RAPID COMMUNICATIONSPHYSICAL REVIEW B 75, 220401共R兲 共2007兲

Indirect magnetic interaction mediated by a spin dimer in Cu2Fe2Ge4O13
T. Masuda,1,* K. Kakurai,2 M. Matsuda,2 K. Kaneko,3 and N. Metoki3
1International Graduate School of Arts and Sciences, Yokohama City University, Yokohama, Kanagawa, 236-0027, Japan
2Quantum Beam Science Division, JAEA, Tokai, Ibaraki 319-1195, Japan
3

Advanced Science Research Center, JAEA, Tokai, Ibaraki 319-1195, Japan
共Received 26 March 2007; published 1 June 2007兲

Cu2Fe2Ge4O13 is a bicomponent compound that consists of Cu dimers and Fe chains with separate energy
scale. By inelastic neutron scattering technique with high-energy resolution we observed the indirect Fe-Fe
exchange coupling by way of the Cu dimers. The obtained parameters of the effective indirect interaction andrelated superexchange interactions are consistent with those estimated semistatically. The consistency reveals
that the Cu dimers play the role of nonmagnetic media in the indirect magnetic interaction.
DOI: 10.1103/PhysRevB.75.220401

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

Some types of magnetic interaction in solid are indirectones by way of nonmagnetic media. The superexchange interaction among the most common type in insulating metaloxides is derived by perturbative treatment of electron transfer between nonmagnetic anions and magnetic ions.1 Anothertype is found in Ruderman-Kittel-Kasuya-Yoshida 共RKKY兲
interaction in magnetic alloys2 where free electrons in thenonmagnetic metal transfer exchange integral between magnetic atoms. Anions or nonmagnetic metal behaves as themedia for the indirect interaction. As the exotic types spinsinglet entities such as spin dimers, ladders, spin integer
chains3 can play a similar role. The theoretical treatment wasconsidered in nonmagnetic impurity doped S = 1 chains,4,5 S= 1 / 2 ladders,6,7 or S = 1 / 2 dimerized chains.8,9 Around theimpurity the induced staggered spins generate an effective
spin of S = 1 / 2 in the sea of nonmagnetic state. The low
energy excitation are described by the effective spins thatindirectly interact with the similar formula to the RKKY
interaction. This scenario is experimentally realized in doped
spin gap compounds such as CuGeO3,10 SrCu2O3,11 and
PbNi2V2O8.12 However, randomness of the impurity distribution makes the system rather complex. For the basic understanding a bicomponent spin compound with uniformly
distributed magnetic ions is necessary.
Recently, Fe-doped CuGeO3 with uniform Fe distribution
was identified in insulating metal oxide Cu2Fe2Ge4O13;
Cun−2Fe2GenO3n+1 with n = 4 in a hypothetical series of homologous structures with n = 3 , 5 , 6 , . . . , ⬁.13 Here the compound with n = ⬁ corresponds to well known spin-Peierls cuprite CuGeO3 共Ref. 10兲, where uniform 1D chains of Cu2+ions run in the c direction as is illustrated in Fig. 1共a兲. In
Cu2Fe2Ge4O13 50% of Cu2+ 共S = 1 / 2兲 ions are periodicallyreplaced by Fe3+ 共S = 5 / 2兲 and the uniform chains become
distorted alternating chains of S = 1 / 2 and S = 5 / 2 dimers in
Fig. 1共b兲. Consequently, the lattice unit is quadrupled along
the chain direction and the c axis in CuGeO3 changes to thea axis in Cu2Fe2Ge4O13. Bulk susceptibility and neutron inelastic scattering 共NIS兲 measurements indicated that theJCu-Fe are weak14–17 and, instead, the strength of the interdimer interaction of Fe ions 共Jb⬘兲 is the same as that of theintradimer one 共Jb兲 within the experimental error. Hence,
Cu2Fe2Ge4O13 is a weakly coupled quantum dimers of Cu
ions and classical uniform chains of Fe ions in the b direc1098-0121/2007/75共22兲/220401共4兲

