# Constraints on Extragalactic Background Light using TeV Observations of BL
Lacertae objects

Longhua Qin1, Jiancheng Wang2,3, Quangui Gao1, Weiwei Na1,
Huaizhen Li1,Ao Wang1, Chuyuan Yang2,3, and Jianping Yang4
1Department of Physics, Yuxi Normal University, Yuxi, Yunnan, 653100, China
2Yunnan Observatory, Chinese Academy of Sciences, Kunming, Yunnan, 650011,
China.
3Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese
Academy of Sciences, Kunming, Yunnan, 650011, China.
4Yunnan Agricultural University, Kunming, Yunnan, 650201, China
<EMAIL_ADDRESS>
###### Abstract

The extragalactic background light (EBL) in the IR to UV bands partly absorbs
very high energy (VHE, $E\geq$ 100GeV) $\gamma-$ray photons travelling over
cosmological distances via pair production. In this paper, to get stronger
constraints on EBL, we use the deliberate selection of the EBL model and data
of five BL Lacs with better statistics and the harder spectra to limit the EBL
density and the radiation mechanism of BL Lacs. We constrain the upper limit
of the EBL density by fitting the spectral energy distributions (SEDs) of TeV
BL Lacs and find that our results are compatible with the published
measurement, reaching 50 $\rm{nWm^{-2}sr^{-1}}$. We also obtain that the EBL
is not necessarily transparent to high VHE photons. We fix the intrinsic
spectral index $\Gamma_{i}$ of TeV BL Lacs as 1.0 and 1.5 under observation
evidence and model assumption. Comparing the EBL density given by galaxy count
and $Spitzer$ observations, we then obtain that 1ES 1101-232 has $\Gamma_{i}$
$\leq$ 1.0 and 1ES 0229+200 should have $\Gamma_{i}$ not harder than 1.0. We
demonstrate that the common radiation assumption of BL Lacs, in which the
$\Gamma_{i}$ is softer than 1.5, should be revisited. Furthermore, we propose
that the upper EBL density could be given by fitting the hardest energy
spectra of TeV BL Lacs.

###### keywords:

Gamma-rays: galaxies– BL Lac objects: general – diffuse radiation

††pagerange: Constraints on Extragalactic Background Light using TeV
Observations of BL Lacertae objects–LABEL:lastpage††pubyear: 2023

##  1 INTRODUCTION

The extragalactic background light (EBL) is the diffuse photon field formed
mainly by thermal processes resulting from structure formation. It exists
today at wavelengths from the cosmic optical background (COB) at $\lambda_{\rm
EBL}=0.1-10\mu m$ to the cosmic infrared background (CIB) at $\lambda_{\rm
EBL}=10-1000\mu m$. Thus, EBL’s strength and spectral shape hold fundamental
information about the history of star formation and cosmological evolution
(Inoue et al., 2013; Saldana-Lopez et al., 2021). In addition, the EBL photons
can attenuate the distant VHE (E $\geq$ 100GeV) $\gamma$-rays via the
$\gamma\gamma\rightarrow e^{+}e^{-}$ process that forms an insurmountable
obstacle to the high-energy cosmic astronomy (Partridge & Peebles, 1967;
Peebles, 1993; Hauser et al., 1998; Dwek & Krennrich, 2013). Therefore, the
knowledge of EBL intensity and spectra is necessary and relatively important
but remains controversial.

Directly measuring the EBL faces several challenges, and the most difficult
task is removing the pollution of the foreground photons like zodiacal light
from the Milky Way and the diffuse galactic light (Hauser et al., 1998; Hauser
& Dwek, 2001; Mattila, 2006). These attempts are engrossed in the optical
(Bernstein, 2007; Lauer et al., 2021) and the IR(Bernstein, 2007), while the
reliability of data sets needs to be in general agreement. In addition,
calibration issues also exist in practice. A strict lower limit on EBL
intensity can be acquired by collecting the light emitted from resolved
galaxies (Madau & Pozzetti, 2000; Hauser & Dwek, 2001). This approach,
available in many bands, including UV, optical/NIR far IR, and submillimeter,
favours the local EBL determination. The acquired EBL intensity in optical and
near-IR are generally below direct photometry estimates. Basic descriptions of
EBL and its measurement are elaborated in the context of recent studies by
Saldana-Lopez et al. (2021).

An indirect way to constrain the EBL is to obtain the EBL optical depth as the
functions of wavelength and redshift if both the intrinsic and observational
$\gamma$-rays are available from distant extragalactic sources. Blazars are
the most common VHE gamma-ray sources, classified into flat-spectrum quasars
(FSRQ) and BL Lacs based on the width of the emission line. Besides, blazars
exist at various redshifts, and their VHE spectrum is modified by EBL
absorption, becoming the natural candidate for studying EBL. This approach has
been tested by several authors (Stecker & de Jager, 1993; Dwek & Slavin, 1994;
Mazin & Raue, 2007; Finke & Razzaque, 2009; Yang, & Wang, 2010; Domínguez et
al., 2013, 2019; Fermi-LAT Collaboration et al., 2018; Desai et al., 2019;
Longhua et al., 2020; Singh & Meintjes, 2020; Singh et al., 2021), and has
been remarkably successful in practice. However, this method still faces the
uncertainty of the intrinsic spectral shape and optical depth model in the
fitting process. In fact, the VHE $\gamma$-rays of blazars can be generated
through one-zone leptonic synchrotron self-Compton model (SSC) (Zhang et al.,
2012), hadronic process (Zheng et al., 2016) or electromagnetic cascades (Chen
et al., 2015). The intrinsic/unattenuated energy spectra of blazars are
usually assumed to constrain the EBL (Domínguez et al., 2019; Fermi-LAT
Collaboration et al., 2018; Desai et al., 2019).

Unlike previous studies focusing on each distant object (e.g., Acciari et al.
(2019)), we only consider three main types of TeV BL Lacs. To get lower
statistics and the FIR limits due to a pile-up at energies, we use two near
and well-measured BL Lacs, Mkn421 and Mkn501. Besides, we take one distant
source, 1ES0229-200, and two sources with the hard spectra, 1ES1101-232 and
1ES1218+304, because their TeV emissions are more subject to EBL absorption,
and their hard spectra are natural candidates to study the lower limit on EBL.
We obtain the VHE data compiled by (Biteau & Williams, 2015) and derive the
updated data in an online database 111https://gamma-cat.readthedocs.io/. Note
that we exclude the $Fermi$-LAT data in this work since the origin of photons
in the GeV bands may not be the same as in the TeV band, and fitting intrinsic
spectra also requires the (quasi)-simultaneous observations.

For the BL Lacs, the intrinsic spectra in the TeV regime are characterized by
$dN/dE\propto E^{-\Gamma_{i}}$. One can expect the same photon index
$\Gamma_{i}=1.5$ as the synchrotron emission spectrum if the inverse-
Compton(IC) occurs in the Thomson regime and $\Gamma_{i}\geq 1.5$ if the
scattering happens in Klein-Nishima regime. Furthermore, if the VHE comes from
the hadronic process, then $\Gamma_{i}=2.0$ (Mazin & Raue, 2007). However,
Katarzyński et al. (2006) argue that if synchrotron emission and IC do not
occur in the region of electron acceleration, the electron spectrum may become
truncated due to propagation, such as $\Gamma_{i}<1.0$. Besides, at least some
distant TeV BL Lacs, e.g., 1ES 1101-232 and 1ES 0229+200, exhibit a much
harder spectral index. Krennrich et al. (2008) point out that their spectral
index $\Gamma_{i}$ is 1.28 or harder even under the weakest EBL model. The
high density of EBL photons will lead to more absorption, requiring the harder
intrinsic spectrum to get the measured spectrum that is not very soft.
Considering the above mentioned, we can set $\Gamma_{i}$ as 1.5 and 1.0 in the
fitting procedure to determine the EBL density’s lower limits.

In this paper, a flat $\Lambda$CDM cosmology with $\Omega_{\Lambda}=0.7$,
$\Omega_{M}=0.3$, $H_{0}=70\rm{km\ s^{-1}\ Mpc^{-1}}$ is assumed. Besides, the
unprimed and primed quantities represent the quantities in the observer and
the co-moving frames respectively.

