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)
2DEG
a
Magnetization pattern of ferromagnet determines stray field
Transport in a periodic magnetic field
Commensurability (Weiss) Oscillations :
(
1
4
)a
SdH oscillations
ρxx (Ω
Ω)
2RC = n ±
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 Einstein
1905: On the electrodynamics of moving bodies
Magnetic field stems from Lorentz contraction of
moving charges
Magnetism and Spin-Orbit Interaction:
Magnetism
Basic concepts
magnetic moments, susceptibility, paramagnetism, ferromagnetism
exchange interaction, domains, magnetic anisotropy,
Examples:
detection of (nanoscale) magnetization structure
using 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 structure
using 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 dipole
S
area S
μ = IS
I
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 −ev
1
1e
L
I=
=
⇒ µ = − evr = −
L = − − µB
T
2πr
2
2m
ℏ
µ
eℏ
μ = − B L with µB =
= Bohr Magneton
ℏ
2m
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 momentum
of all occupied electron states L and
of all corresponding spins S
Susceptibility
Connection between field H (e.g. generated by coil) and magnetization M:
Linear materials:
H
M = χH
diamagnetic if χ < 0
paramagnetic 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 a
magnetic 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  M
Nzz 
with Nxx + Nyy + Nzz = 1
Some examples:
Nxx = 0, Nyy = Nzz =
1
2
Nxx = Nyy = Nzz =
1
3
long cylindrical rod
z
y
x
sphere
H
Nxx = Nyy = 0, Nzz = 1
flat plate
M
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 ⋅ B
Classically: 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 ℏ
µz = −
gµB
Jz = −mJgµB
ℏ
Jz = 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 + 1
gJµBB
 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 x
x << 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=N
µ0H ⇒ χ = ≈ N
3k B T
3k B T
H
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
χ
Taking into account diamagnetism of free electrons (Mdia = − 31 Mpara )
χ=
N
gµB2µ0
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 same
spin orientation out of a region with diameter λ →
exchange hole forms
Consequences of spin arrangement for total energy
λ
Exchange 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 hydrogen
molecule like system:
r1
2
ℏ
Hˆ ψ = −
∇12 + ∇22 ψ + V(r1,r2 )ψ = Eψ
2m
(
R1 r2
R2
)
Fermi-Dirac statistics requires that
ψ(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 )] χS
symmetric
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)] χT
anti-symmetric
symmetric 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 symmetric
χS anti-symmetric
see, 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
= 2∫ dr1dr 2 ϕ1(r1 )ϕ2 (r 2 )  
+
−
−
 r1 − r 2
R1 − R2
r1 − R1 r 2 − R2

S
1
energy
ms

 ϕ1(r1 )ϕ2 (r 2 ) 



exchange integral J
1
0
-1
J/4
sign of J determines spin ground state
J
0
-3J/4
0
B
J > 0 ⇒ E S > ET ⇒ spins parallel
J < 0 ⇒ E S < E T ⇒ spins anti-parallel
Heisenberg Hamiltonian
Expressing spin Hamiltonian with Eigenvalues ES and ET in terms
of 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 ↓↓ ) and
ES for singulett ( ↑↓ or ↓↑ )
Shift of origin results in usual form of Heisenberg Hamiltonian
Hˆ = −(ES − E T ) Sˆ 1 ⋅ Sˆ 2 = − J Sˆ 1 ⋅ Sˆ 2
Generalisation for many spins:
Hˆ = − ∑ J ijSˆ i ⋅ Sˆ j
i,j
Exchange 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 i
magnetic moment
μi of spin S i
Eex = −2Si ⋅ ∑ JijS j ,
μi =
j
Eex
−gµBSi
μℏ
⇒ Si = − i
ℏ
gµB
 2ℏ2
μi ℏ
=2
⋅ ∑ JijS j = −μi ⋅ 
μi
2

gµB j
 (gµB )

