biexciton-exciton cascades in graphene quantum dots cap 2014, sudbury isil ozfidan i.ozfidan, m....
Post on 14-Dec-2015
229 Views
Preview:
TRANSCRIPT
Biexciton-Exciton Cascades in Graphene Quantum Dots
CAP 2014, Sudbury
Isil Ozfidan
I.Ozfidan, M. Korkusinski,A.D.Guclu,J.McGuire and P.Hawrylak, PRB89,085310 (2014).
Motivation
• Biexciton-Exciton Cascades in semiconductor quantum dots for entangled photon generation.-Benson et al, PRL 84, 2513 (2000).
XX
X1X2
GS
σ+
σ+
σ-
σ-
Two paths for radiative recombination
Motivation
• Biexciton-Exciton Cascades in semiconductor quantum dots for entangled photon generation. -Korkusinski et al, Phys. Rev. B 79, 035309 (2009).
XX
X1X2
GS
V
H
H
V
But in semiconductor qdots, due to anisotropy the X levels are not degenerate.
Post-growth tuning of excitonic splitting.
XX
X1X2
GS
σ+
σ+
σ-
σ-
Two paths for radiative recombination
Motivation
• Biexciton-Exciton Cascades in graphene quantum dots for entangled photon generation.
C168
XX
X1X2
GS
σ+
σ+
σ-
σ-
Outline
1. Theory
2. Introducing C168
3. Band-Edge Excitons and Biexcitons
4. Auger Coupling
5. Conclusion
Theory
Tight Binding + Hartree Fock + CI
Tight-binding Hamiltonian, τij is the tunelling element
sp2
pz
sp2
sp2
I.Ozfidan, M. Korkusinski,A.D.Guclu,J.McGuire and P.Hawrylak, PRB89,085310 (2014).
Mobile electrons occupy the spin-degenerate pz orbitals
k: dielectric constantScreening by sigma electrons and surrounding fluid is introducedas the dielectric constant
Theory
Tight Binding + Hartree Fock + CI
Electron-electron interactions
Slater pz orbitals
Coulomb elements
Theory
Tight Binding + Hartree Fock + CI
Mean Field – Hartree Fock Hamiltonian
Density MatrixDirect Exchange
Theory
Tight Binding + Hartree Fock + CI
Mean Field – Hartree Fock Hamiltonian
ci+ → bi+
Tight-binding states → Hartree Fock states Rotating the basis!
Theory
Tight Binding + Hartree Fock + CI
Rewrite the full Hamiltonian
in the HF basis:
Theory
Tight Binding + Hartree Fock + CI
Corellated groundAnd excited states
Configuration – Interaction Hamiltonian
Outline
1. Theory
2. Introducing C168
3. Band-Edge Excitons and Biexcitons
4. Auger Coupling
5. Conclusion
C3 Symmetry of C168
Atom j in section A;
¿ 𝑗𝐵⟩
¿ 𝑗𝐶⟩
We can characterize the C168eigenstates according to their rotational symmetry
Then the Hamiltonian becomes block diagonalw.r.t. the phase; angular momentum, m.
and m is the angular momentum m={0,1,2}
3 identical segments. Create states by combining the same atom from each segment with a phase.
C3 Symmetry of C168
78 80 82
-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.60.60.81.01.21.41.61.82.02.22.42.62.83.0
Eig
enva
lues
(eV
)
0 20 40 60 80-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
Ener
gy (e
V)
Eigenstate index
Since m=1 and m=2 states are conjugates of each other,we have degenerate m=1,2 subspaces.
m=0 m=1 m=2
C3 Symmetry of C168
m=1 m=2
m=2 m=1
78 80 82
-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.60.60.81.01.21.41.61.82.02.22.42.62.83.0
Eig
enva
lues
(eV
)
0 20 40 60 80-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
Ener
gy (e
V)
Eigenstate index
m=0 m=1 m=2
Degenerate band edgedue to symmetry!
