Antiferromagnetic Dimers of Ni(II) in the S=1 Spin-Ladder Na_2Ni_2(C_2O_4)_3(H_2O)_2

1.5 MB·English

by  C. Mennerich

Antiferromagnetic Dimers of Ni(II) in the S = 1 Spin-Ladder Na Ni (C O ) (H O)
2 2 2 4 3 2 2
C. Mennerich1, H.-H. Klauss1, M. Broekelmann1,2, F.J. Litterst1, C. Golze1,2, R. Klingeler2,4, V. Kataev2, B.
Bu¨chner2, S.-N. Grossjohann3, W. Brenig3, M. Goiran4, H. Rakoto4, J.-M. Broto4, O. Kataeva5, and D.J. Price6
1Institut f¨ur Physik der Kondensierten Materie, TU Braunschweig,
Mendelssohnstr.3, D-38106 Braunschweig, Germany
2Leibniz-Institute for Solid State and Materials Research IFW Dresden, P.O. Box 270116, D-01171 Dresden, Germany
6
0 3 Institut f¨ur Theoretische Physik, TU Braunschweig,
0 Mendelssohnstr.3, D-38106 Braunschweig, Germany
2 4 Laboratoire National des Champs Magn´etiques Puls´es, 31432 Toulouse Cedex 04, France
5 Arbuzov Institute of Organic and Physical Chemistry, RAS, 420088 Kazan, Russia and
n 6 WestCHEM,Department of Chemistry, University of Glasgow, Glasgow, G12 8QQ, UK
a
(Dated: 13th January 2006)
J

4 We report the synthesis, crystal structure and magnetic properties of the S=1 2-leg spin-ladder
1 compound Na2Ni2(C2O4)3(H2O)2. The magnetic properties were examined by magnetic suscep-
tibility and pulsed high field magnetization measurements. The magnetic excitations have been

] measured in high field high frequency ESR. Although the Ni(II) ions form structurally a 2-leg lad-
i
c der, an isolated dimer model consistently describes the observations very well. The analysis of the
s temperaturedependentmagnetizationdataleadstoamagneticexchangeconstantofJ =43Kalong
-
l therungs of theladder and an average value of the g-factor of 2.25. From theESR measurements,
r we determined the single ion anisotropy to D = 11.5 K. The validity of the isolated dimer model
t
m is supported by Quantum Monte Carlo calculations, performed for several ratios of interdimer and
intradimer magnetic exchange and taking into account the experimentally determined single ion
.
t anisotropy. Theresultscanbeunderstoodintermsofthedifferentcoordinationandsuperexchange
a
angles of theoxalate ligands along the rungsand legs of the2-leg spin ladder.
m
- PACSnumbers: 75.50.Xx,75.10.Pq,76.30.-v,75.30.Gw
d
n
o
I. INTRODUCTION transition metal ions) show a strong axial crystal field
c
[ anisotropyD whichmodifiesthegroundstateofthespin
The physical properties of one-dimensional spin sys- system11. Totheknowledgeoftheauthorsnorealizations
1
of an S=1 spin ladder have been reported so far.
tems have attracted a lot of attention. For isotropic
v
5 magnetic interactionsdescribed in the Heisenbergmodel Inthispaperwereportthesynthesisandstructureand
0 andlowspinquantumnumbersthe groundstate proper- in particular a full experimental and theoretical charac-
3 ties arestronglyinfluenced by quantumfluctuations and terization of the magnetic properties of a Ni(II) based
1 often pure quantum groundstates are found in a macro- S=1 2-leg spin-ladder material, Na2Ni2(C2O4)3(H2O)2.
0 scopic system. For the ideal S=1/2 antiferromagnetic Weperformedmagneticsusceptibilityandhighfieldmag-
6
Heisenberg chain (S=1/2 AFHC) with uniform nearest netizationstudies (up to55T)to determine the thermo-
0
neighbor exchange coupling the ground state is a spin dynamic properties. In order to study the spectrum of
/
at singlet with gapless excitations1,2. Introducing an ad- spin states in this system, we have used tunable high-
m ditional interchain magnetic exchange coupling between frequency high-field ESR-spectroscopy at frequencies up
adjacent spin chains the so-called n-leg spin-ladders are to 740GHz in magnetic fields up to 30T.
