Typeset with jpsj2.cls <ver.1.2.2b>
Full Paper
Frustrated magnetism and cooperative phase transitions in spinels
S.-H. Lee1,2 ∗ , H. Takagi3 , D. Louca1,2 , M. Matsuda4, S. Ji1 , H. Ueda5 , Y. Ueda5 , T.
Katsufuji6 , J.-H. Chung7 , S. Park8 , S-W. Cheong9 , C. Broholm10
1
2
Advanced Institute for Materials Research, Tohoku University, Katahira, Sendai 980-8577, Japan
3
4
Department of Physics, University of Virginia, Charlottesville, VA 22904, USA
Department of Advanced Materials, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan
Quantum Beam Science Directorate, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan
5
The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan
6
Department of Physics, Waseda University, Tokyo 169-8050, Japan
7
8
9
10
Department of Physics, Korea University, Seoul 136-701, Korea
HANARO Center, Korea Atomic Energy Research Institute, Daejeon, Korea
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218. USA
The spinel crystal system, AB2 O4 , has been fertile ground in studying the effects of magnetic and orbital frustration. The experimental findings, with primary focus on neutron and
synchrotron x-ray scattering techniques, are hereby reviewed in spinels with magnetic B
ions. Highlighted are novel collective phenomena, such as the zero-energy excitation mode
in the spin liquid phase, zero-field and field-induced novel phase transitions, the emergence
of complex local spin entities, and heavy fermionic behaviors. Such a diversity in the exotic
properties stems from their close proximity to critical points among degenerate states and a
delicate balance among different degrees of freedom such as spin, orbital, and lattice.
KEYWORDS: Frustration, spinels
1. Introduction
Exotic magnetic states have been greatly sought after in strongly correlated electron systems ever since a resonating singlet state was proposed by Anderson in 1972 as a possible
ground state for a two dimensional triangular antiferromagnet.1, 2) Over the last two decades
or so, geometrically frustrated magnets have received considerable attention in an effort to
identify and characterize novel quantum paramagnetic states.3, 4) In geometrically frustrated
magnets, the spins are arranged on a lattice with a triangular motif that prevents them from
satisfying all their magnetic interactions simultaneously regardless of the nature of the spin
i.e. quantum or classical. This leads to a macroscopic ground state degeneracy where unusual
low temperature properties may emerge.
In two dimensions, if the triangular plaquettes are arranged in an edge-sharing network
∗
shlee@virginia.edu
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(a)
(b)
Fig. 1. (a) Crystal structure of spnel AB2 O4 . The blue (grey) polygon represents BO6 octahedron
(AO4 tetrahedron). The neighboring BO6 octahedra share an edge. (b) The B ions form the
three-dimensional network of corner-sharing tetrahedra.
they form a triangular lattice or if in a corner-sharing network, they form a kagome lattice.
In the case of the triangular antiferromagnet it has been shown theoretically that even for
quantum spins the ground state is the classical 120◦ spiral structure. Recently interests in the
field focus on further neighbor interactions, multi-spin interactions, and spatial anisotropy
that may lead to more exotic states.5–15) The kagome antiferromagnet formed by the cornersharing triangles lacks any Neel ordering due to higher degeneracy, which makes it a likely
candidate for spin liquid states.16) Experimentally, however, a real material that realizes a
perfect quantum kagome antiferromagnet has not been found so far.17–23)
In three dimensions, a frustrating lattice is achieved when spins interact antiferromagnetically in a network of corner sharing tetrahedra.24–26) A tetrahedron with four Heisenberg
classical spins has an infinite number of spin configurations as its ground state. When arranged
in a corner sharing network, such tetrahedral spin arrangements will lead to a macroscopic
ground state degeneracy. This can give rise to properties that are qualitatively different from
conventional low temperature properties associated with the long-range ordered states that
are commonly observed in ordinary magnets. Among the novel properties are the cooperative
paramagnetic (or spin liquid) state that persists down to temperatures much lower than the
Curie-Weiss temperature and unconventional phase transitions that occur upon further cooling. There are three chemical compositions that can realize the network of the corner-sharing
tetrahedra: spinels AB2 O4 , pyrochlores A2 B2 O7 ,24, 27) and AB2 .28, 29)
In spinels AB2 O4 , the B-site forms a network of corner-sharing tetrahedra.16, 30) The B site
cations are octahedrally coordinated by six oxygen ions and neighboring BO6 octahedra are
edge shared (see Fig. 1). Thus, when the B site is occupied by a transition metal ion with t2g
electrons, the system can realize a simple and most frustrating Heisenberg spin Hamiltonian,
!
H = J Si · Sj with dominant uniform nearest neighbor interactions.31) The spinel system
will be the focus of this review paper due to its rich physics that range from the realization of
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-3
-1
90
0.1T
-1
40 CdCr2O4
70 2T
65 1T
0.5T
60
0
2
H=0.1T
4
6
T ( K)
ZnCr2O4
0
0
50
(b)
HgCr2O4
75 5T
60
20
85
80
HgCr2O4
-3
χ (10 emu mol )
80
(a)
χ (10 emu mol )
100
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100
150
200
8
250
10
300
T ( K)
Fig. 2. (a) Temperature dependence of the bulk susceptibility, χ, taken from ACr2 O4 (A = Hg, Cd,
Zn). The inset shows the field dependence of χ up to 5 T for HgCr2 O4 . (b) Bulk magnetization
as a function of an external magnetic field, taken from ACr2 O4 (A = Hg, Cd, Zn, Mg), ZnCr2 S4
and ZnCr2 Se4 . The data were taken at 1.8 K (for A = Hg and Cd) and 4.2 K for others. These
figures were taken from Ref.34)
the zero-energy excitation in the spin-liquid phase, zero-field or field-induced phase transitions,
and heavy fermionic behavior. In section II, chromium based spinels will be discussed without
orbital degeneracy. This will be followed by spinels with orbital degeneracy and mixed valence
in section III, followed by the conclusion in section V.
