B. A. Ivanov 1 and C. E. Zaspel , USA. PACS: a, Gb, Hk

Transcription

1 Excitation of pin dynamic by pin-polarized current in vortex tate magnetic dik B. A. Ivanov 1 and C. E. Zapel 1 Intitute of agnetim, 0314 Kiev, Ukraine; Univerity of ontana-wetern, Dillon, T 5975, USA PACS: a, Gb, Hk 1

2 A pin-polarized current with the polarization perpendicular to the plane of a vortex-tate dik reult in renormalization of the effective damping for a given magnetization mode, and the effective damping become zero if the current exceed a threhold value, I c. The lowet I c correpond to the lowet frequency, ω G vortex gyrocopic mode. For I Ic >, the dynamic magnetization tate i characterized by preceion of the vortex around the dot center with nonmall amplitude and higher frequency, ω > ω. G

3 The injection of a pin current in magnetic nanotructure can reult in intereting new effect having potential application in pintronic. For example, it wa firt theoretically predicted by Slonczewki [1] and Berger [] that a dc pin polarized current in a uniformly magnetized thin film renormalize the pin wave damping and above a critical value of the pin current, I c can reult in negative damping and generation of pin wave. For a more complete theoretical decription of the nature of the mode excited by a pin-polarized current it i neceary to take nonlinear effect into account. In the thin film it wa hown [3] that nonlinearity from the four-wave interaction will limit the growth of the excitation. oreover, it wa predicted [4] that the table tructure will be a elf-localized pin wave bullet. In nanomagnet uch a thin dik the ground tate tructure i vortex-like providing an ideal ytem to tudy pin-tranfer effect in confined nonuniform tate. For thee ytem it wa recently hown [5] that a pin-polarized current with the polarization in the vortex plane reult in a diplacement of the vortex core owing to the lower ymmetry introduced by the current. ore recently for the imple model of the eay-plane magnet it wa theoretically predicted that the vortex core polarity can be witched [6] by a pin current perpendicular to the vortex plane, however, ome of the important magnetotatic effect were neglected. Thi particular ituation ha direct application to information torage ince the vortex core polarization a well a the vortex chirality can potentially each tore one bit of information. In thi letter the nonlinear dynamic effect from a dc pin polarized current are invetigated including dominant magnetotatic effect from both urface and volume magnetotatic charge. Since the dynamic effect from a pin polarized current are theoretically invetigated here, we begin with a hort introduction to the pectrum of vortex-tate dik. The lowet frequency (ub GHz) excitation correpond to gyrotropic ocillation of the vortex core a a reult of an initial diplacement induced by an in-plane magnetic pule [7,8]. There are alo higher frequency (GHz) mode excited [9-11] and their tructure depend on the ymmetry of the initial pule. In particular, the radially ymmetric mode can be excited by an out-of-plane pule. 3

4 To determine how a current will affect thee excitation, the Landau-Lifhitz equation are analyzed including effect from the pin-polarized current and diipation. For the particular cae were the polarization direction, ˆp i perpendicular to the vortex plane, and the cylinder thickne, L i mall compared to the radiu, R the magnetization can be aumed to be uniform in the perpendicular ( z -direction) and the Slonczewki form [1] for the pin torque i applicable, τ = σ I ( ˆ m m p), (1) where I i the current and σ = ε gµ / e LA where ε i the pin-polarization efficiency, e i B the magnitude of the electric charge, g i the Lande factor, µ B i the Bohr magneton and A = π R c i the area of the circular nanocontact, where R c = R in the following. Alo thi i a function of the unit magnetization vector, m= / where i the aturation magnetization, and thi can be expreed ( m= inθcoϕrˆ + inθinϕχˆ + co θ zˆ ) in term of polar and azimuthal angle, θ and ϕ, repectively. Taking into account diipation and the Slonczewki torque, the equation of motion for the magnetization have the following form ϕ δw δq inθ = + γ t δθ δθ, () θ δw δq σiin θ inθ = +, γ t δϕ δϕ γ (3) for the particular cae when ẑ i the polarization direction. Here Q i the diipation function which reult in Gilbert damping. Q r ( θ in θϕ ) α = + dr, γ (4) Firt it i hown that there will be a critical current where damping will be balanced by the Slonczewki torque. Thi i eaily done through an energy balance etimate, by multiplication of () by ϕ, multiplication of (3) by θ and integration over the dik volume to get 4

