Optical signatures of type-II Weyl fermions in the noncentrosymmetric semimetals $R$AlSi ($R$=La, Ce, Pr, Nd, Sm)
J. Kunze M. Köpf Weizheng Cao Yanpeng Qi C. A. Kuntscher




Experimental Physics II, Institute of Physics, University of Augsburg, 86159 Augsburg, Germany

Experimental Physics II, Institute of Physics, University of Augsburg, 86159 Augsburg, Germany


caowzh@shanghaitech.edu.cn
School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China

qiyp@shanghaitech.edu.cn
School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China
ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 201210, China
Shanghai Key Laboratory of High-resolution Electron Microscopy and ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 201210, China

christine.kuntscher@physik.uni-augsburg.de
Experimental Physics II, Institute of Physics, University of Augsburg, 86159 Augsburg, Germany



Weyl semimetals with magnetic ordering provide a promising platform for the investigation of rare topological effects such as the anomalous Hall effect, resulting from the interplay of nontrivial bands with various spin configurations. The materials RAlSi, where R represents a rare-earth element, are prominent representatives of Weyl semimetals, where the Weyl states are
induced by space inversion symmetry breaking and in addition, for several rare-earth elements R, enhanced by time-reversal symmetry breaking through the formation of a magnetic order at low temperature. We report optical signatures of Weyl fermions in the magnetic compounds CeAlSi, PrAlSi, NdAlSi, and SmAlSi as well as the non-magnetic family member LaAlSi by broad-frequency infrared spectroscopy at room temperature, i.e., in the paramagnetic phase. A similar profile of the optical conductivity spectrum and a metallic character are observed for all compounds, with LaAlSi showing the strongest free charge carrier contribution. Furthermore, the linear-in-frequency behavior of the optical conductivity of all investigated compounds indicates the presence of Weyl nodes in close vicinity of the Fermi energy, resulting from inversion symmetry breaking in noncentrosymmetric structures. According to the characteristics of these linear slopes, the RAlSi compounds are expected to host mainly type-II Weyl states with overtilted Weyl cones. The results are compared to the optical response of the closely related RAlGe materials, which are considered as potential hybridization-driven Weyl-Kondo systems.



Optical signatures of type-II Weyl fermions in the noncentrosymmetric semimetals 
 RAlSi (R=La, Ce, Pr, Nd, Sm)
    C. A. Kuntscher
    April 29, 2024
===============================================================================================================






§ INTRODUCTION

In recent years, the investigation of topological phases in condensed matter has attracted a lot of interest. A prominent example is the Dirac semimetal phase, where linear band crossings occur near the Fermi energy E_F, which are protected by spatial and/or time-reversal symmetry <cit.>.
Other than high-symmetric Dirac points, the existence of Weyl points requires either inversion or time-reversal symmetry to be broken, creating pairs of Weyl points with different chirality <cit.>. One famous representative of such Weyl semimetals (WSMs) with broken inversion symmetry is the noncentrosymmetric and nonmagnetic compound TaAs <cit.>.
When the degeneracy of the Dirac cones is lifted by the broken inversion symmetry, the Weyl cones are shifted in energy relatively to each other, whereas breaking of time-reversal symmetry through magnetic moments, causes a shift in k-space <cit.>. WSMs with broken time-reversal symmetry are rare as compared to those with broken inversion symmetry, yet promising since in inversion symmetric systems there is no shift of the nodes in energy, resulting in a vanishing density of states at the Fermi level <cit.>. In addition, the combination of topological phases and magnetism enables the realization of rare quantum mechanical effects, such as the intrinsic anomalous Hall effect or the topological Hall effect <cit.>. EuCd_2As_2 represents this group as a famous member, where the alignment of Eu spins magnetically induces Weyl nodes via exchange coupling to create a magnetic WSM candidate <cit.>.

The family of rare-earth based compounds RAlX (R = La, Ce, Pr, Nd, Sm; X=Si and Ge) also has attracted a lot of attention recently as WSM candidates <cit.>. The noncentrosymmetric RAlX family allows the investigation of WSM states in magnetic and nonmagnetic phases in isostructual compounds, as the magnetic contributions depend on the rare-earth element R <cit.>. Among the La, Ce, Pr, Nd and Sm elements, only La does not exhibit a magnetic moment in the RAlSi compounds <cit.>. Therefore, varying the rare-earth ion affects the electronic topology through different magnetic ground states <cit.>. In the case of PrAlSi, a ferromagnetic order sets in at T_C=17.8K, followed by two spin-glass-like reentrant magnetic transitions at lower temperatures <cit.>. There are also several critical temperatures for NdAlSi, as this compound undergoes an antiferromagnetic transition at T_N=7.2K and a ferrimagnetic transition at T_C=3.3K <cit.>.  SmAlSi orders antiferromagnetically at T_N=10.7K, while CeAlSi also shows a ferromagntic magnetic structure below T_C=17.8K <cit.>. RAlSi compounds crystallize for the most part in the space group I4_1md, yet with a certain percentage also the I4_1/amd phase has been found experimentally <cit.>. In the main space group I4_1md, the unit cell parameters of RAlSi vary between a=4.1566Å for Sm and a=4.3069Å for La, and c=14.4552Å for Sm and c=14.6494Å for La <cit.>.

Regarding the electronic structure, the RAlX compounds have been shown to be promising candidates for the WSM phase, as numerous Weyl points are reported to exist near the Fermi level, e.g., 40 Weyl points
in the paramagnetic phase of NdAlSi <cit.>, CeAlSi <cit.>, and LaAlGe <cit.> upon inclusion of spin-orbit coupling.
The electronic band structure of RAlSi is very similar among the different compounds and the density of states at the Fermi level is expected to be vanishingly small, which leads to a semi-metallic ground state <cit.>.
Resistivity measurements have confirmed the (semi-)metallic behaviour of the RAlSi materials as dc conductivity values between 11000 and 18000 Ω^-1cm^-1 have been found in recent publications <cit.>.

