A modified LLG equation of motion, that is derived from more fundamental Dirac equation.

1. Relativistic theory of spin relaxation
mechanisms in the Landau-Lifshitz
equations of spin dynamics
Ritwik Mondal, M. Berritta, P. Maldonado, A. Aperis and
P. M. Oppeneer
Uppsala University, Sweden
DPG Spring, Regensburg
March 9, 2016

2. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Spin dynamics
Landau-Lifshitz-Gilbert equation of motion:
∂M
∂t
= −γ [M × Heff] + M ×
[
←→α ·
∂M
∂t
]
γ = ge
2m
M = Magnetization
Heff = Effective magnetic field
←→α = damping parameter
(tensor)

3. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Existing theories
▶ Breathing Fermi Surface → V. Kamberský, Can. J. Phys. 48, 2906
(1970)
▶ Torque correlation model → V. Kamberský, Phys. Rev. B 76,
134416 (2007)
▶ Spin-orbit coupling → M. C. Hickey et al., Phys. Rev. Lett. 102,
137601 (2009)
▶ Linear response formalism → H. Ebert et al., Phys. Rev. Lett. 107,
066603 (2011)
▶ Effective field theories → M. Fähnle et al., J. Phys.: Condens.
Matter 23, 493201 (2011)

4. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Relativistic Theory: Modified Dirac Hamiltonian
▶ Dirac Hamiltonian for electrons in a (ferro)magnetic
materials, excited by an external field:
H = cα · (p − eA) + (β − )mc2 + V − µBβΣ · Bxc
α, β, Σ → Dirac matrices
▶ Foldy-Wouthuysen transformed Hamiltonian:
HFW =
(p − eA)2
2m
+ V − µBσ · B
Pauli
− µBσ · Bxc
Exchange
−
(p − eA)4
8m3c2
Mass correction
+ HDarwin + HSOC + HRCXC
A, E, B → applied fields
R. Mondal, M. Berritta, K. Carva and P. M. Oppeneer, Phys. Rev. B 91, 174415
(2015)

5. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Remaining Hamiltonian(s)
HDarwin = −
1
8m2c2
(
p2
V
)
−
eℏ2
8m2c2
· E
HSOC = −
eℏ
8m2c2
σ · {E × (p − eA) − (p − eA) × E}
+
i
4m2c2
σ · (pV ) × (p − eA)
HRCXC = +
µB
8m2c2
{ [
p2
(σ · Bxc
)
]
+ 2σ · (pBxc
) · (p − eA)
+ 2(p · Bxc
) σ · (p − eA) + 4[Bxc
· (p − eA)] σ · (p − eA)
}
+
iµB
4m2c2
[(p × Bxc
) · (p − eA)]

6. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Approximations and definitions
To derive the spin Hamiltonian:
▶ Wavelength of incident light, λ ∼ 800 nm
▶ Sample thickness, d ∼ 20 nm
▶ λ d
▶ Within Coulomb gauge ( · A = 0), slowly varying
magnetic field: A = B×r
2
▶ Spherically symmetric crystal potential V (r) = V (r)
▶ Spin angular momentum S = ℏ
2 σ
▶ Orbital angular momentum L = r × p

7. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Spin Hamiltonian
The spin Hamiltonian: HS(t) = H0 + HSOC
H0
= −
e
m
S · Beff = −
e
m
S · (B + Bxc
eff)
▶ Crystal-field spin-orbit coupling:
Hint
soc =
1
2m2c2
1
r
dV
dr
S · L −
er
4m2c2
dV
dr
S · B
+
e
4m2c2
1
r
dV
dr
(S · r) (r · B)
▶ External-field spin-orbit coupling (Hermitian):
Hext
soc =
ieℏ
4m2c2
S ·
∂B
∂t
(
1 −
(r · p)
iℏ
)
+
e
4m2c2
(S · r)
(
∂B
∂t
· p
)

8. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Equation of motion
▶ Magnetization dynamics:
dM
dt
=
gµB
V
1
iℏ
∑
j
Tr
{
ρ[Sj
, HS
(t)]
}
▶ External-field magnetization dynamics:
dM
dt
ext
soc
= −
ieℏ
4m2c2
M ×
∂B
∂t
(
1 −
⟨
r · p
⟩
iℏ
)
−
e
4m2c2
M ×
⟨
r
(
∂B
∂t
· p
) ⟩
Relation of magnetization and magnetic induction:
∂tB = µ0(1 + ←→χ −1
m ) · ∂tM ←→χm → susceptibility tensor

9. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Gilbert damping parameter
intrinsic Gilbert damping parameter:
←→α = −
ieℏµ0
4m2c2
(
1 −
⟨
r · p
⟩
iℏ
)
[
1 + ←→χm
−1
]
▶ interband case:
−
⟨r · p⟩
iℏ
=
1
2m
∑
k
n=occ.
n′=unocc.
|⟨n, k|p|n′, k⟩|2
(
εn,k − εn′,k
)
▶ intraband case:
−
⟨r · p⟩
iℏ
=
1
2m
∑
k,n
(
∂f
∂ε
)
EF
|⟨n, k|p|n, k⟩|2
▶ New expression for Gilbert damping parameter!

10. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Gilbert damping parameter
intrinsic Gilbert damping parameter:
←→α = −
ieℏµ0
4m2c2
(
1 −
⟨
r · p
⟩
iℏ
)
[
1 + ←→χm
−1
]
▶ interband case:
−
⟨r · p⟩
iℏ
=
1
2m
∑
k
n=occ.
n′=unocc.
|⟨n, k|p|n′, k⟩|2
(
εn,k − εn′,k
)
▶ intraband case:
−
⟨r · p⟩
iℏ
=
1
2m
∑
k,n
(
∂f
∂ε
)
EF
|⟨n, k|p|n, k⟩|2
▶ New expression for Gilbert damping parameter!

11. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Angular momentum dynamics (without exchange)
Describe systems with unquenched orbital momentum
▶ individual orbital (L) and spin (S) angular momentum
operator dynamics
▶ total angular momentum dynamics: J = L + S
dJ
dt
=
Precession
−
e
2m
B × (L + 2S) +
er
4m2c2
dV
dr
(S × B) +
B × Lp2
4m3c2
−
iℏe
4m2c2
(S × ∂tB)
{
1 −
(r · p)
iℏ
}
Gilbert damping
+
e
4m2c2
(S · r) (∂tB × p)
other damping
▶ B → 0 ⇒ J conserved.

12. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Angular momentum dynamics with exchange
▶ Exchange interaction of Heisenberg type:
HHei = −
∑
i̸=j
Jij(R) Si · Sj
▶ Total angular momentum dynamics of Heisenberg exchange
including relativistic field-spin interaction:
d
dt
∑
i̸=j
(Li + Lj + Si + Sj)
xc
= 0
▶ Relativistic interaction of applied field with exchange field
does not change conservation of total angular momentum.

13. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Conclusions
▶ Intrinsic Gilbert relaxation parameter is a tensor, which
depends on the susceptibility and the value of ⟨r · p⟩
▶ Total angular momentum dynamics (without exchange)
comprises the precession and their relativistic counter-parts
and relaxations
▶ Exchange angular momentum dynamics is not affected by
the external EM field and it is conserved

14. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Acknowledgement
Thank you for your attention!

15. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Susceptibility
Inverse of susceptibility in Fourier space:
χ−1
ij (q, ω) = −
ωex
γµ0M0
δij + χ−1
⊥ (q, ω)
The damping parameter:
α = −
eℏµ0
4m2c2
(
1 −
⟨
r · p
⟩
iℏ
)
Im
(
χ−1
⊥
)

16. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
ab initio values for ⟨r · p⟩
Elements
interband intraband
fcc Ni
103 0.44
bct Fe
89 0.34
▶ Gilbert damping parameter for Ni in atomic units:
α ∼
µ0
4c2
⟨
r · p
⟩
× Im(χ−1
) ∼
0.00066 × 103
4 × 1372 × 0.0001
∼ 0.01
µ0 = 0.00066, c = 137, Im(χ) = 0.0 − 0.0001 at low frequency,
133 Hz and low temperature, T → 0.

17. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
Exchange angular momentum dyanmics
dSi
dt
xc
0
=
e
m
∑
i̸=j
Jij
[
Si × Sj −
1
2m2c2
(Si × pi) (Sj · pi)
]
dSj
dt
xc
0
=
e
m
∑
i̸=j
Jij
[
−Si × Sj −
1
2m2c2
(Si · pi) (Sj × pi)
]
dLi
dt
xc
0
=
e
2m3c2
∑
i̸=j
Jij
[
(Si × pi) (Sj · pi) + (Si · pi) (Sj × pi)
]
dLj
dt
xc
0
= 0

18. Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz equations of spin dynamics
exchange field-spin interaction dynamics
dSi
dt
xc
I
=
e2
2m3c2
∑
i̸=j
Jij
{
(Sj · Ai) (Si × pi) + (Si × Ai) (Sj · pi)
}
dSj
dt
xc
I
=
e2
2m3c2
∑
i̸=j
Jij
{
(Sj × Ai) (Si · pi) + (Si · Ai) (Sj × pi)
}
dLi
dt
xc
I
= −
e2
2m3c2
∑
i̸=j
Jij
{
(Sj · Ai) (Si × pi) + (Si × Ai) (Sj · pi)
}
−
e2
2m3c2
∑
i̸=j
Jij
{
(Sj × Ai) (Si · pi) + (Si · Ai) (Sj × pi)
}