Models in Spintronics - The European School on Magnetism

<strong>Models</strong> <strong>in</strong> sp<strong>in</strong>tronics (Part I) 

OUTLINE : 

Sp<strong>in</strong>-dependent transport <strong>in</strong> metallic magnetic multilayers 

-Introduction to sp<strong>in</strong>-electronics 

-Sp<strong>in</strong>-dependent scatter<strong>in</strong>g <strong>in</strong> magnetic metal 

-Current-<strong>in</strong>-plane Giant Magnetoresistance 

-Modell<strong>in</strong>g transport <strong>in</strong> CIP sp<strong>in</strong>-valves 

-Current-perpendicular-to-plane Giant Magnetoresistance 

-Sp<strong>in</strong> accumulation, sp<strong>in</strong> current, 3D generalization. 

B.Dieny « <strong>Models</strong> <strong>in</strong> sp<strong>in</strong>tronics » 2009 <strong>European</strong> <strong>School</strong> on Magnetism, Timisoara

Birth of sp<strong>in</strong> electronics : Giant magnetoresistance (1988) 

Fe/Cr multilayers 

Fert et al, PRL (1988), Nobel Prize 2007 

Two limit geometries of measurement: 

I 

~ 80% 

V 

Current-<strong>in</strong>-plane 

I 

I 

V 

Magnetic field (kG) 

I 

I 

Current-perpendicular-to-plane 

Antiferromagnetically coupled multilayers 

GMR 

 

R 

AP 

 

R 

P 

R 

P 


Low field GMR: Sp<strong>in</strong>-valves 

FeMn 90Å 

NiFe 40Å 

Cu 22Å 

Antiferromagnetic 

p<strong>in</strong>n<strong>in</strong>g layer 

Ferromagnetic 

p<strong>in</strong>ned layer 

Non magnetic spacer 

M (10 -3 emu) 

1 

0 

-1 

4 

(a) 

(b) 

NiFe 70Å 

Ta 50Å 

Soft ferromagnetic layer 

Buffer layer 

R/R (%) 

3 

2 

1 

substrate 

IBM Almaden 

Ultrasensitive magnetic field sensors (MR heads) 

Sp<strong>in</strong> eng<strong>in</strong>eer<strong>in</strong>g 

R/R (%) 

0 

-200 

4 

3 

2 

1 

(c) 

0 200 400 600 800 

H(Oe) 

B.Dieny et al, Phys.Rev.B.(1991)+patent US5206590 (1991). 

B.Dieny « <strong>Models</strong> <strong>in</strong> sp<strong>in</strong>tronics » 2009 <strong>European</strong> <strong>School</strong> on Magnetism, Timisoara 

0 

-40 

-20 

0 20 40 

H(Oe)

Benefit of GMR <strong>in</strong> magnetic record<strong>in</strong>g 

GMR sp<strong>in</strong>-valve heads 

from 1998 to 2004 


Two current model (Mott 1930) for transport <strong>in</strong> magnetic metals 

As long as sp<strong>in</strong>-flip is negligible, current can be considered as carried <strong>in</strong> 

parallel by two categories of electrons: sp<strong>in</strong> and sp<strong>in</strong> (parallel and 

antiparallel to quantization axis) 

 

 

 

 

1 

 

 

 

1 

 

 

 

 

 

1 

 

 

 

 

 

 

 

 

 

 

Sources of sp<strong>in</strong> flip: magnons and sp<strong>in</strong>-orbit scatter<strong>in</strong>g 

Negligible sp<strong>in</strong>-flip often crude approximation (sp<strong>in</strong> diffusion length <strong>in</strong> 

NiFe~4.5nm, 30% sp<strong>in</strong> memory loss at Co/Cu <strong>in</strong>terfaces) 


Sp<strong>in</strong> dependent transport <strong>in</strong> magnetic metals (1) 

Band structure of 3d transition metals 

In transition metals, partially filled bands which participate to conduction are s and d bands 

Non-magnetic Cu : Magnetic Ni : 

E 

s 

(E) 

s 

(E) 

E 

s 

(E) 

s 

(E) 

D 

(E) 

D 

(E) 

D 

(E) 

D 

(E) 

E 

E 

D 

(E F 

) = D 

(E F 

) 

D 

(E F 

) D 

(E F 

) 

Most of transport properties are determ<strong>in</strong>ed by DOS at Fermi energy 

Sp<strong>in</strong>-dependent density of state at Fermi energy 


Sp<strong>in</strong> dependent transport <strong>in</strong> magnetic metals (2) 

m*(d) >> m*(s) 

J mostly carried by s electrons <strong>in</strong> transition metals 

Scatter<strong>in</strong>g of electrons determ<strong>in</strong>ed by DOS at E F 

: 

E 

Fermi Golden rule : 

P 

iW 

f 

2 

D f ( E F 

) 

s 

(E) 

s 

(E) 

s 

s 

d 

D 

(E) 

D 

(E) 

s 

s 

d 

Most efficient scatter<strong>in</strong>g channel 

Sp<strong>in</strong>-dependent scatter<strong>in</strong>g rates <strong>in</strong> 

magnetic transition metals 

Example: 

Co 10nm; 

 

