mesoscopic anisotropic magnetoconductance fluctuations in ferromagnets

Mesoscopic Anisotropic Magnetoconductance

Fluctuations in Ferromagnets

Shaffique Adam

Cornell University

PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 

For details: S. Adam, M. Kindermann, S. Rahav and P.W. Brouwer,

Phys. Rev. B 73 212408 (2006)

Mesoscopic Anisotropic Magnetoconductance

Fluctuations in Ferromagnets

PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 

Mesoscopic Anisotropic Magnetoconductance

Fluctuations in Ferromagnets

PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 

Quantum Magnetism

Mesoscopic Anisotropic Magnetoconductance

Fluctuations in Ferromagnets

PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 

Quantum Magnetism

Electron Phase Coherence

Mesoscopic Anisotropic Magnetoconductance

Fluctuations in Ferromagnets

PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 

Quantum Magnetism

Electron Phase Coherence Ferromagnets

Mesoscopic Anisotropic Magnetoconductance

Fluctuations in Ferromagnets

PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 

Quantum Magnetism

Electron Phase Coherence Ferromagnets

Phase Coherent Transport in Ferromagnets

Mesoscopic Anisotropic Magnetoconductance

Fluctuations in Ferromagnets

PiTP/Les Houches Summer School on Quantum Magnetism, June 2006 

Quantum Magnetism

Electron Phase Coherence Ferromagnets

Phase Coherent Transport in Ferromagnets• Motivation (recent experiments)

• Introduction to theory of disordered metals• Analog of Universal Conductance Fluctuations in nanomagnets

Picture taken from Davidovic groupPicture taken from Davidovic group

nmL 30~

nm5~Cu-Co interface are good contactsCu-Co interface are good contacts

regime disordered ,L

Motivation:

Physical System we are Studying [2]

•Aharanov-BohmAharanov-Bohm contributioncontribution

• Spin-Orbit EffectSpin-Orbit Effect

Data/Pictures: Y. Wei, X. Liu, L. Zhang and D. Davidovic, PRL (2006)Data/Pictures: Y. Wei, X. Liu, L. Zhang and D. Davidovic, PRL (2006)

]2/,0[

mV)(V

T)(B

10

10-

Physical System we are Studying [3]

2~ c

Aharanov-Bohm Aharanov-Bohm contributioncontribution

Spin-Orbit EffectSpin-Orbit Effect

10~ c

anglen correlatio is c

am

~mkk

SkkVSO

)(~

)(~

Introduction to Phase Coherent Transport

A

I V

Sample dependent fluctuations areSample dependent fluctuations arereproducible (not noise)reproducible (not noise)

Ensemble AveragesEnsemble Averages

Need a theory for the mean <G> and Need a theory for the mean <G> and fluctuations <GG>fluctuations <GG>

Smaller and colder!Smaller and colder!

[Mailly and Sanquer, 1992]

Image Courtesy (L. Image Courtesy (L. Glazman)Glazman)

Introduction to Phase Coherent Transport [2]

Impurities

Electron

Electron diffusing in a dirty metal

*2

*

,

2

||

||

AAAAAA

BA

BA

P

P

eCAi

Classical ContributionClassical ContributionQuantum Interference

path

DiffusonDiffuson Cooperon

fluxpath,~

Introduction to Phase Coherent Transport [3]

Weak Localization in Pictures

Introduction to Phase Coherent Transport [3]

Weak Localization in Pictures

Introduction to Phase Coherent Transport [3]

For no magnetic field, the phase depends only on the path.

Every possible path has a twin that isexactly the same, but which goes around in the opposite direction.

Because these paths have the same fluxand picks up the same phase, they caninterfere constructively.

Therefore the probability to return to thestarting point in enhanced (also calledenhanced back scattering).

