Dynamical anisotropic magnetoelectric effects at ferroelectric/ferromagnetic insulator interfaces
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Li Yaojin, Koval Vladimir, Jia Chenglong. Dynamical anisotropic magnetoelectric effects at ferroelectric/ferromagnetic insulator interfaces. Chinese Physics B, 2019, 28(9): 097501 
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Dynamical anisotropic magnetoelectric effects at ferroelectric/ferromagnetic insulator interfaces
† Corresponding author. E-mail:
Abstract
The interfacial magnetoelectric interaction originating from multi-orbital hopping processes with ferroelectric-associated vector potential is theoretically investigated for complex-oxide composite structures. Large mismatch in the electrical permittivity of the ferroelectric and ferromagnetic materials gives rise to giant anisotropic magnetoelectric effects at their interface. Our study reveals a strong linear dynamic magnetoelectric coupling which genuinely results in electric control of magnetic susceptibility. The constitutive conditions for negative refractive index of multiferroic composites are determined by the analysis of light propagation.
Keyword:interfacial magnetoelectric effect;ferromagnetic insulator;magnetic susceptibility;refractive index
1. Introduction
The study on the magnetic response of materials subjected to an external electric fields has generated great interest from both fundamental physics and potential application perspectives.[1–8] In particular, the direct coupling between electrical and magnetic polarization allows for purely electrical manipulation of magnetism with promising multi-functionality and low energy consumption. From the symmetry point of view, the magnetoelectric (ME) interactions can only develop in systems with both time-reversal and spatial-inversion symmetry breaking,[2] such as single phase magnetoelectric hexaferrites,[9] multiferroic bismuth ferrites[10] or ferroelectric/ferromagnetic (FE/FM) composite structures.[11] For single phase multiferroics, the mechanisms of ferroelectricity with spin origin have been identified[9,12] and the multiferroic response is well documented experimentally and theoretically.[13] On the other hand, a comprehensive understanding of coupling between spin and charge degrees of freedoms in the insulating interface state of composite multiferroics is still missing. In artificial FE/FM multiferroics, the studies on the interfacial ME effect have thus far remained interesting such as the electrostatic screening effect,[5,14–17] ionic liquid gating,[18–20] orbital reconstruction,[20–24] etc. However, this electric field effect can be ruled out in case of the FM insulator (FMI) interfaces due to the absence of mobile charge carriers.
In this paper, the non-local FE-associated vector potential is demonstrated to give rise to an anisotropic ME interaction through the multiorbital hopping occurring in FMI. The large electrical permittivity mismatch between FE and FMI is found to bring about a significant enhancement of the interfacial ME effects. The dynamic magnetic response driven by electromagnetic waves via strong linear ME coupling is investigated in detail. The electrical tuning of magnetic permeability in FE/FMI is discussed as well. Moreover, the negative refractive index is possible in the FM system with small electric permittivity. All these findings not only contribute to the current understanding of the magnetoelectric effects in composite structures but also fetch in a new perspective to nanoscale composite multiferroics.
2. Theoretical model
where α and β denote two
orbitals, respectively.
is the destruction operator of an itinerant electron with spin
. tij describes the hoping between i and j sites, more precisely, between filled and empty orbitals at different sites, which is essential for the ME interaction.
is the Hund coupling,
is the spin of the itinerant electron on the orbital α at site i, and the local spin
is approximated by a classical spin.
is energy splitting between the two orbitals at site i, and
describes the Coulomb repulsion with
being the total number of itinerant electrons on the site. For
, an electron jumping between nearest-neighbor sites does not require energy (
), if all the spins are parallel. Hence, double exchange and FM ordering occur in the system.[40–42] On the other hand, the hoping terms involving only single-orbital per site with the limit of large U result in the t–J model with antiferromagnetic interaction.[43] Here, to second order perturbations of multi-orbital hopping, one gets an effective ferromagnetic interaction between the spins[38]
where
and
is the unit vector along the local spin
.
where
is a unit vector normal to the FE/FMI interface (hereafter we take
, the z axis has its origin at the FMI interface),
and
are the displacement field in FE and FMI, respectively, the vanishing of free charges (i.e.,
) at the insulator interfaces leads to
where
is the dielectric constant of FE and FMI. Considering
with
being an applied electric field by gate voltage, the normal component of the effective electric field
in FMI is significantly enhanced up to several orders (
) compared with the other two in-plane components
. As a consequence, giant anisotropic electric-field effects are expected in composite FE/FMI heterostructures.
where
is the dimensionless coupling strength and
![]()
describes the spin energy change due the electron hopping from site i to site j.
is related to the spin interactions that do not originate from exchange processes.
where the exchange constant
.
represents the effective ME interaction,
in which the first and the second terms describe the static and dynamic ME contributions, respectively.
Let us consider lightly doped manganese oxides La1−xSrxMnO3 (LSMO with 
 | Table 1.
