Origin, criterion, and mechanism of vortex-core reversals in soft magnetic nanodisks under perpendicular bias fields

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Origin, criterion, and mechanism of vortex-core reversals in soft magnetic nanodisks under perpendicular bias fields

Myoung-Woo Yoo, Ki-Suk Lee, Dae-Eun Jeong, and Sang-Koog Kim

*

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

共

Received 14 September 2010; published 24 November 2010兲

We studied dynamics of vortex-core reversals driven by circular rotating fields along withstatic perpen- dicular magnetic fieldsof different direction and strength. We found that the application of perpendicular fields H_p modifies the starting ground state of vortex magnetizations, thereby instigating the development of a magnetization dip m_z,dip in the vicinity of the original core up to its threshold value, m_z,dip^cri

⬃

−p, which is necessary for vortex-core reversals, where p is the initial core polarization. We found the relationship of the dynamic evolutions of the m_z,dip and the out-of-plane gyrofields h_z, which was induced, in this case, by vortex-core motion of velocity␷, thereby their critical value relation␷cri⬀h_z^cri. The simulation results indicated that the variation of the critical core velocity ␷cri withH_p can be expressed explicitly as␷cri/␷cri 0

=

共␳/␳

0

兲兩

−p−m_z,dip^g

兩

, with the core size␳and the starting ground-state magnetization dipm_z,dip^g variable with H_p, and for the values of␷_cri⁰ and ␳0atH_p= 0. This work offers deeper and/or new insights into the origin, criterion and mechanism of vortex-core reversals under application of static perpendicular bias fields.

DOI:10.1103/PhysRevB.82.174437 PACS number

共

s

兲

: 75.40.Gb, 75.40.Mg, 75.60.Jk

Vortex magnetization configurations^1–4are of growing interest owing to their potential applications to nonvolatile in- formation storage^5–9and nano-oscillator devices,^10–12 not to mention the fundamental interest in their nontrivial dynamic characteristics including vortex-core reversals^13–19 and its gyrotropic mode.^20–23 It is well known that vortex-core reversals in ferromagnetic dots occur via the specific serial dynamic processes of the nucleation and annihilation of a vortex-antivortex共VAV兲pair.13,15,17–19 Recently, Leeet al.²⁴ found that the so-named critical velocity ␷cri of the vortex- core motion is required for switching events. In fact, ␷criis independent not only of the driving forces typical of oscil- lating in-plane fields, rotating fields or currents, but also of their parameters and dot dimensions, while more generally, critical exchange energy has been considered as a criterion for VAV-mediated core reversals.²⁵The measurable␷criquan- tity has been experimentally measured,²⁶and has been found to be expressed explicitly, on the basis of micromagnetic simulation results, as␷cri=␩␥^Aex

1/2

, with the proportional constant ␩= 1.66⫾0.18, the gyromagnetic ratio ␥^{= 2}␲

⫻2.8 MHz/Oe, and the exchange stiffness A_ex.²⁴ Inserting the numerical value ofA_exfor a given material,␷criis readily determined 共e.g., for Permalloy 共Py; Ni₈₀Fe₂₀兲, 330 m/s兲. Moreover, in cases where any driving forces are tuned to the eigenfrequency of a vortex in a given dot, the threshold field strength and current density are sufficiently small, since the vortex-core velocity can reach ␷crithrough the resonant gyrotropic motion.⁷ ␷cri thus is considered from the technical application point of view to be a useful quantity for the pre- diction of vortex-core reversal events and the determination of efficient driving force parameters such as frequency and strength.

Contrarily, Khvalkovskiy et al.²⁷ quite recently reported that ␷cri can vary with the direction and strength of static fieldsH_papplied perpendicularly to the dot plane. Still other scenarios for a vortex-core reversal event have been found for some specific cases. For example, Kravchuk et al.²⁸ re-

ported that nucleation of a VAV pair is possible for a subse- quent vortex-core switching event in the case of an immobile vortex core, which physical origin they ascribed to softening of the dipole magnon mode.²⁹ Also, Gliga et al.²⁵ reported possible vortex-core switching by application of spatially lo- calized pulse fields in proximity to the vortex core; they found in this case that the vortex-core velocity is not necessary to reach␷cri. Therefore, not only does the physical origin of the variation of ␷cri withH_p remain to be unraveled, but also the origin, criterion and mechanism of the relevant vortex-core reversals.

