Giant Magnetoresistance

N.H.Mott, Proc. Roy. Soc. A

153

, 699 (1936)

Mott Two-Current Model (2)

The two current model Assume that scattering probabilities can be added

     

ss

       

sd

      

Conductivities satisfy

  ,      

n s e

2 

s

 , 

m

*

s

   1

Small portion of foreign atoms -> not only the availability of states is relevant, but the scattering potential of inclusions in host Spin-polarized densities of states for the elemental metals Fe, hcp-Co, Ni and Cu

Mott Two-Current Model (3)

Scattering potentials are different for majority and minority condiction electrons Schematic representation of the matching of the d bands of the magnetic elements in the middle (b) and elements in columns at the left (a) and right (c) of it in the periodic table d-bands of elements at the left from the host in periodic table resemble the minority d bands, majority – different



majorities are scattered more strongly

  

1

to the right

  

1

               1  1 2   

Applies not very high temperatures

Interlayer Coupling

Cu Layer Thickness (nm) Different types of coupling in a layered magnetic structure

 

the interlayer thickness x of glass/Fe(6nm)/[Co(1nm)/Cu x A]50 superlattices In 1986 was identified and characterized in Fe/Cr superlattice structures and rare earth yttrium multilayers Transport of spin along the interfaces results in torque acting on the magnetization which is due to the fact that majority and minority electrons have different reflection coefficients at the interfaces. The torque alignes the magnetization according to the associated ratio of reflection coefficients

P.Grünberg, R.Schreiber, Y.Pang, M.B.Brodsky and H.Sowers, Phys. Rev.Lett.

57

, 2442 (1986)

First Evidence of GMR

Resistivity versus applied field for Fe/Cr multilayers Relative resistance change as a function of the external magnetic field for Fe/Cr/Fe and 250A thick Fe film Discovered in 1988 in antiferromagnetically coupled magnetic multilayers by Baibich

et al

and on Fe/Cr superlattices In Fe/Cr multilayers the low field antiparallel configuration was induced by antiferromagnetic coupling between Fe layers across Cr Does not depend on the angle between the current and magnetization



resulting anisotropy play a minor role MR=79% at T=4.2K and 20% at room temperature

MR



R AP



R P R P

  

spin-orbit coupling and

P AP

 1 M.Baibich, J.Brote, A.Fert, F.Nguyen Van Dau, F.Petroff, P.Etienne, G.Greuzet, A.Friederich and J.Chazelas, Phys.Rev.Lett

61

, 2472 (1988)

Qualitative Picture of GMR

Were made with electrical current parallel to the plane of the layers – CIP geometry In CIP geometry GMR arises when layer averaged electron mean-free path for at least one spin direction is larger than the multilayer period Qualitative expression for MR if the mean free path for both spin directions is much larger than the multilayer period



P



AP

   1        1      4   1

MR

  1   4   2

In magnetic layered structures - dependence of the resistance on the angle between the magnetization directions of successive magnetic layers Phenomenological expression

R

  

  0



 

R GMR



1  cos 



2

Spin-dependent electron scattering for parallel and antiparallel alignment of magnetic films

CPP Geometry

For CPP the length scale is not determined by the mean free paths of diffusive scattering, but is given by the spin diffusion length



sd



the multilayer repetition period



MR can be analyzed with above simple approach In CPP MR is larger than in CIP Schematic representation of the array of nanowires in an insulating polymer matrix

Tunnel MR

Obtained in tunnel junctions: two ferromagnetic layers are separated by thin insulating layer (barrier) Approximation: Spin is conserved in the tunneling process Tunnel resistance is different in Parallel and Antiparallel configuration of electrodes

TMR



R AP



R P R P

TMR of Co/AL2O3/Permalloy (coersive fields of electrodes are different) Transition metall electrodes TMR 65% at T=4.2K, 40% at room temperature Half-metallic ferromagnets (mixed valence Mn oxides) TMR more 400% at T= 4.2K

Colossal MR

Found in mixed valence Mn oxides, I.e. La 1 x Sr x MnO 3 Conduction is by hopping electrons between Mn3+ and Mn4+ sites, magnetic moments must be parallel !



ferromagnetic state is needed At Tc transition from metal to insulator



maximum of resistivity At T>Tc increase of thermally exited



decrease fo resistivity carriers Applied magnetic field increase ferromagnetic ordering



decrease resistivity Large fields (several Tesla) are needed Top:Magnetization against temperature for La 0.75

Ca 0.25

MnO 3 for various field values Middle: resisitivity against temperature Bottom: magnetoresistance against temperature

Semiclassical Model of Conductivity

Electrons are essentially regarded as point like particles (“classical”), but consequences of quantum mechanics are taken into account (“semi”)



n

 

r



k



In presence of electric and magnetic fields coordinate wave vector and band are changing according to rules: 1 band number is integral of motion -> no interband jumps 2 with a given n

r

  

v n

 1    

n k

   

k

  

e

 

