rashba
TRANSCRIPT
Rashba effect: Spin splitting ofsurface and interface states
Ulrich Zuelicke
Institute of Fundamental Sciences and
MacDiarmid Institute for Advanced Materials and Nanotechnology
Massey University, Palmerston North, New Zealand
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 1
OutlineIntroduction
electron spin
Zeeman effect
spin-orbit coupling
Rashba effect in semiconductor heterostructures
structural inversion asymmetry results in spin splitting
basis for spin-dependent transport effects
Rashba effect at (metal) surfaces
basic setup & discovery
STM study of effect
Discussion
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 2
Introduction: Spin, Zeemaneffect & spin-orbit coupling
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 3
Electron spinelectron = charge + spin
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 4
Electron spinelectron = charge + spin
spin behaves like angular momentum, butis not related to any real rotational motion
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 4
Electron spinelectron = charge + spin
spin behaves like angular momentum, butis not related to any real rotational motion
quantum discreteness of spin projection: electronscome in two flavours (spin-up ↑ or spin-down ↓)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 4
Magnetic moment due to spinmicroscopic magnetic dipole associated with spin
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 5
Magnetic moment due to spinmicroscopic magnetic dipole associated with spin
spin interacts w/ magnetic fields (Stern-Gerlach expt)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 5
Magnetic moment due to spinmicroscopic magnetic dipole associated with spin
spin interacts w/ magnetic fields (Stern-Gerlach expt)
spin degeneracy at zero field lifted in finite fields
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 5
Magnetic moment due to spinmicroscopic magnetic dipole associated with spin
spin interacts w/ magnetic fields (Stern-Gerlach expt)
spin degeneracy at zero field lifted in finite fields
energy splitting between the two spin states in a magneticfield: Zeeman spin splitting of electron states in atoms
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 5
Electric fields interact with spinelectron spin affected by magnetic field (Zeeman effect)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 6
Electric fields interact with spinelectron spin affected by magnetic field (Zeeman effect)
electric field affects spin state of moving electrons
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 6
Electric fields interact with spinelectron spin affected by magnetic field (Zeeman effect)
electric field affects spin state of moving electrons
relativistic effect: electric field in lab frame causesmagnetic field in rest frame of moving electron
v
E
laboratory frame
B= E x vE c21
electron rest frame
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 6
Electric fields interact with spinelectron spin affected by magnetic field (Zeeman effect)
electric field affects spin state of moving electrons
relativistic effect: electric field in lab frame causesmagnetic field in rest frame of moving electron
‘Zeeman effect’ of electron moving in an electric field!
v
E
laboratory frame
B= E x vE c21
electron rest frame
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 6
Electric fields interact with spinelectron spin affected by magnetic field (Zeeman effect)
electric field affects spin state of moving electrons
relativistic effect: electric field in lab frame causesmagnetic field in rest frame of moving electron
‘Zeeman effect’ of electron moving in an electric field!
