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)
39
Embed
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). - PowerPoint PPT Presentation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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