Table Of ContentQuantum-magneto oscillations in a supramolecular Mn(II)-[3 × 3] grid
O. Waldmann,1,∗ S. Carretta,2 P. Santini,2 R. Koch,3 A. G. M.
Jansen,4,5 G. Amoretti,2 R. Caciuffo,6 L. Zhao,7 and L. K. Thompson7
1Department of Physics, The Ohio State University, Columbus, OH 43210, USA
2INFM and Dipartimento di Fisica, Universit`a di Parma, I-43100 Parma, Italy
3Physikalisches Institut III, Universit¨at Erlangen-Nu¨rnberg, 91058 Erlangen, Germany
4CEA-Grenoble, DRFMC/SPSMS, 38054 Grenoble, France
5Grenoble High Magnetic Field Laboratory, CNRS and MPI-FKF, 38042 Grenoble, France
6INFM and Diprtimento di Fisica, Universit`a Politecnica della Marche, I-60131 Ancona, Italy
7Department of Chemistry, Memorial University, St. Johns, Newfoundland, Canada A1B 3X7
(Dated: February 2, 2008)
4 The magnetic grid molecule Mn(II)-[3 × 3] has been studied by high-field torque magnetometry
0 at 3He temperatures. At fields above 5 T, the torque vs. field curves exhibit an unprecedented
0 oscillatory behavior. A model is proposed which describes these magneto oscillations well.
2
PACSnumbers: 33.15.Kr,71.70.-d,75.10.Jm
n
a
J
Among the known magneto-oscillatory effects, the de total spin S = 5/2 ground state at zero field. At a field
6
Haas-vanAlphen(dHvA)effectinmetals(andthegroup ofabout7T,thegroundstatechangesabruptlytoanew
1
of effects related to it) is the most prominent example, state, a S = 7/2 level, accompanied by a change of the
1 having had a formative influence on our modern picture sign of the magnetic anisotropy. Notably, intermolecu-
v ofsolidstatephysics[1]. ThedHvAeffectoriginatesfrom lar magnetic interactions are at best on the order of few
4
aquantizationofclosedelectronorbitsintoLandaulevels 10 mK. This is evident from the crystal structure (the
0
in a magnetic field, so that the density of states at the smallestseparationbetween two molecules in the crystal
3
1 Fermi energy exhibits a markedly oscillatory evolution is > 8 ˚A), but has been checked also experimentally [9].
0 as a function of magnetic field. Currently, it finds appli- Accordingly, even at 3He temperatures the magnetism
4 cation in a wide range of materials, e.g. heavy fermion of a macroscopic crystal sample reflects that of a single
0
compounds[2],low-dimensionalorganicmetals [3], high- molecule.
/
t temperature superconductors [4], two-dimensional elec- SinglecrystalsofMn-[3×3]werepreparedasreported
a
m trongases[5], orinthe recentlydiscoveredsuperconduc- [10]. They crystallize in the space group C2/c. The
- tor MgB2 [6]. cation[Mn9(2POAP-2H)6]6+ exhibitsaslightlydistorted
d In this work, we report on a new magneto- S4 molecular symmetry, the C2 axis is perpendicular to
n oscillatory effect, discovered in the molecular nanomag- thegridplane. TheaveragedistancebetweentheMn(II)
o
net [Mn (2POAP-2H) ](ClO ) · 3.57 MeCN · H O, the
c 9 6 4 6 2
: so called Mn-[3 × 3] grid. Molecular nanomagnets are
v
compoundswithmanymagneticmetalionslinkedbyor- 0.02 S=5/2 T = 0.38 K
i
X ganicligandstoformwelldefinedmagneticnanoclusters. j » 0.5°
)
s
ar Tfahsceiynahtaivnegaqtutraancttuemdmeffuecchtsinattertehsetsminecseostchoepyiccasncaelex.hiFboirt unit 0.01
instance, quantum tunneling of the magnetization has b.
r
a
been observed in the clusters Mn12 or Fe8 [7]. We have e ( 0.00
performed high-field torque magnetometry on the Mn- u
q
[3 × 3] grids at 3He temperatures and observed striking or
t
magneto oscillations in the torque signal. We show that -0.01
they arise from the interplay between antiferromagnetic
interactions within a molecule and Zeeman splitting on 0 5 10 15 20
B (T)
the one hand and the magnetic anisotropy on the other
hand.
