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Designing a Quantum Dot Implemented in a Photonic Crystal Cavity

Designing a Quantum Dot Implemented in a Photonic Crystal Cavity. By : Majid Sodagar Supervisor : Dr. Sina Khorasani Faculty : Electrical Engineering Date : Nov. 2008. Synopsis. Literature Review Main Theme Exciton Transfer Matrix Method Matrix Diagonalization

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Designing a Quantum Dot Implemented in a Photonic Crystal Cavity

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  1. Designing a Quantum Dot Implemented in a Photonic Crystal Cavity By : MajidSodagar Supervisor : Dr. SinaKhorasani Faculty : Electrical Engineering Date : Nov. 2008

  2. Synopsis • Literature Review • Main Theme • Exciton • Transfer Matrix Method • Matrix Diagonalization • Photonic Crystal Cavity • Finite Difference Time Domain • Quality Factor • Photon-Exciton Interaction • Time Domain Evolution • Energy Splitting • Conclusion

  3. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Literature Review Nature, Vol. 445, 2007 Switzerland, US Investigating the strong coupling regime using self assembled InAsQD g = 76 meV = 8.5GHz=35meV Q = 13000 = 24.1GHz=100meV

  4. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Literature Review Phys. Rev. Lett. 101, 113903 (2008) Germany Quantum dots that couple to a photonic crystal waveguide are found to decay up to 27 times fasterthan uncoupled quantum dots. This shows the promising potential of photonic crystal waveguides for efficient single-photon sources. DR SE

  5. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Literature Review J. Phys.: Condens. Matter 20, 454209 (2008) Germany Discuss the recently discovered non-resonantcoupling mechanism between quantum dot emission and cavity mode for large detuning. Spectral dot–cavity detuning is discussed on the basis of shifting either the quantum dot emission via temperature tuning or the cavity mode emission via a thin film deposition technique.

  6. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Literature Review Nature photonics, VOL 2, (2008) Japan fully confined electrons and photons using a combination of three dimensional photonic crystal nanocavities and quantum dots. Important due to polarization issue. Applications : Triggered single-photon sources quantum logic gate for optical fibre-based quantum cryptography communication and quantum repeater systems

  7. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Applications • CQED: • All optical quantum information and computation • Quantum cryptography • Realization of quantum repeaters • Single photon sources • Qbit realization • Strong coupling regime: • Fabrication ofhigh-efficiencymicrocavity LEDs • Low-threshold vertical-cavity surface emitting lasers • Microsphere lasers • Entanglement • Weak coupling regime: • Modification of the emission diagram • Enhancement or inhibition of the SE rate • Funneling of SE photons into a single mode • Control of the SE process on the single photon level D-Wave 16-bit Q-Computer

  8. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion System Structure • Methods of Photon Confinement : • In plane : Distributed Bragg Reflection • Normal: Total Internal Reflection Disk-Like Quantum Dot Photonic Crystal Slab (PCS) Electron-Hole Pair (Exciton)

  9. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion QD Structure R0=150nm Vze= 300 meV Vzh= 150 meV X=0.36 Ga1-xAlxAs • Strain for GaAs : • exx=eyy= -9×10-4 • ezz= 8.3×10-4 Z0=4nm V≈∞ GaAs av= -1.116 eV Hydrostatic Strain Ga1-xAlxAs b = -2 eV Uniaxial Strain Pikus-Bir Deformation Potentials

  10. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Exciton • Exciton types: • Frenkel : • Localized near single atom • Smaller Bohr radius • Strong coupling • Wannier : • Electron holes are far apart in CV and VB • Larger Bohr radius • Weak coupling λ e Eexciton Eg λ e Exciton Binding Energy Electron-Hole No Binding Energy Electron

  11. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Envelope Function Appr. Using real potential in Schrödinger equation makes it unwieldy. Macroscopic Potential Contributing to Envelope Part Microscopic Potential Contributing to Bloch Part

  12. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Policy • In contrast to CB, there is atwofolddegeneracy in VB besides the closeness of SO Band structure for typical III-V and IV group semiconductor • Electron wave function • Influenced by only one band (CB) • Simple Schrödinger equation • S-Likeorbital was taken as Bloch part • Hole wave function • Influenced by three bands (HH,LH,SO) • using 6×6 LuttingerHamiltonian • Combination of Px,Pyand Pzincluding spin was taken as Bloch part Energy <111> <001> 0 Wave Vector

  13. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Wave Functions Disk like Quantum Dot Thickness << Area For holes In plane treated as single band Bessel Sinusoidal

  14. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Luttinger Hamiltonian Can be reduced to two3×3block diagonal matrix via unitary transformation on basis set. Strain effect can be added directly by updating P, Q, R and S using Pikus-Birdeformation potentials. J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869, (1955).

