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