spin dependent electron transport in nanostructures

04/22/2304/22/23 A. Ali Yanik, Purdue UniversityA. Ali Yanik, Purdue University 11

Spin Dependent Electron Spin Dependent Electron Transport in Nanostructures Transport in Nanostructures

A. Ali YanikA. Ali Yanik††

DissertationDissertation

††Department of Physics Department of Physics &&

Network for Computational NanotechnologyNetwork for Computational NanotechnologyPurdue University, West Lafayette, IN 47907Purdue University, West Lafayette, IN 47907

April 2007April 2007


Spin + Electronics = Spin + Electronics = SpintronicsSpintronics


Spintronic DevicesSpintronic Devices

Non volatile RAM, Freescale,2006

Devices: GMR (read heads), TMR (MRAM), BMR Devices, etc..

MagnetoelectroniMagnetoelectronics cs

GateFM FM

2DEG

GateFM FM

2DEG

S. Datta & B. Das, APL. 56, 665 (1990)

Gate Voltage Control / Rashba Effect

Field Controlled SpintronicsDevices: Spin-FET (Datta), etc..

Contact Contact Injection/DetectionInjection/Detection

Gate Gate ContactContact

External B External B FieldField

Spin Spin DephasingDephasing


NEGF Formalism

Motivation-IMotivation-I

DeviceDevicess

Concepts

Physics Community

Spin Decoherence + QM

Equilibrium

Engineering Community

Transport + QM

Non-Equilibrium

Decoherence Physics Quantum Transport

Ph.D. Thesis: First formalized treatment of Quantum-Transport with Spin-Decoherence in NEGF


Motivation-IIMotivation-II

ContactsChannelElectrons

Ballistic Transport / NEGF FORMALISM

Phononsel

NEGF FORMALISM (Inelastic Transport)

Electron-phononrelaxation time

LocalizedSpins

es sl Spin-latticerelaxation time

EQUILIBRIUM PHYSICSEQUILIBRIUM PHYSICS

Challenges:

Physics Based Unified Treatment (not specialized for each device, geometry, etc)

Conservation Laws (angular momentum, total energy, particles)

Numerically Treatable

Benchmark against experiment.

State of Art Modelling

Averaging of Coherent Processes

Doesn’t Capture the Physics

Not straightforward to include dissipative interactions

NON-EQUILIBRIUM TRANSPORT


A Unified Quantum A Unified Quantum Transport Model Transport Model


Unified Approach to Nanoscale DevicesUnified Approach to Nanoscale Devices

Quantum

Device

Source Drain

Gate

Scatterer

Molecule (Gosh et al)

MTJ (Yanik et al)

L R

Scattering

Source Drain

Gate

QuantumDevice

H U

Spin Torque (Prabhakar et al)

Nanotubes (IBM, Kosawatta et al)

Nuclear Spin Polarization(Salahuddin et al)

MOSFET (Damle et al)

RTD (Klimeck et al)


Magnetic Tunnel Magnetic Tunnel JunctionsJunctions

Availability of Experimental DataAvailability of Experimental Data

Technological ImportanceTechnological Importance


Coherent RegimeCoherent Regime


Junction MagnetoresistanceJunction Magnetoresistance Potential Barrier + Magnetic Contacts

Soft Layer & Hard Layer (fixed)

Exchange shifted two current model

Parallel Contacts Anti-parallel Contacts

0

0F AF F AF

AF F

I IR R G GRJMR

R R G I

Soft Layer

Hard Layer

Tunneling Oxide

F

F

Soft Layer

Hard Layer

Tunneling Oxide

F

F

Δ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

ΔΔ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

Δ Δ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

Δ


0

0F AF F AF

AF F

I IR R G GRJMR

R R G I

Soft Layer

Hard Layer

Tunneling Oxide

F

F

Soft Layer

Hard Layer

Tunneling Oxide

F

F

Δ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

ΔΔ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

Δ Δ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

Δ

F AF F AF

AF AF

R R G GRJMR

R R G

Parallel ContactsAntiparallel Contacts

T.M. Maffit et al IBM J. Res. & Dev. 50, 25 (2006)

Barrier

FE

minoritycE

majoritycE

FE

minoritycE

majoritycE

barrU

Barrier

FE

minoritycE

majoritycE

FE

minoritycE

majoritycE

barrU

Stearns M. B., J. Magn. Magn. Mater. 5, 167 (1977)


