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Volume 75, 2015, Pages 49-55 20th International Conference on Magnetism

Electric tuning of magnetization dynamics and negative magnetic permeability in nanoscale composite multiferroics

Min Chen, Tongli Wei, and Chenglong Jia*

Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou

University, Lanzhou 730000, China

Abstract

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 qualitatively new functionalities. FM/FE heterostructures allow, for instance, the electrical manipulation of magnetic anisotropy via interfacial magnetoelectric (ME) couplings. Here we present a magnon-driven, strong interfacial ME effect acting within the spin-diffusion length of the order of nanometers. This type of linear ME interaction allows for electrical control of simultaneously the magnetization precession and its damping, both of which are key elements for magnetic switching and spintronics. The ferromagnetic resonance unravels further an electric-field-induced negative magnetic permeability effect, pointing so to a new class of optical negative-index metamaterials based on composite multiferroics.

Keywords: Multiferroics, Magnetoelectric interaction, Magnetization dynamics, FMR

1 Introduction

Motivated by the potential of novel technological applications, the emerging effects at the interface of heterostructures have attracted a multitude of investigation in recent years [1, 2, 3, 4]. In particular, controlling magnetism via applying electric fields based on interfacial magnetoelectric (ME) effects in ferroelectric (FE)/ferromagnetic (FM) composites[5, 6, 7, 8, 9] holds great promise in multifunctional low-energy consumption devices. Indeed, along this line a series of findings have been reported, such as voltage-controllable magnetic anisotropy, i.e., the effective magnetic field that is responsible for magnetization procession, in thin FM metal films and synthesized functional multiferroic FM/FE heterostructures [3, 4]. It is however difficult to manipulate the precessional damping, especially by electrical means. Here we demonstrated that the spin-current-based mechanism of magnetically driven ferroelectricity in single-phase multiferroics [10, 11, 12, 13, 14, 15] is a key element for interface ME coupling as well: Interfacing FM with FE triggers in the FM low-energy (coherent magnonic) excitations near the

* E-mail address: cljia@lzu.edu.cn

Selection and peer-review under responsibility of the Scientific Programme Committee of ICM 2015 49

(gi The Authors. Published by Elsevier B.V. doi:10.1016/j.phpro.2015.12.008

interface, which builds up a spiral spin density within the spin-diffusion length (on the order of nanometers). Considering the dynamics of the nonequilibrium spin density at the interface, two electrically controllable feedback contributions to the magnetization dynamics are to be introduced in the linear response approximation: (i) A linear ME interaction and thus an effective gate-controlled magnetic field resulting from the adiabatic component of surface spin density. (ii) An additional electrical tunable spin torque that is directly related to the magnetic damping and can be traced back to the transverse deviations from adiabaticity.

2 Interfacial Spin density.

Typically, several ME coupling mechanisms in composite FM/FE heterostructures have been considered; for instance spin-polarized screening effects, strain effects, and exchange bias [16, 17]. A detailed analysis show, however, that the latter two indirect ME interactions, i.e., the interfacial strain-mediated and exchange-mediated couplings, do contribute to magnetic-anisotropy-driven phenomena, i.e., they alter in effect the effective magnetic field only, but have no direct influence on the relaxation (damping) of the magnetization dynamics.

What if the ME coupling originates from pure screening effects?. Bringing a FM in contact with a FE material a spin polarized nonequilibrium charge density s = tt' ^t(r)6tt'(r), develops in the FM interface in response to the adjacent FE polarization. and ^t are the electron creation and annihilation operators, respectively, which satisfy the normal anticommutation relation {^t (r),^t (r')} = J(r — r')5tt'. In the mean-field approximation, s interacts with localized spins S via s — d exchange interaction [11, 18, 19, 20], Hsd = JexS • with eM|| = M/Ms and the classical magnetization M = — ^^r S, where ¡iB, g and a are the Bohr magneton, g-factor, and lattice constant, respectively. Ms is the intrinsic saturation magnetization. M is distributed with some spatial period in ordered system. After taking into account the kinetic energy and the electrostatic potential V(r)n(r) with n(r) = (—e) ^t ^t(r)^t(r) being the charge density operator, we write the total Hamiltonian for non-interacting surface electron as

f-2 t> H =2^E/ drWt (r) -V^t (r) t J

+ y dr[V (r)n(r) + Hsd].

