of 2
Supplemental Material
High pressure control of optical nonlinearity in the polar Weyl semimetal TaAs
Chen Li,
1, 2
Xiang Li,
1
T. Deshpande,
1
Xinwei Li,
1, 2
N. Nair,
3, 4
J.
G. Analytis,
3, 4
D. M. Silevitch,
1
T. F. Rosenbaum,
1
and D. Hsieh
1, 2,
1
Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
3
Department of Physics, University of California, Berkeley, CA 94720, USA
4
Materials Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA
I. SHG-RA DATA IN PARALLEL POLARIZATION GEOMETRY
FIG. S1: SHG-RA patterns at select pressure in parallel polarization geometry measured upon compression.
The most general form of the electric-dipole SHG-RA intensity in parallel polarized geometry under normal incidence
is given by
I
(
φ
)
∝|
B
1
cos
3
(
φ
) +
B
2
cos
2
(
φ
) sin(
φ
) +
B
3
cos(
φ
) sin
2
(
φ
) +
B
4
sin
3
(
φ
)
|
2
, similar to the case for crossed
polarized geometry described in the main text. From the (112) face of TaAs in the tetragonal 4
mm
phase,
B
2
=
B
4
= 0 by symmetry,
B
1
=
γ
(2
a
2
χ
zxx
+ 4
a
2
χ
xzx
+
c
2
χ
zzz
) and
B
3
=
γ
(4
a
2
+ 2
c
2
)
χ
xzx
+ (2
a
2
+
c
2
)
χ
zxx
, where
γ
= 1
/
(
2
a
3
c
+
ac
2 +
c
2
a
2
)
and
a
and
c
are the TaAs lattice constants.
Figure S1 shows SHG-RA patterns at various pressures measured in parallel polarization geometry. For all pressures,
the patterns are dominated by the
χ
zzz
containing
B
1
term, which was previously shown to be an order of magnitude
larger than both
χ
zxx
and
χ
xzx
[1]. Nonetheless, careful examination of the patterns shows that the nodes at 90
and 270
are lifted at high pressure, evidencing appearance of a
B
4
contribution that is only allowed in the hexagonal
phase.
II: DEPARTURE FROM 4
mm
POINT GROUP AT AMBIENT PRESSURE
For an ideal tetragonal crystal with point group 4
mm
, the
A
1
term in
I
(
φ
) is forbidden by symmetry (see main
text) and the pattern should exhibit nodes at 0
, 90
, 180
and 270
. Yet we observe a small but finite
A
1
component
in our ambient pressure SHG-RA patterns taken outside the DAC , manifested as a lifting of the nodes at 0
and
180
. Our simulation results show that this is not due to misalignment, implying that the crystals may exhibit a
slight departure from 4
mm
symmetry.
2
FIG. S2: Typical cross-polarized SHG-RA pattern from TaAs (112) acquired at ambient pressure outside the DAC (left). A
zoom-in (middle) shows a lifting of the nodes at 0
and 180
, indicating a finite
A
1
term. Fits to the general form for
I
(
φ
)
are overlaid as black lines. A decomposition of the fits into its
A
1
A
4
components is shown to the right. Filled versus empty
lobes represent opposite signs of the associated trigonometric function. The fitted amplitude of the various terms is shown
below, which are normalized to
A
2
for
P < P
c
.
III: PRESSURE DEPENDENCE OF
A
1
AND
A
4
The SHG-RA patterns measured in crossed polarized geometry were fit to
I
(
φ
)
∝|
A
1
cos
3
(
φ
)+
A
2
cos
2
(
φ
) sin(
φ
)+
A
3
cos(
φ
) sin
2
(
φ
) +
A
4
sin
3
(
φ
)
|
2
. The fitted amplitudes of
A
2
and
A
3
are shown in the main text. Figure S3 shows
the pressure dependence of the fitted amplitudes of
A
1
and
A
4
. Subtle kinks appear in
|
A
1
|
and
|
A
4
|
around the
critical pressure
P
c
. However, as expected,
|
A
4
|
does not change as drastically as
|
A
2
|
because
A
4
does not contain
χ
zzz
. Also, as expected,
|
A
1
|
does not exhibit a clear a kink at
P
c
as
|
A
3
|
because
A
1
is present in both the low and
high pressure phases.
FIG. S3: Pressure dependence of the fitted amplitudes of
A
1
(green) and
A
4
(purple) normalized to the maximum value of
A
2
for
P < P
c
Corresponding author.
dhsieh@caltech.edu
[1] Wu, L.
et al.
Giant anisotropic nonlinear optical response in transition metal monopnictide Weyl semimetals.
Nature Physics
13
, 350–355 (2017). URL
https://www.nature.com/articles/nphys3969
. Number: 4 Publisher: Nature Publishing Group.