of 7
ARTICLE
A continuous metal-insulator transition driven by
spin correlations
Yejun Feng
1,2
, Yishu Wang
2,3
, D. M. Silevitch
2
, S. E. Cooper
1
, D. Mandrus
4,5
, Patrick A. Lee
2,6
&
T. F. Rosenbaum
2
While Mott insulators induced by Coulomb interactions are a well-recognized class of metal-
insulator transitions, insulators purely driven by spin correlations are much less common, as
the reduced energy scale often invites competition from other degrees of freedom. Here, we
demonstrate a clean example of a spin-correlation-driven metal-insulator transition in the all-
in-all-out pyrochlore antiferromagnet Cd
2
Os
2
O
7
, where the lattice symmetry is preserved by
the antiferromagnetism. After the antisymmetric linear magnetoresistance from conductive,
ferromagnetic domain walls is removed experimentally, the bulk Hall coef
fi
cient reveals four
Fermi surfaces of both electron and hole types, sequentially departing the Fermi level with
decreasing temperature below the Néel temperature,
T
N
=
227 K. In Cd
2
Os
2
O
7
, the charge
gap of a continuous metal-insulator transition opens only at
T
~10K<<
T
N
. The insulating
mechanism parallels the Slater picture, but without a folded Brillouin zone, and contrasts
sharply with Mott insulators and spin density waves, where the electronic gap opens above
and at
T
N
, respectively.
https://doi.org/10.1038/s41467-021-23039-6
OPEN
1
Okinawa Institute of Science and Technology Graduate University, Onna Okinawa 904-0495, Japan.
2
Division of Physics, Mathematics, and Astronomy,
California Institute of Technology, Pasadena, CA 91125, USA.
3
The Institute for Quantum Matter and Department of Physics and Astronomy, The Johns
Hopkins University, Baltimore, MD 21218, USA.
4
Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA.
5
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA.
6
Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02138, USA.
email:
yejun@oist.jp
;
tfr@caltech.edu
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1
1234567890():,;
U
nlike insulators or semiconductors derived from simple
metals such as sodium and lithium
1
,
2
, metal-insulator
transitions in correlated-electron systems reside outside
the paradigm of the single-electron band structure. Factors such
as reduced dimensionality and randomness
3
,
4
enrich the
description of the critical behavior, with possible separation of
spin and charge, and deep connections to exotic states such as
high
T
c
superconductivity and quantum spin liquids
5
. In the limit
of large electronic correlations, the starting point for discussion is
usually the opening of the Hubbard gap
6
, with antiferromagnetic
order being a subsidiary effect.
There exist several major experimental challenges in estab-
lishing a convincing example of a metal-insulator transition dri-
ven by spin correlations
7
. First, the spin-correlation energy is
typically much smaller than the direct Coulomb interaction, often
at a scale comparable to that of the structural modi
fi
cations
induced by the antiferromagnetic order through magnetostrictive
effects and symmetry changes, leading to a chicken-and-egg
conundrum between ascribing the insulating transition to the
lattice and the magnetism
8
,
9
. It is thus preferable to search for
candidate systems in which the magnetism would preserve the
crystalline symmetry. Materials that demonstrate an all-in
all-
out (AIAO) antiferromagnetic order on a pyrochlore lattice, with
all four spins on the corner of a tetrahedron pointing either
towards or away from the center (Fig.
1
a, b), meet this criterion.
The AIAO order induces no symmetry-breaking magnetostric-
tion and causes an isotropic expansion of the cubic unit cell by a
minimal
Δ
a
/
a
~10
4
, only becoming experimentally resolvable
when
T
N
drops below 40 K
10
,
11
. For AIAO order with
T
N
> 100 K,
this overall magnetostriction is fully camou
fl
aged by the thermal
lattice contraction
10
, which broadens the bandwidths but intro-
duces no band splitting. The family of AIAO antiferromagnets
thus provides a highly desirable model system to investigate spin
correlations with little lattice interference.
Identifying an AIAO system with a metal-insulator transition
represents the next experimental challenge. Accompanying the
AIAO order, many 5
d
oxides, such as
R
2
Ir
2
O
7
(
R
=
Eu, Sm, and
Nd) and Cd
2
Os
2
O
7
(ref.
12
and references in Refs.
11
,
13
,
14
), also
demonstrate a change of temperature dependence in the resis-
tivity at
T
~
T
N
9
,
13
,
15
,
16
. However, the resistive behavior of
R
2
Ir
2
O
7
(
R
=
Nd, Sm, Eu), especially in the paramagnetic phase,
is often inconsistent and raises concerns about their intrinsic,
disorder-free behavior
13
,
17
,
18
.Cd
2
Os
2
O
7
has presented consistent
behavior in both the electron correlation and metal-insulator