In fact the quantum probability to returnis exactly twice the classical probability

Weak Localization in Pictures

Introduction to Phase Coherent Transport [4]

Weak Localization in Equations

fluxpath,~

path

hreversepat

*

,

2

*

,

*2

*

,

2

2

||||

||

||

AAAAAAAAA

AAA

BA

BA

BA

P

P

P

eCAi

*2

*

,

2

||

||

AAAAAA

BA

BA

P

P

eCAi

Classical ContributionClassical ContributionQuantum Interference

DiffusonDiffuson Cooperon

Universal Conductance Fluctuations

[Mailly and Sanquer (1992)][Mailly and Sanquer (1992)]

Theory: Lee and Stone (1985), Altshuler (1985)Theory: Lee and Stone (1985), Altshuler (1985)

[C. Marcus][C. Marcus]

Review of Diagrammatic Perturbation Theory (Kubo

Formula)

DiffusonDiffuson

CooperonCooperon

Conductance G ~Conductance G ~

)(

12 iDq

)~(

12 iqD

Cooperon defined similar to Diffuson upto normalizationCooperon defined similar to Diffuson upto normalization

Calculating Weak Localization and UCF

G

G

Weak LocalizationWeak Localization

cGG

Universal Conductance FluctuationsUniversal Conductance Fluctuations

Calculation of UCF Diagrams

nn

DCh

e2, 4

22 14

2

3

][2

32

22

JCJCJDJDTrLh

eGG

z

Sum is over the Diffusion Equation Eigenvalues scaled by Sum is over the Diffusion Equation Eigenvalues scaled by Thouless EnergyThouless Energy

2

zT L

DE,

2

2

2

22

Ty

zy

x

zxzn E

Ei

L

Ln

L

Lnn

,2,1,0,

,2,1

yx

z

nn

n

Quasi 1D can be done analytically, Quasi 1D can be done analytically, and 3D can be done numerically: Var G = 0.272and 3D can be done numerically: Var G = 0.272

90

1 4

4

n n 15

2~

15

1,

22

DCh

eGG x 4 for spinx 4 for spin

Effect of Spin-Orbit (Half-Metal example) [1]

Ferromagnet

DOS(E)DOS(E)DOS(E)DOS(E)

Fermi EnergyFermi Energy

EnergyEnergy

Spin DownSpin DownSpin UpSpin Up

Half Metal

Effect of Spin-Orbit (Half-Metal example) [2]

H

Without S-OWithout S-O With S-OWith S-O

V

qqVV

V

qq

kk

2

)'('

'

V

qqVV

kkkmiVV

so

soq

soq

Fkkso

kk

2

)'(

/)'(

'

2''

)(

12 iDq

so

mmiDq

'1

1

2

)(

12 iDq

so

mmiDq

'1

1

2

NOTE: For m=m’, Spin-Orbit does not affect the Diffuson (classical motion)NOTE: For m=m’, Spin-Orbit does not affect the Diffuson (classical motion)but large S-O kills the Copperon (interference)but large S-O kills the Copperon (interference)

Calculation of C(m,m’) in Half-Metal

Without S-OWithout S-O With S-OWith S-O

GG mm’

==

)( )(

m

m’

m

m’

90

1 4

4

n n

)](sinh)coth(2[4)(

1 224

4

22xxxx

xAnn

soTEx

)cos(1

)(

12 iDq

==

so

iDq

cos1

1

2

==2)()()()( mGmGmGC

soTEA

)cos(1

Results for Half-Metal

D=3, Done NumericallyD=1, Analytic Result

00

22 cos1cos1

2

3)(

sTsT EF

EF

h

eC

4

22 )(sinh)coth(2)(

x

xxxxxF

We can estimate correlation anglefor parameters and find about fiveUCF oscillations for 90 degree change

, of definition changes ,

Full Ferromagnet

H

Half MetalHalf Metal FerromagnetFerromagnet

V

qqVV

kkkmiVV

so

soq

soq

Fkkso

kk

2

)'(

/)'(

'