Co 

1nm 


Potential experienced by conduction electrons <strong>in</strong> magnetic metallic multilayers 

Parallel magnetic configuration 

Antiparallel magnetic configuration 

(a) 

Co/Cu majority electrons : parallel magnetic configuration 

Co Cu Co 

Cu 

(c) 

Co/Cu majority electrons : antiparallel magnetic configuration 

Co/Cu m<strong>in</strong>ority electrons : parallel magnetic configuration 

Co/Cu m<strong>in</strong>ority electrons : antiparallel magnetic configuration 

(b) 

U(x,z) 

(d) 

x 

•Lattice potential modulation due to difference between Fermi energy and 

bottom of conduction band (reflection, refraction) 

•Sp<strong>in</strong>-dependent scatter<strong>in</strong>g on impurities, <strong>in</strong>terfaces or gra<strong>in</strong> boundaries 

(Dom<strong>in</strong>ant effect <strong>in</strong> GMR) 


z

Simple model of Giant Magnetoresistance 

Parallel config 

Antiparallel config 

Fe 

Cr Fe 

Fe Cr Fe 

 

ap 

 

 

 

 

 

 

2 

 

 

 

2 

 

1 

1 

Equivalent resistances : 

 

 

 

 

 

 

 

 

 

 

 

P 

 

 

2 

 

 

 

 

 

 

 

 

AP 

 

 

 

 

2 

 

Key role of 

scatter<strong>in</strong>g contrast 


Model<strong>in</strong>g current-<strong>in</strong>-plane transport (semi-classical Boltzman theory of GMR) 

Approach <strong>in</strong>itiated by Camley and Barnas, PRL, 63, 664 (1989) 

Gas of <strong>in</strong>dependent particles described by distribution f(r, v, t), submitted 

to force field F (=-eE for electrons <strong>in</strong> electrical field E). 

Time evolution of the distribution described by Boltzman equation: 

t+dt 

E 

 

 

 

df 

dt 

 

 

 

t 

F 

Equilibrium function conserved <strong>in</strong> a volume element drdv 

along a flow l<strong>in</strong>e. 

df df df 

In presence of scatter<strong>in</strong>g, 0 

dt dt dt 

f 

f 

f 

. vx 

. vy 

t 

x 

y 

f 

 

v. 

 

r 

f a. 

v 

f 

t 

f 

. vz 

z 

f 

 

v 

x 

. a 

x 

f 

 

v 

y 

F 

. a 

y 

f 

 

v 

scatter<strong>in</strong>g 

Balance between acceleration due to force 

and relaxation due to scatter<strong>in</strong>g 

with 

 

ma 

 

F 

z 

. a 

z 


df 

dt 

f 

t 

df 

dt 

In stationary regime, 0 

 

 

 

F 

 

 

 

scatt 

f 

t 

In s<strong>in</strong>gle relaxation time approximation () 

 

 

 

 

 

F 

v. 

 

r 

f . 

m 

 

 

 

df 

dt 

 

 

 

scatt 

v 

f 

 

 

 

f 

 

Where f 0 is the equilibrium distribution (Fermi Dirac for electrons). 

f 

0 

 

Boltzmann equation for electron gas <strong>in</strong> electrical field E: 

 

v. 

 

 

r 

f 

 

 

eE 

m 

 

. 

v 

f 

 

 

 

f 

 

f 

0 

 


I 

x 

Model<strong>in</strong>g current transport <strong>in</strong> bulk metals 

V 

E 

Spatially homogeneous transport 

I 

 

r f 

f 

 

v. 

 

 

r 

 

 

 

r 

, v f r 

, v 

g r 

, v 

0 

f 

0 

f 

0 

 

 

r, 

v 

 

 

eE 

 

m 

 

Fermi-Dirac 

1 

 

 

exp 

 

kBT 

 

. 

v 

f 

 

 

 

 

Perturbation 

due to electric field 

F 

 

 

1 

 

f 

 

f 

0 

 

g( 

v 

x 

) 

eE f 

m v 

x 0 

 

x 

k z 

In k-space, shift <strong>in</strong> Fermi surface by 

k 

eE x 

 

k y 

k 

k x 


Model<strong>in</strong>g current transport <strong>in</strong> bulk metals (cont’d) 

Current density: 

j 

j 

 

 

e 

 

v 

x 

g( v 

) d 

3 

v 

2 

ne E E 

 

1 E 

m 

2 

ne 

m 

Well-known expression of conductivity <strong>in</strong> Drude model 

n=density of conduction electrons 

n~1/atom <strong>in</strong> noble metals such as Cu, Ag, Au 

n~0.6/atom <strong>in</strong> metals such as Ni, Co, Fe 


Typical resistivity of sputtered metals 

Material 

Cu a 

Ag f 

Au g 

Pt 50 

Mn 50 

e 

Ni 80 

Cr 20 

e 

Ru c 

Measured 

resistivity 

4K/300K 

0.5-0.7.cm 

3-5 

1.cm 

7 

2.cm 

8 

160.cm 

180 

140.cm 

140 

9.5-11.cm 

14-20 

Material 

(ferro) 

