Quantitative understanding of magnetic vortex oscillations driven by spin-polarized out-of-plane dc current: Analytical and micromagnetic numerical study

Quantitative understanding of magnetic vortex oscillations driven by spin-polarized out-of-plane dc current: Analytical and micromagnetic numerical study

Youn-Seok Choi, Ki-Suk Lee, and Sang-Koog Kim

*

Research Center for Spin Dynamics and Spin-Wave Devices, Nanospinics Laboratory, Research Institute of Advanced Materials, Department of Materials Science and Engineering, College of Engineering, Seoul National University, Seoul 151-744, Republic of Korea

共Received 21 January 2009; revised manuscript received 20 March 2009; published 21 May 2009兲 We studied magnetic vortex oscillations associated with vortex gyrotropic motion driven by spin-polarized out-of-plane dc current by analytical and micromagnetic numerical calculations. Reliable controls of the tun- able eigenfrequency and orbital amplitude of persistent vortex oscillations were demonstrated. This work provides an advanced step toward the practical application of vortex oscillations to persistent vortex oscillators in a wide frequency共f兲range of 10–2000 MHz and with high values off/⌬f.

DOI:10.1103/PhysRevB.79.184424 PACS number共s兲: 75.75.⫹a, 72.25.Ba, 75.40.Gb, 75.40.Mg

I. INTRODUCTION

In 2007, spin-polarized dc-current-driven self-sustained oscillators based on magnetic vortex oscillations were ex- perimentally demonstrated using a nanoscale spin valve structure.¹ Since then, magnetic vortex oscillators have be- gun to attract considerable attentions owing to several advan- tages over spin-transfer-torque共STT兲-driven nano-oscillators associated with the precessional motion of uniform magneti- zation 共M兲 in a nanomagnet. For examples, Shibata et al.² observed a vortex gyrotropic motion, the so-called vortex- core共VC兲translation mode, through STT driven by in-plane current passing through a single vortex. Kasaiet al.³reported on resonant vortex oscillations in soft magnetic nanodots driven by in-plane harmonic oscillating current. Krüger et al.^4,5and Leeet al.^6,7also reported more quantitative studies of harmonic vortex oscillations in nanodots driven by har- monic oscillating currents and magnetic fields applied in the dot plane. Meanwhile, several other groups reported on the use of out-of-plane dc current for the dynamic excitation of vortex oscillations in different types of nanostructures.^8–11 More recently, Mistral et al.¹² experimentally demonstrated current driven sub-GHz oscillators caused by VC orbital mo- tions outside a metallic nanocontact area. In addition, Ruo- tolo et al.¹³ showed experimentally the coherent synchroni- zation of multivortex oscillations in multi-point-contact systems.

From these studies mentioned above, it is known that the vortex oscillators have the narrow width of the eigenfrequen- cies of vortex oscillations and that the phase and orbital am- plitude of VC gyrotropic motions are reliably controllable without the application of additional large magnetic fields.

Although a new concept of nano-oscillators based on such a vortex translation mode excited in magnetic nanodots has been proposed, the quantitative understandings of the under- lying physics of this phenomenon and associated new phe- nomena have yet been explored. In this paper, we report on quantitative interpretations of vortex oscillations in a free standing soft magnetic nanodot driven byspin-polarized out- of-plane dc current studied by analytical calculations and numerical simulations. In this study, we consider both the STT effect of spin-polarized currents acting directly on vor- tex nonuniform M structure and the comparable Oersted

field共OH兲effect accompanying the current flow. The results obtained from this work reveal a reliable means of manipu- lating the eigenfrequency and the orbital amplitude of VC translation motions in an oscillating manner in a dot of a different vortex state, by out-of-plane dc currents flowing through a perpendicularMpolarizer. The quantitative under- standing of dc current driven vortex oscillations and its ma- nipulation by key driving parameters, as found in this study, can offer an advanced step in its practical applications to persistent vortex oscillators with a tunable eigenfrequency 共f兲 in its broad range of 10–2000 MHz and a high f/⌬f value, without applying additional large in-plane and perpen- dicular magnetic fields.

