Denis D. Sheka¼MAGelm2013/talks/Sheka.pdf · Dynamics of Magnetic Vortices in Nanoparticles Denis...

Dynamics of Magnetic Vortices in Nanoparticles

Denis D. Sheka

Taras Shevchenko National University of Kiev, Ukraine

Workshop “Domain microstructure and dynamics in magnetic elements”

(Heraklion, Crete, April 8 — 11, 2013)

Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 1 / 24


Denis D. Sheka




Vortex

Vortex Meron Swirl



Denis D. Sheka




Vortex Meron

Meron Swirl



Denis D. Sheka




Vortex Meron Swirl

Swirl



Denis D. Sheka




Vortex Meron Swirl Magnetic Vortex



Denis D. Sheka





Collaborators

Ukraine

Volodymyr Kravchuk, Yuri GaidideiBogolyubov Institute for Theoretical Physics, Kiev

Olexandr Pylypovskyi, Olexii Volkov, Mykola SloykaTaras Shevchenko National University of Kiev, Ukraine

Our team website :: http://slasi.rpd.univ.kiev.ua/


http://slasi.rpd.univ.kiev.ua/

Collaborators

Ukraine

Volodymyr Kravchuk, Yuri GaidideiBogolyubov Institute for Theoretical Physics, Kiev

Olexandr Pylypovskyi, Olexii Volkov, Mykola SloykaTaras Shevchenko National University of Kiev, Ukraine

Our team website :: http://slasi.rpd.univ.kiev.ua/

Germany

Franz G. MertensUniversitat Bayreuth

Denys Makarov, Robert StreubelInstitute for Integrative Nanosciences, IFW Dresden


http://slasi.rpd.univ.kiev.ua/

Outline

1 Vortices in planar magnets

Motivation

Statics of vortices

Vortex dynamics

2 Vortices in spherical shells

Model

In–surface and out–of–surface vortices

Polarity–chirality coupling

3 Vortices in spherical caps


Outline


Motivation

Statics of vortices

Vortex dynamics


Model





Outline


Motivation

Statics of vortices

Vortex dynamics


Model





Vortices in planar magnets

Soft nanomagnets

E =

∫d3x

[ A2M2

S

(∇M)2⏟ ⏞ exchange

−1

2M · Hms⏟ ⏞

magnetostatic

]

Hms magnetostatic field{∇× Hms = 0,∇ · Hms = −4𝜋∇ · M

Exchange length: ℓ =

√A

4𝜋M2S∼ 5 ÷ 10 nm

Landau-Lifshitz equation

𝜕M𝜕t

= −𝛾[M × Hef

]+

𝛼

MM ×

𝜕M𝜕t

Hef = −𝛿E

𝛿Meffective magnetic field

length (m)

single-domain state

exchange

nanomagnets

exchange /dipolar

multi-domain state

anisotropy10−8 10−5

“Flower state” “Landau state” “Diamond state”

“C–state” “Onion–state” “Vortex–state”



Soft nanomagnets

E =

∫d3x

[ A2M2

S


−1

2M · Hms⏟ ⏞

magnetostatic

]



√A

4𝜋M2S∼ 5 ÷ 10 nm


𝜕M𝜕t

= −𝛾[M × Hef

]+

𝛼

MM ×

𝜕M𝜕t

Hef = −𝛿E


length (m)

single-domain state

exchange

nanomagnets

exchange /dipolar

multi-domain state






Soft nanomagnets

E =

∫d3x

[ A2M2

S


−1

2M · Hms⏟ ⏞

magnetostatic

]



√A

4𝜋M2S∼ 5 ÷ 10 nm


𝜕M𝜕t

= −𝛾[M × Hef

]+

𝛼

MM ×

𝜕M𝜕t

Hef = −𝛿E


length (m)

single-domain state

exchange

nanomagnets

exchange /dipolar

multi-domain state






Soft nanomagnets

E =

∫d3x

[ A2M2

S


−1

2M · Hms⏟ ⏞

magnetostatic

]



