Spin Excitations in the Vortex pState Ferromagnetic Dots

Konstantin Y. Guslienko

Dpto. Fisica de Materiales, Universidad del Pais Vasco, 20018 San Sebastian, Spain

IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain

Outline of the talk• Classification of the vortex dynamic excitations

V i d i• Vortex gyrotropic dynamics

• Vortex core reversal (gyrofield and critical velocity) o e co e e e sa (gy o e d a d c ca e oc y)

• Radial & azimuthal spin waves (frequency splitting)

• Recent advances: vortex-spin wave coupling and vortex massspin torque induced vortex dynamicsvortex modes in coupled dot arraysp y

• Summary

Spiral galaxy in the universe Atmospheric vortex

Vortex in nature

Vortex in the oceanMagnetic vortex

Vortices (skyrmions) in ferroelectrics

a) b) c)

S. Seki et al., Science 336, p. 198, April 2012

a) b) c)

(a) Lorentz TEM imaging of thin film single crystal Cu2OSeO3 Lateral magnetization distribution (Cu+2) on 10 nm scale(b) Si l ti k i(b) Single magnetic skyrmion(c) A magnetic skyrmion crystal

Magnetic vortex state stability

Magnetic vortices in soft magnetic dots

Magnetic vortex state stability

In-plane magnetization

Vortex core

M ti V t I t

Vortex

Magnetic Vortex Integers:Vorticity (topological charge): q = 1Chirality (CCW, CW): C = 1

Single-

y ( , )Polarization: p = 1

Magnetic ground stateMagnetic ground statedepends on: depends on: •• Geometry: L and RGeometry: L and R

M t i l A d MM t i l A d M

S g edomain

Phase diagram for magnetically-soft nanodots:LL

•• Material: A and MMaterial: A and Mss

K. Metlov and K. Guslienko, JMMM, 2002. Scale LE=18 nm for FeNi2R2R

Vortex and antivortex structures

Antiortex

Vortex

Vortex structure : +Z : -Z-curling structure of magnetization (M) -perpendicular component (Mz) at the core

Vortex structure

Antivortex structure

-Cross-like lines of magnetization (M)

pe pe d cu a co po e t ( z) at t e co eTopological charge:+1 (V) or -1 (AV)

Cross like lines of magnetization (M) -perpendicular component (Mz) at the core Core polarization: up/down

MFM Ti

Magnetic vortices in soft magnetic particles (dots)

Magnetic flux

MFM Tip

0.5

1.0(e)

(d)

Magnetizat

-0.5

0(c)

(b)(a)

tion, M/M

s

Field mT

R ~ 100R ~ 100--500 nm500 nmL ~ 10L ~ 10--50 nm50 nm

MFM image of magnetic vortex -150 -100 -50 0 50 100 150-1.0 (b)(a) Field, mT

Guslienko et al PRB 2002

L 10L 10 50 nm50 nm

H(a) (b) (c) (d) (e)

Magnetization reversal via vortex movement

Experimental observation of magnetic vortices

Guslienko et al., PRB 2002

Magnetization reversal via vortex movementg

Spin Excitations of Magnetic Vortex: two kinds of eigenmodes

High-frequency spin-waves, GHz range

Low-frequency vortex eigenmodes, sub-GHz range

• Radial modes• Azimuthal modes

• Translation (gyrotropic) modes

Single vortex dynamics:

Vortex Pair Dynamicsin elliptic dots

Dynamic vortexinteractions in:

• Cylindrical• Square/rectangular

Elli ti l d t

p• Tri-layer F/N/F dots• Dense 2D dot

**

• Elliptical dots arrays

** Magnetostatic interactions dominate in sub-micron and micron-size dots **

Governing equation 16Vortex equation of motion: topological solitons

g q

t = 0 t = tM M M

v Some objectis moving !t 0 t t

rX(0) X(t) r r - X(t)

is moving !

