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Magnetoresistance – Giant MagnetoResistance (GMR) and Tunnelling MagnetoResistance (TMR) (http://www.almaden.ibm.com/st/ gives more information) Reading heads in magnetic data storage media Multibillion dollar industry!

Future applications include spintronics ...

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Fundamentals of ferromagnets Pair-interaction between atomic magnetic moments ijJ j

jiiijex ssJE ⋅∑−=

,2 , Jij [J]

with the magnetic moments given by the relation iBi sgm µ= . Short-range interaction, often enough to consider nearest-neighbour (nn) interactions. For ferromagnetic 3d elements, the magnitude of the pair-interaction can be estimated from mean-field theory (z = number of nn's,

if nn-interaction, otherwise) JJ ij = 0=ijJ

2zJTk cB ≈ if T , this implies an energy ~ 0.1 eV K 1000≈c

The Density Of States (DOS), N (E), in a ferromagnet is split into majority and minority bands due to the exchange interaction.

N (E) majority

4s

3d ( )EN↑

( )EN

EF

E

s- and d-eelectrons imass) than

minority ↓

lectrons contribute to electrical conduction. The mobility of 3d s smaller (flat energy bands ⇒ low velocity/high effective for 4s electrons.

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Since -electrons have more empty states to scatter to, the resistivity will be higher for these electrons; two independent conduction electron channels.

The resistivity of -electrons (majority electrons) will be ss→↑ ≈ ρρ , while the resistivity of -electrons (minority electrons) can be written as

dsss →→↓ += ρρρ , where ssds →→ > ρρ . The essence of this is that in a ferromagnetic metal there exists two current channels, one that can conduct a current better than the other.

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Giant MagnetoResistance (GMR ) Spin dependent scattering, two current channels, one for majority spins (often low resistivity) ↑ρ and one for minority spins (often high resistivity) ↓ρ Systems/geometries displaying GMR A Multilayers CIP (Current In Plane) multilayers

H

Me

Me

FM

FM

FM

E , J CPP (Current Perpendicular Plane) multilayers

Me

Me

FM

FM

FM

E , J

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(FM tFM / Me tMe) n multilayers FM = Fe, Ni, Co or some 3d alloy, Me = Cu, Ag, V, Cr, tFM (Me) = layer thickness ≈ a few monolayers

CIP geometry

0.5

0.6

0.7

0.8

-40 -30 -20 -10 0 10 20 30 40 H (kG)

-

R / R(H=0)

(Fe 30Å / Cr 9Å)60

(Fe 30Å / Cr 12Å)35

(Fe 30Å / Cr 18Å)35

Hs

High resistance for antiferromagnetically (AF) coupled layers, low resistance for ferromagnetically (F) aligned layers. Hs corresponds to the field at which all layer magnetizations point along the field direction.

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( ) RHRR −∆

s

s

RR=

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Magnetic coupling Phenomenologically, the coupling between the magnetizations iM in nearest-neighbor FM layers can be expressed as

1

11

+

+⋅−=i

i

i

iex M

MMM

JAE

where J1 is given in units of [J/m2]. Adding a Zeeman term, the energy for two FM layers is

θ

M1

M2

FMt

H

Minimiz

The mic(RudermRev. B 4FM1 pothe Me stwo loca

( )

( )

( ) ( )θµθ

θπµθπ

µ

sin 22 cos

2 cos22 cos

01

01

21021

211

sFM

sFM

FM

HMtJ

HMtJ

MMHMMMM

tJ

VE

−=

=

−−−−=

=+⋅−⋅

−=

ing with respect to θ and using ( )θsinsMM = one obtains

2or

2 ;

20

10

1s

1

20 FMss

FMs

FMs tHMJ

tMJ

HJ

tHMM

µµ

µ−=−=−=

roscopic origin of the AF coupling can be explained using the RKKY an-Kittel-Kasuya-Yosida) model (P. Bruno and C. Chappert, Phys. 6, 261 (1992)), indirect type of interaction between two FM layers,

larizes the conduction electrons and the polarization propagates across pacer layer and interacts with FM2. The RKKY interaction between lized spins is

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( ) jiijij SSRJH ⋅−= where the exchange integral is

( ) ( ) ( )∫ ⋅ ijij RqiqqdRJ exp ~ 3 χ where ( )qχ is the q -component of the carrier spin susceptibility of the Me spacer layer ( q = scattering vector, connects two Fermi surface points).

