Foldover, quasi-periodicity, spin-wave instabilities in ultra-thin films subject to RF fields

ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 316 (2007) e523–e525 www.elsevier.com/locate/jmmm

Foldover, quasi-periodicity, spin-wave instabilities in ultra-thin films subject to RF fields M. d’Aquinoa,, G. Bertottib, C. Serpicoa, I.D. Mayergoyzc, R. Boninb, G. Guidaa a

Department of Electrical Engineering, University of Napoli ‘‘Federico II’’, Napoli I-80125, Italy b Istituto Nazionale di Ricerca Metrologica (INRIM), I-10135 Torino, Italy c ECE Department and UMIACS, University of Maryland, College Park, MD 20742, USA Available online 12 March 2007

Abstract We study magnetization dynamics in a uniaxial ultra-thin ferromagnetic disk subject to spatially uniform microwave external fields. The rotational invariance of the system is such that the only admissible spatially uniform steady states are periodic (P-modes) and quasiperiodic (Q-modes) modes. The stability of P-modes versus spatially uniform and nonuniform perturbations is studied by using spinwave analysis and the instability diagram for all possible P-modes is computed. The predictions of the spin-wave analysis are compared with micromagnetic simulations. r 2007 Elsevier B.V. All rights reserved. PACS: 76.50.þg; 75.30.Ds Keywords: Landau–Lifshitz–Gilbert equation; Ferromagnetic resonance; Spin-waves; Foldover

The problem of ferromagnetic resonance and spin-wave instability in magnetic bodies of nanoscale dimensions has lately received considerable attention [1]. The traditional theory of spin-wave instabilities [2] is based on the assumption of small magnetization motion around a saturated equilibrium state and it is generally applicable to relatively large bodies. In this paper, we analyze, both theoretically and with micromagnetic simulations, the case when large precessional motion is driven by high power microwave external fields. We consider a ferromagnetic ultra-thin disk (thickness dH5 nm, diameter DH100 nm), with uniaxial crystal anisotropy along its symmetry axis and subject to spatially uniform and circularly polarized external fields. Magnetization dynamics is governed by the LLG equation: qm qm  am  ¼ m  heff , qt qt

(1)

where m ¼ M=M s , heff ¼ Heff =M s (normalized effective field), time is measured in units of ðgM s Þ1 , M s is the Corresponding author. Tel.: +39 081 7683253; fax: +39 081 2396897.

E-mail address: [email protected] (M. d’Aquino). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.03.049

saturation magnetization, g is the absolute value of the gyromagnetic ratio, a is the damping constant. The effective field is heff ¼ ha? ðtÞ þ hm þ ðhaz þ kmz Þez þ r2 m, where hm is the magnetostatic field, k is the anisotropy constant, ez is the unit vector along the z-axis, subscript ‘?’ denotes components of vectors in the plane orthogonal to the z-axis, and spatial coordinates are measured in units of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the exchange length l ex ¼ 2A=m0 M 2s (A is the exchange stiffness constant). The applied field consists of the circularly polarized microwave component ha? ðtÞ ¼ ha? ðcos otex þ sin otey Þ of amplitude ha? and angular frequency o, and of the DC component haz ez ((ex ; ey ; ez ) are cartesian unit vectors). No surface anisotropy is considered. In the limit of magnetic disks of small dimension (diameter Dbl ex and thickness d  l ex ) and under spatially uniform excitation conditions, spatially uniform modes are expected to be the main modes of oscillation of the system. By using the rotational invariance of the system with respect to the axis ez , one can prove that there are two admissible spatially uniform steady states: periodic and uniformly rotating solutions, termed P-modes, and quasiperiodic magnetization solutions, termed Q-modes [3].

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Each P-mode can be simply identified by the deviation y0 of m with respect to ez and by the lag angle f0 of m? with respect to ha? . These angles can be determined analytically through the equations [3] n0 ¼

haz  o þ keff ; cos y0

n20 ¼

h2a?  a2 o2 , sin2 y0

(2)

