Rashba effect: Spin splitting ofsurface and interface states

Ulrich Zuelicke

u.zuelicke@massey.ac.nz

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