In this paper, we present dynamics of vortex-core reversals driven by circular rotating fields along with the application of static perpendicular magnetic fields, and reveal the physical origin of the variation in␷criwithH_p, as elucidated by micromagnetic simulations combined with quantitative interpretation. We calculated an out-of-plane gyrofieldh_zthat is produced by the deformation of vortex-core magnetizations induced by vortex-core motion,^14,30 and we revealed how it builds up a magnetization dipm_z,dipnear the original vortex core, until the critical value necessary for vortex-core reversal events is achieved. We found the correlations of the critical values and dynamic evolutions ofm_z,dip,h_zand␷^{, and} discovered that the modification of the initial ground state of a vortex structure byH_pgives rise to theH_pdependences of h_z^criand␷cri.

We chose micromagnetic simulation as our approach to this investigation, because it is a well-established, optimized tool for the study of magnetization dynamics on scales of a few nanometers spatially and ⬃10 ps temporally. For the simulations, we employed the OOMMF code³¹ that incorpo- rates the Landau-Lifshitz-Gilbert 共LLG兲 equation³² of motion of magnetization M:⳵^M/⳵^t^{= −}␥M⫻H_eff+共␣/M_s兲M

⫻⳵^M/⳵^{t, where}^Heffis the total effective field, which in the present case includes, the demagnetizing, exchange and ex- ternal static perpendicular fields, and additionally rotating field as a driving force of vortex excitations. We used a Py

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nanodisk of radius R= 300 nm and thickness L= 30 nm.

Uniform static fields were applied perpendicularly to the disk plane, the strength of which fields varied fromH_p= −3 kOe to +4 kOe at an interval of 1 kOe.³³The positive共negative兲 sign corresponds to the parallel 共antiparallel兲 orientation of the vortex-core magnetization 共polarization p兲 in the field direction 关Fig. 1共a兲兴. We chose, as the driving force, the counterclockwise circular rotating field H_CCW

=H₀关cos共␻Ht兲xˆ+ sin共␻Ht兲yˆ兴 because its application at rotating angular frequency ␻H tuned to the angular eigenfrequency␻0is the most effective means of vortex-polarization selective resonant excitations of, in this case, only the up- ward core, as demonstrated in earlier theoretical, experimen- tal and simulation studies.^7,34,35Because␻0changes withH_p as well,␻Hwas varied as well.³⁶ The value of H₀= 20 Oe, being larger than the threshold fields for all of the cases of H_p, was used to achieve the vortex-core reversals. The typical Py material parameters were used as follows: saturation magnetization M_s= 8.6⫻10⁵ A/m and exchange stiffness A_ex= 1.3⫻10⁻¹¹ J/m with zero magnetocrystalline aniso- tropy. The size of each individual cell was 2⫻2⫻30 nm³, and the Gilbert damping constant was␣^{= 0.01.}

First we examined how application of different values of H_p and with opposite directions changes the initial ground state of the vortex structure, as reported earlier.³⁷ Such ground-state variation would be expected to be essential to vortex-core reversal dynamics. As can be seen from the per- spective images of local magnetizations’ spatial distribution in Fig. 1共a兲, there were distinct changes in the ground state.

TheH_pdependence of them_z共=Mz/M_s兲profile on the radial axis crossing the core center关Fig.1共b兲兴 reveals clearly that the in-plane magnetizations in the ground state are remark- ably deviated from the in-plane at the edge and the interme-

diate area around the core due to the Zeeman energy term of the perpendicular field. The modification of the m_z component at the very core center was negligible; the initial core magnetization m_z= +1 is maintained, due predominantly to the strong exchange coupling at the center, until much stron- ger perpendicular fields were applied.³⁸ Outwards from the core center, the Zeeman contribution became dominant, be- ginning to modify them_zprofile around the core, specifically 共1兲the m_z component level of the magnetization dip,m_z,dip 共in the opposite direction with respect to the core magneti- zation兲, just outside the core and共2兲the variation in the core size, represented by the full width at half maximum 共FWHM兲 of the core m_z component 关Figs. 1共b兲 and 1共c兲兴.