orbital motion and spin intertwined: spin-orbit coupling
v
E
laboratory frame
B= E x vE c21
electron rest frame
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 6
Spin splitting: Vacuum vs. solidsquantum electron dynamics described by Hamiltonian H
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 7
Spin splitting: Vacuum vs. solidsquantum electron dynamics described by Hamiltonian H
orbital motion: Horb = p2
2m0+ Vext(~r)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 7
Spin splitting: Vacuum vs. solidsquantum electron dynamics described by Hamiltonian H
orbital motion: Horb = p2
2m0+ Vext(~r)
Zeeman effect: HZ = g µB
~
~B · ~S
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 7
Spin splitting: Vacuum vs. solidsquantum electron dynamics described by Hamiltonian H
orbital motion: Horb = p2
2m0+ Vext(~r)
Zeeman effect: HZ = g µB
~
~B · ~S
spin-orbit coupling: Hso = −[
~p
m0× ~∇
(
Vext
2m0c2
)]
· ~S
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 7
Spin splitting: Vacuum vs. solidsquantum electron dynamics described by Hamiltonian H
orbital motion: Horb = p2
2m0+ Vext(~r)
Zeeman effect: HZ = g µB
~
~B · ~S
spin-orbit coupling: Hso = −[
~p
m0× ~∇
(
Vext
2m0c2
)]
· ~S
quasi-free electrons in solids: effect of periodic crystalpotential incorporated by effective-mass Hamiltonian
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 7
Spin splitting: Vacuum vs. solidsquantum electron dynamics described by Hamiltonian H
orbital motion: Horb = p2
2m0+ Vext(~r)
Zeeman effect: HZ = g µB
~
~B · ~S
spin-orbit coupling: Hso = −[
~p
m0× ~∇
(
Vext
2m0c2
)]
· ~S
quasi-free electrons in solids: effect of periodic crystalpotential incorporated by effective-mass Hamiltonian
Horb = p2
2m∗
+ Vext(~r) ; HZ = g∗ µB
~
~B · ~S
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 7
Spin splitting: Vacuum vs. solidsquantum electron dynamics described by Hamiltonian H
orbital motion: Horb = p2
2m0+ Vext(~r)
Zeeman effect: HZ = g µB
~
~B · ~S
spin-orbit coupling: Hso = −[
~p
m0× ~∇
(
Vext
2m0c2
)]
· ~S
quasi-free electrons in solids: effect of periodic crystalpotential incorporated by effective-mass Hamiltonian
Horb = p2
2m∗
+ Vext(~r) ; HZ = g∗ µB
~
~B · ~S
Hso = −
[
~p
m∗
× ~∇(
Vext
Eg
)]
· ~S, where Eg is the band gap
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 7
Spin splitting: Vacuum vs. solidsquantum electron dynamics described by Hamiltonian H
orbital motion: Horb = p2
2m0+ Vext(~r)
Zeeman effect: HZ = g µB
~
~B · ~S
spin-orbit coupling: Hso = −[
~p
m0× ~∇
(
Vext
2m0c2
)]
· ~S
quasi-free electrons in solids: effect of periodic crystalpotential incorporated by effective-mass Hamiltonian
Horb = p2
2m∗
+ Vext(~r) ; HZ = g∗ µB
~
~B · ~S
Hso = −
[
~p
m∗
× ~∇(
Vext
Eg
)]
· ~S, where Eg is the band gap
typically Egap ∼ 1 eV, compare with 2m0c2 ≈ 1 MeV
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 7
Spin splitting: Vacuum vs. solidsquantum electron dynamics described by Hamiltonian H
orbital motion: Horb = p2
2m0+ Vext(~r)
Zeeman effect: HZ = g µB
~
~B · ~S
spin-orbit coupling: Hso = −[
~p
m0× ~∇
(
Vext
2m0c2
)]
· ~S
quasi-free electrons in solids: effect of periodic crystalpotential incorporated by effective-mass Hamiltonian
Horb = p2
2m∗
+ Vext(~r) ; HZ = g∗ µB
~
~B · ~S
Hso = −
[
~p
m∗
× ~∇(
Vext
Eg
)]
· ~S, where Eg is the band gap
typically Egap ∼ 1 eV, compare with 2m0c2 ≈ 1 MeV
spin-orbit effects are drastically enhanced in solids!