II. MICROMAGNETIC SIMULATION AND RESULTS To quantitatively understand and explore the underlying physics of vortex oscillations in nanodots driven by spin- polarized out-of-plane dc currents, we chose two comple- mentary approaches: micromagnetic numerical and analyti- cal calculations using a model system of the circular shaped Permalloy 共Py: Ni₈₀Fe₂₀兲 nanodot of 2R= 300 nm diameter and L= 20 nm thickness, as shown in Fig. 1. The ground states of energetically equivalent four different vortex struc- tures can be characterized by two vortex integers: the chiral- itycand the polarizationp. The termc= + 1共−1兲corresponds to the counter-clockwise 共CCW兲 关clockwise共CW兲兴 rotation

j0

j0

(ip=+1)

150 0 -150

0 20

150

-150

x(nm) y(nm)

z(nm)

X(t)=(Xx(t),Xy(t))

c= +1 c= -1

FIG. 1. 共Color online兲Vortex state withp= + 1共upwardMori- entation at the VC兲 and c= + 1 共CCW in-plane curling M兲 or c= −1共CW in-plane curlingM兲in a Py nanodot with the indicated thickness and diameter. The color and height indicate the in-plane orientation of localMs and the out-of-planeMcomponents, respec- tively. The direction of current flow is indicated by the large arrow pointing in the +zdirection.

sense of the in-plane curling M, and the term p= + 1共−1兲 corresponds to the upward共downward兲Morientation of the VC. Figure1shows a specific vortex state havingc=⫾1 and p= + 1. In the present simulations, we used the

LLG COMMERCIAL code¹⁴ that utilizes the Landau-Lifshitz- Gilbert equation of motion,¹⁵ including the STT term¹⁶ ex- pressed as T_STT=共aSTT/Ms兲M⫻共M⫻mˆ_P兲 with a_STT

=_2␲¹ h␥^Pj0/共␮02eM_sL兲. The mˆ_P is the unit vector of spin- polarization direction, his the Plank’s constant,␥is the gy- romagnetic ratio, P is the degree of spin polarization, j₀ is the current density, ␮0 is the vacuum permeability, e is the electron charge, and M_s is the constant magnitude of M.

Out-of-plane dc currents were applied toward the +z direc- tion for sufficiently long time共100 ns in this study兲through the polarizer with perpendicular M pointing in either +z or

−z direction. Nonignorable OHs accompanying the out-of- plane dc currents were taken into account using Biot-Savart’s formulation. Here, we define the directions of the applied current and the spin polarization asip andS_pol, respectively:

the sign of ip= + 1共−1兲 and S_pol= 1共−1兲 corresponds to the +z共−z兲 direction. Thus, the rotation sense of the circumfer- ential OH is determined simply by the sign of i_p, i.e., i_p

= + 1共−1兲represents the CCW共CW兲rotation sense of the OH in-plane orientation.

Figure2 shows examples of the characteristic features of the translational motions of a VC driven by specific values of j₀= 1.5, 0.1, 0.52, and 0.48⫻10⁷ A/cm², which were ob- tained from simulations with considering both the STT and the accompanying OH with its circumferential in-plane ori- entation parallel 共P兲 to c= + 1 and antiparallel 共AP兲 to c

= −1. By the definitions ofcandi_p, the P共AP兲configuration

can also be denoted as c·ip= + 1共−1兲. The first and middle rows in Fig. 2 represent the observed trajectories of the or- bital motions and the␥components, respectively, of the VC position vector, X共t兲=关Xx共t兲,Xy共t兲兴 in the dot 共x-y兲 plane, where Xx and Xy are the x and y components of X共t兲. The initial VC position X₀ at t= 0 was displaced to 关−1.5 nm, 25.5 nm兴 for c= + 1 or 关1.5 nm, −25.5 nm兴for c= −1 by a static field of 100 Oe along the x direction before applica- tions of out-of-plane dc currents.

