Motivation         NEGF Formulation             Calculation Results   Conclusion




         Non-equilibrium Green’s Func...
Motivation                    NEGF Formulation   Calculation Results   Conclusion


Speed of Light Motivates Research on E...
Motivation                     NEGF Formulation   Calculation Results   Conclusion


Speed of Light Motivates Research on ...
Motivation                     NEGF Formulation     Calculation Results   Conclusion


Speed of Light Motivates Research o...
Motivation                  NEGF Formulation         Calculation Results         Conclusion


Motivation Studying Electron...
Motivation                  NEGF Formulation         Calculation Results         Conclusion


Motivation Studying Electron...
Motivation                  NEGF Formulation         Calculation Results         Conclusion


Motivation Studying Electron...
Motivation                  NEGF Formulation         Calculation Results         Conclusion


Motivation Studying Electron...
Motivation                  NEGF Formulation         Calculation Results         Conclusion


Motivation Studying Electron...
Motivation          NEGF Formulation   Calculation Results   Conclusion


We Study Optical Absorption in Quantum Well Infr...
Motivation          NEGF Formulation       Calculation Results      Conclusion


We Study Optical Absorption in Quantum We...
Motivation    NEGF Formulation   Calculation Results   Conclusion



NEGF Framework with Electron-Photon Interaction
Motivation            NEGF Formulation       Calculation Results    Conclusion


The Device is Represented by its Hamilton...
Motivation                    NEGF Formulation       Calculation Results         Conclusion



Self-Enery Matrix for Elect...
Motivation     NEGF Formulation   Calculation Results   Conclusion



Calculation Steps
Motivation                 NEGF Formulation                Calculation Results   Conclusion



Photocurrent Calculation


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Motivation                  NEGF Formulation                                                       Calculation Results    ...
Motivation                 NEGF Formulation                                                    Calculation Results        ...
Motivation                  NEGF Formulation                                  Calculation Results   Conclusion



Plot of ...
Motivation               NEGF Formulation         Calculation Results   Conclusion



Conclusion



       1     We study ...
Derivation of Self-Energy Matrices   Device Simulator Approach       Photocurrent Response from Absorption Coefficient



Ph...
Derivation of Self-Energy Matrices   Device Simulator Approach     Photocurrent Response from Absorption Coefficient



Inte...
Derivation of Self-Energy Matrices       Device Simulator Approach      Photocurrent Response from Absorption Coefficient


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Derivation of Self-Energy Matrices       Device Simulator Approach     Photocurrent Response from Absorption Coefficient



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Derivation of Self-Energy Matrices   Device Simulator Approach   Photocurrent Response from Absorption Coefficient



Device...
Derivation of Self-Energy Matrices   Device Simulator Approach   Photocurrent Response from Absorption Coefficient



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Non-equilibrium Green's Function Calculation of Optical Absorption in Nano Optoelectronic Devices

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Slides presentation for 13th International Workshop on Computational Electronics 27-29 May 2009 in Beijing

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Non-equilibrium Green's Function Calculation of Optical Absorption in Nano Optoelectronic Devices

