Heavy-Hole Spin Resonance in Quantum Dots Denis 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 a quantum dot [1]. Due to spin-orbit coupling between states with different orbital momenta and opposite spin orientations, an applied rf electric field induces transitions between spin-up and spin-down states. This scheme can be used for detection of heavy-hole spin resonance signals, for the control of the spin dynamics in two-dimensional systems, and for determining important parameters of 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 2 y )+ mω 2 0 2 (x 2 + y 2 ) − 1 2 g ⊥ µ B B ⊥ σ z + H SO , H SO = iαP 3 − σ + + βP − P + P − σ + + γB − P 2 − σ + + h.c., where α =3γ0αREz/2m0∆, β =3γ0γ1P 2 z /2m0η∆, γ =3γ0κµB/m0∆, and ∆ = E hh 0 − E lh 0 . |+ = ˛ ˛ ˛ ˛ 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) 3 ω±(ω 2 − + ω 2 + )/ω ± D , γ ± =3 √ 2γ0κµB(ml) 2 ω 2 ± /m0∆ω ± , ω± =Ω ± ωc/2, and Ω= q ω 2 0 + ω 2 c /4. Spin Relaxation and Decoherence 1 T 1 = 1 T DSO 1 + 1 T 1 + 1 T RSO 1 , 1 T DSO 1 = β 2 ω 3 Z m 2 2 4 πρ N ωZ + 1 2 ω − ω − − ω Z − ω + ω + + ω Z 2 α e −ω 2 Z l 2 /2s 2 α s 5 α I (3) , 1 T 1 = γ 2 B 2 ω 5 Z 2 7 πρΩ 4 N ωZ + 1 2 ω 2 − 2ω − − ω Z + ω 2 + 2ω + + ω Z 2 α e −ω 2 Z l 2 /2s 2 α s 7 α I (5) , 1 T RSO 1 = α 2 3 ω 7 Z 2 8 πρΩ 6 N ωZ + 1 2 ω 3 − 3ω − − ω Z − ω 3 + 3ω + + ω Z 2 α e −ω 2 Z l 2 /2s 2 α s 9 α I (7) . At low temperatures (ω ph T ) [2], T 2 =2T 1 . 1 10 10 2 10 3 10 4 10 5 10 6 10 7 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1 / T 1 (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 Energy (meV) B ⊥ (T) Fig. 1. Heavy hole spin relaxation rate 1/T 1 in a GaAs QD versus an applied perpendicular magnetic field B ⊥ (the height of a QD is h = 5nm, the lateral size l 0 = /mω 0 = 40 nm, κ =1.2, γ 0 =2.5, g ⊥ =2.5). Inset: Energy differences of lowest excited levels with respect to the ground state E 0,0,+3/2 . Interaction of HHs with RF Electric Fields E(t) = E(sin ωt, − cos ωt, 0), H E (t) = Tr(ρH E (t)) = −d SO · E(t), (coupling energy) d SO = β|e|mω 2 0 ωΩ 2 ω 2 − ω − − ω Z + ω 2 + ω + + ω Z (effective dipole moment). RF Power Absorbed by the System P = − dH E (t) dt = 2ω(d SO E) 2 T 2 ρ T z / 1+ δ 2 rf T 2 2 + (2d SO E/) 2 T 1 T 2 . 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 perpendicular magnetic field B ⊥ and rf frequency ω (T 2 =2T 1 , E =2.5V/cm, B = 1 T). B r,1 ⊥ = ω/g ⊥ µ B ,B r,2 ⊥ = ω 0 /g ⊥ µ B 1+2m 0 /g ⊥ m, B r,3 ⊥ = 4ω 0 /g ⊥ µ B 1+4m 0 /g ⊥ m, B d ⊥ =(ω 0 /2g ⊥ µ B ) 2m 0 /g ⊥ m. Rabi Oscillations S z = S T z + e −(T −1 1 +T −1 2 )t/2 3 2 − S T z cos ω R t + (d SO E) 2 T 2 2 ω R S T z − T −1 1 − T −1 2 2ω R 3 2 − S T z sin ω R t , where ω R = (d SO E/) 2 − (T −1 1 − T −1 2 ) 2 /4 is the Rabi frequency and S T z = (3/2)ρ T z /[1 + (d SO E/) 2 T 1 T 2 ]. -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 the perpendicular magnetic field: B ⊥ = 0.8T (damped fast oscillations), B ⊥ =0.865 T (dotted line), and B ⊥ =0.5T(solid line). B = 0, δ rf = 0, E =1.5V/cm. Conclusions • Spin-orbit coupling is suppressed for flat QDs • Spin relaxation time T 1 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 Holes in Quantum Dots, cond-mat/0608410. [2] Denis V. Bulaev, Daniel Loss. Spin Relaxation and Decoherence of Holes in Quantum Dots, Phys. Rev. Lett. 95, 076805 (2005).