form the photonic membrane, they determined a GaN rout
mean square (RMS) top surface roughness of 1.4 nm, whereas the AlN backside
exhibited an RMS roughness of 0.6 nm, which was extracted from 500 x 500
$\text{nm}^{2}$ AFM scans. The backside roughness of the nitrogen-polar AlN
was accessed by exfoliation and subsequent flipping of a large
($\approx\,80\,\times\,80\,\mu\,\text{m}^{2}$) square of an entire membrane
similar to the GaN pad shown in Fig. 1(b). In addition to this surface
roughness, a minor waviness was observed for the particular membrane in
$25\,\times\,25\,\mu\text{m}^{2}$ AFM scans, which yields an RMS waviness of
4.9 nm and $\approx$ 30 nm peak-to valley height variation over distances of
5-10 $\mu\text{m}$. These height variations lead to the color contrast shown
in Fig. 1(d) due to thin film interference. In addition, the AFM scans on the
AlN surface on the backside of the photonic membrane yielded a high density of
pin pricks forming dips that correspond to dislocations with a density of
$3\,-\,4\,\times\,10^{10}\,\text{cm}^{2}$, which is a typical value for AlN
grown on (111) silicon Eric Feltin (2003). Similar AFM scans of the top GaN
yield the same range for the dislocation density, which corresponds to a
distance of $50\,-\,60\,\text{nm}$ between these defects. For the sake of
simplicity, in the following the epilayer from Fig. 1(a) will be described as
the GaN pad (with silicon support) or membrane (without silicon support), as
its main constituent GaN will dominate the thermal properties that lie in the
scope of this work.

The processing that leads to the GaN pad structure illustrated in the
secondary electron microscopy (SEM) image from Fig. 1(b) is based on electron
beam lithography, dry etching, and chemically selective vapor phase etching.
First, hydrogen silsesquioxane is applied as negative tone resist for the
electron beam lithography, which also acts as a hard mask, enabling the
transfer of written patterns into the underlying epilayer via dry etching.
Exactly during this pattern transfer, any potential edge roughness of the etch
mask is transferred to the underlying epilayer. As no apparent edge roughness
can be observed within the spatial resolution limits of the applied electron
microscope, only an upper bound for the RMS sidewall roughness of
$\lesssim\,5\,\text{nm}$ can be approximated. It should be noted that this
sidewall roughness does not represent a general limit as even atomically flat
sidewalls can be achieved by anisotropical wet etching of GaN by
tetramethylammonium hydroxide Palacios  _et al._ (2000). To this end, the
corresponding sidewalls of any etched structure must be aligned to the
crystallographic axes as it is the case for the present GaN pad, cf. Fig.
1(b). The edge region of this GaN pad is then released from the silicon
substrate over $\approx\,8\,\mu\text{m}$ by chemically selective vapor phase
etching using XeF2. The resulting structural situation is sketched in Fig.
1(b) and further detailed by the SEM image from Fig. 1(c). Additional details
regarding the sample growth and processing can be found in Ref. Rousseau
(2018) and the Supplemental Material of Ref. Rousseau  _et al._ (2017),
reporting on high quality one-dimensional photonic crystal cavities based on
the epilayer in use for the present work. For comparison, we also perform 1LRT
measurements on a freestanding piece of a state-of-the-art c-plane GaN wafer
with a dislocation density $\sim\,10^{6}\,\text{cm}^{-2}$, which already
formed the basis for detailed optical studies on InGaN-based light-emitting
diodes Haller  _et al._ (2017).

###  I.2 The customized optical setup

Figure 1(e) introduces the spectroscopic setup that enables spatially resolved
$\mu\text{PL}$ and $\mu$Raman mapscans by the independent displacement of two
laser spots on the surface of the epilayer. Whereas the basic features of this
setup are described in the following, its full description is given in S-Sec.
I. First, an ultraviolet (UV) laser that provides above bandgap excitation of
GaN (purple), is guided towards the sample via a suitable beamsplitter and a
microscope objective. We use either a 266 nm or a 325 nm continuous wave (cw)
laser for this purpose. Both lasers can then directly be used as excitation
lasers for $\mu\text{PL}$ spectroscopy. The excitation spot of both lasers can
be translated across the surface of the GaN pad over a range of
$\approx\,100\,\mu\text{m}$ by actuating the closed-loop X/Y-piezo stage
labeled as scanner I in Fig. 1(e). In addition, coarser movements of the
entire sample structure are enabled by a long-range X/Y/Z-piezostage (not
shown) that holds the sample in a cryostat, where it is located at a base
pressure of $\approx\,1\times\,10^{-6}\,\text{mbar}$. Even though the cooling
option of this cryostat with its built-in X/Y/Z-piezostage is not in use for
the present work, it still acts as the heat-sink for then entire sample,
keeping it at an ambient temperature of $295\,\pm\,1\,\text{K}$. Upon UV
excitation, the resulting $\mu\text{PL}$ signal is guided backwards through
the microscope objective towards the detection system comprising a suitable
monochromator and charge-coupled device (CCD) detector with a spectral
resolution of $\approx\,1\,cm^{-1}$ in the visible region around 532 nm.

Resonant Raman spectra based on one of the UV lasers (266 nm or 325 nm) are
measured in the same manner. Here, varying the power of these lasers gives
direct access to Raman thermometry based on one laser (1LRT), which originates
the labeling of these lasers as "heating lasers". However, low signal strength
in addition to the underlying $\mu\text{PL}$ signal arising from various
optical transitions in GaN, can significantly challenge the data recording as
further detailed in Sec. III.1. In addition, non-resonant Raman mapscans of
the epilayer can be recorded based on two visible (VIS) lasers (488 nm and 532
nm), which we describe as probe lasers. Therefore, any movement of the visible
laser spot can either be achieved by scanner I or the angular tilting mirror
(scanner II) as soon as the two scanning lenses are part of the beampath, cf.
Fig. 1(e). These lenses are present to ensure that the visible laser beam
enters the objective in its center, and the angle between the beam and the
microscope axis creates the beam displacement on the sample surface.
Especially the scanning option based on scanner II is of high importance for
2LRT measurements. Such measurements can be best described by the following
four measurements steps that form the basis for Secs. II \- V:

Step A - $\mu$PL spectra & positioning the heating laser (Sec. II): The
positioning of the heating laser based on $\mu\text{PL}$ mapscans yields a
spatially and spectrally resolved intensity distribution based on scanner I
[Figs. 2(a) & 2(b)]. During such measurements the spot size of the heating
laser is determined as described in S-Sec. I. As the spot size of the heating
laser is a crucial parameter for the subsequent data analysis, we made it
common practice to probe the laser spot size for every sequence of
measurements. Small laser focus deviations can lead to non-tolerable laser
spot size variations that scale with the numerical aperture of the microscope
objective in use. Commonly, after positioning the heating laser on the desired
sample location, excitation power-dependent series of $\mu\text{PL}$ spectra
are recorded as a first optical and thermal characterization of the sample.
For this step it is crucial that both scanning lenses illustrated in Fig. 1(e)
are already in place in preparation of step B. Measuring series of
$\mu\text{PL}$ spectra also supports the determination of the maximal
applicable excitation power range for the heating laser. For this, the
reproducibility of the $\mu\text{PL}$ spectra is checked after cycling the
excitation power of the heating laser. The positioning and focal lengths of
the two scanning lenses are such that both lenses do not impact the heating
laser besides minor wavefront distortions. See S-Sec. I in the SM for further
details.

Step B - Raman pre-characterization & positioning the probe laser (Sec. III):
While the position of the heating laser’s focus spot is kept on the sample
(fixed position for scanner I), the probe laser’s focus spot is scanned across
the sample by scanner II in combination with the scanning lenses, which is the
normal operation mode of a laser scanning microscope Stock (2010). For the
scanning, only the probe laser beam is tilted in the entrance aperture of the
microscope objective. During the scanning action of this laser, non-resonant
Raman spectra are recorded for each desired point on the sample. Due to the
presence of the angular tilting mirror, the detection beam path for such non-
resonant Raman spectra must deviate from the detection beam path for resonant
Raman and $\mu\text{PL}$ spectra described under step A, cf. S-Sec. I. As a
result, a mapscan of non-resonant Raman spectra is recorded, which, e.g., can
illustrate the distribution of strain across the sample’s surface. As the
heating laser is still off during this recording, we call this result an
unheated Raman mapscan (Fig. 3a).

Step C - 1LRT characterization & 2LRT0 (Sec. IV): After positioning the
heating and probe lasers, conventional resonant Raman thermometry (1LRT) is
performed. For such measurements, the heating laser simultaneously acts as the
temperature probe laser, meaning that no spatially resolved temperature maps
can be recorded. Thus, the heating laser is the source of the resonant Raman
signal, which serves as a local thermometer. In this work, we show 1LRT
results as a first step towards 2LRT measurements, since such measurements
commonly appear in literature. Later in the paper (Sec. VII), we will show
that such temperature measurements directly at the heat spot imply several
complications for our photonic membrane. In addition to the 1LRT
characterization and based on the non-resonant Raman scan from step B and the
$\mu\text{PL}$ mapscan from step A, it is now possible to overlap the spots of
the heating and probe lasers. As a result, non-resonant Raman spectra can be
recorded, while the heating power is varied. We call this technique two-laser
Raman thermometry without spatial displacement (2LRT0).

Step D - 2LRT mapscans (Sec. V): Spatial resolution can now be introduced by
controlling scanner II, which exclusively allows movements of the probe but
not the heating laser spot across the sample in our setup. A well-suited UV-
VIS microscope objective must be utilized, featuring the same focal length for
two significantly different laser wavelengths and a sufficient planarity of
the corresponding focal planes. As a result, 2LRT mapscans can be recorded for
varying powers of the heating laser. Here, the unheated Raman mapscan can be
used to free the heated mapscans from any Raman mode shifts and mode
broadenings induced by, e.g., strain or varying defect concentrations as
further explained in Sec. V. Finally, 2LRT mapscans comprising Raman mode
shifts or Raman mode broadenings can be translated into two-dimensional
temperature maps based on a particular temperature calibration, cf. S-Sec.
III. These temperature maps will then serve as the basis for the extraction of
key parameters like the thermal conductivity $\kappa$.

##  II Characterization of the photonic membrane by $\mu\text{PL}$
spectroscopy & positioning of the heating laser (step A)

Figure 2: Characterization of the photonic membrane by $\mu\text{PL}$
spectroscopy at an ambient temperature $T_{amb}=\,295\,\text{K}$ under
excitation with a cw UV laser with a wavelength $\lambda_{exc}=266\,\text{nm}$
at a power $P_{exc}\,=\,0.8\,\text{mW}$. (a) Intensity distribution of the QW
emission centered around $E_{1}=2.75\,\text{eV}$ with a bandpass $\Delta
E_{1}=0.25\,\text{eV}$, aiming to approximate the FWHM of the QW emission.
Here, monolayer thickness and indium content fluctuations lead to intensity
variations, which prominently appear as localization centers in the mapscan.
(b) In contrast, the GaN bandedge emission at around $E_{2}=3.41\,\text{eV}$
shows a more homogeneous intensity distribution in the region of the
freestanding GaN membrane with a bandpass matching its FWHM ($\Delta
E_{2}=0.14\,\text{eV}$). Here, the out-coupled signal strength only
significantly rises in the top left corner, where the underlying silicon
support structure begins, cf. Fig. 1(b). The target cross in (a) and (b)
indicates the final desired position of the heating laser. (c) Comparison of
two local $\mu\text{PL}$ spectra recorded at the color-matched indicators in
(a) and (b).

Figure 2 shows the $\mu$PL signal of the III-nitride epilayer introduced in
Fig. 1, which is required for step A. A UV laser
($\lambda_{exc}\,=\,266\,\text{nm}$), later described as the heating laser, is
scanned over a corner region of the GaN pad and a $\mu$PL spectrum is recorded
for each point, yielding polychromatic $\mu$PL mapscans as shown in Figs. 2(a)
and 2(b) for the InxGa1-xN ($x=0.15$) QW emission (bandpass centered at
$E_{1}\,=\,2.75\,\text{eV}$, full width at half maximum (FWHM) $\Delta
E_{1}\,=0.25\,\text{eV}$) and the GaN bandedge emission
($E_{2}\,=\,3.41\,\text{eV}$, $\Delta E_{1}\,=0.14\,\text{eV}$). Two exemplary
local $\mu$PL spectra are shown in Fig. 2(c) for the two positions 1
(supported on silicon) and 2 (freestanding) identified in Fig. 2(a) and (b).
In addition to the two main emission features (QW and bandedge emission), the
onset of GaN-related defect luminescence is visible below
$\sim\,2.4\,\text{eV}$, which is of relevance for the 2LRT measurements
introduced in Sec. V.

