Present address: ]Physics Department, California Polytechnic University, San
Luis Obispo, California 93407, USA.

# Structural tunability and origin of two-level systems in amorphous silicon

H. C. Jacks ID [  Physics Department, University of California, Berkeley,
California 94720, USA  M. Molina-Ruiz ID<EMAIL_ADDRESS>H. C. Jacks
and M. Molina-Ruiz contributed equally to this work.  Physics Department,
University of California, Berkeley, California 94720, USA  M. H. Weber ID
Physics and Astronomy Department, Washington State University, Pullman,
Washington 99164, USA  J. J. Maldonis ID Materials Science and Engineering
Department, University of Wisconsin - Madison, Wisconsin 53706, USA  P. M.
Voyles ID Materials Science and Engineering Department, University of
Wisconsin - Madison, Wisconsin 53706, USA  T. Metcalf ID Naval Research
Laboratory, Washington, DC 20375, USA  X. Liu ID Naval Research Laboratory,
Washington, DC 20375, USA  F. Hellman ID Physics Department, University of
California, Berkeley, California 94720, USA  Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA

(August 27, 2024)

###### Abstract

Amorphous silicon films prepared by electron beam evaporation have
systematically and substantially greater atomic density for higher thickness,
higher growth temperature, and slower deposition rate, reaching the density of
crystalline Si when films of thickness greater than $\sim 300$ nm are grown at
425 °C and at $<1$ Å/sec. A combination of spectroscopic techniques provide
insight into atomic disorder, local strains, dangling bonds, and nanovoids.
Electron diffraction shows that the short-range order of the amorphous silicon
is similar at all growth temperatures, but fluctuation electron microscopy
shows that films grown above room-temperature show a form of medium-range
order not previously observed in amorphous silicon. Atomic disorder and local
strain obtained from Raman spectroscopy reduce with increasing growth
temperature and show a non-monotonic dependence on thickness. Dangling bond
density decreases with increasing growth temperature and is only mildly
dependent on thickness. Positron annihilation Doppler broadening spectroscopy
and electron energy loss spectroscopy show that nanovoids, and not density
variations within the network, are responsible for reduced atomic density.
Specific heat and mechanical loss measurements, which quantify the density of
tunneling two-level systems, in combination with the structural data, suggest
that two-level systems in amorphous silicon films are associated with
nanovoids and their surroundings; which are in essence loosely bonded regions
where atoms are less constrained.

###### pacs:

61.43.Dq, 68.37.-d, 78.66.Jg

##  I Introduction

Amorphous silicon ($a$-Si) is used in photovoltaic, thin-film transistor,
quantum computing and gravitational-wave detection technologies [1, 2, 3, 4,
5], to name a few, since it is much easier to implement than its crystalline
counterpart. The performance of amorphous silicon in these technologies is
dependent on its properties, which change with the material structure obtained
during the preparation process. $a$-Si is typically found 4-fold coordinated,
with little variability in bond length, but variability in bond angle and
occasional 3-fold coordination leading to dangling bonds. 5-fold coordinated
atoms are found in the liquid phase, but not identified in the solid phase [6,
7]. This limited flexibility in solid phase makes it more controllable than
systems with highly variable coordination, e.g., amorphous carbon, or
multiatomic materials. Four-fold coordination yields an over-constrained
atomic environment, which was postulated by Phillips in 1972 to suppress
tunneling two-level systems (TLSs) [8], later confirmed by our work on $a$-Si
[9, 10, 11, 12].

The suppression of TLSs has been of interest to the glass community because
they are the source of energy loss at low temperatures and the origin of
anomalous acoustic, thermal, and dielectric properties [13]. The ability to
control the TLS density in glasses is crucial in applications for which $a$-Si
is a potential material, including phase-resonant qubits [14, 15, 16] and
coatings for gravitational-waves detection [17, 18]. Electron-beam (e-beam)
evaporated $a$-Si shows very low TLS density in high atomic density films, as
opposed to high TLS density found in low atomic density films, suggesting that
low-density regions are the source of TLSs in e-beam prepared $a$-Si [10, 11,
12]. Therefore, a careful understanding of the structural differences between
these films should lead to a deeper knowledge of the nature of the TLSs. It
has been shown that coupled dangling bonds create tunneling states on the
surface of hydrogen terminated crystalline silicon [19], but no correlation
between dangling bond density and TLS density has been seen in $a$-Si films
[12].

In this work we present extensive structural characterization of $a$-Si films
prepared by e-beam evaporation under ultra-high vacuum (UHV) conditions, and
show how differences in growth temperature, deposition rate, and film
thickness affect the growth mechanisms and film properties. We discuss
connections between growth parameters and physical properties, as well as
between TLS density and structural features. We conclude that nanovoids are
the most likely structures responsible for TLSs in e-beam evaporated $a$-Si
films.

##  II Sample preparation and characterization methods

Samples were grown by e-beam physical-vapor deposition (PVD) in an UHV chamber
with a base pressure $<5\times 10^{-9}$ Torr, and substrate temperature
T${}_{\textrm{S}}$ ranging from 50 to 450 °C, growth rate r${}_{\text{t}}$
from 0.5 to 2.5 Å/s, and thickness t from 10 to 750 nm. Samples were grown
either onto low-stress amorphous silicon nitride ($a$-SiN${}_{\textrm{X}}$),
or onto crystalline silicon with the native oxide layer left intact. A
naturally occurring and continuous self-passivating native oxide layer, not
thicker than 3 nm [20], was allowed to form on the surface of most $a$-Si
films. In some cases, a thin capping layer of Al was used, which forms an
exellent passivation layer on exposure to air that protects the film against
oxygen and water subsequent to growth.

Structural characterizations, detailed below, span length scales from $\sim$
0.1 to 10 nm. These techniques provide information on both the network and
structural defects of the films, specifically surface topography via atomic
force microscopy (AFM), atomic density via Rutherford backscattering
spectrometry (RBS) together with profilometry and AFM, short-range order (SRO)
and medium-range order (MRO) via electron microscopy, atomic bond angle
deviation and local strain via Raman spectroscopy, dangling bond density via
electron paramagnetic resonance (EPR), and empty space or nanovoids volume via
positron annihilation Doppler broadening spectroscopy (DBS) measurements. RBS
was also used to set limits on impurities concentration. Hydrogen forward
scattering (HFS) was used to quantify the films’ water concentration.

Substrate curvature measurements were taken using a Tencor FLX-2320 thin-film
stress measurement instrument. The stress of $a$-Si films grown on 2-in
diameter silicon wafers was determined at room-temperature using Stoney’s
equation [21] with no corrections, since the thicknesses ratio between films
and substrate ($\sim 10^{-4}$) fulfils the condition $\ll 1$ [22].

With the aim of looking for depth dependence of atomic density and bond angle
deviation, and in particular to investigate the origin of the differences seen
in properties of thin and thick films [10, 12], select thick films were etched
in order to study structural properties as a function of depth. Films were wet
etched to avoid densification or atomic reconstruction via energy transfer,
since a dry etch may alter the samples underlying atomic structure [23]. To
improve thickness control and homogeneity, samples were etched under mild
sonication in an isopropanol-saturated 6M KOH bath at 80 °C for several
minutes until specific target thicknesses were reached [24].

###  II.1 Ion beam analysis and atomic force microscopy

The measured atomic density $n_{at}$ of the $a$-Si films is the combined
result of profilometry, RBS and AFM measurements. Profilometry was done using
a KLA Tencor ASIQ profilometer, with an error between 0.1% to 4% for the
thicker and thinner films, respectively. Ion beam analysis was performed by
RBS and HFS in an NEC model 5SDH Pelletron tandem accelerator, with $\alpha$
particles beam energy of 3040 keV. Resonant RBS and HFS techniques were used
to set limits on impurities, including specifically oxygen and hydrogen.
SIMNRA analysis software was used to obtain the samples composition and areal
density. A Digital Instruments AFM Nanoscope Dimension 3100 equipped with
Budget Sensors Tap300-G silicon probes with a resonant frequency of 300 kHz
and force constant of 40 N/m was used to characterize the samples’ surface
roughness. The scanned areas are $1\times 1$ $\upmu$m2 yielding root mean
square (RMS) surface roughness $R_{q}$ and topography of the $a$-Si films, as
well as of their substrates. The thickness obtained by profilometry was
corrected by the roughness measured by AFM in order to accurately determine
the films’ volume. Although the roughness was small (less than 2 nm even in
the roughest films), this correction is very important for thinner films.

