Upload
others
View
8
Download
0
Embed Size (px)
Citation preview
Dieter Weiss
Experimentelle und Angewandte Physik
Magnetism and Spin-Orbit Interaction:
Some basics and examples
Magnetization pattern of ferromagnet determines stray field
ferromagnet (Ni)
2DEG
a
Transport in a two-dimensional electron gas with superimposed periodic magnetic field (le >> a)
(Our initial) Motivation
B (T)
ρρ ρρ xx
( ΩΩ ΩΩ)
( )= ± 1C 4
Commensurability (Weiss) Oscillations :
2R n a SdH oscillations
Transport in a periodic magnetic field
see, e.g., P.D. Ye PRL 74, 3013 (1995)
A little bit of historyHans Christian Ørsted
Albert Einstein
SciencephotoLIBRARY
1820: Electric current generates magnetic field
1905: On the electrodynamics of moving bodiesMagnetic field stems from Lorentz contraction of moving charges
Magnetism
Basic conceptsmagnetic moments, susceptibility, paramagnetism, ferromagnetismexchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basicsRashba- and Dresselhaus contribution, SO-interaction and effectivemagnetic field
Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces
Magnetism and Spin-Orbit Interaction:
Magnetism
Basic conceptsmagnetic moments, susceptibility, paramagnetism, ferromagnetismexchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basicsRashba- and Dresselhaus contribution, SO-interaction and effectivemagnetic field
Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces
Magnetism and Spin-Orbit Interaction:
Magnetic moment and angular momentum
I
S
Current in a loop generates magnetic moment µµµµ, constitutes magnetic dipole
area SI=μ S
As mass is transported around the loop ⇒ µµµµ is connected to angular momentum
2 [Am ] gyromagnetic ratio= γ γ =μ L
Size of atomic magnetic moment
− −= = ⇒ µ = − = − = − − µπ
µ= − µ = =
ℏ
ℏ
ℏ
B
BB
e ev 1 1 e LI evr L
T 2 r 2 2me
with Bohr Magneton2m
μ L-e-
v
µµµµ
L
Magnetization M: magnetic moments / volume [A/m]
http://www.orbitals.com
/orb/ov.htmAngular momentum: quantum mechanical picture
Total angular momentum of an atomis composed of angular momentumof all occupied electron states L and of all corresponding spins S
Total angular momentum (L-S coupling):
= +J L S
µ= −ℏ
Bgμ J
Connection between µµµµ and J
with Landé factor
( ) ( ) ( )( )
+ + + − += +
+J J 1 S S 1 L L 1
g 12J J 1
Susceptibility
Connection between field H (e.g. generated by coil) and magnetization M:
= χM Hχ <
χ >diamagnetic if 0paramagnetic if 0
( )0Connection between B and H field : (for M H)= µ + <<B H M
( )0 0 (1 ) ⇒ = µ + χ = µ + χB H H H
Not the whole story!
( )= µ + +0 dB H H M
demagnetizing field or stray field(important for ferromagnets)
H Linear materials:
magnetic susceptibility
permeability of free space
+
+
+
M
Hd [ ]∇ ⋅ = ∇ ⋅ µ + =d0 ( ) 0HB M
=∇ ⋅ −∇ ⋅dH M
Any divergence of M creates a magnetic field – can be seen assort of (bound) magnetic charges
Demagnetizing field
Easy expression for strayfield (demagnetizing field) Hd only for ellipsoids:
= −
+ + =
xx
yy
zz
xx yy zz
d
N 0 0
0 N 0
0 0 N
with N N N 1
MH
Some examples:
= = = 1xx yy zz 2N 0, N N = = = 1
xx yy zz 3N N N
= = =xx yy zzN N 0, N 1
x
y
z
M
( )− ⇒
==µ + + = µ0 0
d
d
M
B H M
H
H H
H
long cylindrical rod
sphere
flat plate
Paramagnetism (of isolated magnetic moments)Requisite: Atoms with non-vanishing angular momentum J