transition. Samples from several groups
9
,
15
,
16
always demonstrate
magnetic ordering at
T
N
=
225
227 K, metallic behavior above
T
N
, and a three-to-four-decade rise of the resistivity for
T
<
T
N
in
the best samples. This repeatability from crystal to crystal likely
indicates a low level of disorder because of the 2
+
/5
+
valence
condition of Cd and Os ions as well as the chemical transport
growth procedure at low temperature
9
,
15
,
16
.
There exists an additional challenge arising from complications
in modeling and understanding the transport data in many 5
d
AIAO antiferromagnets due to the intrinsically conductive and
highly coercive ferromagnetic domain walls
9
,
19
21
. Ferromagnetic
domain walls introduce a Zeeman shift in the metallic para-
magnetic band structure, but they are not expected to gap the
Fermi surface like the antiferromagnetic bulk, and they are
expected to remain metallic down to
T
=
0. As was pointed out
recently
21
, the highly coercive metallic ferromagnetic domain
walls generate antisymmetric linear magnetoresistance (MR) of
the same functional form as the Hall resistance. Moreover, the
antisymmetric linear MR is detectable in Hall channels due to the
distorted current paths through the domain walls
21
. This effect is
likely the root cause of the widely varying Hall coef
fi
cient
reported in the literature for Cd
2
Os
2
O
7
9
,
16
, as the standard
procedure for extracting Hall resistance through anti-
symmetrization with respect to magnetic
fi
eld direction leads to
erroneous results.
Here we present high-
fi
delity resistivity and Hall coef
fi
cient
measurements on single-crystal Cd
2
Os
2
O
7
after employing an
intricate procedure to eliminate the in
fl
uence of conductive fer-
romagnetic domain walls. Unlike the common understanding of a
concurrent metal-insulator transition with the AIAO magnetic
transition at
T
N
, our results reveal that Cd
2
Os
2
O
7
is metallic for a
broad temperature range below
T
N
, despite an increasing resistivity
with decreasing
T
. It only becomes an insulator at
T
MIT
~10 K,
when four sets of Fermi surfaces have sequentially left the Fermi
level to open a true electronic gap. This large separation in tem-
perature for spin order (
T
N
) and the charge gap (
T
MIT
), with
T
N
>>
T
MIT
, unambiguously establishes spin ordering as the driving
force in Cd
2
Os
2
O
7
s metal-insulator transition. Our methodology
in separating the Hall behavior of the bulk from the in
fl
uences of
the domain walls should provide a generic approach to parsing
spin and charge effects in correlated antiferromagnetic insulators
with metallic domain walls.
Results
Insulating behavior at zero
fi
eld
. Our single-crystal Cd
2
Os
2
O
7
samples demonstrate a monotonic rise of resistivity over three
decades (~3000×) from
T
N
to 1.8 K at zero
fi
eld (Fig.
1
c), con-
sistent with the best samples reported in the literature
9
,
15
,
16
.
Instead of using bar-shaped samples for both resistivity and Hall
measurements, our key approach is to utilize the van der Pauw
(vdP) con
fi
guration of electrical lead placement on plate-shaped
samples (ref.
22
, schematics in Figs.
1
c,
2
c). This choice of putting
four leads on an equal footing aims to comprehensively evaluate
the effects of the electrical current paths, which constantly
redistribute between the conductive domain walls and an
increasingly insulating bulk as the temperature decreases. As
demonstrated by our samples, the vdP ratio, de
fi
ned as
R
vdP2
/
R
vdP1
at zero
fi
eld, stays constant only in the paramagnetic phase
above
T
N
. Below
T
N
, the vdP ratios of two samples have a strong
temperature dependence (Fig.
1
c, inset) but still evolve con-
tinuously as expected from the lack of lattice symmetry breaking
in Cd
2
Os
2
O
7
. The resistivity in Fig.
1
c is calculated from
R
vdP2
and
R
vdP1
at each
T
according to the standard vdP procedure
22
,
an issue we will revisit below.
Extracting the bulk Hall coef
fi
cient
. The highly coercive,
fi
eld-
independent magnetization
M
of the conductive ferromagnetic
domain walls necessarily introduces antisymmetric linear MR
21
that makes the Hall resistances of the two reciprocal channels,
R
12,34
(
H
,
M
) and
R
43,12
(
H
,
M
), have different linear slopes with
H
(Fig.
2
c). Due to the cubic symmetry of AIAO order, it is not
possible to create a single antiferromagnetic domain and remove
the domain walls altogether by
fi
eld cooling. The separation of
galvanomagnetic responses of the bulk domains and the domain
walls is instead carried out by introducing a variable
M
(
φ
)
through
fi
eld-cooling along 24 angular directions
φ
within the
sample surface plane (Fig.
2
a, b, Methods, and ref.
21
). Here we
fi
rst examine the
φ
-dependences of the Hall resistivity slopes from
two reciprocal channels
R
Hall1
H
φ

