*4.3. Magneto-Electric (ME) E*ff*ect*

The magneto-electric (ME) effect had been observed only for inorganic multiferroic materials such as YMnO3, TbMnO3, and so forth, which showed both ferroelectric and magnetic order (ferromagnetism or anti-ferromagnetism) at cryogenic temperatures [17–20].

In this context, it was expected that a unique ME effect might occur in the second-harmonicgeneration (SHG)-active [55] and ferroelectric LC (SmC\*) phase of the (*S*,*S*)-**1** (m = n = 13), which displayed both an excellent ferroelectricity [*P*<sup>S</sup> = 24 nC/cm2, τ10-90 = 213 μs, θ = 29◦, η = 73.0 m Pa s at 74 ◦C by the triangle wave method] in a rubbed surface-stabilized liquid-crystal sandwich cell (4 μm thick) [29] and a positive magneto-LC effect (super-para-magnetic interactions) in the bulk SmC\* phase [35]. Such an assumption was the case. The temperature dependences of relative paramagnetic susceptibility (χrel), *g*-value, and Δ*H*pp were measured by EPR spectroscopy using the thin sandwich cell into which the most appropriate sample of (*S*,*S*)-**1** (m = n = 13, 65% ee) was loaded (Figure 20).

**Figure 20.** (**a**) Experimental setup to monitor the variable-temperature or electric field dependent EPR spectra of (*S*,*S*)-**1** (m = n = 13, 65% ee) confined in a long 4-μm thin sandwich cell. (**b**) Principal axes of inertia and *g*-values of the *trans*-**1** molecule. (Reprint with permission [37]; Copyright 2020 the Royal Society of Chemistry).

Since the magnetization measurement of the sample in the LC cell by SQUID magnetometry was technically difficult, χpara was derived from the Bloch equation (Equation (8)) [56] by using the EPR parameters, such as *g*, Δ*H*pp, and maximum peak height (*I* m and −*I* m) as described earlier [35,36],

$$
\lambda \chi\_{\text{para}} = 2 \mu\_{\text{B}} \text{g} \mathbf{l}'\_{\text{m}} \Delta \mathbf{H}^2 \llcorner \text{\(\sqrt{3} \, h \nu H\_1\)} \tag{8}
$$

where μ<sup>B</sup> is the Bohr magneton, *h* is Planck's constant, ν is the frequency of the absorbed electromagnetic wave, and *H*<sup>1</sup> is the amplitude of the oscillating magnetic field. The relative paramagnetic susceptibility (χrel,*T*) in the absence of an electric field is defined as:

$$
\chi\_{\text{rel},T} \equiv \chi\_{\text{para}}/\chi\_0 \tag{9}
$$

where χ<sup>0</sup> is the standard paramagnetic susceptibility at 30 ◦C in the heating process. First, the generation of positive magneto-LC effect and ferroelectric switching in the liquid crystal cell was confirmed. By measuring the temperature dependence of χrel,*<sup>T</sup>* under the conditions in which the applied magnetic field was parallel (Configuration A) and perpendicular (Configuration B) to the cell surface and the rubbing direction in the absence of an electric field, considerable χrel,*<sup>T</sup>* increases (ca. 40%) together with large Δ*H*pp ones, which were noted at the crystal-to-LC phase transition in both cases (Figure 21). The ferroelectric switching at 25 V was verified by polarized optical microscopy in the absence of a magnetic field. The bright fan-shaped texture at −25 V and the dark one at +25 V were distinctly observed [37].

**Figure 21.** Temperature dependence of (a and d) χrel,*T*, (b and e) Δ*H*pp and *g*-value (c and f) for (*S*,*S*)-**1** (m = n = 13, 65% ee) confined in a thin rubbed sandwich cell by EPR spectroscopy at a magnetic field of around 0.33 T. The magnetic field was applied (**a**–**c**) parallel and (**d**–**f**) perpendicular to the rubbing direction. The LC transition temperatures in the heating and cooling runs are shown in the lower and upper sides of the panels, respectively. (Reprint with permission [37]; Copyright 2020 the Royal Society of Chemistry).

