Moderne Konzepte der Festkörperphysik
Giant Magnetoresistance
Alexey Dick
Fritz-Haber Institut der MPG Berlin
2001
Outline
Overview of magnetoresistance effects
Anisotropic MR
Normal MR
Mott two current model
Interlayer coupling
Giant MR (first experiments and qualitative picture)
Tunnel MR
Colossal MR
GMR theory
Semiclassical model of conductivity
Boltzmann equation
Semiclassical Camley-Barnas theory
Ab Initio calculations of GMR
Applications of GMR
Summary
Anisotropic MR (1)
Magnetoresistance – change in electrical resistance of a material in response to a
magnetic field
AMR- dependence of the resistivity on the angle
Positive MR
between the magnetization direction and the current
density
Spontaneous resistivity anisotropy can be expressed
using the resisitivity tensor for monodomain
policrystal with magnetization along z axis
Negative MR
 

 ij    EH
 0

  EH

0
0

0
|| 
Corresponding electric field is
Schematic magnetoresistance curves for a
ferromagnet


  
 
E    j  ||       j     EH  j
Dependence of the resistivity on the angle between the field and the current is

||  2  
3
1

 ||      cos 2   
3

Magnetic multilayers and giant magnetoresistance : fundamentals and industrial applications, Springer 2000
Magnetische Schichtsysteme in Forshung und Anwendung, Materie und Material, band 2, 1999
Solid State Physics, vol.47, Acad. Press 1994
Anisotropic MR (2)
Spontaneous resistivity anisotropy ratio generally defined as
SRA 
SRA 


||   
|| 3  2   3
Origin – spin-orbit interaction -> coupling adds some orbital contribution to the spin
moment, gives rise to a dependence of the electron scattering on the angle between the
electron wave vector and the magnetization direction
The largest AMR effect at room temperature is found for Ni1-xCox alloys with x close to 0.2,
for which SRA ~ 6%
For permalloy Ni80Fe20
SRA ~ 4%
Effect disappears above Tc
Normal MR
Lorenz force acting on the charge carriers
 increase of the resistance in an applied
magnetic field
All metals have an inherent normal
(ordinary) magnetoresistance
In ferromagnetic magnetic field is





B  0 H a  H d  M

Normal MR obeys the Kohler rule


 F B  0 
 0 at B=0;
In ferromagnetic materials
0
 NMR can be very large when
is
0
 ||
or

depending on the relative orientation of
current and magnetization
is very small  pure single crystal at low temperature
In thin films the concentration of defects is high 
 0 is high  generally NRM is neglected
Mott Two-Current Model (1)
Sir Neville Mott explained the sudden decrease in
resistivity of ferromagnetic metals as they are cooled
through the Curie point
Model for electrical conductivity in metals 
conduction current in ferromagnetic metals can be
decomposed into two carrier types–current
Total conductivity can be expressed as sum of
separate contributions from majority and
minority electrons
Assumptions:
spin is preserved
spin-flip transitions take place at
collisions with magnons, low magnon
density at T<Tc, no spin-flip processes
occur while scattering on defects
conduction is almost by means of s-electrons
low effective masses compare to delectrons
Schematic band structure for magnetic
transition metal
At T<Tc scattering is mostly dominated by foreign atoms and defects (I.e. no phonons and
magnons scattering)
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

,
ns e 2 s , 

ms*
    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 1
bands, majority – different  majorities are scattered more strongly 
to the right 
 1
At higher temperatures momentum transfer between two channels are relevant.

Electron-magnon scattering increase  . Fert derived:
       1  
 

 

     1
2
Applies not very high temperatures
Interlayer Coupling
Cu Layer Thickness (nm)
Different types of coupling in a layered magnetic
structure
The saturation field Hs and   as a function of
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  spin-orbit coupling and
resulting anisotropy play a minor role
R R

MR=79% at T=4.2K and 20% at room temperature
MR 
AP
RP
P

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 layeraveraged 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  1    1   
1
 AP        4
2

1 
MR 
4
In magnetic layered structures - dependence of the
resistance on the angle between the magnetization
directions of successive magnetic layers
Phenomenological expression
R   R  0  RGMR 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
Average distance between subsequent spin-flip scattering processes. If  sd is larger than
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
RAP  RP
TMR 
RP
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. La1xSrxMnO3
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  carriers
decrease fo resistivity
Applied magnetic field increase ferromagnetic
ordering  decrease resistivity
Large fields (several Tesla) are needed
Top:Magnetization against temperature for
La0.75Ca0.25MnO3 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”)
Having
  the state of electron defined by

n k
 
,r
k and n.
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



 
1  n k

r  vn k 
 k

 

