PHYSICAL REVIEW B 72, 094434 共2005兲

Magnetic excitations in the weakly coupled spin dimers and chains material Cu2Fe2Ge4O13
T. Masuda,* A. Zheludev, and B. Sales
Condensed Matter Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6393, USA

S. Imai† and K. Uchinokura‡
Department of Advanced Materials Science, The University of Tokyo, 5-1-5, Kashiwa-no-ha, Kashiwa 277-8581, Japan

S. Park§
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
共Received 2 June 2005; revised manuscript received 8 August 2005; published 28 September 2005兲
Magnetic excitations in a weakly coupled spin dimers and chains compound Cu2Fe2Ge4O13 are measured byinelastic neutron scattering. Both structure factors and dispersion of low-energy excitations up to 10 meVenergy transfer are well described by a semiclassical spin wave theory involving interacting Fe3+共S = 5 / 2兲chains. Additional dispersionless excitations are observed at higher energies, at ប␻ = 24 meV, and associatedwith singlet-triplet transitions within Cu2+ dimers. Both types of excitations can be understood by treating
weak interactions between the Cu2+ and Fe3+ subsystems at the level of the mean-field random phase approximation. However, this simple model fails to account for the measured temperature dependence of the 24 meV
mode.
DOI: 10.1103/PhysRevB.72.094434

PACS number共s兲: 75.10.Jm, 75.25.⫹z, 75.50.Ee

I. INTRODUCTION

The ground states of low-dimensional magnets arestrongly affected by quantum spin fluctuation. In so-called
quantum spin liquids spin correlation remains short range
even at zero temperature, and the excitation spectrum isgapped. This disorder is relatively robust and such systems
resist long-range ordering even in the presence of residual
three-dimensional interactions or anisotropy. In contrast,
gapless low-dimensional magnets are very sensitive to external perturbations that can easily drive them toward longrange ordering. New physics is found in bicomponent systems that combine these two distinct types of lowdimensional spin networks. An example is R2BaNiO5materials where Haldane spin chains weakly interact withmagnetic rare earth ions.1–3 More recently, we reported thediscovery and study of the quantum ferrimagnet
Cu2Fe2Ge4O13.4,5 We showed that this compound can beviewed as a system of antiferromagnetic 共AF兲 Cu dimers thatweakly interact with almost classical Fe3+ chains. This weakcoupling leads to a rather unusual cooperative ordering
phase transition at low temperatures.
The crystal structure of Cu2Fe2Ge4O13 is monoclinic
P21 / m with a = 12.101 Å, b = 8.497 Å, c = 4.869 Å, and ␤= 96.131°.5 The arrangement of magnetic ions and likely exchange pathways is shown in Fig. 1共a兲. Fe3+ ions form
crankshaft-shaped chains that run in the b direction. These
chains are separated by GeO4 tetrahedra in the c direction.
The Cu2+ dimers are located in between Fe3+ chains along
the a direction. Simultaneous cooperative long-range magnetic ordering of the two magnetic subsystems occurs at
40 K.5 The magnetic structure is roughly collinear, with
spins lying in the crystallographic a-c plane. In addition,
there are small out-of-plane spin components 共canting兲. The
saturation magnetic moment on Cu2+ ions is anomalously
small, being suppressed by residual quantum fluctuations in
1098-0121/2005/72共9兲/094434共8兲/$23.00

the dimers: mCu = 0.38共4兲␮B. This suggests that the coupling
JCu-Fe between the Cu dimers and the Fe subsystem is weakcompared to intradimer AF interactions JCu. The pairs of Cu
spins remain in a spin-singlet state that is only partially polarized by interactions with the long-range order in the Fe
subsystem. The data collected in preliminary inelastic neutron scattering experiments for energy transfers up to
10 meV could be well explained by fluctuations of Fe3+
spins alone. The Fe3+ spins form weakly coupled S = 5 / 2
chains, the corresponding effective exchange constants being
JFe = 1.60共2兲 meV, JFe⬘ = 0.12共1兲meV, as shown in Fig. 1.
These values are consistent with rough estimates of exchange constants based on magnetic susceptibility data: JFe= 1.7 meV, JCu = 25 meV. To date, no excitations associatedwith the Cu dimers could be identified.
While the very nature of the crankshaft-shaped chains implies some alternation of bond strength, we find no evidence
thereof in the measured dispersion curves. In fact, they are
well reproduced by a model involving uniform magnetic Fe
chains with weak interchain interactions along the c direction. In the remainder of this work we shall therefore disregard the bond alternation and assume the Fe chains to bemagnetically uniform.
In the present paper we report a more detailed inelastic
neutron scattering study of Cu2Fe2Ge4O13. Our main result isan observation of a complete separation of energy scales for
Cu2+- and Fe3+-centered magnetic excitations. An analysis of
the intensity pattern for low-energy spin waves allows us tounambiguously associate them with the dynamics of S = 5 / 2
chains. In addition, separate narrowband excitation originating from Cu subsystem is observed at higher-energy transfers. The layout of the paper will be as follows. In Sec. II wewill describe the characteristics of our samples and experimental setups. Section III describes the results for lowenergy excitations in single-crystal samples, as well as measurements of the much weaker high-energy excitations in a

