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