The difference in temperature dependence of *g*-values along two direction shows that the magnetic susceptibility of the (*S*,*S*)-**1** clearly has an anisotropy with respect to the molecular axis. Since the Curie constant revealed *S* = 1/2 nature of the system, the anisotropy stems from high-order perturbation term of exchange interactions with respect to spin-orbit interactions, such as Dzyaloshinkii-Moriya (DM) interactions. Moreover, the temperature dependence in the case of Configuration A (Figure 21c) shows anomalies in the vicinity of ferroelectric LC (FLC) transition. That implies some changes in magnetism

in connection with ferroelectricity, while little anomaly was found in the case of configuration B (Figure 21f).

On the other hand, the relative paramagnetic susceptibility (χrel,*E*) in the presence of an electric field is defined as:

$$
\lambda \chi\_{\text{rel}, \text{E}} = \chi\_{\text{parra}} / \chi\_{\text{1}} \tag{10}
$$

where χ<sup>1</sup> is the standard paramagnetic susceptibility at the initial potential of +25 V and at 75 ◦C. Next, the electric field dependences of χrel,*E*, Δ*H*pp, and *g*-value were plotted for the (*S*,*S*)-**1** (m = n = 13, 65% ee), displaying each hysteresis between +25 V and −25 V when the magnetic field was applied only perpendicularly to the electric field and parallel to the rubbing direction (Figure 22).

**Figure 22.** Electric field dependence of *g*-value, χrel,E, and Δ*H*pp for the FLC phase of (*S*,*S*)-**1** (m-n = 13, 65% ee) confined in a thin rubbed sandwich cell at 75 ◦C by EPR spectroscopy at a magnetic field of around 0.33 T. The magnetic field was applied (**a**–**c**) perpendicularly to the electric field and parallel to the rubbing direction and (**d**–**f**) parallel to the electric field and perpendicular to the rubbing direction. Open and filled circles represent the application of electric fields from +25 V to −25 V and from −25 V to +25 V, respectively. (Reprint with permission [37]; Copyright 2020 the Royal Society of Chemistry).

It is considered that the molecules with the spontaneous electric polarization is inversed by sweeping the external electric field. Therefore, it is noteworthy that the spin inversion accompanies the inversion of the molecule, which can be regarded as a nonlinear ME effect. Since the usual paramagnetic components cannot show such a nonlinear hysteresis loop, the hysteresis indicates the spin flipping of superparamagnetic domains. This result clearly shows the predominance of the electric field over a magnetic field in controlling the super-para-magnetic behavior, since the spin flipping by the electric field occurs even under a large magnetic field of 0.33 T with a constant direction used for this EPR measurement. The direction-of-magnetic field dependence of the *g*-value, χrel,E, and Δ*H*pp indicates that the super-para-magnetic domains are directed almost perpendicularly to the spontaneous electric polarization. For both the nonlinear ME effect and the direction-of-magnetic field dependence, the high-order perturbation term of exchange interactions with respect to spin-orbit interactions is usually necessary for this *S* = 1/2 system.

In summary of Section 4, the unique magnetic properties observed for LC nitroxide radicals are characterized in connection with SG-like magnetic features (see Section 3.1) as follows:


As shown in Figures 18 and 19, these results imply the formation of the super-para-magnetic domains, most likely due to the local inhomogeneity in LC phases. Therefore, this picture is analogous to the cluster glass (See Section 3.3.2), or super-para-magnetic system without magnetic interactions. A partially broken degree of freedom in LC phases is considered to affect the magnetic properties of the domains. The super-para-magnetic response to the external magnetic field is due to the domains' rotational degree of freedom. Likewise, the factors of molecular mobility and intermolecular interactions resulting in a favorable molecular correlation in LC phases are likely to contribute to the long-time scale growth of superparamagnetic domains. To comprehend the microscopic state of the superparamagnetic domains, we have to investigate the dynamics and interactions between the domains.

