Relaxation Dynamics and Magnetic Anisotropy in a Low-Symmetry DyIII Complex

Article abstract of DOI:10.1002/chem.201505211

The magnetic behaviour of a Dy(LH)₃ complex (LH^- is the anion of 2-hydroxy-N'-[(E)-(2-hydroxy-3-methoxyphenyl)methylidene]benzhydrazide) was analysed in depth from both theoretical and experimental points of view. Cantilever torque m

Full text of DOI:10.1002/chem.201505211

DOI: 10.1002/chem.201505211
Full Paper
&
Single-Molecule Magnets |Hot Paper|
Relaxation Dynamics and Magnetic Anisotropy in a Low-
Symmetry DyIII Complex
Eva Lucaccini,[a] Matteo Briganti,[a, b] Mauro Perfetti,[a, c] Laure Vendier,[d] Jean-Pierre Costes,[d]
Federico Totti,*[a] Roberta Sessoli,[a] and Lorenzo Sorace*[a]
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Abstract: The magnetic behaviour of a Dy(LH)3 complex
(LHÀ is the anion of 2-hydroxy-N’-[(E)-(2-hydroxy-3-methoxy-
phenyl)methylidene]benzhydrazide) was analysed in depth
from both theoretical and experimental points of view. Can-
tilever torque magnetometry indicated that the complex has
Ising-type anisotropy, and provided two possible directions
for the easy axis of anisotropy due to the presence of two
magnetically non-equivalent molecules in the crystal. Ab
initio calculations confirmed the strong Ising-type anisotropy
and disentangled the two possible orientations. The com-
puted results obtained by using ab initio calculations were
then used to rationalise the composite dynamic behaviour
observed for both pure DyIII phase and YIII diluted phase,
which showed two different relaxation channels in zero and
non-zero static magnetic fields. In particular, we showed
that the relaxation behaviour at the higher temperature
range can be correctly reproduced by using a master matrix
approach, which suggests that Orbach relaxation is occur-
ring through a second excited doublet.
an MJ value that is as large as possible.[9] Although it has been
shown that complexes with low symmetries can also possess
axial eigenstates,[10–12] in these situations it is not possible to
predict a priori the composition of the ground state because
several different jMJi values can in principle contribute to it. It
has further been noted that the relaxation of the magnetisa-
tion can be due to different mechanisms.[13–15] For example, at
low temperature and for small values of applied magnetic
field, quantum tunnelling of magnetisation (QT) can be strong-
ly effective in accelerating the relaxation. In this regard, mole-
cules that contain Kramers’ ions are clearly preferred because
in the absence of further interactions QT is forbidden for semi-
integer spins. However, this process can be mediated by dipo-
lar and hyperfine interactions, so fast relaxation of the magnet-
isation is often observed under a zero field. Suppression of QT
is then achieved by diluting paramagnetic complexes within
an isostructural diamagnetic matrix, which reduces dipolar in-
teractions, and/or by applying a static magnetic field. However,
on increasing the magnetic field, the direct process gains
importance due to the larger number of available phonons of
correct energy and can become the dominant contribution.[16]
Finally, relaxation may occur through a Raman process that
gives a more marked temperature dependence of the magnet-
ic relaxation time (t) because an interaction with phonons
from the thermal bath and virtual energy states is involved.[17]
It is quite clear that to correctly describe the relaxation pro-
cesses in these systems, a detailed picture of the electronic
structure of the lanthanide ion and its relation to the molecular
structure is needed. An approach that combines spectroscopic
characterisation (electron paramagnetic resonance, lumines-
cence, inelastic neutron scattering) and ab initio theoretical
studies is now becoming the standard procedure for this
scope.[10,18–22] Single-crystal magnetic measurements are still
not very widespread,[23,24] even though they can cast light on
magnetic anisotropy by providing access to the preferred ori-
entations of magnetisation and to the degree of axiality of the
low-lying levels of the J multiplet.[25–27] However, most crystal
structures are comprised of symmetry-related but magnetically
non-equivalent sites. In these cases, single-crystal magnetome-
try provides an averaged response and deconvolution of the
different contributions to anisotropy is often impossible.[28,29]
Cantilever torque magnetometry (CTM), which only probes the
anisotropic part of the magnetisation, is less affected by this
Introduction
The discovery by Ishikawa and co-workers,[1] that a lanthanide
bisphthalocyaninato molecule shows slow relaxation of mag-
netisation at low temperature, has ignited considerable inter-
est in the dynamics of the magnetisation of lanthanide-based
complexes.[2,3] Following that seminal report, these systems (re-
ferred to as single-ion magnets) have been proposed as candi-
dates for applications in high-density data storage, molecular
spintronics and quantum information processing.[4–7] The origin
of the peculiar magnetic behaviour of lanthanide ions lies in
their strong magnetic anisotropy, which stems from the com-
bined action of the spin-orbit coupling and the crystal field
(CF) induced by the donor atoms of the ligand(s), and by the
large total angular momentum J (for the second half of the 4f
series).[8]
When the two states with the largest projection of J are the
ground state, this results in an anisotropy energy barrier for
the reversal of the magnetisation. In the absence of other
efficient relaxation paths, it is then possible to observe slow
relaxation of the magnetisation through an Orbach process,
with the thermal dependence of the relaxation rate following
an Arrhenius-like behaviour. This result requires both a highly
symmetric axial disposition of the ligands around the lantha-
nide centre, which reduces the mixing between states with dif-
ferent MJ values, and a resulting ground state characterised by
[a] E. Lucaccini, M. Briganti, Dr. M. Perfetti, Dr. F. Totti, Prof. Dr. R. Sessoli,
Prof. Dr. L. Sorace
Dipartimento di Chimica “U.Schiff” and UdR INSTM
Università degli Studi di Firenze
Via della Lastruccia 3-13, 50019 Sesto Fiorentino (Italy)
E-mail: federico.totti@unifi.it
lorenzo.sorace@unifi.it
[b] M. Briganti
Current address: Instituto de Química, Universidade Federal Fluminense
Outeiro de S¼o Jo¼o Batista, 24020-141 Niterói, Rio de Janeiro (Brazil)
[c] Dr. M. Perfetti
Current address: Institut für Physikalische Chemie
Universität Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart (Germany)
[d] Dr. L. Vendier, Dr. J.-P. Costes
Laboratoire de Chimie de Coordination CNRS
205 Route de Narbonne, Toulouse F-31077 (France)
Supporting information for this article can be found under http://
dx.doi.org/10.1002/chem.201505211.