√A

4𝜋M2S∼ 5 ÷ 10 nm


𝜕M𝜕t

= −𝛾[M × Hef

]+

𝛼

MM ×

𝜕M𝜕t

Hef = −𝛿E


length (m)

single-domain state

exchange

nanomagnets

exchange /dipolar

multi-domain state






Vortex as a ground state of a submicron-sized disc

Vortex in–plane structure: 𝜑 ≡ arctanmymx

= arctan yx + C𝜋

2

Chirality: counterclockwise or clockwise

C = −1 C = +1[Wachowiak et al, Science (2002)]

Vortex out–of–plane structure: mz ∼ pe−r2/ℓ2

Polarity: up or down

p = +1 p = −1

[Shinjo et al, Science

(2000)][Chou et al, Bac, APL (2007)]


Vortices in planar magnets Motivation

Motivation

Typical particle sizes

Diameter 100 ÷ 1000 nm

Thickness 5 ÷ 100 nm

Vortex core ∼ 10 nm

Array of vortex state dots

Huge data storage (Tbit/inch2)

Very fast MRAM (Tbit/sec)

How can we decrease the

vortex size?

How can we control the vortex

state fast enough?


[Raabe et al (2000)]



Motivation









vortex size?


state fast enough?

Vortex Random Access Memory (VRAM)

[Bohlens et al (2008)]



Motivation









vortex size?


state fast enough?


[Yu et al APL (2011)]



Motivation









vortex size?


state fast enough?


[Yu et al APL (2011)]


Vortices in planar magnets Statics of vortices

Equilibrium magnetisation distribution in nanodisk

Micromagnetic simulations

Experiment

Vortex state: minimal diameter > 50 nm

[Chung, McMichael, Pierce, Unguris, PRB (2010)]



Equilibrium magnetisation distribution in nanoring

Vortex state:

minimal diameter > 20 nm

[Kravchuk, Sheka, Gaididei, JMMM (2007)]




Vortex state:






Vortex state:




Vortices in planar magnets Vortex dynamics

Vortex dynamics

Thiele equations [Thiele, PRL (1973)]

Fgyro + Fms = 0

Fgyro = 4𝜋Q

[dXdt

× z

], Fms ∝ −kX

Q = −qp/2 (𝜋2–topological charge)

q = 1 (vorticity), p = ±1 (polarity)

Vortex trajectory

[Kovalev, Mertens, Schnitzer, EPJB (2003)]

Linear problem

Linear equations on X

Linear analysis: magnons on the vortex

[Ivanov, Zaspel, PRL (2005)]



Vortex dynamics


Fgyro + Fms = 0

Fgyro = 4𝜋Q

[dXdt

× z

], Fms ∝ −kX



Vortex trajectory


Linear problem



Low–frequency Gyroscopic mode




Vortex dynamics


Fgyro + Fms = 0

Fgyro = 4𝜋Q

[dXdt

× z

], Fms ∝ −kX



Vortex trajectory


Linear problem



Magnon modes




Vortex dynamics

Magnons on Vortex

Generalized Schrodinger equation for the

linearized magnetization on the vortex

background [Sheka et al, PRB (2004)]

H Ψ+ WΨ⋆ = i𝜕tΨ,

H =[(−i∇− A

)2+ U

]Specific long–range magnetic field

A ∝ Q∇𝜒: Aharonov–Bohm type of

soliton–magnon scattering

[Sheka et al, PRB (2004)]

Unusual scattering results: generalized

Levinson theorem

[Sheka et al, PRA (2004); Sheka et al, PRA (2006)]

Additional “singular force” in effective

equations of motion [Sheka, JPhysA (2006)]

Linear problem



Magnon modes




Control of the vortex polarity

Perpendicular field (symmetrical switching)

DC field: 1T

[Okuno et al, JMMM (2002)]

[Thiaville et al, PRB (2003)]

[Kravchuk et al, Sol.St.Phys., (2007)]

AC field: 30mT

[Wang and Dong, APL (2012)]

[Yoo et al, APL (2012)]

[Pylypovskyi et al, arXiv (2012)]


Linear polarized field (asymmetrical switching)

Field amplitude 1 mT

Field frequency 400 MGHz

[Lee et al, PRB (2007)]

[Kim et al, APL (2008)]

CPP structure (asymmetrical switching)

Currents: 106 ÷ 108 A/cm2

[Caputo et al, PRL (2007)]

[Sheka et al, APL (2007)]

[Choi et al, APL (2010)]

Field burst (asymmetrical switching)

