Optical Distinctions Between Weyl Semimetal TaAs and DiracSemimetal Na3Bi: An Ab Initio Investigation

MEHRDAD DADSETANI1,2 and ALI EBRAHIMIAN1,3

1.—Department of Physics, Lorestan University, Khorramabad, Iran. 2.—e-mail: dadsetani.m@lu.ac.ir. 3.—e-mail: ebrahimian.Al@fs.lu.ac.ir

We present ab initio a study on linear and nonlinear optical properties oftopological semimetal Tantalum arsenide and Sodium bismuthate. The realand imaginary part of the dielectric function in addition to the energy lossspectra of TaAs and Na3Bi have been calculated within random phaseapproximation (RPA); then, the electron–hole interaction is included by solv-ing the Bethe–Salpeter equation for the electron–hole Green’s function. Inspite of being in the single category of topological materials, we have foundobvious distinction between linear optical responses of TaAs and Na3Bi at ahigh energy region where, in contrast to Na3Bi, Tantalum arsenide has exci-tonic peaks at 9 eV and 9.5 eV. It is remarkable that the excitonic effects inthe high energy range of the spectrum are stronger than in the lower one. Thedielectric function is overall red shifted compared with that of RPA approxi-mation. The resulting static dielectric constants for Na3Bi are smaller thancorresponding ones in TaAs. At a low energy region, the absorption intensityof TaAs is more than Na3Bi. The calculated second-order nonlinear opticalsusceptibilities vijk

(2)(x) show that Tantalum arsenide acts as a Weyl semimetal,and has high values of nonlinear responses in the low energy region whichmakes it promising candidate for the second harmonic generation in the ter-ahertz frequency region. In the low energy regime, optical spectra are domi-nated by the 2x intra-band contributions.

Key words: Weyl semimetal, Dirac semimetal, Na3Bi, excitonic effects, TaAs,optical properties

INTRODUCTION

Nowadays, Dirac materials and topological effectshave become one of the most active research areasin condensed matter physics.1–9 Topologicalsemimetals are a new class of Dirac material, theinterest in which has grown dramatically recentlyfollowing the experimental observation of Weyl andDirac semimetals.3–5 Dirac and Weyl semimetalsare two kinds of topological semimetals of whichtheir low energy bulk excitations are described byDriac and Weyl equations, respectively. To get aWeyl semimetal, either time-reversal (TR) or inver-sion symmetry needs to be broken.6 In a Weyl

semimetal, the low energy physics near the Weylpoint is given as 3D two-component Weyl fer-mion,10–12 H = ± vr.k where r is the pauli matrixand k is the momentum as measured from bandtouching (Weyl point). In the case with both TR andinversion symmetry, a 3D Dirac semimetal state isdescribed as a four-component Driac fermion whichis composed of two Weyl fermions of oppositechirality. This Dirac semimetal can be protectedby additional symmetry like crystal rotational sym-metry.12–14 When time reversal symmetry is bro-ken, the Weyl points are separated in momentum.On the other hand, when inversion symmetry isbroken, they are separated in energy. In both cases,the Fermi surface of surface states also split intoopen segments which are Fermi arcs discussed inWeyl semimetals.7–15 Therefore, there could be

(Received February 19, 2016; accepted June 18, 2016)

Journal of ELECTRONIC MATERIALS

DOI: 10.1007/s11664-016-4766-0� 2016 The Minerals, Metals & Materials Society

qualitative and quantitative differences in the prop-erties of Dirac and Weyl semimetals.

Despite being a gapless metal, a topologicalsemimetal is characterized by topological invari-ants, broadening the classification of topologicalphases of matter beyond insulators. In contrast totopological insulators where only surface states areinteresting,16 a topological semimetal features anunusual band structure in the bulk and on thesurface. The bulk conduction and valence bands of atopological semimetal touch linearly at pairs ofdiscrete points (the Weyl points), through which thebands disperse linearly along all the three momen-tum directions. Topological semimetals exhibit avariety of unusual phenomena, including topologi-cal surface states, chiral anomalies, quantumanomalous Hall effects and unusual optical conduc-tivity.17,18 The existence of linear dispersions, lowenergy excitations and poor screening make a Weylfermion a good candidate to show novel opticalproperties in linear and nonlinear responses. There-fore, fairly recently, it has been proposed theoreti-cally that the Weyl fermion in two dimensionsshows giant nonlinear responses to electromagneticfields in the terahertz (THz) region.19 Nonlinearelectromagnetic responses in the THz region are thefocus of investigation due to good coupling betweenTHz fields and free carriers in semiconductors, anddue to the promotion of the intensive THz excita-tions.19–23 On the other hand, ab initio calcula-tions24–27 have predicted, and subsequentexperimental studies28 have confirmed, the exis-tence of bound excitons in one-dimensional (1D)metallic carbon nanotubes and graphene which hasWeyl Fermions29,30 and can be considered as twodimensional (2D) analogs to Weyl semimetals interms of electronic dispersions. Furthermore, theelectrostatic screening is weak for Weyl fermionssince the density of states vanish at the Fermienergy.19 Therefore, it is considerably interesting toexplore that there are significant excitonic effects intopological semimetals.