##  2 MODEL AND STRATEGY

The distant $\gamma$-ray photon of energy $E_{\gamma}$ at $z$ will interact
with a background photon of energy $\epsilon$ by the cross section
$\sigma_{\gamma\gamma}\left(E_{\gamma},\epsilon,\mu\right)$:

$\displaystyle\sigma_{\gamma\gamma}\left(E_{\gamma},\epsilon,\mu\right)=$
$\displaystyle\frac{3\sigma_{\mathrm{T}}}{16}\left(1-\beta^{2}\right)$
$\displaystyle\times\left[2\beta\left(\beta^{2}-2\right)+\left(3-\beta^{4}\right)\ln\left(\frac{1+\beta}{1-\beta}\right)\right],$
$\displaystyle\beta\equiv$
$\displaystyle\sqrt{1-\frac{\epsilon_{\mathrm{th}}}{\epsilon}},$
$\displaystyle\epsilon_{th}(E_{\gamma},\ \mu)=$
$\displaystyle{2(m_{e}c^{2})}^{2}\over E_{\gamma}(1-\mu),$ (1)

where $\sigma_{\rm T}$ is the Thompson cross section,
$\epsilon_{th}(E_{\gamma},\ \mu)$ is the threshold energy dictated by
kinematics, $\mu$ are the cosine of the angle for the incident photons and the
mass of electron, respectively. The corresponding optical depth is given by

$\displaystyle\tau_{\gamma}\left(E_{\gamma},z\right)=$
$\displaystyle\int_{0}^{z}\left(\frac{dl}{dz^{\prime}}\right)dz^{\prime}\int_{-1}^{+1}d\mu\frac{1-\mu}{2}$
$\displaystyle\times\int_{\epsilon_{\mathrm{th}}^{\prime}}^{\infty}d\epsilon^{\prime}n_{\rm{EBL}}\left(\epsilon^{\prime},z^{\prime}\right)\sigma_{\gamma\gamma}\left(E_{\gamma}^{\prime},\epsilon^{\prime},\mu\right),$
(2)

where $dl/dz^{\prime}$ is the differential distant of redshift written as

$\frac{dl}{dt}=c(H_{0}(1+z)\sqrt{(\Omega_{M}(1+z)^{3}+\Omega_{\Lambda})})^{-1},$
(3)

$\epsilon_{\mathrm{th}}^{\prime}=\epsilon_{\mathrm{th}}(E_{\mathrm{\gamma}}^{\prime},\mu)$,
$E_{\mathrm{\gamma}}^{\prime}=E_{\mathrm{\gamma}}(1+z^{\prime})$, and
$n_{\rm{EBL}}$ is the EBL density in the observer frame given by

$\displaystyle\epsilon^{2}n_{\rm{EBL}}(\epsilon,\ z)$ $\displaystyle=$
$\displaystyle\left({4\pi\over c}\right)\nu I_{\nu}(\nu,\ z)$ (4)
$\displaystyle=$
$\displaystyle\int_{z}^{\infty}\nu^{\prime}{L}_{\nu}(\nu^{\prime},\
z^{\prime})\left|{{\rm d}t\over{\rm d}z^{\prime}}\right|{{\rm
d}z^{\prime}\over 1+z^{\prime}}.$

Following Dwek & Krennrich (2005), we set the EBL intensity in
$\rm{nWm^{-2}sr^{-1}}$ as the 9 degrees of polynomial:

$\log\left[\nu
I_{\nu}(\lambda)\right]=\sum_{j=0}^{8}a_{j}[\log(\lambda)]^{j}.$ (5)

The form of the intrinsic spectrum $(\frac{dN}{dE})_{\rm int}$ used in this
paper is taken from very general considerations about the emission mechanisms.
The SEDs of blazars are commonly thought to be originated from the leptonic
model, e.g., the SSC (Band & Grindlay, 1985). In addition, the VHE spectrum
may result from hadronic scenarios like emission from leptonic models
demonstrating a smooth and concave spectrum. Furthermore, the spectrum may
have an addition of cut-off components caused by the Klein-Nishina effect or a
cut-off in the underlying electron distribution. Considering the above
reasons, the intrinsic spectrum, following H. E. S. S. Collaboration et al.
(2013), can be expressed as a power law (PL), $N_{0}(E/E_{0})^{-\Gamma}$, a
power law with the exponential cut-off (PLE), $N_{0}(E/E_{0})^{-\Gamma}{\rm
exp}(-E/E_{cut})$, a log parabola (LP), $N_{0}(E/E_{0})^{{\rm-a-
bln}(E/E_{0})}$, and a log parabola with the exponential cut-off (LPE),
$N_{0}(E/E_{0})^{{\rm-a-bln}(E/E_{0})}{\rm exp}(-E/E_{cut})$. In the above
formulas, $\Gamma$ and $N_{0}$ are the photon index and flux normalization,
$a$ is the photon index at $E_{0}$, and $b\geq 0$ is the curvature parameter.
Here, the reference energy $E_{0}$ is set to the decorrelation energy of the
spectrum. In this work, following Biteau & Williams (2015), $E_{0}$ is fixed
to the central value of the energy range of each spectrum and obtained by
$\sqrt{E_{\rm{min,VHE}}E_{\rm{max,VHE}}}$.

The distant VHE $\gamma$-rays arriving at Earth can be attenuated by the EBL
photons, and the observed spectrum is given by

$(\frac{dN}{dE})_{\rm obs}=(\frac{dN}{dE})_{\rm
int}e^{-\alpha\rm{\tau}{(E_{\rm{\gamma}},z)}},$ (6)

where $\alpha$ is the scale factor determined by spectral fitting.

In this paper, four widely accepted EBL templates of Franceschini et al.
(2008)(Fr), Finke & Razzaque (2009)(Fi), Domínguez et al. (2011)(Do) and
Gilmore et al. (2012)(Gi) are used to fit each source. The first template by
Franceschini et al. (2008) is based on data sets available from $Spitzer$ IR
observations and ground-based telescopes under an empirical backward evolution
model to estimate EBL contribution. The distant observables at near IR bands
are obtained from the contribution of the evolution of spheroidal, spiral, and
merger galaxies. The rest of the EBL bands are from extrapolation (for shorter
wavelengths) or the thermal dust emission from the interstellar galaxy medium
(for longer IR wavelengths). The second template proposed by Finke & Razzaque
(2009) predicts the intensity of EBL directly from the stellar radiation and
re-emission by dust in the stellar medium. In this template, the emission in
the shorter wavelength is obtained from the main and off-main sequence stars
under the assumption of the star formation rate, initial mass function, and
dust extinction. At the larger wavelength, the emission mainly originates from
the dust, similar to a combination of three black bodies. The third template
by Domínguez et al. (2011) is mainly based on the observations of 6000
galaxies. The EBL’s evolving spectrum comes from the observed rest-frame
$K$-band galaxy luminosity. The final template by Gilmore et al. (2012) is
based on the semi-analytical approach describing the evolving radiation
released by galaxies and quasars starting from initial cosmological
conditions.

We perform the spectral fitting with a minimum chi-square given by the LMFIT
package (Newville et al., 2016) under MCMC method (Lewis & Bridle, 2002;
Mackay, 2003; Foreman-Mackey et al., 2013). LMFIT is a Least-Squares
Minimization routine and provides a simple and flexible approach to explore
the high-level parameters’ space. We input the $\chi^{2}$, which comes from
$\chi^{2}=\sum_{i=1}^{N}(\frac{f_{obs}-f_{model}}{\sigma_{i}})^{2}$, where
$f_{obs}$,$f_{model}$ are observed and modelled spectra of TeV BL Lacs,
$\sigma_{i}$ is the observed error. We thereby derive the intrinsic spectrum
of a chosen source by adopting the function to obtain the lowest value of
$\chi^{2}(\alpha)$ when $\alpha=0$. Based on the adopted model, we then fit
$\alpha$ that should satisfy $\chi^{2}(1)\geq\chi^{2}(\alpha)$. We derive the
final optical depth and uncertainty by calculating the mean of four scaled
individual optical depths. Upon those procedures, we employ the MCMC algorithm
(Lewis & Bridle, 2002; Mackay, 2003) to do the EBL density fitting based on
Equations from Eq.2 to Eq.5.

##  3 APPLICATION AND RESULTS

To get the EBL shapes at the redshift of z=0, we use the TeV data to fit the
observed SEDs of 5 BL Lacs based on the assumed intrinsic spectra. At first,
following the formalism described in section 2, we derive the best fitting
SEDs, the scale factors, and their uncertainties. We present the results in
Figure 1 \- 4 and the Table 1. Then we determine the nine parameters of the
EBL model by fitting average scaled optical depth. We provide the best fitted
spectra of the EBL and the 68% confident level (C.L.) results in Figure 5 \- 6
and the Table 2. Finally, as the comparison, we also give the upper limits of
the EBL density derived by direct measurements (Dwek & Krennrich, 2013),
$spitzer$ observations in mid-IR bands (Fazio et al., 2004), galaxy counts
(Madau & Pozzetti, 2000), and the lower limits (LW08) at 3.6$\mu m$ band
(Levenson & Wright, 2008).

###  3.1 Upper limits on EBL from TeV BL Lacs

The $\gamma$-rays from the TeV BL Las attenuated by the soft photon of the EBL
at redshift z via pair production process. In this paper, the VHE photons
covering 100 GeV to 10 TeV emitted by TeV BL Lacs correspond to the EBL
photons with 1-5$\mu m$ (0.1-8eV). The relation in dimensional units can
roughly be expressed as E(TeV)$\sim$0.5$\lambda$($\mu m$), and the soft photon
obtained contains only a portion of the EBL energy spectrum.