The simulation results reveal that spirally rotating mo- tions of a VC with theexponentially increasing, decreasing, and almost constant orbital radii关shown in Figs. 2共a兲–2共c兲, respectively兴, along with the corresponding eigenfrequencies 共third row of Fig.2兲are remarkably contrasting for different j₀ values chosen here. The first thing to stress here is the observed blue and redshifts of the eigenfrequency from 580 MHz at j₀= 0 共i.e., without application of current兲 for c·ip

= + 1 and −1, respectively, as reported in Ref.17. The second one is the remarkable variation in the orbital amplitude of VC motions with different j₀ values. For j₀= 1.5共0.1兲

⫻10⁷ A/cm², the orbital amplitude exhibits its increase共de- crease兲 with time from the corresponding X₀. By contrast, for the application of j₀= 0.48共0.52兲⫻10⁷ A/cm² for the c·ip= −1共+1兲 configuration, the VC is allowed to be main- tained around the initial orbit, being analogous to the reso- nant motion of a VC on a steady-state circular orbit that is driven by harmonic oscillating in-plane magnetic fields and currents.^6,7,18,19Also, the numerical estimates of the eigenfre- quency f, the full width at half maximum共FWHM:⌬f兲, and f/⌬f values in frequency spectra are given in Table I. Such remarkable variations in the eigenfrequency and the orbital amplitude with j₀ have not been understood quantitatively

0 00 100

0 00 100

P AP

j₀(10⁷A/cm²)= 1.5

Time (ns)

0 5 10

-150 0 150 Xy(nm)

f(GHz) P 0.1 AP

0 0.4 0.8 0.6

0.3

FFT(a.u.) 0 0.66 0.50 0.59 0.57

(a) (b)

P AP

0.52

0.55 0.61

(c) _0.48

0 0.4 0.8 0 0.4 0.8 0 0.4 0.8 0 0.4 0.8 0 0.4 0.8 0 5 10 0 15 30 0 15 30 0⁰ 15¹⁵ 30 0³⁰ 15 30

FIG. 2.共Color online兲VC translation motions driven by spin-polarized out-of-plane dc current of 共a兲 j₀= 1.5⫻10⁷ A/cm², 共b兲 0.1⫻10⁷ A/cm², and 共c兲 0.48⫻10⁷ and 0.52⫻10⁷ A/cm² for indicated antiparallel 共AP:c·i_p= −1兲 and parallel 共P: c·i_p= + 1兲 configurations, as noted in text. The orbital trajectories of a moving VC are shown in the top row as well as the time variations of theycomponents of the VC position in the middle row and their FFT power spectra in the bottom row. The FFT power spectra obtained from the VC motion duringt= 0 – 100 ns.

TABLE I. Estimates of the eigenfrequency共f兲, FWHM共⌬f兲, andf/⌬ffactor obtained from Gaussian fits to the simulation results shown in Fig.2. The FFT power spectra shown in Fig.2were obtained from time oscillations of theycomponent of a moving VC position for a time durationt= 0 – 100 ns.

j₀

共10⁷ A/cm²兲 1.5 0.1 0.52 0.48

ci_p +1 −1 +1 −1 +1 −1

f共MHz兲 660⫾0.7 500⫾0.6 586⫾0.7 572⫾0.7 612⫾0.3 550⫾0.06

⌬f 共MHz兲 60⫾1.4 60⫾1.3 38.7⫾1.5 33.4⫾1.5 14.9⫾0.5 6.5⫾0.04 f/⌬f 11⫾2.7 8.3⫾0.2 15.1⫾0.7 17.1⫾0.9 41.1⫾1.4 84.6⫾0.5

and analytically, whereas the quantitative understanding of these dynamic properties are crucial in the determination of key parameters to control vortex oscillations that are practi- cally applicable to self-sustained nano-oscillators. Also, the phase of a moving VC position and the f/⌬f values are crucial factors to be understood from an application point of view.