Recently, several materials have been experimen-tally identified to be crystal-symmetry-protectedtopological semimetals. Among them, the Diracmaterial is Na3Bi, whose Dirac points are protectedby rotation symmetry.31,32 On the other hand, theWeyl semimetal is realized in a system whichbreaks time-reversal or inversion symmetry.5,15,33

Recently, the non-centrosymmetric and non-mag-netic transition metal monoarsenides/phosphides(including TaAs, NbP, TaP, NbP) have been pre-dicted to be Weyl semimetals and twelve pairs ofWeyl nodes are expected in their 3D Brillouinzone.33,34 Soon after, experimental realization of aWeyl semimetal in TaAs has been reported by B. Lvet al.35 Unlike the previously proposed Weylsemimetals, these isostructural compounds areWeyl semimetals in their natural states33 whichmake them a good platform for studying andmanipulating novel properties of Weyl semimetals

with promising application potential. Optical tech-niques as a contact-free probe can be used to searchsuch an application potential and exotic propertiesof Weyl fermions. Recent investigation reveals thatoptical conductivity of Weyl semimetals in the lowenergy part is mainly attributed to inter-bandtransitions in the vicinity of Weyl points and freecarriers (Drude peak).36,37 The inter-band transi-tions part grows linearly with frequency (energy)38

while the Drude peak decreases as the temperatureis reduced.39,40 These behaviors have been observedin TaAs by measuring optical conductivity at differ-ent temperatures.40

In this article, we focus on the optical propertiesof crystalline materials Tantalum arsenide (TaAs)and Sodium bismuthate (Na3Bi) which are absent inthe literature and call for further investigations. Wehave compared their optical properties andattempted to find unique characteristics that dis-tinguish Weyl semimetals from Dirac semimetals.To the best of our knowledge, there have been nofirst principles of studies to date of the opticalproperties of TaAs and Na3Bi including linear andnonlinear responses. We describe the detailed cal-culations of the band structure, linear optical prop-erties including the electron–hole interaction andsecond harmonic generation (SHG) for Weyl semi-metal TaAs and Dirac semimetal Na3Bi by usingdensity functional theory (DFT)-based methods,presently, the most successful and also the mostpromising approach to compute the electronic struc-ture of matter. DFT calculations have been found tomatch well with experimental results. However, it iswell known that excitons present in the excitationspectra are not obtained within DFT and randomphase approximation (RPA). Therefore, in thiswork, in addition to RPA, we apply the firstprinciple Bethe–Salpeter equation (BSE) approachto study quasiparticle energy and optical excitationsof TaAs and Na3Bi.41,42 The rest of the article isorganized as follows. In the ‘‘Calculation Method’’section, we outline the theoretical framework inwhich the calculations have been performed. In the‘‘Results and Discussion’’ section, we present anddiscuss the results of study concerning the struc-tural, electronic and optical properties of TaAs andNa3Bi. Finally, in the ‘‘Summary and Conclusions’’section, we summarize our calculations.

CALCULATION METHOD

The electronic and linear optical properties ofNa3Bi and TaAs have been calculated based on thehighly accurate all-electron full potential linearizedaugmented plane wave (FP-LAPW) method asimplemented in Exciting code.43 The linearizedaugmented plane wave (LAPW) basis functions areconstructed by connecting plane waves in theinterstitial regions to linear combinations ofatomic-like functions inside non-overlappingspheres at the atomic sites (muffin-tin spheres).