FI G. 1: Torque vs. magnetic field of a Mn-[3 × 3] single
IntheMn-[3×3]gridmolecules,ninespin-5/2Mn(II)
crystal at low temperature. Asthemagnetic field lies almost
ions occupy the positions of a regular3 × 3 matrix,held
intheplaneofthegridmolecule, thesignal isverysmalland
in place by a lattice of organic ligands [inset of Fig. 1]. might be affected by a significant, but smooth background
Thesegridswerecharacterizedrecentlybymagnetization (e.g. due to Faraday forces). The peak at ≈ 2 T stems from
and torque measurements and found to exhibit unusual thezero-fieldsplittingoftheS =5/2groundstate. Theinset
magnetic properties [8]. The Mn ions within a molecule showsthestructureofthecation[Mn9(2POAP-2H)6]6+(with
Mn in black and H atoms omitted).
experience an antiferromagnetic interaction leading to a
2
(a)
part of the signal stems from the S = 5/2 ground state
experiment
0.06 and shows the expected behavior [12]. At higher fields,
s) however, above about 5 T, the overall torque signal de-
nit creases with the superimposition of pronounced oscilla-
u
b. 0.04 0.2 tions,whichfadeoutwithincreasingfieldstrength. Such
ar magnetooscillationshavenotbeenobservedinmolecular
e ( 0.1 nanomagnets so far [13], and their observationis a main
u
q 0.02 result of this work.
or 2.8 29.8 0.0
t 4.1 38.8 An overview of the dependence of the torque curves
7.3 56.8 0 5 10 15 on the magnetic field orientation is given in Fig. 2(a).
20.3 70.3
0.00 The oscillations are clearly visible for angles smaller
0 5 B (T) 10 15 than about 10◦. For larger angles, the torque decreases
(b) 2.0 stronglyathigherfieldsandactuallycrossesthezeroline,
theory consistent with the observations in Ref. [8] (the torque
measurements reported there did not show the oscilla-
T) 1.5 tionsbecausethey wereperformedattoohighatemper-
B
µ ature).
A
N
( 1.0 4 Themagneticpropertiesofmolecularmetalcomplexes
e are generally described by a spin Hamiltonian, which,
u 2
q
r assuming an idealized structure, for Mn-[3 × 3] becomes
to 0.5 0
0 5 10 15
7
0.0 H = −J S ·S +S ·S
R i i+1 8 1
0 5 10 15 !
B (T) −J (SXi=+1 S +S +S )·S
C 2 4 6 8 9
8
Fa sIGfu.n2c:tio(an)oEfxmpaegrinmeteinctafileladndof(ba)Mcanl-c[u3la×te3d]tsoinrqguleeccruyrsvteasl +DR Si2,z +DCS92,z+gµBS·B (1)
i=1
at 0.4 K for various orientations of the magnetic field (the X
legend gives the angle ϕ between field and grid plane). The with S = S . It consists of the isotropic nearest-
◦ i i
main panel shows the curves for ϕ = 2.8 . The dashed line neighbor exchange terms, the second order ligand-field
in panel (b) represents the calculated torque curve for 0.4 K terms and thPe Zeeman terms [17]. J characterizes the
◦ R
and2.8 ,butwitheffectsoflevelmixingartificially forced to
couplings of the eight peripheral Mn ions, J those in-
zero. C
volving the central Mn ion (spins at ”corners” are num-
bered 1, 3, 5, 7; those at ”edges” 2, 4, 6, 8; and the
central spin 9).