With increasing negativeH_p, the magnetization dip became deeper and the FWHM, smaller. These results are in good agreement with other reports.²⁷

Next, in order to clarify how the above-mentioned ground-state modification by H_p affects vortex-core reversals, we plotted, as shown in Fig. 2共a兲, the vortex-core velocity before and after the core reversal events initiated by application of the fieldH_CCW. The instantaneous core velocity ␷was estimated from the trajectories of the vortex-core motions. Prior to the core reversals, the velocity increased, finally reaching its threshold value, ␷cri. Then, the deformation of the entire magnetization structure of the vortex core was maximized and, which was accompanied by the serial processes of the nucleation of a newly formed VAV pair hav- ing the magnetization direction共p= −1兲opposite to the original magnetization orientation 共p= +1兲, and of the annihilation of the newly formed antivortex and the original

H_P(kOe) (a)

x

z y

+4 +3 +2 +1 0

Fielddirection

H_p(kOe)

= +4 +3 +2+1

−−−−10

−−−−2

−−−−3 FWHM

1

0 0.5

-0.5 mz (b)

(c)

−300 −150 0 150 300 r(nm)

-1 -2 -3

F 30

FWHM(nm) 10 20

c

兲

H_p dependence of FWHM of the m_z profiles of cores.

10.0197 15.7833 19.0146

10.0178 17.4510 19.0125 t(ns) = 10.0190 14.4790 19.0156

0 5 10 15 20 0.5

υ(km/s) 0 0.5

0 0.5 0

Time (ns) H_P=−1 kOe

H_P= 0 kOe

H_P= 1 kOe

(c) 0.8

Hp=−1kOeHp=0

i l i υcri

20nm

(a) (b)

0 0.4 0.8

υcri(km/s)

−4 −2 0 2 4 H_P(kOe)

Hp=+1kOe

reported in Ref.27.

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vortex,13,15,17–19 for complete vortex-core reversals, as seen in Fig. 2共b兲. From the three H_p cases, the core reversal mechanism was the same as that in the known VAV-pair- mediated reversals; the only difference was the remarkable change in ␷criwithH_p. In Fig.2共c兲, the resultant H_p dependence of␷criis plotted, showing an almost linear dependence onH_pover a wide but limited range. The simulation result is in good agreement with the explicit expression共solid line兲of

␷cri共Hp兲=␷_cri⁰ 共1 −Hp/H_c兲共Ref.27兲in the negative field region with a static perpendicular field H_c⬃−6.5 kOe, although there are some discrepancies in the positive field region. The value of H_c⬃−6.5 kOe, which was obtained from simulations on our dot dimensions, is the threshold strength required for core reversals only using the static perpendicular field. The ␷cri variation with H_p is surprising, in that, for a given material and cases under H_p= 0, ␷cri is known to be independent of dot geometry and dimensions as well as type of the driving forces and their variable parameters.^15,24

To elucidate the underling physics of such changes, we employ the dynamic evolution of gyrofields induced by the deformation of the magnetizations of the entire vortex structure due to vortex-core motion. The out-of-plane gyrofield is expressed as hz= −共1/␥^Ms2兲共M⫻dM/dt兲z, as derived from the time-derivative term of the LLG equation.^14,39 From the numerical data of micromagnetic simulations, one can di- rectly calculate the spatial distribution of the localh_zvalues as a function of time during vortex-core motion and even after the core’s reversal event is complete. Figure 3共a兲 pro- vides snapshot plane-view images of both the local h_z and m_z, taken att= 3.501 ns prior to the reversal. Them_zandh_z are strongly concentrated locally near the original vortex

core. The maximum of兩h_z兩is found at about 6 nm away from the core center共the maximum of兩mz兩兲. The serial profiles of the m_zandh_zshown in Fig.3共b兲show how the highly con- centratedh_zinfluences the formation and consequent development of the indicated magnetization dip, m_z,dip in the direction opposite to the original core magnetization 共in this casep= +1兲.^14,30As the兩hz兩 grew larger, the兩mz,dip兩did like- wise, finally reaching the critical value m_z,dip^cri ⬃−p for the initial polarization, here p= +1. As seen in them_zprofile of the magnetization dip, the width was larger than the original core width. The dip width included a newly formed antivortex as well, reflecting the fact that, when a new vortex is formed near the original vortex, the energetically stable structure is the vortex-antivortex-vortex configuration.⁴⁰

Figure3共c兲shows the distinct time evolution ofh_z,m_z,dip with␷^{. As}␷becomes faster,h_zgrows more negative, making m_z,dip further deepen, as explained above. When m_z,dip reaches the threshold valuem_z,dip^cri ⬃−p, vortex-core reversals occur via the known mechanism, as depicted in Fig. 2共b兲.

The critical valueh_z^criis achieved whenm_z,dip^cri ⬃−p, and at the same time,␷crialso is found becauseh_zis induced by vortex- core motion, and thus via h_z⬀␷, as explained in Ref. 30.

