Supplementary Materials
Approaching the quantum critical point
in a highly-correlat
ed all-in-all-out
antiferromagnet
Yishu Wang
1,2
, T. F. Rosenbaum
1
, D. Prabhakaran
3
, A. T. Boothroyd
3
, Yejun Feng
1,4,*
1
Division of Physics, Mathematics, and Astronomy, California Institute of Technology,
Pasadena, California 91125, USA
2
The Institute for Quantum Matter and Department of Physics and Astronomy, The Johns
Hopkins University, Baltimore, Maryland 21218, USA
3
Department of Physics, University of Ox
ford, Clarendon Laboratory, Oxford, OX1 3PU,
United Kingdom
4
Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-
0495, Japan
Experimental Methods:
Over the past decade, a set of measurement procedures has been developed to detect
AIAO antiferromagnetic order using resonant
x-ray magnetic diffract
ion [12-14, 22, 24].
Following the typical protocols, we directly probe the AIAO order in Sm
2
Ir
2
O
7
under
pressure at beamline 4-ID-D of the Advanced Photon Source [22-23]. The
L
-edge resonance
in Ir compounds is limited to the
L
3
edge (
E
=11.215 keV) due to the presence of strong spin-
orbit coupling. In order to match up to the typical sample mosaic under pressure, x-ray
polarization analysis was provided by a highly
oriented pyrolytic graphite crystal of 0.35
o
FWHM mosaic at the (0, 0, 8) reflection. The analyzer crystal also provides the necessary
uniformity in reflectivity across its surface. From the literature [12-14, 22, 24], diffraction
geometries between horizontal and vertical were extensively compared. As the dominant
polarization of the incident synchrotron-based x-ra
ys lies parallel to the horizontal scattering
plane, it is defined as
π
-polarization. For horizontally
diffracted x-rays, the in-plane
polarization is defined as
π
’-polarization. By comparison,
an out-of-scattering-plane x-ray
polarization would be
σ
nature [22]. For resonant studies at the Ir
L
3
edge using graphite
analyzers, it is more advantageous to use th
e horizontal scattering geometry to minimize the
leakage of anisotropic tensor susceptibility
(ATS) resonance from the polarization-preserving
channel to the polarization switching channel [12, 22, 24].
Using a transmission (Laue) geometry in the horizontal plane, and plate-shaped single
crystal samples of 15-20
μ
m thickness with a surface normal of (1, -1, 0), we could access
reflections such as (0, 0, 6), (1, 1, 1), (2, 2, 0), (0, -2, 4), and (2, 4, 0) within the confined
geometry of a diamond anvil cell. The charge resonance, known as ATS, has an energy
profile with contributions from both 5
d
t
2g
and
e
g
bands across the
L
3
edge (Fig. 1d) [13, 14].
On the other hand, the magnetic resonance, with a contribution only from the 5
d
t
2g
band, has
a different profile mostly confined below the
L
3
edge (Fig. 2). Our transmission geometry
generates a more prominent dip feature at 11.220 keV in comparison to resonance spectra
measured under a reflection geometry [13], most
ly due to a large absorption effect at the Ir
L
3
edge. Nevertheless, no correction was
carried out in our data for the Ir
L
3
edge absorption.
After verifying the continuous evolution of the lattice within the cubic symmetry
(Fig.1a), the least amount of symmetry breaking, if it exists, would be from
Fd-3m
to one of
three maximal non-isomorphic subgroups
F-43m
(no. 216),
F4
1
32
(no. 210) or
Fd-3
(no. 203),
all maintaining the face-centered cubic Bravais la
ttice. The two reflections (0,-2,4) and (0,0,6)
in the
π
-
π
’ channel of our measurements would have
non-vanishing diffraction intensity in
the
F-43m
space group, and reflection (0,-2,4) would have non-vanishing intensity in the
F4
1
32
space group. Since they are both vanishingly small and constant (Fig. 4b), no further
measurement on reflections such as (3,0,0), (5,0
,0), and (2,1,1) are needed in order to be
time-efficient at a synchrotron beam line. Th
ose reflections would become nonvanishing only
when the face-centered Bravais lattice is broken further beyond the maximal non-isomorphic
subgroups
F-43m
,
F4
1
32
, and
Fd-3
. The
Fd-3
space group does not break the selection rules
of the
Fd-3m
space group for Sm
2
Ir
2
O
7
, with the highest site degeneracy at 48. A full
refinement is thus necessary, which is extremely difficult with the limited number of
reflections that are accessible with our high
-pressure diffraction set
up. Nevertheless, the
Fd-3
space group would not break the inversion symmetry. A transition from
Fd-3m
to
F-43m
(no.