094434-1©2005 The American Physical Society

PHYSICAL REVIEW B 72, 094434 共2005兲

MASUDA et al.

FIG. 1. 共Color online兲 共a兲 A schematic view of the crystal structure of Cu2Fe2Ge4O13. Only magnetic ions are shown. Arrows symbolize the spin structure in the magnetically ordered phase. Possible
exchange pathways are also shown. 共b兲 Relative orientations ofreciprocal-lattice vectors for the two domains in bulk crystalline
samples with twinning.

large-volume powder sample. In Sec. IV we shall interpretthe observed separation of energy scales by a mean-field random phase approximation 共MF RPA兲 treatment of interactions between Cu2+ and Fe3+ spins. The discussion and conclusion will be given in Secs. V and VI, respectively.
II. EXPERIMENT

High-quality single crystals with the dimension of 3 ⫻ 4⫻ 35 mm3 were grown by the floating zone method. All
samples were found to be twinned. The two twins share acommon 共b , c兲 plane, the monoclinic structure allowing two
independent orientations of the a axis. In reciprocal space thedomains share a common 共a* , b*兲 plane, but have distinct c*axes, as illustrated in Fig. 1共b兲. Two separate single crystals
were coaligned to obtain a larger sample of cumulative mosaic spread 0.44°. For single-crystal inelastic neutron scattering experiments we exploited three different setups. Setup 1
employed the SPINS cold neutron spectrometer at the NIST
Center for Neutron Research 共NCNR兲. The scattering plane
was defined by the b* and the bisector c⬘* of the c* axes inthe two crystallographic domains, as shown in Fig. 1共b兲. The
direction of c⬘* is the same as that of the c axis in the real
space. In this geometry the scattering planes are the same inboth domains. It is convenient to define c⬘* ⬅ c*cos共␤− 90° 兲. The momentum transfer in the scattering plane of thespectrometer is then indexed by Miller indices h⬘, k⬘, and l⬘of a fictitious orthorhombic structure. For the two domains
h = ± 共c⬘* / a*兲l⬘ tan共␤ − 90° 兲, k = k⬘ and l = l⬘. Since ␤ is close
to 90°, h is almost zero for most measurements using setup
1, where l is small. In setup 2 the scattering plane was
共a* , b*兲, which is also common for the two domain types. In
both setups we used 共guide兲-80⬘-80⬘-共open兲 collimation witha Be filter positioned after the sample and a fixed final energy E f = 5 or 3 meV. The data were collected at T = 1.4 K
using a standard He-flow cryostat.
Intensity was the main limiting factor for studies ofhigher-energy magnetic excitations. To maximize sample
volume the measurements were performed on a 50 g polycrystalline Cu2Fe2Ge4O13 powder that was prepared by thesolid state reaction method. This experiment was performedon the HB3 three-axis spectrometer at HFIR with 48⬘-40⬘-60⬘-120⬘ collimations and a Pyrolytic graphite filter after
the sample 共setup 3兲. The final neutron energy was fixed at
E f = 14.7 meV. A closed-cycle refrigerator was used toachieve low temperatures.
Setup 4 was used for wide surveys in reciprocal space on
single crystals. We employed the HB1 thermal neutron spectrometer at the High Flux Isotope Reactor at ORNL. The
scattering plane was 共a* , b*兲, as in setup 2. The collimation
was 48⬘-40⬘-40⬘-240⬘. Neutrons with E f = 13.5 meV wereused in conjunction with a pyrolytic graphite filter positioned
after the sample. The experiments in setup 3 were performed
at T = 6.4 K maintained by a closed-cycle He refrigerator.
III. EXPERIMENTAL RESULTS
A. Low energies

