The complexity of the systems investigated in molecular magnetism is continually increasing and the number of independent parameters is so large that measuring the temperature dependence of the magnetic susceptibility is no longer sufficient to characterize the materials. In fact several sets of parameters can often equally well simulate the experimental mT vs. T curves and additional information are therefore necessary. For instance, the manifold nature of magnetic levels in molecular systems can be probed in considerable detail by exploitingZeeman interactions with a strongmagnetic field at very low temperature. Under these conditions, the evolution of the ground spin state properties as a function of applied field provides a wealth of information about quantities of interest in the spin Hamiltonian, such as intramolecular exchange-coupling inter- actions and magnetic anisotropies. For this purpose, the field dependence of the magnetization (M), differential susceptibility (dM/dB), or magnetic torque ( ) is usually measured as a function of applied field B on powder or single-crystal samples. High-field magnetization or differential susceptibility measurements have been generally applied to polycrystalline samples and only occasionally to single crystals, due to their intrinsic low sensitivity. By contrast, single crystals are now very conveniently investigated by torque magnetometry (TM). This technique was used at the beginning of magnetochemistry, but the available apparatus required large single crystals, which were difficult to prepare. Now the sensitivity of modern torquemeters allows one to handle very small samples (down to the microgram size).47In the followingsections, we will describe two
Mo Mo
Mo Mo
N N
N
N
Figure 11 Sketch of the mechanism of spin polarization responsible for the antiferro- (up) and ferro- magnetic (down) coupling between molybdenum(V) ions bridged by 4,40- and 4,50-bipyridine, respectively
(after Baylyet al.).46
leading techniques for the high-field characterization of molecule-based magnetic materials, namely the magnetization step (MST) and the torque step (TST) methods. They rely on a thermodynamic technique for determiningthe energies and splittingof individual spin levels and can be regarded as ‘‘thermodynamic spectroscopies.’’48,49
2.31.4.1 High-field Magnetization: The MST Method
Though mainly applied to powder samples, the MST method well serves to illustrate the principles underlying the high-field characterization of molecular magnetic systems. The MST technique was initially developed for investigating exchange-coupled metal pairs, triplets, quartets, etc. in dilute magnetic semiconductors and related materials.50–58 These paramagnetic clusters embedded in a nonmagnetic matrix bear direct structural and magnetic resemblance to molecular systems. The MST method can be applied to antiferromagnetically coupled clusters, in which Zeeman interaction of the spins with an external magnetic field is competitive with intramolecular exchange-coupling interactions. Hence, sufficiently strong magnetic fields induce the progressive decoupling of the spins, resulting in crossovers in the nature of the ground spin state. The simplest molecular compounds to which this approach has been applied are antiferro- magnetic dimers, such as [Fe2(salen)2Cl2]59 and [Fe2(C2O4)(acac)4].60 Here, isotropic intradimer exchange interactions are dominant and the spin levelsE(S,M) are well described by the formula
E S;ð Mị ẳ J
2S Sð ỵ 1ị ỵgBMB ð11ị
which follows directly from the spin Hamiltonian
HẳJS1S2ỵ gBSB ð12ị
for two exchange-coupled spins S1and S2in a magnetic fieldB(Bẳ|B|). In Equation (11), Sis the total-spin quantum number of the dinuclear species while Mẳ–S,Sỵ1,. . ., S1,S labels the total spin component along B. For two high-spin ferric ionsS1ẳS2ẳ5/2 and Sthus ranges from 0 to 5. In the presence of antiferromagnetic interactions (J>0) the ground state hasSẳ0 in zero applied field, as shown in Figure 12a. By contrast, in a strongmagnetic field theSẳ5 state must lie lowest, the high magnetic field limit being reached when the external field overcomes the AF interaction. By sweepingthe magnetic field, it is thus possible to observe the crossovers from Sẳ0 to 1, from 1 to 2, etc. as depicted inFigure 12b. These occur at evenly spaced magnetic field valuesBngiven by:
Bnẳn J
gB with nẳ1;2. . .5 ð13ị At each crossover the value of |M| in the ground state changes by one unit, so that the magnetization exhibits a sudden step-like increase at low temperature (kBTJ). Alternatively, when dM/dB is measured (as in pulsed-field experiments) each MST shows up as a peak in the dM/dB vs. B plot (Figure 12c). Because the position of the steps is directly related to the magnitude of the exchange constant through Equation (13), the MST method represents a useful alternative to the traditional approach based on the temperature variation of magnetic suscept- ibility for the determination ofJvalues, provided that the experiment is performed at sufficiently low temperature (typically below 1 K). In fields up to 55 T, [Fe2(salen)2Cl2] exhibits two peaks in the dM/dBvs.Bcurve,59whereas [Fe2(C2O4)(acac)4] shows all five predicted peaks because of its comparatively smaller couplingconstant (Figure 13).60The valuesJẳ16.8(4) cm1and 7.2(2) cm1 have been obtained for the two compounds, respectively, by reasonably assuminggẳ2.00 for the high-spin ferric ion.