2. ACr2 O4
From bulk magnetization measurements, it is shown that the magnetic Cr3+ ions in
ACr2 O4 interact antiferromagnetically with each other, evidenced by the negative CurieWeiss temperature, ΘCW = −32 K for Hg, -88 K for Cd, and -390 K for Zn containing
compounds.16, 32–34) Despite the large value of ΘCW that represents the strength of the magnetic interactions, the system remains in a spin-liquid state well below the |ΘCW |. As shown
in Fig. 2, upon further cooling, a transition is evident at TN = 6 K (Hg), 7.8 K (Cd), and 12.5
(Zn).16, 32, 34) Of interest is the nature of the cooperative paramagnetic (or spin liquid) phase
that exists over a wide range of temperatures TN < T < |ΘCW |. More specifically, how are the
fluctuating spins correlated. There must be a zero-energy excitation among the degenerate
ground state; what is the real space representation of the zero-energy mode?35, 36) The second
issue is why the system orders at lower temperatures when the theory based on the spin degree
of freedom predicts the system should not. More specifically, what other degrees of freedom
are involved in the phase transition?37, 38) How do the relevant degrees of freedom couple to
drive the system to order below TN ? What is the nature of the ordered state? Is the ordered
state universal in ACr2 O4 or does it depend on the A ion?
In addition, when an external magnetic field is applied to the ordered phase, the Cr spinel
undergoes a field-induced phase transition into a magnetization plateau phase at Hc1 = 10 T
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for Hg, 28 T for Cd, and 120 T for Zn33, 34, 39) (see Fig. 2). The value of the magnetization, 1.5
µB /Cr3+ , which is half of the full saturated moment, 3µB /Cr3+ , suggests that each tetrahedron has three up (majority) and one down (minority) spins (3:1 constraint). There are many
ways to arrange tetrahedra holding the 3:1 constraint on the pyrochlore lattice because there
is considerable freedom in choosing the location of the down spin in each tetrahedron, leading
to many ways of organizing the majority and minority spins over the entire lattice.40–42) One
question that arises is how a certain structure can be stabilized over a wide range of H. Another is whether or not an universal ground state for the plateau phase exists for each type
of frustrated lattice and Hamiltonian. These issues have been experimentally addressed by
scattering experiments using neutrons and synchrotron x-ray, as discussed below.
2.1 Zero-energy magnetic excitations in the spin liquid phase of ACr2 O4
There have been intense theoretical and experimental works on the nature of the spin liquid
phase of the pyrochlore lattice.25, 26) The theoretical works showed that the ground state is
a spin liquid state. A characteristic property of the spin liquid state is the existence of zeroenergy excitations among the degenerate ground states that prevent the system from ordering.
Their presence was postulated by an entropy argument whose real-space representation was
not determined.35) The first breakthrough came when the momentum wave vector dependence
of the low energy inelastic magnetic neutron scattering intensity was measured on single
crystals of ZnCr2 O4 .36) Neutron scattering directly probes the squared structure factor of
the fundamental spin degree of freedom in the spin liquid phase. As shown in Fig. 3 (a)
and (b) the data, collected for wave vector transfer, Q, in the high symmetry (hk0) and
(hhl) planes, exhibit broad maxima at the Brillouin zone boundaries. Similar zone boundary
scattering patterns were observed in CdCr2O4 as well.43) At a fixed low energy transfer, spin
waves produce scattering that is localized in Q-space around magnetic Bragg peaks. In the
geometrically frustrated ACr2 O4 low energy scattering is extended in Q-space indicating that
spin correlations are localized in real space. The data in Fig. 3 (a) and (b) can be considered
as a measurement of the form factor for the composite spin degrees of freedom.36) The second
breakthrough was a theoretical work that showed that the Q-pattern of the structure factor
can be reproduced when antiferromagnetic spin waves of six ferromagnetic spins in hexagons
in the lattice were considered.38) The theoretical model had, however, a few problems: (a) the
model assumes a magnetically ordered state, which is absent in the spin liquid state. (b) The
entire pyrochlore lattice cannot be covered by the ferromagnetic hexagons, which means that
there will be two sets of spins: spins participating in the antiferromagnetic spin waves and
others that do not. (c) The antiferromangetic spin waves of the six ferromagnetic spins do not
have zero-energy. Such excitations require an energy of the order θ 4 when θ is the deviation
angle from the ferromagnetic state.
It turned out that a zero-energy excitation mode in the pyrochlore lattice involves an4/24
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(e)
(f )
Fig. 3. (a)-(b) Color images of inelastic neutron scattering intensities from single crystals of ZnCr2 O4
in the (hk0) and (hkk) symmetry planes obtained at T = 15K > TN = 12.5 K for !ω = 1 meV.
The data are a measure of the dynamic form factor for self-organized nano-scale spin clusters
in the material. (c)-(d) Color images of the form factor squared calculated for antiferromagnetic
hexagon spin loops (see (e)) averaged over the four hexagon orientations in the spinel lattice. The
excellent agreement between model and data identifies the spin clusters as hexagonal spin loops.
These figures were taken from Ref.36) (e) A group of six fluctuating spins that self-organize into
an antiferromagnetic hexagonal loop. (f) The hexagon directors, represented by arrows located at
the centers of the hexagons, are decoupled from each other. Hence their reorientations embody
the long-sought local zero energy modes for the pyrochlore lattice.
tiferromagnetically arranged hexagonal spins that cover the entire lattice at any moment.
