Spin waves and the origin of commensurate magnetism in Ba2CoGe2O7
ABSTRACT The square-lattice antiferromagnet Ba2CoGe2O7 is studied by means of neutron diffraction and inelastic scattering. This material is isostructural to the well-known Dzyaloshinskii-Moriya helimagnet Ba2CuGe2O7, but exhibits commensurate long-range Neel order at low temperatures. Measurements of the spin wave dispersion relation reveal strong in-plane anisotropy that is the likely reason for the suppression of helimagnetism. Comment: 5 pages 3 figures, submitted to Phys. Rev. B
arXiv:cond-mat/0302592v1 [cond-mat.str-el] 27 Feb 2003
Spin waves and the origin of commensurate magnetism in Ba2CoGe2O7.
Condensed Matter Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6393, USA.
T. Sato, T. Masuda†, and K. Uchinokura
Department of Advanced Materials Science, The University of Tokyo, Tokyo 113-8656, Japan.
Physics Department, Brookhaven National Laboratory, Upton, NY 11973-5000, USA.
Laboratory for Neutron Scattering, ETH Zurich and Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland.
(Dated: February 2, 2008)
The square-lattice antiferromagnet Ba2CoGe2O7is studied by means of neutron diffraction and in-
elastic scattering. This material is isostructural to the well-known Dzyaloshinskii-Moriya helimagnet
Ba2CuGe2O7 but exhibits commensurate long-range N´ eel order at low temperatures. Measurements
of the spin wave dispersion relation reveal strong in-plane anisotropy that is the likely reason for
the suppression of helimagnetism.
Ba2CuGe2O7was recognized as an extremely interesting
material for studying Dzyaloshinskii-Moriya (DM) off-
diagonal exchange interactions. A great deal of attention
was given to the incommensurate nature of the magnetic
ground state,1a unique field-induced incommensurate-
to-commensurate (IC) transition,2,3and the field de-
pendence of the spiral spin structure.4Studies of the
spin wave spectrum in the incommensurate5,6
commensurate6phases led to the first direct observa-
tion of a new type of magnetic interactions in insulators,
the so-called KSEA term.7,8,9,10,11Additional theoretical
studies shed light on the nature of the so-called “inter-
A recent study indicated that an IC transition in
Ba2CuGe2O7 can be induced not only by applying an
external magnetic field, but also by a partial chem-
ical substitution of the spin-carrying Cu2+ions by
Co2+.13For all Co-concentrations x the solid solution
Ba2(CoxCu1−x)Ge2O7 orders magnetically at tempera-
tures between TN = 3.2 K (x = 0) and TN = 6.7 K
(x = 1). Magnetization data suggest that the helimag-
netic state realized at x = 0 gives way to a canted weak-
ferromagnetic structure at some critical concentration
xc, estimated to be between 0.05 and 0.1. The mech-
anism of this transition or crossover is poorly under-
stood. One possible explanation was proposed in Ref. 13.
The structure of Ba2(CoxCu1−x)Ge2O7is tetragonal, the
S = 1/2 Cu2+or S = 3/2 Co2+ions forming a square
lattice within the (a,b) crystallographic plane. The dom-
†Present address: Condensed Matter Sciences Division, Oak Ridge
National Laboratory, Oak Ridge, TN 37831-6393, USA.
inant interaction is the antiferromagnetic (AF) coupling
J between nearest-neighbor (NN) sites along the (1,1,0)
direction.In the x = 0 compound the helimagnetic
distortion is caused by the in-plane component Dxy of
the Dzyaloshinskii vector D associated with the same
Cu-Cu bonds.2,3This component retains its direction
from one bond to the next, and thus favors a spin spi-
ral state.14In contrast, the out-of-plane component Dz
is sign-alternating and stabilizes a weak-ferromagnetic
structure. The z-axis component of D was never detected
in Ba2CuGe2O7, where it was assumed to be weak. In
Ref. 13 it was tentatively suggested that the out-of-plane
component is dominant in the Co-based x = 1 mate-
rial, and stabilizes a commensurate magnetic structure.
