LETTERS
PUBLISHED ONLINE: 5 DECEMBER 2016 | DOI: 10.1038/NPHYS3969
Giant anisotropic nonlinear optical response in
transition metal monopnictide Weyl semimetals
Liang Wu
1,2
*
, S. Patankar
1,2
, T. Morimoto
1
, N. L. Nair
1
, E. Thewalt
1,2
, A. Little
1,2
, J. G. Analytis
1,2
,
J. E. Moore
1,2
and J. Orenstein
1,2
*
Although Weyl fermions have proven elusive in high-energy
physics, their existence as emergent quasiparticles has been
predicted in certain crystalline solids in which either inversion
or time-reversal symmetry is broken
1–4
. Recently they have
been observed in transition metal monopnictides (TMMPs)
such as TaAs, a class of noncentrosymmetric materials that
heretofore received only limited attention
5–7
. The question that
arises now is whether these materials will exhibit novel, en-
hanced, ortechnologically applicable electronic properties. The
TMMPs are polar metals, a rare subset of inversion-breaking
crystals that would allow spontaneous polarization, were it
not screened by conduction electrons
8–10
. Despite the absence
of spontaneous polarization, polar metals can exhibit other
signatures of inversion-symmetry breaking, most notably
second-order nonlinear optical polarizability, χ
(2)
, leading to
phenomena such as optical rectification and second-harmonic
generation (SHG). Here we report measurements of SHG
that reveal a giant, anisotropic χ
(2)
in the TMMPs TaAs, TaP
and NbAs. With the fundamental and second-harmonic fields
oriented parallel to the polar axis, the value of χ
(2)
is larger by
almost one order of magnitude than its value in the archetypal
electro-optic materials GaAs
11
and ZnTe
12
, and in fact larger
than reported in any crystal to date.
The past decade has witnessed an explosion of research investi-
gating the role of band-structure topolog y, as characterized for ex-
ample by the Berry curvature in momentum space, in the elec tronic
response functions of crystalline solids
13
. While the best established
example is the intrinsic anomalous Hall effect in time-reversal
breaking systems
14
, several nonlocal
15,16
and nonlinear effects related
to Berry curvature generally
17,18
and in Weyl semimetals (WSMs)
specifical ly
19,20
have been predicted in crystals that break inversion
symmetry. Of these, the most relevant to this work is a theoretical
formulation
21
of SHG in terms of the shift vector, which is a quantity
related to the difference in Berry connection between two bands that
participate in an optical transition.
Figure 1a and its caption provide a schematic and description
of the optical set-up for measurement of SHG in TMMP crystals.
Figure 1b,c shows results from a (112) surface of TaAs. The SH
intensity from this surface is very strong, allowing for polarization
rotation scans with signal-to-noise ratio above 10
6
. In contrast, SHG
from a TaAs (001) surface is barely detectable (at least six orders
of magnitude lower than the (112) surface). Below, we describe
the use of the s et-up shown in Fig. 1a to characterize the second-
order optical susceptibility tensor, χ
ijk
, defined by the relation,
P
i
(2ω) =
0
χ
ijk
(2ω)E
j
(ω)E
k
(ω).
As a first step, we determined the orientation of the high-
symmetry axes in the (112) surface, which are the [1,−1,0] and
[1,1,−1] directions. To do so, we simultaneously rotated the linear
polarization of the generating light (the generator) and the polarizer
placed before the detector (the analyser), with their relative angle
set at either 0
◦
or 90
◦
. Rotating the generator and analyser together
produces scans, shown in Fig. 1b,c, which are equivalent to rotation
of the sample about the surface normal. The angles at which we
observe the peak and the null in Fig. 1b and the null in Fig. 1c allow
us to identify the principal axes in the (112) plane of the surface.
Having determined t he high-symmetry directions, we character-
ize χ
ijk
by performing three of types of scans, the results of which are
shown in Fig. 2. In scans shown in Fig. 2a,b, we oriented the analyser
along one of the two high-symmetry directions and rotated the plane
of linear polarization of the generator through 360
◦
. Figure 2c shows
the circular dichroism of the SHG response—that is, the difference
in SH generated by left and right circularly polarized light. For all
three scans the SHG intensity as a function of angle is consistent
with the second-order optical susceptibility tensor expe cted for the
4mm point group of TaAs, as indicated by the high accuracy of the
fits in Fig. 1b,c and Fig. 2a–c.
