Denis Bulaev and Daniel Loss Department of Physics University of Basel, Switzerland Anticrossing and Spin Anticrossing and Spin Relaxation Relaxation of Electrons and Holes of Electrons and Holes in Quantum Dots in Quantum Dots
Jan 11, 2016
Denis Bulaev and Daniel Loss
Department of PhysicsUniversity of Basel, Switzerland
Anticrossing and Spin Relaxation Anticrossing and Spin Relaxation of Electrons and Holes of Electrons and Holes
in Quantum Dotsin Quantum Dots
OutlineOutline
• Dresselhaus and Rashba SO Coupling
• Electrons in Quantum Dots
• Heavy holes in Quantum Dots
HDc = g s ◊k, g =
bc
(2mel )3Eg
, bc =4mel
3mcv
h 1-h
3
Ê
ËÁÁÁ
ˆ
¯˜̃˜
- 1/2
,
HDv = -
g
hJ ◊k, h =
D so
Eg + D so
,k z = Pz Px2 - Py
2( ).
Dresselhaus spin-orbit couplingDresselhaus spin-orbit couplingK
an
e m
od
el
conduction band
valence band
⎭⎬⎫
bulk
Qu
an
tum
Wel
l
HDc = HD
c,L + HDc,C ,
HDc,L = g Pz
2 - s xPx + s yPy( ), HDc,C = g s xPxPy
2 - s yPyPx2( ),
HDv = HD
v,L + HDv,C ,
HDv,L = -
g
hPz
2 - JxPx + JyPy( ), HDv,C = -
g
hJxPxPy
2 - JyPyPx2( ).⎭⎬⎫
2D
Dresselhaus spin-orbit couplingDresselhaus spin-orbit couplingQ
ua
ntu
m W
ell
HDc = HD
c,L + HDc,C ,
HDc,L = g Pz
2 - s xPx + s yPy( ), HDc,C = g s xPxPy
2 - s yPyPx2( ),
HDv = HD
v,L + HDv,C ,
HDv,L = -
g
hPz
2 - JxPx + JyPy( ), HDv,C = -
g
hJxPxPy
2 - JyPyPx2( ).⎭⎬⎫
2D
Qu
an
tum
Do
t
HDL
HDC
ªPz
2
Px2
ªlxlz
Ê
ËÁÁÁÁ
ˆ
¯˜̃˜̃
2
= 36 fiGD
L
GDC
ª 103 (lx = 30 nm, lz = 5 nm).⎭⎬⎫
0D
Rashba spin-orbit couplingRashba spin-orbit coupling
HRc = a c P¥ E◊£m,
a c =e h
2mel
D so(2Eg + D so )
Eg (Eg + D so )(3Eg + 2D so ) [1].
HRv = a1P¥ E◊J + a 2P¥ E◊n , n = (Jx
3, Jy3, Jz
3) [2].
[1] E. A. de Andrada e Silva, G. C. La Rocca, and F. Bassani, PRB 50, 8523 (1994).[2] R. Winkler, H. Noh, E. Tutuc, and M. Shayegan, Phys. Rev. B 65, 155303 (2002).
From the Group Theory:
a1
a 2
due to the coupling between the conduction band ( ) and the valence band ( ).
c6Γ
v8Γ
due to the coupling between more remote bands ( and ).c7Γ c
8Γ
From the Kane model, a 2 = 0!
H = H 0 + H SO ,
H 0 =P2
2mel
+melw0
2
2x2 + y2( ) +
1
2gelmBBs z ,
H SO = a Rel s xPy - s yPx( ) + bD
el - s xPx + s yPy( ),
P = p +e
cA(r), B ||Oz
Effective Hamiltonian of electronsEffective Hamiltonian of electrons
( )
( )
( ) ( ) ./4
,2
,2
,2
21
221
13
12
1
ωλωω
ω
ω
ω
RZ
Z
lw
wE
wE
E
+−=
++Ω=
−+Ω=
−Ω=
hh
hh
hh
Dresselhaus SO coupling Rashba SO coupling
W= w0
2 + wc2 / 4, w1,2 = Wm
wc
2, l = h / melW, l R = h / mela R
el .
.2
,2
,2
13
2
1
Z
Z
Z
E
E
E
ωω
ω
ω
hhh
hh
hh
−+Ω=
+Ω=
−Ω=
Anti-crossing (crossing) of the levels E2 and E3 at w1 = wZ = mB | gelB | /h.
Three lowest electron energy levelsThree lowest electron energy levels
D = 2hwZ (l / l R ) ª 0.5 meV = 6 mK = 0.02 T = 1.3¥ 10- 9 s l R = 8 mm( ).
0
0.05
0.10
0.15
0.20
0.25
2 4 6 8 10
Energy [meV]
B [T]
E2 – E1
E3 – E1
E1 – E1
T 2.50 ≈B
Δ
orbital
Zeeman
[1] C. F. Destefani, S. E. Ulloa, and G. E. Marques. Phys. Rev. B 69, 125302 (2004).
Anticrossing due to Rashba couplingAnticrossing due to Rashba coupling
Ø
≠
≠
Ø ≠
Γ Γ
Γ
Anticrossing due to Rashba couplingAnticrossing due to Rashba coupling
D = 2hwZ (l / l R ) ª 0.5 meV = 6 mK = 0.02 T = 1.3¥ 10- 9 s l R = 8 mm( ).
0
0.05
0.10
0.15
0.20
0.25
2 4 6 8 10
Energy [meV]
B [T]
E2 – E1
E3 – E1
E1 – E1
T 2.50 ≈B
Δ
[1] C. F. Destefani, S. E. Ulloa, and G. E. Marques. Phys. Rev. B 69, 125302 (2004).
