Journal Club 2012. február 16. Tóvári Endre

Journal Club 2012. február 16.Tóvári Endre

Resonance-hybrid states in a triple quantum dot

PHYSICAL REVIEW B 85, 081301(R) (2012)

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 simulationsHere: resonance-hybrid states in a few-electron TQD, exploring the origin of the hybrid bond stability focusing on spinModel: 3-site Hubbard modelPhys. Rev.B 65, 085324 (2002) Phys. Rev. Lett. 90, 166803 (2003)

2012.02.16. JC: Resonance-hybrid states in a triple quantum dot

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Valence bond theory;• multiple contributing structures (bonds)• the bonding cannot be expressed by one single Lewis formula• delocalized electrons (or superposition of wavefunctions)• lower energyhybrids are more stable than any of the contributing structures

Resonance hybrid molecules

http://en.wikipedia.org/wiki/Resonance_(chemistry)#Resonance_hybrids

Valence bond theory;• multiple contributing structures (bonds)• the bonding cannot be expressed by one single Lewis formula• delocalized electrons (or superposition of wavefunctions)














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DC current from S to D, QDs in parallel

Vg1 Vg3

Vg2

Vg2’

=Vg2

Al0.3Ga0.7As/GaAs double-barrier resonant tunneling structure

Size: adjusted to attain the few-electron regime, 100 mKDC current, B=0, Vg2=Vg2’


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Determining charge configurations: from the slope (ΔVg1/ΔVg2) of each Coulomb oscillation line away from the anticrossing regionsseparation between, and “rounding”of lines at anticrossing regions (X,Y,Z): interdot Coulomb interaction and tunnel coupling

Vsd=300μV

Vg1 Vg3

Vg2

Vg2’

=Vg2

the levels in dots 1 and 3 are aligned near Z...

On increasing Vsd , the Coulomb oscillation lines broaden into current stripes and excited states within the energy window eVsd become accessible

Vsd=1mV

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=δε energy detuning between QD1 and QD3

side viewtop view

drain


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110: ground state , excited state12S 12T

011: ground state , excited state23S 23T

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) *

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=δε energy detuning between QD1 and QD3

S-S hybridization is strongerweaker curvature,

more stable resonance

μg(2)

μe(2)

μe(3)

μg(3)

CALCULATION (3-site Hubbard)

|ε|>0: 110 or 011 is dominant, energy ~ -| ε|

* μ(N) is the energy of the N electron state minus the energy of the N −1 electron ground state

N=3 doublet states2 levels, Stot=1/2 D1, D0 doublet statesμD1= μg(3) < μD0= μe(3)

μe(3) excited state D0: S’=S1+S3=0μg(3) ground state D1: S’=1


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Tuning E2=δ (and thus t31 and the resonance) with Vg3 ↑

010

N=1: 010 becomes more stabilized (the 1st line shifts ), δ decreasesN=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

μg(2)

μe(2)

μe(3)

μg(3)


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Tuning E2=δ (and thus t31 and the resonance) with Vg3 ↑

010

μg(2)

μe(2)

μe(3)

μg(3)

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


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μg(2)

μe(2)

Calculated charge state contributions in the ground state: N=2

QD1QD2

QD3

δ is reduced → the weight of 020 increases, stronger hybridization and resonance, so μg(2) flattens

δ=-1.9meV δ=-2.2meV δ=-2.5meV

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


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δ is reduced: (n,2,m) configurations more and more preferable

μe(3)

μg(3)

• 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

Calculated charge state contributions in the ground state: N=3


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if 111 is dominant: doublet and quadruplet states:• Stot=1/2 - D1, D0 doublet statesμe(3) excited state D0: S’=S1+S3=0μg(3) ground state D1: S’=1• Stot=3/2 - Q quadruplet

doublet states

symmetric and asymmetric states of 120 and 021

120 and 021 hybridize at δQ < δ: D1, D0, Q

N=3

μe(3)

μg(3)

π phase gain

no phase gain

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

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 phasehybridization of D1 and AS, and D0 and S→ the separation between two doublet states remains small


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doublet states

symmetric and asymmetric states of 120 and 021

N=3

μe(3)

μg(3)

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


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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

Journal Club 2012. február 16. Tóvári Endre

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