Ni 80 

Fe 20 

a 

Ni 66 

Fe 13 

Co 2 

b 

1 

Co a,d 

Co 90 

Fe 10 

h 

Co 50 

Fe 50 

h 

Measured resistivity 

4K/300K 

10-15.cm 

22-25 

9-13.cm 

20-23 

4.1-6.45.cm 

12-16 

6-9.cm 

13-18 

7-10.cm 

15-20 

<strong>The</strong>rmal variation of resistivity due to phonon scatter<strong>in</strong>g and magnon scatter<strong>in</strong>g (<strong>in</strong> 

magnetic metals) 


g 

z 

0 

z, 

v g z, 

v eE f 

v 

 

 

 

v F 

v 

General solution : 

Model<strong>in</strong>g current transport <strong>in</strong> metallic th<strong>in</strong> films 

I 

 

f r, v f r, 

v g r, 

v 

0 

z 

 

V 

eE f f 

v. 

 

r 

f . v 

f 

I 

m 

 

E 

x 

Due to scatter<strong>in</strong>g at outer surfaces, the perturbation g is no longer homogeneous: g(z) 

z 

 

mv 

z 

v 

x 

= elastic mean free path 

g 

 

 

f 

 

0 

z, v 

eE 

vx 

1 

A 

 

 

+(-) refer to electrons travel<strong>in</strong>g towards z>0 (z

Boundary conditions for a th<strong>in</strong> film : 

g 

 

 

z sup 

g 

z sup 

g 

 

z 

 

Rg 

sup 

z sup 

z sup 

z <strong>in</strong>f 

g 

g 



Proba R Proba 1-R 

g 

 

z 

Rg 

<strong>in</strong>f 


Proba of specular reflection R 

Proba of diffuse reflection 1-R 

Two boundary conditions, two unknowns A+, A-, solvable problem 

Current density: 

j( 

z) 

 

1 

 

0 

 

 

 

j(z) e 

 

v 

x 

g 

t 

 

 

 

v 

z 

, z 

 

2 

1 

2 A exp 

A exp 

d 

t=layer thickness, = cos<strong>in</strong>e of electron <strong>in</strong>cidence 

 

 

 

d 

3 

v 

t 

 

 


G 

G 

G 

Model<strong>in</strong>g current transport <strong>in</strong> metallic th<strong>in</strong> films (cont’d) 

G 0 

G 1 

2 

ne t 

v m 

0 

 

F 

3 

1 

1 

 

4 

0 

t 

2 

d 

1 

 

 

thickness 

Same conductance as <strong>in</strong> bulk 

 

A 

 

 

t 

1 

exp 

 

A 

 

 

t 

1 

exp 

 

 

G 1 

conta<strong>in</strong>s all f<strong>in</strong>ite size effects. 

Characteristic length <strong>in</strong> current-<strong>in</strong>-plane transport=elastic mfp 

Fuchs-Sondheimer approximate expressions: 

j(z) 

 

1 3 

for 

2 

 

ne 8t 

 

t 

Assum<strong>in</strong>g diffuse surface scatt 

 

mv F 

 

4mv F 

1 

 

for 

2 

3ne 

tln 

/ t 0.423 

t 



OUTLINE : 





-Modell<strong>in</strong>g CIP Giant Magnetoresistance 




Layer i+1 

Model<strong>in</strong>g current <strong>in</strong> plane GMR <strong>in</strong> metallic multilayers 

V 

I 

I 

CIP 

Proba T i 

of 

transmission 

Same as for th<strong>in</strong> films but separately 

consider<strong>in</strong>g sp<strong>in</strong> and sp<strong>in</strong> 

electrons and solv<strong>in</strong>g the Boltzmann 

equation with<strong>in</strong> each layer with 

appropriate boundary conditions. 

•Bulk sp<strong>in</strong>-dependent scatter<strong>in</strong>g 

described by sp<strong>in</strong> dependent mfp 

•Interfacial scatter<strong>in</strong>g described by sp<strong>in</strong> 

dependent R and T 

Interface i 

Layer i 

Proba R i 

of reflexion 

Proba 1-R i 

-T i 

of diffusion 

g 

g 

 

i1, 

 

Ti 

gi, 

 

Ri 

gi 

1, 

 

 

i, 

Ti 

gi 

1, 

Ri 

g 

i, 

 


G 

Model<strong>in</strong>g current <strong>in</strong> plane GMR <strong>in</strong> metallic multilayers 

G 0 

G 1 

G 

n 

N N 

2 i i i 

0 

e ti 

/ 

i1, 

vFmi 

i1, 

 

t 

 

 

i 

Same conductance as if all layers 

were connected <strong>in</strong> parallel 

G 

1 

 

3 

4 

 

i 

 

i 

1 

 

, 

i 0 

 

d 

1 

 

2 

 

A 

 

 

 

i, 

 

t 

i 

1 

exp 

 

 

 

 

i 

 

 

A 

 

i, 

 

t 

i 

1 

exp 

 

 

 

 

i 

 

 

G 1 

conta<strong>in</strong>s all f<strong>in</strong>ite size effects and is responsible for GIP GMR. 

To obta<strong>in</strong> the GMR, G 1 

is calculated <strong>in</strong> parallel and antiparallel configurations. 