When comparing the intensity fluctuations from Fig. 2(a) and 2(b), it is
apparent that the QW emission is less homogeneously distributed across the
freestanding part of the GaN membrane compared to the GaN bandedge emission.
For the QW intensity distribution, numerous, so-called, localization centers
Schömig  _et al._ (2004) with higher emission intensity can be observed, which
often can also be correlated to emission energy shifts (not shown). Here,
local QW thickness fluctuations, structural defects (e.g., local assemblies of
dislocations), and interrelated indium content fluctuations can lead to
carrier localization Schömig  _et al._ (2004); Hangleiter  _et al._ (2005);
Piccardo  _et al._ (2017); Callsen  _et al._ (2019). For instance, a QW
thickness fluctuation of $\pm\,1$ monolayer can already lead to emission
energy shifts of $\approx\pm\,60\,-\,70\,\text{meV}$ for the given indium
concentration Bai  _et al._ (2000) at room temperature and low carrier
injection conditions that are met in Fig. 2. Clearly, such thickness, indium
content, and defect concentration fluctuations naturally occur during growth
of III-nitride epilayers on silicon (111), cf. Sec. I.1. Details regarding
these carrier localization effects that are also linked to the high luminosity
of this photonic membrane, despite the apparent structural defects, can, e.g.,
be found in Ref. Rousseau (2018).

The GaN bandedge emission from Fig. 2(b) shows a comparably homogeneous
intensity distribution as the GaN thickness of $\approx\,200\,\text{nm}$ does
not lead to any carrier confinement that would make thickness fluctuations a
pronounced source of intensity variations. At the given ambient temperature
$T_{amb}\,=\,295\,\text{K}$, both, excitons and band-to-band transitions
already contribute to the bandedge emission of GaN as the thermal energy
($\approx\,26\,\text{meV}$) has already surpassed the binding energy of the
free A-exciton and B-exciton of $\approx\,23\,\text{meV}$ Callsen  _et al._
(2018). Here, the mapscans shown in Figs. 2(a) and 2(b) already enable a first
estimation of an upper bound for the effective mean exciton and carrier
diffusion length $l_{\text{diff}}\,\leq\,250\,\text{nm}$, as features with the
size of individual pixels appear in the mapscan. Spatially resolved
cathodoluminescence (CL) mapscans with higher spatial resolution recorded
exactly in the corner region of the GaN membrane for low injection conditions
are discussed in S-Sec. II. Here, an upper bound of
$l_{\text{diff}}\lesssim\,115\,\text{nm}$ is derived, which will be of
relevance for the interpretation of the thermal analyses discussed in Secs.
VII and VIII.

The two local spectra 1 and 2 shown in Fig. 2(c) illustrate the difference
between the supported epilayer (1) and the freestanding membrane (2).The
emission intensity for these locations differs by approximately a factor of
two. Since the underetching process (see Sec. I.1) is selective to the AlN
layer, meaning that the InGaN/GaN layer stack remains unaffected, the
reduction of intensity cannot be attributed to material damage. The increase
in intensity here is solely associated to the difference in refractive index
contrast between AlN and silicon (111) or vacuum. In addition, this
observation explains the motivation for a freestanding photonic membrane that
requires most light to travel along in-plane directions for commonly applied
one- Triviño  _et al._ (2015) and two-dimensional Vico Triviño  _et al._
(2012) cavity designs. Less apparently, but highlighted in Fig. 2(c), the QW
emission and the bandedge emission exhibit an energy shift of
$\approx\,10\,\pm\,3\,\text{meV}$ (QW, limited by the FWHM) and
$4\,\pm\,1\text{meV}$ (bandedge), when comparing positions 1 and 2. This is a
sign of tensile strain in the epilayer grown on silicon (111) due to lattice
mismatch Feltin  _et al._ (2001), which is released upon formation of the
photonic membrane. The strain state of our III-nitride epilayer is further
quantified based on $\mu$Raman spectroscopy in Sec. III.

After the $\mu$PL mapscan is recorded, the UV heating laser can be positioned
at any desired position. We choose the position that is indicated by the
purple target in Figs. 2(a) and 2(b), while the positioning precision is given
by the stepping size (here 0.5 $\mu$m). Once the first of the required two
lasers for 2LRT0 and 2LRT measurements has reached its position, the power can
be varied to induce heating in the sample that will be probed by the second
laser. Subsequent to this, the temperature probe laser needs to be positioned
(step B) by recording an unheated, non-resonant Raman map as described in the
following Sec. III.

##  III Recording of unheated, non-resonant Raman maps & positioning of the
probe laser (step B)

Figure 3: (a) Raman mapscan based on non-resonant Raman spectra, showing the
energetic position of the $E_{2}^{\text{high}}$ Raman mode of wurtzite GaN for
the corner region of the GaN pad. Two local non-resonant Raman spectra for the
positions 1 and 2 are shown in (b). Here, a small $E_{2}^{\text{high}}$ mode
shift occurs due to the strain release in the freestanding part of the
epilayer. Please note that for experimental reasons, this Raman mapscan is
rotated by 135∘ in comparison to the $\mu\text{PL}$ mapscans from Figs. 2(a)
and 2(b), showing exactly the same corner of the GaN pad. (c) Three Raman
spectra of GaN, from bottom to top: non-resonant Raman spectrum of bulk GaN
(black), equivalent non-resonant spectrum for the GaN pad (blue, position 2),
and resonant Raman spectrum for the same GaN pad (purple, position 2). Any
laser-induced heating was excluded for all Raman spectra by measuring at
sufficiently low laser powers. The Raman mode of CaF in the resonant Raman
spectrum (purple) originates from the microscope objective in use.

A $\mu$Raman mapscan based on non-resonant Raman spectra
($\lambda_{\text{probe}}=488\,\text{nm}$) is shown in Fig. 3(a). Here, exactly
the same corner of the GaN membrane that was already probed by $\mu$PL is now
analyzed by $\mu$Raman spectroscopy, cf. Fig. 2. As a result, control over the
spatial displacement in between the laser spots of the heating and the probe
laser with respect to the position of the structured GaN membrane (corner) is
achieved (step B), representing a prerequisite for the Raman thermometry
presented in Secs. IV and V (steps C and D). To form the mapscan from Fig.
3(a), first, non-resonant Raman spectra are recorded with a step resolution of
650$\,\pm\,50\,\text{nm}$ without any effect on the positioning of the
disengaged heating laser that was achieved via the $\mu$PL mapscan from Fig.
2. Second, the position of the $E_{2}^{\text{high}}$ Raman mode of wurtzite
GaN is extracted from the entire set of non-resonant Raman spectra. Plotting
these Raman mode positions over the spatial coordinates lead to Fig. 3(a).
Here, again similar to Figs. 2(a) and 2(b), the freestanding and supported
part of the GaN pad can clearly be distinguished as growth on silicon (111)
leads to the formation of tensile strain in the entire epilayer, responsible
for the additional Raman shift.

Two local spectra were recorded to illustrate the shift, at positions 1
(supported) and 2 (freestanding) as indicated in Figs. 3(a) and 3(b). Here, in
addition to a strong contribution from the silicon underneath the GaN pad, the
$E_{2}^{\text{high}}$ Raman mode of GaN can be seen for the freestanding and
supported part of the epilayer. Based on fitting such spectra with a suitable
model, a small mode shift of $0.8\,\pm\,0.2\,\text{cm}^{-1}$ can be extracted,
cf. Fig. 3(b). For such fitting, we employed a Lorentzian peak with an
exponential background. Such an exponential background arises from the tail of
the silicon Raman mode visible in Fig. 3(b).

In general, growth of GaN on silicon (111) leads to the build-up of tensile
strain in the epilayer due to lattice mismatch, which can additionally be
affected by the incorporation of AlN interlayers that also allow lowering the
dislocation density Feltin  _et al._ (2001). As such dislocations are a source
of relaxation, the final tensile strain state of the epilayer is given by the
balance between layer thickness and the dislocation density. Based on the
bisotropic (i.e, biaxial and isotropic) phonon pressure coefficient of the
$E_{2}^{\text{high}}$ Raman mode (2.86 cm-1/GPa) Callsen  _et al._ (2011), we
find a minor stress release in our epilayer upon removal of the silicon (111)
substrate of $\approx\,0.3\,\pm\,0.1\,\text{GPa}$. This minor stress release
maintains the overall quality of the GaN membrane as, e.g., no cracks or
additional structural defects are formed upon release. However, even though
the assumption of a bisotropic strain situation is questioned by the
particular geometry of the GaN pad corner, the value for the stress release
still holds as an approximation. Only minor strain variations that are induced
by the corner geometry of the freestanding GaN membrane can be observed in
Fig. 3(a). Please note that the color scale chosen in Fig. 3(a) enables direct
comparison with 2LRT mapscans introduced in Sec. V.

###  III.1 Non-resonant and resonant Raman spectra of GaN

Three different non-resonant and resonant Raman spectra of bulk, c-plane GaN
and the c-plane GaN membrane are shown in Fig. 3(c) on a larger energy range
that covers all first order Raman modes of GaN except for the longitudinal-
optical (LO) modes. For a bulk piece of c-plane GaN (bottom, black spectrum),
the Raman selection rules Gil (2014) are obeyed as the $E_{2}^{\text{low}}$
and $E_{2}^{\text{high}}$ Raman modes dominate this spectrum (allowed modes)
in the chosen non-polarized, backscattering geometry, cf. Fig. 1(e). Thus,
only minor traces of the transversal-optical (TO) Raman modes $A_{1}(TO)$ and
$E_{1}(TO)$ can be observed on the lower wavenumber side of the
$E_{2}^{\text{high}}$ Raman mode in addition to a broad background that arises
from a defect-related $\mu$PL signal arising from sub-bandgap optical
excitation. Analyzing the GaN membrane in position 2, i.e., in the
freestanding part of the GaN epilayer, yields the blue and purple spectra
shown in Fig. 3(c). Despite of the large airgap ($\geq\,8\,\mu\text{m}$), for
the middle spectrum, the Raman signal of silicon dominates the signal, while
only the $E_{2}^{\text{high}}$ Raman mode of GaN remains visible as again the
$\mu$PL background signal limits the dynamic range of the entire Raman
spectrum, inhibiting the visibility of the $E_{2}^{\text{low}}$ Raman mode.
Thus, only the $E_{2}^{\text{high}}$ Raman mode remains for Raman thermometry
based on non-resonant spectra. In general, this situation can be improved by
using a microscope objective with a numerical aperture (NA) in excess of 0.55.
However, this choice is often not compatible with 2LRT thermometry employing
UV heating lasers Rousseau (2018). For 2LRT measurements, a dedicated UV-VIS
microscope objective is required with a sufficiently large working distance
for measurements in a vacuum chamber or cryostat to exclude thermal transport
by convection. Furthermore, the focal planes for the two selected wavelengths
need to overlap, while providing a sufficiently low curvature for the position
mapping schemes for the probe laser spot as described in Sec. I.2.
Consequently, all these experimental requirements hinder the utilization of
microscope objectives with higher NA and consequently lower depth of field in
our experimental setup.

A similar situation regarding the balance between Raman and $\mu$PL signals
occurs for the resonant Raman spectrum shown in the top of Fig. 3(c) in
purple. As most of the probe light ($\lambda_{probe}\,=\,325\,\text{nm}$) is
absorbed in the $\approx\,250\text{-nm-thick}$ membrane (light penetration
depth $p^{325\,\text{nm}}_{\text{GaN}}\,=\,74\,\text{nm}$ Kawashima  _et al._
(1997)), the Raman signal of silicon is reduced when switching from the 488 nm
laser (non-resonant Raman) to the 325 nm laser (resonant Raman). Please note
that every value for $p_{\text{GaN}}$ given in this paper is based on
spectroscopic ellipsometry. For this method, a suitable dispersion model is
fitted to experimental data and we estimate the corresponding error of
$p_{\text{GaN}}$ in the wavelength range of interest with $\pm\,5\%$. A strong
luminescence background occurs, which is caused by band-to-band transitions in
GaN and additional contributions belonging to the resonant Raman spectrum of
GaN (e.g., LO Raman modes and multiple combinations of them). Therefore,
resonant Raman spectroscopy on GaN represents a challenge, as often Raman
modes are covered by inadvertent signal. In addition, the integration time of
the resonant Raman spectrum (purple spectrum), surpasses the integration time
of its non-resonant counterpart (red spectrum) in Fig. 3(c) by a factor of
$\approx\,100$. Clearly, this renders such non-resonant Raman spectra more
promising for mapping applications, while also any accidental laser heating is
less likely due to the high transparency of the entire GaN membrane at the
laser probe wavelength used in this study, cf. Sec. I.1.

##  IV One- and two-laser Raman thermometry (step C)

After full positioning control is achieved for the probe and heat laser spots
in regard to the corner of the GaN membrane based on steps A and B from Sec.
II and III, the 1LRT and 2LRT0 measurements are performed at the same sample
location. The position of the probe and heat laser spots coincides for 2LRT0
measurements. In the spirit of a step-by-step approach, the 2LRT0 measurements
serve first as a direct comparison to the common 1LRT technique, before the
introduction of spatial resolution through 2LRT measurements presented in Sec.
V.

