Electronic transport Electronic transport simulation of spintronic simulation of spintronic

devicesdevices

Keith McKennaKeith McKenna

Condensed matter groupCondensed matter groupE.C. Stoner LaboratoryE.C. Stoner LaboratorySchool of Physics and School of Physics and AstronomyAstronomyUniversity of LeedsUniversity of LeedsUKUK

(phykm@phys-irc.leeds.ac.uk)(phykm@phys-irc.leeds.ac.uk)

AcknowledgementsAcknowledgements

Jim MorganJim MorganBryan Hickey Bryan Hickey Lisa MichezLisa Michez

OverviewOverview

SpintronicsSpintronics ApplicationsApplications Giant Magnetoresistance (GMR)Giant Magnetoresistance (GMR)

Quantum simulationQuantum simulation Tight-bindingTight-binding TechniqueTechnique

ResultsResults Homogeneous systemsHomogeneous systems Inhomogeneous systems (interface resistance)Inhomogeneous systems (interface resistance) Mean-free-path effects in CPP GMRMean-free-path effects in CPP GMR

Spin relaxation calculationsSpin relaxation calculations Spin-orbit and Elliott-Yafet mechanismSpin-orbit and Elliott-Yafet mechanism Spin-diffusionSpin-diffusion

Electronics: charge transportElectronics: charge transport Spin introduces factor of twoSpin introduces factor of two

Spintronics: charge + spin transportSpintronics: charge + spin transport Introduce spin dependenceIntroduce spin dependence Ferromagnetic materialsFerromagnetic materials

ExamplesExamplesTunnel junctionsTunnel junctions

Fe\MgO\Fe, for example (MRAM)Fe\MgO\Fe, for example (MRAM)Giant magnetoresistive devices (Hard disk read heads)Giant magnetoresistive devices (Hard disk read heads)QubitsQubitsMagnetic semiconductors devicesMagnetic semiconductors devicesMagnetic tunnel transistorMagnetic tunnel transistor

Spin-electronicsSpin-electronics

Itinerant ferromagnetismItinerant ferromagnetism

Transition metalsTransition metals Balance:Balance:

Reduce correlation energies Reduce correlation energies (exchange)(exchange)Reduce kinetic energy Reduce kinetic energy (exclusion principle)(exclusion principle)

Stoner criterion:Stoner criterion:d band - high DOS near Fermi d band - high DOS near Fermi energyenergy

CobaltCobalt Majority – free electronMajority – free electron Minority – low mobilityMinority – low mobility Spin polarised transport Spin polarised transport

propertiesproperties

Majority

Minority

Giant MagnetoresistanceGiant Magnetoresistance

Thin film multilayer systems (e.g. Co/Cu/Co) Thin film multilayer systems (e.g. Co/Cu/Co) Large change in R – small HLarge change in R – small H

Moments can be aligned AP @ H=0Moments can be aligned AP @ H=0 RKKY type of interactionRKKY type of interaction Exchange bias (pin) with AF layerExchange bias (pin) with AF layer

Very small applied field: APVery small applied field: AP→P→P Transport properties alteredTransport properties altered

Co

Cu

M

M

J

VGMR

R AP RP

R P

100

GMR effect ~100%GMR effect ~100%

CPP geometryCPP geometry Spin dependent bulk Spin dependent bulk

scatteringscattering Spin dependent Spin dependent

interface resistanceinterface resistance Series resistor modelSeries resistor model

CIP geometryCIP geometry Spin dependent bulk Spin dependent bulk

scatteringscattering Spin dependent Spin dependent

transmissiontransmission Mean-free-path important Mean-free-path important

scaling lengthscaling length

Conductance calculationConductance calculation

Complicated quantum problemComplicated quantum problem NanoscaleNanoscale Millions of atomsMillions of atoms Complex geometryComplex geometry Complex spin-dependent electronic structureComplex spin-dependent electronic structure

Approaches within one-electron approximation:Approaches within one-electron approximation: Boltzmann, Density matrix or Wigner DFBoltzmann, Density matrix or Wigner DF Landauer-BLandauer-Büüttikerttiker Kubo (layer-by-layer recursive Green’s functions)Kubo (layer-by-layer recursive Green’s functions) NEGF (Keldysh)NEGF (Keldysh) Simulation (QMC, EOM)Simulation (QMC, EOM)

