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PHYSICAL REVIEW B 89, 134106 (2014)
Magnetic and structural phase transitions in the spinel compound Fe
1+x
Cr
2−x
O
4
J. Ma,
1
V. O. Garlea,
1
A. Rondinone,
2
A. A. Aczel,
1
S. Calder,
1
C. dela Cruz,
1
R. Sinclair,
3
W. Tian,
1
Songxue Chi,
1
A. Kiswandhi,
4,5
J. S. Brooks,
4,5
H. D. Zhou,
3,4
and M. Matsuda
1
1
Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
2
Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
3
Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
4
National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306, USA
5
Department of Physics, Florida State University, Tallahassee, Florida 32306-3016, USA
(Received 7 January 2014; revised manuscript received 28 February 2014; published 17 April 2014)
Neutron and x-ray diffraction, magnetic susceptibility, and specific heat measurements have been used to
investigate the magnetic and structural phase transitions of the spinel system Fe
1+x
Cr
2−x
O
4
(0.0 x 1.0).
The temperature versus Fe concentration (x) phase diagram features two magnetically ordered states and four
structural states below 420 K. The complexity of the phase diagram is closely related to the change in the spin
and orbital degrees of freedom induced by substitution of Fe ions for Cr ions. The systematic change in the
crystal structure is explained by the combined effects of Jahn-Teller distortion, spin-lattice interaction, Fe
2+
-Fe
3+
hopping, and disorder among Fe
2+
,Fe
3+
,andCr
3+
ions.
DOI: 10.1103/PhysRevB.89.134106 PACS number(s): 61.05.fm, 75.25.Dk, 75.30.Cr, 75.40.Cx
I. INTRODUCTION
Due to the strong interactions among the spin, orbital,
and lattice degrees of freedom, the transition metal oxides
with spinel structure, AB
2
O
4
, present complicated magnetic
and structural phase transitions and have attracted extensive
attention in past years [1–3]. In the system, the octahedrally
coordinated B-site cations form a geometrically frustrated
network of corner shared tetrahedra, while the A-site cations
form a diamond lattice and are located at the center of oxygen
tetrahedra. The 3d orbitals of the B-site cation split into the
triply degenerate low-energy t
2g
states and doubly degenerate
high-energy e
g
states, while the A-site cation has two low
energy e
g
states with three high energy t
2g
states [3,4]. Since
the properties of both A and B cations are driven by the electron
occupancies on 3d orbitals, which determine the magnetic
and orbital degrees of freedom, it is challenging to obtain the
original driving forces for those magnetic and structural phase
transitions in the spinel oxides.
In order to get insightful information on the phase transition
mechanism, one can occupy the B
3+
site with a spin only
cation, (such as the chromite spinels ACr
2
O
4
). The electronic
configuration for Cr
3+
cation is 3d
3
[S(Cr
3+
) = 3/2], which
leads to half filled t
2g
and empty e
g
orbitals. In addition, it has
been found that different A-site cations could yield different
ordered states: (i) If A
2+
ions are magnetically neutral,
such as ZnCr
2
O
4
[1,5–8], MgCr
2
O
4
[7–9], CdCr
2
O
4
[8,10],
and HgCr
2
O
4
[8,11,12], a transition from the paramagnetic
cubic phase to the N
´
eel-ordered tetragonal or orthorhombic
one at low temperatures is obtained. (ii) If A
2+
ions are
magnetic with spin only, such as MnCr
2
O
4
[S(Mn
2+
) =
5/2] [13–17], and CoCr
2
O
4
[S(Co
2+
) = 3/2] [13–21], the
lattice remains cubic, with a paramagnetic-to-ferrimagnetic
transition at high temperature, followed by a transition to
spiral ordering at lower temperature due to weak magnetic
geometrical frustration. (iii) If A
2+
ions are magnetic with
the orbital degree of freedom, such as FeCr
2
O
4
[17,20–25],
NiCr
2
O
4
[17,21–23,26,27], and CuCr
2
O
4
[17,21,22,27–29],
a cubic-tetragonal phase transition is observed at a higher
temperature, followed by magnetic order, which indicates that
the magnetic ordering is stabilized by reducing the lattice
symmetry through a spin-lattice coupling. The long-range
ordered collinear ferrimagnetic state can eventually evolve into
different noncollinear ferrimagnetic states at a lower temper-
ature, such as conical ordering in FeCr
2
O
4
and NiCr
2
O
4
, and
Yafet-Kittel-type magnetic ordering in CuCr
2
O
4
. Moreover,
the multiferroic ordering and the dielectric response induced
by the magnetic field also have been found in several chromium
spinel oxides [17,20–22,30,31]. Substantial experimental and
theoretical works have been performed to study the intriguing
properties of ACr
2
O
4
. Previously, a largely separate line of
research has been devoted to the spin-lattice interaction which
is related t o the spin frustration and the cooperative Jahn-Teller
distortion especially for the compounds involving orbitally
active A-site cations [17,20–22,27,31–35]. However, very few
materials have been studied from the viewpoint of the coupling
between frustration and Jahn-Teller effects by changing the
orbital configuration of B-site cation.
