Magnetic interactions in the multiferroic phase of CuFe1 xGaxO2 (x = 0.035) refined by inelastic neutron scattering with uniaxial-pressure control of domain structure

Abstract

We have performed inelastic neutron scattering measurements in the ferroelectric noncollinear-magnetic phase of CuFe{sub 1-x}GaâOâ (CFGO) with x = 0.035 under applied uniaxial pressure. This system has three types of magnetic domains with three different orientations reflecting the trigonal symmetry of the crystal structure. To identify the magnetic excitation spectrum corresponding to a magnetic domain, we have produced a nearly single-domain multiferroic phase by applying a uniaxial pressure of 10 MPa onto the [1{bar 1}0] surfaces of a single crystal CFGO sample. As a result, we have successfully observed the single-domain spectrum in the multiferroic phase. Using the Hamiltonian employed in the previous inelastic neutron scattering study on the multi-domain multiferroic phase of CFGO (x = 0.035) [Haraldsen et al. Phys. Rev. B 82 020404R (2010)], we have refined the Hamiltonian parameters so as to simultaneously reproduce both of the observed single-domain and multi-domain spectra. Comparing between the Hamiltonian parameters in the multiferroic phase of CFGO and in the collinear four-sublattice magnetic ground state of undoped CuFeOâ [Nakajima et al, Phys. Rev. B 84 184401 (2011)], we suggest that the nonmagnetic substitution weakens the spin-lattice coupling, which often favors a collinear magnetic ordering, as a consequence of the partial release of the spin frustration.

Full text

PHYSICAL REVIEW B 85, 144405 (2012)
Magnetic interactions in the multiferroic phase of CuFe1−xGaxO2(x=0.035) refined by inelastic
neutron scattering with uniaxial-pressure control of domain structure
Taro Nakajima*and Setsuo Mitsuda
Department of Physics, Faculty of Science, Tokyo University of Science, Tokyo 162-8601, Japan
Jason T. Haraldsen
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA and Center for Integrated Nanotechnologies,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Randy S. Fishman
Material Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
Tao Hong
Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
Noriki Terada
National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan
Yoshiya Uwatoko
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 903-0213, Japan
(Received 3 February 2012; revised manuscript received 23 March 2012; published 9 April 2012)
We have performed inelastic neutron scattering measurements in the ferroelectric noncollinear-magnetic phase
of CuFe1−xGaxO2(CFGO) with x=0.035 under applied uniaxial pressure. This system has three types of
magnetic domains with three different orientations reflecting the trigonal symmetry of the crystal structure.
To identify the magnetic excitation spectrum corresponding to a magnetic domain, we have produced a nearly
“single-domain” multiferroic phase by applying a uniaxial pressure of 10 MPa onto the [1¯
10] surfaces of a
single-crystal CFGO sample. As a result, we have successfully observed the single-domain spectrum in the
multiferroic phase. Using the Hamiltonian employed in the previous inelastic neutron scattering study on the
“multi-domain” multiferroic phase of CFGO (x=0.035) [Haraldsen et al. Phys. Rev. B 82, 020404(R) (2010)],
we have refined the Hamiltonian parameters so as to simultaneously reproduce both of the observed single-domain
and multidomaim spectra. Comparing between the Hamiltonian parameters in the multiferroic phase of CFGO
and in the collinear four-sublattice magnetic ground state of undoped CuFeO2[Nakajima et al.,Phys. Rev. B
84, 184401 (2011)], we suggest that the nonmagnetic substitution weakens the spin-lattice coupling, which often
favors a collinear magnetic ordering, as a consequence of the partial release of the spin frustration.