 

 

In AP configuration, , R ,T and 

, R ,T 

are <strong>in</strong>verted <strong>in</strong> every other layer. 

CIP GMR comes from second order effect <strong>in</strong> conductivity (<strong>in</strong> contrast to 

CPP GMR) 

Characteristic lengths <strong>in</strong> CIP GMR are the elastic mean-free paths 


Influence of feromagnetic layer thickness on GMR <strong>in</strong> sp<strong>in</strong>-valves 

G ( -1 ) 

G ( -1 ) 

0 .3 

0.25 

0 .2 

0.15 

0 .1 

0.05 

0 

0.00 3 

0.00 2 

0.00 1 

0 

R=1 

p = 1 

R=1 

p = 1 

p = 0 

R=0 

p = 0 

R=0 

Example of calculated CIP curves 

For a NiFe t NiFe 

/Cu 2nm/NiFe t NiFe 

Sandwich. 

R=coeff of specular reflection at lateral edges 

 

 

T 

NiFe 

 

Cu 

7nm, 

 

12nm, 

NiFe / Cu 

 

T 

NiFe 

 

NiFe / Cu 

 

.8nm, 

0.9 

Absolute change of sheet conductance 

most <strong>in</strong>tr<strong>in</strong>sic measure of CIP GMR 

F/Cu25Å/NiFe50Å/FeMn100Å 

R/R p 

(%) 

10 

5 

p = 1 

R=1 

p = 0 

R=0 

0 

0 50 100 1 50 200 250 300 

t (nm) F (Å) 

B.Dieny 

JMMM (1994) 


Influence of nature of ferro materials on GMR <strong>in</strong> sp<strong>in</strong>-valves 

5 

5 

R/R (%) 

4 

3 

2 

R/R p 

(%) 

4 

3 

2 

NiFe 

Fe 

Co 

1 

1 

0 

0 10 20 30 40 50 

t F (nm) 

0 

0 10 20 30 40 50 

t F 

(nm) 

F tF/Cu 2.5nm/NiFe 5nm/FeMn 10nm, 

with F=Ni80Fe20, Co and Fe (Dieny, 1991). 

 

 

 

 

NiFe 

 

Co 

 

Fe 

 

7nm, 

 

 

9nm, 

 

Co 

NiFe 

 

4.5nm, 

 

0.9nm, 

T 

Fe 

F t F 

/Cu 2.5nm/NiFe5nm/FeMn 10nm 

0.7nm, 

T 

 

Co / Cu 

4.5nm, 

T 

 

NiFe / Cu 

 

Fe / Cu 

0.85, T 

0.95, T 

 

Co / Cu 

0.70, T 

 

Fe / Cu 

 

NiFe / Cu 

 

0.30, 

 

 

0.30. 

0.30, 


Influence of non-magnetic spacer layer thickness on GMR <strong>in</strong> sp<strong>in</strong>-valves 

20 

NiFe 3nm/Cu t Cu 

/NiFe 3nm 

NiFe50Å/NM t NM 

/NiFe50Å/FeMn100Å 

R/R p 

(%) 

15 

10 

5 

p = 1 

p = 0 

0 

0 50 100 150 

t 

t (nm) 

CuF 

Semi-classical theory 

Experiments 

B.Dieny 

JMMM (1994) 

Two effects as t NM 

<strong>in</strong>creases: 

•Reduced number of electrons travell<strong>in</strong>g from one ferro layer to the other 

•Increas<strong>in</strong>g shunt<strong>in</strong>g of the current <strong>in</strong> the spacer layer 


Local current density and magnetic field due to current <strong>in</strong> a CIP sp<strong>in</strong>-valve 

 

j( z) 

 

, 

, 

i 0 

i 

 

1 

2 

 

t 

 

t 

i 

i 

1 

2 

A 

i 

A 

, 

exp 

d 

 

i 

exp 

 

 

i 

 

Oersted field due to sense current 

<strong>in</strong> a CIP sp<strong>in</strong>-valves 

J 

B( z) 

0 j( 

z') 

dz' 

0 

z 

cst 

Oe field plays an important role <strong>in</strong> the bias of sp<strong>in</strong>-valve heads 

Local current density (u.a) 

=2.10 7 A/cm² 

Magnetic field due to sense current (Oe) 


Vertical head with CIP MR reader 

reader 

writer 

1m 

Planar coil 

substrate 

shield 1 

MR 

shield 2 

P<strong>in</strong>ned 

Free 

Fly height 10nm, 

100nm 

J 

Field to be 

measured 

disk rotat<strong>in</strong>g at 5000 to 10000 rpm. 


L<strong>in</strong>ear response of sp<strong>in</strong>-valves MR heads 

Energy terms <strong>in</strong>fluenc<strong>in</strong>g the orientation 

of free layer magnetization: 

•Zeeman coupl<strong>in</strong>g to H, to coupl<strong>in</strong>g field 

through Cu spacer and to Oersted field: 

E 

M1( 

H H SV 

H J 

).s<strong>in</strong>( ) 

•Uniaxial and shape anisotropy 

E 

( 

K 

2 

NM s 

).cos ²( ) 

•Dipolar coupl<strong>in</strong>g with p<strong>in</strong>ned layer 

E M1H dip 

.s<strong>in</strong>( ) 

M<strong>in</strong>imiz<strong>in</strong>g energy yields s<strong>in</strong>() 

l<strong>in</strong>ear <strong>in</strong> H. 

