1. 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).

2. 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+ σ-σ- σ-σ- Two paths for radiative recombination

3. 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+ σ-σ- σ-σ- Two paths for radiative recombination

4. Motivation Biexciton-Exciton Cascades in graphene quantum dots for entangled photon generation. C168 XX X1X2 GS σ+σ+ σ+σ+ σ-σ- σ-σ-

5. Outline 1.Theory 2.Introducing C168 3.Band-Edge Excitons and Biexcitons 4.Auger Coupling 5.Conclusion

6. Theory Tight Binding + Hartree Fock + CI Tight-binding Hamiltonian, τ ij is the tunelling element sp 2 pzpz I.Ozfidan, M. Korkusinski,A.D.Guclu,J.McGuire and P.Hawrylak, PRB89,085310 (2014). Mobile electrons occupy the spin-degenerate pz orbitals

7.  : dielectric constant Screening by sigma electrons and surrounding fluid is introduced as the dielectric constant Theory Tight Binding + Hartree Fock + CI Electron-electron interactions Slater pz orbitals Coulomb elements

8. Theory Tight Binding + Hartree Fock + CI Mean Field – Hartree Fock Hamiltonian Density Matrix Direct Exchange

9. Theory Tight Binding + Hartree Fock + CI Mean Field – Hartree Fock Hamiltonian ci+ → bi+ci+ → bi+ Tight-binding states → Hartree Fock states Rotating the basis!

10. Theory Tight Binding + Hartree Fock + CI Rewrite the full Hamiltonian in the HF basis:

11. Theory Tight Binding + Hartree Fock + CI Corellated ground And excited states Configuration – Interaction Hamiltonian

12. Outline 1.Theory 2.Introducing C168 3.Band-Edge Excitons and Biexcitons 4.Auger Coupling 5.Conclusion

13. C3 Symmetry of C168 We can characterize the C168 eigenstates according to their rotational symmetry Then the Hamiltonian becomes block diagonal w.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.

14. C3 Symmetry of C168 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

15. C3 Symmetry of C168 m=1 m=2 m=2 m=1 m=0 m=1 m=2 Degenerate band edge due to symmetry!

16. C3 Symmetry of C168 Optical Selection rule! ∆m!=0 m=1 m=2 m=2 m=1 m=0 m=1 m=2 Looking at the dipole element between these eigenfunctions;

17. Outline 1.Theory 2.Introducing C168 3.Band-Edge Excitons and Biexcitons 4.Auger Coupling 5.Conclusion

18. Band edge is robust Only C168?

19. Band edge is robust Any GQD with C3 symmetry! Triangle

20. Band edge is robust Any GQD with C3 symmetry! Hexagon

21. Band edge is robust Any GQD with C3 symmetry! Star

22. Band edge is robust Any GQD with C3 symmetry! The Superman

23. Band Edge Excitons Δm=0 Excitons Δm=1 Excitons Δm=-1 Excitons Dipole allowed Transitions σ+σ+ σ-σ- X1X1 X2X2 X 0A X 0B Dark Transitions TOTAL = 8

24. σ-σ- Band Edge Excitons Δm=±1 σ-σ- σ+σ+ σ+σ+ triplet singlet

25. Band Edge Excitons Δm=0 σ+σ+ σ-σ- Only optically active BE-X Singlet ∆m=±1 σ-σ- σ+σ+

26. Δm=-2 Δm=2 Δm=0 Δm=1Δm=-1 Band Edge-Biexcitons Total=18

27. Band Edge Biexcitons Too many to talk about!

28. Band Edge-Biexcitons Only Interested in the Cascade ones emit to the bright excitons?

29. Δm=-2 Δm=2 Δm=0 Δm=1Δm=-1 Band Edge-Biexcitons Only Interested in the Cascade ones emit to the bright excitons?

30. Band Edge Biexcitons Δm=±2 σ+σ+ σ-σ-

31. Band Edge Biexcitons Δm=0 σ-σ- σ+σ+

32. GREAT CANDIDATE! σ-σ- σ+σ+

33. Outline 1.Theory 2.Introducing C168 3.Band-Edge Excitons and Biexcitons 4.Auger Coupling 5.Conclusion

34. CI-Space & Auger Coupling Eg Smallest CI-Space to properly understand auger coupling of BE-XXs ??

35. Eg CI-Space & Auger Coupling

36. Eg CI-Space & Auger Coupling

37. Eg CI-Space & Auger Coupling

38. Eg CI-Space & Auger Coupling

39. Eg CI-Space & Auger Coupling

40. Eg GS+X+XX in this 15 valence (v), 23 conduction (c) level – space we have: 172846 states CI-Space & Auger Coupling

41. 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

42. Evolution of the band-edge XXs 58.29meV 47.94meV XX binding energies

43. Spectral Function of XX Turn on XX – X interactions: XX & X correlation

44. Outline 1.Theory 2.Introducing C168 3.Band-Edge Excitons and Biexcitons 4.Auger Coupling 5.Conclusion

45. XX-X cascade identified We’ve got a candidate! but how stable is he? σ-σ- σ+σ+ σ-σ- σ+σ+ E XX -E X =2.07eV E X -GS=2.13eV Conclusion