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Journal of ELECTRONIC MATERIALS DOI: 10.1007/s11664-016-4766-0 2016 The Minerals, Metals & Materials Society Optical Distinctions Between Weyl Semimetal TaAs and Dirac Semimetal 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: [email protected] We present ab initio a study on linear and nonlinear optical properties of topological semimetal Tantalum arsenide and Sodium bismuthate. The real and imaginary part of the dielectric function in addition to the energy loss spectra of TaAs and Na3Bi have been calculated within random phase approximation (RPA); then, the electron–hole interaction is included by solving the Bethe–Salpeter equation for the electron–hole Green’s function. In spite of being in the single category of topological materials, we have found obvious distinction between linear optical responses of TaAs and Na3Bi at a high energy region where, in contrast to Na3Bi, Tantalum arsenide has excitonic peaks at 9 eV and 9.5 eV. It is remarkable that the excitonic effects in the high energy range of the spectrum are stronger than in the lower one. The dielectric function is overall red shifted compared with that of RPA approximation. The resulting static dielectric constants for Na3Bi are smaller than corresponding ones in TaAs. At a low energy region, the absorption intensity of TaAs is more than Na3Bi. The calculated second-order nonlinear optical susceptibilities v(2) ijk(x) show that Tantalum arsenide acts as a Weyl semimetal, and has high values of nonlinear responses in the low energy region which makes it promising candidate for the second harmonic generation in the terahertz frequency region. In the low energy regime, optical spectra are dominated by the 2x intra-band contributions. Key words: Weyl semimetal, Dirac semimetal, Na3Bi, excitonic effects, TaAs, optical properties INTRODUCTION Nowadays, Dirac materials and topological effects have become one of the most active research areas in condensed matter physics.1–9 Topological semimetals are a new class of Dirac material, the interest in which has grown dramatically recently following the experimental observation of Weyl and Dirac semimetals.3–5 Dirac and Weyl semimetals are two kinds of topological semimetals of which their low energy bulk excitations are described by Driac and Weyl equations, respectively. To get a Weyl semimetal, either time-reversal (TR) or inversion symmetry needs to be broken.6 In a Weyl (Received February 19, 2016; accepted June 18, 2016) semimetal, the low energy physics near the Weyl point is given as 3D two-component Weyl fermion,10–12 H = ± vr.k where r is the pauli matrix and k is the momentum as measured from band touching (Weyl point). In the case with both TR and inversion symmetry, a 3D Dirac semimetal state is described as a four-component Driac fermion which is composed of two Weyl fermions of opposite chirality. This Dirac semimetal can be protected by additional symmetry like crystal rotational symmetry.12–14 When time reversal symmetry is broken, the Weyl points are separated in momentum. On the other hand, when inversion symmetry is broken, they are separated in energy. In both cases, the Fermi surface of surface states also split into open segments which are Fermi arcs discussed in Weyl semimetals.7–15 Therefore, there could be Dadsetani and Ebrahimian qualitative and quantitative differences in the properties of Dirac and Weyl semimetals. Despite being a gapless metal, a topological semimetal is characterized by topological invariants, broadening the classification of topological phases of matter beyond insulators. In contrast to topological insulators where only surface states are interesting,16 a topological semimetal features an unusual band structure in the bulk and on the surface. The bulk conduction and valence bands of a topological semimetal touch linearly at pairs of discrete points (the Weyl points), through which the bands disperse linearly along all the three momentum directions. Topological semimetals exhibit a variety of unusual phenomena, including topological surface states, chiral anomalies, quantum anomalous Hall effects and unusual optical conductivity.17,18 The existence of linear dispersions, low energy excitations and poor screening make a Weyl fermion a good candidate to show novel optical properties in linear and nonlinear responses. Therefore, fairly recently, it has been proposed theoretically that the Weyl fermion in two