Traveling wave ansatz(rigid motion)

M (r,t )=M (r− X (t ) )(rigid motion)

( )Gd d

M MM M H( )Geff

Sdt M dt

M M HX(t) – soliton center

W(X) – potential

( )ˆ 0WD

XG X X

X

( ) p

Eq. of motion

A. Thiele, Phys. Rev. Lett, 30, 230 (1973)

XTopol. charge q=1

Vortex Dynamics: Core TrajectoryM (r,t )=M (r,X ( t ))Core position X(t) Thiele’s equation

ˆ 0Wd dD

dt dt

XX XGX

XX

Core polarization p=+1

Magnetic vortex in a potential wellp=+1 p=-1

Thiele’s equation is essentially more general than Thiele assumed. It’s validnot only for steady motion V=const

Vortex Dynamics: Core TrajectoryVortex moves along circular trajectory with angular frequency ~100 MHz

Vortex core

Vortex coretrajectory X(t)

Vorticity q=1

ω0

Snap shot of the dynamical out-of-plane M-component

X = (X,Y)

M-component

ωM =γ4πM s≈ 30 GHzFor thin dots L<<R ω0=5

9πLRωM

Guslienko et al., JAP 91, 8037, 2002; PRL 96, 067205 (2006)

9π R0 ~ 100-500 MHz

Comparison with FMR and PEEM experiments

Narrow linewidth ~ 2-11 MHz !Be ond rigid orte model Di t t d iBeyond rigid vortex model: Distorted moving vortex - first stage of the vortex core reversal

70FeNiM

Hz

50

60

FeNi=2.95 MHz/OeMs=720 G

uenc

y M

Exp.

0 008 0 010 0 012 0 014

40

50

TM fr

equ

Calc.

FMR: Novosad et al., Phys. Rev. B (2005)Excellent agreement of theory/exp. for thin dots

0,008 0,010 0,012 0,014 Dot aspect ratio, L/R

PEEM: Guslienko et al., PRL (2006)

Vortex core polarization reversal: X-ray experimentSquare Permalloy dots in variable in-plane magnetic field – TR-X-ray STM

Core “up”,CCW rotat.

Core “down”,CW rotatCW rotat.

Small burst (15 Oe) of alternating magnetic field with ω ≈ω0 leads to p-reversal

B. Waeyenberge et al., Nature 444, Nov. 2006 – challenge to theory

Gyrovector ~(pq) is not conserved! It changes sign

Vortex dynamics in cylindrical ferromagnetic dot under oscillating field

H

H t it t ti d t

xy

z

Linearly polarized in-plane field (SP

How to excite vortex motion and get core reversal ?

xcurrent)

H=H (cos(ωt ) sin (ωt ))Circularly polarized field

H y =H0 sin(ωt )H H 0(cos(ωt ),sin (ωt ))Circularly polarized field

Ultrafast vortex core reversal ~ 50 ps in relatively small a.c. field 10-20 Oe(Static core reversal in perp. field of 3-4 kOe)

R. Hertel et al., PRL 2007; K.-S. Lee et al., Phys. Rev B 2007

Spin-wave radiation process related to Vortex core reversal (linear field (polarization)

Magnetic field changeMagnetic field change

V t ti t l tiVortex-antivortex nucleation

Vortex-antivortex pair annihilation

Spin-wave radiation

V-AV pair nucleation V-AV pair annihilationChoi et al., PRL 2007

Scheme of the strong Spin Waves generation with wave vectorsk>105 cm-1 by vortex core p-reversal in magnetic nanoelement

Origin of dynamic vortex core reversal: gyrofield

Critical core deformation -> Crit. Field -> Critical velocity

Large, spatially non-uniform field h near VC

Critical core deformation Crit. Field Critical velocity

uniform field h near VC leads to its deformation and to core p-reversal at critical value h ~ vcritical value hc ~ vc(critical velocity)

Gyrotropic field induced by vortex motion acts as origin of vortex coreorigin of vortex core dynamical reversal

