Magnetism and Spin-Orbit Interaction:
Some basics and examples

Dieter Weiss
Experimentelle und Angewandte Physik

(Our initial) Motivation
Transport in a two-dimensional electron gas with superimposed periodic
magnetic field (le >> a)ferromagnet (Ni)2DEGaMagnetization pattern of ferromagnet determines stray fieldTransport in a periodic magnetic field
Commensurability (Weiss) Oscillations :()

2RC = n ± 41 a

ρxx (ΩΩ)

SdH oscillations

B (T)see, e.g., P.D. Ye PRL 74, 3013 (1995)

A little bit of history

Hans Christian Ørsted

SciencephotoLIBRARY

1820: Electric current generates magnetic field
Albert Einstein1905: On the electrodynamics of moving bodies
Magnetic field stems from Lorentz contraction ofmoving charges

Magnetism and Spin-Orbit Interaction:
Magnetism
Basic concepts
magnetic moments, susceptibility, paramagnetism, ferromagnetism
exchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basics
Rashba- and Dresselhaus contribution, SO-interaction and effective
magnetic field
Example:Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial
Fe/GaAs interfaces

Magnetism and Spin-Orbit Interaction:
Magnetism
Basic concepts
magnetic moments, susceptibility, paramagnetism, ferromagnetism
exchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basics
Rashba- and Dresselhaus contribution, SO-interaction and effective
magnetic field
Example:Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial
Fe/GaAs interfaces

Magnetic moment and angular momentum
Current in a loop generates magnetic moment µ, constitutes magnetic dipoleS
area Sμ = ISI
As mass is transported around the loop ⇒ µ is connected to angular momentum
μ = γL [Am2 ]γ = gyromagnetic ratio

Size of atomic magnetic moment−e −ev11e
LI==⇒ µ = − evr = −L = − − µB
T
2πr
2
2m
ℏ
µ
eℏ
μ = − B L with µB == Bohr Magnetonℏ2m

Magnetization M: magnetic moments / volume [A/m]

L
v
-e

µ

Angular momentum: quantum mechanical picture

Total angular momentum (L-S coupling):
J =L+S

Connection between µ and J
gµB
μ=−
J
ℏ

with Landé factor
g = 1+J ( J + 1) + S ( S + 1) − L (L + 1)
2J ( J + 1)

http://www.orbitals.com/orb/ov.htm

Total angular momentum of an atom
is composed of angular momentumof all occupied electron states L andof all corresponding spins S

Susceptibility
Connection between field H (e.g. generated by coil) and magnetization M:
Linear materials:H

M = χHdiamagnetic if χ < 0paramagnetic if χ > 0
magnetic susceptibility

Connection between B and H field : B = µ0 (H + M)

(for M << H)⇒ B = µ0 (H + χH ) = µ0 (1 + χ)H

Not the whole story!

permeability of free space

B = µ0 (H + Hd + M)

demagnetizing field or stray field(important for ferromagnets)

Demagnetizing field∇ ⋅ B = ∇ ⋅ [µ0 (Hd + M)] = 0

Hd
M

+
+
+

∇ ⋅ Hd = −∇ ⋅ M

Any divergence of M creates amagnetic field – can be seen as
sort of (bound) magnetic charges

Easy expression for strayfield (demagnetizing field) Hd only for ellipsoids: Nxx
Hd = −  0
 0


0
Nyy
0

0 
0  MNzz 

with Nxx + Nyy + Nzz = 1

Some examples:
Nxx = 0, Nyy = Nzz = 21

Nxx = Nyy = Nzz = 31

long cylindrical rodzyx

sphereH

Nxx = Nyy = 0, Nzz = 1

flat plateM
Hd = −M ⇒B = µ0 (H + Hd + M) = µ0H

Paramagnetism (of isolated magnetic moments)
Requisite: Atoms with non-vanishing angular momentum J

Energy of magnetic moment E = −μ ⋅ BClassically: arbitrary values and orientation of µ
11
U mB