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 7
Rashba effect: Spin splitting &structural inversion asymmetry
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 8
Semiconductor heterostructuressemiconductor: conduction/valence band edges Ec & Ev
E
Ev
c
InA
lAs
zz
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 9
Semiconductor heterostructuressemiconductor: conduction/valence band edges Ec & Ev
Ev
c
InA
lAs
InG
aAs
zz
InA
lAs
++++
++
+++
E2Dbound state
combine two different materials in a heterostructure:spatial variation of band edges induced (band bending)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 9
Semiconductor heterostructuressemiconductor: conduction/valence band edges Ec & Ev
Ev
c
InA
lAs
InG
aAs
zz
InA
lAs
++++
++
+++
E2Dbound state
combine two different materials in a heterostructure:spatial variation of band edges induced (band bending)
realises textbook example of 2D quantum well
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 9
Semiconductor heterostructuressemiconductor: conduction/valence band edges Ec & Ev
Ev
c
InA
lAs
InG
aAs
zz
InA
lAs
++++
++
+++
E2Dbound state
combine two different materials in a heterostructure:spatial variation of band edges induced (band bending)
realises textbook example of 2D quantum well
spatial asymmetry of band edges mimics electric field:
gives rise to a spin-orbit coupling HR = 2[
~~km∗
× ksoz]
· ~S
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 9
Semiconductor heterostructuressemiconductor: conduction/valence band edges Ec & Ev
Ev
c
InA
lAs
InG
aAs
zz
InA
lAs
++++
++
+++
E2Dbound state
combine two different materials in a heterostructure:spatial variation of band edges induced (band bending)
realises textbook example of 2D quantum well
spatial asymmetry of band edges mimics electric field:
gives rise to a spin-orbit coupling HR = 2[
~~km∗
× ksoz]
· ~S
wave-vector scale kso is measure for the structuralinversion asymmetry of heterostructure: tuneable!
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 9
Rashba spin splitting
spin-orbit effects from structural inversion asym-metry: studies pioneered by Emmanuel Rashba
Rashba splitting
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 10
Rashba spin splitting
spin-orbit effects from structural inversion asym-metry: studies pioneered by Emmanuel Rashba
in a 2D electron system: HR = 2~
mkso [Sxky − Sykx]
Rashba splitting
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 10
Rashba spin splitting
spin-orbit effects from structural inversion asym-metry: studies pioneered by Emmanuel Rashba
in a 2D electron system: HR = 2~
mkso [Sxky − Sykx]
Rashba term causes momentum-dependent spinsplitting, which is different from Zeeman effect!
kkk
spin−degenerate Zeeman splitting
E E E
y y y
Rashba splitting
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 10
Rashba effect & spin electronics
full picture: 2D electron eigenstates have ~S ⊥ ~k
2k
yk
kx
so
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 11
Rashba effect & spin electronics
full picture: 2D electron eigenstates have ~S ⊥ ~k
2k
yk
kx
so
gate-tuneable kso: spinFET
Ferromagnet Ferromagnet2D electron system
Vg
Vsd
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 11
Rashba effect & spin electronics
full picture: 2D electron eigenstates have ~S ⊥ ~k
2k
yk
kx
so
gate-tuneable kso: spinFET
Ferromagnet Ferromagnet2D electron system
Vg
Vsd
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 11
Rashba effect & spin electronics
full picture: 2D electron eigenstates have ~S ⊥ ~k
2k
yk
kx
so
gate-tuneable kso: spinFET
Ferromagnet Ferromagnet2D electron system
Vg
Vsd
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 11
Rashba effect & spin electronics
full picture: 2D electron eigenstates have ~S ⊥ ~k
2k
yk
kx
so
gate-tuneable kso: spinFET
Ferromagnet Ferromagnet2D electron system
Vg
Vsd
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 11
Rashba effect & spin electronics
full picture: 2D electron eigenstates have ~S ⊥ ~k
2k
yk
kx
so
gate-tuneable kso: spinFET
Ferromagnet Ferromagnet2D electron system
g
Vsd
"OFF" V
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 11
Rashba effect & spin electronics
full picture: 2D electron eigenstates have ~S ⊥ ~k
2k
yk
kx
so
gate-tuneable kso: spinFET "ON"
Ferromagnet2D electron system
V’g
Vsd
Ferromagnet
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 11
Rashba effect of surface states
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 12
Surface statespotential for electrons in solids terminates at surface
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 13
Surface statespotential for electrons in solids terminates at surface
bulk states (bands) and surfaces states (discrete!) exist
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 13
Surface statespotential for electrons in solids terminates at surface
bulk states (bands) and surfaces states (discrete!) existequilibration between bulk and surface: band bending!