The materials CeAlGe and PrAlGe are furthermore discussed as potential hybridization-driven Weyl-Kondo semimetals <cit.>. According to optical conductivity studies, electronic correlations due to the hybridization of the localized f electrons of the rare-earth element (Ce 4f^1 and Pr 4f^2) with the conduction electron states lead to a reduction of the Fermi velocity and an enhancement of the charge carriers effective mass, in contrast to LaAlGe lacking f electrons (La 4f^0). It was furthermore suggested that the onset of magnetic ordering in CeAlGe and PrAlGe significantly affects the electronic band structure, as observed in the temperature-dependent optical response <cit.>, suggesting a coupling between magnetism and the electronic degrees of freedom. In contrast, recent angle-resolved photoemission spectroscopy results combined with first-principles calculations for PrAlSi and SmAlSi revealed only a weak coupling between the localized 4f electrons and the conduction electrons and a negligible effect of the magnetic ordering on the electronic band structure <cit.>. Accordingly, whether the RAlX are indeed Weyl-Kondo systems is still a matter of debate.


In this work, the optical response of RAlSi (R = La, Ce, Pr, Nd, Sm) is investigated by comparing the optical functions such as the optical conductivity and dielectric function of the various rare-earth based compounds. The results are discussed in terms of the transport and topological properties discussed in the literature <cit.>. In particular, a comparison to the optical conductivity of the closely related RAlGe materials is given.




§ METHODS


Single crystals of RAlSi were synthesized using the self-flux method, as described elsewhere <cit.>.
The dimensions of these plate-like samples are in the range of 0.5 and 3 mm and high-quality surfaces have been prepared by careful polishing, as they are essential for quantitative infrared reflectivity measurements. We have performed these reflectivity measurements from 100 to 20000  (0.01 to 2.48 eV) at room temperature using Fourier-Transform Infrared spectroscopy (FTIR) with the use of a Bruker 80v spectrometer, which is coupled to a Hyperion microscope. In order to obtain the absolute reflectance of our samples, these have been aligned perpendicular to the incoming IR beam, which also holds for an aluminum mirror next to it serving as reference. In the case of LaAlSi, additional low-temperature reflectivity measurements were carried out.
By using resistivity values and volumetric data from the literature <cit.>, these reflectance data were extrapolated in the low and high energy ranges, which is necessary to calculate the optical functions, such as the optical conductivity σ_1, the real part of the dielectric function ε_1, and the loss function -Im(1/ε), from Kramers-Kronig relations using programs by David Tanner <cit.>.
The optical functions have been analyzed with the RefFIT software <cit.> by fitting with the Drude-Lorentz model.





§ RESULTS AND DISCUSSION


The reflectance spectra displayed in Fig. <ref>(a) show a high value towards the low-energy range, indicating the metallic characteristics of the investigated materials of the RAlSi compound family. Also, a plasma edge is observed at around 900  for all five compounds. For LaAlSi the plasma edge is located at the highest frequency, indicating that it has the strongest metallic character among the studied samples (see also the discussion of the plasmon peak position below). Towards higher energy, the reflectance spectra show a very similar behavior for all compounds and are just varying slightly in their absolute value, therefore hinting at a similar electronic structure of the RAlSi materials in the higher energy regime. Especially noticeable is the merging of the reflectance spectra from approximately 16000 to 20000 .
Only the compound LaAlSi shows a broad bump at around 2500 , and hence marks an exceptional behavior among the studied RAlSi materials.







The corresponding optical conductivity spectra σ_1 are depicted in Fig. <ref>(b).
It is important to note that the optical conductivity spectra have a similar profile for all five compounds with only quantitative differences. This is in contrast to results reported for the closely related materials LaAlGe, CeAlGe, and PrAlGe, which exhibit different optical conductivity profiles already at room temperature <cit.>.
Among the studied compounds, LaAlSi has the highest σ_1 value at the lowest studied wavenumber 100 . NdAlSi exhibits the second highest value and CeAlSi and SmAlSi do have very similar values at the low-frequency limit, although the value of CeAlSi is slightly higher, and PrAlSi has the lowest value. Besides the difference for the value of σ_1 at low energy (100  or 0.01 eV) and the varying distinction of the first peak in the spectrum (2000 to 5500 ), the spectra of the five different compounds resemble a similar profile towards higher energy. This leads to the conclusion, that the main differences in the materials lie in the low-energy regime, where the excitations of free charge carriers (Drude contribution) and the interband transitions between electronic bands in direct vicinity of the Fermi energy contribute.
The Lorentz contributions in the higher-energy range are very similar for all investigated materials and show no dependence on the rare earth element R.


As the reflectance and optical conductivity data already suggest, the compound LaAlSi stands out in comparison to the other four studied compounds.
This is also revealed by the position of the plasmon peak in the loss function (LF) [see inset in Figure <ref>(a)]. Although no absolute value can be obtained for the position of the plasmon peak, since the peak is not distinct enough, for the compounds of CeAlSi, NdAlSi, PrAlSi and SmAlSi the plasmon peak is located at approximately the same frequency. However, for LaAlSi the peak position is obviously shifted to a higher value. The presence of a plasmon peak in the loss function also indicates the metallic character of all studied compounds.

