Heavy-Hole Spin Resonance in Quantum DotsDenis Bulaev and Daniel Loss

Department of Physics and Astronomy, University of Basel, Switzerland

ABSTRACT

We propose and analyze a new method for manipulation of a heavy hole spin in aquantum dot [1]. Due to spin-orbit coupling between states with different orbitalmomenta and opposite spin orientations, an applied rf electric field inducestransitions between spin-up and spin-down states. This scheme can be usedfor detection of heavy-hole spin resonance signals, for the control of the spindynamics in two-dimensional systems, and for determining important parametersof heavy-holes such as the effective g-factor, mass, spin-orbit coupling constants,spin relaxation and decoherence times.

Effective Hamiltonian for Heavy Holes

H =1

2m(P 2

x + P 2y ) +

mω20

2(x2 + y2) − 1

2g⊥µBB⊥σz + HSO,

HSO = iαP 3−σ+ + βP−P+P−σ+ + γB−P 2

−σ+ + h.c.,

where α = 3γ0αR〈Ez〉/2m0∆, β = 3γ0γ1〈P 2z 〉/2m0η∆, γ = 3γ0κµB/m0∆, and ∆ = Ehh

0 − Elh0 .

|+〉 =

˛˛0, 0, +

3

2

fl+ iβ

+1

˛˛1, +1,− 3

2

fl+ β

+2

˛˛3, +1,− 3

2

fl+ α

+˛˛3, +3,−3

2

fl+ γ

+B+

˛˛2, +2,− 3

2

fl,

|−〉 =

˛˛0, 0,−3

2

fl+ iβ

−1

˛˛1,−1, +

3

2

fl+ β

−2

˛˛3,−1, +

3

2

fl+ α

−˛˛3,−3, +

3

2

fl+ γ

−B−

˛˛2,−2, +

3

2

fl,

where β±1 = β(ml)3ω±(ω2

− + ω2+)/ω±

D , γ± = 3√

2γ0κµB(ml)2ω2±/m0∆ω±

‖ , ω± = Ω ± ωc/2, and

Ω =q

ω20 + ω2

c/4.

Spin Relaxation and Decoherence

1T1

=1

TDSO1

+1

T‖1

+1

TRSO1

,

1TDSO

1

=β2

ω3Zm2

24πρ

(NωZ +

12

)(ω−

ω− − ωZ− ω+

ω+ + ωZ

)2 ∑α

e−ω2Zl2/2s2

α

s5α

I(3),

1

T‖1

=γ2B2

‖ω5Z

27πρΩ4

(NωZ +

12

)(ω2−

2ω− − ωZ+

ω2+

2ω+ + ωZ

)2 ∑α

e−ω2Zl2/2s2

α

s7α

I(5),

1TRSO

1

=α2

3ω7

Z

28πρΩ6

(NωZ +

12

)(ω3−

3ω− − ωZ− ω3

+

3ω+ + ωZ

)2 ∑α

e−ω2Zl2/2s2

α

s9α

I(7).

At low temperatures (ωph T ) [2], T2 = 2T1.

1

10

102

103

104

105

106

107

0.3 0.5 0.7 0.9 1.1 1.3 1.5

1 /

T1

(1/s

)

B⊥ (T)

B|| = 0 T

B|| = 0.5 T

B|| = 3 T

0

0.1

0.2

0.3

0.3 0.7 1.1 1.5

Ene

rgy

(meV

)

B⊥ (T)

Fig. 1. Heavy hole spin relaxation rate 1/T1 in a GaAs QD versus anapplied perpendicular magnetic field B⊥ (the height of a QD is h = 5nm,the lateral size l0 =

√/mω0 = 40 nm, κ = 1.2, γ0 = 2.5, g⊥ = 2.5).

Inset: Energy differences of lowest excited levels with respect to theground state E0,0,+3/2.

Interaction of HHs with RF Electric Fields

E(t) = E(sin ωt,− cos ωt, 0),〈HE(t)〉 = Tr(ρHE(t)) = −dSO · E(t), (coupling energy)

dSO =β|e|mω2

0

ωΩ2

(ω2−

ω− − ωZ+

ω2+

ω+ + ωZ

)(effective dipole moment).

RF Power Absorbed by the System

P = −d〈HE(t)〉dt

=2ω(dSOE)2T2ρ

Tz /

1 + δ2rfT

22 + (2dSOE/)2T1T2

.

1e-8 1e-6 1e-4 1e-2 1 1e2 1e4 1e6

ω (GHz)

B⊥

(T

)

50 100 150 200

0.4

0.6

0.8

1.0

1.2

Fig. 2. Absorbed power P (meV/s) as a function of perpendicularmagnetic field B⊥ and rf frequency ω (T2 = 2T1, E = 2.5 V/cm,B‖ = 1 T).

Br,1⊥ = ω/g⊥µB, Br,2

⊥ = ω0/g⊥µB

√1 + 2m0/g⊥m,

Br,3⊥ = 4ω0/g⊥µB

√1 + 4m0/g⊥m, Bd

⊥ = (ω0/2g⊥µB)√

2m0/g⊥m.

Rabi Oscillations

〈Sz〉 = STz + e−(T−1

1 +T−12 )t/2

(32− ST

z

)cos ωRt

+[(dSOE)2T2

2ωRST

z − T−11 − T−1

2

2ωR

(32− ST

z

)]sinωRt

,

where ωR =√

(dSOE/)2 − (T−11 − T−1

2 )2/4 is the Rabi frequencyand ST

z = (3/2)ρTz /[1 + (dSOE/)2T1T2].

-3/2

0

3/2

0 0.2 0.4 0.6 0.8 1

S z

t (µs)Fig. 3. Rabi oscillations at three different values of theperpendicular magnetic field: B⊥ = 0.8 T (damped fastoscillations), B⊥ = 0.865 T (dotted line), and B⊥ = 0.5 T (solidline). B‖ = 0, δrf = 0, E = 1.5 V/cm.

Conclusions

• Spin-orbit coupling is suppressed for flat QDs

• Spin relaxation time T1 can be milliseconds

• Coherent spin manipulation by RF electric fields

• Strong dependence of Rabi oscillations on B⊥

References[1] Denis V. Bulaev, Daniel Loss. Electric Dipole Spin Resonance for Heavy Holesin Quantum Dots, cond-mat/0608410.[2] Denis V. Bulaev, Daniel Loss. Spin Relaxation and Decoherence of Holes inQuantum Dots, Phys. Rev. Lett. 95, 076805 (2005).