ions is 3.93 ˚A, the smallest distance between clusters is
The dimension of the Hilbert space is huge for Mn-
larger than 8 ˚A. Consistent with the planar structure,
[3 × 3]. However, recently it has been shown that the
the Mn-[3 × 3] grid exhibits a practically uniaxial mag-
field dependent low-temperature properties can be very
neticbehavior,anduniaxialandmolecularC symmetry
2 welldescribedbyaneffectivespinHamiltonian,inwhich
axes coincide [8]. Magnetic torque was measured with a
thespinoperatorsforthecornerandedgespinsarecom-
homemade silicon cantilever device [8, 11] inserted into bined to sublattice spin operators S and S , respec-
A B
the M10 magnet at the Grenoble High Magnetic Field
tively, with S = S = 4 ×5/2 [18]. Physically, this
Laboratory using a 3He evaporation cryostat. Resolu- A B
approach works well because the internal spin structure
tionofthesensorwasabout10−11 Nm,theweightofthe
duetothe dominantHeisenberginteractionisessentially
crystals investigated was about 20 µg, nonlinearity was
classical [19]. In this approach, Mn-[3 × 3] is described
lessthan2%,andtheaccuracyoftheinsitualignmentof
by
theC crystalaxiswithrespecttothemagneticfieldwas
2
0.3◦. Only raw torque data are shownhere. Background
signals were, if not noted otherwise, negligible. Temper- H3×3 = −J˜ S ·S +D˜ (S2 +S2 )
eff R A B R A,z B,z
ature was measured with a RuO2 thick film resistor. −J S ·S +D S2 +gµ S·B, (2)
C B 9 C 9,z B
Figure1presentsthe fielddependence ofthe torqueof
a Mn-[3 × 3]singlecrystalat0.38K.The magnetic field with J˜ = 0.526J , D˜ = 0.197D [18]. In Ref. [8],
R R R R
was applied almost perpendicular to the C axis of the |J |≪|J | wassuggested. With this parameterregime,
2 C R
molecule; the angle ϕ between magnetic field and plane however, the experimental results could not be repro-
of the grid was about 0.5◦. At low fields, the torque duced satisfactorily. We thus tried J = J ≡ J, and
C R
rises quickly to reach a maximum at about 2 T. This also set D = D ≡ D, as suggested by recent inelastic
C R
3
neutron scattering (INS) measurements [20]. As long as 15
J andJ arenottoodifferent,thecalculatedproperties -20
C R
S=5/2
werefoundtobeonlyweaklyaffected. Asimilarsituation 10 -30
S=7/2
holds for DC and DR. J and D were determined such K) -40 S=9/2
that the position of the first ground state level crossing y ( 6 8 10
andthezero-fieldsplittingoftheS =5/2groundstatein g 5
r
e
the calculated energy spectrum match the experimental n
e
values of ≈ 7 T and ≈ 3 K, respectively [8]. The result
0
is J = -5.0 K and D = -0.14 K, in agreement with INS
experiments [20]. S = 5/2 9/2 11/2 13/2
The torque curves calculated with H3×3 using the -5
eff
0 5 10 15
givenparametersarepresentedinFig.2(b). Apparently,
B (T)
the model Hamiltonian reproduces the data, and in par-
ticular the oscillations, very well [21]. Their origin shall
FI G. 3: Energy spectrum vs. magnetic field for Mn-[3 × 3]
be discussed in the following in more detail. at ϕ = 2.8◦ (energy of lowest state was set to zero at each
For illustration, Fig. 3 shows the calculated energy field). The arrows indicate the ground state level crossings,
spectrum as a function of magnetic field for an angle of andthenumbersthe(approximate)totalspinquantumnum-
ϕ = 2.8◦. As magnetic anisotropy is weak, |D| ≪ |J|, ber S of the respective ground states. The inset details the
the total spin S is almost conserved. It is convenient to energy spectrum near the first two level crossing, where the
rotate the spin operators such that S′ is parallel to the ground state changes as S =5/2→7/2 and S =7/2→9/2.
magnetic field B. Levels may then bze classified by the In contrast to the main panel, energies are given here with
respect to theground state energy at zero field.
eigenvalues of S2 and S′, S and M, and be written as
z
|S,Mi.
At zero field, the dominant isotropic exchange results energy spectrum can be described by a two-level Hamil-
inaS =5/2groundstatefollowedbyexcitedstateswith ǫ ∆/2
tonian H = S . Here, ǫ (B,ϕ) and
S = 7/2, 9/2, 11/2, .... Upon application of a mag- S,S+1 ∆/2 ǫS+1 S
(cid:18) (cid:19)
netic field, the levels are shifted by the Zeeman energy, ǫS+1(B,ϕ) describe the field and angle dependence of
gµ MB, leading to a series of ground state level cross- the two levels excluding level mixing at the LC, but in-
B
ings (LCs) at characteristic fields at which the ground cluding the effects of the zero-field splitting. The level
state switches abruptly from 5/2 → 7/2, 7/2 → 9/2, mixing itself is parameterized by a possibly field and
9/2 → 11/2, and so on (we abbreviated |S,−Si by the angle dependent ∆. Up to first order, the energies
valueofS). ThisisshownforthefirsttwoLCsinthein- ǫ can be written as ǫ (B,ϕ) = −bS + ∆ (ϕ) with
S S S
setofFig.3. Thechangeoftheslopeofthe groundstate the reduced field b = gµ B [15, 16]. ∆ (ϕ) accounts
B S
from −5/2gµ to −7/2gµ and −9/2gµ is apparent. for the magnetic interaction and the zero-field splitting
B B B
Theeffectoftheligand-fieldtermsistomixspinlevels and thus does not depend on magnetic field. The LC
with |∆S|=0,1,2[22, 23]. The ∆S =0 mixing leads to field is given by bc(ϕ) = ∆S+1 − ∆S. Importantly,
a zero-field splitting of each spin multiplet, which, e.g., in first order the level mixing is also independent of
isclearlyvisibleinFig.3forthesixlevelsoftheS =5/2 the field: It is determined by the matrix element be-
ground state. Importantly, because of |∆S| = 1, the tween the two states and the ligand-field terms in Eq. 1,
′
ligand field terms also mix the levels which are involved ∆=hS,−S|H |S+1,−S−1i. Thus ∆=∆(ϕ).