and
R
Hall2
H
φ

, the vdP ratio,
and resistivity
ρ
(
H
=
0), at temperatures 195, 30, and 1.8 K
(Fig.
2
d
f), where the ferromagnetic domain walls have different
levels of in
fl
uence as gauged by the conductance and the bulk Hall
coef
fi
cient.
At all three temperatures (Fig.
2
d
f), the average slopes of the
two Hall resistivities,
R
H
ð
φ
Þ¼½
R
Hall1
H
φ

þ
R
Hall2
H
φ


=
2, follow a
constraint that (1)
R
H
ð
φ
Þ
are always
φ
-independent. On the other
hand, the resultant
φ
-dependent Hall resistivity slopes of the
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reciprocal channels,
Δ
R
Hall1
;
2
H
φ

¼
R
Hall1
;
2
H
φ


R
H
ð
φ
Þ
, behave
differently. At 195 and 30 K,
Δ
R
Hall1
H
φ

and
Δ
R
Hall2
H
φ

follow a
second constraint that (2) they are identical at
φ
and
φ
+
π
respectively
Δ
R
Hall1
H
φ

¼
Δ
R
Hall2
H
φ
þ
π

, leading to identical
φ
-
averaged resistivity slopes of both Hall channels. From the
previous analysis
21
, constraint (1) re
fl
ects the voltage
current
reciprocity, while constraint (2) manifests Onsager
s reciprocity
relation with regard to the ferromagnetic domain wall
M
. At 1.8
K, the
φ
-dependent Hall resistivity slopes
Δ
R
Hall1
H
φ