dimensions shows giant nonlinear responses to electromagnetic fields in the terahertz (THz) region.19 Nonlinear electromagnetic responses in the THz region are the focus of investigation due to good coupling between THz fields and free carriers in semiconductors, and due to the promotion of the intensive THz excitations.19–23 On the other hand, ab initio calculations24–27 have predicted, and subsequent experimental studies28 have confirmed, the existence of bound excitons in one-dimensional (1D) metallic carbon nanotubes and graphene which has Weyl Fermions29,30 and can be considered as two dimensional (2D) analogs to Weyl semimetals in terms of electronic dispersions. Furthermore, the electrostatic screening is weak for Weyl fermions since the density of states vanish at the Fermi energy.19 Therefore, it is considerably interesting to explore that there are significant excitonic effects in topological semimetals. Recently, several materials have been experimentally identified to be crystal-symmetry-protected topological semimetals. Among them, the Dirac material is Na3Bi, whose Dirac points are protected by rotation symmetry.31,32 On the other hand, the Weyl semimetal is realized in a system which breaks time-reversal or inversion symmetry.5,15,33 Recently, the non-centrosymmetric and non-magnetic transition metal monoarsenides/phosphides (including TaAs, NbP, TaP, NbP) have been predicted to be Weyl semimetals and twelve pairs of Weyl nodes are expected in their 3D Brillouin zone.33,34 Soon after, experimental realization of a Weyl semimetal in TaAs has been reported by B. Lv et al.35 Unlike the previously proposed Weyl semimetals, these isostructural compounds are Weyl semimetals in their natural states33 which make them a good platform for studying and manipulating novel properties of Weyl semimetals with promising application potential. Optical techniques as a contact-free probe can be used to search such an application potential and exotic properties of Weyl fermions. Recent investigation reveals that optical conductivity of Weyl semimetals in the low energy part is mainly attributed to inter-band transitions in the vicinity of Weyl points and free carriers (Drude peak).36,37 The inter-band transitions part grows linearly with frequency (energy)38 while the Drude peak decreases as the temperature is reduced.39,40 These behaviors have been observed in TaAs by measuring optical conductivity at different temperatures.40 In this article, we focus on the optical properties of crystalline materials Tantalum arsenide (TaAs) and Sodium bismuthate (Na3Bi) which are absent in the literature and call for further investigations. We have compared their optical properties and attempted to find unique characteristics that distinguish Weyl semimetals from Dirac semimetals. To the best of our knowledge, there have been no first principles of studies to date of the optical properties of TaAs and Na3Bi including linear and nonlinear responses. We describe the detailed calculations of the band structure, linear optical properties including the electron–hole interaction and second harmonic generation (SHG) for Weyl semimetal TaAs and Dirac semimetal Na3Bi by using density functional theory (DFT)-based methods, presently, the most successful and also the most promising approach to compute the electronic structure of matter. DFT calculations have been found to match well with experimental results. However, it is well known that excitons present in the excitation spectra are not obtained within DFT and random phase approximation (RPA). Therefore, in this work, in addition to RPA, we apply the first principle Bethe–Salpeter equation (BSE) approach to study quasiparticle energy and optical excitations of TaAs and Na3Bi.41,42 The rest of the article is organized as follows. In the ‘‘Calculation Method’’ section, we outline the theoretical framework in which the calculations have been performed. In the ‘‘Results and Discussion’’ section, we present and discuss the results of study concerning the structural, electronic and optical properties of TaAs and Na3Bi. Finally, in the ‘‘Summary and Conclusions’’ section, we summarize our calculations. CALCULATION METHOD The electronic and linear optical properties of Na3Bi and TaAs have been calculated based on the highly accurate all-electron full potential linearized augmented plane wave (FP-LAPW) method as implemented in Exciting code.43 The linearized augmented plane wave (LAPW) basis functions