Generally, Raman spectra enable at least three methods to measure the
temperatures that are required for Raman thermometry. One can either measure
i) the Raman mode shift, ii) the Raman mode broadening, or iii) the intensity
ratio of Stokes and Anti-Stokes Raman modes upon variations of the power of
the heating laser. As approach iii) typically results in uncertainties of the
measured temperature that are significantly larger compared to i) and ii) as
described in Refs. Beechem  _et al._ (2008, 2015), we will discard an analysis
of the Stokes/Anti-Stokes intensity ratios in this work. In general, an
analysis of these Stokes/Anti-Stokes intensity ratios is not particularly
promising to map temperatures, as we found that the required integration times
exceed the integration times of the Raman spectra that exclusively focus on
the Stokes side by a factor of $\sim\,100$ in our experimental setup around
$T_{amb}$.

###  IV.1 One-laser Raman thermometry on GaN

Figure 4: (a) Scheme of the 1LRT measurements showing the position of the
heating laser spot (purple) on the corner of the GaN membrane. (b) Resonant
Raman spectra showing the $E_{2}^{\text{high}}$ Raman mode of GaN with rising
power of the heating laser ($\lambda_{heat}=\lambda_{probe}=325\,\text{nm}$)
at an ambient temperature $T_{amb}\,=\,295\,\text{K}$. (c) Temperature rise
induced by the heating laser plotted over its normalized absorbed power
($P_{abs}$). The reasoning behind this normalization of $P_{abs}$ is given in
the main text. Here, the Raman mode shift and its broadening yield different
temperature rises ($T_{rise}$) due to the build-up of thermally induced strain
in the GaN membrane. For comparison, the corresponding $T_{rise}$ values for
bulk GaN are also shown. The dashed green line illustrates our modeling for
$T_{rise}\,\lesssim\,150\,\text{K}$ based on a thermal conductivity
$\kappa^{\,1LRT}_{\,\,memb}\,=\,180_{-7}^{+12}\,$$\frac{\text{W}}{\text{m}\cdot\text{K}}$
. For more elevated powers of the heating laser, both temperature rises follow
parabolas, as $\kappa$ continuously decreases. The dashed horizontal and
vertical lines enable the comparison to the 2LRT0 data, cf. Fig. 5

One-laser Raman thermometry is based on resonant Raman spectroscopy. The
heating laser spot is positioned at the target position in the center of the
freestanding GaN pad corner as sketched in Fig. 4(a) and the wavelength
utilized for heating and temperature probing is the same
($\lambda_{heat}=\lambda_{probe}=325\,\text{nm}$). Resulting resonant Raman
spectra are displayed in Fig. 4(b) for rising absorbed laser power
$P_{\text{abs}}$. The $E_{2}^{\text{high}}$ Raman mode continuously shifts and
broadens towards lower wavenumbers, indicating a local, laser-induced heating.
The corresponding Raman mode positions and FWHM values are extracted by
fitting the peaks with a Lorentzian peak over a linear background, as
illustrated for three $P_{\text{abs}}$ values in Fig. 4(b) based on overlaying
the experimental result (datapoints) with a solid line (fitting model).
Position and FWHM values related to the $E_{2}^{\text{high}}$ Raman mode can
then be translated into local temperatures based on an equally local
temperature calibration. Details regarding this local temperature calibration
are given in S-Sec. III. The resulting temperature rise ($T_{rise}$) values,
either based on Raman mode position or FWHM values, are plotted in Fig. 4(c).
Here, the horizontal axis represents $P_{abs}$ normalized by the product of
the heat spot radius $r_{\text{abs}}$ and the temperature probe volume to
enable direct comparison to the 2LRT0 data presented in Fig. 5 from Sec. IV.2,
where we employ a heating laser with $\lambda_{heat}\,=\,266\,\text{nm}$. The
additional normalization by $r_{\text{abs}}$ originates from the approximation
that $T_{rise}\sim\frac{P_{\text{abs}}}{r_{\text{abs}}}$ holds Lax (1977). The
spot sizes of the probe and heat lasers are determined via the knife edge
method (S-Sec. I A), and the probe volume is approximated by the product of
the probe laser spot size and its penetration depth.

Up to $T_{rise}\,\approx\,150\,\text{K}$, the temperature rises originating
from Raman mode positions and FWHM values overlap within the error bars.
However, upon further increase in laser power, the FWHM of the
$E_{2}^{\text{high}}$ Raman mode systematically indicates higher $T_{rise}$
values. As described by Beechem et al. Beechem  _et al._ (2008), such
discrepancy is caused by the built-up of thermally induced strain in the
epilayer due to the local laser heating. As the heated volume tends to expand
in the middle of the corner of the GaN membrane, the surrounding colder
material hinders this expansion, mostly, for in-plane directions. The result
is thermally induced compression in the plane of the GaN membrane around the
heat spot, which is accompanied by a shift of the $E_{2}^{\text{high}}$ Raman
mode to higher wavenumbers Callsen  _et al._ (2011), thus, counteracting its
evolution towards lower wavenumbers due to laser-induced local heating. The
temperature induced evolution of a Raman mode is not only caused by a volume
effect (i.e. the thermal expansion), but also by phonon-phonon scattering
events Li  _et al._ (2000) that scale with the phonon population. Eventually,
the Raman mode position tends to become a poor temperature sensor for
$T_{rise}\,>\,150\,\text{K}$ in GaN at $T_{amb}\,=\,295\,\text{K}$, while the
FWHM values are not affected by thermally-induced stress as a first order
approximation Beechem  _et al._ (2008); Callsen  _et al._ (2011). The
quantification of the stress that is thermally induced by the local laser
heating for $T_{rise}\,>\,150\,\text{K}$ of the GaN membrane remains a task
for future work.

Consequently, $T_{rise}$ values extracted from the evolution of the FWHM
values of the $E_{2}^{\text{high}}$ Raman mode are best suited for determining
the thermal conductivity of the GaN membrane $\kappa^{\,1LRT}_{\,\,memb}$. For
the given normalization of $P_{abs}$ in Fig. 4(c), the evolution of $T_{rise}$
should follow a linear function for a fixed $\kappa^{\,1LRT}_{\,\,memb}$
value. However, for $T_{rise}\,>\,150\,\text{K}$ the temperature trend follows
a parabola, indicating a reduction of $\kappa^{\,1LRT}_{\,\,memb}$ with
increasing $T_{rise}$ at the given $T_{amb}\,=\,295\,\text{K}$. Commonly, it
is assumed that such an evolution of $\kappa^{\,1LRT}_{\,\,memb}$ can be
described by a power law like $\kappa(T)=aT^{-b}$ Graczykowski  _et al._
(2017) with the fitting parameters $a$ and $b$. This approach has already
allowed the description of the decrease of the thermal conductivity $\kappa$
with increasing temperature in bulk silicon, bulk germanium Glassbrenner and
Slack (1964), and silicon membranes Graczykowski  _et al._ (2017).

However, in this work, we will not focus on this particular dependence of
$\kappa(T)$. The reasoning for this is mostly related to the determination of
the two fitting parameters $a$ and $b$. These need to be obtained from
extended numerical simulations and fitting efforts beyond the scope of this
work, as we do consider a more complex sample geometry, i.e., the corner of a
GaN membrane and will constitute the basis of further investigations. In this
work, we focus on the low $T_{rise}$ limit
($T_{rise}\,\lesssim\,150\,\text{K}$) to approximate a constant thermal
conductivity $\kappa_{\,\,memb}$ for our GaN membrane, which we derive based
on 1LRT and 2LRT0, as well as 2LRT measurements. Already the comparison of
these three values for $\kappa_{\,\,memb}$ bears interesting physical insight
as described in Sec. VII.

####  IV.1.1 Determination of the thermal conductivity

As no straightforwardly accessible analytical solution exists for modeling the
heating laser induced temperature distribution in the corner region of our GaN
membrane Carslaw and Jaeger (1959), we directly turn towards finite element
simulations based on the COMSOL MultiphysicsⓇ software, which were already
successfully applied to InP microcrystals with dimensions in the few- and sub-
micrometer regime Jaramillo-Fernandez  _et al._ (2018). We operate COMSOL
through scripts in MATLABⓇ using LiveLinkTM. In doing so, we enable the
analysis of a large amount of simulations and are able to compute errors for
all $\kappa$ values we derive. The procedure for extracting
$\kappa^{\,1LRT}_{\text{\,\,memb}}$ based on the 1LRT data from Fig. 4(c) is
described as follows. Details regarding the thermal model and the numerical
simulations can be found in S-Sec. IV.

Step I - Experimental data analysis: We start by fitting a linear function to
the datapoints from Fig. 4(c) for $T_{rise}\,\lesssim\,150\,\text{K}$. The
least square regression is restricted to this range of $T_{rise}$ values for
two reasons: First, here the mode shift and the mode broadening can be applied
as valid equivalent thermometers. Second, limiting the fitting to lower values
of $T_{rise}$ motivates the assumption of a constant $\kappa_{memb}$ value. As
a result, we obtain the slope of T${}_{rise}^{exp}$(P${}_{abs})$ in addition
to the symmetric slope error.

Step II - Modeling the experiment: The numerical simulation aims to model the
laser-induced heating, the thermal dissipation in the sample, and the
temperature probing by a laser. To this end, the entire GaN pad along its
silicon support is created inside of the simulation software with a size of
220 $\mu$m $\times$ 220 $\mu$m $\times$ 10.25 $\mu$m. The heating laser is
introduced as a surfacic Gaussian heating source with an integrated power
density equal to Pabs, while the bottom of the silicon support acts as the
heat sink remaining at ambient temperature (295 K). When comparing a surfacic
heating source with a volumetric heating source that considers the light
penetration depth of the heating laser ($p_{abs}$), we found a maximal
temperature deviation of less than 0.5%. For the temperature probe volume, we
define a cylindrical volume over which temperatures are averaged to
$T_{rise}$, according to Eq. 1. Here, the height of the cylinder in the cross-
plane direction is given by the light penetration depth of the probe laser
($p_{abs}$). More details are provided in S-Sec. IV. For the case of 1LRT
measurements ($\lambda_{heat}\,=\,\lambda_{probe}\,=\,325\,\text{nm}$ and
$p_{abs}\,=\,p_{probe}\,=\,74\,\text{nm}$) Kawashima  _et al._ (1997), the
heating and probe lasers are identical, hence, they are modeled with the same
parameters. For the in-plane direction $T_{rise}$ is obtained from a Gaussian-
weighted average of temperatures, considering the probe laser spot radius
($r_{probe}$) and its centering around the heat spot. Subsequently, the
simulation is performed for all experimental $P_{abs}$ values, yielding a
linear trends T${}_{rise}^{sim}$(P${}_{abs})$. In addition, these simulations
are repeated for a suitable range of $\kappa^{\,1LRT}_{\,\,memb}$. The spacing
between two values of $\kappa_{memb}^{1LRT}$ is mainly determined by the slope
error of the linear fit function obtained in step I.

Step III - Deducing the thermal conductivity: We deduce $\kappa_{memb}^{1LRT}$
by comparing the slopes of the simulated and measured trends for
Trise(P${}_{abs})$. Once the best matching simulation is found, the error of
$\kappa_{memb}^{1LRT}$ is determined via the experimental slope error obtained
under step I. For a given slope and slope error of
T${}_{rise}^{exp}$(P${}_{abs})$, one obtains the lower bound
($\kappa_{memb}^{1LRT}\,-\,\Delta\kappa_{1}$) and the upper bound
($\kappa_{memb}^{1LRT}\,+\,\Delta\kappa_{2}$) of the thermal conductivity with
$\Delta\kappa_{1}\neq\Delta\kappa_{2}$. Hence, we adopt the following notation
for asymmetric errors: $\kappa_{-\Delta\kappa_{1}}^{+\Delta\kappa_{2}}$ Barlow
(2006). These asymmetries on the error intervals can be understood based on
the standard analytical relation for bulk material that relates $T_{rise}$ to
the inverse of $\kappa^{\,1LRT}_{\,\,bulk}$ as follows:
$\frac{T_{rise}}{P_{abs}}\,\propto\,1/(r\,\cdot\,\kappa^{\,1LRT}_{\,\,bulk})$.
Here, the precise prefactor of this proportionality depends on the
experimental conditions as described by Lax Lax (1977).

