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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 !
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