are constructed by connecting plane waves in the interstitial regions to linear combinations of atomic-like functions inside non-overlapping spheres at the atomic sites (muffin-tin spheres). Optical Distinctions Between Weyl Semimetal TaAs and Dirac Semimetal Na3Bi: An Ab Initio Investigation Spin–orbit coupling is included by a second variational procedure. The exchange correlation functional within a generalized gradient approximation (GGA) parametrized by Perdew, Burke and Ernzerhof has been used.44 The muffin-tin radii for Sodium (Na), Bismuth (Bi), Tantalum (Ta) and Arsenide (As) have been set to 2.7, 3.1, 2.62 and 2.24 bohr, respectively. The interstitial plane wave vector cut off Kmax is chosen in a way that RMTKmax equals 7. The valence wave functions inside the atomic spheres are expanded up to lmax = 20. The Brillouin zone (BZ) was sampled with K-mesh up to 12 9 12 9 12. The optical properties of matter can be described by means of dielectric function. The influences of excitonic effects are important in order to correctly account for quantitative as well as qualitative features of optical spectra. Therefore, in order to include the electron–hole interaction, which is absent in the RPA, we apply many-body perturbations theory on top of DFT calculations. The BSE for a two-particle Green’s function41,42 is solved using the Exciting code. The matrix eigenvalue form of the BSE is given by45,46 X Hveff0 c0 k0 ;vck Ajm0 c0 k0 ¼ Ej Ajvck : ð1Þ m 0 c0 k 0 The indices vðcÞ and k stand for valence (conduction) band and vector k in the irreducible part of the Brillouin zone. Eigenvalues Ej and eigenvectors Ajmck represent the excitation energy of the jthcorrelated e–h pair and the coupling coefficients used to construct the exciton wave function, respectively. Heff describes all interaction in the optical processes, which consists of three interaction terms: diag eff dir x Hvck;m 0 c0 k 0 ¼ H vck;m0 c0 k0 þ Hvck;m0 c0 k0 þ cx Hvck;m0 c0 k0 : ð2Þ The kinetic term Hdiag is determined from the quasiparticle energies. By considering only the first term vinter ijk ð2x; x; xÞ ¼ 1X X in the right hand side of Eq. 2 corresponds to the independent particle approximation. The attractive direct and the repulsive exchange interaction matrix elements Hdirand Hx are responsible for the formation of bound excitons. The pre-factor cx allows one to choose different levels of approximation and to distinguish between spin-singlet (cx = 2) and spin-triplet channels (cx = 0). Using eigenvalues and eigenvectors of the BSE, the long wavelength limit of the imaginary part of the dielectric function eii ðxÞ is given42 Im eii ðxÞ ¼ 2 8p2 X X hvkjPi jcki j j A dðE xÞ: j vck vck e e X ck mk ð3Þ where X and x stand for the crystal volume and the frequency, respectively. hmk|Pi|cki is the optical matrix element of the momentum operator. The valence and conduction state energies evk and evk are approximated by Kohn–Sham eigenvalues. We note that via derivation of Eq. 3, the BSE is solved based on Tamm–Dancoff approximation,42 in which the excited state is expanded only in electron–hole states. This approximation has given accurate results for optical absorption spectra of other metallic systems such as graphite, metallic carbon nanotubes (CNTs) and graphene.24–26 The excitonic effects 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 susceptibility tensor have been calculated within independent particle approximation.47,48 The complex secondorder nonlinear optical susceptibility tensor v(2) ijk(2x; x, x) can be written as the sum of following three terms49–55 n*k *i o39 > = r nl r lm r mn 7 1 6 W 4 5 k nmlk > ; :ðxln xml Þðxmn 2xÞ ðxmn xÞ ðxnl xmn Þ ðxlm xmn Þ > 8 > < n *j *k o 2rinm r ml r ln 2 n *i *j o rlm r mn rnl *k *j ð4Þ n o n*j *k o 9 ( *i *i j *k h *j n*k *i o n*i *j oi X X r nm rml rln ðxml xln Þ= X x2 X rnm Dmn r nm k 1 * intra mn xln r nl rlm r mn xml r lm rmn rnl 8i þ2 vijk ð2x; x; xÞ ¼ Wk k nm x2 ðx nml ; X x2mn ðxmn 2xÞ ðxmn xÞ mn 2xÞ mn nml ð5Þ n*j o9 *i k X = h n o n o i r r D X X mn nm nm i j k i j k 1 1 * * * * * * x ð 2x; x; x Þ ¼ W r r r r r r vmod x i k nl lm lm nl ijk mn nl lm mn k nml x2 ðx nm x2 ðx 2X mn xÞ mn xÞ ; mn mn ð6Þ Dadsetani and Ebrahimian Fig. 1. The crystal structures of Na3Bi (left) and TaAs (right). intra where vinter ijk (2x; x, x) and vijk (2x; x, x) are inter-band transitions and intra-band transitions, respectively, while vmod ijk (2x; x, x) stands for modulation of inter-band terms by intra-band