DOI: 10.1103/PhysRevB.85.144405 PACS number(s): 75.30.Ds, 78.70.Nx, 75.80.+q, 75.85.+t
I. INTRODUCTION
A triangular lattice antiferromagnet CuFeO2(CFO) is
known as a spin-driven magneto-electric multiferroic.1While
the ground state of CFO is a nonferroelectric collinear four-
sublattice (4SL) magnetic phase with a magnetic modulation
wave vector of ( 1
4,1
4,3
2),2the spin-driven ferroelectricity
appears in the first magnetic-field induced phase from the 4SL
phase.1By substituting only a few percent of nonmagnetic ions
(Al3+,Ga
3+,orRh
3+) for the magnetic Fe3+sites, the transi-
tion field from the 4SL phase to the multiferroic phase is readily
reduced down to zero field;3–6for example, in CuFe1−xGaxO2
(CFGO) with x=0.035, the 4SL phase disappears, and
instead the multiferroic phase shows up as a ground state.5This
implies that the 4SL and the multiferroic phases are almost
degenerate in energy owing to the magnetic frustration in this
system, and thus a small perturbation such as a few percent
of nonmagnetic substitution can affect the magnetic ground
state. Previous elastic neutron scattering measurements using
CFGO and CuFe1−xAlxO2(CFAO) samples have revealed that
the magnetic structure in the multiferroic phase is a screw-type
structure having a magnetic modulation wave vector of (q,q,3
2)
with q=0.203.7,8Recent theoretical studies by Haraldsen
et al. have pointed out that the turn angles of the screw-type
helical arrangement of the spins have anharmonicity.9,10
To elucidate how a small amount of nonmagnetic sub-
stitution changes the magnetic ground state of this system,
it is important to determine the Hamiltonian parameters in
both of the 4SL and multiferroic phases. In a previous
study, Ye et al. have performed inelastic neutron scattering
measurements on CFO to determine the parameters in the 4SL
phase.11 They have observed spin-wave excitation spectra in
the multi-domain 4SL phase, in which three types of magnetic
domains with modulation wave vectors of (1
4,1
4,3
2), ( 1
4,−1
2,3
2),
and (−1
2,1
4,3
2) coexist. The formation of these three domains is
due to the trigonal symmetry of the original crystal structure of
CFO. Hence the magnetic excitation spectra observed in the
multi-domain 4SL phase are mixtures of the spectra arising
from the three domains having the different orientations.
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TARO NAKAJIMA et al. PHYSICAL REVIEW B 85, 144405 (2012)
Fe
3+
Cu
+
O
2-
(a)
(c) (d) (e)
(b)
(110)-domain
am
bma
b
Q
Q
a*
b*a*
b*
ab
bm
am
a*
b*
Q
a
b
bm
am
(210)-domain(120)-domain
b
a
c
J3J2
Jz1 Jz2
Jz3
FIG. 1. (Color online) (a) Crystal structure of CuFeO2with a
hexagonal basis. (b) Paths of the exchange interactions. For the
nearest-neighbor exchange interactions, J1,seeFigs.4(a) and 4(b).
(c)–(e) The three types of magnetic domains with the monoclinic
bases (am,bm) and the hexagonal bases (a,b). Dashed arrows denote
c∗-plane projections of the magnetic modulation wave vectors in each
domain.
On the other hand, Nakajima et al. have recently demon-
strated that volume fractions of the three domains can be
controlled by application of uniaxial pressure.12,13 Since the
formation of the three magnetic domains accompanies with
monoclinic lattice distortions in each domain as illustrated in
Figs. 1(c)–1(e), an application of uniaxial pressure, which is
directly coupled with the lattice degree of freedom, results
in significant changes in the volume fractions of the mag-
netic/crystal domains. Actually, in Ref. 13, a single-domain
4SL phase is achieved by applying a uniaxial pressure of only
10 MPa onto the [1¯
10] surfaces of the single-crystal CFO
sample. Recent inelastic neutron scattering measurements
using the slightly pressurized CFO sample have successfully
identified the spin-wave dispersion relations belonging to a
magnetic domain in the 4SL phase.14 Starting from the Hamil-
tonian used in the previous theoretical spin-wave analysis by
Fishman,15 Nakajima et al. have refined the parameters, so as
to reproduce the spin-wave dispersions in both of the single-
and multi-domain 4SL states.14
As for the multiferroic phase, Haraldsen et al. have
recently performed inelastic neutron scattering measurements
using CFGO (x=0.035) samples, and have deduced the
Hamiltonian parameters from magnetic excitation spectra in
the multi-domain multiferroic phase.9In the present study, we
have performed inelastic neutron scattering measurements on
CFGO (x=0.035) under applied uniaxial pressure, in order
to observe single-domain magnetic excitation spectra, from
which we can more accurately determine the Hamiltonian
parameters in the multiferroic phase.