Field burst 1.5 mT by 4ns

[Waeyenberge et al, Nature (2006)]

[Hertel et al, PRL (2007)]

Rotating field (asymmetrical switching)

Field amplitude 0.4 mT


[Kravchuk et al, JAP (2007)]


[Curcic et al, PRL (2008)]

CIP structure (asymmetrical switching)

Current intensity: 107 A/cm2

Current frequency: 300 GHz

[Yamada et al, Nat. Mat. (2007)]






DC field: 1T




AC field: 30mT


































DC field: 1T




AC field: 30mT
































Criterion of switching

General criterion for the vortex core switching

The vortex velocity should reach the critical velocity

vc ∼ 1.7𝛾√

A ∼ 300 m/s

[Lee, Kim, Yu, Choi, Guslienko, Jung, Fischer, PRL (2008)]

Is it necessary for vortex to

move at all in order to

switch its polarity?

Vortex switching by immobile vortex

Vortex pinned by an impurity + rotating field [Kravchuk, Gaididei, Sheka, PRB (2009)]

Permalloy disc: D = 150nm, h = 20nmField: Bx + iBy = B0e−i𝜔t

B0 = 20mT, 𝜔 = 8GHz

v-av paircreation

2 GHz

7 GHz

9 GHz

11 GHz

14 GHz

20 GHz

119

14

720

2






vc ∼ 1.7𝛾√

A ∼ 300 m/s








B0 = 20mT, 𝜔 = 8GHz

v-av paircreation

2 GHz

7 GHz

9 GHz

11 GHz

14 GHz

20 GHz

119

14

720

2






vc ∼ 1.7𝛾√

A ∼ 300 m/s








B0 = 20mT, 𝜔 = 8GHz

v-av paircreation

2 GHz

7 GHz

9 GHz

11 GHz

14 GHz

20 GHz

119

14

720

2



Phenomenological model of the nanomagnet

Energy of the nanomagnet

E =

∫d3r

(Wex + W f + Wms

)Wex =

ℓ2

2(∇m)2, W f = −m · B

Wms = −1

2m · Hms

Hms =

{∇× Hms = 0,∇ · Hms = −4𝜋∇ · m

Lagrangian formalism

m =M

Ms= (sin 𝜃 cos 𝜙, sin 𝜃 sin 𝜙, cos 𝜃)

Lagrangian

L =

∫d3r(1 − cos 𝜃)�� − E

Dissipative Function

F =𝜂

2

∫d3r

(��

2+ sin2

𝜃��2)

Rotating reference frame

Rotating field: Bx + iBy = Be−i𝜔t

Energy: W f = B sin 𝜃 cos(𝜑− 𝜔t)The invariance of the magnetic energy

of cylindrical nanodots under two

simultaneous rotations:

in the spin space 𝜑 → 𝜑− 𝜔t

in the real space 𝜒 → 𝜒− 𝜔t

In the rotating frame of reference the

magnetic energy is time–independent





E =

∫d3r

(Wex + W f + Wms

)Wex =

ℓ2

2(∇m)2, W f = −m · B

Wms = −1

2m · Hms

Hms =

{∇× Hms = 0,∇ · Hms = −4𝜋∇ · m


m =M


Lagrangian

L =

∫d3r(1 − cos 𝜃)�� − E


F =𝜂

2

∫d3r

(��

2+ sin2

𝜃��2)














E =

∫d3r

(Wex + W f + Wms

)Wex =

ℓ2

2(∇m)2, W f = −m · B

Wms = −1

2m · Hms

Hms =

{∇× Hms = 0,∇ · Hms = −4𝜋∇ · m


m =M


Lagrangian

L =

∫d3r(1 − cos 𝜃)�� − E


F =𝜂

2

∫d3r

(��

2+ sin2

𝜃��2)












Center–manifold approach

Our explanation for the complicated nonlinear dynamics of externally

perturbed magnetic nanodots is the following:

One has to study the magnon modes on the vortex background

Due to rotation there is a ’softening’ of magnon modes𝜔m = 𝜔m − m𝜔

Only few of them are effectively excited by the external driving, all

others are damped and slaved

The dynamics of the dip creation can be then described by an

attractor consisting of a few modes

Dip is the stationary state of the system in the rotating frame of

reference













reference













reference













reference













reference



Center-manifold approach in action

Partial wave expansion

Magnetization

M = MS{sin 𝜃 cos𝜑, sin 𝜃 sin𝜑, cos 𝜃}Magnon modes:

cos 𝜃 = cos 𝜃v(r) +∑

m=0,±1

𝛼m(t)f|m|(r)eim𝜒,

𝜑 = 𝜑v(𝜒) +∑

m=0,±1

𝛽m(t)g|m|(r)eim𝜒,


Vortex + Magnon modes ⇒ L [𝜃, 𝜑] ⇒ L ef[𝛼, 𝛽]

Coupling between modes: m = 0,±1

��0 = −𝜔0𝛽0 − 𝜂𝜔0A0𝛼0

��1 = −𝜔1𝛽1 + i𝛼1(k𝛼0 + 𝜔)− Bef − 𝜂𝜔1A1𝛼1

��0 = 𝜔0𝛼0 + ik(𝛼1𝛽*1 − 𝛼*

1𝛽1)− 𝜂𝜔0B0𝛽0

��1 = 𝜔1𝛼1 − i𝛽1(k𝛼0 − 𝜔)− 𝜂𝜔1B1𝛽1

Dip depth: numerics

[Kravchuk, Gaididei, Sheka, PRB (2009)]

v-av paircreation

2 GHz

7 GHz

9 GHz

11 GHz

14 GHz

20 GHz

119

14

720

2



Center-manifold approach in action

Partial wave expansion

Magnetization

M = MS{sin 𝜃 cos𝜑, sin 𝜃 sin𝜑, cos 𝜃}Magnon modes:

cos 𝜃 = cos 𝜃v(r) +∑

m=0,±1

𝛼m(t)f|m|(r)eim𝜒,

𝜑 = 𝜑v(𝜒) +∑

m=0,±1

𝛽m(t)g|m|(r)eim𝜒,


Vortex + Magnon modes ⇒ L [𝜃, 𝜑] ⇒ L ef[𝛼, 𝛽]

Coupling between modes: m = 0,±1

��0 = −𝜔0𝛽0 − 𝜂𝜔0A0𝛼0

��1 = −𝜔1𝛽1 + i𝛼1(k𝛼0 + 𝜔)− Bef − 𝜂𝜔1A1𝛼1

��0 = 𝜔0𝛼0 + ik(𝛼1𝛽*1 − 𝛼*

1𝛽1)− 𝜂𝜔0B0𝛽0

��1 = 𝜔1𝛼1 − i𝛽1(k𝛼0 − 𝜔)− 𝜂𝜔1B1𝛽1

Dip depth: numerics + analytics

[Kravchuk, Gaididei, Sheka, PRB (2009)]



Switching by immobile vortex

Nucleation by nonhomogeneous rotating magnetic field[Gaididei, Kravchuk, Sheka, Mertens, PRB (2010)]

Field: B = Bx + iBy = B0ei(𝜇+1)𝜒+i𝜔t ⇒ mode with m = 𝜇 is excited

B0 = 40mT, 𝜔 = 6GHz; Py: D = 300nm, h = 20nm

Excitation of linear mode with

m = 3Nonlinear excitation of higher

modes

Vortex-antivortex pairs

generation


Vortices in spherical shells

Spherical shell

Disk Small spherical shell

Demo



Spherical shell


Demo



Spherical shell


Demo



Hairy Ball Theorem

Hairy Ball Theorem

Any continuous tangent vector field on S2 must vanish

somewhere

[Poincare (1885)]

You cannot comb a hairy ball flat without creating a

cowlick

Meteorological consequences: the eye of a cyclone



Hairy Ball Theorem

Hairy Ball Theorem


somewhere

[Poincare (1885)]


cowlick




Hairy Ball Theorem

Hairy Ball Theorem


somewhere

[Poincare (1885)]


cowlick




Vortices on curved surfaces

2D in–surface vector field

[Turner, Vitelli, Nelson, “Vortices on curved surfaces”, Rev. Mod. Phys. (2010)]

Homotopy group 𝜋1(S1):In–surface vortices are characterized by the vorticity q ∈ Z

Examples:

