Transcript Transport in Solids Peter M Levy Email: Room 625 Meyer

Transport in Solids
Peter M Levy
Email: [email protected]
Room 625 Meyer
Phone:212-998-7737
Material I cover can be found in
General:
Solid State Physics, N.W. Ashcroft and N.D. Mermin (Holt,
Rinehardt and Winston, 1976)
Electronic Transport in Mesoscopic Systems, S. Datta (Cambridge
University Press, 1995).
Transport Phenomena, H. Smith and H.H. Jensen ( Clarendon Press,
Oxford, 1989).
J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).
Ab-initio theories of electric transport in solid systems with reduced
dimensions, P. Weinberger, Phys. Reports 377, 281-387 (2003).
Electrical conduction in
magnetic media
How we got from 19th century concepts to applications
in computer storage and memories.
1897- The electron is discovered by J.J. Thomson
~1900 Drude model of conduction
based on kinetic theory of gases {PV=RT}
~1928 Sommerfeld model of conduction
in metals
l ~ 100 A
n  neff ~ N ( F )  
N( F ) 
n
F
;
  k BT or eEl
V  IR  RI
Ohm' s law
j = I A ; E  V L ;   AR L
 E  j
j  nev
vavg 
eE
;  is the time between collisions
m
ne2 
j  
E  E
 m 

