mesoscopic anisotropic magnetoconductance fluctuations in ferromagnets
DESCRIPTION
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 PresentationTRANSCRIPT
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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)
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Mesoscopic Anisotropic Magnetoconductance
Fluctuations in Ferromagnets
PiTP/Les Houches Summer School on Quantum Magnetism, June 2006
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Mesoscopic Anisotropic Magnetoconductance
Fluctuations in Ferromagnets
PiTP/Les Houches Summer School on Quantum Magnetism, June 2006
Quantum Magnetism
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Mesoscopic Anisotropic Magnetoconductance
Fluctuations in Ferromagnets
PiTP/Les Houches Summer School on Quantum Magnetism, June 2006
Quantum Magnetism
Electron Phase Coherence
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Mesoscopic Anisotropic Magnetoconductance
Fluctuations in Ferromagnets
PiTP/Les Houches Summer School on Quantum Magnetism, June 2006
Quantum Magnetism
Electron Phase Coherence Ferromagnets
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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
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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
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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:
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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-
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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
)(~
)(~
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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)
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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,~
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Introduction to Phase Coherent Transport [3]
Weak Localization in Pictures
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Introduction to Phase Coherent Transport [3]
Weak Localization in Pictures
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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
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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
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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]
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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
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Calculating Weak Localization and UCF
G
G
Weak LocalizationWeak Localization
cGG
Universal Conductance FluctuationsUniversal Conductance Fluctuations
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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
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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
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Effect of Spin-Orbit (Half-Metal example) [2]
H
Without S-OWithout S-O With S-OWith S-O
V
qqVV
V
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)
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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
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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 ,
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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
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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
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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-
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Magnetic Properties of Nanoscale Conductors
Shaffique Adam
Cornell University
Backup Slides
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Backup Slide
2~ c
Aharanov-Bohm contribution
Spin-Orbit Effect
10~ c
anglen correlatio is c
20 ~~
Lc1~~~ L
LE soTsoc
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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
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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
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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
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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
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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
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Backup SlideWeak Localization and UCF in
Pictures
<G><G>
<G G><G G>
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Backup Slide
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Backup Slide
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Backup Slide
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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)