The upper limits of the EBL SED $\nu I_{\nu}$ in different bands given by five
TeV BL Lacs are between 10$\sim$40 $\rm{nWm^{-2}sr^{-1}}$. Figure 5 shows the
local EBL SED and its 68% C.L. model uncertainties (shaded area with colors).
This result is consistent with the previous studies (Yang, & Wang, 2010;
Longhua et al., 2020). However, as shown in Figure 5, for 1ES 1218+304(z=0.18)
and 1ES 0229+200(z=0.14), the derived EBL SEDs are close to or lower than the
integrated ones of resolved galaxies. The results agree with those obtained by
Abeysekara et al. (2019), in which the EBL SED is calculated by reproducing
the SEDs of 14 VERITAS-detected blazars under the generated EBL shapes. In
addition, from Table 1, we find that the optical depths obtained by
MKn421(z=0.03) and Mkn501(z=0.03) are systematically higher than those
obtained by the common models and the observations. The results also supported
by previous studies (Aharonian et al., 2006) imply that the EBL is more
transparent for the high redshift VHE photons than the local ones. However,
the optical depth of 1ES 1101-232 (z=0.186) is at odds with the above
conclusions and has a similar value at lower redshift.

###  3.2 Lower limits on EBL from the fixed spectral index of TeV BL Lacs

The intrinsic spectra of BL Lacs follow the power-law form ($E^{-\Gamma_{i}}$)
in the TeV band based on the theoretical analysis (Aharonian et al., 2006).
Besides, as discussed in the previous section, ${\Gamma_{i}}$ looks much
harder than 1.5, and several spectral indices in the flaring statement are
around 1.5 with large uncertainties recorded by the second $Fermi$-LAT flare
catalogue (Abdollahi et al., 2017), which enable us to consider the hardest
index in the study. Here, the fixed ${\Gamma_{i}}$ set as 1 and 1.5 have been
used to detect the lower limits of the EBL intensity and test the common
radiation mechanisms of the BL Lacs. The result are shown in Table 1-2 and
Figure 6.

The harder index could require the lower EBL intensity Aharonian et al.
(2007); Krennrich et al. (2008), meaning that the hardest index of
${\Gamma_{i}}=1$ could result in the lowest limits of EBL spectra. In this
paper, we have set the spectral indices of two BL Lacs as 1 and 1.5 to limit
the EBL. From Table 1-2 and Figure 6, we can find that, for the 1ES 0229+200,
the EBL SED is slightly larger than that obtained by the $Spitzer$ mid-IR and
galaxy count when ${\Gamma_{i}}=1.5$ is fixed, while the upper limit of the
derived EBL SED is close to the direct measurements if ${\Gamma_{i}}=1.0$ is
given. Interestingly, we find that, from the left panel of Figure 6, the lower
limit given by $Spitzer$ at 3.5$\mu m$ is between the upper
(${\Gamma_{i}}=1.0$) and the lower (${\Gamma_{i}}=1.0$) limits of the derived
EBL SED. Meanwhile, for 1ES 1101-232, we can see that the upper and lower
limits of the derived EBL SED are at the top and bottom of the Finke 2010
model. However, the upper limit of the derived EBL SED is slightly lower than
that measured by $Spitzer$ at the 3.5$\mu m$ band.

Table 1: The scaled parameter $\alpha$ constrained by VHE data in four EBL
models

EBL Model | Mkn 421 | Mkn 501 | 1ES 1218+304 | 1ES 0229+200 | 1ES 0229+200(1.0) | 1ES 0229+200(1.5) | 1ES 1101-232 | 1ES 1101-232(1.0) | 1ES 1101-232(1.5)
---|---|---|---|---|---|---|---|---|---
| [1] | [2] | [3] | [4] | [5] | [6] | [7] | [8] | [9]
Fr |  3.69$\pm$ 1.03 |  4.42$\pm$ 1.64 |  0.99$\pm$ 0.53 | \- - |  1.48$\pm$ 0.30 |  0.95$\pm$ 0.28 |  4.95$\pm$ 1.61 |  1.55$\pm$ 0.17 |  1.15$\pm$ 0.17
Fi |  4.50$\pm$ 0.90 |  4.19$\pm$ 1.43 |  0.88$\pm$ 0.55 | \- - |  1.11$\pm$ 0.25 |  0.72$\pm$ 0.21 |  3.20$\pm$ 1.66 |  1.43$\pm$ 0.16 |  1.06$\pm$ 0.16
Do |  4.22$\pm$ 1.06 |  2.34$\pm$ 1.59 |  0.96$\pm$ 0.49 |  2.71$\pm$ 1.62 |  1.41$\pm$ 0.26 |  0.92$\pm$ 0.25 |  3.73$\pm$ 1.58 |  1.42$\pm$ 0.15 |  1.06$\pm$ 0.15
Gi |  3.87$\pm$ 0.85 |  3.71$\pm$ 1.62 |  0.87$\pm$ 0.45 | \- - |  1.16$\pm$ 0.30 |  0.71$\pm$ 0.26 |  3.42$\pm$ 1.61 |  1.43$\pm$ 0.16 |  1.07$\pm$ 0.16

  * •

.NOTES.–Fr,Fi,Gi,Do, four EBL templates are obtained from Franceschini et al.
(2008),Finke & Razzaque (2009),Domínguez et al. (2011) and Gilmore et al.
(2012), respectively.

Table 2: The polynomial parameters $a_{\rm j}$ derived from optical depths by
Eq.5

TeV BL Lacs | Polynomial parameters $a_{\rm j}$, j=0, 8
---|---
MRK421 | {1.56, 0.68, -2.37, -1.90, 4.62, 0.76, -4.64, 2.77, -0.53 }
(68% CL) | {1.49 - 1.57, 0.39 - 0.78,-2.19 - -0.95,-2.39 - -0.61,
| 0.62 - 3.54,-1.18 - 2.34,-5.00 - -1.58, 1.42 - 5.65,-2.25 - -0.39}
MRK501 | {1.53, 0.71, -1.37, -2.35, 1.41, 2.78, -2.23, 0.48, -0.06}
(68% CL) | {1.49 - 1.62, 0.41 - 0.93,-2.50 - -0.98,-2.57 - -0.69,
| 0.00 - 3.44,-2.00 - 2.21,-5.00 - -1.23, 1.76 - 7.16,-3.32 - -0.66}
1ES 1218+304 | {0.93, 0.45, -2.10, -1.47, 3.49, 1.01, -3.47, 1.76, -0.29}
(68% CL) | {0.82 - 0.99, 0.36 - 0.98,-2.83 - -1.02,-3.00 - -0.69,
| 0.00 - 3.54,-1.24 - 2.43,-5.00 - -1.25, 1.94 - 7.41,-3.52 - -0.75}
1ES 0229+200 | {1.76, 0.56, -2.44, -1.80, 4.11, 1.19, -4.32, 2.39, -0.43}
(68% CL) | {1.63 - 1.82, 0.35 - 0.96,-3.14 - -1.18,-3.00 - -0.82,
| 0.65 - 3.65,-0.98 - 2.58,-4.22 - -0.97, 1.80 - 7.49,-4.01 - -0.78}
1ES 0229+200(1.0) | {1.40, 0.54, -2.28, -1.48, 3.62, 0.80, -3.13, 1.56, -0.26}
(68% CL) | {1.32 - 1.40, 0.30 - 0.67,-2.28 - -1.05,-2.23 - -0.50,
| 0.77 - 3.77,-1.03 - 2.51,-5.00 - -1.68, 1.19 - 4.80,-1.74 - -0.28}
1ES 0229+200(1.5) | {1.21, 0.55, -2.48, -1.72, 4.66, 0.90, -4.60, 2.44, -0.38}
(68% CL) | {1.11 - 1.21, 0.30 - 0.74,-2.25 - -0.92,-2.40 - -0.57,
| 0.00 - 3.55,-1.19 - 2.28,-5.00 - -1.54, 1.49 - 6.12,-2.55 - -0.44}
1ES 1101-232 | {1.56, 0.45, -2.28, -1.14, 3.59, 0.80, -3.61, 1.87, -0.29}
(68% CL) | {1.46 - 1.60, 0.34 - 0.86,-2.44 - -0.93,-2.53 - -0.62,
| 0.00 - 3.55,-1.26 - 2.27,-5.00 - -1.41, 1.67 - 6.68,-2.94 - -0.56}
1ES 1101-232(1.0) | {1.14, 0.64, -2.37, -2.17, 4.65, 1.47, -4.82, 2.33, -0.32}
(68% CL) | {1.10 - 1.15, 0.37 - 0.62,-2.44 - -1.40,-2.19 - -0.60,
| 1.53 - 4.21,-0.72 - 2.88,-5.00 - -1.85, 0.83 - 2.98,-0.78 - -0.09}
1ES 1101-232(1.5) | {1.03, 0.50, -2.53, -0.71, 3.62, -0.36, -2.08, 1.14, -0.17}
(68% CL) | {0.97 - 1.03, 0.32 - 0.62,-2.38 - -1.25,-2.10 - -0.48,
| 1.11 - 4.06,-0.84 - 2.72,-4.44 - -1.80, 0.88 - 3.47,-1.02 - -0.13}

Figure 1: Best-fit SEDs of TeV BL Lacs under four EBL templates Figure 2:
Continue–As Fig.1 Figure 3: Continue–As Fig.1 Figure 4: Continue–As Fig.1

##  4 DISCUSSION AND CONCLUSION

In this paper, we study the EBL property by fitting the observed spectra of
five TeV BL Lacs, including close-by and well-studied sources, distant
sources, and hard spectral sources. We thoroughly test four shapes of
intrinsic gamma-ray spectra to fit the observed SEDs using the commonly used
EBL templates. Unlikely the stacking analysis (Fermi-LAT Collaboration et al.,
2018; Desai et al., 2019) that requires a large amount of data, we get the
average scale factor $\alpha$ of EBL templates and its uncertainty by finding
the lowest value of $\chi^{2}(\alpha)$ during the fitting procedures. From
Table1, we can see that the scale factors are from 0.8 to 4.0, although their
uncertainties are around 50%, which are similar to those obtained by (Sinha et
al., 2014). Finally, we get the EBL SED from Eq.2 to Eq.5 based on the average
scaled EBL templates. It is stressed that the evolution of the EBL with
redshift must be considered because both the universe expansion and the
evolution of radiation sources affect the EBL intensity. Therefore, to compare
with the observed EBL SED, we convert the EBL SED at the redshift z to one at
z=0.