To verify this hypothesis and better understand the un-
derlying physics, a detailed knowledge of magnetic inter-
actions not only in Ba2CuGe2O7 (x = 0), but also in
Ba2CoGe2O7(x = 1) is required. In the present paper
we report the results of neutron diffraction and inelastic
neutron scattering measurements on the x = 1 material
To date, the exact crystal structure of Ba2CoGe2O7
has not been determined. However, powder data15in-
dicate that the material is very similar to its Cu-based
counterpart and is characterized by the P421m crys-
tallographic space group.
Ba2CoGe2O7 are a = b = 8.410˚ A and c = 5.537˚ A,
as measured at T = 10 K. In each crystallographic unit
cell the magnetic Co2+ions are located at (0,0,0) and
(0.5,0.5,0) positions. The NN Co-Co distance is thus
along the (1,1,0) direction and equal to a/√2 ≈ 5.9˚ A.
For the present study we utilized two single crystal sam-
ples prepared using the floating zone technique. Both
The lattice parameters for
k(r. l. u.)
magnetic Bragg peak intensity in Ba2CoGe2O7 (symbols).
The solid line is a guide for the eye. Left inset: transverse
scans (rocking curves) across the (1,0,0) Bragg reflection
measured below (open circles) and above (solid circles) the
N´ eel temperature TN ≈ 6.7 K. The residual intensity seen at
T > TN is of non-magnetic origin and due to multiple scatter-
ing. Right inset: the proposed model for the spin structure of
Ba2CoGe2O7. All spins are within the (a,b) crystallographic
Measured temperature dependence of the (1,0,0)
crystals were cylindrical, roughly 5 mm diameter ×
50 mm long, with a mosaic spread of about 0.4◦.
The first series of experiments was carried out at the
HB1 3-axis spectrometer installed at the High Flux Iso-
tope reactor at Oak Ridge National Laboratory (Setup
I). Its main purpose was to determine the spin arrange-
ment in the magnetically ordered state.
was mounted with the b axis vertical making (h,0,l) re-
flections accessible for measurements. Neutrons with a
fixed incident energy of 13.5 meV were used in combina-
tion with a pyrolytic graphite (PG) monochromator and
analyzer, 30′− 40′− 20′− 120′collimation, and a PG
higher-order filter. Sample environment was a closed-
cycle refrigerator that allowed measurements at temper-
atures down to 3.5 K. To isolate the magnetic contribu-
tion, integrated intensities were measured in a series of
rocking curves at T = 3.5 K< TNand T = 10 K> TN.
While using Setup I, it became apparent that the study
of magnetic excitations could be much better carried out
using of a cold neutron instrument, the relevant energy
scale for Ba2CoGe2O7 being about 2 meV. These mea-
surements were therefore performed at the TASP 3-axis
spectrometer installed at the SINQ spallation source at
Paul Scherrer Institut (Setup II). Neutrons with a fixed
final energy of 5.5 meV were used with PG monochroma-
tor and analyzer, and a PG filter after the sample. The
beam collimation was (guide) − 80′− 80′− (open). The
sample was mounted with the c axis vertical, making mo-
mentum transfers in the (h,k,0) reciprocal-space plane
accessible for measurement. Spin wave dispersion curves
were measured along the (h,0,0) and (h,h,0) directions
using constant-Q scans in the energy range 0–4 meV.
The sample environment was a standard “ILL Orange”
He-4 flow cryostat, and most of the data were taken at
T = 2 K.
A.A model for the spin structure
In the Ba2CuGe2O7 system magnetic ordering gives
rise to incommensurate peaks surrounding the integer
h, k and l reciprocal-space points.1In contrast, in
Ba2CoGe2O7 magnetic Bragg scattering was detected
below TN = 6.7 K at strictly commensurate positions
h, k and l-integer. Due to their location, the mag-
netic reflections, except those on the (h,0,0) and (0,k,0)
reciprocal-space rods, coincide with nuclear ones. Fig-
ure 1 shows the measured temperature dependence of
the (1,0,0) peak intensity (Setup II). The insert shows
rocking curves measured above and below the ordering
temperature. The appreciable residual intensity seen at
T > TNat the (1,0,0) forbidden nuclear peak position is
due to multiple scattering. 19 non-equivalent magnetic
Bragg intensities measured using Setup I were normalized
by the resolution volume, which in a 3-axis experiment
plays the role of the Lorentz factor. In our case it was cal-
culated using the Cooper-Nathans approximation. The
resolution-corrected magnetic intensities Iobs are listed
in Table I. The observed intensity pattern indicates a
planar spin arrangement, with all spins confined to the
(a,b) plane, and nearest-neighbor spins aligned antipar-
allel with each other. The alignment of nearest-neighbor
spins along the c direction is “ferromagnetic”. Such a
spin structure is identical to the one in the commensu-
rate spin-flop phase of Ba2CuGe2O7 stabilized by an ex-
ternal magnetic field applied along the c axis.2,3As can
be seen from Table I, where Icalcare the calculated mag-
netic intensities, this simple collinear model reproduces
our limited diffraction data for Ba2CoGe2O7rather well.