E

 1

c



v n

 

H

 

 

Gives rules how in absence of collisions coordinate and wave vector are changing when external electrical and magnetic fields are applied. Gives relation between known band structure and kinetic characteristics Quasi impulse – determined only by externally applied fields, not by periodic lattice field

Solid State Physics, N.Ashcroft and N.Mermin

Boltzmann Equation

How to find distribution function if that for previous infinitely close moment of time is known

f

(

r

 ,

k

 ,

t

) 

f

(

r

 

t v

 ,

k

  

F dt

 ,

t



dt

)

Liouville theorem

f

(

r

 ,

k

 ,

t

) 

f

  (

r

      

f



f t



v r

 , ,

k

 , 

t k



r

 

t

      

out in



F dt dt dt

  ,

t



dt

) 

Collisionless movement

 

collisions

 

only because of collisions Leaving only linear dt terms in limit dt



0

 

f t



v

  

f r

 

F

1   

g k

  

f t coll

Drift Collisions If collision term is in relaxation time approximation -> linear differential equation

 

f

   

t



f coll

0 ( 

k

)  |

-

f

(

k

 )    (

k

)

f

0 (

k

 ) |

local equilibrium distribution function

Steady State Boltzmann Equation

g f

(

k

  (

k

) )  

f

0

f

0 (

k

 (

k

)  ) 

g

(

k

 )

f k

 0  exp  

E k

  1

E F



k B T

  1

Assuming no magnetic field , time independent electric field and temperature gradient in steady state

g k



e

 

f k

 0 

E k

  

v k

 

E k

 

j

   2

e V



k

 

v k



g k



j

  2

e

2

V

 

k

  

f k

 0 

E k



v



k

  

k



j

   ˆ 

E

  ˆ  2

e

2

V

  

k

 

E



k



E F

 

v



k



v



k

 ˆ  2 2

e

 

3  

dS k

 

v k

 

v k

 

v k



For metals the changes in distribution function can be restricted to energies close to Fermi surface because for metals df o /dE is sharply peaked around the Fermi energy

Semiclassical Camley-Barnas Theory

Hark back to the Fuchs-Sondheimer theory



metallic plane-parallel slab is considered as free electron gas with electrons scattering at outer boundaries

f

(

k

 ) 

f

0 (

k

 ) 

g

(

k

 )

Where f 0 is Fermi distribution function



v



g

 

 

r v

  ,  

e E

 

v



f

0 

E

  

g

 

v

  ,

r



j

   

e m

 3 

d

3

v



v g



g

 

g

  

e

 

E x v x



f



E

0    1 

A

  exp  

a



v z z

     

e

 

E x v x



f



E

0    1 

A

  exp  

a



v z z

    

CB model involves spin dependent probability for specular reflection at the outer boundaries.

At the interfaces three cases are distinguished: transmission with probability T i,s Specular reflection with probability R i,s Diffuse scattering with probability D i,s Probabilities depend on interface I and spin s, T+R+D=1 Systems with different magnetization directions in the different layers are dealt with spin-transmittion coefficients The coefficients A are determined from the boundary conditions for each layer and spin direction. From resulting g function the z-dependent current can be calculated



contribution to the conductivity from each spin follows

K.Fuchs, Proc.Roy.Soc.

34

, 100 (1938) E.H.Sondheimer, Advan.Phys.

1

,1 (1952) R.E.Camley and J,Barnas, Phys.Rev.Lett.

63

, 664 (1989)

Intrinsic GMR (1)

 ˆ   ˆ    ˆ 

Mott two current model

 ˆ 

e

2

V

  

k

 

E



k



E F



v

 

k

 

v



k



AP

     

ratio is determined only by electronic structure

GMR

 

k

 

E k

  2 

k E F

 

v k i

 2 

E k AP

  

k



E F

 

E k



v k i AP

2 

E F



v k i

 2  1

Density of states at the Fermi surface Averaged over Fermi surface

GMR



N



 

F

2

N v k i

 2 

AP

 

F N



 

F v k i AP

2

v k i

 2  1

GMR is completely determined by the Fermi velocities and Fermi surface as function of the magnetization configuration



pure band-structure effect

Details of calculation: P. Zahn, I. Mertig, M.Richter and H.Eschrig, Phys. Rev. Lett.

75

, 2996 (1995)

Intrinsic GMR (2)

In Co/Cu multilayered structure in parallel configuration: Differences in potentials are minimal for majority electrons, as Co and Cu d-bands are very similar and there is no large differences in potentials For minority electrons there are potentials steps while transition between Co Cu layers, as d bands are very different On boundary electrons will be scattered



Fermi surface for minority carriers is lower



lower conductivity



conductivity is determined by majority channel of fast electrons In antiparallel configuration both channels are degenerated



decrease of mean Fermi speed



decrease of conductivity

Extrinsic GMR (1)

Any defect diffusively scatters Bloch waves



additional resistivity of the system Defects: foreign atoms, clusters, boundary roughness etc.