Electron diffusionElectron diffusionCurrents driven by electro-chemical (EC) potential:Currents driven by electro-chemical (EC) potential:

Chemical (density)Chemical (density) Electrostatic (electric fields)Electrostatic (electric fields) Connected through Poisson’s equationConnected through Poisson’s equation

Currents must balance in equilibrium:Currents must balance in equilibrium: Einstein relationEinstein relation Valid for Linear responseValid for Linear response

For small fields:For small fields: Electrons at EElectrons at EF F cause variation in cause variation in μμ

Can neglect electrostatic and simulate diffusion at ECan neglect electrostatic and simulate diffusion at EFF Poisson’s equation numerically intensivePoisson’s equation numerically intensive In linear response composition of EC potential irrelevantIn linear response composition of EC potential irrelevant

The EOM methodThe EOM method

Simulate electron diffusion at ESimulate electron diffusion at EFF using TDSE using TDSE

Represent contacts by allowing net current to flowRepresent contacts by allowing net current to flowCalculate conductance using steady state wavefunctionCalculate conductance using steady state wavefunction

Inject electrons at the Fermi enegry

Electrons transmitted into contact

K.P. McKenna et al, Phys. Rev. B 72, 054418 (2005)

Tight-bindingTight-binding Localised basisLocalised basis O(N) computationO(N) computation Represent complex electronic structures Represent complex electronic structures

(parameterised)(parameterised) Very good for transition metalsVery good for transition metals Structural disorder and complex geometryStructural disorder and complex geometry

2 centre and orthogonal2 centre and orthogonal

TB ParameterisationTB ParameterisationPapaconstantopolous:Papaconstantopolous:

Separate parameters for each elementSeparate parameters for each element Polynomial separation dependencePolynomial separation dependence Slater-Koster angular dependenceSlater-Koster angular dependence Scalar relativistic fits with exchangeScalar relativistic fits with exchange Volume dependent fitsVolume dependent fits

Cu: sp3d5 Co: s2p6d10

Ge: sp3 (top)sp3d5 (bottom)

Inject electrons @ EF

Boundary conditionsBoundary conditions

Negative imaginary site energy causes decay in probability (transmission)

Imaginary part of self-energy

•Broadens eigenstates•Finite lifetime

PBC or BOX

Absorb electrons

Optimise transmission:

E γ E γ iη

η V

Source term•Diffusion equation(heat sources)•Retarded Greens function @ EF

Simulation procedureSimulation procedure

t=0:t=0:Evolve TDSE (LF unstable - 2Evolve TDSE (LF unstable - 2ndnd order Runge- order Runge-Kutta)Kutta)Steady state:Steady state:

Calculate time averaged density and currentCalculate time averaged density and currentCalculate density of statesCalculate density of statesConductance determined:Conductance determined:

Four pointFour point Everywhere in systemEverywhere in system

aγ 0

Homogeneous systemHomogeneous systemSimple test system:Simple test system:

Simple cubicSimple cubic Nearest neighbours onlyNearest neighbours only 15x15x110 (24,750 atoms)15x15x110 (24,750 atoms) Diagonal Anderson disorder (W)Diagonal Anderson disorder (W)

ε

P(ε)

W

Comparison:Comparison: Dispersion relation is simple:Dispersion relation is simple:

Born approximation for relaxation time:Born approximation for relaxation time: Evaluate Boltzmann:Evaluate Boltzmann:

Van Hove singularities removed by disorderVan Hove singularities removed by disorder

Simulation gives correct resistanceSimulation gives correct resistance

Inhomogeneous systemsInhomogeneous systems

Varying local disorder (impurities)Varying local disorder (impurities) CPPCPP Simple cubic modelSimple cubic model Mean-free-path ~ 34aMean-free-path ~ 34a00 (outer) (outer) Series combinationSeries combination 0

300

600

900

1200

10 40 70 100

Plane number in z-direction

(e

V)