Energy of magnetic moment E = − ⋅μ B
Classically: arbitrary values and orientation of µµµµ
0 p€€€€€2
p 3 p€€€€€€€€€€2
2 p
f
- 1
- 0.5
0
0.5
1
UmB
02
π2π3
2
ππ
1
0
-1
U/µ
B
φ
Bz z J B
gJ m g
µµ = − = − µℏ
z JJ m= ℏ
ℏ32
ℏ12
ℏ52
− ℏ12
− ℏ32
− ℏ52
J: discrete values and orientations of , e.g. J 5 / 2=Quantum mechanically μ
Jm J,J 1,... (J 1), J= − − − −
2J+1 discrete values(and hence energy levels)
Jm J BAssociated energy levels E m g B get thermally occupied
= µ
z−µ
Magnetization (thermal average)
Magnetization ( )B JM NgJ J: total angular mome uB x nt m= µ
Brillouin function ( ) BJ
B
2J 1 2J 1 1 x gJ BB x coth x coth ; with x
2J 2J 2J 2J k T
+ + µ = − =
J
B S
Saturation:
for x 1: B 1
and M NgJ M
N :Density of magnetic atoms
>> →→ µ =
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Bx gJ B / kT= µ
B J(x) 1 3
J ,1, ,2...2 2
= ∞
( )
( )
−
<< ⇒ ≈ + −+
⇒ ≈
+ µ + µ + µ µ= µ ≈ = µ ⇒ χ = ≈
J
2 2 2 2 2 2B B B 0
B J 0B B B
"High T":1 x
x 1 coth x ...x 3
J 1xB
J 3g J(J 1) g J(J 1) g J(J 1)M 1
M NgJ B x N B N H N3k T 3k T H 3k T
Paramagnetism (of free electrons)
D(E)
E
EF
2gµBB
( ) B BMagnetization: M N N N↑ ↓= − µ = ∆ µ
( )F B BF
1 3 NN D E 2g B g B
2 2 E∆ = ⋅ µ = µ
( )FF
3 ND E in 3D
2 E=
Resulting magnetization:
2 2B B 0
F B F
3 N 3 NM g B g H
2 E 2 k T= µ = µ µ
χ1
dia para3Taking into account diamagnetism of free electrons (M M )= −
2B 0
B F
Ng
k Tχ = µ µ Independent of temperature!
Exchange Interaction (a la Heisenberg)
Exchange due to Pauli exclusion principle and Coulomb repulsion
An electron pushes a second electron with the samespin orientation out of a region with diameter λ →exchange hole forms
Exchange holeConsequences of spin arrangement for total energy
↑↑
↑↓
For parallel orientation, e.g., potential energy gets reduced compared
to anti-parallel arrangement, e.g., since Coulomb-repulsion is reduced
due to larger average interparticle distance.
On the ↑↑ other hand kinetic energy is larger for since Fermi-energy increases due to fixed electron number.
delicate interplay!
λ
Increase of kinetic energy (example)
Fermi energy EF of a free electron gas increases if, for fixedelectron density only one spin species is permitted
D (E)↓↑D (E) D (E)↑ D (E)↓
δE
FE
FE
EE
1R 2R
1r
2r
( )2
2 21 2 1 2
ˆ V( , ) E2m
ψ = − ∇ + ∇ ψ + ψ = ψH r rℏ
Exchange Interaction (a la Heisenberg)
Two electron wave function of hydrogenmolecule like system:
Fermi-Dirac statistics requires that
ψ = −ψ1 2 2 1( , ) ( , ) exchange of electronsr r r r
Construction of Ψ with Heitler-London approximation:
[ ]s 1 2 1 1 2 1 1 S2 2 2( , ) ( ) ( ) ( ) ( )ψ = ϕ ϕ χ+ ϕ ϕr r r r r r
Spin of both electrons parallel (Triplett, S=1)
Spin of both electrons anti-parallel (Singulett, S=0)
symmetric anti-symmetric under particle exchange
[ ]T 1 2 1 1 2 1 1 T2 2 2( , ) ( ) ( ) ( ) ( )ψ = ϕ ϕ χ− ϕ ϕr r r r r r
anti-symmetric symmetric under particle exchange
χ
↑↑
↑↓ + ↓↑
↓↓
↑↓ − ↓↑
1
0
-1
0
1
1
1
0
S ms
Triplett
Singulett
χT symmetric
χS anti-symmetric
1 2⋅S S
1
4
1
4
1
4
3
4−
Spin wave function χχχχ
see, e.g. Stephen Blundell Magnetism in Condensed
Matter Oxford University Press
1R 2R
1r
2r
1 2 1 2
S S T TS T
T TS S
2 2 2 2
1 2 1 1 2 2 1 1 2 21 21 2 1 1 2 2
S T S T( , ) and ( , ) have different energy E and E
ˆ ˆH HE E .........