In this regard, Fe
1+x
Cr
2−x
O
4
is a remarkable compound.
The Fe
3+
ions (3d
5
) are not orbitally active and have a large
spin, S = 5/2, while Fe
2+
ions have an orbital degree of
freedom with 3d
6
and S = 2. Since the Cr
2+
energy level
lies well above the Fe
2+/3+
energy levels, the valence of Cr
3+
is stable. Although electrons have been reported to be hopping
between A and B sites for x ∼0.8[36,37], the Cr
3+
ions
always stay at the B site and the only electron transfer should be
between Fe
2+
and Fe
3+
ions. This arrangement leads to the 3d
electronic ground state of Fe
2+
ion changing from e
g
on the A
site to t
2g
on the B site and the type of the Jahn-Teller distortion
effect being alternated in the system [37–40]. Therefore, the
primary effect of Fe doping in this system is to change the
average moment on the B sites and alter the competition of
antiferromagnetic interactions between A-B and B-B.For
this reason, studying the magnetic and structural lattice of
Fe
1+x
Cr
2−x
O
4
can help to uncover the origin of the orbital
ordering effect on the structural transition.
The structure was first discussed by Verwey et al. in
1947 [38]; then several techniques have been applied to study
1098-0121/2014/89(13)/134106(9) 134106-1 ©2014 American Physical Society
J. MA et al. PHYSICAL REVIEW B 89, 134106 (2014)
the physical properties. In 1964, G. Shirane et al. [25] mea-
sured the magnetic structures of the parent compound FeCr
2
O
4
by neutron powder diffraction (NPD). In 2008, Tomiyasu
et al. [23] reported the dynamical spin-frustration effect on
the magnetic excitations of FeCr
2
O
4
. Both composition and
temperature dependencies of cubic-tetragonal-orthorhombic
structure transitions had been reported by x-ray powder
diffraction (XPD) and specific heat [37,39–43]. The magnetic
properties have been studied by magnetic and M
¨
ossbauer effect
measurements [36,42–45]. However, a systematic study of the
doping effect on magnetic structures is still missing, and the
T -x phase diagram is still under debate [41,42].
In this paper, the magnetic and crystal structure of
Fe
1+x
Cr
2−x
O
4
(0.2 x 1.0) are studied by NPD, XPD,
magnetic susceptibility, and specific heat measurements. The
phase diagram for 0.0 x 1.0 is also constructed. The
measurements confirm the existence of a paramagnetic-to-
collinear ferrimagnetic phase transition for the entire x range
and a conical ferrimagnetic state at low temperature in the low
Fe-doping region (x 0.6). The structural phase transition
is complicated: Although the cubic-to-tetragonal transition
(0.0 x 0.8) and a short-range tetragonal distortion is
suggested (0.8 x 1.0), the related structural transition
temperature (T
S
1
) decreases at x 0.3, then increases grad-
ually at 0.3 x 0.8, and decreases again at x 0.8. In
addition, a tetragonal-to-orthorhombic transition is detected
at T
S
2
and disappears at the high Fe-doping region (x 0.7).
We extend the study of phase diagram to compositions beyond
0.4 and temperatures below 80 K. These observations not only
emphasize the competitions among the spin-lattice interac-
tion, the Jahn-Teller distortion, the cooperative spin-orbital
coupling, and the disordered states of the A- and B-site ions,
but also reveal the effect of the magnetic moment magnitude
and electron hopping on this frustrated spinel system.
II. EXPERIMENT
Polycrystalline samples of Fe
1+x
Cr
2−x
O
4
(0 x 1.0)
were synthesized by solid state reaction. Stoichiometric
mixtures of Fe
2
O
3
, Fe, and Cr
2
O
3
were ground together and
calcined under flowing Ar at 1150
◦
C for 20 h. The magnetic
susceptibility was measured with a SQUID (Quantum Design)
with an applied field H = 100 Oe with zero field (ZFC) and
field cooling processes (FC). The specific heat measurements
were performed on a Quantum Design physical property
measurement system (PPMS).