Hydrodynamic vortices on different surfaces

Vortices in liquid crystals on a curved surface

Vortices in curved helium films

3D out–of–surface vector field

Homotopy group 𝜋2(S2,S1):Out–of–surface vortices in magnets are characterized by the

vorticity q ∈ Z and by the polarity p = ±1

magnetic vortices



Vortices on curved surfaces

2D in–surface vector field

[Turner, Vitelli, Nelson, “Vortices on curved surfaces”, Rev. Mod. Phys. (2010)]

Homotopy group 𝜋1(S1):In–surface vortices are characterized by the vorticity q ∈ Z

Examples:

Hydrodynamic vortices on different surfaces

Vortices in liquid crystals on a curved surface

Vortices in curved helium films

3D out–of–surface vector field

Homotopy group 𝜋2(S2,S1):Out–of–surface vortices in magnets are characterized by the

vorticity q ∈ Z and by the polarity p = ±1

magnetic vorticesmagnetic vorticesmagnetic vortices


Vortices in spherical shells Model

Model

Heisenberg easy–surface ferromagnet

E = A

∫dr

[−m · ∇2m⏟ ⏞

exchange

+(m · n)2

ℓ2⏟ ⏞ anisotropy

]

Thin shell

Eex = Ah∫

d2𝜎√

|g|gij𝜕m𝛼

𝜕𝜎i

𝜕m𝛽

𝜕𝜎j𝜎i = (𝜗, 𝜒), i, j = 1, 2𝛼, 𝛽 = x, y, z

Local spherical reference frame

m = (mr,m𝜗,m𝜒)mr = cosΘm𝜗 = sinΘ sinΦm𝜒 = sinΘ cosΦ y

x

z

m

mr

m𝜗

m𝜒

𝜗

𝜒

E = E0 + Ecrv,

E0 =A2

∫dr

[(∇Θ)2 + sin2 Θ(∇Φ)2 +

cos2 Θ

ℓ2

]Ecrv = A

∫drr2

[1 + sin2 Θ

cos 2𝜗

2 sin2 𝜗+ cosΦ𝜕𝜗Θ− sinΘ cosΘ sinΦ𝜕𝜗Φ+

sinΦ

sin𝜗𝜕𝜒Θ

+(cosΘ cosΦ+cot𝜗 sinΘ) sinΘ𝜕𝜒Φ

sin𝜗

].



Model


E = A

∫dr

[−m · ∇2m⏟ ⏞

exchange

+(m · n)2


]

Thin shell

Eex = Ah∫

d2𝜎√

|g|gij𝜕m𝛼

𝜕𝜎i

𝜕m𝛽

𝜕𝜎j𝜎i = (𝜗, 𝜒), i, j = 1, 2𝛼, 𝛽 = x, y, z



x

z

m

mr

m𝜗

m𝜒

𝜗

𝜒

E = E0 + Ecrv,

E0 =A2

∫dr

[(∇Θ)2 + sin2 Θ(∇Φ)2 +

cos2 Θ

ℓ2

]Ecrv = A

∫drr2

[1 + sin2 Θ

cos 2𝜗


sinΦ

sin𝜗𝜕𝜒Θ


sin𝜗

].



Model


E = A

∫dr

[−m · ∇2m⏟ ⏞

exchange

+(m · n)2


]

Thin shell

Eex = Ah∫

d2𝜎√

|g|gij𝜕m𝛼

𝜕𝜎i

𝜕m𝛽

𝜕𝜎j𝜎i = (𝜗, 𝜒), i, j = 1, 2𝛼, 𝛽 = x, y, z



x

z

m

mr

m𝜗

m𝜒

𝜗

𝜒

E = E0 + Ecrv,

E0 =A2

∫dr

[(∇Θ)2 + sin2 Θ(∇Φ)2 +

cos2 Θ

ℓ2

]Ecrv = A

∫drr2

[1 + sin2 Θ

cos 2𝜗


sinΦ

sin𝜗𝜕𝜒Θ


sin𝜗

].


Vortices in spherical shells In–surface and out–of–surface vortices


High easy–surface anisotropy (ℓ → 0)

In–surface vortices: Θ =𝜋

2, Φ = Φ0 = const.

Finite anisotropy (0 < ℓ ≪ L)

Out–of–surface vortices: Θ = Θ(𝜗), Φ = Φ(𝜗).