  1. 1. Motivation NEGF Formulation Calculation Results Conclusion Non-equilibrium Green’s Function Calculation of Optical Absorption in Nano Optoelectronic Devices Oka Kurniawan, Ping Bai, Er Ping Li Computational Electronics and Photonics Institute of High Performance Computing Singapore 28th May 2009
  2. 2. Motivation NEGF Formulation Calculation Results Conclusion Speed of Light Motivates Research on Electron-Photon Interaction 1 1 Images courtesy of IBM.
  3. 3. Motivation NEGF Formulation Calculation Results Conclusion Speed of Light Motivates Research on Electron-Photon Interaction 2 2 Images courtesy of Intel.
  4. 4. Motivation NEGF Formulation Calculation Results Conclusion Speed of Light Motivates Research on Electron-Photon Interaction 2 Six Building blocks 2 Images courtesy of Intel.
  5. 5. Motivation NEGF Formulation Calculation Results Conclusion Motivation Studying Electron-Photon Interaction with Non-equilibrium Green’s Function (NEGF) Framework 1 Commonly used for nanoscale transport with phase-breaking phenomena. 2 Electron-photon interaction is important for optoelectronics. 3 Takes into account open systems with complex potentials and geometries. 4 no prior assumptions on the nature of the transitions. 5 Other interaction can be included, such as electron-phonon.
  6. 6. Motivation NEGF Formulation Calculation Results Conclusion Motivation Studying Electron-Photon Interaction with Non-equilibrium Green’s Function (NEGF) Framework 1 Commonly used for nanoscale transport with phase-breaking phenomena. 2 Electron-photon interaction is important for optoelectronics. 3 Takes into account open systems with complex potentials and geometries. 4 no prior assumptions on the nature of the transitions. 5 Other interaction can be included, such as electron-phonon.
  7. 7. Motivation NEGF Formulation Calculation Results Conclusion Motivation Studying Electron-Photon Interaction with Non-equilibrium Green’s Function (NEGF) Framework 1 Commonly used for nanoscale transport with phase-breaking phenomena. 2 Electron-photon interaction is important for optoelectronics. 3 Takes into account open systems with complex potentials and geometries. 4 no prior assumptions on the nature of the transitions. 5 Other interaction can be included, such as electron-phonon.
  8. 8. Motivation NEGF Formulation Calculation Results Conclusion Motivation Studying Electron-Photon Interaction with Non-equilibrium Green’s Function (NEGF) Framework 1 Commonly used for nanoscale transport with phase-breaking phenomena. 2 Electron-photon interaction is important for optoelectronics. 3 Takes into account open systems with complex potentials and geometries. 4 no prior assumptions on the nature of the transitions. 5 Other interaction can be included, such as electron-phonon.
  9. 9. Motivation NEGF Formulation Calculation Results Conclusion Motivation Studying Electron-Photon Interaction with Non-equilibrium Green’s Function (NEGF) Framework 1 Commonly used for nanoscale transport with phase-breaking phenomena. 2 Electron-photon interaction is important for optoelectronics. 3 Takes into account open systems with complex potentials and geometries. 4 no prior assumptions on the nature of the transitions. 5 Other interaction can be included, such as electron-phonon.
  10. 10. Motivation NEGF Formulation Calculation Results Conclusion We Study Optical Absorption in Quantum Well Infrared Photodetector Zero bias with a terminating barrier on the right. Henrickson, JAP, (91) 6273, 2002.
  11. 11. Motivation NEGF Formulation Calculation Results Conclusion We Study Optical Absorption in Quantum Well Infrared Photodetector Zero bias with a terminating Biased and no terminating barrier barrier on the right. at the contacts. Henrickson, JAP, (91) 6273, 2002.
  12. 12. Motivation NEGF Formulation Calculation Results Conclusion NEGF Framework with Electron-Photon Interaction
  13. 13. Motivation NEGF Formulation Calculation Results Conclusion The Device is Represented by its Hamiltonian, and the Interaction by its Self-Energy Matrices G (E ) = [ES + ıη − H0 − diag(U) − Σ1 − Σ2 − Σph ]−1
  14. 14. Motivation NEGF Formulation Calculation Results Conclusion Self-Enery Matrix for Electron-Photon Interaction Σ< (E ) = rs < < Mrp Mqs [NGpq (E − ω) + (N + 1)Gpq (E + ω)] pq 1 N is the number of photon. 2 G < is the less-than Green’s function, giving us the electron distribution. 3 Mij is the coupling matrix obtained from the Interaction Hamiltonian, and is a function of photon flux.
