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Spin Dependent Electron Transport
in Nanostructures
A. Ali Yanik†
Dissertation
†Department
of Physics
&
Network for Computational Nanotechnology
Purdue University, West Lafayette, IN 47907
April 2007
5/24/2017
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Spin + Electronics = Spintronics
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Spintronic Devices
Field Controlled Spintronics
Devices: Spin-FET (Datta), etc..
Magnetoelectronics
Devices: GMR (read heads),
TMR (MRAM), BMR Devices, etc..
Contact Injection/Detection
Gate Contact
External B Field
Spin Dephasing
Gate Voltage Control / Rashba Effect
FM
Gate
FM
2DEG
S. Datta & B. Das, APL. 56, 665 (1990)
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Non volatile RAM, Freescale,2006
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Motivation-I
Concepts
Devices
Physics Community
Engineering Community
Spin Decoherence + QM
Transport + QM
Equilibrium
Non-Equilibrium
Decoherence Physics
Quantum Transport
NEGF Formalism
Ph.D. Thesis: First formalized treatment of Quantum-Transport with
Spin-Decoherence in NEGF
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Motivation-II
NEGF FORMALISM (Inelastic Transport)
Ballistic Transport / NEGF FORMALISM
Electron-phonon
relaxation time
Contacts

 el
Channel
Electrons
 sl
 es
Challenges:
State
of Art Modelling
NON-EQUILIBRIUM
TRANSPORT
 Physics Based Unified Treatment (not
specialized
device,
geometry, etc)
 Averagingfor
ofeach
Coherent
Processes
EQUILIBRIUM
PHYSICS
Conservation
Laws
(angular
Doesn’t Capture
the
Physicsmomentum,
total energy, particles)
 Not straightforward to include dissipative
interactions
Numerically Treatable
Phonons
Spin-lattice
relaxation time
Localized
Spins
EQUILIBRIUM PHYSICS
 Benchmark against experiment.
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A Unified Quantum
Transport Model
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Unified Approach to Nanoscale Devices
MOSFET
Nanotubes (IBM, Kosawatta et al)
(Damle et al)
GateGate
Source
(Salahuddin et al)
U 
 HQuantum
L
Nuclear Spin Polarization
R
Quantum
Device
Device
Drain
Drain
Source
Scatterer
Scattering
RTD (Klimeck et al)
MTJ
(Yanik et al)
Molecule (Gosh et al)
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Spin Torque (Prabhakar et al)
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Magnetic Tunnel Junctions
Availability of Experimental Data
Technological Importance
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Coherent Regime
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Junction Magnetoresistance
Parallel
Parallel
ContactsContacts
Antiparallel
Contacts
Anti-parallel
Contacts
F
Δ
E
Tunneling Oxide
Hard Layer
Hard Layer
EF
minority
c
Δ
majority
c
Δ
E
minority
c
E
minority
c
E
majority
c
EF
Δ
majority
c
RF  RAF GF  GAF I   0   I    


RAF
GF
I   0 
Δ
Barrier
JMR 
U barr
EF
Soft Layer
Tunneling Oxide
 Exchange
shifted two current
Hard Layer
model
EF
T.M.E MaffitE et al IBM J. Res. & Dev. E50, 25 (2006)
majority
c
F
Soft Layer
Tunneling Oxide
EF
E
 Soft
Layer & Hard Layer
F
(fixed)
F
Soft Layer
de
minority
c
Anti-parallel Contacts
F
F
EF
 Potential Barrier + Magnetic
Contacts
E
minority
c
E cmajority
JMR

E
F
E
minority
c
E
majority
c
R RF  RAF GF  GAF

E
R
RAF
GAF
F
Δ
Δ
E cminority
E cmajority
 GF  GAF I   0   I    
R RF  R
AF



R
RAF
GF
I   0 

Ecminority
EF
Ecminority


majority
c
E
Ecmajority
Stearns M. B., J. Magn. Magn. Mater. 5, 167 (1977)
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Junction Magnetoresistance
Parallel
Parallel
ContactsContacts
Antiparallel
Contacts
Anti-parallel
Contacts
F
F
de
EF
Δ
E
minority
c
E
F
Soft Layer
F
Soft Layer
Tunneling Oxide
Tunneling Oxide
Hard Layer
Hard Layer
EF
majority
c
 Soft
Layer & Hard Layer
F
(fixed)
Δ
Δ
E
minority
c
EF
E
minority
c
E
majority
c
Δ
majority
c
RF  RAF GF  GAF I   0   I    