As a result of our modeling procedure based on step I - III, we derive
$\kappa^{\,1LRT}_{\,\,memb}\,=\,180_{-7}^{+12}\,\text{W/mK}$, illustrated by
the linear function (dashed, green) shown in Fig. 4(c). This value of
$\kappa^{\,1LRT}_{\,\,memb}$ that we extracted here for our thin photonic
membrane, comprising around 200 nm of GaN and a 50-nm-thick AlN interlayer, is
highly doubtful. Only GaN samples of very high crystalline quality (low
structural and point defect concentrations) with a thickness of several
micrometers grown on bulk GaN substrates reach such high $\kappa$ values
Beechem  _et al._ (2016). A careful analysis regarding the reliability of
$\kappa^{\,1LRT}_{\,\,memb}$ follows in Sec. VII along with a comparison to
the results of 2LRT0 and 2LRT measurements. Here, we will reveal a systematic
problem of 1LRT measurements for the given heating and probe laser spot sizes
and laser penetration depths. As a precise knowledge of $P_{abs}$ is of high
importance for all 1LRT and 2LRT measurements, the following Sec. IV.1.2 is
dedicated to its determination. Furthermore, also the case of 1LRT
measurements on bulk GaN is briefly shown in Sec. IV.1.4, before 2LRT
measurements are introduced in Sec. IV.2, following a rigorous step-by-step
approach.

####  IV.1.2 Determination of the absorbed and the heating power

Determining a robust value for the power absorbed by the layer $P_{abs}$ and
its fraction that goes into heating $P_{heat}$ is of high relevance for the
Raman thermometry presented in this work. Generally, for the acquisition of
Raman spectra, we measure the power of the heating laser between the two
scanning lenses illustrated in Fig. 2(e). This position for the power
measurements proved most convenient during the operation of the experimental
setup. However, the power of the heating laser that reaches the surfaces of
our GaN pad can only be derived based on a precise knowledge of all losses
introduced by the optical elements in the beampath (lenses, mirrors,
microscope objective, cryostat window). Additionally, the power of the heating
laser that is reflected from the surface of the photonic membrane is
determined to provide the power of the heating laser entering into the sample.
In our experimental setup, the power of the reflected heating laser $P_{refl}$
can directly be measured for every step in power of the heating laser. We then
cross-checked this method by both direct, angular-dependent reflection
measurements at $\lambda_{heat}\,=\,325\,\text{nm}$ and $266\,\text{nm}$ in a
dedicated and calibrated UV-VIS spectrophotometer and by straightforward
calculations based on Fresnel equations. Even for elevated excitation powers
of the heating laser, we never observed any significant thermally induced
change in reflectivity. Any reflectivity changes we measured were always
directly linked to surface damage of the GaN material that only occurred at
excitation powers beyond the values presented in this work. Any power that is
attributed to elastic (Rayleigh scattering) and inelastic (Brillouin and Raman
scattering) light scattering $P_{scat}$ is neglected in this work as in
previous works on 2LRT Reparaz  _et al._ (2014); Graczykowski  _et al._
(2017). Commonly, inelastic scattering is orders of magnitude weaker than
elastic scattering, while the latter is strongly reduced in our sample due to
its monocrystallinity.

After the determination of the power entering the GaN pad, its transmitted
fraction must be estimated to find the absorbed laser power $P_{abs}$. For the
sake of simplicity, we treat the GaN/InGaN/GaN membrane as one effective GaN
membrane with a thickness of 203 nm (3-nm-thick InGaN QW). At such GaN layer
thicknesses, only 0.5% (325 nm) or 0.3% (266 nm) of the light of the heating
laser are transmitted through the membrane. Thus, even though the backside of
our sample is not experimentally accessible as in measurements on silicon
membranes Reparaz  _et al._ (2014), we can safely assume that the power of the
heating laser that enters the sample is also fully absorbed, enabling the
direct determination of $P_{abs}$.

Furthermore, it shall be noted that the absorption of GaN at 325 nm or 266 nm
is so high Muth  _et al._ (1997) that Fabry-Perot interference effects can be
neglected for our GaN membrane despite its sub-wavelength thickness. In
contrast, for 1LRT measurements on silicon membranes such interference effects
can complicate the calculation of the absorbed power, rendering direct
transmission measurements at the wavelength of the heating laser of high
importance Chávez-Ángel  _et al._ (2014). This is also especially relevant for
2LRT measurements on silicon membranes that use visible lasers for the heating
(405 nm) and the temperature probing (488 nm) as demonstrated in Ref. Reparaz
_et al._ (2014). At such wavelength in the visible spectral range, the
absorption of silicon does not suffice to suppress light propagation for in-
plane directions in silicon membranes. Depending on the thickness of such
silicon membranes, Fabry-Perot interference effects can occur that strongly
alter the overall membrane transmission. Such silicon membranes therefore
require precise transmission measurements when heated with visible lasers,
which in turn requires full optical access from the front- and backside of the
membrane. Alternatively, a suitable modeling can be applied Chávez-Ángel  _et
al._ (2014), however, this approach requires precise knowledge about the top
and bottom surface roughnesses.

Figure 5: (a) Scheme of the 2LRT0 measurements without a spatial displacement
in between the heating laser spot (purple) and the probe laser spot (green),
both positioned in the corner region of the GaN pad. (b) Non-resonant Raman
spectra showing the $E_{2}^{\text{high}}$ Raman mode of GaN with rising power
of the heating laser ($\lambda_{heat}\,=\,266\,\text{nm}$) at an ambient
temperature $T_{amb}\,=\,295\,\text{K}$. Here a probe laser wavelength
$\lambda_{probe}\,=\,488\,\text{nm}$ was used. (c) Temperature rise induced by
the heating laser plotted over normalized absorbed laser power ($P_{abs}$).
Here, the given normalization of $P_{abs}$ allows direct comparison of this
plot with Fig. 5(c). The Raman mode shift and its broadening yield different
temperature rises due to the build-up of thermally induced strain in the GaN
membrane. The dashed green line illustrates our modeling for
$T_{rise}\,\lesssim\,150\,\text{K}$ and a fixed thermal conductivity yielding
$\kappa^{\,2LRT_{0}}_{\,\,memb}\,=\,110_{-8}^{+11}\,\text{W/mK}$. For more
elevated powers of the heating laser, both temperature rises follow parabolas,
as $\kappa$ continuously decreases. The intersection of the dashed horizontal
and vertical lines enable the comparison to the 1LRT data, cf. Fig. 4.

The last step is to determine which fraction of $P_{abs}$ leads to a local
heating of the GaN membrane. In principle, one can assume that a certain
fraction of $P_{abs}$ leads to the formation of, e.g., excitons and/or
electron-hole pairs, which in turn can emit light with a radiated power
$P_{rad}$. Clearly, such light emission is one of the main aims of a photonic
membrane comprising a QW as built-in light source. In the best case scenario,
the external quantum efficiency $\eta_{EQE}$ of the GaN membrane would be
available to evaluate the following equality:
$P_{abs}\,=\,P_{heat}\,+\,P_{rad}$, with $P_{heat}$ being the fraction of the
absorbed power that went into heating.

In a light emitting diode $\eta_{EQE}$ is defined by the number of photons
emitted to free space per second divided by the number of injected electrons
per second. However, despite the numerous $\mu$PL results shown in Fig. 2,
$\eta_{EQE}$ of the present sample is low at $T_{amb}\,=\,295\,K$. In Ref.
Rousseau (2018) time-resolved PL measurements were performed between 5 K and
295 K to measure the internal quantum efficiency $\eta_{IQE}$ of exactly the
GaN membrane structure used for the present work. Here $\eta_{IQE}$ is again
defined in a light emitting diode by the number of photons emitted from the
active region per second divided by the number of injected electrons per
second. Consequently, the time-resolved PL analysis yielded $\eta_{EQE}$ <
$\eta_{IQE}\,\approx\,0.1\%$ for $T_{amb}\,=\,295\,\text{K}$. Thus, we can
safely assume $P_{abs}\,\approx\,P_{heat}$ for the present sample at room
temperature and beyond. However, this situation would change, if the sample
was cooled towards, e.g., 5 K or if in general a sample with less non-
radiative defects would be analyzed, which remains a task for future work.
Adding an InGaN underlayer to our structure grown on silicon as described by
Haller et al. would already drastically boost $\eta_{IQE}$ at room temperature
Haller  _et al._ (2017). Nevertheless, already the present sample’s PL signal
rises by up to four orders of magnitude when transitioning from 295 K to 5 K,
which is accompanied by $\eta_{IQE}$ reaching several tens of percent at 5 K.

####  IV.1.3 Experimental determination of the temperature

The temperature derived from our experiments are either based on the Raman
mode shift or broadening, which are in turn acquired from every point under
the laser spot. Strictly speaking, the technique does not probe the local
temperature but its average under the laser spot, weighted by the beam profile
intensity and its penetration depth Beechem  _et al._ (2015); Očenášek and
Voldřich (2015):

$T_{meas}=\int_{V}\,dr\,d\theta\,dz\,\,r\cdot T(r,\,\theta,\,z)\cdot
f_{probe}(r,\,\theta,\,z)$ (1)

with $f_{probe}(r,\,\theta,\,z)$ being the weighting function of the probe
laser and $T(r,\,\theta,\,z)$ the local temperature of the sample. The weight
function $f_{probe}(r,\,\theta,\,z)$ can be written as product of the probe
laser beam profile (assumed as a Gaussian) and the attenuation of the back-
scattered probe laser beam:

$f_{probe}(r,\,\theta,\,z)=\frac{1}{w_{e}\sqrt{\pi}}\cdot
e^{-\frac{(r-r_{0})^{2}}{w_{e}^{2}}}\cdot e^{-\frac{2z}{p_{probe}}}$ (2)

with $w_{e}$ the beam waist as the beam intensity is reduced to $\frac{1}{e}$
of its initial value, $r_{0}$ the location of the probe beam, $p_{probe}$ the
penetration depth of the laser, and $r,\,\theta$, and $z$ a cylindrical
coordinate system centered at the probe beam spot location. The factor of $2$
in the attenuation term accounts for the probe beam travel to depth $z$ into
the material and its return to the material surface. To extract the
coefficient of thermal conductivity $\kappa$, we compare $T_{meas}$ to
numerical simulations. As the outcome of such simulations is usually the
simulated local temperature $T_{sim}(r,\,\theta,\,z)$, it needs to be averaged
by a function that models the probe laser:

$f_{probe}^{sim}(r_{sim},\,\theta_{sim},\,z_{sim})=\frac{1}{w_{e}\sqrt{\pi}}\cdot
e^{-\frac{(r_{sim}-r_{0})^{2}}{w_{e}^{2}}}\cdot
e^{-\frac{2z_{sim}}{p_{probe}}}$ (3)

with $r_{sim},\,\theta_{sim}$ and $z_{sim}$ being the cylindrical simulated
coordinates, to distinguish from the physical ones. $w_{e}$ and $r_{0}$ are
measured and inserted in the model and $p_{probe}$ comes from Ref. Kawashima
_et al._ (1997).

####  IV.1.4 One-laser Raman thermometry on bulk GaN

1LRT measurements are frequently reported in literature and were, e.g.,
applied to bulk germanium Jaramillo-Fernandez  _et al._ (2018), bulk silicon
Stoib  _et al._ (2014), and related membranes Chávez-Ángel  _et al._ (2014).
However, performing such measurements on bulk GaN already represents a
challenge for the reasons given in Sec. III.1. Nevertheless, we recorded the
required resonant Raman spectra of bulk GaN with
$\lambda_{heat}=\lambda_{probe}=325\,\text{nm}$, showing the
$E_{2}^{\text{high}}$ Raman mode of GaN, cf. S-Sec. IV. Even when raising the
laser power to the maximum laser power available, no Raman mode shift or Raman
mode broadening as a sign of local heating occurred. The corresponding
temperatures are shown for bulk GaN in Fig. 4(c). Thus, the thermal
conductivity of high quality bulk GaN $\kappa_{bulk}$ cannot be measured by
1LRT in our setup as described in S-Sec. IV. Even a further increase of the
power of the heating laser does not represent a promising pathway for 1LRT
measurements on bulk GaN with very high thermal conductivities, as also any
heating laser-induced material damage must always be avoided.

###  IV.2 Two-laser Raman thermometry on GaN

Figure 6: (a) Raman map illustrating the position of the $E_{2}^{high}$ Raman
mode extracted from non-resonant Raman spectra
($\lambda_{probe}\,=488\,\text{nm}$) at an ambient temperature of 295 K. The
effect of the simultaneously engaged heating laser
($\lambda_{probe}\,=266\,\text{nm}$) can directly be observed close to the
corner of the GaN membrane. There, an additional Raman mode shift can be seen,
which is induced by the heating laser. See Fig. 3(a) for the same, but non-
locally heated Raman map. (b) Based on directly the unheated Raman map from
the locally heated map, one can derive a temperature map based on a suitable
calibration (see main text). (c) Cut lines showing the evolution of the
temperature rise along the a\- (circles) and m-direction (triangles) in the
c-plane of the GaN membrane. Here, the sample edge as well as the location and
spot size of the heating laser are illustrated.