Dadsetani and Ebrahimian

Spin–orbit coupling is included by a second varia-tional procedure. The exchange correlation func-tional within a generalized gradient approximation(GGA) parametrized by Perdew, Burke and Ernz-erhof has been used.44 The muffin-tin radii forSodium (Na), Bismuth (Bi), Tantalum (Ta) andArsenide (As) have been set to 2.7, 3.1, 2.62 and 2.24bohr, respectively. The interstitial plane wave vec-tor cut off Kmax is chosen in a way that RMTKmax

equals 7. The valence wave functions inside theatomic spheres are expanded up to lmax = 20. TheBrillouin zone (BZ) was sampled with K-mesh up to12 9 12 9 12. The optical properties of matter canbe described by means of dielectric function. Theinfluences of excitonic effects are important in orderto correctly account for quantitative as well asqualitative features of optical spectra. Therefore, inorder to include the electron–hole interaction, whichis absent in the RPA, we apply many-body pertur-bations theory on top of DFT calculations. The BSEfor a two-particle Green’s function41,42 is solvedusing the Exciting code. The matrix eigenvalue formof the BSE is given by45,46

Xm0c0k0

Heffv0c0k0;vckA

jm0c0k0 ¼ EjAj

vck: ð1Þ

The indices vðcÞ and k stand for valence (conduction)band and vector k in the irreducible part of theBrillouin zone. Eigenvalues Ej and eigenvectorsAmckj represent the excitation energy of the jth-

correlated e–h pair and the coupling coefficientsused to construct the exciton wave function, respec-tively. Heff describes all interaction in the opticalprocesses, which consists of three interaction terms:

Heffvck;m0c0k0 ¼ Hdiag

vck;m0c0k0 þHdirvck;m0c0k0 þ cxH

xvck;m0c0k0 : ð2Þ

The kinetic term Hdiag is determined from the quasi-particle energies. By considering only the first term

in the right hand side of Eq. 2 corresponds to theindependent particle approximation. The attractivedirect and the repulsive exchange interactionmatrix elements Hdirand Hx are responsible for theformation of bound excitons. The pre-factor cx allowsone to choose different levels of approximation andto distinguish between spin-singlet (cx = 2) andspin-triplet channels (cx = 0). Using eigenvaluesand eigenvectors of the BSE, the long wavelengthlimit of the imaginary part of the dielectric functioneiiðxÞ is given42

Im eiiðxÞ ¼8p2

X

Xj

Xvck

Ajvck

hvkjPijckieck � emk

����

����2

�dðEj � xÞ:

ð3Þ

where X and x stand for the crystal volume and thefrequency, respectively. hmk|Pi|cki is the opticalmatrix element of the momentum operator. Thevalence and conduction state energies evk and evk areapproximated by Kohn–Sham eigenvalues. We notethat via derivation of Eq. 3, the BSE is solved basedon Tamm–Dancoff approximation,42 in which theexcited state is expanded only in electron–holestates. This approximation has given accurateresults for optical absorption spectra of other metal-lic systems such as graphite, metallic carbon nan-otubes (CNTs) and graphene.24–26 The excitoniceffects in the dielectric function of TaAs (Na3Bi)was converged by including 15(13) valence and 8(10)conduction states.

To calculate nonlinear optical response of TaAs,the second-order nonlinear optical susceptibilitytensor have been calculated within independentparticle approximation.47,48 The complex second-order nonlinear optical susceptibility tensorvijk

(2)(�2x; x, x) can be written as the sum offollowing three terms49–55

vinterijk �2x;x;xð Þ ¼ 1

X

Xnmlk

Wk

2rinm r*j

mlr*k

ln

n o

ðxln � xmlÞðxmn � 2xÞ �1

ðxmn � xÞr*k

lm r*i

mnr*j

nl

n o

ðxnl � xmnÞ�

r*j

nl r*k

lmr*i

mn

n o

ðxlm � xmnÞ

264

375

8><

>:

9>=

>;ð4Þ

vintraijk �2x;x;xð Þ¼ 1

X

XkWk

X

nml

x�2mn

xmn�xð Þ xlnr*j

nl r*k

lmr*i

mn

n o�xmlr

*k

lm r*i

mnr*j

nl

n oh i�

(8iX

nm

r*i

nm Djmnr

*k

nm

n o

x2mnðxmn�2xÞþ2

Xnml

r*i

nm r*j

mlr*k

ln

n oðxml�xlnÞ

x2mnðxmn�2xÞ

9=

;

