quasi-one-dimensional excitations of copper metaborate in the commensurate phase 10 k

Journal of Magnetism and Magnetic Materials 269 (2004) 106–112

Quasi-one-dimensional excitations of copper metaboratein the commensurate phase 10 KoTo20K

S. Martynova,*, G. Petrakovskiia, B. Roesslib

a Institute of Physics, SB RAS, Laboratory of Resonance, Properties of Magnetically Ordered Subst., 660036 Krasnoyarsk, RussiabPaul Scherrer Institute, CH-5232 Villigen, PSI, Switzerland

Received 18 June 2003

Abstract

The low-energy branch of the spin-wave spectrum of CuB2O4 was investigated theoretically by means of linear spin-

wave theory in the commensurate magnetic phase. We have found that these excitations are associated with the

magnetic subsystem Cu(B). The exchange interactions in this subsystem between nearest and next nearest neighbours,

which forms the quasi-one-dimensional ‘‘zig-zag’’ ladder structure, were determined.

r 2003 Elsevier B.V. All rights reserved.

PACS: 75.25.+z; 75.10.Hk; 75.30.Gw

Keywords: One-dimensional magnetic structure; Spin-wave spectrum; Inelastic neutron scattering

1. Introduction

Intensive investigations of the magnetic proper-ties of copper metaborate CuB2O4 single crystalhas revealed a series of magnetic phase transitionsbetween different magnetic phases [1–6]. In parti-cular, according to the elastic neutron scatteringexperiments, below T� ¼ 10 K the magnetic struc-ture becomes incommensurate along the tetrago-nal axis with a propagation vector which risecontinuously from ~kk0 ¼ ð0; 0; 0Þ to ~kk0 ¼ ð0; 0; 0:15Þ(r.l.u.) when the temperature decreases. In spite ofabundance of the experimental dates and theexistence of phenomenological description ofincommensurate phase transition, the microscopic

mechanism of this behavior of propagation vectorremains undetermined. In paper [7] the high-energy part of the spin-wave spectrum wasidentified as a spectrum of excitation of the copperions subsystem with a large magnetic momentmB1 mB (Cu(A)) and the parameters of exchangeinteraction between the ions of ‘‘strong’’ subsys-tem A were determined. However the low-energypart of spectrum which was observed for ~kkJ~cc of asubsystem B of the magnetic ions Cu(B) with asmall magnetic moment mB0:2 mB remain theore-tically unexplored. The sufficient difference of theantiferromagnetic ordering temperature TN ¼20 K and the incommensurate phase transitiontemperature T� ¼ 10 K allow to assume that the‘‘weak’’ subsystem B plays a large role in theformation of the magnetic incommensuratestructure. Therefore it is important to determine

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*Corresponding author. Tel.: 73912432635; fax: 73912438923.

E-mail address: [email protected] (S. Martynov).

0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S0304-8853(03)00571-7

the exchange interactions in the subsystem Bfor understanding of the microscopical mech-anism of the incommensurability. The aim ofthis work is the determination of the mainmagnetic interactions in the ‘‘weak’’ subsystemB of copper metaborate from the analysisof the inelastic neutron scattering spectrum atT ¼ 12 K:

2. Exchange interactions

Copper metaborate, CuB2O4; crystallizes in thetetragonal space group I %42d (D12

2d) [8] with 12formula units in the chemical cell and the latticeparameters a ¼ 11:528 (A; c ¼ 5:607 (A: The com-plicated magnetic behavior is due to the existenceof two nonequivalent positions of copper ions,labelled Cu(A) and Cu(B) in Fig. 1. The Cu(A)ions are located at site 4b (point symmetryS4; 0012) and are at the center of a square unitformed by four oxygen ions in basal plane. Cu(B)are at site 8d (point symmetry C2;x 1