U/µB

0.5

00

- 0.5
-1

-1 00

p
π
€€€€€
22

p
π
fφ

3πp
€€€€€€€€€
22

2p
2π

Quantum mechanically: discrete values and orientations of μ J , e.g. J = 5 / 2
5
ℏ
2
3
ℏ
2
1
ℏ
2
− 12 ℏ− 32 ℏ− 52 ℏµz = −gµB
Jz = −mJgµBℏJz = mJ ℏ

mJ = J,J − 1,... − (J − 1), −J

2J+1 discrete values(and hence energy levels)

Associated energy levels EmJ = mJgµBB
get thermally occupied−µzMagnetization (thermal average)
Magnetization M = NgJµBBJ ( x )Brillouin function BJ ( x ) =J: total angular momentum

2J + 1gJµBB
 2J + 1  1
x
coth x  − coth   ; with x =2J
kB T 2J  2J
 2J 

Saturation:

1

for x >> 1: B J → 1

0.8

N : Density of magnetic atoms

BJ(x)and M → NgJµB = MS

0.6
0.4

1 3
J = , 1, ,2...∞2 2

0.2"High − T" :0
1 xx << 1 ⇒ coth ( x ) ≈ + − ...
0
1
2
3
4
x 3
J +1x
x = gJµBB / kT
⇒ BJ ≈J 3
g2J(J + 1)µB2
g2J(J + 1)µB2
g2J(J + 1)µB2µ0 1
M
M = NgJµBBJ ( x ) ≈ N
B=Nµ0H ⇒ χ = ≈ N
3k B T
3k B TH
3kB
T

5Paramagnetism (of free electrons)
Magnetization: M = (N↑ − N↓ ) µB = ∆NµB

E
EF

1
3N
∆N = D (EF ) ⋅ 2gµBB =
gµBB
2
2 EF
D (EF ) =

3N
in 3D
2 EF

Resulting magnetization:

2gµBB
D(E)

M=

3N 2
3 N
gµBB =
gµB2µ0H
2 EF
2 kB TF
χTaking into account diamagnetism of free electrons (Mdia = − 31 Mpara )χ=NgµB2µ0
kB TF

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
Consequences of spin arrangement for total energy

λExchange hole

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!

Increase of kinetic energy (example)
E

E
EF

δE
EF

D↑ (E)

D↓ (E) D↑ (E)D↓ (E)

Fermi energy EF of a free electron gas increases if, for fixed
electron density only one spin species is permitted

Exchange Interaction (a la Heisenberg)
Two electron wave function of hydrogenmolecule like system:r12ℏHˆ ψ = −∇12 + ∇22 ψ + V(r1,r2 )ψ = Eψ
2m

(

R1 r2

R2

)

Fermi-Dirac statistics requires thatψ(r1,r2 ) = −ψ(r2 ,r1) exchange of electrons
Construction of Ψ with Heitler-London approximation:
Spin of both electrons anti-parallel (Singulett, S=0)

ψ s (r1,r2 ) = [ ϕ1(r1 )ϕ2 (r2 ) + ϕ1(r2 )ϕ2 (r1 )] χSsymmetric

anti-symmetric under particle exchange

Spin of both electrons parallel (Triplett, S=1)

ψ T (r1,r2 ) = [ϕ1(r1)ϕ2 (r2 ) − ϕ1(r2 )ϕ2 (r1)] χTanti-symmetricsymmetric under particle exchangeSpin wave function χ

r1

Triplett

R1 r2

S

ms

χ

S1 ⋅ S2

1

1

↑↑

1
4

1

0

↑↓ + ↓↑

1
4

↓↓

1
4

Singulett

1
0

-1
0

↑↓ − ↓↑

−

3
4

R2

χT symmetricχS anti-symmetricsee, e.g. Stephen Blundell Magnetism in Condensed
Matter Oxford University Press

Exchange splitting
exclusively due to the exchange of the two electrons

ψ S (r1,r2 ) and ψ T (r1,r2 ) have different energy ES and E T
ES − E T =ψS Hˆ ψ S
ψS ψS−

ψ T Hˆ ψ T
ψT ψT.........