In this section we concentrate on the low-energy excitations at energy transfers up to 10 meV. As was explained in
our preliminary report5 and will be discussed in detail below,
this part of the spectrum can be associated with conventional
spin waves from the Fe subsystem. The Cu dimers only provide an effective Fe-Fe interaction, but contribute nothing to
the dynamics at low energies.
Typical energy scans at q = 共0 k 0.5兲 measured using setup
1 are shown in Fig. 2. White and gray circles correspond todata collected with E f = 5 and 3 meV, respectively. Welldefined resolution-limited peaks are observed in the entire
Brillouin zone. Solid lines are Gaussian fits to the data after
a subtraction of a linear background. The excitation energy isa minimum at the magnetic zone center at q = 共0 2.0 0.5兲. A
small gap of about 1.6 meV is observed at this wave vector.
The apparent shoulder structure can be attributed to a splitting of the spin wave branch into two components withsomewhat different gap energies. The gaps are most likely
anisotropy related. The zone-boundary energy at q
= 共0 3.0 0.5兲 is about 9 meV. The dispersion relation was
obtained by Gaussian fits to individual scans and is shown in
Fig. 3共a兲. The k dependence of the measured energyintegrated peak intensity is plotted in symbols in Fig. 3共b兲.
The intensity scales roughly as 1 / ␻. The inset in Fig. 3共a兲 is
a sketch of the dispersion relation of mode I that is derived inSec. IV.

094434-2

PHYSICAL REVIEW B 72, 094434 共2005兲

MAGNETIC EXCITATIONS IN THE WEAKLY COUPLED…

FIG. 2. Energy scans collected in Cu2Fe2Ge4O13 for momentum
transfers q = 共0 k 0.5兲. White and gray circles correspond to measurements with Ef = 5 and 3 meV, respectively. Solid lines are
Gaussian fits.

Energy scans at q = 共0 2.0 l兲 and 共0 2.9 l兲 are shown in
Fig. 4. Constraints on experimental geometry prevented us
from reaching the more symmetric 共0 3.0 l兲 reciprocal-spacerods at higher energy transfers. These two sets of data reveal
two distinct excitation branches, that are strongest near evenand odd k values, respectively. The former branch is dispersive along the c axis, while the latter is almost flat. Dispersion relations and the integrated intensity plots for bothmodes were obtained using Gaussian fits and are shown in
Figs. 5共a兲 and 5共b兲. Even for the more dispersive lowerenergy branch the boundary energy along the c⬘* direction is
only about 5 meV: about half of that in the b* direction.
No dispersion of low-energy excitations could be detectedalong the a* direction. This is illustrated by the energy scanscollected on the 共h 2.0 0兲 reciprocal-space rod using setup 2
共Fig. 6兲. Two peaks are observed at 5 and 9 meV, respectively. The excitation energies are plotted in Fig. 7共a兲. Interestingly, the peak intensity is strongly dependent on h, asshown in Fig. 7共b兲.
From the data presented above one can immediately conclude that the most relevant magnetic interactions are within
the Fe3+ layers, parallel to the 共b , c兲 plane. Interlayer coupling along the a direction, which involves the Cu2+ dimer
coupling, is considerably weaker.
B. Higher-energy excitations

The model that we previously proposed for Fe2Cu2Ge4O13
共Ref. 5兲 implies a separation of energy scales of spin-wavelike excitations on the Fe subsystem and triplet excitations of

FIG. 3. Measured energies 共a兲 and integrated intensities 共b兲 ofmagnetic excitations as a function of momentum transfer along theb* direction 共symbols兲. Lines are as described in the text.