###  II.2 Electron microscopy

Electron nanodiffraction and electron energy loss spectroscopy (EELS)
experiments were performed in a ThermoFisher Titan scanning transmission
electron microscope (STEM) operated at 200 kV. The diffracted electron
intensity $I$ as a function of the scattering vector $\mathbf{k}$ was measured
using energy-filtered electron diffraction on a GIF 865ER with a Gatan US1000
CCD camera at a camera length of 840 mm, an energy filtering with a slit width
of 20 eV, and a high coherence, 2 nm diameter nanoprobe beam with a 0.6 mrad
semi-convergence angle. One hundred $512\times 512$ pixels nanodiffraction
patterns were acquired in a $10\times 10$ grid of positions $r$ covering a
$25\times 25$ nm2 area from ten different regions of each sample. The
acquisition time for each diffraction pattern was 6 seconds. Each diffraction
pattern was averaged over the azimuthal angle to produce the intensity as a
function of the scattering vector magnitude $k$.

The nanodiffraction data set was used to evaluate short- and medium-range
order. SRO was evaluated from $I(k)$, which is the average of $I(k,r)$ over
$r$. This experiment is the equivalent of a large-area diffraction experiment,
but with somewhat worse $k$ resolution due to the convergent probe. MRO was
evaluated using the fluctuation electron microscopy (FEM) normalized variance
$V$, defined as

$V(k)=\frac{\langle I^{2}(k,r)\rangle}{\langle I(k,r)\rangle^{2}}-1$ (1)

where values within $\langle\,\rangle$ are averaged over the position $r$ on
the sample. $V$ measures the magnitude of spatial fluctuations in the
diffracted intensity, which are sensitive to three and four-body correlation
functions [25, 26, 27, 28]. In $a$-Si, MRO spans length scales from
approximately the fourth coordination shell ($\sim 0.8$ nm) to just under what
is detectable by Bragg diffraction ($\sim 3$ nm) [29]. The position in $k$ of
peaks in $V$ is controlled by the interatomic spacing inside nanoscale
structural heterogeneities (such as deviations in the distribution of ring
sizes, see Ref. 30), and the magnitude of $V$ is controlled by the size,
density, and internal order of the heterogeneities. $V$ was corrected for
Poisson noise in $I$ as described in [26].

EELS was performed at a camera length of 160 mm, a probe convergence angle of
25 mrad, and an EELS collection angle of 52 mrad. The energy dispersion was
0.05 eV/channel, and the energy resolution was 0.8 eV, measured as the full
width at half maximum (FWHM) of the zero-loss peak. EELS experiments focused
on the bulk plasmon loss, which is sensitive to the volume number density of
electrons in the material, $n$. In the Drude model, the peak plasmon energy
$\lambda_{0}=\hbar\sqrt{ne^{2}/\epsilon_{0}m}$, where $e$ is the electron
charge, $m$ is the electron mass, and $\epsilon_{0}$ is the permittivity of
free space. At constant composition, shifts in $n$ result from shifts in the
volume number density of atoms $\rho$, which has been used, for example, to
measure thermal expansion in Al as a means of thermometry [31] and the change
in volume of Al on melting [32]. Introducing voids into the sample does not
shift $\lambda_{0}$. Instead, it introduces new plasmon modes at different
energies associated with the surfaces of the voids. As a result, EELS
measurements of $\lambda_{0}$ report the average density of the sample
excluding voids, unlike density measurements by RBS or Archimedes principle,
and hence report the average interatomic spacing of the material network.

High resolution transmission electron microscopy (HRTEM) was performed on a
FEI Tecnai-TF30 microscope operated at 300 kV. Images with resolution
$2048\times 2048$ pixels were recorded by a Gatan Ultrascan CCD with 1 second
exposure time. $a$-Si films for all TEM experiments were $\sim 30$ nm thick
grown at different temperatures on $50\times 50$ $\upmu$m2
$a$-SiN${}_{\textrm{X}}$ membranes 50 nm thick and measured in plan view.

###  II.3 Raman spectroscopy

The distribution of atomic bonds, or bond angle deviation, $\Delta\theta$ and
the fractional volume change, or local strain, $tr(\epsilon)$ were determined
by Raman spectroscopy performed using an inVia Renishaw micro-Raman/PL system
equipped with a 488 nm laser. The laser power was set to $\sim 250$ $\mu$W on
a spot with area of $\sim 2$ $\mu$m2, low enough to prevent degradation or
crystallization of the amorphous films. Under these conditions, most of the
Raman signal comes from the top $\sim 30$ nm of the film due to the laser
intensity attenuation.

In $a$-Si the bond angles have a variability around the tetrahedral angle of
109.5°, the canonical tetrahedral angle in c-Si, characterized by
$\Delta\theta$. Beeman et al. [33], and later confirmed by others [34, 35],
established an empirical correlation between $\Delta\theta$ and the FWHM of
the $a$-Si transverse optical (TO) peak, observed at $\sim 480$ cm-1, which
yields $\Delta\theta=(\textrm{FWHM}-15)/6$. $tr(\epsilon)$ manifests in c-Si
Raman spectra through shifts in the TO peak position $\omega$ [36, 37].
Strubbe et al. [38] demonstrated an empirical relationship between TO peak
shifts and local strain in $a$-Si:H, which is valid for isotropic amorphous
vibrational mode frequencies. In vibrational density of states calculations,
the TO peak vibrations remain almost unaffected by the presence of H atoms,
which indicates that atomic interactions are mostly determined by the nature
of Si–Si bonds [38]. Experimental Raman spectra of $a$-Si and $a$-Si:H around
the TO peak region are quite similar. Therefore, the empirical relationship
obtained for $a$-Si:H is also valid for $a$-Si, and establishes
$\omega=s\leavevmode\nobreak\ tr(\epsilon)+\omega_{0}$, where $s=-460\pm 10$
cm-1 is a fitting parameter and $\omega_{0}\approx 480$ cm-1 is the bulk
$a$-Si TO peak position [35].

###  II.4 Double-paddle oscillator

Transverse sound velocity ($v_{t}$) measurements of $a$-Si films were taken at
300 mK using the double-paddle oscillator (DPO) technique described elsewhere
[39, 40]. The resonant frequency of the second antisymmetric torsional
resonance mode (AS2) at $\sim 5500$ Hz is measured to an accuracy of $<$10-5
Hz on both the bare and film-laden oscillator. The shear modulus G is measured
and $v_{t}$ is then calculated as $v_{t}=\sqrt{G/\rho}$, where $\rho$ is the
film mass density.

###  II.5 Electron paramagnetic resonance

Electron paramagnetic resonance measurements were used to determine the
dangling bond density of the films, using a Bruker ELEXSYS E580 EPR
spectrometer with an X-band ER 4123D CW-Resonator at 9.36 GHz. EPR determines
the density of dangling bond defects, or unpaired electrons, by measuring the
signal strength of the resonant transition between the Zeeman split energy
levels of the paramagnetic dangling bond defect. Microwave power (1.5 mW) and
magnetic field modulation amplitude (5 G) were adjusted for optimum intensity
without line shape distortion. Spectra were measured from 3282 to 3383 G. A
bare substrate was used to determine the background contribution, whereas the
samples’ spin density $N_{S}$ was determined by double integration of the
experimental absorption first derivative spectra and by comparison to a KCl
weak pitch with $N_{S}$ = 9.5$\times$10${}^{12}\pm 5\%$ spins/cm3 and g =
$2.0028\pm 0.0002$. These experimental conditions yielded $N_{S}$ with a
systematic error of $\sim 10\%$. The EPR spectra obtained for the $a$-Si
samples studied in this work are isotropic with a Landé g-factor of 2.0055,
typical of $a$-Si dangling bonds [41].

###  II.6 Doppler broadening spectroscopy

Positron annihilation experiments using DBS were performed to obtain the
volume of empty space, or total volume of nanovoids, as a function of sample
depth. Amorphous silicon films were grown on c-Si substrates with the native
oxide left intact on the substrate. In order to avoid attenuation from the
native oxide on the $a$-Si surface, the films were etched in a buffered oxide
etch 10:1 solution for 10 minutes, then introduced into the measurement
chamber and brought below $10^{-5}$ Torr within 25 minutes. The vacuum
pressure during typical measurements was less than $10^{-7}$ Torr. The
positron incident beam energy was varied from 50 eV to 70 keV, which in c-Si
yields implantation depths around 1 nm to 15 $\upmu$m, respectively. In the
present study, 25 keV yielded an implantation depth in the films’ substrate;
and data analysis is limited to below 40 keV to avoid systematic error from
backscattered positrons that annihilate from the steel vacuum chamber walls.