Ashcroft, Mermin



2
2
2
2
e
e
e
e
 ϕ1(r1 )ϕ2 (r 2 ) = 2∫ dr1dr 2 ϕ1(r1 )ϕ2 (r 2 )  +−−
 r1 − r 2
R1 − R2
r1 − R1 r 2 − R2  S
1energy

ms

J/4

1
0
-1

exchange integral J

sign of J determines spin ground state
J
0

-3J/4

0
B

J > 0 ⇒ E S > ET ⇒ spins parallelJ < 0 ⇒ E S < E T ⇒ spins anti-parallel

Heisenberg Hamiltonian
Expressing spin Hamiltonian with Eigenvalues ES and ET in termsof S1 and S2:+ 41 S = 1
Sˆ 1 ⋅ Sˆ 2 =Hˆ = 41 (ES + 3ET ) − (ES − E T ) Sˆ 1 ⋅ Sˆ 2
− 43
S=0
using that Sˆ 2 = S(S + 1)

Hˆ has Eigenvalue E T for triplett ( ↑↑ or ↓↓ ) andES for singulett ( ↑↓ or ↓↑ )Shift of origin results in usual form of Heisenberg HamiltonianHˆ = −(ES − E T ) Sˆ 1 ⋅ Sˆ 2 = − J Sˆ 1 ⋅ Sˆ 2

Generalisation for many spins:

Hˆ = − ∑ J ijSˆ i ⋅ Sˆ ji,jExchange integral between spin i and spin jForms of exchange interaction
Direct exchange interaction
from direct overlap

Superexchange
mediated by paramagnetic atoms
or 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

Nature Physics
6, 955 (2010)

Mn (S=5/2) in two-dimensional hole gas

see: WP-20 Anomalous Hall effect in Mn
doped,p-type InAs quantum wells
C. Wensauer, D. Vogel et al.

Wurstbauer et al.

Weiss molecular field

Pierre Weiss 1907

Idea: Replace interaction of a given spin with all other spins by interaction with
an effective field (molecular field)
Sum of all S j act on
one particular spin S imagnetic moment
μi of spin S i

Eex = −2Si ⋅ ∑ JijS j ,

μi =

j

−gµBSiμℏ⇒ Si = − i
ℏ
gµB

 2ℏ2

μi ℏ
Eex = 2
⋅ ∑ JijS j = −μi ⋅ 
μi ∑ Jij  = −μi ⋅ BM
2


gµB j
(g
µ
)
j
B


= λM
Weiss molecular field BM
Sj replaced by constant moment:

μ j = −gµB S j /ℏ ⇒ S j = −ℏ μ j / gµB

Magnetic moments get aligned in an effective field Beff = B0+BM >>B0

Ferromagnetism within the molecular field picture
Molecular field Bm=λM allows to map ferromagnetism onto paramagnetism

2J + 1  1x  2J + 1
M = NgJJµB 
cothx  − coth 
 2J  2J
 2J 
 2J
M on both sides of equation:
graphical or numerical solution
TC =gJµB (J + 1)λMS
3k B

Für TC ~ 103 K und J = 1 / 2 ⇒BM = λMS ~ 1500 Twith x =gJJµB (B + λM)
kTBand magnetism and Stoner criterion
Heisenberg Model unable to describe all aspects of ferromagnetism e.g. momentof iron is 2.2 µB. Stoner picture:
E
Redistribution of spins: Energy cost Ekin
1
∆Ekin = D(EF )δE2
EF
δE
2
Resulting magnetization
M = (n↑ − n↓ ) µB = D(EF )µB δE

Energy of electron‘s magnetization dM in
effective (Weiss) field Beff of other electronsD↑ (E)

D↓ (E)

M

M2
1
dEpot = −dM ⋅ Beff → ∆Epot = − ∫ dM′ λM′ = −λ
= − λµB2D(EF )2 δE2
2
20
1
Energy gain if ∆Ekin + ∆Epot = D(EF )δE2 [1- λµB2D(EF )] < 02Ferromagnetism occurs if λµB2 ⋅ D(EF ) ≥ 1 Stoner criterion (holds for Fe,Ni,Co)

Ferromagnet: Density of States D(E)
Schematic

Nickelmajority spins
majority spinss-electrons
minority spins

minority spinsd-electrons

Magnetic Domains
1
1
Energy of stray field Hd: Ed = µ0 ∫ H2d dV = − µ0 ∫ Hd ⋅MdV
2 space2 sample

Formation of domains reduces stray field energy!

