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Heavy-Hole Spin Relaxation and Decoherence in Quantum Dots
Denis Bulaev and Daniel Loss
Department of Physics and Astronomy, University of Basel, Switzerland Bulk Hamiltonian for the Valence Band
H where H
LK v D v R Z bulk

ABSTRACT
We investigate heavy-hole spin relaxation and decoherence in quantum dots in perpendicular magnetic fields [1]. We show that at low temperatures the spin decoherence time is two times longer than the spin relaxation time. We find that the spin relaxation time for heavy holes can be comparable to or even longer than that for electrons in strongly two-dimensional quantum dots. We discuss the difference in the magnetic-field dependence of the spin relaxation rate due to Rashba or Dresselhaus spin-orbit coupling for systems with positive (i.e., GaAs quantum dots) or negative (i.e., InAs quantum dots) g -factor.
10
el 1

Low B , Comparison with Electrons
T1 16 el 9 T1 GaAs QD ( = 0.18) T1 1.6 в 10-2 T el
1

=H

LK

v v + HD + HR + HZ ,

gel gz z

4

mel m

4

l0 d

4

2 .

H

H

is the Luttinger­Khon Hamiltonian, = - J · , (Dresselhaus term) = R P в E · J, (Rashba term) = -2µB B · J - 2q µB B · J , (Zeeman term)

InAs QD ( = 0.48) T1 1.5 T el
1

1 10
-1

H

T1/T

is due to BIA, =

so

/(Eg +

so

10

-2

Spin Relaxation and Decoherence
10

-3

Effective Hamiltonian for Heavy Holes
H= where
10
7 6 5

1 =W T1

n-1 n1

+
i=1

Win ,

1 1 1 = + T2 2T1 2

n-1

10

-4

0.01

0.1 B (T)

1

10

T1/T

2 2 3 3 3 ), z = Pz (Px - Py ), and J = (Jx , Jy , Jz ).

el 1

1

0.1 0.01

0.1 B (T)

1

10

Wi1 ,
i=2

2 1 m0 2 2 (P 2 + P y ) + (x + y 2 ) + H 2m x 2

hh D

+H

hh R

1 - g z z µ B B z z , 2

where Wij is the transition rate from state j to state i. At low temperatures T ), ( ph

Fig. 3. Ratio between the heavy hole (T1 ) and electron (T ) spin relaxation time due to Dresselhaus SO coupling.

el 1

T 2 = 2T 1 .
(a) 10 10 10 1/T1 (1/s) 10 10 10 0.2 10 0 0 100 2 4 B (T) 6 2 B (T) 8 4 10 10 10
8 7 6 5 4 3

(b)

hh HD

= - (+ P- P+ P- + - P+ P- P+ ), (Dresselhaus term)
3 3 = i(+ P- - - P+ ), (Rashba term [2])
1/T1 (1/s)
2 z

10 10

H

hh R

Conclusions · Anticrossing and spin mixing (GaAs QD)
10

104 Energy (meV) 10 10
3 2

= 30 R Ez /2m0 , = 30 P hh lh = E 0 - E0 .

/2m0 ,

±

= (

x

± iy )/2, P

±

= Px ± iPy , and

0.4

DSO
2 1 0

RSO electrons 2 4 B (T) 6 8

101

Energy Levels
Without spin-orbit (SO) interaction E
n1 n2 ()

· Cusp-like behavior of the spin relaxation (GaAs QD) · No cusp in spin relaxation (InAs QD) · Rashba B 9 Dresselhaus B
5

FIG.1 Fig. 2. Spin relaxation rate 1/T1 in BulaevGaAs (a) and an InAs (b) QD a /m0 = 30 nm, and T = 0.1 K; for an InAs QD (b), (d = 5 nm, l0 = m = 0.115m0, gzz = -2.2, / 3 = 130 eV °3 , and = 150 meV). A

=

-

1 n1 + 2
2 0 2 c

+

+

1 n2 + 2

Z , 2

where ± = ± c /2, = + /4, and Z = gzz µB B / . SO interaction leads to level anticrossings for gzz > 0.

Low B , Field Dep endence Electrons
1 B T1

· Spin relaxation time for heavy holes CAN BE longer than for electrons · T2 = 2T1 at low temperatures

0.4

0.2

Fig. 1. Energy levels GaAs QD relative to state (m = 0.14m0 , A / 3 = 28 eV °3 , 40 meV).
0 2 B (T) 4

of HHs in a the ground gzz = 2.5, and =

H

so

B

2+3

(2N



Z

+ 1) [B < 4 T]

Energy (meV)

Heavy holes
Dresselhaus Rashba H H
so

References
1 B 2+3 (2NZ + 1) [B < 0.5 T] T1 1 B3 B 6+3 (2NZ + 1) [B < 0.5 T] T1 B [1] D. V. Bulaev and D. Loss, cond-mat/0503181 (to be published in Phys. Rev. Lett.). [2] R. Winkler, H. Noh, E. Tutuc, and M. Shayegan, Phys. Rev. B 65, 155303 (2002).

0

so