Next, in order to find the concrete correlation of the dynamic evolution ofh_zand the development ofm_z,dip, we plotted the variation in the temporal evolution ofm_z,dip withh_zfor dif- ferentH_pvalues, in Fig.3共d兲. With increasing time, increases in h_z means increasingly negative values ofm_z,dip, until the critical valuem_z,dip^cri ⬃−p. From the results, it was confirmed that the critical valueh_z^criis reached whenm_z,dip^cri ⬃−p, regard- less of the values of H_p applied. Thus, more generally, m_z,dip^cri ⬃−pis the criterion for vortex-core reversal switching.

As seen in Fig. 1, the applications of H_p modify the initial ground state of the magnetization dip, m_z,dip^g . With increasingly negative H_p, the value ofm_z,dip^g is already close to −p.

m_z,dip^g is a function ofH_p, and as its value grows more negative,h_z^cribecomes smaller in order to reachm_z,dip^cri ⬃−p. Thus

⌬mz共Hp兲⬃m_z,dip^cri −m_z,dip^g 共Hp兲is the crucial factor in the determination ofh_z^crias a function ofH_p.

In Fig.4共a兲, we plotted the dependences ofh_z^criand␷crion H_p. The decrease in␷criwith decreasingH_pis associated with the reduction inh_z^cri:h_z^criis smaller, and so␷crialso is smaller, according to the relation of ␷cri⬀h_z^cri, reported in our earlier work.³⁰As mentioned above, the decrease ofh_z^criis related to the reduction of ⌬mz共Hp兲⬃−p−m_z,dip^g 共Hp兲 as well. The applications ofH_pmodify not onlym_z,dipbut also the core size, denoted as␳= FWHM. The variation of␳共Hp兲⌬mz共Hp兲plot- ted in Fig.4共a兲revealed mutual linear relationships between

␷cri共Hp兲,h_z^cri共Hp兲, and␳共Hp兲⌬mz共Hp兲. Further, the simulation results showed the correlations of ␷cri共Hp兲⬀h_z^cri共Hp兲 and h_z^cri共Hp兲⬀⌬mz共Hp兲␳共Hp兲, along with the result, ␷cri共Hp兲

⬀␳共Hp兲⌬mz共Hp兲, as shown in Fig.4共b兲. Finally we found the universal relation for anyH_p

␷cri共Hp兲

␳共Hp兲兩⌬mz共Hp兲兩= ␷cri 0

␳0兩⌬m_z⁰兩. 共1兲 From Eq. 共1兲, we obtained the following equation for 兩⌬m_z⁰兩= 1:

␷cri共Hp兲=共␷cri

0 /␳0兲␳共Hp兲兩⌬mz共Hp兲兩. 共2兲

0 200

|X| (nm) 1

0.5 0

−0.5

−1 mz

0 200 0 200

t= 3.5010 ns 4.1325 ns 4.7234 ns H_p= 0 kOe

H0= 20 Oe

3.2

0

−3.2 hz(kOe) m_z

h_z

m_z,dip h_z

mzhz(kOe)

−4 4

−1 1

t= 3.5010 ns

(a) (b)

−30 −60

15

−15 15

−15 x(nm)

y(nm)y(nm)

υ(km/s)

0 0.2 0.4

mz,dip

−1 0

−0.5

−4

−2 0

hz(kOe)

0 4 8

Time (ns) (c)

treversal

h_z^cri υυυυ_cri

m_z,dip h_z

−6

−4

−2 0 hz(kOe)

1 0.5 0 −0.5 −1 m_{z, dip}

H_p(kOe) +4+3 +2+1

−10

We next plotted␷criversusH_p, as obtained from the simulations of vortex-core reversals underH_pfor different dimensions, in Fig. 5. Surprisingly, the results 共closed symbols兲 agreed well with the explicit expression 共solid line兲 of Eq.

共2兲, where␷_cri⁰ ⁼共1.66⫾0.18兲␥^A_ex^1/2^and␳0are the critical velocity and FWHM of the core, respectively, at H_p= 0. The analytical form predicts well the dynamic quantity ␷cri as a function of H_p even in both negative and positive field re- gions, informing us that␷crican be predicted simply with the values of ␷_cri⁰ and␳0 for H_p= 0 from the magnetization profiles in the static ground state for differentH_p values,␳共Hp兲 and m_z,dip^g 共Hp兲, without conducting further micromagnetic simulations of vortex-core reversal dynamics under H_p. In- terestingly, the values of␷_cri⁰ andH_pdependences of␳/␳0and 兩⌬mz兩are independent of the dot dimensions; thus the expression of␷cri共H_p兲=共␷_cri⁰ /␳0兲␳共H_p兲兩⌬m_z共H_p兲兩is not variable with the dot dimensions.