Width of a Bloch wall
Classical Heisenberg Hamiltonian
Eex = −2JS1 ⋅ S2 = −2JS2 cos θθ = 0 → Eex = −2JS2

Energy to rotate spin by angle θθ ≠ 0 → ∆Eex = JS2θ2

for θ << 12π
2 πSpin rotation by π over N sites ⇒ θ = ⇒ energy cos t per line : ∆Eex = JS
N
N

Energy per unit area εex =

∆Eex
2

a

a: lattice constant

2

= JS

π2
Na2

⇒ best N → ∞ ?

Magnetocrystalline anisotropy
Ferromagnetic crystals have easy and hard axes. Along some crystallographic
directions it is easy to magnetize the crysal, along others harder. Reflects the
crystal symmetry.
Extra energy associated with magnetocrystalline anisotrpy:
uniaxial anisotropy (e.g. Co)

cubic anisotropy (e.g. Fe)()

E = K1 41 sin2 (θ) sin2 (2ϕ) + cos2 (θ) sin2 (θ) + ....

E = K1 sin2 (θ) + K 2 sin4 (θ) + ...

ϕ

θ

M

Example: GaMnAs core-shell nanowires
system with very strong uniaxial
anisotropy

see: WP-22, C. Butschkow et al.

Width of a Bloch wall, continued
assume simple form of anisotropy energy
Eani = K sin2 ( θ); K > 0aenergy cost of rotation out of easy direction
N

π

N
NK
2
∑ K sin θi ≈ π ∫ K sin θdθ = 2
i=102

energy per unit area=

NKa
2

2
NKa
2 π
exchange energy + anisotropy energy εBW =
+ JS
2
Na2

From dεBW / dN = 0 ⇒ N = πS

2J
2J
⇒
=
π
S
Na
Ka
Ka3

domain wall widthReversal of magnetization: Hysteresis
macroscopic bulk material consisting of many domains
M
MS
Mr
-HC

HS
H

Processes
1. Shift of domain walls (reversible and irreversible)
2. Domain rotation towards nearest easy axis
3. Coherent rotation at high fields

Magnetization of a cube with uni-axial anisotrpy

sample size

micromagnetic simulation

multi-domain-state

vortex-state
two-domain-state

single-domain-statenormalized anisotropy constant Ku / K danisotropy
W. Rave, K. Fabian, A. Hubert, JMMM 190, 332 (1998)

Magnetism and Spin-Orbit Interaction:
Magnetism
Basic concepts
magnetic moments, susceptibility, paramagnetism, ferromagnetism
exchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basics
Rashba- and Dresselhaus contribution, SO-interaction and effective
magnetic field
Example:Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial
Fe/GaAs interfaces

Measurement of domains: magnetic force microscopy
cantilever withmagnetic tip

lift height

1. topography scan

2. MFM scan (lift mode)

MFM measures „magnetic charges“

magnetization M

div M

sim. MFM picture

real MFM pictureDomains, measured by MFM
Perpendicular magnetization

in-plane magnetization

4Å Fe / 4Å Gd (75 layers)

50 Mbyte hard driveBetween multidomain and single-domain state:
vortex structure

4 ground state configurations
in-plane and out-of plane magnetization component can beswitched independently!

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)

300 nmDetection of in-plane vortex

Lorentz-microscopy probes in-plane magnetization…
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

M

M

Measures directly the stray field,
averaged over the central area ofthe Hall junction
UH =

I
B
ens

Hysteresis trace of a vortex state

M. Rahm et al., J. Appl. Phys. 93, 7429 (2003)

Vortex-Golf
1.8 mmMagnetism and Spin-Orbit Interaction:
Magnetism
Basic concepts
magnetic moments, susceptibility, paramagnetism, ferromagnetism
exchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basics
Rashba- and Dresselhaus contribution, SO-interaction and effective
magnetic field
Example:Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial
Fe/GaAs interfaces

