Spin electronics at the nanoscale

Michel Viret

Service de Physique de l’Etat Condensé

CEA Saclay

France

Principles of spin electronics:

ferromagnetic metalsspin accumulation

Resistivity of homogeneous materials:

AMRDWR

Spin torque in DWs

Reduced dimensions:

Mesoscopic transportAtomic MR

Different DOS for upand down spins :

Spin dependent electrical transport in ferromagnetic metals

s electrons : low density of states + high mobility

d electrons : large density of states + low mobility

Transport is dominated by s electrons scattered into d bandsd bands split by the exchange energy→ diffusion is spin dependent

→ Two current model :Two conduction channels in parallel

with ρ↑≠ ρ↓

Resistivity :

or (with spin-flip) :

ρρρρρρρρ

↑↓↓↑

↓↑↑↓↓↑

++

++=ρ

4

)(

ρρρρ

↓↑

↓↑

+=ρ

E

d bands

s bands

Spin

down

spin

up

↑↑↑↑

↑↑↑↑↓↓↓↓

↓↓↓↓

Current generated spin accumulation at a Ferro (Co) / Normal metal (Cu) interface:

Typically, at 4.2 K :lsf (Co) ≈ 60 nmlsf (Cu) ≈ 500 nm

H (kOe)

80%

I

V

Multilayers

F metal / NM metal(ex : Fe / Cr, Co / Cu, etc )

4 K

Conf igurat ion P

M MNM

-

+

M MNM

+

-

Conf igurat ion AP

r r

r

r

R R R

RR+= r

R- = R

R+ = (r+R)/ 2

R- = (R+r)/ 2

P

PAP

R

RR −=GMR

rrR

RrRP ≈≈≈≈

++++====

4

rRRAP

++++====<

Introduction to spin electronics

Domain wall resistance

Example of FePd:

MFM image 2 x 2 µm2 at zero field, in the virgin state and after saturation. The up domains are black andthe down ones are white

Resistive

measurements : on a nanostructure withstripe width = 300nm

0.5

7.992

8.016

8.040

8.064

8.088 ρ

CPW

ρCIW

ρ (µΩ

cm

)

H

∆R/R = 8 % within the DWs

Why is it so small?

(R. Danneau et al., Phys. Rev. Lett. 88, 157201 (2002))

Spin transfer from the conduction electrons to the DW

Current

direction

Theory

Two kinds of electrons:

• Localised• Conduction electrons

→ s-d Hamiltonian

Action of a current:

Globally, the conduction electrons transfer gµB to the DW

s-d Hamiltonian :

: localised spins

Precession equation :

s : conduction electrons

⇒⇒⇒⇒

Simple model : the particle approach

( )

⇒⇒⇒⇒

In the rotating frame:

Loc

a l M

omen

t

Electron spinφ0

In the frame of the local moment :

In the laboratory frame

Spin evolution during DW crossing

Loc

a l M

omen

t

Electron spinφ0

In the frame of the local moment :

In the laboratory frame

Spin evolution during DW crossing

exw

F

J

v

δφ

η=0

2wex

F

2

w 1

J

v

)p1(

p2

R

R

δ

−=

∆ η

Loc

a l M

omen

t

Electron spinφ 0

In the frame of the local moment direction :

Loc

a l M

omen

t

Electron spinφ0

In the frame of the local moment :

In the laboratory frame

Spin evolution during DW crossing

exw

F

J

v

δφ

η=0

2wex

F

2

w 1

J

v

)p1(

p2

R

R

δ

−=

∆ η

Loc

a l M

omen

t

Electron spinφ 0

In the frame of the local moment direction :

Loc

a l M

omen

t

Electron spinφ0

In the frame of the local moment :

In the laboratory frame

Spin evolution during DW crossing

exw

F

J

v

δφ

η=0

2wex

F

2

w 1

J

v

)p1(

p2

R

R

δ

−=

∆ η

Loc

a l M

omen

t

Electron spinφ 0

In the frame of the local moment direction :

Loc

a l M

omen

t

Electron spinφ0

In the frame of the local moment :

In the laboratory frame

Spin evolution during DW crossing

exw

F

J

v

δφ

η=0

2wex

F

2

w 1

J

v

)p1(

p2

R

R

δ

−=

∆ η

Loc

a l M

omen

t

Electron spinφ 0

In the frame of the local moment direction :

Loc

a l M

omen

t

Electron spinφ0

In the frame of the local moment :

