Spin Coherent Phenomena in Quantum Dots Driven by Magnetic Fields Gloria Platero Instituto de Ciencia de Materiales (ICMM), CSIC, Madrid, Spain María Busl (ICMM), Rafael Sánchez,Université de Genève Toulouse, november 010
Motivation and outline Semiconductor QD s: Small number of electrons, discrete levels: level system: q-bit. Manipulation of q-bits in QD s: driven coherent spin rotations Electron Spin Resonance in Double Quantum Dots (ESR): Coherent Spin Rotations ti TQD s in magnetic fields: interplay between different coherent spin phenomena Triple Quantum Dots as step towards a network of QD s Semiconductor QD s in magnetic fields: Spintronic devices: spin filters, spin inverters
Spin Blockade and Current Rectification by Pauli effect B=0 Spin Blockade K. Ono et al., Science, 97 (0) Processes: (0,1) (1,1) 1) (0,) (0,1). Transport state
Removing Spin Blockade by an external magnetic field B parallel to the current K. Ono et al., Science, 97 (0)
ESR in Quantum Dots 5
ESR: Driven coherent oscillations of an e spin in a DQD Koppens et al. Nature 006 driven coherent oscillations of a single e spin in a DQD 6
Coherent Control of a Single Electron Spin with Electric Fields: Electron Dipole Spin Resonance (EDSR) K. C. Nowack et al., Science 007 ESR induced by AC electric fields in materials with a spin orbit interaction: Spin-Orbit
EDSR M. Pioro-Ladrière et al., Nature Physics 008 Hyperfine-mediated gate-driven ESR E. A. Laird et al., PRL 07 8
ESR:Theoretical models: ESR lifts spin blockade 1 QD: Engel et al., PRL, 001 Rudner et al. PRL, 007 Danon et al., PRL, 008 QDs: R. Sánchez et al. PRB, 008 J. Danon et al., PRL 09 3 QDs: M. Busl et al. PRB (RC) 010; PRB 010 9
Electron Spin Resonance (ESR) in DQDs electrons: electrons: Bac only acts on triplet states
Hamiltonian Master equation: Born-Markov Approximation Coherent dynamics Transition rates and decoherences the state which contributes to the current is : 0,
electrons: Singlets and Triplets 1 S 0 S T = 0 = ( ) Coherently coupled,, by inter-dot tunneling H Tunnel S g T 1 = 0, + 1 =, Transport state = ( ) spin T 0, +, conserves total B ac acts on the triplet subspace if 1 T 1 =, S T =1 Left Right B = B No S-T mixing Left Righ B B Ψ S = 1 + δ = 0 T S T B ( Right Left B B ) δ = g μ
Δ Δ L R 1 1 (, +, ) mixes with singlet state : R. Sánchez et al., PHYSICAL REVIEW B 008 Δ L = ω S 0 Sg 1 =, = 0, (,, ) Transport state B ac 0 removes SB,, B ac 1 0 (, +, ) I 0 Current drops to zero by Spin Blockade
Δ zl Δ zr BAC removes SB, but also: B AC 0 B AC induces spin blockade at ω = Δ L + Δ R a coherent superposition of and,, is formed: 0.9,, Δ Δ zl zr Δ zr = 0.9Δ zl
Hamiltonian Unitary Transformation ˆ ' H (,, ) = 0 for ω = Δ + L Δ R ( Δ +Δ ) t ( Δ +Δ ) ( ) back transformed state: Ψ t = e, e, i zl This state verifies the time-dependent Schrodinger equation and: =0 zr i zl zr t 15
=ω 16
Bichromatic magnetic field At resonace conditions Spin Blockade is induced: Δ Δ = = 1 ω 1 ω 17
Coherent Trapping of Atomic Populations H. R. Gray et al., Optics Letters, 1978 E. Arimondo et al., Nuevo Cimento Lett 76 G. Alzetta et al., Nuovo Cimento 76 Superposition of two splitted ground states which is decoupled from the light Dark state CPT in QD s: Brandes et al., PRL 00 18
B. Michaelis, C. Emary and C. Beenakker, EPL 06 Φ = ( A B ) Dark State Destructive interference of the two reversible transitions traps an electron in a coherent superposition of the states on dots AandB B.
