Theory of Spin Dependent Tunnelingmagnetism.eu/esm/2015/slides/chshiev-slides.pdf · AFM metals AFM...

Mairbek Chshiev European School on Magnetism

M. Chshiev «Theory of spintronic phenomena in magnetic tunnel junctions»

GMR

&

TMR

Spin Transfer

Torques

(STT)

Spin Hall

Effect

(SHE)

Rashba

Effect

Spin-orbit phenomena -> Spin orbitronics

Interlayer

Exchange

(IEC)

Spin

Filtering

(SF)

Magneto

crystalline

anisotropy

Conventional spintronics

Dzyaloshinskii-

Moriya (DMI) Spin-orbit

Torques

(SOT)

Spin

valves

Alloys

(BiCu,

IrCu…)

Single &

double

barrier

MTJs

AFM

metals

AFM

insulators

Frustrated

magnets

Rutiles Heusler

alloys

Graphene Garnets

(Y3Fe5O12)

Layered

structures

(Pt/Co/AlOx,

Ta/CoFe/MgO Interfaces perovskites

Domain walls

…

…

Ferrites

(CoFe2O4)

This lecture:

Tunnel magnetoresistance (TMR)

Spin transfer torques (STT)

• Quantum origin of spin transfer torque

• Description of spin currents and spin transfer torques

- Free electron model

- Tight-binding model

• Voltage dependence of STT

- symmetric MTJs

- asymmetric MTJs

Interlayer exchange coupling

…

Spintronics


)(kn 0|| k

FE

FE

Fe M’

M H

magnetization acts on current

Spintronics

Huge TMR in crystalline MTJ if:

• Good epitaxial fit between FM and I(SC)

• Evanescent states in I(SC) with the same

Bloch state symmetry

• High symmetry Bloch state (1) for one of two e-

spin states in FM electrodes (“half-metallic”-like)

dxy

dxz, dyz

px, py

1 in MgO

dz2 pz s

The 1st Brillouin

zone with high

symmetry k-points

TMR~600%

Fe|MgO|Fe

(Tohoku,2008)

Tunnel (TMR) magnetoresistance:

W. H. Butler et al, PRB (2001)


based Spin Filtering (SF)


M’ M H

TMR~600%

Fe|MgO|Fe

(Tohoku,2008)

Spintronics







The 1st Brillouin

zone with high

symmetry k-points

0 5 10 15 20 25

e-20

e-16

e-12

e-8

e-4

Fe Fe MgO

layer


based Spin Filtering (SF)


IEEE Trans. Mag., 41 (2005) 2645

Sci. Technol. Adv. Mater. 9 (2008) 014106

magnetization acts on current Tunnel (TMR) magnetoresistance:



M’ M j

ref free

)(HR

Field (Oe)

)( jR

Current (mA)

Spin Transfer Torque (STT): current acts on magnetization

R(H)

Sensors

“1” “0” (Rhigh)

Freescale 4Mbit

(Rlow)

MRAM

Memories RF components

Cu

PtMn

Cu

CoFe CoFe

MgO

(Pt/Co)

J

Si Logic

MTJs

Logics

Functional

devices

GMR, TMR, STT

M’ M H

P

PAP

R

RR TMR

Spintronics

Tunnel (TMR) magnetoresistance: magnetization acts on current


Spintronics

Giant

magnetoresistance

(GMR)

metallic

nanostructures

Keldysh formalism

Tight - binding model

Kubo formalism

Free electron model Ab - initio (DFT )

Calculation techniques

Tunnel

m agnetoresistance

(TMR )

amorphous and

crystalline , single

and double barrier

tunnel junctions

Spin transfer

torque (STT),

exchange coupling

(IEC), anisotropy

Spin

filtering

Spin Hall Effect (SHE)

alloys

Materials

for spintronics

amorphous and

crystalline tunnel

junctions

crystalline

magnetic tunnel

junctions

chalcogenides ,

rutiles , spinels ,

Heusler alloys,

Graphene ,

frustrated

magnets

Computational Materials Science Condensed Matter Theory +

Boltzmann approach

(drift - diffusion)

Quantum transport theory and electronic structure of materials for spintronics


RLRL

parallel DDDDJ RLRL

elantiparall DDDDJ

300K

P>0 (~50%) in Fe, Co R/R~40 - 70% with alumina barriers at low T

Julliere 1975 -> Moodera 1995

R/R~ 15%

P+0.26

H

Fe Co AlOx

Different

coercive fields

For Co, Fe

)()(

)()(

)()(

)()(

)(

FRLFRL

FRLFRL

RLEDED

EDEDP

RL

RL

P PP

PP

R

RTMR

1

2

Moodera et al, PRL 74, 3273 (1995)

Parallel configuration Antiparallel configuration

Miyazaki


Julliere’s model is insufficient!

Stearns’ polarization

Stearns first explanation in this way

)()(

)()(

)()(

)()(

)(

FRLFRL

FRLFRL

RLEDED

EDEDP

))()(

)()(

)(

RLRL

RLRL

RLkk

kkP

TMR remians to be defined this way?