We stress the following points: At first, we carefully select the samples that
contain the well-observed gamma-ray data, the moderate redshifts, and the hard
$\gamma$ spectra. Commonly, a source with many data can minimize the
uncertainty of the derived EBL, and the appropriate distant source can balance
EBL’s extinction study and the re-emission of $\gamma$-rays during the
propagation process (Zheng et al., 2016). Secondly, the sources with hard
gamma-ray spectra could reveal the lower limit of the EBL. Under simple
assumptions, we can obtain the lowest limits of the EBL SED and test the
radiation mechanism of TeV BL Lacs. Finally, the choice of $\Gamma$ as 1.0 and
1.5 is somewhat arbitrary. However, the previous studies revealed that the
hardest index ($\sim 1.28$) is required to fit the gamma-ray spectra for some
TeV BL Lacs (Krennrich et al., 2008) and the second $Fermi$-LAT flare
catalogue (2FAV) could contain the sources with large uncertainties (Abdollahi
et al., 2017).

We derive the final optical depth and uncertainty by calculating the mean of
four scaled individual optical depths. Upon these procedures, we employ MCMC
algorithm (Lewis & Bridle, 2002; Mackay, 2003) to fit the EBL SED based on
Equations from Eq.2 to Eq.5. We can see that the maximum EBL SED is roughly
consistent with the published measurement, around 50 $\rm{nWm^{-2}sr^{-1}}$.
We also find that the ”bump-like” spectral feature in our derived EBL SED
around 1 eV band is similar to ones obtained by (Korochkin et al., 2020),
demonstrating that the excess features in EBL flux could have physical
significance. Furthermore, inspired by theory and observation, we use the
fixed spectral index to detect the lower limit of the EBL SED that is most
likely compatible with direct observations and galaxy count. Interestingly, we
know that two spectral windows mostly favour finding a faint extragalactic
background; one is near the infrared band at 3.5 $\mu m$. We find that this
faint value is also higher than the derived EBL density by 1ES 1101-232 under
the case of $\Gamma_{i}=1$, implying this TeV BL Lacs may have much harder
spectra. We summarize the results as following:

(1) Four TeV BL Lacs of MKn421, MKn501, 1ES1101-232, and 1ES0229+200 can well
constrain the upper limits of EBL SED. However, for 1ES0229+200 with lower
$\alpha$, the EBL SED is higher than that obtained by another source with
large scale factor, such as 1ES1101-232, because the fitting optical depths
have large uncertain (the error to be $\geq$ 1.6) and the EBL templates are
only given by Finke & Razzaque (2009);

(2) The upper limits of EBL density is around 50 $\rm{nWm^{-2}sr^{-1}}$. at 1
eV, and the lower density could be at ones between given by $spitzer$ (Fazio
et al., 2004) and galaxy counts (Madau & Pozzetti, 2000);

(3) 1ES1218+304 favours the scenario that the EBL is more transparent to VHE
$\gamma$-rays. However, 1ES1101-232 with a similar redshift does not favour
this collusion;

(4) Using the fixed spectral index to limit the EBL, for the 1ES1101-232, we
doubt that its spectral index could be harder than that we considered before
and below 1.0.

From above, the behavior of EBL is still complex, and the EBL SEDs constrained
by TeV objects vary with redshifts and directions from our results. Mirizzi &
Montanino (2009) singled out that axion-like particles (ALPs) could affect the
EBL distribution in different sky directions. However, ALPs work at the
redshift z$>$0.2 (Mirizzi & Montanino, 2009). The EBL inhomogeneity could
exist at only the order of 1%, which is much smaller than the results in this
paper. The proper reason is that the evolution of the EBL with redshift must
be fully considered. In fact, the evolution of the EBL number density $n_{\rm
EBL}$ is scaled redshift as $(1+z)^{3-k}$ rather than the empirical
$(1+z)^{3}$, and Singh et al. (2021) point out that $k$ relates to redshift.
Another mechanism to explain our results is that the origin of TeV
$\gamma$-rays is still unclear, the intrinsic spectrum used here does not
fully cover, and the VHE could be much harder than that in previous studies
when the spectral index is fixed at 1.0. Besides, the recent VHE observations
of distant gamma-ray sources show that the EBL is more transparent to
$\gamma$-rays than we thought before.

Figure 5: Spectral intensity of the EBL constrained by VHE measurements.

Figure 6: Spectral intensity of the EBL constrained by VHE measurements under
the fixed $\Gamma$.

## Acknowledgments

The authors would like to thank the anonymous referee for the helpful
comments, which improved the manuscript significantly. The authors gratefully
acknowledge the financial supports from the National Natural Science
Foundation of China (grants 12063005, 12063006 and U2031111), the Special
Basic Cooperative Research Programs of Yunnan Provincial Undergraduate
Universities (grants 2019FH001-012, 2019FH001-076 and 202001BA070001-031), the
Program for Innovative Research Team (in Science and Technology) in University
of Yunnan Province (IRTSTYN), the Science Research Foundation of Yunnan
Education Department of China (grants 2020J0649), the program for Reserve
Talents of Young and Middle-aged Academic and Technical Leaders in Yunnan
Province (grants 202205AC160087) and the National Natural Science Foundation
of Yunnan Province (grants 202101AU070010). The authors (QLH) gratefully
acknowledge the financial supports from the Hundred Talents Program of Yuxi
(grants 2019).

## Data availability

The data underlying this article are derived from gammacat: https://gamma-
cat.readthedocs.io, and will be shared on reasonable request to the
corresponding author.

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#  Weak antilocalization effect and triply degenerate state in Cu-doped CaAuAs

Sudip Malick  Department of Physics, Indian Institute of Technology, Kanpur
208016, India  Arup Ghosh  Saha Institute of Nuclear Physics, HBNI, 1/AF
Bidhannagar, Calcutta 700 064, India  Chanchal K. Barman  Department of
Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India
Aftab Alam  Department of Physics, Indian Institute of Technology, Bombay,
Powai, Mumbai 400076, India  Z. Hossain<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology, Kanpur 208016, India  Institute of
Low Temperature and Structure Research, Polish Academy of Sciences, ulica
Okolna 2, 50-422 Wroclaw, Poland  Prabhat Mandal<EMAIL_ADDRESS>Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Calcutta 700 064,
India  Department of Condensed Matter Physics and Material Sciences, S. N.
Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake,
Kolkata 700106, India  J. Nayak<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology, Kanpur 208016, India

###### Abstract

The effect of 50% Cu doping at the Au site in the topological Dirac semimetal
CaAuAs is investigated through electronic band structure calculations,
electrical resistivity, and magnetotransport measurements. Electronic
structure calculations a suggest broken-symmetry-driven topological phase
transition from the Dirac to triple-point state in CaAuAs via alloy
engineering. The electrical resistivity of both the CaAuAs and CaAu0.5Cu0.5As
compounds shows metallic behavior. Nonsaturating quasilinear magnetoresistance
(MR) behavior is observed in CaAuAs. On the other hand, MR of the doped
compound shows a pronounced cusplike feature in the low-field regime. Such
behavior of MR in CaAu0.5Cu0.5As is attributed to the weak antilocalization
(WAL) effect. The WAL effect is analyzed using different theoretical models,
including the semiclassical $\sim\sqrt{B}$ one which accounts for the three-
dimensional WAL and modified Hikami-Larkin-Nagaoka model. Strong WAL effect is
also observed in the longitudinal MR, which is well described by the
generalized Altshuler-Aronov model. Our study suggests that the WAL effect
originates from weak disorder and the spin-orbit coupled bulk state.
Interestingly, we have also observed the signature of chiral anomaly in
longitudinal MR, when both current and field are applied along the $c$ axis.
The Hall resistivity measurements indicate that the charge conduction
mechanism in these compounds is dominated by the holes with a concentration
$\sim$1020 cm-3 and mobility $\sim 10^{2}$ cm2 V-1 S-1.