III. ANALYTICAL CALCULATION AND RESULTS To elucidate the underlying physics of the observed dy- namic behaviors and to search for key parameters for reliably controlling the eigenfrequency and the orbital amplitude of VC oscillations in a nanodot of a different size and of a different vortex state characterized by c and p, we analyti- cally calculatedX共t兲in the dot共x-y兲plane. In this analytical calculation, we used the linearized Thiele’s²⁰equation of mo- tion by employing the force term, F_STT= 2␲^SpolaTj₀共zˆ⫻X兲 with the STT coefficient aT=a_STTLMs/共␥^j0兲=_2␲¹hp/共2␮0e兲 共Refs. 8 and21兲. The governing equation of VC translation 共gyrotropic兲motions in the linear regime is written in terms of a moving VC position vectorX共t兲as

−G⫻X˙ +⳵W/⳵^{X−DX˙ −F}STT= 0, 共1兲 whereG= −p兩G兩zˆ is the gyrovector with its constant G, and D⬍0 is the damping constant. The potential energy of a displaced VC in a circular dot can be expressed as W共X,t兲

=W共X= 0兲+␬^X2共t兲/2 with the stiffness coefficient␬^{. Equa-} tion 共1兲is rewritten in the matrix form

−

冋

^{−p兩G兩D p兩G兩D}

册

^{X˙ +}

冋

^{− 2␲S␬polaTj0 2␲Spol␬aTj0}

册

^{X= 0.}

共2兲 The general solution of Eq. 共2兲 is simply given as X共t兲

=X₀exp共−i␻t兲with the angular eigenfrequency␻. Inserting this solution into Eq. 共2兲 leads to the analytical form of the eigenfrequency ␻⁼共␬+i2␲^SpolaTj₀兲/共p兩G兩−iD兲, and the re- lation of the XxandXy, i.e., Xy=iXx. Since ␻⁼␻R+i␻I is a complex function, the real and imaginary terms can be ex- pressed as ␻R=共␬p兩G兩− 2␲^SpolaTj₀D兲/共G²+D²兲 and ␻I

=共␬^{D+ 2}␲^Spola_Tj₀p兩G兩兲/共G²+D²兲, respectively. The VC position vector as a function of time in response to the out-of-plane dc current is thus given by X共t兲

=X₀exp共␻It兲exp共−iRt兲. Here ␻R corresponds to the true eigenfrequency of VC gyrotropic motion and nonzero values of the imaginary term ␻I indicate that the orbital amplitude changes with time as in the form of X₀exp共␻It兲. From the above results, the orbit radius of the VC motion as a function of time is given asR_orb=兩X0兩exp共␻It兲, and the phase relation betweenX_xandX_yisX_y/X_x=e^␲/2, which reflect a circularly rotating motion of a VC inside the dot with ␻R and with CCW 共CW兲 rotation sense for ␻R⬎0共␻R⬍0兲. Therefore, the obtained results from the simulations shown in Fig.2can be more quantitatively understood from the analytical equa- tions of ␻R and ␻I that vary with p=⫾1 and c=⫾1, the magnitude and direction of j₀, andS_pol=⫾1 for the material and the dimensions of a given nanodot.

Moreover, the circumferential OH generated by the flow of out-of-plane currents should be considered to understand how this type field affects both␻Rand␻I. As reported in our previous work,¹⁷ it is known that the OH influences the variation of␬, such that␬⁼␬0+␬OH, where␬0is the intrinsic stiffness coefficient without considering the OH contribution, and␬OHis the additional term newly introduced by the OH contribution to the effective potential energy of a displaced VC. The␬OHterm is proportional toj₀with a constant value of ␩^=cip␵共where␵⬎0兲,¹⁷ so that the sign of␩ ^{can switch} depending on the sign ofc·ip. Note that the OH contribution gives rise to the increase共decrease兲in␬for the configuration of c·ip= + 1共−1兲. Based on the “surface charge free”

model,²²␵can be analytically derived in terms of dot dimen- sional parameters R andL and a material parameter M_s, as

