Spin-orbit effects in semiconductor quantum dots

Departament de Física, Universitat de les Illes BalearsInstitut Mediterrani d’Estudis Avançats IMEDEA (CSIC-UIB)

Palma de Mallorca (SPAIN)

Llorenç Serra

Outline: Introduction: experimental motivation Level structure in horizontal B Vertical B: spin precession Far Infrared absorption Confinement induced by SO

Collaborators: Manuel Valín-Rodríguez (Mallorca) Antonio Puente (Mallorca) Enrico Lipparini (Trento)

Introduction: experimental motivation

Experiments: level splittings of 1-electron quantum dots in B||

Hanson et al, PRL 91,196802 (2003)

Potok et al, PRL 91, 016802 (2003)

spli

ttin

g ( e

V )

B|| (T)

| g | = 0.37

| g | = 0.44

Origin of the deviations ?

* Extension of the wf’s in AlGaAs region (g=+0.4)

* Nuclear polarization effects (hyperfine)

* Non parabolicity of the bands

What is the role of typical spin-orbit couplings of semiconductors?

I. QD levels in a horizontal B

Model of spatial confinement: 2D representation (strong z confinement) effective mass model (GaAs conduction band) parabolic potential in xy plane

The Zeeman term: bulk GaAs gyromagnetic factor Bohr magneton Pauli matrices

)( *2

1

* 2222

0

22

yxmm

ppH yx

xy

)(* 2

1yyxxBZ BBgH

meV

067.0*

0

emm

44.0* g

cme

eB 2

yx ,

B

x

y

z

The Zeeman scenario

number) quantum(spin 1

number) quantum( ... , 2 , 1 , 0

number) quantum (principal ... , 2 , 1 , 0

*2

1 )12( 0

s

L

n

sBgn

z

Bsn

eigenstates: Laguerre polynomials eigenspinors in direction of B

sp energy levels

Bg Bs * spin splitting

Natural units:

e

cm

m

* field

*length

energy

0

00

0

Bg Bs *

),,( sn

The SO coupling terms

σVcm

p

4 22

conduction band (3D)

ii

zyxippp iiii

3

; ),,( 3,2,1 ; )( 2

22

1

* linear Dresselhaus term (bulk asymmetry)

in 2D quantum wells [001]:

2

0

2

zk zD

)( yyxxD

D pp

( z0 vertical width )

coupling constant

* Rashba term (nanostructure z asymmetry) )( yxxyR

R pp

eR 0 ( vertical electric field )

Rashba and Dresselhaus terms:

* used to analyze the conductance of quantum wells and large (chaotic) dots

R and D uncertain in nanostructures (sample dependent!)in GaAs 2DEG’s: 5 meVÅ - 50 meVÅ

* tunability of the Rashba strength with external fields (basis of spintronic devices)

We shall treat R and D as parameters

No exact solution with SO, but analytical approximations in limits:

a) Weak SO in zero field 0, ; 0 DRZ

smm

n RDRDsn

*

*

)12( 222

2220

fine structure: zero-field up-down splitting !

Kramers degeneracy

*

2 222 RDs

m

),,(),,( snsn

2nd order degenerate pert. theory

an alternative method: unitary transformation

)O( todiagonal is

~

) ()( exp

3

UU

yxxyiU yxDyxR

ZDRxy )(

b) Weak SO in large field

)12( 1 1

2

*

22)0(

zns

z

FG

msnsn

)2(sin 2

)2(sin 2

22

22

0

DRDR

DRDR

B

F

G

ω

Bg*μz

definitions

- new fine structure of the major shell- ( dependence) anisotropy!

Intermediate cases only numerically, - xy grid - Fock-Darwin basis

ZDR ,

ZDR ,

Parameters:

00

00

1.0

2.0

deg45

R

D

θ

00R

0

R0

15.0

T 58.0 Å; 330

meVÅ 05 and meV 1 if Typical?