Low-temperature XPD patterns were collected using a
PANalytical Multi-Purpose Diffractometer (MPD) equipped
with an Oxford Cryosystems PheniX cryostage closed-cycle
helium refrigerator. The MPD was configured with copper
K
α
1,2
radiation and fixed slits, a diffracted-beam monochroma-
tor to minimize background fluorescence from iron, and a high-
speed X’celerator position-sensitive detector. The powder
samples were either pressed in a stainless-steel cup or solution
cast from ethanol onto an anodized flat holder, depending on
the amount of material available. The instrument alignment
was verified prior to the data collection using NIST 660a
LaB
6
standard, but no internal standards were used in order
to prevent contamination of the samples. The cryostage was
operated under a vacuum of approximately 10
−6
Torr. Data
were collected over broad and limited diffraction angles in
order to verify structure and to carefully examine the structural
transitions over a broad temperature range from 15 to 300 K.
The X’Pert HighScore Plus software was employed to identify
possible phases and determine the lattice parameters.
NPD experiments were performed at the High Flux Iso-
tope Reactor (HFIR) of the Oak Ridge National Laboratory
(ORNL). For each composition, about 5 g of powder was
loaded in a vanadium-cylinder can. A closed-cycle refrigerator
was employed for samples with x 0.8, while a cryofurnace
was used for samples with 0.9 x 1.0. Preliminary neutron
diffraction data were obtained from the wide angle neu-
tron diffractometer (WAND). High-resolution neutron powder
diffraction measurements were performed using the neutron
powder diffractometer, HB2A. Data were collected at selected
temperatures using two different wavelengths, λ = 1.538
and 2.406
˚
A, and collimation of 12
-open-6
. The shorter
wavelength gives a greater intensity and higher Q coverage that
was used to investigate the crystal structures, while the longer
wavelength gives lower Q coverage with better resolution that
was important for investigating the magnetic structures of each
material. The diffraction data were analyzed using the Rietveld
refinement program FullProf [46].
The magnetic order parameter measurements were carried
out using the HB1A triple-axis spectrometer at HFIR. HB1A
was operated with an incident neutron wavelength of λ =
2.359
˚
A. A pyrolytic graphite (PG) (002) monochromator and
analyzer were used together with collimation of 40
-40
-40
-
80
. Contamination from higher-order beams was removed
using PG filters.
III. RESULTS
A. Magnetic susceptibility and specific heat
Figure 1 shows the temperature dependence of magnetic
susceptibilities for Fe
1+x
Cr
2−x
O
4
(0.0 x 0.8). As the
temperature decreases, the curves of ZFC and FC split at
the paramagnetic-to-ferrimagnetic transition temperature T
C
,
and an additional magnetic phase transition becomes apparent
at low temperature T
N
for the low Fe-doped compounds
(x 0.6). The composition dependent T
C
agree with the
previous reports [36,42,43]. As Fe content increases, T
C
increases gradually. For the second magnetic phase transition
temperature T
N
, it decreases with more Fe
3+
ions introduced to
the B site. As described in the following section, this transition
corresponds to a spin reorientation into a noncollinear conical
state.
For the high Fe-doped compounds (x 0.9), T
C
is above
room temperature, as shown in Fig. 2. Although there is no
obvious anomaly observed from the FC data, there are two
peaks obtained from the temperature derivative of the ZFC
susceptibility, which might be related to Jahn-Teller effects
from the nondegenerate e
g
and t
2g
orbitals and will be discussed
in the following sections.
In order to check the effect of the magnetic transitions on
the lattice, the specific heat was measured from 2 to 300 K for
several compounds, such as x = 0.0, 0.1, 0.4, 0.5, and 0.7, as
shown in Figs. 1(a)–1(e). For x 0.4, the structural transition
temperatures are comparable to the data of Kose et al. [42].
134106-2
MAGNETIC AND STRUCTURAL PHASE TRANSITIONS IN . . . PHYSICAL REVIEW B 89, 134106 (2014)
T
C
T
C
T
C
T
C
T
N
T
T
C
T
N
T
C
T
N
T
S1
T
S1
T
S1
T
S2
T
S1
T
S2
T
S1
T
T
T
T
T
T
N
FIG. 1. (Color online) The temperature dependence of magnetic susceptibilities and specific heat for Fe
1+x
Cr
2−x
O
4
(0.0 x 0.8). Black
and red lines are the results of the field-cooled (FC) and zero-field-cooled (ZFC) measurements, respectively. The blue line represents the
specific heat data. For x 0.3, T
S
2
is close to T
C
; for 0.3 x 0.8, T
S
1
is close to T
C
. Insets present the enlarged view of the specific heat data
around T
S
1
and T
C
.
The apparent structural transition becomes less distinct with
Fe doping. In addition, the data of specific heat clearly present
the collinear-to-noncollinear ferrimagnetic transition for x
0.6. Therefore, there are two distinct structural transitions for
the low Fe concentration (x 0.3). One is observed in the
paramagnetic state, above T
C
, and the other emerges in the
vicinity of the magnetic ordering temperature. Those structural
transitions are also in agreement with the diffraction data,
which we will discuss the details in the following sections.