II. EXPERIMENTAL DETAILS AND
PRELIMINARY RESULTS
Single crystals of CFGO (x=0.035) of nominal compo-
sitions were grown by the floating zone method.16 Before
the present measurements with uniaxial pressure, we have
performed inelastic neutron scattering measurements in the
multi-domain multiferroic phase using three as-grown crystals
(total mass 4.8 g). We used a triple-axis neutron spectrometer
HER(C1-1) at JRR-3, Japan Atomic Energy Agency. The
energy of the scattered neutrons was fixed at Ef=3.5meV,
and the horizontal collimation was open 80’-80’-80’. The three
single-crystal CFGO (x=0.035) samples were co-aligned
and mounted in a pumped 4He cryostat with the (H,H,L)
scattering plane. Figure 2(b) shows the magnetic excitation
spectrum along the (H,H,3
2) line at 2.0 K in the multiferroic
phase. We have confirmed that the observed spectrum is the
same as that in previous measurements by Haraldsen et al.9
After this measurement, we cut one of the three crystals into
a plate shape (3.0×3.2×17 mm3) with the widest surfaces
normal to the [1¯
10] direction. The mass of the plate-shaped
sample is 0.9 g. The sample was set into a uniaxial pressure
cell, which is almost the same as the pressure cells used in
Refs. 13,14, and 17. A uniaxial pressure of 10 MPa was applied
on the [1¯
10] surfaces of the sample at room temperature,
and was kept throughout the present experiment by CuBe
disk springs set in the bottom of the pressure cell. We have
performed inelastic neutron scattering measurements using the
pressurized CFGO sample at another cold neutron triple-axis
spectrometer CTAX(CG-4C) at the High Flux Isotope Reactor
(HFIR), Oak Ridge National Laboratory. The pressure cell
was loaded into a pumped 4He cryostat with the (H,H,L)
scattering plane. The energy of the scattered neutrons was
mainly fixed at Ef=4.0 meV. To measure the excitation
spectra below 0.5 meV, we also used Ef=2.9 meV. The
energy resolutions at elastic conditions in the measurements
with Ef=4.0 and 2.9 meV are 0.2 and 0.1 meV (full width at
half maximum), respectively. The higher-order contaminations
were removed by a cooled Be filter placed between the sample
and the analyzer.
III. EXPERIMENTAL RESULTS
We have estimated the volume fractions of the three
magnetic domains, in the same manner as in Ref. 12.Using
the notations in Ref. 12, we refer to the three domains as
(110), (1¯
20), and (¯
210) domains [see Figs. 1(c)–1(e)]. In
the multiferroic phase at 2 K, we have measured integrated
intensities of three magnetic Bragg reflections located on
(or near) the (H,H,L) scattering plane; specifically (q,q,3
2),
(1
2−q,2q,1
2), and (2q,1
2−q,−1
2) reflections, which belong to
the (110), (1¯
20), and (¯
210) domains, respectively. Comparing
the observed integrated intensities with the calculated structure
factors for these reflections, we have determined the volume
fraction of the (110), (1¯
20), and (¯
210) domains as 0.78, 0.09,
and 0.13, respectively. This indicates that the (110) domain,
144405-2

MAGNETIC INTERACTIONS IN THE MULTIFERROIC ... PHYSICAL REVIEW B 85, 144405 (2012)
3.0
2.5
2.0
1.5
1.0
0.5
0.0
2.5
2.0
1.5
1.0
0.5
0.0
Energy (meV) Energy (meV)
0.0
(a)
(b)
0.1 0.2 0.3 0.4 0.5
H (r. l. u.)
(c)
(1/2,1/2,0)
(0,0,0)
(0,0,3) (1/2,1/2,3)
(H,H,3/2)
140
80
0
120
100
60
40
20
200
50
0
150
100
140
FIG. 2. (Color online) (a) Calculated and (b) observed intensity
map of the magnetic excitation spectra along the (H,H,3
2) line in the
multi-domain multiferroic phase. (c) Reciprocal-lattice map of the
(H,H,L) scattering plane. The dashed arrow denotes the direction of
the the (H,H,3
2) line.
whose propagation wave vector of (q,q,3
2) lies in the (H,H,L)
scattering plane, dominates over the others.
In the (nearly) single-domain multiferroic phase at 2 K,
we have performed constant-wave-vector (constant-q) scans
with Ef=4.0 meV, and have obtained an intensity map
along the (H,H,3
2) line as shown in Fig. 3(b).Wehave
found that intensities around H≈0.3 and E≈1.7 meV are
reduced in the single-domain state, as compared to those in
the multi-domain state. This is consistent with the previous
calculation by Haraldsen et al.;9they have pointed out that the
intensities around H≈0.3 and E≈1.7 meV are attributed
to the magnetic excitation spectra arising from the (1¯
20) and
(¯
210) domains. We also found that the intensities around the
shoulder at H≈0.08 are also reduced in the single-domain
state. While the previous calculation has suggested that the
3.0
2.5
2.0
1.5
1.0
0.5
0.0
2.5
2.0
1.5
1.0
0.5
0.0
Energy (meV) Energy (meV)
0.0 0.1 0.2 0.3 0.4 0.5
H (r. l. u.)
