Boltzmann Theory of spin transfer torque in

magnetic textures

Frédéric PiéchonAndré Thiaville

Laboratoire de Physique des solides, CNRS, Univ. Paris­sud, 91405 Orsay

Phys. Rev. B 75, 174414 (2007) 

   

● Current induced spin transfer torque● Zhang & Li macroscopic approach● Boltzmann theory● Summary: Spin current & accumulation● Perspectives

Outline:

   

Current induced spin transfer torque

●Modified LLG equation:

●transverse wall in nanowire+DC current :  

microscopic derivation of transfer torque and parameters  

proportional to current densitysimilar to damping 

Slonczewski term

G. Tatara et al, Phys. Rev. Lett. 92 (2004) 086601, H. Kohno et al, J. Phys. Soc. Jpn. 75  (2006) 113706S. Zhang and Z. Li,  Phys. Rev. Lett. 92 (2004) 207203, ibid 93 (2004) 127204

   

Zhang & Li macroscopic approach

● s-d ferromagnetic exchange model

– action of d-electron magnetization on the dynamics of itinérant s-electron

– retro-action of s-electron magnetization on LLG equation of d-electron magnetization

s   

d

S. Zhang and Z. Li,  Phys. Rev. Lett. 92 (2004) 207203, ibid 93 (2004) 127204

   

s-d ferromagnetic exchange model

● itinérants s­electron:

 

●localized  d­electron:

 s­d magnetic exchange term

spin flip scattering

Larmor time:

   

spin accumulation of itinerant s-electron

equilibrium magnetization:

adiabatic spin current:

● spin accumulation:

currentinduced

● hypothesis

   

Retro-action on d-electron magnetization

●exchange torque:

correction to

correction to

currentinduced

   

Boltzmann transport theory

● why doing that ?● Boltzmann transport equations ● first order gradient Ansatz ● solutions and interpretations:

– without scattering– spin conserving scattering– spin flip scattering

   

Motivations for a Boltzmann theory

●distribution function

charge spin magnetization

spin currentnot colinear ?

coupling with charge degrees of freedom ?

   

Spin-charge Boltzmann transport equations

Drift Diffusion charge­spin coupling

spin­charge coupling

s­d exchange

collisionintegral

second order gradient 

first order gradient 

domain wall resistance

   

first order gradient Ansatz

●projected transport equations 

first order gradient corrections

   

equilibrium in absence of scattering

●spin distribution function

magnetization equilibrium spin current

   

the Figure...

   

Spin conserving scattering

●equilibrium distribution●momentum scattering time

magnetization rotated equilibriumspin current

adiabatic spin current spin accumulation

●out of equilibrium

●effective magnetic field

   

spin-flip scattering

●effective scattering time: 

●equilibrium distribution

magnetization rotated equilibriumspin current

adiabatic spin current spin accumulation

●out of equilibrium

Much more complicated !

yes !

   

summary: spin current & accumulation

no scattering:      current

               magnetization

spin conserving: current

                magnetization

spin flip:           current

               magnetization

equilibriumout of equilibrium

yes 

yes 

yes 

yes

yes

yes  no

yes

 no

 no  no

yes

yes 

yes  yes 

yes yes

no  

   

Perspectives

● finite frequency: linear and non-linear response (rectification)

● temperature gradient

● general finite frequency,finite momentum longitudinal and transverse screening properties

● second order gradient in

● quantum Boltzmann (Keldysh): correction to collision integral ( work in progress with Yann LeMaho)

● spin valve (preliminary work): microscopic boundary conditions new torque terms