Rashba

Rashba effect: Spin splitting ofsurface and interface states

Ulrich Zuelicke

[email protected]

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


Introduction: Spin, Zeemaneffect & spin-orbit coupling


Electron spinelectron = charge + spin



spin behaves like angular momentum, butis not related to any real rotational motion



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 ↓)


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




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


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





‘Zeeman effect’ of electron moving in an electric field!

v

E

laboratory frame

B= E x vE c21

electron rest frame





‘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


Spin splitting: Vacuum vs. solidsquantum electron dynamics described by Hamiltonian H



orbital motion: Horb = p2

2m0+ Vext(~r)




2m0+ Vext(~r)

Zeeman effect: HZ = g µB

~

~B · ~S




2m0+ Vext(~r)


~

~B · ~S

spin-orbit coupling: Hso = −[

~p

m0× ~∇

(

Vext

2m0c2

)]

· ~S




2m0+ Vext(~r)


~

~B · ~S


~p

m0× ~∇

(

Vext

2m0c2

)]

· ~S

quasi-free electrons in solids: effect of periodic crystalpotential incorporated by effective-mass Hamiltonian




2m0+ Vext(~r)


~

~B · ~S


~p

m0× ~∇

(

Vext

2m0c2

)]

· ~S


Horb = p2

2m∗

+ Vext(~r) ; HZ = g∗ µB

~

~B · ~S




2m0+ Vext(~r)


~

~B · ~S


~p

m0× ~∇

(

Vext

2m0c2

)]

· ~S


Horb = p2

2m∗


~

~B · ~S

Hso = −

[

~p

m∗

× ~∇(

Vext

Eg

)]

· ~S, where Eg is the band gap




2m0+ Vext(~r)


~

~B · ~S


~p

m0× ~∇

(

Vext

2m0c2

)]

· ~S


Horb = p2

2m∗


~

~B · ~S

Hso = −

[

~p

m∗

× ~∇(

Vext

Eg

)]


typically Egap ∼ 1 eV, compare with 2m0c2 ≈ 1 MeV




2m0+ Vext(~r)


~

~B · ~S


~p

m0× ~∇

(

Vext

2m0c2

)]

· ~S


Horb = p2

2m∗


~

~B · ~S

Hso = −

[

~p

m∗

× ~∇(

Vext

Eg

)]


typically Egap ∼ 1 eV, compare with 2m0c2 ≈ 1 MeV

spin-orbit effects are drastically enhanced in solids!


Rashba effect: Spin splitting &structural inversion asymmetry


Semiconductor heterostructuressemiconductor: conduction/valence band edges Ec & Ev

E

Ev

c

InA

lAs

zz



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)



Ev

c

InA

lAs

InG

aAs

zz

InA

lAs

++++

++

+++

E2Dbound state


realises textbook example of 2D quantum well



Ev

c

InA

lAs

InG

aAs

zz

InA

lAs

++++

++

+++

E2Dbound state



spatial asymmetry of band edges mimics electric field:

gives rise to a spin-orbit coupling HR = 2[

~~km∗

× ksoz]

· ~S



Ev

c

InA

lAs

InG

aAs

zz

InA

lAs

++++

++

+++

E2Dbound state



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!


Rashba spin splitting

spin-orbit effects from structural inversion asym-metry: studies pioneered by Emmanuel Rashba

Rashba splitting




in a 2D electron system: HR = 2~

mkso [Sxky − Sykx]

Rashba splitting




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


Rashba effect & spin electronics

full picture: 2D electron eigenstates have ~S ⊥ ~k

2k

yk

kx

so




2k

yk

kx

so

gate-tuneable kso: spinFET

Ferromagnet Ferromagnet2D electron system

Vg

Vsd




2k

yk

kx

so



Vg

Vsd




2k

yk

kx

so



Vg

Vsd




2k

yk

kx

so



Vg

Vsd




2k

yk

kx

so



g

Vsd

"OFF" V




2k

yk

kx

so

gate-tuneable kso: spinFET "ON"

Ferromagnet2D electron system

V’g

Vsd

Ferromagnet


Rashba effect of surface states


Surface statespotential for electrons in solids terminates at surface



bulk states (bands) and surfaces states (discrete!) exist



bulk states (bands) and surfaces states (discrete!) existequilibration between bulk and surface: band bending!


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 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)


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)




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)


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)







image spin-dependent quantum-interference patternsPascual et al., PRL (04)

Walls & Heller, Nano Lett (07)

is graphite surface special (Dirac-electron quasiparticles)


Rashba

Documents

6th annual

bix pb1x ag

solidsquantum

structural

2d electron

2d electron

spinmicroscopic

magnetic moment