Download Optical Distinctions Between Weyl Semimetal TaAs and Dirac

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).