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
equilibrium in absence of scattering
●spin distribution function
magnetization equilibrium spin current
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