S<strong>in</strong>ce R varies as M 

 

1. M 

 

2 

s<strong>in</strong>( ) 

l<strong>in</strong>ear R(H) 

M H 

 

RH 

R R 

R 1 

 

 

H 

SV 

H 

I 

H 

dip 

 

 

K 

NM 

1 

 

P AP P 

2 

2 

1 


Experimental optimization of sp<strong>in</strong>-valves 

Sp<strong>in</strong>-valves greatly optimized <strong>in</strong> 1994-2003 for HDD MR heads but replaced <strong>in</strong> 2004 by 

TMR heads 

NiFeCr/PtMn 120Å/ CoFe 15Å /Ru 7Å/CoFe 5Å/NOL/CoFe 15Å /Cu 20Å/CoFe10Å/NiFe 20Å /NOL 

25 

p<strong>in</strong>ned 

soft 

20 

GMR (%) 

15 

10 

5 

0 

-1000 -800 -600 -400 -200 0 200 400 600 800 1000 

H (Oe) 

GMR amplitude significantly improved by <strong>in</strong>creas<strong>in</strong>g specular reflection at 

boundaries of active part of the sp<strong>in</strong>-valve //(p<strong>in</strong>ned/spacer/free)// 


Conclusion on CIP GMR : 

CIP GMR well modeled by semi-classical Boltzmann theory. 

Local conductivity can only vary on length scale of the order of the elastic 

mean free path. 

Quantum mechanical theory of CIP transport were proposed to properly 

take <strong>in</strong>to account the quantum conf<strong>in</strong>ement/reflection/refraction effects 

<strong>in</strong>duced by lattice potential modulation. Predictions of oscillations <strong>in</strong> 

conductivity and GMR versus thickness but hardly seen <strong>in</strong> experiments due 

to <strong>in</strong>terfacial roughness 

CIP GMR amplitude up to 20% <strong>in</strong> specular sp<strong>in</strong>-valves. 

Sp<strong>in</strong>-valves were used <strong>in</strong> MR heads of hard disk drives from 1998 to 2004. 

Later on, they were replaced by Tunnel MR heads. 



OUTLINE : 





-Modell<strong>in</strong>g CIP Giant Magnetoresistance 




Current Perpendicular to Plane GMR 

I 

I 

I 

V 

Much more difficult to measure, 

Either on macroscopic samples (0.1mm diameter) with superconduct<strong>in</strong>g leads 

(R~ . thickness / area ~ 10 ) 

or on patterned microscopic pillars of area

Serial resistance model for CPP-GMR 

Without sp<strong>in</strong>-filp, serial resistance network can be used for CPP transport 

CPP transport through F/NM/F sandwich described by: 

(a) Parallel magnetic configuration : 

 

Ft F 

 

AR 

F / NM 

NM 

t NM 

 

AR 

F / NM 

 

Ft 

F 

 

t F F 

 

AR 

F / NM 

NM 

t NM 

 

AR 

F / NM 

 

t F 

F 

(b) Antiparallel magnetic configuration : 

 

t F F 

 

AR 

F / NM 

NM 

t NM 

 

AR 

F / NM 

 

t F 

F 

 

Ft F 

 

AR 

F / NM 

NM 

t NM 

 

AR 

F / NM 

 

Ft 

F 


Parallel resistance model for CIP-GMR: 

CIP GMR of (Co t Co /Cu t Cu ) 2 multilayers 

Serial resistance model for CPP-GMR: 

CPP GMR of (Co t Co 

/Cu t Cu 

) 2 

multilayers 

Parallel magnetic configuration : 

Parallel configuration : 

Antiparallel configuration : 

 

 

t Co 

/ Cot Co 

Cu 

Cu t Cu 

/ t Cu 

 

 

t Co 

/ 

Co 

t Co 

Cu t 

Cu Cu / t Cu 

 

 

Co t Co 

/ tCo 

Co 

 

Cut Cu / t Cu 

 

 

t Co 

/ Co t Co 

Cu t Cu / 

Cu 

t Cu 

CIP GMR cannot be described 

by a parallel resistance network only 

(no change of R between parallel and 

antiparallel magnetic configuration) 

Co t Co 

Co t Co 

Cu t Cu 

Sp<strong>in</strong> up channel 

Sp<strong>in</strong> down channel 

Antiparallel magnetic configuration : 

Co t Co 

Co t Co 

Co t Co 

Cu t Cu 

Cu t Cu Co t Co 

Cu t Cu 

Cu t Cu 

Sp<strong>in</strong> up channel 

Sp<strong>in</strong> down channel 

Cu t Cu 

Co t Co 

Co t Co 

Cu t Cu 

Cu t Cu 

CPP GMR can be « qualitatively » described 

by a serial resistance network 

B.Dieny « <strong>Models</strong> (without <strong>in</strong> sp<strong>in</strong>tronics sp<strong>in</strong>-flip, i.e. » no 2009 mix<strong>in</strong>g <strong>European</strong> between <strong>School</strong> up on and Magnetism, down channels) Timisoara

Sp<strong>in</strong> accumulation – sp<strong>in</strong> relaxation <strong>in</strong> CPP geometry 

F1 

F2 

In F1: 

In F2: 

e flow 

J 

1 

2 

J 

1 

2 

J 

J 

l SF 

l SF 

e flow 

Valet and Fert 

theory of CPP-GMR 

(Phys.Rev.B48, 

7099(1993)) 

z 

Different scatter<strong>in</strong>g rates for sp<strong>in</strong> and sp<strong>in</strong> electrons 

different sp<strong>in</strong> and sp<strong>in</strong> currents. 

