elementary excitations in antiferromagnetic heisenberg spin segments

8
Elementary excitations in antiferromagnetic Heisenberg spin segments A. Ghirri, 1,2, * A. Candini, 1 M. Evangelisti, 1,3 M. Affronte, 1,2 S. Carretta, 4,1 P. Santini, 4,1 G. Amoretti, 4,1 R. S. G. Davies, 5 G. Timco, 5 and R. E. P. Winpenny 5 1 National Research Center on nanoStructures and bioSystems at Surfaces S 3 , INFM-CNR, via Campi 213/a, 41100 Modena, Italy 2 Dipartimento di Fisica, Università di Modena e Reggio Emilia, via Campi 213/a, 41100 Modena, Italy 3 CNISM, Unità di Modena, via Campi 213/a, 41100 Modena, Italy 4 Dipartimento di Fisica, Università di Parma, via Usberti 7/A, 43100 Parma, Italy 5 Department of Chemistry, University of Manchester, Oxford Road, Manchester M139PL, United Kingdom Received 26 June 2007; published 7 December 2007 We report on ac susceptibility, low-temperature magnetization, and specific heat measurements on molecular compounds, shortly named Cr 6 2 , Cr 7 2 , Cr 8 Cd, and NiCr 6 2 Zn, that comprise different variants of spin arrays. These systems constitute real examples of collections of identical antiferromagnetic Heisenberg spin segments. We indeed show that this picture, with dominant exchange term in the spin Hamiltonian J / k B ranging from 13 to 16 K in all compounds and weak anisotropy term, fits well the measured physical prop- erties. The character of energy spectra and the low-lying magnetic excitations are discussed accordingly. The direct comparison of experimental results and of the energy spectra of these variants with those of similar cyclic spin systems evidences effects associated with i the breaking the cyclic boundary conditions and ii odd and even nuclearities of spin segments. DOI: 10.1103/PhysRevB.76.214405 PACS numbers: 75.50.Xx, 75.30.Ds, 75.10.Pq, 75.75.a I. INTRODUCTION Infinite spin chains have attracted interest of physicists for long time since they are model systems in which different types of excitations can be created depending, for instance, on the character of the exchange interaction, on anisotropy, or on the spin. A remarkable example is the Haldane predic- tion: a gap opens in the energy spectrum when the spin value of the magnetic centers is integer but not in the case of half- integer spin chains. 1,2 In the past decades the study of the excitations of Heisenberg spin chain has received a large interest from both the theoretical and experimental points of view. 3 More recently the attention was drawn to finite spin chains or “segments.” For short enough chains, the low-lying energy levels can be directly evaluated by diagonalizing the spin Hamiltonian and the finite size leads to a discrete energy spectrum. Examples of spin segments are spin chains broken by impurities or engineered atomic structures. 4 Here we present an alternative molecular route to obtain well ordered spin segments. For low spin values, these systems have purely quantomechanichal character and, while chemists simply talk about “molecular” levels and states, physicists like to call these systems nano- or quantum spin wires, in analogy with semiconducting artificial structures. One inter- esting issue is the localization of excitations in finite spin chains. It is known that edge states in both integer and half- integer chains arise from exchange interactions, 59 while lo- calized states 10,11 can be observed when nonlinear effects are significant. Spin segments, or finite spin chains, may also have special boundary conditions imposed at the termina- tions that may strongly influence the character of excitations. 12 A variety of molecular crystalline compounds comprising collections of identical and independent ring-shaped spin clusters have been synthesized and characterized in the last two decades. Relatively well known are the ferric wheels Fe 6 , 13 Fe 10 , 14,15 Fe 12 , 16,17 Fe 18 , 18 with Fe 3+ s =5 / 2; but also Cr-based molecular rings such as Cr 8 Ref. 19 and Cr 10 Ref. 20comprising Cr 3+ s =3 / 2 and, to minor extent, V-based rings such as V 8 and V 10 Ref. 21V 3+ s =1. They all have a dominant Heisenberg exchange coupling be- tween spin centers that leads to a characteristic S = 0 ground state, due to the exact compensation between the spin centers within the ring and common features of the excited states. The low-lying energy spectrum is typically structured in three bands: the lowest one, named L band, is a rotational band that follows the Landè rule ES SS +1. 2224 These excited states are characterized by the rotation of the oppo- sitely oriented total spins of each sublattice. The second group, named E band, is generated by higher energy excita- tions that have analogies with spin waves. The remaining states form a quasicontinuum. 23,25 The strategy to produce molecular spin segments is de- fined as follows: Starting from the synthetic approach that yields Cr 8 , a class of heterometallic derivatives is obtained 26,27,29 and they can be regarded as “broken” or “open” spin rings. These systems are obtained by bridging trivalent metal mainly Cr 3+ ions with carboxylate and flou- ride ligands that results in antiferromagnetic exchange cou- pling between two neighboring spin centers. In a first syn- thetic route, nonmagnetic Cd 2+ or Zn 2+ were substituted to a Cr 3+ in Cr 8 thus breaking the symmetry of the ring. The final result, Cr 7 Cd or analogously Cr 7 Zn, 26 was investigated in order to evaluate the magnetocaloric effect 30 and effects of S mixing. 31 Broken Cr 8 M, with M =Ni, Cd, rings are also ob- tained and they comprise an even number of spin centers. 27,32 In a further synthetic route, finite Cr 3+ segments have been synthesized with characteristic shape of two “horseshoe” fac- ing one another. 26 Each horseshoe may result either even membered or odd membered. Furthermore, heterometallic chains have also been synthesized were a Ni 2+ is positioned at the end of a Cr 6 segment. This compound, NiCr 6 2 Zn, is PHYSICAL REVIEW B 76, 214405 2007 1098-0121/2007/7621/2144058 ©2007 The American Physical Society 214405-1

Upload: r-e-p

Post on 07-Apr-2017

213 views

Category:

Documents


0 download

TRANSCRIPT

Elementary excitations in antiferromagnetic Heisenberg spin segments

A. Ghirri,1,2,* A. Candini,1 M. Evangelisti,1,3 M. Affronte,1,2 S. Carretta,4,1 P. Santini,4,1 G. Amoretti,4,1 R. S. G. Davies,5

