Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Ultrafast spin dynamics in ferromagnetic filmsVlasovia 2016

Jérôme Hurst

Institut de Physique et Chimie des Matériaux de Strasbourg

31 Mai 2016

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Outlines

1 Introduction : The Vlasov equation in solid state physics

2 Vlasov equation with spins

3 Applications to ferromagnetic films

4 Conclusions and future investigations

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Introduction: Motivations

Theoretical aspect :

Ultrafast (∼ 100 fs = 10−13 s) magnetization dynamics atthe nanometric scaleSpin and relativistic effects (Zeeman + Spin-orbitcoupling)Non linear effectsMean field model

Experimental aspect :

Ultrafast demagnetization in ferromagnetic films inducedby a femtosecond laser pulse : (E. Beaurepaire et al. Phys.Rev. Lett. 76, 4250 1996)

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Introduction: Description of the system

Thin ferromagnetic films of Nickel : L = 5 − 20 nm (1Dproblem)

Two types of charges: electrons & ions :

Electrons Ions

Charge -e +2eDensity (1029m−3) 1.8 0.9Temperature (K) 300 300

mass (a.u) 1 1.1 ∗ 105

ω−1p (fs) 0.25 330

Magnetic moment (µb) 0.066 0.54

Ultrafast dynamics (∼ 100fs) → Ions fixed(1fs = 10−15s)Electrons ↔ Quantum plasmas

Shukla and Eliasson, Rev. Mod. Phys.,

2011

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Introduction: Description of the system

Interactions:

- Coulomb

- Magnetic exchangeions-ions (J)

- Magnetic exchangeions-electrons (K)

Effective magnetic field(100 - 1000 Tesla)

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Introduction: Wigner and Vlasov equations

Phase-space quantum mechanics (Equivalent to Schrödinger)

Wigner without spins:

∂f

∂t+ v

∂f

∂x=

ime

2π~2

∫

ds dv′e

−im

(

v′−v

)

s/~

[

VH

(

x +s

2, t

)

− VH

(

x −s

2, t

)]

f

(

x, v′, t

)

∇2VH =

e

ǫ0

(∫

f (x, v, t) dv − ni

)

Semi-classical limit(

~

m〈v〉e<< L

)

→ Vlasov equation :

∂f

∂t+ v ∂x f +

e

m∂xVH · ∂v f = 0

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Wigner function with spin

Generalisation of the Wigner function :

f (r , v , t) → F(r , v , t) F =

(

f ↑↑ f ↑↓

f ↓↑ f ↓↓

)

Projection onto the Pauli matices:

F =1

2σ0f0 +

1

~f · σ

with

f0 = tr {F} = f ↑↑ + f ↓↓ f =~

2tr (Fσ)

Physical observable : i , α = {x , y , z}

n(r , t) =

∫

f0(r , v , t)dv mi (r , t) =

∫

fi(r , v , t)dv

Ji(r , t) =

∫

vi f0(r , v , t)dv Jsiα(r , t) =

∫

vi fα(r , v , t)dv

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Vlasov with spin

Zeeman + Spin-orbit hamiltonian :

H =

[

(P + eA)2

2m+ V (r , t)

]

σ0 + µBσ · B (r , t)

+µB

8mc2σ · [E × (p + eA) − (p + eA) × E ]

Wigner + semi-classical limit i = {x , y , z}

∂f0

∂t+ v · ∇f0 −

e

m(E + v × B) · ∇v f0 +

µB

2mc2(E × ∇)

ifi

−µB

m∇[

Bi −(

v × E

2c2

)

i

]

· ∇v fi −µBe

2m2c2[E × (B × ∇v)]i fi = 0

∂fi

∂t+ v · ∇fi −

e

m(E + v × B) · ∇v fi +

µB

2mc2(E × ∇)

if0

−µB

m∇[

Bi −(

v × E

2c2

)

i

]

· ∇v f0 −µBe

2m2c2[E × (B × ∇v )]i f0

−2µB

~

{[

B −1

2c2(v × E)

]

× f

}

i

= 0.

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Maxwell equations

Maxwell equations with spin corrections

∇ · E =ρ

ǫ0

−∇ · P

ǫ0

∇ · B = 0

∇ × E = −∂B

∂t

∇ × B = µ0j + µ0ǫ0

∂E

∂t+ µ0

∂P

∂t+ µ0∇ × M

Source terms depend on the spin (Semi-relativistic expansion)

ρ = −e

∫

f0dv , j = −e

[∫

v f0dv +E × M

2mc2

]

M = −µB

∫

f dv , P = −µB

2c2

∫

v × f dv

A Dixit, Y. Hinschberger, J. Zamanian, G. Manfredi, et P.-A

Hervieux, Phys. Rev. A 88, 032117 (2013).

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Minimal self consistent model

1d-1v distribution function (x , vx ) → (no spin-orbite)

∂f0

∂t+ v ∂x f0 +

e

m(∂x φ) (∂v f0) −

µb

m∂x

(

Bi −KniS0

2µb

Si

)

∂v fi = 0

∂fi

∂t+ v ∂x fi +

e

m(∂xφ) (∂v fi ) −

µb

m∂x

(

Bi −KniS0

2µb

Si

)

∂v f0

−KN0S0

~{f × S}

i−

2µB

~{B × f }

i= 0

+ Electrostatic limit of Maxwell equations : i = {x , y , z}

∂2φ

∂x2=

e

ǫ0

(n(x) − nions )∂Bi

∂x= −µBµ0

∂

∂xmi (x) n =

∫

f0dv mi =

∫

fi dv

Landau Lifchitz Gilbert (LLG) equation for the magneticmoment of ions :

∂S(x)