For a free-electron gas, the RKKY exchange integral for two atomic magnetic moments, embedded in a non-magnetic host and being a distance R apart is

( ) ( )RkR

RJ F2cos13

∝

The interaction decays as 3R1 and oscillates with the period Fkπλ = . In a superlattice, the interlayer coupling is obtained by summing over all magnetic pairs ij (i and j referring to FM1 and FM2, respectively, the coupling energy per unit area for any magnetic moment in FM1 is

( ) ( ) ( )12FM2

121 coscos θθ ∑−∝−=∈j

ojex RJJAE ,

where 12θ is the angle between the magnetizations in FM1 and FM2. Summing all pair interactions one obtains

( ) ( )zkz

zJ F2sin121 ∝

Here the coupling strength decays with thickness of the Me layer as 1/z2.

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Mechanism of GMR Two current channels with different resistivities, the difference is mainly explained by the electronic structure and a difference in the density of states for the majority and minority conduction electrons. In addition, we need to consider scattering centers, here we distinguish between bulk scattering and interface (FM/Me) scattering.

↑

↓

↓ FM elements

e-

b

ulk

Scattering cinterdiffusibottom of thwhile scatte An importaGMR mateand at RT lscale is the "sample" dihomogeneo

↑

↓

FMal

enters at interfaces may be suon, spin dependent band offsee conduction band and the Fering in the bulk of a layer is d

nt length scale in GMR materrials it is important that l >>

Å. For the CIPmean free path l , since thefferent FM layers. If > laus limit, "alloy limit", while i

sdl

mfp

43 1010 −≈sdl

mfp

l

↑

interfac

loys

assumed 1>= ↑↓ RR

rfact (dirmiue t

ials th ge coyer f lm

α

e-

e roughness, regions of fference in energy between the energy in adjacent layers), etc., o impurity atoms.

is the spin diffusion length l . In e layer thickness. In 3d elements ometry, another important length nduction electrons should be able to thickness we are in the

< layer thickness we are in the

sdl

fp

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local limit. For 3d elements, ~ 50 - 300 Å, while a permalloy layer, l for minority electrons is 10 - 20 Å.

mfpl mfp

MeR2

R

In the homogeneous limit, the resistance of the different current channels can be described using simple resistor models. Similar results are expected for the CIP and CPP geometries in this limit. AF configuration

( ) ( )MeRRRRRRR 2 and ++=++= ↓↑−↑↓+ where R+ (R-) is the resistance for electrons with S=+1/2 (S=-1/2), while

are the resistances for the two conduction channels. ↑↓ RR and F configuration

RR 2=+

R

( M

↑

↓R

R2+↑

↑

↓R2

↓

2↑↓ +

=RR

R AF

R

)e

↑

(RR 2 and += ↓−

)MeR2

R2

↑↓

↑↓

+=

RRRR

RF2

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The magnetoresistance thus is

( )( )

( )( )2

2

2

2

11

+

−=

+

−=

∆

↑↓

↑↓

αα

RR

RRR

R

AF (*)

In the local limit, there is a difference between CIP and CPP geometries. For the CIP geometry, all layers will conduct in parallel and independently and there will be no difference between the AF and F configurations; i.e. there is no GMR effect. For the CPP geometry, we instead have a similar situation as for the homogeneous limit and a magnetoresistance given by (*). B Sandwich structures In the absence of AF coupling, there are (at least) two possible approaches to obtain different relative orientation of the magnetization in successive FM layers: - Use two ferromagnetic materials exhibiting different coercivities, either as

building block in a multilayer or as part of a sandwich structure. Si / 100 Å Ta / 40 Å Ni80Fe20 / 60 Å Cu / 40 Å Ni80Co20 / 50 Å Ta.