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problem.[29] Furthermore, its high sensitivity and ability to mea-
sure the magnetic anisotropy up to relatively high tempera-
tures allow one to get an independent estimate of the gap be-
tween the ground and first excited doublet in lanthanide com-
plexes.[29,30] The outcome of this experimental characterisation
can then be used to validate the results of ab initio calcula-
tions that, in their turn, help to unravel the knot and to avoid
misinterpretation of the observed dynamic behaviour by pro-
viding information on the composition of the eigenstates and
the corresponding energy gaps. Following this strategy, we
present herein a complete experimental characterisation, ob-
tained by using X-ray diffractometry, electron paramagnetic
resonance (EPR) spectroscopy, single-crystal CTM and ac and
dc susceptibility, alongside a theoretical analysis based on ab
initio methods, of a new mononuclear DyIII complex that be-
haves as a single-ion magnet in zero field. The complex has
been synthesised by using a potentially pentadentate ligand;
however, only three of its binding sites are used to bind to the
DyIII ion, which results in a neutral molecule that is, in principle,
sublimable.[31]
Figure 1. View of the asymmetric unit of Dy, including one molecule of the
complex and one DMF molecule. The dashed lines indicate the three intra-
molecular hydrogen bonds that stabilise the structure. The remaining hydro-
gen atoms are not shown for the sake of clarity. Ellipsoids are drawn at the
40% probability level, with the exception of the three hydrogen atoms, for
which a ball-and-stick representation is used.
(Figure 1) is composed of one mononuclear [Dy(LH)3] neutral
molecule and one DMF molecule. The DyIII ion is surrounded
by a N6O3 coordination environment provided by three tris-
chelating LHÀ ligands. The hydrazide function is deprotonated
whereas the phenol function remains protonated and not in-
volved in coordination with the DyIII ion. Each ligand remains
essentially planar, with the exception of the phenolic residues,
due to the presence of an intramolecular hydrogen bond that
involves the non-deprotonated phenol function and the hydra-
zide nitrogen atom of each ligand. The DyÀN(pyridine) bond
lengths (2.574(3)–2.696(4) Š) are slightly longer than the DyÀ
N(hydrazide) bond lengths (2.496(3)–2.538(4) Š) and the short-
er bonds that involve the DyÀO of the hydrazone part of the
ligand (2.322(3)–2.368(3) Š; selected bonds and angles are re-
ported in Table S1 in the Supporting Information). Analysis of
the nine-coordinate Dy-centred N6O3 polyhedron by using the
SHAPE program[33] (see Table S2 in the Supporting Information)
suggests that the coordination polyhedron is intermediate be-
tween the different possible choices for a nine-coordinate ar-
rangement, with a slight preference for a spherical capped-
square antiprism. For this option, the capping atom is the N8
hydrazide nitrogen and the two square faces of the antiprism
have at their vertices the N9-N3-O5-N6 and O1-N2-O3-N5
atoms (Figure S1 in the Supporting Information). The packing
of the molecules in the lattice is such that the closest DyÀDy
intermolecular separation is 10.73 Š. Phenoxo- rings on mole-
cules related by the inversion centre are parallel; however, the
corresponding centroids are separated by 6.75 Š so the
corresponding p–p stacking interactions are expected to be
very weak.[34]
Results and Discussion
Synthesis and structure determination
The
2-hydroxy-N’-[(E)-(2-hydroxy-3-methoxyphenyl)methyli-
dene]benzhydrazide ligand, LH2 (Scheme 1), was obtained by
Scheme 1. LH2 ligand with the numbering scheme used in the NMR
spectroscopy data (1H and 13C).
treating 2-hydroxybenzohydrazide with pyridine carboxalde-
hyde, in accordance with the procedure reported in the Experi-
mental Section. This is a potentially pentadentate ligand that
possesses two functional groups that can be deprotonated; its
coordination chemistry has, until now, only been investigated
with regard to nickel or cobalt complexes.[32] Its complete char-
acterisation, including assignment of 1D and 2D 1H and
13C NMR spectra and chemical analysis are also reported in the
Experimental Section, and partially differ from those previously
reported.[32] When dissolved in dimethylformamide (DMF), this
LH2 ligand reacts with Dy(NO3)3 in the presence of piperidine
to give a yellow complex, which from analytical data and X-ray
analysis (see below) was found to be neutral, with the DyIII ion
being coordinated to three monodeprotonated ligands.