In the 4mm structure xz and yz are mirror planes but reflection
through the xy plane is not a symmetry; therefore, TaAs is an
acentric crystal w ith an unique polar (z) axis. In crystals with 4mm
symmetry there are three independent nonvanishing elements of
χ
ijk
: χ
zzz
, χ
zxx
= χ
zyy
and χ
xzx
= χ
yzy
= χ
xxz
= χ
yyz
. Note that each has
at least one z component, implying null electric dipole SHG when
all fields are in the xy plane. This is consistent with observation of
nearly zero SHG for light incident on the (001) plane. Below, we
follow the convention of using the 3 × 6 second-rank tensor d
ij
,
rather than χ
ijk
, to express the SHG response, where the relation
between the two tensors for TaAs is: χ
zzz
= 2d
33
, χ
zxx
= 2d
31
and
χ
xzx
= 2d
15
(ref. 22) (see Methods and Supplementary Section A).
Starting with the symmetry-constrained d tensor, we de-
rive expressions, specific to the (112) surface, for the angu-
lar scans with fixed analyser shown in Fig. 2a,b (Methods and
Supplementary Section A). We obtain |d
eff
cos
2
θ
1
+ 3d
31
sin
2
θ
1
|
2
/27
and |d
15
|
2
sin
2
(2θ
1
)/3 for analyser parallel to [1,1,−1] and [1,−1,0],
respectively, where d
eff
≡ d
33
+ 2d
31
+ 4d
15
. Fits to these expres-
sions y ield two ratios: |d
eff
/d
15
| and |d
eff
/d
31
|. Although we do
not determine |d
33
/d
15
| and |d
33
/d
31
| directly, it is clear from the
extreme anisotropy of the angular scans t hat d
33
, which gives the
SHG response when both generator and analyser are parallel to the
polar axis, is much larger than the other two components. We can
place bounds on |d
33
/d
15
| and |d
33
/d
31
| by s etting d
15
and d
31
in
and out of phase with d
33
. We note that the observation of circular
dichroism in SHG, shown in Fig. 2c, indicates that relative phase
between d
15
and d
33
is neither 0
◦
or 180
◦
, but rather closer to 30
◦
(Supplementary Section A).
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
1
Department of Physics, University of California, Berkeley, California 94720, USA.
2
Materials Science Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA.
*
e-mail: liangwu@berkeley.edu; jworenstein@lbl.gov
350 NATURE PHYSICS | VOL 13 | APRIL 2017 | www.nature.com/naturephysics
NATURE PHYSICS DOI: 10.1038/NPHYS3969
LETTERS
2
ω
ω
2
θ
1
θ
TMMPs
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Normalized SHG
Normalized SHG
3002001000
Incident polarization angle (°)
Parallel
Full fit
Full fit
d
33
term only
20 K
|d
e
|
2
/27
b
0
45
90
135
180
225
270
315
0
45
90
135
180
225
270
315
0.30
0.25
0.20
0.15
0.10
0.05
0.00
3002001000
Incident polarization angle (°)
Perpendicular
d
33
term only
|d
31
|
2
/3
ca
Figure 1 | Second-harmonic generation versus angle as TaAs is eectively rotated about the axis perpendicular to the (112) surface. a, Schematic of the
SHG experimental set-up. To stimulate second-harmonic light, pulses of 800 nm wavelength were focused at near-normal incidence to a 10-µm-diameter
spot on the sample. Polarizers and waveplates mounted on rotating stages allowed for continuous and independent control of the polarization of the
generating and second-harmonic light that reached the detector. θ
1
and θ
2
are the angles of the polarization plane after the generator and the analyser,
respectively, with respect to the [1,1,−1] crystal axis. b,c, SHG intensity as a function of angle of incident polarization at 20 K. In b,c, the analyser is,
respectively, parallel and perpendicular to the generator. In both plots the amplitude is normalized by the peak value in b. The solid lines are fits that include
all the allowed tensor elements in 4mm symmetry, whereas the dashed lines include only d
33
. The insets are polar plots of measured SHG intensity versus
incident polarization angle.
