Angular dependence of ferromagnetic resonance as indicator of the nature of magnetoelectric coupling in ferromagnetic-ferroelectric heterostructures

A. Sukhov, C.-L. Jia, L. Chotorlishvili, P. P. Horley, D. Sander, and J. Berakdar

Phys. Rev. B 90, 224428 – Published 31 December 2014

Abstract

We investigate theoretically how ferromagnetic resonance (FMR) in a multiferroic heterostructure provides quantitative information on multiferroic coupling. In particular, we analyze the asymmetry of the angular dependence of FMR on the direction of resonance magnetic fields as experimentally reported for ${Co / BaTiO}_{3}$ [Jedrecy et al., Phys. Rev. B 88, 121409(R) (2013)] and for ${permalloy / Pb (MgNb) O}_{3} - {PbTiO}_{3}$ [Nan et al., Sci. Rep. 4, 3688 (2014)]. Based on both analytical expressions for the dependence of FMR on the magnetic-field orientation and full numerical simulations which account for mutually coupled polarization $P$ and magnetization $M$ dynamics, we conclude that it is the linear magnetoelectric coupling in $P$ and $M$ originating from spin-dependent screening that leads to the experimentally observed asymmetry in FMR angular behavior. This suggests that angular resolved FMR is well suited to quantify the type and the strength of magnetoelectric coupling. In addition, we predict a pronounced thickness dependence of the resonance asymmetries and show how the FMR angular asymmetry can be tuned by an external static electric field. The possibility to observe electric-field-assisted FMR in the absence of an external static magnetic field is also discussed.

Received 23 September 2014
Revised 28 October 2014

DOI:https://doi.org/10.1103/PhysRevB.90.224428

Authors & Affiliations

A. Sukhov¹, C.-L. Jia², L. Chotorlishvili¹, P. P. Horley³, D. Sander⁴, and J. Berakdar¹

¹Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle/Saale, Germany
²Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, 730000, China
³Centro de Investigacion en Materiales Avanzados (CIMAV S.C.), Chihuahua/Monterrey, Mexico
⁴Max Planck Institute of Microstructure Physics, 06120 Halle/Saale, Germany

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Vol. 90, Iss. 22 — 1 December 2014

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Figure 1
Schematic of the effect of finite ME coupling on FMR spectra when: (a) The net magnetization $M$ and the magnetic static field $B$ are both aligned parallel to the electric polarization $P$ and (b) are both antiparallel with the direction of $P$ . Note, that the resonance field $B_{res}$ differs for parallel and antiparallel orientations between $M$ and $P$ .
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Figure 2
(a) Schematic of the plane of variation for the static magnetic field $B$ and the magnetization $M$ . (b) Angular dependence of the positions of resonance fields based on Eqs. (6, 7, 8) in the case of an ultrathin FM film such that $K_{S} / d_{FM} \approx μ_{0} M_{S}^{2} / 2$ and therefore $K_{eff} = K_{1}$ for different cases: I–III. (c) Angular dependence of the positions of resonance fields based on Eqs. (6, 7, 8) in the case of a nanometer-sized FM film such that $K_{S} / d_{FM} \approx 0$ and the demagnetizing contribution $μ_{0} M_{S}^{2} / 2$ becomes dominant, i.e., $K_{eff} = - μ_{0} M_{S}^{2} / 2$ for different cases: I–III. Case I represents the situation without ME coupling, resulting in a fully symmetric angular dependence (b) and (c). Cases II and III demonstrate the changes in response of the homogeneously magnetized sample for different types of ME coupling with powers $n = 1$ and $n = 2$ , respectively. Remarkable asymmetry can only be observed in the case of a linear ME coupling [case II, both (b) and (c)]. For a biquadratic ME coupling the angular dependence changes the strength of $B_{res}$ shifts with no visible asymmetry (b) and (c). The fields are expressed in units of the anisotropy field $B_{A} = 2 K_{1} / M_{S}, ω / γ = B_{A}$ . Polarization is along the $- z$ direction, $P_{z} < 0$ .
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Figure 3
Angular dependence of the positions of resonance fields based on Eq. (10) for different values of the uniaxial anisotropy angle $θ_{u}$ , which is varied in the $x z$ plane as to simulate the situation of a polycrystalline magnetic sample. The fields are expressed in units of the anisotropy field $B_{A} = 2 K_{1} / M_{S}, ω / γ = B_{A}$ . Azimuthal angles $ϕ, ϕ_{u}$ and $ϕ_{B}$ were set to zero.
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Figure 4
Numerically calculated angular dependence of the positions of resonance fields for angles $θ_{B}$ varied in the $x z$ plane for different values of the ME coupling for ${Co / BaTiO}_{3} . Δ B_{res} = B_{res} (θ_{B} = π) - B_{res} (θ_{B} = 0)$ . Further parameters: $ω / (2 π) = 35 GHz$ , uniaxial anisotropy constant of Co $K_{1} = 4.1 \times 10^{5} {J / m}^{3}, θ_{u} = 0, M_{S} = 1.44 \times 10^{6} A / m, α_{FM} = 0.1, d_{FE} = 8, d_{FM} = 1.3 nm$ , and $λ_{MS} = 0$ . No external electric field is applied, and the polarization is kept negative $P_{z} < 0$ . The inset shows the dependence of the asymmetry $Δ B_{res}$ on $λ_{1}$ (orange squares) and a linear fit $Δ B_{res} (λ_{1})$ (black line).
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Figure 5
Numerically calculated angular dependence of the positions of resonance fields for angles $θ_{B}$ for various FM thicknesses $N_{FM} = d_{FM} / a_{FM}$ varied in the $x z$ plane for different values of the ME coupling for ${Co / BaTiO}_{3} . Δ B_{res} = B_{res} (θ_{B} = π) - B_{res} (θ_{B} = 0)$ . Further parameters: $ω / (2 π) = 35 GHz$ , the strength of the ME coupling $λ_{1} = 0.5 s / F$ , uniaxial anisotropy constant of Co $K_{1} = 4.1 \times 10^{5} {J / m}^{3}, θ_{u} = 0, M_{S} = 1.44 \times 10^{6} A / m, α_{FM} = 0.1, d_{FE} = 8, a_{FM} = 1.3 nm$ , and $λ_{MS} = 0$ . No external electric field is applied, and the polarization is kept negative $P_{z} < 0$ . The inset represents the dependence of the FMR asymmetry on the thickness of the FM sample (orange squares) which was fitted by the $Δ B_{res} (N_{FM}) = 0.237 (T) / N_{FM}$ function (black solid curve).
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Figure 6
Numerically calculated FMR-asymmetry $Δ B_{res} = B_{res} (θ_{B} = π) - B_{res} (θ_{B} = 0)$ as a function of applied electric field $E_{z}$ applied along the $z$ axis. Further parameters: $ω / (2 π) = 35 GHz$ , uniaxial anisotropy constant of Co $K_{1} = 4.1 \times 10^{5} {J / m}^{3}, θ_{u} = 0, M_{S} = 1.44 \times 10^{6} A / m, α_{FM} = 0.1, d_{FE} = 8, d_{FM} = 2.6 nm$ , and $λ_{MS} = 0$ .
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