The MST has allowed precise measurements of exchange-coupling interactions in higher- nuclearity spin clusters as well. In the manganese(II) cluster [Mn3(CH3CO2)6(bpy)2],61 featuring a linear spin topology, the crossover between the Sẳ5/2 and Sẳ7/2 states at about 16 T was used to determine the nearest-neighbor coupling constant, Jẳ4.4 cm1. This value is in very good agreement with that deduced from m vs. Tdata. In larger exchange-coupled systems, the MST method becomes an even more powerful tool because the magnetic fields required to
404 Magnetism: General Introduction
induce successive crossovers are often smaller than for dimers with comparable J values. For instance, in ring-like clusters comprising 2Nantiferromagnetically coupled spins Si, the energies of the first excited exchange multiplets (S) and the crossover fields are approximated by the expressions:62–67
Sẳ J
NS Sð ỵ 1ị ð14aị
Bn ẳ n 2J
NgB withnẳ1;2. . .2NSi ð14bị S
5
5
5
5
6
6 15
10 5 0
0
0
0 –5
–10 –15
4
4
4
4 3
3
3
3 2
2
2
2 1
1
1
1 0
B1 B2 B3 B4 B5
(a)
(c)
(b)
gàBB (J)
gàBB (J)
dM/dB(a.u.)
M/(NAgàB)E(J)
Figure 12 (a) Calculated spin levels for a dimer of antiferromagnetically coupled Siẳ5/2 spins in zero magnetic field. (b) Evolution of the spin levels in a magnetic fieldB. The crossover fieldsBnare marked by dashed vertical lines. (c) Molar magnetization (M) and differential susceptibility (dM/dB) of the dimer at low
temperature.
For a given J value, S and Bnare thus expected to decrease with increasing N, followingthe evolution of the ringtoward a chain-like gapless system. Pulsed-field experiments up to 52 T on the compounds [NaFe6(OCH3)12(dbm)6]Cl (Nẳ3,Jẳ20 cm1)62,68 and [Fe12(OCH3)24(dbm)12] (Nẳ6, Jẳ22 cm1)64,69 revealed three and four steps in the magnetization, respectively, at T<1 K. In the field range from 0 T to 42 T,nineMSTs have been observed in the decairon(III) cluster [Fe10(OCH3)20(CH2ClCO2)10] (Nẳ5,J10 cm1) due to the combined effects of a small couplingconstant and a large number of spins in the ring(Figure 14).63From the position of the MSTs approximate J values can be obtained by using Equation (14b), more accurate estimates requiringexact diagonalization of HeisenbergHamiltonian.
For a given spin topology the largest exchange constants which can be measured by the MST method are dictated by the maximum available magnetic field. On the other hand, very weak exchange interactions require very low temperatures to resolve the MSTs, so that the use of dilution refrigerators is often mandatory. Utilizing magnetic explosion generators70,71 magnetic fields of megagauss strength can now be achieved which allow the complete decoupling of the spins, yieldinga field-induced quasiferromagnetic state. An outstandingexperiment on the dodecanuclear manganese cluster [Mn12O12(CH3CO2)16(H2O)4]4H2O2CH3COOH, Mn12Ac, in fields up to 600 T has revealed a cascade of discrete quantum jumps of the magnetization at low temperature, reflectingthe stepwise modulation from the ground stateSẳ10 toSẳ22. The latter corresponds to the state of higher spin multiplicity for four manganese(IV), Siẳ3/2, and eight manganese(III), Siẳ2, ions of the cluster.72
The amount of information that can be extracted from high-field magnetization measurements on single-crystals is considerably larger than that obtained with powder measurements. For a given orientation of the magnetic field, the position of the MSTs is not only determined by the exchange constants, but also by anisotropic terms such as single-ion zfs and dipolar and aniso- tropic-exchange interactions.53,55,56 The resultingangular dependence of crossover fields is com- pletely lost in polycrystalline samples due to powder averaging. The results of angle-resolved measurements on single-crystals of Pb1xEuxS have been interpreted in terms of exchange
Fe Fe
1.5 K
(a)
(b)
0 10 20 30 40
dM/dB (a.u.)
