bdth f e i tlitband theory for experimentalists, for mgo...

B d Th f E i t li tBand Theory for Experimentalists,for MgO TMRfor MgO TMR

인하대학교물리학과인하대학교물리학과

유천열

[email protected]

Spin Summer School July. 2008 2

ContentsContents

1. Introduction2. Simple Band Theoryp y

Let’s try to read & understand the band diagram !!

3. Origin of Magnetism (Quantum) g g (Q )Magnetism in 3d metals (practical)

4. Giant (MgO) TMR4. Giant (MgO) TMR


Recent Hot issues in MagnetismRecent Hot issues in Magnetism

Spin-Torque

CIMSDomain Wall

BMRSpin Transistor

DMS Spin InjectionDomain Wall

Microwave

GMRTMR

BMRPMA Spin Transistor

Spin-FET

Microwave

MgO-TMR

1990 1995 2000 2005

GMR

1985

MOMgO TMR

Spin-LED


1990 1995 2000 20051985

Recent two important progressesRecent two important progressesSpin Transfer Torque p q

CIMS: Current Induced Magnetic SwitchingNew way to switching magnetization directiony g gSTT (Spin Transfer Torque)-RAM

CIDWM: Current Induced Domain Wall MotionRace-Track Memory

Microwave generationTunable oscillator, high Q-factor

Giant TMR (MgO barrier)Fe/MgO/FeTheory & Experiments

Spin Summer School July. 2008 인하대학교 5

FerromagnetFerromagnetEE

EF , ,n k n k EXE E E± = Δm

( )2

( ) ( )FE

M D E D E dE

IM

+ −= −∫N↑(E) N↓(E)

Ferromagnetic metal( )

2

( ) ( )2

FEFM IME D E D E EdE+ −= − −∫I S E h I l

, when ( ) 1FM PM FE E ID E< >

I : Stoner Exchange Integral

, ( )FLarge Density of State causes ferromagnetism in 3d, 4d, 5d.


Conductivity of 3d metals

2

*nemτσ =*m

1 2| | ( )scatt FV N Eτ− =

Effective mass of conduction electron decreasedEffective mass of conduction electron decreased due to the large effective mass of d-electron: m*Get more scattering with localized d electron

Spin Summer School July. 2008

Get more scattering with localized d electron.인하대학교 7

Two current modelTwo-current model

Spin is conserved after many collisions:

Well below TC (~ 1000K)Spin wave excitation is not easy

Each spin current behave independently R↑

R↓


↓

인하대학교 8

TMR (Tunneling Magneto-Resistance)TMR (Tunneling Magneto Resistance)


Properties of d orbitalProperties of d-orbital

Well localized : important magnetic property!!In transition metals, d orbit is not important in the , pchemical properties


ContentsContents






Simple Band TheorySimple Band Theory

Vector Space Hydrogen Moleculey g1-dim. chain moleculesB dBand energy2, 3-dim. band structureBrillouin zonesDOS in 2 3 dimDOS in 2, 3-dim.Fermi surface


Vector Space IVector Space I

Example of 3 dim.k̂ A

r1 2 3

ˆˆ Â Ai A j A k= + +r

kx̂ ⋅ ŷ = ŷ ⋅ ẑ = ẑ ⋅ x̂ = 0ˆ ˆ ˆ ˆ ˆ ˆ

î ĵAr

x̂ ⋅ x̂ = ŷ ⋅ ŷ = ẑ ⋅ ẑ = 1

k̂ A1 2 43

ˆˆ ˆ Â Ai A j A A lk= + + + +r

L

î ĵ

l̂4

Â A l= ⋅r


i j A4: projection of A to l.

Vector Space IIVector Space II

Ex) Fourier Series

a ∞ ( ) ( )01

( ) cos sin2

ˆ ˆc s

n nn

af t a n t b n t

a b

ω ω=

∞

= + +⎡ ⎤⎣ ⎦

= +

∑

∑0

c sn n nnn

a b=

= +∑

anˆ ˆc sdnn n are orthonormal basis vector

Projection to the is Integral is the inner product

ĉ nn1 ( ) cos( )ma f t t d t

π

πω

π −= ∫

Integral is the inner product.