M. B. Stearns, JMMM 5, 1062 (1977)

RL

RL

P PP

PP

R

RTMR

1

2

- Effective masses are different for up

and down bands

- Should lead to different tunneling

probabilities and polairsation meaning

M² different for differnt bands Not overall density of states important but

specific bands at the Fermi level and their

properties


-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

En

erg

y,

Wa

ve

Fu

nc

tio

n

y

Energy

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

En

erg

y,

Wa

ve

Fu

nc

tio

n

y

Energy

Matching Boundary Conditions (4 linear equations with 4 unknowns)

allows the solution for the transmission probability.

y=0 y=a

yyBeAe 00

yikRteyikyik LL ree

Free electron model for tunneling (no spin)

Boundary conditions :

continuity of |Y> and its derivative

t transmission amplitude

EF at equal potential


a

RL

a

R

R

L

L

RL

a

RL

a

RLRLRL

E

eTTek

k

k

kT

kk

ekkT

e

kkkkakk

kkT

Th

e

V

IG

F

00

0

0

'||

22

2

0

2

0

2

0

2

0

2

0

22

0

2

22

0

2

2

0

22

0

22

00

2

0

22

0

2

21

2

0

,

||

2

)(

4

)(

4

:as written becan that thisNote

))((

16

:1t enough tha thick is systemWhen

))((4)2cosh())((

8

:yprobabiliton Transmissi

eConductancfor FormulaLandauer )(

>>

kk

'

||

||

k,kFree electron model for tunneling (no spin)

- Depends on barrier thickness + height

- k=kF

- Given by transmission probabilities L, R

T=|t|²

=T(k//)

a

RL

a

R

R

L

L

RL

a

RL

a

RLRLRL

E

eTTek

k

k

kT

kk

ekkT

e

kkkkakk

kkT

Th

e

V

IG

F

00

0

0

22

2

0

2

0

2

0

2

0

2

0

22

0

2

22

0

2

2

0

22

0

22

00

2

0

22

0

2

21

2

0

||

2

)(

4

)(

4

:as written becan that thisNote

))((

16


))((4)2cosh())((

8

:yprobabiliton Transmissi

eConductancfor FormulaLandauer )(

>>

||k

k


U

EF

E E

Slonczewski (1989) Delocalized electrons contribute most to the current (sp states)

Localized states don’t (d states) free electron tunneling

))((

162

0

2

'

2

0

2

2

'

2

0'0

kk

ekkT

a

Spin and spin channels conduct in parallel (two current model): GGGGGG elAntiparallParallel and

2

22

0

22

0

2

022

0016

kk

kkkkeGG

a

elantiparallParallel

Tunnel magnetoresistance

²

2

1

2

P

P

G

GG

G

G

parallel

elantiparallParallel

Parallel

FF

FF

FF

FF

kk

kk

kk

kkP

2

0

2

0

Slonczewski model for tunneling (with spin)

- Depends on properties of FM and barrier!

- Bandstructure details are important!

for simplicity L=R material

Slonczewski Polarisation

Note integration over all k// (or all E) to get G


Case of high barrier:

>> FF kk ,

0>

FF

FF

kk

kkP

Free electrons: kmk

EDOS 22

)(

DD

DDP Back to Julliere formula

With P defined via DOS

In Julliere’s model, only the polarization within the magnetic electrodes influences

the TMR. In Slonczewski’s model, the barrier height also plays a role.

²

2

1

2

P

P

G

G

R

R

ParallelelAntiparall

FF

FF

FF

FF

kk

kk

kk

kkP

2

0

2

0

2

||20

)(2k

EUm

Slonczewski model for tunneling (with spin)

Not overall density of states

important but specific bands

at fermi level and their

properties

kf gives strongest

contribution since electron

with highest velocity

DOS does not provide

information on the velocity

Electrons with highest

velocity give strongest

contribution to tunneling


RR

RRR

LL

LLL

RL

RL

RLRL

RLRLRLRL

AP

APP

TT

TTP

TT

TTP

PP

PP

TTTT

TTTTTTTT

TT

TTTT

T

TT

where1

2TMR

TMR

Jullière

model

a

RL

a

eTTkk

ekkT 0

0

2'

2

0

2

'

2

0

2

2

'

2

0'

))((

16

)(

4;

)(

42

0

2

'0

2

0

2

0

R

RR

L

LL

k

kT

k

kT

Generalized model for tunneling (with spin)

Generalization: Write TMR in terms of the transmission probability

Provides defintion that can be generalized to complex bandstructures

U

EF

E E


z=0 z=a

V1

V0

V2

Ve

az /VV(z)

Voltage dependence of tunnel current


>>

EUL

Dkk

ekkT

e

L

L

002

0

2

2

2

0

2

1

2

21

2

0

2

2exp

))((

16


0

0

But what if the barrier is not rectangular???

2

1

)(2

exp

:ionapproximat WKBuse We

0

z

z

EzUdzDT

• de Broglie wavelength is much smaller than (z2-z1)

• U(z) should vary slowly over (z2-z1)

zp/

Simmons model based on WKB is usually used to estimate potential barrier height and width

Voltage dependence of tunnel current

Slonczewski

What for TMR at high voltages

Fowler Nortdheim regime (deformation below Ef)


1

1

ik zre

k

2

2

ik zte

k

dzzdzz

BeAe)()(

1

1

ik ze

k

z=0 z=a

V1

V0

V2

Ve

V2

2

2

||

2

22 em

kkk E

)(2

222 VEm

k E

2

||

2

11 kkk E

)(2

121 VEm

k E

)(V2

)(2

2

||

2

0 zem

kz

)(2

020 EVm

az /VV(z)

|||||| ),V,()V()(2

dkkkETeEfEfdEh

ej

2

2

22

1

2

21

0

2

2

22

1

2

21||

)()0()()0(4)(2cosh)()0(

)()0(8)V,,(

kakakkdzzkak

akkkET

a

Current density:

voltage dependence appears

Equation for j

comes from eq first

slide taking

tunneling LR-RL

and Spin

dependence

presentation of j up j

down tunneling

I(V)

Voltage dependence tunnel current


|||||| ),V,()V()(2

dkkkETeEfEfdEh

ej

Current density: I(V)

Fermi Dirac, gives window ~V

I/V=G = conductance =Integral of T

Note 1) increase of G with V since more states available for tunneling 2) For symmetric barriers G~V2

3) For asymmetric barrier only G~V+const*V2

Voltage dependence of TMR and tunnel current


Exemple of experimental I(V) characteristics in Co|AlOx|Co tunnel junction

T. Dimopoulos et al

Dynamic conductance=dI/dV

See lectures of C. Tiusan and S. Valenzuela for epitaxial Fe|MgO MTJs

Voltage dependence of TMR and tunnel current

3 effects for bias dependence of tunnel current and TMR

- More states to tunnel into

- when barrier is tilted, thickness reduced for high levels

- spin flip scattering

-Here only first point considered in the model?