  15. 15. Motivation NEGF Formulation Calculation Results Conclusion Calculation Steps
  16. 16. Motivation NEGF Formulation Calculation Results Conclusion Photocurrent Calculation q < < I = t(Gp,q (E ) − Gq,p (E ))dE π and I RI = qIω 1 t is the off-diagonal coupling element of the Hamiltonian. 2 Iω is the photon flux at energy ω. 3 RI is the photocurrent response.
  17. 17. Motivation NEGF Formulation Calculation Results Conclusion Our Calculation Agrees Well with Published Result Photocurrent Response, RI (nm2/photon) 0 10 Our Simulation 10 -1 Henrickson’s 10-2 10-3 10-4 -5 10 -6 10 10-7 10-8 0 0.5 1 1.5 2 2.5 Photon Energy (eV) 1 LE = LC = 2 nm and LW = 5nm. 2 Barrier height is 2.0 eV, and terminating barrier height on the right is 0.2 eV. 3 We use a uniform GaAs effective mass for all region. 4 First peak location agrees pretty well with the result from Henrickson, JAP, (91) 6273, 2002.
  18. 18. Motivation NEGF Formulation Calculation Results Conclusion Effect of Bias on Photocurrent Spectral Response Peak Locations is not Significant Photocurrent Response, RI (nm2/photon) 10-1 Vb = 0.05 V Vb = 0.10 V Vb = 0.20 V 10-2 10-3 10-4 0.4 1.9 1.1 10-5 0 0.5 1 1.5 2 2.5 Photon Energy (eV) 1 Peak Locations do not change significantly. 2 Magnitude seems to be affected.
  19. 19. Motivation NEGF Formulation Calculation Results Conclusion Plot of Transmission Curves Under Various Bias 100 -1 10 -2 10 -3 Transmission 10 -4 10 -5 10 -6 10 -7 10 Vb = 0.05 V 10-8 Vb = 0.10 V -9 Vb = 0.20 V 10 0 0.5 1 1.5 2 2.5 Energy (eV) 1 Resonant peak locations are shifted to the left for higher bias. 2 Distance between resonant peaks, however, does not change significantly.
  20. 20. Motivation NEGF Formulation Calculation Results Conclusion Conclusion 1 We study electron-photon interaction using the NEGF framework. 2 Our calculation agrees with the previously published result. 3 Peak locations of photocurrent spectral response under various bias does not change significantly. 4 Transmission curves show the shift in the peaks of the resonant energies.
  21. 21. Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient Photon Flux We assume that the photon flux is a constant and is given by Nc Iω ≡ √ (1) V µr r Since the photocurrent response is normalized I RI = (2) qIω hence, we can set Iω = 1.
  22. 22. Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient Interaction Hamiltonian The vector potential is given by A(r, t) = ˆ a (be −ıωt + b † e ıωt ) exp(ık · r) (3) 2ω V We also assume dipole approximation, i.e. e k·r ≈ 1. The interaction Hamiltonian in the second quantized form is † H1 = r |H 1 |s ar as (4) rs q r |H 1 |s = r |A · p|s (5) m0
  23. 23. Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient Interaction Hamiltonian We assume that the field is polarized in the ˆ direction. Therefore, z the interaction Hamiltonian can be shown to be iq H1 = (zr − zs ) (be −iωt + b † e iωt ) × ˆzr r H 0 s ar as a † (6) rs If we use finite difference, it can be shown that H1 = Mrs be −ıωt + b † e ıωt (7) rs where ∗  √  +1/ms , s = r + 1 q µr r Prs = ∗ −1/ms , s = r − 1 Mrs = Iω Prs ı2a 2Nω c  0 , else
  24. 24. Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient Self-Energy Matrices And the self-energy matrices is given by Σrs (t1 , t2 ) = Gpq (t1 , t2 )Drp;qs (t1 , t2 ) (8) pq and > 1 1 Drp;qs (t1 , t2 ) ≡ Hrp (t1 )Hqs (t2 ) (9) < 1 1 Drp;qs (t1 , t2 ) ≡ Hqs (t2 )Hrp (t1 ) (10) Hence, we can write the self-energy matrices as Σ< (E ) = rs < < Mrp Mqs [NGpq (E − ω) + (N + 1)Gpq (E + ω)] pq
  25. 25. Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient Device Simulator Approach to Photogeneration Simulator calculate the change in carrier density from the continuity equations. ∂n 1 = Jn + Gn − Rn (11) ∂t q where Jn is the electron current density, Gn is the generation rate and Rn is the recombination rate. The generation is calculated from Pλ G = η0 α exp (αy ) (12) hc where η0 is the internal quantum efficiency, P is the intensity, α is the absorption coefficient, and y is distance.
  26. 26. Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient From Photogeneration to Photocurrent Once we know the change in carrier density, we can calculate the current from the Drift-Diffusion equation. Jn = qnµn En + qDn n (13)

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