∂t=

a2JS0

~

[

S(x) ×∂2

∂x2S(x)

]

+K

2~

[

S ×

∫

f dv

]

− γS × Bext

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Ground state : Thin film of Nickel

Collinear magnetism + Pauli exclusion principle :f

ground0 = FD

+ + FD−, f

groundz = FD

+ − FD−

FD± =

2πkbT

m

(

m

2π~

)

ln[

1 + exp(

−1

kbT

(

m

2v

2 − eφ ± µBBz − µ(T )))]

Ions : 〈S〉 = SBS (βSheff ) , heff = K2

me(r) + JNv 〈S〉 + gµBH

-4 -2 0 2 4x (nm)

0

0,5

1

Den

sitie

s / n

i

ni

nup

ndown

Density

200 400 600T (K)

0,01

0.061

0,1

0.54

1

Spin

pol

ariz

atio

n (

µ b )

me

mi

Magnetization

Tc= 631 K

Curietemperature :

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Linear regime : Film of Nickel (∼ 5nm)

Laser Excitation : E (t) = E0 sin (ωt) e−[(t−t0)/∆t]2 E0 ∼ 108V /m

Two modes : Plasmon ωp =√

e2n/mǫ0 & Ballistic ωb = (2L/vF )−1

Observable : Delectric =∫

x f0dxdv ; Dmagnetic =∫

x fzdxdv

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Linear regime : Film of Nickel (∼ 5nm)

Laser Excitation : E (t) = E0 sin (ωt) e−[(t−t0)/∆t]2 E0 ∼ 108V /m

Two modes : Plasmon ωp =√

e2n/mǫ0 & Ballistic ωb = (2L/vF )−1

Observable : Delectric =∫

x f0dxdv ; Dmagnetic =∫

x fzdxdv

0 25 50 75 100Time (fs)

1e-06

1e-05

0,0001

0,001

∆t = 5 fs ; ω = 2.73 PHz (800 nm)

(1)

(1)

0 10 20 30 40Frequency (PHz)

1e-09

1e-06

0 10 20 30Time (fs)

-0,02

-0,01

0

0,01

0,02

Ele

ctri

c fi

eld

(1010

V/m

)

∆t = 5 fs

ωb

(1)

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Linear regime : Film of Nickel (∼ 5nm)

Laser Excitation : E (t) = E0 sin (ωt) e−[(t−t0)/∆t]2 E0 ∼ 108V /m

Two modes : Plasmon ωp =√

e2n/mǫ0 & Ballistic ωb = (2L/vF )−1

Observable : Delectric =∫

x f0dxdv ; Dmagnetic =∫

x fzdxdv

0 25 50 75 100Time (fs)

1e-06

1e-05

0,0001

0,001

∆t = 5 fs ; ω = 2.73 PHz (800 nm)

(1)

(1)

0 5 10 15 20 25Time (fs)

1e-05

0,0001

0,001

∆t = 1 fs ; ω = ωp = 24.4 PHz (77 nm)

(1)

(2)

0 10 20 30 40Frequency (PHz)

1e-09

1e-06

0 10 20 30Time (fs)

-0,02

-0,01

0

0,01

0,02

Ele

ctri

c fi

eld

(1010

V/m

)

∆t = 5 fs∆t = 1 fs

ωp

(2)

ωb

(1)

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Initial distribution function : Water bag model

fup

x/Lf

v/vf

−40 −30 −20 −10 0 10 20 30 40−1.5

−1

−0.5

0

0.5

1

1.5

fdown

x/Lf

v/vf

−40 −30 −20 −10 0 10 20 30 40−1.5

−1

−0.5

0

0.5

1

1.5

-2 -1 0 1 2v / v

F

0

0,5

1

1,5

2

Vel

ocit

y di

stri

buti

on

fup

fdown

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Initial distribution function : Water bag model

f0

x/Lf

v/vf

−40 −30 −20 −10 0 10 20 30 40−1.5

−1

−0.5

0

0.5

1

1.5

fz

x/Lf

v/vf

−40 −30 −20 −10 0 10 20 30 40−1.5

−1

−0.5

0

0.5

1

1.5

-2 -1 0 1 2v / v

F

0

0,5

1

1,5

2

Vel

ocit

y di

stri

buti

on

fup

fdown

f0

fz -1 0

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Non linear regime

E (t) = E0 sin (ωt) exp(

− [(t − t0)/∆t]2)

E0 ∼ 1.1 1010V /m ; ∆t = 5fs ; ω = 2.73 PHz (800 nm)

Plasmon mode → Non linear effectsThe magnetic dipole remains

0 25 50 75 100Time (fs)

-0,1

-0,05

0

0,05

0,1

Electric dipoleMagnetic dipole

0 15 30

-101

E (

1010

V/m

)

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Conclusions and future investigations

Conclusions :

We built a self consistant model to study ultrafastmagnetization dynamics in thin ferromagnetic films

We show the existence of an oscillating magnetic dipole atultra high frequency (PHz) NEW

Futur investigations:

Non collinear dynamics

Non magnetic/magnetic bilayers

Extension to 4D (Spin orbite)

-20 -10 0 10 20x (nm)

0

0,5

1

Den

sity

NO

N M

AG

NE

TIC

MA

GN

ET

IC

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

Acknowledgement

Giovanni Manfredi

Paul-Antoine Hervieux

David Coulette

Omar Morandi

Ultrafast spin

dynamics in

ferromagnetic

films

J.Hurst

Introduction :

The Vlasov

equation in

solid state

physics

Vlasov

equation with

spins

Applications

to

ferromagnetic

films

Conclusions

and future

investigations

THANK YOU FOR YOUR

ATTENTION !