Complete basis setComplete basis set

In the vector space, if we have complete basis set, we can represent any functions.The wave function formed complete basis set.

∞

0

( ) ( )n k kk

r a rψ φ∞

=

= ∑r r

Bra-ket notation:

∫ ( ) ( )*volume

r r dVψ φ ψ φ= ∫r r


Basis vectors for Hydrogen IBasis vectors for Hydrogen I

Solution of Schrödinger Eq.S h i l h i Y (θ φ)Spherical harmonics Ylm(θ,φ)Linear combinations of Ylm(θ,φ) is more convenient.


Basis vectors for Hydrogen IIBasis vectors for Hydrogen IIxy yz zx

Linear combinations of Ylm(θ,φ).2 2 2

2 2 2

2 2 2 2

2 2

, ,

3,2

xy yz zx

x y z

xy yz zxd d dr r r

x y z rd d

= = =

− −= =2 2,2 2 3x y zr r−


Basis vectors for Hydrogen IIIBasis vectors for Hydrogen IIIExample of linear combination of Yl (θ φ)Example of linear combination of Ylm(θ,φ).

a 1a a= + +1a− 1a+ 0a= + +

1 1, 1

1 1, 1

y

y

a Y p

a Y p

− −

+ +

=

=i ⎡ ⎤

1, 1 1, 1 1,01 1 0y a a Yp aY Y− +− ++ +=

Angular momentum of p and d orbit from the

,

0 1,0

y

ya Y p=py =

i2

Y1,−1 + Y1,1⎡⎣ ⎤⎦

Angular momentum of p- and d-orbit from the orbital shape. (Sutton p. 15)


Hydrogen MoleculeHydrogen Molecule

Two hydrogen atoms1 21

1ˆ |1 |1sH E= 2

ˆ | 2 | 2sH E=

2

ˆ | |H E

1 2

| |H Eψ ψ=

1 2| |1 | 2c cψ = +


Solve H MoleculesSolve H2 Molecules

1 2

1

ˆ

1

, 1 2ˆ 1 2

H E c c

H c c

ψ ψ ψ

ψ

= = +

= ⎡ + ⎤⎣ ⎦

1ˆ |1 |1sH E=ˆ | 2 | 2H E=1 2

1 2

1 1

2

1 2ˆ 1 22

H c c

H c c

ψ

ψ

= ⎡ + ⎤⎣ ⎦

= ⎡ + ⎤⎣ ⎦

2 | 2 | 2sH E=

0 1,1 1,1

1 1 2 1sE s s E

β

= ≠ E0: On-site matrix elementβ: Hopping matrix element1,1 2,1s sβ = β: Hopping matrix element

c E c Ecβ+ = E E β E β+⎧1 0 2 11 2 0 2

c E c Ecc c E Ec

ββ

+ =

+ =0

0

0E E

E Eβ

β−

=−

0

0

EE

Eββ

+⎧= ⎨ −⎩


Bonding & anti bonding statesBonding & anti-bonding states

0

0

EE

Eββ

+⎧= ⎨ −⎩

β is always negative

0 β⎩

Corresponding wave functions

( )1 1 22a

Ψ = −

( )2

1 1 22b

Ψ = +( )2


Orbits of HOrbits of H2( )1 1 2

2bΨ = +

http://wps.prenhall.com( )

2

( )1 1 2aΨ = −Spin Summer School July. 2008 22

( )2a

1 dimensional chain molecules1-dimensional chain molecules1 2 3 + + + j N

Assumptions

1 − 2 − 3 − + − + − + − j − N −…… ……

Assumptions 1. known j-th atomic orbit:

H ilt ij

Ĥ H H H H H2. Hamiltonian3. Each states are orthonoromal iji j δ=

1 2 12 13 1NH H H H H H= + + + + +L L

4. We want to solve

ˆ ˆN

∑N

jΨ ∑1

jj

H E H c j=

Ψ = Ψ = ∑1

jj

c j=

Ψ =∑


Matrix for the 1D chainMatrix for the 1D chain

0 1i j ±⎧0, 1ˆ ,

1

i ji H j i j

i jαβ

≠ ±⎧⎪= =⎨⎪ ±⎩

Nearest hopping only

, 1i jβ⎪ = ±⎩N x N matrix for N chains

1

ˆ ˆN

jj

i H i H c j=

⎛ ⎞Ψ = ⎜ ⎟

⎝ ⎠∑

1

2

cEcE

α ββ α β

β

− ⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟− ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ MO O