The results of 2LRT0 measurements without a spatial displacement are shown in
Fig. 5. Again the heating laser (purple, $\lambda_{heat}\,=\,266\,\text{nm}$)
is positioned at the same position as for the 1LRT measurements, cf. Fig.
4(a). However, now this heating laser is overlapped with the probe laser
(green, $\lambda_{probe}\,=\,488\,\text{nm}$) as sketched in Fig. 5(a). Upon
varying the power of the heating laser, non-resonant Raman spectra are
recorded with the probe laser and are shown for the wavenumber range around
the $E_{2}^{high}$ Raman mode in Fig. 5(b). Again a suitable fitting model is
applied to the Raman spectra to extract the position and FWHM of the
$E_{2}^{\text{high}}$ Raman mode of GaN with rising power of the heating
laser. A Lorentz peak with an exponential background is best suited for
fitting the $E_{2}^{high}$ Raman mode in close energetic vicinity of the
omnipresent Raman mode of silicon. Resulting fitting functions are illustrated
in Fig. 5(b) for selected powers of the heating laser.

From the Raman mode positions and FWHM values, two local temperatures can be
extracted based on our local temperature calibration, cf. S-Sec. III. Similar
to the results of the 1LRT measurements, the two trends for the local
temperatures shown in Fig. 5(c) start to deviate from each other at a
$T_{rise}$ value of about 150 K. Our subsequent method to extract the thermal
conductivity from our 2LRT0 measurements $\kappa^{\,2LRT_{0}}_{\,\,memb}$ is
similar to our evaluation of the 1LRT measurements described in Sec. IV.1
based on steps I - III. The only major difference is now related to the heat
and probe volume due to the different wavelengths of the heating and probe
lasers. As the GaN membrane is transparent to the probe laser, the
corresponding temperature probe volume for our simulations now corresponds to
a cylinder with a height of 203 nm and a radius of 600 nm, which includes the
QW for sake of simplicity. See S-Sec. IV for details regarding the modeling.

For $T_{rise}\,\lesssim\,150\,\text{K}$ we derive
$\kappa^{\,2LRT_{0}}_{\,\,memb}\,=\,110_{-8}^{+11}\,\text{W/mK}$ as
illustrated by the linear function (dashed, green) illustrated in Fig. 5(c).
Again, here the sub- and superscript of this value denote the lower and upper
bound of the asymmetric error interval. The reliability of this value for
$\kappa^{\,2LRT_{0}}_{\,\,memb}$ and its obvious discrepancy to the higher
value of $\kappa^{\,1LRT}_{\,\,memb}\,=\,180_{-7}^{+12}\,\text{W/mK}$ will be
discussed in Sec. VII. As the increase in the temperature probe volume will be
key to an understanding of
$\kappa^{\,2LRT_{0}}_{\,\,memb}<\kappa^{\,1LRT}_{\,\,memb}$, it is promising
to further increase the temperature probe volume by scanning the probe laser
across the focus of the heating laser during 2LRT measurements as described in
the following Sec. V.

Please note that the slope of the green dashed line (simulated $T_{rise}$
values for a fixed $\kappa$) in Fig. 4(c) is steeper than its equivalent in
Fig. 5(c). This could appear counter-intuitive as in general one expects
higher values of the thermal conductivity $\kappa$ to result in smaller
temperature rises. However, the slope additionally depends on the heat laser
spot radius and the temperature probe volume. The heat laser spot radius is of
600 nm for the 1LRT measurements from Fig. 4(c), which surpasses the
corresponding value for the 2LRT0 measurements (250 nm), which explains the
aforementioned slope discrepancy. See Sec. IV.1 for details regarding the
normalization of the x-axis in Figs. 4(c) and 5(c).

##  V Spatially resolved two-laser Raman thermometry (step D)

To derive spatially resolved temperature mapscans by 2LRT measurements, it is
necessary to displace the probe laser ($\lambda_{probe}\,=488\,\text{nm}$)
from the heating laser ($\lambda_{heat}\,=266\,\text{nm}$). One then obtains
Raman mapscans showing, e.g., the position of the $E_{2}^{high}$ Raman mode
extracted from non-resonant Raman spectra, depending on the spatial
coordinates as shown in Fig. 6(a). Here, the position of the focus spot of the
heating laser is clearly visible close to the corner of the membrane, as
locally an additional Raman mode shift towards lower wavenumbers is induced.
This additional Raman mode shift was missing in the unheated Raman mapscan
from Fig. 3(a). By subtracting the locally heated Raman mapscan in Fig. 6(a)
from this unheated Raman mapscan in Fig. 3(a), one can now determine the
spatial distribution of the Raman mode shift that is exclusively induced by
the local heating. By applying a suitable local temperature calibration (see
S-Sec. III), we can translate these Raman mode shifts to temperatures as shown
in Fig. 6(b). Strain variations that induced Raman mode shifts in the not
locally heated Raman mapscan are eliminated by this approach. This is
observable in Fig. 6(a), when looking at the area enclosed by a black dashed
line. Here, the area of fully supported GaN is visible due to the occurrence
of tensile strain. Exactly this area is no longer visible in the temperature
mapscan of Fig. 6(b) due to the subtraction of mapscans. Please note that the
zig-zag shape of the upper edge of the corner shown in this temperature map is
exclusively caused by the step resolution in our Raman mapscans of
$\approx\,650\,\text{nm}$. From a temperature mapscan as shown in Fig. 6(b),
one can now extract temperature profiles for any in-plane direction.
Corresponding temperature profiles are shown in Fig. 6(c) for the a\- and
m-directions in the c-plane of the GaN membrane. Exactly such cut lines
through a temperature mapscan resulting from 2LRT measurements can form the
basis for extracting the thermal conductivity for all in-plane directions.

Figure 7:  (a) Temperature profile plotted over the logarithmized distance to
the center of the heat spot $u$ (black rectangles). This experimental data
matches the temperature data relying on the Raman mode shifts introduced in
Fig. 6(c) for the m-direction. A linear least square fit to the data is shown
by the dashed, black line, while also the corresponding 1$\sigma$\- and
2$\sigma$ confidence intervals are shown. The solid, color-coded lines show a
subset of the the simulated temperature trends for different
$\kappa^{\,2LRT}_{\,\,memb}$ values ranging from $20\,-\,165\,\text{W/mK}$ in
steps of $15\,-\,20$ $\frac{\text{W}}{\text{m}\cdot\text{K}}$ . We do not show
the full set of simulated curves, in steps of 5
$\frac{\text{W}}{\text{m}\cdot\text{K}}$ , to lighten the image. Here, the
horizontal dashed line illustrates an infinite thermal conductivity. (b)
Similar data for the temperature profile extracted from the Raman mode shift
broadening. Here, a much larger fluctuation is visible for the experimental
data (red rectangles), which challenges any meaningful determination of
$\kappa^{\,2LRT}_{\,\,memb}$ at the given value of
$P_{abs}\,=\,4.0\,\text{mW}$.

In Fig. 7 we exemplify the determination of the thermal conductivity
$\kappa^{\,2LRT}_{\,\,memb}$ based on 2LRT measurements. Therefore, Fig. 7(a)
shows the spatial evolution of temperatures extracted from the shift of the
$E_{2}^{high}$ Raman mode for $P_{abs}\,=\,4.0\,\text{mW}$. The data shown
matches the experimental one depicted in Fig. 6(c) for the m-direction in our
c-plane GaN membrane. First, we obtain a linear least square fit (dashed black
line) to the experimental temperature values (black rectangles), which are
plotted over logarithmized distances $u$ to the heat spot in Fig. 7(a). Here a
linear fit to the experimental data is adequate due to the analytical solution
of the Fourier equation for an infinite two-dimensional membrane Reparaz  _et
al._ (2014). In addition, we also obtained corresponding 1$\sigma$\- and
2$\sigma$ confidence intervals for the slopes from the linear least square
fit, which are illustrated as gray-shaded areas in Fig. 7(a). Note that the
intercept in the fit is not considered for the analysis.

Subsequently, we fit our already spatially resolved thermal model described in
Sec. IV.1.1 and S-Sec. IV to the experimental temperatures by scanning through
a range of $\kappa^{\,2LRT}_{\,\,memb}$ values in steps of 5
$\frac{\text{W}}{\text{m}\cdot\text{K}}$ . This stepping matches half of the
error of the final value of $\kappa^{\,2LRT}_{\,\,memb}$, aiming to avoid the
introduction of uncertainties from the data analysis. In Fig. 7(a) we
illustrate this procedure by showing the simulated temperature trends for
different $\kappa^{\,2LRT}_{\,\,memb}$ values ranging from $20\,-\,165$
$\frac{\text{W}}{\text{m}\cdot\text{K}}$ in steps of $15\,-\,20$
$\frac{\text{W}}{\text{m}\cdot\text{K}}$ by the solid, color-coded lines.

To determine the error of the $\kappa^{\,2LRT}_{\,\,memb}$ value that reaches
the best agreement with the experimental data, we match the slopes of the
numerical temperature trend to the boundaries of the 2$\sigma$ confidence
interval shown in Fig. 7(a). As a result, we obtain asymmetric error bars for
$\kappa^{\,2LRT}_{\,\,memb}$ by our numerical fitting model similar to the
error bars belonging to $\kappa^{\,2LRT_{0}}_{\,\,memb}$ and
$\kappa^{\,1LRT}_{\,\,memb}$, cf. Sec. IV. This approach resembles the fitting
procedure described in Sec. IV under step III. As a result, for the
experimental temperature values that were extracted from Raman mode shifts, we
obtain $\kappa^{\,2LRT}_{\,\,memb}\,=\,95_{-7}^{+11}$
$\frac{\text{W}}{\text{m}\cdot\text{K}}$ based on our numerical fitting
approach. Please note that we restrict our numerical fitting approach to
$T_{rise}\,\lesssim\,150\,\text{K}$ to avoid a thermally-induced lowering of
$\kappa^{\,2LRT}_{\,\,memb}$, similar to the data analysis of the 1LRT and
2LRT0 measurements presented in Sec. IV.

Similar experimental data along with the corresponding modeling is shown in
Fig. 7(b) for temperature values extracted from the broadening of the
$E_{2}^{high}$ Raman mode with temperature. However, for the given
$P_{abs}\,=\,4.0\,\text{mW}$ value, we observe a large scatter of the
temperature values in Fig. 7(b) and consequently in the thermal conductivities
presented in Fig. 8. That is because the determination of the mode width by
fitting relies on finding the maximum of the peak and its baseline. Each of
these steps introduces errors that rely on the signal-to-noise ratio of the
Raman signal, while temperatures obtained from the Raman mode shift only rely
on finding the maximum of the peak, cf. Fig. 7(a). Thus, the numerical fitting
of cut lines extracted from the mode broadening (Fig 7(b)), leads to an
uncertainty on the thermal conductivity so large that within the standard
2-$\sigma$ confidence interval, the $T$ vs. $\text{log}(u)$ slope can be equal
to zero. This results in an upper bound for the thermal conductivity of
$+\infty$, cf. Fig. 8.

In Fig. 8 we summarize all our $\kappa^{\,2LRT}_{\,\,memb}$ values for three
different values of $P_{abs}$ and the two selected crystal directions in the
c-plane of our GaN membrane. Here, we also state the maximal value of
$T_{rise}$ among the range of the experimental temperatures that were fitted
by our numerical approach. As Fig. 8 shows, values for
$\kappa^{\,2LRT}_{\,\,memb}$ that are based on the Raman mode broadening (red
symbols) show significantly larger error bars compared to their counterparts
arising from the Raman mode shift (black symbols). For this reason for
$P_{abs}\,=\,2$ mW we can only obtain meaningful values for
$\kappa^{\,2LRT}_{\,\,memb}$ based on the mode shift. Interestingly, all
$\kappa^{\,2LRT}_{\,\,memb}$ values that originate form Raman mode shifts in
Fig. 8 indicate thermal conductivities around $95^{+11}_{-7}$
$\frac{\text{W}}{\text{m}\cdot\text{K}}$ . In contrast, while for
$P_{abs}\,=\,4.0\,\text{mW}$ the $\kappa^{\,2LRT}_{\,\,memb}$ values
originating from Raman mode broadenings still overlap with their counterparts
originating from Raman mode shifts, an increasing deviation can tentatively be
observed towards $P_{abs}\,=\,6.7\,\text{mW}$. This is connected to the
corresponding $T_{rise}$ values at our highest heating power, which yield
$T_{rise}\,\leq\,170\,\text{K}$ for the Raman mode shift (black symbol) and
$T_{rise}\,\leq\,300\,\text{K}$ for the Raman mode broadening. For the latter
case, such drastic increase in temperature yields a decline of the measured
$\kappa^{\,2LRT}_{\,\,memb}$ value, which matches our observation from Figs. 4
and 5, where we observed a parabolic evolution of $T_{rise}$ with increasing
power of the heating laser for the same reason. Furthermore, within the given
error bars, no directional anisotropy can be determined for
$\kappa^{\,2LRT}_{\,\,memb}$ in the c-plane of our GaN membrane.

