Spin-orbit torques from first principles : example of Pt/FePt/Pt thin films

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Spin-orbit torques from first principles : example of Pt/FePt/Pt thin films

Guillaume Géranton, Frank Freimuth, Stefan Blügel and Yuriy Mokrousov Peter Grünberg Institut and Institute for Advanced Simulation,

Forschungszentrum Jülich and JARA, 52425 Jülich, Germany

Hˆ_ext (t) = F(t) ˆB

ρˆ^(t) ≈ ρˆ_eq − i dt ' ˆ^#_$B(t − t ')_H, ˆρ_eq^%_&

−∞

t

∫

^{F(t ') + O(F)}

Aˆ

t ≈ Aˆ

eq − i dt ' [ ˆA_H (t), ˆB(t ')]

−∞ eq t

∫

^{F(t ') +O(F)}

d ! M

dt = −2µ_BΤ^!

t_ij = e

h dE df (E)

−∞ dE

+∞

∫

^{ℜTr TˆiGˆ R(E) ˆvj( ˆGR(E)−Gˆ A(E))}

+ e

h dEf (E)

−∞

+∞

∫

^{ℜTr TˆiGˆ R(E) ˆvj dGˆ}

R(E)

dE −Tˆ_i dGˆ ^R(E)

dE vˆ_jGˆ ^R(E)

Inversion

Ferromagnet (FM)

Heavy metal (HM)

Τ! _SOT = t E^→

Gˆ ^R(E) = !

E − Hˆ +iΓ

Gˆ ^A(E) = !

E − Hˆ −iΓ

t_ij^{odd Γ→}=

0 e!

2ΓN_k

"

e_i ψ_kn ^Tˆ_i ψ_kn

k,n

∑

^{ψkn vˆj ψkn ∂f}_∂⁽_E^{E) E=εkn}

t_ij^even =

Γ→0 2e N_k

e!_i f (ε_kn^{) e!}_φℑ ∂u_kn

∂θ

∂u_kn

∂k_j − !

e_θ 1

sin(θ^{) ℑ}

∂u_kn

∂θ

∂u_kn

∂k_j

&

' ((

)

* ++

k,n

∑

Spin accumulation (Rashba effect)

Spin current (Spin-Hall effect) A torque is exerted on the ferromagnet

FerromagnetHeavy metal

B^//even (mT) B odd (mT) FePt⁽⁵⁾ / Pt⁽⁶⁾ 2,1 -0,62 Pt⁽⁶⁾ / FePt⁽⁵⁾ / Pt⁽⁶⁾ 0 0 Pt⁽⁶⁾ / FePt⁽⁵⁾ / Pt⁽¹²⁾ 0,35 -0,39 Pt⁽⁶⁾ / FePt⁽⁵⁾ / Pt⁽¹⁸⁾ 0,54 -0,42

µs (µb) B^//even (mT) B odd (mT)

FePt⁽⁵⁾ / Pt⁽⁶⁾ 10,2 2,1 -0,62

Co⁽³⁾ / Pt⁽¹⁰⁾ 5,4 3,2^[5] 0,15^[5]

O⁽¹⁾ / Co⁽³⁾ / Pt⁽¹⁰⁾ 4,02 5,1^[5] -3,0^[5]

AlOx / Co / Pt (exp) 5 ± 0.2^[8] -3.2 ± 0.2^[8]

(x) = N_atoms B fields for j = 10⁷ A/m² and ρ = 36 µΩ.cm

E"

M"

δM^⊥"

" δM^//"

E"

M"

δM^⊥"

" δM^//"

E"

M"

δM^//"

δM^⊥"

"

E"

M"

δM^⊥"

" δM^//"

E"

M"

δM^⊥"

" δM^//"

E"

M"

δM^⊥"

" δM^//"

E"

M"

δM^//"

δM^⊥"

"

E"

M"

δM^⊥"

" δM^//"

E"

M"

δM^⊥"

" δM^//"

E"

M"

δM^⊥"

" δM^//"

E"

M"

δM^//"

δM^⊥"

"

E"

M"

δM^⊥"

" δM^//"

Inversion center

1 10 100 1000

Γ [meV]

0 0.2 0.4 0.6 0.8

teven [ea 0]

Even torkance for Pt6 / FePt5

18 WF per atom / N_bands ~ 1.4 x N_WF / frozen up to Ef + 5eV ~ -0.6 eV / 8 x 8 k-points in BZ

-0.4 -0.2 0 0.2 0.4

∆E_F [eV]

-0.5 0 0.5 1

teven [ea 0]

Broadening Γ = 2 meV Broadening Γ = 25 meV

-0.4 -0.2 0 0.2 0.4

∆E_F [eV]

-10 -5 0

todd [ea 0]

Broadening Γ = 2 meV Broadening Γ = 25 meV

1 10 100 1000

Γ [meV]

-5 -4 -3 -2 -1

0

todd [ea 0]

Pt6 / FePt5 (2048x2048 kpts)

Odd torkance for Pt / FePt / Pt thin films

18 WF per atom / N_bands ~ 1.4 x N_WF / frozen up to Ef + 5eV ~ -0.6 eV / 8 x 8 k-points in BZ

1a 1b

1c 1d

0.9$nm$

1.2$nm$

0.9$nm$

1.2$nm$

2.1$nm$

0.9$nm$

1.2$nm$

3.4$nm$

0.9$nm$

1.2$nm$

1.2$nm$

0.9$nm$1.2$nm$ L$

0.9$nm$ 1.2$nm$

L$

-0.4 -0.2 0 0.2 0.4

∆E_F [eV]

-0.8 -0.4 0 0.4

todd [ea 0]

L = 2,1 nm L = 3,4 nm

1 10 100 1000

Γ [meV]

-5 -4 -3 -2 -1

0

todd [ea 0]

L = 2,1 nm L = 3,4 nm

Even torkance for Pt6 / FePt5

18 WF per atom / N_bands ~ 1.4 x N1 _WF / frozen up to Ef + 5eV ~ -0.6 eV / 8 x 8 k-points in BZ10 100 1000

Γ [meV]

0 0.1 0.2

teven [ea 0]

L = 2,1 nm L = 3,4 nm

-0.4 -0.2 0 0.2 0.4

∆E_F [eV]

-0.4 -0.2 0 0.2

teven [ea 0]

L = 2,1 nm L = 3,4 nm

sign change

0.9$nm$

1.2$nm$

0.9$nm$

1.2$nm$

2.1$nm$

0.9$nm$

1.2$nm$

3.4$nm$

0.9$nm$

1.2$nm$

1.2$nm$

0.9$nm$1.2$nm$

L$

0.9$nm$

1.2$nm$

L$