Junction MagnetoresistanceJunction Magnetoresistance Potential Barrier + Magnetic Contacts

Soft Layer & Hard Layer (fixed)

Exchange shifted two current model


0

0F AF F AF

AF F

I IR R G GRJMR

R R G I

Soft Layer

Hard Layer

Tunneling Oxide

F

F

Soft Layer

Hard Layer

Tunneling Oxide

F

F

Δ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

ΔΔ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

Δ Δ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

Δ


0

0F AF F AF

AF F

I IR R G GRJMR

R R G I

Soft Layer

Hard Layer

Tunneling Oxide

F

F

Soft Layer

Hard Layer

Tunneling Oxide

F

F

Δ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

ΔΔ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

Δ Δ

FE

minoritycE

majoritycE

FEminoritycE

majoritycE

Δ

F AF F AF

AF AF

R R G GRJMR

R R G

Parallel ContactsAntiparallel Contacts

T.M. Maffit et al IBM J. Res. & Dev. 50, 25 (2006)

Spin polarization is conserved

Rectangular potential barrier & exchange shifted parabolic bands.

Qualitatively correct and widely used by experimentalists

2

2F F F F

FM

F F F F

k k k kP

k k k k

22 barr Fm U E

Slonczewski’s Formula:

Fails for Thin Tunneling Barriers!!!

J.C. Slonczewski PRB 39, 6995 (1989)

Practical Interest


Coherent Regime (NEGF)Coherent Regime (NEGF)

F z AF zz

F z

I E I EJMR E

I E

Weighting Factor

JMR for Different Incoming Energies

z

z F z F zE

E I E I E

majoritycE

minoritycE FE

EF=2.2eV, ∆=1.45eV and Vbias=1meV after Stearns et al.

ω(Ez) shifts towards higher energies with increasing barrier thicknesses


Coherent Regime (NEGF)Coherent Regime (NEGF)

F z AF zz

F z

I E I EJMR E

I E

Weighting Factor

JMR for Different Incoming Energies

z

z F z F zE

E I E I E

EF=2.2eV, ∆=1.45eV and Vbias=1meV after Stearns et al.


( )z z zJMR E JMR E dE

Experimentally Measured JMR



Incoherent RegimeIncoherent Regime

Impurity Concentration

Barrier Thickness

Barrier Height


MTJs with Magnetic Impurity LayersMTJs with Magnetic Impurity LayersR. Jansen & J. S. Moodera, J. Appl. Phys. 83, 6682 (1998)

Hard Layer

FTunneling Oxide

Impurity Layer

Tunneling Oxide

Soft Layer

F

Barrier

FE

minoritycE

majoritycE

FE

minoritycE

majoritycE

barrU

Impurity Layer

Barrier

FE

minoritycE

majoritycE

FE

minoritycE

majoritycE

barrU

Barrier

FE

minoritycE

majoritycE

FE

minoritycE

majoritycE

barrU

Impurity Layer


MTJs with Magnetic Impurity LayersMTJs with Magnetic Impurity Layers

Normalized JMR ratios are barrier thickness independent

JMR(Ez) ratios reduces at all energies

Elastic spin scattering doesn’t effect normalized ω(Ez)

Decreasing JMRs with increasing impurity concentrations


MTJs with Magnetic Impurity LayersMTJs with Magnetic Impurity Layers

A universal trend independent from the barrier heights

Minimal Fitting Parameters


Pd & Ni Impurity LayersPd & 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.


High-Spin/Low-Spin Phase TransitionHigh-Spin/Low-Spin Phase Transition

J exchange coupling used as a fitting parameter

Large temperature dependence

Thermally driven low-spin/high-spin phase transitions

d4-d7 systems:t2g set → low spin state

eg set → high spin case.

S. W. Biernacki et al, PRB. 72, 024406 (2005).

Crystal Field Theory

-The Pairing energy (P) Coulombic repulsion Exchange Energy-The eg - t2g Splitting


Details of the TheoryDetails of the Theory


Spin Array

L RSource

LΣ

SΣ

RΣ

Drain

Gate

QuantumDevice

H U

Spin Array

L RSource

LΣ

SΣ

RΣ

Drain

Gate

QuantumDevice

H U

Exchange Interaction Spin ScatteringExchange Interaction Spin Scattering

Hamiltonian: , ch L R IH H H H

chH Effective mass description†

k k kk

c c

,L RH Modeled through contact self energy

IHModeled using self consistent Born approximation S

, L R

Magnetic Impurity Magnon ScatteringAranov-Bir-Pikus (Electron-Hole)Nuclei (Hyperfine Interaction)