Guslienko et al., PRL 2008

Vortex core reversal: critical velocity

Lee et al., PRL 2008PRL 2008

υc≈ γM s Lex =γ(2A)1/2Critical velocity is only determined by exchange

Vortex core reversal: switching field & timeΩ=ω/ω0Ω ω/ω0

t s(H 0)=− (dω0)− 1 ln (1− H 0C / H0) d=α[1+ ln (R/Rc)/2]Switch. time

H 0C (Ω= 1)= 3 dvC /γR H (t )=H0 exp (iωt )Excit. field

U i l it i f th VC l i hi th iti l l it

Min Switch. field

Universal criterion of the VC reversal is reaching the critical velocity

Guslienko, PRL’08; Lee, PRL’08 Exp. confirmed by Vansteenkiste, PRL 2008

Vortex core switching field vs. frequencyX-Ray imaging experimentX-Ray imaging experiment

A Vansteenkiste et alA. Vansteenkiste et al., Nature Phys. 2009

Minimum of switching field vs. frequency and critical velocity were detected

Vortex core polarization switching: the mainV f t (10 ti l ! ) i ll i bl fi ld HVery fast (10 ps time scale ! ), occurs in small variable field H

VC switching is pure dynamic process - no overcoming an energy barrier

Mechanism - creation and annihilation of the Vortex-AntiVorex pairs (V escape)p )Origin of vortex core p‐ reversal: Gyrofield induced by the vortex motion

Criterion of the reversal: Critical velocity of the vortex core

Choi et al. PRL 2007, Lee et al. PRB 2007, PRL 2008, Guslienko et al. PRL 2008

Criterion of the reversal: Critical velocity of the vortex core

Vortex core reversal in perpendicular fieldh(rf)

1H d.c. perp. p=+1p=-1

Single dot Magnetic resonance force microscopy + r.f. in-plane field

Frequency jumps correspond toVortex core polarization reversal

Mag. resonance force microscopy:ω0(H )=ω0 (0 )[1+pH / H s]

Vortex frequency vs. perpendicular field.NiMnSb, L=44 nm, R=130 and 520 nm G. de Loubens et al. PRL, 2009

Vortex core dynamics and spin wavesSW are classified by number of nodes along radial (n) and azimuthal (m) direct.

Vortex core

Spin waves

Moving vortex core strongly interacts with someMoving vortex core strongly interacts with some spin waves (azimuthal) and influences their frequencies

Spin Wave Modes in the Vortex ground state

μM TR Kerr experimentM (ρ,t )=M υ(ρ)+μ(ρ,t ) μ<< M s

μM υ TR Kerr experimentBuess et al. PRL 2004

Static vortex Spin waves

( )

2.8 GHzn=0

- in plane radius vectorρ=(ρ,)

( )3.9 GHz

μz (ρ,t )=bn(ρ)sin(ωt− m)

μ=(μρ ,μ) - Dynamical magnetization n=1

μρ (ρ,t )=an(ρ)cos(ωt− m)μz (ρ, ) n(ρ) ( )

4.5 GHzn=2

Radial mode profile, index n Azimuthal index, m

m=0 - Radial spin waves (standing) m=+1/-1, +2/-2, … - Azimithal spin waves (propagating)

Splitting of azimuthal spin-wave frequencies: TR Kerr experiments

1) Park and Crowell PRL 95 167201 (2005):1) Park and Crowell, PRL 95, 167201 (2005):ω0=0.15-0.25 GHz, Δω=0.5-0.8 GHz

2) Zhu et al., PRB 71, 180408 (2005):ω0=0.44 GHz, Δω=1.25 GHz

3) Hoffmann et al., PRB 76, 014416 (2007):Δω=0 6 GHz (R=1 μm)Δω=0.6 GHz (R=1 μm)

(from Hoffmann )et al., PRB 2007)

(from Park et al., PRL 2005)

Splitting of azimuthal spin-wave frequencies: the eigenmodes profiles

Splitting of the azimuthal SW (m=+1/-1) due to moving vortex

2 2 ∫ ( ) ( ) ( )

T i l di l SW i f ti

Δω ~ I2ωR2/ωM

m (ρ)

I=∫ dρρmG (ρ)mR(ρ)

l l t d i ll

mG (ρ)= (1− ρ2)/ ρ

Trial radial SW eigenfunctions mR(ρ)

20

u.),

were calculated numerically

Strong pinning at the dot border r=R

Hybridization of gyrotropic mode15

and

mR (a

.