the Cu dimers. From magnetic susceptibility measurements
we have estimated the intradimer exchange constant to bearound 25 meV. Due to the small sample size, in singlecrystal experiments we failed to observe any clear magnetic
inelastic features in this energy transfer range. To search for
the Cu-triplet mode we performed additional measurements
on a large-size powder sample by employing the setup 3
experimental configuration.
The powder data collected at T = 12 K are summarized in
the false color plot in Fig. 8共a兲. The data were obtained by
combining 17 separate constant-q scans. For each such scana linear background was subtracted from the measured intensity. Figure 8共a兲 clearly shows a narrow excitation band at
ប␻ ⬃ 24 meV. The measured intensity of the 24 meV peakdecreases with the increase of q, as expected for magnetic
scattering. The observed energy width at q = 2.3 Å−1 is somewhat larger than experimental resolution, but still small compared to the central energy 关Fig. 8共b兲兴. Such energy dependence in powder samples typically indicates a narrowdispersion bandwidth 共see, for example, Refs. 6 and 7兲. The
observed spectrum is consistent with the excitations originating from the structural Cu dimers, indicated by the thicksolid bonds in Fig. 1共a兲. However, the momentum transfer
range covered in the experiment, especially at 兩q兩 → 0, is in-

094434-3

PHYSICAL REVIEW B 72, 094434 共2005兲

MASUDA et al.

FIG. 4. Energy scans collected in Cu2Fe2Ge4O13 for several
momentum transfers along the c* direction. Symbols and lines are
as in Fig. 2.

sufficient for a more detailed analysis of the structure factor.
In particular, the size of the dimers cannot be independently
extracted from the experimental data.
To obtain additional information on the 24 meV excitation we studied its temperature dependence at q = 2.3 Å−1.
The main challenge was dealing with a large background thatoriginates from 共i兲 temperature-independent scattering, including spurious scattering from an “accidental” Bragg powder line at 24.8 meV and 共ii兲 temperature-dependent phonon
scattering. These two contributions can be removed from thedata if one assumes that the useful magnetic signal is relatively weak and/ or temperature independent above T= 200 K. This would indeed be true if the magnetic scattering
originated from effectively isolated Cu dimers with an intradimer exchange constant of 24 meV. The phonon contribution, which is assumed to scale with the Bose factor, isthus estimated from comparing scans at T = 300 K and T= 200 K. It is then appropriately scaled and subtracted fromall scans, leaving only the true magnetic signal and the
T-independent background. The two cannot, in principle, bereliably separated. However, we can follow the change inmagnetic signal from T = 200 K by using the phononsubtracted T = 200 K scan as “background.”The result of this elaborate background subtraction isplotted in Fig. 9. The well-defined peak seen at low temperature broadens progressively with increasing T and practicallydisappears at T ⲏ 40 K. In this range there also seems to be adownward shift in the peak’s central energy.
IV. ANALYSIS
A. Separation of energy scales

To understand the dynamics of the coupled Fe3+ and Cu2+
subsystems we shall make the central assumption that the

FIG. 5. Measured energies 共a兲 and integrated intensities 共b兲 ofmagnetic excitations as a function of momentum transfer along thec* direction. The solid and open circles correspond to the higherand lower-energy branches, respectively. Lines are as described in
the text.

exchange constant JCu that binds pairs of Cu2+ spins into AF
dimers is the largest energy scale in the system. Under thesecircumstances the degrees of freedom associated with Cu
spins can be effectively integrated out at low energies. Indeed, the isolated Cu2+ subsystem has a spin singlet groundstate and a large energy gap ⌬ = JCu. At ប␻ Ⰶ ⌬ it lacks any
intrinsic dynamics, i.e., its dynamic susceptibility is purely
real and almost energy independent. In the spirit of RPA,
from the point of view of Fe3+ spins, the Cu dimers merely
act as a polarizable medium that can transfer magnetic interactions between the Fe layers. The staggered spin susceptibility of each S = 1 / 2 dimer being 2 / 共JCu兲, this effective coupling, labeled as Jeff in Fig. 1 is given by2
Jeff = JCu-Fe/共2JCu兲.