e e e e2 d d ( ) ( ) ( ) ( )
ψ ψ
ψ ψ ψ ψ− = −
ψ ψψ ψ = ϕ ϕ + − − ϕ ϕ −− − −
∫
r r r r
r r r r r rR Rr r r R r R
exchange integral J
Exchange splitting
sign of J determines spin ground state
exclusively due to the exchange of the two electrons
Ashcroft, Mermin
S energy ms
1
0
10-1
J
J/4
-3J/4
B
0
S T
S T
J 0 E E spins parallelJ 0 E E spins anti-parallel
> ⇒ > ⇒
< ⇒ < ⇒
Heisenberg Hamiltonian
Expressing spin Hamiltonian with Eigenvalues ES and ET in termsof S1 and S2:
1S T S T 1 24
ˆ ˆˆ (E 3E ) (E E )= + − − ⋅H S S 1 2ˆ ˆ⋅ =S S
14
34
S 1
S 0
+ =
− =
T
S
ˆ has Eigenvalue E for triplett ( or ) and
E for singulett ( or )
↑↑ ↓↓
↑↓ ↓↑
H
Shift of origin results in usual form of Heisenberg Hamiltonian
S T 1 2 1 2ˆ ˆ ˆ ˆˆ (E E ) J= − − ⋅ = − ⋅H S S S S
Generalisation for many spins: i ji,j
ijˆ ˆˆ J= − ⋅∑H S S
Exchange integral between spin i and spin j
2ˆusing that S(S 1)= +S
Forms of exchange interaction
Direct exchange interaction
from direct overlap
Superexchange
mediated by paramagnetic atomsor ions
RKKY Interaction(After M.A. Rudermann, C. Kittel, T. Kasuya, K. Yosida)
coupling mediated by mobile charge carriers
Example: p-d exchage interaction in a 2DHG
Mn (S=5/2) in two-dimensional hole gas
Nature Physics
6, 9
55 (201
0)
Wurstbauer et al.
see: WP-20 Anomalous Hall effect in Mn doped,p-type InAs quantum wellsC. Wensauer, D. Vogel et al.
Weiss molecular field Pierre Weiss 1907
j j B j j j BS replaced by constant moment: g / / g= − µ ⇒ = − µμ S S μ
ℏ ℏ
i i
magnetic moment of spin μ S
j
i
Sum of all act on
one particular spin
S
S
= λM
MWeiss molecular field B
Magnetic moments get aligned in an effective field Beff = B0+BM >>B0
B i iex i ij j i i
Bj
2i
ex ij j i i ij i M2B j jB
gE 2 J ,
g
2E 2 J J
g (g )
− µ= − ⋅ = ⇒ = −
µ
= ⋅ = − ⋅ = − ⋅ µ µ
∑
∑ ∑
S μS S μ S
μS μ μ μ B
ℏ
ℏ
ℏ ℏ
Idea: Replace interaction of a given spin with all other spins by interaction withan effective field (molecular field)
Ferromagnetism within the molecular field picture
Molecular field Bm=λM allows to map ferromagnetism onto paramagnetism
J BJ B
g J (B )2J 1 2J 1 1 xM Ng J coth x coth with x
2J 2J 2J 2J kTMµ+ + = µ − =
+ λ
M on both sides of equation:graphical or numerical solution
BC
B
SJg (J 1)T
3k
Mµ + λ=
3C
M S
Für T ~ 10 K und
B M ~1500
J 1
T
/ 2== λ
⇒
Heisenberg Model unable to describe all aspects of ferromagnetism e.g. momentof iron is 2.2 µB. Stoner picture:
Band magnetism and Stoner criterion
2kin F
1E D(E ) E
2∆ = δ
Redistribution of spins: Energy cost EkinE
EF EδResulting magnetization
( ) B F BM n n D(E ) E↑ ↓= − µ = µ δ
Energy of electron‘s magnetization dM ineffective (Weiss) field Beff of other electrons
M 22 2
eff p2Bot F
0
M 1dEpot dM B E dM M D(E ) E
2 2λ′ ′= − ⋅ → ∆ = − λ = −λ = µ− δ∫
2kin pot F
2B F
1Energy gain if E E D(E ) E [1- D(E )] 0
2∆ + ∆ λµ= δ <
2FBFerromagnetism occurs if D(E ) 1 (holds for Fe,Ni,C o)⋅ ≥λµ Stoner criterion
D (E)↓D (E)↑
Ferromagnet: Density of States D(E)
majority spins
minority spins
s-electrons
d-electrons
majority spins
minority spins
Schematic Nickel
Magnetic Domains
Formation of domains reduces stray field energy!