tion. In this geometry Cu dimers seem to behave as nonmagnetic media for effective exchange interaction 共Jeff兲 of neighboring Fe ions as is described in Fig. 1共c兲. A set of exchange
parameters were determined by NIS and neutron diffraction
共ND兲 in the previous study as is summarized in the first lineof Table I. In spite of the rigorous experiments the direct
measurement of Jeff was not performed because of the twinning of the available crystal. The values in the square bracket
is the reference estimates by a semistatic method.
In this paper we performed cold neutron inelastic scattering experiment with high-energy resolution on the twinned
but high-quality crystal. Through elaborate data analysis we
successfully obtained the low energy excitation in the a* direction where the Jeff makes a main contribution. We analyze
whole dispersion in combination with previously reported
data17 by both the conventional spin wave 共SW兲 and theeffective SW model based on the energy scale separation. We
found the latter model more appropriate in quantitative level.
The obtained consistency between the Jeff’s estimated by the
dynamics measurement and the semistatic method reveals
that the Cu dimers play the role of the nonmagnetic media in
the indirect interaction between Fe spins.
The high quality crystal with the dimension of 7 ⫻ 5⫻ 40 mm3 was obtained by floating zone method. NIS experiment was performed on LTAS spectrometer installed in
JRR-3 in JAEA. Open-80⬘-80⬘-open with sollar collimationand fixed final energy of E f = 3.5 meV were employed.
Closed He cycle refrigerator was used to achieve T = 3.0 K.
The crystal structure of Cu2Fe2Ge4O13 is monoclinic. The
obtained crystal was twinned so that a* and b* axes areshared as is illustrated in Fig. 1共d兲. Here the reciprocal
spaces for crystal A and B are defined as a*b*c* 共dashed grid兲and a⬘*b⬘*c⬘* 共dashed dotted grid兲 coordinates, respectively.
In Cu2Fe2Ge4O13 a reciprocal vector 共h k l兲 in crystal A corresponds to (−共h + 0.53l兲 − k l) in crystal B. The a*b*c* coordinate will be used hereafter.
A series of constant q scans were performed on the trajectory of the thick solid line in Fig. 1共d兲. Peak profiles of thetypical scans are shown in Fig. 2共a兲. At antiferromagnetic
zone center of h = 2.0, well-defined spin wave excitation withan anisotropy gap of about 1 meV was observed in crystal A.
The peak is wider than experimental resolution and seems to220401-1©2007 The American Physical Society

RAPID COMMUNICATIONSPHYSICAL REVIEW B 75, 220401共R兲 共2007兲

MASUDA et al.
(a)

CuGeO 3

Jb(a) L T A S o p e n - 8 0  - 8 0  - o p e n , - F = 3 . 5 m e V , 6 = 3 . 0 K

Ja

Jc

Jc

Cu 2+

a

Cu 2 Fe 2 Ge 4 O 13

Jb

Jc

Jb

J b’

J Cu-FeJ Cu-Fe
J Cu

a

Fe 3+

J Cu

Cu 2+
c’ *

(d)

+ K 

J Cu-Fe
S Cu2
J Cu
S Cu1

J eff

a’ *

1 0 0

D = 2 .20
1 0 0

D = 2 .1

2 3

4

a*
b*

3

b’ *
Jc
Jb

J eff

a

1

(b)

c

FIG. 1. 共Color online兲 共a兲 Schematic structure of CuGeO3 projected onto c-b 共left panel兲 and c-a 共right panel兲 planes. 共b兲 Schematic structure of Cu2Fe2Ge4O13. Lattice unit cell and magnetic
unit cell are shown by solid and dashed lines, respectively. Main
exchange path is crank-shaft Fe chains along b direction and Cu
dimers. 共c兲 Indirect Fe-Fe coupling by way of Cu dimers. 共d兲 The
structure of the twinned crystal of Cu2Fe2Ge4O13 projected onto thea*-c* plane. Dot-dashed and dashed grids are for crystal A and B,
respectively. 共e兲 Schematic structure of Cu2Fe2Ge4O13 based on theeffective SW model 共see text兲.