Ø
≠
≠
Ø ≠
Relaxation, Decoherence, and DephasingRelaxation, Decoherence, and Dephasing
Spin dephasing rate
Orbital dephasing rate
1
T1
=2p
h
V
2p( )3 dq3Ú
a 2Nqa + 1( ) f Uqa
ph i2d | E f - Ei | - hwqa( ),
1
T2
=1
2T1
+1
Tj
,
1
Tj
µ O a so4( ),
1
Tj
µ limqÆ 0
q3,
↑↓Γ
↑↓Γ32
Dresselhaus coupling Rashba coupling
hw0 = 1.1 meV, d = 5 nm, gel = - 0.44, l R = l D = 8mm.
At kBT < < hwc < < hw0 , G≠Ø µ wZ5 .
Electron relaxation ratesElectron relaxation rates
21Γ31Γ
Zeeman energy for holesZeeman energy for holes
H Z = - 2kmBB ◊J - 2qmBB ◊J, [1]
H hhZ = -
1
2ghhmBBs z , B||Oz, ghh = 6k +
27
2q
Ê
ËÁÁÁ
ˆ
¯˜̃˜
ghh ª7, (GaAs),
46, (InAs).
ÏÌÔÔ
ÓÔÔ
ghh ª2.5, (GaAs) [2],
- 2.21, (InAs) [3].
ÏÌÔÔ
ÓÔÔ
[1] J. M. Luttinger, Phys. Rev. 102, 1030 (1956).[2] H.W. van Kestern, et al., PRB 41, 5283 (1990). [3] M. Bayer,et al., PRL 82, 1748 (1999).
⎭⎬⎫
bulk
⎭⎬⎫
2D case
Heavy holes and SO interactionsHeavy holes and SO interactions
Hh = H LK + U(x, y) + HD + H R + HZ ,
H LK =1
2m0
F H I 0
H * G 0 I
I * 0 G - H
0 I * - H * F
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃
,
F = (g1 + g0 )(Px2 + Py
2 ) + (g1 - 2g0 )Pz2 ,
G = (g1 - g0 )(Px2 + Py
2 ) + (g1 + 2g0 )Pz2 ,
H = - 2 3g0PzP- ,
I = - 3g0P-2 ,
HD = bDh - JxPx + JyPy( ),
H R = a Rh JxPy - JyPx( ).
Jx =
03
20 0
3
20 1 0
0 1 03
2
0 03
20
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃
, Jy =
0 -3
2i 0 0
3
2i 0 - i 0
0 i 0 -3
2i
0 03
2i 0
Ê
Ë
ÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃˜̃
.
Effective Hamiltonian of heavy holesEffective Hamiltonian of heavy holes
H = H0 + H R + HD ,
H0 =P2
2mhh
+mhhw0
2
2x2 + y2( ) -
1
2ghhmBBs z ,
H R = ia s + P-3 - s - P+
3( ),
HD = - b s + P- P+ P- + s - P+ P- P+( ),
mhh = m0 / g1 + g0( ), a = 3g0a Rh / 2m0D hl ,
b = 3g0bRh / 2m0D hl , P± = Px ± iPy ,s ± = s x ± is y .
Dresselhaus coupling Rashba coupling
,8/)(
,/)(200
2
nm. 5 nm, 30
33
33
22ph
0
h
h
ZhhR
ZhhD
lm
lm
dl
dl
ωαωωβω
λ
≈
≈
−≈
==
Γ
Γ
Heavy-hole relaxation rates (GaAs)Heavy-hole relaxation rates (GaAs)
the maximum in the rate
310 ≈⇒⎭⎬⎫
Γ
ΓR
D
ωω
Dresselhaus coupling Rashba coupling
nm. 5 nm, 300 == dl
Heavy-hole relaxation rates (InAs)Heavy-hole relaxation rates (InAs)
Magnetic-field dependence of ratesMagnetic-field dependence of rates
H so µ B fi
1
T1
µ B2+ 3 2NwZ+ 1( ) [B < 4 T, hwZ = 1.2 K]
Electrons
HD µ B fi
1
T1
µ B2+ 3 2NwZ+ 1( ) [B < 0.5 T = 0.8 K]
Heavy holes
Hsophonons
Dresselhaus SO coupling
H R µ B3 fi
1
T1
µ B6+ 3 2NwZ+ 1( ) [B < 0.5 T = 0.8 K]
Rashba SO coupling
Electrons vs. heavy holesElectrons vs. heavy holes
Gel
Ghh
ª16
9
gel
ghh
Ê
ËÁÁÁÁ
ˆ
¯˜̃˜̃
4mel
mhh
Ê
ËÁÁÁÁ
ˆ
¯˜̃˜̃
4l0d
Ê
ËÁÁÁ
ˆ
¯˜̃˜
4D so
2
(Eg + D so )2 (low B, hwZ < < kBT).
GaAs QD with l0=30 nm, d=5 nm
Gel
Ghh
ª 0.01
InAs QD with l0=30 nm, d=5 nm
Gel
Ghh
ª 1.6 (0.4 with DA-phonons)
SummarySummary
• Anticrossing and spin mixing
• Cusp-like behavior of the spin relaxation
Electrons
• Anticrossing and spin mixing (GaAs QD)
• Cusp-like behavior of the spin relaxation (GaAs QD)
• No cusp in spin relaxation (InAs QD)
• Rashba Dresselhaus
• Spin relaxation time for heavy holes CAN BE longer
than for electrons
Heavy holes
µ B9 µ B5