Larger scatter<strong>in</strong>g rates for sp<strong>in</strong> : J >>J far from the <strong>in</strong>terface. 

Larger scatter<strong>in</strong>g rates for sp<strong>in</strong> : J 

>>J 

far from the <strong>in</strong>terface. 

Majority of <strong>in</strong>com<strong>in</strong>g sp<strong>in</strong> electrons, majority of outgo<strong>in</strong>g sp<strong>in</strong> electrons 

Build<strong>in</strong>g up of a sp<strong>in</strong>accumulation around the <strong>in</strong>terface balanced <strong>in</strong> steady 

state by sp<strong>in</strong>-relaxation 


Start<strong>in</strong>g po<strong>in</strong>t : Valet and Fert theory of CPP-GMR (Phys.Rev.B48, 7099(1993)) 

 

: sp<strong>in</strong>-dependent chemical potential 

Sp<strong>in</strong>-relaxation at F/F <strong>in</strong>terface: 

In homogeneous material, = F 

-e 

J 

1 

2 

J 

Sp<strong>in</strong>-dependent current driven by 

J 

 

 

1 

e 

 

 

 

z 

 

J 

1 

2 

J 

l SF 

l SF 

Generalization of Ohm law 

z 

Sp<strong>in</strong> relaxation : 

e 

F * 

l 

F 

SF 

J 

e 

 

J 

z 

 

 

 

 

 

2 

2l 

SF 

 

 

l SF 

=sp<strong>in</strong>-diffusion length (~5nm <strong>in</strong> NiFe, ~20nm <strong>in</strong> Co)) 

l SF 

l SF 


z

Interfacial boundary conditions 

 

( 

) 

( 

) 

( 

) 

z 

 

z 

r J z 

 

( 

) 

i1 i1 

i i1 

i1 

i i1 

(Ohm law at <strong>in</strong>terfaces) 

J 

 

( 

) 

z J z 

 

( 

) 

i1 i1 

 

i i1 

(if no <strong>in</strong>terfacial sp<strong>in</strong>-flip is considered) 

Note: Interfacial sp<strong>in</strong> memory loss can be <strong>in</strong>troduced by : 

J 

 

( 

) 

z J z 

 

( 

) 

i1 i1 

 

i i1 

30% memory loss as at Co/Cu <strong>in</strong>terface yields =0.7 


Input microscopic transport parameters to describe macroscopic CPP properties : 

With<strong>in</strong> each layer : 

-<strong>The</strong> measured resistivity . 

-<strong>The</strong> scatter<strong>in</strong>g asymmetry . 

-<strong>The</strong> sp<strong>in</strong> diffusion length l sf 

. 

At each <strong>in</strong>terface : 

 

 

 

 

2 

* 1 

( ) 

 

measured 

 

 

 

 

 

 

 

 

 

 

 

 

* 1 

 

2 

 

-<strong>The</strong> measured <strong>in</strong>terfacial area*resistance product 

r measured 

-<strong>The</strong> <strong>in</strong>terfacial scatter<strong>in</strong>g asymmetry . 

r 

r 

 

2r 

* 1 

( ) 

 

 

measured 

 

r 

 

r 

 

r 

 

 

r 

 

r * 

 

1 

2 

 


Material 

Cu 

Au 

Measured 

resistivity 4K/ 

300K 

0.5-0.7.cm 

3-5 

2.cm 

8 

Ni 80 

Fe 20 

10-15 

22-25 

Ni 66 

Fe 13 

Co 21 

9-13 

20-23 

Co 4.1-6.45 

12-16 

Co 90 

Fe 10 

6-9 

13-18 

Co 50 

Fe 50 

7-10 

15-20 

Pt 50 

Mn 50 

160 

180 

Examples of bulk parameters 

Ru 9.5-11 

14-20 

 

Bulk 


asymmetry 

0 

0 

0 

0 

0.73-0.76 

0.70 

0.82 

0.75 

0.27 – 0.38 

0.22-0.35 

0.6 

0.55 

0.6 

0.62 

0 

0 

0 

0 


l SF 

500nm 

50-200nm 

35nm 

25nm 

5.5 

4.5 

5.5 

4.5 

60 

25 

55 

20 

50 

15 

1 

1 

14 

12

Examples of <strong>in</strong>terfacial parameters 

Material 

Measured R.A 

<strong>in</strong>terfacial 

resistance 

Co/Cu 0.21m.m 2 

0.21-0.6 

Co 90 

Fe 10 

/Cu 0.25-0.7 

0.25-0.7 

Co 50 

Fe 50 

/Cu 0.45-1 

0.45-1 

NiFe/Cu 0.255 

0.25 

NiFe/Co 0.04 

0.04 

Co/Ru 0.48 

0.4 

Co/Ag 0.16 

0.16 

 