B. Neutron and x-ray diffraction
The s tructural and magnetic phases of Fe
1+x
Cr
2−x
O
4
system are identified by NPD and XPD on several different
compositions. The temperature dependencies of the (004)
Bragg peak intensity of Fe
1+x
Cr
2−x
O
4
(x = 0.4 and 0.9)
measured by XPD are plotted in Fig. 3.
For the low Fe-doped compound, such as x = 0.4, the
structure changes from cubic to tetragonal at T
S
1
∼ 150 K and
then to orthorhombic at T
S
2
∼ 110 K, as shown in Fig. 3(a);
For the high Fe-doped compound, such as x = 0.9, there
is no obvious structural phase transition observed and the
symmetry is cubic, as shown in Fig. 3(b). The composition
dependence of the diffraction pattern observed at 5 K is shown
in Fig. 4. The three different patterns were well described by
the space groups Fd
¯
3m (one peak), I 41/amd (two peaks),
and Fddd (three peaks). This is consistent with the previ-
ous analysis and reflecting the cubic-tetragonal-orthorhombic
sequence of structural phase transitions upon decreasing the
Fe amount [37,39–43]. Because of the limited instrumental
resolution, the two peaks of the tetragonal phase (x = 0.8) are
observed as one broad peak, and two peaks (one is sharp
and the other is broad) are observed i n the orthorhombic
phase instead of three sharp peaks for x = 0.2,0.3, and
0.6, Fig. 4. For the high Fe-doped compound (x 0.9),
although the phase transition related peak splitting is not fully
perceived with the current XPD experimental resolution, the
peak widths of (004) at 20 K are broader than at 300 K.
The two characteristic temperatures indicated by the magnetic
susceptibility measurements are marked by arrows shown in
Fig. 3(d). The temperature dependent anomaly at ∼210 K is
strong on both integrated intensities and peak widths, hence a
tetragonal phase is suggested, which agrees with Francombe
et al. [43]. On the other hand, there is no apparent anomaly
at ∼60 K. It is possible that the anomalies in the magnetic
susceptibilities originate from the transition/crossover to the
different ground state of e
g
orbitals due to the structural
transition/distortion. Therefore, we speculate that the anomaly
at ∼60 K is also related to a structural distortion. Due to the
instrumental resolution, the difference between a structural
134106-3
J. MA et al. PHYSICAL REVIEW B 89, 134106 (2014)
FIG. 2. (Color online) The temperature dependence of magnetic
susceptibilities for Fe
1+x
Cr
2−x
O
4
[x = 0.9 (a) and 1.0 (b)]. Black
and red lines are the results of FC and ZFC measurements,
respectively. The insets are the related temperature derivative of
the ZFC susceptibilities and the arrows mark the peak positions in
temperature.
distortion and a structural transition cannot be fully resolved,
i.e., a change that is observed as a peak broadening rather
than a distinct peak splitting. So far, we cannot confirm the
phase around 60 K with laboratory XPD. Synchrotron or
36.1
34.9
35.3
35.7
(a)
140
90
165
190
115
(b)
(c)
(d)(d)
FIG. 3. (Color online) The 2θ dependence of the (004) Bragg
peak of Fe
1+x
Cr
2−x
O
4
[x = 0.4 (a) and 0.9 (b)] as a function of
temperature by XPD. The XPD data around the cubic (004) Bragg
position of x = 0.4 (c) and 0.9 (d) at selected temperatures. Inset
shows the width (open circle) and integrated intensity (filled square)
of the (004) Bragg reflections of x = 0.9 by XPD.
FIG. 4. (Color online) The NPD data around the cubic (004)
Bragg position for different compositions at 5 K.
single crystal diffraction could make it possible to investigate
the structural transitions/distortions more accurately, however,
this is beyond the scope of the current investigations.
Figure 5 presents the order parameters of the (111) and
incommensurate satellite reflections measured by NPD for
different Fe concentrations. The rise of the (111) magnetic
Bragg intensities indicate a collinear ferrimagnetic order set
in at ∼150 K ( x = 0.4) and ∼410 K (x = 1.0), respectively,
which are in good agreement with the previous reports [36,41–
43] and the bulk magnetization agrees well for x = 0.4, as
showninFig.1. Similar to the parent compound, FeCr
2
O
4
[25],
the collinear-to-noncollinear ferrimagnetic transitions are also
observed by the appearance of incommensurate magnetic
reflections at T
N
, with an onset of ∼28 K for x = 0.2 and
∼18 K for x = 0.4, respectively. Actually, these incommen-
surate peaks have also been reported in other magnetic A-site
(c)
(b)
40 8
0
120 160
0
30
60
90
(a)
(d)
240
280
320
360
5
1
0
20 25
15
FIG. 5. The neutron diffraction data of Fe
1+x
Cr
2−x
O
4
upon
warming. The paramagnetic-to-collinear ferrimagnetic transitions of
x = 0.4 (a) and 1.0 (b) are presented by the integrated intensities
of (111) Bragg peaks, and the integrated intensities of the incom-
mensurate reflection at 19.197
◦
presents the collinear-to-conical
ferrimagnetic transitions of x = 0.4 (c) and 0.2 (d), respectively.