In the laboratory frame

Spin evolution during DW crossing

Loc

a l M

omen

t

Electron spinφ0

In the frame of the local moment :

In the laboratory frame

Spin evolution during DW crossing

Loc

a l M

omen

t

Electron spinφ0

In the frame of the local moment :

In the laboratory frame

Spin evolution during DW crossing

Loc

a l M

omen

t

Electron spinφ0

In the frame of the local moment :

In the laboratory frame

Spin evolution during DW crossing

Loc

a l M

omen

t

Electron spinφ0

In the frame of the local moment :

In the laboratory frame

Spin evolution during DW crossing

For a long wall and

Precession around the effective field :

Rotating frameMagnetisation

Magnetic

moment

Direction of

electron

propagation

Laboratory frame

→ The mistracking angle is small (a few degrees) and theinduced spin scattering is weak

The total moment is conserved →

Reaction on the wall

The torque can be decomposed into a constant and periodic partFor long walls, the periodic part averages to zero and the constant part reads:

distortion (steady state)

P = polarisation, j = current density pressure

• Torques: non-homogeneous within the walls + small ‘pressure’ term

• Importance of the magnetic structure of the DW

• Very thin DWs: Enhanced pressure oscillating with thickness

Conclusions :

Effect of the current : Globally, the conduction electrons transfer gµB to the DW→→→→ Spin torque

Preliminary conclusions

Question: Can we build spin electronics devices with domain walls?

DW: topological soliton which can be moved by a current and which scatters electronsRussel Cowburn (Durham/London) : magnetic logic based on DWsStuart Parkin (IBM San Jose) : registery memory

Problems: DWs are not very resistive and cannot be pushed by small currents…

Solution : 1D structures

Ohm’s law: I = G VG = σ S / L if L >>

• lF Fermi wavelength (quantum)

• ℓ mean free path (elastic)

• Lj coherence length (inelastic)

macroscopicdiffusiveatomic ballistic

Different transport regimes

Magnetoresistance in reduced dimensions: the constriction

+ magnetism :

Spin degeneracy removed by the exchange energy

≠ Fermi wavelength for up and down electrons

→ the size of the constriction at which the

number of transmitted channels changes

is spin dependent

↓↑

↓↑

≠→

−±=

=

++∇−=

NN

kEEm

k

dnk

JrVm

H

perpexFlong

perp

ex

)(2

2.

.)(2

2

2

*,

22

η

η

π

σ

⇒ conductance quantized :

G = ΣΣΣΣTi↑↓↑↓↑↓↑↓G0 (G0 = e2/h = 1/26kOhm)

2D electron gases : σ quantized in units of 2G0 , but large H : quantization in G0

Perfect case: continuous materials of cross section close to λF (ex: 2DEG) : k⊥ quantized

Closer to real life :

Metals : λF ≈ 2Å→ quantization requires atomic contacts !Conducting channels defined by overlap of atomic orbitals→ atomic calculation needed+ In the single atomic regime more than one conduction channels are opened for 3d elements. Ni: 4s2 3d8→ potentially 4↓ + 6↑ channels

Magnetic problems :DWs not infinitely thin: Micromagnetic configuration of the relevant atoms?

Experimental problems : Magnetostriction ??For Ni : λ ≈ -40 ppm → 100 µm wire shrinks by 4 nm !

Introduction of a DW in a constriction (ideal case) :DW width = size of constriction (P. Bruno, PRL83, 2425 (1999), Y. Labaye et al., J.A.P.91, 5341 (2002))

Constricted DW width ≈constriction diameter or length

+ Potential barrier of magnetic (exchange) origin (Imamura et al. PRL 84, 1003 (2000)) : large MR effects because a DW can close the conductance channel.

Break junction technique :Samples : ferromagnetic bridges suspended with pads of different shapes made in polycrystalline Ni, Co, Fe :

MR in ferromagnetic atomic contacts

Micromagnetics:

Program OOMMF (NIST)

Experimental setup

Bending system in a cryostatPulling ≈ 1 nm/turn

H. Ohnishi, Y. Kondo and K. Takayanagi,

Nature 295, 780 (1998)

TEM pictures while pulling an Au tip from a Au film :

Measurements : breaking

01234567

89

101112131415

-1 -0.8 -0.6 -0.4 -0.2 0 0.2

d (nm)

(e2/h)

Conductance steps : atomic reorganization (+ quantization)