Interfacial 


assymetry 

0.77 

0.7 

0.77 

0.7 

0.77 

0.7 

0.7 

0.63 

0.7 

0.7 

-0.2 

-0.2 

0.85 

0.80 


Sp<strong>in</strong>-dependent currents : 

e 

J 

z 

 

2l 

SF 

Sp<strong>in</strong> relaxation : 

2 

 

Solution with<strong>in</strong> each layer: 

 

 

z 

 

Aexp 

B exp 

 

l 

Diffusion equation of sp<strong>in</strong>-accumulation 

z 

sf 

l sf 

J 

 

 

 

 

 

 

 

e 

 

1 

 

 

z 

 

Diffusion equation diffusion for sp<strong>in</strong> accumulation 

 

 

( generalized Ohm law <strong>in</strong> the bulk of the layer) 

2 

 

 

2 

z 

A, B <strong>in</strong>tegration constants determ<strong>in</strong>ed from <strong>in</strong>terfacial boundary conditions 

2 

l SF 

J 

 

 

1 

 

1 

 

 

grad 

 

 

e z 

 

Drift due to 

electrical field 

Diffusion due 

to local sp<strong>in</strong> 

accumulation 


F<strong>in</strong>al expression of CPP resistance for any multilayered structures (N layers) : 

 

AA 

i 

 

AB 

i 

R 

CPP 

Where : 

 

 

 

1 

 

 

 

i 

1 

 

N 

 

i1 

* 

i 

1 li1 

* 

i 

li 

* 

1 li1 

* 

i 

li 

2 * * 

1 

 

d r 1 

 

1 

 

A 

 

B 

 

 

i 

i 

i 

 

i 

1 

 

J 

 

i 

A 

i 

 

 

 

i 

 

r 

 

 

i i 1 

i 

1 

1 

e 

i 

A 

i 

i 

 

l 

1 

B 

J 

i 

i 

i 

 

 

 

 

e 

 

d 

l 

 

 

 

i i i li 

1 

1 

e e 

i 

AA AB 

1 

i 

i JJˆ 

ˆ 

 

i 

B 

 

 

i 

i 

with : ˆ 

 

i 

BA BB 

1 

i 

i 

i 

 

di 

r 

i li 

e 

* 

i l 

i 

di 

r 

i li 

e 

* 

 

AJ 

 

 

 

i l 

i 

i 

and Jˆ 

i 

 

BJ 

with : 

di 

 

r 

i 

i li 

e 

* 

i l 

i 

 

BJ 1 * 

* 

 

d 

i r i i li 

i 

i 

i1 

1 

1 

i 

i 

 

r 

2 

i li 

e 

AJ 1 * 

* 

* 


i l 

i 

r i i li 

i i 

i 

i 

i1 

1 

1 

 

i 

i 

 

 

r 

1 

* 

BA 

i 

li1 

i 

1 1 

 

* 

i 

li 

 

 

* 

BB 

i1li1 

i 

1 

 

* 

i 

li 

 

2 

 

l 

i 

i 

 

 

 

d 

 

 

i

Comparison of NiFe and FeCo free layer : <strong>in</strong>fluence of l SF 

Free layer : 

NiFe/FeCo 

or FeCo only 

.cm m.m² nm nm 

F 

t 

4 

Ni 80 

Fe 20 

l SF NiFe 

=4.5nm 

Smooth maximum <strong>in</strong> CPP GMR versus t F 

R/R (%) 

3 

2 

Fe 50 

Co 50 

l SF FeCo 

=15nm 

Phenomenologically, 

R 1-exp(-tF/lSF F) 

and 

R/R [1-exp(-tF/lSF F]/(tF+cst) 

1 

(c) 

Data from Headway Technologies 

0 

0 5 10 15 20 25 30 


Free layer thickness (nm)

Comparison of Cu and Au spacer layers : <strong>in</strong>fluence of l SF 

3 

2.5 

R/R (%) 

2 

1.5 

1 

Cu 

l SF Cu 

=50nm 

0.5 Au 

l =10nm 

SF Au 

0 

0 10 20 30 40 50 

Spacer layer thickness (nm) 

Data from Headway 

Technologies 

Phenomenologically, decrease of R as exp(-tspacer/lSF spacer) 

and 

R/R as exp(-tspacer/lSF spacer)/(tspacer+cst) 


Generalization of sp<strong>in</strong>-dependent transport to any geometry (col<strong>in</strong>ear magnetization) 

Inhomogeneous current distributions <strong>in</strong> numerous experimental situations… 

Current conf<strong>in</strong>ed paths (CCP) GMR 

Current crowd<strong>in</strong>g effects <strong>in</strong> low RA MTJ 

K.Nagasaka et al, JAP89 (2001), 6943 

H.Fukazawa et al, IEEE Trans.Mag.40 (2004), 2236 

See for <strong>in</strong>stance: J.Chen et al, 

JAP91(2002), 8783. 