A summary of DBS measurement and analysis procedures can be found elsewhere
[42]. Photons emitted by positron annihilation were detected with a high
purity germanium detector from EG&G Ortec, with an energy resolution of 1.45
keV FWHM at the photon energy equivalent to the rest mass of positrons and
electrons at 511 keV. To analyze Doppler broadening due to annihilations from
nanovoids, we examined the photoelectric peak from 5$\times$104 to
6$\times$104 detected annihilations after suitable background subtraction. The
accumulated events $N_{c}$ in a narrow 1.45 keV window around the center and
in the two wing regions $N_{w}$ are compared to the total event number in the
full photoelectric peak $N_{tp}$. The DBS parameters $S$ and $W$ are the
ratios of $N_{c}/N_{tp}$ and $N_{w}/N_{tp}$, respectively. The beam energy $E$
in keV is converted to implantation depth $d$ using the empirical formula
$d[nm]=40\leavevmode\nobreak\ E^{1.6}/\rho$, where $\rho$ is the mass density
in g/cm${}^{3}\leavevmode\nobreak\ $ [43]. VepFit software is used to simulate
the empty volume depth profiles optimizing $S$, layer thickness and positron
diffusion lengths [44]. The $S$ parameter yields information about the number
of nanovoids and their size, from atomic vacancies to nanovoids [45]. The $W$
parameter is sensitive to higher momentum electrons and probes the elemental
type of the nearest neighbor [46].

##  III Results

In this section we present the results for different properties as a function
of growth temperature, growth rate and thickness. We follow the same structure
presented in II, showing first the results that provide insight into the
structure and network, and later the structural defects: dangling bonds and
nanovoids.

###  III.1 Topography and roughness

Figure 1 shows AFM images of samples grown at different temperatures and
thicknesses. These images show the topography of the films’ surface, where the
roughness is considerably increased for thick films grown at high temperature.
The statistical analysis of the images yields an average in plane grain
diameter $\sim 12$ nm for all films, except for the thick films grown at 425
°C that show grains of $\sim 20$ nm. These grains measured by AFM are likely
to be the end of amorphous columns observed previously in similar $a$-Si films
[12].

Figure 1: AFM images of the surfaces of a subset of $a$-Si films: 10 and 300
nm thick, left and right, respectively, grown at 50 °C and 425 °C, bottom and
top, respectively. All images are $1\times 1$ $\upmu$m2.

The root mean square roughness $Rq$ is shown in Fig. 2 as a function of
thickness for films grown at 50 °C, 225 °C and 425 °C. $Rq$ is less than 0.5
nm, the same as the roughness of the substrate, for films grown at 50 °C at
all thicknesses. At higher growth temperature, $Rq$ increases proportionally
to growth temperature and with thickness up to $\sim 100$ nm, where it
plateaus or slowly increases. A similar dependence of the surface roughness
with thickness has been reported in hydrogenated $a$-Si films grown above
room-temperature [47, 48].

Figure 2: RMS roughness $Rq$ of $a$-Si films as a function of thickness t for
samples grown at 50 °C (brown squares), 225 °C (red circles) and 425 °C
(orange triangles). All samples are grown at 0.5
$\mathrm{\SIUnitSymbolAngstrom}$/s. The gray area shows the substrate RMS
roughness. Lines are guides to the eye.

###  III.2 Atomic density

Atomic density $n_{at}$ of $a$-Si films grown at different temperatures and
rates was determined by RBS measurements in combination with profilometry,
where the thicknesses were corrected after the AFM measured roughness, as
discussed in II.1. The resulting data are shown in Fig. 3 as a function of
thickness, which shows a remarkable and systematic dependence on thickness,
growth temperature and rate. Error bars are dominated by the thickness
uncertainty (profilometry) for thin films, and by areal atomic density
uncertainty (RBS) for thick films. No observable discontinuities are seen. The
lowest density films are quite underdense, $\sim 22\%$ less than c-Si, whereas
the highest density films are $\sim 1\%$ less dense than c-Si. Previous work
on vapor deposited $a$-Si films reported densities 19% smaller than that of
c-Si [49], which was there suggested to be caused by by low-density regions or
nanovoids found in $a$-Si [50].

Figure 3: Atomic density $n_{at}$ of $a$-Si films as a function of thickness t
for (a) growth temperatures of 50 °C (brown squares), 225 °C (red circles) and
425 °C (orange triangles), and (b) growth rates of 0.05
$\mathrm{\SIUnitSymbolAngstrom}$/s (lilac hexagons), 0.5
$\mathrm{\SIUnitSymbolAngstrom}$/s (brown squares) and 2.5
$\mathrm{\SIUnitSymbolAngstrom}$/s (green diamonds). In (a), all films are
grown at 0.5 $\mathrm{\SIUnitSymbolAngstrom}$/s, and in (b), all films are
grown at 50 °C. In both plots the c-Si density is given for reference (dashed
horizontal line). Solid lines are guides to the eye.

Atomic density of $a$-Si films grown by e-beam is thus highly dependent on
thickness, growth temperature and rate. We report density reductions with
thickness for $a$-Si samples grown at room-temperature between 25% and 10% for
the thinnest and thickest films, respectively, when compared to c-Si density.
Higher growth temperature and lower growth rate yield films with higher atomic
density at any given thickness. Thick $a$-Si films, around 600 nm thick, and
grown at different temperatures were subsequently annealed at 425 °C in UHV
for 3 hours. The film grown at 50 °C shows a thickness reduction from 598.6 nm
to 593.1 nm ($-5.5\pm 2.1$ nm, or -0.92%), whereas the thickness variation for
films grown at 225 and 425 °C is within error bars. Annealing at temperatures
up to 425 °C thus does very little to the atomic density of $a$-Si films.

Atomic density provides a proxy to evaluate the thermodynamic and kinetic
stability of amorphous solids [51], where it is known that stability can be
tuned by means of growth temperature and rate on vapor-deposited glasses [52].
Increasing growth temperature increases surface mobility during deposition and
allows atoms to reach lower energy positions before being deposited over [53].
Similarly, slower growth rates allow these positions to be reached before
mobile atoms are deposited over. In the particular case of $a$-Si and for the
data presented in this work, atomic density shows a stronger dependence on
growth temperature and thickness rather than growth rate. For this reason, we
focus the present study on growth temperature and thickness dependence.

Resonant Rutherford backscattering spectrometry was performed on the films to
determine the oxygen content through the samples. Films grown at 50 and 425 °C
show 4% and 1% oxygen content, respectively, at all depths below the native
oxide layer. Oxygen content did not increase with film aging (after 4 months).
Neither contaminants nor water were observed via RBS and HFS, respectively, in
the films. Thickness measurements were also performed both immediately and
four months after deposition, no changes were observed in thickness as a
function of time.

###  III.3 Short- and Medium-range order

In this section we present electron microscopy results, specifically
diffraction, EELS, FEM, and high-resolution imaging results for $\sim 30$ nm
thick $a$-Si samples grown at 0.5 $\mathrm{\SIUnitSymbolAngstrom}$/s, and at
50 °C, 250 °C and 450 °C.

Figure 4 shows the average diffracted intensities $I(k)$. ($k$ defined as
$k=\theta/\lambda$, where $\lambda$ is the electron wavelength.) The peak
positions are typical of $a$-Si and do not shift significantly from sample to
sample. The first broad peak sits at $3.08\pm 0.05$ nm-1, which matches the
c-Si $\langle 111\rangle$ peak at 3.13 nm-1. The second $a$-Si peak position
at $5.54\pm 0.05$ nm-1 sits between the c-Si reflections $\langle 220\rangle$
at 5.2 nm-1 and $\langle 311\rangle$ at 6.1 nm-1. None of the data show the
sharp peaks that would indicate nanocrystallization. Isolated nanocrystals
were observed in the 450 °C film, indicating that under these growth
conditions crystallization of $a$-Si starts at a temperature between 425 °C
and 450 °C. Electron microscopy characterization was performed between the
nanocrystals on fully amorphous regions. Differences in peak heights between
films are not well quantified in this data. For more quantitatively accurate
$I(k)$ and structure factor $S(q)$, synchrotron experiments are needed.
Similarities between positions and widths for both low and high-$k$ peaks seen
in Fig. 4, however, suggest that there are no significant differences in SRO
for the films grown at different temperatures, consistent with previous work
on $a$-Si [26].