0x(-

4M

s)

Hybridization of gyrotropic mode mG and azimuthal spin waves mRwith m=+1/-1 is strong

10 mR

prof

iles,

mG

field

Hz d, 5

0

Interaction of azimuthal spin waves mR with static dipolar field H of the core is weak0

5

genm

ode

p

Dip

olar

mG

Hzd

Hd of the core is weak0.0 0.2 0.4 0.6 0.8 1.00E

i

Relative radial coordinate, r/RRC

Guslienko et al., PRL 2008

Splitting of azimuthal SW frequencies: Comparison of theory with experiment

1 2

1.4 Pump-probe Kerr experiments:

0 8

1.0

1.2

cy, G

Hz

Kerr experiments:

Park & Crowell, PRL 2005

0 4

0.6

0.8

Freq

uenc

Zhu et al., PRB 2005

0.2

0.4F

G+ Hoffmann et al., PRB

2007

0.00 0.02 0.04 0.06 0.08 0.100.0

Dot aspect ratio, L/R

The calculated SW frequencies splitting (solid line) and Gyrotropic frequency (dashed line): good agreement

The splitting increases as Δω ~ (L/R) and ~1 GHz Guslienko et al., PRL, 2008

SW frequencies in V state: azimuthal SW doublets and evolution with in-plane field

Broadband FMR set‐up andhysteresis loop

Azimuthal m=1/-1 modes (two doublets observed !)

SW frequencySW frequency spectra vs. field H:parallel (c) & perp.(d) rf pumping(d) rf pumping

Theory: Guslienko et al., PRL 2008 FMR measurements: Aliev et al., PRB 2009

Vortex reversal induced by high-frequency driving field

V i i i h

Guslienko et al. PRB 2010

Vortex interaction with azimuthal spin waves(n, m=+1/-1)

RF pumping at SW frequencies leads to VCqReversal at Ho~5-10 Oe

Experimental (X-ray microscopy) phase diagram of vortex core reversal.Blue/red colors correspond to reversal after CW/CCW in-plane filed bursts.Minima in the switching fields correspond to frequencies of the azimuthalMinima in the switching fields correspond to frequencies of the azimuthal spin waves with indices m=+1/-1

M. Kammerer et al., Preprint ArXiv: condmat 2010

Tri-layer dots: quasi-spin valve structure Magnetostatic interaction plays important role:

intralayer + interlayerintralayer interlayer

• Magnetostatic interactions between

− 1γ∂M∂ t

=M × H eff

Magnetostatic interactions between dots strongly affect the dynamic excitations

• Two gyrotropic eigenfrequencies and complicated vortex core trajectories p jfor tri-layer F/N/F dots

T f i (F) l i h i (N)Two ferromagnetic (F) layers with nonmagnetic (N) spacer

Different polarizations and chiralities of F-layers

Guslienko et al., Appl. Phys. Lett., 2005 GMR Exp.: Locatelli et al., 2010

SpinSpin--transfer torque (tritransfer torque (tri--layers)layers)

MMFMFM

MM“Free” layer“Free” layer

N ti lN ti l

ppPerpendicular Perpendicular spin polarizedspin polarized

FMFM

NMNM

ee--MMfixedfixed“Fixed” layer“Fixed” layer

Nonmagnetic layerNonmagnetic layer spin polarizedspin polarizedcurrent Icurrent I