Static magnetic properties
The molecular structure obtained by using single-crystal X-
ray diffractometry indicates that Dy(LH)3·DMF (Dy) crystallises
in the monoclinic P21=n space group. Its asymmetric unit
The static magnetic properties of a microcrystalline powder
sample of Dy were investigated by using dc measurements
and the behaviour of the magnetisation was studied both as
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a function of field (H) and temperature (T). The cT versus T
curve is reported in Figure S2 in the Supporting Information;
the room-temperature experimental value (cT =13.97 emuK
molÀ1) is consistent with the free ion expectation for DyIII
( 6H15=2, gJ ¼ 4=3, cT =14.17 emuKmolÀ1). A smooth decrease
was observed on lowering the temperature, which is attributed
thogonal crystallographic reference frame in a monoclinic
*
space group is abc and rotations in two orthogonal planes
have been performed: in the first one (Rot1) the rotation is
performed around b, whereas in the second one (Rot2) the ro-
*
tation axis lies in the ac plane (see Figure S4 in the Support-
ing Information for details of the reference frames and rotation
geometries). Interestingly, in the first rotation all molecules in
the unit cell provide the same contribution, being related by
the twofold axis that lies along b, whereas two families of dif-
ferently oriented molecules are detected in the second rota-
tion. We measured the torque signal at T=5 and 10 K with B
ranging from 30 to 70 kG. The resulting torque data were
fitted by modelling the electronic structure of the Dy complex
by using an effective spin-Hamiltonian:
6
to the depopulation of excited levels of the H15=2 multiplet,
split by the CF. The magnetisation versus field was measured
at 2 and 4 K and is reported in Figure S3 in the Supporting In-
formation. The saturation value was 5 mB, as already observed
for other molecular complexes that contain DyIII.[35,36]
Study of the magnetic anisotropy
EPR spectroscopy is widely used to investigate the magnetic
anisotropy of Kramers’ systems, however, in our case the
complex was revealed to be silent. This can, in principle, be
attributed to either fast relaxation or a low intra-doublet
transition probability due to the ground-state composition,
whereas the excited doublets are too high in energy to ob-
serve inter-doublet transitions. The latter explanation points to
a large axiality of the ground state, which was then investigat-
ed by using CTM. This technique exploits the magnetic torque
of a molecule immersed in a homogenous magnetic field and
has already proven to be extremely useful to determine the
anisotropic features of lanthanide-based single-molecule mag-
nets.[29,30,37] In this experimental setup, the variation in the ca-
pacitance of the capacitor due to the deflection of the cantile-
ver is proportional to the magnetic torque experienced by the
Hs ¼ mBSeff Á g Á B
ð2Þ
in which g is the g tensor, diagonal in the molecular reference
frame, mB is the Bohr magneton and Seff is an effective S ¼ 1=2
spin operator. To correctly reproduce the torque shape, three
further parameters have to be taken into account: a scale
factor (Fsc, which must be the same for all rotations) and two
Euler angles, namely 1 and x (by using the extrinsic x-conven-
tion), which describe the orientation of the molecular reference
frame with respect to the crystallographic one. Indeed, in the
case of uniaxial anisotropy the third Euler angle is irrelevant. In
Figure 2 we report two torque curves measured at the same
applied magnetic field and temperature for the two rotations
with corresponding fits (the complete set of experimental data
and the corresponding fits are reported in Figure S5 in the
Supporting Information). The fitting procedure gave the
following best fit parameters: gx ¼ gy ¼ 0:010ð5Þ, gz ¼ 16ð1Þ,
1 ¼ 212ð1Þꢁ, x ¼ 71ð1Þꢁ and Fsc ¼ 2:6ð3Þ Â 10À2, which indi-
cates the strong Ising character of the ground doublet. The
resulting director cosines of the easy axis (z in the molecular
sample. This torque
tween the magnetisation (M) and the magnetic field (B). Exper-
imentally, only the component of the torque that lies on
is defined as the vector product be-
y
the rotation axis (chosen as the y axis of the XYZ laboratory
reference frame) is detectable, thus we can write Equation (1):
¼ MZBX À MX BZ ¼ B2sinfcosfðcZZ À cXX Þ
*
reference frame), with respect to abc are cosa1 ¼ À0:501,
ð1Þ
Y
cosa2 ¼ 0:801, cosa3 ¼ 0:326 (a1 ¼ 120ꢁ, a2 ¼ 37ꢁ, a3 ¼718).
in which an additional B2ð2sin2f À 1ÞcXZ term was omitted
because it is always possible to set it to zero by using a proper
shift in the rotation. In Equation (1), which is only valid for
gmBSeffB ꢀ kBT, cIJ are the components of the magnetic sus-
ceptibility tensor and f is the angle between the magnetic
field B and the projection of the easy axis (z axis in the xyz mo-
lecular reference frame) on the rotation plane. This should not
be confused with the rotation angle (the angle between the
cantilever plane and the magnetic-field direction), hereafter de-
noted as q. If B is increased enough, Equation (1) breaks down
and the torque curves assume a different shape if B is parallel
or perpendicular to the easy direction of anisotropy, which
makes it possible to disentangle non-collinear contributions. To
reduce the effect of intermolecular interactions and demagnet-
ising fields due to shape anisotropy, we chose to use a sample
of the isostructural YIII derivative doped with approximately
10% DyIII, hereafter denoted YDy. Because the compound crys-
tallises in the monoclinic space group P21=n and the molecules
do not sit on any symmetry element, the crystal anisotropy
will be in general different from the molecular one. The or-
Due to the presence of two magnetically non-equivalent mole-
Figure 2. Torque curves for the two rotations in a magnetic field of 70 kG
and at 5 K. The solid lines represent the best fit of the magnetic torque by
using the parameters reported in the text; the symbols are the experimental
data for Rot1 (!) and Rot2 ( ).