(a)
(b)
0.16 0.18 0.2 0.22 0.24
0
50
0
50
100
0
50
Intensity (counts/ ~ 4 min)
H (r. l. u.)
(c)
0.26 0.28 0.3 0.32 0.34
0
50
Intensity (counts/ ~4 min)
H (r. l. u.)
0
50
(d)
(c) (d)
E = 0.5 meV
E = 0.4 meV
E = 0.3 meV
E = 0.5 meV
E = 0.3 meV
200
80
0
160
120
40
140
80
0
120
100
60
40
20
140
FIG. 3. (Color online) (a) Calculated and (b) observed intensity
map of the magnetic excitation spectra along the (H,H,3
2) line in the
single-domain multiferroic phase. (c),(d) Profiles of the constant-E
scans below 0.5 meV around (c) H=0.2 and (d) 0.3.
shoulder belongs to the (110) domain,9the present results
indicate that it belongs to the other two domains.
To investigate the low-energy magnetic excitations be-
low 0.5 meV, we have performed constant-energy-transfer
(constant-E) scans with Ef=2.9 meV, as shown in Figs. 3(c)
and 3(d). These scattering profiles are almost the same as those
in the previous high-resolution inelastic neutron scattering
measurements in the multi-domain state,9indicating that in
the (H,H,3
2) line, the dispersions of the magnetic excitations
in the (1¯
20) and (¯
210) domains lie above 0.5 meV.
IV. CALCULATIONS AND DISCUSSIONS
To find a set of parameters reproducing both of the
single-domain spectrum and the multi-domain spectrum in
144405-3

TARO NAKAJIMA et al. PHYSICAL REVIEW B 85, 144405 (2012)
(a) (b)
(c)
bm
am
J1
(3)
J1
(1)
J1
(2)
bm
amJ1
(3) J1
(2)
J1
(1)
Fe (up spin) Fe (down spin)
O (upper layer)
O (lower layer)
FIG. 4. (Color online) The relationship among the magnetic
structure, the displacements of the oxygen ions and the nearest-
neighbor exchange interactions of J(1)
1,J(2)
1,andJ(3)
1in (a) the
4SL phase and (b) the multiferroic phase with the monoclinic bases.
(c) Calculated spin arrangements in the multiferroic phase.
Ref. 918 we started from the Hamiltonian used in the previous
calculation for the multi-domain spectra in the multiferroic
phase:9,10
H=−1
2
i=j
Jij Si·Sj−D
iSz
i2,(1)
in which three in-plane exchange interactions (J1,J2, and J3),
three interplane exchange interactions (Jz1,Jz2, and Jz3), a
single ion uniaxial anisotropy (D), and a lattice distortion
parameter (K) are employed.9,10 The Kparameter splits the
nearest-neighbor exchange interactions, J1, into two strong
interactions, J(1)
1=J(2)
1=J1−K
2and a weak interaction,
J(3)
1=J1+K, as shown in Fig. 4(b). It should be noted
here that the application of the uniaxial pressure of 10 MPa
does not result in significant changes in the magnetic-phase
transition temperatures in this system.12,13 Therefore we have
considered that the Hamiltonian parameters in the slightly
pressurized single-domain sample are the same as those in
the multi-domain sample.
Since Fe3+is S=5/2, the energy can be minimized
classically. Therefore, to assess the precise magnetic ground
states and their energies, we employ a variational method on a
large lattice of 2.0×104sites, where the spin harmonics are
incorporated by defining Szwithin the plane as
Sz(R)=A
l=0
C2l+1cos[Q(2l+1) ·x]
−
l=0
B2l+1sin[(2π−Q)(2l+1) ·x],(2)
where the C2l+1and B2l+1harmonics are produced by the
anisotropy energy Dand the lattice distortion K, respectively.
The observed elastic intensities at odd multiples of Qand
2π−Qare proportional to the square of these harmonics.