RAF
GF
I   0 
Soft Layer
Tunneling Oxide
 Exchange
shifted two current
Hard Layer
model
EF
Δ
T.M.E MaffitE et al IBM J. Res. & Dev. E50, 25 (2006)
majority
c
Anti-parallel Contacts
F
EF
minority
c
 Potential Barrier + Magnetic
Contacts
E
minority
c
E cmajority
JMR

E
F
E
minority
c
E
majority
c
R RF  RAF GF  GAF

E
R
RAF
GAF
F
Δ
Δ
E cminority
E
Practical Interest
majority
c
R RF  RAF GF  GAF I   0   I    



R
RAF
GF
I   0 
Formula:
JMR 
 Spin polarization is conserved Slonczewski’s
 Rectangular potential barrier &
exchange shifted parabolic bands.
PFM
 Qualitatively correct and widely
used by experimentalists

kF  kF  2  kF kF
 

kF  kF  2  kF kF
 2m  U
2
barr
 EF 
PRB 39,
6995 (1989)
FailsJ.C.
forSlonczewski
Thin Tunneling
Barriers!!!
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Coherent Regime (NEGF)
JMR for Different Incoming Energies
JMR  Ez  
I F  Ez   I AF  Ez 
I F  Ez 
Weighting Factor
  Ez   I F  Ez 
I E 
F
z
Ez
ω(Ez) shifts towards higher energies with
increasing barrier thicknesses

Ecmajority
Ecminority
EF
EF=2.2eV, ∆=1.45eV and Vbias=1meV after Stearns et al.
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Coherent Regime (NEGF)
EF=2.2eV, ∆=1.45eV and Vbias=1meV after Stearns et al.
JMR for Different Incoming Energies
JMR  Ez  
I F  Ez   I AF  Ez 
I F  Ez 
Weighting Factor
  Ez   I F  Ez 
I E 
F
z
Ez
ω(Ez) shifts towards higher energies with
increasing barrier thicknesses
Experimentally Measured JMR
JMR     Ez  JMR ( Ez )dEz
ω(Ez) shifts towards higher energies with
increasing barrier thicknesses
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Incoherent Regime
Impurity Concentration
Barrier Thickness
Barrier Height
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MTJs with Magnetic Impurity Layers
R. Jansen & J. S. Moodera, J. Appl. Phys. 83, 6682 (1998)
Soft Layer

Hard Layer
F
Tunneling Oxide
Tunneling Oxide
F
Impurity Layer
Barrier
U barr
EF


Ecminority
EF
Ecminority


majority
c
E
Ecmajority
Impurity Layer
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MTJs with Magnetic Impurity Layers
JMR(Ez) ratios reduces at all
energies
Elastic spin scattering doesn’t effect
normalized ω(Ez)
Decreasing JMRs with increasing
impurity concentrations
Normalized JMR ratios are barrier
thickness independent
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MTJs with Magnetic Impurity Layers
A universal trend
independent from
the barrier heights
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Minimal Fitting
Parameters
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Pd & Ni Impurity Layers
<J2>2D exchange coupling used as
a fitting parameter
Minimal temperature dependence
Close <J2>2D coupling constants
estimated for Pd and Ni impurities
+1 spin state is believed to be the
dominant state.
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High-Spin/Low-Spin Phase Transition
J exchange coupling used as a
fitting parameter
Large temperature dependence
Thermally driven low-spin/highspin phase transitions
S. W. Biernacki et al, PRB. 72, 024406 (2005).
Crystal Field Theory
-The Pairing energy (P)
Coulombic repulsion
Exchange Energy
-The eg - t2g Splitting
d4-d7 systems:
t2g set → low spin state
eg set → high spin case.
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Details of the Theory
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Exchange Interaction Spin Scattering
Gate
H U 
 L  ΣL 
Quantum
Device
 ΣR 
 ΣS 
Source
Drain
Magnetic Impurity
Magnon Scattering
Aranov-Bir-Pikus (Electron-Hole)
Nuclei (Hyperfine Interaction)

