# Fundamental physics with the Lyman-alpha forest: constraints on the growth
of structure and neutrino masses from SDSS with effective field theory

Mikhail M. Ivanov<EMAIL_ADDRESS>Center for Theoretical Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Michael W.
Toomey<EMAIL_ADDRESS>Center for Theoretical Physics, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  Naim Göksel Karaçaylı
<EMAIL_ADDRESS>Center for Cosmology and AstroParticle Physics, The Ohio
State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA
Department of Astronomy, The Ohio State University, 4055 McPherson Laboratory,
140 W 18th Avenue, Columbus, OH 43210, USA  Department of Physics, The Ohio
State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA

###### Abstract

We present an effective field theory (EFT) approach to extract fundamental
cosmological parameters from the Lyman-alpha forest flux fluctuations as an
alternative to the standard simulation-based techniques. As a first
application, we re-analyze the publicly available one-dimensional Lyman-alpha
flux power spectrum data from the Sloan Digital Sky Survey (SDSS). Our
analysis relies on informative priors on EFT parameters which we extract from
a combination of public hydrodynamic simulation and emulator data. Assuming
the concordance cosmological model, our one-parameter analysis yields a $2\%$
measurement of the late time mass fluctuation amplitude $\sigma_{8}=0.841\pm
0.017$, or equivalently,the structure growth parameter $S_{8}=0.852\pm 0.017$,
consistent with the standard cosmology. Combining our EFT likelihood with
Planck + baryon acoustic oscillation data, we find a new constraint on the
total neutrino mass, $\sum m_{\nu}<$ 0.08 eV (at 95% CL). Our study defines
priorities for the development of EFT methods and sets the benchmark for
cosmological analyses of the Lyman-alpha forest data from the Dark Energy
Spectroscopic Instrument.

††preprint: MIT-CTP/5721

Introduction. The distribution of galaxies and matter on cosmic scales, known
as large-scale structure, is a key probe of cosmological physics. In
particular, the cosmic web has been instrumental in discoveries and subsequent
studies of dark matter and dark energy, thereby helping establish the standard
$\Lambda$ Cold Dark Matter ($\Lambda$CDM) model of cosmology. Within
$\Lambda$CDM, structure formation has a hierarchical pattern: small structures
form in the early Universe at high redshifts while larger structures form at
later times. Some of the earliest and smallest clustering structures available
for observations are neutral hydrogen clouds that fill the intergalactic
medium at redshifts $z\gtrsim 2$. These clouds absorb light from distant
quasars, creating characteristic combs in their spectra, known as the Lyman-
alpha (Ly$\alpha$) forest Gunn and Peterson (1965); Croft  _et al._ (1997,
1998, 2002); Hui and Gnedin (1997); Gnedin and Hui (1998); McDonald  _et al._
(2000, 2006); Viel and Haehnelt (2006). Fluctuations in the transmitted flux
from quasars trace the underlying dark matter density, providing a probe of
fundamental physics at scales of a few Megaparsec and redshifts $2\lesssim
z\lesssim 5$. Thanks to this advantageous property, the Ly$\alpha$ forest data
have provided an excellent probe of $\Lambda$CDM, dark matter, neutrino
masses, primordial features, and many other extensions of the concordance
model McDonald  _et al._ (2006); Seljak  _et al._ (2006); Viel and Haehnelt
(2006); Viel  _et al._ (2005, 2013); Slosar  _et al._ (2011, 2013); Boyarsky
_et al._ (2009); Palanque-Delabrouille  _et al._ (2015a, b); Baur  _et al._
(2016, 2017); Iršič  _et al._ (2017a); Palanque-Delabrouille  _et al._ (2020);
Boyarsky  _et al._ (2019); Goldstein  _et al._ (2023).

Decades of observational and theoretical efforts in Ly$\alpha$ forest physics
have culminated in percent-level precision measurements of the baryon acoustic
oscillations, flux power spectrum (FPS) and three-dimensional correlations of
the Ly$\alpha$ forest flux with the Sloan Digital Sky Survey (SDSS) Slosar
_et al._ (2013); Palanque-Delabrouille  _et al._ (2013); Chabanier  _et al._
(2019); du Mas des Bourboux  _et al._ (2020a), and FPS measurements Iršič  _et
al._ (2017b); Walther  _et al._ (2018); Day  _et al._ (2019); Karaçaylı  _et
al._ (2022) with high-resolution spectrographs such as X-Shooter Vernet  _et
al._ (2011); López  _et al._ (2016), HIRES Vogt  _et al._ (1994); O’Meara  _et
al._ (2017), and UVES Dekker  _et al._ (2000); Murphy  _et al._ (2019). The
number of quasar spectra with the Ly$\alpha$ forest will soon increase
multifold with the ongoing Dark Energy Spectroscopic Instrument (DESI)
Karaçaylı  _et al._ (2024); Ravoux  _et al._ (2023).

With great statistical power comes great responsibility for modeling accuracy.
This is particularly important given that the effective slope and amplitude of
the linear matter power spectrum inferred from the SDSS 1D FPS are found to be
in some tension with the $\Lambda$CDM model Palanque-Delabrouille  _et al._
(2015a); Chabanier  _et al._ (2019); Goldstein  _et al._ (2023); Rogers and
Poulin (2023). To reconcile this potential discrepancy new physics
interpretations of this tension have been suggested, e.g. in Hooper  _et al._
(2022); Goldstein  _et al._ (2023); He  _et al._ (2023); Rogers and Poulin
(2023). Given the significance of these results, it is imperative to
independently test the modeling assumptions behind Ly$\alpha$ forest power
spectrum analyses.

The standard approach to the Ly$\alpha$ physics has relied on emulators
trained on hydrodynamic simulations Borde  _et al._ (2014); Palanque-
Delabrouille  _et al._ (2015a); Rossi  _et al._ (2014); Bolton  _et al._
(2017); Bird  _et al._ (2019); Pedersen  _et al._ (2020, 2021); Fernandez  _et
al._ (2023). These simulations are based on detailed models of underlying
astrophysics, e.g. star formation, stellar feedback, reionization. Powerful as
they are, these simulations are currently limited by resolution issues.
Emulators also face additional uncertainties in the numerical interpolation
over a grid of simulations.

In this Letter, we explore an alternative approach to the Ly$\alpha$ forest
analysis based on ideas borrowed from particle physics. The perturbative or
effective field theory (EFT) approaches to large-scale structure develop the
description of cosmic web dynamics that is agnostic about the underlying
astrophysical processes Baumann  _et al._ (2012); Carrasco  _et al._ (2012);
Desjacques  _et al._ (2018a); Ivanov (2022) (see Seljak (2012); Cieplak and
Slosar (2016); Garny  _et al._ (2018, 2021); Givans and Hirata (2020); Chen
_et al._ (2021); Givans  _et al._ (2022); Fuß and Garny (2022); Ivanov (2024)
for perturbative calculations of the Ly$\alpha$ forest). In our context, the
only two assumptions that EFT makes are that the flux absorption fluctuations
trace the underlying dark matter density and non-linear corrections are
perturbative Ivanov (2024). EFT does not make additional assumptions about gas
physics. In particular, it does not impose any temperature-density relation.
The relationship between flux fluctuations and the dark matter density field
should be described by an analytic function, which can be Taylor expanded on
large scales. Then the terms in this Taylor series are parameterized using the
fundamental symmetries of the problem, such as the rotations around the line-
of-sight direction and the equivalence principle. Perturbative EFT approaches
have been especially successful in galaxy clustering analyses, where they have
become a standard tool for parameter estimation Alam  _et al._ (2017, 2021);
Ivanov  _et al._ (2020a); D’Amico  _et al._ (2019); Chen  _et al._ (2022).
This Letter presents an application of the similar approach to the Ly$\alpha$
data.

An important advantage of EFT is its flexibility: a typical calculation of
relevant non-linear corrections to the Ly$\alpha$ power spectrum takes about 1
second Chudaykin  _et al._ (2020), which allows one to efficiently explore
cosmological models without the need to re-run hydrodynamical simulations for
new parameters added. In contrast to emulators, the EFT theory model can be
consistently computed for every point in parameter space, thereby removing any
uncertainty associated with the grid interpolation. This is especially
valuable for extended cosmological models proposed as a resolution to the
above tension between $\Lambda$CDM and the SDSS Ly$\alpha$ data. A rigorous
interpretation of the Ly$\alpha$ data in the context of such models requires
one to self-consistently recompute the FPS, which can be easily done with EFT.

This Letter lays the groundwork for systematic full-shape analyses of
Ly$\alpha$ data with EFT. Here, we re-analyze the publicly available SDSS data
on the 1D FPS using the EFT for Ly$\alpha$ data developed in Ivanov (2024).
Our work can be seen as an extension of earlier efforts to use perturbative
techniques for the Ly$\alpha$ forest 1D data analysis Garny  _et al._ (2018,
2021); Givans and Hirata (2020); Givans  _et al._ (2022); Fuß and Garny
(2022). The main novelties of our work are formulating a self-consistent
theory for Ly$\alpha$ in 3D, and fitting cosmological parameters from
Ly$\alpha$ data just like in the full-shape analyses of galaxy surveys.

Our work pursues several goals. First, we develop an efficient toolkit to
probe fundamental cosmology with the Ly$\alpha$ forest and test it against the
real data. Second, we investigate if the SDSS Ly$\alpha$ data are consistent
with $\Lambda$CDM within the EFT approach. Third, we identify key directions
for the future development of the Ly$\alpha$ EFT full-shape analysis.

Data. Our analysis is based on quasar spectra from the fourteenth data release
(DR14) of SDSS Chabanier  _et al._ (2019), which consists of the Baryon
Oscillation Spectroscopic Survey (BOSS) and extended-BOSS (eBOSS) samples. The
1D FPS measurements from the SDSS quasars in the redshift range $2.2<z<4.6$
are publicly available. In our analysis, we will focus on a more narrow
redshift range $3.4\leq z\leq 4.2$, encompassing 6 redshift bins with centers
$3.2,3.4,3.6,3.8,4.0,4.2$. We do not use the lower redshift bins because their
description requires an accurate calibration of the bias parameters that is
not possible with publicly available simulations. We discard the highest two
redshift bins in this work in order to be more conservative in light of a
potential sensitivity to inhomogeneous reionization Cain  _et al._ (2024). To
build the power spectrum likelihood, we use the public covariance matrices
that include additional diagonal corrections to account for systematic effects
due to finite resolution, active galactic nuclei feedback, damped Ly$\alpha$
systems etc. Chabanier  _et al._ (2019).

