Spin-orbit effects in semiconductor quantum dots Departament de Física, Universitat de les Illes Balears Institut 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)
Transcript
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)
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.