2''

so

mmiDq

'1

1

2

221'

'

/)'()( Fyxz

kkso

zzkk

kkkeemiV

EV

soTEA

cos1

EquationDiffusion of

sEigenvalue are )(2 An

EquationDiffusion 22 of

sEigenvalue are )(2

ann

Results for C(m,m’) in Ferromagnet

)()(2

3)(

22

aFaF

h

eC 4

22 )(sinh)coth(2)(

x

xxxxxF

Limiting Cases for m = m’

m=m’ SO C D spin Total

Normal Metal - 1/15 1/15 4 8/15

Half Metal No 1/15 1/15 1 2/15

Half Metal Strong 0 1/15 1 1/15

Ferromagnet Weak 1/15 1/15 2 4/15

Ferromagnet Strong 0 1/15 1 1/15

Conclusions:

Showed how spin-orbit scattering causes Mesoscopic Showed how spin-orbit scattering causes Mesoscopic Anisotropic Magnetoconductance Fluctuations in half-Anisotropic Magnetoconductance Fluctuations in half-metals (This is the analog of UCF for ferromagnets)metals (This is the analog of UCF for ferromagnets)

This effect can be probed experimentallyThis effect can be probed experimentally

mkk

SkkVSO

)(~

)(~

2~5 cmV)(V

T)(B

10

10-

Magnetic Properties of Nanoscale Conductors   

Shaffique Adam

Cornell University

Backup Slides

Backup Slide

2~ c

Aharanov-Bohm contribution

Spin-Orbit Effect

10~ c

anglen correlatio is c

20 ~~

Lc1~~~ L

LE soTsoc

Backup Slide

Density of States quantifies how closely packed are energy levels.DOS(E) dE = Number of allowed energy levels per volume in energy window

E to E +dE

DOS can be calculated theoretically or determined by tunneling experiments

Fermi Energy is energy of adding one more electron to the system (Large energy because electrons are Fermions, two of which can not be in the same quantum state).

DOS(E)DOS(E)

EnergyEnergy

Fermi EnergyFermi Energy

Backup Slide

DOS(E)DOS(E)

EnergyEnergy

Fermi EnergyFermi Energy DOS(E)DOS(E)DOS(E)DOS(E)

Fermi EnergyFermi Energy

EnergyEnergy

Spin DownSpin DownSpin UpSpin Up

Backup Slide

• Magnetic Field shifts the spin up and spin down bands

Spin DOS

Ferromagnet

DOS(E)DOS(E)

Fermi Energy

Energy

Spin DownSpin Up

Half Metal

Backup SlideWeak localization (pictures)

For no magnetic field, the phase depends only on the path.

Every possible path has a twin that isexactly the same, but which goes around in the opposite direction.

Because these paths have the same fluxand picks up the same phase, they caninterfere constructively.

Therefore the probability to return to thestarting point in enhanced (also calledenhanced back scattering).

In fact the quantum probability to returnis exactly twice the classical probability

Backup SlideWeak localization (equations)

*2

*

,

2

||

||

AAAAAA

BA

BA

P

P

eCAi

Classical ContributionClassical ContributionQuantum InterferenceQuantum Interference

fluxpath,~

path

hreversepat

*

,

2

*

,

*2

*

,

2

2

||||

||

||

AAAAAAAAA

AAA

BA

BA

BA

P

P

P

eCAi

DiffusonDiffuson CooperonCooperon

Backup SlideWeak Localization and UCF in

Pictures

<G><G>

<G G><G G>

Backup Slide

Backup Slide

Backup Slide

Backup Slide

Introduction to Quantum Mechanics Energy is Quantized Wave Nature of Electrons (Schrödinger Equation)

Wavefunctions of electrons in the Hydrogen Atom (Wikipedia)

Scanning Probe Microscope Image of Electron Gas (Courtesy A. Bleszynski)