  
1 
 
k  e E r , t   vn k  H r , t 
c


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  tv , k  F dt  , t  dt )
 
   
f (r , k , t )  f (r  tv , k  F dt  , t  dt ) 
 
 f r , k , t 
 dt 
 

t

 out


 f r , k , t 
 dt
 

t

in




Liouville theorem
Collisionless movement
 
Some electrons
  can not
reach r , k because of
collisions
 
 
r,k
Some electrons reach
only because of collisions
Leaving only linear dt terms in limit dt0
f  f  1 g  f 
v  F
 
t
r
 k  t coll
Drift
Collisions
If collision term is in relaxation time
approximation -> linear differential
equation


 f k

 t




| f (k )  f 0 (k ) |
 


 (k )
 coll

f 0 (k ) - local equilibrium
distribution function
Steady State Boltzmann Equation



f ( k )  f 0 (k )  g ( k )


g (k )  f 0 (k )
1
f k0 
exp Ek  EF  k BT   1
Assuming no magnetic field , time independent electric field and temperature gradient in
steady state
 f k0   
v  E 
g k  e
 E   k k
 k

2e   
j  
vk g k

V k

 f k0    
 2e 2

 v v E
j

 
V
Ek  k k
k 
 



j  ˆE

 
2e 2
  E  v v

ˆ 


E
F
k
k k

V
k
dS k  
2e 2
ˆ 

vk vk
3 
2  vk
For metals the changes in distribution function can be restricted to energies close to Fermi
surface because for metals dfo/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 f0 is Fermi distribution function
 
 
 g v , r    f 0
g  v , r 
v
 eE  v


r
E


3
 

 
m
j r   e   d 3vv g  v , r 

g

 a  z 

f 0 


v , z   e Ex vx 1  A exp   
E 
  v z 
 a  z 

f 
g   v , z   e  E x v x 0 1  A  exp   
E 
  v z 
CB model involves spin dependent
probability for specular reflection at the
outer boundaries.
At the interfaces three cases are
distinguished: transmission with
probability Ti,s
Specular reflection with probability Ri,s
Diffuse scattering with probability Di,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
 
e2
  E v  v

ˆ   

E
F
k
k
k
V k
Using relaxation time approximation with isotropic relaxation time
ratio is determined only by electronic structure
GMR 




  Ek  EF vk    Ek  EF vk
k
2
i

2  E
k
AP
k

 EF v
AP 2
ki
i
 AP      
GMR
2
1
k
Density of states at the Fermi surface
Averaged over Fermi surface
N E F  v

GMR 
2
ki
 N E F  v

2 N AP EF  vkAP
i
2
ki
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 dbands 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:
   

1 3
Tkk    d r  k r  V r   k  r 
V
Microscopic probability of transition is equal according to the Fermi golden rule
Pkk  
2
2
cN Tkk   Ek  Ek  

Proportional to the concentration of scatterers c (deluted case), delta-function – elastic
scattering
Extrinsic GMR (2)
 
Pkk 
 Pk
k
  
 Pkk 

Pk
k 

Pk
k 
In ferromagnetic systems transition matrix contains two spinconserving 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
    P
 1
k
k

kk 
Depends on spin and state. For simplicity take average over states:
 




E

E

 k F k
k
  E
k
 EF 
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 k

 t


   f  1  f k P  f  1  f  P
k
k k
k
k
kk 

 coll k 


 
Describes scattering-out from k
state
Describes scattering-in in k state
Principe of microscopic reversibility
Quasiclassical equation is
Pkk  Pkk


  
  
 k   k v k   Pkk   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
e2
ˆ 
V
 
  E k  EF v k  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)


  
  
 k   k v k   Pkk   k 
k


Solution of Boltzmann equation
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  2
 
2e
  E  v  v



E
F
k
k
k

V
k
ˆ 
This is boundary case of isotropic scattering – all electrons have the same relaxation time
Taking spin-dependent isotropic lifetime
e2
ˆ 
V




k





   E

RL
RL

k

k


k
 EF
  

 k RL

The most general case (when scattering-in term is omitted)
e2
ˆ 
V




  E  EF  k v   v 

k

k

k
k
Relaxation time depends on spin and layer
e2
ˆ 
V

    E   EF v   v 

k

k
2
 
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 spinvalve layered structure
F – magnetically very soft
GMR in a Co/Au/Co-layered structure due to
different coercitivities Hc of the two Co films
Schematic curves of the magnetic moment (a) and
resistance (b of a “simple” exchange-biased spinvalve layered structure
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
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
Increase with time of the areal bit density
in hard-disk recording, for commercial
IBM systems
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