Elucidation of the microscopic mechanism for the formation of super-para-magnetic domains in the LC nitroxide radicals is a challenging subject. In this context, we discovered that the nonlinear ME effect occurs in the LC nitroxide radicals (Figure 22). Namely, the super-para-magnetic domains turned out to be controllable by external electric fields. This result demonstrates the ferroelectric aspects of super-para-magnetic domains. Accordingly, in Section 5, we propose and discuss the mechanism of the positive magneto-LC effect on the basis of the nonlinear ME effect.

Meanwhile, quite recently another mechanism has been suggested, in which the positive magneto-LC effect can be accounted for microscopically in terms of the molecular mobility and the resulting inhomogeneity of intermolecular interactions without assuming the formation of superparamagnetic domains by means of MD simulation and DFT calculations (Figure 16) [47]. Namely, the magnetic features seem to reflect the macroscopic inhomogeneity of the liquid crystal orientation field, even though the magnetic properties look like those resulting from the superparamagnetic domains. Since the macroscopic studies on this mechanism is under investigation, only the former mechanism based on the ME effect is discussed in Section 5.

In addition, it is fairly possible that both mechanisms operate complementarily. The latter mechanism originating from the molecular mobility is predominant in the LC and isotropic phases, while the formation of super-para-magnetic domains by the former mechanism results in the preservation of χTIM in the supercooled LC and solid phases during the cooling process.

#### **5. Proposed Mechanism**

The existence of superparamagnetic domains in rod-like LC nitroxide radicals means the ferromagnetic spin arrangement inside the domains. However, ferromagnetic ordering through the direct exchange interactions between radical spins at ambient or higher temperatures in metal-free organic radicals had been believed to be unrealistic. Moreover, the molecular rotation of as fast as around 1010 s−<sup>1</sup> in a rod-like LC phase may also make the average magnetic interactions much weaker than in a crystalline phase [25,26]. Contrary to such general believing, the positive magneto-LC effect has been observed in the rod-like LC nitroxide radicals with a specific molecular structure, but not in the homologous non-LC nitroxide radicals.

Here, we propose the parasitic ferromagnetism [57] in terms of the nonlinear ME effect, that is, the induction of ferromagnetism by the emergence of the domains with spontaneous electric polarization (Figure 23). If there is a cross-correlation term between magnetization (*M*) and electric polarization (*P*) in the free energy, it is plausible that the emergence of *M* can lower the free energy of the system in the presence of *P*.

**Figure 23.** Two possible mechanisms for the emergence of ferromagnetism in LC nitroxide radicals induced via DM interactions. (**a**) The induction of weak ferromagnetism (*M*) directed perpendicularly to the electric polarization (*P*). *L* denotes the antiferromagnetic vector *S*<sup>1</sup> − *S*2. (**b**) A helical magnetic structure induced by helical molecular alignment in the LC phase. The following are the explanation of individual symbols and signs in panel b—Spheroid: the rod-like LC nitroxide radical molecule, *pi*: the electric polarization of the *i*-th molecule, *pij*: the summed electric polarization between the *i*-th and *j*-th molecules, and *eij*: translational vector directed to the screw axis.

For the operation of this mechanism, the existence of spontaneous polarization (*P*s) is necessary, which emerges as the result of breaking the space inversion symmetry. As described in Section 4.3, the *P*s is already present in the LC phases composed of chiral nitroxide radical molecules. We could obtain the following three experimental results that indicated the correlation of super-para-magnetic behavior and the breaking of space inversion symmetry. Firstly, the super-para-magnetic domains grow during the crystal-to-LC-to-Iso phase transition sequence in the heating process, in which the birth and growth of domains without space inversion symmetry was revealed by the gradual increase in the SHG signal (Figure 14). Second, the result that the super-para-magnetic domains augmented with the increasing *ee* value of the LC compounds (See Section 4.1.1) is likely to demonstrate the correlation of the positive magneto-LC effect and the breaking of space inversion symmetry. Third, and more importantly, it was the external electric field that flipped the *g*-value of the EPR signal to draw a ferromagnetic-like hysteresis loop (Figure 22a). These results suggest that the positive magneto-LC effect is driven by collective alignment of LC molecules and the subsequent breaking of its space inversion symmetry.