G. Timco,5 and R. E. P. Winpenny5

1National Research Center on nanoStructures and bioSystems at Surfaces �S3�, INFM-CNR, via Campi 213/a, 41100 Modena, Italy2Dipartimento di Fisica, Università di Modena e Reggio Emilia, via Campi 213/a, 41100 Modena, Italy

3CNISM, Unità di Modena, via Campi 213/a, 41100 Modena, Italy4Dipartimento di Fisica, Università di Parma, via Usberti 7/A, 43100 Parma, Italy

5Department of Chemistry, University of Manchester, Oxford Road, Manchester M139PL, United Kingdom�Received 26 June 2007; published 7 December 2007�

We report on ac susceptibility, low-temperature magnetization, and specific heat measurements on molecularcompounds, shortly named �Cr6�2, �Cr7�2, Cr8Cd, and �NiCr6�2Zn, that comprise different variants of spinarrays. These systems constitute real examples of collections of identical antiferromagnetic Heisenberg spinsegments. We indeed show that this picture, with dominant exchange term in the spin Hamiltonian �J /kB

ranging from 13 to 16 K in all compounds� and weak anisotropy term, fits well the measured physical prop-erties. The character of energy spectra and the low-lying magnetic excitations are discussed accordingly. Thedirect comparison of experimental results and of the energy spectra of these variants with those of similarcyclic spin systems evidences effects associated with �i� the breaking the cyclic boundary conditions and �ii�odd and even nuclearities of spin segments.

DOI: 10.1103/PhysRevB.76.214405 PACS number�s�: 75.50.Xx, 75.30.Ds, 75.10.Pq, 75.75.�a

I. INTRODUCTION

Infinite spin chains have attracted interest of physicists forlong time since they are model systems in which differenttypes of excitations can be created depending, for instance,on the character of the exchange interaction, on anisotropy,or on the spin. A remarkable example is the Haldane predic-tion: a gap opens in the energy spectrum when the spin valueof the magnetic centers is integer but not in the case of half-integer spin chains.1,2 In the past decades the study of theexcitations of Heisenberg spin chain has received a largeinterest from both the theoretical and experimental points ofview.3 More recently the attention was drawn to finite spinchains or “segments.” For short enough chains, the low-lyingenergy levels can be directly evaluated by diagonalizing thespin Hamiltonian and the finite size leads to a discrete energyspectrum. Examples of spin segments are spin chains brokenby impurities or engineered atomic structures.4 Here wepresent an alternative molecular route to obtain well orderedspin segments. For low spin values, these systems havepurely quantomechanichal character and, while chemistssimply talk about “molecular” levels and states, physicistslike to call these systems nano- or quantum �spin� wires, inanalogy with semiconducting artificial structures. One inter-esting issue is the localization of excitations in finite spinchains. It is known that edge states in both integer and half-integer chains arise from exchange interactions,5–9 while lo-calized states10,11 can be observed when nonlinear effects aresignificant. Spin segments, or finite spin chains, may alsohave special boundary conditions imposed at the termina-tions that may strongly influence the character ofexcitations.12

A variety of molecular crystalline compounds comprisingcollections of identical and independent ring-shaped spinclusters have been synthesized and characterized in the lasttwo decades. Relatively well known are the ferric wheels

Fe6,13 Fe10,14,15 Fe12,

16,17 Fe18,18 with Fe3+ �s=5 /2�; but also

Cr-based molecular rings such as Cr8 �Ref. 19� and Cr10�Ref. 20� �comprising Cr3+ �s=3 /2�� and, to minor extent,V-based rings such as V8 and V10 �Ref. 21� �V3+�s=1��.They all have a dominant Heisenberg exchange coupling be-tween spin centers that leads to a characteristic S=0 groundstate, due to the exact compensation between the spin centerswithin the ring and common features of the excited states.The low-lying energy spectrum is typically structured inthree bands: the lowest one, named L band, is a rotationalband that follows the Landè rule E�S��S�S+1�.22–24 Theseexcited states are characterized by the rotation of the oppo-sitely oriented total spins of each sublattice. The secondgroup, named E band, is generated by higher energy excita-tions that have analogies with spin waves. The remainingstates form a quasicontinuum.23,25

The strategy to produce molecular spin segments is de-fined as follows: Starting from the synthetic approach thatyields Cr8, a class of heterometallic derivatives isobtained26,27,29 and they can be regarded as “broken” or“open” spin rings. These systems are obtained by bridgingtrivalent metal �mainly Cr3+� ions with carboxylate and flou-ride ligands that results in antiferromagnetic exchange cou-pling between two neighboring spin centers. In a first syn-thetic route, nonmagnetic Cd2+ or Zn2+ were substituted to aCr3+ in Cr8 thus breaking the symmetry of the ring. The finalresult, Cr7Cd �or analogously Cr7Zn�,26 was investigated inorder to evaluate the magnetocaloric effect30 and effects of Smixing.31 Broken Cr8M, with M =Ni, Cd, rings are also ob-tained and they comprise an even number of spin centers.27,32

In a further synthetic route, finite Cr3+ segments have beensynthesized with characteristic shape of two “horseshoe” fac-ing one another.26 Each horseshoe may result either evenmembered or odd membered. Furthermore, heterometallicchains have also been synthesized were a Ni2+ is positionedat the end of a Cr6 segment. This compound, �NiCr6�2Zn, is

PHYSICAL REVIEW B 76, 214405 �2007�

1098-0121/2007/76�21�/214405�8� ©2007 The American Physical Society214405-1

composed by two NiCr6 moities attached through a nonmag-netic Zn2+ in a S-shaped “seahorse” cluster. It turns out thatthese molecules are beautiful examples of collections ofidentical spin segments, thus opening a promising route toperform targeted studies of finite spin chains. With respect tostatistical doping of infinite chains, this bottom-up approachhas the obvious advantage that there is no size dispersion.Also, the possibilities of chemical control over length, dop-ing, and spin values are remarkable and still in part unex-plored.

