Coupling Maxwell\'s Equations to Particle-Based Simulators

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Coupling Maxwell\'s Equations to Particle-Based Simulators

  1. 1. Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  2. 2. •  Introduction / Motivation •  Full Band Simulator •  Finite-Difference Time Domain Method (FDTD) / Maxwell Solver •  Coupling of Maxwell/Monte Carlo methods •  Simulation results / ConclusionsNanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  3. 3. Traditional methods of simulating semiconductor devices involve a solution of Poisson’s equation on a discrete mesh. However, the static field distribution that results is unable to fully account for the time-varying nature of the total electromagnetic environment that exist within and surrounding the device. •  As operating frequencies increase, must treat signals as electromagnetic waves propagating along transmission lines in devices. •  Must take into account absorption /emission of EM energy throughout system.Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  4. 4. Development of devices operating in this new high-frequency regime is occuring on two separate ends of a gap. Terahertz GapNanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  5. 5. Current research motivated by a desire to accurately simulate and capture radiated EM field patterns emanating from ultrafast, high -frequency devices Experimental measurements of high-field transport in GaAs and InP under extreme non-equilibrium conditions have been reported by Leitenstorfer et. al.1 Recent numerical experiments of transient responses in GaAs and InP by Wigger et. al.2 have provided further motivation for this work 1A. Leitenstorfer, S. Hunsche, J. Shah, M.C. Nuss, and W.H. Knox: Phys. Rev. Lett. 82 5140 (1999). 2S. Wigger, M. Saraniti, S. Goodnick, A. Leitenstorfer: J. Comp. Elec. 1:475-480 (2002)Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  6. 6. •  Introduction / Motivation •  Particle-Based Simulator •  Finite-Difference Time Domain Method (FDTD) / Maxwell Solver •  Coupling of Maxwell/Monte Carlo Methods •  Simulation results / ConclusionsNanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  7. 7. Use a semiclassical description of carrier transport via stochastic solution ofthe Boltzmann Transport Equation (BTE),Boltzmann Transport Equation: drift diffusion where, Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  8. 8. initializationFlowchart of Simulator: calculate charge Poisson solver free flight particle dynamics NO end simulation time ? YES calculate averages end Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  9. 9. initializationparabolic dispersion: fullband dispersion: 6 4 energy [eV] 2 0non-parabolic dispersion: -2 -4 -6 L X U,K L L wave vector Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  10. 10. Initialization: initialization Empirical Pseudopotential Method density of states [10 cm eV ] 5 density of states [10 cm eV ] -1 -1 7 InP GaAs -3 -3 6 4 22 5 22 3 4 3 2 2 1 1 L L L L wave vector wave vector X X U,K U,K L L -10 -5 0 5 10 -10 -5 0 5 10 energy [eV] energy [eV]Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  11. 11. Initialization: initialization Phonon Dispersion Valence Shell Method 0 .0 5 0 .0 4 LO InP LO GaAs 0 .0 4 TO 0 .0 3 TOenergy [eV] energy [eV] 0 .0 3 0 .0 2 0 .0 2 LA LA 0 .0 1 0 .0 1 LA LA TA TA TA TA 0 L L 0 L X U,K L L L X U,K wave vector wave vector Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  12. 12. Free Flight: Drift free flightNewton’s Equations of motion: Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  13. 13. Ensemble Monte Carlo vs.“Cellular Monte Carlo”The Ensemble Monte Carlo The Cellular Monte Carlo methodmethod tabulates the scattering computes and tabulates therate integrated over the entire scattering rates from an initialmomentum space. The final state momentum state to all possibleis then obtained by inverting the final states, which satisfy theenergy-momentum dispersion appropriate conservation laws.relation, which is also tabulatedfor full band. choose scattering choose new k new energy   computationally fast find new k with   high memory requirements dispersion relation Scattering Mechanisms:   computationally slow   polar scattering   low memory requirements   deformation potential scattering Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH   impact ionization
  14. 14. Hybrid EMC/CMC Idea: use MC scattering in regions of band structure where scattering is low.   Nearly as fast as CMC.   Reduces memory usage. Hybrid/MC performance ratio time per iter. [sec/5000 e ] - 6 4energ y [eV] 2 0 -2 -4 EMC -6 CMC X U,K L L L wave vector field [V/m] Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  15. 15. •  Introduction / Motivation •  Full Band Simulator •  Finite-Difference Time Domain Method (FDTD) / Maxwell Solver •  Coupling of Maxwell/Monte Carlo Methods •  Simulation results / ConclusionsNanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  16. 16. “Finite-Difference Time-Domain” • First introduced by K. S. Yee in 1966. • Method remained relatively unused for ~10 yrs.   Inadequate processing power.   Method lacked proper boundary conditions.Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  17. 17. Curl form of necessary Maxwellequations are: Constitutive Relationships Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  18. 18. By applying the curl operator and equating the vector components of the previous (2) equations, we arrive at the following set of (6) scalar equations:Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  19. 19. Now, using a centered-difference scheme, each expression can be rewritten in appropriate finite-difference form shown here: Note that E & H Fields are offset Magnetic Field Update Equation from each other. Electric Field Update EquationNanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  20. 20. •  Scheme is often referred to as a “Leapfrog” Method i-2 i-1 i i+1 i+2 Ex t – Δt/2 i-1 1/2 i-1/2 i+1/2 i+1 1/2 Hy t i-2 i-1 i i+1 i+2 Ex t +Δt/2Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  21. 