216) was previously observed in Cd
2
Os
2
O
7
under pressure [24], in which the scheme of space
group detection was also discussed.
For the structural analysis, especially with regard to the coordinate
x
, we use
PowderCell software to generate the diffractio
n intensities of individual reflections, and
analyze the relative intensities between them. It is also possible to tune the coordinate
parameter
x
of the pyrochlore lattice in the software in order to simulate the diffraction
intensities’ dependence on
x
. After generating the relative ratio between (2,2,0) and (4,4,0)
reflection intensities in Fig. 1c, the slope
dI/dx
was extracted near
x
=0.338 for Sm
2
Ir
2
O
7
.
Fitting Fig. 1b of the experimental data, it is possible to extract
dI/dP
. In turn, the pressure
dependence of
dx/dP
can be calculated, together with
Δ
x
at
P
c
.
While both spin and charge resonances are
observed at forbidde
n lattice diffraction
orders such as (0, 2, 4) and (0, 0, 6), the magnetic resonance signal can be isolated from the
charge resonance at (0, 0, 4n+2) orders with a special azimuthal
ψ
~45
o
relative to the (1, 0, 0)
vector [12-14, 22, 24]. Our measurement sensitiv
ity of the magnetic signal at (0, 0, 6) is
limited by sample imperfections such as disloc
ations and voids. This is demonstrated through
Figs. 2-4 as the residual diffraction intensity both beyond
P
c
at 4K and beyond
T
N
at 6.26
GPa. In Fig. 4a, the constant relative intensity at a 4 10
-7
level is on par with the constant
intensity at a 6 10
-7
level above
T
N
in Fig. 3b. Given the similar shapes of three mosaic
profiles of the (0, 0, 6) reflection at azimuthal
ψ
=137
o
~140
o
for 6.26 GPa in Fig. 2d, the
residual mosaic form is likely due to dislocatio
ns and voids, instead of multiple scattering,
and could be attributed to a small
σ
component in the
π
-polarized synchrotron light, sharing
the same origin as the minimal (0,0,6) diffraction intensity in the
π
-
π
’ channel (Fig. 4b). On
the other hand, for resonance profiles, the residual spectral weight above
T
N
at 11.225 keV is
due to enhanced multiple scattering above the absorption edge.
Our general high-pressure techniques have
been discussed in Refs. [22-24]. A
methanol/ethanol 4:1 mixture was used as the pressure medium. Pressure was calibrated by a
Ag manometer
in situ
at
T
= 4 K using a two-parameter Birch equation of state, with bulk
modulus
B
0
=108.85 GPa and its derivative
B
’=
dB
/
dP
=5.7 over a 20 GPa range. Data
presented here were collected from three Sm
2
Ir
2
O
7
samples, prepared from three different
octahedral shaped single crystals in the same growth batch, and all pressurized across the
AIAO magnetic phase boundary with both sp
in and charge resonances explored.
Supplementary Table I:
A compendium of lattice parameters of
R
2
Ir
2
O
7
plotted in Fig. 4d.
Element
R
: Lattice constant (Å): Pyroch
lore parameter x:
Reference:
Gd 10.2609
0.3490
21
Eu 10.3020
0.3476
21
Eu 10.290
0.330
30
Eu 10.2857
0.336
25
Eu 10.2740
0.339
34
Eu 10.2880
0.331
36
Sm 10.3097
0.3450
21
Sm 10.3063
0.339
37
Sm 10.3110
0.329
36
Nd 10.3768
0.330
25
Nd 10.383
0.3347
35
Pr 10.4105
0.330
25
Pr 10.3960
0.329
34
Pr 10.4159
0.3310
35