Values of the magneto-electric coupling coefficient, α, reported in the literature for selected multiferroic heterostructures. T is the temperature and LSMO stands for La1−xSrxMnO3. . |
In these perovskite systems, the five (Mn) d-orbitals split under the octahedral crystal field into three lower 
 |
 |
By interfacing FMI layer with FE films, the effective electric field develops in FMI. Taking into account the boundary conditions for the electric field at the interfaces between materials[39]
 |
 |
In presence of electric field, the hopping amplitude can be modified in a gauge-invariant manner with the Peierls substitution,[44,45] 
 |
 |
The low dimensionality of the FM film would lead to a uniaxial magnetic anisotropy, 
 |
 |
3. Results and discussion
3.1. Static ME effects
where z0 is the initial position and
with λ being the coherence length of the ME coupling. It should be pointed out that an initial ferromagnetic configuration is considered in the derivation of the above interfacial magnetization. Note that the sign of λ depends on the anisotropy
and the external electric field;
. Along with the boundary conditions determined by the magnetic anisotropy: (i)
for out-of-plane magnetization with
or (ii)
for in-plane magnetization with
, one can find that the interfacial nonlinear magnetization can be triggered by only and if only an outward external electric field from the FMI interface, i.e.,
. This selection rule of the interfacial ME coupling holds promising ways for the electric field-control of domain-wall excitations. On the other hand, following the arguments of ferroelectricity of spin origin,[15,46,47] an inhomogeneous charge distribution can exist due to nonlinear magnetic moments at the interface.[15] As for the typical values:
meV,
,
meV,
mC/cm2,
/cm, and the lattice constant of FMI
, we obtain a weak (∼0.1 meV) but long-range (
nm) static interfacial ME interaction, which is most pronounced in the vicinity of FE/FMI interface acting with the ME coherence length λ of the order of submicrometers.
By rewriting the magnetization 
 |
3.2. Dynamic ME effects
where γ is the gyro-magnetic ratio defined as
with the G-factor g and the Bohr magneton
. Introducing the above Landau–Lifshitz equation into the ME interaction, Eq. (8 ), a linear dynamic ME coupling in the lowest order can be expressed as
where
is the driving rf magnetic field and
is the magnetic permeability of FMI.
(
) is the rf electric field. According to Eq. (4 ), the most pronounced dynamic ME coupling occurs if as the rf electric field normal to the interface, i.e.,
, then, the coupling constant reads
Without losing of generality, we assume an out-of-plane magnetization, which can be excited by applied (transversal) rf fields 
 |
 |
 |
It should be pointed out that such linear ME effects can be enhanced further in the resonance regime through the magnetic permeability. For typical vaule 
3.3. Transformable magnetic susceptibility
where
is the magnetic susceptibility of FMI,
for isotropic materials. The rf electric fields satisfy the Maxwell–Faraday equation,
. Taking the propagation of microwave as the
direction and
, one gets
where c is the speed of light in vacuum,
is the complex refractive index in FMI, and
is an antisymmetric tensor. By introducing
into Eq. (13 ), one obtains
which can be measured by a very sensitive technique, for instance, ferromagnetic resonance (FMR) spectroscopy. Due to the linear ME interaction, the magnetization dynamics is simultaneously influenced by the ac magnetic field
and the ac electric field
.[49] The FMR spectrum, thus, is not of a pure absorptive Lorentz mode but an interference mode between the in-phase dispersive (
) and out-of-phase absorptive (
) modes. Furthermore, the mixture phase φ is angular-dependent and electric field-dependent as well, which offers electrical tuning of transformable magnetic susceptibility. More importantly, possible negative magnetic susceptibility could be obtained as
. Our recent direct permeability measurements confirmed these dynamic ME effects,[50] in which the linear ME coupling is around
s/m in Co/BaTiO3[51] and CoZr/PMN-PT films.[49]
Phenomenologically, the linear ME coupling may result in an additional ME contribution to the magnetization[2,28]
 |
 |
 |
Assuming that 
 |
 |
3.4. Negative refraction
where
is the Fourier transformation of the electromagnetic vector potential with the magnetic flux density field
and electric field
. In presence of linear ME effects (cf. Eq. (13 ), there are three different contributions to the bound current: (i) the magnetization current
![]()
(ii) the polarization current
with
being the electric susceptibility of FMI, and (iii) the magnetoelectric current
![]()
Assuming the propagation of light in the y direction,
,
, the bound current then reads
The Maxwell equation
![]()
in the Lorentz gauge yields the following equation for the refractive index
:
For the case of
, similar equation is obtained but with the replacement of
by
. It is obvious that we get a standard refractive index
in the absence of any ME coupling. Considering ME interactions at the FE/FM interface, a large modification in the refraction index is obtained. For
and the limit of
, equation (20 ) gives two approximate solutions,
Note that
(with
being the phase velocity of light in materials), thus we might have negative refraction
in natural nanoscale FE/FM heterostructures. Such negative refraction can exist as well in natural ferromagnetic metals such as nickel, iron, or cobalt at a frequency close to the ferromagnetic resonance if the strength of the mode is large enough.[52]
In order to explore the effects of the FE/FM interface on the refractive index 
 |
 |
 |
 |
4. Conclusion
In summary, we have shown that magnetoelectric interactions may develop at the FE/FMI interface due to the non-local hopping between multi-orbitals. The large mismatch in dielectric constant of FE and FMI was found to give rise to giant anisotropic electric field effects on magnetization. The interfacial linear dynamic magnetoelectric coupling together with the transformable magnetic susceptibility and the negative refractive index were successfully described. It should be emphasized that the proposed dynamic ME effects are intrinsic phenomena occurring in multiferroic systems with strong linear magnetoelectric interaction and small enough electric permittivity. The predicted properties are of a huge importance in designing in a new class of optical negative-index metamaterials and the obtained results provide new perspectives for construction of the next-generation electronic components based on nanoscale composite multiferroics.
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