C3 Symmetry of C168
Optical Selection rule! ∆m!=0
m=1 m=2
m=2 m=1
78 80 82
-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.60.60.81.01.21.41.61.82.02.22.42.62.83.0
Eig
enva
lues
(eV
)
0 20 40 60 80-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
Ener
gy (e
V)
Eigenstate index
m=0 m=1 m=2
Looking at the dipole element between these eigenfunctions;
Outline
1. Theory
2. Introducing C168
3. Band-Edge Excitons and Biexcitons
4. Auger Coupling
5. Conclusion
Band edge is robust
Only C168?
78 80 82
-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.60.60.81.01.21.41.61.82.02.22.42.62.83.0
Eig
enva
lues
(eV
)
Band edge is robustAny GQD withC3 symmetry!
78 80 82
-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.60.60.81.01.21.41.61.82.02.22.42.62.83.0
Eig
enva
lues
(eV
)
Triangle
Band edge is robustAny GQD withC3 symmetry!
78 80 82
-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.60.60.81.01.21.41.61.82.02.22.42.62.83.0
Eig
enva
lues
(eV
)
Hexagon
Band edge is robustAny GQD withC3 symmetry!
78 80 82
-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.60.60.81.01.21.41.61.82.02.22.42.62.83.0
Eig
enva
lues
(eV
)
Star
Band edge is robustAny GQD withC3 symmetry!
78 80 82
-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.60.60.81.01.21.41.61.82.02.22.42.62.83.0
Eig
enva
lues
(eV
)
The Superman
Band Edge Excitons
Δm=0 Excitons
Δm=1 Excitons
Δm=-1 Excitons
Dipole allowed Transitions
σ+
σ-
X1
X2
X0A X0B
Dark Transitions
TOTAL = 8
-2 -1 0 1 2
1.9
2.0
2.1
3.8
3.9
4.0
4.1
4.2
Exc
itat
ion
Ene
rgy
(eV
)
m
C3.75 Band Edge X & XX
σ-
Band Edge Excitons
Δm=±1σ-
σ+
σ+
triplet
singlet
-2 -1 0 1 2
1.9
2.0
2.1
3.8
3.9
4.0
4.1
4.2
Exc
itat
ion
Ene
rgy
(eV
)
m
C3.75 Band Edge X & XX
Band Edge Excitons
Δm=0
σ+σ-Only optically active BE-X
Singlet ∆m=±1
σ- σ+
Δm=-2 Δm=2
Δm=0 Δm=1 Δm=-1
Band Edge-Biexcitons
Total=18
-2 -1 0 1 2
1.9
2.0
2.1
3.8
3.9
4.0
4.1
4.2
Exc
itat
ion
Ene
rgy
(eV
)
m
C3.75 Band Edge X & XX
Band Edge Biexcitons
Too many to talk about!
Band Edge-Biexcitons
Only Interested in the Cascadeones emit to the bright excitons?
Δm=-2 Δm=2
Δm=0 Δm=1 Δm=-1
Band Edge-Biexcitons
Only Interested in the Cascadeones emit to the bright excitons?
-2 -1 0 1 2
1.9
2.0
2.1
3.8
3.9
4.0
4.1
4.2
Exc
itat
ion
Ene
rgy
(eV
)
m
C3.75 Band Edge X & XX
Band Edge Biexcitons
Δm=±2
σ+σ-
-2 -1 0 1 2
1.9
2.0
2.1
3.8
3.9
4.0
4.1
4.2
Exc
itat
ion
Ene
rgy
(eV
)
m
C3.75 Band Edge X & XX
Band Edge Biexcitons
Δm=0σ-σ+
-2 -1 0 1 2
1.9
2.0
2.1
3.8
3.9
4.0
4.1
4.2
Exc
itat
ion
Ene
rgy
(eV
)
m
C3.75 Band Edge X & XX
Band Edge Biexcitons
Δm=0
GREAT CANDIDATE!
σ-σ+
Outline
1. Theory
2. Introducing C168
3. Band-Edge Excitons and Biexcitons
4. Auger Coupling
5. Conclusion
1
2
3
4
5
6
7
8
9
Eig
enva
lue
(eV
)
Eigenvalue Index
HF eigenvalues
CI-Space & Auger Coupling
Eg
Smallest CI-Space to properly understand auger coupling of BE-XXs ??