terms, where n = m = l and i, j and k correspond to 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 first one is mcc¢ which contains one valence band and two conduction bands, and the second transition is mm¢c which contains two valence bands and n one conduco * * * tion band. The symbols Dinm ðkÞ and rinm ðkÞrjml ðkÞ are defined as follows * * * Dinm k ¼ vinn k vimm ðkÞ ð7Þ n * * * * o 1 i * j * rnm k rml k þ rjnm k riml k rinm ðkÞrjml ðkÞ ¼ 2 ð8Þ *i where vnm is the i component of the electron velocity * *i * given as vnm ¼ ixnm ðkÞrinm ðkÞ. rinm ðkÞ, position matrix elements are calculated by using the *i momentum matrix element Pnm , from the relation56 * rinm * Pinm ðkÞ k ¼ * imxnm ðkÞ ð9Þ where the energy difference between the states n and m are given by hxnm ¼ hðxn xm Þ. Secondorder nonlinear optical susceptibility tensors have been calculated by using FP-LAPW as implemented in Elk code.57 RESULTS AND DISCUSSION Structural and Electronic Properties The crystal structure of TaAs58 and Na3Bi are shown in Fig. 1. Tantalum arsenide crystallizes in a body-centered-tetragonal structure with a nonsymmorphic space group I41md (No. 109), which lacks inversion symmetry. The measured lattice Fig. 2. The band structures of Na3Bi (left) and TaAs (right) calculated by GGA with spin–orbit coupling. constants58 are a = b = 3.434 Å and c = 11.641 Å. 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 the four-fold screw rotation along the z-axis and two mirror reflections with respect to the x-axis and yaxis. On the other hand, Sodium bismuthate (Na3Bi) is a semimetal that crystallizes in the hexagonal P63/mmc (No. 194) crystal structure with a = b = 5.448 Å and c = 9.655 Å.59 In this structure, there are two nonequivalent Na sites [Na(1) and Na(2)]. Na(1) and Bi form simple honeycomb lattice layers which stack along the c axis, while Na(2) atoms are inserted between the layers, making connection with Bi atoms.13 Lattice constants and internal coordinates of TaAs were fully optimized and we have obtained a = b = 3.429 Å, c = 11.670 Å for lattice constants and optimized u = 0.4176 for the Ta site. In a similar procedure, we have obtained a = b = 5.449 Å, c = 9.610 Å for Na3Bi. These results are consistently perfect with the ones in experiment.35,58 The electronic band structures of TaAs have been depicted in Fig. 2. Calculations show that in the absence of spin–orbit coupling, the valence and conduction bands cross and form closed rings. This band behavior indicates that TaAs is a semimetal. In the presence of spin–orbit coupling, the valence and conduction bands become fully gapped along the high symmetry lines with the Optical Distinctions Between Weyl Semimetal TaAs and Dirac Semimetal Na3Bi: An Ab Initio Investigation 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-polarized incident light for Na3Bi (top) and TaAs (bottom) in RPA. Fig. 5. Real and imaginary parts of the dielectric tensor to z-polarized incident light for Na3Bi (top) and TaAs (bottom) in RPA. exception of one point along the ZN line and pairs of Weyl nodes have appeared. In fact, there is a pseudogap centered above Fermi energy with a very small density of states which is in agreement with previously reported calculations based on norm-conserving pseudopotential.5 Because of the lack of inversion symmetry, the double spin degeneracy splits the band structure as shown in Fig. 2. Recent first-principle calculations have predicted that TaAs has 12 pairs of Weyl points.34 Four pairs of these Weyl points are exactly in the kz = 0 plane and the other eight pairs of Weyl points are located off the kz = 0 plane. The bulk band structure of Na3Bi in Fig. 2 shows that there is a Dirac point along the CA line and a band inversion at the BZ center, which are similar to previous study.13 The band inversion is mostly due to Bi, which has 6p states and large spin–orbit coupling. Therefore, Na3Bi is a semimetal with two nodes (band crossing) exactly at the Fermi energy. Due to the protection of an additional three-fold rotational symmetry along the [001] crystalline direction, Dirac band touching (Dirac node) is preserved in the presence of spin– orbit coupling. Since both time-reversal and inversion symmetries are present, there is four-fold degeneracy at the Dirac node. The calculated partial density of states (p-DOS) in Fig. 3 shows that the major contribution to the density of states of TaAs Dadsetani and Ebrahimian Fig. 6. The x-component of the energy loss function of Na3Bi (top) and TaAs (bottom) in RPA and in BSE. around the Fermi energy is of the Ta 5d character. In fact, Ta 5d states are hybridized strongly with As 4p to construct upper conduction and lower valence bands. 