where n0 ¼ ao cot f0 and keff ¼ k þ N ?  N z (N z and N ? are the body demagnetizing factors). The angle f0 is in one-to-one correspondence with n0 because 0pf0 pp under all circumstances [3], i.e., in a P-mode the magnetization always lags the field. It is important to underline that a given field ðhaz ; ha? Þ is associated with either two or four P-modes [3]. Which P-mode is effectively reached by the system depends on the history of both magnetization and external field. The stability of P-modes can be studied in rigorous terms by using linearization of Eq. (1) around the P-mode solutions m0 ðtÞ specified by formulas (2). The linearized equation is obtained by assuming that m0 ðtÞ is slightly distorted by the perturbation Dmðr; tÞ. In ultra-thin films d  l ex and one can assume that magnetization is spatially uniform across the disk thickness: Dmðr; tÞ ¼ Dmðx; y; tÞ. Since at first order of approximation, m0 ðtÞ  Dm ¼ 0, we introduce unit vectors e1 ðtÞ and e2 ðtÞ parallel to ½ez  m0 ðtÞ  m0 ðtÞ, and ez  m0 ðtÞ, respectively, as timevarying basis lying on the plane perpendicular to m0 ðtÞ. In this basis, Dmðx; y; tÞ ¼ Dm1 ðx; y; tÞe1 ðtÞ þ Dm2 ðx; y; tÞe2 ðtÞ. By linearizing Eq. (1) with respect to Dm and projecting along the unit vectors (e1 ,e2 ), after appropriate manipulations, one arrives to the following equation: ! ! !   Dm1 1 a q Dm1 Dhm2 ¼ C0 þ , (3) Dm2 a 1 qt Dm2 Dhm1 R d=2 where f denotes the average ð1=dÞ d=2 f ðx; y; zÞ dz, ! ao cos y0 n0 þ N ? þ r2 C0 ¼ , (4) n0  N ?  k sin2 y0  r2 ao cos y0 and Dhm1 ¼ Dhm  e1 ðtÞ, Dhm2 ¼ Dhm  e2 ðtÞ. Eq. (3) must be complemented with the following conditions at the body surface: qDm1;2 =qn ¼ 0. For spatially uniform perturbations Eq. (3) is simply reduced to ! ! Dm1 d Dm1 ¼ A0 , (5) Dm2 dt Dm2 where A0 ¼ LðaÞ

LðaÞ ¼

ao cos y0

n0

n0  keff sin2 y0

ao cos y0

1 1 þ a2



1 a

 a . 1

! ,

(6)

(7)

The instabilities due to uniform perturbations occur when the determinant det A0 varies from positive to negative

values (saddle-node bifurcation) and when the trace tr A0 varies from negative to positive values (Hopf bifurcation). For generic spatially nonuniform perturbation, we study Eq. (3) by using the spin-waves analysis [2,4] which is based on the following Fourier’s expansion: X Dmðx; y; tÞ ¼ c ðtÞejqr , (8) q q with q  ez ¼ 0 to take into account the very small thickness of the disk. By neglecting the small effect due to the lateral surface of the ultra-thin disk, and after appropriate manipulations, one can derive that the Fourier coefficients of the averaged magnetostatic field inside the region d=2pzpd=2 are given by   q  cq ðtÞ Dhm;q ¼ q  ½1  sq ðdÞ  ez cqz sq ðdÞ , (9) q2 where Dhm;q ðr; tÞ is the component of Dhm produced by the qth Fourier’s mode in Eq. (8), and sq ðdÞ ¼ ð1  eqd Þ=qd. By substituting the Fourier expansion of Dm and Dhm in Eq. (3) one can derive the following equation for cq ðtÞ: ! ! ! cq1 cq1 d cq1 ¼ Aq þ ½1  sq ðdÞRq ðtÞ , (10) cq2 cq2 dt cq2 where cq1 ¼ cq ðtÞ  e1 ðtÞ, cq2 ¼ cq ðtÞ  e2 ðtÞ, ! ao cos y0 nq Aq ¼ LðaÞ , nq  kq sin2 y0 ao cos y0 LðaÞ Rq ðtÞ ¼ 2

cos y0 sin 2ot cos2 y0 cos 2ot

! cos 2ot ,  cos y0 sin 2 ot

(11)

(12)

where nq ¼ n0  N ? þ q2 þ 12½1  sq ðdÞ, 1 kq ¼ k  sq ðdÞ þ 2 ½1  sq ðdÞ. The key information about spin-wave instabilities is carried by the one-period map [5] associated with Eq. (10). Given the matrix solution C q ðtÞ of Eq. (10), with C q ð0Þ ¼ dij , the one-period map M q is defined as M q ¼ C q ð2p=oÞ. Stability is controlled by the eigenvalues of M q , the characteristic multipliers m . The system becomes unstable whenever jmþ j41 or jm j41. Therefore, one can immediately determine the stability of all P-modes with respect to any particular spin-wave perturbation by numerical integration of Eq. (10) and for all possible values of q. The integration is performed for any value of ðcos y0 ; n0 Þ and for qXqmin (qmin ¼ pl ex =D). The shaded region in Fig. 1 is the region corresponding to P-modes which are unstable for at least one value of q, reported in the ðhaz ; ha? Þ-plane. In order to test the theory, we have performed numerical simulations of LLG equation for an ultra-thin permalloy disk (l ex ¼ 5:71 nm,m0 M s ¼ 1 T, with thickness d ¼ 3 nm and diameter D ¼ 200 nm). The magnetic body is subdivided into a collection of rectangular prisms with dimensions 5  5  3 nm with edges parallel to the coordinate axes. The magnetization is uniform within each cell. The LLG equation is numerically integrated by using the mid-point rule technique [6], which is particularly suitable for long simulation time owing to its