The function Sz(R) is normalized so that the maximum of
|Sz(R)|is S=5/2. The perpendicular spin Syis given by
Sy(R)=S−Sz(R)2sgn[g(R)],(3)
where
g(R)=sin(Q·x)+B1
C1
cos[(2π−Q)·x].(4)
Based on the provided magnetic ground state, the spin-wave
(SW) dynamics are evaluated using a Holstein-Primakoff
transformation with the spin operators given by Siz =S−
a†
iai,Si+=√2Sai, and Si−=√2Sa†
i(aiand a†
iare bosonic
destruction and creation operators). The local spin operators
account for the noncollinearity through a general rotation
matrix.19,20
The spin-wave frequencies ωqare determined by the
equations of motion, which are solved for the vectors vq=
[a(1)
q,a(1)†
−q,a(2)
q,a(2)†
−q,...]. This can be in terms of the 2N×2N
matrix M(q)asidvq/dt =−[H2,vq]=M(q)vq, where Nis
the number of the spin sites in the large lattice employed in the
present calculation, specifically 2.0×104.19 The spin-wave
frequencies are calculated from the condition Det[M(q)−
ωqI]=0, where the SW frequencies must be real and positive
and all SW weights must be positive to assure the local stability
of a magnetic phase.
To determine the spin-wave intensities, we examine the
coefficients of the spin-spin correlation function:
S(q,ω)=
αβ
(δαβ −qαqβ)Sαβ (q,ω),(5)
where αand βare x,y,orz.21 A more detailed discussion of
this method is contained in Ref. 19. The total intensity I(q,ω)
for a constant-qscan is given by
I(q,ω)=S(q,ω)F2
qe(−(ω−ωq)2/2δ2),(6)
where δis the energy resolution and Fqis the Fe3+ionic
form factor, which is given as Fq=j0(q), where j0(q)=
A0ea0s2+B0eb0s2+C0ec0s2+D0and s=sin θ/λ =q/4π.
The coefficients are A0=0.3972 (a0=13.2442), B0=0.6295
(b0=4.9034), C0=−0.0314 (c0=0.3496), and D0=0.0044
from Ref. 22. The simulated energy resolution is based on a
Gaussian distribution, which is standard for constant-qscans
on a triple-axis spectrometer.23,24
The simulated spectra, produced by the energy-minimized
state for select parameters, is then compared to the observed
multi-domain and single-domain spectra. The parameters are
manually adjusted and the spectra are recalculated until the
calculated distributions of the intensities in the q-ωspace
reproduce the overall features of the observed multi-domain
and single-domain spectra.
Figures 2(a) and 3(a) show the results of the calculations for
the multi-domain and single-domain states, respectively. These
calculations have well captured the overall features of the
observed spectra. In particular, the reduction of the intensities
of the shoulder around H≈0.08 in the single-domain state
is successfully reproduced. This reduction is not explained
by the previous Hamiltonian parameters determined only
from the multi-domain spectra.9This demonstrates that the
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MAGNETIC INTERACTIONS IN THE MULTIFERROIC ... PHYSICAL REVIEW B 85, 144405 (2012)
TABLE I. The Hamiltonian parameters in the 4SL phase (taken from Ref. 14) and the multiferroic phase (in meV). Note that in Ref. 14,
the distant interactions, Ji(i=2,3,z), are also assumed to split into two Jiand J
iowing to the monoclinic lattice distortion, although the
splits are found to be relatively small (|Ji−J
i|/Ji∼0.1). In this table, we show the mean values of them.
Composition J(1)
1J(2)
1J(3)
1J2J3Jz1Jz2Jz3DK
CuFeO2−0.182 −0.169 −0.060 −0.041 −0.142 −0.071 0.064 0.077
CuFe1−xGaxO2(x=0.035) −0.169 −0.169 −0.066 −0.070 −0.098 −0.070 0.014 −0.006 0.007 0.071
uniaxial-pressure control of the magnetic domain structure
is highly effective to investigate the magnetic excitations in
this system. The refined parameters are shown in Table I.