Tunnelling : R = R0 exp(x/x0) , x0=0.045 nm

Tunneling regime - H=0.4T perp

3

4

5

6

7

8

9

10

11

0 0.01 0.02 0.03 0.04 0.05gap (nm)

ln(R)

Two types of measurements

R(θ) → AMR:

0

0.2

0.4

0.6

0.8

1

1.2

-90 -60 -30 0 30 60 90 120 150 180 210 240 270

Angle (degree)

Resistance

R(Η) → ‘DWR’ :

- 9 0 - 6 0 -3 0 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0

6 9 8

7 0 0

7 0 2

7 0 4

7 0 6

A n g l e ( d e g r e e )

Atomic contact:

∆∆∆∆R/Rmin= 75%

9 0 0 0

1 0 0 0 0

1 1 0 0 0

1 2 0 0 0

1 3 0 0 0

1 4 0 0 0

1 5 0 0 0

1 6 0 0 0

1 7 0 0 0

1 8 0 0 0

Atomic contact:

∆∆∆∆R/Rmin= 21%

7 2 0 0

7 4 0 0

7 6 0 0

7 8 0 0

8 0 0 0

8 2 0 0

8 4 0 0

8 6 0 0

8 8 0 0

R (

Oh

m)

Nanostructure:

∆∆∆∆R/Rmin= 1.1%

Atomic contact regime (Fe, 4.2K)

R(θθθθ) curves :

• Significant effect• Clear departure from cos2(θ)

7000

7200

7400

7600

7800

8000

8200

8400

8600

8800

-100 -50 0 50 100 150 200

angle (°)

R (Ohm)

2T, Vdc=0mV

3e2/h

3.5e2/h

→ Channels closing?

Parallel with evolution of conductance with stretching

Tentative explanation of the AMR effects :

Orbitals overlap responsible for the

opened channels

Distortion with field because of spin-orbit

coupling?

+ enhanced effects in reduced dimensions

Ab-initio calculations

Pseudo potential plane wave method + spin-orbit coupling

Fe atomic chain: magnetisation parallel and perpendicular

Tight binding calculations

fcc (111)

bcc (001)

The exact geometry of the contact is

important, especially the coordination

number of the central atom.

Tunnelling regime, Fe (4.2K) :

→AMR still large in tunnelling

Measurements quite ‘erratic’ because different configurations can give the same resistance

Tunnelling defined by the overlap of evanescent orbitals→ Spin-orbit coupling of the same nature but on the evanescent orbitals

Summary of AMR measurements in Fe

6.2

6.25

6.3

6.35

6.4

6.45

6.5

6.55

6.6

-60 -40 -20 0 20 40 60 80 100 120 140 160 180 200Angle(degrees)

R(kohms)

6

6.2

6.4

6.6

6.8

7

7.2

7.4

7.6

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

H(T)

R(kohms)

180°

90°

90°<--180°

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7

7.1

-2.5 -1.5 -0.5 0.5 1.5 2.5H(T)

R(kohms)

← Fe (4.2K)‘Domain wall’ effects

12%

27

28

29

30

31

32

33

34

35

36

37

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

H(T)

R(kohms)

180°

90°

Co (4.2K)↓

DWR ≈ 10% – 20%

21%

E-field efects

3700

3800

3900

4000

4100

4200

4300

4400

4500

4600

4700

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

H (T)

R (Ohm)

Vdc=100mV

Vdc=50mV

Vdc=0

field at 0°

Electric field effects, Fe (4.2K):

Effect of the electric field:

Evidence for spin torque?

At 50mV, j ≈≈≈≈ 5 108 A/cm2

differential conductance for two different magnetic configurations

8000

9000

10000

11000

12000

13000

14000

15000

16000

17000

-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

Vbias (V)

dI/dV (Ohm)

H=0

H=1T

Conclusions for the MR at the atomic scale

conductance depends on orbital overlapS-O coupling = atomic AMR effectIn reduced dimensions: large S + large O !

Contact:

AAMR : 50 %

DWR : 20%

AAMR>DWR

Tunnelling

TAMR : 100 %

TMR : 35%

TAMR>TMR

tunnelling AMR : same with evanescent orbitals

See: M. Viret et al., Eur. Phys. J. B 51, 1 (2006)

Conclusions

Can DWs be used for spin electronics?

In the bulk :

DW resistance too small + current induced pressure too weak

In atomic constrictions:

Importance of orbital effects (AMR)

DW Resistance enhanced (20%)

Pushing with a current impossible (destruction of the contacts)

The ideal : 1D atomic chains…