have a structure probably due to xy two ion anisotropy. The
shape of the scan profiles are well explained by consideringresolution function in the Cooper-Nathans approximation18that is shown by solid curves. At ប␻ ⬃ 2.4 meV, broad peakfrom crystal B is detected. These two peaks move with thechange of h on the identical trajectories shown by dasheddotted and dashed curves. We fitted the data by a pair of

3

q = (h 1 -0.5)

1

Crystal A
Crystal B
1.0

Jb
b

2

Ñw(meV)

2

0

J eff

C ry s ta l B

D = 2 .0
0

1

(e)

a

D = 2 .3

0

0

c*

S Fe1

. A 

1 0 0

1

S Fe2
K

0

1 0 0

c

C ry s ta l A

D = 2 .4

0

b

(c)

1 0 0

Ñw (meV)(b)

Intensity (counts/1min.)

c

b

. A

D = 2 .50ca

(D 1 -0 .5 )

1 0 0

1.5
h (r. l. u.)

2.0

2.5

FIG. 2. 共a兲 Constant q scans at 共h 1 − 0.5兲. The dashed-dottedand dashed curves indicate excitations from crystal A and B, respectively. Solid curves are fitting curves by resolution convoluted function. 共b兲 Dispersion curves of crystal A 共dashed-dot兲 and crystal B
共dashed兲.

simple Gaussian functions and we obtained the dispersion
relations in Fig. 2共b兲. Two curves from crystals A and B are
separately observed, which are identical by translational displacement of h⬘ = −共h + 0.53l兲. Thus the dispersion in the a*direction with the boundary energy of about 3 meV is successfully obtained in Cu2Fe2Ge4O13. In combination with theprevious data17 the whole dispersions in three independent
directions are drawn in Fig. 3共a兲. Filled symbols are the databy single crystal NIS and the gray shaded area around24 meV is a narrow band excitation observed in powder NIS
experiment.
In the first scenario we analyze the data by the conventional SW model. Since all the spins are confined in almost

TABLE I. Exchange parameters for Cu2Fe2Ge4O13 determined by neutron inelastic scattering 共NIS兲 and neutron diffraction 共ND兲. The
values in the square bracket are the estimates by the semistatic method.

NIS and ND 共Refs. 14 and 17兲
NIS 共Conventional SW兲
NIS 共Effective model兲

JCu 共meV兲Jb 共meV兲

Jc 共meV兲

Jeff 共meV兲JCu-Fe / JCu

JCu-Fe 共meV兲

⌬ 共meV兲

2.4共2兲 ⫻ 10
2.共4兲 ⫻ 102

1.60共2兲
1.59共2兲
1.59共2兲

0.12共1兲
0.13共2兲
0.13共2兲关0.13共4兲兴关0.10共5兲兴关2.4共2兲兴
0.9共6兲
2.0共5兲

1.0共5兲
1.0共5兲

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Energy transfer (meV)

INDIRECT MAGNETIC INTERACTION MEDIATED BY A…
Cu Dimer

(a)

25
20

( h 1 -0.5)( h +1 2 0)
( h 2 0)(0 2 l )15(0 k 0.5)

10
5
0

0

0.25
0.5 1
l (r. l. u.)
5

4

10

1.5
h (r. l. u.)Ñw (meV)
15

2 2

2.5
k (r. l. u.)

20

25

3

J Cu c ’( p , w )(b)
3

Fe spinnetwork2

dimer
susceptibility
0

1
0 .0

0 .2

0 .4

0 .6

Ñw/J Cu

0 .8

dimer
J
Cu
1 .0

FIG. 3. 共Color online兲 共a兲 The dispersion relation in
Cu2Fe2Ge4O13. The lower energy excitation in a* direction 共filled
circles兲 is determined in the current study. Other data are from Ref.
17. Solid lines and small open circles are the fit by conventional
SW and effective SW model 共see text兲. 共b兲 Energy scales of Fe
chains and Cu dimers. Dynamic susceptibility of dimers is indicatedby a solid curve.

a-c plane and the canting is small,14 we start from collinear
structure. We assume isotropic interactions for Fe-Fe, Cu-Fe,
and Cu-Cu spins for the simplicity. Hence the Hamiltonian isdefined as
H=

1
共J␣␤S␣Fe · S␤Fe + J␥␦S␥Fe · T␦Cu兺
2 ␣,␤,␥苸8 Fe sublattices,␦,⑀,␨苸8 Cu sublattices+ J⑀␨T⑀Cu · T␨Cu兲.