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Normalized SHG
3002001000
Incident polarization angle (°)
Fixed analyser 0°
Fit
Fixed analyser 90°
Fit
Incident polarization angle (°)
|d
31
|
2
/3
|d
e
|
2
/27
0
45
90
135
180
225
270
315
0
45
90
135
180
225
270
315
a
0.06
0.04
0.02
0.00
−0.02
−0.04
−0.06
Circular dichroism
3002001000
Analyser angle (°)
+
−
−
σσ
Fit
c
0.04
0.03
0.02
0.01
3002001000
Temperature (K)
|d
15
/d
33
|
|d
31
/d
33
|
d
12
10
8
6
4
2
0
Normalized SHG (× 10
−3
)
SHG coecient ratio
3002001000
|d
15
|
2
/3
b
Figure 2 | Second-harmonic intensity with fixed analysers, circular dichroism and temperature dependence on a TaAs (112) sample. a,b, SHG as function
of generator polarization with analysers fixed along 0
◦
(a) and 90
◦
(b), which are the [1, 1, −1] and [1, −1, 0] crystal axes, respectively. Solid lines are fits
considering all three non-zero tensor elements. The insets are polar plots of measured SHG intensity. c, Circular dichroism (SHG intensity dierence
between right- and left-hand circularly polarized generation light) normalized by the peak value in Fig. 1b versus polarization angle of the analyser at 300 K.
The solid line is a fit. d, Temperature dependence of |d
15
/d
33
| and |d
31
/d
33
| of a TaAs (112) sample. The error bars are determined by setting d
15
and d
31
in
phase and out of phase with respect to d
33
in the fits.
The results of this analysis are plotted in Fig. 2d, where it is shown
that |d
33
/d
15
| falls in the range ∼25–33 for all temperatures, and
|d
33
/d
31
| increases from ∼30 to ∼100 with increasing temperature.
Perhaps because of its polar metal nature, the anisotropy of the
second-order susceptibility in TaAs is exceptionally large compared
with what has been obser ved previously in crystals with the same
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351
LETTERS
NATURE PHYSICS DOI: 10.1038/NPHYS3969
+0.5
+1
+0.5
+1
0
45
90
135
180
225
270
315
0
45
90
135
180
225
270
315
0
45
90
135
180
225
270
315
0
45
90
135
180
225
270
315
Parallel
Perpendicular × 4
TaAs (112) ZnTe(110) × 4
Fit
a
0
+0.5
+1
+0.5
+1
Parallel × 4
Perpendicular × 4
b
+0.5
+1
+0.5
+1
GaAs (111) × 6.6
Parallel × 6.6
Perpendicular × 6.6
c
+0.5
+1
+0.5
+1
TaAs (112)
TaP (112)
NbAs (112)
d
Figure 3 | Benchmark SHG experiments on TaAs (112), TaP (112), NbAs (112), ZnTe (110) and GaAs (111). a–c, SHG polar plots in the same scale for TaAs
(112) (a), ZnTe (110) (b) and GaAs (111) (c) in both the parallel and perpendicular generator/analyser configurations at room temperature. For TaAs, data in
the perpendicular configuration is magnified by a factor of 4 for clarity. The SHG intensity of ZnTe and GaAs are multiplied by a factors of 4 and 6.6,
respectively, to match the peak value of TaAs. d, SHG polar plots for TaAs (112), TaP (112) and NbAs (112) in the parallel configurations at room
temperature, with plots of TaP and NbAs rotated by 60
◦
and 120
◦
for clarity.
set of non-zero d
ij
. For example, α-ZnS, CdS and KNiO
3
have
|d
31
|
∼
=
|d
15
|
∼
=
d
33
/2 (ref. 23), while in BaTiO
3
the relative sizes are
reversed, with |d
31
|
∼
=
|d
15
| ≈ 2|d
33
| (ref. 24).