Out–of–surface distribution: Ansatz function

cosΘ = p1e− 1

2

(𝜗𝜗c

)2

+ p2e− 1

2

(𝜋−𝜗𝜗c

)2

In–surface distribution:

𝜕uuΦ = −g(u) sinΦ, g(u) = − 4eu𝜕u cos Θ1+e2u , u = ln tan(𝜗/2)

g(u) consists of two peaks localized near the vortex cores at

uc ≈ ln cot(𝜗c/2).

The interplay between out–of–surface and in–surface structure

ΔΦ = −4𝜋𝜌, where 𝜌 is caused by cosΘVortex core plays the role of a charge for the vortex phase structure






2, Φ = Φ0 = const.




cosΘ = p1e− 1

2

(𝜗𝜗c

)2

+ p2e− 1

2

(𝜋−𝜗𝜗c

)2












2, Φ = Φ0 = const.




cosΘ = p1e− 1

2

(𝜗𝜗c

)2

+ p2e− 1

2

(𝜋−𝜗𝜗c

)2







[Kravchuk, Sheka, Streubel, Makarov, Schmidt, Gaididei, PRB (2012)]






2, Φ = Φ0 = const.




cosΘ = p1e− 1

2

(𝜗𝜗c

)2

+ p2e− 1

2

(𝜋−𝜗𝜗c

)2













2, Φ = Φ0 = const.




cosΘ = p1e− 1

2

(𝜗𝜗c

)2

+ p2e− 1

2

(𝜋−𝜗𝜗c

)2













2, Φ = Φ0 = const.




cosΘ = p1e− 1

2

(𝜗𝜗c

)2

+ p2e− 1

2

(𝜋−𝜗𝜗c

)2













2, Φ = Φ0 = const.




cosΘ = p1e− 1

2

(𝜗𝜗c

)2

+ p2e− 1

2

(𝜋−𝜗𝜗c

)2


𝜕uuΦ = −g(u) sinΦ, g(u) = − 4eu𝜕u cos Θ1+e2u𝜕uuΦ = −g(u) sinΦ, g(u) = − 4eu𝜕u cos Θ1+e2u , u = ln tan(𝜗/2)




ΔΦ = −4𝜋𝜌ΔΦ = −4𝜋𝜌, where 𝜌 is caused by cosΘVortex core plays the role of a charge for the vortex phase structure


𝜕uuΦ = −g(u) sinΦ, g(u) = − 4eu𝜕u cos Θ1+e2u

ΔΦ = −4𝜋𝜌


Vortices in spherical shells Polarity–chirality coupling

Out–of–surface vortices structure[Kravchuk, Sheka, Streubel, Makarov, Schmidt, Gaididei, PRB (2012)]

0.2

0.4

0.6

0.8

1.0cos(Θ)

12345

0.1 0.2 0.3 0.4

1.0

0.8

0.6

0.4

0.2

π/4 π/2 3π/4 π π/4 π/2 3π/4 π

2.0

1 — exact numerical solution (p1 = p2)

2 — analytics (p1 = p2): Φ(𝜗) ≈ ±𝜋2

(1 − p𝜗c𝛼 ln tan 𝜗

2

), where 𝛼 = cos(𝛼𝜗cuc𝜋/2)

3 — micromagnetic simulations: exchange + anisotropy (p1 = p2)

4 — micromagnetic simulations: exchange + magnetostatics (p1 = p2)

5 — out–of–surface structure (p1 = −p2)

The role of the magnetostatics

p1 = −p2 ⇒ Φ ≈ ±𝜋/2

p1 = p2 ⇒ Φ = Φ(𝜗)

Polarity–chirality coupling!

Chiral symmetry breaking

Asymmetry in switching thresholds for

vortices with opposite polarities in flat

magnets by by sample roughness[Curcic et al, PRL (2008)]

[Vansteenkiste et al, New J. Phys (2009)]




0.2

0.4

0.6

0.8

1.0cos(Θ)

12345

0.1 0.2 0.3 0.4

1.0

0.8

0.6

0.4

0.2

π/4 π/2 3π/4 π π/4 π/2 3π/4 π

2.0




2






p1 = −p2 ⇒ Φ ≈ ±𝜋/2

p1 = p2 ⇒ Φ = Φ(𝜗)










0.2

0.4

0.6

0.8

1.0cos(Θ)

12345

0.1 0.2 0.3 0.4

1.0

0.8

0.6

0.4

0.2

π/4 π/2 3π/4 π π/4 π/2 3π/4 π

2.0




2






p1 = −p2 ⇒ Φ ≈ ±𝜋/2

p1 = p2 ⇒ Φ = Φ(𝜗)

Polarity–chirality coupling!Polarity–chirality coupling!