ð5Þ

vmodijk �2x;x;xð Þ ¼ 1

2X

XkWk

Xnml

1

x2mn xmn �xð Þ xnlr

*i

lm r*j

mnr*k

nl

n o�xlmr

*i

nl r*j

lmr*k

mn

n oh i�

�iX

nm

r*i

nm r*j

nmDkmn

n o

x2mnðxmn �xÞ

9=

;

ð6Þ

Optical Distinctions Between Weyl Semimetal TaAs and Dirac Semimetal Na3Bi:An Ab Initio Investigation

where vijkinter(�2x; x, x) and vijk

intra(�2x; x, x) areinter-band transitions and intra-band transitions,respectively, while vijk

mod(�2x; x, x) stands formodulation of inter-band terms by intra-bandterms, where n = m = l and i, j and k correspondto Cartesian indices. Here, n(m) represents valence(conduction) state and l denotes all states ðl 6¼ m;nÞ.Two kinds of transitions can take place. The firstone is mcc¢ which contains one valence band and twoconduction bands, and the second transition is mm¢cwhich contains two valence bands and one conduc-

tion band. The symbols Dinmðk

*

Þ and rinmðk*

Þrjmlðk*

Þn o

are defined as follows

Dinm k

*� �¼ vinn k

*� �� vimmðk

*

Þ ð7Þ

rinmðk*

Þrjmlðk*

Þn o

¼ 1

2rinm k

*� �rjml k

*� �þ rjnm k

*� �riml k

*� �� �

ð8Þwhere v

*i

nm is the i component of the electron velocity

given as v*i

nm ¼ ixnm kð Þrinmðk*

Þ. rinmðk*

Þ, positionmatrix elements are calculated by using the

momentum matrix element P* i

nm, from the relation56

rinm k*� �

¼ Pinmðk

*

Þ

imxnmðk*

Þð9Þ

where the energy difference between the states nand m are given by �hxnm ¼ �hðxn � xmÞ. Second-order nonlinear optical susceptibility tensors havebeen calculated by using FP-LAPW as implementedin Elk code.57

RESULTS AND DISCUSSION

Structural and Electronic Properties

The crystal structure of TaAs58 and Na3Bi areshown in Fig. 1. Tantalum arsenide crystallizes in abody-centered-tetragonal structure with a nonsym-morphic space group I41md (No. 109), which lacksinversion symmetry. The measured lattice

constants58 are a = b = 3.434 A and c = 11.641 A.Both Ta and As are at the 4a Wyckoff position (0, 0,u) with u = 0 and 0.417 for Ta and As, respectively.The symmetry elements of this space group are thefour-fold screw rotation along the z-axis and twomirror reflections with respect to the x-axis and y-axis. On the other hand, Sodium bismuthate(Na3Bi) is a semimetal that crystallizes in thehexagonal P63/mmc (No. 194) crystal structure witha = b = 5.448 A and c = 9.655 A.59 In this struc-ture, there are two nonequivalent Na sites [Na(1)and Na(2)]. Na(1) and Bi form simple honeycomblattice layers which stack along the c axis, whileNa(2) atoms are inserted between the layers, mak-ing connection with Bi atoms.13

Lattice constants and internal coordinates ofTaAs were fully optimized and we have obtaineda = b = 3.429 A, c = 11.670 A for lattice constantsand optimized u = 0.4176 for the Ta site. In asimilar procedure, we have obtaineda = b = 5.449 A, c = 9.610 A for Na3Bi. Theseresults are consistently perfect with the ones inexperiment.35,58 The electronic band structures ofTaAs have been depicted in Fig. 2. Calculationsshow that in the absence of spin–orbit coupling, thevalence and conduction bands cross and form closedrings. This band behavior indicates that TaAs is asemimetal. In the presence of spin–orbit coupling,the valence and conduction bands become fullygapped along the high symmetry lines with the

Fig. 1. The crystal structures of Na3Bi (left) and TaAs (right).

Fig. 2. The band structures of Na3Bi (left) and TaAs (right) calcu-lated by GGA with spin–orbit coupling.