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 in Fig. 3 indicates that the valence and conduction bands are dominated by Bi 6p and Na 3s states. In comparison to TaAs, the density of states of Na3Bi is negligible at the Fermi level. TaAs has a wider bandwidth relative to the narrower bandwidth of Na3Bi which reflects that 5d electrons are more delocalized than 6p electrons. Near the Fermi level, the density of states of Ta-d is higher than Bi-p in Na3Bi. Optical Properties The dielectric tensor of tetragonal TaAs (hexagonal Na3Bi) is diagonal and has two independent components: exx = eyy perpendicular to the C axis and ezz along C axis. In Figs. 4 and 5, the optical response to x- and z-polarized incident light described in terms of the dielectric tensor Im eii and Re eii are depicted for TaAs and Na3Bi. These components have been calculated within RPA approximation of Eq. 3. This optical absorption does Fig. 7. The z-component of the energy loss function of Na3Bi (top) and TaAs (bottom) in RPA and in BSE. not include intra-band transitions because the Drude peak vanishes at low temperature as mentioned before and the Weyl fermions properties can be determined from inter-band transitions in the low energy part of optical absorption. In general comparison, the optical spectra of both Im eii and Re eii show considerable anisotropy between x and z components. In other words, in both crystals, the component Im exx displays different dispersions from Im ezz, which reveals the polarization 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 more main peaks, distributed throughout the wider range of energy. For TaAs, there are two main peaks below 2 eV while Na3Bi has just one peak. In fact, at the low energy region, the absorption intensity of TaAs is more than Na3Bi. This absorption behavior can be related to high density of Ta-d states near the Fermi level. On the contrary, above 2 eV, the absorption intensity of Na3Bi is more than TaAs. A comparison of the absorption spectra of both crystals shows that in TaAs, the region of the principal absorption of the Im exx is narrower than Im ezz while they are at the same level in Na3Bi. The component Im exx of Na3Bi has a high peak at 2.3 eV. In considering Re eii of TaAs, Fig. 4 shows the negative values of the x (z) Optical Distinctions Between Weyl Semimetal TaAs and Dirac Semimetal Na3Bi: An Ab Initio Investigation Fig. 8. Imaginary parts of the dielectric tensor to x-polarized incident light for Na3Bi (top) and TaAs (bottom) in RPA and BSE. Fig. 9. Imaginary parts of the dielectric tensor to z-polarized incident light for Na3Bi (top) and TaAs (bottom) in RPA and BSE. component of the real part of the dielectric function from 3 eV up to 7.07 (7.46) eV which corresponds to a high reflectivity region. The component Re ezz of TaAs is negative from 3 eV to 7.43 eV while the component Rezz of Na3Bi has high reflectivity from 2.84 eV to 6.01 eV. In Na3Bi, Re exx is negative in two regions, from 2.44 to 2.63 and between 3 eV and 6.13 eV. Moreover, in the low energy region, the component Re exx of TaAs decreases faster than Re exx of Na3Bi. In the negative region of Re exx, the oscillator strength of TaAs is more than Na3Bi. Below 3 eV, Re ezz of TaAs (Na3Bi) has three peaks at 0.61 (1.15), 1.47 (1.70) and 2.54 (2.51) eV while Re exx has a main peak at 0.64(1.11) eV in addition to the high reflectivity region around 2(2.5) eV. As can be seen in Figs. 4 and 5, the result of static dielectric constants for Na3Bi are smaller than corresponding ones in TaAs. The static dielectric constants along the x- and z-axis are 13.53 (5.99) and 11.42 (5.73) for TaAs (Na3Bi), respectively. To study the collective excitations of TaAs and Na3Bi, we have shown their energy loss functions in Figs. 6 and 7. The energy loss function [L(x) = Im (1/e)] which shows the energy loss of a fast electron moving across a medium, is a complicated mixture of inter-band transitions and plasmons. Inter-band transition peaks are related to the peaks of Im eii while Plasmon peaks correspond to zeros of the Re eii. The plasmon peak of the loss function is large if Re eii is zero and Im eii is small. For Na3Bi (TaAs), there is a main peak around 6 (7) eV which corresponds to the