Figure 4: Diffracted electron intensity $I(k)$ for $a$-Si thin films, $\sim
30$ nm thick, grown at 0.5 $\mathrm{\SIUnitSymbolAngstrom}$/s, and at 50 °C
(brown), 250 °C (red) and 450 °C (orange). The diffraction peaks of c-Si are
shown for reference. Inset: Bulk plasmon relative change
$\Delta\lambda/\lambda_{0}$ from EELS as a function of growth temperature
T${}_{\textrm{S}}$ for the same samples. No significant change in average bond
length between the films is observed.

EELS measurements performed on the same samples find an average bulk plasmon
peak energy $\lambda_{0}$ of 16.6 eV. The relative change in $\lambda_{0}$,
$\Delta\lambda/\lambda_{0}$ is +0.2%, +0.5%, and -0.7% for samples grown at 50
°C, 250 °C and 450 °C, respectively (Fig. 4 inset). At constant composition
and temperature, $\Delta\lambda/\lambda_{0}\propto\sqrt{\Delta\rho/\rho_{0}}$,
so the maximum change in density, not counting contributions from voids, is
negligible. Together, these results show that the average bond length for
atoms within the network is the same for the different samples. The SRO
structure, specifically bond length and coordination, is independent of the
growth conditions, and there is no change in atomic number density except what
is created by introducing voids, to be discussed below.

FEM data of the same films shown in Fig. 4 are shown in Fig. 5. $V(k)$ peaks
in amorphous materials typically occur at the same $k$ as peaks in $I(k)$.
That is the case for the film grown at 50 °C, but the films grown at 250 °C
and 450 °C show a splitting of the first peak into a contribution near 3.09
nm-1 and another peak at lower $k$, closer to 2.6 nm-1. This peak indicates
that local structures exist in the higher substrate temperature samples with a
larger interatomic spacing than has been observed in any previous FEM
experiments on $a$-Si. There are no larger interatomic spacings in diamond
structure Si, but a different crystal structure for Si, Si24, which has
diffraction features at lower $k$ corresponding to the peak position in $V(k)$
[54]. The low-$k$ diffraction in Si24 arises from 8-membered rings. In that
work [54], the Si24 crystal was synthesized at high pressure with Na atoms
embedded in the crystal, filling the 8-atom Si rings. The material was brought
to room pressure, and the Na atoms were removed using thermal degassing. The
resulting crystal is stable at room-temperature and atmospheric pressure. We
do not suggest that there are nanocrystals of Si24 in the material, or even
that we have created a paracrystalline analog to Si24. Instead we suggest that
the low-$k$ feature in $V(k)$ of these $a$-Si films grown at higher
temperatures (225 °C and 425 °C) could be caused by structures in the
amorphous network involving large rings of 8 atoms or more, significantly
larger than those believed to typically be present in $a$-Si [55]. These could
even be viewed as extremely tiny nanovoids.

Figure 5: FEM data $V(k)$ as a function of wavevector $k$ of $a$-Si thin
films, $\sim 30$ nm thick, grown at 0.5 $\mathrm{\SIUnitSymbolAngstrom}$/s,
and at 50 °C (brown squares), 250 °C (red circles) and 450 °C (orange
triangles). Grey stars show MRO typically observed in $a$-Si films from Ref.
[29]. The data for the 250 °C and 450 °C films, however, reveal a MRO
structure not previously reported.

Fig. 6 shows under-focus, in-focus, and over-focus HRTEM images of the same
samples. Under- and over-focus image pairs are a classical method for
identifying voids in materials from the switch in contrast of the Fresnel
fringe surrounding the void. The 50 °C sample shows clear evidence of columnar
microstructure with nanometer-scale voids forming part of the column
boundaries. The higher temperature films do not have the columnar
microstructure and do not have voids detectable by this method. In our
previous work [12], cross-sectional TEM images (XTEM) of $a$-Si films shown a
columnar structure whose diameter increases with growth temperature. Such
structure is common in amorphous films grown by PVD techniques [56], including
e-beam evaporation [57]. These images also reveal a region close to the
substrate interface that lacks columnar microstructure; this region is thicker
for higher growth temperature. The columnar structure observed for the film
grown at 50 °C has not developed yet for films grown at 250 °C and 450 °C;
therefore, the thickness at which columns develop in $a$-Si thin films is
growth temperature dependent.

Figure 6: HRTEM images of $a$-Si thin films $\sim 30$ nm thick grown at 0.5
$\mathrm{\SIUnitSymbolAngstrom}$/s and at 50 °C, 250 °C and 450 °C (top to
bottom row, respectively). Columns from left to right show under-focus, in-
focus, and over-focus images (see description in text). The film grown at 50
°C shows columnar structure, visible as wiggly lines and voids visible as
white features in the under-focus image and black features in the same place
in the over-focus image. Films grown at higher temperatures show no notable
features.

###  III.4 Bond angle deviation and local strain

Electron microscopy was used as described above to characterize SRO. $I(k)$
and $\Delta\lambda/\lambda_{0}$ are mostly sensitive bond length and
coordination number, and less to bond angles. We here turn to Raman to
characterize the deviations from the tetrahedral angle $\theta$ that are found
in $a$-Si. Figure 7(a) shows the bond angle deviation $\Delta\theta$ as a
function of thickness for a series of growth temperatures. The range of
$\Delta\theta$ found for these samples, approximately 9° to 13°, is similar to
values reported for $a$-Si model structures [33]. Films grown at 425 °C show
$\Delta\theta$ that ranges from 9° to 10°. These values fall on the low end of
those obtained from the radial distribution function of $a$-Si [6]; and to our
knowledge are amongst the lowest reported by experiment. Note that
$\Delta\theta=0\degree$ for c-Si, and has been shown theoretically to be $\geq
6.6\degree$ for $a$-Si [33], evidence that the transition from crystalline to
amorphous structures is not continuous. $\Delta\theta$ is considered a measure
of disorder at very short length scales; nearest neighbor distances, or around
2 Å. At each thickness, $\Delta\theta$ is lower for films grown at higher
temperatures, indicating that disorder decreases with increasing growth
temperature.

Figure 7: (a) Bond angle deviation $\Delta\theta$, (b) TO peak position
$\omega$, and (c) trace of strain $tr(\epsilon)$ as a function of thickness t
for samples grown at 50 °C (brown squares), 225 °C (red circles) and 425 °C
(orange triangles). Lines are guides to the eye.

Fig. 7(b) shows the TO peak position $\omega$ and Fig. 7(c) shows the trace of
the local strain $tr(\epsilon)$ calculated from $\omega$ as described in II.3.
$\Delta\theta$ and $tr(\epsilon)$ shown in Figs. 7(a) and 7(c), respectively,
are correlated. Higher growth temperature thus yields less disordered and less
strained films at all thicknesses. However, the monotonic trend observed for
atomic density as a function of thickness is not seen for these properties:
both $\Delta\theta$ and $tr(\epsilon)$ increase with thickness up to a maximum
value, then decrease. This behavior is most clear in films grown at 50 °C,
where the maximum values of $\Delta\theta$ and $tr(\epsilon)$ occur near 30
nm. For films grown at higher temperature, the peak in $\Delta\theta$ and
$tr(\epsilon)$ is more subtle, and may occur at a different thicknesses.

Raman data analysis shows a relaxation process in which disorder and local
strain reduce above a critical thickness; however, it is unclear whether this
process implies a structural change in the atoms underneath, i.e., the
reorganization of the atoms already deposited.

Following the procedure described in II, we study $n_{at}$ and $\Delta\theta$
of several $a$-Si etched films. In Fig. 8 we compare these results to
equivalent as-deposited films to probe for reorganization during growth. The
larger error bars are due to increased surface roughness and thickness
uncertainty after the etch process. All films etched and measured were grown
at 50 °C because they exhibit the largest change in atomic density and bond
angle deviation as a function of thickness, as seen in Figs. 3(a) and 7(a),
respectively. The gray areas labeled ‘reorganization’ in Fig. 8 are the
extrapolated $n_{at}$ and $\Delta\theta$ values of thick as-deposited films. A
complete reorganization would yield etched thin films with the higher $n_{at}$
and lower $\Delta\theta$ of the as-deposited thicker films, whereas a lack of
reorganization would yield etched thin films with the lower density and higher
disorder of the as-deposited thin films.

