[PDF] - 1 Linear momentum transfer Angular momentum transfer Linear PDF Document

SLIDE 1

1 Spin torques and spin pumping Gerrit Bauer, ESM Part I.2 F|N|F spin valves: non-collinear

L

sd 

 

L

I

Current-induced magnetization reversal

Electrical currents can controllably reverse the magnetization in small (< 200 nm) magnetic devices (Slonczewski, Berger): Cu Co Pt

Current-induced magnetization reversal

1.0
0.5

0.0 0.5 3.50 3.55 3.60 3.65

Current (mA) dV/dI (Ohms) Electrical currents can controllably reverse the magnetization in small (< 200 nm) magnetic devices (Slonczewski, Berger):

Parallel

(P)

Antiparallel

(AP)

Cu Co Kiselev et al. (2003) Pt

Ic

Positive Ic

2013 Oliver E. Buckley Condensed Matter Physics Prize

Luc Berger Carnegie Mellon University John Slonczewski IBM Research Staff Emeritus

Citation: "For predicting spin-transfer torque and

pening the field of current-induced

control over magnetic nanostructures."

Charged spinning billiard ball

B

 

2 2 e e d V m     



v M r r L

 

2 2 e e d V m     



v M r r L v gyromagnetic ratio M L 2 e m  

SLIDE 2

2

Linear momentum transfer

Angular momentum transfer Angular momentum transfer Angular momentum transfer

SLIDE 3

3

Angular momentum transfer Angular momentum pump Angular momentum pump Angular momentum addition Angular momentum addition Angular momentum addition

SLIDE 4

4 Rotation in quantum mechanics

x y z  x x dx x yd y y dy y xd            

   

ˆ , ˆ 1

z z

f f R d f f x dx y dy f dx dy x y i f y x f d L d f x y                                    

Finite rotation:

 



 lim

n

nd

 

 



         ˆ ˆ lim 1

n z z n

i R d L

   

log 1 lim log 1 lim

n n n

x x x nx

 

    

     



   



   



 

ˆ

ˆ ˆ ˆ log li ˆ m

z

z z z L n i z

R e i i R nd L L  n ( , ,0) x y  n

Rotation in quantum mechanics

1 , 2 2

x

             R  x y z n  

 

cos ,sin ,0

With

   n

    1

, ,             R generates all possible spin states. Example:

 

ˆ

R

R    

n

 

ˆ ˆ exp / R i  

n

n L    Rotation operator: Electron spin: ˆ 2   L s σ 

 

/2

ˆ

i

R e







n σ n

Rotation of a state  by an angle  around an axis with unit vector n:

Spins on the Bloch sphere

1 1 1 2

x

       

y

1 /2 /2 1

1 1 , 2 ! 2 1 cos sin 2 2 sin cos 2 2

m m i m

e e m

  

      

        

                                   



R

 

1,0,0 2

x x 

  s  Check: Spin-up state in x-direction is obtained by rotation about y-axis with   . 1 ,2 2 1                 R



m

Normal  

     / 2

HMF

Slonczewski torque in half metals

  / 2  / 2

transverse spin current = torque longitudinal spin current m

 / 2  / 2

 

   

 

         2 2 Ne V V N = number of incoming channels

 

 =

m

Spin-mixing conductance

 

 =

m

   

2

ˆ ˆ e N x z V V e h

 

   m  

2 2 Sharvin

2 2 1

F

E

e e G N h h





 



k

ˆ z ˆ x



k incoming wave vectors

    

ˆ I e ˆ R m ˆ V V G G e

    

    z y m x   Scattering theory:

 

2 2 *

1

F

E

e e N G r r h h

  



  

        



k k k

 

2 2 *

1

F

E

e e N G r r h h

  



  

        



k k k

Spin mixing conductance:

Exchange-field torque

ˆ z ˆ x

 

m

ˆ y

Precession by spins in evanescent states

SLIDE 5

5

Switching by spin transfer Switching by spin transfer Ohm’s and Kirchhoff’s Laws

2 2 2 nm nm

e e G g t h h  



2 2 2 nm nm

e e G g t h h  



Landauer formula

V  Ic

V1 V2 V3

,1 2 c

I



 

 

 

1 2 1 2 , 1 2 c

V I V G

conductance Charge current:

1 2

G 

, c j i j

I

 