collective excitations (plasmons). Weak peaks below the plasmon peak can be assigned to inter-band transitions in accordance with peaks in Im eii. In TaAs, there are two weak peaks at the shoulder of the plasmon peak, which are absent in Na3Bi and related to the large intensity of states of TaAs near the Fermi level. As mentioned before, recent studies have confirmed the existence of bound excitons in metallic systems.24–26 Therefore, by exploring the excitonic effects, we have calculated optical properties of TaAs and Na3Bi by solving the full BSE for the e–h two-particle Green’s function Eq. 3. The components of the imaginary part of the dielectric function, including excitonic features, have been depicted in Figs. 8 and 9. Regarding to electron– hole interactions, we observe a red shift of the optical absorption feature. The red shift of the absorption spectra is more strongly pronounced for Na3Bi compared to TaAs. This is the general excitonic effect on the dielectric function which has been previously reported for graphene.26 In TaAs, e–h interactions change the intensity of absorption and create an excitonic peak at a high Dadsetani and Ebrahimian Fig. 10. Imaginary parts of v(2) ijk (x) for TaAs. Fig. 11. Imaginary and real parts of v(2) 311(x) for TaAs and its 2x intraband contributions. Fig. 12. Calculated x/2x inter-band and x intra-band contributions of v(2) 311(x) for TaAs. energy region. This peak cannot be seen in absorption spectra of Na3Bi. The excitonic features in the dielectric function of TaAs show that the oscillator strength of the first optical transition of the x- and z-component increases while the second one decreases. In both x- and z-components, a weak peak rises up at 9.5 eV in the region where singleparticle oscillator strengths were vanishing. To analyze this more accurately, we have shown loss function including e–h interaction in Figs. 6 and 7. It is remarkable that the excitonic effects in the high energy range of the spectrum are stronger than in the lower one, in contrast to ordinary semiconductors or insulators where excitonic effects mainly affect the lowest energy part of optical spectra. With respect to RPA, e–h effects decrease the main plasmon peak and shift it towards lower energies. The x-component of loss function displays two excitonic peaks at 9 eV and 9.5 eV while the z-component shows an excitonic peak at 9.5 eV with lower intensity. These peaks correspond to excitonic peaks in the imaginary part of the dielectric tensor around 9.5 eV. We have used different levels of approximation in Eq. 2 to understand better the origin of relevant excitonic states. Once all contributions of Eq. 2 are included in the calculations, excitonic peaks appear at high energy. Taking diagonal and direct terms into account results in elimination of the excitonic peaks. Therefore, our calculations show that these excitonic effects arise mainly from the repulsive exchange interaction term in the electron–hole kernel, with the attractive direct term playing a negligible role. Recent theoretical investigation19 of nonlinear optical responses of Weyl fermions in two dimensions has encouraged us to explore the possible giant nonlinear optical responses of Weyl fermions in three dimensions. Second-order nonlinear optical interactions can occur only in non-centrosymmetric crystals. Therefore, optical second harmonic generation is allowed in TaAs, while Na3Bi cannot support nonlinear effects of even order. SHG can yield additional information about the structure of TaAs. In order to evaluate the nonlinear optical response of TaAs, we have calculated its non-linear The second-harmonic susceptibilities v(2) ijk(x). ð2Þ response vijk ðxÞ involves a 2x resonance in addition to the normal x resonance which can be separated into inter-band and intra-band contributions. Therefore the analysis of nonlinear spectra is a demanding job. For TaAs, which is a body-centeredtetragonal crystal with space group I41md, there are four second-order nonlinear susceptibilities: v(2) 131(x), (2) (2) (x), v (x), v (x). The imaginary parts of the v(2) 113 311 333 susceptibilities are presented in Fig. 10. These results, calculated within independent particle approximation, show that the Weyl semimetal TaAs has high values of nonlinear response which make it promising for SHG in THz region. We find that ð 2Þ below 0.38 eV,v311 ðxÞ is a dominant component with (2) a peak at 0.1 eV where both v(2) 113(x) and v131(x) have a peak