418; x ¼ 0:0815)

and are surrounded by a slightly distorted squareof oxygen ions, which turned at the same anglefrom (1 0 0) and (0 1 0) planes (labelled B1 and B2in Fig. 1).In order to analyze the main exchange interac-

tions we consider the possible superexchangepathways. The exchange interactions between allcopper ions are realized by hybridized s–p orbitalsof boron-oxygen tetrahedrons. So, for all exchangeinteractions we may consider only the Cu–O–B–O–Cu bonds. In this assumption the exchangeinteraction between ions Cu(A) in A-subsystem isrealized by the single pathways Cu–O–B–O–Cu,whereas in B-subsystem both single and doublepathways through two boron tetrahedrons arepossible (Fig. 2). An analysis of exchange interac-tions bonds between A- and B-subsystems showsthat the interactions between Cu(B) ions andCu(A) ions of two sublattices of the A-subsystemare completely frustrated. It is allowed to considerthe two subsystems in copper metaborate asnoninteracting at first approximation. Such as-sumption leads to the coincident values of theexchange interaction between Cu(A) ions obtainedfrom the spin-wave spectrum and the mean field

evaluations of TN [7]. This result confirms theassumption that the antiferromagnetic ordering atTN is due to the exchange interactions within theA-subsystem. The magnetic state of the B-sub-system at temperatures ToTN remains undefined.The elastic neutron scattering data shows thatthere is the average value of the magnetic momenton B-sites is mBB0:2 mB at T ¼ 12 K [9]. Theexistence of the dispersion dependence of the low-energy spin-wave branch for ~kkJ~cc allows to assumethat the B-subsystem has the some magnetic orderalong tetragonal axis in the high-temperaturephase T�oToTN also. Neglecting all frustratedbonds between the antiferromagnetically orientedCu(B) ions, B-subsystem can be considered as a set

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Fig. 1. The chemical cell of CuB2O4: Cu(A) and Cu(B)

positions are represented by black and open symbols, respec-

tively.

Fig. 2. Main exchange interactions within B-subsystem.

S. Martynov et al. / Journal of Magnetism and Magnetic Materials 269 (2004) 106–112 107

of the double chains of ions along the c-axis—‘‘zig-zag’’ ladder structures (Fig. 2). This structure hasthe double Cu–O–B–O–Cu exchange bonds be-tween nearest and single ones between next nearestneighbour spins and the above-mentioned disper-sion dependence may be considered as a spectrumof the collective spin excitations in the quasi-one-dimensional magnetic chains.

3. Spin-wave theory

We analyzed the low-energy branch of thecollective excitations spectrum, as it was madefor A-subsystem in Ref. [7] neglecting the interac-tions between subsystems and taking into accountonly the exchange in double (ladder) chains ofCu(B)ions. For the analysis of spin excitationswith energies EB1 meV we ignore other possibleweak interactions such as the antisymmetricanisotropic exchange interaction, three-dimen-sional interchain exchange interactions and di-pole–dipole interactions. In both coordinatespheres the exchange interactions are consideredas the single-axis anisotropic interactions withcommon axis z: The Hamiltonian of B-subsystemfor S ¼ 1

2has the form

H ¼ Jz

Xij

Szi Sz

j þ Jxy

Xij

ðSxi Sx

j þ Syi S

yj Þ

þ J 0z

Xii0

Szi Sz

i0 þ J 0xy

Xii0

ðSxi Sx

i0 þ Syi S

yi0 Þ

þ J 0z

Xjj0

Szj Sz

j0 þ J 0xy

Xjj0

ðSxj Sx

j0 þ Syi S

yj0 Þ; ð1Þ

where the sum is carried out over interactionsbetween B1ðiÞ and B2ðjÞ ions and within B1ðiÞ andB2ðjÞ chains separatively (Fig. 2). One of the mainaim of this work is the determination of theexchange anisotropy type : ‘‘easy axis’’ or ‘‘easyplane’’. So we consider both possibilities of theexchange anisotropy.