We have found that the magnitudes of J1,J2, and J3slightly
decrease, on the contrary, the magnitude of Jzincreases, as
compared to the results of the previous analysis using only
the multi-domain spectra.9Figure 4(c) shows the refined spin
configuration, which remains almost the same as that in the
previous study.9
We now discuss the relationship between the Hamiltonian
parameters in the 4SL phase of CFO and the multiferroic
phase of CFGO. As shown in Table I, the exchange interactions
(J(1)
1∼Jz) in the 4SL and multiferroic phases are close to each
other. On the other hand, the uniaxial single-ion anisotropy D
in the multiferroic phase is significantly smaller than that in the
4SL phase. As was discussed in the previous works,9,10,15 this
reduction of Daccounts for the change in magnetic excitations
from the gapped spin-wave excitations observed in the 4SL
phase to the Goldstone modes emerging from H≈0.2 and
0.3 in the multiferroic phase. Therefore this must be one of the
reasons for the disappearance of the collinear 4SL magnetic
ground state. However, at this moment, we have no clear
explanations on the reduction of D. Further investigations to
elucidate electronic states of Fe3+ions in CFO and CFGO will
be needed.
We subsequently focus on the small differences in exchange
interactions, especially in nearest-neighbor interactions, be-
tween the 4SL and multiferroic phases. Although we could
not deduce errors of the refined Hamiltonian parameters from
the present analysis, we suggest the possibility that the small
differences in exchange interactions can also contribute to the
drastic change in the magnetic ground state, as discussed in
the following.
In the previous spin-wave analysis in the 4SL phase,14
Nakajima et al. have assumed that J1splits into three different
interactions, J(1)
1,J(2)
1, and J(3)
1. This assumption is based
on the “scalene triangle distortion model” proposed in the
synchrotron radiation x-ray-diffraction study by Terada et al.25
They have observed superlattice reflections in the 4SL phase of
CFO, suggesting that the equilateral symmetry of the triangular
lattice is broken due to displacements of oxygen ions, as
illustrated in Fig. 4(a). These displacements are explained in
terms of “magnetostriction,” and the resultant splitting of J1is
expected to stabilize the 4SL magnetic order.26 The spin-wave
fitting analysis for the 4SL phase has revealed that J(1)
1and
J(2)
1are comparable to each other, and are rather stronger
than J(3)
1, as shown in Table I. This relationship is similar to
the splitting of J1in the previous theoretical works on the
multiferroic phase.9,10,27 This indicates that the effects of the
lattice distortion, namely the oxygen displacements, on the
nearest-neighbor interactions can be well described by the K
parameter in both of the 4SL and multiferroic phases. This is
reasonable because the superlattice reflections corresponding
to the displacements of the oxygen ions have been observed not
only in the 4SL phase but also in the multiferroic phases.26,28,29
In order to quantitatively compare the effects of the lattice
distortions in the 4SL and the multiferroic phases, we have
estimated Kand J1for the 4SL phase, as follows:
K=J(3)
1−J(1)
1+J(2)
1
22
3,(7)
J1=J(1)
1+J(2)
1
2+K
2.(8)
As a result, Kand J1in the 4SL phase are estimated to be
0.077 and −0.138 meV, respectively. We found that J1in
the 4SL phase is almost the same as that in the multiferroic
phase (J1=−0.140 meV). On the other hand, the value of
K/|J1|, which shows the magnitude of the effect of the lattice
distortion, is 0.56 in the 4SL phase; this value is slightly
larger than that in the multiferroic phase (K/|J1|=0.51).
This implies that the effect of the lattice distortion on J1is
slightly weakened in the Ga-doped system. Taking account
of the fact that CFO exhibits the lattice distortion in order
to relieve the spin frustration, it is natural to consider that
the nonmagnetic substitution, which partially releases the spin
frustration, reduces the lattice distortions and their effects on
the exchange interactions. In other words, the nonmagnetic
substitution weakens the spin-lattice coupling in this system.
Recent theoretical studies have pointed out that strong
spin-lattice coupling favors a collinear magnetic ordering.30,31
The spin-lattice coupling in a localized spin system can be
explained as a bond-length (or bonding-angle) dependence
of a exchange interaction Jbetween two spins Siand Sj;
specifically, a lattice distortion changes the energy of the bond
by −Jαρ
i,j Si·Sj+Eela, where αis a spin-lattice coupling
constant, ρi,j is the change in distance between the two spins,
and Eela is the loss of the elastic energy due to the distortion.