with lower intensity. Up to 1.43 eV, v(2) 113(x) and v(2) 131(x) have exactly the same intensity while Optical Distinctions Between Weyl Semimetal TaAs and Dirac Semimetal Na3Bi: An Ab Initio Investigation they are dominant responses with considerable intensities between 0.38 and 1.43 eV. As shown in the inset of Fig. 10, from 1.43 eV up to 3 eV, all components have different dispersion and appreciable intensities. Therefore, there are giant nonlinear responses in the low energy region (terahertz frequency region). The dispersion of real and imagð2Þ inary parts of v311 ðxÞ and its corresponding intraand inter-band contributions of x and 2x resonance have been depicted in Figs. 11 and 12. Comparing ð 2Þ v311 ðxÞ with its intra-band part in Fig. 11 shows ð 2Þ that the dominant peak of v311 ðxÞ comes from the 2x intra-band contribution. Figure 12 shows that x intra- and inter-band have nearly the same intensity values but are opposite in signs. Therefore, their overall intensity is negligible. The 2x interband contribution has intensity fluctuation and decreases rapidly. We have found the same features for other components. Therefore, the dominant peak of the nonlinear response of TaAs comes from the 2x intra-band contribution. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. SUMMARY AND CONCLUSIONS In this work, based on density-functional theory, we have investigated and compared the optical properties of TaAs and Na3Bi and attempted to find unique characteristics that distinguish a Weyl semimetal from a Dirac semimetal. Using different levels of approximation in the BSE, we have studied the excitonic effects on the optical absorption of Na3Bi and TaAs. The calculated optical spectra of TaAs show excitonic effects at a high energy. We have found excitonic peaks at 9 eV and 9.5 eV in the optical absorption of TaAs. In both crystals, the dielectric function is overall red shifted compared with that of RPA approximation. Our calculations show that these excitonic effects arise mainly from the repulsive exchange interaction term in the electron–hole kernel, with the attractive direct term playing a negligible role. At low energy, the absorption intensity of TaAs is more than Na3Bi, related to the 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 excitations (plasmons). The static dielectric constants along the x- and z-axis are 13.53 (5.99) and 11.42 (5.73) for TaAs (Na3Bi), respectively. The secondharmonic responses v(2) ijk(x) have been calculated for TaAs, within independent particle approximation. These results show that the Weyl semimetal TaAs has high values of nonlinear response which make it promising for SHG in the THz region. The dominant peak of the nonlinear response of TaAs comes from a 2x intra-band contribution. REFERENCES 1. D. Ciudad, Nat. Mater. 14, 863 (2015). 2. Z.K. Liu, J. Jiang, B. Zhou, Z.J. Wang, Y. Zhang, H.M. Weng, D. Prabhakaran, S.-K. Mo, H. Peng, P. Dudin, T. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. Kim, M. Hoesch, Z. Fang, X. Dai, Z.X. Shen, D.L. Feng, Z. Hussain, and Y.L. Chen, Nat. Mater. 13, 677 (2014). M. Neupane, S.Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. Liu, I. Belopolski, T.R. Chang, H.T. Jeng, H. Lin, A. Bansil, F. Chou, and M.Z. Hasan, Nat. Commun. 5, 3786 (2014). S. Jeon, B.B. Zhou, A. Gyenis, B.E. Feldman, I. Kimchi, A.C. Potter, Q.D. Gibson, R.J. Cava, A. Vishwanath, and A. Yazdani, Nat. Mater. 13, 851 (2014). L.X. Yang, Z.K. Liu, Y. Sun, H. Peng, H.F. Yang, T. Zhang, B. Zhou, Y. Zhang, Y.F. Guo, M. Rahn, D. Prabhakaran, Z. Hussain, S.-K. Mo, C. Felser, B. Yanand, and Y.L. Chen, Nat. Phys. 11, 728 (2015). L. Balents, Physics 4, 36 (2011). G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Phys. Rev. Lett. 107, 186806 (2011). A.M. Turner and A. Vishwanath, Contemp. Concept Condens. Matter Sci. 6, 293 (2013). O. Vafek and A. Vishwanath, Annu. Rev. Condens. Matter Phys. 5, 83 (2014). H. Weyl, Z. Phys. 56, 330 (1929). G.T. Volovik, JETP Lett. 75, 55 (2002). S.M. Young, S. Zaheer, J.C.Y. Teo, C.L. Kane, E.J. Mele, and A.M. Rappe, Phys. Rev. Lett. 108, 140405 (2012). Z.J. Wang, Y. Sun, X.Q. Chen, C. Franchini, G. Xu, H.M. Weng, X. Dai, and Z. Fang, Phys. Rev. B 85, 195320 (2012). Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Phys. Rev. B 88, 125427 (2013). X.G. Wan, A.M. Turner, A. Vishwanath, and S.Y. Savrasov, Phys. Rev. B 83, 205101 (2011). M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010). P. Hosur, Phys. Rev. B 86, 195102 (2012). A.C. Potter, I. Kimchi, and A. Vishwanath, Nat. Commun. 