*
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cules in the unit cell, these direction cosines can identify two
possible orientations for the magnetic anisotropy axis with re-
spect to the molecular structure, see Figure 3a. It is interesting
to note that neither of the two possibilities correlate to the
highest symmetry axis (C4 or C3) of the best fit coordination
polyhedra, capped-square antiprism or tricapped-trigonal
prism (see Figure S1 in the Supporting Information). This is in
agreement with the fact that the idealised structural geometry
does not take into account the heterolepticity of the complex,
which results in a CF symmetry completely different from the
structural one. A simple electrostatic model[38] has been used
to calculate the orientation of the easy axis of magnetisation:
the angle between experimental (drawn as a black rod in Fig-
ure 3a) and the calculated direction resulted in a value of
about 138 (see Tables S3–S4 and Figure S7 in the Supporting
Information). Because the error of a visual crystal alignment
can be estimated to be about 58 (see the Experimental Sec-
tion), we can consider the calculated orientation to be in fair
agreement with the experimental one. This confirms that the
coordination environment of DyIII in the complex can provide
a strong Ising character, but an accurate prediction of the ori-
entation of the anisotropy axis, as well as an estimation of the
low-symmetry components of the CF, requires a more detailed
approach.
Ab initio calculations
Since we had indications of the strong Ising character of the
complex, we chose to perform an accurate ab initio characteri-
sation to get more detailed insights into the DyIII electronic
structure. A first relevant test of the correct reproduction of ex-
perimental properties concerns calculation of the orientation
and magnitude of the magnetic anisotropy compared with the
results of CTM. In agreement with the latter, the calculated
ground Kramers’ doublet shows an almost-pure Ising character
with a principal value of gz ¼ 19:8 (see Table 1) and a contribu-
tion from MJ ¼ Æ15=2 only.
Table 1. Results of the calculations by using the RCC basis sets for Dy:
6
energy splitting of the ground H15=2 multiplet and principal values of
the corresponding g tensor for each Kramers’ doublet. See Table S6 in the
Supporting Information for the calculated splitting of the first excited
multiplet (6H13/2).
Energy level
[cmÀ1
]
gx
gy
gz
E0
E1
E2
E3
E4
E5
E6
E7
0
195
237
289
324
371
430
478
0.0
0.5
0.1
1.5
0.6
2.6
3.4
1.1
0.0
1.8
2.2
3.7
2.8
4.5
4.1
4.7
19.8
14.9
14.3
12.1
13.8
8.8
6H15/2
7.4
15.9
The computed easy axis orientation for the ground doublet
(Figure 3b) lies approximately on the line that connects two
carbonyl groups of two LHÀ ligands (cosa1 ¼ À0:531454,
cosa2 ¼ 0:788731, cosa3 ¼ 0:308967). It is evident that the cal-
culated direction is almost coincident with one of the two
possible choices provided by the CTM analysis (see above and
Figure 3a), with an angle of 2.98 formed by the calculated and
experimental orientations, well below the estimated experi-
mental uncertainty (58). Interestingly, the quality of the fit of
the torque data is not very sensitive to the actual magnitude
of the gz value, as long as the system has strong Ising charac-
ter (i.e., gz > 15, gx; gy < 1), whereas it depends strongly on
the Euler angles. As a consequence, if the torque fit is per-
formed with the g values fixed to the computed ones
(gz ꢂ 20), the Euler angles are not affected (see Figure S6 in
the Supporting Information). The calculated ab initio anisotrop-
ic properties are then perfectly consistent with the results of
the CTM investigation.
Figure 3. a) The two possible directions of the easy axis of magnetic aniso-
tropy of Dy as determined by using CTM: the one identified by the black
rod is substantially coincident with the calculated direction (compare with
Figure 3b). b) The computed direction of the magnetic easy axis for the first
three energy doublets: ground (black), first excited (grey), second excited
(light grey).
With this proof of the reliability of ab initio calculations, we
used the set of calculated Stevens’ spin-Hamiltonian parame-
ters[39] (see Table S5 in the Supporting Information) that pertain
to the ground J ¼ 15=2 state to simulate the static magnetic
properties of the complex by using the home-developed soft-
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ware EVALUCF.[13] In particular, the correct simulation of the M
versus H curves at low temperature confirms that the ground
doublet properties are well reproduced by ab initio calcula-
tions, whereas an indication of the correct evaluation of
energy splitting and eigenstates of the ground J ¼ 15=2 state
can be provided by the simulation of the cT versus T curve.
For this quantity, the agreement between experimental and
calculated curves (see Figure S2 in the Supporting Informa-
tion), except for a small scaling factor within experimental
error (<5%), lends further support to the electronic structure
obtained by using ab initio calculations. This is reported in
term of computed energy splitting[39] between the Kramers’
doublets and their composition (jMJi contributions larger than
0.1) in Figure S8 in the Supporting Information. The calculated
energy separations between ground doublet and the first and
second excited doublets are 195 and 237 cmÀ1, respectively.
The first excited doublet shows a prominent contribution from
the jMJi ¼ Æ13=2 components with only jMJi ¼ Æ11=2 and
jMJi ¼ Æ7=2 ones as minor contributions, which are neverthe-
less enough to induce appreciable deviations from the axiality
of the g tensor. A much larger mixing between different jMJi
components is observed for the third excited doublet, whereas
for the fifth excited doublet the axiality is completely lost.
It is to be stressed that in addition to the increased rhombic-
ity, the non-collinearity of the easy axis with respect to that of
the ground state also increases with the energy of the
doublets (see Table S7 in the Supporting Information and Fig-
ure 3b). Indeed, the easy axis of the ground doublet forms an
angle of 68 with the easy axis of the first excited doublet and
an angle of about 608 with that of the second excited doublet.
In this framework, the calculations suggest that if magnetic re-
laxation occurs only through an Orbach two-phonon mecha-
nism, this should likely involve the second excited state. In
such a case the energy barrier to be overcome would be of
the order of 330 K, which indicates the possible observation of
an overall slow relaxation rate at relatively high temperatures.