In what follows, we present low-temperature specificheat, susceptibility, and magnetization measurements of themolecular �Cr6�2 �Cr7�2, Cr8Cd, and �NiCr6�2Zn antiferro-magnetic spin segments. The results are used to determinethe parameters of the spin Hamiltonian, including thestrength and role of the anisotropy terms, and in turn to cal-culate the energy spectra. For the sake of simplicity and topermit a direct comparison between the different systems, weintroduce an exchange constant J solely. This approximationdescribes satisfactorily the behavior of the discussedsystems.28

II. EXPERIMENTAL DETAILS

The crystal structures of the molecules studiedin this work are shown in Fig. 1.��H2NR2�3Cr6F11�O2CCMe3�10�H2O��2 with R=C3H7,in short �Cr6�2, was prepared as reported.26

��H2NR2�3Cr7F12�O2CCMe3�12�H2O��2 with R=isC3H7, inshort �Cr7�2, was prepared similarly to �Cr6�2 by usingdiisopropylamine instead of dipropylamine and fulldetails of the synthesis and the structure of �Cr7�2 will bepublished shortly.33 �H2NtBuisPr��Cr8CdF9�O2CCMe3�18�,in short Cr8Cd, was obtained by the method given inRef. 27. �Ni�cyclen�2Cr12ZnF18�O2CCMe3�24�, in short�NiCr6�2Zn: �Ni�cyclen��O2CCMe3�H2O��O2CCMe3� wassynthesized by adding drop wise cyclen �24 mmol�dissolved in ethanol �60 ml� to a solution of

�Ni2�H2O��O2CCMe3�4�HO2CCMe3�4� �11 mmol�34 in etha-nol �100 ml� at 80 °C. After 1 h at reflux the ethanol wasremoved in vacuo. The product was washed with hexane anddried under N2. �Ni�cyclen��O2CCMe3�H2O��O2CCMe3��2 mmol� was then heated in pivalic acid �10 g� to 140 °C.Basic zinc carbonate �0.3 mmol� and chromium trifluoridetetrahydrate �19 mmol� were added and after 15 min thetemperature was increased to 160 °C. The mixture wasstirred at 160 °C for 5 h before cooling to room temperature.Next day the dark green solid was diluted with acetone, fil-tered, and washed with a small amount of acetone. The prod-uct was recrystallized from the mixture of THF and acetoni-trile. Yield: 0.88 g, 5%. Elemental analysis calculated forC136H256Cr12F18N8Ni2O48Zn1: C, 41.66, H, 6.58, N, 2.86, Cr,15.91, Zn, 1.67, Ni, 2.99 Found: C, 40.50, H, 6.66, N, 2.77,Cr, 15.37, Zn, 1.62, Ni, 2.97. If crystallized from a mixtureof THF and MeCN the �NiCr6�2Zn is isostructural with�Ni�cyclen�2Cr12NiF18�O2CCMe3�24�, in short �NiCr6�2Ni,reported in Ref. 35.

As a common feature, each Cr+3 ion has quenched orbitalmomentum and carries spin s=3 /2,36 while each Cr-Cr pairis linked by one fluorine ion and two carboxylates bonds.The O-C-O angle range between 120° and 130° while theCr-F-Cr between 121° and 125° from one derivative to an-other. This relatively small variation suggests that the mag-netic coupling between two nearest neighboring Cr ionsshould be similar and/or comparable in all these compounds.In �Cr6�2 and �Cr7�2 �Fig. 1�a� and 1�b�, respectively�, we canrecognize two horseshoes facing one another with H bondsin between: the latter are known to be ineffective in produc-ing magnetic coupling, hence the two horseshoes are ex-pected to be magnetically uncoupled as confirmed by mag-netic data analysis �see below�. �NiCr6�2Zn �Fig. 1�d�� can beregarded as two independent NiCr6 moieties, if we neglectthe coupling between the nonmagnetic Zn2+ and the nearest-neighbor Cr ions. Each NiCr6 moiety presents six Cr+3 anti-ferromagnetically coupled with an external Ni2+ ion �s=1�bridged by one carboxylate and one fluorine. The remainingCr8Cd �Fig. 1�c�� can be regarded as a Cr-based ring brokenby a nonmagnetic Cd ion.27

Measurements were performed on microcrystalline pow-ders by means of a Quantum Design PPMS-7T system. acsusceptibility is measured applying a sinusoidal field of10 Oe at different frequencies 90, 1730, and 9300 Hz. Mag-netization data are taken by using extraction method andstatic magnetic fields. The specific heat is measured in thetemperature range 0.3–20 K with a 3He refrigerator and therelaxation method with two-tau data fitting. Thin pellets��2 mg� of pressed powders were measured.

III. RESULTS

On the basis of the structural features of �Cr6�2, �Cr7�2,and �NiCr6�2Zn previously discussed, it is convenient to nor-malize and compare values referred to single moieties,hence, in the following, data per formula unit have beenfurther divided by 2, so we may simply refer to Cr6, Cr7, andNiCr6. Figure 2 shows results of ac-susceptibilty measure-

FIG. 1. �Color online� Structures of �a� �Cr6�2, �b� �Cr7�2, �c�Cr8Cd, and �d� �NiCr6�2Zn. Colors: Cr, large green spheres; Ni,purple; Cd, pale blue; Zn, orange; F, yellow; O, red; C, black; andN, blue. H atoms omitted for clarity.

GHIRRI et al. PHYSICAL REVIEW B 76, 214405 �2007�

214405-2

ments plotted as �T vs T. The room temperature �T valuesapproach that expected for noncorrelated spins with gCr�2,i.e., �T�11.2, 13.1, 15.0, and 12.2 emu K mol−1 for Cr6,Cr7, Cr8Cd, and NiCr6, respectively.

At low temperature, the susceptibility curves evidence thedifferent behavior of the even and odd spin segments: whilefor the even ones, i.e., Cr6 and Cr8Cd, �T falls to zero indi-cating a singlet S=0 ground state, in the case of odd spinsegments, i.e., Cr7 and NiCr6, a S�0 ground state is ob-served. The fact that �T in Cr7 tends to saturate forT�4 K suggests that the ground state is energetically wellseparated from the higher ones and the susceptibility value�T�2 K�=2.1 emu K mol−1 points to S=3 /2 ground state ofeach moiety. The low-temperature susceptibility value�T�2 K�=1.2 emu K mol−1 of NiCr6 is consistent with aS=1 ground state as expected for one uncompensated s=1 ofNi2+.