1

ˆ

j

N

j ij

c i H j Ec=

⎝ ⎠

= =∑1

0

NcE

ββ

β α β −

⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

MO O

MO O

j 1N

NcEβ β

β α⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠


Ring or N⇒¶Ring or N⇒¶

Periodic boundary condition, ignore surface effect.In real system N~ 1023

( )E α0 ,N j N jc c c c += =

1 1( ) 0j j jEc c cαβ− +−

− + =

T i l l ti [ ]i jθ2 , where 0,1, 1m Nmπθ = = −L

Trial solution: [ ]exp ijc jθ= − Bloch Theorem

( ) ( )

, where 0,1, 1

1 2 2exp i m , 2 cosmjm

NN

c EN

mN

j

mθ

π πα β⎡ ⎤ ⎛ ⎞= = + ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠Spin Summer School July. 2008 25

pj N NNβ ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠

Energy SpectrumEnergy Spectrum

For infinite (or large) N θFor infinite (or large) N, θis continuous variable.It is related withIt is related with momentum of electron.

Spin Summer School July. 2008 2622 cos , mE

Nπα β θ θ= + =

For N=5 casesFor N=5 cases

Remember the hydrogen casey gMore node, less stable states

Energy

stable statesβ is always negative


From atom to solidsFrom atom to solids


More hoppingMore hopping …

1 2 32 cos 2 cos 2 2 cos3E α β θ β θ β θ= + + +1 2 3β β β


2 3 Dim cases2, 3-Dim. cases

Almost similar to the 1D case.Using translation symmetry Bloch theoremUsing translation symmetry, Bloch theorem

( )expk kr T ik T r+ Ψ = ⋅ Ψr rrr rr r

( ) ( )mj kc c R→ rr( )

( )k kR

c R RΨ =∑r rr r

( )j kj R→

r

( )( ) expc R N ik R= rr r( )( )

( ) exp

1 exp

k

k

c R N ik R

ik R RN

= ⋅

Ψ = ⋅∑

r

rr

r r r


RN

For simple cubic latticesFor simple cubic lattices

( )2E k k kβ ( )2 cos cos cosx y zE k a k a k aα β= + + +

2mπ2

2

xx

m

n

kN a

k

π

π

=

=

2

yy

z

kN a

k lπ

=

=zzN a

Momentum (k-) space: Fourier transform of the real spaceB ill i f k t


Brillouin zone: range of proper k-vectors

Brillouin ZoneBrillouin Zone

( )2E k k kβ

The range of kx y z⇒ BZ

( )2 cos cos cos zx yE k a k ka aα β= + + +

g x,y,zIn 2D case

ky πa

,πa

⎛⎝⎜

⎞⎠⎟

−πa

,πa

⎛⎝⎜

⎞⎠⎟

kxπΕ

πa

,− πa

⎛⎝⎜

⎞⎠⎟

−πa

,− πa

⎛⎝⎜

⎞⎠⎟

kx

ky−π

Spin Summer School July. 2008 32π −π

Free electron model in 2DFree electron model in 2D

2D square latticeReduced BZReduced BZ


1st Brillouin Zones1st Brillouin Zoneshttp://cst-www.nrl.navy.mil/bind/kpts/index.html

Simple cubicFace centered cubic

Spin Summer School July. 2008 34Hexagonal Body centered cubic

Limit of free electron modelLimit of free electron model


Density of State (DOS) IDensity of State (DOS) I

Definition of DOS: D(E): Number of state per unit energy at give E.

S: total number of state up to energy E( ) dSD E

dE=

S: total number of state up to energy E.D(E): Number of seat at E.


