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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
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)
Vortex
Vortex Meron Swirl
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 1 / 24
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)
Vortex Meron
Meron Swirl
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 1 / 24
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)
Vortex Meron Swirl
Swirl
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 1 / 24
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)
Vortex Meron Swirl Magnetic Vortex
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 1 / 24
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
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/
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 2 / 24
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 2 / 24
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 3 / 24
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 3 / 24
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 3 / 24
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”
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 4 / 24
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”
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 4 / 24
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”
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 4 / 24
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”
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 4 / 24
Vortices in planar magnets
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)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 5 / 24
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?
Array of vortex state dots
[Raabe et al (2000)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 6 / 24
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?
Vortex Random Access Memory (VRAM)
[Bohlens et al (2008)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 6 / 24
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?
Vortex Random Access Memory (VRAM)
[Yu et al APL (2011)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 6 / 24
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?
Vortex Random Access Memory (VRAM)
[Yu et al APL (2011)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 6 / 24
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)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 7 / 24
Vortices in planar magnets Statics of vortices
Equilibrium magnetisation distribution in nanoring
Vortex state:
minimal diameter > 20 nm
[Kravchuk, Sheka, Gaididei, JMMM (2007)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 8 / 24
Vortices in planar magnets Statics of vortices
Equilibrium magnetisation distribution in nanoring
Vortex state:
minimal diameter > 20 nm
[Kravchuk, Sheka, Gaididei, JMMM (2007)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 8 / 24
Vortices in planar magnets Statics of vortices
Equilibrium magnetisation distribution in nanoring
Vortex state:
minimal diameter > 20 nm
[Kravchuk, Sheka, Gaididei, JMMM (2007)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 8 / 24
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)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 9 / 24
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
Low–frequency Gyroscopic mode
[Ivanov, Zaspel, PRL (2005)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 9 / 24
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
Magnon modes
[Ivanov, Zaspel, PRL (2005)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 9 / 24
Vortices in planar magnets Vortex dynamics
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
Linear equations on X
Linear analysis: magnons on the vortex
Magnon modes
[Ivanov, Zaspel, PRL (2005)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 9 / 24
Vortices in planar magnets Vortex dynamics
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)]
[Pylypovskyi et al, arXiv (2013)]
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
Field frequency 500 MGHz
[Kravchuk et al, JAP (2007)]
[Kim et al, APL (2008)]
[Curcic et al, PRL (2008)]
CIP structure (asymmetrical switching)
Current intensity: 107 A/cm2
Current frequency: 300 GHz
[Yamada et al, Nat. Mat. (2007)]
[Kim et al, APL (2007)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 10 / 24
Vortices in planar magnets Vortex dynamics
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)]
[Pylypovskyi et al, arXiv (2013)]
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
Field frequency 500 MGHz
[Kravchuk et al, JAP (2007)]
[Kim et al, APL (2008)]
[Curcic et al, PRL (2008)]
CIP structure (asymmetrical switching)
Current intensity: 107 A/cm2
Current frequency: 300 GHz
[Yamada et al, Nat. Mat. (2007)]
[Kim et al, APL (2007)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 10 / 24
Vortices in planar magnets Vortex dynamics
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)]
[Pylypovskyi et al, arXiv (2013)]
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
Field frequency 500 MGHz
[Kravchuk et al, JAP (2007)]
[Kim et al, APL (2008)]
[Curcic et al, PRL (2008)]
CIP structure (asymmetrical switching)
Current intensity: 107 A/cm2
Current frequency: 300 GHz
[Yamada et al, Nat. Mat. (2007)]
[Kim et al, APL (2007)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 10 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 11 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 11 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 11 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 12 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 12 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 12 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 13 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 13 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 13 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 13 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 13 / 24
Vortices in planar magnets Vortex dynamics
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,
Lagrangian formalism
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 14 / 24
Vortices in planar magnets Vortex dynamics
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,
Lagrangian formalism
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)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 14 / 24
Vortices in planar magnets Vortex dynamics
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 15 / 24
Vortices in spherical shells
Spherical shell
Disk Small spherical shell
Demo
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 16 / 24
Vortices in spherical shells
Spherical shell
Disk Small spherical shell
Demo
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 16 / 24
Vortices in spherical shells
Spherical shell
Disk Small spherical shell
Demo
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 16 / 24
Vortices in spherical shells
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 17 / 24
Vortices in spherical shells
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 17 / 24
Vortices in spherical shells
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 17 / 24
Vortices in spherical shells
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 18 / 24
Vortices in spherical shells
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 18 / 24
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
].
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 19 / 24
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
].
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 19 / 24
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
].
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 19 / 24
Vortices in spherical shells In–surface and out–of–surface vortices
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 20 / 24
Vortices in spherical shells In–surface and out–of–surface vortices
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 20 / 24
Vortices in spherical shells In–surface and out–of–surface vortices
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
[Kravchuk, Sheka, Streubel, Makarov, Schmidt, Gaididei, PRB (2012)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 20 / 24
Vortices in spherical shells In–surface and out–of–surface vortices
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
[Kravchuk, Sheka, Streubel, Makarov, Schmidt, Gaididei, PRB (2012)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 20 / 24
Vortices in spherical shells In–surface and out–of–surface vortices
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
[Kravchuk, Sheka, Streubel, Makarov, Schmidt, Gaididei, PRB (2012)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 20 / 24
Vortices in spherical shells In–surface and out–of–surface vortices
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
[Kravchuk, Sheka, Streubel, Makarov, Schmidt, Gaididei, PRB (2012)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 20 / 24
Vortices in spherical shells In–surface and out–of–surface vortices
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𝜕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 = −4𝜋𝜌ΔΦ = −4𝜋𝜌, where 𝜌 is caused by cosΘVortex core plays the role of a charge for the vortex phase structure
[Kravchuk, Sheka, Streubel, Makarov, Schmidt, Gaididei, PRB (2012)]
𝜕uuΦ = −g(u) sinΦ, g(u) = − 4eu𝜕u cos Θ1+e2u
ΔΦ = −4𝜋𝜌
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 20 / 24
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)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 21 / 24
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)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 21 / 24
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!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)]
Polarity–chirality coupling!