Sp<strong>in</strong> transfer oscillators <strong>in</strong> po<strong>in</strong>t-contact geometry 

Metallic CPP heads 

Shield 1 

J 

Shield 2 

S.Kaka et al, Nat.Lett.437, 389 (2005) 

F.B.Mancoff et al, Nat.Lett.437, 393 (2005) 

20nm 

… almost all models of sp<strong>in</strong>-dependent transport assume uniform current 


Extend<strong>in</strong>g Valet-fert theory at 3D <strong>in</strong> col<strong>in</strong>ear geometry 

J 

J 

 

 

 

 

 

 

 

 

 

 

grad 

 

grad 

 

1 

 

Out-of-equilibrium magnetization 

 

e z 

m 

 

 

1 

 

B 

 

e z 

energy 

DOS 

Electrical current: 

J 

el 

 

J 

 

 

J 

 

 

 

 

 

 

 

grad 

 

With 1 1 

 

 

 

 

 

 

e 

 

B 

 

 

m 

z 

J 

el 

2 

grad 

 

2 

m 

e 

z 

3D expression of electron current: 

B 

 

J 

e 

 

2 

 

 

2 

 

m 

e 

 

B 


Sp<strong>in</strong> current: 

J 

J 

J 

s 

s 

m 

1 

 

e 

 

 

 

 

 

 

 

1 

 

 

e 

 

J 

J 

 

 

 

 

2 

 

grad 

 

e 

2 

e 

B 

 

grad 

 

 

2 

m 

2 

e z 

B 

2 

m 

2 

e z 

 

 

grad 

 

 

 

 

: Sp<strong>in</strong> current 

 

e 

 

: Moment current 

B 

 

 

m 

z 

 

 

 

3D expression of moment current: 

 

J 

m 

 

 

 

2 

e 

B 

 

 

 

 

2 

 

m 

2 

e 

In col<strong>in</strong>ear magnetization J m 

is a vector 


J 

 

J 

e 

m 

 

2 

 

 

 

 

 

2 

e 

B 

2 

 

m 

eB 

2 

 

 

m 

2 

e 

 

m 

2 Unknowns: z 

Transport equations: 

divJ e 

div 

J m 

0 

 

 

2 

l 

2 2 

sf 

derived from Valet/Fert: 

1 

m 

0 

e 

 

J 

z 

 

 

 

 

2 

2l 

SF 

2 Equations: 1 diffusion of e + 1 diffusion of m 

 

 

Diffusion of charge (with 

conservation of charge) 

Diffusion of sp<strong>in</strong> (without 

sp<strong>in</strong> conservation due to 

sp<strong>in</strong>-torque and sp<strong>in</strong>relaxation 

lSF=sp<strong>in</strong>-diffusion length 

Can be solved <strong>in</strong> complex geometry with a f<strong>in</strong>ite element solver 


<strong>in</strong> 

Cu Cu 

out 

2D CPP pillar with extended electrodes 

Mapp<strong>in</strong>g of charge current 

+2.03·10 -5 

200nm 

Charge current amplitude 

J e 

Co3/Cu2/Co3nm 

10 times higher current density at 

corners than <strong>in</strong> center of CPP- 

GMR stack 

20nm 


2D CPP pillar with extended electrodes 

z 

y 

x 

Mapp<strong>in</strong>g of y-sp<strong>in</strong> current 

( J m,yx 

, J m,yy 

) 

Cu Co/Cu/Co Cu Cu Co/Cu/Co Cu 

Very large excess of e 

Large 

excess 

of e 

Large 

excess 

of e 

Large <strong>in</strong>plane 

sp<strong>in</strong>current 

+2.14·10 -4 

m y 

Parallel 

Weak excess of e 

Antiparallel 

-2.14·10 -4 


Conclusion on CPP transport 

-Serial resistance model can be used at lowest order of approximation to 

describe CPP transport. However, does not take <strong>in</strong>to account sp<strong>in</strong>-flip. 

-With sp<strong>in</strong>-flip, sp<strong>in</strong> accumulation and sp<strong>in</strong>-relaxation play an important 

role <strong>in</strong> CPP transport. 

-Semi-classical theory of transport describes CPP transport fairly well. 

CPP macroscopic transport properties (R, R/R) can be calculated from 

microscopic transport parameters ( 

, l SF 

, r 

) 

-In complex geometry, charge and sp<strong>in</strong> current can have very different 

behavior. 

Two contributions to j 

: drift along electrical field and diffusion along 

gradient of sp<strong>in</strong> accumulation. 



Signal/noise ration <strong>in</strong> CIP MR heads 

Signal : 

V R. 

I 

Noise : 

V 

Johnson 

4k 

B 

TRf 

Signal/Noise : 

SNR 

 

V 

V 

Johnson 

R 

 

R 

 

 

 

power 

4k 

B 

Tf 

To maximize the SNR <strong>in</strong> MR heads, the power dissipated <strong>in</strong> the head 

must be as large as possible compatible with reasonable heat<strong>in</strong>g and 

electromigration 

~4.10 7 A/cm² used <strong>in</strong> heads with CIP-GMR~15%

Models in Spintronics - The European School on Magnetism

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