Figure 4. Imaginary susceptibility of Dy at 10 K as a function of the applied
field.
in zero and applied field were fitted according to a Debye
model[44,45] and the corresponding relaxation times are report-
ed in Figure 5a as Arrhenius plots. We note that the relaxation
rate of the channel that dominates in zero field is almost tem-
perature-independent up to 12 K. Above this temperature, it
starts to increase with temperature similarly to the behaviour
observed for the relaxation channel activated with an applied
Dynamic magnetic properties
To gain more insights into this important point, the dynamic
magnetic properties of Dy were investigated by performing ac
magnetic susceptibility experiments in a broad range of
frequency (0.02–10000 Hz), temperature and dc applied field
(see Figures S9–S12 in the Supporting Information). This inves-
tigation revealed a composite dynamic magnetic behaviour
between 2 and 20 K. The pure complex showed two different
relaxation channels in zero and non-zero magnetic static fields;
the application of an external field allows the suppression of
one channel and the activation of the other (see Figure 4).
Further, field-dependent measurements indicated that the
slow relaxation process reaches a minimum rate for Hdc =
1000 Oe, whereas at higher fields the rate starts to increase
again. This led us to choose this field for the measurements
under an
Figure 5. a) Relaxation times vs. TÀ1 for Dy, in zero ( ) and applied static
*
*
field ( ). The relaxation in zero field is almost temperature independent up
to 12 K. The grey dotted line represents the best fit curve with D=270 K.
b) Relaxation times vs. TÀ1 for YDy in zero ( ) and applied static field (3).
The dotted lines represent the fit of the relaxation time; see Equation (3)
with D=270 K, the solid line is the relaxation time simulated by using the
master matrix equation.
~
external field. This behaviour is reminiscent of that previously
reported by some of our group for the Dy(DOTA) derivative
and other SMMs.[35,40–43] The imaginary susceptibility curves c00
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magnetic field. This trend is present up to 20 K, at which they
become too fast to be measured. Conversely, the rate ob-
served for the relaxation channel activated by applying a mag-
netic field is clearly temperature-dependent even at lower tem-
peratures (see Figure 5a and Figures S11 and S12 in the Sup-
porting Information). As a whole, this behaviour suggests that
the dominant process in zero field is directly related to the QT
of magnetisation, whereas the slower process, which occurs in
an applied field, is dominated by thermally activated relaxa-
tion. The curved nature of the Arrhenius plot for the relaxation
time in a field, however, clearly points to a combination of
processes that contribute to the relaxation, which include the
persistence of a temperature-independent process at the
lowest temperatures.[13]
and QT dominate the relaxation, the Orbach process is active
in promoting the relaxation in the high-temperature regime. It
is, however, not completely clear whether the latter process
occurs via the first or the second excited doublet. For the ob-
tained t0 values, it is known that, for an Orbach process, t0
should be ð10À3 Ä 10À5ÞDÀ3 (D expressed in K);[49] substitution
of the two different values for D gives a value of t0 in line with
the findings. Moreover, the differences in t0 values are qualita-
tively consistent with expectations, t0 being smaller in the case
of larger D.
Quantitative rationalisation of magnetisation dynamics
To clarify whether the intermolecular (dipolar) interactions
between magnetic centres play any role in this behaviour, we
have repeated the investigation of the dynamics on the mag-
netically diluted YDy sample (DyIII concentration of 10.5%; see
Figure S3 in the Supporting Information). Its cT versus T curve
is reported in Figure S2 in the Supporting Information, and
shows the same temperature behaviour as the pure com-
pound. Even in the diluted sample, two different relaxation
channels were found to contribute to relaxation in zero and
applied fields (see Figures S13–S16 in the Supporting Informa-
tion). However, for YDy the relaxation of magnetisation is char-
acterised by a much longer relaxation time t, which reaches
a few seconds at 4.5 K. Accordingly, at 2 K the diluted sample
exhibited a butterfly-shaped hysteresis (see Figure S17 in the
Supporting Information) that is not visible above 4 K. The ob-
served temperature dependence of the relaxation rate of YDy
for both relaxation channels was reproduced by including
three contributions; Raman and Orbach processes and a
temperature-independent process:[19,46–48]
To obtain some more hints on the mechanism of the single-ion
relaxation, transition moments between the states were com-
puted[9,50] (see Figure 6). Based on these transition moments,
no efficient QT relaxation is expected for the ground Kramers’
doublet. The underestimation of the probability of QT relaxa-
tion by transition moments with respect to the experimental
results is clearly due to the fact that this method takes into ac-
count only purely electronic, single-molecular properties,
whereas zero-field QT in Kramers’ systems needs residual dipo-
lar interactions and hyperfine coupling to occur (in the present
case the magnetic nuclei are Dy (161Dy, rel. ab.=18.9% and
163Dy rel. ab.=24.9%, both with I ¼ 5=2).[51] On the basis of
the ab initio results, a thermally assisted QT is likely to occur al-
ready for the first excited state. However, the magnetic
moment matrix element computed for a quantum tunnelling
mechanism between the two components of the first excited
Kramers’ doublet (j1; Æ i) suggests that this is not the most
likely process to occur. Indeed, from the first excited state both
Orbach and thermally assisted QT processes are more proba-
ble; the former provides access to the second excited state
whereas the latter would allow a reversal of the magnetisation.
Based on the above considerations, a relaxation via the second
excited state (E2 =237 cmÀ1) seems to be more likely, in fairly
good agreement with the phenomenological value of the
tÀ1 ¼ CTn þ tÀ0 1expðÀD=TÞ þ B
ð3Þ
We performed a simultaneous fit of the relaxation time in
a field and in zero field (the corresponding parameters are in-
dicated by f and zf, respectively, below), by using the same set
of parameters for the Orbach process (t0 and D), whereas the
Raman contribution was left free to vary for the two situations.