Theory model overview. Let us start our theory background with a brief
discussion of the galaxy clustering in redshift space at the lowest (linear)
order in dark matter perturbations Bernardeau  _et al._ (2002); Desjacques
_et al._ (2018a, b). Causality, the equivalence principle, and rotational
symmetry imply that on the largest scales, the galaxy overdensity field
$\delta_{g}=n_{g}/\bar{n}_{g}-1$ ($n_{g}$ being the galaxy number density, and
overbar denotes spatial averaging), should be linearly proportional to the
dark matter overdensity $\delta^{(1)}$ computed in linear theory,

$\delta_{g}=b_{1}\delta^{(1)}\,.$ (1)

The coefficient of proportionality $b_{1}$ is called the linear bias
parameter. It encapsulates details of the galaxy formation physics, and its
prediction is possible given a galaxy formation model. In EFT, $b_{1}$ is
treated as a nuisance parameter to marginalize over. Since Eq. (1) is written
in the galaxy rest frame, to describe the galaxy distribution seen by the
observer $\delta^{(s)}_{g}$, one has to transform $\delta_{g}$ in Eq. (1) to
the observer’s frame, which introduces redshift-space distortions Kaiser
(1987); Scoccimarro (2004). At the linear level, the mapping between the two
frames yields

$\delta^{(s)}_{g}=b_{1}\delta^{(1)}-\frac{\partial_{\parallel}v^{(1)}_{\parallel}}{aH}\,,$
(2)

where $v^{(1)}_{i}$ is the peculiar velocity of galaxies, $H$ is the Hubble
parameter, $a$ is the metric scale factor, and the subscript $\parallel$
denotes the projection onto the line-of-sight. Note that Eq. (2) does not
introduce any new free parameters. The coefficient in front of the line-of-
sight velocity gradient is protected by the equivalence principle, because the
galaxies should “fall” in the gravitational potential the same way as dark
matter. This fact has been the rationale behind using redshift-space
distortions as a cosmological probe Alam  _et al._ (2017).

In the context of the Ly$\alpha$ forest, the direct observables are absorption
lines that depend on the local physics along the observer’s line-of-sight. In
particular, the probability of absorption depends on peculiar velocity
gradients along the line of sight, making it necessary to include such terms
in the expansion already in the rest frame of hydrogen clouds McDonald  _et
al._ (2000). In EFT, this implies that the Ly$\alpha$ forest bias expansion
has only the azimuthal symmetry w.r.t. rotations around the line-of-sight, as
opposed to the galaxy distribution, which enjoys the full spherical symmetry,
imposing the linear bias relation Eq. (1). In particular, in linear theory the
Ly$\alpha$ flux fluctuations may depend now on an operator
$\partial_{\parallel}^{2}\Phi\propto\partial_{\parallel}v^{(1)}_{\parallel}$
($\Phi$ stands for the Newton’s potential), i.e. the linear bias relation
takes the form Desjacques  _et al._ (2018b); Ivanov (2024)

$\delta^{(s)}_{g}=b_{1}\delta^{(1)}+b_{\eta}\frac{\partial_{\parallel}v^{(1)}_{\parallel}}{aH}\,,$
(3)

where $b_{\eta}$ is a new bias parameter that captures the correlation of the
Ly$\alpha$ fluctuations with projections of the dark matter line-of-sight
velocity gradient. Eq. (3) is still subject to the redshift-space mapping,
which will produce an unobservable shift of $b_{\eta}$ by 1.

EFT provides a systematic program for the description of non-linear effects in
the Ly$\alpha$ forest by higher-order operators constructed from the relevant
degrees of freedom, i.e. the density, velocity, and tidal fields. The
structure of these operators is fixed by symmetries only, just like in Eq.
(3). As such, EFT automatically implements constraints dictated by the
equivalence principle for non-linear operators stemming from the redshift-
space mapping Ivanov (2024).

We describe now the EFT for non-linear Ly$\alpha$ correlations in three
dimensions, derived in Ivanov (2024). (See also Desjacques  _et al._ (2018b)
for an earlier related work in the context of galaxy bias with line-of-sight
selection effects.) The non-linear flux fluctuation field
$\delta_{F}=F/\bar{F}-1$ in redshift space at the cubic order in the linear
matter overdensity $\delta^{(1)}$ is expressed as

$\begin{split}\delta^{(s)}_{F}({\bf
k})=&\sum_{n=1}^{3}\Big{[}\prod_{j=1}^{n}\int\frac{d^{3}{\bf
k}_{j}}{(2\pi)^{3}}\delta^{(1)}({\bf k}_{j})\Big{]}K_{n}({\bf k}_{1},...,{\bf
k}_{n})\\\ &\times(2\pi)^{3}\delta^{(3)}_{D}({\bf k}-{\bf k}_{1}-...-{\bf
k}_{j})\,,\end{split}$ (4)

with $K_{1}({\bf k})=b_{1}-b_{\eta}f\mu^{2}$ ($f$ is the logarithmic growth
factor, $\mu\equiv k_{\parallel}/k$, $k_{\parallel}$ is the line-of-sight
wavevector component), and other non-linear kernels $K_{n}$ given in
Supplemental Material. The EFT for Ly$\alpha$ features a number of free bias
parameters. All together, there are two linear bias parameters, $b_{1}$ and
$b_{\eta}$, and 11 non-linear bias coefficients,

$\begin{split}\\{b_{2}\equiv
b_{\delta^{2}},b_{\mathcal{G}_{2}},b_{(KK)_{\parallel}},b_{\delta\eta},b_{\eta^{2}},b_{\Pi^{[2]}_{\parallel}}\\}\,,\\\
\\{b_{\Gamma_{3}},b_{\delta\Pi^{[2]}_{\parallel}},b_{\eta\Pi^{[2]}_{\parallel}},b_{(K\Pi^{[2]})_{\parallel}},b_{\Pi^{[3]}_{\parallel}}\\}\,,\end{split}$
(5)

where the first and second lines contain coefficients of operators quadratic
and cubic in linear density field, respectively. In what follows we will refer
to the above bias parameters as $b_{\mathcal{O}}$,
$\mathcal{O}=\delta^{2},\mathcal{G}_{2}$ etc. Assuming that the linear density
field is Gaussian-distributed with the power spectrum $P_{\rm lin}(k)$, we
obtain the 3D one-loop power spectrum

$\begin{split}&P^{\rm 1-loop}_{\rm 3D}({\bf k}=\\{k,\mu\\})=K_{1}^{2}({\bf
k})P_{\rm lin}(k)\\\ &+2\int_{\bf q}K_{2}^{2}({\bf q},{\bf k}-{\bf
q})P_{\text{lin}}(|{\bf k}-{\bf q}|)P_{\text{lin}}(q)\\\ &+6K_{1}({\bf
k})P_{\text{lin}}(k)\int_{\bf q}K_{3}({\bf k},-{\bf q},{\bf
q})P_{\text{lin}}(q)\,.\end{split}$ (6)

We have implemented the calculation of the one-loop model above as a module in
the CLASS-PT code Chudaykin  _et al._ (2020). The one-loop integrals are
decomposed into powers of $\mu$ and computed using the FFTLog technique
presented in Ivanov (2024); Simonovic  _et al._ (2018). All together, 135 one
loop integrals are computed in $\approx 1$ second on one CPU core for a single
point in cosmological parameter space. This speed is sufficient for Markov
chain Monte Carlo (MCMC) cosmological parameter sampling. Note that sampling
nuisance parameters is much faster as it does not require a re-calculation of
the loop corrections.

In principle, higher derivative counterterms should be added to Eq. (6) to
renormalize the loop integrals. These counterterms also capture the effects of
Jeans smoothing. However, the UV-sensitivity is mild and the Jeans effects are
weak at the one-loop order so that the counterterm contributions can be
ignored given the current range of scales and the precision level of the SDSS
data Ivanov (2024).

The 1D power spectrum of Ly$\alpha$ fluctuations measured along the line-of-
sight is given by Lumsden  _et al._ (1989)

$P_{\rm{1D}}(k_{\parallel})=\frac{1}{2\pi}\int_{k_{\parallel}}^{\infty}dk~{}kP_{\rm
3D}(k,k_{\parallel})\,.$ (7)

Importantly, the 1D power spectrum involves integration over hard momenta that
are not in the perturbative regime. In EFT, one should do the above integral
up to a certain cutoff $k_{\rm max}$ and then add suitable counterterms that
remove the cutoff dependence. As was shown in Garny  _et al._ (2021); Ivanov
(2024), the leading order counterterms take the following form:

$P^{\rm ctr}_{\rm 1D}(k_{\parallel})=C_{0}+C_{2}k^{2}_{\parallel}\,.$ (8)

This increases the number of free parameters up to 13. For each point in
cosmological parameter space, we compute the integral Eq. (7) numerically,
assuming $k_{\rm max}=3~{}h{\text{Mpc}}^{-1}$ as in Ivanov (2024).

Finally, the Ly$\alpha$ data from SDSS is modulated by several systematic
effects. These cannot be described by EFT, and we follow the standard approach
for their theoretical modeling Chabanier  _et al._ (2019), see Supplemental
Material for more detail.

Figure 1: Left panel: constraints on the mass fluctuation amplitude
$\sigma_{8}$ from the EFT-based full-shape Ly$\alpha$ likelihood
(EFTxLy$\alpha$) presented in this work. We also display Planck 18 CMB + BOSS
DR12 BAO (P18B) results for $\sigma_{8}$ and the sum of neutrino masses $\sum
m_{\nu}$, and the results of the joint analysis of the P18B + EFTxLy$\alpha$
data. Right panel: 1-dimensional Ly$\alpha$ flux power spectrum data from SDSS
DR14 in the range $3.1<z<4.3$ (central values of redshifts are quoted), along
with EFT best-fit curves. Errorbars shown correspond to diagonal elements of
covariance matrices with systematic weights included.