-
d formed. These have been discussed very intensively for Combining the experimental results from low field
n the S=1/2 case since they exhibit completely different magnetic susceptibility, high field magnetization and
o groundstateproperties3,4. Inparticularforeven-leglad- ESR we are able to determine all relevant parameters
c
dersanon-magneticspin-liquidgroundstatewithafinite of the spin system. In particular we find that the struc-
:
v energy gap is found5. turally nearly isotropic S=1 spin ladder is magnetically
i
X For S=1 already for a single antiferromagnetic chain very well described by isolatedspin dimers with an anti-
the ground state is a non-magnetic spin-liquid with a ferromagnetic exchange of J = 43 K on the rungs and a
r
a finite energy gap as proposed by Haldane6 and several large axial anisotropy of D = 11.5 K. In order to quan-
experimental realizations are known7. Only very little is tify the degree of intrachain versus interchain magnetic
known about the ground state and excitations of a S=1 exchangeinthissystemweperformedrealisticQMCcal-
2-legspinladder. Frombosonizationstudies8,9andquan- culationsincluding the experimentallydetermined single
tum Monte-Carlo simulations10 a continuous crossover ion anisotropy which support the relevance of the iso-
from the delocalized Haldane spin gap state to the case lated dimer model for this system. The calculated field
of localized antiferromagnetic dimers, which also shows dependence of the spin states suggests that strong mag-
a spin excitation gap, is predicted. netic fields allow to tune the ground state from a non-
Experimentallythesituationiscomplicatedbythefact magnetic S = 0 to a magnetic S = 1 state and at even
thatmostS=1systems(usuallybasedonhighspinNi(II) higher fields to the fully polarized S = 2 state. This is
2
confirmed by our magnetization data which provide di- = 5.294(1) ˚A) forming the rungs of the ladder. The sec-
rect evidence for a magnetic field induced switching of ondoxalateisunsymmetricallycoordinated,chelatingto
the S=0 ground state to an S=1 state at B ≈ 30T and one Ni(II) ion and forming a monodentate coordination
to S=2 at B ≈ 60T . to a second Ni ion, with 1,3-syn anti geometry through
thebridgingcarboxylate,thismodeformstheladderlegs
(d =5.810(1)˚A).Thesestructurallinkagesare
Ni···Ni(leg)
II. RESULTS AND DISCUSSION likely to provide the only significant magnetic superex-
change pathways.
A. Synthesis and crystal structure
The compound Na Ni (C O ) (H O) (SNOX here-
2 2 2 4 3 2 2
after) is synthesized in a hydrothermalreactionfrom so-
lutions containing a very high sodium halide concentra-
tion. Typically Na C O , Ni(C O )(H O) , NaBr and
2 2 4 2 4 2 2
distilled water are mixed in a 1:1.9:140:500 molar ratio,
heatedinasealedautoclaveto225◦Cfor5hoursandal-
lowed to cool slowly to room temperature. The product
occurs as green crystals of Na Ni (C O ) (H O) em-
2 2 2 4 3 2 2
bedded in a matrix of solid NaBr. Phase homogeneity
and purity of SNOX were established by a combination
of optical microscopy,powder X-Ray diffraction and ele-
mental analysis. By slowing down the cooling rate, and
reducing the concentration of Ni2+ and C O 2− we can
2 4
grow single crystals up to 2 mg in lower yielding reac-
tions. Figure 1: Crystal structure of Na2Ni2(C2O4)3(H2O)2. For
clarity,NaandHatomsarenotshown. Thenumbersindicate
Single crystal X-ray structure determination reveals
the distance (in ˚A) between opposing oxygen atoms on the
SNOXtocrystallizeinthemonoclinicspacegroupP2 /c
(#14) with a=5.8144(2), b=15.6474(5), c=6.8357(3)1˚A, distorted NiO6 octahedra which are tilted by 17 degree with
respect to thea axis.
β=100.56(60) ◦ and V=611.37(4) ˚A3 at 120 K. Refine-
ment of 118 parameters gave a goodness-of-fit S=1.074,
R1=0.244 and wR2=0.506 on all data, with the largest
peak (and hole) in the difference Fourier map 0.383 (-
0.390) e˚A3. B. Magnetic susceptibility
SNOX is an end member of the series of isostructural
compounds with the composition Na T (C O ) (H O) We examined several single crystals in magnetic sus-
2 2 2 4 3 2 2
with T = Mn(II), Fe(II)13, Co(II)14 and Ni. Although ceptibility measurements along the a, b and c axes. The
this structure type has been described previously13, it measurements were performed using a Quantum Design
is useful to recap certain key features here. The Ni(II) Magnetic Properties Measuring System (MPMS) in ex-
ion lies on a general position, and experiences a pseudo- ternal fields of Bex = 2 T and 5 T in the temperature
octahedral coordination environment. It is coordinated range 2-300 K. In addition we performed measurements
in a cis geometry by two chelating and crystallographi- at different angles between the a axis and the external
cally independent oxalatedianions. The remaining coor- field by rotating the crystal about an axis perpendicular
dination sites are filled by a monodentate oxalate oxy- to the a axis.
