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Journal Club 2012. február 16. Tóvári Endre

Journal Club 2012. február 16. Tóvári Endre. PHYSICAL REVIEW B 85 , 081301(R) (2012). Resonance-hybrid states in a triple quantum dot.

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Journal Club 2012. február 16. Tóvári Endre

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  1. Journal Club 2012. február 16. Tóvári Endre PHYSICAL REVIEW B 85, 081301(R) (2012) Resonance-hybrid states in a triple quantum dot • Using QDs as building blocks: exploring quantum effects seen in real molecules and solids (but with tunable parameters, # of electrons, arrangement of QDs in an arbitrary structure, even lattice of artificial atoms) • QD arrays (flat band ferromagnetism, GMR, superconductivity calculations) • quantum information processors (seperating entangled electrons, topological quantum computation for fault-tolerant quantum computers) • modelling chemical reactions • quantum simulations • Here: resonance-hybrid states in a few-electron TQD, exploring the origin of the hybrid bond stability focusing on spin • Model: 3-site Hubbard model Phys. Rev.B 65, 085324 (2002) Phys. Rev. Lett. 90, 166803 (2003)

  2. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot Resonance hybrid molecules • Valence bond theory; • multiple contributing structures (bonds) • the bonding cannot be expressed by one single Lewis formula • delocalized electrons (or superposition of wavefunctions) • lower energyhybrids are more stable than any of the contributing structures • Valence bond theory; • multiple contributing structures (bonds) • the bonding cannot be expressed by one single Lewis formula • delocalized electrons (or superposition of wavefunctions) http://en.wikipedia.org/wiki/Resonance_(chemistry)#Resonance_hybrids

  3. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot Vg2’ =Vg2 Vg1 Vg3 Vg2 Al0.3Ga0.7As/GaAs double-barrier resonant tunneling structure DC current from S to D, QDs in parallel Size: adjusted to attain the few-electron regime, 100 mK DC current, B=0, Vg2=Vg2’

  4. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot Vg2’ =Vg2 Vg1 Vg3 Vg2 Vsd=1mV Vsd=300μV side view top view drain Determining charge configurations: from the slope (ΔVg1/ΔVg2) of each Coulomb oscillation line away from the anticrossing regions the levels in dots 1 and 3 are aligned near Z... separation between, and “rounding” of lines at anticrossing regions (X,Y,Z): interdot Coulomb interaction and tunnel coupling 3-site Hubbard model: Ui intradot Coulomb-energies Vij interdot Coulomb-en. tij interdot tunnel coupling Ei lowest single-e- level: E1=0.5ε=-E3, E2=δ On increasing Vsd , the Coulomb oscillation lines broaden into current stripes and excited states within the energy window eVsdbecome accessible ε energy detuning between QD1 and QD3

  5. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot hybridization: first-order (direct) tunneling: (1,1,0) → (0,1,1) second-order tunneling via intermediate virtual states (important if ΔE(δ) between intermediate and initial states is small): 110→101→011 and 110→020→011 Ei lowest single-e- level: E1 = 0.5ε = - E3, E2=δ ε energy detuning between QD1 and QD3 |ε|>0: 110 or 011 is dominant, energy ~ -| ε| N=3 doublet states 2 levels, Stot=1/2 D1, D0 doublet states μD1= μg(3) < μD0= μe(3) μe(3)excited state D0: S’=S1+S3=0 μg(3)ground state D1: S’=1 110: ground state , excited state 011: ground state , excited state Near Z: ε→0, E1=E3 S12 and S23, T12 and T23 become resonant, (1,1,0) and (0,1,1) hybridize, neither is dominant, the energy separation between S and T (ground and excited) levels increases: μg(2) < μe(2) * μe(3) * μ(N) is the energy of the N electron state minus the energy of the N −1 electron ground state μg(3) μe(2) CALCULATION (3-site Hubbard) μg(2) S-S hybridization is strongerweaker curvature, more stable resonance

  6. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot Tuning E2=δ (and thus t31 and the resonance) with Vg3↑ N=1: 010 becomes more stabilized (the 1st line shifts ),δ decreases N=2: the separation between the N=2 singlet and N=2 triplet levels increases due to stronger tunneling and hybridization, the former’s curvature weakens further 010 μe(3) μg(3) μe(2) μg(2)

  7. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot Tuning E2=δ (and thus t31 and the resonance) with Vg3↑ N=3: the sign of the curvature of level μg(3) changes from + to -, while the separation of the doublet levels at ε=0 remains small 010 μe(3) μg(3) μe(2) μg(2)

  8. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot Calculated charge state contributions in the ground state: N=2 δ is reduced → the weight of 020 increases, stronger hybridization and resonance, so μg(2) flattens QD1 QD3 QD2 stronger resonance-hybrid bond between the 110 and 011 singlet states compared to the hybridized triplet states (the former can hybridize with tunneling through 020, which is promoted by lowering E2=δ) → |μe(2) – μg(2)| increases δ=-1.9meV δ=-2.2meV δ=-2.5meV μe(2) μg(2)

  9. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot Calculated charge state contributions in the ground state: N=3 δ is reduced: (n,2,m) configurations more and more preferable • δP < δ: positive curvature of μg(3) because: 111 is still dominant (its energy is independent of ε), and μg(2) varies as ε or –ε (N=2 dominant config.: 110 or 011)* • δQ < δ < δP: μg(3) flattens because: 020 gains weight and the 111-energy is ε-independent • δ < δQ: μg(3) has negative curvature because: 120 and 021 gain weight, their energy varies as ε or –ε * * μ(N) is the energy of the N electron state minus the energy of the N −1 electron ground state μe(3) μg(3)

  10. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot N=3 • if 111 is dominant: doublet and quadruplet states: • Stot=1/2 - D1, D0 doublet states • μe(3)excited state D0: S’=S1+S3=0 • μg(3) ground state D1: S’=1 • Stot=3/2 - Q quadruplet symmetric and asymmetric states of 120 and 021 doublet states • the separation between two doublet states remains small (ε=0): • both doublet states are stabilized (Q is not) • geometrical phase from the single electron in QD2 120 and 021 hybridize at δQ < δ: D1, D0, Q One might expect that D1 should hybridize with the S state from the permutation process of electrons in dots 1 and 3, but this is not so due to additional geometrical phase π phase gain no phase gain hybridization of D1 and AS, and D0 and S→ the separation between two doublet states remains small μe(3) μg(3)

  11. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot N=3 symmetric and asymmetric states of 120 and 021 doublet states A quantum computation aspect: Changing δ adiabatically, using the level crossing: going from a charge qubit to a spin qubit (S→D1 or AS →D0) hybridization of D1 and AS, and D0 and S→ the separation between two doublet states remains small μe(3) μg(3)

  12. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot Conclusions Enhanced stability of the 110 ↔ 011 singlet resonance over the triplet resonance was observed due to the difference in accessibility of the (0,2,0) intermediate state Evolution of the three-electron ground and excited-state energies: from the accessibility of (1,2,0) and (0,2,1) intermediate states with the resonance-hybrid picture and geometrical phase in the electron hopping process

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