∑ Jij  = −μi ⋅ BM
j

= 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  1
x 
 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 T
with x =
gJJµB (B + λM)
kT
Band magnetism and Stoner criterion
Heisenberg Model unable to describe all aspects of ferromagnetism e.g. moment
of 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 electrons
D↑ (E)
D↓ (E)
M
dEpot = −dM ⋅ Beff → ∆Epot
Energy gain if ∆Ekin + ∆Epot
M2
1
= − ∫ dM′ λM′ = −λ
= − λµB2D(EF )2 δE2
2
2
0
1
= D(EF )δE2 [1- λµB2D(EF )] < 0
2
Ferromagnetism occurs if λµB2 ⋅ D(EF ) ≥ 1 Stoner criterion (holds for Fe,Ni,Co)
Ferromagnet: Density of States D(E)
Schematic
Nickel
majority spins
majority spins
s-electrons
minority spins
minority spins
d-electrons
Magnetic Domains
1
1
Energy of stray field Hd: Ed = µ0 ∫ H2d dV = − µ0 ∫ Hd ⋅MdV
2 space
2 sample
Formation of domains reduces stray field energy!
Width of a Bloch wall
Classical Heisenberg Hamiltonian
Eex = −2JS1 ⋅ S2 = −2JS2 cos θ
θ = 0 → Eex = −2JS2
Energy to rotate spin by angle θ
θ ≠ 0 → ∆Eex = JS2θ2
for θ << 1
2
π
2 π
Spin 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
E = K1 sin2 (θ) + K 2 sin4 (θ) + ...
ϕ
θ
M
(
1
4
)
sin2 (θ) sin2 (2ϕ) + cos2 (θ) sin2 (θ) + ....
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 > 0
a
energy cost of rotation out of easy direction
N
π
N
NK
2
∑ K sin θi ≈ π ∫ K sin θdθ = 2
i=1
0
2
energy per unit area=
NKa
2
exchange energy + anisotropy energy εBW
From dεBW / dN = 0 ⇒ N = πS
2
NKa
2 π
=
+ JS
2
Na2
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-state
normalized anisotropy constant Ku / K d
anisotropy
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 structure
using 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 with
magnetic 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
4Å Fe / 4Å Gd (75 layers)
in-plane magnetization
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 be
switched independently!
Picture: Ref 3)
1)
J. Raabe, R. Pulwey et al., J. Appl. Phys. 88, 4437 (2000)
T. Shinjo et al., Science 289, 930 (2000)
3) A. Wachowiak et al., Science 298, 577 (2002)
2)
300 nm
Detection 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 of
the 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 structure
using 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 Pauli
spin 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"
BBIA
ky
kx
 kx 
∝ γ

−
k
y


BBIA
ψ (z)
2
V(z)
Structure inversion asymmetry (SIA)
due to macroscopic confining potential:
"Rashba contribution α". Tunable by
external electric field!
BSIA
ky 
∝ α

−
k
 x
ky
kx
BSIA
SO interaction in 2DEG: Rashba & Dresselhaus terms
2 2
ℏ
ˆH = k + Hˆ SO with
2m
tunable by gate voltage
Pauli spin matrix
Hˆ SO = α( σ xk y − σ yk x ) + γ ( σ xk x − σ yk y )
Rashba
Dresselhaus
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 y
αk y   0
−
0   iαk x
0
α(k y + ik x ) 
−iαk x  
=

0
0   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
Rashba
Dresselhaus
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 and
Dresselhaus
ℏ2k 2
E=
± (α + γ ) k
2m
]
ℏ2k 2
E=
± (α − γ ) k
2m
SO-interaction in a InGaAs quantum well
B
Vx
∝ ρ xx
Beating of SdH-oscillations
Vg
I
Nitta et al., Phys. Rev. Lett 78, 1335 (1997)
Quantum oscillations (SdH) reflect k-space area
ky
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 structure
using 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 Å
Ga
aFe=2.867 Å
Device fabrication
Substrate
Au
Fe
As-Cap
8 nm
GaAs
Substrate
Fe
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 dependence
Spin-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=0.5T
M
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=0.5T
M
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,r
Hz = −
∆(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=−1,1
particle 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 || )
z
zl
zr
Anisotropy det ermined by
T(k || ) ≈ f([n ⋅ w(k || )]2 ) → R / R[110] − 1 ~ αγ(cos2φ − 1)
Anisotropy vanishes for
αγ → 0
y
 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 dependence
R( 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)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)
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 et
al.
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 structure
using 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