In this mode, spins on a hexagon maintain their antiferromagnetic arrangement (see Fig. 3
(e)), and move collectively forming a spin director.36) The director is decoupled with other
neighboring directors (see Fig. 3 (f)), and its orientation realizes a truly zero-enegy mode
for the pyrochlore antiferromagnet. The structure factor of the antiferromagnetic hexagons
reproduces the experimental data well as shown in Fig. 3. It is an open question if this mode is
unique for the pyrochlore lattice and why other theories of the pyrochlore antiferromagnet44)
produce the so-called pinch point scattering intensity in Q-space that is inconsistent with the
data observed in ACr2 O4 .
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CdCr2O4
ZnCr2O4
(a)
Fig. 4. Magnetic Bragg intensities (squares) and lattice strains for (a) ZnCr2 O4 and (b) CdCr2 O4 .
(a) and (b) were taken from Ref.32) and,43) respectively. The tetragonal distortion involves a c-axis
contraction (a = b > c) in ZnCr2 O4 while it involves a c-axis elongation (a = b < c). The magnetic
structure is commensurate for Zn while it is incommensurate for Cd.
2.2 Spin-lattice coupling in ACr2 O4
Upon cooling, the Cr-spinels do order at low temperatures du to spin-lattice coupling.
As shown in Fig. 4, at the phase transition a lattice distortion and a magnetic long range
order simultaneously occur. The nature of the magnetic structure and lattice distortion are
different for different A site ions. Synchrotron x-ray measurements showed that the symmetry
of the low T crystal structure is orthorhombic F ddd for Hg,34) tetragonal I41 /amd with
c > a = b for Cd,43, 45) and tetragonal I 4̄m2 with a = b > c for Zn.46, 47) Their magnetic
structures determined by neutron diffraction also have different characteristic wave vectors:
two commensurate wave vectors Qm = (1,0,1/2), (1,0,0) are obtained for Hg,48) a single
incommensurate Qm = (0, δ, 1) for Cd,43) and two commensurate Qm = (1/2,1/2,0), (1,0,1/2)
for Zn.47) What is common is the fact that in all the ordered states the ordered moment of the
Cr3+ ion seems to be less than the expected value of gSµB /Cr 3+ when it is fully polarized:
< M >= 1.74(6) for Hg and 2.03(2) µB /Cr3+ for Zn. This indicates that frustration is not
fully lifted and strong spin fluctuations exist even in the ordered phase.
Theoretical efforts to understand the nature of the phase transition have focused on
magneto-elastic coupling that involves symmetric isotropic nearest neighbor (NN) exchange
interactions.37, 38) To test the theoretical predictions requires the determination of the crystal distortion and the magnetic structure. In the case of CdCr2 O4 , the tetragonal distortion
with the I41 /amd symmetry involves an elongation of the lattice along the c-axis without
any change in the atomic coordinates from their high symmetry positions, while its magnetic
structure is a complex spiral structure. Dzyaloshinskii-Moriya interactions can be the origin
of the spiral structure,49) but the one-to-one correspondence between the observed crystal
distortion and spiral structure is yet to be fully understood.
In the case of ZnCr2 O4 , determination of the magnetic and crystal structures was very
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Fig. 5. (a) ab-projection of the Cr sites in ZnCr2 O4 . Spheres in different colors represent different Cr
sites in the tetragonal phase: CrI (red), CrII (violet), CrIII (dark blue), CrIV (green), CrV (orange),
and CrVI (light blue). The yellow bar and the grey line between Cr ions are short (strong) and
long (weak) bonds, respectively. The red and blue shaded squares are the cubic unit cells that have
√
√
different pattern of distortions as shown in (c), expanding the tetragonal unit cell by 2 × 2 × 2
compared to the cubic unit cell. (b) Decoupled sublattices that emerge when only the strong bonds
are considered. (c) Different distortion patterns of the red and blue cells in close-up. The black
arrows represent the directions of the Cr distortions in the tetragonal phase. The magnitudes
of the distortions are listed in Table I. (d) Distorted tetrahedrons in the tetragonal phase. The
numbers are the distance in angstrom between Cr ions. This figure is taken from Ref.47)
recently reported using neutron and synchrotron x-ray scattering. The synchrotron x-ray
measurements showed that {0.5,0.5,0.5}c type superlattice peaks appear below TN , indicating
that the tetragonal symmetry is I 4̄m2 and the chemical unit cell is doubled along all three
crystallographic directions, compared to that of the cubic phase above TN . By analyzing ∼
140 superlattice peaks, it was shown that in I 4̄m2, Cr3+ ions occupy six crystallographically
distinct sites: four 8i and two 16j sites and that there are 19 different nearest Cr-Cr bond
lengths, varying from 2.9228 Å (CrIV -CrIV bonding in the ab-plane) to 2.9649 Å (CrI-CrI
bonding in the ab-plane) (see Fig. 5 (d) and Table I). Two aspects of the tetragonal structure
should be noted: (1) Each tetrahedron has either two strong and four weak bonds (type I) or
four strong and two weak bonds (type II) or one strong and five weak bonds (type III). This
breaks the frustrating triangular motif of the tetrahedral bonds. (2) If only the strong bonds
are considered, the entire pyrochlore lattice is divided into four different sublattices that are
connected with each other by weak bonds, as shown in Fig. 5 (b).