Figure 8: Summary of all $\kappa^{\,2LRT}_{\,\,memb}$ values found for
different values of $P_{abs}$ based on 2LRT measurements either relying on
Raman mode shifts (black circles) or Raman mode broadenings (red squares).
Here, the a\- and m-direction in the c-plane of our photonic membrane are
considered, as illustrated in the inset of Fig. 6(c). In addition, the upper
bound for the $T_{rise}$ interval is given, which formed the basis for the
extraction of $\kappa^{\,2LRT}_{\,\,memb}$. Comparably large error bars for
the $\kappa^{\,2LRT}_{\,\,memb}$ values based on Raman mode broadening (red
symbols) originate from the large scatter among the spatially resolved
temperatures as shown in Fig. 7(b). This scatter is so large that it made it
impossible to extract a meaningful thermal conductivity when
$P_{abs}\,=\,2.0$mW, as the maximal $T_{rise}$ observed was almost completely
within the noise level. Finally, we determine
$\kappa^{\,2LRT}_{\,\,memb}\,=95^{+11}_{-7}$
$\frac{\text{W}}{\text{m}\cdot\text{K}}$ from our experimental temperature
trends solely based on the Raman mode shift.

##  VI Ab initio computation of phonon transport

To theoretically compute the thermal conductivity $\kappa$, we use an ab
initio solution of the linearized phonon Boltzmann transport equation (BTE),
which furnishes the mode (branch $s$, wave vector $\mathbf{q}$) resolved
steady state distribution function

$n_{s\mathbf{q}}=n^{0}_{s\mathbf{q}}\left[1-(1+n^{0}_{s\mathbf{q}})\dfrac{\mathbf{\nabla}T\cdot\mathbf{F}_{s\mathbf{q}}}{k_{\text{B}}T}\right],$
(4)

where $n^{0}$ is the equilibrium Bose-Einstein distribution, $T$ is the
crystal temperature, $\mathbf{\nabla}T$ is the applied temperature gradient,
$k_{\text{B}}$ is the Boltzmann constant, and
$k_{\text{B}}^{-1}\mathbf{F}_{s\mathbf{q}}$ is a vectorial “mean free
displacement” Li  _et al._ (2014) carrying the unit of length. This quantity
is essentially a measure of how far each phonon mode is moved out of
equilibrium due to the presence of the temperature gradient field.

The phonon BTE can be conveniently cast in the following form:

$\mathbf{F}_{s\mathbf{q}}=\mathbf{F}^{0}_{s\mathbf{q}}+\Delta\mathbf{F}_{s\mathbf{q}}^{\text{S}}[\mathbf{F}],$
(5)

where the first term on the right hand side is the mode-resolved relaxation
time approximation (RTA) expression and the second, non-local term gives the
in-scattering corrections. The expressions for these terms have been given
before in, e.g., Ref. Protik  _et al._ (2022). A full self-consistent solution
of this equation takes us beyond the RTA.

From the solution of the BTE, a scalar, phonon mean free path (MFP) is
constructed in the following manner:

$l^{\text{MFP}}_{s\mathbf{q}}=\dfrac{\mathbf{F}_{s\mathbf{q}}\cdot\mathbf{v}_{s\mathbf{q}}}{k_{\text{B}}v_{s\mathbf{q}}},$
(6)

where $\mathbf{v}$ is the phonon group velocity.

The phonon thermal conductivity tensor is given by

$\kappa=\dfrac{1}{Vk_{\text{B}}T}\sum_{s\mathbf{q}}\mathchar
22\relax\mkern-8.0muh\omega_{s\mathbf{q}}n^{0}_{s\mathbf{q}}(1+n^{0}_{s\mathbf{q}})\mathbf{v}_{s\mathbf{q}}\otimes\mathbf{F}_{s\mathbf{q}},$
(7)

where $\mathchar 22\relax\mkern-8.0muh$ is the reduced Planck constant,
$\omega$ is the phonon angular frequency, and $V$ is the crystal volume.

In wurtzite crystals, the thermal conductivity has the following structure:
$\kappa_{xx}=\kappa_{yy}\neq\kappa_{zz}$, with the off-diagonal terms
identically zero. In the harmonic approximation, phonons are infinitely long-
lived collective excitations of the crystal. Anharmonicity is captured via the
quantum mechanical interactions between the phonons (ph). In this work, we
included 3ph and 4ph interactions. We have also included the ph-(iso)tope
scattering within the Tamura model Tamura (1983). Phonon-boundary scattering,
hereafter referred to as _ph-thin-film_ , is included empirically via Li  _et
al._ (2020)

$\tau^{-1}_{s\mathbf{q},\text{ph-thin-
film}}=\dfrac{2v^{\perp}_{s\mathbf{q}}}{h},$ (8)

where $v^{\perp}$ is the phonon group velocity component perpendicular to the
plane of the film and $h$ is the film thickness. Equation 8 includes diffuse
scattering with the top and bottom facets of the GaN membrane (see Fig. 1). We
note that it is possible to derive more refined expressions for boundary
scattering, including in-plane Carrete  _et al._ (2017) (commonly known as the
Fuchs-Sondheimer model) and cross-plane directions Vermeersch  _et al._
(2016). The development of these models is beyond the scope of this work, and
will be considered in future works.

While the real-life sample contains dislocation defects and charge carriers,
we have not included the effects of the presence of these in our current
calculations. Each of these excluded phonon scattering channels degrades the
thermal current further and, as such, the theoretical prediction made here
should be interpreted as an upper limit of the actual $\kappa$. Nevertheless,
our computation of $\kappa$ and the $l^{MFP}$ values will prove valuable for
the qualitative comparison to the experimental results that we obtain from our
different thermometric techniques, cf. Sec. VII.

###  VI.1 Computational details

We use the Quantum Espresso suite Giannozzi  _et al._ (2009, 2017, 2020) for
our density functional theory (DFT) and density functional perturbation theory
(DFPT) calculation. We employ the Optimized Norm-Conserving Vanderbilt (ONCV)
Hamann (2013) pseudopotential with the local density approximation for the
exchange correlation function. The relaxed lattice constants are $a=3.15$ Å
and $c=5.14$ Å.

The 2nd order force constants (2FCs) required for the harmonic properties are
computed using DFPT with a $6\times 6\times 6$ wave vector mesh. The 3rd order
force constants (3FCs) needed for the 3-phonon scattering are calculated using
the displaced supercell method. For this, the thirdorder code Li  _et al._
(2014) is used to generate an irreducible set of $4\times 4\times 4$ ($256$
atoms) supercells with two displaced atoms. Following the self-consistent-
field (SCF) DFT calculations on these supercells using a $\Gamma$-point
sampling of the electronic reciprocal space, the 3FCs are read-out using again
the thirdorder code. In this work, we include 4-phonon scattering since, in
general, it is not possible to know a priori whether this will significantly
limit the phonon transport or not. To calculate the 4-phonon scattering rates,
we need the 4th order force constants (4FCs). For this, we use the fourthorder
Han  _et al._ (2022) code to generate $256$-atom supercells with $3$ atoms
displaced in each. Following the SCF calculations, the same code is used to
read-out the 4FCs. Furthermore, the FourPhonon code Han  _et al._ (2022) is
used to generate the 4-phonon scattering rates on a coarse $12\times 12\times
12$ phonon wave vector ($\mathbf{q}$) mesh, $\tau^{-1}_{\text{4ph, coarse}}$.

The 2FCs, 3FCs, and $\tau^{-1}_{\text{4ph, coarse}}$ are fed into elphbolt
Protik  _et al._ (2022) which calculates the phonon transport properties by
solving the BTE on a converged $36\times 36\times 36$ $\mathbf{q}$-mesh. The
code calculates harmonic properties and 3-phonon scattering rates using the
2FCs and 3FCs, respectively. The energy conserving delta functions are
calculated using the analytic tetrahedron method Lambin and Vigneron (1984).
The $\tau^{-1}_{\text{4ph, coarse}}$ are calculated on the fine transport
$\mathbf{q}$-mesh using an (in general tri-) linear interpolation method. We
use this indirect method for computing the 4ph scattering rates on the
transport $\mathbf{q}$-mesh since the direct computation of this quantity on
such a fine mesh is prohibitively expensive at the moment. The RTA term,
$\mathbf{F}^{0}$, is constructed using Mathiessen’s rule where the total
scattering rate is given by $\tau^{-1}_{\text{total}}=\sum_{x}\tau^{-1}_{x}$,
where $x=\\{\text{ph-iso, ph-thin-film, 3ph, 4ph}\\}$. In the full solution of
the BTE, the in-scattering correction due to the 3ph interactions is added
back in iteratively. The thin-film is modeled as having a $250$ nm height
along the crystal $c$-axis and an infinite extent in the $c$-plane.

###  VI.2 Presentation of the theoretical results

Figure 9: Calculated phonon dispersions along high-symmetry paths in the
Brillouin zone for GaN at 0 K. We observe a total of 12 branches: 3 acoustic +
3 low-lying optical phonon modes separated by an energy gap from the 6 high-
lying optical phonon modes. The experimental energies of the $E_{2}^{high}$
and $E_{2}^{low}$ optical modes measured at 10 K are indicated by the red
dashed arrows close to the $\Gamma$ point.

We present in Fig. 9 the calculated phonon dispersions at 0 K. The 3 acoustic
+ 3 low-lying optical phonon branches are separated from the 6 high-lying
optical phonon branches by a large energy gap which, as we shall see shortly,
has important consequences for the thermal transport properties of this
material. The two red arrows indicate the experimental energies of the
$E_{2}^{high}$ and $E_{2}^{low}$ optical modes, measured at 10 K, in
comparison to the calculated ones.

Figure 10 shows the 300 K, RTA scattering rates of phonons due to their
interactions in the various scattering channels included in the simulation. We
discuss here some salient features of the acoustic scattering rates that
highlight the roles of the different competing interactions. First, the 3ph
scattering rates (empty red circles) show a pronounced dip in the 20 - 30 meV
energy range. As such, the phonon-isotope scattering (black circles) becomes
dominant in this energy window. The large dip can be connected to the phonon
dispersions of the material. Specifically, the presence of the large gap
between the acoustic+low-lying optic and the high-lying optic sectors results
in a reduction in the 3ph scattering phase space, explaining the corresponding
dip in the scattering rates. One can understand this by noticing in Fig. 9
that two $\sim$ 25 meV phonons cannot coalesce since there is no 50 meV state
available in the spectrum. In contrast, three $\sim$ 25 meV phonons can indeed
coalesce, explaining that fact that no such dip in the 4ph scattering rates
(empty green circles) is present. As such, the 4ph scattering rates become
important, even at 300 K. This suggests that at higher temperatures, the 4ph
interaction will play a progressively stronger role and will cause a stronger
than $T^{-1}$ scaling of $\kappa$. Observation of such a deviation from the
$T^{-1}$ scaling of $\kappa$ has been reported in Ref. Zheng  _et al._ (2019).
Next, we focus on the high energy optic phonons. As for the low energy optic
phonons, because of scattering phase-space restrictions of the 3ph scattering
rates, the 4ph scattering rates become important in the 70 - 85 meV energy
range. However, these phonons do not directly transport heat since they have
high scattering rates and/or have low group velocities. Lastly, The ph-thin-
film scattering (blue crosses) rates are weakly dependent on energy and are
high for the heat carrying phonons. This, as we shall see below, makes them a
strong limiter of the $\kappa$ even at 300 K.

Figure 10: Room temperature phonon scatterings rates in the RTA for the
various scattering channels included in the simulation. Because of the energy
gap around 50 meV, two phonons with energies around 25 meV cannot coalesce
together, creating a dip around 25 meV in the 3ph scattering channel. This dip
is completely absent in the 4ph scattering process since no such restriction
exists.

Next, we plot the energy spectrum of the $\kappa_{\text{in-plane}}$ in Fig.
11. The ph-iso scattering is included in all the cases presented. For the 3ph
limited case (green dot-line), we see two dominant peaks in the 3 - 12 and 20
- 30 meV range. The lower energy peak is due to high velocity acoustic phonons
while the second peak is owing to the dispersive low-lying optic phonons. The
low energy peak follows from the classical Slack criteria for high $\kappa$
materials Lindsay  _et al._ (2013). The second peak, on the other hand, is
enabled by the fact that the 3ph scattering rates show an anomalous dip, as
explained in the previous paragraph, reminiscent of the material BAs Lindsay
_et al._ (2013). The optic phonons across the energy gap contribute negligibly
to the thermal conductivity. Next, with the 4ph scattering turned on, we see a
general reduction in the $\kappa$ contribution (blue curve), especially in the
20-30 meV energy range. Lastly, the thin-film scattering causes a significant
reduction in the spectral $\kappa$ (dashed blue line), especially for the low
energy region where this is the dominant scattering channel. The second peak
is also significantly reduced, albeit to a lesser extent than the first,
evidently owing to the anomalous weakening of the 3ph scattering rates. The
strong suppression of the low energy peak due to the thin-film scattering
suggests that the low energy acoustic phonons are carrying heat ballistically.
In contrast, the moderate suppression of the second peak suggests diffusive
transport of the phonons in that energy range since the isotope, 3ph, and 4ph,
all play an important role in degrading the heat current.