Even more striking than the extreme anisotropy of χ
ijk
is the
absolute size of the SHG response in TaAs. The search for materials
with large second-harmonic optical susceptibility has been of
continual interest since the early years of nonlinear optics
25
. To
determine the absolute magnitude of the d coefficients in TaAs,
we us ed GaAs and ZnTe as benchmark materials. Both crystals
have large and well-characterized second-order optical response
functions
11,12
, with GaAs regarded as having a SH susceptibility
among the largest of any known crystal. GaAs and ZnTe are also
ideal as benchmarks, because their response tensors have only one
nonvanishing co efficient, d
14
≡ 1/2χ
xyz
.
Figure 3a–c shows polar plots of SHG intensity as TaAs (112),
ZnTe (110), and GaAs (111) are (effectively) rotated about the optic
axis with the generator and analyser set at 0
◦
and 90
◦
. A lso shown
(as solid lines) are fits to the polar patterns obtained by rotating
the χ
(2)
tensor to a set of axes that includes the surface normal,
which is (110) and (111) for our ZnTe and GaAs crystals, respectively
(Methods and Supplementary Section A). Even prior to analysis
to extract the ratio of d coefficients between the various crystals,
it is clear that the SHG response of TaAs (112) is large, as the
peak intensity in this geometry exceeds ZnTe (110) by a factor
of 4.0(±0.1) and GaAs (111) by a factor of 6.6(±0.1). Figure 3d
compares the parallel polarization data for TaAs shown in Fig. 3a
with SHG measured under the same conditions in t he (112) facets
of two other TMMPs: TaP and NbAs. The strength of SHG from
the three crystals, which share the same 4mm point group, is clearly
very similar, with TaP and NbAs intensities relative to TaAs of
0.90(±0.02) and 0.76(±0.04), respectively. The SHG response in
these compounds is also dominated by the d
33
coefficient. Finally,
we found that the SHG intensity of all three compounds does not
decrease after exposure to atmosphere for several months.
352
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To obtain the response of t he TMMPs relative to the two
benchmark materials we used the Bloembergen–Pershan formula
25
to correct for the variation in specular reflection of SH light t hat
results from the small differences in the index of refraction of
the three materials at the fundamental and SH frequency. (See
Methods. Detai ls concerning this correction, which is less than 20%,
can be found in Supplementary Section B.) Table 1 presents the
results of this analysis, showing that |d
33
|
∼
=
3,600 pm V
−1
at the
fundamental wavelength 800 nm in TaAs exceeds the values in the
benchmark materials GaAs
11
and ZnTe
12
by approximately one order
of magnitude, even when measured at wavelengths where their
response is largest. The d coefficient in TaAs at 800 nm exceeds the
corresponding values in the ferroelectric materials BiFeO
3
(ref. 26),
BaTiO
3
(ref. 24) and L iNbO
3
(ref. 23) by two orders of magnitude.
In the case of the ferroelectric materials, SHG measurements have
not been performed in their spect ral regions of strong absorption,
typically 3–7 eV. However, ab initio calculations consistently predict
that t he resonance-enhanced d values in this region do not exceed
roughly 500 pm V
−1
(refs 27,28).
The results described ab ove raise the question of why χ
zzz
in
the TMMPs is s o large. Answering this question quantitatively
will require further work in which measurements of χ
(2)
as a
function of frequency are compared with theory based on ab initio
band structure and wavefunctions. For the present, we describe a
calculation of χ
(2)
using a minimal model of a WSM that is based
on the approach to nonlinear optics proposed by Morimoto and
Nagaosa (MN)
21
. This theor y clarifies the connection between band-
structure topology and SHG, and provides a concise expression with
clear geometrical meaning for χ
(2)
. Hopefully this calculation will
motivate the ab initio theory that is needed to quantitatively account
for the large SH response of the TMMPs and its possible relation to
the existence of Weyl nodes.