Scattering process is described by scattering T-matrix:

T k k

  1

V



d

3

r



k

 

V



r



k



r



Microscopic probability of transition is equal according to the Fermi golden rule

P k k

  2  

cN T k k

 2 



E k



E k





Proportional to the concentration of scatterers c (deluted case), delta-function – elastic scattering

Extrinsic GMR (2)

P



k k

    

P k



k



P k



k



P k



k



P k



k

 

In ferromagnetic systems transition matrix contains two spin conserving and two spin-flip terms Spin-flip terms are generally small-> consider only spin-conserving part Electron life-time reverse proportional to the sum over probabilities to scatter in any possible state

  

k

 1   

k P



k k



Depends on spin and state. For simplicity take average over states:

   

k

 

k





E k



E k

 

E F E F







k



Depending on the potential perturbations and electronic structute life-times are different for different spin directions



spin anisotropy

     

Extrinsic GMR (3)



P

 

P

  

P

 

P

  

P

 

P

  

AP

 

P



GMR is complex effect determined by interplay of intrinsic and extrinsic properties

Boltzmann Equation and Scattering (1)

Including microscopic probabilities of scattering in Boltzmann equation obtain



f



t k



coll

  

k



f k

   1 

f k

 

P k

 

k



f k

  1 

f k

  

P k k

  

Describes scattering-out from k state Describes scattering-in in k state Principe of microscopic reversibility Quasiclassical equation is

  

k P



k k

   

k



P



k



k

  

v k

   

k P k k

    

k

  

Considering both spin-conserving and spin-flip processes get system of intercoupled equations. Dismissing spin-flip obtain two independent systems for majority and minority carriers

 ˆ 

e

2

V



k

 

E



k



E F

 

v k

  

k

P.Zahn, and I.Mertig, Phys. Rev. Lett.

72

, 2996 (1995) J.Binder, P.Zahn, and I.Mertig, J. Appl. Phys.

87

, 5182 (2000) F.Erler, P.Zahn and I.Mertig, Phys. Rev. B

64

, 944081 (2001)

Boltzmann Equation and Scattering (2)

Solution of Boltzmann equation

 

k

  

k

  

v k

   

k P k k

   

k

  

Is very hard



several approximations are used If one omits scattering-in term in collision term of the equation and take relaxation time constant and spin-independent

  ˆ  2

e V

2  

k

  

E k

 

E F



v



k

 

v



k



This is boundary case of isotropic scattering – all electrons have the same relaxation time Taking spin-dependent isotropic lifetime

 ˆ 

e

2

V

    

k

  

E



k

 

E F



v

  

k



v

  

k

Relaxation time depends on spin and layer

 ˆ 

e V

2   

R L

   

L

 

k

 

E



k

 

E F

 

k

 2

The most general case (when scattering-in term is omitted)

 ˆ 

e

2

V

 

k

  

E



k

 

E F

  

k v

  

k



v

  

k v

  

k



v

  

k

Applications of GMR

Sensors on GMR are very sensitive and can be made very small. For CPP sensors the performance even improved upon miniaturization

Use:

direct field measurements



in disk drives in compute systems, tape heads in consumer products (audio, video), magnetometers, compass systems Position detection



permanent magnetization pattern is attached to the object that has to be detected. A sensor detects the change in the field as a result of object displacement



sensor field modulation



speed, acceleration, force, rotational speed, torque etc.



automobiles (e.g. ABS), robotics, assembly lines Comparison of performance of magnetic-field sensors based on GMR and AMR effects Expected turnover for Europe auto sensors, bil.US-$

Spin Valves

AF coupling is not a necessary prerequisite for the GMR effect !

AP configuration can be obtained in multilayers in which consecutive layers have different coercitivities Another way

 

Combining hard and soft magnetic layers exchange-biased spin-valve layered structure Schematic cross-section of a “simple” exchange-biased spin valve layered structure F – magnetically very soft Schematic curves of the magnetic moment (a) and resistance (b of a “simple” exchange-biased spin valve layered structure GMR in a Co/Au/Co-layered structure due to different coercitivities H c of the two Co films Only the soft layer is affected by the external magnetic field, magnetization of the pinned layer is fixed



relative orientation of the magnetization is changed by external field

Increase with time of the areal bit density in hard-disk recording, for commercial IBM systems

Read Heads

GMR – higher sensitivity and better signal-to-noise ratio Contactless sensors do not exibit mechanical wear Thin film MR heads are produced by photolithographic processing Prototype GMR devices based on exchange-biased spin-valve have been offered by IBM, Hitachi, INESC, Fujitsu

Summary Briefly considered different MR effects

Anisotropic MR Normal MR Giant MR Tunnel MR Colossal MR

Ideas of GMR theory are introduced

Semiclassical Camley-Barnas theory Ab Initio calculations of GMR

Short notes on GMR applications are done