1
2
3
4
5
6
7
8
9
Eig
enva
lue
(eV
)
Eigenvalue Index
HF eigenvalues
Eg
CI-Space & Auger Coupling
1
2
3
4
5
6
7
8
9
Eig
enva
lue
(eV
)
Eigenvalue Index
HF eigenvalues
Eg
CI-Space & Auger Coupling
1
2
3
4
5
6
7
8
9
Eig
enva
lue
(eV
)
Eigenvalue Index
HF eigenvalues
Eg
Eg
CI-Space & Auger Coupling
1
2
3
4
5
6
7
8
9
Eig
enva
lue
(eV
)
Eigenvalue Index
HF eigenvalues
Eg
Eg
CI-Space & Auger Coupling
1
2
3
4
5
6
7
8
9
Eig
enva
lue
(eV
)
Eigenvalue Index
HF eigenvalues
Eg
Eg
Eg
CI-Space & Auger Coupling
1
2
3
4
5
6
7
8
9
Eig
enva
lue
(eV
)
Eigenvalue Index
HF eigenvalues
Eg
Eg
Eg
GS+X+XX in this 15 valence (v), 23 conduction (c)level – space we have: 172846 states
CI-Space & Auger Coupling
1
2
3
4
5
6
7
8
9
Eig
enva
lue
(eV
)
Eigenvalue Index
HF eigenvalues
Eg
Eg
Eg
GS+X+XX in this 15 valence (v), 23 conduction (c)level – space we have: 172846 states
Introduce cut-offs, check convergence.
CI-Space & Auger Coupling
Evolution of the band-edge XXs
-0.14-0.12-0.10-0.08-0.06-0.04-0.020.00
1.84
1.85
1.86
1.87
0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030
3.74
3.75
3.76
3.77
GS
Ene
rgy
(eV
)
GS eigenvalue Interpolated GS
GS, LX, LXX Convergence
-0.12169
LX
Ene
rgy
(eV
)
LX eigenvalue Interpolated LX
1.83665
LX
X E
nerg
y (e
V)
1/N
LXX eigenvalue Interpolated LXX3.74204
2.01
2.02
2.03
2.04
0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030
4.09
4.10
4.11
4.12
HX
Ene
rgy
(eV
)
HX eigenvalue Interpolated HX
HX, HXX Convergence
2.01125
HX
X E
nerg
y (e
V)
1/N
HXX eigenvalueInterpolated HXX
4.0859
v2c2 v3c3 v6c6 C3.0 C3.25 C3.50 C3.75 C4.0 C4.25 C4.50 Fit
3.80
3.85
3.90
3.95
4.00
4.05
4.10
4.15
4.20
4.25
4.30
2LX 2HX
Exci
tatio
n En
ergi
es (e
V)
C168 GS+X+XXEvolution of BE-XX excitation energy
10
100
1000
10000
100000
Hilb
ert S
pace
58.29meV
47.94meV
XX bindingenergies
0 1 2 3 4 51E-5
1E-4
1E-3
0.01
0.1
1
Spectr
um
Eigenvalues (eV)
BEXX-HXX, C3.75HXX spectrum on GS+X+XX
HXX Energy
Spectral Function of XX
𝐴𝑖¿𝑖⟨ (𝐺𝑆+𝑋+𝑋𝑋 )∨𝐻𝑋𝑋 ⟩
Turn on XX – X interactions: XX & X correlation
Outline
1. Theory
2. Introducing C168
3. Band-Edge Excitons and Biexcitons
4. Auger Coupling
5. Conclusion
XX-X cascade identified
We’ve got a candidate!
buthow stable is he?
-2 -1 0 1 2
1.9
2.0
2.1
3.8
3.9
4.0
4.1
4.2
Exc
itat
ion
Ene
rgy
(eV
)
m
C3.75 Band Edge X & XX
σ-σ+
σ- σ+
EXX-EX=2.07eV
EX-GS=2.13eV
Conclusion
top related