  15. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Stress • Strain Effect Strain due to lattice mismatch • Solving Methods • Plane Wave Expansion (PWE) • Transfer Matrix Methods (TMM) • Finite Difference (FD)

  16. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Transfer Matrix Method a4 b4 a3 b3 3 Forward-Backward Waves a2 b2 2 1 a1 b1 Bounded states Condition: T22 should has eigenvalue equal to 0 for some E ! Total Transfer Matrix: • Continuity of • envelope function • probability current B. Chen, M. Lazzouni, Phys. Rev. B ,45, 1024, (1992).

  17. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion TMM Output • Applied parameters : • Sweeping over system energiesso that the coefficients of all non-damping terms vanish. 10Log|D(T22)| • Zero Crossing points denote holebound states energies quantized in the zdirection |D(T22)| HH1 LH1 HH2 Energy (eV)

  18. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion TMM Output HH1 LH1

  19. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion TMM Output HH2 Derivative were not conserved

  20. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Matrix Diagonalization Interaction Monte Carlo Non-uniform Stratified Big Challenge Using VEGAS Algorithm Using Cluster Computer Using MPI Libraries in C++ G.P. Lepage, “VEGAS: An Adaptive Multidimensional Integration Program”, Publication CLNS-80/447, (1980).

  21. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Excitonic States Without interaction Exciton space basis set consists of 243 vectors, so 59049 matrix elements have been calculated. Eg = 1424 meV Exciton wave form: With binding energy

  22. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Photonic Crystals • Periodic structure capable of exhibiting gap in frequency domain • PC were used to realize a photon cavity • Simulation Tools : • Plane Wave Expansion (PWE) • Finite Difference • Time Domain • Frequency Domain • Finite Element • Time Domain • Frequency Domain • Multiple Multi pole (MMP) • Wannier Functions Method First 3D PC realization E.Yablonovitch , Phys. Rev. Lett. 67, 2295 - 2298 (1991)

  23. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion FDTD Maxwell’s Equation Discretized form Yee Cube Stability Condition Minimum Simulation time Spatial Maximum Divisions

  24. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion FDTD • Specifications • Fully vectorial in 2D and 3D • Dispersivematerial • Dissipative material • Various boundary condition were implemented • Periodic (Reducing simulation time) • Bloch (Virtual physical problems) • PML {Mur and SPML} (Finite domain simulations) • PEC, PMC • Yee algorithm was implemented (Simple V. MD-WDF) • Implemented in C++ (Speed) • Equipped with harmonicinversion(Extraction efficiency) • Subpixel averaging K. S. Yee, IEEE Trans. Antennas Propagat. 14, 302 (1966).

  25. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion PC Cavity Slab Photonic crystal Triangle air hole lattice 1 Ky 0.9 Light Cone X J G 0.8 Kx PEC Applied 0.7 0.6 0.5 0.4 unguided modes TE Like Gap 0.3 Guided modes 0.2 0.1 0 J y z G G X x x Band folding due to bigger unit cell 12 division per a,d=0.2a, c=0.5a, r=0.4a PEC Applied

  26. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion PC Gaps Gap Size Versus Slab Thickness Gap Size Versus Hole Radius 40% 0.14 35% 0.12 30% 0.10 25% 0.08 20% 0.06 15% 0.04 10% 0.02 5% 0 0% 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Slab thickness (a) Hole Radius (a) 12 division per a, r=0.3a Contour 20 Division Per a, d=0.6a Contour It seams reasonable to choose slab thickness between 0.6a and 0.9a . Maximum gap is achieved with holeradiusaround 0.35a .