共1兲

Here ␣, ␤, and ␥ denote Fe sublattices and ␦, ⑀, and ␨ denoteCu sublattices. SFe and TCu are the spin operators of S = 5 / 2
and S = 1 / 2 on Fe and Cu ions, respectively. We consideredJb, Jc, JCu-Fe, and JCu in Fig. 1共b兲 as the contributed exchange
parameters in Eq. 共1兲. In addition the split anisotropy gaps ⌬xand ⌬y are empirically introduced as 关ប␻x,y共q兲兴2 = 关ប␻共q兲兴2+ ⌬2x,y. ⌬x,y = 0.95 and 1.16 meV were obtained by fitting theresolution convoluted function to the constant q scan at q
= 共2 1 − 0.5兲, where the xy anisotropy is the most enhcnaed.
The split is small enough and we will use the mean value⌬ = 1.0共5兲 meV as fixed parameter hereafter.
The numerical fit to the dispersion is indicated by small
open circles in Fig. 3共a兲. The fit to the data is very good.

关ប␻共q兲兴2 =再While doing simulation it is found that dispersionless excitation at ប␻ ⬃ 24 meV, where four modes are overlapped, issensitive to the change of parameter JCu. This means thatthey are from Cu centered spins. Similarly, four modes in thelower energy are from Fe centered spins. The estimated exchange parameters relating to Fe spins are summarized in thesecond line in Table I. They are reasonably consistent withpreviously estimated values in the first line. However, JCu of
2.共6兲 ⫻ 102 meV is very different from the previous value
24.共2兲 meV in Ref. 17. This is because of the different assumption for the ground state; classical Néel state for thepresent SW and singlet dimer state for the previous analysis.
To discuss the JCu, the isostructural compound
Cu2Sc2Ge4O13,15 where Fe ions in Fig. 1共b兲 are replaced bynonmagnetic Sc ions would give good insight. In these compounds the angle ⬔Cu-O-Cu and the bond distance are very
close to each other and so would the magnitude of superexchange constant be. The intradimer interaction in
Cu2Sc2Ge4O13 was reliably determined as JSc = 24.共5兲⬃ 25.共4兲 meV by the isolated dimers model.16 The value isclose to the previous estimate of JCu 24.共2兲 meV 共Ref. 17兲
rather than the present one 2.共6兲 ⫻ 102 meV. Hence, the singlet dimer ground state is a better assumption and we adopt
the JCu = 24.共2兲 meV as the reasonable value hereafter.
In the second scenario we will consider the effective SW
model based on the energy scale separation of the magnetic
excitations.17 The absence of dispersion in the Cu centeredmode and the suppressed magnetic moment means that Cu
spins are strongly bound to spin dimers singlet state. Onto
the Cu1 spin in the dimer the field hCu1 is applied throughJCu-Fe by the Fe1 spins in Fig. 1共c兲. Then the polarized spins
SCu = ␹共␲ , ␻兲hCu, are induced on the both Cu1 and Cu2 ionsthat transfer the exchange integral from one Fe spin to another. Here the dynamic susceptibility of the dimer ␹共␲ , ␻兲 ispurely real and expressed by ␹共␲ , ␻兲 = 1 / 共JCu − ប␻兲 at T
= 0 K 关solid curve in Fig. 3共b兲兴. In the energy region of Fe
spin networks 共gray shaded area兲 ␹共␲ , ␻兲 is almost 1 / JCu
and constant. This suggests that static treatment of MF-RPA
works well when we consider the spin dynamics at ប␻ⱗ 10 meV. Then the indirect coupling Jeff can be approxi22
␹共␲ , ␻兲 / 2 ⬃ JCu-Fe/ 2JCu.
mately expressed as Jeff共␻兲 = JCu-Fe
Thus the effective spin Hamiltonian for the low energy region is
H=