In general, the position of the plasmon peak yields a value for the screened plasma frequency ω_pl^scr, which can serve as a measure for the metallic strength of a material.
The value of ω_pl^scr can also be extracted from the zero crossing of the real part of the dielectric function ε_1 (see Fig. <ref>).
The so-obtained ω_pl^scr values for all studied materials are depicted in the inset of Fig. <ref>. Firstly, it can be seen that for low frequencies ( < 1000 ) ε_1 takes large negative values, evidencing the metallic character of the investigated materials. Secondly, the negative values of ε_1 at high energies are caused by the rather pronounced interband transitions due to the presence of multiple energy bands in the vicinity of E_F. Thirdly,
it is obvious that the screened plasma frequency is the highest for LaAlSi, since the zero crossing point is clearly set apart from the other four compounds towards higher energies. For the compounds CeAlSi, PrAlSi, NdAlSi, and SmAlSi the frequency of the zero crossing is roughly at the same position.
The ω_pl^scr value for LaAlSi is approximately 1500 , whereas for the compounds CeAlSi, NdAlSi and SmAlSi, a value of around 700   is obtained. The compound PrAlSi has the smalles value of 650  . Most importantly, LaAlSi has by far the highest value of the screened plasma frequency, which is in accordance with the loss function, reflectance, and optical conductivity.






For quantitative analysis of the optical response functions, a Drude-Lorentz fitting model with one Drude component and six
Lorentz oscillators was applied for all five RAlSi compounds. As an example, we show in Fig. <ref> the fit of the optical conductivity σ_1 for PrAlSi together with the various fitting contributions.
According to Hall resistivity measurements the main charge carrier type in the material LaAlSi are electrons <cit.>, which justifies the implementation of only one Drude oscillator term in the fitting model. For other RAlSi compounds, the contribution of both electron and hole pockets to the transport, i.e., the multiband nature, has been pointed out. Nevertheless, we keep the same Drude-Lorentz fitting model (one Drude term only) for all studied materials for consistency reason. We also point out that for none of the compounds it was necessary to include a second Drude term in the model, in order to obtain a very good fit of both reflectivity and optical conductivity data simultaneously.

The optical conductivity spectrum has a similar profile for all five compounds, however, there are quantitative differences: In LaAlSi the Lorentz oscillators L1 and L2 are shifted to lower energies as compared to the other compounds, implying slight differences in the electronic band structure close to E_F. The most pronounced differences are observed in the low-energy range, where the Drude term due to itinerant carrier excitations is located.

The plasma frequency ω_pl, as obtained from the spectral weight of the Drude term, is displayed in Fig. <ref> as a function of rare earth element.
In comparison, the compounds CeAlSi, PrAlSi, NdAlSi, and SmAlSi have a very similar value of the plasma frequency (∼7500 ), LaAlSi exhibits a much higher value (∼12000 ).


It is important to note, that the metallic characteristics of the studied RAlSi compounds, as evidenced by the pronounced Drude term, is rather surprising, since electronic band structure calculations reveal an almost vanishing electronic density of states at the Fermi energy <cit.>.


We furthermore note that the values of the screened plasma frequency calculated from the Drude spectral weight
(taking into account the ϵ_1 value at around 1500 cm^-1) are slightly higher than the values
as obtained from the zero crossing of ϵ_1 (see inset of Fig. <ref>). Such a discrepancy has been reported in the literature in several cases <cit.>, and is due to the choice of the frequency for extracting
the ϵ_1 value and the error bar of the optical parameters like the Drude spectral weight.







For all studied compounds one notices that the optical conductivity σ_1 follows an ω-linear behavior above
∼1000 cm^-1, i.e., in the frequency range of the interband excitations. For further analysis and interpretation of
the interband transitions, we subtracted the Drude term from the total optical conductivity spectrum σ_1 and thus
obtained the interband conductivity σ_1,interband, which are plotted in Fig. <ref>.

There are two linear regimes in σ_1,interband, as shown for PrAlSi and NdAlSi in Fig. <ref> as examples.
The steep linear increase in σ_1,interband at low frequencies  (ω>Ω_1) is followed by another linear frequency dependence for ω>Ω_2, however, with a smaller slope, leading to a kink in the spectrum. This higher-energy linear frequency dependence (above Ω_2) extrapolates to a finite σ_1,interband value in the ω →0 limit. The two regimes with the linear frequency dependences in σ_1,interband are indicated in Fig. <ref> by blue and orange areas, respectively. For LaAlSi the higher-energy regime is least developed among the studied materials.

A linear-in-frequency behavior of the interband optical response is a characteristic fingerprint for the existence of Dirac and Weyl fermions in condensed matter.
Namely, the interband optical response of such linear Dirac/Weyl cones is predicted to follow a linear dependence in frequency ω for a three-dimensional (3D) system according to  <cit.>

    σ_1(ω)=N e^2/24 πħ v_F·ω·Θ(ω - 2 E_F)    ,

where N is the number of Weyl points (lifting the spin degeneracy) and v_F the average Fermi velocity.
A possible shift of the Dirac node away from the Fermi energy is taken into account by the Heaviside step function Θ, describing the Pauli blocking edge. For 3D Weyl semimetals, one or two quasilinear regions with different slopes are predicted <cit.>, as described in more detail below.
The characteristic linear frequency dependence of the interband optical response was indeed observed in the 3D Dirac semimetal ZrTe_5 and in the 3D Weyl semimetal TaAs:
While for ZrTe_5 a linear behavior in σ_1 was observed in the low-frequency range <cit.>, for TaAs two ω-linear components with different slopes were found <cit.>. In TaAs, the two ω-linear components in the σ_1 spectrum are visible already at room temperature and persist down to low temperatures <cit.>.









According to theoretical predictions <cit.>,
for a time-reversal symmetry breaking WSM the Weyl cones are shifted along the direction in k-space, whereas for a space-inversion breaking WSM the cones are shifted in energy but stay situated at the same position in k-space. The latter situation results in an optical conductivity spectrum, where two linear-in-frequency ranges with different slopes are formed. The first linear slope is caused by excitations within the same Dirac cone, which requires less energy for electronic interband excitations in comparison. When the energy is high enough, these excitations can also be found between the two shifted Weyl cones. Thus, two linear ranges with different slopes are observed <cit.>. Accordingly, from the number of linear-in-frequency regimes in σ_1 one could in principle distinguish between time-reversal symmetry breaking and space-inversion breaking WSMs.

