Spin and charge oscillation properties of semiconductor quantum
dots from real time simulations
Llorenç Serra
Departament de Física, Universitat de les Illes Balears,
IMEDEA (CSIC-UIB)
Outline:
* Model of 2d quantum dot
* Theoretical framework
* Some results
* Spin-orbit coupling effects on spin precession
Collaborators:
A. Puente (Mallorca)
M. Valín-Rodríguez (Mallorca)
E. Lipparini (Trento)
V. Gudmundsson (Reykjavik)
1
Semiconductor quantum dots
Vertical quantum dots
z
L  10-100 nm
AlxAsGa1-x
 1 nm
GaAs
* System of confined electrons in 2D
* Possibilities: Control of
Geometry (circular, elliptic, rectang..)
Size
Number of electrons
ARTIFICIAL ATOMS
2
A SIMPLE MODEL: N electrons in 2D confined by Vext(r)
conduction electrons in GaAs
effective mass m = 0.067 me
dielectric constant k = 12.4
interaction e2/kr
confinement potential:
jellium disk
square well
harmonic potential
1
Vext (r )  V0  m* 02 r 2
2
V0  2 N p rs
 02  1

N p rs3

H*  m*e 4 /  2 (4 0k ) 2  12 meV
effective atomic units
a0*
o
  (4 0k ) / m e  98 A
* 
2

H
*
* 2
 55 fs
3
Mean field Ground State

Set of orbitals and single-particle energies i (r ) ,  i
 i  1,...,N;  , 
* Kohn-Sham version of density-functional-theory:
exchange-correlation in LSDA (or CDFT)
h[  (r),  (r)] i (r )   i i (r )
Spin densities:      
    
m    
*Hartree-Fock theory:
exact exchange but no correlation
h  ' s i (r )   i i (r )
Numerical methods:
Discretization of the xy plane in a grid
Iterative solution
4
Time evolution
td-LSDA i
 

i (r ; t )  h ,   i (r ; t )


t
td-HF
 


i (r ; t )  h ' s i (r ; t )
t
i
After an initial perturbation on the GS
keep track of observables in ‘real’ time
Fourier analysis O( )
i (r;0)  P i (r)
O (t )
oscillation frequencies
eigenmodes:
FIR (charge dipole)
spin modes
Alternative to perturbation theory
not restricted to small amplitudes or by symmetry
5
Example: Spin-density oscillation
6
Dipole Excitation
L
i
b
r
e
X
Y
S 
Free
Collective excitations in
deformed systems
N=20 electrons in a deformed parabola
y= 0.75 x = 0.218 H*

 1
V (r )  m *  2x x 2  2y y 2
2
E
s
p
н
n p-h transitions
Spin
S 
X
Y
0
.
0
0
.
1
X
Y

.
2
 0
D
e
n
s
i
d
a
d
S 
Density
Landau damping
atract. residual interaction
Generalized Kohn theorem x,y
 x  0.291H*
 y  0.218 H*
0
.
0
0
.
2

0
.
4
7
* td-(mean field) includes correlation effects
* Defines a new correlated ground state (RPA)
Applications:
* Orbital modes (Lz)
* Quadrupole modes (xy)
* Absorption patterns in triangular (square) dots
* Large amplitude motions
* ...
8
FIR Absorption in polygonal dots
9
Local absorption patterns:
* amplitude of oscillating
density
B=1T
* corner and side modes
10
Large amplitude motion in tdHF:
* non parabolic confinement
* CM trajectory
* initial rigid displacement
* 3 intervals of 9000 steps (12 ps each)
* Amplitude shrinks
11
Energy goes to internal modes
12
Spin-orbit coupling and spin precession in quantum dots
Two sources :
*Dresselhaus (bulk asymmetry)
hR 
*Rashba (nanostructure asymmetry)
hD 
R

D

( P y  x  Px y )
( P x  x  Py y )
Coupling constants for 2D bulk:
R   0 e Ε
D   k z 2
( E vertical electric field )
 
   
 z0 
2
( z0 vertical width )
*  and 0 known from calculations for the bulk (k.p, tight binding)
* R and D uncertain in nanostructures (sample dependent)
in GaAs 2DEG’s: 5 meVA - 50meVA
* Tunability of the Rashba strength
13
*assume given ’s
2
e
H   h(i )  
i
i  j rij
h  h0  hZ  hR  hD
*sp hamiltonians
2
P
h0 
 Vext ( x, y )
2m *
1
hZ  g *  B ( Bx x  B y x  Bz x )
2
14
  i (r, ) 

 i (r, )  
  i (r, ) 
* spinorial orbitals:
* noncollinear SDFT:

E[  (r ), m(r )]
 (r )     i (r, )
2
i 


m(r )     i *(r, )  ( , ' )  i (r, ' )
i  '
* spin textures on the ground state
* time evolution
 h(r; ' )  i (r; ' )   i  i (r, )
'

i  i (r, )   h(r; ' )  i (r, ' )
t
'
15
*Analytical solution:
neglect interactions
vertical magnetic field (Bz)
Aleiner-Falk’o transformation
 
U  exp  i  R ( y x  x y )   D ( x x  y y )
~

3
h  U h U is diagonal to O( )

* Spin precession: dl = 0 quasi spin-flip
 P   L  2 D 2
 (2n    1) D 2
c
02  c 2 / 4
* Larmor precession:
 L  g *  B B
16
1
5
0
N
=
7
=


1
0
0
P / 2 (GHz)
* Dreselhaus SO
* strong (z0=50A, blue symbols)
weak (z0 =85A, green symbols)
=

5
0

L
0
N
=
9


P / 2 (GHz)
1
0
0
5
0
=
0
0
=


3
N
=
1
1
1
0
0
P / 2 (GHz)
* zero field offset
* rearrangements jumps
* results within LSDA
0
5
0
0
0
1
2
B
(T
)
3
17
time simulation of spin precession in LSDA:
18
In a horizontal B:
*numerical calculation
*first and second levels
*circular parabolic confinement
19


1
2
2
Vext ( x, y )   x 2 x   y 2 y
2
y
transition between Kramers conjugates

x
Elliptical dots:
20
21