Note symmetric barriers G~V+V3

(increase of G with V since more states available for tunneling)

Asymmetric barrier only V2


Pros/Cons of Julliere/Slonczewski Models

• AlOx tunnel barriers are amorphous.

• In amorphous materials, all electronic effects related to crystal symmetry

are smeared out.

• Evanescent waves in alumina have “free like” character.

• Free electron models work OK in this case.

• However, they fail with crystalline barriers. Additional band structures effect

in the electrodes and barrier must be taken into account (Bloch state

symmetry based spin filtering).

1995-2005 AlOx barriers >2005 MgO barriers

Best MR~80%

Best MR~600%

See lectures of C. Tiusan, S. Valenzuela for cristalline Fe|MgO MTJs


This lecture:

Tunnel magnetoresistance (TMR)

Spin transfer torques (STT)

• Landau Lifshitz equation (LLG)

• Quantum origin of spin transfer torque

• Description of spin currents and spin transfer torques

- Free electron model

- Tight-binding model

• Voltage dependence of STT

- symmetric MTJs

- asymmetric MTJs

Interlayer exchange coupling

…


Spin Transfer Torque (STT): current acts on magnetization

Prediction:

J. Slonczewski (1996)

L. Berger (1996)

First observations:

M. Tsoi et al, PRL 80, 4281 (1998)

J. Katine et al, PRL 84, 3149 (2000)

Y. Huai et al, APL 84, 3118 (2004)

G. Fuchs, et al., APL 85, 1205 (2004)

j T

||T

free ref

M 'M

M 'M

APP PAP

MMMMMM

MHMM

''||0 TTg

dt

d

Mdt

d Be

s

precession damping parallel field-like

• exchange of angular momentum between conduction and localized electrons

• conservation of total angular momentum

Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation:

Quantum

origin? )V(10

ΤΤInterlayer Exchange Coupling (IEC)

J. Slonczewski (1989), R.P.

Erickson (1993)

S. Zhang, P. M. Levy and A. Fert (2002)

M. D. Stiles and A. Zangwill (2002)

sspin valve tallicmfor 0~ eT

A. Kalitsov et al, JAP 99, 08G501 (2006)

MTJsfor 0)V(

T


Unpolarized

Electrons

q

Polarizer P Free Layer M

Polarized

Electrons

Transmitted

Electrons

t

Electron

Flow

Transfer of spin angular

momentum m

=

Spin Torque incoming transmitted

m

Conduction Electrons

M

m

Mo

Local Magnetization

Spin Momentum Transfer - Concept

Flow of angular momentum has a source or sink

Local exchange

interaction between

conduction electron

spins and local

magnetization M


Spin Transfer Torque

particle density

Particle transport

current density

Spin transport

Continuity equation

conserved spin density

spin current density not conserved

Torque

M.D.Stiles and A.Zangwill, PRB 66 (2002) 014407


Particle density:

Particle transport

Current

density:

Continuity equation:

where

Spin density:

Spin transport

Spin current

density:

where Vector of Pauli matrices

Continuity equation:

0

t

sQ

)(rs


Spin density:

iiiii

**

x2

s

iiiii

ii **

y2

s

iiiii

**

z2

s

Spin current density:

i

iiii vvQ ˆˆRe **

x

iiiii

vvQ ˆˆRe **

z

iiiii

ii vvQ ˆˆRe **

y

Tensor quantity with elements with i=x,y,z in spin space and j=x,y,z in real space ijQ

0xy Q

0yy Q

0zy QCurrent flows in y direction

)(rs


Spin current tensor:

0zy Q

0xy Q

0yy Q

Current flows

in y direction

ikkQ QT

i=x, y, z in spin space k=x, y, z in real space

ikQ

Spin current and spin torque in non-collinear case

Spin torque:

not conserved

conserved

JJ

JJ• Current matrix:

z

y x (||)

Barrier FM’

M’

FM

M zyQ

yyQ

xyQ

parallel (Slonczewski)

0 ,0 )((||) yx TT

perpendicular (field-like)

))()((2

)(

))()((2

)(

qqq

qqq

JJe

J

JJe

Qzy

)(

)()0(TMR

J

JJ

T

T|| q

M. D. Stiles and A.Zangwill, PRB 66 (2002) 014407 A. Manchon et al, JPCM 20 (2008) 145208


Physical origin of Spin transfer torque (sd model)

Consider two populations of electrons:

-1) s conduction electrons (spin-polarized)

-2) d more localized electrons responsible for magnetization

The spin-polarized conduction electrons and localized d electrons

interact by exchange interactions

)()(2

2

d.Sσ

sdJrUm

pH Hamiltonian of propagating s electrons:

In non-colinear geometry, exchange of angular momentum takes place between

the two populations of electrons but total angular moment is conserved.