To maximize the energy gain due to the distortion, the value
of Si·Sjshould be ±1(the±sign depends on the signs of
J,ρi,j , and α), and consequently the spins favor a collinear
arrangement when the spin-lattice coupling is strong. From
the above considerations and the present results, we suggest
that the reduction of the spin-lattice coupling can be also
one of the reasons for the disappearance of the collinear
4SL magnetic ground state in this system. This is consistent
with the theoretical study on the CFO system by Haraldsen
and Fishman; they have pointed out that the ground state
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TARO NAKAJIMA et al. PHYSICAL REVIEW B 85, 144405 (2012)
changes from the anharmonic screw-type magnetic order to
the collinear 4SL magnetic order with increasing K/|J1|.10
It is worth mentioning here that in previous synchrotron
radiation measurements on CFO under applied magnetic
field, Terada et al. have reported that the intensity of the
superlattice reflection corresponding to degree of the scalene
triangular lattice distortion starts to decrease at the magnetic-
field induced phase transition from the 4SL phase to the
multiferroic phase.29 This implies that the spin-lattice coupling
also plays an important role in the magnetic-field induced
phase transitions in the CFO system.
We should also mention the distant interactions. As seen in
Table I, there are small differences in the values of J2and J3
between the multiferroic phase and the 4SL phase, although the
relationship of J3>J
2holds in both of the phases. This might
be because Jz2and Jz3are neglected in the spin-wave analysis
in the 4SL phase in Ref. 14. Another possibility is the effect of
the monoclinic lattice distortion. Nakajima et al. have pointed
out that the monoclinic lattice distortion affects the distant
exchange interactions, although this effect is relatively small
as compared to the effect of the oxygen displacements on J1.14
Because the previous x-ray-diffraction studies have revealed
that the degree of the monoclinic distortion in the 4SL phase
is larger than that in the multiferroic phase,29,32 the difference
in the monoclinic lattice constants could be the reason for the
differences in J2and J3.
V. CONCLUSION
We have performed inelastic neutron scattering measure-
ments on a multiferroic CFGO (x=0.035) in order to refine
the Hamiltonian parameters in the multiferroic phase. By
applying a uniaxial pressure of 10 MPa onto the [1¯
10] surfaces
of the single-crystal CFGO sample, we have produced a nearly
single-domain multiferroic phase. We have successfully ob-
served the magnetic excitation spectrum in the single-domain
state. Using the Hamiltonian employed in the previous inelastic
neutron scattering study on a multi-domain multiferroic phase
of CFGO (x=0.035),9we have refined the Hamiltonian
parameters so as to simultaneously reproduce both of the
observed single-domain and multi-domain spectra. Comparing
the refined Hamiltonian parameters in the multiferroic phase
with those in the collinear 4SL magnetic ground state of
undoped CFO,14 we have found that the single-ion uniaxial
anisotropy Dis significantly reduced and the lattice distortion
parameter Kis slightly reduced by the nonmagnetic substitu-
tion. Although the disappearance of the collinear 4SL magnetic
ground state can be mainly ascribed to the reduction of D,we
have suggested that the reduction of the spin-lattice coupling,
which reflects the partial release of the magnetic frustration
due to the nonmagnetic substitution, can also contribute to
the emergence of the noncollinear incommensurate magnetic
ground state.
ACKNOWLEDGMENTS
This work was supported by a Grant-in-Aid for Young
Scientist (B) (Grant No. 23740277), from JSPS, Japan. The
neutron scattering measurement at JRR-3 was carried out along
Proposal No. 8588B. The neutron scattering measurement at
HFIR was conducted (Proposal No. IPTS-5820) under the
emergency-proposal-transfer program from JRR-3 to HFIR
with the approval of Institute for Solid State Physics, The
University of Tokyo (Proposal No. 11571), Japan Atomic
Energy Agency, Tokai, Japan. The work by T.H. at the
High Flux Isotope Reactor was partially supported by the
US Department of Energy, Office of Basic Energy Sciences,
Division of Scientific User Facilities. We are grateful to
M. Matsuda and J. A. Fernandez-Baca for fruitful discussions.
The images of the crystal and magnetic structures in this
paper were depicted using the software VESTA developed by
K. Momma.33 Research by JTH was supported by the Center
for Integrated Nanotechnologies, a US Department of Energy,
Office of Basic Energy Sciences user facility. Los Alamos
National Laboratory, an affirmative action equal opportunity
employer, is operated by Los Alamos National Security,
LLC, for the National Nuclear Security Administration of
the US Department of Energy under Contract No. DE-AC52-
06NA25396. Research by R.F. was sponsored by the US
Department of Energy, Office of Basic Energy Sciences,
Materials Sciences and Engineering Division.
*nakajima@nsmsmac4.ph.kagu.tus.ac.jp
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