Spin Array
Hamiltonian:
R
H  H ch  H L, R  H I
H ch
†
n, p

c

   Gn ,p mass
r , r '; E description
 d  
 r , r '; E  
 k  kDck ;   r , r '; Effective
H L, R
Modeled through contact self energy
in ,out
S , i j
k  k , l
i
j
k
l
k
l
 L, R
Analogous to the
Rate at which electrons/holes
S
Modeled using self consistent
Born approximation
Electron/Hole
Density
H
are
I scattered in/out of a state
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Spin Scattering Self Energy
Interaction Hamiltonian:



H int  r    R J r - R j   S j
j
Spin Array
Spin Exchange Interaction
*
D n  ,  '  H int  r , t  H int
 r ', t '
Channel
Preserves Angular
Momentum
1
1
1


H int  r , t   J  r  R  aS  t   a † S  t    a †a   S z  t  
2
2

2


  a† 0
  0
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Electron Operators
Jordan-Wigner
1
1
 x          a †  a 
2
2
y 
1 
1
       a†  a 

2i
2i
 z  a†a 
1
2
A. Ali Yanik, Purdue University
Impurity Operators

0 eiq t 
S t   d  

0
0




 0
S  t   d    iqt
e

Sz  d d 
0

0
1
2
22
Inelastic Spin Flip Scattering
 Fu
0
Dn, p  r , r ';     D n, p  r , r ';     D n, p  r , r ';  
sf
nsf

 k , l 
q
 q 
I




Spin Flip Scattering

 i , j 
q
Non-Spin Flip Scattering
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


0
Fu ,d   q 

 Fd ,u   q 
0

0
0


0
0

 k , l 
 Dn,p  r , r ';       r  r '     J 2 N I q 
nsf
0
Fd 

 i , j 
 D n,p  r , r ';       r  r '   J 2 N
sf
Impurity Density
Matrix




A. Ali Yanik, Purdue University
1
0

0

0


0
1
0
0
0
0
1
0
0
0
0
0

0

0

0
0 

0
0 
0

 1
23
Elastic Spin Flip Scattering
a
  r  r '  J 2 N I q  
q
a
1 2
J
a
2D
nI
2-D Translational
Symmetry
Elastic Spin Flip Scattering
 k l 

 i j 

 D n,p    0  
sf
 J2
2D
nI a



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 0

F
 d,u
 0

 0


F
u, d
0
0
0
0
0
0
0

0

0

0

0
24
Unpolarized Spin Ensemble
Gate
 L  ΣL 
H U 
Quantum
Device
 ΣR 
 ΣS 
Source
R
Drain
Magnetic Impurity Layer

Spin Array
0.5 0 

 0 0.5

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Direct Sol
Self-consitent Sol.
Fixed at the Outset
Self-consistent Solution
Channel:
Hamiltonian
Hz 
Regular Contacts:
inL, R  Ez   f 2 D  Ez   L , R   L , R  Ez 


  N ln 1  exp    E
out
L , R  Ez   1  f 2 D  E z   L , R   L , R  E z 
f 2 D  Ez  L, R
s
z

 L, R  kBT 

Incoherent Scattering:
 S ; i j    D i k ; k l  G k l  E 
 i j
Green’s Function
G  E    EI  H  U  L  R  S 
2D

G n  Ez   I  P  Ez 
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
1
2D
S n  Ez 
Transport Equations:
q
I L   tr  2 D inL  Ez  A  Ez   tr  L  Ez  2 DG n  Ez  dEz
h Ez

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
26
Summary
Electron-phonon
relaxation time
Challenges:
Contacts
 Physics Based Unified Treatment

 Conservation Laws (angular momentum,
total energy, particles)
 Numerically Treatable
 el
Channel
Electrons
 sl
 es
NON-EQUILIBRIUM TRANSPORT
Phonons
Localized
Spins
 Benchmarking against experiment
Spin-lattice
relaxation time
Magnetic Impurity Layer
0.5 0 


 0 0.5
Contributions:
 A Non-Equilibrium Quantum Transport model with Spin Decoherence
is developed.
 A Self Energy Calculation scheme is derived for Exchange Interaction
Scattering.
 A numerical implementation is shown in MTJ devices.
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Acknowledgement

Professors Supriyo Datta and Gerhard Klimeck

Dr. Dmitri Nikonov – Intel corporation

Sayeef Salahuddin, Prabhakar Srivastava

NSF funded Network for Computational Nanotechnology
(NCN) and MARCO
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