共1兲Thus, to a good approximation, the low-energy spin dynamics of Fe2Cu2Ge4O13 is simply that of the Fe3+ subsystemwith an additional exchange coupling. From the experiment,
where no dispersion of spin waves could be observed along
the a axis, the effective exchange constant must be rather
small. Nevertheless, as explained in our previous paper,5 it isabsolutely crucial in completing a three-dimensional spin
network and allowing long-range magnetic ordering at anonzero temperature.
At high energy transfers, comparable to the Cu dimer gap,
it is the Fe3+ degrees of freedom that can be effectively in-

094434-4

PHYSICAL REVIEW B 72, 094434 共2005兲

MAGNETIC EXCITATIONS IN THE WEAKLY COUPLED…

FIG. 6. Energy scans measured in Cu2Fe2Ge4O13 for momentum transfers q = 共h 2.0 0兲. Symbols and lines are as in Fig. 2.

tegrated out. At T ⬍ TN their effect is reduced to producing astatic staggered exchange spin field that acts on the Cu2+
spins and is proportional to the ordered Fe3+ moment:
hCu = 具SFe典JCu-Fe .

共2兲

In our approximation at high energy transfers Cu2Fe2Ge4O13
behaves as a collection of 共possibly interacting兲 Cu dimers inan effective staggered field.5

FIG. 7. 共a兲 Measured energies 共a兲 and integrated intensities 共b兲
of magnetic excitations as a function of momentum transfer along
the a* direction. Symbols and lines are as in Fig. 5.

Fig. 1 is that of a regular Bravais lattice. The relation between real lattice and the magnetically equivalent lattice isillustrated in Fig. 10. The actual structure factor S共q , ␻兲 and
S0共q , ␻兲 for the equivalent lattice is exactly related as follows:
S共q, ␻兲 = S0共q, ␻兲cos2 2␲␦ah cos2 2␲␦bk
1+ S0共q + 共0,1,0兲, ␻兲共1 + sin2 2␲␦ah sin2 2␲␦bk兲
2

B. Spin waves and dynamic structure factor for low energies

As explained above, the low-energy spectrum of
Cu2Fe2Ge4O13 can be understood by considering the Fe subsystem in isolation, and even Jeff can be ignored for its smallness. For the magnetically ordered state the dynamic structure factor of such a system can be calculated using
conventional spin wave theory 共SWT兲. For simplicity weshall assume a collinear magnetic structure, ignoring thesmall canting of the Fe3+ spins out of the 共a , c兲 plane. In thefirst step of our analysis we fitted the dispersion relation withthe coupled alternating chains by numerical method.8 The
obtained exchange parameters were the same for both alternating bonds in our experimental error. For the further calculation of the structure factor we can approximately employ
magnetically uniform chains that will give us intuitive physical meanings of the observed modes.
There are four Fe ions in each unit cell of Cu2Fe2Ge4O13,
even though we assume magnetically uniform chains and the
topology of exchange interactions defined by JFe and JFe⬘ in+ S0共q + 共0,2,0兲, ␻兲cos2 2␲␦ah sin2 2␲␦bk
1+ S0共q + 共0,3,0兲, ␻兲共1 + sin2 2␲␦ah sin2 2␲␦bk兲,
2
共3a兲␦a = 0.1239,␦b = 0.0629.

共3b兲

Here ␦a and ␦b are the displacements of Fe3+ ions from highsymmetry positions, as shown in Fig. 1共b兲.
The SWT dynamic structure factor for a collinear antiferromagnet on a Bravais lattice is well known:9094434-5

S0共q, ␻兲 = 共uq + vq兲2␦共ប␻q − ប␻兲,uq2 =S共ប␻q + 2Sj共0兲兲
,
ប␻q

共4a兲
共4b兲

PHYSICAL REVIEW B 72, 094434 共2005兲

MASUDA et al.

FIG. 10. Spatial arrangement of Fe3+ sites in the structure of
Cu2Fe2Ge4O13 共open circles兲 and an equivalent Bravais lattice ofspins 共solid circles兲.

u qv q = −2S2 j共q兲,ប␻q

ប␻q = S冑 j共0兲2 − j共q兲2 + ⌬2 .

共4c兲

共4d兲

FIG. 8. 共Color online兲 共a兲 False-color plot of inelastic intensity
measured in Cu2Fe2Ge4O13 powder samples as a function of energy
and momentum transfer. 共b兲 Typical energy scans measured at T
= 12 K for different momentum transfers. Heavy solid lines are
Gaussian fits. The shaded Gaussians represent experimental energy
resolution.

Here j共q兲 is the Fourier transform of exchange interactions,and ⌬ empirically accounts for the anisotropy gap. In ourparticular case of the Fe3+ subsystem in Cu2Fe2Ge4O13 wehave

FIG. 9. Typical energy scans at measured in Cu2Fe2Ge4O13
powder at q = 2.3 Å−1 for different temperatures. Solid lines are
Gaussian fits. The background has been subtracted, as explained in
the text.