LF
in a LC. This results in a repulsion of the two states Atzerotemperature,themagneticmomentisgivenby
whenever they come close, and the LCs split, as is seen m=−∂ǫ/∂B, where ǫ is the lowest energy as calculated
by the energy gaps at the LC fields in Fig. 3 (which are from HS,S+1. We decompose the magnetic moment into
about 0.3 K at the angle of 2.8◦). m = mS +δm. Here, mS is the magnetic moment for
The abrupt changes of the magnetic character of the fields well below the LC field bc so that δm describes
ground state at the LCs lead to a step-like field depen- the changes of m at the LC. Similarly, the torque τ =
dence of the magnetization and torque. These magneti- −∂ǫ/∂ϕ is decomposed into τ = τS +δτ. In our model,
zationand torque steps are frequently observed[24], e.g. one obtains
inmolecularferricwheels[13,14,15,16]. However,aswe
will show now, the level mixing at the LCs lead also to 1
peak-like contributions in the torque (but not the mag- δm(b,ϕ)= [1+(b−bc)G(b,ϕ)] (3)
2
netization). For the sake of completeness, we mention ∂b 1 ∂∆
c
that mixing of levels with |∆S| = 2 is pronounced only δτ(b,ϕ)=δm − G(b,ϕ)∆ , (4)
∂ϕ 2 ∂ϕ
for excited states [23], and is of little relevance here.
Near each LC only two states, |S,−Si and |S + with G(b,ϕ) = [(b − b)2 + ∆2]−1/2. Therefore, as
c
1,−S − 1i, are relevant at low temperatures, and the function ofmagnetic field the magnetic momentexhibits
4
a step at the LC which is broadened by the level mixing Energy (Grant No. DE-FG02-86ER45271),and through
(and by temperature if T > 0). In contrast, the torque theTMRProgramoftheEuropeanCommunityisgrate-
consists of two contributions. The first term, being pro- fully acknowledged.
portionaltoδm,leadstoabroadenedtorquestep. These
magnetization and torque steps, as mentioned already,
are well known [14, 15, 16, 24]. The second term, how-
ever,whosefielddependenceiscontrolledbyG(b,ϕ)only,
∗
Corresponding author.
leadsto apeak-likecontributioncenteredatthe LCfield
E-mail: [email protected]
b . This is apparent by noting that G(b,ϕ) is basically
c [1] D. Shoenberg, Magnetic oscillations in metals (Cam-
the square root of the Lorentz function. Importantly, as
bridge University Press, Cambridge, 1984).
this peak arises only for nonzero level mixing it is a di- [2] R. Joynt and L. Taillefer, Rev. Mod. Phys. 74, 235
rectsignature of it. It is absentin the magnetic moment (2002).
sincethelevelmixing∆ismarkedlyangledependentbut [3] J. Wosnitzaet al.,Phys.Rev.Lett.67, 263(1991); J.S.
virtually field independent. Brooks et al., ibid. 69, 156 (1992); M. V. Kartsovnik et
al., ibid.77, 2530 (1996).
Thus,the torquecurveconsistsofa seriesofsteps and
[4] C. M. Fowler et al.,Phys. Rev.Lett. 68, 534 (1992).
peaks at each LC field. For Mn-[3 × 3], the steps are
[5] J. P. Eisenstein et al., Phys. Rev. Lett. 55, 875 (1985);
smallandthepeaksateachLCfieldsuperimposetopro- J. G. E. Harris et al.,ibid. 86, 4644 (2001).
duce anoscillatoryfield dependence. This is clearly seen [6] E.A.Yellandet al.,Phys.Rev.Lett.88, 217002 (2002);
inFig.2(b), whichalsopresents the 2.8◦ torquecurveas A. Carrington et al.,ibid. 91, 037003 (2003).
it is calculated with the level mixing at the LCs set to [7] R. Sessoli et al., Nature 365, 141 (1993); L. Thomas et
zero artificially (i.e., only mixing of levels with ∆S = 0 al., ibid. 383, 145 (1996); J. R. Friedman et al., Phys.