gen atom and a water molecule. Ni-O bond lengths Fig. 2 shows the temperature dependence of the mag-
lie within the range 2.020(1) to 2.114(1) ˚A. We note netizationalongthea,bandcaxesinanexternalfieldof
that opposite pairs of O have similar Ni-O lengths, and B = 2 T. One can clearly see a pronounced maximum
ex
that the O···O separations between opposite pairs, for for all directions with a strong downturn below ≈ 50 K
which the bond vectors are mostly directed along the suggestinganon-magneticspin-singletgroundstate. No-
crystallographic axes, are (d =4.163(1), d =4.044(1), ticethatthemaximumalongtheaaxes(theladderdirec-
a b
d =4.082(1) ˚A along the a, b and c-axes, respectively, tion) is locatedat a lowertemperature than the maxima
c
Fig. 1). The overall structure can be regarded as an along the b and c axes (39 K, 47 K and 45 K for the a,
anionic [Ni (C O ) ]2n− network with a 1-D ladder like b and c axes respectively). The difference between the b
2 2 4 3 n
topology(Fig. 1),wherethemetalionsformthe vertices and c axes magnetization is interpreted mainly due to a
and the bridging oxalate ions form the bonds. The two g-factoranisotropy. This behavior in generalis expected
oxalate ions bridge nickel ions in quite different modes. aswell foran isotropictwo-legspin ladder asfor the two
One oxalate forms a symmetric bis-chelating bridging limiting cases, an isolatedS = 1 Haldane chain or a sys-
mode(withacrystallographicinversionlocatedattheC- tem of antiferromagnetically coupled dimers5,6,8,9,10. At
C centroid) and links pairs of nickel ions (d very low temperatures a small Curie-tail due to a very
Ni···Ni(rung)
3
low concentration of paramagnetic centers (less than 1 To analyse the magnetization data, we developed a
%) masks the approachto zero. fitroutinewhichnumericaldiagonalizestheHamiltonian
(2), calculates the free energy A and magnetization M
for a particular magnetic field orientation γ and varies
theparametersintheHamiltoniantominimizethemean
square deviation between the data and this model. We
performed a combined fit of M = χB for B = 2 T
ex ex
alongthe a,bandc axesusinganisotropicmagnetic ex-
changecouplingconstantJ, afixedabsolutevalueofthe
axialanisotropyof|D|=11.5K(determinedbytheS=1
zerofieldsplittingmeasuredbyESRasdescribedbelow).
The total susceptibility χ consists of the main contribu-
tion (3) and comprises also a temperature independent
term χ = χ + χ which includes a diamagnetic
0 dia vv
contributionχ anda Van-Vleckparamagneticsuscep-
dia
tibilityχ ofNi(II),aswellasaCuriecontributionC/T
vv
with a Curie constant C owing to paramagnetic impuri-
ties:
χ=χ +C/T +χ . (4)
Figure 2: Temperature dependence of the magnetization of 0 ladder
a SNOX single crystal (2.04 mg) at an external field of
Inaddition,weusedanorientationdependentg-factorg
Bex = 2 T along the a (circles), b (diamonds) and c axes γ
and,since the orientationofthe principalaxis(z axis)of
(triangles).
the magnetic anisotropy tensor is not known, the angle
γ as parameters for the different orientations. The fit
Taking into account the two different exchange path-
results for the B = 2 T measurements are shown in
way topologies, we assume a stronger magnetic interac- ex
Fig. 2 as solid lines.
tion J along the rungs than the magnetic interaction K
We obtain an intradimer coupling constant of
along the legs. This leads to an isolated dimer approx-
J = 43 K, an interdimer coupling constant of K = 0 K
imation where each dimer consists of two S=1 spins on
and g-values of g = 2.206, g = 2.291 and g = 2.256 .
the rungs of the ladder. Since all Ni ions on a ladder a b c
The anglesto the anisotropyaxisareγ = 18◦, γ = 90◦
are crystallographically equivalent they share the same a b
andγ = 117◦, a Curie constantC = 0.0056µ K/dimer
strength and orientation of the single ion anisotropy D. c B
and a temperature independent contribution χ =
Including the Zeeman energy in the external magnetic 0
0.0087µ /dimer has been used. The variations of the g-
field, the spin Hamiltonian of the system in the dimer B
values are consistent with the variation of the distances
approximation is given by
to the adjacent oxygens,leading to a larger g-value with
H =JS1S2+gµBB(S1+S2)+ X SiDSi. (1) smaller distance. Using gγ2 = gk2cos2γ + g⊥2cos2γ 12
with g = g and g = 2.206 we find g = 2.197. The
i=1,2 ⊥ b a k
knowledgeof γ , γ and γ allows a determinationof the
a b c
Here, we assume that the magnetic field is applied along orientationoftheanisotropyaxisinthecrystal. γ =18◦
a
the z-axis, the main axis of the crystal field anisotropy is explained by an anisotropy axis parallel to the line
tensor D. For pure axial anisotropy this tensor is di- connectingbothoppositeoxygenneighborionsalongthe
agonal, defined as Dxx= - 1/3 D, Dyy= - 1/3 D and aaxis(Oa axishereafter). Thislineistiltedwithrespect
Dzz=+2/3D. Tocalculatetheenergylevelsforapplied totheaaxisforallNisites(seeFig.1,rightpanel)about
fields along different directions, we rotate the axis of the anangleof17◦determinedbyX-raycrystallography. Re-
axialanisotropytensorbyanangleγ aboutaperpendic- garding γ we have to consider that with respect to the
b
ular axis. Then the spin Hamiltonian can be expressed b-axistheO axisistiltedalternatingfromladdertolad-
a
as der, which leads to anaveragetilting of 90◦, also in very
goodagreementwith the fitted value. Finally, γ = 117◦
H =JS1S2+gγµBB(S1z+S2z)+X SiUTγDUγSi (2) can be explained by taking into account the mocnoclinic
i=1,2 structurewith∠(a,c)=β =100.6◦. Thetiltingof117◦is
with the rotation matrix U and an angle dependent g- approximatelythesumoftheanglebetweenaandcaxes
γ
and an additional tilting of the anisotropy axis with re-
factor g .