Analysis details. Our EFT theory involves many nuisance parameters whose
redshift-dependence is not determined within the EFT itself. To estimate the
impact of the EFT coefficients on parameter estimation, we run a series of
Fisher matrix analyses, described in detail in Supplemental Material. We first
fix all cosmological parameters except $\sigma_{8}$ and study the impact of
the EFT parameters. Focusing on a single redshift bin, we find that the error
on $\sigma_{8}$ grows quickly with the number of free parameters in the fit.
This is because the shape of the 1D FPS does not have enough features to break
degeneracies between different EFT one-loop shapes, given the SDSS noise
level. This effect can be mitigated with informative priors on the EFT
parameters. We have found that we can achieve good constraints on $\sigma_{8}$
and maintain a significant level of flexibility if we keep the linear bias
parameters $\\{b_{1},b_{\eta}\\}$ free in the fit, while the non-linear EFT
parameters are assumed to be deterministic functions of $b_{1}$. This approach
is motivated by the fact that in EFT the signal is dominated by linear
fluctuations, and hence the linear bias parameters are most important to fit
the data. This is the strategy that we adopt in what follows.

The non-linear bias parameters can be extracted from large sets of high-
fidelity hydrodynamic simulations of 3D Ly$\alpha$ forest data. 3D information
is crucial in order to break degeneracies between the EFT parameters that are
present for 1D FPS. We extract the EFT parameters from the public power
spectra measurements from the state-of-the-art public Sherwood hydrodynamic
simulation Bolton  _et al._ (2017); Givans  _et al._ (2022). We will use
relationships between non-linear bias parameters and $b_{1}$,
$b_{\mathcal{O}}(b_{1})$, which are more robust than the bias parameter values
themselves. Setting priors on bias parameters in terms of functions of $b_{1}$
also makes our analysis more flexible to possible inaccuracies in astrophysics
models used in simulations. This approach is motivated by galaxy clustering
analysis, where semi-analytic functions $b_{\mathcal{O}}(b_{1})$ are often
used in cosmological analyses, see e.g. Beutler  _et al._ (2017); Cabass  _et
al._ (2022a, b); Ivanov  _et al._ (2024). In line with these arguments, we
have found that the EFT parameters from the Sherwood data exhibit significant
correlations with $b_{1}$, which can be well captured with simple
interpolating functions, presented in Supplemental Material. In what follows
we refer to functions $b_{\mathcal{O}}(b_{1})$ as “Sherwood priors.” We found
that the Sherwood FPS data cannot constrain $b_{\Gamma_{3}}$ well, so we set
it to zero in all analyses following Ivanov  _et al._ (2020a).

The power spectra from Sherwood simulations do not match those of SDSS for
overlapping redshifts. As a result, we do not expect the Sherwood priors alone
to provide a good description of the data. In order to overcome this issue we
have produced mock data on 1D FPS covering the redshift range of the SDSS data
using the LaCE emulator Pedersen  _et al._ (2021). The resulting synthetic
spectra roughly match the real SDSS data.

Our 1D mocks do not provide 3D information needed to constrain the EFT bias
parameters, but we can use them for an additional re-calibration of the 1D
counterterms $C_{0}$ and $C_{2}$ that appear at the 1D level. We find that the
EFT model with 3D non-linear bias parameters fixed to priors from Sherwood and
1D counterterms determined by functions $C_{n}(b_{1})$ ($n=0,2$) extracted
from a combination of Sherwood and LaCE simulations provides a good
description of the SDSS data, as estimated based on the $\chi^{2}$ statistic.
Therefore, we adopt this modeling choice as our baseline. It is always
possible to widen or remove some priors, which will lead to more conservative
but less constraining results.

Several options are available for the description of the redshift evolution of
EFT parameters. The most conservative choice is to fit the EFT parameters
independently in each bin, see e.g. Ivanov  _et al._ (2020a). An alternative
is to assume a smooth evolution of the free EFT parameters as a function of
redshift as was done e.g. in Garny  _et al._ (2021). We found that the former
option works better (in terms of the $\chi^{2}$ statistic), and it also
provides more flexibility. This choice, which we adopt as a baseline, results
in 12 free parameters for 6 redshift bins in the fit.

As far as the cosmological parameters are concerned, our Fisher matrix
analysis suggests that with the above baseline settings we can obtain
constraints competitive with other probes only for the mass fluctuation
amplitude $\sigma_{8}$, at the level of few percent. Adding other parameters
to the fit leads to strong degeneracies that cannot be broken at the level of
the 1D FPS data. Thus, our main analysis will have free $\sigma_{8}$ and other
cosmological parameters fixed to a fiducial cosmology consistent with the
Planck baseline $\Lambda$CDM model. These are
$\\{\omega_{b},\omega_{cdm},n_{s},h,\sum
m_{\nu}\\}=\\{0.02215,0.12,0.961,0.678,0\\}$.

Results. To start off, we validate our pipeline on mock data generated with
the LaCE emulator. Applying our pipeline to the LaCE mocks in the range
$3.2\leq z\leq 4.2$, we have found that after re-calibration of $C_{n}$, it
recovers the input value of $\sigma_{8}$ with $0.3\%$ accuracy, which
corresponds to $\approx 15\%$ of the statistical error. This suggests that the
theory systematic error of our baseline analysis is negligible.

Applying the same pipeline to the actual SDSS data we find the marginalized
constraint $\sigma_{8}=0.841\pm 0.017$ in the baseline analysis. The best-fit
$\chi^{2}=170$ for $197$ degrees of freedom, indicating an excellent fit to
data. (Note that our covariance matrix includes diagonal systematic weights,
so the $\chi^{2}$ statistic is expected to be smaller than the number of
degrees of freedom.) The best-fit models and data for relevant redshift bins
are shown in the right panel of Fig. 1. Using our fiducial value
$\Omega_{m}=0.3082$, our $\sigma_{8}$ constraint can be converted into a limit
on $S_{8}=(\Omega_{m}/0.3)^{1/2}\sigma_{8}$. We find $S_{8}=0.852\pm 0.017$,
which agrees well with the Planck CMB result $0.834\pm 0.016$ Aghanim  _et
al._ (2018).

Combining our EFTxLy$\alpha$ measurement with the Planck 2018 base likelihood,
and adding the BAO data from BOSS DR12 as in Ref. Ivanov  _et al._ (2020b), we
obtain the limit on the sum of neutrino masses $\sum m_{\nu}<0.080$ eV (at
$95\%$CL), see the left panel of Fig. 1 for the 1D and 2D marginalized
contours. (Other cosmological parameters of $\nu\Lambda$CDM model were also
consistently varied but not shown on the plot for compactness.) This is one of
the strongest cosmological constraints on the neutrino mass to date, c.f.
Aghanim  _et al._ (2018); Ivanov  _et al._ (2020b); Alam  _et al._ (2021);
Adame  _et al._ (2024).

Finally, we include additional cosmological parameters in our analysis.
Varying both $\sigma_{8}$ and $n_{s}$ we find a strong degeneracy between
these parameters, in agreement with our Fisher matrix analysis. The data are
not able to produce competitive constraints on either of these parameters
individually with our baseline analysis settings. It will be interesting to
see if informative priors on $b_{1}$ and $b_{\eta}$, or more constraining data
sets, e.g. 1D FPS from DESI, could break this degeneracy.

Discussion and conclusions. We have presented the first EFT-based full-shape
cosmological parameter analysis of the Ly$\alpha$ forest 1D clustering data
from SDSS. From SDSS 1D Ly$\alpha$ data alone, we find a $2\%$ constraint on
the mass clustering amplitude $\sigma_{8}$. Combining our likelihood with data
on CMB anisotropies from Planck 2018 and BAO, we derive one of the strongest
constraints on the total neutrino mass to date, $\sum m_{\nu}<0.08$ eV. These
results show the potential power of the 1D FPS when analyzed with EFT.

Our results indicate that within the EFT framework, the SDSS Ly$\alpha$ data
are in good agreement with the baseline Planck $\Lambda$CDM cosmological
model. This weakens the case for new physics interpretations of the apparent
tension between $\Lambda$CDM and Ly$\alpha$ data in the literature. This
suggests that new ingredients beyond the standard models of gas physics used
in Ly$\alpha$ emulators may be needed to reconcile the SDSS 1D Ly$\alpha$
forest data with the $\Lambda$CDM.

We emphasize that our analysis should be taken in conjunction with our
assumptions about the EFT nuisance parameters. We have used priors on EFT
parameters from Sherwood simulations, and priors on 1D counterterms from the
1D power spectrum mocks produced with the LaCE emulator. These priors play a
significant role in our analysis. Their removal, i.e. marginalization over EFT
parameters within conservative priors, significantly degrades constraining
power. Further studies of the EFT parameters with large sets of high-fidelity
hydrodynamical simulations are needed in order to fully validate our priors.
Similar precision measurements of EFT parameters have been previously carried
out for halos and galaxies Lazeyras and Schmidt (2018); Abidi and Baldauf
(2018); Barreira (2020); Barreira  _et al._ (2021); Lazeyras  _et al._ (2021);
Ivanov  _et al._ (2024); Cabass  _et al._ (2024). In particular, recent
studies have used informative priors on galaxy EFT parameters to improve
constraints on primordial non-Gaussianity Cabass  _et al._ (2022a); Ivanov
_et al._ (2024); Cabass  _et al._ (2024). Performing a similar study is the
main priority of the EFT program in the Ly$\alpha$ context. That said, we note
that our model still has a significant level of flexibility as we keep linear
bias parameters $b_{1}$ and $b_{\eta}$ free for each redshift bin in our fits.