##  I INTRODUCTION

Topological quantum materials have attracted considerable attention from the
scientific community because they exhibit several novel electronic phenomena
Hasan and Kane (2010); Armitage  _et al._ (2018). The presence of distinct
nontrivial electronic bands in these materials strongly influences transport
behavior, resulting in extremely large linear and nonsaturating
magnetoresistance, chiral anomalies, the topological Hall effect and the weak
antilocalization effect (WAL) of the charge carrier Kumar  _et al._ (2017);
Lou  _et al._ (2017); Xu  _et al._ (2017); Singha  _et al._ (2017); Huang  _et
al._ (2015); Laha  _et al._ (2021); Puphal  _et al._ (2020); Yan  _et al._
(2019); Singh  _et al._ (2020); Chiu and Lin (2013). These unconventional
electronic bands introduce an additional nontrivial $\pi$ Berry phase around
the time-reversal closed loop. In a weakly disordered system and quantum
diffusion regime, the electronic conductivity is modified by the quantum
interference of the carriers. This quantum interference leads to weak
localization or weak antilocalization depending on the value of the phase
shift of the carriers around the time-reversal loop. The nontrivial $\pi$
phase yields destructive interference, which is manifested through the WAL
effect Liu  _et al._ (2017); Hikami  _et al._ (1980); Bergmann (1984) in
several three-dimensional (3D) topological insulators and semimetals He  _et
al._ (2011); Li  _et al._ (2019); Naumann  _et al._ (2020); Chamorro  _et al._
(2019); Hou  _et al._ (2015).

Very recently, the WAL effect was realized in several ternary topological
semimetals such as CaAgBi Sasmal  _et al._ (2020), YbCdGe Laha  _et al._
(2019), YbCdSn Laha  _et al._ (2020a) and ScPdBi Zhang  _et al._ (2019). This
particular ABC-type hexagonal family of materials provides various topological
phases, including the Dirac semimetallic phase in CaAuAs, the topological
insulating phase in CaAgAs, the nodal-line semimetallic phase in CaCdSn, and
the correlated nodal-line semimetallic phase in YbCdGe Tsetseris (2016); Singh
_et al._ (2018); Nayak  _et al._ (2018); Laha  _et al._ (2020b, 2019).
Interestingly, various broken-symmetry-driven topological phases have also
been observed in the ternary pnictide BaAgAs, and SrAgAs Mardanya  _et al._
(2019); Mondal  _et al._ (2019). Likewise, CaAuAs belonging to the
$P6_{3}/mmc$ space group is an excellent example of a topological material
that exhibits various topological ground states depending on the existing
symmetry. According to a first-principles calculation, when the spin-orbit
coupling (SOC) is ignored, CaAuAs hosts a nodal-line semimetallic state Singh
_et al._ (2018). Strikingly, a Dirac semimetallic state can also be achieved
by considering SOC. Furthermore, in the absence of $C_{3}$ rotational symmetry
and inversion symmetry, it leads to the topological insulating and Weyl
semimetallic states, respectively Singh  _et al._ (2018). A recent report on
angle-resolved photoemission spectroscopy (ARPES) corroborated the nontrivial
band topology and Dirac semimetallic state in CaAuAs. However, the Dirac band
crossing is observed about 0.25 eV above the Fermi level Nakayama  _et al._
(2020). It has been suggested that chemical doping in CaAuAs may shift the
Fermi level, and the Dirac band crossing can be probed directly by studying
ARPES and magnetotransport properties. Apart from the shift in the Fermi
level, doping also introduces disorders in the crystal which may result in WAL
or weak localization of the charge carrier. The above mentioned aspects
encourage us to explore the effect of chemical doping on the magnetotransport
properties of the topological Dirac semimetal CaAuAs.

In this work, we study the band structure and magnetotransport properties of a
CaAuAs single crystal and examine the effect of chemical doping on CaAuAs by
replacing 50% of Au with Cu. The first-principles calculations reveal
alloying-induced symmetry breaking in CaAuAs, and a phase transformation from
the Dirac point to triple point (TP) occurs when 50% Cu is doped at the Au
site. The magnetotransport measurements unfold strong a WAL effect up to room
temperature in the doped compound. Furthermore, a negative longitudinal MR is
observed along the $c$ axis, which indicates a chiral anomaly effect. The Hall
resistivity confirms the dominant hole type charge carrier with high mobility.

##  II SAMPLE PREPARATION AND EXPERIMENTAL TECHNIQUES

Figure 1:  (a) and (b) Rietveld refinement profile of the powder XRD pattern
of crushed single crystals for CaAuAs and CaAu0.5Cu0.5As, respectively. Red
dots are the experimental intensity, the black line is the calculated
intensity, and the blue line is the difference between the experimental and
calculated intensities. The green vertical lines display the Bragg positions.
(c) and (d) The XRD pattern for CaAuAs and CaAu0.5Cu0.5As single crystals,
respectively. Only (00$l$) peaks are observed.

Single crystals of CaAuAs and CaAu0.5Cu0.5As were grown by the flux method
using bismuth as a flux. High-purity Ca (99.9%, Alfa Aesar) shots, Au
(99.999%, Alfa Aesar) shots, Cu (99.9999%, Alfa Aesar) shots, As (99.999%,
Alfa Aesar), and Bi (99.99% Alfa Aesar) pieces were taken in 1:1:1:10, and
1:0.5:0.5:1:10 molar ratios for CaAuAs and CaAu0.5Cu0.5As, respectively, and
mixed thoroughly. The mixture was loaded into an alumina crucible and then
sealed in an evacuated quartz ampule. The whole assembly was heated to 1050 ∘C
and soaked at this temperature for 20 h and then slowly cooled down to 400 ∘C
at a rate 3 ∘C/h. Finally, the crystals were separated from the Bi flux by
centrifuging. The typical size of the plate-like crystals is 1.5 mm$\times$1.5
mm$\times$0.2 mm. Further details of the crystal growth method can be found
elsewhere Xu  _et al._ (2020). The grown crystals were characterized by x-ray
diffraction (XRD) in a PANalytical X’Pert PRO diffractometer with Cu
K$\alpha_{1}$ radiation and an energy dispersion x-ray spectroscopy (EDS) in a
JEOL JSM-6010LA spectrometer. The electrical transport and magnetotransport
measurements were carried out in a 9 T physical property measurement system
(Quantum Design) using the standard four-probe technique using the ac-
transport option. Electrical contacts were made using a thin gold wire and
conducting silver paste. For the angle dependence of the MR between 0∘ and
360∘, a sample rotator was used.

The electronic band structure calculations were carried out using density
functional theory implemented within the Vienna $ab$-initio simulation package
(VASP) Kresse and Hafner (1993); Kresse and Joubert (1999). The plane-wave
basis set using the projected augmented wave Blöchl (1994) method was used
with an energy cut-off of 500 eV. Generalized-gradient approximation (GGA)
Kresse and Joubert (1999) by Perdew, Burke, and Ernzerhof was employed to
describe the exchange and correlations. The effect of spin-orbit coupling is
explicitly included in all the calculations. The total energy (force) was
converged up to 10-5 eV (0.01 eV/Å). The Brillouin zone (BZ) integrations were
carried out using a $12\times 12\times 6$ $\Gamma$-centered $k$-mesh.

##  III RESULTS AND DISCUSSION

Table 1: Lattice parameters $a$ and $c$ obtained from the refinement of the powder XRD data. |  $a$ (Å) |  $c$ (Å)
---|---|---
CaAuAs | 4.3776(4) | 7.939(3)
CaAu0.5Cu0.5As | 4.2587(4) | 7.912(3)

###  III.1 Crystal Structure

The powder x-ray diffraction pattern of crushed single crystals of CaAuAs and
CaAu0.5Cu0.5 are shown in Figs. 1 (a) and 1 (b), respectively. Within the
resolution of XRD, we did not see any peak due to the impurity phase. The
diffraction pattern was analyzed by the Rietveld refinement method using
FullProf software, which indicates that both the samples crystallize in a
hexagonal structure with space group $P6_{3}/mmc$ (No. 194), and the
corresponding lattice parameters are listed in Table 1. The lattice parameters
of pure compound agree well with the previous report Tsetseris (2016). The
lattice constants $a$ and $c$ are reduced by 2.72%, and 0.34% ,respectively,
after Cu doping in CaAuAs, which is expected as Cu has a smaller atomic radius
than Au. In order to check the crystalline quality, we performed XRD on single
crystals [Figs. 1 (c) and (d)]. It is clear from the Fig. 1 that only (00$l$)
peaks are present and they are extremely sharp. This confirms high crystalline
quality of the grown single crystals. For the detailed information about the
overall chemical composition, we determined the elemental concentrations at
different randomly selected regions of the grown crystals using EDS. The EDS
data confirm that the atomic compositions of CaAuAs and CaAu0.5Cu0.5As are
close to the expected stoichiometries of 1:1:1 and 1:0.5:0.5:1, respectively.

###  III.2 Electronic structure

Figure 2: Crystal structure of (a) CaAuAs and (c) ordered CaAu0.5Cu0.5As. (b)
and (d) show the electronic band structure of CaAuAs and CaAu0.5Cu0.5As
respectively, including the SOC effect. The insets in (b,d) show a zoomed-in
view of the bands along $\Gamma$-A. $\Lambda_{i},s$ are the irreducible
representations of band characters. The Dirac point (DP) and triply degenerate
nodal points (TP) are shown by arrows.