As one of the origins of nonlinear ME effect, DM interactions, *E*DM ∼ *ij Dij*·*S<sup>i</sup>* × *Sj*, are known.

The Dzyaloshinkii vector (*Dij*) can be present when the space inversion symmetry is broken. The existence of DM interactions in LC nitroxide radicals were indicated by the anisotropic *g*-value (Figure 21c). Next, we discuss two possibilities for parasitic ferromagnetism on the basis of DM interactions.

The first possibility is the emergence of weak ferromagnetism accompanying the electric polarization (Figure 22a). Such weak ferromagnetism is suggested when the contribution of the energy term, *E***PLM**∼ *P*·(*L* × *M*), exists in the free energy of the system [58]. This is a phenomenological term resulting from DM interactions, where *P* denotes the electric polarization, *L* is antiferromagnetic vector *S*<sup>1</sup> − *S*2, and *M* is the remanent magnetization perpendicular to *L* (i.e., weak ferromagnetism). If *P* is present in the system, the emergence of *M* leads to a finite *E***PLM** value. Therefore, the canting of anti-ferromagnetic pairs of *S*<sup>1</sup> and *S*<sup>2</sup> to the direction perpendicular to the electric polarization can lower the total energy of the system by *E*PLM (Figure 23a). As expressed in *L*, large anti-ferromagnetic correlations between spins by sufficient intermolecular contacts are needed for this mechanism. It is highly plausible that the large molecular mobility in the LC and isotropic phases is likely responsible for the inhomogeneous intermolecular contact (Figure 16) [47]. This picture is consistent with the results obtained from the experiment on the ME effect (Figure 22). Both the *g*-value and the line width (Δ*H*PP) are switched by the external electric field (*E*) when it is perpendicular to the external magnetic field *H*<sup>0</sup> (Figure 24).

**Figure 24.** Image of domains with electric polarization and the resulting weak ferromagnetism proposed in Figure 23a. The light blue-colored background denotes the paramagnetic LC region. Response of domains to the external electric field rather than the perpendicular magnetic field in the experiment on ME effect (Figure 22). Note that the correspondence of the direction of magnetic moments with observed *g*-value depends on the internal magnetic field at resonant spin sites.

The second possibility is the emergence of helical magnetism by helical alignment of LC molecules (Figure 23b). It is known that, in the helical magnetic structures, the spins *S<sup>i</sup>* and *S<sup>j</sup>* can induce the electric polarization (*pij*) to satisfy *pij* ∼ *eij* × *S<sup>i</sup>* × *S<sup>j</sup>* due to inverse DM interactions, where *eij* is the unit vector connecting the *i*-th and *j*-th sites [59]. In contrast, in the case of LC nitroxide radicals, the helical structure formed by local *pij* from the *i*-th and *j*-th molecule might induce the helical structure of *S<sup>i</sup>* and *Sj*. This helical magnetic structure can have the net magnetization *M* along the

direction of the screw axis. To the best of our knowledge, although there is no example for the helical electric polarization-induced helical magnetization, our proposed mechanism is likely to explain the unique magnetic properties observed in chiral helical LC phases (Figures 10 and 11) or achiral LC phases containing partial chiral helical domains (Figure 14).

As for the correlation of super-para-magnetism and electric polarization, similar magnetic properties can be found in relaxors [60]. A relaxor can be regarded as an electric version of SG because it has randomly directed ferroelectric domains called Polar Nano Regions (PNR) due to the inherent randomness and frustration in the system (Figure 25). Recently, the emergence of super-para-magnetism has been observed at high temperatures with respect to some relaxors (or the system with relaxor-like behavior) containing magnetic metal ions [61,62] (Figure 26). For the mechanism of such super-para-magnetic relaxors, the presence of DM interactions are suggested, which will give weak ferromagnetism perpendicular to the polarization of PNR [61]. In addition, the MD simulation showed that the addition of spherical apolar impurity to spheroidal polar particles produces PNR to show the typical behavior of relaxors [63]. This seems analogous to the formation of ferroelectric domains in LC nitroxide radical samples by an impurity effect (See Section 4.2).