ARTICLE IN PRESS M. d’Aquino et al. / Journal of Magnetism and Magnetic Materials 316 (2007) e523–e525

C

0.7

m⊥

0.235 GHz 100

D

0.5

101

f [GHz]

time [ns]

0.4



0.2

0.3

B

0

0.2

-0.2

0.1

preservation of the properties of LLG dynamics. The numerical results are reported in Fig. 2. The disk is initially saturated along the z-axis by a very strong DC field haz ¼ 1:5. Then, the RF field of amplitude ha? ¼ 0:0056 and angular frequency o ¼ 0:25 is applied. Afterwards, the RF field is kept constant and the DC field is linearly decreased till a value haz ¼ 0:5 in a time interval of 2000 ns. Finally, the DC field is increased back to the initial value haz ¼ 1:5, again in 2000 ns. During the whole simulation the quantity m? ¼ ðhmx i2 þ hmy i2 Þ1=2 is stored (hi denotes spatial average). In the first part of the simulation the branch ABC is traced where P-modes are observed. The numerical solution is reasonably close to the curve (dotted line) given by P-mode theory. An irreversible jump between points C and D is observed. This instability can be justified by taking into account that the point C represented in the ðhaz ; ha? Þ-plane (see Fig. 1) is just inside the spin-wave instability region. After the jump the system reaches Q-modes regimes (branch DEF) which disappear through an Hopf bifurcation at the value of haz  0:95. This is slightly larger than the value predicted by the theory haz ¼ keff  0:92, but still in a very reasonable agreement. In the branch FG the solution follows the P-mode foldover lower branch up to the point G where again the system (see Fig. 1) enters the spin-wave instability region and the solution is driven into a new regime in which spatial nonuniformities are slightly more important.It is interesting to notice that magnetization jumps back to the

1010 101

0.6

Fig. 1. Portion of the spin-wave instability (shaded) region in the ðhaz ; ha? Þ-plane. The value of parameters are m0 M s ¼ 1 T, a ¼ 0:02, o ¼ 0:25 (which corresponds to 7 GHz), l ex ¼ 5:71 nm, d ¼ 3 nm, N z ¼ 0:9498 keff ¼ 0:9248. The horizontal thick line represents the external field history. The line labeled with det A0 ¼ 0 refers to instability due to uniform perturbations (foldover phenomenon). The line labeled with tr A0 ¼ 0 refers to the disappearing of Q-modes regime (Hopf bifurcation). The points A,B,C,F,G,H are indicated in order to compare this figure with Fig. 2.

7 GHz

105

E

0.8

spectrum of

0.9

e525

2786 2788 2790 2792

0

0.5

F

G 1

H

A 1.5

haz [Ms] Fig. 2. Numerical simulations of foldover and instability processes. Plot of m? as function of DC field haz . The value of the parameters are the same as Fig. 1. The RF field is constant ha? ¼ 0:0056. The continuous line with symbols ‘’’ represents numerical simulations. The dotted line represents the spatially uniform P-mode foldover theoretical curve. The points A,B,C,F,G,H are also indicated in Fig. 1. The shaded region around the branch D,E,F indicates numerically observed Q-modes. Inset (a) and (b): hmx iðtÞ; hmy iðtÞ and computed spectrum of hmx i in the case of a Q-mode. The quasi-periodicity derives from the combination of the microwave frequency f ¼ 7 GHz and the frequency f ¼ 0:235 GHz selfgenerated by the system.

foldover upper branch at point H, exactly at the point where the jump is expected by the P-mode theory. The spatial uniformity has been monitored during the simulation by means of the value ðhmx i2 þ hmy i2 þ hmz i2 Þ1=2 . This value is very close to unity on the branches ABC, FG, BA, whereas it differs from 1% up to 5% on CD and GH, near the irreversible jumps. We can conclude that, for ultra-thin disks, the spin-wave theory can reasonably predict the excitation conditions for which deviations from P-modes theory have to be expected. This work was partially supported by the BURC Project (Regione Campania, L.R.5 28/03/02, Annualita` 2003) and the MIUR-PRIN Project N.2006098315. References [1] [2] [3] [4] [5]

G.N. Kakazei, et al., Appl. Phys. Lett. 85 (2004) 443. H. Suhl, J. Phys. Chem. Solids 1 (1957) 209. G. Bertotti, et al., Phys. Rev. Lett. 86 (2001) 724. G. Bertotti, et al., Phys. Rev. Lett. 87 (2001) 217203. L. Perko, Differential Equations and Dynamical Systems, Springer, Berlin, 1996. [6] M. d’Aquino, et al., J. Comput. Phys. 209 (2005) 730.