On the other hand, also a tilting of the Weyl cones in reciprocal space can cause two linear-in-frequency regimes in the optical conductivity spectrum. The difference between type-I and type-II WSMs was described by Carbotte <cit.> based on the degree of tilting of the Weyl cones, given by the tilting parameter ŵ. Without tilting, the value ŵ=0 is assigned. For a small tilt, the values of ŵ vary between 0 and 1. These states describe the case of a type-I WSM according to Ref. <cit.>. For a strong tilt, where the value of ŵ exceeds 1, a type-II WSM is formed. The value ŵ=1 describes the case for a tilt of π/2 into the plane perpendicular to the energy axis <cit.>.

A type-II WSM is also referred to as an overtilted Weyl cone in energy-momentum space.
Based on the optical conductivity, the two types of WSMs can be differentiated by looking at the linear ranges in the spectrum: For a type-I WSM the linear-in-frequency behavior at frequencies ω>Ω_2 can be extrapolated to zero frequency, so that it crosses the origin. This is different for type-II WSM: These are also described by two linear slopes in the optical conductivity, where the extrapolation of the linear range at ω>Ω_2 yields an interception of the y-axis at a finite conductivity value. Furthermore, the slope of the higher-energy linear range (ω>Ω_2) is expected to decrease with an increasing tilt of the Weyl-nodes <cit.>.
Please note that Ref. <cit.> considers Weyl nodes shifted away from the Fermi energy (see below).


According to the optical data obtained in this study, two linear ranges with different slopes are found in the σ_1,interband spectra of the studied RAlSi materials (see Fig. <ref>). Furthermore, the ω→0 extrapolation of the linear slope in the higher-energy regime shows the crossing of the y-axis at a finite conductivity value, indicating that the investigated materials of the RAlSi compound family are in fact type-II WSMs formed by the space-inversion symmetry breaking. The data therefore suggest that the studied materials are type-II Weyl semimetals, with mainly overtilted ŵ > 1 Weyl nodes in the energy-momentum space (see also below). However, it is important to note, that in the case of LaAlSi both type-I and type-II Weyl nodes have been found and reported at ∼0.1 eV above E_F and exactly at E_F, respectively <cit.>. Indeed, for LaAlSi the higher-energy linear-in-frequency regime in the σ_1 spectrum (see Fig. <ref>) is less developed, which suggests a significant contribution of type-I Weyl nodes to excitations in the optical response. We also note that in the closely related compounds NdAlSi, CeAlSi, and LaAlGe 40 type-I and type-II Weyl nodes in the vicinity of the Fermi level have been theoretically predicted upon inclusion of spin-orbit coupling <cit.>.


In a recent study, optical conductivity data of the related materials LaAlGe and CeAlGe revealed the typical optical signatures of type-II Weyl fermions, namely two linear frequency dependences in the interband optical conductivity <cit.>.
Interestingly, for PrAlGe only a single linear frequency dependence in σ_1,interband was observed <cit.>, which is a typical signature for type-I Weyl semimetals, as discussed above.
For all three materials (LaAlGe, CeAlGe, PrAlGe), these signatures appear at much lower energies as compared to RAlSi, hence being not well separated from the free charge carrier excitations, and were much less developed at room temperature.
For CeAlGe and PrAlGe, the role of f-electrons has been pointed out, namely mediating electronic correlations (Kondo coupling between
f-electrons and conduction bands), which cause a renormalization of the topological bands, with the reduction of the Fermi velocity and the increase in the effective mass of charge carriers at low temperature <cit.>. Therefore, CeAlGe and PrAlGe have been suggested as correlated Weyl-Kondo systems. In contrast, LaAlGe does not possess f electrons and therefore is expected to be non-correlated <cit.>.


Comparing the optical response of RAlSi and RAlGe materials, we note that all RAlSi compounds of the present study show a similar profile of the optical conductivity spectrum at room temperature, i.e., above the magnetic ordering, revealing the typical signatures of type-II Weyl fermion excitations. In particular, we cannot confirm a type-I Weyl fermion behavior for PrAlSi in contrast to PrAlGe. Like LaAlGe, LaAlSi is expected to be non-correlated as there are no f electrons, consistent with the observed highest Drude spectral weight (squared unscreened plasma frequency) among the RAlSi compounds (see Fig. <ref>).
Following the analysis of the interband optical conductivity spectra in Ref. <cit.>, we can extract the tilting parameter ŵ of the Weyl cones, the chemical potential μ, and the average Fermi velocity v_F. According to Ref. <cit.>, the parameters μ and v_F can be obtained from the limiting frequencies Ω_1 and Ω_2 using the equations Ω_1=2μ/(1+ŵ) and Ω_2=2μ/(ŵ-1). The so-obtained parameters for the studied RAlSi compounds at room temperature are listed in Table <ref>.
For all studied materials RAlSi the tilting parameter ŵ is 1 or greater than 1, suggesting an overtilt of the Weyl cones.
Furthermore, from the slope of the linear-in-frequency behavior of the interband optical conductivity at ω>Ω_2 (see Fig. <ref>) the Fermi velocity can be estimated according to <cit.>

    σ_1(ω)=N e^2/24 πħ v_F·1+3ŵ^2/4ŵ^3·ω   .

For all studied compounds we set N=40 <cit.>.
The so-obtained values of the Fermi velocity v_F are included in Table <ref>. The values of v_F for all compounds lie in the range (35 - 61) ·10^4 ms^-1, in reasonable agreement with results from Shubnikov-de Haas oscillation measurements on SmAlSi and LaAlSi at low temperature <cit.>. For LaAlSi without f electrons and hence no low-temperature magnetic ordering, we also extracted the
low-temperature parameters from corresponding optical data and found them to be independent of temperature.
Finally, it is important to note that the v_F values of RAlSi are a factor of ∼10 higher than for LaAlGe and CeAlGe <cit.>, suggesting that the RAlSi materials are less correlated than RAlGe.






