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Center for Materials for Information Technologyan NSF Materials Science and Engineering Center

Modeling of Electronic Structure and Transport

Bill ButlerJulian Velev

Amrit BandyopadhyaySanjoy Sarker


CrO2, TiO2 and RuO2 have the Rutile Structure(tetragonal with 2 forumla units/cell)

Lattice constants differ by less than 4%.


Electronic Structure of CrO2, TiO2 and RuO2

EF

•RuO2 is a non-magnetic metal

•TiO2 is an insulator

•CrO2 is a half-metalic ferromagnet

•Multilayers with CrO2 electrodes and TiO2 or RuO2 spacer layers should have interesting transport properties.

EF


Solid Solutions of Ilmenite (FeTiO3) and α-Hematite (Fe2O3)

Ti

Fe

O FeO

Hematite

•Hematite has corundum structure.

•Replacing alternate layers of Fe atoms by Ti yields Ilmenite.

•Continuous solid solutions of composition:(1-x)(FeTiO3)·x (Fe2O3) can be grown.

•Disorder on cation lattice affects magnetic and electrical properties.Ilmenite


Related systems•Ti2O3 has corundum structure and is antiferromagnetic!•0.2µB/Ti•TN = 660K•Some other corundum/ilmenite systems:

•Cr2O3•V2O3•VTiO3•MnTiO3•CoTiO3•NiTiO3

•LiNbO3•Al2O3


Calculated Electronic Structure of α-Hematite

-10

-5

0

5

10

-4 -2 0 2 4

DO

S (S

tate

s/eV

)

Energy (eV)

DOS for HematiteEFMajority

Minority-4-3-2-101234

-4 -3 -2 -1 0 1 2 3 4 5

DO

S (S

tate

s/eV

)

Energy (eV)

d-DOS for Fe Atom

EFMajority

Minority

First-Principles LSDA calculations predict that α-Hematite is an anti-ferromagnetic insulator. The majority DOS for an Fe atom equals minority DOS for Fe atom on neighboring layer. Calculated moment is ~ 4µB/Fe.


a

b

b

a

a

b

Ilmenite-Hematite Solid Solutions30 atom supercell calculations

95

100

105

110

115

120

125

4 4.5 5 5.5 6 6.5 7

Inte

grat

ed D

OS

Energy (eV)

EF

Integrated DOS for Fe11TiO18

Minority

Majority

no Ti

Ti impurity adds 4µB and 1 electron.


The calculated maximum net moment of 4µB/Ti agrees qualitatively with experiment for ordered ilmenite-hematite

solid solutions.

Bozorth et al. 1957

Annealed

Quenched

Ilmenite magnetic structure


Impurities in α-Fe2O3

Sc Ti V Cr Mn Fe Co Ni Number of up electrons

102 102 102 102 102 102 103 104

Number of down electrons

97 98 99 100 101 102 102 102

Magnetic moment

5 4 3 2 1 0 1 2

Number of p-type carriers

0 0 Gap state

0 Gap state

0 0 1

Number of n-type carriers

0 1 Gap state

0 Gap state

0 Gap state

0


Electronic Structure Calculations

• DFT seems to work well for Rutile oxides predicting structure and properties correctly– Future: Transport properties of multilayers

• For corundum-structure TM oxides, DFT predicts physical and magnetic structure (including magnetic anisotropy and tendency for Ti to sit on alternate layers) correctly.– Future: Investigate transport properties

• Future –– Magnetic anisotropy and magnetostriction.– Layered systems (continuation of established work)


Spin Transport in Confined Geometries

Niwires

1-10 nm

• Nanochannels– CPP GMR

with Confined Current Paths

Insulating layer with nanochannels Pinned layer

Electrodes

Free layer

• Nanowires– Spin transfer

effect

Cu lead

CoJσ

Cu lead

Co

• Nanocontacts– Very large

magnetoresistancefrom nanoscaleconstrictions


Transport Modeling Tools

• Classical effective circuit model – Is being used to investigate CCP-CPP in collaboration

with H. Fujiwara and K. Nagasaka.• First-principles based semi-classical Boltzmann

approach• Semi-empirical tight-binding approach• Layer Korringa-Kohn-Rostoker Approach

– Kubo (Quantum Linear Response)– Landauer (Conductance from Transmission Probability)


Classical Effective Circuit Model for CPP

•Each element (nm-scale) is replaced by four (or six) resistors for each spin channel.

•Interfacial resistances modeled by effective resistors.

•Allows treatment of two or three dimensional current flow as needed for CCP.

V2

V1


Why is a classical resistor model sensible for CPP?The solution to the CPP-BTE differs from the Series Resistor

model in two ways.

• There is an interfacial resistance that arises from electron reflection at the interfaces.

