Transcript Document

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)
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
http://en.wikipedia.org/wiki/Resonance_(chemistry)#Resonance_hybrids
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2012.02.16. JC: Resonance-hybrid states in a triple quantum dot
Al0.3Ga0.7As/GaAs double-barrier
resonant tunneling structure
Vg2’
=Vg2
Vg1
DC current from S to D, QDs in parallel
Size: adjusted to attain the fewelectron regime, 100 mK
DC current, B=0, Vg2=Vg2’
Vg3
Vg2
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2012.02.16. JC: Resonance-hybrid states in a triple quantum dot
Vg2’
=Vg2
Vg1
drain
top
view
side view
Vsd
=1mV
Vsd
=300μV
Vg3
Vg2
Determining charge configurations:
from the slope (ΔVg1/ΔVg2) of each
Coulomb oscillation line away from the
anticrossing regions
separation between, and “rounding”
of lines at anticrossing regions (X,Y,Z):
interdot Coulomb interaction and tunnel
coupling
the3-site
levelsHubbard
in dots 1model:
and 3 are aligned near Z...
Ui intradot Coulomb-energies
Vij interdot Coulomb-en.
tij interdot tunnel coupling
On increasing Vsd , the Coulomb
oscillation lines broaden into current stripes
Ei lowest single-e- level: E1=0.5ε=-E3, E2=δ
and excited states within the energy
ε energy detuning between QD1 and QD3 4
window eVsd become accessible
2012.02.16. JC: Resonance-hybrid states in a triple quantum dot
Ei lowest
level: E1 = 0.5ε = - E3, E2=δ
ε energy detuning between QD1 and QD3
hybridization:
single-e-
|ε|>0: 110 or 011 is dominant, energy ~ -| ε|
110: ground state S12 , excited state T12
011: ground state
S23 , excited state T23
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)
μg(3)
μe(2)
μg(2)
first-order (direct) tunneling:
(1,1,0) → (0,1,1)
N=3 doublet
second-order
tunnelingstates
via
2
levels, Stot=1/2
intermediate
virtual states (important
D
, D0 doublet
states
if 1ΔE(δ)
between
intermediate and
μ
μD0
= μe(3)
initial
small):
D1= μstates
g(3) < is
μ
D0110→020→011
: S’=S1+S3=0
110→101→011
and
e(3) excited state
μg(3) ground state D1: S’=1
* μ(N) is the energy of the N electron state minus the energy of
the N −1 electron ground state
CALCULATION
(3-site Hubbard)
S-S hybridization is
strongerweaker curvature,
more stable resonance
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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)
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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)
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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
μe(2)
δ=-2.5meV
μg(2)
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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)
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2012.02.16. JC: Resonance-hybrid states in a triple quantum dot
symmetric and
asymmetric states
of 120 and 021
doublet states
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
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 might
and 021
hybridize
δQ < δ:hybridize with the S state from
One
expect
that Dat
1 should
D1, D
0, Q
the permutation
process
of electrons in dots 1 and 3, but this is
not so due to additional geometrical phase
hybridization of D1 and AS, and D0 and S→ the separation
between two doublet states remains small
π phase gain
no phase gain
μe(3)
μg(3)
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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)
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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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