␵=⁴⁵₆₈RLMs 共for detail, see Ref. 23兲. Consequently, the ␻R

and␻I terms are parametrized as

␻R= p␵兩G兩

G²+D²

冋

^{␬␵0+cipj0}

册

^{, 共3a兲}

␻I= B

G²+D²

冉

^j0+␬0DB

冊

^{with B=cip␵D+ 2␲SpolaTp兩G兩.}

共3b兲 For a given dot dimension and a material,␻Rand␻Iare both controllable with only external driving force parameters j₀ andip, for a given vortex state characterized byp andcand for a given S_pol. Here p·S_pol= + 1共−1兲 corresponds to the P 共AP兲orientation betweenp andS_pol.

For different combinations of p·S_pol=⫾1 andc·ip=⫾1, we plotted the numerical values 共solid lines兲 of ␻R and ␻I

calculated from Eqs.共3a兲and共3b兲as a function of the vari- able j₀ and compared them with the corresponding simula- tion results共symbols兲, as shown in Fig.3. The values of␻R

and ␻I vary with j₀. For j₀= 0 the values of␻R and␻I be- come ␻R,0=p␬0兩G兩/共G²+D²兲 and ␻I,0=␬0D/共G²+D²兲, re- spectively. More specifically, for p= + 1 ␻R increases 共de- creases兲linearly with j₀ for c·i_p= + 1共−1兲, independently of the sign of S_pol关see left panel of Fig.3共a兲兴. In addition, we obtain the relation of ␻R共p= −1兲= −␻R共p= + 1兲 关see Fig.

3共a兲兴. Regardless of what signs p and S_pol have, for c·ip

= −1 there exists a critical value of j₀=j_max= −␬0/共ci_p␵兲 where␻R becomes zero, as indicated by the black thick ar- rows in Fig.3共a兲. According to the analytical calculation, the sign of␻Rswitches crossing j₀=j_max, such that VC gyrotro- pic motion for a given p= + 1共−1兲 is CCW共CW兲in the re- gion of j₀⬍j_maxand switches to CW共CCW兲in the region of j₀⬎j_max. However, the numerically estimated value ofj_maxis as extremely large as an order of 10⁸ A/cm², and hence in such largej₀values vortex polarization and chirality switch- ing events can take place additionally according to simula- tion results共not shown here because these switching events are beyond the scope of the present paper兲. Consequently, the analytical results in the region of j₀⬎j_max are physically meaningless. The corresponding simulation results共indicated by symbols兲for relatively small values ofj₀are in the same trends as the analytical results, although they show some discrepancies in magnitude.²¹

Instead, as shown in Fig.3共b兲, the linear variations of␻I

with j₀ are contrasting and of opposite slope between p·S_pol= + 1 and −1, but their dependence on the sign ofc·i_p is ignorable in the region of small j₀ values. For p·S_pol

= + 1,␻I⬍0 in the entire region of j₀, which reflects the fact that R_orb共t兲 decreases exponentially with time, as R_orb共t兲

=兩X0兩exp共␻It兲, and consequently reachesX= 0. For the other case ofp·S_pol= −1, ␻Ilinearly increases with j₀, but its sign changes from negative to positive one crossing j₀=j_criwhere

␻I= 0. The value of j_cri is analytically derived as j_cri

= −␬0D/B withB=cip␵D+ 2␲^SpolaTp兩G兩 from Eq.共3b兲. It is clear that for j₀⬍j_cri, ␻I⬍0, but for the other region j₀

⬎j_cri,␻I⬎0. The fact of␻I⬎0 implies thatR_orb共t兲exponen- tially increases with time, as 兩X0兩exp共␻It兲. The important point we have to stress here is that VC gyrotropic motions driven by j₀=j_criare maintained on an initially displaced VC orbit radius 兩X0兩 and with a characteristic value of ␻R

⬘

=␻R共j₀=j_cri兲. The analytical form of␻R

⬘

was obtained to be

␻R

⬘

⁼␻R,0共1 −cip␵D/B兲 by putting j₀=j_cri into Eq. 共3a兲. The value of j_criis a crucial parameter for controlling persistent vortex oscillations by applications of out-of-plane dc current.