0.2

0.4

0.6

0.8

1.0cos(Θ)

12345

0.1 0.2 0.3 0.4

1.0

0.8

0.6

0.4

0.2

π/4 π/2 3π/4 π π/4 π/2 3π/4 π

2.0




2






p1 = −p2 ⇒ Φ ≈ ±𝜋/2

p1 = p2 ⇒ Φ = Φ(𝜗)

Polarity–chirality coupling!Polarity–chirality coupling!








Vortices in spherical caps

Spherical Caps

Disk

Spherical caps

Do we have homogeneous magnetisation for

the small caps?



Spherical Caps

Disk

Vortex state: minimal diameter ∼ 50

nm

Phase diagram for the Py caps

[Streubel, Kravchuk, Sheka, Makarov, Kronast, Schmidt,

Gaididei, APL (2012)]

Vortex state: minimal diameter ∼ 40 nm



Spherical Caps

Py caps

[Streubel, Makarov, Kronast, Kravchuk, Albrecht, Schmidt,

PhysRevB (2012)]

Phase diagram for the Py caps

[Streubel, Kravchuk, Sheka, Makarov, Kronast, Schmidt,

Gaididei, APL (2012)]

Vortex state: minimal diameter ∼ 40 nm



Future

Vortex dynamics on a curved surface

Gyroscopical vortex motion

Nonlinear vortex dynamics, including switching phenomena



Future






Future






Conclusions

Possibility to excite and to control the vortex dynamics by

magnetic fields, and by spin–polarized current

Importance of nonlinear effects for the vortex dynamics

Center manifold approach for the vortex dynamics description

Magnetic vortex naturally appears as a ground state in spherical

shells (topology!)

Curvature causes polarity–chirality coupling

Thank you for the attention!



Conclusions

Possibility to excite and to control the vortex dynamics by

magnetic fields, and by spin–polarized current

Importance of nonlinear effects for the vortex dynamics

Center manifold approach for the vortex dynamics description

Magnetic vortex naturally appears as a ground state in spherical

shells (topology!)

Curvature causes polarity–chirality coupling

Thank you for the attention!


Symmetrical switching by DC field in details

What is the mechanism of the switching?

Switching is mediated by a Bloch point


Theoretical description of symmetrical switching

[Kravchuk, Sheka, Sol.St.Phys., (2007)]

Bloch point

Bloch Point structure in a nanosphere:

[Pylypovskyi, Sheka, Gaididei, PRB (2012)]

DC field: 1T




AC field: 30mT

Resonance excitation of symmetrical

mode by AC field

Regular and chaotic dynamics



[Pylypovskyi et al, arXiv

(2012)]


(2013)]








Bloch point



DC field: 1T




AC field: 30mT


mode by AC field





(2012)]


(2013)]








Bloch point



DC field: 1T




AC field: 30mT


mode by AC field





(2012)]


(2013)]








Bloch point



DC field: 1T




AC field: 30mT


mode by AC field





(2012)]


(2013)]


Asymmetrical switching in details

Dip creation

Vortex-antivortex pair creation

Annihilation of a new-born antivortex with original vortex

[Waeyenberge et al, Nature(2006)]





Dip creation








Dip creation








Dip creation







Out–of–surface vortices structure: case of p1 = −p2 = p

p = +1 outward

p = −1 inward

3D onion state

Φ = 𝜋 for p = 1Φ = 0 for p = −1


Out–of–surface vortices structure: case of p1 = p2 = p

Φ(𝜗) ≈ ±𝜋2


2

), where 𝛼 = cos(𝛼𝜗cuc𝜋/2).

p = -1

p = 1

p = 1 p = -11.9

1.3

1.8

1.7

1.6

1.5

1.4

π/4 π/2 3π/4 π


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