5, 5161 (2014). T. Morimoto and N. Nagaosa, Phys. Rev. B 93, 125125 (2016). doi:10.1103/PhysRevB.93.125125. S.D. Ganichev and W. Prettl, Intense Terahertz Excitation of Semiconductors (Oxford: Oxford Univ. Press, 2006). M.C. Hoffmann, N.C. Brandt, H.Y. Hwang, K.-L. Yeh, and K.A. Nelson, Appl. Phys. Lett. 95, 231105 (2009). D. Turchinovich, J.M. Hvam, and M.C. Hoffmann, Phys. Rev. B 85, 201304 (2012). M. Cornet, J. Degert, E. Abraham, and E. Freysz, J. Opt. Soc. Am. B 31, 1648 (2014). C.D. Spataru, S. Ismail-Beigi, L.X. Benedict, and G. Steven, Louie. Phys. Rev. Lett. 92, 077402 (2004). J. Deslippe, C.D. Spataru, D. Prendergast, and S.G. Louie, Nano Lett. 7, 1626 (2007). L. Yang, J. Deslippe, C.H. Park, M.L. Cohen, and S.G. Louie, Phys. Rev. Lett. 103, 186802 (2009). M. Dvorak and Z. Wu, Phys. Rev. B 92, 035422 (2015). F. Wang, D.J. Cho, B. Kessler, J. Deslippe, P.J. Schuck, S.G. Louie, A. Zettl, T.F. Heinz, and Y.R. Shen, Phys. Rev. Lett. 99, 227401 (2007). A.K. Geim and K.S. Novoselov, Nat. Mat. 6, 183 (2007). A.H.C. Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, Rev. Mod. Phys. 81, 109 (2009). S.-Y. Xu, C. Liu, S.K. Kushwaha, R. Sankar, J.W. Krizan, I. Belopolski, M. Neupane, G. Bian, N. Alidoust, T.-R. Chang, H.-T. Jeng, C.-Y. Huang, W.-F. Tsai, H. Lin, F. Chou, P.P. Shibayev, R.J. Cava, and M.Z. Hasan, Science 347, 294 (2015). Z.K. Liu, B. Zhou, Y. Zhang, Z.J. Wang, H.M. Weng, D. Prabhakaran, S.K. Mo, Z.X. Shen, Z. Fang, X. Dai, Z. Hussain, and Y.L. Chen, Science 343, 864 (2014). S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B.K. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia, A. Bansil, H. Lin, and M.Z. Hasan, Nat. Commun. 6, 7373 (2015). H. Weng, C. Fang, Z. Fang, B.A. Bernevig, and X. Dai, Phys. Rev. X 5, 011029 (2015). B.Q. Lv, H.M. Weng, B.B. Fu, X.P. Wang, H. Miao, J. Ma, P. Richard, X.C. Huang, L.X. Zhao, G.F. Chen, Z. Fang, X. Dadsetani and Ebrahimian 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. Dai, and T. Qian, and H. Ding Phys. Rev. X 5, 031013 (2015). P.E.C. Ashby and J.P. Carbotte, Phys. Rev. B 89, 245121 (2014). R.Y. Chen, S.J. Zhang, J.A. Schneeloch, C. Zhang, Q. Li, G.D. Gu, and N.L. Wang, Phys. Rev. B 92, 075107 (2015). T. Timusk, J.P. Carbotte, C.C. Homes, D.N. Basov, and S.G. Sharapov, Phys. Rev. B 87, 235121 (2013). P. Hosur, S.A. Parameswaran, and A. Vishwanath, Phys. Rev. Lett. 108, 046602 (2012). B. Xu, Y.M. Dai, L.X. Zhao, K. Wang, R. Yang, W. Zhang, J.Y. Liu, H. Xiao, G.F. Chen, A.J. Taylor, D.A. Yarotski, R.P. Prasankumar, and X.G. Qiu, Phys. Rev. B 93, 121110 (2016). doi:10.1103/PhysRevB.93.121110. G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002). M. Rohlfing and S.G. Louie, Phys. Rev. Lett. 81, 2312 (1998). A. Gulans, S. Kontur, C. Meisenbichler, D. Nabok, P. Pavone, S. Rigamonti, S. Sagmeister, U. Werner, and C. Draxl, J. Phys.: Condens. Matter 26, 363202 (2014). J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). P. Puschnig and C. Ambrosch-Draxl, Phys. Rev. B 66, 165105 (2002). M. Rohlfing and S.G. Louie, Phys. Rev. B 62, 4927 (2000). J.E. Sipe and E. Ghahramani, Phys. Rev. B. 48, 11705 (1993). C. Aversa and J.E. Sipe, Phys. Rev. B. 52, 14636 (1995). 49. S. Sharma, J.K. Dewhurst, and C. Ambrosch-Draxl, Phys. Rev. B. 67, 165332 (2003). 50. S. Sharma and C. Ambrosch-Draxl, Phys. Scr. T. 109, 128 (2004). 51. A.H. Reshak (Ph.D. thesis, Indian Institute of TechnologyRookee, India, 2005). 52. A.H. Reshak, J. Chem. Phys. 125, 014708 (2006). 53. A.H. Reshak, J. Chem. Phys. 124, 014707 (2006). 54. S.N. Rashkeev and W.R.L. Lambrecht, Phys. Rev. B. 63, 165212 (2001). 55. S.N. Rashkeev, W.R.L. Lambrecht, and B. Segall, Phys. Rev. B 57, 3905 (1998). 56. C. Ambrosch-Draxl and J. Sofo, Comput. Phys. Commun. 175, 1 (2006). 57. K. Dewhurst, S. Sharma, L. Nordstrom, F. Cricchio, F. Bultmark, H. Gross, C. Ambrosch Draxl, C. Persson, C. Brouder, R. Armiento, A. Chizmeshya, P. Anderson, I. Nekrasov, F. Wagner, F. Kalarasse, J. Spitaler, S. Pittalis, N. Lathiotakis, T. Burnus, S. Sagmeister, C. Meisenbichler, S. Lebegue, Y. Zhang, F. Kormann, A. Baranov, A. Kozhevnikov, S. Suehara, F. Essenberger, A. Sanna, T. McQueen, T. Baldsiefen, M. Blaber, A. Filanovich, and T. Bjorkman, Elk FP-LAPW code http://elk.sourceforge.net/. Accessed 22 Jan 2016. 58. S. Furuseth, K. Selte, and A. Kjekshus, Acta Chem. Scand. 19, 95 (1965). 59. T.B. Massalski, Binary Alloy Phase Diagrams (Materials Park: ASM, 1990).