Finally, we should note that in our previous report, the critical velocity was found not to change with dot dimensions and driving force parameters.²⁴This is the case where the ground state of a vortex is not affected initially by other factors and the reversal is assisted with the motion of a mov- able vortex core. In the case of the application of H_p, the initial magnetization dip varies with H_p, and so the critical velocity, because it modifiesm_z,dip.

In conclusion, we studied vortex-core reversal dynamics in ferromagnetic dots under static perpendicular magnetic

fields, by micromagnetic simulations. It was found that the modification of the ground state of the vortex structure byH_p gives rise to variations in h_z^cri as well as ␷cri with H_p. The results provide clear correlations of the dynamic evolution of

␷^, ^hz andm_z,dip for the relevant core reversals. The critical values␷criandh_z^criare attained whenm_z,dipreaches the critical value,m_z,dip^cri ⬃−p. The initial value ofm_z,dipin the ground state, which can be modified byH_p, determines those critical values of ␷cri andh_z^cri for differentH_p. The threshold value m_z,dip^cri ⬃−pis the criterion for VAV-pair-mediated core reversals. The previously found criterion,␷cri, is the criterion only where the VAV-mediated vortex-core reversals occur as assisted by vortex-core motion and where the initial ground state does not vary. We also found the concrete relations of

␷cri⬀h_z^criandh_z^cri⬀␳⌬m_zover a wide range ofH_p, as well as

␷cri/␷cri0 =共␳/␳0兲兩⌬mz兩. This work offers, in the form of the results here presented, deeper insights into the fundamentals of vortex-core reversal dynamics.

We thank Andrei Slavin for his careful reading of this manuscript. This work was supported by the Basic Science Research Program through the National Research Founda- tion of Korea 共NRF兲 funded by the Ministry of Education, Science and Technology 共Grant No. 20100000706兲.

*Author to whom correspondence should be addressed;

.

30

0.8

s)

−6

−4

−2 hzcri(kOe) 0.8

0.4

0 υcri(km/s) 0 20 40

ρ|∆mz|(nm)

h_z^cri υ_cri ρ|∆m_z|

−4 −2 0 2 4

H_p(kOe) (a)

−6

e)

40

m)

(b)

30 20 10

h_z^cri(kOe) 0.4

0 υcri(km/s

10 20 30 40

−2 −4 −6

0 0.4 0.8

−4

−2 hzcri(kOe

υcri(km/s) ρ|∆mz|(nm

ρ|∆m_z|(nm) FIG. 4.

共

Color online

兲共

a

兲

0.8 0.6

(k m /s )

i 0.4 ρ/ρ0

2

−4 0 4 H_p(kOe) 0.5

1.0 1.5

(R,L) = (300 nm, 30 nm) (150 nm, 20 nm) (150 nm, 40 nm)

Simulation Analytical

0

υ

cri

0.2

−4 −2 0 2 4

H

_p

(kOe)

−4 0 4 H_p(kOe) 0

∆mz2

1

FIG. 5.

共

Color online

兲

␷criversusH_p, obtained from simulations

共

closed symbols

兲

for indicated different dot dimensions, as com- pared with the results from analytical expression of ␷cri

=

共␷

_cri⁰ /␳0

兲␳兩⌬

m_z

兩共

solid lines

兲

. For the different dimensions, the simulation results are within the error associated with the finite cell size employed for finding the vortex-core velocity. For all of the dimensions, the variations in

兩⌬

m_z

兩

and ␳/␳0 with H_p reflect the same trend, as shown in the insets.

.

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2008

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共

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兲

.

31The version of the OOMMF code used is 1.2a4. See http://

math.nist.gov/oommf

, which occurs as follows: ␻0

共

−3

兲

= 2␲⫻290 MHz,

= 2␲

⫻430 MHz, ␻0

共

+1

兲

= 2␲⫻470 MHz, ␻0

共

+2

兲

= 2␲

.

39A. A. Thiele,Phys. Rev. Lett. 30, 230

共

1973

兲

.

40E. E. Huber, Jr., D. O. Smith, and J. B. Goodenough, J. Appl.

Phys. 29, 294

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; R. D. Gomez, T. Luu, A. Pak, K. Kirk, and J. Chapman,ibid. 85, 6163

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Origin, criterion, and mechanism of vortex-core reversals in soft magnetic nanodisks under perpendicular bias fields