FIG. 11. 共Color online兲 共a兲 Constant energy scans at ប␻= 8.5 meV in wide q = 共h k 0兲 range. 共b兲 Simulation based on ourmodel cross section convoluted with the experimental resolutionfunction.

094434-6

PHYSICAL REVIEW B 72, 094434 共2005兲

MAGNETIC EXCITATIONS IN THE WEAKLY COUPLED…

TABLE I. Exchange parameters for Cu2Fe2Ge4O13.
JFe

JFe
⬘

Jeff

JCu-FeJCu

1.60共2兲 meV

0.12共1兲 meV

0.13共4兲 meV

2.54共3兲 meV

24.共2兲 meV

冉

j共q兲 = 2 JFe cos

冊␲⬘ cos 2␲l .
k + JFe2

共5兲

The cross section for inelastic neutron scattering from spinwaves is given by冋 冉 冊册

qy 2
d 2␴
具nq + 1典S共q, ␻兲.
⬀ 兩F共q兲兩2 1 +
d⍀ dE
q

共6兲

In this formula F共q兲 is the magnetic form factor for Fe3+,
具nq典 is the Bose factor, and qz is the projection of the scattering vector onto the direction of ordered Fe3+ moments.
C. Fits to data

The model cross section given by Eq. 共6兲 can accurately
reproduce the observed low-energy spectra in Cu2Fe2Ge4O13.
Due to the presence of four terms in Eq. 共3兲, there are four
distinct spin wave branches, which we shall denote as modes
I through IV, respectively. The dispersion relation given by
Eq. 共4d兲 was fitted to the experimental data shown in Figs.
3共a兲 and 5共a兲 using a least-squares algorithm. A good fit isobtained with JFe = 1.60 meV, JFe⬘ = 0.12 meV, and ⌬= 2.02 meV. The result is shown in lines in Figs. 3共a兲 and
5共a兲. With these parameters our model also agrees well withthe measured dispersion 共or, rather, absence thereof兲 along
the a axis, as shown in Fig. 7共a兲.
What is important is that not only the energies but also theintensities of the observed excitations are well reproduced byour model. Calculated intensities for each mode are shown inlines in Figs. 3共b兲, 5共b兲, and 7共b兲. In cases where experimental energy resolution is insufficient to resolve individual
branches, combined intensities of a couple of modes areshown. This quantitative agreement between the measured
and observed structure factors confirms that the scattering is
indeed due to Fe3+ spins. An excellent illustration of this was
obtained by mapping out the scattering intensity in a wide q
range using setup 4, as shown in Fig. 11共a兲. These data correspond to a fixed energy transfer ប␻ = 8.5 meV, and have acharacteristic checkerboard pattern. Our spin wave modelwith the parameters quoted above reproduces this behavior
very well, as shown in Fig. 11共b兲. In this calculation themodel cross section was numerically convoluted with the
known experimental resolution function. The apparent periodicity along k is due to a steep dispersion that takes theexcitations in and out of the probed energy range. However,the periodicity along h is related to the trigonometric coefficients in Eq. 共3a兲. These, in turn, are determined by the geometry of the crankshaft-shaped Fe3+ chains in
Cu2Fe2Ge4O13.

