Electric tuning of magnetization dynamics and electric field-induced negative magnetic permeability in nanoscale composite multiferroics

Steering magnetism by electric fields upon interfacing ferromagnetic (FM) and ferroelectric (FE) materials to achieve an emergent multiferroic response bears a great potential for nano-scale devices with novel functionalities. FM/FE heterostructures allow, for instance, the electrical manipulation of magnetic anisotropy via interfacial magnetoelectric (ME) couplings. A charge-mediated ME effect is believed to be generally weak and active in only a few angstroms. Here we present an experimental evidence uncovering a new magnon-driven, strong ME effect acting on the nanometer range. For Co92Zr8 (20 nm) film deposited on ferroelectric PMN-PT we show via ferromagnetic resonance (FMR) that this type of linear ME allows for electrical control of simultaneously the magnetization precession and its damping, both of which are key elements for magnetic switching and spintronics. The experiments unravel further an electric-field-induced negative magnetic permeability effect.


INTERFACE MAGNETOELECTRIC EFFECT
Here we give a summary of the main implications following from an interfacial magnetoelectric (ME) coupling based on the formation of a spiral magnetic order that is coupled to the polarization in a ferroelectric (FE) interfaced with ferromagnet FM [1]. In a composite FM/FE multiferroics a spin polarized nonequilibrium densityŝ = ∑ σσ ′ ψ † σ (r)σ σσ ′ ψ σ (r) develops in FM interface due to the surface FE polarization. ψ † σ and ψ σ are respectively the electron creation and annihilation operators, satisfying the anti-commutation relation In the mean-field approximation,ŝ interacts with localized spins S via s-d exchange interaction [2], H sd = J exŝ · e M ∥ with e M ∥ = M/M s and the classical magnetization M = − gµ B a 3 S, where µ B , g and a are the Bohr magneton, g-factor, and lattice constant, respectively. Together with the kinetic energy and the electrostatic potential V (r)n(r) withn(r) = (−e) ∑ σ ψ † σ (r)ψ σ (r) being the charge density operator, the total Hamiltonian for non-interacting surface electrons reads . (1) Upon considering the dynamics of the spin density operatorŝ based on the Heisenberg , we obtain a Bloch equation for the spin density s = ⟨ŝ⟩ in the semiclassical approach [3], ⟩ is the spin current density with nonequilibrium surface electron charge buildup. No steady charge currents occur in the present FM/FE setup. The spin current is thus related only to the nonequilibrium spin density s along the interface and τ sf being the spin-flip relaxation time due to scattering with impurities, electrons, and phonons, etc. In general, the spin polarization η of electron density in transition FM metals is less than 1 within the Stoner mean-field theory [4], it is therefore instructive to separate the induced spin density into two parts, where s ∥ represents the spin density whose direction follows approximately the intrinsic magnetization M due to local exchange couplings at an instantaneous time t, i.e., s ∥ = s ∥ e M ∥ .
s ⊥ describes the transverse deviation from M. Then eq. (2) can be rewritten as As suggested by first principle calculation [5] the amplitude of the adiabatic spin density s ∥ is frequency-insensitive and thus ∂s ∥ /∂t is disregarded here. In the FMR dynamics, the transverse deviation s ⊥ is found to be mainly dominated by the time variation of the magnetization M, [3] the contribution of ∇ 2 z s ⊥ to the spin dynamics can be ignored, as well. ∂s ⊥ /∂t is on the order of ∂ 2 M/∂t 2 and can be omitted in the linear response approximation.
Under the above approximations, we obtain thus a closed form for the spin-density dynamics, The diffusion equation (5) results in an exponentially decaying surface spin density, s ∥ = determined by the electrical neutrality constraint, C ∥ = ηP s /λ m , where P s is the surface electron density due to the electrostatic screening. Clearly, s ∥ can penetrate into the FM system within the spin diffusion length λ m , which is over tens of nanometers in typical transition metals and alloys [6], giving rise to a marked interfacial ME interaction on overall thin FM films via the s-d exchange interaction, wheres ∥ = ηP s /d FM and d FM is the FM film thickness. A contribution to the effective magnetic field is then given by H ME eff = −δF ME /δM as Considering that with e and ϵ being the electron charge and the dielectric permittivity at the interface, respectively, the induced magnetizations ∥ is linearly determined by the applied electric field E, so does the effective magnetic field.
The dynamic equation of s ⊥ , Eq. (6) deduces that with ξ = τ ex /τ sf . s ⊥ in turn exerts a spin torque on the magnetization, Comparing the functional structure of ME torque T ME with the terms appearing in the LLG , we conclude that when taking this ME torque into account it effectively renormalizes the LLG equation as with β = s ∥ Ms 1 1+ξ 2 . This makes evident that in general ME changes the effective magnetic field and the precessional damping, allowing to vary both by electric means, in as much as ME coupling is tunable electrically.
Especially, in the event that ξ ≪ 1 ands ∥ /M s ≪ 1, e.g., in typical transition metal layer, one infers thatγ ≈ γ andα where the averageds ∥ has been exploited to derive the effective damping constantα. It should be noted that, ass ∥ becomes increasingly negative by tuning the applied gate-voltage, it results in a linear growth of the effective magnetic field H ME eff according to equation (8). By the same token we expect a decrease of the effective dampingα given by equation (13), and even a positive-to-negative transition inα for sufficiently small intrinsic damping α at the critical point E c = (αM s ed FM ) / (ξηϵµ B ).

STATIC MAGNETOELECTRIC RESPONSE
The static magnetic properties of CoZr(20 nm)/PMN-PT response to external electric field are demonstrated in Fig.s1. M s is not changed obviously when applying the electric field, which means that the change of the remnant magnetization M r is caused by the induced magnetic anisotropy via the interfacial magnetoelectric effect. The dynamic components m = [0, m y , m z ] has a time variation e −iωt and its magnitude is much smaller than M s . For low excitation rf field h, a linearization of the LLG equation leads two coupled equations, which yields the dynamic magnetic susceptibility χ z : The measured microwave power, P (H), absorbed by a magnetic film is given by [7] P (H) = 1 2 ωℑχ z |h| 2 . FMR therefore occurs when the imaginary part of the magnetic susceptibility ℑχ y is maximal. For small damping one can set α ≈ 0 the maximum is given then by (Kittel formula) Under certain the conditions FMR spectra are found to have almost perfect Lorentzian shape [8]. This can be shown by replacing H eff with H r + δH. In linear δH − iα ω γ the dynamic susceptibility becomes Rationalizing the denominator yields the imaginary and the real parts of χ z , Considering the profile, the imaginary part ℑχ z represents a symmetric Lorentzian with a half-width at half-maximum (HWHM) ∆ = α ω γ . Whereas, the real part ℜχ z is antisymmetric and has a zero-crossing point at H eff = H r . It should be noted that, in general, the magnetization is probed with a certain phase correlation with respect to the microwave excitation, the FMR spectrum does not correspond to only imaginary part of the susceptibility, but represents a mixture of imaginary and real parts [9]. Hence, the actual fit function can be given by an asymmetric Lorentzian function: where ϕ is the phase which mixes the real and imaginary parts of the dynamic susceptibility.
∆ eff = ∆ cos ϕ is the effective inhomogeneous line broadening of FMR spectra.
The angular dependence of FMR spectra of CoZr(20nm)/PMN-PT when varying the applied magnetic field in plane is presented in Fig. s2.