Kinetic Potential

Exchange sd

Pauli matrices vector

Unit vector//M

),(electrons s todue on Torque trJ sd

d sSS d

s=local spin-density of s electrons A. Manchon et al, JPCM 20 (2008) 145208


Physical origin of Spin transfer (sd model) (cont’d)

Local spin density at r and t :

Electron wave-function ),( tr

),(2

),(),( * trtrtr σs

Temporal variation of local spin density:

σσs**

2),( tr

Schrödinger equation : ),(),( trHi

tr

(1)

(2)

Substitution (2) in (1) : σσs

**

2

1),( HH

itr

),(,),( trJ

trtr sd sSQs d

…

Q is the spin density current

3x3 tensor

Spin space x real space trtr

m,,Im

2

*2

rσQ

),(

),(

tr

tr

A. Manchon et al, JPCM 20 (2008) 145208


Physical origin of Spin transfer (sd model) (cont’d)

),(, trJ

tr sd sSQT d

In ballistic systems:

The exchange interaction between spin-polarized s electrons and more

localized d electrons is responsible for spin-transfer torque. This interaction

yields a precessional motion of spin-density of s electrons around the local

magnetization. In ballistic regime, the spin-transfer torque is also equal to the

divergence of spin-current.

In diffusive systems: ),(, trJ

tr sd

SF

sSs

QT d

Takes into account the spin-memory loss by scattering with spin lifetime SF

can be fully calculated by solving Schrodinger equation in non-colinear geometry T

See lectures of G. Bauer, T. Jungwirth, S. Valenzuela for metallic spin valves


Let’s derive STT expressions

1 1,k k 2 2,k k

0k

z

y x (||)

Barrier FM’

M’

FM

M f

1 2 0

1 1

11

W.F. in layer 1 W.F. in layer 2: W.F. in layer 3:

(quant. axis || M): (quant. axis || M') (quant. axis || M')

ik y ik y

ik y

e r e

r e

Y

20 0

0 02

0 2

ik yk y k y

k y k y ik y

t eA e B e

A e B e t e

Y Y

1

1

2

component of majority spin

ˆ momentum with M as quantization axis

component of minority spin


component of majority spin

k z

k z

k z

2


component of minority spin


k z


Rotate Y1 in respect to direction of M’

1 1

1

1 1 1

1 1 1

1

cos / 2 sin / 2

sin / 2 cos / 2

cos / 2 sin / 2

sin / 2 cos / 2

ik y ik y

ik y

ik y ik y ik y

ik y ik y ik y

e r e

r e

e r e r e

e r e r e

Y

Rotate Y1 about y axis through angle f.

See G. Bauer’s lecture

1 1,k k 2 2,k k

0k

z

y x (||)

Barrier FM’

M’

FM

M f

1 2 0


Wave functions for non-collinear MTJ

z

y x (||)

Barrier FM’

M’

FM

M

1 3 2

4 systems of

8Eqs. and

8Uknowns


where),()()(

)V()(')(sin)(')(cos))((esin)0())()(()0(

)(

10

2)(2

2

222

||0

ETETET

eEfEfykkkkaqykkkkaqDen

kkqkkkkaqqET RLRRRRRRRR

dyyq

LLRRLL

a

z

y x (||)

Barrier FM’

M’

FM

M zyQ

yyQ

xyQ

T

T|| q

Lk

Lk

Rk

Rk)(yq

VeU

EF

E E

2 2

')(sin)0())(()())(0(')(cos)()0())()(()0(

)V(2esin))()(()0(

)(

2222

)(2

2

220 0

ykkkkqkkaqkkaqkkqykkkkaqkkqkkkkaqq

eEfDen

kkkkaqqET

RRLLRRRRLLRRRRLLRRLL

R

dyyq

RRLL

a

)V()(')(cos)(')(sin))((esin)0())()(()0(

)( 2)(2

2

2221 0 eEfEfykkkkaqykkkkaq

Den

kkqkkkkaqqET RLRRRRRRRR

dyyq

LLRRLL

a

a

Free electron model

• Wave vectors depends on V and band positions

• Boundary conditions for Green functions and derivatives

Explicit analytical expressions for STT

In the right FM layer (as an example):

M. Chshiev, A. Manchon, A. Kalitsov et al, Phys. Rev. B (2015)


z

y x (||)

Barrier FM’

M’

FM

M zyQ

yyQ

xyQ

T

T|| q

Lk

Lk

Rk

Rk)(yq

VeU

EF

E E

2 2

a

Free electron model

ii

iii

kk

kkP

iii

iiii

kkq

kkq2

2

iii

iiii

kkq

kkq2

)(

Local torques (derivatives of Q):

Oscillations of different periods

Shorter period for RKKY coupling

Longer period for voltage induced


z

y x (||)

Barrier FM’

M’

FM

M zyQ

yyQ

xyQ

T

T|| q

Lk

Lk

Rk

Rk)(yq

VeU

EF

E E

2 2

a

Free electron model

ii

iii

kk

kkP

iii

iiii

kkq

kkq2

2

iii

iiii

kkq

kkq2

)(

Local torques (T = ∇Q):

Oscillations of different periods

Shorter period for RKKY coupling

Longer period for voltage induced

Transverse characteristic length scales:

• Larmor spin precession length, 𝐿

• Transverse spin decay length, 𝑑

• Applied voltage dependent

M. Chshiev, A. Manchon, A. Kalitsov et al, Phys. Rev. B (2015)


z

y x (||)

Barrier FM’

M’

FM

M zyQ

yyQ

xyQ

T

T|| q

Lk

Lk

Rk

Rk)(yq

VeU

EF

E E

2 2

a

Free electron model

ii

iii

kk

kkP

iii

iiii

kkq

kkq2

2

iii

iiii

kkq

kkq2

)(

Total torques:

Deviations from sine dependence

in case of low/thin barrier

Eventually for metallic spin valves


z

y x (||)

Barrier FM’

M’

FM

M zyQ

yyQ

xyQ

T

T|| q

Lk

Lk

Rk

Rk)(yq

VeU

EF

E E

2 2

a

Free electron model

ii

iii

kk

kkP

iii

iiii

kkq

kkq2

2

iii

iiii

kkq

kkq2

)(

Total torques:

Case of thick barrier

Parallel torque related to longitudinal

spin current

iii

ii

S

i

PP

PP

- Slonczweski polarization

- out-of-plane polarization

22

ii

iiT

- Spin averaged interfacial

transmission probability M. Chshiev, A. Manchon, A. Kalitsov et al, Phys. Rev. B (2015)


z

y x (||)

Barrier FM’

M’

FM

M zyQ

yyQ

xyQ

T

T|| q

Lk

Lk

Rk

Rk)(yq

VeU

EF

E E

2 2

a

Free electron model

x x x x x x x x x x x x x x x x x a ’ ’ ’ i j b

VeB

2 2

Tight-binding model

Model parameters: B ,)(- on-site energies

t, tB – hopping

ta, t’b – couplings


z

y x (||)

Barrier FM’

M’

FM

M zyQ

yyQ

xyQ

T

T|| q

VeB

2 2

Tight-binding model


t, tB – hopping


V ,10

01)(

..10

01

..10

01

)V(

cossin

sincos

210

01

2

BBB

''

'

''''

)(

)()()()(

eN

icctccH

chcctH

chcctH

cctcceH

HHHHH

cctccH

HHHHHH

B

i

i NN

jiii

i

B

bbRB

aaLB

NN

R

RL

NN

L

RBLBBRL

RLRLRLRL

Hamiltonian

I. Theodonis et al, PRL 97, 237205 (2006)

M. Chshiev et al, IEEE Trans. Mag. 44, 2543 (2008)

A. Kalitsov et al, PRB 79, 174416 (2009)



z

y x (||)

Barrier FM’

M’

FM

M zyQ

yyQ

xyQ

T

T|| q


VeB

2 2

Tight-binding model


t, tB – hopping


Charge current:

Keldysh formalism with non-equilibrium

Green functions

Total torque:

1','',1'' QQΤTorque:

2 x 2 matrices

yJ ˆ''8

1','',1'||3

TGTGrTdkdEe

h

σQ

''16

11','',1'||3',1' TGTGrTdkdE

Spin current:

)()'()',( '

†

''' tctcittG

Non-collinear M and M’

GG

GGG

0'

,10,11','',1'

0'

'

QQQQΤΤ

D. M. Edwards et al, PRB 71, 054407 (2005)



'

'

'ˆ

GOiO


1. Using divergence of spin current Q:

2. Using magnetic moment and exchange splitting:

Two ways to find spin torque in ballistic regime

0xy Q

0yy Q

0zy Q

Current flows

in y direction

ikkQ QTi=x, y, z in spin space and k=x, y,

z in real space

ΔμT B

1zΔ ˆ

2

In ballistic regime:

),(, trJ

tr sd sSQT d


aμT B

1

Let’s check relation between T and using its’ angular dependence.

Suppose they are related via unknown vector a. Then:

),0,0( zaa 0

||

||

z

z

z

T

aT

aT

||

||

TT

az

Δa

The two methods give quantitative

agreement and are connected directly via

exchange splitting

zΔ ˆ2

Q Δ μ T

B

1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

T

orq

ue

(e

V)

rad

T||

T

||

V=0.1 V, N=5, i=1 (at FM/I interface)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.0

2.0x10-1

4.0x10-1

Tx/

y=

-T

y/

x

(

)/2=-0.1

(

)/2=-0.2

(

)/2=-0.3

A. Kalitsov et al, JAP 99, 08G501 (2006)



),(, trJ

tr sd sSQT d

In ballistic systems:

The exchange interaction between spin-polarized s electrons and more

localized d electrons is responsible for spin-transfer torque. This interaction

yields a precessional motion of spin-density of s electrons around the local

magnetization. In ballistic regime, the spin-transfer torque is also equal to the

divergence of spin-current.

In diffusive systems: ),(, trJ

tr sd

SF

sSs

QT d

Takes into account the spin-memory loss by scattering with spin lifetime SF

A. Manchon et al, JPCM 20 (2008) 145208



In the right FM layer site ’

Coupling amplitude decreases

with exchange splitting

0 6 12 18 24 30

-4.0x10-11

-2.0x10-11

0.0

2.0x10-11

4.0x10-11

6.0x10-11

8.0x10-11

1.0x10-10

V=0,

Torq

ue (

eV

/a.u

.)

'

T,

T,


Period of oscillations is )/(1 kk

Perpendicular torque component T is

not zero @ zero bias describing

exchange coupling between FM layers

J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989);

ibid.71, 6995 (2005);

D. M. Edwards et al, Phys. Rev. B 71, 024405 (2005)

A. Manchon et al, JPCM 20, 145208 (2008)

e-

Local torques in the right FM at zero voltage

Note RKKY period (summation of k)


0 5 10 15 20 25 30

-1.0x10-10

0.0

, V=0.5V,

Torq

ue (

eV

)

'

T||

T

0 5 10 15 20 25 30

-1.0x10-10

0.0

1.0x10-10

, V=0.5V,

To

rqu

e (

eV

)

'

T||

T

Local torques in the right FM at positive/negative bias

Period of oscillations is )/(1 kk


A. Kalitsov et al., JAP 99, 08G501

A. Manchon et al, JPCM 20, 145208

M. Wilczyński et al., PRB 77, 054434

D. Ralph and M.Stiles, JMMM 320, 1190

e- Note voltage (current) induced STT period (difference of k)


Precession of spin state in exchange field of the magnet

Case 1: insulating barrier (tunnel junctions) precession period 2/(k↑ + k↓) for RKKY IEC

precession period 2/(k↑ - k↓) for STT voltage induced

only electrons with k interface tunnel selection of k-vectors

integration over k-vectors does not average to zero

NM M2

Qtrans

M1

q

M2 x

y z

T

T|| Non-zero X and Y –

component of

torque!!!