134106-4
MAGNETIC AND STRUCTURAL PHASE TRANSITIONS IN . . . PHYSICAL REVIEW B 89, 134106 (2014)
FIG. 6. (Color online) Plots of raw NPD data (black dots) for
Fe
1.2
Cr
1.8
O
4
measured at T = 5, 75, and 150 K for 2θ 120
◦
.
Solid lines are results of Rietveld refinements described in the main
text. Differences between observed and calculated intensities are
shown directly below the respective patterns. The stars indicate an
incommensurate phase.
chromites such as CuCr
2
O
4
,MnCr
2
O
4
, and CoCr
2
O
4
[13–
15,18,19,34].
Rietveld analyses were employed to determine precisely
the changes in both the crystal and magnetic structures for
each composition. The Rietveld fitted patterns on Fe
1.2
Cr
1.8
O
4
at 150, 75, and 5 K are shown in Fig. 6. Figure 7 displays
the Rietveld profile fitting results at 5 K for selected Fe
FIG. 7. (Color online) Plots of NPD data (black dots) for
Fe
1+x
Cr
2−x
O
4
(x = 0.3, 0.6, 0.8, and 1.0) measured at T = 5Kfor
2θ 120
◦
. Solid lines are results of Rietveld refinements described in
the main text. Differences between observed and calculated intensities
are shown below the respective patterns. The stars indicate an
incommensurate phase and the gray arrows are the signal from Al-can.
compositions. The data at 150 K for Fe
1.2
Cr
1.8
O
4
confirm
the cubic spinel structure (defined by Fd
¯
3m space group)
without impurity phases. The refinement results indicate that
less than 0.5% dislocations between the A and B site in
Fe
1.2
Cr
1.8
O
4
, which confirms the statement of the Cr
2+
energy
level lying well above the Fe
2+/3+
energy level [37,39]. Similar
to FeV
2
O
4
[47], the diffraction patterns are well described by
the space groups Fd
¯
3m for the cubic lattice and the Fddd
for orthorhombic lattice with decreasing temperature, while
the magnetic phases are well described by the collinear and
conical magnetic states, respectively. The conical magnetic
phase is an incommensurate phase [with the corresponding
reflections labeled by stars in Figs. 6(a) and 7(a)]. Upon
Fe doping, the positions of incommensurate peaks do not
change significantly, but the intensities decrease. In order to
model these reflections, we have tried the conical model with
ferrimagnetic order along [110] (as in MnCr
2
O
4
[13,14]) or
along the c axis (as found for CuCr
2
O
4
[28]). The quality of
the fits is not very satisfactory for either model, being affected
by an anisotropic peak broadening which might come from
the microstrains or other structural distortions in the sample.
Furthermore, the lack of enough unique magnetic peaks in this
powder data hinders the reliable determination of the direction
and magnitude of the Fe
2+
and Fe
3+
/Cr
3+
magnetic moments.
Single crystal neutron diffraction measurements are clearly
needed to determine the exact canting angles and ordered
moments. Figures 6(a) and 7(a) present the best fits from
the refinements with the propagation vector of k = [0.391,
0.391, 0] for the centered cell Fddd. Detailed information
about the structural refinement and the atomic coordinates
is summarized in Tables I and II. The doping effect on the
tetragonal splitting is also confirmed by the change of the
c/a ratio [25,37,39–43]. The cubic-tetragonal-orthorhombic
sequence of structural phase transitions is also captured by
the XPD measurements with 5 K/ step, which is consistent
with the previous analysis [42] and similar to other FeB
2
O4
spinels, such as Mn-doped chromite Fe
1−x
Mn
x
Cr
2
O
4
[34] and
Fe-vanadate Fe
1+x
V
2−x
O
4
[47–49].
IV. DISCUSSION
Combining the diffraction and the magnetic susceptibility
measurements, a T -x phase diagram including both the crystal
and magnetic structures can be constructed as shown in
Fig. 8. The complicated phase diagram clearly presents the
Fe
3+
-doping effects on the Jahn-Teller distortion, spin-lattice
interaction, orbital-lattice interaction, Fe
2+
-Fe
3+
hopping, and
disordering effect of Fe
2+
,Fe
3+
, and Cr
3+
ions in the system.
There are three major regions, which will now be discussed
separately.
(i) x 0.3: The doped Fe
3+
ions only occupy the B site of
the spinel, and the compounds have the normal type of struc-
ture with the formula Fe
2+
[Cr
3+
2−x
Fe
3+
x
]O
4
[36,37,39–43,45].