Gaudreau et al., PRL 006 Experiments in TQDs TQD s in series Schröer et al., PRB 07 Rogge et al., PRB 08 Equivalent circuit Laterally coupled triple vertical quantum dots in triangular arrangement, S. Amaha et al., APL 09
TQD triangular configuration: 1 electron 1 1 ( 0,,0,0,0 ) ( 0,,0,0,0 ) Dark states: do not contribute to the current due to destructive ti interference of the tunneling to dot 3 Aharonov-Bohm current oscillations Φ = n Φ 0 Dark states t remain and I=0 Φ 0 Φ n Dark states are destroyed Φ = Φ 0 1
Triangular TQD with 1 electron: ESR B ac 1 ( ) 1 0,,0,0,0 0 ( ) 0,,0,0,0 Δ1 = Δ = Δ3 =ω ac I = 0 ρ ii Φ = Φ 0 Time π Ω t Coherent rotations of 1 single electron Φ 0 Φ n Dark States are destroyed and I 0
electrons: Spin Blockade 3
1 3 TQD with electrons Transport state: Spin Blockade s, 0, s 0, s, s S = Δ = Δ Δ 1 3 0,0, I ( t) Γρ S, S ( t) I = 0 1 Dark State and SB ( 0, s, s s,0, s ) Ψ s s = s ' s Ψ I 0, s H tun S = 0 s I = 0
Δ 1 = Δ Δ3 AB phase breaks DS Dark State Spin Blockade
t 1 = t13 = t3 1 t13 t3 1 t13 t3 t = t Eigenstate at: 1 Ψ = Φ = 0.5 Φ0 1 = Δ Δ Bac induces Spin Blockade 3 ω 0 = Δ 1 + Δ 3 ( i,0,, i,0,, 0,, + 0,, ) Δ Δ3 = 0. 75Δ M. Busl et al., PRB (RC) 010 1
Other TQDs configurations.. 7
Current Rectification Double occupation just in QD B ac = 0 ε 1 = ε + U t 13 = t 3 In this configuration no SB 1 Δ 1 = Δ = Δ3 Dark State Ψ = (,,0,,0 + 0,,0 ) 3 Eigenstate of DM with eigenvalue 1 Eigenstate of the closed system I ( t) Γρ ( t) 3, 3 Singlet state 8
There is not trapping: Electrons tunnel from dot 1 to the left contact Dark state: Singlet state Ψ = 1 3 Trapping occurs for positive bias voltage (,,0,,0 + 0,,0 )
Spin filters and inverters AlGaAs-InGaAs-AlGaAs-GaAs-AlGaAs g factors engineering Spin Bottleneck Maximal current Spin Bottleneck Bdc 0 one electron δ = Δ Δ 1 S.M. Huang et al., PRL 010 30
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quasi-degenerate levels whichdifferinbac in δ = Δ Δ 1 at ψ ψ + 1 δ + Bac + δ = + + N B ac 1 δ Bac + δ = + N B ac + ψ ψ 3
Δ 1 =ωac Peaks at δ ε = ± DQD as spin filter ε Fully spin polarized current with opposite polarization at different detunings Levels are renormalized by the BAC 33
I ε 34
Spin bottleneck at δ ε = ± Strong spin level mixing in the drain dot: unpolarized current at: ε =0 35
DQD as Spin Filter by tuning ω and Vgate I I 36
37
ω = Δ + Δ 1 Δ > Δ 1 38
Δ + Δ s = 1 Δ Δ > Δ 1 I = (δ ) In resonance: Off resonance I = f ( Δ 1, Δ ) f δ ( = Δ Δ 1 39
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ESR in TQD in series: from spin filter to spin inverter Double occupation allowed just in the left dot Δ L = Δ R = 0. 13meV ΔC = 0. 06meV
TQD as spin inverter: I in I out I in I out 4
M. Busl et al., Phys. Rev. B, 8, 05304 (010). 43
Conclusions DQD s and TQD s 0, 0 B dc B ac Application as spin filters and spin inverters: Spin up or down polarized current tuned by electric gates: DQD s as spin filters TQD s also as spin inverters TQD s in triangular configuration: 1 e: coherent oscillations of one electron spin e s: es: Bac induces Spin Blockade in DQD s and TQD s, both in series and triangular configurations: Bac a tool for tuning SB