For the cases of j₀⫽j_cri, the vortex oscillations cannot per- severe because the orbital amplitude either decreases or in- creases for those cases. This phenomenon can be applicable to self-sustained vortex oscillators. Some simulation results 共noted by symbols兲 are in similar trends with the analytical calculations, but their discrepancy in magnitude becomes in- creased with j₀.

It is worthwhile to address more physical pictures on the observed steady-state vortex oscillations. The oscillation be- TABLE II. Comparison of the numerical values ofj_criand␻R⬘/2␲between the analytical calculation and

micromagnetic simulation results for several dot dimensions, as indicated by the small red circles in Fig.4.

ci_p

Dot size

j_cri

共10⁶ A/cm²兲 ␻R⬘/2␲

共GHz兲 R

共nm兲

L

共nm兲 Analytical Micromagnetic Analytical Micromagnetic

+1 105 20 9.29 6.4 1.05 0.83

150 20 6.21 5.2 0.74 0.61

150 10 1.47 1.6 0.36 0.34

−1 105 20 8.79 6.1 0.99 0.78

150 20 5.77 4.8 0.69 0.55

150 10 1.42 1.5 0.35 0.32

0 5 10 15

8 6 4 2 0

0 5 10 15

8 6 4 2 0

1.5 1.0 0.5 0.0

(GHz)(GHz)

j₀(10⁷A/cm²)

0 2 4 6 8

(a)

(b)

1, pol 1 p= + S = ±

p 1

⋅ = + c i

p 1

⋅ = −

c i c i⋅ = +^p 1

p 1

⋅ = − c i 0

-2 -4 -6 -8

0 0 0 0.00 -0.5 -1.0 -1.5

pol 1

p S⋅ = − p S⋅ pol= +1

0 5 10 15

0 2 4 6 8

j₀(10⁷A/cm²)

1, pol 1 p= − S = ±

p 1

⋅ = + c i

p 1

⋅ = − c i

p 1

⋅ = + c i

p 1

⋅ = − c i

j

max

j

max

j

cri

FIG. 3. 共Color online兲 Estimated values of the real 共␻R兲 and imaginary共␻I兲 terms of the eigenfrequency 共␻兲 versus j₀ for the indicated cases of p·S_pol=⫾1 and c·i_p=⫾1. Symbols and solid lines represent the results of micromagnetic simulation and analyti- cal calculations, respectively.

40 30 20 10

L(nm) 0 ^0.05

R(nm)

0 250 500 750 40

30 20 10 0

0.01 1

100

10 (x10⁵A/cm²)

0.1 1

0.1 (GHz) 0.25 0.5

0.05 0.01 1

0.1(GHz) 0.25

0.5 5

50 25 100

10 (x10⁵A/cm²) 0.1 1 5 50 25

0 250 500 750 1000

p

1

⋅ + c i

1

10001000 2 −

2

j_cri(10⁵A/cm²) ω_R′/ 2π(GHz) 10^-510^-310 10³ 10^-2 10^-1 1

(a) (b)

FIG. 4.共Color online兲Contour plot ofj_criversus dot thicknessL and radiusRand plot of␻R⬘/2␲obtained atj=j_crifor both cases of c·i_p=⫾1 and forp·S_pol= −1. Both j_cri and␻R⬘/2␲ were obtained from numerical calculation of the corresponding analytical forms:

j_cri= −␬0D/共ci_p兩␩兩D+ 2␲S_polpa_T兩G兩兲 and ␻R⬘⁼␻R,0共1 −ci_p兩␩兩D/B兲 with␬0⬇⁴⁰₉␲M_s²L²/R共Ref.22兲. The region above the dashed line corresponds to stable vortex states obtained using an analytical equation of Ref.25.