Figure 11: Room temperature spectrum of $\kappa_{\text{in-plane}}$ for an
infinite GaN crystal and a 250-nm-thick GaN membrane. Phonon-isotope
scattering is included in all three cases. Of the two peaks visible for the
infinite crystal (solid green and blue line), the one corresponding to the
acoustic modes is the most affected by the introduction of the top and bottom
facets boundary scattering caused by our membrane (dashed blue line).

The effect of the thin-film on the phonon transport can be seen further in
Fig. 12 where we plot the phonon MFP vs. phonon energy for the transport
active acoustic+low-lying optic states. When the 250 nm thin-film scattering
is turned on, a significant reduction in the MFP occurs, especially for the
low energy acoustic phonons. This is the reason for the corresponding
reduction in the spectral $\kappa$ in that energy range. We notice that for
some states, the thin-film scattering does not affect the MFP. These are the
states with a vanishing velocity component along the $c$-axis. Consequently,
these phonons continue to carry heat along the $c$-plane despite the
imposition of the thin-film boundary.

Lastly, in Fig. 13 we show the accumulation of the c-plane $\kappa$ as a
function of MFP. The ph-iso scattering is included in all these calculations.
The inclusion of the 4ph scattering on top of the 3ph scattering leads to a
reduction of the total $\kappa_{\text{in-plane}}$ from $270$ to $248$
$\frac{\text{W}}{\text{m}\cdot\text{K}}$ , the last value being in good
agreement with earlier experimental measurements on high purity, bulk samples
with a natural isotopic mix Jeżowski  _et al._ (2003). For reference, we also
give the corresponding $\kappa_{\text{cross-plane}}$ values for the 3ph and
3+4ph cases: $275$ and $254$ $\frac{\text{W}}{\text{m}\cdot\text{K}}$ ,
respectively. The anisotropy between the bulk thermal conductivities is low,
with the in-plane value smaller than the cross-plane value by 1.8% and 2.4%
for the 3ph and 3+4ph calculations, respectively. The 3+4ph calculation for
the $250$-nm-thin film case (dashed blue curve) is more relevant to our
experimental observations. The calculated total $\kappa_{\text{in-plane}}$ in
this case is $136$ $\frac{\text{W}}{\text{m}\cdot\text{K}}$ . Again, for
reference, the calculated total $\kappa_{\text{cross-plane}}$ is $89$
$\frac{\text{W}}{\text{m}\cdot\text{K}}$ in this case. The significantly
stronger reduction of the $\kappa_{\text{cross-plane}}$ is due to the fact
that the thin-film preferentially scatters phonons with large $c$-axis
velocity component.

Figure 12: Room temperature phonon mean free path $l^{MFP}$ for the acoustic +
low-lying optical phonons, with and without the introduction of the top and
bottom facets boundary scattering. Phonon-isotope scattering is included in
both cases. We observe a strong reduction of the acoustic modes mean free
paths, due the large cross-plane component of their mean free paths. Some
phonons are however barely affected by the introduction of boundary
scattering; these are phonons with large cross-plane component of their
velocities.

In Fig. 13 we mark with vertical lines the various limiting length scales
$r_{lim}$ of our experimental temperature probe volumes. This either
corresponds to the penetration depth of the laser ($p_{abs}$) for the 1LRT
measurements, the probe spot radius ($r_{probe}$) for the 2LRT0 measurements,
and the scanning range ($r_{range}$) of the 2LRT measurements beyond the
limits of Fig. 13. Physically, in our experimental methods, these various
lengths are phonon mean free path visibility cut-offs. That is, the heat
dissipation effects of any phonon with a mean free path larger than $r_{lim}$
cannot be detected. Thus, any measurement with $r_{lim}$ shorter than the
maximum phonon mean free path will measure too large values of $\kappa$. Our
theoretical calculations predict that in order to capture the heat dissipation
effects of all the phonons in this material, $r_{lim}$ must be at least $3$
$\mu$m, which can only be met by 2LRT measurements as summarized in Sec. VII
via Fig. 14.

Figure 13: $\kappa_{\text{in-plane}}$ accumulation with respect to phonon mean
free path $l^{MFP}$ for an infinite crystal and a 250-nm-thick membrane.
Phonon-isotope scattering is included in both cases (3+4ph and 3+4ph including
boundary scattering). Vertical lines mark the relevant length scales $r_{lim}$
of the 1LRT (purple) and 2LRT0 (green) measurement techniques, while the gray
arrow indicates the corresponding $r_{lim}$ of 2LRT measurements. As $r_{lim}$
of the 2LRT measurement is $9$ $\mu$m, all relevant phonon mean free paths are
encompassed by the temperature probe volume in the GaN membrane. Additionally,
a dashed vertical brown line indicates the GaN (without the AlN) membrane
thickness.

##  VII Comparison of 1-laser and 2-laser Raman thermometry

Interestingly, based on 1LRT, 2LRT0, and 2LRT thermometry we determined a
continuous decline of the measured thermal conductivities from
$\kappa^{\,1LRT}_{\,\,memb}\,=\,180^{+12}_{-7}\,$$\frac{\text{W}}{\text{m}\cdot\text{K}}$
, over
$\kappa^{\,2LRT_{0}}_{\,\,memb}\,=\,110^{+11}_{-8}\,$$\frac{\text{W}}{\text{m}\cdot\text{K}}$
, to
$\kappa^{\,2LRT}_{\,\,memb}\,=\,95^{+11}_{-7}\,$$\frac{\text{W}}{\text{m}\cdot\text{K}}$
as summarized in Fig. 14. In the following, we will first discuss the
relevance of the absolute values that we determined for $\kappa_{memb}$ in
Sec. VII.1, before the evolution of $\kappa_{memb}$ that depends on the Raman
thermometry technique in use is discussed in Sec. VII.2.

###  VII.1 Absolute value of the thermal conductivity

Based on our theoretical results presented in Sec. VI and literature values,
only $\kappa^{\,2LRT}_{\,\,memb}$ represents a realistic value for our 253-nm-
thick photonic membrane due to the dominance of phonon boundary scattering
Beechem  _et al._ (2016) that inhibits thermal transport. For GaN layers with
a similar thickness, Beechem et al. Beechem  _et al._ (2016) predicted a
thermal conductivity of
$\approx\,70\,$$\frac{\text{W}}{\text{m}\cdot\text{K}}$ if phonon boundary
scattering represents the main limiting factor for $\kappa$. However, even
drastically lower $\kappa$ values were predicted for the dislocation density
in our photonic membrane of $3\,-\,4\,\times\,10^{10}\,\text{cm}^{-2}$. In our
membrane, anharmonic scattering, phonon-boundary-scattering, i.e., size
effects, isotope-scattering, and phonon-dislocation scattering are potentially
relevant limitations for thermal transport. Phonon-impurity-scattering can
only be of minor relevance due to the low impurity density
($<\,10^{17}\,\text{cm}^{-3}$) in our sample, cf. Sec. I.1. Regarding the
phonon-dislocation-scattering, Li et al. Li  _et al._ (2020) have
experimentally and theoretically demonstrated that the role of phonon-
dislocation-scattering is often overestimated in literature for GaN layers, as
even for a dislocation density of $1.80\,\times\,10^{10}\,\text{cm}^{-2}$ they
measured $\kappa\,\approx\,175\,\text{W/mK}$ for a 3.19-$\mu$m-thick GaN
sample grown on AlN and sapphire. Nevertheless, as our photonic membrane also
includes 50 nm of AlN, we can expect that this interlayer can further boost
thermal transport in our entire photonic membrane. The thermal conductivity of
bulk AlN is commonly around 50% higher compared to bulk GaN at room
temperature, when considering the natural abundance of isotopes Dagli  _et
al._ , while the AlN in our epilayer contributes $\approx\,20\%$ to the
overall volume.

However, as the AlN buffer layer is grown on silicon (111), it is particularly
rich in structural (e.g., dislocations) and point defects (e.g., silicon
originating from diffusion from the substrate), which will further limit its
thermal conductivity advantage compared to the GaN layer grown on top. In
addition, with 50 nm the AlN layer is comparably thin, meaning that phonon-
boundary scattering at the AlN/vacuum interface and phonon-interface
scattering at the AlN/GaN interface, will significantly lower its thermal
conductivity. Here, only future work can provide insight into the particular
contribution of this AlN interlayer to the overall
$\kappa^{\,2LRT}_{\,\,memb}$ value of $95^{+11}_{-7}\,\text{W/mK}$, which we
determined for our photonic membrane. Either depth-resolved thermometry based
on frequency-domain thermal reflectance can be performed, or the AlN bottom
layer must be removed by chemically selective wet etching. However, Rousseau
Rousseau (2018) has shown that the removal of this tensilely strained AlN
layer leads to the bowing of the rest of the photonic membrane, which renders
it unusable for any photonic applications. Thus, such modification of the
photonic membrane would counteract the motivation of this work to study
structures with a high relevance for photonic applications from an optical and
thermal viewpoint. In addition, the particular smooth top and bottom facets of
our photonic membrane described in Sec.I.1, potentially lower the impact of
phonon-boundary scattering on $\kappa^{\,2LRT}_{\,\,memb}$.

Regarding the absolute comparison between our experimental value
$\kappa^{\,2LRT}_{\,\,memb}\,=\,95^{+11}_{-7}\,$$\frac{\text{W}}{\text{m}\cdot\text{K}}$
and its theoretical counterpart $\kappa_{\text{in-plane}}\,=\,136$
$\frac{\text{W}}{\text{m}\cdot\text{K}}$ from Sec. VI, it is important to
mention that our theory only predicts an upper bound for the thermal
conductivity. In this work, we refrain from implementing the impact of
dislocations for the following reasons: a) The impact of dislocations on the
thermal conductivity of GaN was lately strongly debated in literature Beechem
_et al._ (2016); Li  _et al._ (2020) and theoretical work by Wang et al. Wang
_et al._ (2019) has predicted a minor impact of dislocations in GaN layers up
to $\approx\,10^{10}\,\text{cm}^{-2}$. Thus, in a photonic membrane with a
thickness of $\approx$ 250 nm one can expect that phonon-boundary scattering
will dominate scattering in samples with a dislocation density of
$3\,-\,4\,\times\,10^{10}\,\text{cm}^{-2}$, cf. Sec I. b) Additionally the
preferential alignment of dislocations in our c-plane GaN membrane would need
to be considered in a more advanced theory, which represents an interesting
task for a future sample series with varying dislocation densities.
Furthermore, we can expect that not only the consideration of phonon-
dislocation scattering but also the theoretical treatment of thermal transport
across the interfaces of our pad structure (GaN/InGaN, GaN/AlN, AlN/Si
interfaces) would further close the gap between our theoretical value
$\kappa_{\text{in-plane}}$ and its experimental counterpart
$\kappa^{\,2LRT}_{\,\,memb}$. Here a model based on Green’s function Dai and
Tian (2020) could be employed to model the interfacial thermal transport.
Future work may be devoted to understand and quantify the relative
contributions of these effects.

###  VII.2 Scaling of the thermal conductivity

The particular scaling for the derived $\kappa$ values as one of the main
results of this work can be explained by considering the temperature probe
volume. For this matter, Fig. 14 summarizes the experimental situation for
1LRT, 2LRT0, and 2LRT measurements in three sketches that highlight the main
length scales that limit the temperature probe volume.

Figure 14: Comparison of all thermal conductivities $\kappa$ that were
determined in this work. When increasing the probe volume, the measured
$\kappa$ values continuously decrease, starting with the highest value for
1LRT and ending with the lowest value for 2LRT measurements. The corresponding
experimental situation for all three experimental techniques, namely 1LRT,
2LRT0, and 2LRT, is sketched along with the related laser spot radii for the
heating ($r_{abs}$) and the probe ($r_{probe}$) lasers. As the balance between
the $l^{MFP}$ values and the limiting length scales ($p_{abs}$, $r_{probe}$,
and $r_{range}$) of the associated temperature probe volumes varies, the
measured $\kappa$ values are altered. In the end, only 2LRT measurements allow
enclosing most $l^{MFP}$ values in the thermalization volume, which leads to a
realistic measure of $\kappa^{\,2LRT}_{\,\,memb}$. Due to the limited
temperature probe volume, 1LRT and 2LRT0 only probe effective and artificially
enhanced $\kappa$ values as only a fraction of $P_{abs}$ is probed.

#### 1-laser Raman thermometry

For 1LRT measurements the volume in which the power of the heating laser is
absorbed coincides with the temperature probe volume as a first approximation.
This volume is the origin of the resonant Raman spectra that are used to
determine the local temperature. Here, the heating laser spot radius
($r_{abs}$ measures $600\,\pm\,50\,\text{nm}$, while the light penetration
depths $p_{abs}$ yields $74\,\text{nm}$ Kawashima  _et al._ (1997). Thus, the
main size limit for the temperature probe volume for 1LRT measurements is
oriented in the cross-plane direction, as a significant fraction of heat
carrying phonons with $l^{MFP}\,>\,p_{abs}$ exists that still contributes to
the overall $\kappa$ of the sample, cf. Fig. 13. A certain fraction of all the
generated heat carrying phonons travels quasi ballistically and the first
scattering event that starts their thermalization occurs outside the 1LRT
temperature probe volume $V_{probe}^{exp}$, creating a thermalization volume
$V_{therm}^{exp}$ that is larger than the light absorption volume
$V_{abs}^{exp}$, with $V_{abs}^{exp}\,=\,V_{probe}^{exp}$ for 1LRT.

