Transcript Spectral Properties of 2D Spin

Spectral Properties of
a 2D Spin-Orbit Hamiltonian
Denis Bulaev
Department of Physics
University of Basel, Switzerland
Outline
• Motivation
• k.p method
• 2DEG
• Quantum Dots
• Summary
Motivation
• …..
• Quantum Computing
Supercoducting
[A.Shnirman, G.Shön, Z.Herman, PRL 79, 2371 (1997)]
Quantum-Dot-based
[D.Loss and D.P.DiVincenzo, PRA 57, 120 (1998)]
Nano’ll make $1T/yr by 2015
k.p method
P2
h
H=
+ V (r) + mBs ³B +
—V (r)¥ P³s
2 2
2m0
4m0 c
Pauli Hamiltonian
B = 0, V (r) = V (r + R)
Thomas term
(s-o coupling)
y nks (r) = eikr unk (r) ƒ s = eikr  cn ¢s ¢un ¢(r) ƒ s ¢ = eikr  cn ¢s ¢ n¢s ¢
n ¢s ¢
n ¢s ¢
2 2×
ÏÔÈ
¸Ô
h
k
h
Ô
Ô
Í
Ý
E
(0)
+
d
d
+
k
³P
d
+
D
c = cns En (k),
Ì
Â
n¢
nn ¢ s s ¢
nn ¢ s s ¢
ns n ¢s ¢ð n ¢s ¢
Í
Ý
Ô
2m0 Þ
m0
n ¢s ¢ Ô
Ô
Ô
ÓÎ
ý
h
Pns n ¢s ¢ = n P n¢ , D ns n ¢s ¢ =
ns —V (r)¥ P ³s n¢s ¢ .
2 2
4m0 c
Inversion asymmetric strs. (Td)
Td
E
8 C3 3 C2 6 s 6 S4
CB
G1
1
1
1
1
1
l=0 (s)
G2
1
1
1
-1
-1
j=l+s=1/2
G12
2
-1
2
0
0
G25
3
0
-1
-1
1
G15
3
0
-1
1
-1
E
G1
Eg
VB
G15
l=1 (p)
j=3/2 & 1/2
k
Bir and Pikus. Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York, 1974).
Inversion asymmetric strs. (Td)
E
Single group
G1
Double group
l=0
j=1/2
G6
E
D x G1 = G6
D x G15 = G7+ G8
G15 l=1
k
Eg
j=3/2
G8
j=1/2
G7
D
k
Optical Orientation, ed. by Zakharchenya and F. Meier (North - Holland, Amsterdam, 1984)
Bir and Pikus. Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York, 1974).
Kane Hamiltonian
Ê
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
HK = Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Ë
h 2k2
2m0
0
0
h 2k2
2m0
0
0
F - Eg
H
I
0
H*
G - Eg
0
I
I*
0
G - Eg
- H
3 *
H
2
0
I*
- H*
F - Eg
2I *
1
Pk+*
2
2 *
Pkz
3
-
1
Pk-*
6
1
Pk-*
3
1
Pk+*
6
2 *
Pkz
3
1
Pk-*
2
0
1
Pkz*
3
1
Pk+
2
-
-
1
Pk+*
3
1 *
H
2
-
1
Pkz*
3
ÊH A
H=Á
Á
Á
ËVBA
-
-
2I *
-
2
Pkz
3
1
Pk6
0
-
1
Pk+
6
2
Pkz
3
1
Pk2
-
G- F
2
3 *
H
2
3
H
2
-
G- F
2
-
1
Pkz
3
1
Pk+
3
1
H
2
-
G- F
2
2I
G+ F
- Eg - D
2
1
H
2
0
Folding down
VAB ˆ˜
V V
˜˜ , H Aeff ª H A + AB BA
H B - E˜¯
E
ˆ˜
˜˜
˜˜
˜˜
˜˜
1
˜˜
Pkz
˜˜
3
˜˜
˜˜
˜˜
- 2I
˜˜
˜˜
˜˜
˜˜
3
˜˜
H
˜˜
2
˜˜
˜˜
G- F
˜˜
˜˜
2
˜˜
˜˜
1 *
˜˜
H
˜˜
2
˜˜
˜˜
˜˜
0
˜˜
˜˜
˜˜
G+ F
˜
- Eg - D ˜˜˜
¯
2
-
1
Pk3
Electron effective Hamiltonian
h 2k2
He =
+ g k ³s ,
Dresselhaus SO (DSO) coupling
*
2m
k x = kx (ky2 - kz2 ), k y = ky (kz2 - kx2 ), k z = kz (kx2 - ky2 )
Dresselhaus, Phys. Rev. 100, 580 (1955).
(GaAs, InAs, InSb, etc - inversion asymmetry)
For Ge, Si - inversion symmetric strs (point group Oh = Td x Ci )
DSO = 0!
Remark No. 1
DSO is due to bulk inversion asymmetry (BIA)
AlyGa1-yAs
AlxGa1-xAs
GaAs
AlxGa1-xAs
AlxGa1-xAs
2DEG
GaAs
V(z)
V(z)
z
z
D2d (E; C2; 2C2; 2sd; 2S4)
C2v (E; C2; 2sv)
Dresselhaus SO interaction
H Dbulk = g k ³s ,
kz Æ - iŽ / Žz,
k x = kx (ky2 - kz2 ), k y = ky (kz2 - kx2 ), k z = kz (kx2 - ky2 )