1
J␣␤S␣Fe · S␤Fe .
兺
2 ␣,␤苸8 Fe sublattices

共2兲

We considered Jb, Jc, and Jeff as the exchange parameters as
is illustrated in Fig. 1共e兲. The calculated dispersion relation isS2 兵共A2 − D2兲 − 共B2 + C2 + 2BC cos ␲k兲 ± 2D冑B2 + C2 + 2BC cos ␲k其,S2 兵共A2 − D2兲 − 共B2 + C2 − 2BC cos ␲k兲 ± 2D冑B2 + C2 − 2BC cos ␲k其.
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冎

共3兲

RAPID COMMUNICATIONSPHYSICAL REVIEW B 75, 220401共R兲 共2007兲

MASUDA et al.
2Here A = 2Jb + Jeff + 2Jc, B = 冑J2b + Jeff
+ 2JbJeff cos 2␲h, C = Jb,
and D = 2Jc cos 2␲l. Reasonable fit to the data is shown by
the thick solid curves in Fig. 3共a兲. They are perfectly overlapped on the open small circles calculated in the first scenario. The obtained exchange parameters are summarized in
the third line in Table I.
Excellent consistency was found in the independently estimated exchange parameters of Fe-centered spins. Particularly interesting is the effective indirect coupling Jeff’s. In thecurrent study we obtained Jeff = 0.09共6兲 meV by the direct
measurement of the spin dynamics. In the previous study14
JCu-Fe / JCu was estimated statically in MF-RPA level from thestaggered magnetization curves of Cu dimers obtained by
ND. Combining the parameter with JCu = 24.共2兲 meV in
a separate experiment,17 Jeff = 0.13共4兲 meV was estimatedsemistatically. The reasonable consistency in Jeff’s meansthat the static MF-RPA treatment works well and, thus, Cu
dimers behave as the nonmagnetic media in the indirect
Fe-Fe coupling. Due to the simplicity of the model, all the
related exchange constants, JCu, Jeff, and JCu-Fe are experimentally obtained.
There are a few other compounds reported as bicomponent systems with spin-singlet entity and additional spin.
Among them R2BaNiO5 共R = rare earth metals兲 have beenstudied the most in detail.19 The compound is described byreasonably decoupled one dimensional Ni2+ 共S = 1兲 chains
and surrounding rare earth ions. The weak coupling induce

We wish to acknowledge fruitful discussion with A. Zheludev in the early stage of this study. C. Yasuda is appreciated for useful comments. This work was supported by the
grant for Strategic Research Project No. W17003, No.
K17028, and No. K18032 of Yokohama City University,
Japan.

13 T. Masuda, B. C. Chakoumakos, C. L. Nygren, S. Imai, and K.

*Email address: tmasuda@yokohama-cu.ac.jp
1magnetically ordered state of which the nature is similar tothat in Cu2Fe2Ge4O13.20 However, the rare earth moment behaves as Ising type spin and the indirect interaction by way
of Haldane chains has not been studied. Other examples are
Cu2CdB2O6 and Cu3共P2O6OH兲2.21 Spin model of quantum
dimers and nearly free spins is proposed from the observed
magnetization plateaux in the M-H curves. For lack of single
crystal, however, the detailed spin dynamics is still unknown.
As far as we know Cu2Fe2Ge4O13 is the only model compound for the indirect magnetic interaction by way of spin
singlet media.
In conclusion we succeeded to obtain the spin dispersion
in the a* direction in Cu2Fe2Ge4O13 by high-resolution neutron scattering experiment and by the elaborate data analysis.
The whole dispersion is well explained by the effective SW
model. The effective Fe-Fe interaction Jeff is directly estimated and the value is consistent with that obtained semistatically. It is found that Cu2Fe2Ge4O13 is a prototype compound for randomness-free RKKY-like interaction in
insulating metal oxides.

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