Varying electronic structureVarying electronic structure More relevant to devicesMore relevant to devices Variable V (W/V fixed)Variable V (W/V fixed) Density reflects g(EDensity reflects g(EFF)) Interface resistanceInterface resistance Compare to Boltzmann:Compare to Boltzmann:

For CPP many believe mfp not importantFor CPP many believe mfp not important Spin-diffusion lengthSpin-diffusion length Valet-Fert Boltzmann solution:Valet-Fert Boltzmann solution:

GMR independent of the ordering of magnetic layers in a multilayerGMR independent of the ordering of magnetic layers in a multilayer

Experiment: consider two arrangementsExperiment: consider two arrangements Interleaved:[Interleaved:[FM1FM155PP55FM2FM255PP55]]NN Separated: [Separated: [FM1FM155PP55]]NN[[FM2FM255PP55]]NN Each FM has a different coercivityEach FM has a different coercivity

Problems with experimentsProblems with experiments Very difficult as nVery difficult as nΩ – SQUIDΩ – SQUID Do not know spin-diffusion lengthDo not know spin-diffusion length Do not know how collinear the magnetisation isDo not know how collinear the magnetisation is But experiments do indicate a differenceBut experiments do indicate a difference

Mean free path in CPP GMRMean free path in CPP GMR

Can simulate model system Can simulate model system directlydirectly

Infinite spin-diffusion lengthInfinite spin-diffusion length Perfect collinear magnetisationPerfect collinear magnetisation Vary Anderson disorder to modify Vary Anderson disorder to modify

mfpmfp

As disorder decreases a As disorder decreases a significant difference arises in significant difference arises in the parallel configurationthe parallel configuration

However GMR is barely affectedHowever GMR is barely affected

Interpretation:Interpretation: mfp effect is mostly due to proximity of interfaces mfp effect is mostly due to proximity of interfaces More interfaces and larger resistances in parallel up More interfaces and larger resistances in parallel up

spin channelspin channel However more current flows in less resistive parallel However more current flows in less resistive parallel

down channeldown channel Effect on GMR is weak for collinearEffect on GMR is weak for collinear Will become more significant for non-collinearWill become more significant for non-collinear

Can extract (crudely) the non-local part if the Can extract (crudely) the non-local part if the resistance that depends on the mfpresistance that depends on the mfp

0

0.5

1.0

1.5

2.0

0 0.0005 0.0010 0.0015 0.0020 0.0025

W2

AR

(f

m2)

CPP GMRCPP GMR

CuCu1212CoCo44CuCu33CoCo44CuCu1212

fcc (100) in z-directionfcc (100) in z-directionAt interface use average matrix elementsAt interface use average matrix elementsAnderson disorder 0.5eVAnderson disorder 0.5eV

ρρCuCu=3.2μΩcm=3.2μΩcm ρρCoCo=9.7μΩcm (asymmetry 4.7)=9.7μΩcm (asymmetry 4.7)

ARARminmin=3.0fΩm=3.0fΩm22 (1.1fΩm (1.1fΩm22))

ARARmajmaj=0.7fΩm=0.7fΩm2 2

ARARmaj/minmaj/min=2.0fΩm=2.0fΩm22

GMR = 67%GMR = 67%

P

AP

Interface resistanceInterface resistance

Spin-dependent interface Spin-dependent interface resistance at FM/PM interfaceresistance at FM/PM interface

Due to electronic structure Due to electronic structure mismatchmismatch

Calculation:Calculation: APW electronic structure (match APW electronic structure (match

transmitted and reflected waves)transmitted and reflected waves) Resistance evaluated using the Resistance evaluated using the

Landauer-BLandauer-Büüttiker formulattiker formula

Ge2

Ï€T

1 T

Spin-relaxationSpin-relaxation

Decay of spin coherenceDecay of spin coherence Very important for spintronicsVery important for spintronics

Mechanisms:Mechanisms: Magnons (at high temperatures)Magnons (at high temperatures) Domain wallsDomain walls Spin-orbitSpin-orbit

Dirac equation:Dirac equation:

Matrix elements between states of Matrix elements between states of different spindifferent spin

SymmetrySymmetry

Atomic SCHF

Elliott-Yafet mechanismElliott-Yafet mechanism

Spin-orbit alone does not cause spin Spin-orbit alone does not cause spin flipflip