Due to the possibility of antiferromagnetic domains, the
spin orientation within the (a,b) plane could not be de-
termined unambiguously. Neither did we measure the
actual magnitude of the ordered moment, since the crys-
tallographic data needed to bring the measured magnetic
intensities to an absolute scale is not currently available
for Ba2CoGe2O7. It is reasonable to assume that at low
temperatures the sublattice magnetization is close to its
classical saturation value. Indeed, this is the case in the
S = 1/2 Cu2+-system, where quantum fluctuations may
be expected to be even stronger than in the S = 3/2
Thedispersionof spin waveexcitationsin
Ba2CoGe2O7 was found to be quite different from that
in Ba2CuGe2O7.5,6Figure 2 shows typical constant-q
Ba2CoGe2O7 at T = 3.5 K in comparison to those calcu-
lated for the proposed (a,b)-planar collinear antiferromag-
0.0 0.5 1.01.52.02.53.03.54.0
Intensity (counts/ 4 min)
FIG. 2: Typical constant-Q scans measured in Ba2CoGe2O7
at T = 2 K. The solid lines are Gaussian fits to the data.
Shaded areas represent the background level.
0.00.2 0.40.60.8 1.0
?! ?Z (meV)
?! ?Z (meV)
h (r. l. u.)
(1,1,0) reciprocal-space directions measured in Ba2CoGe2O7
at T = 2 K (symbols). The solid and dashed lines represent
the two spin wave branches in the model defined by Eq. 1,
with dispersion relations given by Eq. 2 and parameters cho-
sen to best-fit the data.
Dispersion of spin waves along the (1,0,0) and
scans measured in the Co-compound using Setup II at
T = 2 K. Two distinct sharp excitations are observed.
One branch is acoustic in origin, with excitation energy
linearly going to zero at the AF zone-center (1,1,0).
The second “optical” branch is barely dispersive and
is always seen around 2 meV energy transfer. The two
spin wave branches converge at the AF zone-boundary
(0.5,0.5,0). In all cases the observed energy width of
spin wave peaks is resolution limited.
variation of peak width seen in Fig. 2 is due to instru-
mental “focusing” effects. The data were analyzed using
Gaussian fits (solid line in Fig. 2).
(shaded areas) was assumed to be constant with an
additional Gaussian component at zero energy transfer
to model incoherent elastic scattering. The dispersion
relations along the (1,1,0) and (1,0,0) reciprocal-space
directions deduced from these fits are plotted in symbols
C. Data analysis
The observed dispersion of spin wave excitations can be
understood in the framework of linear spin wave theory.