Furthermore, EFT can consistently model different Ly$\alpha$ measurements such
as 3D Ly$\alpha$ autocorrelation and Ly$\alpha$-quasar cross-correlation
functions du Mas des Bourboux  _et al._ (2020b); DESI Collaboration  _et al._
(2024), 3D power spectrum Abdul-Karim  _et al._ (2023); de Belsunce  _et al._
(2024), and cross-correlations with CMB Doux  _et al._ (2016). Combining these
statistics with 1D FPS can also improve our current results. For example, the
degeneracy between EFT bias parameters and $\sigma_{8}$ can be broken by CMB
lensing map and Ly$\alpha$ cross-correlations, and the inclusion of 3D
information can alleviate the $b_{1}-b_{\eta}$ degeneracy.

We note that our EFTxLy$\alpha$ approach can be readily applied to extended
cosmological models, including the scenario motivated by the Hubble and
$S_{8}$ tensions, see e.g. Abdalla  _et al._ (2022); Goldstein  _et al._
(2023); He  _et al._ (2023). These, and other research directions described
above, are left for future work.

Acknowledgments. We thank Andrei Cuceau, Vid Irsic, Andreu Font-Ribera, and
Sokratis Trifinopoulos for useful discussions. MCMC chains produced as part of
this work are generated with Montepython Audren  _et al._ (2013); Brinckmann
and Lesgourgues (2019). This material is based upon work supported by the U.S.
Department of Energy, Office of Science, Office of High Energy Physics of U.S.
Department of Energy under grant Contract Number DE-SC0012567. MWT
acknowledges financial support from the Simons Foundation (Grant Number
929255).

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Supplemental Material

##  1 Analysis details

Perturbation theory kernels. We start with the perturbation theory kernels for
Ly$\alpha$,

$\begin{split}&K_{1}({\bf k})=b_{1}-b_{\eta}f\mu^{2}\,,\\\ &K_{2}({\bf
k}_{1},{\bf k}_{2})=\frac{b_{2}}{2}+b_{\mathcal{G}_{2}}\left(\frac{({\bf
k}_{1}\cdot{\bf k}_{2})^{2}}{k_{1}^{2}k_{2}^{2}}-1\right)+b_{1}F_{2}({\bf
k}_{1},{\bf k}_{2})-b_{\eta}f\mu^{2}G_{2}({\bf k}_{1},{\bf
k}_{2})-fb_{\delta\eta}\frac{\mu_{2}^{2}+\mu_{1}^{2}}{2}+b_{\eta^{2}}f^{2}\mu_{1}^{2}\mu_{2}^{2}\\\
&+b_{1}f\frac{\mu_{1}\mu_{2}}{2}\left(\frac{k_{2}}{k_{1}}+\frac{k_{1}}{k_{2}}\right)-b_{\eta}f^{2}\frac{\mu_{1}\mu_{2}}{2}\left(\frac{k_{2}}{k_{1}}\mu_{2}^{2}+\frac{k_{1}}{k_{2}}\mu_{1}^{2}\right)+b_{(KK)_{\parallel}}\left(\mu_{1}\mu_{2}\frac{({\bf
k}_{1}\cdot{\bf
k}_{2})}{k_{1}k_{2}}-\frac{\mu_{1}^{2}+\mu_{2}^{2}}{3}+\frac{1}{9}\right)\\\
&+b_{\Pi^{(2)}_{\parallel}}\left(\mu_{1}\mu_{2}\frac{({\bf k}_{1}\cdot{\bf
k}_{2})}{k_{1}k_{2}}+\frac{5}{7}\mu^{2}\left(1-\frac{({\bf k}_{1}\cdot{\bf
k}_{2})^{2}}{k_{1}^{2}k_{2}^{2}}\right)\right)\,,\end{split}$ (S1)

with $\mu_{i}=(\hat{\bf z}\cdot\hat{{\bf k}}_{i})$, $\hat{\bf z}$ denoting the
line-of-sight direction unit vector, $f=d\mathop{\rm ln}\nolimits
D_{+}/d\mathop{\rm ln}\nolimits a$ ($D_{+}$ is the growth factor), and we have
used the usual density and velocity kernels from standard cosmological
perturbation theory:

$\begin{split}F_{2}({\bf k}_{1},{\bf
k}_{2})&=\frac{5}{7}+\frac{2}{7}\frac{({\bf k}_{1}\cdot{\bf
k}_{2})^{2}}{k_{1}^{2}k_{2}^{2}}+\frac{1}{2}\frac{{\bf k}_{1}\cdot{\bf
k}_{2}}{k_{1}k_{2}}\bigg{(}\frac{k_{1}}{k_{2}}+\frac{k_{2}}{k_{1}}\bigg{)}\,\,,\\\
G_{2}({\bf k}_{1},{\bf k}_{2})&=\frac{3}{7}+\frac{4}{7}\frac{({\bf
k}_{1}\cdot{\bf k}_{2})^{2}}{k_{1}^{2}k_{2}^{2}}+\frac{1}{2}\frac{{\bf
k}_{1}\cdot{\bf
k}_{2}}{k_{1}k_{2}}\bigg{(}\frac{k_{1}}{k_{2}}+\frac{k_{2}}{k_{1}}\bigg{)}\,\,,\end{split}$
(S2)

The general expression for the $K_{3}$ kernel is quite cumbersome. Below we
present it only for the kinematic configurations that appear in the one-loop
power spectrum integrals:

$\begin{split}&\int_{{\bf q}}K_{3}({\bf k},{\bf q},-{\bf q})P_{\rm lin}(q)\\\
=\,&b_{1}\int_{{\bf q}}F_{3}({\bf k},{\bf q},-{\bf
q})P_{\text{lin}}(q)-fb_{\eta}\mu^{2}\int_{{\bf q}}G_{3}({\bf q},-{\bf q},{\bf
k})P_{\text{lin}}(q)\\\ &\>+\int_{{\bf
q}}\left[1-\left(\hat{k}\cdot\hat{q}\right)^{2}\right]P_{\rm lin}(q)\\\
&\>\times\Bigg{\\{}\frac{4}{21}(5b_{\mathcal{G}_{2}}+2b_{\Gamma_{3}})\left[\left(\frac{({\bf
k}-{\bf q})\cdot{\bf q}}{|{\bf k}-{\bf
q}|q}\right)^{2}-1\right]-\frac{2}{21}fb_{\delta\eta}\left[\frac{3(k_{\parallel}-q_{\parallel})^{2}}{|{\bf
k}-{\bf q}|^{2}}+\frac{5q_{\parallel}^{2}}{q^{2}}\right]\\\
&\quad\>+\frac{4}{7}f^{2}b_{\eta^{2}}\frac{q_{\parallel}^{2}}{q^{2}}\frac{(k_{\parallel}-q_{\parallel})^{2}}{|{\bf
k}-{\bf q}|^{2}}+\frac{20}{21}b_{(KK)_{\parallel}}\left[\frac{({\bf
k}\cdot{\bf q}-q^{2})(k_{\parallel}-q_{\parallel})q_{\parallel}}{|{\bf k}-{\bf
q}|^{2}q^{2}}-\frac{1}{3}\frac{(k_{\parallel}-q_{\parallel})^{2}}{|{\bf
k}-{\bf
q}|^{2}}-\frac{1}{3}\frac{q_{\parallel}^{2}}{q^{2}}+\frac{1}{9}\right]\\\
&\quad\>+\frac{10}{21}b_{\Pi_{\parallel}^{[2]}}\frac{({\bf k}\cdot{\bf
q}-q^{2})}{|{\bf k}-{\bf
q}|^{2}}\frac{(k_{\parallel}-q_{\parallel})^{2}}{q^{2}}+\frac{10}{21}\left[b_{\delta\Pi_{\parallel}^{[2]}}-\frac{1}{3}b_{(K\Pi^{[2]})_{\parallel}}-fb_{\eta\Pi_{\parallel}^{[2]}}\frac{q_{\parallel}^{2}}{q^{2}}\right]\frac{(k_{\parallel}-q_{\parallel})^{2}}{|{\bf
k}-{\bf q}|^{2}}\\\
&\quad\>+\frac{10}{21}b_{(K\Pi^{[2]})_{\parallel}}\frac{({\bf q}\cdot{\bf
k}-q^{2})}{q|{\bf k}-{\bf
q}|}\frac{q_{\parallel}(k_{\parallel}-q_{\parallel})}{q|{\bf k}-{\bf
q}|}+\frac{10}{21}fb_{\Pi^{[2]}_{\parallel}}\frac{q_{\parallel}(k_{\parallel}-q_{\parallel})^{3}}{q^{2}|{\bf
k}-{\bf q}|^{2}}\\\
&\quad\>+(b_{\Pi_{\parallel}^{[3]}}+2b_{\Pi_{\parallel}^{[2]}})\Bigg{[}\frac{13}{21}\frac{{\bf
k}\cdot{\bf q}-q^{2}}{|{\bf k}-{\bf
q}|^{2}}\frac{q_{\parallel}(k_{\parallel}-q_{\parallel})}{q^{2}}-\frac{5\mu^{2}}{9}\left[\left(\frac{({\bf
k}-{\bf q})\cdot{\bf q}}{|{\bf k}-{\bf
q}|q}\right)^{2}-\frac{1}{3}\right]\Bigg{]}\\\
&\quad\>+\frac{2}{21}fb_{1}\left[5\frac{q_{\parallel}(k_{\parallel}-q_{\parallel})}{q^{2}}+3\frac{q_{\parallel}(k_{\parallel}-q_{\parallel})}{|{\bf
k}-{\bf
q}|^{2}}\right]-\frac{2}{7}f^{2}b_{\eta}\frac{q_{\parallel}(k_{\parallel}-q_{\parallel})}{q^{2}|{\bf
k}-{\bf
q}|^{2}}\left[(k_{\parallel}-q_{\parallel})^{2}+q_{\parallel}^{2}\right]\Bigg{\\}}\,.\end{split}$
(S3)

The expressions for the standard fluid kernels $F_{3}$ and $G_{3}$ can be
found e.g. in Bernardeau  _et al._ (2002).