Figure 8: (a) Atomic density $n_{at}$, and (b) bond angle deviation
$\Delta\theta$ as a function of thickness t for as-deposited (brown closed
squares) and etched (cyan open circles) films. All films were grown at 50 °C
and 0.5 $\mathrm{\SIUnitSymbolAngstrom}$/s; etched films initially were $\sim$
300 nm thick. Gray areas labeled ‘reorganization’ show the extrapolated
$n_{at}$ and $\Delta\theta$ values of the $\sim 300$ nm as-deposited films
prior to the etch process (see text for further details). Lines are guides to
the eye.

Fig. 8(a) shows that atomic density $n_{at}$ from etched films is higher than
that of as-deposited films with similar thickness, most consistent with
reorganization, with etched films lying within error bars of the as deposited
values of thick films and well above the values of as-deposited thin films.
Bond angle deviation $\Delta\theta$ from etched films is lower than that of
as-deposited films with similar thickness, shown in Fig. 8(b), which suggests
a reduction of disorder, and are also most consistent with reorganization of
the early layers of atoms deposited near the substrate.

It has been proposed that due to the large interface energy between substrate
and film, thin films prepared by physical vapor deposition grow via a Volmer-
Weber mode [58, 59]. This proposed growth mode is in agreement with stress
measurements in e-beam evaporated $a$-Si films, which show the nucleation of
columns that correlate with the transition from compressive to tensile stress
regimes as a function of thickness [60].

The results presented in this section show structural relaxation as a function
of thickness that leads to the reduction of disorder and local strain,
compatible with the previously described studies of the stress evolution
during growth of evaporated $a$-Si [60]. From this work [60], we also see that
the dependence with thickness of the compressive-to-tensile stress transition
is in agreement with our structural observations by XTEM, which show the onset
of the columnar growth [12]. The stress of $a$-Si films grown at room-
temperature was measured through the curvature of their substrates reporting
tensile stresses of $694\pm 94$ MPa for a 30 nm film, and $523\pm 20$ MPa for
a 90 nm film.

###  III.5 Transverse sound velocity

Transverse sound velocity $v_{t}$ as a function of thickness for films grown
at different temperatures is shown in Fig. 9. Sound velocity is only weakly
dependent on thickness, perhaps within error bars of constant, whereas it
increases with increasing growth temperature.

Figure 9: Transverse sound velocity $v_{t}$ as a function of thickness t of
$a$-Si films grown at 50 °C (brown squares), 225 °C (red circles) and 425 °C
(orange triangles). Lines are guides to the eye.

Sound velocity in solids, including amorphous solids, is due to phonons that
propagate through the material, with a velocity that is independent of the
frequency of the phonons for relatively long wavelength phonons. The sound
velocity is generally dependent on the interatomic spacing, the atomic mass,
and the interatomic bond strength. In $a$-Si, the increase of sound velocity
$v_{t}$ with increasing growth temperature, shown in Fig. 9, correlates with
the reduction of atomic disorder ($\Delta\theta$) and local strain
($tr(\epsilon)$), shown in Figs. 7(a) and 7(c), respectively. That sound
velocity depends strongly on growth temperature, but is not strongly dependent
on thickness at constant growth temperature, despite significant changes in
atomic density with thickness, shows that sound waves are carried through an
$a$-Si network. Sound velocity, therefore, is not affected directly by the
overall density, i.e., it is not much affected by nanovoids.

###  III.6 Dangling bond density

Dangling bond density $\rho_{DB}$ has long been considered a defect metric and
is correlated with surface-state transitions in silicon [61]. In photovoltaic
and semiconductor technologies, $a$-Si is prepared with hydrogen in order to
passivate dangling bonds and create high quality, or device quality, films.
The dangling bond densities of our films are on the order of 1018 spins/cm3
(see Fig. 10). Standard $a$-Si has dangling bond densities $\sim 10^{19}$
spins/cm3, and device-quality $a$-Si:H shows values $<10^{16}$ spins/cm3 [62].
We report $\rho_{DB}$ per unit mass (g-1) rather than per unit volume (cm-3)
because the atomic density of our samples is not constant, which makes the
former a more accurate metric for direct comparisons.

Figure 10: Dangling bond density $\rho_{DB}$ as a function of thickness t of
$a$-Si films grown at 50 °C (brown squares), 225 °C (red circles) and 425 °C
(orange triangles). All samples were grown at 0.5
$\mathrm{\SIUnitSymbolAngstrom}$/s. Lines are linear fits to the data.

Figure 10 shows that $\rho_{DB}$ decreases significantly with increasing
growth temperature, and mildly increases with thickness for all growth
temperatures. The former statement is consistent with the strong dependence of
atomic density $n_{at}$ on growth temperature, but is the opposite to how
$n_{at}$ depends on thickness, suggesting that the relationship between
$n_{at}$ and $\rho_{DB}$ is not simple. The dependence of $\rho_{DB}$ on
growth temperature suggests that the nucleation of dangling bonds is inversely
proportional to surface diffusion; specifically, that higher growth
temperature yields more 4-fold coordinated atoms. That dangling bond density
mildly increases with thickness for all growth temperatures, suggests that the
formation of this type of defect is not related to the mechanisms responsible
for the films’ atomic density or the atomic reorganization process previously
discussed.

The short- and medium-range order, bond angle deviation, local strain and
dangling bond density results report information about the distribution of
silicon atoms in the films and about specific electronic defects;
specifically, EELS results show that the interatomic distances do not change
with growth temperature. Therefore, these results do not explain the very low
atomic density values, and particularly that of the thinnest films grown at
room-temperature. In the next section, we study the presence of nanovoids,
which in $a$-Si cannot be detected by microscopy techniques, but are
considered a common structural defect in $a$-Si [63].

###  III.7 Nanovoids characterization

Doppler broadening spectroscopy results for films of different thicknesses and
grown at 50 °C are shown in Fig. 11, where the S-parameter is plotted as a
function of energy. Energy is proportional to penetration depth (as described
in II.6) and, for the films reported in this work, it ranges from 0 keV
(surface) to $\sim 30$ keV (substrate). We could not measure films thinner
than 60 nm due to lack of sensitivity. The data acquired from the annihilation
of positrons confirm that all $a$-Si films measured, grown at different
temperatures and for different thicknesses, contain nanovoids. However, the
S-parameter, which is proportional to the empty space volume, is not
calibrated for $a$-Si limiting our ability to quantify the size of its
nanovoids. Electron microscopy cannot detect the presence of nanovoids in
$a$-Si films due to the atomic superposition and the low atomic scattering
factor of silicon. For these reasons, we can only report a qualitative
analysis of the data and establish relative trends for the empty space, or
nanovoids, volume measured in these $a$-Si films.

Figure 11: S-parameter normalized to the c-Si S-parameter
(S${}_{\textrm{{c-Si}}}=0.5324$) as a function of incident positron energy E
of three $a$-Si films grown at 50 °C and 0.5
$\mathrm{\SIUnitSymbolAngstrom}$/s, with thicknesses of 66 nm, 183 nm and 308
nm (light to dark grey, respectively). Data at 0 keV are values from the
$a$-Si surface, and move progressively through the film for increasing
energies until the beam reaches the substrate at $\sim 30$ keV. Inset: detail
at low energies showing the accuracy of the single (dashed lines) and dual
(solid lines) models. The dual model improves its relative accuracy with
respect to the experimental data almost 50% compared to the single model.
Curves have been vertically shifted to better show the fittings.

We used two different models to simulate the experimental data: 1) a single
volume of empty space that is uniformly distributed throughout the film (one
S-parameter), and 2) a dual volume of empty space that are uniformly
distributed throughout the film (two S-parameters), in which the nanovoids
volume closer to the substrate is different than that closer to the surface.
The simulations yield better results for the dual model, which is almost 50%
more accurate than the single model (see inset in Fig. 11). The dual volume of
empty space is the simplest model to probe whether the data suggests a more
complex distribution of nanovoids than a single volume of empty space model.
These results suggest that, at least, two different distributions of nanovoids
are present on $a$-Si films thicker than 60 nm, with the interface between the
two distributions at $31\pm 6$ nm from the substrate for films grown at 50 °C,
and at $59\pm 17$ nm for films grown at 425 °C. This increase of the first
nanovoids layer thickness with growth temperature is in agreement with the
thickness dependence of void tracks formation with growth temperature seen in
$a$-Si films grown by Floro et al. [60]. The dual model also reports that the
first layer, closer to the substrate, has a larger volume of empty space than
the second layer, closer to the surface. However, since the S-parameter to
nanovoid size calibration is not available, we cannot quantify the volume
reduction of empty space between the two distributions and the relative change
in size and concentration of nanovoids.