This was intended to account for the possible contribution of
a direct process in the case of in-field measurements, which
may affect the best-fit value obtained for the Raman relaxa-
tion. A first attempt was made by fixing the value for the
energy barrier D at 270 K to be approximately 195 cmÀ1, which
is the energy of the first excited doublet computed by ab
initio calculations. With this pre-condition, we obtained the fol-
lowing values for the best-fit parameters: t0 ¼ 1:3ð2Þ Â 10À10 s,
above-reported value of the energy barrier of 230 cmÀ1
.
The use of transition moments to evaluate potential relaxa-
tion paths still provides only semi-quantitative indications,[14,52]
despite being increasingly used to rationalise the spin dynam-
ics. A more directly quantitative reproduction of the observed
dynamics, by using the electronic structure derived by ab initio
calculations, can be obtained by using a master-matrix-based
approach.[45] This approach assumes a series of steps of the
direct-process type promoted by a suitable spin phonon cou-
pling Hamiltonian, assumed here to be of the Villain type[45] for
the sake of simplicity. Indeed, in the low symmetry of our
system the dynamic spin phonon coupling Hamiltonian
proposed by Abragam and Bleaney[49] would require the cal-
culation of a huge number of CF parameters corresponding to
different distortions making it practically unfeasible.[53] Not-
withstanding its simplicity, our approach allows us to extract
the relaxation time by calculating the relaxation rate gpq from
a state jqi (eigenstate of the Hamiltonian that describes the
system) to another state jpi:
nf ¼ 6:19ð4Þ,
Cf ¼ 5:0ð5Þ Â 10À5
s
À1KÀn
,
Bf ¼ 0
(fixed),
In a
nzf ¼ 5:5ð2Þ, Czf ¼ 6ð4Þ Â 10À4
s
À1KÀn
,
Bzf ¼ 53ð2Þ sÀ1
.
second step, the D parameter was left free to vary, to give the
following best-fit values: t0 ¼ 1ð2Þ Â 10À11 s, D ¼ 318ð44ÞK
(ꢂ230 cmÀ1), nf ¼ 6:22ð4Þ, Cf ¼ 4:7ð5Þ Â 10À5
s
À1KÀn, Bf ¼ 0
(fixed), nzf ¼ 5:6ð2Þ, Czf ¼ 5ð3Þ Â 10À4sÀ1KÀn, Bzf ¼ 53ð2Þ sÀ1
(see also Figure S18 in the Supporting Information). It is clear
from these results that, although at low temperature Raman
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Figure 6. The ab initio computed magnetisation blocking barrier for Dy. The thick black horizontal lines indicate the Kramers’ doublets as a function of the
projection of the magnetic moment on the chosen quantisation axis (that of the ground multiplet). The light-grey arrows show the possible pathways of the
Orbach process. The dotted black arrows represent the presence of (thermally assisted) quantum tunnelling between the connecting states. The numbers
reported for each arrow are the mean absolute value for the corresponding matrix element of the transition magnetic moment. The thick dotted black lines
show the most probable relaxation pathways for the reversal of magnetisation.
À
Á
3
It is evident that although this approach reproduces the
linear high-temperature region above 15 K, which is consistent
with an energy barrier of about 230 K, it overestimates the re-
laxation time at lower temperature. This is a common feature
for many lanthanide-based molecular magnets and is usually
attributed to the higher effectiveness of the Raman process at
lower temperatures. This process is, however, quite elusive be-
cause the corresponding parameters are usually considered to
be phenomenological.[13,19,46,47,54–56] Indeed, to the best of our
knowledge, no available reports relate the values of the Raman
parameters (C and n) obtained by fitting the dynamic data to
the structure of the investigated molecule. This is a severe
drawback in the search for increased relaxation times for po-
tential applications because this process, which provides
a channel for relatively fast relaxation even at quite low tem-
perature, is currently beyond our control. However, we stress
that, despite the phenomenological fit of the temperature de-
pendence of the relaxation rate that indicates the existence of
a Tn contribution, we cannot be sure that this is associated
with a real Raman process. Indeed, interactions that are not
taken into account by the model, such as hyperfine and residu-
al dipolar intermolecular interactions, may open the possibility
of relaxation through quantum tunnelling and might also
change the expected field- and temperature-dependence of
direct processes.[57]
Ep À Eq
3v
gpq ¼
Á
Á
 ÁÃ
pꢀh4mc5s
exp b Ep À Eq À 1
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
Â
Ã
f D 2 Á hpjJþjqi
þ
hpjJÀjqi
þ
ꢀ
ꢀ
2
2
ð4Þ
2
2
~
ꢀ
ꢀ
ꢀ
a
ꢀ
ꢀ
2
ꢀ
ꢀ
ꢀ
ꢀ
2
2
~
D
Á ½jhpjfJ ; Jzgjqij þ jhpjfJ ; Jzgjqij Šg
b
þ
À
in which b is 1=kBT, v and m are the volume and the mass of
~
~
the unit cell, Da and Db are the spin-phonon coupling parame-
ters. If the energy levels and the eigenstates are known, the
only parameters that need to be adjusted are the spin-phonon
À
Á
coupling parameters and v= mc5s . In our case, the CF eigen-
functions of the ground J multiplet and the corresponding ei-
genvalues in zero applied field obtained by using the ab initio
calculations were used to obtain the CF matrix as RT VR (in
which R is the eigenfunctions matrix and V is the diagonal
matrix of the corresponding eigenvalues). The complete Hamil-
tonian (CF+static magnetic field) was then obtained by
adding the Zeeman interactions in the jMJi basis. This provid-
ed, after diagonalisation, new eigenvalues and eigenfunctions
by which it was possible to calculate the master matrix (G) for
all the investigated temperatures. Diagonalisation of the
master matrix allowed us to extract the relaxation time as
t ¼ À1=l1, in which l1 is the first non-vanishing eigenvalue of
the master matrix. The result, obtained by adjusting both the
spin-phonon coupling parameter to 0.05 and the pre-factor
ð3v=pꢀh4mc5s Þ to 3000, is shown in Figure 5b.