, , ,, ;

,

, '; , '; , '; i j i j k l k l

k l

in out n p n pS r r E D r r G r r E d

Analogous to the Electron/Hole Density

Rate at which electrons/holes are scattered in/out of a state


Spin Scattering Self EnergySpin Scattering Self Energy

*, ' , ', 'nint intD H r t H r t

int -j

j jRH r J r R S

Interaction Hamiltonian:

† †1 1 1,

2 2 2int zH r t J r R aS t a S t a a S t

Spin Array

Spin Array

Channel

Spin Exchange Interaction

† 0a

0

†1 1

2 2x a a

†1 1

2 2y a ai i

† 1

2z a a

0

0 0

qi teS t d

0 0

0qi tS t d

e

1

2zS d d Imp

uri

ty

Op

era

tors

Ele

ctr

on

O

pera

tors

Preserves Angular Momentum

Jordan-Wigner


n,p 2

nsf

,

,

1 0 0 0D , '; '0 1 0 0

0 0 1 0

0 0 0 1

q

k l

i j

I qr r r r J N

Inelastic Spin Flip ScatteringInelastic Spin Flip Scattering

n,p 2 ,

sf

,

,

,

0 0 0D , '; '

0 0 0 0 0 0 0

0 0 0 0

q

k l

i j

u d qqI

d u q

Fr r r r J N

F

, , ,, '; , '; , '; n p n p n p

sf nsfD r r D r r D r r

Spin Flip Scattering

Non-Spin Flip Scattering

0

0u

d

F

F

Impurity Density Matrix


Elastic Spin Flip ScatteringElastic Spin Flip Scattering

2-D Translational Symmetry

Elastic Spin Flip Scattering

2 2

2

1'

q

I q IDr r J N J n

a

n,p 2Isf 2D

0 0 0,

0 0 0,D 0 n a

0 0 0 0

0 0 0 0

k l

i j

Fu d

Fd uJ

a

a


Spin Array

L RSource

LΣ

SΣ

RΣ

Drain

Gate

QuantumDevice

H U

Spin Array

L RSource

LΣ

SΣ

RΣ

Drain

Gate

QuantumDevice

H U

Unpolarized Spin EnsembleUnpolarized Spin Ensemble

0.5 0

0 0.5

Magnetic Impurity Layer


Self-consistent SolutionSelf-consistent Solution

Regular Contacts:

Channel: zH

Incoherent Scattering:

Hamiltonian

Transport Equations:

Green’s Function

L R SG E EI H U

2 , ,ln 1 expD z L R s z L R Bf E N E k T

Fix

ed

at

the

Ou

tset

Self

-con

sit

en

t S

ol.

2 2 z

D in D nL L z z L z z z

E

qI tr E A E tr E G E dE

h

; ; i j i k k l k l

i j

S D G E

Dir

ect

Sol

, 2 , ,inL R z D z L R L R zE f E E

, 2 , ,1outL R z D z L R L R zE f E E

12 2D n D n

z z zG E I P E S E


SummarySummary

Magnetic Impurity Layer

ContactsChannelElectrons Phononsel


LocalizedSpins



0.5 0

0 0.5

Magnetic Impurity LayerMagnetic Impurity Layer

ContactsChannelElectrons Phononsel


LocalizedSpins



0.5 0

0 0.5

ContactsContactsChannelElectronsChannelElectrons PhononsPhononsel


LocalizedSpins


LocalizedSpins

LocalizedSpins



0.5 0

0 0.5

Challenges:

Physics Based Unified Treatment

Conservation Laws (angular momentum, total energy, particles)

Numerically Treatable

Benchmarking against experiment

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.


AcknowledgementAcknowledgement

Professors Supriyo Datta and Gerhard KlimeckProfessors Supriyo Datta and Gerhard Klimeck

Dr. Dmitri Nikonov – Intel corporationDr. Dmitri Nikonov – Intel corporation

Sayeef Salahuddin, Prabhakar Srivastava Sayeef Salahuddin, Prabhakar Srivastava

NSF funded Network for Computational NSF funded Network for Computational Nanotechnology (NCN) and MARCONanotechnology (NCN) and MARCO

spin dependent electron transport in nanostructures

Documents

maffit et

stearns et

rtd klimeck et ala

spin electronics

spinfet datta

coherent regimea

unified treatment

unified approach