Precession of spin state in exchange field of the magnet Case 2: metallic structures Consequence of dephasing

away from the interface, reflected and transmitted spin currents are collinear to M2

The entire transverse spin current is absorbed by M2 at interface

Difference with metallic structures for STT

Only torque in

x-direction

M1

q

M2 x

y z

SMT

NM

In phase Dephased

M2M2

QtransQrefl

Qin

T

T||

X


-3 -2 -1 0 1

-1.0x10-4

0.0

t

Q(a

.u.)

Energy (eV)

QL

, 0V

QR

, 0V

QL

, +0.1V

QR

, +0.1V

QL

, -0.1V

QR

, -0.1V

t E

F-2.0x10-4

0.0

2.0x10-4

Energy (eV)

Q(a

.u.)

QL

||, 0V

QL

||, +0.1V

QL

||, -0.1V

QR

||, 0V

QR

||, +0.1V

QR

||, -0.1V

EF

1

0'

1'1'

0'

'

QQQQQΤΤ

Spin current (total torque) on energy


Q|| Q

VeB

2

0Fz

y x (||)

)()( |||| EQEQ RL

},min{ RL ffE 0V

)(

0V

)( )V()(

>

eEQEQ LRRL

0V0V)V()(

> eEQEQ

e-

RLQQQ


Back to tunnel junctions


1

0'

1'1'

0'

'

QQQQQΤΤ

-2.0x10-4

0.0

2.0x10-4

Energy (eV)

Q(a

.u.)

QL

||, 0V

QL

||, +0.1V

QL

||, -0.1V

QR

||, 0V

QR

||, +0.1V

QR

||, -0.1V

EF


RLQQQ

VeB

2

0Fz

y x (||)

-3 -2 -1 0 1

-1.0x10-4

0.0

t

Q(a

.u.)

Energy (eV)

QL

, 0V

QR

, 0V

QL

, +0.1V

QR

, +0.1V

QL

, -0.1V

QR

, -0.1V

t E

F

Q|| Q

)()( |||| EQEQ RL

},min{ RL ffE 0V

)(

0V

)( )V()(

>

eEQEQ LRRL

0V0V)V()(

> eEQEQ

e-





1

0'

1'1'

0'

'

QQQQQΤΤ


VeB

2

0Fz

y x (||)

T is an even parity function

of applied voltage! ||T is zero at zero voltage and is non

monotonic function of applied voltage

RLQQQ

-2.0x10-4

0.0

2.0x10-4

Energy (eV)

Q(a

.u.)

QL

||, 0V

QL

||, +0.1V

QL

||, -0.1V

QR

||, 0V

QR

||, +0.1V

QR

||, -0.1V

EF

-3 -2 -1 0 1

-1.0x10-4

0.0

t

Q(a

.u.)

Energy (eV)

QL

, 0V

QR

, 0V

QL

, +0.1V

QR

, +0.1V

QL

, -0.1V

QR

, -0.1V

t E

F

Q|| Q

e-

D. Ralph and M.Stiles, JMMM 320, 1190 (2008) M. Chshiev et al, IEEE Trans. Mag. 44, 2543 (2008)




theory

PRL 97, 237205 (2006)

IEEE Trans. Mag. 44, 2543 (2008)

Total torques: Comparison of theory and experiment

H. Kubota et al,

Nature Physics 4, 37 (2008)

||T

T(field-like)

i

iC 2

2

0 VΤΤ


||T~V

H. Kubota et al,


J. C. Sankey et al, ibid. 4, 67 (2008)

~V2


T(field-like)

i

iC 2

2

0 VΤΤ

theory

PRL 97, 237205 (2006)

IEEE Trans. Mag. 44, 2543 (2008)


||T~V

C. Heiliger et al, PRL 100, 186805 (2008)


M. Wilczyński et al., PRB 77, 054434 (2008)

J. Xiao et al., PRB 77, 224419 (2008)

H. Kubota et al,


A. Deac, ibid. 4, 803 (2008)



theory

I. Theodonis et al,

PRL 97, 237205 (2006)

IEEE Trans. Mag. 44, 2543 (2008)

~V2

T(field-like)

i

iC 2

2

0 VΤΤ


||T

C. Heiliger et al, PRL 100, 186805 (2008)


M. Wilczyński et al., PRB 77, 054434 (2008)

J. Xiao et al., PRB 77, 224419 (2008)

H. Kubota et al,


A. Deac, ibid. 4, 803 (2008)



theory

I. Theodonis et al,

PRL 97, 237205 (2006)

IEEE Trans. Mag. 44, 2543 (2008)

~V2

T(field-like)

i

iC 2

2

0 VΤΤ

C. Wang et al., Nature Phys. (2011)


(fA

) )

()

(

s

zI(AP)

(P) 0

(fA

) )

()

(

s

zI(AP)

(P) 0

)(zQ)0(zQ

The parallel (Slonczewski) torque T|| may be found from collinear currents

)()()( ;'')0()(4

)(|| JJQQQe

T zzz MMM

2121

32

21 ,2/)( ),()()()( whereVOVVVJ

0, VQz )0(

Parallel magnetizations:

Brinkman model:

,

2)( VQz

Antiparallel magnetizations:

I. Theodonis et al, PRL 97, 237205, 2006

2

|| V Τ )0(zQ may vanish

In-plane torque (T||) for symmetric MTJ


Field-like torque (T) for (a)symmetric MTJ

VeB

2L

L

L

0F

2,

2

)()(

)(

)()(0

)(

RLRL

RL

RLRL

RL

)(

0

)()(

)(

0

)()(

RLRLRL

RLRLRL

i

i

i

2

2 VCΤ

Symmetric MTJ: T is an even parity function of applied voltage: (I. Theodonis et al, PRL 97, 237205 (2006); M. Chshiev et al, IEEE Trans. Mag. 44, 2543 (2008))

Asymmetric MTJ (deviations from V2): T with odd parity terms appeared : (C1 in A. Manchon et al, JPCM 20, 145208 (2008); M. Wilczyński et al., PRB 77, 054434) ; Xiao et al., PRB 77, 224419 (2008); S.-C. Oh et al, Nature Physics 5, 898 (2009)

i

i

iVCΤ

-1 0 1

-2.0x10-6

0.0

R eV

L

R eV

T (eV/

)

L eV

L eV

L eV

Voltage


P->AP (AP->P)

Spin transfer

MRAM:

(STT-MRAM)

Uncontroled phenomenon during «bit» writing in STT-MRAMs: “back-switching”

ONjSTT

0

Vdd

écriture “1”

ONjSTT

0

freepinned

Vdd

écriture “0”

Spin Transfer Torques in Magnetic Tunnel Junctions

back switching is a

problem for MRAM

experiment Could theory help

understanding a problem?

Yes!


MMMMMM

MHMM

''||0 TTg

dt

d

Mdt

d Be

s

precession damping Slonczewski

Landau-Lifshitz-Gilbert (LLG) equation: field-like

STT: Current acts on Magnetization

M’ M j

T

||T

ref free

P->AP (AP->P)

T>0

T||>0

T>0

T||<0 back switching is a

problem for MRAM





For symmetric MTJ theory predicted:

For V>0:

T(V) =- T||(V)

Could theory provide a

solution for backswitching?

Yes!

experiment

V- Τ ; V Τ //

2



STT: Current acts on Magnetization

M’ M j

T

||T

ref free

P->AP (AP->P)

T>0

T||>0

T>0

T||<0 back switching is a

problem for MRAM

i

i

iVCΤ For asymmetric MTJ theory predicts linear term

experiment

VeB

ref

F

free

ref ≠ free S.-C. Oh, S.-Y. Park, A. Manchon, M. Chshiev et al, Nature Physics 5, 898 (2009)

Theory

Exp.

For asymmetric

MTJs (ref < free)

backswitching voltage

can be shifted up

(blue curves) away

from writing voltage

(which is also observed in experiment)



Spin Transfer Torques and TMR in Magnetic Tunnel Junctions

STT: MTJ with asymmetric barrier

VeB

2 2

'

M’ M

j T

||T

ref free


-1.0 -0.5 0.0 0.5 1.0

0

10

20

30

-1.0 -0.5 0.0 0.5 1.0

-16

-12

-8

-4

0

-1.0 -0.5 0.0 0.5 1.0

-100

-50

0

50

100

T|| (

pe

V/

)

Voltage (V)

clean

left

center

right

(a) (b)

T (

pe

V /

)

Voltage (V)

(c)

I (p

A /

)

Voltage (V)

-1.0 -0.5 0.0 0.5 1.0

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

-1.0 -0.5 0.0 0.5 1.0

-0.2

-0.1

0.0

0.1

0.2

-1.0 -0.5 0.0 0.5 1.0

-60

-40

-20

0

20

40

60

T|| / I (

h / 2

e)

Voltage (V)

clean

left

center

right

(a) (b)

(T -

T0 )

/ I (h

/ 2

e)

Voltage (V)

(c)

TM

R (

%)

Voltage (V)

STT and TMR voltage dependence tuning by Interface engineering

T||(V) and T(V) vs additional layer position in the barrier

STT efficiency and TMR vs additional layer position in the barrier

t6' B (resonant regime) t6' B > (tunneling regime)


Spin Transfer Torques and TMR in Multiferroic Magnetic Tunnel Junctions

A. Useinov, M. Chshiev and A. Manchon, Phys. Rev. B 91, 064412 (2015)

See lecture of A. Barthelemy


A. Useinov, M. Chshiev and A. Manchon, Phys. Rev. B 91, 064412 (2015)

Spin Transfer Torques and TMR in Multiferroic Magnetic Tunnel Junctions


Spin Filtering (SF) based on magnetic insulators (EuO, EuS, ferrites, etc.)

A. V. Ramos1, M.-J. Guittet1, J.-B. Moussy1, R. Mattana2, C. Deranlot2, F. Petroff2, and C. Gatel3

See lecture of S. Valenzuela


C. Ortiz Pauyac et al, Phys. Rev. B 90, 235417 (2014)


Spin filtering in crystalline MTJs (Fe|MgO|Fe)







Spin Transfer Torque (STT)





A. Khalil et al, IEEE Trans. Mag. 46, 1745 (2010)?

Models seems to be ok

M’ M j

T

||T

ref free

W. H. Butler et al, PRB 63, 054416 (2001); IEEE Trans. Mag. 41 (2005) 2645

MgO

3

bcc Fe ↑

• s-like

• low filled

Spin Transfer Torques and TMR in Magnetic Tunnel Junctions


FM MgO FM Cr

FE

FE

F. Greullet et al, PRL 99, 187202 (2007)

What about effect of Cr on STT?