Diagonalise Hamiltonian with unitary Diagonalise Hamiltonian with unitary transformation:transformation:

As weak perturbation still called `up’ and As weak perturbation still called `up’ and `down’`down’

Scattering of these states mixes spinsScattering of these states mixes spins PhononsPhonons ImpuritiesImpurities Dislocations and vacanciesDislocations and vacancies Spin-flip time depends on momentum Spin-flip time depends on momentum

scattering timescattering time

Preliminary results for Cu and CoPreliminary results for Cu and Co

CoCo ρρCoCo=9.7μΩcm (asymmetry 4.7)=9.7μΩcm (asymmetry 4.7) D=0.57.10D=0.57.10-3-3mm22ss-1-1

ΤΤsfsf=0.4ps=0.4ps llsfsf=15nm (20-80nm=15nm (20-80nm****))

CuCu ρρCuCu=3.2μΩcm=3.2μΩcm D=0.0065mD=0.0065m22ss-1-1

ΤΤsfsf=25ps=25ps llsfsf=400nm (350-1000nm=400nm (350-1000nm**))

Copper

***

Future workFuture work

Spin relaxation for many elementsSpin relaxation for many elements EUROPA-HPCEUROPA-HPC

Interleaved / separated for realistic electronic Interleaved / separated for realistic electronic structuresstructures

NanowiresNanowires

Non-linear responseNon-linear response Dynamic self consistent EC potentialDynamic self consistent EC potential

AdditionalAdditional

TemperatureTemperature

Advantage of EOM is can include dynamic interactions Advantage of EOM is can include dynamic interactions (e.g. phonons)(e.g. phonons)

Computational cost is lowComputational cost is low

Simple model:Simple model: Einstein oscillatorEinstein oscillator 1/r1/r22 dependence with cut-off dependence with cut-off

Advanced modelAdvanced model Molecular dynamicsMolecular dynamics Vibration frequencies filtered – BE distributionVibration frequencies filtered – BE distribution

Similarly:Similarly: MagnonsMagnons Spin-glasses (Ising/Heisenburg EOM)Spin-glasses (Ising/Heisenburg EOM)

Interface resistance proximity Interface resistance proximity effectseffects

Difficult to separate bulk and interface in generalDifficult to separate bulk and interface in general Even classically bulk disorder can increase overall Even classically bulk disorder can increase overall

transmissiontransmission QM QM →→ non-local conductivity non-local conductivity

Natural definition of the effect of interfacesNatural definition of the effect of interfaces

LaLb

Configuration I: 2N-1 interfaces

Configuration II: 1 interface

R

I

R

II

Calculate RCalculate RIntInt as a function of L as a function of LAA

Interface resistance indicated by density differenceInterface resistance indicated by density difference

For large separations approaches Boltzmann resultFor large separations approaches Boltzmann resultEffective resistance is reduced as interfaces get closerEffective resistance is reduced as interfaces get closer

Filtering effectFiltering effect

Current in the planeCurrent in the plane

Freedom to simulate any Freedom to simulate any geometry:geometry:

CIP with varying electronic structureCIP with varying electronic structure

Steady state cross section at Steady state cross section at midpoint:midpoint:

I

50 ML

4 ML

Significant mean-free-Significant mean-free-path effect for currentpath effect for currentDensity must change Density must change abruptly to maintain abruptly to maintain equipotentialequipotential

Other geometries that can be investigated:Other geometries that can be investigated: NanowiresNanowires ConstrictionsConstrictions

Spin relaxation Spin relaxation calculationcalculation

Method of calculationMethod of calculation

Random wavefunction initialisedRandom wavefunction initialised Random phasesRandom phases Contains all eigenstates with equal probabilityContains all eigenstates with equal probability

Wavefunction is filtered so just at EWavefunction is filtered so just at EFF::

Spin imbalance createdSpin imbalance created Spin amplitudes multipliedSpin amplitudes multiplied Chemical potential imbalance createdChemical potential imbalance created

TDSE evolved and spin-polarisation relaxesTDSE evolved and spin-polarisation relaxes Average over sets of phasesAverage over sets of phases Fit chemical potential difference to exponential decayFit chemical potential difference to exponential decay