and
Δ
R
Hall2
H
φ

(Fig.
2
f) behave differently. Only constraint (1) is
sustained, while constraint (2) cannot be satis
fi
ed by any choice of
the
φ
-independent components. The result is that the
φ
-averaged
resistivity slopes of individual Hall channels are no longer
necessarily equal to each other and to the
φ
-independent
component
R
H
ð
φ
Þ
.
The breakdown of constraint (2) seemingly implies a
violation of Onsager
s reciprocity relation between
φ
and
φ
+
π
states. However, the established Onsager
s reciprocity in
our system hinges on the inverse relationship between domain
wall magnetizations
M
(
φ
)and
M
(
φ
+
π
) through separate
fi
eld-cooling processes at
φ
and
φ
+
π
. Their inverse relation-
ship is generally robust in that both
ρ
and the vdP ratio at zero
fi
eld demonstrate a
π
-periodicity at all temperatures (Fig.
2
d
f). At 1.8 K, the conduction patterns at
φ
and
φ
+
π
are similar
enough to be reproducibly differentiated from those at
neighboring
φ
positions, judging by
ρ
and the vdP ratio
(Fig.
2
f). Nevertheless, because of the separate cooldowns,
there exist differences beyond an inversion between
M
(
φ
)and
M
(
φ
+
π
). With the increasingly insulating bulk, the electrical
current is more concentrated along a fraction of the domain
walls, as re
fl
ected by the dramatically oscillating vdP ratio
from 3:1 to 1:4 (Fig.
2
f).Whiletheentiresamplevolume
brings better averaged galvan
omagnetic behavior and demon-
strates Onsager reciprocity (Fig.
2
d, e), probing only a small
number of domain walls enhances the relative difference
between
M
(
φ
)and
M
(
φ
+
π
). Even at a
fi
xed
φ
, multiple
fi
eld-
coolings can lead to a signi
fi
cant difference in individual Hall
channels at 1.8 K (Fig.
2
f), despite the consistency at 30 K
(Fig.
2
e). We note that the assumed uniform medium for the
vdP technique
22
is justi
fi
ed
aposteriori
at 195 K and 30 K, as
the calculated
ρ
(
φ
) varies within ±0.1% and ±3%, respectively.
Constraint (2) remains satis
fi
ed (Fig.
2
d, e). At 1.8 K,
ρ
(
φ
)
calculated from the vdP formula
22
varies by ±50% (Fig.
2
f).
Although the uniformity assumption no longer holds at 1.8 K
to legitimize both the vdP-derived resistivity and constraint
(2), our procedure to extract the Hall coef
fi
cient
R
H
through
the average slope of two Hall resistivity channels is protected
by the fundamental principle of voltage
current reciprocity.
With the understanding of how to extract
R
H
through a
φ
-
dependence study at three
fi
xed temperatures, we now explore
the temperature evolution of
R
H
by taking the average Hall
resistivity slope of two reciprocal channels at a
fi
xed
φ
(Fig.
2
g).
The
R
H
(
T
) measured on two different Cd
2
Os
2
O
7
single crystals
(Fig.
3
a) demonstrates a consistent p
icture that represents the
major
fi
nding of this work.
R
H
(
T
) remains stable above
T
N
=
227 K, and all bands only start to evolve at
T
N
, maintaining a
delicate balance between them until the
fi
rst sharp change at
10
-3
10
-2
10
-1
10
0
0
100
200
Cd
2
Os
2
O
7
COO-1
COO-2
T
(K)
(
cm)
T
(K)
(
μ
cm)
vdP ratio
T
(K)
vdP1 vdP2
0
5
10
15
0
100
200
310
330
220
260
c
a
b
Fig. 1 Transport signatures of Cd
2
Os
2
O
7
at zero
fi
eld. a
Schematic of the Os sublattice in the cubic pyrochlore oxide Cd
2
Os
2
O
7
, with all Os atoms forming
a three-dimensional corner-sharing octahedra network.
b
Schematic of AIAO spin arrangement on two adjacent octahedral. Note that this spin
arrangement does not break the underlying lattice symmetry.
c
Resistivity
ρ
(
T
)ofCd
2
Os
2
O
7
measured over two single crystals (COO-1 in red and COO-2
in blue) using the vdP con
fi
guration (Schematics). At each temperature, both vdP channels were measured in order to account for the changing current
path between the insulating bulk and conductive domain walls. (Top inset) The resistance ratio
R
vdP2
/
R
vdP1
between two vdP channels manifests the
changing current path below
T
N
=
227K. (Bottom inset) Details of
ρ
(
T
) near
T
N
, showing both metallic behavior above
T
N
, and also an upturn of
ρ
near but
still above
T
N
, which is often attributed to dynamical spin
fl
uctuation effects without long-range order.
T
N
is determined precisely from the magnetic
susceptibility
χ
(
T
), individual
R
vdP2
(
T
) and
R
vdP1
(
T
) (not shown), and
R
H
(
T
) in Fig.
3
a.
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3
220 K. Despite a rise of nearly three orders of magnitude in
ρ
(
T
)from
T
N
down to 10 K,
R
H
(
T
)remains
fi
nite and oscillates
between positive and negative values. As
ρ
(
T
)re
fl
ects a fast
drop in total carrier density with decreasing temperature
(Fig.
1
c), the oscillating
R
H
(
T
)re
fl
ects the changes in the
bands, with an alternating dominance by either electrons or
holesasthemajorcompositi
on of itinerant carriers. In a
multiband model with both electron and hole types of carriers,
R
H
¼
1
ec
ð
i
2f
h
g
N
i
μ
2
i