The paramagnetic-to-collinear ferrimagnetic and collinear-to-
conical ferrimagnetic phase transitions are observed at T
C
and
T
N
, respectively. With increasing Fe doping, T
C
increases and
T
N
decreases. Although the cubic, tetragonal, and orthorhom-
bic phases are observed in sequence as temperature decreases,
T
S
1
decreases with Fe doping while T
S
2
increases with T
C
.
134106-5
J. MA et al. PHYSICAL REVIEW B 89, 134106 (2014)
TABLE I. Crystallographic information and Rietveld profile reliability factors for Fe
1+x
Cr
2−x
O
4
(x = 0.2, 0.3, 0.6, 0.8, and 1.0) from NPD
data at 5 K.
x = 0.2 x = 0.3 x = 0.6 x = 0.8 x = 1.0
Crystal symmetry orthorhombic orthorhombic orthorhombic tetragonal cubic
Space group Fddd Fddd Fddd I41/amd F d
¯
3m
a (
˚
A) 8.4300(4) 8.4294(4) 8.3945(5) 5.9179(1) 8.3792(1)
b (
˚
A) 8.4710(3) 8.4754(4) 8.4919(4)
c (
˚
A) 8.2343(3) 8.2414(5) 8.3172(5) 8.4193(1)
c/a 0.977 0.978 0.991 1.001 1.000
V (
˚
A
3
) 588.0(1) 588.8(1) 592.9(1) 294.86(1) 588.32(1)
Z 88884
Recording angular range (
◦
) 10.5–131.9 10.5–131.9 10.5–131.9 10.5–131.9 10.5–131.9
calculated density (g/cm
3
) 5.074 5.068 5.067 5.110 5.142
Bragg R factor 11.5 8.5 5.7 5.5 5.6
Magnetic R factor 11.1 8.2 6.9 6.8 4.2
Moreover, the lattice constant a is larger than c in the related
tetragonal phase.
The six outer-shell electrons of Fe
2+
occupy the 3d orbitals
(e
3
g
t
3
2g
), giving one of the three e
g
electrons the orbital degree
of freedom on 3d
z
2
or 3d
x
2
−y
2
. Since the FeO
4
tetrahedra are
generated from a cube where one Fe
2+
ion is located at the
center of four O
2−
ions that occupy two diagonal corners, the
distortion modes can be represented, as discussed in Ref. 34,
by a combination of 3d
z
2
and 3d
x
2
−y
2
, which are described by
Q
2
and Q
3
, respectively [34,50,51],
Q
2
=
1
√
2L
(δX − δY),
(1)
Q
3
=
1
√
6L
(2δZ − δX − δY ),
TABLE II. Refined atomic positions of Fe
1+x
Cr
2−x
O
4
(x = 0.2,
0.3, 0.6, 0.8, and 1.0) from NPD data at 5 K.
atoms site xyz
x = 0.2 Fe(1) 8a 0.125 0.125 0.125
Fe(2) 16d 0.5 0.5 0.5
Cr 16d 0.5 0.5 0.5
O 32e 0.261(2) 0.265(2) 0.259(2)
x = 0.3 Fe(1) 8a 0.125 0.125 0.125
Fe(2) 16d 0.5 0.5 0.5
Cr 16d 0.5 0.5 0.5
O 32e 0.262(2) 0.265(2) 0.258(2)
x = 0.6 Fe(1) 8a 0.125 0.125 0.125
Fe(2) 16d 0.5 0.5 0.5
Cr 16d 0.5 0.5 0.5
O 32e 0.263(2) 0.265(2) 0.260(2)
x = 0.8 Fe(1) 4a 0.0 0.75 0.125
Fe(2) 8d 0.0 0.0 0.5
Cr 8d 0.0 0.0 0.5
O 16h 0.0 0.020(2) 0.262(1)
x = 1.0 Fe(1) 8a 0.125 0.125 0.125
Fe(2) 16d 0.5 0.5 0.5
Cr 16d 0.5 0.5 0.5
O 32e 0.259(2) 0.259(2) 0.259(2)
where L is the length of the related cube, and δX, δY , and
δZ are the modulation of the cube dimensions. Thus, the
Hamiltonian of the coupling between the distortion and orbital
occupation should be
H =−A(τ
x
Q
2
+ τ
z
Q
3
) , (2)
where τ is the Pauli matrix and A is the coupling constant.