TMR in Fe|Cr|MgO|Fe Magnetic Tunnel Junctions


Model (NEGF):

)()(

)()(

dsds

dsds

JJ

JJCurrent matrix:

x

z y

Barrier FM’ FM

M

))()((2

)( )()()( qqq dsdsds JJe

J

)(

)()0(TMR

J

JJ

q

Ve

BdsU )(

)(

)(

dsU

NM

NM

dsU )(

)(

)(

dsU

M’

'

)',(' zz

zzGzz

dEdJ

''

'' )',('

zz

spin zzGzz

dEd sJ

Charge current:

Spin current:

Spin torque:

spinJT

b

ds

NM

dsds mmm )()(

)(

)( ,,

Effective mass:

M. E. Eames et al, APL, 88, 252511 (2006)

J. Callaway et al, PRB 16, 2095 (1977)

J. Schäfer et al, ibid. 72, 155115 (2005)

A. H. Davis et al, JAP 87, 5224 (2000)

W. H. Butler, et al. PRB 63, 054416 (2001)

Refs

for parameters

a

5

1

''

''

d

s

A. Manchon et al, JPCM, 20, 145208 (2008)

A. Vedyaev et al, J. Appl. Phys. 107, 09C720 (2010)


Results for Fe|Cr(a)|MgO|Fe

-40 -30 -20 -10 0-20

0

20

40

60

To

rqu

e, o

EZ, nm

a=0

a=4

-20 -10 0

0.0

5.0x104

1.0x105

To

rqu

e, o

E

Z, nm

a=0

a=4

• Torque is mainly due to s-electrons • It almost vanishes when Cr is inserted (Cr acts as barrier for s-electrons, i.e. 1)

• Torque due to d-electrons is weak and almost insensitive in value to Cr insertion • Phase of oscillations is affected

s-electrons d-electrons

Torque distribution in the left Fe layer


Spin filtering in crystalline MTJs (Fe|MgO|Fe)







Spin Transfer Torque (STT)





A. Khalil et al, IEEE Trans. Mag. 46, 1745 (2010)?

Models seems to be ok

Interlayer Exchange Coupling (IEC)

M’ M j

T

||T

ref free

W. H. Butler et al, PRB 63, 054416 (2001); IEEE Trans. Mag. 41 (2005) 2645

H. X. Yang et al, APL 96, 262509 (2010)

J. Faure-Vincent et al,

PRL 89, 107206 (2002) Tight-binding results with a choice of parameters

used successfully for STT behaviour

• DFT with GGA for Vxc

• PAW pseudopotentials

• VASP

MgO

Interlayer Exchange Coupling in Magnetic Tunnel Junctions

3

bcc Fe ↑

• s-like

• low filled


z

y x (||)

Barrier FM’ FM

x x x x x x x x x x x x x x x x x

Total energy differences for P and AP

• IEC period and amplitude are nonmonotonic functions of the Fermi level

0 5 10 15 20 25 30

-8,0x10-6

-4,0x10-6

0,0

4,0x10-6

8,0x10-6

0=+3.4

0=+3.6

Left layer thickness (ML)

IEC

(a.

u.)

dR=5

5 10 15 20 25 30-1,2x10

-5

-1,2x10-5

-1,1x10-5

dR=5


IEC

(a.

u.)

0=+3.8

5 10 15 20 25 30

-1,4x10-5

-1,4x10-5

dR=5

IEC

(a.

u.)


0=+4

dL dR

q

Equilibrium IEC on FM layers thickness

1S

2S

B

2 2

Period of oscillations T

T~5 ML

T~7 ML

T~5 ML

T~7 ML

P.Bruno, PRB 52, 411 (1995)

L.E.Nistor et al, PRB 81, 220407 (2010)


5 10 15 20 25 30-1.2x10

-5

-1.2x10-5

-1.1x10-5

dR=5

FM layer thickness (ML)

IEC

(a.

u.)

0=+3.8theory

MTJ with perpendicular magnetic anisotropy (pMTJ)

First observation of oscillatory AF IEC in perpendicular MTJs (pMTJ)

FM layers thickness dependence in MTJ with PMA (pMTJ)

L.E.Nistor et al,

Phys. Rev. B 81, 220407 (2010)

Λ~5 ML

en accord avec P.Bruno, PRB 52, 411 (1995)

exp

Interlayer Exchange Coupling in Magnetic Tunnel Junctions


• Spin Transfer Torques and TMR:

• Free electron and a tight-binding descriptions

• Theory predicted STT voltage dependences in MTJs

• Field-like torque is an even-parity function of applied voltage

for symmetric MTJs

• Linear term appears in case of asymmetric MTJs

• provides with a solution for “back-switching” problem

• Models are satisfactory for TMR, STT and IEC

Summary

ONjSTT

0

Vdd

écriture “1”

M’ M

j T

||T

ref free

Thank you!

Acknowledgments:

A. Manchon, A. Kalitsov, H.-X. Yang, A. Schuhl, B. Dieny, W. H. Butler, K.-J. Lee, C. Ortiz Pauyac,

I. Theodonis, J. Velev, L. Nistor, N. Kioussis, A. Vedyayev, N. Strelkov, N. Ryzhanova and all colleagues

Theory of Spin Dependent Tunnelingmagnetism.eu/esm/2015/slides/chshiev-slides.pdf · AFM metals AFM...

Documents

Transcript of Theory of Spin Dependent Tunnelingmagnetism.eu/esm/2015/slides/chshiev-slides.pdf · AFM metals AFM...