Treatment of Systematics. We model the SiIII oscillations as a multiplicative
factor McDonald  _et al._ (2006); Palanque-Delabrouille  _et al._ (2013)

$\kappa_{\rm SiIII}=1+2\left(\frac{f_{\rm SiIII}}{1-\bar{F}_{\rm
fid}}\right)\cos(vk_{\parallel})+\left(\frac{f_{\rm SiIII}}{1-\bar{F}_{\rm
fid}}\right)^{2}\,,$ (S4)

with $f_{\rm SiIII}=8.7\cdot 10^{-3}$, $v=2\pi/(0.0028)$ km/s, $\bar{F}_{\rm
fid}=e^{-0.0025(1+z)^{3.7}}$. Thermal broadening is modeled with the damping
kernel $e^{-(k_{\parallel}/k_{s})^{2}}$ with $k_{s}=0.11$(km/s)-1. We note
that our main results are insensitive to these particular choices, i.e. we
find similar constraints when the parameters describing the above systematic
effects are varied.

Fisher matrix analysis. We now present details of the Fisher matrix analysis
that we use to determine the baseline parameters of our likelihood analysis.
We focus on a single redshift of the data, centered at $z=3.2$. Our analysis
shows that the error on $\sigma_{8}$ grows fast with a number of free
parameters in the fit, see Fig. S1. In particular, the data loose sensitivity
to $\sigma_{8}$ almost entirely if non-linear bias parameters are varied. We
motivates a strategy to keep the linear bias parameters free, and fix the non-
linear ones to values informed by simulations. Note that even in this case we
find a very significant degeneracy between $\sigma_{8},b_{1}$ and $b_{\eta}$,
see the right panel of Fig. S1.

Figure S1: Left panel: Marginalized Fisher errors on the mass fluctuation
amplitude $\sigma_{8}$ as a function of a number of free EFT parameters in the
fit. Results are reported for a single redshift bin $3.1<z<3.3$. Right panel:
Fisher correlation coefficients for a fit with only three parameters:
$\sigma_{8},b_{1},b_{\eta}$.

As far as the cosmological parameters are concerned, our Fisher matrix
analysis suggests that with the above baseline settings we can obtain
constraints competitive with other probes only for the mass fluctuation
amplitude $\sigma_{8}$, at the level of few percent. Specifically, focusing on
one redshift, we find that adding $n_{s}$ to the fit yield constraints
$\\{\frac{\sigma_{\sigma_{8}}}{\sigma_{8}},\sigma_{n_{s}}\\}=\\{0.25,0.06\\}$;
while adding to the fit $\omega_{\rm cdm}$ produces uninformative constraints
$\\{\frac{\sigma_{\sigma_{8}}}{\sigma_{8}},\sigma_{n_{s}},\frac{\sigma_{\omega_{cdm}}}{\omega_{cdm}}\\}=\\{0.67,0.28,0.52\\}$.
In addition, the cosmological parameter contours are extremely degenerate, e.
g. for the analysis with free $n_{s}$ and $\sigma_{8}$ we find a correlation
coefficient $-0.99$, which indicates that measuring these parameters
individually with the current EFT settings is challenging. We have confirmed
this statement in a full MCMC analysis as well. Better priors on $b_{1}$ and
$b_{\eta}$ from hydrodynamical simulations will be needed in order to measure
$n_{s}$ and $\sigma_{8}$ individually from the 1D FPS.

Figure S2: Bias parameters and counterterterms of the Sherwood simulation
data. Solid lines depict the linear fits to the data.

EFT bias parameters from the Sherwood simulations. We fit the public 3D power
spectra of Ly$\alpha$ fluctuations from the Sherwood simulations using the
same methodology as in Ivanov (2024). The public data cover four redshift bins
with centers at $z=[2,2.4,2.8,3.2]$. The $z=2.8$ and $z=3.2$ data were fitted
before in Ivanov (2024). For the remaining two redshifts, we use the parameter
drift plots to estimate the data cut ${k_{\text{max}}}$ for which the EFT
model is applicable. Our measurements of different nonlinear bias parameters
$b_{\mathcal{O}}$ versus corresponding values of $b_{1}$ are presented in Fig.
S2, along with the linear fits to the data points. Error bars on the quadratic
parameters follow from the Gaussian covariance for the power spectrum. The
errors on the EFT parameters depend very strongly on ${k_{\text{max}}}$ used
in the fits, and the possible choices of ${k_{\text{max}}}$ are limited by the
binning of the public Sherwood power spectra. Because of these reasons, for
some cubic bias parameters the nominal errorbars appear to be quite small,
which may bias the $b_{\mathcal{O}}(b_{1})$ relations extracted from the
simulation. In order to be conservative and avoid overfitting, we assume a
flat error $1$ on all cubic bias parameters. Finally, we also assume flat
errors 1 on $C_{0}/(5\cdot 10^{-3}[h^{-1}{\text{Mpc}}])$ and $C_{2}/(5\cdot
10^{-4}[h^{-1}{\text{Mpc}}]^{3})$, because the covariance for the 1D FPS of
Sherwood is not available, so there is no reliable way to estimate the errors
in this case. We found that the dependencies $b_{\mathcal{O}}(b_{1})$ and
$C_{n}(b_{1})$ can be well fitted with linear functions,

$b_{\mathcal{O}}=A_{\mathcal{O}}b_{1}+B_{\mathcal{O}}.$ (S5)

The best-fit values of $A_{\mathcal{O}}$ and $B_{\mathcal{O}}$ are given in
Table 1. For $C_{n}$ we have

$\frac{C_{0}}{[h^{-1}{\text{Mpc}}]}=-0.17b_{1}-0.0148\,,\quad\frac{C_{2}}{[h^{-1}{\text{Mpc}}]^{3}}=-9.7\cdot
10^{-4}b_{1}+9.4\cdot 10^{-5}\,.$ (S6)

Operator $\mathcal{O}$ | $\mathcal{G}_{2}$ | $\delta^{2}$ | $\eta^{2}$ | $\delta\eta$ | $(KK)_{\parallel}$ | $\Pi^{[2]}_{\parallel}$ | $\Pi^{[3]}_{\parallel}$ | $\delta\Pi^{[2]}_{\parallel}$ | $(K\Pi^{[2]})_{\parallel}$ | $\eta\Pi^{[2]}_{\parallel}$
---|---|---|---|---|---|---|---|---|---|---
$A_{\mathcal{O}}$ | 0.089 | -0.32 | -2.0 | 4.4 | 0.45 | 2.2 | -7.0 | 8.2 | 7.8 | -9.9
$B_{\mathcal{O}}$ | -0.18 | -0.52 | 0.19 | 0.063 | -0.14 | -0.074 | 0.0043 | -1.5 | -0.79 | -2.0
Table 1: Numerical values for the fit to the Sherwood EFT parameters as
functions of the linear bias $b_{1}$.

Synthetic 1D FPS data from LaCE. We use the Cabayol23 neural network emulator
available from LaCE111https://github.com/igmhub/LaCE/tree/main to generate
1-dimensional Ly$\alpha$ power spectrum where we vary only $\sigma_{T}$ and
$\gamma$ and scanning over a range from {0.0,0.5} and {0.5,2.0} respectively.
The typical fits to FPS at a single redshift has a reduced $\chi^{2}$ ranging
from $\sim$ 1.2 to 1.4 which indicates a reasonable match, but not a good fit.
We find that the last point largely arises due to discrepancies at smaller
scales as can be appreciated from the r.h.s. of Fig. 1. Note that we have not
optimized the full set of the LaCE parameters to precisely reproduce the data.
This is done in order to avoid double counting of information when we use the
LaCE mocks for re-calibration of 1D EFT counterterms. The best-fit $C_{n}$
parameters extracted from our mocks are:

$\frac{C_{0}}{[h^{-1}{\text{Mpc}}]}=-0.17b_{1}-0.2053\,,\quad\frac{C_{2}}{[h^{-1}{\text{Mpc}}]^{3}}=-9.7\cdot
10^{-4}b_{1}+2.16\cdot 10^{-2}\,.$ (S7)

We see that the typical momentum scales associated with these parameters are
around $4~{}h{\text{Mpc}}^{-1}$, which is consistent with the EFT expectation
that they should be of the order of $k_{\rm
NL}\sim(5-10)~{}h{\text{Mpc}}^{-1}$.


# Electrical transport properties of thick and thin Ta-doped SnO2 films

Zong-Hui Gao  Tianjin Key Laboratory of Low Dimensional Materials Physics and
Preparing Technology, Department of Physics, Tianjin University, Tianjin
300354, China  Zi-Xiao Wang  Tianjin Key Laboratory of Low Dimensional
Materials Physics and Preparing Technology, Department of Physics, Tianjin
University, Tianjin 300354, China  Dong-Yu Hou  Tianjin Key Laboratory of Low
Dimensional Materials Physics and Preparing Technology, Department of Physics,
Tianjin University, Tianjin 300354, China  Xin-Dian Liu  Tianjin Key
Laboratory of Low Dimensional Materials Physics and Preparing Technology,
Department of Physics, Tianjin University, Tianjin 300354, China  Zhi-Qing Li
<EMAIL_ADDRESS>Tianjin Key Laboratory of Low Dimensional Materials
Physics and Preparing Technology, Department of Physics, Tianjin University,
Tianjin 300354, China

###### Abstract

Ta-doped SnO2 films with high conductivity and high optical transparency have
been successfully fabricated using rf-sputtering method and their electrical
transport properties have been investigated. All films reveal degenerate
semiconductor (metal) characteristics in electrical transport properties. For
the thick films ($t\sim 1\,\mu\rm{m}$ with $t$ being the thickness) deposited
in pure argon, the electron-phonon scattering alone cannot explain the
temperature dependent behaviors of resistivity, the interference effect
between electron-phonon and electron-impurity scattering should be considered.
For the $t\lesssim 36$ nm films, both conductivity and the Hall coefficient
show linear relation with the logarithm of temperature ($\ln T$) from
$\sim$100 K down to liquid helium temperature. The $\ln T$ behaviors of
conductivity and Hall coefficient cannot be explained by the Altshuler-Aronov
type electron-electron interaction effect, but can be quantitatively
interpreted by the electron-electron interaction effects in the presence of
granularity. Our results not only provide strong supports for the theoretical
results on the electron-phonon-impurity interference effect, but also confirm
the validity of the theoretical predictions of charge transport in granular
metals in strong coupling regime.