  27. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Cavity Output Spectrum • Cavity is realized by removing one hole. • Applying PML on the top and the bottom of the cavity. • The system is stimulated • with several broad band dipoles. Response from Broad Band Input 3.5 3.0 2.5 2.0 1.5 Log[czt|Amp|] (a.u.) 1.0 0.5 0.0 0.20 0.25 0.30 0.35 0.40 0.45 Normalized Frequency • Applying Chirp-ZT on output time date. • REDCircles : Cavity modes • Green Circles : Due to band edge near zero group velocity

  28. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Cavity Modes Profile |E| |Ex| |Ey| |E| |Ex| |Ey| |E| |Ex| |Ey| |E| |Ex| |Ey| |E| |Ex| |Ey| |E| |Ex| |Ey|

  29. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Mode Quality Factor Frequency Domain Time Domain 3.5 6 4 3.0 E0 2 2.5 E1 Ey(a.u.) 0 Log[czt|Amp|] (a.u.) 2.0 -2 1.5 -4 T0 T1 1.0 -6 0.20 0.25 0.30 0.35 0.40 0.45 Normalized Frequency Log[|Ey|] Evaluation begins as the input vanishes Q is evaluated as 148 1 2 3 4 5 6 Time (10-13) S. Guo and S. Albin, Opt. Express 11, 1080-1089 (2003).

  30. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion PC Mode Q Resonance Frequency and Qare dramatically affected by changing r`. Other modes emerges as r` approaches zero. Freq. Versus r` Q. Versus r` Modes for r`=0 0.20 0.38 1.2×104 0.36 1×104 0.25 0.34 8×103 0.30 0.32 6×103 Normalized Frequency 0.35 0.30 4×103 0.40 0.28 2×103 0 0.45 0.26 0.4 0.35 0.30 0.25 0.20 0.4 0.35 0.30 0.25 0.20 Output Intensity (a. u.) Nearest Hole Radius (a) Nearest Hole Radius (a)

  31. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion • PC Mode Q r = 0.27a r = 0.30a r = 0.33a 21800

  32. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Light-Matter Interaction Interaction SecondQuantization form has been exploited in order to represent the Hamiltonian in term of field operator Exciton Field operator: Minimal Coupling Scheme Ignoring second order term DirectCoupling Scheme K. Rewski, R. W. Boyd, Journal of modern optics, Vol. 51, no. 8, 1137–1147, (2004).

  33. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Simplifications • Approximations • Using Rotating Wave Approximation (RWA) • Ignoring High Frequency Evolutions • Using Dipole Approximation • Ignoring Epsilon inhomogeneity • Integration Simplification • Coupling Coefficients Matrix Element Optical wavelength Excitonic wavelength De Broglie

  34. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Photon Electron Interaction Can Results in : CreatingElectron-Hole Pair Annihilating Electron-Hole Pair Considering Exciton wave expansion There are only triplet integrals. Photon Exciton Binding Energy Summing over all electrons

  35. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Photon Excition Interaction Can Result in : Changing Electron Position Changing Hole Position Considering Exciton wave expansion There are only triplet integrals. Photon Exciton Binding Energy

  36. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Time Evoloution • Hamiltonian is responsible for time evolution • Choose only one Photonic and Excitonic State P Lambropoulos , et al, Rep. Prog. Phys. 63 (2000) 455–503

  37. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Time Evoloution • Lorentzian approximation for DOS Frequency Density of States Full Exciton |U(t)|2 Full Photons • Rabi Oscillation can occur for 2C > gc • High quality factor • Big coupling constant Photon-Exciton Combination Time

  38. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Energy Splitting Detuning g1=0, g2=50, g3=150 GRad Two Eigen frequencies : Our system at resonance : g = 159 GHz = 110 meV Strong Coupling Regime Dressed States = 2~50 meV Uncoupled State For D=0

  39. Literature Review, Main Theme, Exciton, PC Cavity, Photon-Exciton Interaction, Conclusion Conclusion and Further works • Electronic and hole states were found • Excitonic state were evaluated by diagonalization • A relatively high quality factor PC cavity were designed and simulated. • Coupling coefficient between cavity modes and excitonic states were derived • This structure is capable of operating in strong coupling regime • Investigating more complicated system such as bi-excitons and more photons • Realizing the physical structures • Applying the concept in engineering

  40. The End. • Thanks for your Patience

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