Consistently with the ac-susceptibility behavior, the mag-netization M vs H curves of Fig. 3 show different behaviorfor even and odd spin segments. For Cr7 a Brillouin-likecurve is observed at T=2 K, indicating that the excited statesare energetically apart and not populated. The saturationvalue M �3 �B points to a S=3 /2 ground state, in agree-ment with low-temperature ac susceptibility. Conversely, forthe even-membered spin segments, the magnetization at T=2 K is continuously increasing suggesting the presence oflow-lying S�0 excited states close in energy to the groundstate, which are significantly populated at this temperatureand that progressively contribute by increasing the magneticfield strength. The low-temperature specific heat of molecu-lar spin clusters typically exhibits characteristic Schottkyanomalies related to the energy gaps between the lowest-lying energy levels, while at high temperature this magneticcontribution is overwhelmed by the lattice one.37 Just to get

familiar with these curves, we may notice that for Cr6 �Fig.4� and Cr8Cd �Fig. 5� the low-temperature Schottky anomalyin zero field is mainly characterized by the energy gap be-tween the lowest singlet and the first excited �S=1� state.Reminding that the maximum of the anomalies �1.3 and1.1 K for Cr6 and Cr8Cd, respectively� occurs at �0.4 of theenergy gap between the singlet and the barycenter of thetriplet, it is easy to see that excited states are significantlypopulated at T=2 K and this intuitively explains the ob-served behavior of the magnetization �Fig. 3�. Regarding theodd segments, the low-temperature Schottky anomaly mea-sured at H=0 in Cr7 �Fig. 6� is due to the zero field splittingof the ground state. Even before any deeper data analysis, itis worth noting the similarity between the specific heat of Cr7�Fig. 6� and that of Cr7Cd.30 This confirms that, in spite ofthe different structures, these two compounds well representsegments of seven spins s=3 /2 with quite similar energyspectrum.

To obtain a quantitative analysis, we use a model spinHamiltonian of the form

H = �i=1

n−1

Jisi · si+1 + �i=1

n

di�sz2�i� − si�si + 1�/3� + �B�

i=1

n

giH · si,

�1�

where n is the number of spin centers. The first term is thedominant isotropic Heisenberg exchange, where Ji is the ex-change Heisenberg coupling, the second term describes thelocal crystal field, and the last one is the Zeeman term relatedto the applied magnetic field H. Next-nearest-neighbor�NNN� interaction is expected to be weak in these systemsand therefore is neglected. Indeed, in similar antiferromag-

12121212

10101010

8888

6666

4444

2222

acacacac-S

usce

ptib

ility

-Sus

cept

ibili

ty-S

usce

ptib

ility

-Sus

cept

ibili

tyχχχχ����

���������

���������

���������

���������

300300300300250250250250200200200200150150150150100100100100505050500000

TemperatureTemperatureTemperatureTemperature ���� (K)(K)(K)(K)

5555

4444

3333

2222

1111

00003030303020202020101010100000

CrCrCrCr6666CrCrCrCr7777CrCrCrCr8888CdCdCdCdNiCrNiCrNiCrNiCr6666

FIG. 2. �Color online� ac susceptibility plotted as �T vs T. Mea-surements were taken with a 10 Oe ac field and frequencies 90,1730, and 9300 Hz on powders. No dependence on frequency isobserved. Inset: magnification of the low-temperature region. Con-tinuous lines are results of theoretical calculations.

2.52.52.52.5

2.02.02.02.0

1.51.51.51.5

1.01.01.01.0

0.50.50.50.5

0.00.00.00.0

dcdcdcdc-M

agne

tizat

ion

-Mag

netiz

atio

n-M

agne

tizat

ion

-Mag

netiz

atio

n����

(µ(µ(µ(µBBBB/f.

u.)

/f.u.

)/f.

u.)

/f.u.

)

77776666555544443333222211110000

Magnetic FieldMagnetic FieldMagnetic FieldMagnetic Field ���� (T)(T)(T)(T)

���� =2 K=2 K=2 K=2 KCrCrCrCr6666CrCrCrCr7777CrCrCrCr8888CdCdCdCd

FIG. 3. �Color online� Magnetic field dependence of magnetiza-tion M vs H measured with a dc extraction magnetometer on pow-ders. M vs H curves show a different behavior between the evenand odd spin segments at 2 K: The former �Cr6 and Cr8Cd� haveS=0 ground state and low-lying excited levels close in energy andtherefore populated at T=2 K, whereas the second one �Cr7� have awell defined S=3 /2 ground state at 2 K. Lines are results of theo-retical calculations.

ELEMENTARY EXCITATIONS IN ANTIFERROMAGNETIC… PHYSICAL REVIEW B 76, 214405 �2007�

214405-3

netic rings Fe7Zn and Fe7Mn we found by inelastic neutronscattering that NNN interactions are 0.016 times smaller thannearest-neighbor coupling.39 In Eq. �1� we have also ne-glected dipolar intracluster interaction for two reasons: onthe one hand, the addition of such a term requires to knowthe actual spatial orientation of the z axis with respect to the

unit cell axes. Being measurements performed on powders,this piece of information is unavailable. On the other hand,on these measurements virtually all effects of dipolar inter-actions are undistinguishable from those of the local crystalfields and therefore dipolar interactions can be omitted pro-vided the di parameters are understood as renormalized ones.

For T above a few degrees K, the temperature dependen-cies of the �T products are almost completely determined bythe dominant Heisenberg contribution in Eq. �1�. Therefore,�T has been calculated by using the expression

�T =NAg2�B

2

3kB

�i

Si�Si + 1��2Si + 1�exp�− E�Si�/kBT�

Z,

�2�

where the sum is over all the spin-multiplet eigenstates of theHeisenberg term �E�Si��, and Z is the partition function. Incalculating M�H�, anisotropic terms have been included andthe powder magnetization has been numerically obtainedbyaveraging over all possible orientations of the field. Thespecific heat is the sum of magnetic and lattice contributions,C=Cmag+Clatt, where Cmag is given by

Cmag

R�2 =

�i

�i2 exp�− ��i��

i

exp�− ��i� − ��i

�i exp�− ��i�2

��i

�i exp�− ��i�2.