2d d 0 d 0 d
space sample
1 1Energy of stray field : E dV dV
2 2= µ = − µ ⋅∫ ∫H H H M
Width of a Bloch wall
Energy to rotate spin by angle θ
= − ⋅ = − θ2ex 1 2E 2J 2JS cosS S
θ = → = − 2ex0 E 2JS
Classical Heisenberg Hamiltonian
θ ≠ → ∆ = θ θ <<2 2ex0 E JS for 1
π ππ ⇒ θ = ⇒ ∆ =2
2exSpin rotation by over N sites energy cos t per line : E JS
N N
22ex
ex 2 2
EEnergy per unit area JS
a Na
∆ πε = =
a: lattice constant ⇒ → ∞best N ?
Magnetocrystalline anisotropy
Ferromagnetic crystals have easy and hard axes. Along some crystallographicdirections it is easy to magnetize the crysal, along others harder. Reflects thecrystal symmetry.
uniaxial anisotropy (e.g. Co) cubic anisotropy (e.g. Fe)
= θ + θ +2 41 2E K sin ( ) K sin ( ) ... ( )= θ ϕ + θ θ +2 2 2 21
1 4E K sin ( )sin (2 ) cos ( ) sin ( ) ....
Mθ
ϕ
Extra energy associated with magnetocrystalline anisotrpy:
see: WP-22, C. Butschkow et al.
Example: GaMnAs core-shell nanowires
system with very strong uniaxialanisotropy
Width of a Bloch wall, continued
assume simple form of anisotropy energy
2aniE Ksin ( ); K 0= θ >
energy cost of rotation out of easy direction
N2 2
ii 1 0
N NKKsin Ksin d
2
π
=θ ≈ θ θ =
π∑ ∫
NKaenergy per unit area=
2
a
22
BW 2NKa
exchange energy anisotropy energy JS2 Na
π+ ε = +
BW 32J
From d / dN 0 N S NaK
2J
aS
Kaε = ⇒ = ππ ⇒ =
domain wall width
Reversal of magnetization: Hysteresis
-HC HS
MSMr
M
H
macroscopic bulk material consisting of many domains
Processes
1. Shift of domain walls (reversible and irreversible)2. Domain rotation towards nearest easy axis3. Coherent rotation at high fields
Magnetization of a cube with uni-axial anisotrpy
W. Rave, K. Fabian, A. Hubert, JMMM 190, 332 (1998)
u dnormalized anisotropy constant K / K
two-domain-statevortex-state
single-domain-state
multi-domain-state
micromagnetic simulation
anisotropy
samplesize
Magnetism
Basic conceptsmagnetic moments, susceptibility, paramagnetism, ferromagnetismexchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basicsRashba- and Dresselhaus contribution, SO-interaction and effectivemagnetic field
Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces
Magnetism and Spin-Orbit Interaction:
Measurement of domains: magnetic force microscopy
MFM measures „magnetic charges“
1. topography scan 2. MFM scan (lift mode)
lift height
cantilever withmagnetic tip
magnetization M div M sim. MFM picture real MFM picture
Domains, measured by MFM
4Å Fe / 4Å Gd (75 layers) 50 Mbyte hard drive
Perpendicular magnetization in-plane magnetization
4 ground state configurations
in-plane and out-of plane mag-netization component can beswitched independently!
Between multidomain and single-domain state: vortex structure
Picture: Ref 3)
1) J. Raabe, R. Pulwey et al., J. Appl. Phys. 88, 4437 (2000)2) T. Shinjo et al., Science 289, 930 (2000)3) A. Wachowiak et al., Science 298, 577 (2002)
Detection of in-plane vortex
Lorentz-microscopy probes in-plane magnetization…
300 nm
J. Raabe, R. Pulwey et al., J. Appl. Phys. 88, 4437 (2000)
Detection of vortex singularity
J. Raabe, R. Pulwey et al., J. Appl. Phys. 88, 4437 (2000)
Another technique: Micro Hall magnetometry
allows to measure hysteresis traces
Hs
IU B
en=
M
M
Measures directly the stray field, averaged over the central area of the Hall junction
Hysteresis trace of a vortex state
M. Rahm et al., J. Appl. Phys. 93, 7429 (2003)
1.8 mm
Vortex-Golf
Magnetism
Basic conceptsmagnetic moments, susceptibility, paramagnetism, ferromagnetismexchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basicsRashba- and Dresselhaus contribution, SO-interaction and effectivemagnetic field
Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces
Magnetism and Spin-Orbit Interaction:
Origin of spin-orbit (SO) interaction
SO B 2ˆ ˆH
2mc
× = −µ ⋅
E pσ Zeeman B
ˆ ˆH = −µ ⋅σ B
eff 22mc
×= E pBvector of Pauli
spin matrices
= −µB effE BSpin-orbit interation due to orbital motion
magnetic moment of electron
+
+-
-nucleus
electron
electron‘s restframe
Beff
Bulk inversion asymmetry (BIA)Lack of inversion symmetry in III-V semiconductors"Dresselhaus contribution γ"
Structure inversion asymmetry (SIA)due to macroscopic confining potential: "Rashba contribution α". Tunable by external electric field!