The problem occurs when the thermal conductivity $\kappa^{1LRT}_{memb}$ is
extracted based on the comparison to our Fourier-based model described in Sec.
IV.1. Even though we precisely consider all experimental parameters (e.g.,
$p_{abs}$, $r_{probe}$, $P_{abs}$) for such a model, the calculated $T_{rise}$
will always exceed the experimental $T_{rise}$ values due to the reduced
thermalization volume $V_{therm}^{sim}$ in the simulation of 1LRT
measurements. Since no model exists that accounts for the extended
$V_{therm}^{exp}$ (even based on the results presented in Sec. VI), our
analysis of the 1LRT data is always limited to:

$1LRT:\,\,\,\,\,V_{abs}^{sim,exp}\,=\,V_{probe}^{sim,exp}\,=\,V_{therm}^{sim}\,\ll\,V_{therm}^{exp}.$

Consequently, based on our model for 1LRT measurements that aims to match the
calculated $T_{rise}$ values to the experimental ones, we always determine
artificially enhanced $\kappa^{\,1LRT}_{\,\,memb}$ values, cf. Fig. 14.

#### 2-laser Raman thermometry

The situation is improved as soon as we transition from 1LRT to 2LRT0
measurements that probe the entire GaN membrane thickness through non-resonant
Raman scattering. Here the temperature probe volume in the cross-plane
direction is limited by the GaN fraction of the membrane and for all in-plane
directions by the radius $r_{probe}\,=\,600\,\pm\,50\,\text{nm}$ of the
temperature probe laser spot, cf. Fig. 14. As a result, a much larger fraction
of the thermalization volume is measured in the temperature probe volume of
2LRT0 measurements, leading to:

$2LRT_{0}:\,\,\,\,\,\,V_{abs}^{sim,exp}\,=\,V_{therm}^{sim}\,<\,V_{probe}^{sim,exp}\,<\,V_{therm}^{exp}.$

As a result, we obtain a $\kappa^{2LRT_{0}}_{memb}$ value that is
significantly lower than $\kappa^{1LRT}_{memb}$ (see Fig. 14), which already
compares well to previously published results Beechem  _et al._ (2016).
Nevertheless, a small fraction of heat carrying phonons still exists that
exhibits $l^{MFP}\,>\,r_{probe}$, which in turn also artificially enhances
$\kappa^{2LRT_{0}}_{memb}$. However, compared to the strong artificial
increase of $\kappa^{1LRT}_{memb}$, this enhancement is minor as the majority
of heat carrying phonons thermalizes in $V_{probe}^{sim,exp}$ at
$T_{amb}\,=295\,\text{K}$, cf. Figs. 13 and 14.

By further enlarging the temperature probe volume based on 2LRT measurements
over $r_{range}\,\leq\,9\,\mu\text{m}$, all relevant $l^{MFP}$ values are
comprised in the temperature probe volume:

$2LRT:\,\,\,\,\,\,V_{abs}^{sim,exp}\,=\,V_{therm}^{sim}\,<\,V_{therm}^{exp}\,\ll\,V_{probe}^{sim,exp}.$

Now that the temperature probe volume encompasses the entire thermalization
volume, the extraction of the thermal conductivity based on our Fourier model
leads to our final thermal conductivity
$\kappa^{\,2LRT}_{\,\,memb}\,=95^{+11}_{-7}\,$$\frac{\text{W}}{\text{m}\cdot\text{K}}$
. The fraction of $P_{abs}$ that is measured in the temperature probe volume
of all three experimental techniques is reported in Fig. 14, along with the
main extensions $p_{abs}$, $r_{probe}$, and $r_{range}$ that predominantly
limit the temperature probe volume during our Raman thermometry. These
extensions then enable the comparison to our results of ab initio calculations
shown in Fig. 13.

#### Choice of the heating laser

Let us note that we applied a cw 325 nm heating laser for our 1LRT
measurements and a cw 266 nm heating laser for our 2LRT0 and 2LRT
measurements. The reason for this choice is twofold. First, based on our 266
nm laser we were not able to acquire high quality resonant Raman spectra for
1LRT measurements. Despite the high ultraviolet sensitivity of our
experimental setup, even integration times of 10 min per resonant Raman
spectrum did not yield a solid data basis. Thus, we transitioned to a 325 nm
cw laser for 1LRT measurements, utilizing a microscope objective optimized for
this wavelength. However, for 2LRT0 and 2LRT measurements, we still applied
our 266 nm cw laser due to the larger available range for $P_{abs}$, but also
the dual-wavelength range design of our main microscope objective. This
particular microscope objective is optimized around 266 nm and the visible
range around 500 nm (high transmission and overlapping focal planes), but
exhibits poor transmission at 325 nm ($\leq\,15\%$). Thus, the light
penetration depth for the heating laser changes from
$p_{abs}\,=\,74\,\text{nm}$ Kawashima  _et al._ (1997) for 1LRT measurements
to $p_{abs}\,=\,45\,\text{nm}$ Kawashima  _et al._ (1997) for 2LRT0 and 2LRT
measurements. However, this change in $p_{abs}$ will not affect the overall
trend for all three $\kappa$ values in Fig. 14, on the contrary, the
overestimation of $\kappa^{\,1LRT}_{\,\,memb}$ should even be more pronounced
for a 266 nm heating laser.

##  VIII Discussion and outlook

As a result of Fig. 14, 2LRT measurements appear as the only suited method to
measure $\kappa$ in our photonic membrane made from III-nitride material.
Still 1LRT measurements could also be applied to measure realistic values for
$\kappa$. But for this purpose, the laser spot size and laser penetration
depth would need to be tuned in a way such that their corresponding length
scales ($r_{probe}\,\approx\,r_{abs}$ and $p_{probe}\,\approx\,p_{abs}$) are
sufficient to probe the major fraction of heating carrying phonons that
significantly contribute to $\kappa$, cf. Fig. 13. Let us note however that
for direct bandgap semiconductors like GaN whose absorption coefficient
exceeds $1\,\times\,10^{5}\,\text{cm}^{-1}$ above the bandedge Muth  _et al._
(1997), this remains a challenging task. To achieve larger $p_{probe}$ values
for 1LRT measurements in GaN, one would need to tune the laser wavelength
around the onset of absorption in close energetic proximity to the bandedge
Callsen  _et al._ (2018). However, in this energetic regime the absorption
features of band-to-band transitions and excitons still overlap (e.g., at
$T_{amb}\,=\,295\,\text{K}$), rendering any precise determination of $P_{abs}$
a challenging task. This is especially true for the comparably large
$T_{rise}$ values common for Raman thermometry Beechem  _et al._ (2015). As
soon as temperatures are so drastically increased, the energetically steep
absorption onset of excitons and/or band-to-band transitions shifts towards
lower energies Kudlek  _et al._ (1990), which in turn varies the absorption.
In addition, depending on the local temperature, the balance between the
different contributions to the absorption is altered. As a result of these
thermal non-linearities, one lacks a constant value for the absorption
coefficient at a fixed heating wavelength, which prevents any meaningful 1LRT
measurements. In general, changes in the absorption coefficient are less
pronounced deeper in the bands of a direct bandgap semiconductor compared to
the region close to its bandedge. Nevertheless, for large values of
$T_{rise}$, the absorption coefficient at a certain wavelength cannot always
be considered as constant, even when the energetic spacing to the bandedge is
increased. Clearly, such optical non-linearities remain an interesting task
for future work. Potential optical non-linearities related to the absorption
coefficient are another reason why we refrain from a more detailed analysis of
the high temperature data shown in Figs. 4 and 5 ($>\,150\,\text{K}$).
Interestingly, the high absorption coefficient of GaN that complicates the
Raman thermometry is directly connected to the reason why GaN has such a high
relevance for photonic applications, which commonly rely on direct bandgap
semiconductors for emitters.

For the thermal characterization via TDTR measurements, it is common practice
to work with sufficiently large and well defined laser spot diameters Li  _et
al._ (2020) on the order of several tens of micrometers. At the same time, the
light penetration depths is less problematic compared to our Raman thermometry
technique as usually metal transducers made from, e.g., gold, aluminum, and
silver are employed. On the one hand, this makes TDTR measurements a
quantitative, reliable, and consequently well-established thermal technique.
On the other hand, the complications that arise for Raman thermometry also
carry a strong potential for measuring the mean free path length of heat
carrying phonons and their impact on $\kappa$. By controlling the heat and
probe laser spot diameters during transient thermoreflectance (TTR)
measurements, Minnich et al. have pioneered such phonon mean free path lengths
measurements in silicon Minnich  _et al._ (2011). Interestingly, our results
suggest that Raman thermometry could, similarly to TTR measurements, be suited
for measuring phonon mean free path lengths, which is particularly promising
due to the experimental simplicity of Raman spectroscopy compared to pump-
probe techniques. Clearly, this potential for measuring heat carrying phonon
mean free path lengths adds to the overall interesting aspects of 2LRT
measurements that were shown in this work. Not only that a high spatial
resolution can be achieved by our non-invasive experimental technique, but
also the full angular dependence of $\kappa$ in highly non-symmetric
semiconductors like $\beta$-Ga2O3 Guo  _et al._ (2015); Slomski  _et al._
(2017) could readily be explored. In addition, as in TDTR and micro-TDTR Maire
_et al._ (2017) measurements, Raman thermometry carries the potential for
measurements at cryogenic temperatures. Still, the temperature limits of Raman
thermometry for GaN and AlN need to be explored in full detail. Furthermore,
our 2LRT measurements could even represent an interesting alternative to other
non-invasive thermometry techniques like thermal transient grating (TTG)
spectroscopy Duncan  _et al._ (2020). Thanks to the spatial resolution of 2LRT
measurements one could potentially directly witness effects of non-Fourier-
like thermal transport in real-space, such as a negative local effective
thermal conductivity: close to the material’s edge, the heat flux may have the
same sign as the temperature gradient - a clear violation of Fourier’s law Hao
_et al._ (2009). Therefore, our experimental setup may open up exciting
opportunities to probe these effects, while remaining non-intrusive.

###  VIII.1 Comparison to transient thermoreflectance

The direct comparison in between the pioneering work of Minnich et al. Minnich
_et al._ (2011) (TTR) and our Raman thermometry summarized in Fig. 14 implies
some additional insight into two different experimental perspectives on
thermal transport as soon as the length scale of the heated volume in a bulk
semiconductor approaches the mean free path lengths of the heat carrying
phonons $l^{MFP}$.

First, we will start with a brief summary of the findings of Minnich et al..
Below an ambient temperature of 200 K they found that the effective thermal
conductivity measured on high quality, bulk silicon declined with the diameter
of their heater scaling from 60 - 15 $\mu$m (1/e2 definition of the diameter).
For the case of these TTR measurements, the heater is given by an Al
transducer that is heated by a focused laser. Exactly this Al transducer is
then also used to measure the evolution of temperature via its relative
reflectivity changes $\Delta\,R/R$ probed by a second laser. As this is a
transient experiment, $\Delta\,R/R$ declines (specific to the choice of the
metal transducer and probe laser wavelength) over time, which can be recorded
in a, so-called, thermal transient. Subsequently, relevant material parameters
like, e.g., $\kappa$ can be derived from these transients by comparison to a
Fourier-based dynamic model based on the time-dependent heat equation Schmidt
_et al._ (2008). The interpretation of their findings is based on the
assumption that the phonon bath can be divided into a purely diffusive and
purely ballistic fraction, which represents a well-justified simplification.
When lowering the diameter of their laser heat spot on the metal transducer
(i.e, the heater) in comparison to a smaller and constant probe spot diameter
(11 $\mu\text{m}$), they measure a most pronounced decline of effective
$\kappa$ values at, e.g., an ambient temperature of 60 K. This decline is then
attributed to an increasingly large fraction of heat carrying phonons that
propagates ballistically. In the approximation of Minnich et al. this sub-
group of phonons undergoing ballistic transport does not contribute to thermal
transport in contrast to the sub-group of diffusively propagating thermal
phonons that they predominantly probe by their TTR experiment (regulated by
the diameters of the heat and probe laser spots). As a result, the heat flux
from the perspective of their transient temperature probe is lower than
predicted by Fourier’s law, which is then interpreted as a ballistic thermal
resistance. Finally, the increase of this ballistic thermal resistance with
declining diameter of their heater leads to the measurement of an effective
reduction of $\kappa$ at $\leq\,200\,\text{K}$ in silicon.

As good experimental practice, during the reduction of the heater diameter the
laser power density was kept constant on the surface of the metal transducer