�3�

�i�H� are the eigenvalues of Eq. �1�, R=8.314 J mol−1 K−1,and �=kBT. Below T=10 K, the lattice contribution Clatt is

4444

3333

2222

1111

0000

Spe

cific

Hea

tS

peci

ficH

eat

Spe

cific

Hea

tS

peci

ficH

eat�

�����

���

���

666655554444333322221111

TemperatureTemperatureTemperatureTemperature ���� (K)(K)(K)(K)

CrCrCrCr6666���� = 0 T= 0 T= 0 T= 0 T���� = 1 T= 1 T= 1 T= 1 T���� = 3 T= 3 T= 3 T= 3 T���� = 5 T= 5 T= 5 T= 5 T���� = 7 T= 7 T= 7 T= 7 T

FIG. 4. �Color online� The temperature dependence of the spe-cific heat C�H ,T�, normalized to the gas constant R, for Cr6. Thelow-temperature Schottky anomaly is related to the energy gap be-tween the ground states and the excited multiplets. Above T�4 Kthe lattice specific heat is the leading contribution. Lines are resultsof theoretical calculations.

5555

4444

3333

2222

1111

0000

Spe

cific

Hea

tS

peci

ficH

eat

Spe

cific

Hea

tS

peci

ficH

eat�

�����

���

���

666655554444333322221111

TemperatureTemperatureTemperatureTemperature ���� (K)(K)(K)(K)

CrCrCrCr8888CdCdCdCd���� = 0 T= 0 T= 0 T= 0 T���� = 1 T= 1 T= 1 T= 1 T���� = 3 T= 3 T= 3 T= 3 T���� = 5 T= 5 T= 5 T= 5 T���� = 7 T= 7 T= 7 T= 7 T

FIG. 5. �Color online� The temperature dependence of the spe-cific heat C�H ,T�, normalized to the gas constant R, for Cr8Cd. Thelow-temperature Schottky anomaly is related to the energy gap be-tween the ground states and the excited multiplets. Above T�4 Kthe lattice specific heat is the leading contribution. Lines are resultsof theoretical calculations.

5555

4444

3333

2222

1111

0000

Spe

cific

Hea

tS

peci

ficH

eat

Spe

cific

Hea

tS

peci

ficH

eat�

�����

���

���

666655554444333322221111

TemperatureTemperatureTemperatureTemperature ���� (K)(K)(K)(K)

CrCrCrCr7777���� = 0 T= 0 T= 0 T= 0 T���� = 1 T= 1 T= 1 T= 1 T���� = 3 T= 3 T= 3 T= 3 T���� = 5 T= 5 T= 5 T= 5 T���� = 7 T= 7 T= 7 T= 7 T

FIG. 6. �Color online� The temperature dependence of the spe-cific heat C�H ,T�, normalized to the gas constant R, for Cr7. Thelow-temperature Schottky anomaly is related to the zero field split-ting of the S=3 /2 ground state. Above T�4 K the lattice specificheat is the leading contribution. Lines are results of theoreticalcalculations.

GHIRRI et al. PHYSICAL REVIEW B 76, 214405 �2007�

214405-4

mainly due to acoustic branches of phonons but low energyoptical phonons may contribute as well. Debye temperaturesare very low in molecular materials and corrections to thesimple Clatt�T3 law are needed already above 4 K. Clatt canbe fitted by the phenomenological expression

Clatt

R=

234rT3

��D + T2�3 , �4�

as discussed in Ref. 38. The parameter r is the number ofatoms per molecule and �D and are determined by leastsquare fitting. As can be observed by Figs. 4–6, in thesemolecular systems the contribution of the phonons to thespecific heat typically dominates at high temperatures whilethe magnetic contribution can be clearly observed belowT�4 K.

The diagonalization of Eq. �1� allows us to theoreticallyreproduce the properties of the different compounds. Thechoice of the number of free parameters has some degree ofarbitrariness and it depends on the level of required preci-sion. For each compound, we consider only one couplingconstant JCr-Cr between neighboring Cr spin centers since theintroduction of further constants, although it may occasion-ally improve the quality of data fit, does not change the dis-cussion on the spin excitations that will be presented in thenext paragraph. Since measurements were performed onpowder samples, precision on the determination of aniso-tropy parameters is also limited to 30%. Within this level ofaccuracy, it is worth to stress that the simultaneous fit of the�, M vs H, and C�T ,H� curves was proved to be a verypowerful tool and, in the case of Cr6, Cr7, and Cr8Cd, allowsus to identify and fix one set of microscopic parameters foreach derivative, as reported in Table I. It turns out from thecomparison of results that the exchange and anisotropy con-stants are very similar in the molecular spin segments herestudied, confirming what expected from the similar chemicalenvironment and links between metal ions. Recently, inelas-tic neutron scattering experiments have been performed onsome of these derivatives and similar J values have beenobtained,40 consistently with our experiments and data analy-sis. In summary, these results validate the view of these com-pounds as different variants of spin segments and they allowdirect comparison of energy spectra, as presented in the nextparagraph.