As discussed by
¨
Opik and Pryce [51], the total of the orbital-
lattice coupling in Eq. ( 2) and the Q
2
(Q
3
) r elated harmonic
potential energy can be minimized by an infinite number of
distortions, however, the orbital degeneracy can be lifted by the
FIG. 8. (Color online) The temperature versus Fe content (x)
phase diagram of Fe
1+x
Cr
2−x
O
4
. T
C
is the paramagnetic-to-collinear
ferrimagnetic phase transition temperature (black lines and dots),
T
N
is the collinear-to-conical ferrimagnetic phase transition tem-
perature (red lines and dots), T
S
1
is the cubic-to-tetragonal lattice
transition temperature (olive line with open squares), and T
S
2
is the
tetragonal-to-orthorhombic lattice transition temperature (blue line
with open squares). T he solid lines display the structural and magnetic
transitions, while the dashed lines display the possible structural
distortions.
134106-6
MAGNETIC AND STRUCTURAL PHASE TRANSITIONS IN . . . PHYSICAL REVIEW B 89, 134106 (2014)
anharmonic lattice potential term of the total potential energy,
V =
1
2
Mω
2
Q
2
+ A
3
Q
3
cos 3θ +···, (3)
where Q and θ are the polar coordinations for Q
2
-Q
3
space. A
3
is the term describing the anharmonic effect on the tetragonal
distortion. If A
3
> 0, the complex is compressed along the
tetragonal axis; if A
3
< 0, the complex is elongated along the
tetragonal axis [34,51].
For the low Fe-doping FeO
4
tetrahedron (x 0.3), the e
g
orbital shape is deduced to be of 3d
z
2
type in the paramagnetic
phase, thus A
3
is positive and a/c > 1, as shown in Fig. 8.
As more empty e
g
Cr
3+
ions are replaced by the half-filled
e
g
Fe
3+
ions, the Jahn-Teller distortion of Fe
2+
ions becomes
unstable and T
S
1
decreases gradually.
As temperature decreases, the ferrimagnetic state is reached
and the effect of spin-orbit coupling needs to be included in
the total Hamiltonian. Although the first-order perturbation
of spin-orbit coupling is absent, the second-order term λL · S
breaks the degeneracy of the two e
g
orbitals and lowers the
energy of the 3d
x
2
−y
2
relative to the 3d
z
2
orbital [34,51,52].
The second-order perturbation of the Hamiltonian H
SO
can be
presented as
H
SO
=
B
6
3S
2
z
− S
2
τ
z
−
√
3
S
2
x
− S
2
y
τ
x
, (4)
where B is the energy difference between 3d
x
2
−y
2
and 3d
z
2
states. Thus, the sign of A
3
is changed to negative to form the
orthorhombic phase at low temperature, and T
C
is above the
associated structural transition t emperature T
S
2
[34,51].
At the same time, the magnetic transition temperature could
be roughly estimated by mean-field-theory,
3k
B
T
C
= z
i,j
J
ij
S
i
· S
j
, (5)
where z is the number of nearest neighbors, and S and J
ij
are
the related moment and exchange energy, respectively.
Since the extra half-filled e
g
electrons of the doped
Fe
3+
ions increase not only the interaction between A-
and B-site ions (J
AB
), but also the total moment of B-site
ions(S
Fe
3+
/Cr
3+
), T
C
increases with the doping amount of Fe
3+
ions, which drives T
S
2
to increase. Compared to the decreasing
T
S
1
, they meet at around x = 0.3. Hence, x = 0.3isalsothe
boundary of the two tetragonal phases with different c/a,
Fig. 8.
As in the other spinel compounds with magnetic A
2+
ions,
MnCr
2
O
4
and CoCr
2
O
4
, a conical magnetic state is also ob-
served in the low Fe-doped FeCr
2
O
4
at the lower temperature
due to the geometrical magnetic frustration [13–15,18,19].
Furthermore, the transition temperature T
N
decreases with
Fe
3+
doping. Lyons et al. [16] had presented that the conical
state is complicated and deduced the structure from a factor
of u, which is closely related to the properties of both
moments and interactions between the A- and B-site cations,
4J
BB
S
B
/3J
AB
S
A
. They presented that the conical state is
stable as 8/9 u 1.298. This factor is possibly related
with the decreases of T
N
. However, it is hard to obtain the
u value for Fe
1+x
Cr
2−x
O
4
quantitatively because of lack of
information on the exchange energies, although S
B
, J
BB
, and
J
AB
are increasing with Fe doping. Inelastic neutron scattering
measurements using single crystal, similar to what has been
done for MnV
2
O
4
[53], are needed to clarify the situation.
Another possible reason for the decrease in T
N
of Fe-doped
FeCr
2
O
4
is that the extra Fe
3+
on the B site disturbs the
homogeneous frustrated interactions.