3.1. ‘‘Easy axis’’ antiferromagnet: Jz > Jxy

The Holstein–Primakoff transformation withinapproach of the spin wave theory [10] for

sublattices (ið1Þ and jð2Þ) is defined as

Szi;j ¼ �S þ bþ

i;jbi;j ;

Sxi;j ¼

ffiffiffiffiS

2

rðbþ

i;j þ bi;jÞ;

Syi;j ¼ i

ffiffiffiffiS

2

rðbi;j � bþ

i;jÞ; ð2Þ

where bþi;j and bi;j — are the quantization operators

of B1 and B2 sites with local axes shown in Fig. 3.The substitution Eq. (2) in Hamiltonian (Eq. (1)and the Fourier transformation

bi;j ¼1ffiffiffiffiffiN

p Xk

b1k;2k expði~kk~rri;jÞ;

where ~kk is a vector in reciprocal space, lead to themomentum representation of the part of Hamilto-nian, which is quadratic in the quantizationoperators. For ~kkJ~cc we found

H2 ¼ 2SX

k

½ðJz � J 0z þ J 0

xy cos kcÞðbþ1kb1k þ bþ

2kb2kÞ

þ Jxy coskc

2ðbþ

1kbþ2�k þ b1kb2�kÞ�:

The diagonalization of Hamiltonian H2 givestwo the degenerate branches of spectrum withenergy

EðkÞ ¼ 2S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðJz � J 0

z þ J 0xy cos kcÞ2 � Jxy cos

kc

2

� �2s

;

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Fig. 3. The local axes of ‘‘easy-axis’’ antiferromagnet.

S. Martynov et al. / Journal of Magnetism and Magnetic Materials 269 (2004) 106–112108

in accordance to general spin-wave theory for‘‘easy axis’’ antiferromagnet with next nearestneighbors exchange interaction [11]. The maincharacteristic feature of the ‘‘easy axis’’ antiferro-magnet spectrum is the periodicity with the wavevector period 2p=c

EðkÞ ¼ E k þ2pc

� �:

The absence of this periodicity in the experimentalspectrum of the spin excitations allow us toconclude that the B-subsystem is not an ‘‘easyaxis’’ antiferromagnet.

3.2. ‘‘Easy plane’’ antiferromagnet: JzoJxy

The antiferromagnetic interaction between thenext nearest neightbours requires us to performthe analysis of the equilibrium orientation ofthe magnetic moments in the ‘‘easy’’ plane. Theorientation of the local axes with respect to thecrystal ones is shown in Fig. 4. The relationsbetween projections of spins on the crystal (S

x;y;zi;j )

and local (Sxi;j ;yi;j ;zi;j ) axes are given by

Sxi;j ¼Sxi;j xi cos ai;j � Syi;j sin ai;j

Syi;j ¼Sxi;j xi sin ai;j þ Syi;j cos ai;j

Szi;j ¼Szi;j :

If we choose the axes xi and xj as localquantization axes, the Holstein–Primakoff trans-

formation is defined as

Sxi;j ¼ � S þ bþi;jbi;j ;

Syi;j ¼

ffiffiffiffiS

2

rðbþ

i;j þ bi;jÞ;

Szi;j ¼ i

ffiffiffiffiS

2

rðbi;j � bþ

i;jÞ: ð3Þ

The Hamiltonian up to the second power in thequantization operators becomes

H ¼ H0 þ H1 þ H2;

H0 ¼X

i

H0i þX

j

H0j ;

H1 ¼X

i;j

Syi f ðai; ajÞ þX

i;j

Syj f ðaj ; aiÞ;

where f ðai; ajÞ is a function of the equilibriumangles. The requirement that the linear part of theHamiltonian H1 vanishes, f ðai; ajÞ ¼ 0; is equiva-lent to the requirement that the transverse part ofthe local effective fields of the magnetic momentsin the equilibrium state are equal to zero. It leadsto a condition for the equilibrium angles ai; aj

sin ai �aj þ aj�1

2

� �cos

aj � aj�1

2

¼ �jxy sin ai �aiþ1 þ ai�1

2

� cos

aiþ1 � ai�1

2;

ð4Þ

where jxy ¼ J 0xy=Jxy: This equation has two solu-

tions:

(AF) Uniform solution — antiferromagneticorientation of local axes

ai ¼ ai0 ; aj ¼ aj0 ; ai ¼ aj þ p;

(SS) Simple spiral

ai ¼aiþ1 � ai�1

2¼

aj � aj�1

2þ p:

The energy of each spin in the local equilibriumorientation is

E0i ¼ � E0AF cos ai �

aj þ aj�1

2

� �cos

aj � aj�1

2

þ jxy cos ai �aiþ1 þ ai�1

2

� cos

aiþ1 � ai�1

2

�;

E0AF ¼ �S2Jxy:

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Fig. 4. Local axes of ‘‘easy-plane’’ antiferromagnet.