ψ 2(z)
V(z)
∝ γ −
xBIA
y
kB
k
ySIA
x
kB
k
∝ α
−
kx
BBIA
ky
BSIA
kx
ky
Electric field E in solids
2 2
SOkˆ ˆ with
2m= +H Hℏ
SO x y y x x x y yˆ ( k k ) ( k k )= α − + γ −H σ σ σ σ
SO interaction in 2DEG: Rashba & Dresselhaus terms
Pauli spin matrix
Rashba Dresselhaus
tunable by gate voltage
Calculation of Eigenvalue
− = = = −
= α −
RashbaSO x y y x
x y z0 1 0 i 1 0
ˆ ˆ ˆ; ; 1 0 i 0 0
ˆ ( k k )
1σ σ σ
H σ σ
α α +− α = − = α α −α
= − =
α
y y xx
y
RashbaSO x y y x
y xx
0 k 0 (k ik )0 i k
k 0 (k ik ) 0
ˆ ( k k
i k 0
)H σ σ
2 2x yEigenvalues of the matrix : k k k±α + = ±α
Pauli Spin matrices
Description of zero-field spin splitting by Beff
= α σ − σ + γ σ − σ σ ⋅
SO x y y x x x ey fy fˆ H ( k k ) ( ~ ˆ Bk k ) ;
Rashba Dresselhaus
x x y y z zeff B B Bˆ Bσ ⋅ = σ + σ + σ
Comparison of coefficients. E.g. only Rashba contribution:y
effx
kB
k
∝ α −
effB
E.A. de Andrada e Silva PRB 46, 1921 (1992)
Presence of Rashba & Dresselhaus contributions
Rashba orDresselhaus
Rashba and Dresselhaus
2 2kE ( ) k
2m= ± α − γ
ℏ
2 2kE ( ) k
2m= ± α + γ
ℏ
]
I
BVx
SO-interaction in a InGaAs quantum well
Vg
Beating of SdH-oscillations∝ ρxx
Nitta et al., Phys. Rev. Lett 78, 1335 (1997)
Quantum oscillations (SdH) reflect k-space area
1 2
B A
eπ ∆ = ℏ
kx
ky
A
kF
Periodicity of Shubnikov-de Haas (SdH) oscillations
2 2F sNote that A k 2 n= π = π
Origin of beating: two periodicities due to two k-space areas A1 and A2
ρxx
B
A1
A2
Nitta et al., Phys. Rev. Lett 78, 1335 (1997)
Tuning of Rashba coefficient a by gate voltage Vg
Nitta et al., Phys. Rev. Lett 78, 1335 (1997)
yeff
xCorresponds to tuning of spin orbit f
kBield, i.e.
k,
= α −
Magnetism
Basic conceptsmagnetic moments, susceptibility, paramagnetism, ferromagnetismexchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basicsRashba- and Dresselhaus contribution, SO-interaction and effectivemagnetic field
Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces
Magnetism and Spin-Orbit Interaction:
Epitaxial Fe-GaAs interface
[110]
[110]
As Ga
Fe
aGaAs=5.653 Å aFe=2.867 Å
Spin-orbit interaction plays also role in tunneling
Substrate
GaAs8 nm
Fe
Substrate
Fe
GaAs
Substrate
Au
Device fabrication
As-Cap
J. Moser et al. Appl. Phys. Lett. 89, 162106 (2006)
AlGaAs
Fe
GaAs
TEM-micrograph:C.-H. Lai, Tsing Hua University, Taiwan
Are always two ferromagnetic layers necessary to see a magnetizationdependent resistance?