(ii) 0.3 <x 0.7: As in Region I (x 0.3), the doped-Fe
3+
ions occupy the B site of the spinel with the normal spinel
structure in Region II, Fe
2+
[Cr
3+
2−x
Fe
3+
x
]O
4
. Although the
tetragonal-to-orthorhombic phase transition still follows the
cubic-to-tetragonal transition, the lattice constant ratio, a/c,
of the tetragonal phase is l ess than one. In addition, the driving
forces of the two structural transitions are the reverse of those
in Region I, which means that the spin-orbital coupling effect
on the Jahn-Teller distortion leads to the cubic-to-tetragonal
transition, while the tetragonal-to-orthorhombic transition is
due to the B-site disorders of Fe
3+
and Cr
3+
ions. Thus, T
S
1
in-
creases with Fe -doping, and T
S
2
decreases to 0 as x approaches
0.7. For the magnetic ordering transitions, T
C
accompanies the
first structural distortion (T
S
1
), which increases linearly with
Fe doping. The spin reorientation transition, occurring at T
N
,
continues to decrease and disappears at x = 0.6.
(iii) 0.7 <x 1.0: Unlike Regions I and II, the doped-Fe
3+
ions begin to occupy the A site, and the Fe
2+
ions move to
the B site of the spinel at x 0.7, which makes the system
very complicated, Fe
2+
1.7−x
Fe
3+
x−0.7
[Cr
3+
2−x
Fe
3+
0.7
Fe
2+
x−0.7
]O
4
.As
presented in Fig. 8, T
C
increases more sharply than the linear
relationship in Region II due to the electron hopping effect
between A- and B-site Fe
2+
/Fe
3+
ions, which was confirmed
by the reported M
¨
ossbauer measurement [36,45].
The T
S
1
continues the ascending trend for x up to 0.8, after
which it disappears. Nevertheless, some structural distortions
seem to persist up to the high-doping region (0.8 x 1),
as evidenced by the anomalies of the magnetic susceptibility
measurements as shown in Fig. 2, as well as by the different
width of the (004) Bragg peak between 20 and 300 K, as
shown in Fig. 3(d). However, the magnetic ordering and
structural distortion temperature are disconnected and the T
C
rises steeply. If we still use T
S
1
to label the temperature of the
tetragonal distortion, it decreases due to the electron hopping
effect on the orbitals of the Fe
2+
and Fe
3+
ions, as more
Fe
3+
ions are introduced in the system, indicated by the green
dashed line in Fig. 8. Moreover, another structural distortion
is observed at lower temperature, and it is suggested to be
the orthorhombic/monoclinic distortion related to the extra
Jahn-Teller active Fe
2+
on the B site, which is related to the
t
2g
orbital freedom found in inverse spinel Fe
3
O
4
[54]. T
S
2
still
describes this distortion and is presented as the shaded region
in Fig. 8.
V. CONCLUSION
The structural and magnetic phase diagram of
Fe
1+x
Cr
2−x
O
4
is investigated by means of magnetization,
specific heat, x-ray, and neutron scattering measurements. The
substitution of Fe
3+
for Cr
3+
enhances the paramagnetic-to-
collinear ferrimagnetic transition temperature T
C
and reduces
the collinear-to-conical ferrimagnetic transition temperature
T
N
, which is likely due to the complicated interactions between
A- and B-site ions.
Systematic changes in the crystal structure with tem-
perature and composition are observed. In the low Fe
3+
-
doped compound (x 0.7), both cubic-to-tetragonal and
134106-7
J. MA et al. PHYSICAL REVIEW B 89, 134106 (2014)
tetragonal-to-orthorhombic transitions are driven by the Jahn-
Teller distortion and the related spin-orbital couplings. At
x 0.3, the magnetic energy stabilizes the orthorhombic
phase to increase T
S
2
, while the disorder of the B-site ions
(Fe
3+
/Cr
3+
) leads to the decreasing T
S
1
;at0.3 <x 0.7,
the magnetic energy increases T
S
1
, while the disorder of the
B-site ions decreases T
S
2
. In the high Fe
3+
-doped compound
(x>0.7), a strong electron hopping mechanism between Fe
2+
and Fe
3+
ions leads to the orbital-active Fe
2+
ions occupying
both the A and B site of the spinel, and the related e
g
and t
2g
orbital effects are observed, which results in the temperature
dependence of the lattice distortions.
ACKNOWLEDGMENTS
The research at the High Flux Isotope Reactor, Oak Ridge
National Laboratory was sponsored by the Scientific User
Facilities Division (J.M., M.M. C.D.C. V.O.G., S.C., W.T.,
A.A.A., and S.X.C.) and Center for Nanophase Materials
Sciences (A.R.), Office of Basic Energy Sciences, U.S.
Department of Energy. R.S. and H.D.Z. give thanks for the
support from the JDRD program of University of Tennessee.
Work at FSU is supported in part by NSF-DMR 1005293 and
carried out at the National High Magnetic Field Laboratory,
supported by the NSF, the DOE, and the State of Florida.
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