5 the time derivative of the energy dw dt α L γ ( θ in θϕ ) = + d x+ σ I L γ in + θϕd x. (5) Since it i aumed that the dik i thin, the magnetization i aumed to be contant in the z direction. A expected, diipation will reult in energy lo, but the econd term can be poitive o there can be a critical current, I c where the energy diipation i zero, and for I > I the mode c become untable and it amplitude will increae. To proceed further for an arbitrary mode, ue the uual anatz [9] θ = θ0 + ϑ, ϕ = χ + π / + µ / inθ 0, where 0 θ decribe the vortex profile, ϑ ( r, χ, t) = f ( r) co( mχ ω t) and ( r,, t) = g ( r) in ( m t) mn, mn, µ χ χ ω determine the mn, mn, magnon mode with an azimuthal characteritic number, m and radial characteritic number, n. Here r, χ are polar coordinate in a dot plane, the form of function fmn, ( r), gmn( r), are wellknown [11,1]. Then the condition dw / dt = 0, which i the definition of the critical current, I c give To find the value of c αω + = σ θ. (6) f g rdr Ic fg co 0rdr I, we need definite form of the function fmn, ( r), gmn( r),, but ome general remark can be made. Firt, it i noticed that for mall α << 1 the mode damping decrement from Gilbert diipation can be written a Γ= Q/δ E, where δ E i the mode energy, calculated uing the particular mode with α = 0. The imple calculation how that, the expreion for Γ contain the ame combination of integral a in (6), and the threhold value can be expreed through the value of Γ and the mode frequency ω, I c Γ = σω. (7) Therefore, in the following the critical current i proportional to the ratio Γ/ω. The value of ω 5

6 for different mode are well known, and Γ for ome mode are determined experimentally [13], and the lowet Γ/ω correpond to vortex gyrotropic mode [14]. One can expect that a the current increae the mallet critical current will correpond to amplification of the ub-ghz gyrotropic mode. A we will ee below, the value can be a mall a a few ten of microamp; wherea, for other mode (including the m = 0 mode correponding to core ize ocillation, conidered in [6]) our calculation give higher value in the milliamp range. Since the gyrotropic mode ha the lowet value of I c, let u concentrate on quantitative analyi of thi mode, determining the current dependence of the mode amplitude. The gyrotropic mode in the nonlinear regime can by modeled by the Thiele equation [15], and the eaiet way to derive thi equation including the Slonczewki torque i multiplication of () by ϕ, (3) by θ, add and integrate over the dik volume. Then ue of the vortex anatz, m= m r X 0( ), where m0( r ) i the magnetization of the tatic vortex at the origin, give the Thiele equation dx W dx G η + F SPC = 0, (8) dt X dt where G = zˆ π L/ γ i the gyrovector [15,16], the vicoity coefficient i (( ) in ( ) ) L η α θ θ ϕ γ = + d x, (9) and the pin polarized current (SPC) force i L σ ϕin θ γ F SPC = I d x. (10) From previou work the preceion frequency can be obtained from the firt two term of (8) uing V = ω X and the quadratic dependence of the magnetotatic energy, W = κx to get imply ω = κ G. Alo from (8), the critical current can be obtained by equating the damping force to the SPC force, but to accomplih thi it i next neceary to evaluate the integral in (8) and calculate the magnetotatic energy for the diplaced vortex. 6

7 For the evaluation of (9) and (10) it i a good approximation to neglect the contribution from the core tructure and evaluate the radial integral in the range l 0 r R where l = A π i 0 4 the exchange length, A i the exchange contant. For permalloy the core radiu i the order of l 0 which i about 5 nm. In thi cae θ = π / and the vicoity coefficient i given by απ L dθ in θ η = + rdr, γ dr r (11) or, in main logarithmic approximation, η = ( πα / γ )ln( Rl / 0). Uing equation (8), the value of η can be connected with damping and frequency of gyrotropic mode, with the value of Γ found by Gulienko [14]. When the centered ( X = 0) Γ/ ω = η / G, and compared vortex tructure i ued to evaluate (10) one obtain zero o it i neceary to aume a form for the diplaced vortex tructure. Previou work indicate that boundary condition are approximately fixed at the dik edge reulting in no net edge magnetotatic charge [7,8]. For thi reaon it i ueful to ue the vortex-image vortex anatz ϕ y y π, = + +, x a x R / a 1 1 ( xy) tan tan (1) where R i the dik radiu and a i the diplacement of the vortex center on the x axi, and the image vortex i outide the dik at R / a alo on the x axi. The more general anatz [8] for an arbitrary diplacement i lightly more complicated, but thi impler form i ufficient. Uing (1) it i poible to evaluate (10) exactly to obtain ( ˆ ) F SPC = πlσ I z X (13) for an arbitrary diplacement. Thu, for circular vortex motion with contant frequency the SPC force i antiparallel to the friction force, and both are perpendicular to vortex diplacement. In contrat, both the gyroforce and retoring force, for any W = W( X ), have a radial direction. Then, the condition ω a = dw/ da give the frequency for amplitude a = X, and the condition π σ L I = ωη determine the value of current reulting in thi motion. 7