$S$ vs $W$ plots are shown in Fig. 12 for the same films shown in Fig. 11 and
a $\sim 300$ nm film grown at 425 °C. Typical $S$ vs $W$ plots show
annihilated positrons with a constant $S/W$ ratio from surface to substrate (a
single straight line). Our results, however, show a “V” shaped line due to the
presence of different types of nanovoids across the film (nanovoids of
different size). In other words, different $S$-$W$ pairs, or slopes, represent
nanovoids with a chemically distinct surface bonding. Steeper $S$-$W$ slopes
correspond to larger nanovoids. Points at the end of the “V” with moderate $S$
and high $W$ values exist at the surface of the film, and progressively change
by following the “V” shape to the other end with low $S$ and high $W$, where
positrons annihilate at the interface between the two distributions of
nanovoids (see Fig. 12 and corresponding labels). The chemical change of the
nanovoids across the distinct distributions is due to the distinct chemical
nature of electrons present in them, i.e., to the nanovoids sizes and not to
their concentration.

The analysis of the $S$ data suggests that a single distribution is less
likely than a dual distribution of empty space volume, or total nanovoids
volume, as shown in Fig. 11 inset, and the $S$-$W$ data in Fig. 12 shows more
than one type of nanovoid, which significantly differ in the nature of the
bonding at their inner surfaces. Films grown at all temperatures show that the
distribution closer to the surface has a smaller empty space volume than the
distribution closer to the surface. Additionally, the empty space volume in
the distribution closer to the surface is smaller for films grown at 425 °C
compared to films grown at 50 °C.

Figure 12: $S$ vs $W$ plots for films grown at 50 °C, with 66 nm, 183 nm and
308 nm (light to dark grey squares, respectively), and for a film grown at 425
°C with 291 nm (orange triangles). Vertical and horizontal axes are normalized
to the c-Si $S$ and $W$ values, with S${}_{\textrm{{c-Si}}}=0.5324$ and
W${}_{\textrm{{c-Si}}}=0.0237$. Solid lines are $S$-$W$ fits considering a
bilayer distribution of nanovoids. Labels indicate the position where
positrons annihilate from film surface to substrate.

In $a$-Si, dangling bonds are found to occur on the inner surfaces of
nanovoids [64, 65, 66, 67]. However, the relationship in the literature
between dangling bonds and nanovoids is unclear.

The existence of nanovoids in $a$-Si, confirmed by DBS measurements, provides
an explanation for the the low atomic density values of the films reported in
this work.

##  IV Discussion and conclusions

Atomic density $n_{at}$ of $a$-Si films increases with increasing growth
temperature and thickness, and with reducing growth rate (Fig. 3). In all
cases films are underdense, with $n_{at}$ as high as $\sim 99\%$ of c-Si
density for films grown at 425 °C and thicker than 300 nm, and as low as $\sim
78\%$ of c-Si density, which is achieved for the thinner films grown at room-
temperature. FEM results on $\sim 30$ nm thick films show significant
differences in MRO, as shown in Fig. 5, despite no change in SRO (Fig. 4),
which suggests that larger membered rings appear as the growth temperature
increases. Measurements of the plasmon peak energy relative change
$\Delta\lambda/\lambda_{0}$ presented in the inset of Fig. 4, show no
significant changes in atomic density between films as a function of growth
temperature. Therefore, the reduction of $n_{at}$ can only be attributed to
voids. Those voids must be nanometer-scale, as large voids would be visible in
HRTEM.

Raman data analysis reveals a reduction of the atomic disorder, the bond angle
deviation $\Delta\theta$, with increasing growth temperature [Fig. 7(a)]. This
dependence on growth temperature is also observed for the position of the
transverse-optic peak [Fig. 7(b)], which correlates with the local strain
$tr(\epsilon)$ of the $a$-Si films [Fig. 7(c)]. These data show a non-
monotonic behavior of disorder and local strain with thickness, particularly
for films grown at room-temperature, whose magnitudes build up and then reduce
to their initial values. The characterization of nanovoids shows that the best
simulations of the data (Fig. 11) and the analysis of the $S$-$W$ plots (Fig.
12) suggest the presence of at least two different types, or sizes, of
nanovoids, with larger empty space volume near the substrate. The thickness at
which the transition between the two distributions occurs, increasing from
approximately 30 nm to 60 nm with increasing growth temperature, coincides
with the critical thickness seen by Raman. The stress results also report
higher tensile stress for thinner films, $\sim 30$ nm thick, than for thicker
films, $\sim 90$ nm thick. We therefore speculate that at the critical
thickness a structural relaxation process occurs, triggering the formation of
columns observed by electron microscopy [60, 12], and the reduction of
$\Delta\theta$ (atomic disorder) and $tr(\epsilon)$ (local strain). Floro et
al. [60], see similar effects, albeit at slightly different thicknesses.

The study of $n_{at}$ and $\Delta\theta$ from etched samples, Figs. 8(a) and
8(b), respectively, shows reorganization of the films’ structure during
growth. These results do not suggest a direct relationship between
reorganization and critical thickness, even though they might be caused by a
common underlying relaxation process that takes place during growth. We note
that the atomic density increases monotonically with thickness, suggesting
that the reorganization of the films continuously happens during growth,
whereas the critical thickness seems to be triggered by an specific event.
Additionally, DBS results based on a dual distribution of empty space volume
show that the total volume of nanovoids reduces as films grow thicker. Our
data do not allow us to establish whether reorganization and critical
thickness are two processes independent of each other, or whether one is
triggered by the other.

Dangling bond density $\rho_{DB}$ results report up to $\sim 1$ defect in
$10^{4}$ atoms, which implies that Si atoms in $a$-Si films are largely fully
coordinated despite their notably low atomic densities in some films, with up
to $\sim 22\%$ missing atoms (compared to c-Si). These results suggest that
only a very small fraction (less than 1%) of nanovoids contain even one
dangling bond. Additionally, energetic considerations suggest that having two
dangling bonds in a single nanovoid is unlikely because two nearby under-
coordinated silicon atoms will bond together. Therefore, the increase of
$\rho_{DB}$ with thickness (Fig. 10) suggests an increase of the nanovoid
number, which in combination with the densification of the films (Fig. 3)
imply that nanovoids become smaller as films grow thicker. This conclusion is
also supported by DBS results (Fig. 12).

Two distinct processes happen in e-beam evaporated $a$-Si films during growth:
i) a sudden structural relaxation at a critical thickness, which depends on
growth temperature and correlates with the formation of columns [60, 12]
(Figure 6). This process is also captured by the reduction of atomic disorder
and local strain (Figure 7), and by the reduction of nanovoids size and their
total volume (Figure 12). And ii) a continuous reorganization of the atoms
seen in the atomic density and bond angle deviation of etched thicker films
(Figure 8). This process is responsible for the reduction of the total volume
of nanovoids as films grow thicker, which leads to the continuous
densification of the films (Figure 3) and to the reduction of TLSs [10, 68].

As previously reported, the TLS density measured for $a$-Si depends strongly
on growth conditions, with a dependence that can be completely explained by
plotting TLS density as a function of atomic density. Remarkably, and as
previously observed [10], the excess heat capacity at low temperature, below
10 K, of thinner $a$-Si films (112 nm) is larger than that of thicker films
(278 nm). This observation cannot solely be explained by the larger
concentration of nanovoids in the first layer (near the substrate), which
indeed would yield thinner films with larger excess specific heat, but not
heat capacity. This excess heat capacity reduction can only be explained by
atomic reorganization, which effectively reduces the number of TLSs when films
grow thicker. Similarly, the mechanical loss of thinner $a$-Si films (59 nm)
is larger than that of thicker films (299 nm) [68]. Our data suggest that the
structural origin of TLSs in $a$-Si likely occur in highly disordered regions
where nanovoids are present. The reduction of atomic disorder and nanovoids
volume (increase in atomic density) correlates with the reduction of TLSs
observed in thicker $a$-Si films. The reorganization of the atoms as films
grow thicker plays a crucial role in reducing TLSs in $a$-Si, and it is likely
to be related to local structural rearrangements caused by structural
relaxation processes. The reduction of TLSs with growth temperature is likely
to be related to surface diffusion mechanisms during growth, since surface
diffusion is enhanced by temperature. Additionally, Fig. 9 shows that sound
velocity increases with increasing growth temperature, while its dependence on
thickness is within error bars of no dependence, suggesting that the sound
velocity depends on properties of the $a$-Si network, such as bond angle
disorder, and not on the presence of nanovoids; while nanovoids and their
environment, and not the constrained network, are responsible for the
mechanisms that enable TLSs in $a$-Si.