IV. DISCUSSION

In the following, we discuss some general features of thelow-lying energy spectrum of spin segments. We first focuson the eigenstates of the dominating exchange part of theHamiltonian, which are shown in Fig. 7 for the broken Cr8Cdring, even and odd spin segments Cr6 and Cr7, and, for com-parison, for the closed Cr8 ring. Later we will clarify the roleand some effects of the anisotropy term. At first glance, theconcept of rotational �L� band, which was introduced forantiferromagnetic �AF� closed rings, appears to be applicableto the open segments as well. This is suggested by theroughly parabolic form of the energy vs S, i.e., the E�S�curve for the lowest-S multiplets, and it is more quantita-tively demonstrated by the composition of the correspondingeigenstates. L-band states in closed rings are characterized bythe property that the moduli of the total spin of the twosublattices, SA�SA+1� and SB�SB+1�, are to a great extendlocked in their maximum value, i.e., the eigenstates have thestructure SA=NAs ,SB=NBs ,Stot= SA−SB¯ �SA+SB��. Forinstance, for the the lowest-lying S=1 state in Cr8Cd theweight of SA=6,SB=6,Stot=1� is about 69%. In even-numbered rings the remaining weight in L-band eigenfunc-tions is large on states of type SA=SB ,Stot=0¯2SA�, i.e., thesublattice spin lengths are mainly locked into their maximumvalue, and sublattice-spin fluctuations occur jointly. Table IIshows that the lowest eigenstates in the present systems ac-tually have a dominating L-band-type component and thatthis holds for the heterometallic NiCr6 segment as well �seealso the spectrum in Figs. 7 and 8�. The weight of this com-ponent has to reach, of course, the limiting value of one forthe ferromagnetic multiplet but it does not always do so in amonotonic way. In addition, for a given number of spins thering opening makes the Landé picture less precise since AFcorrelations between sublattices are less enforced. Indeed,the L-type component in the ground low-lying states ofCr8Cd is smaller than in Cr8. According to the Landé rule= �2J /N�S�S+1�, the energy separation between levels, isexpected to scale as �2J /N�, thus inversely proportional tothe number N of spin sites. We just notice from Fig. 7 thatthe spectrum of lowest-lying states of Cr6 actually repro-duces that of Cr8Cd, apart from a multiplicative factor8 /6�1.3, in agreement with the ratio between 1 /N=1 /6 and1 /8 for the two spin systems, respectively. Since the zerofield Schottky anomaly in the specific heat is directly relatedto the energy gap between the ground state and the first ex-cited states, it provides a simple but direct test of applicabil-ity of the model on this family of molecular compounds and,in turn, of the flexibility of the molecular route to providemodel systems. In odd-membered spin segments we may no-tice that the lowest state has not the lowest spin value,namely, in Cr7 and NiCr6 the lowest states are S=3 /2 andS=1, respectively �see Fig. 7�. Model calculations show thatthe energy gap between the ground state and first excitedstates of Cr7 �having S=1 /2� is actually of few K while ineven-membered spin segments this is smaller. This explainswhy magnetization and specific heat below 4 K are wellcharacterized by the ground state solely in odd-memberedcases while excited states are well populated and contribute

TABLE I. List of the parameters obtained by the fit of the ex-perimental data. Cr7Cd data are taken from Ref. 30. Cr8 data aretaken from Ref. 25. For NiCr6 gNi=gCr has been assumed.

Ji �K� di �K� gi �D �K� �K−1�

Cr6 JCr-Cr=13.8 −0.3 gCr=1.98 146 0.25

Cr7 JCr-Cr=13.6 −0.3 gCr=1.98 174 0.32

Cr7Cd JCr-Cr=16.9 −0.3 gCr=1.98 157 0.42

Cr8 JCr-Cr=16.9 −0.3 gCr=1.98 154 0.37

Cr8Cd JCr-Cr=14.8 −0.3 gCr=1.98 168 0.25

NiCr6 JCr-Cr=13.8 gCr=2.03

JCr-Ni=13.8 gNi=2.03

ELEMENTARY EXCITATIONS IN ANTIFERROMAGNETIC… PHYSICAL REVIEW B 76, 214405 �2007�

214405-5

to determine the properties of the even-membered spin seg-ments at these temperatures, in spite of the fact that the Jconstants are similar in all these molecular compounds. Asfar as the E band is concerned, in closed rings these states arecharacterized by a dominating contribution in which the totalspin of the two sublattices differ by one. In terms of thetranslational �cyclic� symmetry of rings these states belongto representations with wave vector k�0, � and are there-fore degenerate in k, −k pairs. The main effect of the ringopening is the splitting of this degeneracy, which can besizable in short segments: this is evident, for instance, bydirectly comparing the spectrum of the broken Cr8Cd ringwith that of closed Cr8 ring �Fig. 7�.

In the previous section we determined the single ion an-isotropy parameters d for Cr6, Cr7, and Cr8Cd and it turns outthat these are much smaller than the Heisenberg couplingconstants J. Since this is one relevant feature of the modelthat well describes our experiments, it naturally leads us tothe conclusion that these—or similar—molecules can actu-ally be used as a model system for studying finite Heisenbergchains. In order to see to what extent this picture holds and

clarify the role of anisotropy term, we note that for a genericvalue of the applied field the effect of the anisotropic terms�as determined in the previous section� on the eigenstates isnot qualitatively important and mostly consists in a first-order splitting of the various S multiplets into symmetry-adapted states. If we focus on the ground state, a compactand physically motivated way to quantify how much aniso-tropy modifies its composition is to calculate the weight�WN� in this state of the two Néel components, where allspins are aligned along the z axis but in opposite directionson the two sublattices. Indeed, being anisotropy of easy-axistype, in segments with integer total spin for a large enoughd /J ratio WN→0.5 and the ground state becomes a superpo-sition of the two Néel states. In this limit, a tunneling picturefor the Néel vector holds. In segments with half-integer totalspin, such as Cr7, the ground state is always at least doublydegenerate by Kramers theorem if H=0. For large enough

TABLE II. Component of the lowest-lying eigenstates forthree values of the total spin S onto the Landé-type stateSA=NAs ,SB=NBs ,Stot=S� �SA=11 /2,SB=9 /2,Stot=S� for the het-erometallic NiCr6�. S0 is the ground-state total spin �S0=0 in Cr8,Cr8Cd and Cr6, S0=3 /2 in Cr7, S0=1 in NiCr6.

Cr8 Cr8Cd Cr6 Cr7 NiCr6

S=S0 0.83 0.79 0.9 0.81 0.81

S=S0+1 0.83 0.69 0.8 0.81 0.81

S=S0+2 0.84 0.59 0.71 0.81 0.81

FIG. 7. �Color online� Lowestenergy levels of the exchange partof the Hamiltonian equation �1�for Cr6, Cr7, Cr8, and Cr8Cd. Notethat the lowest L band has a qua-siparabolic S dependence.