MMfixedfixed

T ST = σIM 0

[M × [M × p]]SpinSpin--transfer torque:transfer torque:0

pp -- spinspin--polarization of the current (|| to the magnetization of the “fixed” layer)polarization of the current (|| to the magnetization of the “fixed” layer)

J. Slonczewski, JMMM J. Slonczewski, JMMM 159159, L1 (1996), L1 (1996) L. Berger, PRB L. Berger, PRB 5454, 9353 (1996), 9353 (1996)

Spin polarized current induced vortex dynamics in nanopillarsvortex dynamics in nanopillars

Nanopillar with CPP d.c. spin polarized current. Free layer L~5 nm, R~100 nm

m= (sinΘ cosΦ, sinΘ sinΦ, cosΘ ) - magnetization spherical anglesm=M / M s

Equation of motion: spin torque term q p qm= − γm× H eff +αLLG m× m+γτ s - LLG equation of m motion + extra torque

τ s =σJm× (m× P )

q q

- Spin torque term by Slonczewski

P=Pz - magnetization of polarizer σ - ST efficiency

SP current excites vortex motion in free layer if IpP > 0 (Dussaux et al.,Nature Com 2010)Nature Com., 2010)

Core position in free layer X=(X,Y) and m(ρ,t )=m[ρ, X (t )]

a=M σJF =aL∫ d 2 ρ sin2Θ ∂Φ F =πaL (z× X )Spin Torque force:

a M s σJF ST aL∫ d ρ sin Θ∂ X

F ST πaL (z× X )

Int. over Vortex core to insert to Thiele equation

Spin polarized current induced vortex dynamics in nanopillarsdynamics in nanopillars

When the vortex steady oscillationsWhen the vortex steady oscillationsstate does exist?

Critical currents of existence ofVortex steady oscillations:

Jc1 < J < Jc2

Caputo et al., PRL 2005Khvakovskiy et al., PRB 2009

Vortex steady state motion Scheme of the forces acting on moving vortex in free layer of nanopillar

G× X = − ∂ X W+ D X +FST ¿ F ST =πaL (z× X )

Vortex motion: steady solutions & Jc2How to find upper current Jc2 of Vortex steady motion ?Two scenarios:

ωG (J )s(J )R=υc →J c2

How to find upper current Jc2 of Vortex steady motion ?Two scenarios:

1) VC reversal at critical core velocity of p-reversal(IpP < 0 oscillations stop after reversal)

J c2' = (1+1 / λ2)J c12) VC expelling from the dot (s=1)

(IpP < 0, oscillations stop after reversal)

(no vortex – no oscillations, but SD steady oscillations are possible)

11

12

Jc2

A/c

m2 )

FeNiMs=800 GL=10 nm

Vortex steady oscillation state(dynamic equilibrium) at

8

9

10

J'c2

nsity

, J (1

07

Jc1 < J < min (Jc2, J’c2)

6

7

8

Cur

rent

den

Jc1x10 Guslienko et al., J. Phys. 2010,in press

60 80 100 120 1405

Dot radius, R (nm)

in press

Vortex state free layer in Nanopillar(Aranda et al JAP 2010)(Aranda et al. JAP 2010)

• Free layer of a illnanopillar

• R=100nm, L=4 Centered Vortex:No oscillations inducedby Spin Transfer Torque

nmby Spin Transfer Torque

Spin polarized current, I

Different regimes of Vortex motionDynamics of av. in-plane magnetization simulated

starting from displaced vortex core (H =50 Oe)starting from displaced vortex core (Hy =50 Oe)

0.20.2<my>

0.0

0.1

>0 0

0.1

etiz

atio

n

-0.2

-0.1

0 0

<my

-0.1

0.0

e m

agne

0 100 200 300 400 5000.2

time(ns)0 100 200 300 400 500

-0.2

In-p

lane

time(ns) Going to a steady state Rs=consttime(ns)J=1.0x106 A/cm2

Damped movement, J<Jc1V t t i th d t t

Going to a steady state Rs=constJ=1.3x106 A/cm2 , Jc1<J<Jc2Finite VC orbit radius