B(T) o o
o
o o o
o o
o o o o
Figure 13 (a) Molecular structure of [Fe2(C2O4)(acac)4]. (b) Differential susceptibility measured in pulsed fields at 1.5 K (after Shapiraet al.).60
406 Magnetism: General Introduction
anisotropy between couples of europium(II) ions.53 Weak magnetic interactions can also be revealed by the width and shape of the MSTs.50 Under isothermal conditions, the expected full- width at half-maximum (B) of the peaks in dM/dBvs.B curves is given by
Bẳ3:5255kBT
gB ð15ị
Departures from Equation (15) observed on single-crystal samples of Cd1xMnxSe have been ascribed to Dzyaloshinski-Moriya (DM) interactions56 within noncentrosymmetric couples of manganese(II) ions. The DM interactions (Equation (16)) are indeed able to mix the spin wave functions nearBn, thus leadingto a roundingof the steps.
HDMẳd12S1S2 ð16ị
However, possible extrinsic sources of line broadening, such as strain effects and crystal mosaicity,73must be carefully ruled out before any interpretation of single-crystal measurements is attempted. Finally, it must be noted that the very fast sweepingrates typical of pulsed-field experiments may lead to thermal nonequilibrium conditions at the crossovers. In this case, the observed lineshape is also strongly influenced by magnetocaloric effects.59
0 1 1 2
2 3 3 4
4 5 5 6 6 7 7 8 8 9 0.6 K
(a)
(b)
0 10 20 30 40
dM/dB (a.u.)
B(T)
Figure 14 (a) Molecular structure of [Fe10(OCH3)20(CH2ClCO2)10]. Atom code: large hatched circlesẳFe;
small black circlesẳO; small empty circlesẳC; medium-size empty circlesẳCl. (b) Differential susceptibility measured in pulsed fields at 0.6 K (after Taftet al.).63
2.31.4.2 Torque Magnetometry 2.31.4.2.1 Basic definitions
TM represents a versatile macroscopic method for the characterization of anisotropic magnetic materials. Excellent reviews describe the earliest developments of TM, which was initially used to investigate paramagnetic anisotropy in simple transition-metal complexes.74,75 Prominent modern applications of TM include the characterization of High-Tc superconductors,76–78 organic conductors,79–83 magnetic nanowires,84,85 nanoparticles,84 thin films and multilayers,86–91 as well as chain molecular magnets92 and molecular magnetic clusters.47,49,93–101
Magnetic torque (t) is the mechanical couple actingon a magnetically anisotropic substance in a homogeneous magnetic field B. It originates from the noncollinearity of B and the sample magnetizationM:
tẳMB ð17ị
Without loss of generality, we can assume that M and B lie on the xz-plane of an orthogonal coordinate frame so that thetvector will be necessarily parallel to they-axis, with xẳ zẳ0 and
yẳ|t| (Figure 15). Labelingthe polar angles of Mand Bwith ’and , respectively,Equation (17) can be rewritten in scalar form as follows
yẳMBsinð’ị ð18ị
whereMẳ|M| andBẳ|B|. If the direction ofMwith respect to the crystal axes is known, as in permanently magnetized substances, ’ is simply fixed by the sample orientation and Equation (18) can be directly exploited to measure the magnetization. Notice that the torque signal is maximum for (–’) ẳ/2, 3/2, etc. while it vanishes at (–’)ẳ0,, etc., i.e., whenMis either parallel or antiparallel to B. However, TM is most useful to investigate magnetic anisotropy in paramagnetic materials, which feature a field induced magnetization.47For an anisotropic para- magnet with spinSand principal magnetic directions alongx,y, and z(Figure 15), two limiting regimes can be envisaged which lead to different torque behaviors. In the weak-field limit, y
is simply proportional to B2 and to the difference between the principal susceptibilities in the xz-plane,47,74,75as given by the formula
yẳB2ðzzxxịsincos ð19ị
By contrast, in the strong-field limit ysaturates and becomes field independent due to Zeeman quenchingof magnetic anisotropy. For axial anisotropy alongz and no g-anisotropy, the molecular torque follows the simple law
yẳ 2DS Sð 1=2ịsincos ð20ị
M B
τ y
z
x
ϕ θ
Figure 15 Reference frame used in the discussion of TM, with the definition of theand’angles.MandB lie on thexz-plane.