FIG. 8. �Color online� Lowest energy levels of the exchangepart of the Hamiltonian equation �1� for NiCr6.

GHIRRI et al. PHYSICAL REVIEW B 76, 214405 �2007�

214405-6

d /J ratio each state of the ground doublet approaches eitherof the two Néel states, WN→1, and no tunneling occurs.Figure 9 shows WN calculated for Cr6, Cr7, Cr8, and Cr8Cd. Itis apparent that for the d values determined by experiments,the ground state is very close to that of the isotropic Heisen-berg model in all molecules. The Néel limit is only reachedfor d�J. For anisotropy in order to play a qualitatively im-portant role on the state composition it is necessary that in-termultiplet splittings be comparable or smaller than the an-isotropy strength. In particular, for the ground state thisoccurs whenever the applied field H produces a crossing be-tween levels belonging to different multiplets of the ex-change part of the Hamiltonian �Eq. �1��. It has been shownthat in Cr7M �M =Ni,Cd� the anisotropy turns such crossinginto anticrossings �ACs�, which physically correspond toquantum oscillations of the molecule’s total spin.31,41 On theother hand, crossings are observed in the Cr8 ring. Amongthe present spin segments, Cr8Cd and Cr6 behave like the Cr8ring, whereas Cr7 and NiCr6 are characterized by ACs �seeFig. 10�. This difference in behavior reflects different sym-metry properties of the Landé states in the two cases: inCr8Cd and Cr6, like in Cr8, the L-band states which crossbelong to different irreducible representations of the permu-tation symmetry of the chain and therefore they are not al-lowed to mix. On the contrary, they do belong to the samerepresentation in Cr7 and NiCr6 where they mix and produceACs. This difference could be experimentally demonstratedby single-crystal low-T torque measurements. In fact, the

calculated H dependence of the torque is characterized by theusual steplike behavior at the crossings for Cr6 and Cr8Cd,whereas it peaks for Cr7 and NiCr6.

V. CONCLUSIONS

In conclusion, we have investigated ac susceptibility, low-temperature magnetization, and specific heat of differentmolecules whose magnetic core represent variants of spinsegments. We showed that the spin Hamiltonian equation �1�with dominant Heisenberg antiferromagnetic term fits wellexperimental data. This indicates that from the point of viewof magnetic properties these molecular compounds are actu-ally very close to a collection of identical and noninteractingHeisenberg segments as anisotropy has been shown to play aminor role. We believe that the synthesis of similar andlonger molecular AF chains will offer a unique opportunityto investigate topological size effects characteristic of theone-dimensional Heisenberg model.

ACKNOWLEDGMENTS

This work was carried out within the framework of theEuropean Network of Excellence MAGMANet supported byEC under Contract No. 515767 and supported by the PRINNo. 2006029518 of the Italian Ministry of Research and bythe EPSRC �UK�.

FIG. 9. �Color online� Component WN� �G N� 2 of the groundstate G� onto one of the two Néel states N� as a function of the d /Jratio. For Cr7 the ground state is doubly degenerate and the curverefers to one of the two components. Horizontal dashed curves rep-resent the limit to which the curves tend for large d /J, i.e., 1 for Cr7

and 0.5 for all remaining segments. It is apparent that for the ob-served d /J ratio the ground state composition is almost the same asfor d /J=0.

FIG. 10. Field dependence of the low-lying levels of Cr6 andCr7 calculated from Eq. �1� for a field making an angle of 45° withthe anisotropy axis. While for Cr6 a level crossing is expected, alevel repulsion �anticrossing� is predicted for Cr7.

ELEMENTARY EXCITATIONS IN ANTIFERROMAGNETIC… PHYSICAL REVIEW B 76, 214405 �2007�

214405-7

*Author to whom correspondence should be addressed;[email protected]

1 F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 �1983�.2 K. Katsumata, H. Hori, T. Takeuchi, M. Date, A. Yamagishi, and

J. P. Renard, Phys. Rev. Lett. 63, 86 �1989�.3 M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev. Lett. 70,

3651 �1993�.4 C. F. Hirjibehdin, C. P. Luts, and A. J. Heinrich, Science 312,

1012 �2006�.5 H. Yamazaki and K. Katsumata, Phys. Rev. B 54, R6831 �1996�.6 S. H. Glarum, S. Geschwind, K. M. Lee, M. L. Kaplan, and J.

Michel, Phys. Rev. Lett. 67, 1614 �1991�.7 T. K. Ng, Phys. Rev. B 50, 555 �1994�.8 S. Qin, T. K. Ng, and Z. B. Su, Phys. Rev. B 52, 12844 �1995�.9 S. Miyashita and S. Yamamoto, Phys. Rev. B 48, 913 �1993�.

10 S. Rakhmanova and D. L. Mills, Phys. Rev. B 54, 9225 �1996�.11 M. Sato and A. J. Sievers, Nature �London� 432, 486 �2004�.12 A. Franchini, V. Bortolani, and R. F. Wallis, Phys. Rev. B 73,

054412 �2006�.13 A. Caneschi, A. Cornia, A. C. Fabretti, S. Foner, D. Gatteschi, R.

Grandi, and L. Schenetti, Chem.-Eur. J. 2, 1379 �1996�; G. L.Abbati, A. Caneschi, A. Cornia, A. C. Fabretti, D. Gatteschi, W.Malavasi, and L. Schenetti, Inorg. Chem. 36, 6443 �1997�.

14 K. L. Taft, C. D. Delfs, G. C. Papefthymiou, S. Foner, D. Gatte-schi, and S. J. Lippard, J. Am. Chem. Soc. 116, 823 �1994�.

15 A. Cornia, A. G. M. Jansen, and M. Affronte, Phys. Rev. B 60,12177 �1999�; A. Cornia, A. G. M. Jansen, M. Affronte, G. L.Abbati, and D. Gatteschi, Angew. Chem., Int. Ed. 38, 2264�1999�.

16 M. Affronte, J. C. Lasjaunias, A. Cornia, and A. Caneschi, Phys.Rev. B 60, 1161 �1999�.

17 A. Caneschi, A. Cornia, A. C. Fabretti, and D. Gatteschi, Angew.Chem., Int. Ed. 39, 1295 �1999�; G. L. Abbati, A. Caneschi, A.Cornia, A. C. Fabretti, and D. Gatteschi, Inorg. Chim. Acta 297,291 �2000�.