8 The remaining force in (8) i from the magnetotatic potential originating from the volume magnetotatic charge denity, which will be obtained uing the magnetization expreed by (1). Thi i the critical part where nonlinearitie become important owing to the diplacement of the vortex core. For the region outide the core the normalized magnetization i given by mx = and my = inϕ with m z = 0. Next thee component of the magnetization are expanded in erie on ar /, a a mx = mx0 + mx 1 + mx + (14) R R For the conideration of nonlinear effect, third and fourth order term hould be ued. For evaluation of integral it i convenient to ue a cylindrical ( r, χ, z) coordinate ytem. We preent here few lowet term only, m x0 = in χ, y0 co m x m = χ, = [ ( R r ) ] in χ, m x 1 Rr = [ ( R r ) Rr] χ, and = ( r R ) ( r + r R 3 ) in χ co χ 1 in m y R. The term m x have the imilar tructure to m y, other term are eaily obtained but are too long to include here. by The main contribution to the energy come from the volume magnetotatic charge given W S ( m Φ + m Φ) S L = x x y y d r (15) including the partial derivative of the magnetotatic potential given by dx dy xφ = L + L 4 ( x x ) m( x, y ) ( r r ) ( r r ) (16) where ( r r) = ( x x ) + ( y y ), with a imilar expreion for y Φ. Next ue (14) (16) to obtain the econd and fourth order expreion for the magnetotatic energy a a power erie in a R. It i remarked that all of the odd contribution will be zero becaue of the form of the anatz (1). oreover, the integral over χ i trivial for all order, but 8

9 before thi integral i done it i convenient to make the ubtitution, α = χ χ. Then the econd order contribution can be expreed in dimenionle form by the ubtitution, r r = Rξ a univeral potential integral = Rξ and Φ ( λ) () La WS = π Φ ( λ), λ = L/ R, (17) R ( ξ ξ )( ξ coα ξ ) coα = ξ dξ dξ Π Π + λ where the upper limit are ξ = ξ = 1 and to horten equation the notation Π Π ( ξ, ξ, α) = ξ + ξ ξξ coα i ued. The integral over α can be eaily done for the thin dik approximation where the L /4R contribution in the denominator i neglected giving combination of elliptic integral. The remaining radial integral over ξ and ξ are done numerically. Similar but more complicated expreion appear for the fourth order term finally giving the magnetotatic energy (4) W S W S 4 L a = π Ra (18) R R The retoring force from (18) can be ued with the other force in (8) to get the vector Thiele equation 4πω ω L R e r a R R + 4π σi αω ln( ) e χ = 0 (19) l0 In the abence of diipation and current in the linear approximation the firt two term will give the frequency of the gyrotropic mode, ω = 0. 5 ω ( L/ R ), (0) where ω = 4πγ which i about 30 GHz for permalloy. The dependence on L/ R i the ame a found in by a different method, with a light difference in the numerical coefficient [18]. The 9

10 third term will give the nonlinear frequency hift in a form a ω( a) = ω ( ) (1) R The critical current i etimated uing the lat two term in (19) with a = 0 I c αω R = ln, σ l0 () coinciding with the imple etimate baed on equation (6, 7). To etimate the critical current for a typical dik ( R = 00 nm and L = 0 nm) for 5 permalloy ( = 8 10 A/m) one obtain Ic 0µ A. Larger value of the current I > Ic will reult in preceion motion of the vortex with amplitude a. The dependence of a on the current i determined uing the nonlinear term in (1) a R =. 153 I I c 0 (3) Ic demontrating a oft non-hyteretic regime of excitation for thi mode. In concluion, mall value of a pin-polarized current can counteract diipation and reult in gyrotropic vortex motion. Since the polarization direction i perpendicular to the vortex plane, thi current cannot excite the initial motion, but any mall fluctuation of the vortex poition or weak and hore in-plane field pule can produce the initial vortex diplacement, which will be amplified by the current. Thi effect can be oberved experimentally by imaging technique uch a time-reolved Kerr microcopy. 10

11 Acknowledgement BI wa partly upported by INTAS

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