**Figure 25.** Image of PNR in relaxor systems. The electric polarization of each PNR is directed randomly.

**Figure 26.** (**a**) The hysteretic temperature dependence of magnetic susceptibility and (**b**) the super-para-magnetic behavior in the magnetization curves in ZnO−Co relaxor-like nanocomposite thin films [60]. The labels (**a**,**b**) in the original graphs were moved out of the frames by the authors of this paper. (Reprint with permission [62]; Copyright 2020, American Chemical Society).

In some of these super-para-magnetic relaxors, it was revealed that the super-para-magnetic domains were identical to PNR by estimating their sizes [61]. In the case of LC nitroxide radicals, we must clarify the detailed relationship between the magnetic and electric properties in order to

uncover the mechanism for the birth and grow of super-para-magnetic domains, which will lead to the elucidation of the microscopic mechanism of positive magneto-LC effect that LC nitroxide radicals exhibited.

#### **6. Conclusions and Prospects**

Since 2008, we have reported that a series of chiral and achiral all-organic LC nitroxide radicals having one or two PROXYL units in the core position display SG-like superparamagnetic features, such as a magnetic hysteresis (referred to as positive magneto-LC effect), and thermal and impurity effects during a heating and cooling cycle in weak magnetic fields. In general, the enantiomerically-enriched rod-like LC nitroxide radicals *trans-***1** and their derivatives always showed a distinct positive magneto-LC effect, irrespective of the molecular positive or negative dielectric anisotropy (Δε), whereas the corresponding racemic samples exhibited the positive or negative magneto-LC effect, depending on the negative or positive sign of molecular Δε, respectively. Furthermore, for the first time, the magneto-electric (ME) effect was observed in the ferroelectric (SmC\*) phase of *S*,*S*-enriched **1** (m = n = 13) at temperatures as high as 75 ◦C. Achiral meso diradical compound (*R*,*S*)-**3** became SHG-active gradually in the discotic hexagonal columnar phase by heating in the presence of a magnetic field to form a chiral helical columnar structure and eventually showed a distinct positive magneto-LC effect. It is also noticeable that a very large impurity effect was observed for this compound when 20 mol% of racemic *cis*-diastereomers (*R\**,*R*\*)-**3** was added to the host (*R*,*S*)-**3** as the impurity.

By comparison of these experimental results with the well-known magnetic properties of SG materials and on the basis of the results of nonlinear ME effect, we suggest that the positive magneto-LC effect, i.e., partial formation of super-para-magnetic domains in the major paramagnetic spins in the LC phase, is most likely to originate from the emergence of weak ferromagnetism accompanying the electric polarization due to DM interactions and/or the emergence of helical magnetism by helical alignment of LC molecules.

For the practical application of the positive magneto-LC effect to organic materials science, it is essential to enlarge the ratio of superparamagnetic domains to paramagnetic spins in the LC and isotropic phases. The utilization of the impurity effect to form SG-like inhomogeneous magnetic domains is quite promising because we can select the most suitable compounds among a variety of candidate magnetic and nonmagnetic organic compounds miscible with LC phases as the impurities. The electric field control of ferromagnetic domains is another interesting choice. However, this needs further studies to understand the detailed mechanism of the positive magneto-LC effect. These studies will offer novel access to the realization and application of ferromagnetic materials composed of organic radical spins. At the same time, it will be possible to measure the AC magnetic susceptibility at low or even higher temperatures for elucidating the mechanism of positive magneto-LC effect owing to a considerable increment of the super-para-magnetic region surrounded by the para-magnetic spins.

**Author Contributions:** Conceptualization, S.S. and Y.U.; methodology, Y.U. and R.T.; validation, S.S., Y.U. and R.T.; formal analysis, S.S. and Y.U.; investigation, Y.U. and R.T.; resources, R.T.; data curation, S.S. and Y.U.; writing—original draft preparation, S.S. and R.T.; writing—review and editing, R.T. and S.S.; supervision, R.T.; administration, R.T.; funding acquisition, R.T. and Y.U. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by JSPS KAENHI (Grant number 26248024).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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