1

l  v  mean free path ~ 10 - 100A
Phenomena
While each atom scatters electrons, when they
form a periodic array the atomic background only
electrons from one state k to another with k+K.
This is called Bragg scattering; it is responsible
for dividing the continuous energy vs. momentum
curve into bands.
1911 Superconductivity is discovered
by Kammerlingh-Onnes
The resistance of metals increases with temperature; that’s sort of intuitive: the greater
the thermal agitation the greater the scattering. What was completely unanticipated
was the lose of all resistance at a finite temperature.
When mercury was cooled to 4.18K above absolute zero it lost all resistance; once a
current was started one could remove the battery and it would continue to flow as if
there were no collisions any more.
An understanding of this phenomenon was not fully enunciated till 1958 with the theory
of Bardeen-Cooper and Schreiffer. A key ingredient in understanding superconductivity
is the coupling of motion of the background to that of the electrons. While this is largely
responsible for resistance when the two are not coupled, those electrons that are
responsible for superconductivity are no longer scattered.
n  neff ~ N( F )  ; N( F ) 
n
F
;   kBT or eEl
Provides explanation for negligible contribution of conduction electrons to specific heat
of metals.
What distinguishes a metal from an
insulator
Intrinsic semiconductors
The number of carriers depends on temperature; at T=0K there are none.
Doping with donors and acceptors
many more carriers at lower temperatures
Transistor: a p-n junction
Depletion layer at interface-transfer of charge
across interface
Effect of bias-voltage on depletion layer
~1955 the transistor; rectification action of
p-n junction
Magnetoresistance
Lorentz force acting
on trajectory of
electron;longitudinal
magnetoresistance
(MR).
A.D. Kent et al
J. Phys. Cond.
Mat. 13, R461
(2001)
Anisotropic MR Role of spin-orbit coupling on electron scattering
A.D. Kent et al
J. Phys. Cond.
Mat. 13, R461
(2001)
Domain walls
References
Spin transport:
Transport properties of dilute alloys, I. Mertig, Rep. Prog. Phys. 62,
123-142 (1999).
Spin Dependent Transport in Magnetic Nanostructures, edited by
S. Maekawa and T. Shinjo ( Taylor and Francis, 2002).
GMR:
Giant Magnetoresistance in Magnetic Layered and Granular
Materials, by P.M. Levy, in Solid State Physics Vol. 47,
eds. H. Ehrenreich and D. Turnbull (Academic Press, Cambridge,
MA, 1994) pp. 367-462.
Giant Magnetoresistance in Magnetic Multilayers, by A. Barthélémy,
A.Fert and F. Petroff, Handbook of Ferromagnetic Materials, Vol.12,
ed. K.H.J. Buschow (Elsevier Science, Amsterdam, The Netherlands,
1999) Chap. 1.
Perspectives of Giant Magnetoresistance, by E.Y. Tsymbal and D,G.
Pettifor, in Solid State Physics Vol. 56, eds. H. Ehrenreich and
F. Spaepen (Academic Press, Cambridge, MA, 2001) pp. 113-237.
CPP-MR:
M.A.M. Gijs and G.E.W. Bauer, Adv. in Phys. 46, 285 (1997).
J. Bass, W.P. Pratt and P.A. Schroeder, Comments Cond. Mater. Phys.
18, 223 (1998).
J. Bass and W.P. Pratt Jr., J.Mag. Mag. Mater. 200, 274 (1999).
Spin transfer:
A. Brataas, G.E.W. Bauer and P. Kelly, Physics Reports 427,
157 (2006).
Spintronics- control of current through spin of electron
The two current model of conduction in ferromagnetic metals
1988 Giant magnetoresistance
Albert Fert & Peter Grünberg
Parallel configuration
Antiparallel configuration
Two current model in magnetic multilayers
Data on GMR
M.N. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988).
GMR in Multilayers and Spin-Valves
110
Co95Fe5/Cu
[110]
multi-layer
DR/R~110% at RT
Field ~10,000 Oe
-40
0
H (kOe)
H(kOe)
H // [ 011]
[011]
40
Py/Co/Cu/Co/Py
NiFe
Co nanolayer
Cu
Co nanolayer
NiFe
FeMn
spin-valve
DR/R~8-17% at RT
Field ~1 Oe
-40
0
40
GMR
-metallic spacer
between
magnetic layers
-current flows inplane of layers
HH(Oe)
(Oe)
NiFe + Co nanolayer
S.S.P. Parkin
Polycrystalline
S.S.P. Parkin et al,
Phys. Rev. Lett. 66, 2152
(1991)
Oscillations in GMR:
Polycrystalline vs.
Single Crystal Co/Cu
Multilayers
Single crystalline
S.S.P. Parkin
Sputter deposited on MgO(100), MgO(110)
and Al2O3 (0001) substrates using Fe/Pt seed
layers deposited at 500C and Co/Cu at ~40C
Current in the plane (CIP)-MR
vs
Current perpendicular to the
plane (CPP)-MR
1995 GMR heads
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
From IBM website; 1.swf
2.swf
Tunneling-MR
Two magnetic metallic electrodes separated by an insulator; transport
controlled by tunneling phenomena not by characteristics of conduction
in metallic electrodes
2000 magnetic tunnel junctions used in magnetic random access memory
From IBM website;
http://www.research.ibm.
com/research/gmr.html
PHYSICAL REVIEW LETTERS VOLUME 84, 3149 (2000)
Current-Driven Magnetization Reversal and Spin-Wave Excitations in CoCuCo Pillars
J. A. Katine, F. J. Albert, and R. A. Buhrman
School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853
E. B. Myers and D. C. Ralph
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853
How can one rotate a magnetic layer with a spin polarized current?
By spin torques:
Slonczewski-1996
Berger -1996
Waintal et al-2000
Brataas et al-2000
By current induced interlayer coupling:
Heide- 2001
Current induced switching of magnetic layers by spin polarized
currents can be divided in two parts:
Creation of torque on background by the electric current, and
reaction of background to torque.
Latter epitomized by Landau-Lifschitz equation; micromagnetics
Former is current focus article in PRL:
Mechanisms of spin-polarized current-driven magnetization switching
by S. Zhang, P.M. Levy and A. Fert. Phys. Rev. Lett. 88, 236601 (2002).
Extension of Valet-Fert to noncollinear multilayers
Methodology
To discuss transport two calculations are necessary:
•Electronic structure, and
•Transport equations; out of equilibrium collective electron
phenomena.
Structures
•Metallic multilayers
•Magnetic tunnel junctions
•Insulating barriers
•Semiconducting barriers
•Half-metallic electrodes
•Semiconducting electrodes
different length scales
Prepared by Carsten Heide
Lexicon of transport parameters
Spin independent transport
 F  Fermi energy

1
vF  Fermi velocity 
k
kF  Fermi momentum
 mfp  Mean time between collisions
mfp  Distance travelled between collisions
 G(r  r',  F )  e
 vF  mfp
i(k F  i  ) rr '
Spin dependent transport parameters
 s  Spin dependent relaxation time
 sf  Time between spin flips
s , / M,m
sdl  sf mfp  Spin diffusion length
hv
dJ  F J  Spin coherence length
due to temporal precession; J  exchange constant
tr  Transverse spin coherence length
 J  dJ mfp  transverse spin diffusion length
lc  1
kF  k F
 Transverse spin coherence length
due to spatial precession.
Ballistic transport: see S. Datta Electronic Transport in Mesoscopic
Systems (Cambridge Univ. Press, 1995).
Collisionless regime; transport conditions set by reservoirs
at boundaries. Conductance measured by transmission
through states on Fermi surface
Tk k ' '  t k ' ',k
in units of the quantum of conduction
2
2e2 / h  12.9k1
2e2
G
MT , where M is the number of channels.
h
Critique of the “mantra” of Landauer’s formula; see M.P. Das
and F. Green, cond-mat/0304573 v1 25Apr 2003.
Spin and charge accumulation in metallic systems
Application to magnetic multilayers