γ
spect to the a axis towards the c axis of 16◦. This is
The isolated dimer approximation may be improved
10◦ morethanwhatis actuallyfoundfor the O axis. In
byintroducinganeffectiveexchangeinteractionK along a
view of the expected accuracy for the determination of
the legs treated in mean field approximation in the cal-
the anisotropyaxis this is reasonable. In summary, from
culation of the magnetic susceptibility χ using
an analysis of the angle dependence of the temperature
χ =χ /(1+Kχ ). (3) dependent magnetization, we can determine the orienta-
ladder dimer dimer
4
C. High Field ESR
ESRmeasurementsinstaticmagneticfieldsupto15T
have been performed with a Millimeterwave Vector Net-
work Analyzer (MVNA) from AB Millimetre, Paris.16
Experiments in pulsed magnetic fields up to 30T have
been carried out at the National Laboratory of Pulsed
Magnetic Fields, LNCMP, Toulouse, France, using a
Fabry-Perot cavity optically pumped by a CO laser for
2
generation of microwaves. Details of both experimen-
tal set-ups can be found in Ref. 17. For the ESR mea-
surements a powder sample of SNOX was prepared as a
pressed pellet with a small amount of a diphenyl-picryl-
Figure 3: Relative energies of the spin states, calculated for hydrazyl (DPPH) marker attached for calibration of the
the magnetic field applied perpendicular to the z axis of the magnetic field strength.
uniaxialanisotropytensor: thetripletS = 1andthequintet
At a base temperature of 4.2K no ESR signals are
S = 2 are well separated from the S = 0 ground state by
foundwhichcanbeassociatedwiththeresonanceofbulk
an activation energy of roughly |J| = 43K and 3|J|, respec-
Ni(II) spins in the sample. However, it is possible to
tively. Both multiplets exhibit a zero field splitting due to
assign four modes to the bulk ESR spectrum of SNOX
crystalfieldanisotropy. Notethegroundstatelevelcrossings
around 30 and 53 T. (for details see text) which develop in the spectrum with increasing temper-
ature (Fig. 4). These modes are numbered from 1 to 4
in order of their appearance in ascending magnetic field
andmarkedbyarrowsintheinsetofFig.4. Theposition
tion of the principal anisotropyaxis to be parallelto the
ofthe modesinamagneticfieldBres depends onthe ex-
connecting line of the opposite nearest neighbor oxygen i
citation frequency ν as would be expected for magnetic
ions along the a axis, tilted by 18◦ with respect to the
resonance. Respective resonance branches ν versus Bres
crystallographic a axis mainly towards the b axis. i
areplottedinthemainpanelofFig.4. IntheESRexper-
TherelativeenergiesofthespinstatesofSNOXcalcu-
imentinstaticmagneticfieldswithν < 550GHzmode1
lated in the framework of the Hamiltonian (2) with the
isthestrongestinintensitywhereasmodes2,3and4ap-
parametersyielding the best fit ofthe magnetizationare
pear as weak satellites on the nonlinear background. At
plottedinFig.3foramagneticfieldorientedperpendicu-
a fixed excitation frequency ν = 300GHz the intensity
lartotheaxisoftheaxialanisotropytensor. Onecansee
I of mode 1 has been studied as a function of temper-
thatthespinsingletgroundstateS = 0iswellseparated
ature. The result is shown in Fig. 5. I(T) exhibits an
fromtheS = 1tripletandtheS = 2quintetstate. The
activationtemperaturedependencesimilartothatofthe
separationenergybetweensingletandtripletcorresponds
static magnetization M(T) in Fig. 2. Because the ESR
tothe couplingconstantJ,whereasthatbetweensinglet
intensity is determined by the static susceptibility χ of
and quintet corresponds to 3J. The splitting of the ex- the spins participating in the resonance12 the common
cited S = 1 and S = 2 states in zero magnetic field is
temperaturebehaviorofI(T)andM(T) = χ(T)B gives
induced by the anisotropy D. Both triplet and quintet
evidence that in the ESR experiment the same spins are
levelssplitwithincreasingmagneticfielddueto theZee-
probed which contribute to the static magnetic proper-
man effect. The role of the zero field and the Zeeman
ties. Mode1hasamaximumintensityatT ∼ 30−35K
splitting for the high field magnetic properties of SNOX
which roughly corresponds to the energy separation be-
will be discussed in detail in Section IIC and IID.