haviors of Ms of single domains in spin valve structures caused by STT have been understood by the force balance between the STT and the Gilbert damping term. In analogy, we consider force balance between the Gilbert damping term F_D=DX˙=␻R

⬘

^D共zˆ⫻X兲 and F_STT= 2␲^Spola_Tj_cri共zˆ⫻X兲 for the condition of␻I= 0 atj₀=j_crirequired for steady-state vortex oscillations. Inserting j_cri= −␬0D/B and ␻R

⬘

⁼␻R,0共1

−cip␵D/B兲into the two yieldsFD= −F_STT, verifying that the steady-state vortex oscillations can maintain at j₀=j_criin the case where the spin torque force cancels the Gilbert damping force.

Next, we numerically calculated the values of j_criand␻R

⬘

versus L and R, as shown in Figs. 4共a兲 and 4共b兲. In the calculations, we used the analytical forms of j_cri= −␬0D/ 共cip␵D+ 2␲^SpolpaT兩G兩兲 and ␻R

⬘

⁼␻R,0共1 −cip␵D/B兲 with an approximated function of ␬0=⁴⁰₉␲^Ms

2L²/R 共Ref. 22兲. The terms G and D are also given as G= 2␲^pMsL/␥ and D

= −␣␲^MsL关2 + ln共R/Rc兲兴/␥ with the VC critical radius Rc, which of these equations are also functions ofLandR共Ref.

24兲. As seen in both equations, the values of j_cri and␻R

⬘

^are functions of c·ip, so that they vary with the sign of it. The contour plots of j_cri and␻R

⬘

^{on the}共L-R兲 plane allow us to gain technologically useful phase diagrams for designing the dot dimensions and a magnetic material, in order to control persistent vortex oscillations and their eigenfrequencies. As shown in Fig. 4共a兲, the value of j_cri increases dramatically with the increasing L for a given R, whereas j_cri decreases slowly with increasingRrelatively for a constant value ofL.

The surprising result is that the value of j_cri is as extremely low as the order of 10⁴ A/cm² in the region of L⬍3 nm.

The eigenfrequency obtained at j₀=j_cri varies remarkably withLandR, indicating its tunability by dot dimensions, in a very broad range from 10 MHz to 2 GHz. We also compare the numerically estimated values of j_criand␻R

⬘

/2␲^{using the}

analytical equations 共Fig. 4兲 with those obtained using mi- cromagnetic simulations for several dot dimensions of 关R共nm兲,L共nm兲兴=关105, 20兴,关150,20兴, and关150,10兴, as shown in Table II. Although there are some discrepancies in the results between the analytical and simulation calculations,²¹ their general trends according toLandRare in good agree- ment.

IV. CONCLUSION

We numerically and analytically calculated the depen- dences of the eigenfrequency and the orbital radius ampli- tude of the translation motion of a vortex core in soft mag- netic nanodots driven by spin-polarized out-of-plane dc currents. We found some key parameters to reliably control the persistent vortex oscillations, including the vortex eigen- frequency and orbital amplitude. Using the analytically de- rived equations of the critical current density j_cri for persis- tent vortex motions and their eigenfrequencies ␻R

⬘

^{, we} constructed two phase diagrams ofj_criand␻R

⬘

on the plane of dot thickness and radius for a Py material. These results pro- vide guidance for practical implementation of vortex oscilla- tions in nanodots to a new class of dc-to-ac oscillator with eigenfrequency tunability in a broad range of 10–2000 MHz, with high f/⌬f values, and extremely low current densities as small as 10⁴– 10⁵ A/cm².

ACKNOWLEDGMENTS

We express our thanks to A. Slavin for his careful reading of this manuscript. This work was supported by Creative Research Initiatives共Research Center for Spin Dynamics and Spin-Wave Devices兲of MEST/KOSEF.

*Corresponding author. [email protected]

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