Origin of spin-orbit (SO) interaction

E = −µBBeff

Spin-orbit interation

due to orbital motion

magnetic moment of electron+

electron

nucleus+electron‘s rest
frame

Beff

 E×p Hˆ SO = −µB σˆ ⋅  2mc2 
vector of Paulispin matrices

Hˆ Zeeman = −µB σˆ ⋅ B
Beff =

E×p
2mc2

Electric field E in solids
Bulk inversion asymmetry (BIA)
Lack of inversion symmetry in
III-V semiconductors"Dresselhaus contribution γ"kykx

 kx BBIA ∝ γ 

−
k
y



BBIA
ψ (z)

2

V(z)

Structure inversion asymmetry (SIA)
due to macroscopic confining potential:"Rashba contribution α". Tunable byexternal electric field!
ky BSIA ∝ α 

−
k
 x

ky
kx

BSIA

SO interaction in 2DEG: Rashba & Dresselhaus terms
2 2ℏˆH = k + Hˆ SO with2m

tunable by gate voltage
Pauli spin matrix

Hˆ SO = α( σ xk y − σ yk x ) + γ ( σ xk x − σ yk y )RashbaDresselhausCalculation of Eigenvalue

Hˆ Rashba= α( σ x k y − σ y k x )SO 0 1
 0 −i 
1 0 
σˆ x =  ; σˆ y =  i 0  ; σˆ z =  0 −1
1
0







Pauli Spin matrices

Hˆ Rashba= α( σ x k y − σ y k x ) =
SO
 0
=
 αk yαk y   0−
0   iαk x0α(k y + ik x ) −iαk x  =
00   α(k y − ik x )Eigenvalues of the matrix : ±α k 2x + k 2y = ±αk Description of zero-field spin splitting by Beff
ˆHSO = α( σxk y − σy k x ) + γ( σxk x − σyk y ) ~ σˆ ⋅ Beff ; σˆ ⋅ B = σ B + σ B + σ B
eff
x x
y y
z z
RashbaDresselhaus

Comparison of coefficients. E.g. only Rashba contribution:


Beff

E.A. de Andrada e Silva PRB 46, 1921 (1992) ky 
Beff ∝ α  −k x Presence of Rashba & Dresselhaus contributions
Rashba or
Dresselhaus
Rashba andDresselhaus


ℏ2k 2
E=
± (α + γ ) k
2m

]


ℏ2k 2
E=
± (α − γ ) k
2m

SO-interaction in a InGaAs quantum wellBVx

∝ ρ xxBeating of SdH-oscillations

Vg

I

Nitta et al., Phys. Rev. Lett 78, 1335 (1997)

Quantum oscillations (SdH) reflect k-space areaky

Periodicity of Shubnikov-de Haas (SdH) oscillations

kF

kx

A

 1  2πe∆  =
 B  ℏA

Note that A = πk F2 = 2π2ns

Origin of beating: two periodicities due to two k-space areas A1 and A2
ρxx
A1

A2

B
Nitta et al., Phys. Rev. Lett 78, 1335 (1997)

Tuning of Rashba coefficient a by gate voltage Vg ky 
Corresponds to tuning of spin orbit field, i.e., Beff = α  −k x 

Nitta et al., Phys. Rev. Lett 78, 1335 (1997)

Magnetism and Spin-Orbit Interaction:
Magnetism
Basic concepts
magnetic moments, susceptibility, paramagnetism, ferromagnetism
exchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basics
Rashba- and Dresselhaus contribution, SO-interaction and effective
magnetic field
Example:Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial
Fe/GaAs interfaces

Spin-orbit interaction plays also role in tunneling
Epitaxial Fe-GaAs interface
[110]
Fe

As
[110]

aGaAs=5.653 ÅGa
aFe=2.867 ÅDevice fabrication

Substrate
Au

Fe
As-Cap
8 nm

GaAs
SubstrateFe
GaAs
Substrate

J. Moser et al. Appl. Phys. Lett. 89, 162106 (2006)

Fe

GaAs

AlGaAs
TEM-micrograph:
C.-H. Lai, Tsing Hua University, Taiwan

TAMR: Tunneling Anisotropic Magnetoresistance
Are always two ferromagnetic layers necessary to see a magnetization
dependent resistance?
Our model system: Fe/GaAs/Au with epitaxial Fe/GaAs interface