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tion of a bulky white precipitate that was filtered off after cooling,
washed with MeOH and diethyl ether and dried (yield: 2.7 g, 95%).
1H NMR (400 MHz, [D6]-DMSO): d=6.99 (t+d, J=7 Hz, 2H; CH-3+
CH-5), 7.45 (t+d, J=7.5 Hz, 5H; CH-4+CH-4’), 7.89 (t+d, J=
7.5 Hz, 2H; CH-5’+CH-6’), 8.00 (d, J=7 Hz,1H; CH-6), 8.49 (s,
1H;HC- -N), 8.64 (d, J=7 Hz, 1H;CH-3’), 11.71 (s, 1H; NH), 12.1 ppm
(s, 1H; OH); 13C NMR (100.63 MHz, [D6]-DMSO): d=116.65 (s, ArC-1),
117.71 (s, ArC-3), 119.44 (s, ArC-5), 120.53 (s, ArC-6’), 125.01 (s, ArC-
4’), 129.14 (s, ArC-6), 134.36 (s, ArC-4), 137.37 (s, ArC-5’), 120.53 (s,
ArC-6’), 149.16 (s, ArC-3’), 150.02 (s, NC- -N), 153.54 (s, ArC-6’),
165.44 ppm (s, OCNH); elemental analysis calcd (%) for C13H11N3O2
(241.2 gmolÀ1): C 64.7, H 4.6, N 17.4; found: C 64.4, H 4.5, N 17.2.
Conclusion
We have presented a comprehensive experimental and theo-
retical investigation of the anisotropy and dynamic behaviour
of a novel mononuclear lanthanide-based single-molecule
magnet. We showed that detailed cantilever torque magneto-
metry, which can be used more generally in terms of crystal
symmetry and size than single-crystal magnetometry, can pro-
vide independent confirmation of the results of ab initio calcu-
lations in the absence of further spectroscopic information.
This is particularly relevant for systems that are EPR silent and
for which no detailed luminescent data are available. In turn,
this allowed us to analyse the observed dynamics of the mag-
netisation on the basis of the calculated electronic structure of
the lanthanide centre. For the studied complex, the experi-
mental and theoretical results indicate a strong axiality of both
the ground doublet and the first excited state; the ab initio
prediction of an almost complete collinearity of the ground
and first excited doublet is mirrored by the low-temperature
slow relaxation of the magnetisation of the complex, which
could be phenomenologically modelled by using a combination
of an Orbach and a Raman process. The observed behaviour
could be qualitatively rationalised by using the commonly
used transition probabilities provided by the ab initio suite.[9,50]
In addition to this, we showed that the relaxation behaviour at
the higher temperature range can be correctly reproduced by
using the ab initio computed electronic structure in a statistical
analysis based on the master matrix approach. Conversely, fur-
ther processes are clearly contributing at low temperature, re-
sulting in an experimental relaxation rate that is much faster
than predicted by this approach. This might be due either to
a true Raman process or to the unaccounted-for hyperfine and
dipolar intermolecular interactions, the latter reduced but not
completely quenched by the doping level used herein. As
a whole, these results outline the necessity of a virtuous inter-
play between detailed single-crystal studies and ab initio calcu-
lations. This process allowed us to obtain a detailed under-
standing of the relation between the electronic structure and
the rich low-temperature magnetisation dynamics in this
system, a point of crucial importance for rationally improving
the properties of lanthanide-based single-molecule magnets.
Dy(LH)3 (Dy)
Addition of piperidine (0.17 g,2.0”10À3 mol) to a stirred solution of
the above ligand (0.27 g, 1.0”10À3 mol) and Dy(NO3)3·5H2O (0.43 g,
1.0”10À3 mol) in DMF (10 mL) induced the appearance of a more
intense yellow solution. The solution was filtered off and set aside.
The crystals that appeared after 12 d were isolated by filtration
and dried (yield: 0.16 g, 50.5%). IR (ATR): u˜ =3448l, 3065 w, 2991
w, 2925 w, 2875 w, 2792 w, 2705 w, 2610 w, 1681 w, 1667 m, 1599
m, 1586 m, 1561 m, 1520 m, 1488 m, 1474 m, 1452 m, 1417 w,
1357 s, 1346 s, 1302 m,1252 m, 1230 w, 1147 m, 1090 w, 1065 m,
1008 w, 926 w, 866 w, 830 w, 760 m, 740 w, 702 w, 689 w, 658 w,
632 cmÀ1 w; elemental analysis calcd (%) for C42H37DyN10O7
(956.32 gmolÀ1): C 52.75, H 3.90, N 14.65; found: C 52.45, H 3.78, N
14.43.
YDy(LH)3
Use of the same experimental process described above with
Y(NO3)3·5H2O (0.36 g, 2.0”10À3 mol) with Dy(NO3)3·5H2O (30 mg)
gave crystals that were isolated by filtration and dried (yield:
0.19 g, 64.7%). IR (ATR): u˜ =3440l, 3068 w, 2929 w, 2775 w, 2718 w,
2612 w, 1663 w, 1598 m, 1584 m, 1562 m, 1519 m, 1484 m, 1472
m, 1454 m, 1426 w, 1362 s, 1349 s, 1297 m, 1246 m, 1227 w, 1156
m, 1146 m, 1100 w, 1066 m, 1042 w, 1028 w, 1008 w, 921 w, 866 w,
830 w, 757 m, 741 w, 703 w, 689 w, 680 w, 633 cmÀ1 w; elemental
analysis was carried out at the Laboratoire de Chimie de Coordina-
tion Microanalytical Laboratory in Toulouse, France, for C, H and N.