-0.2

-0.1

0

0.1

0.23

-20 -10 0 10 20

M [a

rb. u

nits

]

H (Oe

[ ]% RR∆

The coercivity of NiFe is 2 Oe, while NiCo has a coercivit

0

1

2 ²R/R

[%]

)

y of 12 Oe.

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- Use two FM layers separated by a Me layer in a sandwhich structure, one FM layer will be constrained by coupling to a adjacent antiferromagnetic layer (exchange anisotropy).

Si / 50 Å / Ta 2 x (60 Å Ni80Fe20 / 22 Å Cu / 40 Å Ni80Fe20 / 70 Å FeMn) / 50 Å Ta).

-2

-1

0

1

2

0

1

2

3

4

-200 0 200 400 600

m [1

0-3 e

mu]

H (Oe)

[ ]% RR∆

Multilayered or sandwich structures in applications? Multilayer

%/Oe 01.01

kOe 10

%100≈

∆

≈

≈∆

sss

s RR

HHR

R

Sandwich

%/Oe 12.01

Oe 105

%52−≈

∆

−≈

−≈∆

sss

s RR

HHR

R

For a summary of experimental results, see eg. B. Dieny (J. Magn. Magn. M359 (1994)).

²R/R

[%]

ater. 136, 335-

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Tunnelling MagnetoResistance (TMR )

Quantum mechanics dictates that an electron in a metallic electrode has a certain probability to tunnel through an (insulating) potential barrier to another metallic electrode. Important parameters – thickness of barrier (d), hight of potential barrier (V0) and density of states (DOS) in the metallic electrodes ( ( )FN ε ) In ferromagnets like Fe, Ni and Co, or alloys of these, the DOS for spin-up and spin-down 3d electrons are exchange-split

parallel alignment

spin polarized transport ⇒ Important parameter – spin polarization ( ( )FENN ↑↑ = )

↓↑

↓↑

+

−=

NNNN

P

antiparallel alignment

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Jullière´s model

- conservation of spin - the conductance is proportional

to products if the type 21 ↑↑ NN

2

1

i u Relation between conductance and resistivity changes ( is the conductance in case of parallel magnetizations in the two FM electrodes)

↑↑G

1

11

↑↑↑↑

↑↑↑↓

↑↓

↑↓↑↑

↑↓

↑↓↑↑

↑↓

∆=

−=

−

=−

=∆

RR

RRR

R

RRG

GGG

G

Conductance when the magnetization in the two FM electrodes are parallel

22↓↑↓↓↑↑↑↑ +=+∝ NNNNNNG

and the correponding result when the magnetizations are in opposite directions

↓↑↑↓ ∝ NNG 2 Using the definition of spin polarization, we obtain

( )( )

( )( )22

22

121

and 121

↓↑↑↓

↓↑↑↑

+−∝

++∝

NNPG

NNPG

and

11

2

1

2

2

2

2

2

≤+

=∆

−=

∆=

−

↑↓

↑↑↑↓

↑↓↑↑

PP

RR

PP

RR

GGG

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Experimental geometry

U

I

Typical dimensions

FM electrodes 0.1 x 0.1 mm2, for future applications smaller, thickness 100-200 Å

Oxide tunnel barrier 10-20 Å thick, barrier 2-3 eV Materials FM electrodes Co, Fe, Ni, NiFe, CoFe Oxide tunnel barrier NiO, CoO, MgO, Al2O3

Junction resistance from < 100 Ω to tens of kΩ Important

Sharp interfaces without interdiffusion, minimum of spin-flip scattering at interfaces

Measured spin polarizations (Meservey and Tedrow, Phys. Rep. 238, 173 (1994)) Material Ni Co Fe NiFe CoFe Polarization +23% +35% +40% +32% ~50%

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R

H-H3 -H2 H1

H-H3 -H2 H1

M

1

↑↑∆ RR -results for “ 20.2%

RT Ferromagnetic materi

-H3 -H2 H

good” Co/Al2O3/NiFe junctions

27.1% 27.3% 77K 4.2K

als with P ~ 1 would give even better results.

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Use of magnetic tunnel junctions