tween the singlet ground state and the first S = 1 ex-
Note, that analyzing the data with a negative citedstate(Fig.3). Thisallowstounambiguouslyassign
D = 11.5 K leads to a similar good agreement between this mode to a resonance excitation within the S = 1
the model and the data, resulting in J = 42 K and g- multiplet. Owing to the weakness of modes 2, 3 and 4
values of ga = 2.160,gb = 2.256 and gc = 2.213 but an theycanbedetectedonlyinalimitedtemperaturerange
inconsistentsetofanglesγa=90◦,γb=25◦andγc =46◦ ∼30 − 60K. At smaller temperatures they are not visi-
isfound. ThepositiveD valueisalsostronglysupported bleduetothestrongdecreaseoftheiramplitudewhereas
by the high field magnetization data described below. at higher temperatures they also broaden. Nevertheless
An appreciable temperature independent contribution their occurrence in the temperature window where the
to the static susceptibility χ can not be ascribed to the staticmagnetizationisstrongest(Fig.2)allowstoassign
0
Van-Vleckparamagnetismχ alonebecausetypicalval- themtothebulkESRresponseoftheSNOXsampletoo.
vv
uesofχ forNi(II)areintherangeof10−4 emu/mole18 A remarkable feature of the frequency/field diagram
vv
leading to 0.0008 µ /dimer at B =2T. We assign it in Fig. 4 is the intercept of the strongest ESR branch 1
B ex
to a very small concentration of ferromagnetic Ni metal with the frequency axis at ν ≈ 239GHz. It implies a
0
clusters, which are formed in the synthesis process as an finite energy gap ∆ = ν h/k = 11.5K for this reso-
0 B
impurity phase. nance excitation. This zero field gap can be straightfor-
5
field B . As a result orientations perpendicular to the
ex
z axis of the uniaxial anisotropy tensor will dominate in
thespectrumwhereasotherdirectionsmayformabroad
nonlinearbackground. Thusforasemiqualitativeunder-
standing of the field dependence of the resonance modes
itissufficienttoconsidertheenergydiagramforB ⊥ z
ex
shown in Fig. 3. The field dependence of branch 1 can
be very well reproduced (solid line in Fig. 4) assuming
that the resonance transition occurs between the |−i
and|+istatesoftheexcitedtriplet(Fig.3). Infact,this
usually ”forbidden” transition12 has the strongest inten-
sity in the spectrum for ν < 550GHz. This is due to
the strong anisotropy of the crystal field yielding a large
zerofieldgap∆andmixingstronglythepurespinstates
|−1i, |0i and |+1i into the |−i and |+i states for
B ⊥ z in the field regime gµ B < ∆. Indeed, in a
ex B ex
Figure 4: Main panel: Relationship ν versus Bires for the pulsedfieldESRexperimentthe relativeintensity ofthis
ESR modes i = 1 ... 4. Solid line is a theoretical fit of the modebecomesappreciablysmaller(seeFig.4,inset). For
data. Dashedlines isa linearapproximation ofbranches2, 3 the excitation frequency of 739GHz the resonance field
and 4 to high fields. Gray error bar indicates a field region amountstoBres = 11.2T.Forthisfieldstrengthnotthe
1
whereabroadabsorptionmodeisobservedinthepulsedfield crystal field but the magnetic field direction determines
ESR measurement. Inset: ESR spectrum in astatic (bottom aquantizationaxis. Thusthemixed|−iand|+istates
curve)andpulsedmagneticfieldESRexperiment(topcurve).
transform into almost pure |−1i and |+1i states which
Sharp lines are due to the DPPH field marker. Resonance
do not fulfill the ESR selection rule ∆S = ±1.
modesassociatedwithabulkESRspectrumofNi-oxalateare z
In contrast to mode 1, the exact assignment of modes
indicated by black arrows and numbered. The anticipated
2, 3 and 4 is difficult owing to their small intensity. The
positions of modes 2, 3 and 4 at high magnetic fields are
marked by gray dashed arrows. (for details see text). relationshipν versusB1res hasasteepslopebecauseboth
energylevels,|−iand|+i,dependonmagneticfielddi-
verging from each other with increasing B . The much
ex
smaller slope of branches 2, 3 and 4 suggests that reso-
nance transitions within the neighboring pairs of states,
like e.g. (|−i ↔ |0i), (|0i ↔ |+i) etc., of the multi-
pletsS = 1andS = 2shouldbeinvolved. Branch2has
a much smaller frequency offset as compared to ∆ and
can be associated with the transition |−−i ↔ |−i in
the quintet S = 2. Branches 3 and 4 might be gapless,
which is the case for transitions |−i ↔ |0i in triplet
as well as in quintet. Unfortunately, it was not possible
to achieve a better resolution of modes 2, 3, and 4 in
the pulsed field measurements. Only a broad absorption
mode is observed in the field region where resonances 2,
3 and 4 are anticipated (Fig. 4).
