Understanding magnetic relaxation
Despite the breakthroughs that have led to record breakingU eff values, T B, and
magnetic hysteresis temperature, an unequivocal understanding of the
relationship between structure and relaxation profile (\(\tau^{-1}\) vsT ) is still missing. The combined efforts of the Mills and
Chilton Groups, together with Manchester-based and international
colleagues, have taken aim at these phenomena by developing a
systematic, and synergic synthetic/computational strategy to address
specific issues affecting relaxation dynamics.
Spin-phonon coupling. Highly axial Dy(III)-based SMMs
that present \(U_{\text{eff}}\) values larger than 1000 K and an energy
gap between the ground- and first excited mJ states of around 500 K are now routinely
reported.[14] With such energy separations between
the electronic states involved in the relaxation of magnetization, the
question is why these systems cannot retain magnetization at
room-temperature and where the energy that bridges those states is
coming from? The answer to that is the coupling between the spin and
motion degrees of freedom (phonons) of the molecule and crystal,i.e. spin-phonon coupling.
The mathematics describing spin-phonon coupling and magnetic relaxation
was pioneered by the likes of van Vleck and Orbach in the in the
post-war years,[26] and until recently only
parametric models where used to interpret magnetic
relaxation.[27] Modern computational methodologies
have advanced to the point where an ab-initio description of
spin-phonon coupling is now possible.[28] Theab-initio spin dynamics formalism to calculate spin-phonon
couplings was initially proposed to explain the extraordinary magnetic
properties of dysprosocenium (2 ).[14a]The theoretical framework was deeply influenced by the experimental
observation that the relaxation dynamics of pure crystalline,
magnetically dilute crystalline, and also amorphous samples were
superimposable in the Orbach regime, exemplifying the tight relationship
between experiment and theory. This observation implies that in this
temperature range, the processes that govern relaxation are of molecular
origin (i.e. nothing intrinsic to the lattice), greatly
simplifying the theoretical treatment of the problem. In a significant
simplification, the method completely ignores acoustic phonons and
phonon dispersion, focusing only on the localized molecular vibrational
modes. Using Fermi’s golden rule, derived from time-dependent
perturbation theory, one can use these normal modes to monitor the
response of the \(m_{J}\) states as the molecule is distorted and
calculate spin-phonon couplings. Then, these couplings can be turned
into a relaxation profile (\(\tau\) vs T ) by constructing and
solving the master matrix containing the coupling elements between all
states.[1] Currently, this approach describes only
the Orbach regime (first term in eq.1), as it uses expressions that
couple states via single-phonons.
During recent years, efforts have been dedicated to the control and
design of static properties, i.e. how to optimize the
coordination environments to stabilize the magnetic states of interest.
The new generation of highly-axial Dy(III) SMMs have brought enormous\(U_{\text{eff}}\) values and thus, design criteria must now shift
towards considering the dynamic properties of molecules and crystals. To
that end, predictive tools such as the one developed by the Chilton
Group, as well as in other groups,[15,25,30,31]are necessary to drive the field forward.
Dipolar interactions. Although the vast majority of
Dy(III) SMMs are monometallic complexes, through space dipolar
interactions between magnetic moments in proximity to each other can
cause state-mixing and consequently faster loss of magnetic information.
A typical strategy to reduce the effect of dipolar magnetic interactions
is to dilute the spin in a diamagnetic matrix, so the magnetic ions do
not “see” each other. For instance, assuming a perfectly axial Dy(III)
(g z = 20, g x =g y = 0) in a crystal lattice whose nearest
neighbor is a co-parallel Dy(III) center sitting at x = 10.4 Å, the
coupling constant is ca. -0.31 cm-1, which
reduces to ca. -0.18 cm-1 when the distance
increases to 12.5 Å. From an experimental point of view this poses some
challenges in terms of material characterization and analysis. One
approach for heavy Ln SMMs (Tb-Yb) is to use a Y analogue as diamagnetic
host.[9,14a,14b,15] This is due to the many
chemical similarities between Dy and Y, such as their dominant oxidation
state (+3) and similar ionic size, which translates to an essentially
superimposable coordination chemistry. Y complexes are doped with a
small amount of Dy (usually < 5%) and recrystallized in order
to ensure homogeneity To validate the effectiveness of doping protocols,
a combination of experimental techniques are employed to determine the
purity of crystalline phase (single-crystal and powder XRD) and metal
composition (elemental analysis, ICP-MS, ICP-OES, X-ray fluorescence).
This can be a challenging task in the case of extremely air- and
moisture sensitive species like the low-coordinate, or organometallic
SMMs above.
To address the relative effect of dipolar interactions on the relaxation
of magnetization, we, along with an international collaborative team,
produced samples of known SMMs at pure Dy(III), and 5%Dy@Y doping
levels.[29] The systems were chosen as the
magnetic properties (temperature at onset of magnetic hysteresis,T H most notably) of each differed significantly,
though all possess high U eff values, and all had
been studied as pure Dy(III) samples (Figure
3a).[29] Comparison of data from both pure, and
doped samples would hint towards the origin of the different behaviori.e. if dipolar interactions were the sole driver of QTM, doping
5%Dy@Y should engender different improvements in each sample as they
each possess different U eff values. Hysteresis
loops measured at 2 K showed that the effect of magnetic dilution was
clearly visible as a reduction in the QTM step at zero field in2 .[29] The same behavior was also
observed in the other systems,
[K(18-crown-6)(THF)2][Dy(BIPM)2]
(3 , BIPM =
C{PPh2NSiMe3}2) and
[Dy(tBuO)Cl(THF)5][BPh4]
(4 ). However, in these examples a significant drop in remnant
magnetization is still present in the absence of an external magnetic
field, which hints at the fact that dipolar interactions are not the
main cause of the zero-field step in these high-barrier
SMMs.[29]