Using Eq. (4) for every solution yields

EAF0i ¼ E0

AFð1� jxyÞ;

ESS0i ¼ E0

AF cosaj � aj�1

2� jxy cos

aiþ1 � ai�1

2

� :

The minimization of the SS-solution leads to theangle of helix

d ¼ aj � aj�1 ¼aiþ1 � ai�1

2;

cosd2¼

1

4jxy

and energy

ESS0i ¼ E0

AF jxy þ1

8jxy

� �:

We can see, that if jxy > 14the SS-solution have the

lower energy than the AF-solution. So, usingthis standard result of the ANNNI-model [12–14]and Fourier transformation one has to considerthe part of Hamiltonian which is quadraticin the quantization operators, i.e. H2: For ~kkJ~ccwe obtain

H2 ¼ HJ2 þ HJ 02

HJ2 ¼X

k

SJxy

2ðbþ

1kb1k þ bþ2kb2kÞ

�

þ Jz �Jxy

4jxy

� �cos

kc

2ðbþ

1kb2k þ h:c:Þ

� Jz þJxy

4jxy

� �cos

kc

2ðbþ

1kbþ2�k þ h:c:Þ

�;

HJ 02 ¼Xk

S J 0xy 1�

1

8j2xy

!ð2� cos kcÞ þ J 0

z cos kc

!bþ1kb1k

þ1

2J 0

xy 1�1

8j2xy

!� J 0

z

!ðbþ1kbþ1�k exp ikc þ h:c:Þ þ ð1-2ÞÞ:

The separation of the collective excitations into theacoustic and the optical branches is obtained bythe following substitutions

b1k ¼1ffiffiffi2

p ðak � ckÞ; b2k ¼1ffiffiffi2

p ðak þ ckÞ;

H2 ¼X

k

AaðkÞaþk ak þ AcðkÞcþk ck

��ðBaðkÞaþ

k aþ�k þ h:c:Þ

þ ðBcðkÞcþk cþ�k þ h:c:Þ�;

Aa;cðkÞ ¼SJxy 81

4jxy

� jz

� �cos

kc

2þ 2jxy

�

þ1

4jxy

� jxy �1

8jxy

� j0z

� �cos kc

�;

Ba;cðkÞ ¼SJxy

2

1

4jxy

þ jz

� �cos

kc

2

�

7 jxy �1

8jxy

þ j0z

� �exp ikc

�;

jz ¼Jz

Jxy

; j0z ¼J 0

z

Jxy

:

The final diagonalization of the quadratic formsleads to the excitations spectrum

Ea;cðkÞ ¼

2SJxy

1

4jxy

18coskc

2

� �þ jxy �

1

8jxy

� �ð1� cos kcÞ

� �

jxy þ1

8jxy

7jz coskc

2þ j0z cos kc

� � ja;cðkÞ

�1=2;

ð5Þ

where

ja;cðkÞ ¼ 1

þðjxy � 1=8jxy þ j0zÞ

2 sin2 kc

ð2jxy þ 1=4jxy7ðjz � 1=4jxyÞcos kc=2� ðjxy � 1=8jxy � j0zÞcos kcÞ2:

4. Comparison with the experiment

The obtained in Ref. [7] excitations spectrum for~kkJ~cc is shown in Fig. 5. The existence of the low-energy dispersion spectrum only for this orienta-tion of the wave vector confirm our conclusionabout quasi-one-dimensional character of themain exchange interactions in the B-subsystem,as it appears from the analysis of the crystalstructure CuB2O4: Strong diffuse neutron scatter-ing for the neutron scattering vector along the

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[0 0 1] direction [4] also confirms the one-dimen-sional character of magnetic excitations. Theoptical branch of spectrum for B-subsystemesnot observed probably due to weak intensity as itwas the case for the A-subsystem [7]. The bestagreement of our theoretical calculation of theacoustic branch (solid circles in Fig. 5) andexperimental points was obtained for the followingvalues of exchange parameters