Figure5: IntensityofthestrongestESRmode1asafunction
D. High field magnetization
of temperature. Note an activating behavior similar to the
static magnetization M(T) (Fig. 2).
The high field magnetization measurements were car-
ried out at the LNCMP, Toulouse, as well. Measure-
wardly identified with the zero field splitting of the first ments of the static magnetization have been performed
excitedtripletinFig.3. NumericalsolutionoftheHamil- in pulsed magnetic fields up to 55T with a pulse dura-
tonian(2)yields∆ =|D|. Therefore,fromtheESRdata tion of 250ms (raising time 40ms) by an inductive tech-
oneobtainsanaccurateestimate ofthe absolutevalueof nique. A system of two concentric pick-up coils in oppo-
the single ion anisotropy parameter |D| = 11.5K which sition to each other equipped with additional compensa-
is essential for the understanding of the static magnetic tioncoilswasutilized. Themeasuredsignal∂M/∂B was
propertiesofSNOXasdiscussedinSectionIIBandIID. integrated numerically. The sample temperature can be
ESR measurements of a powder sample imply the av- varied between 1.4K and 300K.
eraging of all possible orientations of the microcrystal- As can be seen from the energy diagram of the spin
lites in a powder with respect to the applied magnetic states of SNOX (Fig. 3) the splitting of the states in a
6
magnetic field yields a level crossing of the ground state
with the lowest triplet state |−1i at a field B and a
C1
second level crossing of the |−1i state with the |−2i
quintet state at field B with B < B . This leads
C2 C1 C2
to a step-like behavior in high field magnetization mea-
surements. If the field is applied along the z axis, the
critical field B can be estimated using the equation
C1
B = (J − D/3)/(gµ /k ), depending strongly on
C1 B B
the sign of D. For a field perpendicular to the local
anisotropy axis, a negative D (leading to higher criti-
cal fields for B k z) pushes the critical field to lower
fields and vice versa for a positive D. Since in powder
measurementstheperpendicularsituationdominatesthe
spectrum of the spin states, the field dependent magne-
tization, among others, can determine the sign of the
Figure6: HighfieldmagnetizationofaSNOXpowdersample.
anisotropy D. Thefigureshowsmeasurementsat1.47Kand4.2K,theinset
ThecriticalfieldstrengthsB dependonthemagnetic measurementsat10Kand26K.Thesolidlinerepresentsthe
Ci
exchangeJ andan exchangeenergy of43 K corresponds simulation as described in the text.
to roughly 30 T for B . Therefore high magnetic fields
C1
are needed to observe the magnetization steps. We per-
formed measurements at several temperatures (1.47 K,
4.2 K, 10 K and 26 K) on a powder sample with a mass
of 23.2 mg in magnetic fields up to 55 Tesla. The results
are shown in Fig.6. In the low temperature experiments
the first magnetization step is clearly seen, whereas it
smoothes at higher temperatures due to the thermal av-
eraging process. The critical field B can be obtained
C1
fromthederivationofthemagnetizationdM/dBwhichis
showninFig.7forT =1.47KandT =4.2K(inset). At
both temperatures a sharp peak is observed at 29 Tesla.
Additionally a second increase of the magnetization at
much higher fields of about 50 Tesla can be anticipated.
Thisstepisalsoobservedstronglybroadenedinthe10K
measurement. At 26 K a linear field dependence of the
magnetization is found.
The solid lines in Fig.6 and Fig.7 describe simula-
Figure 7: Differential magnetization curves ∂M/∂B for tem-
tions using the dimer model with the calculated g val- peraturesT=1.47KandT=4.2K(inset). Theopencircles
ues gk (see above) and g⊥ = gb in a powder average are calculated from measured data, the solid lines represent
M = (Mk+2 M⊥)/3 . To describe the high field mag- simulations as described in thetext.
netization well we need to use a coupling constant of
J = 44 K instead of J = 43 K. In particular the dif-
ferential magnetization is very well reproduced by the in BC10 = 29 T in a powder average which is shown in
simulation,describingaswellthewidthasthecenterpo- the figure. Notice, that the powder averaging leads to
sitionofthe peak. Asmalldeviationis foundinthe field a broadening of the magnetization steps. For the second
rangeof30-33TintheT =1.47Kmeasurement,which step,wegetBC2 =67.1TandBC2 =51.3TforBparal-
maydependonalittleshiftofthesampleinducedbythe lel and perpendicular, respectively. This difference leads
external field. in a powder sample (average over all angles, which we
calculated for angles from 0◦ to 90◦ in steps of 1◦ with
The simulations for a negative D = -11.5 K did not
varying g from g to g ) at T = 1.47 K to a continu-
describe the data reasonably well for any J. In this case k ⊥
ous double-sigmoid like increase of the magnetization in
the first magnetizationstep is consistentwith J = 43 K,
the field range of 50 - 70 Tesla, of which only the first
whereasthesecondstepispredictedattoolowafield. To
increase is observed in the measurement.
reproduce the second step well one has to use J = 45 K
which results in a too large a value of B .