Our model system: Fe/GaAs/Au with epitaxial Fe/GaAs interface
≠1 2R R
FeGaAsAu
TAMR: Tunneling Anisotropic Magnetoresistance
See also:Gould et al. PRL 93, 117203 (2004)(Ga,Mn)As/Al2O3/Au
R1 R2
MM
easy
Spin-valve like signal
Tunneling magnetoresistance: B dependence
φ = 0
J. Moser et al., Phys. Rev. Lett. 99, 056601, 2007
φ
B
-88 mV
Double step/single step switching dueto magnetic anisotropyFe on GaAs: Cubic + uniaxial anisotropy
[110]
B
[110]
measured at T = 4.2 K and -90 mV
Co/Fe(epi)/GaAs(8nm)/Au
Fe
GaAsAu
J. Moser et al., Phys. Rev. Lett. 99, 056601, 2007
Angular dependence of TAMR: negative bias
B=0.5T
R
M
Co/Fe(epi)/GaAs(8nm)/Au
Angular dependence of TAMR: positive bias
J. Moser et al., Phys. Rev. Lett. 99, 056601, 2007
measured at T = 4.2 K and +90 mV
Fe
GaAsAu
R
M
B=0.5T
Fe Au
zl zr
z
EF
fb
0 z BR DH H H H H= + + +
BR x y y x ii l r
i,
1H ( p p ) (z z )
== σ − σ δ −α∑ℏ
z(z)
H2
∆= − ⋅n σ
D x x y y1
H ( p (zz
)p )z
∂ ∂ = σ − σ ∂ ∂γ
ℏ
2
01
H V(z)2 m(z)
= − ∇ ∇ +
ℏ
2l r||
1|3 |
,1
T (E,ke
I dEd k [f (E) f (E)])(2 )
σσ=−
= −π ∑ ∫ℏ
particle transmissivity
lα rαγA. Matos-Abiague & J. Fabian, PRB 79, 155303 (2009)
Modelling
GaA
s
γγγγ = 0
αααα > γγγγ > 0 αααα < 0
γγγγ > 0
αααα = 0
[100]kx
ky
[100]
kx
ky
ky
[100]
kx
[100]
kx
ky
||w(k )
y x
x y
k - k
- k k
0
α γ α + γ
||( ) =w k
Spin-orbit field w(k)
lα γ
||( ) =w k
y x
x y
k - k
- k k
0
α γ α + γ
2[110]|| ||
Anisotropy determined by
T( ) f([ ( )] ) R / R 1~ (cos2 1)≈ ⋅ → − αγ φ −k n w k
Anisotropy vanishes for 0αγ →
Fe
GaAsAu
n
||w(k )
x
y
z
Origin of anisotropic resistance: superposition of Rashba & Dresselhaus SO-contribution
Fe Au
zl zr
z
EF
φb
J. Moser et al., Phys. Rev. Lett. 99, 056601, 2007
γ in GaAs: 24 eVA3°
2l (eVA )α °
-25.1-17.4
-0.6
42.3
Angular dependence of TAMR: bias dependence
J. Moser et al., Phys. Rev. Lett. 99, 056601, 2007See also:T. Uemura, Appl. Phys. Lett. 98,102503 (2011)
φ −= [110]
[110]
R( ) RTAMR
R
Fe on (Ga,Mn)As
Au
Fe
(Ga,Mn)As
si-GaAs (001)
F. Maccherozzi et al. PRL 101, 267201 (2008)
M. Sperl et al., Phys. Rev B 81, 035211 (2010)
XMCD: antiferromagnetic coupling
RT
US pat 61/133,344
Thin ferromagnetic iron film induces antiferromagnetic coupling in (Ga,Mn)As at room temperature. Minimum thickness of the coupled(Ga,Mn)As layer is at least 1 nm
F. Maccherozzi et al. PRL 101, 267201 (2008)M. Sperl et al., Phys. Rev B 81, 035211 (2010)
2 3 4
5
Proximity effect: exchange coupling betweenFe and Mn at Fe/(Ga,Mn)As interface
Spin injection: Proximity effect enhances TC
FP-34 C. Song et al.WP-30 M. Ciorga et al.WP-105 J. Shiogai et al.
Magnetism
Basic conceptsmagnetic moments, susceptibility, paramagnetism, ferromagnetismexchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basicsRashba- and Dresselhaus contribution, SO-interaction and effectivemagnetic field
Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces
Magnetism and Spin-Orbit Interaction:
In retrospect
The End