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 21 / 24
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!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)]
Polarity–chirality coupling!
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 21 / 24
Vortices in spherical caps
Spherical Caps
Disk
Spherical caps
Do we have homogeneous magnetisation for
the small caps?
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 22 / 24
Vortices in spherical 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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 22 / 24
Vortices in spherical caps
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 22 / 24
Vortices in spherical caps
Future
Vortex dynamics on a curved surface
Gyroscopical vortex motion
Nonlinear vortex dynamics, including switching phenomena
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 23 / 24
Vortices in spherical caps
Future
Vortex dynamics on a curved surface
Gyroscopical vortex motion
Nonlinear vortex dynamics, including switching phenomena
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 23 / 24
Vortices in spherical caps
Future
Vortex dynamics on a curved surface
Gyroscopical vortex motion
Nonlinear vortex dynamics, including switching phenomena
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 23 / 24
Vortices in spherical caps
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!
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 24 / 24
Vortices in spherical caps
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!
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 24 / 24
Symmetrical switching by DC field in details
What is the mechanism of the switching?
Switching is mediated by a Bloch point
[Thiaville et al, PRB (2003)]
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
[Okuno et al, JMMM (2002)]
[Thiaville et al, PRB (2003)]
[Kravchuk et al, Sol.St.Phys., (2007)]
AC field: 30mT
Resonance excitation of symmetrical
mode by AC field
Regular and chaotic dynamics
[Wang and Dong, APL (2012)]
[Yoo et al, APL (2012)]
[Pylypovskyi et al, arXiv
(2012)]
[Pylypovskyi et al, arXiv
(2013)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 25 / 24
Symmetrical switching by DC field in details
What is the mechanism of the switching?
Switching is mediated by a Bloch point
[Thiaville et al, PRB (2003)]
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
[Okuno et al, JMMM (2002)]
[Thiaville et al, PRB (2003)]
[Kravchuk et al, Sol.St.Phys., (2007)]
AC field: 30mT
Resonance excitation of symmetrical
mode by AC field
Regular and chaotic dynamics
[Wang and Dong, APL (2012)]
[Yoo et al, APL (2012)]
[Pylypovskyi et al, arXiv
(2012)]
[Pylypovskyi et al, arXiv
(2013)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 25 / 24
Symmetrical switching by DC field in details
What is the mechanism of the switching?
Switching is mediated by a Bloch point
[Thiaville et al, PRB (2003)]
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
[Okuno et al, JMMM (2002)]
[Thiaville et al, PRB (2003)]
[Kravchuk et al, Sol.St.Phys., (2007)]
AC field: 30mT
Resonance excitation of symmetrical
mode by AC field
Regular and chaotic dynamics
[Wang and Dong, APL (2012)]
[Yoo et al, APL (2012)]
[Pylypovskyi et al, arXiv
(2012)]
[Pylypovskyi et al, arXiv
(2013)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 25 / 24
Symmetrical switching by DC field in details
What is the mechanism of the switching?
Switching is mediated by a Bloch point
[Thiaville et al, PRB (2003)]
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
[Okuno et al, JMMM (2002)]
[Thiaville et al, PRB (2003)]
[Kravchuk et al, Sol.St.Phys., (2007)]
AC field: 30mT
Resonance excitation of symmetrical
mode by AC field
Regular and chaotic dynamics
[Wang and Dong, APL (2012)]
[Yoo et al, APL (2012)]
[Pylypovskyi et al, arXiv
(2012)]
[Pylypovskyi et al, arXiv
(2013)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 25 / 24
Asymmetrical switching in details
Dip creation
Vortex-antivortex pair creation
Annihilation of a new-born antivortex with original vortex
[Waeyenberge et al, Nature(2006)]
[Hertel et al, PRL (2006)]
[Sheka et al, APL (2007)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 26 / 24
Asymmetrical switching in details
Dip creation
Vortex-antivortex pair creation
Annihilation of a new-born antivortex with original vortex
[Waeyenberge et al, Nature(2006)]
[Hertel et al, PRL (2006)]
[Sheka et al, APL (2007)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 26 / 24
Asymmetrical switching in details
Dip creation
Vortex-antivortex pair creation
Annihilation of a new-born antivortex with original vortex
[Waeyenberge et al, Nature(2006)]
[Hertel et al, PRL (2006)]
[Sheka et al, APL (2007)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 26 / 24
Asymmetrical switching in details
Dip creation
Vortex-antivortex pair creation
Annihilation of a new-born antivortex with original vortex
[Waeyenberge et al, Nature(2006)]
[Hertel et al, PRL (2006)]
[Sheka et al, APL (2007)]
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 26 / 24
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 27 / 24
Out–of–surface vortices structure: case of p1 = p2 = p
Φ(𝜗) ≈ ±𝜋2
(1 − p𝜗c𝛼 ln tan 𝜗
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
Denis D. Sheka (Kiev University) Dynamics of Magnetic Vortices Heraklion, April 11, 2013 28 / 24