Typical level spectra with SO ),,( sn

Anisotropy of first two shells at large B

Isotropic when only one source

Symmetry!

σB *2

1

)/exp(-i operator symmetry

us)(Dresselha

(Rashba)

0 ,

r,

,,

BDRxy

zzD

zzR

DRDR

g

SL

SL

Position of gap minima depend on

)(sgn DR

20B

)](min[)](max[

:(blue) anisotropy

:(red) splitting averaged-

0.2948 :Zeeman

20when

0

B

anisotropy + zero field splitting + position of minima QD energy levels could determine the lambda’s

(need high accuracy!)

Systematics of first-shell gap

In physical units:

below Zeeman |g*|B B (level repulsion)

0 dependence|g*|B B

Second shell:

two gaps (inner, outer)zero field value0 dependence

Experimental results from QD conductance: 1 electron occupancy

Potok et al., Phys. Rev Lett. 91, 018802 (2003)

Hanson et al., Phys. Rev Lett. 91, 196802 (2003)

BUT: zero field splitting of 2nd shell? - anisotropies?

spli

ttin

g ( e

V )

B|| (T)

| g | = 0.37

| g | = 0.44

SO effects in GaAs are close to the observations BUT only for a given B orientation.

Determination of the angular anisotropy and zero field splittingsare important to check the relevance of SO in these experiments.

M. Valín-Rodríguez et al. Eur. Phys. J. B 39, 87 (2004)

II. QD levels in a vertical B

As before, the Zeeman term:

DRxy ; ;

zBZ BgH * 2

1

B

x

y

z

BUT now, B also in spatial parts:

xc

eB

yiPp

yc

eB

xiPp

yy

xx

Symmetric gauge

energy levels (without SO)

parabola effective 4

frequencycyclotron *

*2

1

2

1

4 )12(

220

220

c

c

Bcc

sn

cm

eB

sBgn

at large field

bands)(Landau *2

1

2

)12(

0

sBgn

Bcsn

c

SO coupling redefines magnetic field weak SO (unitary tranformation)

sBgn

sm

Bscsc

sn

RDcscc

*2

1

2

1

4 )12(

*2

,

2,2

0

223,

ZDRxy ,

Spin precession without SO: The Larmor theorem

spin precessing *

eq. Heisenberg ,

BgSSdt

d

HSSdt

di

B

The Larmor frequency

*

Bg B

L

sL equals the spin-flip gap

Spin precession with SO

4

)12(

*

*2

220

22

322

3c

cRDRDLsP n

mm

spin-flip (precessional) transition (N = 7, 9, 11)

Real time simulations

No interaction

Real time simulations:

time-dependent LSDA

),(δ

][δ),',(

),(

' e

),(

'

'

2

t

EtV

tdtV

XCXC

H

rr

r'-r

r'rr

M. Valín-Rodríguez et al. Phys. Rev. B 66, 235322 (2002)

ÅmeV 04D

ÅmeV 011D

LSDAmeV 60

Deformation allows the transition between Kramers conjugates at B=0

x

y

yx yxm

2222*2

1

LSDA

M. Valín-Rodríguez et al. Phys. Rev. B 69, 085306 (2004)

Strong variation with tilting angle:

meVÅ 80

meVÅ 35

9.0

meV 62/)(

R

R

yx

Far Infrared Absorption (without Coulomb interaction):

splitting of the Kohn mode

222

00*

)(

mRD

at B=0

Far Infrared Absorption with Coulomb interaction: restores Kohn mode (fragmented)characteristic spin and density oscillation patterns

at B=0

Confinement induced by SO modulation:

Rashba term )(rRR )(iR)(e

R

bulk bands

... ,2

3 ,

2

1

)()(

jJ z

eR

iR

localized states

Conclusions:

* In horizontal fields SO effects are small, but they are close to recent observations. Zero field splittings and anisotropies are also predicted. * In vertical fields the SO-induced modifications of the g-factors are quiteimportant.

* Possibility of confinement induced by SO ?