kz = 0, kz2 š 0
H D2 DEG = H DL + H DC ,
H DL = g kz2 (- kx s x + kys y ),
H DC = g (kx ky2s x - ky kx2s y ).
H DC =
H DL , if k^2 =
kz2 µ 1 / d 2
D'yakonov & Kocharovskii, Sov. Phys. Semicond. 20, 110 (1986)
Rashba SO interaction
H = H K (kx , k y ,- iŽ / Žz)+ V ( z).
After folding down
He =
h 2 kz2
2m*
h 2 k^2
+
+ HD + HR,
*
2m
H R = a R (s x ky - s y kx ), a R µ Ez + Ž zV ( z)
Bychkov & Rashba, JETP Lett. 39, 78 (1984).
Remark No. 2
RSO is due to structure inversion asymmetry (SIA)
Spin degeneracy & splitting
without SO coupling
with SO coupling
time inversion symmetry (Krames degeneracy)
E° (k) = EØ(- k)
space inversion symmetry
space inversion asymmetry
E° (k) = E° (- k)
E° (k) š E° (- k)
spin degeneracy
spin splitting
E° (k) = EØ(k)
E° (k) š EØ(k)
Energy spectrum of 2DEG
h 2 k^2
H=
+ b D (- s x k x + s y k y )+ a R (s x k y - s y k x ),
*
2m
h 2 k^2
2
2
2
E=
m
a
+
b
k
(
)
R
D
^ - 4a R b D k x k y
*
2m
H SO = s ³Beff ,
B eff
D = b D (- k x , k y ),
B eff
R = a R (k y ,- k x )
h 2 k^2
E (k x = k y ) =
m a R + b D k^ ,
*
2m
h 2 k^2
E (k x = - k y ) =
m a R - b D k^ ,
*
2m
Ganichev, et al., PRL 92, 256601 (2004).
Spin decoherence anisotropy
H SO = s ³Beff (k)
S x ± Sy 1
C
2
&
&
S x ± Sy = ,
= (a R ± b D ) ,
T±
T±
2
m
C= 4
h
ÚdE ÈÎF (E) - F (E)×Þt (E) E ,
ÚdE ÈÎF (E) - F (E)×Þ
+
-
+
p
-
1
=
tp
—dJW
Ú
(1- cos J ).
kk ¢
Averkiev & Golub PRB 60, 15582 (1999).
Remark No. 3
SO coupling leads to anisotropy in dispersion
and spin decoherence
Effective Hamiltonian for a QD
H = H 0 + H SO ,
P2
m*w02 2
1
2
H0 =
+
x + y )+ gmB Bs z ,
(
*
2m
2
2
H SO = a R (s x Py - s y Px )+ b D (- s x Px + s y Py ),
e
P = p + A(r), B || Oz
c
Canonical transformation
H0 =
px2 + py2
2m*
wc
m*W2 2
hwZ
2
+
x + y )+
s z,
(xpy - ypx )+
(
2
2
2
W=
w02 + wc2 / 4, wZ = gmB B / h, w1,2 = Wm wc / 2
x=
1
w1 q1 +
(
2W
px =
w2 q2 ),
WÊ
p1
p2 ˆ˜˜
Á
Á
+
˜˜ ,
Á
2Á
w2 ˜¯
Ë w1
y=
1
m*
py = m
*
Êp
Á
1
Á
Á
2WÁ
Ë w1
W
(
2
w1 q1 +
p12 + p22 m* 2 2
hwZ
2 2
H0 =
+
w1 q1 + w2 q2 )+
s z,
(
*
2m
2
2
hwZ
(0)
En,m
= hw1 (n + 1 / 2) + hw2 ( m + 1 / 2) +
s z.
2
Geyler, Margulis, Shorokhov, PRB 63, 245316 (2001).
p2 ˆ˜˜
˜˜ ,
w2 ˜¯
w2 q2 ),
Three lowest electron energy levels
Dresselhaus SO coupling

E1    Z ,
2

E2    Z ,
2

E3    1  Z .
2
Rashba SO coupling
h
E1 = hW- wZ ,
2
h
E2 = hW+ (w1 - w),
2
h
E3 = hW+ (w1 + w),
2
w=
2
(w1 - wZ ) + 4m*a R2 w12 / hW.
Anti-crossing (crossing) of the levels E2 and E3 at w1 = wZ .
Anticrossing due to Rashba coupling
E3 – E1
0.25
°
Ø
Energy [meV]
0.20
orbital
D
0.15
0.10
Zeeman
0.05
Ø
B0  5.2 T
°
0
E2 – E1
°
2
4
6
8
10
E1 – E1
B [T]
D = 2hwZ (l / l R ) ª 0.5 meV = 6 mK = 0.02 T = 1.3¥ 10- 9 s (l R = 8 mm).
Bulaev, Loss, PRB 71, 205324 (2005).
Summary
• SO coupling is due to space inversion asymmetry
• Dispersion anisotropy in a 2DEG
• Anticrossing due to RSO in a QD