##  I Introduction

Transparent conducting oxide (TCO) films possess both high conductivity and
high optical transparency in the visible range, which render them to be widely
used in the fields of flat panel displays, photovoltaic electrochromics, solar
cells, and energy-efficient windows. [1, 2, 3, 4] Among various TCO films,
SnO2-based films have long plasma wavelength, higher thermal stability,
excellent durability, and higher mechanical hardness. [5, 6, 8, 7, 9] Thus
they are the major industrial TCO films used in the fields of low-emissivity
windows and solar cells. Currently, SnO2-based TCO films are mainly made of
fluorine-doped SnO2 (FTO) and antimony-doped SnO2 (ATO) materials. [10, 11]
For example, billions of square feet of FTO-coated window glass have been
installed in buildings around the world. The architectural FTO films are
generally deposited by spray pyrolysis method and their good conductivity is
sacrificed for infrared radiation reflectivity and high visible transparency.
[8, 12, 13] FTO or ATO film with high conductivity is usually prepared by
chemical vapour deposition techniques and not suitable to be deposited by
sputtering methods. [14, 15, 16, 17, 18] Thus to explore SnO2-based TCO films
which could be fabricated by sputtering deposition is nontrivial.

Recently, the first-principle calculation results[19, 20, 21, 21] indicate
that Ta-doped SnO2 (Ta:SnO2) reveals degenerate semiconductor characteristics
in electronic structure and its optical bandgap is wider than that of the
undoped SnO2. It means that Ta:SnO2 could reveal metallic characteristics in
transport property and remain visibly transparent. Using a target composed of
mixture of SnO2 and Ta2O5, one can fabricate Ta:SnO2 film via rf-sputtering
method. Thus it is necessary to investigate the influences of deposition
conditions on the resistivity and optical transparency of the deposited films.
In addition, to understand the conduction mechanisms, it is essential to
investigate the temperature dependence of the resistivity and Hall coefficient
of the material. In this paper, Ta:SnO2 films are successfully fabricated
using rf-sputtering technique and the deposition conditions including
substrate temperature and sputtering atmosphere are researched and optimized.
The electrical transport properties for films with thickness ranging from
$\sim$1000 to $\sim$10 nm are systematically investigated.

##  II Experimental method

Our films were deposited on (0001) Al2O3 single crystal substrates by standard
rf-sputtering method. Besides higher melting point, Al2O3 single crystal
possesses relatively higher thermal conductivity, which makes the thermal
equilibrium between the film and sample holder more easy in electrical
transport measurements. The sputtering source was a commercial ceramic target
of SnO2 (99.9%) and Ta2O5 (99.99%) mixture, in which the atomic ratio of Ta to
Sn is $3:47$. The base pressure of the depositing chamber was less than
$1\times 10^{-4}$ Pa, and the deposition was done at 0.6 Pa in a mixture of
argon and oxygen atmosphere. Four films with thicknesses $\sim$1 $\mu$m were
first deposited at $T_{s}=933$, 953, 963, and 983 K ($T_{s}$ is the substrate
temperature), respectively. During the deposition, the sputtering power was
maintained at 150 W and the volume ratio of oxygen to argon was set to zero.
It was found that the resistivity of film deposited at 963 K was less than
that of other films at a certain measured temperature. Thus the substrate
temperature is kept at 963 K in the subsequent deposition processes. The
volume ratio of oxygen to argon and oxygen ($O_{\rm pp}$) was then adjusted
between 0 and 2.0% and it was found that $O_{\rm pp}=0.5\%$ was the optimal
ratio in terms of the conductivity and optical transparency of the film.
Finally, Ta:SnO2 films with thickness ranging from $\sim$11 to $\sim$36 nm
were deposited at the optimized condition ($T_{s}=963$ K and $O_{\rm
pp}=0.5\%$).

The thicknesses of the $t>200$ nm films were measured using a surface profiler
(Dektak, 6M), while the thicknesses of the $t\lesssim 36$ nm films were
evaluated through growth rate and deposition time. The crystal structure and
quality of the films were determined by x-ray diffraction (XRD, Rigaku
D/MAX-2500). Optical absorption and transmittance spectra were measured in a
UV-VIS-NIR spectrometer (Hitachi V4100). The resistivity and Hall coefficient
were measured in a physical property measurement system (PPMS-6000, Quantum
Design) using the four-probe method. Hall-bar shaped samples, defined by
mechanical masks, were used for the resistivity and Hall effect measurements.

Figure 1: (Color online) Left panel: XRD patterns for the thick Ta:SnO2 films
deposited at different substrate temperatures $T_{s}$; Right panel: XRD
patterns for the thin Ta:SnO2 films with different thicknesses deposited at
$T_{s}=963$ K. Insets in left panel: enlarged view of the (2 0 0) diffraction
for the thick film deposited at $T_{s}=983$ K; Inset in right panel: enlarged
view of the (2 0 0) peak for the 36-nm-thick film deposited at $T_{s}=963$ K.
The solid curves in the insets are least-squares fits to Gaussian function.

##  III Results and Discussion

###  III.1 Crystal structure and optical transparency

Our samples are split into two groups. The films with thickness $t\gtrsim 550$
nm are called thick films, while these $t\lesssim 36$ nm ones are called thin
films. Figure 1 shows the XRD patterns for some representative films. The
deposition condition and the thickness of each film are labeled in the figure.
The diffractions can be indexed based on the structure of undoped tetragonal
SnO2 (powder-diffraction file number: 46-1088). From the left panel of Fig. 1,
one can see that the peaks related to the diffractions of (1 1 0), (1 0 1), (2
0 0), and (2 1 1) planes of tetragonal SnO2 appear in the patterns for films
deposited at 933 K and 953 K. While for the films deposited at $T_{s}\gtrsim
963$ K, only the diffraction of (2 0 0) plane can be observed. Other impurity
phases such as Ta2O5 or Ta-Sn alloys (compounds) are not detected. The thin
films deposited at 963 K retain the texture-growth feature (along [1 0 0]
direction), even when the thickness of the film is as low as $\sim$11 nm. For
the thick films, it is found that the film deposited at $T_{s}=963$ K has the
minimum resistivity at a certain fixed temperature. Thus we take $T_{s}=963$ K
as the optimal substrate temperature.

The mean grain size of each film was determined from the full width at half
maximum (FWHM) of (2 0 0) diffraction peak using Scherrer’s equation[22, 23,
24, 25]

$d=K\lambda/(B\cos\theta)$ (1)

where $d$ is the mean diameter of the grains, $\lambda$ is the x-ray
wavelength, $B$ is the FWHM of the selected diffraction peak, $\theta$ is the
diffraction angle, and $K$ is a constant. The FWHM of (2 0 0) peak can be
evaluated by fitting the experimental data to Gaussian function (see the inset
of Fig. 1). Taking $K=0.89$, [22] we calculate the mean grain sizes of the
films and list them in Table 1. For the thin films, the mean grain size
increases as the thickness of the films increases, while the mean grain size
increases with increasing $T_{s}$ for the thick films. The mean grain sizes of
the thick films were also measured by a scanning electron microscope. It is
found that the difference of the mean sizes determined by the two methods is
less than 10%.

Figure 2: (Color online) Room temperature optical transmission spectra of the
Ta:SnO2 films deposited at different oxygen partial pressures. Inset: Optical
band gap and resistivity versus oxygen partial pressure.

Figure 2 shows the optical transmittance ($T_{R}$) as a function of wavelength
for films deposited at $T_{s}=963$ K and different oxygen partial pressure
($O_{\rm{pp}}$), as indicated. Here $T_{R}$ is the intensity ratio of
transmitted light to incident light, and the thicknesses of these films are in
the range of $\sim$550 to $\sim$1000 nm. For the film deposited in pure argon,
the transmittance in the visible range is less than 60%, while for these
deposited at O2 and Ar mixture, the transmittance is greater than 85%. From
Fig. 2, one can get the optical absorption coefficient $\alpha$ via
$\alpha=-(\ln T_{R})t^{-1}$ with $t$ being the thickness of the film.
According to Tauc _et al_., [26] the relation between $\alpha$ and the
incident photo energy is $(\alpha h\nu)^{1/n}=A(h\nu-E_{g})$, where $n=1/2$
for a direct transition and $n=2$ for an indirect transition, $A$ is a
constant, and $E_{g}$ is the optical band gap. According to the first-
principle calculation result, [20] Ta:SnO2 is a direct gap degenerate
semiconductor, thus the value of $E_{g}$ of each film can be obtained by
extrapolating the linear part of $(\alpha h\nu)^{2}$ versus $h\nu$ plot to
$(\alpha h\nu)^{2}=0$. The inset of Fig.2 shows $E_{g}$ versus the oxygen
partial pressure $O_{\rm{pp}}$ for these Ta:SnO2 films. The optical band gap
$E_{g}$ is $\sim$3.9 eV for the film deposited at pure argon atmosphere. When
0.5% oxygen is introduced, $E_{g}$ increases to $\sim$4.6 eV and then almost
keeps as a constant with further increasing oxygen percentage. The inset of
Fig. 2 also gives the resistivity versus oxygen partial pressure measured at
300 K. The resistivity increases with increasing $O_{\rm{pp}}$ in the whole
testing range. Combining the oxygen partial pressure dependence of optical
transmittance and resistivity, one can see that the optimum oxygen percentage
is $O_{\rm{pp}}\simeq 0.5\%$. Thus the optimal depositing conditions for
Ta:SnO2 film with both high conductivity and high optical transparency are
$T_{s}\simeq 963$ K and $O_{\rm{pp}}\simeq 0.5\%$ in our sputtering system.