The magnetic characteristic wave vectors of k1 = (1/2,1/2,0)c and k2 = (1,0,1/2)c in the
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Table I. The Cr positions in the tetragonal phase determined from Rietveld refinement of x-ray single
crystal diffraction data.47) Displacements of the Cr ions from cubic positions are denoted by dr =
[dx, dy, dz] in direct lattice coordinates. Fitting of the data is not sensitive to the positions of Zn
and O, which is reflected on the huge error bars some of which are larger than their displacements
by an order of magnitude. Thus, one can ignore the Zn and O displacements.
x
y
z
dx (10−4 )
dy (10−4 )
dz (10−4 )
CrI (8i)
0.125 + dx
0
0.1875 + dz
8.946(2)
0
13.80(4)
CrII (8i)
0.375 + dx
0
0.1875 + dz
-6.355(2)
0
-3.3518(7)
CrIII (8i)
0.375 + dx
0
0.6875 + dz
6.355(2)
0
3.3518(7)
CrIV (8i)
0.125 + dx
0
0.6875 + dz
-8.946(2)
0
-13.80(4)
CrV (16j)
0.375 + dx
0.25 + dy
0.4375 + dz
5.480(1)
5.413(1)
2.2527(6)
CrVI (16j)
0.875 + dx
0.75 + dy
0.4375 + dz
-5.480(1)
-5.413(1)
-2.2527(6)
O11 (8i)
0.1382 + dx
0
0.0691 + dz
0.2(82)
0
-5(33)
O12 (8i)
0.3618 + dx
0
0.0691 + dz
3(13)
0
1(9)
O13 (8i)
0.3618 + dx
0
0.5691 + dz
-0.1(4)
0
0.1(5)
O14 (8i)
0.1382 + dx
0
0.5691 + dz
-1(13)
0
-3(16)
O15 (16j)
0.3882 + dx
0.25 + dy
0.3191 + dz
-8(48)
1(8)
1(8)
O16 (16j)
0.1118 + dx
0.25 + dy
0.3191 + dz
0.6(47)
0.9(54)
-2(26)
O21 (8i)
0.1118 + dx
0
0.8059 + dz
-0.2(82)
0
5(33)
O22 (8i)
0.3882 + dx
0
0.8059 + dz
-3(13)
0
-1(9)
O23 (8i)
0.3882 + dx
0
0.3059 + dz
0.1(4)
0
-0.1(5)
O24 (8i)
0.1118 + dx
0
0.3059 + dz
1(13)
0
3(16)
O25 (16j)
0.3618 + dx
0.25 + dy
0.5559 + dz
8(48)
-1(8)
-1(8)
O26 (16j)
0.1382 + dx
0.25 + dy
0.5559 + dz
-0.6(47)
-0.9(54)
2(26)
Zn1 (2a)
0
0
0
0
0
0
Zn2 (2b)
0
0
0.5
0
0
0
Zn3 (4f )
0
0.5
0 + dz
0
0
1(8)
Zn4 (8h)
0.75 + dx
0.25
0.25
-0.5(32)
0
0
Zn5 (8i)
0.25 + dx
0
0.375 + dz
-0.2(14)
0
0.8(11)
Zn6 (8i)
0.75 + dx
0
0.875 + dz
-0.18(14)
0
0.9(13)
cubic notation are equivalent to a single wave vector of km = (1,0,0)t in the tetragonal (I 4̄m2)
notation.47) Previous unpolarized and polarized neutron single crystal diffraction studies indicated that the magnetic structure with k1 and k2 is co-planar in the ab-plane.50, 51) There are
8 (16) representations for the 8i(16j) sites that can produce ab-inplane magnetic moments.
Thus, for four 8i and two 16j Cr3+ sites in ZnCr2 O4 , in total 2.26 ×1012 coplanar magnetic
structures can be generated by linear combinations of those representations. When the con8/24
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Fig. 6. One of 32 coplanar non-collinear spin configurations that give the same best fit to the data
with the ordered moment per each Cr3+ ion . When the spin configurations are decomposed into
a- and b-components, one component forms a collinear spin structure with k1 = (0.5, 0.5, 0)c while
the other forms a collinear spin structure with k2 = (1, 0, 1/2)c. This figure is taken from Ref.47)
straints that every tetrahedra must have zero net moment and every Cr3+ moments the same
magnitude are imposed, only 32 configurations give identically the best fit to all experimental
data including neutron powder diffraction and polarized single crystal diffraction data. The
common feature of the 32 configurations is that, as shown in Fig. 6, the magnetic moments
are along {110}c directions and every tetrahedron has two pairs of antiparallel spins.
It is noted that the antiferromagnetic pairs in the total spin structure shown in Fig. 6 do
not exactly match the pattern of the strong NN bonds shown in Fig. 5 (a). This indicates that
the microscopic mechanism of the phase transition goes beyond the simple magneto-elastic
coupling mechanism involving the symmetric NN exchange interactions only, and it may suggest that asymmetric magnetic interactions such as Dzyaloshinskii-Moriya interactions52) or
further nearest exchange interactions play an important role in selecting the complex magnetic
structure in ZnCr2 O4 .
2.3 Magnetic field-induced half-magnetization phase of ACr2 O4
In order to understand the nature of the field-induced phase transition, elastic neutron and
synchrotron x-ray scattering measurements were performed on a powder sample of HgCr2 O4 48)
and a single crystal of CdCr2 O4 43) that have different N’eel states for H = 0. Fig. 7 (b) shows
that for H < Hc1 , the low temperature Néel state of HgCr2 O4 has two characteristic wave
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Fig. 7. (a),(b) H-dependence of the neutron scattering intensities obtained from a powder sample of
HgCr2 O4 at various reflections measured at 3.2 K, (a) at (2,2,0) and (1,1,1), and (b) at (3/2,0,1),
(0,1,1) and (2,1,1). As H increases up to 9 Tesla, the neutron diffraction pattern does not change.
However, the (111) Bragg intensity increases by ∼ 90 ± 40 counts/5 min. This increase is due to
canting of spins along H, and we estimate the canting angle at 9 Tesla to be 17(4) degrees. The
field-induced transition to the half-magnetization plateau phase occurs abruptly at H ∼ 10 Tesla.