,3 2 c

I



charge conservation

Spin currents

   

, ,

2Re 2Im 2

N N s s s s

G G e

   

       I m V m V m I   in-plane (damping)

ut-of-plane

(effective field) spin accumulation in N spin accumulation in F  magnetization direction m

N s

V

F s

V N F

N s

V

F s

V transverse spin currents collinear/longitudinal

, ,

c s s

I



I I



Spin currents

 

* nm nm nm

g N r r

   

 

 

* nm nm nm

g N r r

   

  complex spin-mixing conductance for transverse spin current (torque + exchange field)

2 2 s s s nm nm nm nm

g t N r   

 

2 2 s s s nm nm nm nm

g t N r   

 

s=, spin-dependent Landauer conductances for charge and collinear spin current transverse spin currents collinear/longitudinal

, ,

s s

I



I I



spin accumulation in N spin accumulation in F  magnetization direction m

N s

V

F s

V N F

N s

V

F s

V

Pauli matrix notation

*

ˆ ˆ ˆ 1

c s

X X X X X X

   

         X σ ˆ ˆ ˆ 1

N N N c s

V V    V σ

 

1 1 1 ˆ ˆ 1, , , , 1 1 1 i i                                 σ 

*

ˆ G G G G G

   

       N F ˆF V ˆ ˆ ˆ 1

F F F c s

V V V    m σ ˆ ˆ ˆ 1

c s

I I    I σ ˆN V ˆ I

SLIDE 6

6

Circuit theory (Brataas et al., 2000)

V

1

m

3

m

1

ˆ G

2

ˆ G

3

ˆ G

4

ˆ G

1

ˆ V

2

ˆ V

3

ˆ V

2

s

1 ,1 ,1 1 2 ,2 ,2 2

ˆ ˆ ˆ 1 ˆ ˆ ˆ 1

c s c s

V V V V V V       m σ s σ

1 2 3 2

ˆ ˆ I I

 

 

1 2

ˆ I 

3 2

ˆ I 

spin & charge conservation:

Angular magnetoresistance

Exp: S. Urazhdin et al. (2005)

Py (12 nm) Cu Py (6 nm) AF Cu Py Py (1.5 nm) AF

 AF

15 1 2

Re 0.5 10 m G

 

  

Kimura et al. (2005)

Spin accumulation-driven magnetization reversal

V

Spin pumping

Spin currents cause magnetization motion (spin transfer torque, Slonczewski, 1996).

Spin torque and spin pumping

Magnetization motion causes spin currents (spin pumping, Tserkovnyak, 2002).

Onsager reciprocals (Brataas et al., 2011)

LLG equation spin conductance spin pumping spin transfer torque

                

mm ms sm ss s s

L L L V H L J M

   



   spin mixing conductance

T sm ms

g L Μ L Μ M  H

 

s

e  

 

  V s

s

J

spin pumping

Spin vs. charge pumping

charge and mass pumping Pumping (Büttiker, Brouwer): Current flows without applied bias, but due to time-dependent modulation of scattering matrix.

SLIDE 7

7

Spin pumping



 

    4

F N pump

d I t g d m m

 m

Dynamics of bilayers

Landau-Lifshitz-Gilbert equation with new torque term: m s

0, s

I I m spin pumping Slonczewski torque

 

 

                                           4 4 4

bias pump s s

t t M t g M g M m m m B m m m I I m m m m s B m s

Sources and sinks of spin currents

            H t t m m m m

    / 1

sf sd N

d F N

            t t m m m H m   

sp stt

   enhanced damping    



                 4

s

g M t t m m m H m

    / 1

sf sd N

d

Mizukami et al. (2001)  2.1

L

g

   

2 2 L B r

g G G d G e Ad 



  

8

10 G Hz

15 1 2

1.0 10

r

G m A

  

     

s

G M Py|Pt:

Py|Pt mixing conductance

Kato et al. (2004)

Ic

InGaAs

Spin current tensor, spin Hall angle

 

, , ,

, ,

s x x y z s s s s s y s z

J             J J J J J J 

,

ˆ

s SH c 

   J β J

x s

J

, s x

J

polarization of current  x current direction

f polarization  x

ˆ

c SH s  

  



J J 

SLIDE 8

8 Spin pumping

 t m t t            H m m m m   t m m