Table 1:  Relevant parameters for some representative Ta:SnO2 films. Here $t$ is the mean film thickness, $T_{s}$ is the substrate temperature, $d$ is mean diameter of the grains, $\rho_{0}$ is the residual resistivity, $n^{\ast}$ is the mean value of carrier concentrations between 180 and 250 K, and $\beta_{\rm{BG}}$, $B$, and $\theta_{D}$ are the adjusting parameters in fitting Eq. (4). The parameters $g_{T}$ is obtained by least-squares fits to Eq. (10), and $c_{d}$ is the adjusting parameter in Eq. (11).  | $t$ | $T_{s}$ | $d$ |  $\rho$(300 K) | $\rho_{0}$ | $\beta_{\rm{BG}}$ | $B$ | $\theta_{D}$ | $n^{\ast}$ | $g_{T}$ | $c_{d}$
---|---|---|---|---|---|---|---|---|---|---|---
Films | (nm) | (K) | (nm) | ($\mu\Omega$ m) | ($\mu\Omega$ m) | (n$\Omega$ m K-1) | (10-7 K-2) | (K) | (m-3) |  |
1 | 958 | 983 | 47.3 | 4.46 | 4.23 | 4.81 | 1.31 | 1024 |  |  |
2 | 992 | 963 | 36.9 | 3.16 | 2.90 | 4.07 | 5.72 | 1116 |  |  |
3 | 962 | 953 | 34.2 | 4.56 | 4.35 | 4.55 | 0.94 | 1080 |  |  |
4 | 960 | 933 | 26.2 | 3.93 | 3.69 | 4.25 | 2.66 | 1061 |  |  |
5 | 36 | 963 | 31.6 | 8.85 | 8.53 |  |  |  | 3.75 | 24.02 | 2.12
6 | 30 | 963 | 24.2 | 10.93 | 10.64 |  |  |  | 3.22 | 14.84 | 1.46
7 | 18 | 963 | 17.7 | 10.25 | 9.96 |  |  |  | 2.95 | 10.38 | 1.64
8 | 11 | 963 | 9.8 | 13.04 | 12.9 |  |  |  | 3.09 | 4.94 | 1.66

###  III.2 Electron-phonon-impurity interference effect in the thick films

We firstly investigate the electrical transport properties of the Ta:SnO2
films deposited in pure argon. Figures 3(a) and 3(b) show the temperature
dependence of resistivity for two representative thick films deposited at
$T_{s}=933$ and 963 K, respectively. Upon cooling from 300 K, the resistivity
decreases with decreasing temperature, and reaches its minimum at $T_{\rm
min}$ (the value of $T_{\rm min}$ depends on the film and is 30 K and 20 K for
the films deposited at $T_{s}=933$ and 963 K, respectively), and then slightly
increases with further decreasing temperature. The slight enhancement in
resistivity with decreasing temperature could originate from weak-localization
(WL) effect and electron-electron interaction (EEI) effect. [27] The reduction
of resistivity with decreasing temperature above $T_{\rm min}$ indicates that
Ta:SnO2 film reveals degenerate semiconductor characteristics in electrical
transport properties. It is also found that the carrier (conduction electron)
concentrations of the thick films are insensitive to temperature from room
temperature down to liquid helium temperature, which further confirms the
degenerate semiconductor characteristics of the films.

Figure 3: (Color online) Temperature dependence of resistivity for the thick
Ta:SnO2 films deposited at 963 K [(a) and (c)], and 933 K [(b) and (d)]. The
solid curves in (a) and (b) are least-squares fits to Eq. (2), while those in
(c) and (d) are least-squares fits to Eq. (4).

Generally, the resistivity for a degenerate semiconductor or metal comes from
two effects: the residual resistivity caused by collisions of electrons with
impurity atoms and mechanical imperfections in lattice and the temperature-
dependent resistivity originated from electron-phonon scattering. According to
Altshuler[28, 29, 30], the temperature dependence of resistivity caused by the
“pure” electron-phonon scattering in disordered metal is similar to the Bloch-
Grüneisen formula. Thus the total resistivity can be written as[31, 32, 33,
34]

$\rho=\rho_{0}+\beta_{\rm
BG}T\left(\frac{T}{\theta_{D}}\right)^{4}\int_{0}^{\theta_{D}/T}\frac{x^{5}\mathrm{d}x}{(e^{x}-1)(1-e^{-x})},$
(2)

where $\rho_{0}$ is the residual resistivity, $\beta_{\rm BG}$ is a constant,
and $\theta_{D}$ is the Debye temperature. The theoretical predictions of Eq.
(2) are least-squares fitted to the $\rho(T)$ data of the Ta:SnO2 films from
$T_{\rm min}$ to 300 K and the results are shown as solid curves in Figs. 3(a)
and 3(b). Clearly, the overall trends of the experimental data can be
described by Eq. (2). However, a close inspection indicates that the $\rho(T)$
data below $\sim$120 K remarkably deviate from the theoretical predictions.
Thus other scattering processes could be neglected in above analysis.

Since 1987, Reizer and Sergeev have realized the importance of the inelastic
electron scattering from vibrating impurities in dirty metals. [35] In this
case, the contribution to the resistivity due to the interference between the
electron-phonon and electron-impurity scattering can be expressed as[36, 37,
38]

$\rho_{\rm
int}=BT^{2}\rho_{0}\left(\frac{6}{\pi^{2}}\right)\int_{0}^{\theta_{D}/T}\left[\frac{x^{2}e^{x}}{(e^{x}-1)^{2}}-\frac{x}{e^{x}-1}\right]{\mathrm{d}}x,$
(3)

where $B$ is a material-related constant and can be approximately written as
$B\approx 2\pi^{2}k_{B}^{2}u_{l}^{2}/(3\epsilon_{F}p_{F}u_{t}^{3})$. Here
$k_{B}$ is the Bolzmann constant, $\epsilon_{F}$ is Fermi energy, $p_{F}$ is
the Fermi momentum, and $u_{l}$ and $u_{t}$ are the sound speed of
longitudinal and transverse phonons. For $T\ll\theta_{D}$, the integral in the
right hand side of Eq. (3) tends to $\pi^{2}/6$, then Eq. (3) is simplified as
$\rho_{\rm int}\simeq\rho_{0}BT^{2}$. Considering the contribution of
interference mechanism, one can obtain

$\rho=\rho_{0}+\rho_{\rm BG}+\rho_{\rm int},$ (4)

where $\rho_{\rm BG}$ is the resistivity caused by “pure” electron-phonon
scattering and is represented by the second term in the right hand side of Eq.
(2). The theoretical predictions of Eq. (4) are least-squares fitted to the
experimental data in Figs. 3(a) and 3(b), and the results are redrawn in Fig.
3(c) and 3(d), respectively, for clarity. The optimal parameters obtained from
the fitting processes are listed in Table 1. In Fig. 3(c) and 3(d), we also
give the temperature dependence of $\rho_{\rm BG}$ and $\rho_{\rm int}$ for
each film. Clearly, the interference mechanism dominates over the “pure”
electron-phonon scattering term at low temperature regime (the exact value of
the up bound temperature below which $\rho_{\rm BG}$ is less than $\rho_{\rm
int}$ is sample dependent). These results indicate that the effect of
interference between the electron-phonon and electron-impurity scattering
plays a key role in the low-temperature electrical transport processes of
Ta:SnO2 films. We note in passing that the electron-phonon-impurity
interference effect can be neglected in the films deposited at $T_{s}\simeq
963$ K and $O_{\rm{pp}}\gtrsim 0.5\%$. The reason needs further
investigations.

Figure 4: (Color online) Normalized resistivity $\rho/\rho(300\,\rm{K})$
varies as a function of temperature for four thin Ta:SnO2 films with different
thickness.

###  III.3 Intergrain electron-electron interaction effect on the transport in
the thin films

Now we discuss the longitudinal and Hall transport properties of Ta:SnO2 films
with thickness ranging from $\sim$11 to $\sim$36 nm. Figure 4 shows the
temperature dependence of resistivity of these thin films deposited at
$T_{s}=963$ K and $O_{\rm{pp}}=0.5\%$. Although the overall variation trends
of $\rho(T)$ curves for these films are similar to those of the thick Ta:SiO2
films, the enhancement of resistivity with decreasing temperature below
$T_{\rm min}$ is more remarkable than that in those thick films. In addition,
the resistivity approximately varies linearly with $\log_{10}T$ (or $\ln T$)
below $\sim$$T_{\rm min}$. This indicates that the increasing in resistivity
with decreasing temperature in the low temperature regime could originate from
corrections of WL and EEI effects in two-dimensional (2D) disordered
conductors. [39]

The WL and EEI effects in homogeneous disordered metals have been intensively
investigated since 1980s. [39, 40, 41, 42] In 2D homogeneous disordered
metals, the Altshuler-Aronov type EEI effect gives the follow correction to
the conductivity[39, 40]

$\delta\sigma_{\square}=\frac{e^{2}}{4\pi^{2}\hbar}(2-2F)\ln(k_{B}T\tau/\hbar),$
(5)

where $F$ is an electron screening factor and $\tau$ is the elastic scattering
time. When a magnetic field is applied, the correction can be expressed as[43,
27, 44, 45]

$\delta\sigma_{\square}=\frac{e^{2}}{4\pi^{2}\hbar}[(2-2F)\ln(k_{B}T\tau/\hbar)-G_{2}(b)],$
(6)

where $b=g\mu_{B}B/k_{B}T$ with $g$ being the Landé g-factor, the function
$G_{2}(b)$ can be computed numerically and has the value $\ln(b/1.3)$ and
$0.084b^{2}$ for $b\gg 1$ and $b\ll 1$, respectively. [43]

Since 2002, it has been realized that the EEI effect in granular metals is
distinct from that in “homogeneous disordered metals”. [47, 48, 49, 50, 51,
52] The term granular metals refers to composite materials consisting of
immiscible metals and insulators. In granular metals, the intergrain electron
and intragrain dynamics both play key roles in the electrical transport
properties. In strong intergrain coupling regime, [i.e., $1\ll g_{T}\ll
g_{0}$, where $g_{T}=G_{T}/(2e^{2}/\hbar)$, $g_{0}=G_{0}/(2e^{2}/\hbar)$,
$G_{T}$ and $G_{0}$ are intergrain tunneling conductance and conductance of a
metal grain, respectively], the EEI effect in the presence of granularity
gives distinct corrections to both logarithmic conductivity and Hall
coefficient. According to the theory, in the temperature range
$g_{T}\delta/k_{B}\lesssim T\lesssim E_{c}/k_{B}$ ($\delta$ is the mean-energy
level spacing in the grain and $E_{c}$ is the charging energy), the
conductivity is determined by the granular structure and incoherent tunneling
processes, and can be written as[47, 48, 49, 50]

$\sigma(T)=\sigma_{0}\left[1-\frac{1}{2\pi
g_{T}\tilde{d}}\ln\left(\frac{g_{T}E_{c}}{k_{B}T}\right)\right],$ (7)

where $\sigma_{0}$ is the conductivity without the EEI effect and $\tilde{d}$
is the dimensionality of the granular array. In the temperature range
$g_{T}\delta/k_{B}\lesssim T\lesssim\min(g_{T}E_{c},E_{\rm Th})/k_{B}$, the
virtual electron diffusion inside individual grains governs the temperature
dependent behavior of the Hall coefficient[51, 52]

$R_{H}=\frac{1}{n^{\ast}e}\left[1+\frac{c_{d}}{4\pi
g_{T}}\ln\left(\frac{\min(g_{T}E_{c},E_{\text{Th}})}{k_{B}T}\right)\right],$
(8)

where $n^{\ast}$ is the effective carrier concentration, $c_{d}$ is a
numerical lattice factor of order unity, $E_{\text{Th}}$ is the Thouless
energy. It should be noted that the $\ln T$ dependent behaviors of $\sigma$ in
Eq. (7) and $R_{H}$ in Eq. (8) are independent of the granular array
dimensionality, and are the consequence of and specific to the granularity.