M
R1

Fe
GaAs
Au

M
R2

R1 ≠ R2
See also:
Gould et al. PRL 93, 117203 (2004)(Ga,Mn)As/Al2O3/Au

Tunneling magnetoresistance: B dependenceSpin-valve like signal-88 mV]0
1
1[

Double step/single step switching due
to magnetic anisotropy
Fe on GaAs: Cubic + uniaxial anisotropy
easy

]
0
1
1
[

B
φ

φ=0

B

J. Moser et al., Phys. Rev. Lett. 99, 056601, 2007

Angular dependence of TAMR: negative bias
Co/Fe(epi)/GaAs(8nm)/Au
measured at
T = 4.2 K and -90 mV

B=0.5TM
R

J. Moser et al., Phys. Rev. Lett. 99, 056601, 2007

Fe
GaAs
Au

Angular dependence of TAMR: positive bias
Co/Fe(epi)/GaAs(8nm)/Au
measured at
T = 4.2 K and +90 mV

B=0.5TM
R

J. Moser et al., Phys. Rev. Lett. 99, 056601, 2007

Fe
GaAs
Au

Modelling
A. Matos-Abiague & J. Fabian, PRB 79, 155303 (2009)

αl γ αr

H = H0 + Hz + HBR + HD

GaAs

ℏ2  1 
H0 = − ∇ ∇ + V(z)2 m(z) 
H BR =1
αi ( σxpy − σypx )δ(z − zi )∑
ℏ i=l,rHz = −

∆(z)
n⋅σ
2

fb
EF

Au

Fe

1
∂
∂
HD = ( σxpx − σypy )  γ(z ) 
ℏ
∂z 
∂z 

I=

z
zl

zr

e
2
dEd
k || Tσ (E,k || )[f l(E) − f r (E)]∑∫3(2π) ℏ σ=−1,1particle transmissivity

Spin-orbit field w(k)
ky

ky

γ=0

α=0

kx

kx
[100]

ky

[100]

 αk y - γk x 


α
k
+
γ
k
w(k || ) = 
x
y


0



ky

w(k || )
kx

kx
[100]

α>γ>0

[100]

α<0
γ>0

Origin of anisotropic resistance: superposition of Rashba &Dresselhaus SO-contribution
αl γ

φb
EF

Au
GaAs
Fe

x
z

n
Fe

Au

w(k || )zzlzr
Anisotropy det ermined byT(k || ) ≈ f([n ⋅ w(k || )]2 ) → R / R[110] − 1 ~ αγ(cos2φ − 1)

Anisotropy vanishes for

αγ → 0y αk y - γk x w(k || ) =  -αk x + γk y 


0



J. Moser et al., Phys. Rev. Lett. 99, 056601, 2007

Angular dependence of TAMR: bias dependenceR( φ) − R[110]TAMR =
R[110]

αl (eVA°2 )
-25.1
-17.4
-0.6

42.3

γ in GaAs: 24 eVA3°

J. Moser et al., Phys. Rev. Lett. 99, 056601, 2007
See also:T. Uemura, Appl. Phys. Lett. 98,102503 (2011)

Fe on (Ga,Mn)AsAu
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 couplingRT

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)

Spin injection: Proximity effect enhances TC
Proximity effect: exchange coupling between
Fe and Mn at Fe/(Ga,Mn)As interface
5

2

3

4

FP-34 C. Song et al.
WP-30 M. Ciorga et al.
WP-105 J. Shiogai etal.

Magnetism and Spin-Orbit Interaction:
In retrospect
Magnetism
Basic concepts
magnetic moments, susceptibility, paramagnetism, ferromagnetism
exchange interaction, domains, magnetic anisotropy,
Examples:detection of (nanoscale) magnetization structureusing Hall-magnetometry, Lorentz microscopy and MFM
Spin-Orbit interaction
Some basics
Rashba- and Dresselhaus contribution, SO-interaction and effective
magnetic field
Example:Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial
Fe/GaAs interfaces

The End