The MN result for the dominant (zzz) response function is
Re{σ
(2)
zzz
(ω, 2ω)}
∼
=
πe
3
2
¯
hω
2
Z
d
3
k
(2π)
3
|v
z,12
|
2
R
zz
(k)
×
−δ(
21
−
¯
hω) +
1
2
δ(
21
− 2
¯
hω)
(1)
In equation (1) the nonlinear response is expressed as a second-
order conductivity, σ
zzz
(ω, 2ω), relating the current induced at
2ω to the square of the applied electric field at ω, that is,
J
z
(2ω) = σ
zzz
E
2
z
(ω). (The SH susceptibility is related to the conduc-
tivity through the relation χ
(2)
= σ
(2)
/2iω
0
). The indices 1 and 2
refer to the valence and conduction bands, respectively,
21
is t he
transition energy, and v
i,12
is the matrix element of the velocity
operator v
i
= (1/
¯
h)∂H/∂k
i
. Bandstructure topology appears in the
form of the ‘shift vector,’ R
zz
≡ ∂
k
z
ϕ
z,12
+ a
z,1
− a
z,2
, which is a
gauge-invariant length formed from the k derivative of the phase
of the velocity matrix element, ϕ
12
= Im{log v
12
}, and the difference
in Berry connect ion, a
i
= −ihu
n
|∂
k
i
|u
n
i, between bands 1 and 2.
Physically, the shift vector is the k-resolved shif t of the intracell
wavefunction for the two bands connected by the optical transition.
We consider the following minimal model for a time-reversal
symmetric WSM that supports four Weyl nodes,
H = t
n
[cos k
x
a + m
y
(1 − cosk
y
a) + m
z
(1 − cosk
z
a)]σ
x
+ [sin k
y
a + ∆ cos(k
y
a)s
x
]σ
y
+ sin(k
z
a)s
x
σ
z
o
(2)
Here, σ
i
and s
i
are Pauli matrices acting on the orbital and spin
degrees of freedom, respectively, t is a measure of the bandwidth,
a is the lattice constant, m
y
and m
z
are parameters that intro-
duce anisotropy, and inversion breaking is introduced by ∆. The
Hamiltonian defined in equation (2) preserves two-fold rotation
Table 1 | Second-harmonic generation coecients of dierent
materials at room temperature.
Material |d
ij
| |d| (pm V
−1
) Fundamental
wavelength (nm)
Reference
TaAs d
33
3600
(±550)
800 This work
GaAs d
14
350
∗
810 Ref. 11
ZnTe d
14
250, 450
∗
800, 700 Ref. 12
BaTiO
3
d
33
15 900 Ref. 24
BiFeO
3
d
33
15–19 1550, 800 Refs 26,27
t
LiNbO
3
d
33
26 852 Ref. 23
BiFeO
3
d
33
130
∗
500 Ref. 27
t
BaTiO
3
d
33
100
∗
170 Ref. 28
t
PbTiO
3
d
33
200
∗
150 Ref. 28
t
Second-harmonic optical susceptibility can be calculated by χ
ijk
= 2d
ij
.
∗
denotes the peak
value of the material.
t
denotes theoretical calculation. The uncertainty of d
33
in TaAs is
determined by setting d
15
and d
31
in and out of phase with respect to d
33
in the fit.
symmetry about the z-axis and the mirror symmetries M
x
and M
y
.
These symmetries form a subset of the 4mm point group which is
relevant to the optical properties of TMMPs.
Figure 4 illustrates the energy levels, topological structure, and
SHG spectra that emerge from this mo del. As shown in Fig. 4a,
pairs of Weyl nodes with opposite chirality overlap at two p oints,
k = (±π/2a,0, 0), in the inversion-symmetric case with ∆ = 0.
With increasing ∆ the nodes displace in opposite direct ions along
the k
y
axis, with ∆k
y
∼
=
∆/a. The energy of electronic states in the
k
z
= 0 plane, illustrating the linear dispersion near the four Weyl
points, is shown in Fig. 4b. Figure 4c shows the corresponding
variation of |v
12
|
2
R
zz
(k) for the s
x
= +1 bands whose Weyl points are
located at k
y
< 0 (the variation of |v
12
|
2
R
zz
(k) for the s
x
= −1 bands
is obtained from the transformation k → −k). The magnitude of
σ
(2)
derived from this model vanishes as ∆ → 0, and is also sensitive
to the anisotropy parameters m
y
and m
z
. Figure 4d shows that
spectra corresponding to parameters t = 0.8 eV, ∆ = 0.5, m
z
= 5,
and m
y
= 1 can qualitatively reproduce the observed amplitude and
large anisotropy of χ
(2)
(ω, 2ω).