C1
From Fig.7, we can determine the critical field B
C1
of the powder sample to B = 29 T. In our simula- E. Quantum Monte Carlo Calculations
C1
tions, we get B = 27.7 T for B parallel to the mag-
C1
netic anisotropy axis and B = 30.7 T for B perpen- To perform a quantitative analysis of the ratio
C1
dicular to the magnetic anisotropy axis, resulting also R=J/(K+|J|) of the intra- to inter-dimer coupling, we
7
have comparedthe observedmagnetic susceptibility and and χ ) to fit the data to the high temperature range
red
the magnetization with Quantum Monte Carlo (QMC) (2≤ T ≤8) of the QMC calculations. Notice, that the
red
calculations. These calculations explicitely include the scaled experimental dataset appreciably depends on the
effects of the finite single ion anisotropy D = 11.5 K. g-factor (in the range of 2.2 to 2.25). Nevertheless, all
This is different from previous QMC studies of spin-1 possible transformations lead to a value of R ≈ 0.95 –
ladders10. The QMC method we employ is the stochas- 1.00fortotalmagneticexchangestrenghtsbetween43K
ticseriesexpansion(SSE)19,20,whichdirectlysamplesan (R = 1) and 45 K (R = 0.95). The solid circles in Fig.8
expansionofthepartitionfunctioninpowersof1/T until show the best result of the transformation using a total
convergence. WithintheSSEthecumulantsofthispower coupling constant J+K = 45 K, a g-factor g = 2.23, a
series areevaluatedusing diagonalandnon-diagonalup- temperature independent term of 0.0085 µ /dimer and
B
dates within the space of exchange operators from eqn. a Curie constant of 0.0045 µ K/dimer. A comparison
B
(2). The latter update is greatly improved by introduc- with the QMC data in the low temperature regime
ing directed loop-updates21. We have carried out SSE of T ≤1.5 gives an R of 0.95, which results in an
red
calculations on systems with up to 200×2 sites and in- intradimer coupling of J = 42.75 K and an interdimer
verse temperatures β ≤ 10. The system sizes have been coupling of K = 2.25 K.
checkedtorepresentthethermodynamiclimitwithinthe
temperature range studied. In all figures depicted the QMCcalculationsforthehighfieldmagnetizationofa
statistical QMC errors are much smaller than the sym- weakly coupled dimer system were done for a reduced
bol sizes and have been omitted. temperature T/J = 0.034 and a reduced anisotropy
D/J = 0.27 corresponding to a sample temperature
T = 1.47 K and D = 11.5 K if J + K = 43 K. The
calculations for R = 1, 0.95 and 0.9 are compared with
the measured data in Fig.9. The measured data were
scaled by setting the magnetization at B = 40 T to
ex
M = 0.5 and using a value J +K ≈ 43 K. A variation
of J +K mainly shifts the position of the step and does
not affect the broadening of the step. The QMC calcu-
lations show a broadening of the magnetization step for
decreasingR,i.e. increasingmagneticexchangestrength
along the legs of the ladder. The QMC calculations are
possible only for B parallel to the anisotropy axis up to
now. Therefore, comparing with the powder measure-
mentsonehastoconsiderthatthemagnetizationstepin
the experimental data set is additionally broadened by
the presence of the axial anisotropy D. Therefore, the
step gradientof the appropriate QMC calculationhas to
Figure8: QuantumMonteCarlosimulationsforR=1(open
be sharper than the measurement performed on a pow-
circles), 0.95 (open triangles) and 0.9 (open squares). We
dersample. Forthisreason,theexperimentaldataarein
plotted the reduced susceptibility versus reduced tempera-
agreement only with the calculation for R = 1, i.e. the
ture. The solid circles show the measured data for B = 2 T
along the a axis corrected by a Curie tail and a temperature uncoupled dimer model.
independent contribution. The inset shows the high temper-
ature results.
III. THE ANISOTROPY OF THE MAGNETIC
EXCHANGE STRENGTH
Our calculations show a significant change of the
shape of the susceptibility at low temperatures as a
function of R. In the plot of the reduced suscepti- An unexpected result of the present study is the very
bility χ = χ (J+K) versus reduced temperature largedifferencebetweenthestrengthofthe magneticex-
red
T = T/(J+K) the peak increases and narrows with change interaction along the rungs (J = 43 K) and legs
red
decreasing R down to R = 0.7 and then decreases and (|K| < 2 K) of the spin ladder. This can be qualita-
broadens with further decreasing R15. The calculations tively understood considering the different coordination
havebeenperformedfor D/J = 0.27,whichcorresponds and superexchangeangles of the exchange mediating ox-
to D = 11.5 K for J = 42.6 K, similar to the fit alate molecules.