Density of State (DOS) IIDensity of State (DOS) IIdS( ) ( )dSD E dS D E dEdE

= → =

By the definition of S:

ky

ydS=D(k) x blue ring area

kEblue ring area

0E E dE

E EdS dk

= +

⊥∫kx

E+dE

Edk⊥

0E E⊥=∫

kdE E dk⊥= ∇ r

⎛ ⎞

( )0

0

3

32

E E dEx y z

E Ek

N N N a dSdS dEEπ

= +

=

⎛ ⎞⎜ ⎟=⎜ ⎟∇⎝ ⎠

∫r

( )0

0

3

3( ) 2

E E dEx y z

E E

N N N a dSD EEπ

= +

=

⎛ ⎞⎜ ⎟=⎜ ⎟∇⎝ ⎠

∫r


( ) 02 k Eπ ⎜ ⎟∇⎝ ⎠r

Density of State (DOS) IIIDensity of State (DOS) III

DOS: Shape of theDOS: Shape of the basket


s p d bands, p, d band

sp-band: Free electron like, parabolic b dband structuresLarge β, large band width, small D(E)D(E)

d band:d-band: Localized characterSmall β small band width largeSmall β, small band width, large D(E)


Fermi surfaceFermi surface

The electron occupy from the lowest energy state.The seat is filling from the best view.

If the number of electron is less than total possible states…

If the # of audience is smaller than # of seat…Fermi energy: the energy of the highest occupied energy states.Fermi surface: the equi-energy (Fermi energy) surface in the k-space.


Small filling caseSmall filling case

( )( )2 2 2, ,

2 cos cos

4

F x y

F x F y F

E k a k a

E a k k

α β

α β β

= + +

= + − +( ), ,

2

4

y

FF

Eka

α ββ

− −=

−Fermi surface is a circle !

ky πa

,πa

⎛⎝⎜

⎞⎠⎟

−πa

,πa

⎛⎝⎜

⎞⎠⎟

kx πΕπa

,− πa

⎛⎝⎜

⎞⎠⎟

−πa

,− πa

⎛⎝⎜

⎞⎠⎟ ky−π

Ε

Spin Summer School July. 2008 41kx

π −π

Half filling caseHalf filling case( )2 cos cosE k a k aα β= + +( )

( ), ,2 cos cos

cos cos 0,

F x y

x F y F F

E k a k a

k a k a E

α β

α

= + +

+ = =

, ,x F y Fk k aπ

= ± ± Fermi surface is a square!!

ky πa

,πa

⎛⎝⎜

⎞⎠⎟

−πa

,πa

⎛⎝⎜

⎞⎠⎟

kxπΕ

πa

,− πa

⎛⎝⎜

⎞⎠⎟

−πa

,− πa

⎛⎝⎜

⎞⎠⎟

kx

ky−π


Almost full filling caseAlmost full filling casek π δ( )

( )2 2 22 cos cos

4 2

F x y

F x y

E k a k a

E a

α β

α β β δ δ

= + +

= − + +

,

1

Fk a a

a

π δ

δ

≈ −

( ) a

Fermi surface is a circle centered at ,π π⎛ ⎞± ±⎜ ⎟⎝ ⎠

ky πa

,πa

⎛⎝⎜

⎞⎠⎟

−πa

,πa

⎛⎝⎜

⎞⎠⎟

a a⎜ ⎟⎝ ⎠

kxπΕ

πa

,− πa

⎛⎝⎜

⎞⎠⎟

−πa

,− πa

⎛⎝⎜

⎞⎠⎟

kx

ky−π


Physics of Fermi surfacePhysics of Fermi surface

In most case, the electrons on the Fermi surface are only interacting with an external perturbationEspecially important in

Transport propertiesTransport propertiesMagnetismTunneling

Imagine the surface of the water in the basket.gIn real system, it is 3D surface.


Real Fermi Surfaces IReal Fermi Surfaces Ihttp://www.phys.ufl.edu/fermisurface/


Real Fermi Surfaces IIReal Fermi Surfaces IIhttp://www.phys.ufl.edu/fermisurface/


Details of Fermi SurfaceDetails of Fermi Surface

( )ConductivityFerromagnetism,

( ) ( )kk k q

F Eq dk

E Eχ

+

∝−∫

r

r r r

rr

g ,antiferromagnetism

Fermi surface nesting

k k q+

Fermi surface nestingInterlayer exchange

licouplingPeriod ~ caliper vector of Fermi surface


Bands of 3d metalsBands of 3d metals


Hybridization of the bandHybridization of the band


ContentsContents






Band Theory for Magnetism IBand Theory for Magnetism I

l d h i h l β hs-electrons are more spread out, so that it has large β than d-elec.Localized d band and free electron like s band are mixedLocalized d-band and free electron like s-band are mixed up.