The dynamics of the spin density is governed by the Heisenberg equation of motion d| = 1 [s, H]. Adopting a semiclassical approach and taking the average over all electron states, we have the Bloch equation for the spin density s = (s) [21],

ds 1 s

dt + = — —S X eM — 77 (2)

dt 'ex 'sf

where J = 2m (9#t 6" ® Vtp]) is the spin current density including the nonequilibrium surface electronic charge buildup and Tex = h/(2Jex). The spin-flip relaxation time Tsf is due to scatterting with impurities, electrons, and phonons. As the spin polarization n of electron density in FM metals is less than 1 within the Stoner mean-field theory, it is instructive to write the induced spin density into two parts,

s(r,t)= s||(r,t)+ s±(r,t) (3)

where S|| is the spin density whose direction follows in an adiabatic sense the magnetization due to local exchange couplings at an instantaneous time t, i.e., S| = вцемц. sx describes the transverse deviation from M.

Introducing s (Eq. 3) into the Bloch equation for the spin density (Eq. 2), we have

dsn de m ds n n

-Ж^ + ^ + Ж - ^ - sX

1 sll s± (Л\ =--S± x eM||--y---• (4)

Tex Tsf Tsf

Here it should be noted that the spin current in the absence of a charge current across the FM/(insulating) FE heterostructure is related only to the nonequilibrium spin density s normal to the interface (hereafter referred to as the ez direction; the z axis has its origin at the interface)

J = -D0VZ s|| and J = -D0VZ s±, (5)

where D0 is the diffusion constant. In the following, let's investigate the static and dynamic ME effects in detail respectively.

3 Static Magnetoelectric Effects

In the case that the magnetization M is not actuated, we obtain the stationary form for the nontrivial spin-density in the linear response limit,

DcVZ s|| = Tf (6)

DqV2s± = Tf s± x eMs + Tf (7)

For long-range magnetic ordering in typical FM metals and alloys at room temperature, Tex/^sf ~ 10-2 [22], the interface spin density s = [s|, Sy, S||] in the rotating frame around M is given by

sp = C^e-z/xm (8)

sx± + is\ « C±e-(1-i)Qm-r+i&o (9)

Q0 is the initial phase. Am = \JD0Tsf is the effective spin-diffusion length at the surface. The spin-wave vector Qm = ,2 D [0, 0,1] normal to the interface indicates a local spiral density, in despite of the direction of M. Generally, the diffusion length Xm differs from the value in typical transition metals and alloys (8.5 ± 1.5nm in Fe and 38 ± 12nm in Co [22]); however, exchange interaction with long-range magnetic ordering enhances the screening spin density s penetration into the FM system within the nanometer scale, thus spins are rearranged within a much larger characteristic length compared with charge screening (several atomic layers ), which leads to a significantly ME interaction on thin FM films. Ci is given by the electric neutrality conditions:

P = f dz\sn\ + f dz\s±\ (10)

V = f dz\sn\/ f dz(|s||| + \s±\) (11)

where Ps is the surface electron density due to the electrostatic screening over \m, and n is the spin polarization of electron density in a FM within the Stoner mean-field theory [23]. we then derive

C = nVsl\m, Ci = (1 - n)QmPs• (12)

Please noted that for a half metal with n = 1, we have Ci = 0. The screening spins are fully polarized without any nonadiabatic deviation. For a FM metal (n < 1), we expect changes of the magnetization on both the easy and hard axes, respectively given as (d is the FM film thickness)

ДМ|| = nPVB, AMi = (1 - n)PVb. (13)

An effective positive charge screening is developed within the FM metal surface when an outward electric field is applied on a FM interface, in the meanwhile the surface magnetization is suppressed, while an inward field imples an effective negative charge screening and induces the surface magnetization.

The surface charge density in the FM subsystem and the relaxed coarse-grained FE polarization Po at the interface obey ePs = . Considering the s — d exchange interaction Hsd we arrive at an effective interfacial ME interaction

1 f J -

Fme = ^ / drJexs ■ eM| = Ms|| ■ M (14)

where S| = nPs/dFM and dFM is the FM film thickness. A contribution to the effective magnetic field from the adiabatic component of the surface spin density is then given by HME = SFme/SM as

H ME Jex_ i1r\

eff = —1ЛГ SlleM,l • (15)

It is obvious that H;effE is ferroelectric tunable.