Although XRD suggests that both the pure and doped compounds form in the same
crystal structure, for electronic structure calculations, we have fully
relaxed the lattice geometry to achieve the minimum energy structure. The
optimized lattice parameters of CaAuAs are found to be $a=b=4.45$ Å and
$c=8.04$ Å. The parent compound CaAuAs has 6 atoms in its primitive cell. To
simulate CaAu0.5Cu0.5As, we have replaced one of the symmetry equivalent Au
sites in the parent compound with a Cu atom. The optimized lattice parameters
of the Cu-substituted compound are found to be $a=b=4.35$ Å and $c=8.00$ Å.
These lattice parameters match the experimental data fairly well. Due to the
underbinding effect of GGA, the theoretical lattice constants are slightly
overestimated as compared to the experimental values. Figures. 2 (a) and (c)
show the crystal structure for the compounds.

The parent compound CaAuAs belongs to the $D_{6h}$ point group symmetry, which
involves space inversion symmetry, and six fold rotational symmetry ($C_{6z}$)
with respect to the _z_ -axis. The time reversal symmetry and space inversion
symmetry together ensure Kramer’s double degeneracy throughout the BZ of
CaAuAs. The stable intercrossing of two doubly degenerate bands in the BZ is
expected to give a four-fold degenerate Dirac point (DP). Along $\Gamma$-A,
the site symmetry group is $C_{6v}$ and every band along this direction can be
differentiated by the irreducible representations (IRREPS) of $C_{6v}$
symmetry. Fig. 2(b) shows the electronic band structure for CaAuAs, including
SOC. Clearly, the two-dimensional IRREPS $\Lambda_{7}$ and $\Lambda_{9}$ along
the $\Gamma$-A direction intercross each other to form a four-fold degenerate
DP. The stability of the DP is ensured by the group orthogonality which states
that the hybridization between two distinct IRREPS is forbidden Dresselhaus
_et al._ .

Insertion of 50% Cu in CaAuAs (to form the alloy CaAu0.5Cu0.5As) breaks the
space inversion symmetry. The alloy now hosts three-fold ($C_{3z}$) rotational
symmetry and $C_{3v}$ site symmetry group along the _z_ -axis ($\Gamma$-A
direction). Since the space-inversion symmetry is absent, the Kramer’s
degeneracy is lifted at generic momenta except time-reversal invariant points.
However, the presence of $C_{3v}$ point group symmetry along the $\Gamma$-A
direction allows both non-degenerate and doubly degenerate bands along this
direction in the BZ. Fig. 2(d) shows the electronic band structure of
CaAu0.5Cu0.5As. The inset shows a zoomed-in view of the bands along the
$\Gamma$-A direction. It clearly shows a doubly degenerate $\Lambda_{4}$ and
two non-degenerate $\Lambda_{5}$ and $\Lambda_{6}$ bands. The intercrossing of
the later two with $\Lambda_{4}$ band gives two triply degenerate nodal points
TP1 and TP2. Using the symmetry analysis, it can be shown that the TPs are
protected by the $C_{3v}$ point group symmetry of CaAu0.5Cu0.5As.

###  III.3 Transport properties

The temperature dependence of the zero-field electrical resistivity
($\rho_{xx}$) for CaAuAs and CaAu0.5Cu0.5As single crystals in the range 2 -
300 K is shown in Fig. 3. With a decrease in temperature, $\rho_{xx}$ for both
the samples deceases monotonically down to the lowest measured temperature,
indicating metallic behavior. It is clear from the figure that the doped
compound has a larger residual resistivity. This is related to the disorder
effect induced by Cu-doping. We have addressed the disorder effect in detail
while discussing the magnetotransport property. The resistivities of both
compounds follow a similar temperature dependence. The overall behavior of
resistivity is quite similar to that observed in the topological materials
CaAgBi Sasmal  _et al._ (2020) and CaAgAs Nayak  _et al._ (2018).
$\rho_{xx}(T)$ is approximately linear at high temperatures but a weak upward
curvature appears in the low-temperature region. We observe that the
resistivity below 30 K exhibits $T^{2}$ dependence (see the inset in Fig. 3).
This behavior indicates a crossover from electron-phonon to electron-electron
scattering mechanism with the decrease in temperature. Thus, the resistivities
for both compounds can be described by the Bloch Grüneisen (BG) scattering
model along with a $T^{2}$ term, which is as follows,

$\rho_{xx}(T)=\rho_{0}+4R\left(\frac{T}{\Theta_{R}}\right)^{5}J_{5}\left(\frac{\Theta_{R}}{T}\right)+aT^{2},$
(1)

where $J_{5}\left(\frac{\Theta_{R}}{T}\right)$ is the Grüneisen integral
function, $\Theta_{R}$ is the Debye temperature, and $R$ and $a$ are the
constants. The first term $\rho_{0}$ of the above equation is the residual
resistivity, second term represents the resistivity due to electron-phonon
scattering and the third term is the contribution to the resistivity due to
electron-electron scattering. The obtained fitting parameters from the Eq. 1
are $\rho_{0}$ = 0.916 $\mu\Omega$ m, $R$ = 0.83 $\mu\Omega$ m, $\Theta_{R}$ =
255 K, and $a$ = 4.40 $\times$10-6 $\mu\Omega$ m K-2, for CaAuAs and
$\rho_{0}$ = 1.481 $\mu\Omega$ m, $R$ = 0.86 $\mu\Omega$ m, $\Theta_{R}$ = 294
K, and $a$ = 3.42$\times$10-6 $\mu\Omega$ m K-2 for CaAu0.5Cu0.5As.

Figure 3:  The electrical resistivities from 2 to 300 K for CaAuAs and the
doped sample are shown. The yellow lines are the fitting with the BG
expression plus an $aT^{2}$ term as given in Eq. 1. $\rho_{xx}$ as a function
of $T^{2}$ of CaAuAs is presented in the inset. The linear fit suggests
quadratic behavior of $\rho_{xx}$ with temperature ($T$ $<$ 30 K). Figure 4:
The magnetic field dependence of TMR ($B\perp I$) and LMR ($B\parallel I$) of
CaAuAs crystal at different fixed temperatures in the range 2 - 60 K. The top
inset in (a) shows the variation of the exponent $m$ with temperature. The
measurement configurations for TMR and LMR are shown schematically in their
respective insets. Figure 5:  The magnetic field dependence of TMR (a) and LMR
(b) of CaAu0.5Cu0.5As crystal in the magnetic field range from -9 T to +9 T at
few representative temperatures in the range 2 K - 300 K. (c) The field-
dependent LMR along the $c$ axis measured in-between -9 T to +9 T at few fixed
temperatures. (d) Field dependence of MR at 4 K for various field
configurations when current is applied along the c axis.

Fig. 4 displays the magnetic field ($B$) dependence of the transverse
magnetoresistance (TMR) and longitudinal magnetoresistance (LMR) of CaAuAs at
different temperatures. The directions of $B$ and the current ($I$) are
schematically shown in the insets. MR is calculated using the expression MR =
$[\rho_{xx}(B)-\rho_{xx}(0)]/\rho(0)$, where $\rho_{xx}(B)$ and $\rho_{xx}(0)$
are the resistivities in the presence and absence of the magnetic field,
respectively. To eliminate the Hall resistivity contribution in MR, all the MR
data were symmetrized using the formula $\rho_{xx}(B)$ =
[$\rho_{xx}(+B)+\rho_{xx}(-B)$]/2. Both TMR and LMR show a weak dependence on
the magnetic field as well as the temperature. The maximum value of MR lies
between 3% and 4% at 9 T. A similar low-value of MR is also found in other
compounds such as CaIr2Ge2, ScPtBi, and LaAlSi Ruszała  _et al._ (2021);
Pavlosiuk  _et al._ (2021); Su  _et al._ (2021). The observed value of MR is
small compared to that reported in many topological semimetals. Extremely
large MR is commonly found in topological materials mainly due to the perfect
electron-hole compensation or very large carrier mobility Gao  _et al._
(2017); Li  _et al._ (2017a); Zeng  _et al._ (2016); Xu  _et al._ (2019); Wang
_et al._ (2019); Yuan  _et al._ (2016). CaAuAs excludes these facts as the
dominant hole type of carrier is present, which has been confirmed from the
Hall resistivity measurements, which will be discussed later. The low carrier
mobility ($\mu$) is the dominating factor for observing low MR in both
compounds as magnetoconductivity $\sigma(B)\sim n\mu/(1+\mu^{2}B^{2})$ for a
single-band model with carrier density $n$ Jacoboni (2010). Furthermore, the
fact that the Dirac band is about 250 meV above the Fermi level, might also be
responsible for the low value of MR as the topological features may have
little influence on the magnetotransport properties Nakayama  _et al._ (2020).
However, MR continuously increases with the applied field without any sign of
saturation. To know the exact dependence of TMR on applied magnetic field, we
have fitted the MR data using the expression $MR\propto B^{m}$ in the high
field region (5 T $\leq B\leq$ 9 T). The obtained values of $m$ are in between
1.22 and 1.25 in the temperature range of 2 to 60 K, which is shown in the
inset in Fig. 4(a). Such a nonsaturating quasi-linear behavior of MR is quite
common in topological materials Liang  _et al._ (2015); Laha  _et al._
(2020b). It is likely that the quasi-linear behavior of MR in CaAuAs is due
the to non-trivial topological state Nakayama  _et al._ (2020); Laha  _et al._
(2020b).