408 Magnetism: General Introduction
where D denotes the axial zfs parameter.47,98 Notice the angular modulation of the torque provided by the sin cos term in both Equations (19) and (20). Zero torque is expected whenever the magnetic field is applied along a principal direction, i.e., for ẳ0, /2, , 3/2, etc., while | y| is maximum for ẳ/4, 3/4, etc. Equation (19) combined with an average (powder) susceptibility measurement has been largely exploited for the determination of the principal magnetic susceptibilities from low-field torque data.74,75 On the other hand, Equation (20) constitutes the basis of the TST method, which will be described in the following section.
The complete field-dependence of yallows one to investigate anisotropy terms in much greater detail. For instance, it has been used to extract both second- and fourth-order anisotropy parameters of the ground Sẳ10 state in Mn12Ac (Figure 16a). When the magnetic field is applied close to the hard magnetic plane of the molecule, a bell-shaped torque curve is recovered (Figure 16b). The torque signal exhibits a quadratic field dependence in low fields, passes through a maximum at 6.2 T and follows an asymptotic behavior in high fields. The best-fit spin
τy(a.u.) τy (a.u.)
(2)
(1)
4.2K 0
0 2 4 6 8 10
5 10 15 20 25
B2(T2)
B(T)
(b) (a)
Figure 16 (a) Molecular structure of [Mn12O12(CH3CO2)16(H2O)4] viewed alongthe easy magnetic axis (S4).
Same atom code as inFigure 14a, with large hatched circlesẳMn. (b) Torque curves recorded at 4.2 K by applyingthe magnetic field atẳ89.2(line 1) and 88.9(line 2) from theS4-axis and best-fit calculated data with (solid lines) and without (dashed lines) fourth-order anisotropy terms. The inset shows theB2depend-
ence of yin low fields.
Hamiltonian parameters leadingto the solid lines in Figure 16bare in excellent agreement with those determined by spectroscopic techniques, such as inelastic neutron scatteringand high- frequency EPR.94 Notice the much worse fit obtained by neglecting fourth-order anisotropy terms (dashed lines).
We conclude this section by noticingthat the design of new torquemeters with increased performance has considerably expanded the potential of modern TM. Micromechanical torque- meters based on miniaturized cantilever84,88,102–105 or rotor106 devices are now available whose sensitivity is considerably higher than that of commercial SQUIDs. In addition, they are extremely easy to handle and particularly suitable to operate in high magnetic fields or in the restricted confines of3He/4He cryostats and dilution refrigerators. Schematic views of cantilever- and rotor- based torquemeters are shown in Figure 17. In both cases, the sample is firmly anchored to the mobile part of the torquemeter, which becomes sensitive to the torque component responsible for cantilever flexion or wheel rotation ( y). In the late 1990s, microfabricated cantilevers were realized which allow the detection oftwotorque components at the same time.103,104Detection of the torque signal can rely on capacitive, piezoresistive, piezoelectric, or optical methods, in either static or dynamic mode. Cantilever torquemeters with capacitive detection are in particularly widespread use due to their simple design and tunable sensitivity.102,105Their functional principle is extremely simple: the cantilever (Si or Cu/Be) is part of a parallel-plate condenser whose
Sample 2mm
x
x d
z
z y
y
Upper plate (cantilever)
(a)
(b) Lower plate
House (stator) C-electrodes Sample space Wheel (rotor) Torsion wire Hysol posts A&B-electrodes Annular ring
Figure 17 Schematic drawingof cantilever (a) and wheel (b) torquemeters. The size of the gap between the electrodes is exaggerated to show more clearly the electrode configuration (after Wiegerset al.).106
410 Magnetism: General Introduction
capacitance is a function of the plate-to-plate separation d (Figure 17a). The sensitivity of the device can be modulated by a proper design of the cantilever (thickness, leg width) or by adjusting the plate-to-plate separation. In a magnetic field, the torque acting on the sample leads to a small deflection of the cantilever which is detected as a change of capacity. For small displacements and a fully elastic response of the spring, the detected capacitance variation is simply proportional to the torque.