Figure 5: (Color online) Variation in sheet conductivity with a logarithm of
temperature for four thin Ta:SnO2 films [(a) $t=36$ nm, (b) $t=30$ nm, (c)
$t=18$ nm, and (d) $t=11$ nm] measured in a perpendicular magnetic field of 7
T. The solid straight lines are least-squares fits to Eq. (10).

To understand which kind of EEI effect governs the low temperature transport
properties of these thin films, we measured the temperature dependence of
conductivity of the films in a perpendicular magnetic field $B$ of 7 T, which
is sufficiently large to suppress the WL effect. [46] Figure 5 shows the sheet
conductivity $\sigma_{\square}$ as a function of logarithm of temperature,
where $\sigma_{\square}$ is the reciprocal of the sheet resistance. Clearly,
the sheet conductivity varies linearly with $\log_{10}T$, or $\ln T$ from 2 to
$\sim$90 K. We firstly compare the $\sigma_{\square}(T)$ data with Eq. (6). At
low temperature and large magnetic field, Eq. (6) can be rewritten as

$\sigma_{\square}(T)=\sigma_{\square}(T_{0})+\frac{e^{2}}{4\pi^{2}\hbar}(2-F)\ln\left(\frac{T}{T_{0}}\right),$
(9)

where $T_{0}$ is an arbitrary reference temperature. Taking $T_{0}=2$ K, we
fit the $\sigma_{\square}(T)$ data to Eq. (9)using least-square method.
Although the experimental $\sigma_{\square}(T)$ data overlap with the
theoretical predictions of Eq. (5), the fitted values of $F$ are all negative,
e.g. $F\simeq-0.29$, $-0.47$, $-0.28$, and $-0.21$ for the $t\simeq 36$, 30,
18, and 11 nm films, respectively. Theoretically, the value of the screening
factor $F$ should be $0\lesssim F\lesssim 1$. Thus the negative value of $F$
indicates that the $\ln T$ behavior of $\sigma_{\square}(T)$ does not
originate from the Altshuler-Aronov type EEI effect.

Next we analyze the $\sigma_{\square}(T)$ data in the framework of the EEI
theory in the presence of granularity. From Table 1, one can see that the mean
grain sizes of the $t\simeq 36$, 30, 18, and 11 nm films are 31.6, 24.2, 17.7
and 9.8 nm, respectively, thus these films are nominally covered with only one
layer of Ta:SnO2 grains and the dimensionality of the grain array is
$\tilde{d}=2$ for each film. For 2D granular system, we rewrite Eq. (7) as

$\sigma_{\square}(T)=\sigma_{\square}(T_{0})+\frac{\sigma_{\square}^{0}}{4\pi
g_{T}}\ln\left(\frac{T}{T_{0}}\right),$ (10)

where $\sigma_{\square}^{0}$ is the sheet conductance [corresponding to the
$\sigma_{0}$ in Eq. (7)] without the EEI effect. Treating $g_{T}$ as an
adjusting parameter, we fit the $\sigma_{\square}(T)$ data to Eq. (10) by
taking $T_{0}=2$ K and $\sigma_{\square}^{0}=\sigma(90\,\rm{K})$ and the
results are plotted as solid lines in Fig. 5. Inspection of Fig. 5 indicates
that the $\sigma_{\square}(T)$ data for the four films can be well described
by Eq. (10). In addition, the fitted values of the adjustable parameter
$g_{T}$ all satisfy, $g_{T}\gg 1$ (see Table 1), the prerequisite for the
validation of Eq. (10).

Figure 6: (Color online) Hall coefficient as a function of logarithm of
temperature for four thin Ta:SnO2 films with (a) $t=36$ nm, (b) $t=30$ nm, (c)
$t=18$ nm, and (d) $t=11$ nm. The solid straight lines are least-squares fits
to Eq. (11).

Figure 6 shows the Hall coefficient varies as a function of temperature for
the four thin films, where the temperature axis is plotted in logarithmic
scale. Clearly, the Hall coefficients vary linearly with $\ln T$. Similarly,
Eq. (8) is rewritten as

$R_{H}(T)=R_{H}(T_{0})+\frac{1}{n^{\ast}e}\frac{c_{d}}{4\pi
g_{T}}\ln\left(\frac{T_{0}}{T}\right),$ (11)

Our measured $R_{H}(T)$ data are least-squares fitted to Eq. (11) and the
fitted results are represented by the solid lines in Fig. 6. In the fitting
processes, $c_{d}$ is the adjustable parameter, $g_{T}$ has been obtained in
fitting the $\sigma_{\square}(T)$ data, $T_{0}$ is set as 2 K, and $n^{\ast}$
is taken the mean value measured between 180 and 250 K, in which the EEI
correction to $R_{H}$ can be neglected. The values of $c_{d}$ and $n^{\ast}$
for each film are listed in Table 1. Clearly, the $R_{H}(T)$ data of these
thin films are overlapped with the theoretical prediction of Eq. (8). The
values of $c_{d}$ are in the reasonable range. Thus Eqs. (10) and (11) are
safely applicable for the thin films.

Before closing this subsection, we summarize the experimental advances in the
study of the influence of Coulomb effects on the transport properties in
granular metals. Soon after the theoretical finding of the
$\delta\sigma\propto\ln T$ law [Eq. (7)], it was tested in several granular
systems, including Pt/C composite nanwires, [53, 54] B-doped diamond films,
[55] and granular Cr films. [56] Thus the $\delta\sigma\propto\ln T$ law was
verified. To verify $\delta R_{H}\propto\ln T$ law [Eq. (8)], one generally
needs to measure the Hall voltage versus magnetic field at each testing
temperature. This is much more difficult than the conductivity versus
temperature measurement (the logarithmic correction term $\delta R_{H}$ is
estimated to be less than $\sim$10% of the total $R_{H}$). Taking advantage of
the low carrier concentration and degenerate semiconductor characteristics of
tin-doped indium oxide (ITO) films, Zhang _et al_. [57] verified Eq. (8) as
well as Eq. (7) in 2D ITO granular films. Shortly afterwards, Yang _et al_.
[58] also demonstrated the validity of Eq. (8) and Eq. (7) in 2D Al-doped ZnO
granular arrays. In 2015, Wu _et al_. measured the temperature dependence of
conductivities and Hall coefficients of in 2D and 3D Agx(SnO2)1-x ($x$ being
the volume fraction of Ag) granular composites. [59] They observed a $\delta
R_{H}\propto\ln T$ law as well as a $\delta\sigma\propto\ln T$ law in a wide
temperature range. These $\ln T$ temperature behaviors are independent of the
dimensionality of granular arrays, and can be explained by the EEI effects in
the presence of granularity. In the thin Ta:SnO2 films (this paper), each
grain is treated as a metallic particle ($g_{T}\ll g_{0}$) and the carrier
concentration in each film is $\sim$2 to 3 orders of magnitude lower than
those in typical metals (which facilitates the measurement of the temperature
dependence of $R_{H}$). The experimental $\sigma(T)$ [$R_{H}(T)$] data are
least-square fitted to Eq. (10) [Eq. (11)] instead of Eq. (7) [Eq. (8)]. This
ensures only one parameter [$g_{T}$ in Eq. (10) or $c_{d}$ in Eq. (11)] can be
adjusted in the fitting process. Thus the uncertainty arisen in estimating
$E_{c}$ or $E_{\rm Th}$ is avoided. In fact, the value of $E_{c}$ or $E_{\rm
Th}$ is difficult to be estimated in 2D granular arrays with disk-shaped
granules.[56, 57, 58] Overall, our results in the thin Ta:SnO2 films further
confirm that the EEI effects on the transport properties in granular metals
are different from that in homogeneous disordered metals.

##  IV Conclusion

In summary, Ta:SnO2 films with low resistivity and high optical transparency
are successfully fabricated by rf-sputtering methods. It is found that the
optimal depositing conditions for low resistivity and high optical
transparency are $T_{s}\simeq 963$ K and $O_{\rm pp}\simeq 0.5\%$. The
electrical transport properties of the thick films and thin films are
systematically investigated. It is found that all films have degenerate
semiconductor characteristics. For the thick films deposited at pure argon,
besides the electron-phonon scattering, the interference effect between
electron-phonon and electron-impurity scattering also plays an important role
in determining the temperature dependent behaviors of the resistivity. For the
thin films, the robust $\Delta\sigma_{\square}\propto\ln T$ law and $\Delta
R_{H}\propto\ln T$ law have been observed from liquid helium temperature to
$\sim$100 K. We have found that the $\ln T$ behaviors of the conductivity and
Hall coefficient originate from the EEI effect in the presence of granularity.
Our results show that the Ta:SnO2 film, although a new TCO material, can be a
model system for exploration new electrical transport phenomena.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
through Grant No. 11774253 and 12174282.

## DATA AVAILABILITY

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

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