Dadsetani and Ebrahimian

exception of one point along the ZN line and pairs ofWeyl nodes have appeared. In fact, there is apseudogap centered above Fermi energy with avery small density of states which is in agreementwith previously reported calculations based onnorm-conserving pseudopotential.5 Because of thelack of inversion symmetry, the double spin degen-eracy splits the band structure as shown in Fig. 2.Recent first-principle calculations have predictedthat TaAs has 12 pairs of Weyl points.34 Four pairsof these Weyl points are exactly in the kz = 0 planeand the other eight pairs of Weyl points are locatedoff the kz = 0 plane. The bulk band structure ofNa3Bi in Fig. 2 shows that there is a Dirac point

along the CA line and a band inversion at the BZcenter, which are similar to previous study.13 Theband inversion is mostly due to Bi, which has 6pstates and large spin–orbit coupling. Therefore,Na3Bi is a semimetal with two nodes (band crossing)exactly at the Fermi energy. Due to the protection ofan additional three-fold rotational symmetry alongthe [001] crystalline direction, Dirac band touching(Dirac node) is preserved in the presence of spin–orbit coupling. Since both time-reversal and inver-sion symmetries are present, there is four-folddegeneracy at the Dirac node. The calculated partialdensity of states (p-DOS) in Fig. 3 shows that themajor contribution to the density of states of TaAs

Fig. 3. Calculated partial densities of states of Na3Bi (left) and TaAs (right).

Fig. 4. Real and imaginary parts of the dielectric tensor to x-polar-ized incident light for Na3Bi (top) and TaAs (bottom) in RPA.

Fig. 5. Real and imaginary parts of the dielectric tensor to z-polar-ized incident light for Na3Bi (top) and TaAs (bottom) in RPA.

Optical Distinctions Between Weyl Semimetal TaAs and Dirac Semimetal Na3Bi:An Ab Initio Investigation

around the Fermi energy is of the Ta 5d character.In fact, Ta 5d states are hybridized strongly with As4p to construct upper conduction and lower valencebands. Due to the lack of particle–hole symmetry,the density of states is not zero at the Fermi energy.The calculated electronic structure of Na3Bi inFig. 3 indicates that the valence and conductionbands are dominated by Bi 6p and Na 3s states. Incomparison to TaAs, the density of states of Na3Bi isnegligible at the Fermi level. TaAs has a widerbandwidth relative to the narrower bandwidth ofNa3Bi which reflects that 5d electrons are moredelocalized than 6p electrons. Near the Fermi level,the density of states of Ta-d is higher than Bi-p inNa3Bi.

Optical Properties

The dielectric tensor of tetragonal TaAs (hexag-onal Na3Bi) is diagonal and has two independentcomponents: exx = eyy perpendicular to the C axisand ezz along C axis. In Figs. 4 and 5, the opticalresponse to x- and z-polarized incident lightdescribed in terms of the dielectric tensor Im eiiand Re eii are depicted for TaAs and Na3Bi. Thesecomponents have been calculated within RPAapproximation of Eq. 3. This optical absorption does

not include intra-band transitions because theDrude peak vanishes at low temperature as men-tioned before and the Weyl fermions properties canbe determined from inter-band transitions in thelow energy part of optical absorption.

In general comparison, the optical spectra of bothIm eii and Re eii show considerable anisotropybetween x and z components. In other words, inboth crystals, the component Im exx displays differ-ent dispersions from Im ezz, which reveals thepolarization dependence of the optical absorption.The oscillator strength of Im exx is more than Im ezz.On the other hand, the component Im ezz has moremain peaks, distributed throughout the wider rangeof energy. For TaAs, there are two main peaks below2 eV while Na3Bi has just one peak. In fact, at thelow energy region, the absorption intensity of TaAsis more than Na3Bi. This absorption behavior can berelated to high density of Ta-d states near the Fermilevel. On the contrary, above 2 eV, the absorptionintensity of Na3Bi is more than TaAs. A comparisonof the absorption spectra of both crystals shows thatin TaAs, the region of the principal absorption of theIm exx is narrower than Im ezz while they are at thesame level in Na3Bi. The component Im exx of Na3Bihas a high peak at 2.3 eV. In considering Re eii ofTaAs, Fig. 4 shows the negative values of the x (z)

Fig. 6. The x-component of the energy loss function of Na3Bi (top)and TaAs (bottom) in RPA and in BSE.

Fig. 7. The z-component of the energy loss function of Na3Bi (top)and TaAs (bottom) in RPA and in BSE.