## Acknowledgements

We thank P. Ci and J. Wu for assistance with Raman measurements; R. Chatterjee
and J. Cooper for assistance with EPR measurements; D. Castells-Graells for
assistance with films growths; and D. Strubbe for fruitful discussions on
$a$-Si:H Raman spectra calculations. Samples preparation, ion beam analysis
and Raman characterization were done at UCB supported by NSF (DMR-1508828 and
DMR-1809498). Electron microscopy characterization was done at UW-Madison
supported by the Wisconsin MRSEC (DMR-1720415). Sound velocity
characterization was done at NRL supported by the Office of Naval Research.
Electron paramagnetic resonance measurements were done at LBNL Joint Center
for Artificial Photosynthesis supported by DOE (DE-SC0004993). Doppler
broadening spectroscopy characterization was done at WSU supported by the late
Dr. Kelvin Lynn and the Center of Materials Research (now Institute of
Materials Research).

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# The 1st Fermi LAT SNR Catalog: Constraining the Cosmic Ray Contribution

###### Abstract

Despite tantalizing evidence that supernova remnants (SNRs) are the source of
Galactic cosmic rays (CRs), including the recent detection of a spectral
signature of hadronic $\gamma$-ray emission from two SNRs, their origin in
aggregate remains elusive. We address the long-standing question of Galactic
CR nuclei origins using our statistically significant GeV SNR sample to
estimate the contribution of SNRs to directly observed CRs. Interactions
between CRs and ambient gas near the SNRs emit photons via pion decay at GeV
energies, providing an in situ tracer for CRs otherwise measured directly with
balloon-borne and satellite experiments near the Earth. To date, the Fermi LAT
SNR Catalog has detected more than 50 SNRs and potential associations in
classes with a variety of properties, yet all remain possible accelerators. We
investigate the GeV and multiwavelength (MW) emission from SNRs to constrain
their maximal contribution to observed Galactic CRs. Our work demonstrates the
need for improvements to previously sufficient simple models describing the
GeV and MW emission from these objects.

##  1 Introduction

Direct measurements of cosmic ray (CR) energetics and composition combined
with our understanding of high energy accelerators in the Galaxy have long
suggested that supernova remnants (SNRs) are likely the source of Galactic
CRs. Yet proof has for a long while remained elusive. With the advent of
$\gamma$-ray telescopes with degree-scale spatial resolution in addition to
good spectral resolution, we have made significant strides towards more firmly
associating SNRs with CR acceleration. These have included individual SNR
spectra tending towards being dominated by hadronic emission, where CRs
interacting with the local medium emit $\gamma$-rays via $\pi^{0}$ decay (e.g.
[1]), as well as recent detection of proton acceleration through the low-
energy $\pi^{0}$ cutoff [2].

Yet while such individual results are necessary, they are not sufficient, for
the direct data we would ultimately like to compare to is comprised of CRs
from sources throughout the Galaxy. Thus, it is also necessary to show that
the aggregate contribution from all sources can produce the observed CRs. In
order to do so, we have leveraged several years’ worth of Fermi Large Area
Telescope (LAT) survey data to study systematically all known Galactic SNRs,
the majority of which are detected in the radio and compiled in [3], as well
as a few identifications from other wavelengths. Details of the analysis
procedure are laid out in [4], along with a discussion of the implications of
a radio-GeV flux correlation. As interstellar $\gamma$-ray emission is quite
prevalent along the galactic plane, where the majority of SNRs lie, we also
explore systematics related to the choice of interstellar emission model in
[5, 6] to ensure that we will have the most robust results possible. These
results point to a separation of SNRs into classes, notably those which are
young and those which are interacting, often with molecular clouds [4].

##  2 Particle Populations

In [4], we showed that the synchrotron radio emission from high energy leptons
tends to be correlated for interacting SNRs, suggesting a physical link,
whereas the young SNRs showed more scatter.

###  2.1 Emission Mechanisms

If radio and GeV emission arise from the same particle population(s), e.g.
leptons and hadrons accelerated at the SNR shock front, under simple
assumptions, the GeV and radio indices should be correlated. For inverse
Compton (IC) emitting leptons, the GeV and radio photon indices ($\Gamma$ and
$\alpha$ respectively) can be related as $\Gamma=\alpha+1$ whereas for
$\pi^{0}$ decay and e± bremsstrahlung, $\Gamma=2\alpha+1$. Figure 1 shows
that, contrary to our radio/GeV flux observations, young SNRs seem consistent
with expectations from these simple models. Several of the known, young SNRs
are more consistent with the IC relation (dashed line), suggesting that they
may be lepton-dominated and emitting via IC in the GeV regime. SNRs emitting
via a combination of mechanisms in this scenario have indices falling between
the two index relations, that is, the region spanned by the
$\pi^{0}$/bremsstrahlung (solid) and IC (dashed) lines. The young SNR RX
J1713-3946 is one example which bears out this case [7]. Other SNRs, including
those observed to be interacting with molecular clouds, are softer than
expected, and in most cases are not even consistent with combinations of
emission mechanisms.

The apparent lack of correlation between the indices and emission mechanisms
for the majority of observed SNRs suggests that the data are now able to
challenge model assumptions for these SNRs. These assumptions include that:

  * •

the underlying leptonic and hadronic populations may have different power law
indices;

  * •

the emitting particle population(s) may not follow a power law but may instead
have break(s);

  * •

or there may be different zones with different properties dominating the
emission at different wavelengths.

Figure 1: GeV-Radio index and expected slope correlation for: $\pi^{0}$ decay
or e± bremsstrahlung (solid line) and inverse Compton (dashed line). As
described in more detail in [4], the colors correspond to different types of
SNRs and SNR candidates, namely, young SNRs are blue; those identified,
interacting SNRs are red; the green points correspond to newly identified
SNRs; and the grey points to point-like candidate SNRs (dark grey) and point-
like candidate pulsars (light grey). This scheme is maintained throughout the
paper.

###  2.2 Spectral Break?

With SNRs studied in TeV, we have the opportunity to explore the second of the
model assumptions: that the emitting particle population(s) may have breaks.
Such a break in the underlying particle population(s) can also cause a break
in the observed spectrum. As TeV emission may arise from the same mechanisms
as the Fermi-observed GeV emission, we might expect to see such a break
reflected in the spectrum combining Fermi data with observations from Imaging
Air Cherenkov Telescopes (IACTs) such as H.E.S.S., VERITAS, and MAGIC. In
Figure 2 we plot the GeV index versus TeV index for all SNRs observed with
both Fermi and an IACT. Several SNRs’ TeV indices are lower than their GeV
index, and few lie above the line of equal index. This suggests a break either
at or between GeV and TeV energies. The former has been observed for e.g.
IC443 [8] and the latter in, e.g. the young SNR RX J1713-3946 [7]. Such a
break would be a tantalizing clue, likely reflecting a break in the underlying
particle spectrum, as it does for IC443 and RX J1713-3946, of many SNRs. We
note however that, as the TeV sources are not uniformly surveyed, inferring
population statements from this observation requires a careful understanding
of the non-TeV observed SNR subsample.

Figure 2: GeV-TeV index. The line shows equal indices. SNRs lying below the
line suggest that their spectra have breaks, potentially reflecting a break in
the underlying particle population(s’) index or indices.

We note that the GeV-TeV index plot (Fig. 2) also shows a distinct separation
between young and interacting, often older SNRs, suggesting an evolution in
index with age, from harder when younger to softer when older. We explore this
further by explicitly investigating the evolution of the GeV index with age in
the next section.