§ CONCLUSION

In conclusion, we studied the optical response of the noncentrosymmetric compounds RAlSi (R=La, Ce, Pr, Nd, Sm) at room temperature, in order to investigate their electronic structure and electronic excitations in the paramagnetic phase, in particular regarding the dependence on the rare-earth element R. Over the whole studied frequency range, the reflectivity and optical conductivity spectra show only minor differences between the materials CeAlSi, PrAlSi, NdAlSi, and SmAlSi, which all exhibit magnetic order at low temperatures.
The main differences can be found in the frequency range below 7500 , where the free charge carrier excitations and low-frequency interband transitions occur.
Especially the nonmagnetic LaAlSi compound stands out, as it exhibits the strongest metallic character according to its (screened) plasma frequency.
Thus, mostly the electronic band structure close to the Fermi level seems to be affected by the variation of the rare-earth element R. Furthermore, two quasi-linear ranges with different slopes are observed in the low-frequency optical conductivity spectrum, which indicates the existence of Weyl nodes in the vicinity of E_F. From the analysis of the linear-in-frequency behavior of the optical conductivity we can conclude that the Weyl cones are mainly of type-II and are overtilted.




This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – TRR 360 – 492547816.
YQ would like to acknowledge the support by the National Natural Science Foundation of China (Grant Nos. 52272265, U1932217, 11974246, 12004252) and the National Key R & D Program of China (Grant No 2018YFA0704300).