results described above. The results for R = 1, 0.95 On the rungs the oxalate forms a µ-1,2,3,4 bridge be-
and 0.9 are shown in Fig.8 compared with the scaled tween two Ni ions. This bridge provides two symmetric
experimental data. For this scaling we converted the superexchange pathways. On each path the Ni 3d x2-y2
magnetization data for B = 2 T parallel to the a axis orbitalshaveanenhancedelectronprobabilitydensityex-
to the reduced representation and vary the g-factor and tending directly towards the corresponding oxygen ions.
the sum of coupling constants J+K (which scales T There is a direct overlap of the Ni 3d x2-y2 and the O
red
8
O 2p orbitals involved in the Ni-O bonds are not over-
lapping with each other and therefore either the carbon
atoms or orthogonal O 2p orbitals are involved which
strongly suppresses the strength of the superexchange
mechanism and may even lead to a weak ferromagnetic
exchange. Similar µ-1,2,3 oxalato bridges are reported
for Cu(II) systems with magnetic exchange strengths in
the range of -0.2 to 0.3 K24,25 consistent with the esti-
mate of |K| < 2 K for SNOX.
IV. CONCLUSION
Inthispaperwereportthesynthesis,crystalstructure
Figure 9: QMC calculations of the field dependent magneti- and magnetic properties of the structurally well defined
zation for different ratios of R with rescpected D/J = 0.27.
S =1spinladdercompoundNa Ni (C O ) (H O) . We
Open circles: R = 1, open triangles: R = 0.95, open 2 2 2 4 3 2 2
performed magnetic susceptibility and pulsed high field
squares:R = 0.9. The calculations were done for a reduced
magnetization measurements to examine the thermody-
temperature T/J = 0.034, corresponding to J +K = 43 K
namic properties of the spin system. The magnetic exci-
and T = 1.47 K. The solid squares are the normalized mea-
surements scaled with J+K = 43K tations have been examined in high field high frequency
ESR. Although the Ni(II) ions form a structural ladder,
an isolated dimer model on the rungs describes the ob-
servations very well. The analysis of the temperature
dependent magnetization data leads to a magnetic ex-
change constant of J = 43 K along the rungs of the lad-
der. We deducethe principalaxisofthe axialanisotropy
tensor. FromESRmeasurements,wedeterminedthesin-
gle ion anisotropy to D = 11.5 K. To prove the validity
of the isolated dimer model, we compare the single crys-
talmagnetic susceptibility measurementswith Quantum
Monte Carlo calculations for a S=1 spin ladder, which
werecarriedoutforseveralratiosbetweeninterdimerand
intradimer exchange and take into account the experi-
mentally determined single ion anisotropy D. It results
in a upper limit of 5 % interdimer exchange coupling.
This strong anisotropy of the magnetic exchange inter-
Figure 10: Ni-(ox)-Ni coordination and proposed magnetic
actioncanbe tracedback to the different superexchange
exchangepathwaysontherungs(leftpanel)andonthelegsof
thespinladder(rightpanel). TheinvolvedNi3dandoxygen pathways on the oxalate molecules along the rungs and
2p orbitals are shown in black. legs of the spin ladder.
2p wave functions forming σ bonds. The polarized 2p
V. ACKNOWLEDGMENTS
orbitals themselves are strongly overlapping. Therefore
the intermediate carbon atom is not involved in the su-
perexchange mechanism resulting in a strong antiferro- We thank A.U.B. Wolter and S. Su¨llow (TU Braun-
magnetic superexchange interaction (see fig. 10). In the schweig) for help and fruitful discussions in the early
literature several µ-1,2,3,4 oxalato-bridged Ni(II) dimer stage of this work. Financial support by the Deutsche
and chain systems are reported (see e.g.22,23 and refer- Forschungsgemeinschaft through SPP 1137 ”Molecular
ences therein) with J values in the range of 20 to 42 K. Magnetism” grant KL 1086/6-1 is gratefully acknowl-
The value of 43 K in SNOX is at the upper limit of the edged. The work of R.K. in Toulouse was supported by
reported range. the DFG through grant KL 1824/1-1. Experiments of
Fortheoxalatebridgealongthelegsofthespinladder C.G. and V.K. in Toulouse were supported by the Eu-
thesituationiscompletelydifferent. Inthiscaseaµ-1,2,3 roMagNET consortium of the European Union through
oxalato bridge is formed. As shown in the right panel of contact FP6 R113-CT2004-506239. D.J.P. is grateful to
fig. 10 only one Ni ion forms two covalent bonds (one the EPSRC of the UK for the award of an Advance Re-
with the x2-y2 orbital and one with the 3z2 - r2 orbital) searchFellowship(Gr/A00836/02). Wearealsoindebted
with the oxalate molecule whereas of the second Ni ion to M.E. Light (University of Southampton, UK) for X-
only the 3z2 - r2 orbital forms one covalent bond. The ray data collection.
9
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