Vortex stops in the dot center

Different regimes of Vortex MotiongGoing out of the dotJ=1 6x106 A/cm2J=1.6x106 A/cm2

I=2 mA (large Oe-field)I 2 mA (large Oe field)J=6.4x106 A/cm2

Change of polarization by Vortex core expelling (core is out and in):

Steady state after t=400 ns correspondsto quasiunform oscillations of frequency b t 3 GH (J J 2 i ) core expelling (core is out and in):

Oscillations stop after p-reversal,J>>Jc2-prime

about 3 GHz (J>Jc2-prime)

Vortex dynamics excited by SP current in nanocontacts

Q Mi l l PRL 100 2 201 (2008)

“Phase-locking of magnetic vortices mediated byantivortices”, Ruotolo et al., Nature Nanotech. 2009

Q. Mistral et al. PRL 100, 257201 (2008)

Vortex motion arround nanocontactConfinement potential W(X) due to Oe-fieldBlue –vortices, red -antivortices

Magnonics: Spin excitations in l d 2D t d tcoupled 2D vortex dot arrays

13000

z)

10000

11000

12000

(MH

z (b)

(1, +1)

(0, 0)

6000

7000

8000

9000

requ

ency

,

(0, +1)

(1, -1)

100200300400

5000

6000

in m

ode

fr

Gyro

(0, -1)

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.00

100

Dot lattice period, a/2RSp

A. Vogel et al., PRL 2010Broadband FMR experiment

A. Awad et al., APL 2010.The detected by broadband FMR spin i f i l b ll d b th dBroadband FMR experiment eigenfrequencies labelled by the mode

indices (n, m) vs. interdot separation

Collective excitations of vortex dot array (beyond dipolar approximation)array (beyond dipolar approximation)

1.10 (k) /0

1.00

1.05

0.90

0.95

k II (1,0)k II (1,1)quen

cy

0 80

0.85

k II (1,0) d = 2 d = 2.1d = 2 5

k II (1,1) d = 2 d = 2.1d = 2 5

Freq

0.75

0.80 d 2.5 d = 3

d 2.5 d = 3

-4 -3 -2 -1 0 1 2 3Wave vector, kd

Dispresion relations for the square array of the vortex magnetic dots.Dispresion relations for the square array of the vortex magnetic dots. The array period is d in units of dot radius R. The frequency is normalisedto the gyrotropic frequency of isolated dot 0 (Sukhostavets et al.,PRB 2013)

Summary 1) The vortex trajectory and VC reversal strongly depend on the

driving field (SP current) amplitude/frequencyg ( ) p q y

2) To describe vortex p-reversal we need to consider internal vortex structure (vortex core deformation, V/AV pairs creation/annihilation)structure (vortex core deformation, V/AV pairs creation/annihilation)

3) There are especial spin waves in the vortex ground state, whichstrongly interact with moving vortex and split their frequencies Thestrongly interact with moving vortex and split their frequencies. Thevortex – SW interaction results in the vortex distortion and finitevortex mass

4) Vortex dynamics is important for interpretation of SP current induced magnetization dynamics in nanopillars and

( d ill i )nanocontacts (steady state vortex oscillations)

Summaryy

1) Mesoscopic magnetic structures provide now a wide testing areaf f i d i li ifor concepts of nanomagnetism and prospective applications

2) Magnetic vortex in nanodot having two stable discrete states of polarization and chirality is a promising candidate for high densitymagnetic recording (non-volatile data storage devices)

3) Vortex dynamics is important now for interpretation of SP current induced magnetization dynamics in nanopillars and nanocontacts

4) The vortex core orientation reversal is extremely fast (~ tens of ps)and can be readily reached in small a.c. field. THz writing ?

5) Understanding the stability and dynamic behavior of magneticvortices is on the forefront of modern science and technology