2.31.4.2.2 The TST method
In antiferromagnetic clusters, the torque method has been extensively used to detect ground spin state crossover (seeSection 2.31.4.1). When the two multiplets involved have different anisotro- pies, as is usually the case, the ground spin change results in an abrupt, step-like variation of the torque signal at low temperature. After each step, the torque signal reaches a plateau whose height is proportional to the D parameter of the ground state (Equation (20)). Consequently, torque vs.Bcurves show a staircase structure similar to that obtained in magnetization measure- ments. In this field, the TST method has proven to be superior to the MST technique because it can be applied to single-crystal samples with a mass of a few micrograms only. In the octachromium(III) ring[Cr8F8(ButCO2)16] (Figure 18) torque data at Tẳ0.4 K and Bẳ0–10 T
7,6 7,4 7,2 7,0 6,8 6,6
–20 0 20 40 60 80 100
τy(a.u.)
4K 0.4K
0 0
1 2 3 4 5 6 7 8 9 10
B (T) B1
B1(T)
(b) (a)
θ=45˚
θ(deg)
Figure 18 (a) Molecular structure of [Cr8F8(ButCO2)16]. Atom code: large hatched circlesẳCr; small black circlesẳO; small empty circlesẳC; medium-size empty circlesẳF. (b) Torque signal measured at 45 from the ringaxis at two temperatures (4 K and 0.4 K) The inset shows thedependence ofB1and the best-fit
curve calculated with the parameters1andD1given in the text (solid line).
have provided the angular variation of the crossover field for the Sẳ1 0 transition. In particular, B1 is larger (7.6 T) when the field is parallel to the tetragonal molecular axis as compared with the perpendicular orientation (6.8 T), which is indicative of a hard-axis type anisotropy. The smooth angular variation of the step position has been elegantly used to extract the singlet-triplet energy gap 1ẳ6.509(8) cm1and the triplet’s zero-field splittingD1ẳ1.59(3) cm1. The latter value compares extremely well with that obtained by high-frequency EPR (1.63 cm1),97 further showingthat high-field TM is competitive with spectroscopic techniques for the evaluation of spin Hamiltonian parameters. In addition, its applicability is far more general, since EPR resonances from excited states may be unobservable due to fast electronic relaxation, excessive dipolar broadening, or large zero-field splitting.47,49The detection of ground- state modulation by TM can be extended to successive crossovers in order to probe the energy and anisotropy of higher excited states. The angular dependence of B2has been used to investi- gate the Sẳ2 spin state in hexa- and octanuclear iron(III) rings.49,98 Alternatively, the DS
parameters of higher excited states can be evaluated from D1 and from the relative height of the torque plateaus usingEquation (20). This approach has been used to determine the zero-field splittingparameters from D2 to D5 in [Fe10(OCH3)20(CH2ClCO2)10] with measurements up to 23 T (Figure 19).99
In some cases, the sign of magnetic anisotropy can be abruptly reversed at the crossover fields.
The magnetic anisotropy of the 33 grid [Mn9(2POAP)6]6þ(Figure 20) changes from a easy-axis to hard-axis type at 7.5 T as the ground state switches fromSẳ5/2 to 7/2.100This result probably reflects the different projection of single-ion anisotropies on theSẳ5/2 and 7/2 states. The cluster can in fact be regarded as a central manganese(II), SCẳ5/2, ion plusan outer ring(R) of eight manganese(II) ions antiferromagnetically coupled to give SRẳ0, 1, 2, etc. The two spin subsets are weakly magnetically coupled affording the total spin states of the grid. Since the groundSẳ5/2 state originates from SRẳ0 and SCẳ5/2, its anisotropy is ruled by the central ion only. By contrast, the first-excitedSẳ7/2 state derives fromSRẳ1 and thus embeds anisotropic contribu- tions from the outer metal ions as well.