Spin relaxation in quantum dots due to e exchange with leads
Spin relaxation in quantum dots due to e− exchange with leads A.B. V ORONTSOV AND M.G. VAVILOV Department of Physics, University of Wisconsin, Madison, Wisconsin, 53706, USA
2
• Quantum dot is a basis for many nano-technological applications. Electronic and magnetic (spin) properties of fewelectron systems are of a particular interest;
3 Results
Model and equations
3.1 Eigenmodes - spin/charge relaxaton
2.1 Spin relaxation due to leads
1.5
◮ We assume that the intrinsic relaxation times are slow and the main spin-flip process is due to electron exchange with the leads : spin up leaves the dot and spin down enters back
• Traditionally considered e spin relaxation channels in a quantum dot are intrinsic: associated with a) hyperfine coupling to nuclei and b) spin-orbit coupling in the dot;
H = Hdot + Hleads + V
−
V
tunn
◮ Consider Left-Right dot in contact with Right lead and Γ = 0 (no escape to Left dot); No magnetic field: introduce P0 = P↑0 = P↓0 and PT1 = PT+ = PT− Remove probability P0 (unoccupied right dot) and diagonalize equations
(thanks
to
C.B.Simmons
et
al.
[3])
Hleads =
X X
ξα k c†α k σ cα k σ
◮ Eigenmodes of double dot system dynamics:
(1,1)
P˙η (t) + Γη Pη (t) = Jη ,
T
α=L,R k,σ
- free electrons L Hdot - localized“energy levels in the dots ” EF X X V = Wα k d†α σ cα k σ + Wα∗ k c†α k σ dα σ α=L,R k,σ
+
(1,1)
dc
S
H
dots
H
Rate equations from density matrix X
◮ Density matrix equation ρ˜˙ i = −
Γ1 = Γs , J1 = 0 Γ2 = Γs , J2 = 0 Γ3 = Γc , J3 = 2¯ γ1
(spin); (spin); (charge).
X
Γc = ΓR [1 + f (∆)] - charge relaxation rate Γif ρ0f
f (in)
f (out)
◮ Note: electron exchange between single dot and a lead is described by P2 = P↑ − P↓ (with rate Γs ) and P3 = P↑ + P↓ (with rate Γc ).
◮ Transition rates between electron states (leads × dot) : | i i = | ei i × | doti i ˛ ˛2 ˛ ˛ X Vf m Vmi ˛ ˛ Γf i = 2πδ(ǫi − ǫf ) ˛Vf i + + ...˛ ˛ ˛ ǫi − ǫm
τ0 = 0.2 ns valley : γ1
2
• Quantum dot is a basis for many nano-technological applications. Electronic and magnetic (spin) properties of fewelectron systems are of a particular interest;
3 Results
Model and equations
3.1 Eigenmodes - spin/charge relaxaton
2.1 Spin relaxation due to leads
1.5
◮ We assume that the intrinsic relaxation times are slow and the main spin-flip process is due to electron exchange with the leads : spin up leaves the dot and spin down enters back
• Traditionally considered e spin relaxation channels in a quantum dot are intrinsic: associated with a) hyperfine coupling to nuclei and b) spin-orbit coupling in the dot;
H = Hdot + Hleads + V
−
V
tunn
◮ Consider Left-Right dot in contact with Right lead and Γ = 0 (no escape to Left dot); No magnetic field: introduce P0 = P↑0 = P↓0 and PT1 = PT+ = PT− Remove probability P0 (unoccupied right dot) and diagonalize equations
(thanks
to
C.B.Simmons
et
al.
[3])
Hleads =
X X
ξα k c†α k σ cα k σ
◮ Eigenmodes of double dot system dynamics:
(1,1)
P˙η (t) + Γη Pη (t) = Jη ,
T
α=L,R k,σ
- free electrons L Hdot - localized“energy levels in the dots ” EF X X V = Wα k d†α σ cα k σ + Wα∗ k c†α k σ dα σ α=L,R k,σ
+
(1,1)
dc
S
H
dots
H
Rate equations from density matrix X
◮ Density matrix equation ρ˜˙ i = −
Γ1 = Γs , J1 = 0 Γ2 = Γs , J2 = 0 Γ3 = Γc , J3 = 2¯ γ1
(spin); (spin); (charge).
X
Γc = ΓR [1 + f (∆)] - charge relaxation rate Γif ρ0f
f (in)
f (out)
◮ Note: electron exchange between single dot and a lead is described by P2 = P↑ − P↓ (with rate Γs ) and P3 = P↑ + P↓ (with rate Γc ).
◮ Transition rates between electron states (leads × dot) : | i i = | ei i × | doti i ˛ ˛2 ˛ ˛ X Vf m Vmi ˛ ˛ Γf i = 2πδ(ǫi − ǫf ) ˛Vf i + + ...˛ ˛ ˛ ǫi − ǫm
τ0 = 0.2 ns valley : γ1