(c),(d) Synchrotron X-ray diffraction data measured at several different temperature with different
Hs: (c) The nuclear (10,4,2) Bragg reflection. (d) The nuclear (4,4,1) and (522) reflections. This
figure was taken from Ref.48)
vectors of (1,0,1/2)c and (1,0,0)c , and in the ordered state each tetrahedron has two pairs of
antiparallel spins as shown in Fig. 8. For H > Hc1 , the {1,0,1/2} magnetic peaks that were
present below Hc1 have completely disappeared, while the {1,0,0} magnetic peaks became
stronger. Furthermore, new magnetic peaks appeared at several nuclear Bragg reflection points
such as (1,1,1), (1,3,1), and (2,2,2) but not at other nuclear Bragg reflection points such as
(2,2,0). The fact that the magnetization plateau phase over 10 Tesla < H < 28 Tesla has bulk
magnetization < M >bulk ∼ 1.5µB /Cr3+ suggests that each tetrahedron has three up spins and
one down spin. With such tetrahedra, we can construct numerous spin structures in the lattice
of corner-sharing tetrahedra with several different characteristic wave vectors. Among the
numerous spin structures that can be constructed with three-up one-down spin arrangement,
there are two nonequivalent spin configurations that will produce magnetic scattering at the
{1,0,0} reflection points: one with rhombohedral R3̄m symmetry and the other with cubic
P 43 32 symmetry. In the R3̄m structure, for every pair of neighboring tetrahedra, the two
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Canted
3:1
P 43 32
3:1
AF
Fig. 8. H − T phase diagram of HgCr2 O4 . The solid lines represent first-order phase transitions while
the dashed line represents a second-order phase transition. AF stands for antiferromagnetic, 3:1
for three-up one-down, canted 3:1 for canted three-up one-down, Ferro for ferromagnetic. This
figure was taken from Refs.33, 48)
minority (down) spins are on a straight line, while in the P 43 32 structure, the line connecting
the two minority (down) spins are bent (see Fig. 8). The R3̄m symmetry can be ruled out
because it does not allow scattering at the observed (1,1,1) reflection, while it should produce
magnetic scattering at the (220) reflection where no field induced magnetic signal has been
observed (see Fig. 7 (a)). On the other hand, all the observed field-induced magnetic reflections
are allowed for P 43 32 symmetry. A quantitative comparison between the experimental data
and the model calculation showed that the staggered moment in the plateau phase is < M >=
2.22(8)µB /Cr3+ . It turned out that the crystal structure also changes from orthorhombic to
cubic simultaneously as the system enters the half-magnetization plateau phase. Furthermore,
as shown in Fig. 7 (d), the new nuclear peaks appeared at (4,4,1)c and (5,2,2)c reflections.
This indicates that the crystal structure of the field-induced half-magnetization state has the
same P 43 32 symmetry as the magnetic structure.48)
The most relevant Hamiltonian for the Cr-spinels is the nearest neighbor exchange interaction that is sensitive to the bond distance minus an elastic energy associated with the
displacements of the magnetic atoms,
E
Hef
f =J
"
NN
Si · Sj −
where the site displacement u is given by
u∗j = −
kE " ∗ 2
|uj |
2
(1)
j
Jγ "
(Si · Sj )&ij
kE
(2)
NN
where kE is a characteristic elastic spring constant. This effective hamiltonian was originally
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counts/100pulse/80 s
counts/150pulse/80 s
counts/200pulse/200 s
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(a)
40
(1.0675, -1.0125, 0.0275)
field_up
field_down
5
10
15
20
(1, -1, 0) magnetic field (T)
25
30
5
25
30
25
30
30
20
10
0
40 0
(b)
30
20
10
0
0
60
(c)
10
15
20
magnetic field (T)
(2, -2, 0)
40
20
0
0
5
10
15
20
magnetic field (T)
Fig. 9. (a),(b) H-dependence of the neutron scattering intensities obtained from a single crystal of
CdCr2 O4 at various reflections measured at 3.2 K, (a) at (2,2,0) and (1,1,1), and (b) at (3/2,0,1),
(0,1,1) and (2,1,1). As H increases up to 9 Tesla, the neutron diffraction pattern does not change.
However, the (111) Bragg intensity increases by ∼ 90(40) counts/5 min. This increase is due to
canting of spins along H, and we estimate the canting angle at 9 Tesla to be 17(4) degrees. The
field-induced transition to the half-magnetization plateau phase occurs abruptly at H ∼ 10 Tesla.
(c),(d) Synchrotron X-ray diffraction data measured at several different temperature with different
Hs: (c) The nuclear (10,4,2) Bragg reflection. (d) The nuclear (4,4,1) and (522) reflections. This
figure was taken from Ref.53)
considered by L. Balents and co-workers as an Einstein phonon model.42) They showed that
maximizing the displacements or minimizing the Hamiltonian occurs when the number of
the bending line connecting neighboring two minority spins becomes maximum, and that can
realized only in a unique bending pattern of tetrahedra that has the P 43 32 symmetry.
Fig. 9 shows that CdCr2 O4 with a different Néel state from that of HgCr2 O4 also shows
the same field-induced behavior in its half-magnetization plateau phase above Hc1 = 28 T
as observed in the plateau phase of HgCr2 O4 : the intensity increase at the (1, −1, 0)c and
no change at the (2, 2, 0)c reflection.53) Under a pulsed magnetic field, the crystal structure
of the half-magnetization phase is cubic as well.54) This clearly demonstrates that the halfmagnetization plateau phase of CdCr2 O4 has the same P 43 32 magnetic structure as that of
HgCr2 O4 . These results suggest that the simple effective Hamiltonian (Eq. (1)) describes the
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physics of the field-induced phase transition into the half-magnetization plateau phase in the
Cr-spinels. That the observed P 43 32 state is the generic ground state of the field-induced
phase of the Cr-spinels, despite their different crystal and magnetic structures observed at
H = 0, as theoretically predicted.