35
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Armitage et al.(2018)Armitage, Mele, and Vishwanath]Armitage.2018
  author author N. P. Armitage, author E. J. Mele,
   and author A. Vishwanath,  title title Weyl and Dirac semimetals
  in three-dimensional solids, 10.1103/RevModPhys.90.015001 journal journal
  Rev. Mod. Phys. volume 90, pages
  015001 (year 2018)NoStop
[Yan and Felser(2017)]Yan.2017
  author author B. Yan and author C. Felser,  title title Topological Materials: Weyl
  Semimetals, 10.1146/annurev-conmatphys-031016-025458
  journal journal Annu. Rev. Condens. Matter
  Phys. volume 8, pages 337–354
  (year 2017)NoStop
[Lv et al.(2015)Lv,
  Weng, Fu, Wang, Miao, Ma, Richard, Huang,
  Zhao, Chen, Fang,
  Dai, Qian, and Ding]Lv.2015
  author author B. Q. Lv, author H. M. Weng,
  author B. B. Fu, author X. P. Wang, author
  H. Miao, author J. Ma, author P. Richard, author X. C. Huang,
  author L. X. Zhao, author G. F. Chen, author
  Z. Fang, author X. Dai, author T. Qian,  and author H. Ding,  title title Experimental Discovery of
  Weyl Semimetal TaAs, 10.1103/PhysRevX.5.031013
  journal journal Phys. Rev. X volume 5, pages 031013 (year
  2015)NoStop
[Soh et al.(2019)Soh,
  de Juan, Vergniory, Schröter, Rahn, Yan, Jiang, Bristow, Reiss, Blandy, Guo, Shi, Kim,
  McCollam, Simon, Chen,
  Coldea, and Boothroyd]Soh.2019
  author author J.-R. Soh, author F. de Juan,
  author M. G. Vergniory, author N. B. M. Schröter, author M. C. Rahn, author
  D. Y. Yan, author J. Jiang, author M. Bristow, author P. Reiss, author J. N.  Blandy, author Y. F.  Guo, author Y. G. Shi, author T. K. Kim,
  author A. McCollam, author S. H. Simon, author
  Y. Chen, author A. I. Coldea,  and author A. T. Boothroyd, title title Ideal Weyl semimetal induced by magnetic exchange, 10.1103/PhysRevB.100.201102 journal journal Phys. Rev. B volume 100, pages 201102 (year 2019)NoStop
[Tabert et al.(2016)Tabert,
  Carbotte, and Nicol]Tabert.2016a
  author author C. J. Tabert, author J. P. Carbotte,  and author E. J. Nicol, title title Optical and
  transport properties in three-dimensional Dirac and Weyl semimetals,  10.1103/PhysRevB.93.085426 journal journal Phys. Rev. B volume 93, pages 085426 (year 2016)NoStop
[Yang et al.(2021)Yang,
  Singh, Gaudet, Lu,
  Huang, Chiu, Huang,
  Wang, Bahrami, Xu,
  Franklin, Sochnikov, Graf,
  Xu, Zhao, Hoffman,
  Lin, Torchinsky, Broholm,
  Bansil, and Tafti]Yang.2021
  author author H.-Y. Yang, author B. Singh,
  author J. Gaudet, author B. Lu, author
  C.-Y. Huang, author
  W.-C. Chiu, author
  S.-M. Huang, author
  B. Wang, author F. Bahrami, author B. Xu, author J. Franklin, author I. Sochnikov,
  author D. Graf, author
  G. Xu, author Y. Zhao, author C. Hoffman, author H. Lin,
  author D. Torchinsky, author C. Broholm, author
  A. Bansil,  and author
  F. Tafti, title title Noncollinear ferromagnetic Weyl semimetal with
  anisotropic anomalous Hall effect, 10.1103/PhysRevB.103.115143 journal journal
  Phys. Rev. B volume 103, pages
  115143 (year 2021)NoStop
[Wang et al.(2022)Wang,
  Dong, Guo, Lv, Huang, Xiang, Ren, Wang,
  Sun, Li, and Chen]Wang.2022
  author author J.-F. Wang, author Q.-X. Dong,
  author Z.-P. Guo, author M. Lv, author
  Y.-F. Huang, author
  J.-S. Xiang, author
  Z.-A. Ren, author Z.-J. Wang, author P.-J.  Sun, author G. Li,  and author G.-F. Chen, title title NdAlSi: A
  magnetic Weyl semimetal candidate with rich magnetic phases and atypical
  transport properties, 10.1103/PhysRevB.105.144435
  journal journal Phys. Rev. B volume 105, pages 144435 (year
  2022)NoStop
[Lyu et al.(2020)Lyu,
  Xiang, Mi, Zhao,
  Wang, Liu, Chen,
  Ren, Li, and Sun]Lyu.2020
  author author M. Lyu, author J. Xiang, author Z. Mi, author
  H. Zhao, author Z. Wang, author E. Liu, author G. Chen, author Z. Ren, author
  G. Li,  and author
  P. Sun, title title Nonsaturating magnetoresistance, anomalous Hall effect,
  and magnetic quantum oscillations in the ferromagnetic semimetal PrAlSi,  10.1103/PhysRevB.102.085143 journal journal Phys. Rev. B volume 102, pages 085143 (year 2020)NoStop
[Cheng et al.(2023)Cheng,
  Yan, Shi, Lou, Fedorov, Behnami, Yuan, Xu,
  Xu, Xia, Pavlovskii,
  Peets, Zhao, Wan,
  Guo, Li, Yang, and  Büchner]Cheng.2023
  author author E. Cheng, author L. Yan, author X. Shi, author
  R. Lou, author A. Fedorov, author M. Behnami, author J. Yuan, author Y. Xu, author Y. Xu, author W. Xia, author
  N. Pavlovskii, author
  D. Peets, author W. Zhao, author Y. Wan, author Y. Guo, author S. Li, author
  W. Yang,  and author
  B. Büchner, title title Tunable positions of Weyl nodes via magnetism and
  pressure in the ferromagnetic Weyl semimetal CeAlSi, 10.48550/arXiv:2301.03800 journal journal
  arXiv  (year 2023),  10.48550/arXiv:2301.03800NoStop
[Wu et al.(2023)Wu,
  Chi, Zuo, Xu, Zhao, Luo, and Zhu]Wu.2023
  author author L. Wu, author S. Chi, author H. Zuo, author
  G. Xu, author L. Zhao, author Y. Luo,  and author Z. Zhu,  title title Field-induced Lifshitz
  transition in the magnetic Weyl semimetal candidate PrAlSi, 10.1038/s41535-023-00537-y journal journal npj Quantum Materials  volume 8
  (year 2023), 10.1038/s41535-023-00537-yNoStop
[Su et al.(2021)Su,
  Shi, Yuan, Wan, Cheng, Xi, Pi, Wang,
  Zou, Yu, Zhao, Li, and Guo]Su.2021
  author author H. Su, author X. Shi, author J. Yuan, author
  Y. Wan, author E. Cheng, author C. Xi, author L. Pi, author X. Wang, author Z. Zou, author
  N. Yu, author W. Zhao, author S. Li,  and author Y. Guo,  title title Multiple Weyl fermions in
  the noncentrosymmetric semimetal LaAlSi, 10.1103/PhysRevB.103.165128 journal journal
  Phys. Rev. B volume 103, pages
  165128 (year 2021)NoStop