##  3 Evolution or Environment?

As we saw a division between young SNRs having harder indices and interacting
SNRs’ tending to softer ones, we explicitly examine the evolution of the GeV
index with age of the SNR. In Figure 3 we see a clear separation of known
young SNRs having lower, harder GeV indices than interacting SNRs, where ages
were drawn from the literature. This separation could be due to decreasing
shock speed, decreasing the maximum acceleration energy as SNRs age. Particles
will be more readily able to escape a slower shock, thereby reducing the
$\gamma$-ray flux. Further, a less energetic shock will no longer be able to
accelerate particles to the highest energy, thereby reducing emission at the
highest energies.

Figure 3: Age versus GeV index. The young (blue) SNRs are separated in GeV
index from the identified interacting SNRs (red).

In [4], we observed that SNRs known to be interacting, in particular with
large molecular clouds, appear to be more luminous in GeV $\gamma$-rays than
young SNRs. The apparent difference in indices and luminosities for the young
and interacting SNRs may also be caused by differing environments: older SNRs
may interact with denser surroundings not yet reached by younger SNRs. Using
MW information such as ambient density estimated from thermal X-rays in the
catalog context will help disentangle the effects of evolution and
environment.

##  4 Constraining CR Acceleration

Recent work examining GeV $\gamma$-ray data at the $\pi^{0}$ rest mass [2] has
added another piece to the accumulating evidence (e.g. [1, 9, 10]) that SNRs
accelerate hadrons: at least two SNRs, IC443 and W44, show evidence of the
$\pi^{0}$ low energy break ($E<100$ MeV), demonstrating that they accelerate
protons. To this necessary evidence that SNRs accelerate hadrons, we must also
add an understanding of the Galactic SNR population’s ability to accelerate
the appropriate composition of hadrons to the energies observed by direct-
detection experiments.

For a SNR at a given distance $d$ interacting with a density $n$ and
accelerating cosmic rays to a maximum energy $E_{CR,max}\gtrsim 200$ GeV with
index $\Gamma_{CR}\approx 2.5$, we can relate the $\gamma$-ray flux above $1$
GeV to the SNR’s energy, $E_{SN}$, and CR acceleration efficiency,
$\epsilon_{CR}\equiv\frac{E_{CR}}{E_{SN}}$, as:

$\begin{split}F(>1\,\textrm{GeV})\approx
10^{-8}&\times\frac{\epsilon_{CR}}{0.1}\times\frac{E_{SN}}{10^{51}\,\textrm{erg}}\\\
&\times\frac{n}{1\,\textrm{cm${}^{-3}$}}\times\left(\frac{d}{1\,\textrm{kpc}}\right)^{-2}\,\textrm{cm${}^{-2}$
s${}^{-1}$}\end{split}$ (1)

which is consistent with e.g. [11]. It is useful to note that this derivation
includes the approximation that the majority of the transfer of SN explosion
energy to hadrons occurs during the Sedov(-like) phase, and that the
efficiency remains roughly constant during this period (see [11] for further
discussion).

Alternatively, we can allow the CR index and maximal energy to vary. Fixing
the acceleration efficiency to a reasonable $\epsilon_{CR}=1\%$, we can plot
the $\gamma$-ray flux as a function of $\Gamma_{CR}$ and $E_{CR,max}$, as seen
in Figure 4.

Figure 4: Under standard assumptions, a SNR’s $\gamma$-ray flux above $1$ GeV
can be related to the accelerated CRs’ maximal energy and index for a given
acceleration efficiency ($0.01$), effective density ($1$ cm-3), and distance
to the SNR ($1$ kpc).

Figure 4 shows that if we know the CR index and maximal energy for a SNR with
a given $\gamma$-ray flux, distance, and density, we can determine its CR
acceleration efficiency and thereby, the amount of energy going into CRs. [12]
compare theoretically calculated efficiencies, including time evolution of the
SNR-interstellar material system, to measured GeV luminosities for several
detected SNRs. With the $1^{st}$ Fermi SNR Catalog, we have measured fluxes or
upper limits for all SNRs in the energy range $1-100$ GeV [4]. Moreover, we
can fix the CR index to, for instance, the GeV $\gamma$-ray index, self-
consistently measured with the flux. Finally we can, for instance, constrain
$E_{CR,max}$ by relating it to either a SNR’s measured break energy or to the
maximum energy inferred from CRs interacting with the interstellar medium and
creating the diffuse Galactic $\gamma$-ray background. [13] and [14]
illustrate the extraction of CR parameters from the diffuse Galactic
$\gamma$-ray background. Combining the flux (upper limit), inferred index, and
upper limit on CRs’ maximum energy, we can place an upper limit on the energy
transferred from a given supernova explosion to its CRs. Doing so for all
known SNRs yields the total energy being transferred to CRs. If this is less
than the observed total CR energy content, within the limits of our
assumptions, another source must contribute to accelerating particles to CR
energies.

We can examine this explicitly for SNRs with GeV flux upper limits and
distances. Assuming that they are merely faint rather than less energetic, we
can use an index of $2.5$, about average for those observed so far. With a
maximum CR energy of $E_{CR,max}\gtrsim 200$ GeV, we find that the GeV flux is
nearly independent of the energy (for indices $\Gamma\gtrsim 2.0$), and scales
equation (1) by $1.5$. Solving for the efficiency,

$\begin{split}\frac{\epsilon_{CR}}{0.1}&\times\frac{n}{1\,\textrm{cm${}^{-3}$}}\\\
&\approx\frac{F(1-100\,\textrm{GeV})}{1.5\times 10^{-8}\,\textrm{cm${}^{2}$
s}}\times\left(\frac{d}{1\,\textrm{kpc}}\right)^{2}\end{split}$ (2)

for SNRs with a canonical energy of $10^{51}$ ergs. We note that if we use an
observed flux, this efficiency is that for particles accelerated up to and
including the moment of observation.

As it is also necessary to know the density of the medium with which the SNR
interacts as well as the distance to the SNR, for our preliminary study, we
turn to the $\sim 175$ SNRs detected in X-rays [15], in search of those with
thermal emission. The thermal X-ray emission from the shock-heated inter-
and/or circumstellar mediums places reasonable constraints on the density.
Likewise, most of these SNRs have a distance estimate. Densities may
subsequently also be obtained from measurements such as IR emission from
collisionally heated dust and hydrodynamics.

We will explore methods and implications for constraining Galactic SNRs’
contribution to the observed CRs by studying the efficiency, including using
flux upper limits under the assumption of entirely hadronic processes.

##  5 Conclusions

By examining correlations between SNRs’ radio and GeV indexes, we observed
that, while several known young SNRs tend to follow the expected radio-GeV
index correlation for standard emission mechanisms, the majority do not. This
challenges the previously sufficient models assumptions. In particular, we
explored the hypothesis that the underlying particle population(s) may have a
spectral break, correlating to a break in their emission spectrum. The GeV-TeV
index correlation does in fact show that several SNRs have spectral breaks at
or between GeV and TeV energies.

The GeV index’s evolution with SNR age suggests that the decreasing shock
speed or decreasing maximum acceleration energy may cause the GeV index to
tend to soften. Combining this with the observed luminosity differences also
allows a scenario where the fainter, harder young SNRs are interacting with
less dense material, while older, interacting SNRs have eg reached nearby
local overdensities, such as molecular clouds. Multiwavelength information in
the context of the $1^{st}$ Fermi SNR Catalog will help disentangle the
effects of evolution and environment.

We also explored a method for constraining SNR’s contribution to the observed
Galactic CR flux using the flux and index measured in the $1^{st}$ Fermi SNR
Catalog, constraining the maximum CR energy using the diffuse Galactic
$\gamma$-ray flux, for SNRs with known distances. Such a constraint may allow
us to infer if another source population may be contributing to the measured
Galactic CR flux or alternatively, if we need to refine the assumptions made.

Acknowledgment: The Fermi LAT Collaboration acknowledges generous ongoing
support from a number of agencies and institutes that have supported both the
development and the operation of the LAT as well as scientific data analysis.
These include the National Aeronautics and Space Administration and the
Department of Energy in the United States, the Commissariat à l’Energie
Atomique and the Centre National de la Recherche Scientifique / Institut
National de Physique Nucléaire et de Physique des Particules in France, the
Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in
Italy, the Ministry of Education, Culture, Sports, Science and Technology
(MEXT), High Energy Accelerator Research Organization (KEK) and Japan
Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg
Foundation, the Swedish Research Council and the Swedish National Space Board
in Sweden.

Additional support for science analysis during the operations phase is
gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy
and the Centre National d’Études Spatiales in France.

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