Dadsetani and Ebrahimian

component of the real part of the dielectric functionfrom 3 eV up to 7.07 (7.46) eV which corresponds toa high reflectivity region. The component Re ezz ofTaAs is negative from 3 eV to 7.43 eV while thecomponent Rezz of Na3Bi has high reflectivity from2.84 eV to 6.01 eV. In Na3Bi, Re exx is negative intwo regions, from 2.44 to 2.63 and between 3 eV and6.13 eV. Moreover, in the low energy region, thecomponent Re exx of TaAs decreases faster than Reexx of Na3Bi. In the negative region of Re exx, theoscillator strength of TaAs is more than Na3Bi.Below 3 eV, Re ezz of TaAs (Na3Bi) has three peaksat 0.61 (1.15), 1.47 (1.70) and 2.54 (2.51) eV while Reexx has a main peak at 0.64(1.11) eV in addition tothe high reflectivity region around 2(2.5) eV. As canbe seen in Figs. 4 and 5, the result of static dielectricconstants for Na3Bi are smaller than correspondingones in TaAs. The static dielectric constants alongthe x- and z-axis are 13.53 (5.99) and 11.42 (5.73) forTaAs (Na3Bi), respectively. To study the collectiveexcitations of TaAs and Na3Bi, we have shown theirenergy loss functions in Figs. 6 and 7. The energyloss function [L(x) = Im (�1/e)] which shows theenergy loss of a fast electron moving across amedium, is a complicated mixture of inter-bandtransitions and plasmons. Inter-band transitionpeaks are related to the peaks of Im eii while

Plasmon peaks correspond to zeros of the Re eii. Theplasmon peak of the loss function is large if Re eii iszero and Im eii is small. For Na3Bi (TaAs), there is amain peak around 6 (7) eV which corresponds to thecollective excitations (plasmons). Weak peaks belowthe plasmon peak can be assigned to inter-bandtransitions in accordance with peaks in Im eii. InTaAs, there are two weak peaks at the shoulder ofthe plasmon peak, which are absent in Na3Bi andrelated to the large intensity of states of TaAs nearthe Fermi level. As mentioned before, recent studieshave confirmed the existence of bound excitons inmetallic systems.24–26 Therefore, by exploring theexcitonic effects, we have calculated optical proper-ties of TaAs and Na3Bi by solving the full BSE forthe e–h two-particle Green’s function Eq. 3. Thecomponents of the imaginary part of the dielectricfunction, including excitonic features, have beendepicted in Figs. 8 and 9. Regarding to electron–hole interactions, we observe a red shift of theoptical absorption feature. The red shift of theabsorption spectra is more strongly pronounced forNa3Bi compared to TaAs. This is the generalexcitonic effect on the dielectric function whichhas been previously reported for graphene.26 InTaAs, e–h interactions change the intensity ofabsorption and create an excitonic peak at a high

Fig. 9. Imaginary parts of the dielectric tensor to z-polarized incidentlight for Na3Bi (top) and TaAs (bottom) in RPA and BSE.

Fig. 8. Imaginary parts of the dielectric tensor to x-polarized incidentlight for Na3Bi (top) and TaAs (bottom) in RPA and BSE.

Optical Distinctions Between Weyl Semimetal TaAs and Dirac Semimetal Na3Bi:An Ab Initio Investigation

energy region. This peak cannot be seen in absorp-tion spectra of Na3Bi. The excitonic features in thedielectric function of TaAs show that the oscillator

strength of the first optical transition of the x- andz-component increases while the second onedecreases. In both x- and z-components, a weakpeak rises up at 9.5 eV in the region where single-particle oscillator strengths were vanishing. Toanalyze this more accurately, we have shown lossfunction including e–h interaction in Figs. 6 and 7.It is remarkable that the excitonic effects in thehigh energy range of the spectrum are strongerthan in the lower one, in contrast to ordinarysemiconductors or insulators where excitonic effectsmainly affect the lowest energy part of opticalspectra. With respect to RPA, e–h effects decreasethe main plasmon peak and shift it towards lowerenergies. The x-component of loss function displaystwo excitonic peaks at 9 eV and 9.5 eV while thez-component shows an excitonic peak at 9.5 eV withlower intensity. These peaks correspond to excitonicpeaks in the imaginary part of the dielectric tensoraround 9.5 eV. We have used different levels ofapproximation in Eq. 2 to understand better theorigin of relevant excitonic states. Once all contri-butions of Eq. 2 are included in the calculations,excitonic peaks appear at high energy. Takingdiagonal and direct terms into account results inelimination of the excitonic peaks. Therefore, ourcalculations show that these excitonic effects arisemainly from the repulsive exchange interactionterm in the electron–hole kernel, with the attractivedirect term playing a negligible role.