2SJxy ¼ 2:670:05 meVðJxy ¼ 31 KÞ;

jxy ¼J 0

xy

Jxy

¼ 0:263;

jz ¼Jz

Jxy

¼ 0:84;J 0

z

J 0xy

¼ 0:9:

The value jxy ¼ 0:263 > jcxy ¼ 0:25: It corresponds

to the equilibrium orientation of the magnetic

moments in the SS-structure. This means that thewave vectors of the magnetic excitations arecalculated from the helix vector ~kk0 ¼ 0:1 r.l.u.,and excitation spectrum is shifted by this vector.So, the B-subsystem is considered as a quasi-one-dimensional ‘‘zig-zag’’ ladder structures with thehelical order along ~cc-axis. The function ja;cðkÞ inEq. (5) is close to one for all ~kk in the wide intervalof the exchange parameters. After this simplifica-tion one can see that if we take into account theexchange interactions between nearest and nextnearest neighbours the excitation energy is thefunction with the two maximum and therefore thedifference of the experimental point at ~kk ¼ 0:9from theoretical curve could not be explained in theframework of considered model. Nevertheless, thegeneral shift of spectrum relatively the reciprocalvectors at ~kk0 ¼ 0:1 r.l.u. is the obvious feature ofmodulated state in the B-subsystem at T ¼ 12K:The ‘‘easy plane’’ anisotropy parameters for the

B-subsystem are greater than anisotropy in the A-subsystem. This may be explained by a distortionof the oxygen ions square, surrounding the Cu(B)-ions which is absent for the Cu(A)-ions. It seemsreasonable that the anisotropy is greater for theinteraction between nonequivalent ions of thenearest neighbours Cu(B1) and Cu(B2) with alocal neighborhood in the form of distortedsquares which are turned one from another.The lowest reliability can expect for an absolute

value of the exchange interaction Jxy: The spinrepresentation by the second quantization opera-tors in form Eq. (3) for magnetic states with amoment per site sufficiently lower than thesaturation one m51 mB is a very rough approx-imation. In such states, which is typical for quasi-one-dimensional magnets, the resonance proper-ties analysis lead to a decreasing of exchange fieldfrom classical value, i.e. h1d

exoh3dex ¼ nSJ; were n —

is a number of neighbours. In used approach thisvalue is a general multiply in a spectrum energy inEq. (5). So, our analysis of the experimentalspectrum leads to the sufficiently lower exchangevalue in comparison with the real exchange in theHamiltonian of B-subsystem. But a direct connec-tion of this field with observed magnetic momentat T ¼ 12 K for a quasi-one-dimensional structureis unfounded.

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.

.

.

.

.

.

.

.. . . . . . .

Fig. 5. Excitation spectrum of CuB2O4 at T ¼ 12K for

(0, 0, q)- direction. Circles — experimental data points obtained

from inelastic neutron scattering, solid and dash lines — result

of the theoretical analysis for acoustic and optic branches,

respectively.


5. Conclusion

The theoretical calculation of the spin-wavespectrum of the copper ions in the ‘‘weak’’subsystem of copper metaborate at T ¼ 12 Kwas carried out on the basis of model with one-dimensional exchange interaction within the ‘‘zig-zag’’ ladder chains. The form of the experimentalspectrum is similar to the spectrum of the ‘‘easyplane’’ antiferromagnet. The parameters of theexchange anisotropy are determined for bothnearest and next nearest neighbours. The questionabout orientation of the ‘‘easy plane’’ in crystalis open and it will be studied later. The analysisof the neutron inelastic scattering allow tomake a conclusion about the co-existence of twomagnetic subsystem with a different effectivemagnetic dimensionality (3d and 1d) and twotypes of magnetic excitations, respectively, in thecopper metaborate in temperature regionT�oToTN:

Acknowledgements

This work is done under partial financialsupport from RFBR (grant 03-02-16701). Theabove cited experimental dates [7] was performedat the SINQ, Paul Scherrer Institut, Villigen,Switzerland and in the Institute Laue-Langevin,Grenoble, France. We also thank M. Popov, V.

Zinenko, A. Pankrats, K. Sablina and M. Boehmfor fruitful discussions.

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quasi-one-dimensional excitations of copper metaborate in the commensurate phase 10 k

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