We now focus on the magnetotransport properties of the doped sample,
CaAu0.5Cu0.5As. Interestingly, the nature of the field dependence of MR after
50% Cu-doping in CaAuAs has changed drastically. The magnetic field variation
of TMR and LMR inthe range of $\pm$9 T are presented in Figs. 5 (a) and (b).
In the low-field region and below $\sim$100 K, both TMR and LMR increase very
sharply with the increase of the magnetic field, and, as a result, a cusp-like
feature appears. Such a cusplike feature in MR at low fields is attributed to
the WAL effect. The WAL phenomenon was recently realized in several
topological insulators and semimetals. However, very few systems show the WAL
effect up to room temperature as we observe in CaAu0.5Cu0.5As Sasmal  _et al._
(2020); Pavlosiuk  _et al._ (2021). Moreover, TMR increases gradually as the
magnetic field increases without saturating, reaching $\sim$ 6% at 2 K in 9 T.
The maximum value of TMR decreases as the temperature rises, eventually
falling to $\sim$ 3% at 300 K. Surprisingly, LMR shows a downturn at low
temperatures for applied field above 3 T [Fig. 5 (b)]. Such behavior in LMR in
the high-field region may arise due to the Adler-Bell-Jackiw anomaly in which
LMR decreases due to the chiral imbalance Li  _et al._ (2017b); Zhang  _et
al._ (2016).

We have also measured LMR along the $c$ axis to understand the downward trends
of MR as seen in Fig. 5(b). Both the current and the magnetic field are
applied along the $c$ axis in this configuration. The field-dependent LMR at
various temperatures in such a configuration is shown in Fig. 5(c). The LMR
exhibits a WAL effect in the low field region, but as the field increases, it
starts to decrease and becomes negative at higher fields. Furthermore, we have
measured field-dependent MR at 4 K for different angular positions of the
magnetic field with respect to the c axis when current is also along the $c$
axis as shown in Fig. 5(d). A negative MR is observed in the higher fields
when $\theta$ = 0∘ (i.e. $B\parallel I\parallel c$ axis). With the increase of
$\theta$, MR becomes less negative, and at $\theta$ $\geq$ 75∘, MR becomes
positive. For $\theta=90^{\circ}$ (i.e. $B\perp I\parallel c$ axis), MR
increases monotonically with the increase of the field. The observed downturn
in LMR below 60 K [see Fig. 5(b)] may be due to the chiral anomaly effect
induced from a small component of the current and magnetic field along the c
axis as a result of a small misalignment which is unavoidable.

The negative MR may originate from the chiral anomaly effect. Our band
structure calculation predicts that CaAu0.5Cu0.5As hosts triply degenerate
points near the Fermi level. It has been demonstrated that when the current
and the magnetic field are applied along the $C_{3}$ rotation axis of a
crystal, a triply degenerate point splits into Weyl points due to Zeeman
coupling, leading to the chiral anomaly effect Zhu  _et al._ (2016); Chang
_et al._ (2017); Ma  _et al._ (2018). So, the chiral anomaly-induced negative
LMR is expected in the case of CaAu0.5Cu0.5As as the $C_{3}$ rotation axis is
along the $c$ axis. Recently, the chiral anomaly-induced negative MR was
observed in triply degenerate semimetals like WC He  _et al._ (2017) and
YRh6Ge4 Zhu  _et al._ (2020). Note that the other possible source of negative
LMR is the current jetting effect. To rule out such a possibility, the Ohmic
contacts were made on the surface of the samples across a line using sliver
paint to ensure homogeneous current distribution. Furthermore, several
independent measurements were performed on the same and different crystals,
and the data were reproducible within the experimental error.

Figure 6:  MC data of CaAu0.5Cu0.5As measured at (a) 2 K and (b) 100 K. The
red line is the fit to the 3D WAL model (Eq. 2).

To analyze the WAL effect, we have calculated magnetoconductivity (MC) from
the TMR data [Fig. 5(a)] using the relation
$\sigma_{xx}(B)=\rho_{xx}/(\rho_{xx}^{2}+\rho_{yx}^{2})$, where $\rho_{xx}$
and $\rho_{yx}$ are the longitudinal and Hall resistivities, respectively.
According to the semiclassical model, the transverse MC in the presence of the
WAL effect for a 3D system can be expressed as Kim  _et al._ (2013)

$\sigma_{T}(B)=\sigma_{WAL}+\sigma_{n},$ (2)

where $\sigma_{WAL}$ is the contribution to conductivity due to WAL, which is
expressed as $\sigma_{WAL}$=$\sigma_{xx}(0)+\eta\sqrt{B}$, and
$\sigma_{n}=[\rho_{xx}(0)+A\cdot B^{2}]^{-1}$ is the correction to
conductivity that comes from the conventional Fermi surface. $\sigma_{xx}(0)$,
$\rho_{xx}(0)$, $\eta$ and $A$ are the zero-field conductivity, resistivity,
and constants, respectively. An excellent agreement of MC data with Eq. 2 in
the field range -1.5 T $\leq$ $B$ $\leq$ 1.5 T, confirms the presence of WAL
effect in CaAu0.5Cu0.5As. As a representative, two fitted curves of MC data
calculated from TMR are presented in Fig. 6 and the values of fitting
parameters are $\sigma_{xx}(0)$ = 0.655 $(\mu\Omega$ m)-1, $\rho_{xx}(0)$ = 54
$\mu\Omega$ m, $\eta$=-7.83$\times$10-3 $(\mu\Omega$ m)-1T-0.5, and $A$ = 4004
$\mu\Omega$ mT-2 at 2 K and $\sigma_{xx}(0)$ = 0.580 $(\mu\Omega$ m)-1,
$\rho_{xx}(0)$ = 106 $\mu\Omega$ m, $\eta$= -9.36 $\times$10-3 $(\mu\Omega$
m)-1T-0.5, and $A$ = 1222 $\mu\Omega$ mT-2 at 100 K, respectively.

Figure 7:  (a) MC of CaAu0.5Cu0.5As is presented in the low field region at 2
and 100 K. The red line is the calculated value from the modified HNL
equation. (b) The variation of $l_{\phi}$ with temperature and the pink line
is the fit to Eq. 4. (c) The dependence of parameter $\alpha$ with
temperature. (d) Parallel field MC fitted with the generalized AA model (green
line).

Further, we have analyzed the WAL effect as observed in TMR with the modified
Hikami-Larkin-Nagaoka (HLN) model. Primarily HLN model explains the WAL effect
in two-dimensional (2D) system, however, it is also noticed that the WAL
effect in several topological materials like YbCdGe, $R$PtBi ($R$ = rare
earth), CaAgBi and LaCuSb2 is in concurrence with the HLN model Laha  _et al._
(2019); Chen  _et al._ (2020); Sasmal  _et al._ (2020); Chamorro  _et al._
(2019). In the modified HLN model, a $B^{2}$ term is introduced to take care
of additional scattering such as spin-orbit scattering and elastic scattering.
The modified HLN equation is given as Assaf  _et al._ (2013); Hikami  _et al._
(1980)

$\Delta\sigma_{xx}(B)=-C\left[\Psi\left(\dfrac{1}{2}+\dfrac{\hbar}{4el^{2}_{\phi}B}\right)-ln\left(\dfrac{\hbar}{4el^{2}_{\phi}B}\right)\right]+\gamma
B^{2},$ (3)

where, $C=\frac{\alpha e^{2}}{\pi h}$ and $\alpha$ is 1/2 in the case of 2D
materials. Here, $\Psi$, $l_{\phi}$ and $\gamma$ are defined as the digamma
function, phase coherent length and coefficient of the $B^{2}$ term,
respectively. A good fitting of MC data
[$\Delta\sigma_{xx}(B)=\sigma_{xx}(B)-\sigma_{xx}(0)$] with the modified HLN
equation for 2 and 100 K is shown in Fig. 7(a) in the field range $\pm$1.5 T.
This confirms the validation of the HLN model in CaAu0.5Cu0.5As. The estimated
values of $l_{\phi}$ are displayed in Fig. 7(b) as a function of temperature.
$l_{\phi}$ decreases with the increase of temperature because the inelastic
scattering increases, which destroys the phase coherence. Thus, the WAL effect
weakens with increasing temperature, which is evident from Fig. 5(a). The
value of $l_{\phi}$ for a disordered metal is usually in the range of a few
hundred nanometers as observed in CaAu0.5Cu0.5As. Similar values of $l_{\phi}$
were also found in other topological materials like Bi2Te3 He  _et al._
(2011), BiSbTeSe2 Li  _et al._ (2019) and LuPtSb Hou  _et al._ (2015). It may
be mentioned that the value of $l_{\phi}$ is much smaller than the crystal’s
thickness, which implies the 3D nature of the WAL effect. The temperature
dependence of $l_{\phi}$ can be interpreted by considering both electron-
electron ($e-e$) and electron-phonon ($e-ph$) scattering, which follows the
relation Shrestha  _et al._ (2017)