Recent theoretical investigation19 of nonlinearoptical responses of Weyl fermions in two dimen-sions has encouraged us to explore the possiblegiant nonlinear optical responses of Weyl fermionsin three dimensions. Second-order nonlinear opticalinteractions can occur only in non-centrosymmetriccrystals. Therefore, optical second harmonic gener-ation is allowed in TaAs, while Na3Bi cannotsupport nonlinear effects of even order. SHG canyield additional information about the structure ofTaAs. In order to evaluate the nonlinear opticalresponse of TaAs, we have calculated its non-linearsusceptibilities vijk

(2)(x). The second-harmonic

response v 2ð Þijk xð Þ involves a 2x resonance in addition

to the normal x resonance which can be separatedinto inter-band and intra-band contributions.Therefore the analysis of nonlinear spectra is ademanding job. For TaAs, which is a body-centered-tetragonal crystal with space group I41md, there arefour second-order nonlinear susceptibilities: v131

(2) (x),v113

(2) (x), v311(2) (x), v333

(2) (x). The imaginary parts of thesusceptibilities are presented in Fig. 10. Theseresults, calculated within independent particleapproximation, show that the Weyl semimetal TaAshas high values of nonlinear response which make itpromising for SHG in THz region. We find that

below 0.38 eV,v 2ð Þ311ðxÞ is a dominant component with

a peak at 0.1 eV where both v113(2) (x) and v131

(2) (x) havea peak with lower intensity. Up to 1.43 eV, v113

(2) (x)and v131

(2) (x) have exactly the same intensity while

Fig. 10. Imaginary parts of vijk(2)(x) for TaAs.

Fig. 11. Imaginary and real parts of v311(2) (x) for TaAs and its 2x intra-

band contributions.

Fig. 12. Calculated x/2x inter-band and x intra-band contributionsof v311

(2) (x) for TaAs.

Dadsetani and Ebrahimian

they are dominant responses with considerableintensities between 0.38 and 1.43 eV. As shown inthe inset of Fig. 10, from 1.43 eV up to 3 eV, allcomponents have different dispersion and apprecia-ble intensities. Therefore, there are giant nonlinearresponses in the low energy region (terahertzfrequency region). The dispersion of real and imag-

inary parts of v 2ð Þ311ðxÞ and its corresponding intra-

and inter-band contributions of x and 2x resonancehave been depicted in Figs. 11 and 12. Comparing

v 2ð Þ311ðxÞ with its intra-band part in Fig. 11 shows

that the dominant peak of v 2ð Þ311ðxÞ comes from the 2x

intra-band contribution. Figure 12 shows that xintra- and inter-band have nearly the same inten-sity values but are opposite in signs. Therefore,their overall intensity is negligible. The 2x inter-band contribution has intensity fluctuation anddecreases rapidly. We have found the same featuresfor other components. Therefore, the dominant peakof the nonlinear response of TaAs comes from the 2xintra-band contribution.

SUMMARY AND CONCLUSIONS

In this work, based on density-functional theory,we have investigated and compared the opticalproperties of TaAs and Na3Bi and attempted to findunique characteristics that distinguish a Weylsemimetal from a Dirac semimetal. Using differentlevels of approximation in the BSE, we have studiedthe excitonic effects on the optical absorption ofNa3Bi and TaAs. The calculated optical spectra ofTaAs show excitonic effects at a high energy. Wehave found excitonic peaks at 9 eV and 9.5 eV in theoptical absorption of TaAs. In both crystals, thedielectric function is overall red shifted comparedwith that of RPA approximation. Our calculationsshow that these excitonic effects arise mainly fromthe repulsive exchange interaction term in theelectron–hole kernel, with the attractive direct termplaying a negligible role. At low energy, the absorp-tion intensity of TaAs is more than Na3Bi, related tothe high density of Ta-d states near the Fermi level.For Na3Bi (TaAs), there is a main peak around 6(7) eV which corresponds to the collective excita-tions (plasmons). The static dielectric constantsalong the x- and z-axis are 13.53 (5.99) and 11.42(5.73) for TaAs (Na3Bi), respectively. The second-harmonic responses vijk

(2)(x) have been calculated forTaAs, within independent particle approximation.These results show that the Weyl semimetal TaAshas high values of nonlinear response which make itpromising for SHG in the THz region. The dominantpeak of the nonlinear response of TaAs comes from a2x intra-band contribution.

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