3. Spinels with mixed valence and orbital degeneracy
When the B site is occupied by vanadium ions with orbital degeneracy, complex electronic and magnetic properties emerge.55) LiV2 O4 , for instance, with monovalent Li ions at
the tetrahedral A site and mixed valent V3.5+ ions exhibits heavy fermion (HF) behavior at
low temperatures with the largest Sommerfeld constant among d-electron systems, γ ≈ 0.42
J/mol K2 .56) AV2 O4 with divalent ions such as Zn,57) Mg,58) Cd,59) at the A site and trivalent
V3+ (3d2 ) ions is a Mott insulator that undergoes two separate phase transitions at low temperatures, in contrast to other insulating spinels without orbital degeneracy such as ACr2 O4
(A = Zn, Cd, Hg, Mg).
Many theoretical efforts have been made to understand the unusual low temperature
behaviors of metallic and insulating vanadates.60–62) The macroscopic ground state degeneracy
induced by the geometrical frustration intrinsic to the magnetic lattice was attributed to
explain the enhancement of the specific heat at low temperatures in LiV2 O4 .78) It was also
used to explain why the Neél temperature, TN , is considerably lower than the Curie-Weiss
temperature, ΘCW , in the insulating vanadates. This section summarizes the experimental
works that have been performed to elucidate the interplay between spin, orbital, and lattice
degrees of freedom in the vanadates and titanates, which in turn can test the theoretical
models.
3.1 AV2 O4 with A = Zn, Cd, Mg
When orbital degrees of freedom exist in a frustrated lattice, one can study orbital as
well as magnetic frustration. The vanadates, AV2 O4 , with doubly degenerate orbital degrees
of freedom provide excellent model systems for the research. The chromates without orbital
degeneracy exhibit simultaneous lattice distortion and magnetic order at the same transition
temperature due to the spin-lattice coupling. On the other hand, insulating vanadates exhibit
two successive phase transitions. For example, as shown in Fig. 10, ZnV2 O4 (S = 1) exhibits
a sharp drop in the bulk susceptibility, χ, at Tc = 50 K that is due to a cubic-to-tetragonal
lattice distortion and a magnetic long range order at TN = 40 K.57) For comparison, for
CdV2 O4 , Tc = 90 K and TN = 30 K.64) Thus there are three distinct phases in AV2 O4 (A =
Zn and Cd) as a function of temperature: the high temperature cubic paramagnetic (I), the
intermediate tetragonal paramagnetic (II), and the low temperature tetragonal Néel phase
(III).
The tetragonal distortion for both A = Zn and Cd involves a c-axis contraction (c < a = b).
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(a)
(b)
Metallic
(c)
Fig. 10. (a) Bulk susceptibility, χ, as a function of temperature, obtained from a powder sample of
ZnV2 O4 under an external magnetic field of H = 0.001 T in zero-field cooling (ZFC) or fieldcooling process. (b) Effects of Li doping on χ, measured on Lix Zn1−x V2 O4 . The data were taken
in the ZFC process. (c) Phase diagram of Lix Zn1−x V2 O4 . Tt , TN and Tsg represent the cubicto-tetragonal, antiferromagnetic and spin-glass phase transition temperature, respectively. This
figure is taken from Ref.57)
Fig. 11. Three theoretical models proposed for the orbital state of a vanadate. This figure is taken
from Ref.65)
In the tetragonal phase, among the triply degenerate t2g orbitals, the dxy orbital is favored
and is occupied by one electron at every V site. There are many different ways for the second
electron of the V3+ (3d2 ) ion to occupy the remaining t2g orbitals. So far, three different
theoretical models have been proposed for the orbital ordering (see Fig. 11. First, the second
electron can be in an antiferro-orbital state that can be described by stacking the ab-planes
along the c-axis with alternating dyz and dzx orbitals.60) This effectively forms straight spin
chains on the ab-planes due to the direct overlap of the neighboring dxy orbitals along the
chain and the lack of overlap of neighboring dyz and dzx between the chains.66) The orbital
state will be stable in a lattice distortion with I41 /a symmetry.61) The second electron may
uniformly occupy a superposed state dyz ± idzx , which will be consistent with I41 /amd crystal
symmetry.61) The orbital of the second electron can order in a more complicated state where
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Fig. 12. Q-dependence of inelastic magnetic neutron scattering intensity obtained from powder samples of (a) ZnV2 O4 and (b) CdV2 O4 at various temperatures. This figure is taken from Ref.64)
it occupies a sequence of zx − yz − zx − yz − zx − yz − .. along the chains in the ab-plane
while it does a sequence of zx − zx − yz − yz − zx − zx − yz − yz − ... along the chains in
both bc- and ca-planes, which will have P 41 21 2 symmetry.62) The latter two models would
yield stronger interchain couplings than the first model. Different orbital states will result in
different magnetic interactions, and thus neutron scattering that probes how the spins are
correlated can provide important information on the nature of the orbital ordering.
Neutron scattering experiments were performed on powder samples of ZnV2 O4 and
CdV2 O4 .64, 66) Fig. 12 shows the energy integrated inelastic magnetic neutron scattering intensity as a function of Q at several temperatures. Two features stand out. (1) In the cubic
phase, the S(Q) has a peak at Qcub
= 1.35(4) Å−1 that is different from the characteristic
c
wave vector of Qhex
= 1.5 Å−1 of the antiferromagnetic hexagonal spin loops observed in the
c
chromates. This indicates that the dynamic spin correlations in the cubic phase of ZnV2 O4 are
different in nature than those in ZnCr2 O4 . (2) The Q-lineshape changes from symmetric to
asymmetric with a sharp increase at low Q and a long tail at high Q, as the system changes
from cubic to tetragonal. This indicates that the spin correlations in the tetragonal phase are
low-dimensional.