21. •  In 3D, the E and H fields can be visualized as existing on separate but interlaced grids over a cubic cell, z Ey Ex Ex Hz Ey Ex Ez Ez Hy Hx Ex Hy “Yee cell” Hx Ex Ey Ex Hz Ey y xNanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  22. 22. •  An upper bound is imposed on the simulation timestep due to Courant-Freidrich-Levy1 (CFL) condition for finite-difference solutions of the wave equation,where c is the wave velocity, Δt is the timestep, and Δx, Δy, and Δz arethe spatial dimensions of each grid cell. 1R. Courant, K. Friedrichs, and H. Lewy. “On the Partial Difference Equations of Mathematical Physics.” IBM Journal, pp 215-234, Mar. 1967Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  23. 23. •  EM wave problems are defined, in general, with OPEN or UNBOUNDED domains that extend out to infinity. •  However, computationally impossible to store unlimited amount of data required Domain must be truncated so that it: •  Fully contains structure of interest. •  Resolves any region of interest within/around device. •  Allows for wave propagation while minimizing reflections of outward traveling waves at boundaries.Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  24. 24. We have chosen to implement the Perfectly Matched Layer (PML) absorbing boundary condition recently developed by Berenger1. •  Formulation involves a “field-splitting” approach creating boundary layer that can:   absorb any kind of traveling wave.   regardless of direction of travel.   without reflection back into domain. 1J. P. Bérenger, “Perfectly matched layer for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Antennas Propagat., vol 44, pp. 110-117, Jan. 1996.Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  25. 25. •  Berenger introduced (1) complex permittivity/permeability (2) split the field components into 2 parts. •  This resulted in the following set of (12) equations,Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  26. 26. Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  27. 27. •  Introduction / Motivation •  Full Band Simulator •  Finite-Difference Time Domain Method (FDTD) / Maxwell Solver •  Coupling of Maxwell/Monte Carlo •  Simulation results / ConclusionsNanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  28. 28. Coupling FDTD solver to EMC Initialize Device Initialize Device Calculate Charge Calculate Current Density Poisson Solver Maxwell Solver Free Flight Free Flight Particle Dynamics Particle Dynamics No No End of End of Simulation? Simulation? Yes Yes Calculate Averages Calculate Averages End End Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  29. 29. Curl form of necessary Maxwell equations are: The current density, J can be calculated directly via temporal and spatial evolution of charge from Ensemble Monte Carlo. Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  30. 30. Current density is computed at every timestep using weighted summationof particle velocities in each grid cell, charge velocity Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  31. 31. Determine carrier distribution EMC/CMC Solver Calculate current density J(i,j,k) Calculate E and H fields using J(i,j,k) Maxwell Solver Determine Lorentz ForceNanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  32. 32. •  Introduction / Motivation •  Full Band Simulator •  Finite-Difference Time Domain Method (FDTD) / Maxwell Solver •  Coupling of Maxwell/Monte Carlo •  Simulation results / ConclusionsNanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  33. 33. In experimental setup, a mode-locked Ti: Sapphire laser with apulse duration of 12fs, central photon frequency of 1.49eV, and bandwidthof 120meV used to optically excite electron-hole pairs in GaAs and InP pinDiodes with intrinsic region 500 nm long 1A. Leitenstorfer, S. Hunsche, J. Shah, M.C. Nuss, and W.H. Knox: Phys. Rev. Lett. 82 5140 (1999). Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  34. 34. 1A. Leitenstorfer, S. Hunsche, J. Shah, M.C. Nuss, and W.H. Knox: Phys. Rev. Lett. 82 5140 (1999).Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  35. 35. pin diode: h p+ i n+ VA momentum space EC real space -EC h EG - + EV EFpEV h qVA EFn h = EG+ + Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  36. 36. EX=-VA/LXempirical generation rate: + - + - + - + -change in carrier density: + - LX ninj=5x1014 cm-3 tp = 10fs t0 = 20 fs t = 0.0167 fs Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  37. 37. simulated region p+ i n+h Ly EX=-VA/LX + - + - LX + + - - 10 grid cells + - h = 1.49 eV 50 grid cells Lx = 500 nm Ly = 100 nm Lz = 100 nm Ne = Nh = 25,000 Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  38. 38. 500 nm 500 nm PML Region 300 nm Filament GaAs AirSnapshot #2 at 480nm from left Snapshot #1 at 50nm from left y GaAscontact surface. contact surface. x z Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  39. 39. Magnitude of Longitudinal Field (Ex) at 50 nm from contact surface.Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  40. 40. Transverse field (Ey) at 50 nm from Displacement of carriers vs. timecontact surface. Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  41. 41. Longitudinal field (Ex) at 50 nm from contact surface.Nanostructures Research GroupCENTER FOR SOLID STATE ELECTRONICS RESEARCH
  42. 42. 500 nm 500 nm PML Region 300 nm Filament GaAs AirSnapshot #2 at 480nm from left Snapshot #1 at 50nm from left y GaAscontact surface. contact surface. x z Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  43. 43. Transverse field (Ey) at 480 nm from contact surface. Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  44. 44. Simulation results for 100 kV/cm Simulation results due to Wigger et. al. Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  45. 45. •  Presented direct simulation and capture of THztransient field patterns from simple device structureusing a fullband simulator coupled with a Maxwell solver.•  Demonstrated usefulness of Global Solver to modelEM characteristics of a simple device.• Future work will involve implementation of FDTDMethod not constrained by timestep criterion. Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH

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