4 Magnetoelectric Dynamics

As First principle calculations suggested that the amplitude of the adiabatic spin density S|| is frequency-insensitive and thus ds^/df in Eq.2 is disregarded. In the FMR dynamics, the transverse deviation s^ is found to be mainly dominated by the time variation of the megneti-zation M, the contribution of s^ to the spin dynamics can be ingnored, as well. ds^/dt is on the order of d2M/dt2 and can be omitted in the linear response approximation. Thus, the spin-density dynamics is decribed by the form-closed equations

DoV2s|| = SL, (16)

sil = — ±si - eM| — % (17)

The Landau-Lifschitz-Gilbert ( LLG ) equation, which describe the magnetic dynamics phe-nomenologically, is

d M ^ TT a dMN

~dt = (M X Heff ) + M(M X ~6t )' (18)

where 7 is the gyro-magnetic ratio. Hef f is the effective magnetic field, governing the procession directly and Heff = "SF/SM. F is the free energy density, which includes the related magnetic

interactions such as the exchange, the demagnetization, the magneto-crystalline anisotropy, and possibly Zeeman energy due to an external magnetic-field. From the dynamic equation of s^, Eq.(17), we deduce that

s± = -

'£s|| dM S| m dM

W,~dt + M2 x ~dt

with e = Tex/Tsf. s^ in turn exerts a spin torque on the magnetization

ME =--^ eM|| x s±

dM £sn dM 1 (20)

Ms dt M2 dt

The similarity between the functional structure of ME torque TME and the terms appearing in the LLG equation manifests that when taking this ME torque into account the LLG equation can be effectively renormalized as

7 = y/(1 + P), a=(a + £P)/(1 + P), (21)

which illustrates that in general ME changes the effective magnetic field and the preccessional damping, both governing the magnetization dynamics, and hence it allows to manipulate both by electric means, for the ME coupling influenced electrically. In the above relation P = M i+1g2. Taking that £ ^ 1 and S^/Ms ^ 1, e.g., in typical transition metal layer, one infers that 7 « 7 and _

a=a+M, (22)

where the averaged s| has been exploited to derive the effective damping constant a.

Considering that ePs = eE with e being the electron charge, the induced magnetization s| is linearly determined by the applied electric field E and the dielectric permittivity e (in the unit of the vacuum permittivity e0) at the FE/FM interface, which can be significantly enhanced up to several orders ( compared with e ~ 1 in bulk FM metals ). Furthermore, depending on an outward/inward electric field ( implying an effective hole/particle screening within the FM metal surface ), one tunes to a negative (suppression)/positive (inducement) surface magnetization s|, which is consistent with the first prinple results and previous experimental observations. This is insofar important, as increasing the outward electric field, s| becomes increasing negative resulting in a linear growth of the effective magnetic field HME according to equation (15). By the same token we expect a decrease of the effective damping a given by equation (22), and probably more detectable, an emergence of positive-to-negative transition in a for sufficiently small intrinsic damping a at the critical point Ec = (aMsedpM)/(£ne^B).

5 Conclusion

We unraveled theoretically an exponential spiral spin density at a FM surface/interface due to an interplay of electrostatic interaction, the s - d exchange interaction, and spin diffusion. The noncollinearity of spins results in a strong interfacial ME interaction, which manifests itself as an effective magnetic field (Eq. 15) and the dynamic coupling possesses substantial influence on the magnetic damping (Eq. 22). Such the direct electric control of magnetic dynamics that offers a qualitatively new way to manipulate multiferroic devices with fast low-power

heterogeneous read/write capability through the interfacial ME interaction. Indeed, our recent direct permeability measurements of polycrystalline CoZr/PMN-PT and Co/PMN-PT show the emergence of positive-to-negative transition in both real and imaginary parts of magnetic permeability [24]. Given that the dielectric permittivity of a metal is negative below its plasma frequency, the electric tuning of the magnetic permeability opens new perspectives to construct optical negative-index metamaterials based on composite multiferroic nano-structures.

Acknowledgments.- This work was supported by the National Natural Science Foundation of China (No. 11474138), the National Basic Research Program of China (No. 2012CB933101), the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT1251), and the Fundamental Research Funds for the Central Universities.

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