As discussed above, our minimal model of an inversion-breaking
WSM is intended mainly to motivate further research into the
mechanism for enhanced SHG in the TMMPs. However, the model
does suggest universal properties of χ
(2)
that arise from transitions
near Weyl nodes between bands with nearly linear dispersion.
According to bulk band-structure measurements
7
, such transitions
are exp ected at energies below approximately 100 meV in the TaAs
family, corresponding to the far-infrared and terahertz regimes. In
these regimes, where the interband excitation is within the Weyl
cones, the momentum-averaged |v
12
|
2
R
zz
(k) tends to a non-zero
value, hv
2
Ri, leading to the prediction that σ
(2)
→ g (ω)hv
2
Ri/ω
2
as ω → 0. Because g (ω), the joint density of states for Weyl
fermions, is proportional to ω
2
, we predict that σ
(2)
approaches
a constant (or alternatively χ
(2)
diverges as 1/ω) as ω → 0, even
as the linear optical conductivity vanishes in proportion to ω
(ref. 29). The 1/ω scaling of SHG and optical rectification is a
unique signature of a WSM in low-energy electrodynamics, as it
requires the existence of both inversion breaking and point nodes.
In real materials, this divergence will be cut off by disorder and
non-zero Fermi energy. Disorder-induced broadening, estimated
from t ransport scattering rates
30
, and Pauli blocking from non-
zero Fermi energy, estimated f rom optical conductivity
30
and band
calculation
3
, each sug gest a low-energy cutoff in the range of a
few meV.
We conclude by observing that the search for inversion-breaking
WSMs has led, fortuitously, to a new class of polar metals with
unusually large second-order optical susceptibility. Although WSMs
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353
LETTERS
NATURE PHYSICS DOI: 10.1038/NPHYS3969
k
y
a/π
k
y
a/π
k
x
a/π
k
x
a/π
+
+
−
−
ab
c
Photon energy (eV)
(×10
3
pm V
−1
)
χ
|
zzz
|
χ
|
xzx
|
χ
d
−1.0
−1.0
−0.5
0.0
0.5
1.0
k
y
a/π
−1.0
−3
−2
−1
0
1
2
3
−0.5
0.0
0.5
1.0
−0.5 0.0 0.5 1.0
k
x
a/π
−1.0 −0.5 0.0 0.5 0.5
0.5
0.0
1.0
1.0
1.5
1.5
2.0
2.0
2.5
2.5
3.0
3.0
1.0
Energy (eV)
−1
1
0
1
−1
0
−2
0
2
Δ
Figure 4 | Numerical results for the second-harmonic response of a WSM. a, Location of Weyl points in the k
z
= 0 plane for ∆ = 0.5. For ∆ = 0,
Weyl points with opposite chiralities are located at (±π/2a, 0,0) (denoted by black dots). b, The band structure for ∆ = 0.5 and k
z
= 0. c, Colour plot of
|v
12
|
2
R
zz
, which appears as an integrand in the formula for σ
(2)
zzz
. For clarity we plot only |v
12
|
2
R
zz
for s
x
= +1 bands with Weyl points located at k
y
< 0.
|v
12
|
2
R
zz
for s
x
= −1 bands is obtained by setting k → −k. |v
12
|
2
R
zz
shows structures at Weyl points. d, |χ
zzz
| and |χ
xzx
| plotted as a function of incident
photon energy for ∆ = 0.5. We adopted parameters t = 0.8 eV, m
y
= 1,m
z
= 5.
are not optimal for frequency-doubling applications in the visible
regime because of their strong absorption, they are promising
materials for terahertz generation and optoelectronic devices such
as far-infrared detec tors because of their unique scaling in the
ω → 0 limit. Looking forward, we hope that our findings will
stimulate further investigation of nonlinear optical spectra in
inversion-breaking WSMs for technological applications and in
order to identify the defining response functions of Weyl fermions
in crystals.
Methods
Methods, including statements of data availability and any
associated accession co des and references, are available in the
online version of this paper.
Received 13 June 2016; accepted 1 November 2016;
published online 5 December 2016
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