1. Theory of current-driven domain wall motion - spin transfer and momentum transfer Gen Tatara 多々良 源 Graduate School of Science, Osaka University Hiroshi Kohno 河野 浩 Graduate School of Engineering Science, Osaka University G. Tatara and H. Kohno, cond-mat/0308465.

2. resistivity 9.8  10 8 (  m) 9.7  10 8 H 4T Domain wall and electron transport Gregg et al. (1996) Co film 5  m Low resistance High resistance current Nano contact Magnetization conductivity(2e 2 /h) Garcia et al. '99 420420 Ballistic magnetoresistance (BMR) Garcia et al. '99  /  1% Domain wall Ni-Ni  /  300% H=20 Oe H=0 and nano

3. Information storage based on nanomagnets Read out is fine large signal How to write? Electric current Magnetic field But not suitable for high density storage

4. Magnetization flip= write Magnetization flip by electric current Domain wall Force Torque Electron current Magnetization Electron spin Low resistance High resistance current Read out (resistance) magnetic memory operated by current Magnetization Write conductivity(2e 2 /h) Garcia et al. '99 420420 Ballistic magnetoresistance (BMR) Magnetic nanocontact

5. Current-driven domain wall motion 1. Spin transfer  ： width of wall magnetization electron spin Change of spin angular momentum 2. Momentum transfer Change of momentum Berger ’ 92, Slonczewski ‘ 96 Berger ’ 84,  =-j s /e Torque on DW j s :Spin current FjFj Force on DW

6. Spin transfer and Momentum transfer This talk Unified theory of current-driven domain wall motion No physical insight required! Mathematically rigorous formulation

7. Theory Exchange interaction Force Torque Momentum transfer Spin transfer n : electron spin density  : damping Equation of motion of domain wall under finite current Collective coordinates Takagi &GT ‘ 96 X,  GT and Kohno 2003 Domain wall and electron Spin part Position and polarization of DW

8. Force by Momentum transfer n : electron spin density  : damping Equation of motion of domain wall under finite current Linear response Torque by Spin transfer Adiabatic wall  (hSN/ ) v el DW velocity (Adiabatic limit:  w  0) (Vanishes in thin wall limit:  0 ) Dominates in thin wall Dominates in adiabatic wall

9. 0 Results Thick wall Spin transfer dominates j s cr  K  Threshold spin current K  : hard axis anisotropy Strong pinning case j s cr  V 0 V 0 : pinning energy,  : damping V0>K/V0>K/ Pinning is not essential if weak K  ~0.1K ~1000Å J s cr ~10 12 [A/m 2 ] DW velocity (Full angular momentum conversion)

10. Thin wall Momentum transfer dominates j cr  V 0 /  w Threshold current  w : wall resistivity Nanocontacts: J s cr ~10 7 [A/m 2 ] Very good for application

11. Experiment Yamaguchi, Ono, Nasu, Miyake, Mibu,Shinjo (2003) Threshold current  10 12 A/m 2 seems O.K. if K  ~0.1K DW speed  3-6 m/s is much smaller than full spin transfer case  300 m/s Strong dissipation of angular momentum ? cf. Barnes&Maekawa, cond-mat/0311039 ~1000Å Adiabatic limit Ni 81 Fe 19 0 GT&Kohno

12. Summary Motion of domain wall driven by current Metallic wires (thick wall) Spin transfer by spin current j s cr  K  j s cr  V 0 weak pinning strong pinning Nanocontacts, magnetic semiconductors (thin wall) Momentum transfer by charge current j cr  V 0 /  w G. Tatara and H. Kohno, cond-mat/0308465. Next: effect of dissipation (spin-wave)