Microwave magnetoelectric effect in structures based on ferromagnetic metals

The results of a theoretical calculation of the resonant magnetizing field values for ferromagnetic resonance in thin films of nickel, iron and cobalt for frequencies of 3, 10, 30 GHz for various orientations of this field are presented. Analytical dependences of the ferromagnetic line shifts on the electric field strength are obtained for twolayer magnetoelectric composites in which PZT, PMN-PT or PZN-PT disks are used as piezoelectrics on which thin films of Ni, Fe or Co are deposited.


Introduction
In connection with the development of spintronics, interest in the magnetoelectric (ME) effect in nanostructures has increased. The ME effect in multiferroics in the terahertz range based on various manganites has recently been studied [1]. It is of interest to study the magnetoelectric effect in the ferromagnetic resonance (FMR) region in nanostructures based on ferromagnetic metals [2]. This paper presents the results of studies of the features of FMR in thin films of nickel, iron, and cobalt (film thickness ~ 50 nm). The objects of theoretical research were two-layer ME composites, in which PZT, PMN-PT or PZN-PT disks with a thickness of 0.5 mm was used as piezoelectric materials, on which thin films of Ni, Fe or Co were deposited. The values of the resonance bias field for frequencies of 3, 10, 30 GHz with different orientations of this field are calculated. Cases are considered when the magnetizing field is perpendicular to the plane of a thin metal film and also directed in the plane of a thin metal film along or perpendicular to easy magnetization axis. The main attention is paid to the theoretical study of the FMR line shift in two-layer ME composites based on thin films of ferromagnetic metals under the influence of an electric field.

Calculation methods of the FMR line shift in two-layer magnetoelectric composites
The calculation method for the FMR line shifting in two-layer ME composites consists of two stages. First, the values of the resonant magnetizing field are calculated at frequencies of 3, 10, 30 GHz for various orientations of this field. So, cases were considered when the magnetizing field is perpendicular to the plane of a thin metal film, and also directed in the plane of a thin metal film along or perpendicular to the axis of easy magnetization. Then, two-layer ME composites based on thin films of ferromagnetic metals Ni, Fe or Co are examined. In case when a constant electric field directed perpendicular to the plane of the disk to influence at ME composites, the FMR line shifts as a result of the microwave ME effect [3]. The calculated dependences of the FMR line shift on the electric field strength were obtained by using the method of effective demagnetizing factors. The use of layered structures based on nanofilms of ferromagnetic metals will allow to proceed to further miniaturization of devices and design a number of new nanoelectronic microwave devices [4].

FMR in a ferromagnetic metal nanofilm
Let us consider a ferromagnetic metal nanofilm to which a constant magnetizing field is applied perpendicular to the plane of the film. We assume that the magnitude of this field is sufficiently large and the ferromagnetic metal film is uniformly magnetized to saturation. We direct the axis 3 along H 0 , then H 0 has components (0, 0, H 0 ), and the equilibrium magnetization has components (0, 0, M 0 ). Axis 1 is directed along the axis of easy magnetization of the nickel (iron) film.
The equation of the motion of magnetization in a ferromagnetic metal nanofilm under the action of a high-frequency magnetic field h in the presence of a constant magnetic field H, taking into account dissipation is where the full magnetization consists of the equilibrium and high-frequency components 0 M M m   (2) and the density of free energy is Then we write the dissipative term component wise The magnetic field strength also consists of two components Then we find the derivatives of free energy with respect to the components of magnetization Next, let's us write the equation of motion for the components Linearizing these equations and preserving only the first-order terms of smallness in them, we obtain a system of two linear inhomogeneous equations for two unknowns m 1 , m 2 Having solved this system, we can write the form of the high-frequency magnetic susceptibility tensor 11 12 Where components of this tensor are     We will be interested in the imaginary parts of the complex components of highfrequency susceptibility  2  2  2  2  0  0  0  0  22  33  0  11  22  33  0  0  11  0  2  2  2  2  2 2 2  0  0  0  11  22  33  0   2  2  2  2  0  0  0  11  33  0  11  22  0  22  0   1  2  2   1  2  2 1 2 Demagnetizing factors associated with sample shape Substituting (12), (13) into (15), we obtain the values of the constant magnetizing field for nickel H 0 4439, 6850, 13745 Oe at resonant frequencies f=3, 10, 30 GHz, respectively. For iron we obtain the values of the constant magnetizing field for nickel H 0 2414, 4824, 11720 Oe at same frequencies, respectively.
We similarly consider the case when a constant magnetizing field is directed in the film plane along the axis of easy magnetization of the nickel (iron) film. For nickel, we have H 0 98, 1973, 8626 Oe and for iron H 0 381, 2672, 9530 Oe at the same frequencies, respectively.
For the case when a constant magnetizing field is directed in the plane of the film perpendicular to the axis of easy magnetization of the nickel (iron) film, For nickel, we have H 0 483, 2317, 8942 Oe and for iron H 0 730, 2986, 9831 Oe at the same frequencies, respectively.
For cobalt, there are only demagnetizing factors associated with the shape of the sample  Similarly, we consider the case when the constant magnetizing field is directed in the plane of the cobalt film, and the values obtained are H 0 69, 730, 5102 Oe at resonant frequencies f=3, 10, 30 GHz, respectively.

FMR line shift under an applied electric field
First, we write (14) without taking into account the effective demagnetizing factors arising from the action of the electric field strength E Now we write (14), taking into account the small correction δH E associated with small effective demagnetizing factors arising from the action of the electric field strength E Subtract (17) From this condition we find the shift of the resonance line     Next, we find the demagnetizing factors for ferromagnetic metal/piezoelectric structures associated with the action of a constant electric field. Consider the case when a constant electric field is directed along axis 3.
We substitute the previously found demagnetizing factors (12) and (13) into (20)     Due to the specific symmetry of the problem, k does not depend on the directivity of the magnetizing field and the magnitude of the resonant frequency f.

Conclusion
A calculation method for determining the FMR line shift in two-layer ME composites based on thin ferromagnetic metals films under the influence of an electric field is proposed.
Analytical dependences of the FMR line shift in such composites on the electric field strength are obtained. This work was supported by the Russian Federation Grant RFBR 19-57-53001 NNSF_a.