The observed behaviors can be understood when we take the orbital degeneracy of
V
3+ (3d2 )
ions into account.66) In the cubic phase the three t2g orbitals, dxy , dyz , and dzx ,
are almost equivalent, thus at each V site their occupancy will fluctuate with time with an
equal probability of 1/3. As a result, at one instant of time, two out of the three orbitals will
be randomly occupied at all V3+ sites, as illustrated in Fig. 13 (a). When all possible magnetic
interactions due to direct overlap of the orbitals are considered, the effective fluctuating spin
objects emerge to form three-dimensionally tangled antiferromagnetic spin chains shown as
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Fig. 13. (a)-(b) Illustrations of the orbital states of ZnV2 O4 in one cubic unit cell. Balloons represent
the t2g orbitals of the V3+ (3d2 ) ions: dxy (blue), dyz (red), and dzx (yellow) orbitals located at the
vanadium site. The four different sizes of the ballons represent four different ab-planes with different
z-coordinates. (a) The cubic phase above 50 K. The three orbitals are randomly distributed. The
blue rods connect possible dynamic magnetic interactions at a snap shot due to direct overlap
!
of the neighboring orbitals. (b) Antiferro-orbital model for the tetragonal phase. J, J , and J3
represent coupling constants for the nearest neighbor intrachain, the nearest neighbor interchain
(interplane), and the second nearest neighbor intrachain interactions, respectively. In this model,
J " is negligible because dyz and dzx orbitals do not overlap. This figure is taken from Ref.66)
thick blue lines in Fig. 13 (a). This model reproduces the data well including the characteristic
wave vector Qcub
c .
In the tetragonal phases II and III, S(Q) is very asymmetric for ZnV2 O4 , which can be
reproduced by an 1D spin chain model, while S(Q) is less asymmetric for CdV2 O4 . The onedimensionality of the magnetic interactions in ZnV2 O4 can be understood if two orbitals per
V ion are occupied in a striated form along the c-axis, as shown in Fig. 13 (b). One electron
from every V ion resides in the dxy orbital while the occupancy of the second electron can be
described by stacking the ab-planes with alternating dyz and dzx orbitals along the c-axis.60)
The direct overlap of neighboring t2g orbitals occurs only between dxy orbitals, yielding orbital
chains and thereby one dimensional antiferromagnetic spin chains in the ab-planes. The V3+
magnetic moments do not order even in the orbitally ordered state below 50 K because
of the one-dimensionality of the magnetic interactions until weak further nearest neighbor
interactions set in at 40 K. In the case of CdV2 O4 the less asymmetric S(Q) indicates that a
different type of orbital order such as the ferro-orbital ordering61) where the second electron
of every V3+ ion resides on the
dyz ±idzx
√
2
orbitals, may be realized in CdV2 O4 .
The crystal symmetry of the tetragonal structure of ZnV2 O4 is presumed to be I41 /amd
that seems to be inconsistent with the proposed antiferro-orbital order with I41 /a. Recently,
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Fig. 14. Synchrotron x-ray scattering data obtained from a single crystal of MnV2 O4 . The scattering
angle scan at the (a) (802)c and (b) (421)c nuclear reflections. (c) Temperature dependence of
the two Bragg intensities. (d) Schematic picture of the orbital state of the tetragonal phase of
MnV2 O4 . This figure is taken from Ref.65)
the tetragonal structure of a related vanadate, MnV2 O4 , was found to be I41 /a, as shown
in Fig. 14. It is an open question what the actual orbital and crystal structures are of the
tetragonal phase of AV2 O4 (A = Zn and Cd). Very recently, B. Lake and her coworkers
succeeded to grow large single crystals of MgV2 O4 and performed neutron and synchrotron
x-ray scattering measurements.67) Surprisingly, their results showed that its high T cubic and
low T tetragonal structures have lower symmetry than F d3̄m and I41 /amd, respectively. This
indicates that different vanadates can have different lattice distortion and orbital states.
3.2 AlV2 O4
When the A site is occupied by trivalent Al ions, the valence of V becomes 2.5+. This
AlV2 O4 exhibits a structural phase transition from cubic to trigonal at 700 K, and both
resistivity and magnetization shows anomalies at this phase transition.68, 69) By the Rietveld
analysis of the synchrotron x-ray powder diffraction data, it was found that V “heptamers”
are formed, where 7 V ions (V2 and V3 in Fig. 15(a)) are connected by short bonds, and
there also appear “lone” V ions (V1 in Fig. 15(a)) that are not connected to any other V ions.
Therefore, the low temperature trigonal phase can be regarded as the alternate stacking of
the V heptamer and a lone V ion along the (111) direction of the spinel structure.
Hepetamerization of the V ions in AlV2 O4 can be explained by the orbital ordering of the
t2g states. Along each side of the heptamer, either dxy , dyz , or dzx orbitals form a bonding
state, as shown in Fig. 15(b), and there are 9 bonding states in one heptamer. Accordingly,
18 electrons can be accommodated in the bonding states of the heptamer as a spin singlet
state. On the other hand, 2 electrons occupy the lone V ion with a S=1 state. This state is
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Fig. 15. (Color online) (a) Crystal structure of AlV2 O4 in the trigonal phase. Red thick lines represent
the short V-V bonds. (b) Schematic electronic bonds in the heptamer. Each bond consists of dxy ,
dyz , or dzx orbitals, as illustrated by different colours. These figures were taken from Ref.68) .
