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