$\frac{1}{l_{\phi}^{2}(T)}=\frac{1}{l_{\phi}^{2}(0)}+A_{e-e}T+A_{e-ph}T^{2},$
(4)

where, $l_{\phi}(0)$ is the phase coherence length at 0 K, $A_{e-e}$ and
$A_{e-ph}$ are the coefficient of $e-e$ and $e-ph$ scattering, respectively.
According to Fig. 7(b), the estimated value of $l_{\phi}$ in the low-
temperature region is consistent with Eq. 4, when the parameters are
$l_{\phi}(0)$ = 209 nm, $A_{e-e}$ = 1.48 $\times$10-7 (nm2 K)-1 and $A_{e-ph}$
= 1.01 $\times$10-8 (nm K)-2. Fig. 7 (c) discloses the large value of $\alpha$
($\sim$105) calculated from the Eq. 3, which indicates that WAL effect mainly
originates from the 3D bulk state with multiple conduction channels. Similar
behavior has also been observed in other 3D systems such as LuPtSb, LuPdBi and
ScPdBi Hou  _et al._ (2015); Xu  _et al._ (2014); Zhang  _et al._ (2019). On
the other hand, it is predicted that CaAu0.5Cu0.5As hosts a triply degenerate
topological state. Interestingly, the WAL effect was also recently observed in
another triply degenerate nodal semimetal YRh6Ge4 Zhu  _et al._ (2020). We
believe that the WAL effect in CaAu0.5Cu0.5As is originates from a strong
spin-orbit coupled nontrivial topological state like in Dirac/ Weyl
semimetals.

Figure 8:  The angular dependence of TMR of CaAuAs and CaAu0.5Cu0.5As measured
at 5 K and 9 T are shown. The schematic diagram of experimental configuration
is illustrated in the right panel.

In order to address the low-field characteristic of LMR, we analyze the
parallel field MC ($\sigma_{\parallel}$) data [Fig. 5(b)] with the help of the
generalized Altshuler–Aronov (AA) model. According to this model, the nature
of the low-field longitudinal MC data can be described by the expression Lin
_et al._ (2013).

$\Delta\sigma_{\parallel}\simeq-\frac{\alpha
e^{2}}{2\pi^{2}\hbar}~{}ln\left[1+\beta\left(\frac{e^{2}d^{2}l_{e}^{2}}{\hbar^{2}}\right)B^{2}\right],$
(5)

where $\Delta\sigma_{\parallel}=\sigma_{xx}(B_{\parallel})-\sigma_{xx}(0)$.
Here, $d$ (= 0.19 mm) is the sample thickness, and $l_{e}$ is the electron
mean free path. Depending on the value of parameter $\beta$, parallel field
transport can be classified into several scattering regimes. For instance, in
a disordered system with $d\gg l_{e}$ and $\beta$ = 1/3, it is considered to
be the Altshuler–Aronov regime Altshuler and Aronov (1981), whereas a clean
metal with $d\ll l_{e}$ and $\beta$ = 1/6 implies the Dugaev–Khmelnitskii
regime Dugaev and Khmelnitskii (1984). As the magnetotransport measurement has
been performed on a bulk single crystal, it is fair to consider $d\gg l_{e}$
Shon  _et al._ (2019). This further gives the indication that the low-field
LMR of CaAu0.5Cu0.5As is in the AA regime. To verify the above fact, we have
fitted the experimental MC data with Eq. 5, as demonstrated in Fig. 7(d). An
excellent fitting with the AA model supports our assumption. The obtained
fitting parameters are $\alpha$ = 4.2$\times$104 and $l_{e}$ = 0.3 nm. Thus,
$d\gg l_{e}$ and $l_{\phi}\gg l_{e}$, which indicates that the electron
transport occurs in the quantum diffusive regime, which is consistent with the
WAL effect Hikami  _et al._ (1980); Lu and Shen (2014, 2016).

Resistivity measurements along different crystallographic directions are used
to probe the anisotropic nature of the charge conduction mechanism associated
with the electronic band structure. Often, the dimensions of the grown single
crystals restrict the direction dependent resistivity measurements. In such a
case, studying the direction-dependent TMR, by rotating the magnetic field,
can be used to shed some light on the nature of charge conduction. We have
studied the angular ($\theta$) dependence of TMR of the CaAuAs and
CaAu0.5Cu0.5As crystals. Fig. 8 (left panel) shows TMR measured at 5 K and 9 T
as a function of $\theta$. The experimental setup is also illustrated
schematically in the right panel of Fig. 8. In this configuration, the
direction of the current is maintained along the $x$-axis within the plane of
the rectangular crystal while the magnetic field is rotated slowly about the
current direction ($x$-axis), i.e., from out-of-plane (along $z$-axis;
$\theta$ = 0) to in-plane direction ($y$-axis; $\theta$ = 90∘). It can be seen
from Fig. 8 that the MR reaches the maximum, when field is perpendicular to
the crystal plane ($\theta$ $\approx$ 0∘) and minimum when field is along the
plane ($\theta$ $\approx$ 90∘). The polar plot revels a two-fold symmetric
pattern, which is prominent in the case of CaAu0.5Cu0.5As. The estimated
anisotropy ratio in TMR of CaAu0.5Cu0.5As at 5 K and 9 T is about 2.4,
indicating a weak anisotropy in the electronic structure.

Figure 9: Field dependent Hall resistivity of CaAuAs and CaAu0.5Cu0.5As for 2
K and 300 K is shown in (a) and (b), respectively. The inset of Fig. (a) shows
the linear fit to the Hall resistivity data at 2 K. The variation of carrier
concentration and mobility of both the compounds CaAuAs and CaAu0.5Cu0.5As are
shown in (c) and (d), respectively.

To get a quantitative idea about the density of the charge carrier and its
mobility, we measured Hall resistivity ($\rho_{yx}$). The field dependences of
$\rho_{yx}$ for CaAuAs and CaAu0.5Cu0.5As crystals at different fixed
temperatures are shown in Figs. 9(a) and (b), respectively. Both compounds
show linear Hall resistivity with a positive slope, suggesting a dominant hole
type of carrier. It can also be noticed that $\rho_{yx}$($B$) is almost
independent of temperature. The carrier concentration and mobility are
extracted using the relations $n=1/eR_{H}$ and $\mu=R_{H}/\rho_{(B=0)}$, where
$R_{H}$ is the slope of the linear fit of $\rho_{yx}$ vs. $B$ curve [see inset
of Fig. 9(a)]. For both compounds, the estimated values of carrier
concentration and mobility as a function of temperature are shown in Figs.
9(c) and (d), respectively. The observed carrier concentration in both
compounds is nearly independent of temperature. The estimated carrier
concentration is $n\sim 10^{20}$ cm-3, which is much smaller than in typical
metals ( $n\sim 10^{22}$ cm-3 to 1023 cm-3). This indicates a low density of
states at the Fermi level, which further implies the semimetallic nature of
both compounds as observed in similar topological semimetals like CaAgAs Nayak
_et al._ (2018), CaAgBi Sasmal  _et al._ (2020), YbCdGe Laha  _et al._ (2019),
and PrAlGe Meng  _et al._ (2019). The $\mu$ varies in between 352 cm2 V-1 S-1
and 155 cm2 V-1 S-1 for CaAuAs, and 355 cm2 V-1 S-1 and 195 cm2 V-1 S-1 for
CaAu0.5Cu0.5As in the temperature range form 2 to 300 K. The estimated value
of the mobility is a few orders of magnitude smaller than that observed in
typical topological Dirac/Weyl semimetal Liang  _et al._ (2015); Shekhar  _et
al._ (2015). However, the value of $\mu$ is comparable to that observed in
topological materials like YbCdGe and CaAgAs.

##  IV SUMMARY

We have successfully doped 50% Cu at the Au site of the topological Dirac
semimetal CaAuAs. Interestingly, a significant change in the nature of MR is
noticed due to Cu-doping in CaAuAs. The low-field behavior of MR of the doped
system gives a clear indication of the WAL effect. MC of CaAu0.5Cu0.5As in the
low-field regime was found to fit well with the 3D WAL model and the modified
HNL model. The WAL effect in longitudinal MC was well described by the
Altshuler-Aronov model. Our electronic band structure calculation suggested
that the replacement of Au with 50% Cu breaks the space inversion symmetry,
which gives rise to a phase transformation from a fourfold degenerate Dirac
point to two threefold degenerate triple points. The signature of a chiral
anomaly further supports the existence of triple-point state in
CaAu0.5Cu0.5As. ARPES measurements can confirm the non-trivial band topology
of the alloy system.

##  V ACKNOWLEDGMENT

We acknowledge IIT Kanpur, Science and Engineering Research Board, India
(Projects No. SRG/2019/001686 and No. CRG/2018/000220), and Department of
Science and Technology, India, for financial support. We thank S. Roy for the
magnetoresistance measurement along the $c$ axis. ZH acknowledges the Polish
National Agency for Academic Exchange for the Ulam Fellowship. A.A.
acknowledges the support from the SPACE-TIME supercomputing facility hosted at
IIT Bombay, where the electronic structure simulations are performed.

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