Band Theory for Magnetism IIBand Theory for Magnetism II

F f i i i d d hi h iFor ferromagnetism, spin is ordered which is seems violate thermodynamics law. (entropy)Why does the spin up and down band shift?Why does the spin-up and down band shift?By Hund’s rule, electron start to fill up the state with parallel spin (minimize Coulomb Energy)parallel spin. (minimize Coulomb Energy).But, there is energy cost to fill up higher band state with parallel spins.parallel spins.The exchange split energy can lower the energy.Competition between exchange energy and kinetic energyCompetition between exchange energy and kinetic energy

Stoner Criterion


Stoner CriterionStoner CriterionE

±

EF, ,n k n k EXE E E± = Δm

( )( ) ( )FEM D E D E dE+ −∫N↑(E) N↓(E)

( )

( )2

( ) ( )

( ) ( )FEFM

M D E D E dE

IME D E D E EdE

+

+ −

= −

= − −

∫

∫N↑(E) N↓(E)Ferromagnetic metal

( )( ) ( ) 2E D E D E EdE∫I : Stoner Exchange Integral

, when ( ) 1FM PM FE E ID E< >

Large DOS causes ferromagnetism in 3d, 4d, 5d.DOS & lattice spacing is important in magnetism.


Lattice spacing and DOSLattice spacing and DOS

Surface and interface magnetism Lattice spacing is very important. p g y p

In simple band model, D(EF) ~ 1/β.


Slater Pauling Curve ISlater-Pauling Curve I

Co=1.7Fe Ni ~1 7

Fe=2.2

Fe50Ni50~1.7Slope ~ 1

Slope ~-1

Ni=0.6

Fe

Co NiM Co NiMnCr


Slater Pauling Curve IISlater-Pauling Curve II

Rigid band model:Band shape is not changed with alloying. p g y gNot a good approximation, but …


Band structure of 3d metalsBand structure of 3d metals

Good guide line for the materials selections.


ContentsContents






Importance of Giant TMRImportance of Giant TMR

Physical point of view:Theoretical prediction ⇒ ExperimentsMore understanding of TMRStill many questions

Applications point of view :Bigger signals in MRAM, HDD headSpin torque, Spin diode (Nature 438 (2005)).

H. Ohno

FeCoB/MgO/FeCoB system shows the best performances350 % TMR @ RT


Requirement of giant TMRRequirement of giant TMR

TMR of F/AlO /F : 50 %TMR of F/AlOx/F : ~ 50 %Large but not enough for practical applications

Butler et al. : Predict giant gTMR in Fe/MgO/Fe system

More than Julliere’s ModelMore than Julliere s ModelBand symmetry of Fe and MgO is key issuekey issueEpitaxial crystal growth is essential


Theoretical predictions l l (2000)Butler et al. PRB (2000).

J. Mathon et al. PRB (2001)(2001).

More than 1000 % TMR is possible in theory.possible in theory.It is not easy to grow epitaxial Fe/MgO/Feepitaxial Fe/MgO/Fe system


Spin filteringSpin filtering

ΔΔ2’

Δ5

Δ1

MgO Barrier


MgO Barrier

인하대학교 62

AIST : MBE, Oscillatory TMRIBM : SputteringCoFeB gives good results


ReferencesReferences

M d M ti M t i l R C O’H dlModern Magnetic Materials, R. C. O’HandleyPhysics of Ferromagnetism, S. ChikazumiIntroduction to Magnetic Materials, B. D. CullityIntroduction to Magnetic Materials, B. D. CullityMagnetism, Principles and Applications, D. CraikFundamentals of Physics, D. Halliday금속원자계의다체이론, 김덕주Electronic Structure of Materials, A. P. SuttonIntroduction to Solid State Physics C KittelIntroduction to Solid State Physics, C. Kittelhttp://hyperphysics.phy-astr.gsu.edu/hbase/hph.htmlLecture note of “Introduction to solid state physics” @ Orange State

iUniv. I took many figures from various web sites and references.


bdth f e i tlitband theory for experimentalists, for mgo...

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