V. DISCUSSION AND CONCLUSIONAs demonstrated above, the low-energy spin dynamics of
Cu2Fe2Ge4O13 is well described by an effective spin wave
theory for the Fe3+ spin chains. Based on the available data it
is impossible to unambiguously associate the observed
24 meV mode with the Cu2+ dimers. However, much confidence in this assumption can be drawn from a recent study of
Cu2Sc2Ge4O13.10 In this isostructural compound11 only the
Cu2+ ions are magnetic. Indeed the magnetic susceptibility
was fitted by S = 1 / 2-dimers with JCu ⬃ 26 meV.10 The consistency of the energy scales strongly suggests that the originof the high-energy excitation is the same for both isostructural compounds.
We can now estimate all the relevant exchange interactions in the system. Using Eq. 共2兲 in combination with thestaggered susceptibility of an isolated antiferromagnetic
dimer, from the known saturation moments 具SCu典 = 0.18共8兲
and 具SFe典 = 1.77共4兲 for Cu2+ and Fe3+, respectively,5 we get
JCu-Fe = 2.54共3兲meV. The effective coupling is then Jeff
= 0.13共4兲 meV. The results for all exchange parameters are
summarized in Table I.
Our model for Cu2Fe2Ge4O13 is qualitatively consistent
with the observed slight increase of the energy of the Cudimer mode with decreasing temperature. Indeed, the gap
energy of isolated dimers, as that of other gapped systems, isknown to increase with the application of a staggered
field.12,13 In Cu2Fe2Ge4O13 this field is generated by the ordered moment on the Fe3+ sites, and at T ⬍ TN increases proportionately to 具SFe典. In spite of the qualitatively consistent
explanation, the observed T dependence of the 24 meV modeis quantitatively different from that for isolated dimers. In thecase of isolated dimers, the intensity would remain almost
constant below T = 40 K. Moreover, the peak would remain
sharp at all temperatures. The discrepancy may be related tointrinsic limitations of the MF RPA approach, and merits
further investigation. In particular, it is tempting to somehow
associate the observed emergence and sharpening of the
24 meV inelastic peak with the onset of long-range order.
Our data bring solid quantitative support to the concept of
separation of energy scales in the mixed-spin quantum antiferromagnet Cu2Fe2Ge4O13. Using a simple MF RPA approach we are able to determine all the relevant exchangeinteractions. However, certain features of the temperature dependence of spin excitations require further theoretical andexperimental study.
ACKNOWLEDGMENTS

Work at ORNL was carried out under Contract No. DEAC05-00OR22725, U.S. Department of Energy. Experiments
at NIST were supported by the NSF through Grants No.
DMR-0086210 and No. DMR-9986442.

094434-7

PHYSICAL REVIEW B 72, 094434 共2005兲

MASUDA et al.

*Present address: International Graduate School of Arts and Sciences, Yokohama City University, 22-2, Seto, Kanazawa-ku,
Yokohama city, Kanagawa, 236-0027, Japan. Electronic address:
tmasuda@yokohama-cu.ac.jp†Present address: Semiconductor R & D Center, Yokohama R & D
Laboratories, The Furukawa Electric Co., Ltd., Japan.
‡Present address: The Institute of Physical and Chemical Research
共RIKEN兲, Wako, Saitama 351-0198, Japan.
§
Present address: HANARO Center, Korea Atomic Energy Research Institute, Daejeon, South Korea.
1 A. Zheludev, J. P. Hill, and D. J. Buttrey, Phys. Rev. B 54, 7216
共1996兲.
2 A. Zheludev, E. Ressouche, S. Maslov, T. Yokoo, S. Raymond,and J. Akimitsu, Phys. Rev. Lett. 80, 3630 共1998兲.
3
J. V. Alvarez, R. Valenti, and A. Zheludev, Phys. Rev. B 65,
184417 共2002兲.
4 T. Masuda, B. C. Chakoumakos, C. L. Nygren, S. Imai, and K.

Uchinokura, J. Solid State Chem. 176, 175 共2003兲.
T. Masuda, A. Zheludev, B. Grenier, S. Imai, K. Uchinokura, E.
Ressouche, and S. Park, Phys. Rev. Lett. 93, 077202 共2004兲.
6 A. Zheludev, G. Shirane, Y. Sasago, M. Hase, and K. Uchinokura,Phys. Rev. B 53, 11642 共1996兲.
7
T. Masuda, A. Zheludev, H. Kageyama, and A. Vasiliev, Europhys. Lett. 63, 757 共2003兲.
8 A. W. Sáenz, Phys. Rev. 125, 1940 共1962兲.
9 See, for example, S. W. Lovesey, Theory of Neutron Scatteringfrom Condensed Matter 共Clarendon, Oxford, 1984兲, Vol. 2.
10
T. Masuda et al. 共unpublished兲.
11 G. J. Redhammer and G. Roth, J. Solid State Chem. 177, 2714
共2004兲.
12
Th. Jolicœur and O. Golinelli, Phys. Rev. B 50, 9265 共1994兲.
13
S. Ma, D. H. Reich, C. Broholm, B. J. Sternlieb, and R. W. Erwin,
Phys. Rev. B 51, R3289 共1995兲.
5094434-8