To construct a model spin Hamiltonian, we assumed that
the dominant magnetic interactions in Ba2CoGe2O7are
those between nearest-neighbor spins in the (a,b) plane,
as is the case in Ba2CuGe2O7. Given that the S = 3/2
Co2+ions are frequently associated with a large mag-
netic anisotropy, in our model we allowed this coupling to
be anisotropic, and also included a single-ion anisotropy
term. The resulting model Hamiltonian is written as:
ˆ H =
n + J⊥S(x)
Here m and n label the spins on the two antiferro-
magnetic square sublattices with origins at (0,0,0) and
(0.5,0.5,0), respectively, and˜?stands for summation
over nearest neighbors. Note that the interactions along
the c axis are not included in the above expression. They
were not measured directly in this work, and are likely
to be ferromagnetic, due to the value of the magnetic or-
dering vector. The corresponding coupling constant was
previously found to be extremely weak in Ba2CuGe2O7,1
and was assumed to also be small in the Co-system. The
c-axis coupling should thus have no considerable effect
on spin wave dispersion in the (h,k,0) reciprocal-space
The spin wave Hamiltonian is obtained from Eq. 1
through a Holstein-Primakoff transformation. After lin-
earization it can be easily diagonalized by a Fourier-
Bogolyubov transformation. This straightforward yet te-
dious calculation for our particular case yields two spin
wave branches with the following dispersion relations:
q ]2= S2(8J⊥∓ 4J⊥Cq)(8J⊥± 4JzCq+ 2A), (2)
Cq≡ cos(πh + πk) + cos(πh − πk). (3)
From Eq. 2 it follows that exchange anisotropy and
single-ion anisotropy can not be distinguished based on
dispersion measurements alone. Indeed, it can be rewrit-
ten using only two independent parameters:
q ]2= (4JS)2(2 ∓ Cq)(2A ± Cq),
where J ≡√J⊥Jz and A ≡J⊥
is unity in the isotropic case and is a generalized mea-
sure of easy-plane anisotropy in our model. Excellent
fits to the data are obtained assuming S = 3/2 and using
J = 0.103(1) meV and A = 2.58(3). Dispersion curves
calculated using these parameters are shown in dashed
and solid lines in Fig. 3. Unlike its Cu-based counter-
part, Ba2CoGe2O7is characterized by very strong mag-
netic easy-plane anisotropy.
4Jz. The quantity A
The strong anisotropy effects in Ba2CoGe2O7 push
all spins in the system into the (a,b) crystallographic
plane. This effect is similar to that of a magnetic field ap-
plied along the c axis that favors an (a,b)-planar state in
Ba2CuGe2O7.5,6Since the helimagnet-forming uniform
component Dxy of the Dzyaloshinskii vector is itself in
the (a,b) plane, forcing the spins into the (a,b) plane
makes the corresponding triple-product in the Hamilto-
nian Dxy(Sm× Sn) vanish. Only the non-helimagnet-
forming sign-alternating z-axis component of D remains
relevant. As a result, the spin structure may be slightly
canted, but is, nevertheless, commensurate.
It is important to stress that effective easy-plane
anisotropy was previously detected in Ba2CuGe2O7 as
well. However, in this S = 1/2 Cu-based system any
single-ion term is reduced to a constant and is therefore
irrelevant. The only source of anisotropy is a two-ion
term, which was shown to be caused by the so-called
KSEA interactions.5,6The latter are a very weak ef-
fect with an energy scale of D2/J ∼ 3 · 10−2J.
a square lattice KSEA interactions happen to be just
strong enough to distort the helical structure, but not
to fully destroy incommensurability.5In contrast, as fol-
lows from the present study, easy-plane anisotropy in
Ba2CoGe2O7 is much stronger, of the order of J itself.
The anisotropy is probably due to single-ion effects that
are only allowed for S > 1/2, and its magnitude is well
beyond the critical value needed to destroy the helimag-
The results discussed above allows us to speculate
about the IC transition in Ba2(CoxCu1−x)Ge2O7 that
occurs with increasing Co-concentration x.
impurity strongly “pins” the original spiral at the im-
purity site, firmly confining the corresponding spin to
the (a,b) plane. Helimagnetic correlations are totally de-
stroyed when the characteristic distance between such
strong-pinning locations becomes comparable with the
period of the unperturbed spiral, which in Ba2CuGe2O7
is roughly 40 nearest-neighbor bonds.1This suggests a
critical concentration of about x ≈ 2%, in reasonable
agreement with bulk magnetization data of Ref. 13.
To summarize, the commensurate nature of the ground
state in Ba2CoGe2O7 is primarily due not to a domi-
nant staggered component of the Dzyaloshinskii vector,
but to easy-plane anisotropy effects that are orders of
magnitude stronger than typical Dzyaloshinskii-Moriya
or KSEA interactions. As a result, the destruction of he-
limagntism in Ba2(CoxCu1−x)Ge2O7occurs very rapidly
with increasing Co-concentration, as soon as the mean
distance between impurities becomes comparable to the
period of the spin spiral.
This work is supported in part by the Grant-in-Aid for
COE Research “SCP coupled system” of the Ministry
of Education, Culture, Sports, Science and Technology,
Japan. Work at ORNL and BNL was carried out un-
der DOE Contracts No. DE-AC05-00OR22725 and DE-
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