[Xu et al.(2017)Xu,
  Alidoust, Chang, Lu,
  Singh, Belopolski, Sanchez,
  Zhang, Bian, Zheng,
  Husanu, Bian, Huang,
  Hsu, Chang, Jeng,
  Bansil, Neupert, Strocov,
  Lin, Jia, and Hasan]Xu.2017
  author author Su-Yang Xu, author Nasser Alidoust,
  author Guoqing Chang, author Hong Lu, author
  Bahadur Singh, author
  Ilya Belopolski, author
  D. S. Sanchez, author
  Xiao Zhang, author
  Guang Bian, author
  Hao Zheng, author Marious-Adrian Husanu, author Yi Bian, author Shin-Ming  Huang, author Chuang-han  Hsu, author Tay-Rong  Chang, author Horng-Tay  Jeng, author Arun Bansil, author Titus Neupert,
  author Vladimir N. Strocov,
  author Hsin Lin, author Shuang Jia,  and author M. Zahid Hasan, title title Discovery of Lorentz-violating type II
  Weyl fermions in LaAlGe, 10.1126/sciadv.1603266
  journal journal Sci. Adv. volume 3, pages e1603266 (year
  2017)NoStop
[Dhital et al.(2023)Dhital,
  Dally, Ruvalcaba, Gonzalez-Hernandez, Guerrero-Sanchez, Cao,
  Zhang, Tian, Wu,
  Frontzek, Karna, Meads,
  Wilson, Chapai, Graf,
  Bacsa, Jin, and DiTusa]Dhital.2023
  author author C. Dhital, author R. L. Dally,
  author R. Ruvalcaba, author R. Gonzalez-Hernandez, author J. Guerrero-Sanchez, author H. B. Cao, author
  Q. Zhang, author W. Tian, author Y. Wu, author M. D. Frontzek,
  author S. K. Karna, author A. Meads, author
  B. Wilson, author R. Chapai, author D. Graf, author J. Bacsa, author R. Jin,  and  author J. F. DiTusa, title title Multi-k magnetic structure and large
  anomalous Hall effect in candidate magnetic Weyl semimetal NdAlGe,  10.1103/PhysRevB.107.224414 journal journal Phys. Rev. B volume 107, pages 224414 (year 2023)NoStop
[Wang et al.(2021)Wang,
  Guo, and Wang]Wang.2021
  author author T. Wang, author Y. Guo,  and  author C. Wang, title title Structure and magnetic properties of
  RAlSi (R = light rare earth), 10.1088/1674-1056/abe3e8
  journal journal Chinese Phys. B volume 30, pages 075102 (year
  2021)NoStop
[Cao et al.(2022a)Cao, Wang, Pei, Gao,
  Zhao, Li, Yu, Wang, Chen, Li, and Qi]Cao.2022
  author author W. Cao, author Q. Wang, author C. Pei, author
  L. Gao, author Y. Zhao, author C. Li, author N. Yu, author J. Wang, author
  Y. Chen, author J. Li,  and author Y. Qi, title title
  Quantum oscillations in Noncentrosymmetric Weyl Semimetal SmAlSi,  110.1088/0256-307X/39/4/047501 journal
  journal Chinese Phys. Lett. volume
  39, pages 047501 (year
  2022a)NoStop
[Gaudet et al.(2021)Gaudet,
  Yang, Baidya, Lu,
  Xu, Zhao, Rodriguez-Rivera,
  Hoffmann, Graf, Torchinsky,
  Nikolić, Vanderbilt, Tafti,  and Broholm]Gaudet.2021
  author author J. Gaudet, author H.-Y. Yang,
  author S. Baidya, author B. Lu, author
  G. Xu, author Y. Zhao, author J.-A.  Rodriguez-Rivera, author C.-M.  Hoffmann, author D.-E.  Graf, author D.-H. Torchinsky, author P. Nikolić, author D. Vanderbilt, author F. Tafti,   and author C.-L. Broholm,  title title Weyl-mediated helical
  magnetism in NdAlSi, 10.1038/s41563-021-01062-8
  journal journal Nat Mater. volume 20(12), pages 1650–1656 (year
  2021)NoStop
[Sakhya et al.(2023)Sakhya,
  Huang, Dhakal, Gao,
  Regmi, Wang, Wen,
  He, Yao, Smith, Sprague, Gao, Singh, Lin,
  Xu, Tafti, Bansil, and  Neupane]Sakhya.2023
  author author Anup Pradhan  Sakhya, author Cheng-Yi  Huang, author Gyanendra  Dhakal, author Xue-Jian  Gao, author Sabin Regmi, author Baokai Wang,
  author Wei Wen, author R.-H. He, author
  Xiaohan Yao, author
  Robert Smith, author
  Milo Sprague, author
  Shunye Gao, author
  Bahadur Singh, author
  Hsin Lin, author Su-Yang Xu, author Fazel  Tafti, author Arun Bansil,  and author Madhab Neupane, title title Observation of
  fermi arcs and weyl nodes in a noncentrosymmetric magnetic weyl semimetal,  10.1103/PhysRevMaterials.7.L051202 journal
  journal Phys. Rev. Mater. volume 7,  pages L051202 (year 2023)NoStop
[Lou et al.(2023)Lou,
  Fedorov, Zhao, Yaresko,
  Büchner, and Borisenko]Lou.2023
  author author Rui Lou, author Alexander Fedorov, author Lingxiao Zhao, author Alexander Yaresko, author Bernd Büchner,  and author Sergey Borisenko, title title Signature of
  weakly coupled f electrons and conduction electrons in magnetic weyl
  semimetal candidates pralsi and smalsi, 10.1103/PhysRevB.107.035158 journal journal
  Phys. Rev. B volume 107, pages
  035158 (year 2023)NoStop
[Corasaniti et al.(2021)Corasaniti, Yang, Hu, Abeykoon, Petrovic, and Degiorgi]Corasaniti.2021
  author author M. Corasaniti, author R. Yang,
  author Z. Hu, author
  M. Abeykoon, author
  C. Petrovic,  and author
  L. Degiorgi, title title Evidence for correlation effects in noncentrosymmetric
  type-ii weyl semimetals, 10.1103/PhysRevB.104.L121112
  journal journal Phys. Rev. B volume 104, pages L121112 (year
  2021)NoStop
[Yang et al.(2022)Yang,
  Corasaniti, Le, Hu,
  Hu, Petrovic, and Degiorgi]Yang.2022
  author author R. Yang, author M. Corasaniti,
  author C. C. Le, author Z. Hu, author
  J. P. Hu, author C. Petrovic,  and author L. Degiorgi, title title Charge dynamics of a noncentrosymmetric magnetic weyl semimetal,  10.1038/s41535-022-00507-w journal journal npj Quantum Mater. volume 7, pages 101 (year 2022)NoStop
[Cao et al.(2022b)Cao, Zhao, Pei, Wang,
  Zhang, Ying, Zhao,
  Gao, Li, Yu, Gu, Chen, Liu, and Qi]Cao.2022a
  author author Weizheng  Cao, author Ningning  Zhao, author Cuiying  Pei, author Qi Wang, author Qinghua Zhang,
  author Tianping Ying, author Yi Zhao, author
  Lingling Gao, author
  Changhua Li, author
  Na Yu, author Lin Gu, author Yulin  Chen, author Kai Liu,  and author Yanpeng Qi, title title Pressure-induced
  superconductivity in the noncentrosymmetric Weyl semimetals LaAlX