• There are exponential terms in the current induced by the interfacial reflections• Empirical interfacial resistance would include effects of disorder at interface.


First-Principles Based Semi-Classical Transport Theory

Co CoCu

τ⇑ τ⇑

τ⇓ τ⇓τ

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P PS S

PR

T

S = R + T

• Electrons are treated as classical objects that obey Fermi statistics and correct quantum dispersion.

• First-Principles parameters:– Fermi surface, kz– electron velocity, vx,vy,vz– Transmission and Reflection

Probabilities, Tij and Rij

• Empirical Parameters:– Electron lifetimes: τ– P and S are probabilities for

an electron to reflect specularly at the boundaries and interfaces respectively.


Current work emphasizes calculation of specularity parameters

in

TR

RS=SRTS=ST

• The scattered wave will have specular and diffuse components. – Specular component conserves k||

• Usual Assumption: – A fraction S of reflected and

transmitted beams at interfaces is specular

– A fraction P of reflected wave at boundary is specular

– The diffusely scattered electrons are scattered isotropically

– These assumptions are not accurate

in

RS=P


Atom Probe Field Ion Microscopy supports a model for interfaces in which layers interfaces are structurally regular, but interdiffused.

Co-Cu multilayer

D. J. Larson and N. Tabat -SeagateNi-Co-Cu multilayer


If we can obtain the composition of each layer, either from experiment or from analysis of simulations, how do we obtain

S, the interfacial diffuse scattering parameter?

• Treat the disordered layers assubstitutional alloys.

• Treat the disorder using the coherent potential approximation.

• Obtain T and R in the presence of the disordered interface


Independent Configuration Averaging of retarded and advanced Green functions is equivalent to retaining only specular part of the transmitted or

reflected beams.

CopperCobalt

' '|| || || ||

' '|| || || ||

' '|| || || ||

( , , ) ( , , ) ( , ) 0, 0

( , , ) ( , , ) ( , , ) ( , , ) vertex corrections

( , ) ( , ) diffuse transmission

G I J k G I J k T k k I J

G I J k G I J k G I J k G I J k

T k k k kδ

+ −

+ − + −

∝ < >

≈ +

∝ +


In the free electron limit, this approach agrees with the approach of Dugaev et. al. PRB -1995. However, the free electron approximation is not accurate even for the

majority channel.Sp

ecul

arity

Para

met

er fo

r Tra

nsm

issi

on

0.9989

0.999

0.9991

0.9992

0.9993

0.9994

0.9995

0.9996

0

Comparison Between Analytic Solution and CPA Result for Free Electron System

's.cocu.mixstep.bhb.free''s_fit5.cocumix.dat'

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

kx


Calculated Effect of Disordered Interface on transmission Probability for Majority Co Bloch electrons incident on Cu (100)

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Tran

smis

sion

Pro

babi

lity

kx

Transmission Probability vs. k||

no intermixing

10%intermixing

20%intermixing

30% intermixing50%intermixing

• Transmission probability in majority channel is much larger than expected from free electron model for perfect interface.

• S parameter should depend on k||.• Two interfacial layers are intermixed in this example.


Calculated Reflection Probability for Majority Co Bloch electrons incident on Cu (100)

• S can be quite different for reflected and transmitted beams.• Disorder can actually increase probability of specular reflection.• Multiple reflections become important for higher intermixing.

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Ref

lect

ion

Prob

abili

ty

kx

Reflection at a Majority Co-Cu Interface

0.00.10.20.30.40.5

Fraction two layers intermixed


A small amount of disorder can cause a large amount of diffuse scattering in the minority channel

• Two atomic layers are assumed to be 10% interdiffused.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Tran

smis

sion

Pro

babi

lity

kx

Effect of Disorder on Minority Band by Band Transmission

Band-1

No intermixing

10% intermixing

Band-2

Band-3


The reflection probability is also sensitive to disorder in the minority channel.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Ref

lect

ion

Prob

abili

ty

kx

Effect of 10% Disorder on Minority Reflection

no mixing

10% mixing on two layers


Classical and Semiclassical Models

• Classical model– Tool ready for use by Fujiwara group

• Semiclassical Model– Calculation of specular beam for disordered interface understood

• Test using semi-empirical tight-binding approximation– Use tight-binding approximation to simulate diffuse scattering

from disordered interface• Test CPA based theory for diffuse scattering

– Incorporate into semi-classical Boltzmann transport equation


The semi-empirical tight-binding model will be used to treat nanocontacts

Nanocontact


The Layer KKR Approach allows first-principles based calculation of transport properties of magnetic multilayers

with non-collinear moments.

Nanowire

•Both Layer KKR and tight-binding will be applied to treat spin-transfer. We will calculate the spin-flux. Pieter Visscher’s group will use the calculated spin-flux as input to calculate how magnetization changes.

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