Fig. 16. Heavy fermion behavior of LiV2 O4 : (a) resistivity ρ, (b) specific heat C, (c) magnetic susceptibility χ, (d) hall coefficient RH , and (e) the resistivity coefficient A vs the specific heat coefficient
γ (C = γT ). The inset of (a) shows that ρ = ρ0 + AT 2 at low temperatures. This figure is taken
from Ref.71)
consistent with the behavior of the magnetic susceptibility in this compound.68) An NMR
measurement also confirmed such coexistence of the spin-singlet cluster and the nearly free
spins.70) This result demonstrates that the orbital degrees of freedom on the quite simple
spinel crystal structure can give rise to a new state with complexity.
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Fig. 17. (a)-(d) Inelastic neutron scattering intensity obtained from a powder sample of LiV2 O4 ,
averaged over 0.2 meV < !ω < 0.8 meV at various temperatures. (e)-(g) Dynamic susceptibility
!!
χ (Q0 , ω) at Q0 = 0.6 Å−1 derived from magnetic neutron scattering data via the fluctuation
dissipation theorem. This figure is taken from Ref.82)
3.3 LiV2 O4
When the A site is occupied by monovalent Li ions, the vanadate, LiV2 O4 , with mixed
valent V3.5+ ions exhibits heavy fermion (HF) behavior at low temperatures with the largest
Sommerfeld constant among d-electron systems, γ ≈ 0.42 J/mol K2 (see Fig. 16, as heavy as
many Ce- and U-based heavy fermion (HF) systems.56, 71) Bulk susceptibility data indicate
an effective localized spin < S >ef f = 1/2 with g = 2.2 at high temperature (T ) and a
saturated moment < M >= g < S >sat µB = 0.6µB at low temperatures in an external
magnetic field (50 T ≤ H ≤ 60 T).72) This led to theoretical models based on band structure
calculations that point to two distinct types of 3d electrons due to the trigonal splitting of the
t2g orbital into a half-filled A1g singlet and a 1/8 filled Eg doublet.73–75) The two bands play
the role of localized and itinerant electrons that hybridize to form a heavy fermi liquid at low
temperatures.73, 76) However, strong on-site Hund’s coupling complicates full quenching of the
localized spin at the low temperatures.77) Another group of theoretical work focuses on the
frustrating magnetic lattice that induces macroscopic ground state degeneracy and therefore
enhances the specific heat at low temperatures.78) Formation of one-dimensional chains/rings
due to frustrated charge order or orbital order has also been proposed.79, 80)
Previous magnetic neutron scattering studies on LiV2 O4 81, 82) is summarized in Fig. 17. For
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T < 80 K, the magnetic inelastic neutron intensity exhibits a prominent peak centered at finite
wave vector (Qc = 0.6 Å−1 at 1.4 K), indicting that LiV2 O4 enters the heavy fermion phase
as an antiferromagnetically correlated paramagnet. Interestingly, Qc is close to values found
in Y2 Mo2 O7 and ZnFe2 O4 , both materials that are expected to have longer range interactions
than insulating Cr- and V-spinels. This may suggest that longer range interactions are also
present in metallic LiV2 O4 . The !ω-dependence of the dynamic susceptibility, χ(!ω), shown in
Fig. 17 (e)-(g), that the spectral weight moves to low energies as temperature decreases. When
we fit the data to a single Lorentzian with a characteristic relaxation rate, Γ, the temperature
dependence of Γ turned out to be close to linear, Γ(T ) = Γ(0) + CkB T (T /θ)α−1 with the
residual Γ(0) = 1.4(2) meV and α = 0.9(2) from 1.4 K to 80 K. The linear T dependence of
Γ is generally associated with spin liquid states of insulating frustrated magnets, while the
residual Γ(0) is a feature of a strong correlated metal. Thus, LiV2 O4 seems to have features
both of a strongly correlated metal and of frustrated magnetism. How the magnetic frustration
and correlated electrons are mixed to exhibit the heavy fermionic behaviors in LiV2 O4 is yet
to be fully understood.
3.4 MgTi2 O4
MgTi2 O4 can be a candidate material with a quantum spin singlet state if the single unpaired electron of the Ti3+ ion can be localized. Upon cooling, MgTi2 O4 undergoes a metalto-insulator phase transition at 260 K, evidenced by a sharp decrease of the bulk susceptibility
and a cubic-to-tetragonal structural transition.83) A previous synchrotron x-ray and neutron
powder diffraction study showed that in the tetragonal structure the Ti atoms dimerize helically along the c-axis probably due to an orbital ordering.84) The magnetic ground state
is supposed to be a quantum spin singlet formed by the dimerized Ti3+ (3d1 ) moment. The
singlet-to-triplet excitations which would be a direct experimental evidence for the spin singlet are yet to be observed. Previous inelastic neutron scattering experiments have not yielded
any clear magnetic signal up to 25 meV. It is an open question if the electrons are partially
delocalized even in the tetragonal phase, which will work against the singlet formation.
4. Conclusions
Spinels AB2 O4 with magnetic B-ions provide excellent model systems to study the physics
of magnetic and orbital frustration. Several novel phenomena have been observed, such as the
zero energy excitation mode in the spin liquid phase of the chromates, zero-field and fieldinduced novel phase transitions in chromates and vanadates, and heavy fermionic behaviors
in LiV2 O4 . Lots of progress have been made to understand the unconventional collective
phenomena. There are still questions to be answered. For instance, in the insulating spinels
that undergo phase transitions due to either spin-lattice or orbital ordering, the frozen moment
is much smaller than the value that is expected if the moment is fully saturated. Why do the
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moments partially order? What is the nature of the spin excitations in the ordered phase?
We showed in this paper that the spinels are very close to critical points surrounded
by several different phases, and selection of which state depends on delicate balancing acts
between spin, orbital, and lattice degrees of freedom that vary with particular systems. New
phenomena are expected to emerge as new spinel systems are being synthesized using high
pressure techniques and with materials beyond the 3d transition metal ions.
Acknowledgment
SHL was supported by the U.S. DOE through DE-FG02-07ER46384.
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