18 S. P. Watton, P. Fuhrmann, L. E. Pence, A. Caneschi, A. Cornia,G. L. Abbati, and S. J. Lippard, Angew. Chem., Int. Ed. Engl.36, 2774 �1997�.

19 J. van Slageren, R. Sessoli, D. Gatteschi, A. A. Smith, M. Helli-well, R. E. P. Winpenny, A. Cornia, A. L. Barra, A. G. M.Jansen, E. Rentschler, and G. A. Timco, Chem.-Eur. J. 8, 277�2002�.

20 D. M. Low, G. Rajaraman, M. Helliwell, G. Timco, J. vanSlageren, R. Sessoli, S. T. Ochsenbein, R. Bircher, C. Dobe, O.Waldmann, H. U. Güdel, M. A. Adams, E. Ruiz, S. Alvarez, andE. J. L. McInnes, Chem.-Eur. J. 12, 1385 �2005�.

21 R. H. Laye, M. Murrie, S. Ochsenbein, A. R. Bell, S. J. Teat, J.Raftery, H. U. Güdel, and E. J. L. McInnes, Chem.-Eur. J. 9,6215 �2003�.

22 J. Schnack and M. Luban, Phys. Rev. B 63, 014418 �2000�.23 O. Waldmann, Phys. Rev. B 65, 024424 �2001�.24 O. Waldmann, T. Guidi, S. Carretta, C. Mondelli, and A. L.

Dearden, Phys. Rev. Lett. 91, 237202 �2003�.25 S. Carretta, J. van Slageren, T. Guidi, E. Liviotti, C. Mondelli, D.

Rovai, A. Cornia, A. L. Dearden, F. Carsughi, M. Affronte, C.

D. Frost, R. E. P. Winpenny, D. Gatteschi, G. Amoretti, and R.Caciuffo, Phys. Rev. B 67, 094405 �2003�; M. Affronte, T.Guidi, R. Caciuffo, S. Carretta, G. Amoretti, J. Hinderer, I.Sheikin, A. G. M. Jansen, A. A. Smith, R. E. P. Winpenny, J. vanSlageren, and D. Gatteschi, ibid. 68, 104403 �2003�.

26 F. K. Larsen, J. Overgaard, S. Parsons, E. Rentschler, A. A.Smith, G. A. Timco, and R. E. P. Winpenny, Angew. Chem., Int.Ed. 42, 5978 �2003�.

27 G. A. Timco, A. S. Batsanov, F. K. Larsen, C. A. Muryn, J.Overgaard, S. J. Teat, and R. E. P. Winpenny, Chem. Commun.�Cambridge� 2005, 3649.

28 The only exception is the �NiCr6�2Zn seahorse, for which a fitimprovement can be achieved by introducing additional ex-change terms. Since, however, this refinement is not relevant tosubstantiate our discussion, we rather limit our model to thesingle J that better fits the temperature dependence of the acsusceptibility.

29 M. Affronte, S. Carretta, G. A. Timco, and R. E. P. Winpenny,Chem. Commun. �Cambridge� 2007, 1789.

30 M. Affronte, A. Ghirri, S. Carretta, G. Amoretti, S. Piligkos, G.Timco, and R. E. P. Winpenny, Appl. Phys. Lett. 84, 3468�2004�.

31 S. Carretta, P. Santini, G. Amoretti, M. Affronte, A. Ghirri, I.Sheikin, S. Piligkos, G. Timco, and R. E. P. Winpenny, Phys.Rev. B 72, 060403�R� �2005�.

32 O. Cador, D. Gatteschi, R. Sessoli, F. K. Larsen, J. Overgaard, A.L. Barra, S. J. Teat, G. Timco, and R. E. P. Winpenny, Angew.Chem., Int. Ed. 43, 5196 �2004�.

33 S. T. Ochsenbein, R. S. G. Davies, G. A. Timco, C. A. Muryn, O.Waldmann, R. Bircher, A. Sieber, G. Carver, H. Mutka, F.Fernandez-Alonso, A. Podlesnyak, H. U. Gudel, and R. E. P.Winpenny �unpublished�.

34 G. Chaboussant, R. Basler, H.-U. Gudel, S. Ochsenbein, A. Par-kin, S. Parsons, G. Rajaraman, A. Sieber, A. A. Smith, G. A.Timco, and R. E. P. Winpenny, J. Chem. Soc. Dalton Trans.2004, 2757.

35 S. L. Heath, R. H. Laye, C. A. Muryn, N. Lima, R. Sessoli, R.Shaw, S. J. Teat, G. A. Timco, and R. E. P. Winpenny, Angew.Chem., Int. Ed. 43, 6132 �2004�.

36 V. Bellini, A. OlivierI, and F. Manghi, Phys. Rev. B 73, 184431�2006�.

37 M. Evangelisti, F. Luis, L. J. de Jong, and M. Affronte, J. Mater.Chem. 16, 2534 �2006�.

38 M. Affronte, J. C. Lasjaunias, and A. Cornia, Eur. Phys. J. B 15,633 �2000�.

39 T. Guidi, J. R. D. Copley, Y. Qiu, S. Carretta, P. Santini, G.Amoretti, G. Timco, R. E. P. Winpenny, C. L. Dennis, and R.Caciuffo, Phys. Rev. B 75, 014408 �2007�.

40 S. T. Ochsenbein, O. Waldmann, A. Sieber, G. Carver, R. Bircher,H. U. Gudel, R. S. G. Davis, G. A. Timco, R. E. P. Winpenny, H.Mutka, and F. Fernandez-Alonso, Europhys. Lett. 79, 17003�2007�.

41 R. Caciuffo, T. Guidi, G. Amoretti, S. Carretta, E. Liviotti, P.Santini, C. Mondelli, G. Timco, C. A. Muryn, and R. E. P. Win-penny, Phys. Rev. B 71, 174407 �2005�.

GHIRRI et al. PHYSICAL REVIEW B 76, 214405 �2007�

214405-8