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 13
Giant Rashba effect at surfacesstructural inversion asymmetry due to surface could resultin Rashba spin splitting of surface-state dispersions
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 14
Giant Rashba effect at surfacesstructural inversion asymmetry due to surface could resultin Rashba spin splitting of surface-state dispersions
seen in ARPES for Au(111) LaShell, McDougall & Jensen, PRL (96)Henk, Ernst & Bruno, PRB (03)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 14
Giant Rashba effect at surfacesstructural inversion asymmetry due to surface could resultin Rashba spin splitting of surface-state dispersions
seen in ARPES for Au(111) LaShell, McDougall & Jensen, PRL (96)Henk, Ernst & Bruno, PRB (03)
enhanced by surface alloying, eg, Bi/Ag(111) Ast et al. PRL (07)
or BixPb1−x/Ag(111) Ast et al. PRB (08)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 14
Giant Rashba effect at surfacesstructural inversion asymmetry due to surface could resultin Rashba spin splitting of surface-state dispersions
seen in ARPES for Au(111) LaShell, McDougall & Jensen, PRL (96)Henk, Ernst & Bruno, PRB (03)
enhanced by surface alloying, eg, Bi/Ag(111) Ast et al. PRL (07)
or BixPb1−x/Ag(111) Ast et al. PRB (08)
enhanced surface potential and/or high-Z atom content?Bentmann et al., Europhys. Lett. (09)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 14
STM detection of spin splittingAst et al., PRB (07)
STM measures electron density of states (DOS)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 15
STM detection of spin splittingAst et al., PRB (07)
STM measures electron density of states (DOS)
DOS of Rashba-spin-split 2D electron system exhibitsvan-Hove-like divergence Winkler, Springer book (03)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 15
STM detection of spin splittingAst et al., PRB (07)
STM measures electron density of states (DOS)
DOS of Rashba-spin-split 2D electron system exhibitsvan-Hove-like divergence Winkler, Springer book (03)
measuring STM differential conductance dI/dV allowsextraction of local spin-splitting energy (unlike ARPES)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 15
Discussionare there advantages of surface-state Rashba systems ascompared with those in semiconductor heterostructures?
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 16
Discussionare there advantages of surface-state Rashba systems ascompared with those in semiconductor heterostructures?
accessibility to local (STM) studies
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 16
Discussionare there advantages of surface-state Rashba systems ascompared with those in semiconductor heterostructures?
accessibility to local (STM) studies
possibly can more easily create spin-splitting gradientby introducing a spatially varying surface alloying
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 16
Discussionare there advantages of surface-state Rashba systems ascompared with those in semiconductor heterostructures?
accessibility to local (STM) studies
possibly can more easily create spin-splitting gradientby introducing a spatially varying surface alloying
can novel device geometries be achievedusing nanostructuring of surface layers?
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 16
Discussionare there advantages of surface-state Rashba systems ascompared with those in semiconductor heterostructures?
accessibility to local (STM) studies
possibly can more easily create spin-splitting gradientby introducing a spatially varying surface alloying
can novel device geometries be achievedusing nanostructuring of surface layers?
which novel experiments are possible using STM?
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 16
Discussionare there advantages of surface-state Rashba systems ascompared with those in semiconductor heterostructures?
accessibility to local (STM) studies
possibly can more easily create spin-splitting gradientby introducing a spatially varying surface alloying
can novel device geometries be achievedusing nanostructuring of surface layers?
which novel experiments are possible using STM?
image spin-dependent quantum-interference patternsPascual et al., PRL (04)
Walls & Heller, Nano Lett (07)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 16
Discussionare there advantages of surface-state Rashba systems ascompared with those in semiconductor heterostructures?
accessibility to local (STM) studies
possibly can more easily create spin-splitting gradientby introducing a spatially varying surface alloying
can novel device geometries be achievedusing nanostructuring of surface layers?
which novel experiments are possible using STM?
image spin-dependent quantum-interference patternsPascual et al., PRL (04)
Walls & Heller, Nano Lett (07)
is graphite surface special (Dirac-electron quasiparticles)
6th Annual Clusters and Nanoparticles Meeting, Lake Tekapo, New Zealand, 30 Nov – 1 Dec 2009 – p. 16