j
2f
e
g
N
j
μ
2
j
Þ
=
ð
i
2f
h
g
N
i
μ
i
þ
j
2f
e
g
N
j
μ
j
Þ
2
,
with carrier densities
N
i
;
j
and mobilities
μ
i
;
j
¼
e
τ
i
;
j
=
m
i
;
j
determined by the carrier relaxation times
τ
i
;
j
and masses
m
i
;
j
of individual band
i
(of holes) or
j
(of electrons), respectively
(Eq. 2.78 of ref.
23
with multiple bands). Here, carrier
mobilities are expected to have only a weak temperature
dependence. On the other han
d, when a band moves out from
the Fermi level, the relative contributions of charge carriers
(electron or hole) dramatically change, signaled by a rapid turn
(over a temperature range ~20 K) in the trend followed by
R
H
(
T
) and similar to other band-gap-opening scenarios
exhibited, for example, by c
harge and spin density waves
24
,
25
.
The
fi
rst surface moves out from the Fermi level at ~220 K
(Fig.
3
a, top inset), and with
R
H
(
T
) turning more electron-like,
the departed carriers are of hole
type. As thermal excitation at
fi
nite temperature still populates the departed bands close to
the Fermi level,
R
H
(
T
) remains a smooth function of
T
.Here
Fig. 2 Separating the in
fl
uence of metallic ferromagnetic domain walls. a
,
b
Schematics of the measurement procedure using the two-rotator setup.
Every galvanomagnetic Cd
2
Os
2
O
7
sample (gray plate) is
fi
eld-cooled in two stages:
a
fi
eld
H
is aligned parallel to the sample surface at room temperature
before cooling down through
T
N
, and
b
the sample is rotated below
T
N
to have a
fi
eld perpendicular to its surface for galvanomagnetic measurements. The
in-plane magnetizing direction is de
fi
ned by angle
φ
at 24 discrete positions, set outside the cryostat at room temperature.
c
Typical raw data of Hall
resistances
R
(
H
) between two reciprocal channels, measured for sample COO-2 at a speci
fi
ed temperature and magnetization angle
φ
. The difference in
slopes indicates the in
fl
uence of asymmetric linear magnetoresistance from ferromagnetic domain walls
21
. (inset) vdP con
fi
guration for the two reciprocal
Hall channels.
d
f
φ
-dependence of Hall resistivity slopes
R
H
in two reciprocal channels (red and blue) and their average (fresh green), in addition to the
resistivity
ρ
(green) and vdP ratio (orange), measured in sample COO-2 at three temperatures 195, 30, and 1.8 K. (Methods) (
g
) Hall resistivity slopes
R
H
from two reciprocal channels (red and blue) are plotted alongside their average (fresh green) for two different single-crystal Cd
2
Os
2
O
7
samples COO-1 and
COO-2. Although the resistance slopes of an individual Hall channel are very different for each sample, with occasional crossings at various tempera
tures,
the averages are similar in shape between the two samples and determine the bulk Hall coef
fi
cient
R
H
(
T
). This irregular behavior of resistance slopes of
individual Hall channels highlights potential experimental de
fi
ciencies in previous Hall measurements. All statistical uncertainties are smaller than the
symbols.
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