Semi-classical approaches to electron dynamics
External fields are treated classically,
background is not.

r  vn (k)  1
while potential of periodic
 n (k) k 


1
k  eE(r,t)  vn (k)  H (r,t)


c
1
f ( n (k))   n (k ) / k BT
e
1

Validity
As long as one does not try to localize electron on length scale
of unit cell, and wavelength of applied fields long compared to
lattice constant.
2
eEa, c   gap
(k)  F .
Diffusive transport
Collisions assure local equilibrium of current; however
a  mfp  L, where a is lattice constant . Also,
mfp  phase coherence length of wavefunctions.
In the diffusive regime processes that occur on a length scale
long with repsect to the mean free path have to be averaged,
e.g., the distance traversed by an electron undergoing random
scattering is
2
L ~

 mfp
(mfp cos  )2  1/3 vF2  mfp .
By definning a diffusion constant D  1/ 3vF2  mfp we find
L2  D .

Simple derivation
df (k,r,t) dk
dr
f (k,r,t)

 k f 
 r f 
dt
dt
dt
t
From semiclassical electron dynamics :

r  v n (k)  1
n (k) k 


1
k  eE(r,t)  v n (k)  H(r,t)


c

and from :
 k f  n (k) k  f 0 n  v n (k)  (n  F ) at T = 0K.
Thus we find :
f t  v  f  eE  v (   F )
 1  f  f

Derivation of Landauer formula (see Datta)
I  nev
For a conductor of length L (one dimensional electron gas)
the electron density for each k state is 1/L. Thus the current
carried by k states travelling in one direction is
e
e 1  


I   vf ()M()  
f ()M()
L k
L k k
  2(spin)
k
I 
2e
h
 df
L
2

 dk
()M()
2e
2e 2
I  I (1 )  I (2 )  M(1  2 ) 
MV
h
h
2e 2
Gc 
M
h
Therefore the contact resistance of a ballistic conductor is
12.9k
1
Gc 
M


Landauer reasoned that when the conductor is not perfectly
ballistic, i.e., has a transmission probability T<1 that
2e 2
G
MT
h
so that
h 1
h
h 1 T
G1  2
 2  2
2e M T 2e M 2e M T
1
1
 Gc  Gs
In other words when T < 1 in addition to the contact resistance
there is a reistance due to the scattering in the conductor.
While the latter is independent of the length L of the conductor,
it can be directly related to " ohmic" resistance as follows.
For a wide conductor W with many channels or modes of conduction
M ~ W/(/k F ), so that the conductance is
G  e 2W (m /  2 )(v F T(L) /  )
How does T depend on L ?
If we neglect treat the quantum interference between electrons,
the
transmission probability through a conductor of length L which contains
scatterers is :
L0
T(L) 
L  L0
where L 0 is the average distance travelled between scatterings. This
is derived as follows :
When one has a sequence of two scatterings with probabilities of transmission
T1 and T2 , then the joint transmission probability is not simply the product of them;
rather one have to take into account the multiple reflections :
2
2
T1T2 R1 R2 ,T1T2 R1 R2 ,..... so one arrive at,
T1T2
1 R1 R2
where Ti  1 Ri . This can be rewritten as
T12 
1- T12 1- T1 1- T2


,
T12
T1
T2
which for N scatterings in series can be extended to
1- T (N )
1- T
T
N
 T (N ) 
.
T (N )
T
N(1 T )  T
If we denote the linear density of scatters in a sample as  ; then the number of
scatterers in a conductor of length
equation we find
L0
T(L) 
L  L0
L is N  L. Placing this in the preceeding
T
where L0 
.
 (1 T)
To identify L0 we note that the mean free path, Lm , is the average distance an electron
travels before being scattered; as the probability of scattering is ( 1- T) we can
write
(1 T)Lm ~ 1 
when T ~ 1.
Lm 
1
~ L0 ,
 (1 T)
By placing this result
L0
T(L) 
L  L0
into the conductance
G  e 2W (m /  2 )(v F T(L) /  )
we find
G  e 2 (m /  2 )(v F L0 /  )
W
L  L0
 G1 
L  L0
.
W
We thereby arrive at resistance made up of the combination
of an actual (Ohmic), and a contact resistance :
1
Gs 
L
L
1
, and Gc  0 .
W
W
Conclusion
In general we can always write
h 1
h
h 1 T
G  2
 2  2
2e m T 2e m 2e m T
1
 Gc1  actual resistance.
The
 contact resistance is also known as the Sharvin resistance.