Rev. Lett. 76, 3830 (1996); C. Sangregorio et al., ibid.
wasretained). Thiscurveexhibitsonlythermallybroad-
78, 4645 (1997).
enedstepsattheLCfields,demonstratingtheconnection
[8] O. Waldmann , L. Zhao, and L. K. Thompson, Phys.
of the oscillations with nonzero level mixing. Rev. Lett.88, 066401 (2002).
InviewofthesimplicityofourmodelHamiltonian,the [9] Because of the large seperation of individual molecules,
agreement between theoretical and experimental curves Mn-[3 × 3] crystals are good insulators showing that a
isreallygood,thoughitisnotperfect. Mostnotably,the dHvA type of effect cannot be responsible for the mag-
experimentalpeaks are significantly broaderandsmaller neto oscillations observed here.
[10] L. Zhao, C. J. Matthews, L. K. Thompson, and S. L.
than the theoretical ones. The agreement can be im-
Heath, Chem. Commun. 2000, No. 4, 265 (2000).
proved by using a larger temperature value in the calcu-
[11] R. Koch et al.,Phys.Rev. B 67, 094407 (2003).
lation, Teff = T +x. We call x the Dingle temperature [12] A. Cornia et al.,Chem. Phys.Lett. 322, 477, (2000).
because of the analogy to the practice in dHvA work [13] ”Oscillations” were presented for e.g. molecular ferric
[1], but emphasize the different underlying physics. The wheels in the field derivative of the magnetization and
experimentsuggestsx≈80mK.Severalintrinsicandex- torque, dm(B)/dB and dτ(B)/dB [14, 16]. They reflect
trinsic effects can be envisaged to be responsible for this stepsinthemagnetizationandtorque[15,16],andshould
not be confused with the oscillations observed here.
additional broadening and more detailed investigations
[14] K.L.Taftet al.,J.Am.Chem.Soc.116,823(1994); D.
are needed.
Gatteschi et al.,Science 265, 1054 (1994).
In conclusion, in the molecular grid Mn-[3 × 3] we [15] A.Corniaetal.,Angew.Chem.Int.Ed.38,2264(1999).
have observed a new type of quantum-magneto oscilla- [16] O.Waldmannet al.,Inorg.Chem.38, 5879 (1999); ibid.
tionsinthe fielddependence ofthe magnetictorque. We 40, 2986 (2001).
have shown that they are associated with level crossings [17] Intramolecular dipole-dipole interactions cannot be ne-
which appear regularly as a function of field due to the glected, but have similar effects as the single-ion
anisotropy and are well grasped by S2 terms. The D
combinedeffect ofthe magnetic interactionand the Zee- i,z
values should be thus understood as to include both
manterm. Theselevelcrossingsaresplitbecause ofpro-
anisotropy contributions.
nouncedlevelmixinginducedbythemagneticanisotropy
[18] O. Waldmann (unpublished).
whichareaccompaniedbypeak-likesignalsinthetorque [19] O.Waldmannet al.,Phys.Rev.Lett.91,237202(2003).
producing the spectacular, oscillatory field dependence. [20] T. Guidi et al. (unpublished).
This effect should be observable in a number of other [21] Theoscillations werenotobtainedwiththeanalysispre-
molecular magnetic clusters, like e.g. dimers of two dif- sented in [8], since, following the accepted approach for
systems in the strong exchange limit like Mn-[3 × 3],
ferent magnetic centers with an antiferromagnetic cou-
mixing between different spin levels was neglected.
pling. The magnetic oscillations demonstrated in this
[22] A.BenciniandD.Gatteschi,ElectronParamagneticRes-
studyprovidedirectaccesstoquantumproperties,which
onance of Exchange Coupled Cluters (Springer, Berlin,
will help our understanding of molecular nanomagnets. 1990).
We thank N. Magnani, E. Liviotti and T. Guidi [23] O. Waldmann, Europhys.Lett. 60, 302 (2002).
for enlightning discussions. Financial support by the [24] Y. Shapira and V. Bindilatti, J. Appl. Phys. 92, 4155
Deutsche Forschungsgemeinschaft, the Department of (2002).