IR spectra were recorded by using a Spectrum 100 FT-IR Perkin-
Elmer spectrophotometer in the ATR mode.
Crystallographic data collection and structure determination
Crystals of Dy were kept in the mother liquor until they were re-
moved and dipped in oil. The chosen crystals were mounted on
a Mitegen micromount and quickly cooled to 180 K. The selected
crystals of Dy (yellow, (0.18”0.10”0.04 mm3) were mounted on an
Oxford Diffraction Xcalibur diffractometer that used graphite-mon-
ochromated MoKa radiation (l ¼ 0:71073 Š) and equipped with an
Oxford Instrument Cooler Device. Data were collected at low tem-
perature (180 K). The final unit cell parameters were obtained by
using least-squares refinements. The structures were solved by
using direct methods in SIR92,[59] and refined by using a least-
squares procedures on F2 with the program SHELXL97[60] included
in the software package WinGX version 1.63.[61] The atomic scatter-
ing factors were taken from the International Tables for X-Ray Crys-
tallography.[62] All non-hydrogen atoms were anisotropically re-
fined, and in the last cycles of refinement a weighting scheme was
used, in which weights were calculated from the following formu-
Experimental Section
Synthesis
Dy(NO3)3·5H2O, Y(NO3)3·5H2O, pyridine carboxaldehyde and piperi-
dine (Aldrich) were used as purchased. 2-Hydroxybenzohydrazide
was prepared as previously described.[58] High-grade solvents
(diethyl ether, dimethylformamide, methanol) were used for the
syntheses of ligands and complexes.
2-Hydroxy-N’-[(E)-(2-hydroxy-3-methoxyphenyl)methyl-
idene]benzhydrazide
Addition of pyridine carboxaldehyde (1.07 g, 1.0”10À3 mol) to
a stirred solution of 2-hydroxybenzhydrazide (1.52 g,1.0”10À3 mol)
in MeOH (30 mL) followed by heating for 30 min induced forma-
2
2
2
2
la: w ¼ 1=½s2ðFo Þ þ ðaPÞ þ bPŠ in which P ¼ ðFo þ 2Fc Þ=3.
CCDC 1433866 (Dy) contains the supplementary crystallographic
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data for this paper. These data are provided free of charge by The
Cambridge Crystallographic Data Centre.
tions with definite projection of the total moment jJ; MJi were
computed by using the SINGLE_ANISO package. The quantisation
axis was chosen to be the main magnetic axis of the ground
doublet.
Crystal data for Dy
C42H7DyN10O7; Mr =956.32 gmolÀ1; monoclinic P21/n; Z=4; a=
9.8585(3), b=22.1870(6), c=19.1215(5) Š; a=g=90, b=
94.160(3)8; V=4171.3(2) Š3; 35087 collected reflections, 8511
unique reflections (Rint =0.0416); R factor=0. 0368, weighted R
factor=0.0921 for 6996 contributing reflections [I>2s(I)].
Acknowledgements
We acknowledge the financial support from MIUR through the
project Futuro in Ricerca 2012 (RBFR12RPD1), from the Europe-
an Research Council through the Advanced Grant MolNanoMas
(grant no. 267746). We also acknowledge the computational
facilities at the National High-Performance Computing Center
in Sao Paulo (Centro Nacional de Processamento de Alto
Desempenho em S¼o Paulo-CENAPAD-SP) for the awarded
computational time (proj627).
Magnetic characterisation
The purity of the polycrystalline powders of Dy used for magnetic
characterisation was checked by using a Bruker D8 Advance
powder diffractometer equipped with
a Cu source (CuKa,
l ¼ 1:54 Š; see Figure S19 in the Supporting Information). The iso-
morphicity of diluted compound YDy compared with the pure Dy
compound was checked by using a Xcalibur3 single-crystal diffrac-
tometer equipped with a MoKa source (l ¼ 0:71073 Š). Xcalibur3 is
a four-cycle kappa geometry diffractometer equipped with a Sap-
phire 3 CCD detector. dc magnetic measurements were performed
by using a Quantum Design MPMS SQUID magnetometer on pow-
ders pressed in a pellet to avoid field-induced orientation of the
crystallites. The concentration of DyIII in YDy was estimated to be
10.5% by the scaling factor between the isothermal magnetisation
curves at low temperature of the pure and diluted compounds. ac
susceptibility was measured by using a Quantum Design PPMS in
ac mode for the frequency range of 10–104 Hz. The Quantum
Design MPMS SQUID magnetometer was used for low frequencies
(0.02–103 Hz). EPR spectra (not shown) were recorded at 3 and 5 K
by using a E500 Bruker spectrometer for the X band (n ꢂ 9:4 GHz).
The CTM measurements were performed by using a home-built
two-legged CuBe cantilever separated by 0.1 mm from a gold
plate. The cantilever was inserted into an Oxford Instruments
MAGLAB2000 platform with automated rotation of the cantilever
chip in a vertical magnet. The capacitance of the cantilever was de-
tected by using a Andeen-Hegerling2500A Ultra Precision Capaci-
tance Bridge. The faces of the measured crystal were indexed by
using X-ray diffraction in the above-described setup, then fixed on
the cantilever with glue. This procedure resulted in an estimated
uncertainty of the actual orientation of the crystal of about 58.
Keywords: ab initio calculations
magnetometry lanthanides magnetisation relaxation
single-molecule magnets
·
cantilever torque
·
·
·
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