  (X = Si,Ge), 10.1103/PhysRevB.105.174502 journal journal
  Phys. Rev. B volume 105, pages
  174502 (year 2022b)NoStop
[Tanner(2015)]Tanner.2015
  author author D. B. Tanner, title title Use of x-ray
  scattering functions in Kramers-Kronig analysis of reflectance, 10.1103/PhysRevB.91.035123 journal journal Phys. Rev. B volume 91, pages 035123 (year 2015)NoStop
[Kuzmenko(2005)]Kuzmenko.2005
  author author A. B. Kuzmenko, title title
  Kramers–Kronig constrained variational analysis of optical spectra,  10.1063/1.1979470 journal journal Rev. Sci. Instrum. volume 76,  pages 083108 (year 2005)NoStop
[Piva et al.(2023)Piva,
  Souza, Brousseau-Couture, Sorn, Pakuszewski, John, Adriano, Côté, Pagliuso, Paramekanti, and Nicklas]Piva.2023
  author author M. M. Piva, author J. C. Souza,
  author V. Brousseau-Couture,
  author Sopheak Sorn, author K. R. Pakuszewski, author Janas K. John, author
  C. Adriano, author M. Côté, author P. G.  Pagliuso, author Arun  Paramekanti,  and author M. Nicklas, title title Topological features in the ferromagnetic Weyl semimetal CeAlSi:
  Role of domain walls, 10.1103/PhysRevResearch.5.013068
  journal journal Phys. Rev. Res. volume 5, pages 013068 (year
  2023)NoStop
[Li et al.(2007a)Li, Hu, Dong, Qian,
  Hsieh, Hasan, Morosan,
  Cava, and Wang]Li.2007
  author author G. Li, author W. Z. Hu, author J. Dong, author
  D. Qian, author D. Hsieh, author M. Z.  Hasan, author E. Morosan, author R. J. Cava,   and author N. L. Wang,  title title Anomalous metallic state of
  Cu_0.07TiSe_2: An optical spectroscopy
  study, 10.1103/PhysRevLett.99.167002 journal journal Phys. Rev. Lett. volume 99, pages 167002 (year
  2007a)NoStop
[Li et al.(2007b)Li, Hu, Qian, Hsieh,
  Hasan, Morosan, Cava, and  Wang]Li.2007a
  author author G. Li, author W. Z. Hu, author D. Qian, author
  D. Hsieh, author M. Z. Hasan, author E. Morosan, author R. J.  Cava,  and author N. L.  Wang, title title
  Semimetal-to-semimetal charge density wave transition in
  1TTiSe_2, 10.1103/PhysRevLett.99.027404 journal journal
  Phys. Rev. Lett. volume 99, pages
  027404 (year 2007b)NoStop
[Tabert and Carbotte(2016)]Tabert.2016b
  author author C. J. Tabert and author J. P. Carbotte, title title Optical
  conductivity of Weyl semimetals and signatures of the gapped semimetal phase
  transition, 10.1103/PhysRevB.93.085442 journal journal Phys. Re. B volume
  93, pages 085442 (year 2016)NoStop
[Ashby and Carbotte(2014)]Ashby.2014
  author author P. E. C.  Ashby and author J. P.  Carbotte, title title
  Chiral anomaly and optical absorption in Weyl semimetals, 10.1103/PhysRevB.89.245121 journal journal Phys. Rev. B volume 89, pages 245121 (year 2014)NoStop
[Hosur et al.(2012)Hosur,
  Parameswaran, and Vishwanath]Hosur.2012
  author author Pavan Hosur, author S. A. Parameswaran,  and author Ashvin  Vishwanath, title title
  Charge transport in weyl semimetals, 10.1103/PhysRevLett.108.046602 journal journal
  Phys. Rev. Lett. volume 108, pages
  046602 (year 2012)NoStop
[Timusk et al.(2013)Timusk,
  Carbotte, Homes, Basov, and  Sharapov]Timusk.2013
  author author T. Timusk, author J. P. Carbotte, author C. C. Homes,
  author D. N. Basov,  and  author S. G. Sharapov,  title title Three-dimensional dirac
  fermions in quasicrystals as seen via optical conductivity, 10.1103/PhysRevB.87.235121 journal journal Phys. Rev. B volume 87, pages 235121 (year 2013)NoStop
[Chen et al.(2015)Chen,
  Zhang, Schneeloch, Zhang,
  Li, Gu, and Wang]Chen.2015
  author author R. Y. Chen, author S. J. Zhang,
  author J. A. Schneeloch,
  author C. Zhang, author Q. Li, author
  G. D. Gu,  and author
  N. L. Wang, title title Optical spectroscopy study of the three-dimensional Dirac
  semimetal ZrTe_5, 10.1103/PhysRevB.92.075107 journal journal
  Phys. Rev. B volume 92, pages
  075107 (year 2015)NoStop
[Xu et al.(2016)Xu,
  Dai, Zhao, Wang,
  Yang, Zhang, Liu,
  Xiao, Chen, Taylor,
  Yarotski, Prasankumar, and Qiu]Xu.2016
  author author B. Xu, author Y. M. Dai,
  author L. X. Zhao, author K. Wang, author
  R. Yang, author W. Zhang, author J. Y.  Liu, author H. Xiao, author G. F. Chen,
  author A. J. Taylor, author D. A. Yarotski, author
  R. P. Prasankumar,  and author
  X. G. Qiu, title title Optical spectroscopy of the Weyl semimetal TaAs,  10.1103/PhysRevB.93.121110 journal journal Phys. Rev. B volume 93, pages 121110 (year 2016)NoStop
[Kimura et al.(2017)Kimura,
  Yokoyama, Watanabe, Sichelschmidt, Süß, Schmidt, and  Felser]Kimura.2017
  author author Shin-ichi  Kimura, author Hiroko  Yokoyama, author Hiroshi  Watanabe, author Jörg  Sichelschmidt, author Vicky  Süß, author Marcus  Schmidt,  and author Claudia  Felser, title title
  Optical signature of weyl electronic structures in tantalum pnictides
  TaPn (Pn= P, As), 10.1103/PhysRevB.96.075119 journal journal Phys.
  Rev. B volume 96, pages 075119
  (year 2017)NoStop
[Carbotte(2016)]Carbotte.2016
  author author J. P. Carbotte, title title Dirac cone
  tilt on interband optical background of type-I and type-II Weyl
  semimetals, 10.1103/PhysRevB.94.165111 journal journal Phys. Re. B volume
  94, pages 165111 (year 2016)NoStop
[Xu et al.(2022)Xu,
  Niu, Bai, Zhu, Yuan, X., Han, Zhao,
  Yang, Xia, Liang, and  Tian]Xu.2022
  author author L. Xu, author H. Niu, author Y. Bai, author
  H. Zhu, author S. Yuan, author He.  X., author Y. Han, author L. Zhao, author Y. Yang, author
  Z. Xia, author Q. Liang,  and author Z. Tian, title title
  Shubnikov–de Haas oscillations and nontrivial topological states in Weyl
  semimetal candidate SmAlSi, 10.1088/1361-648X/ac987a
  journal journal J. Phys.: Condens. Matter  volume 34, pages 485701 (year 2022)NoStop



