1. Semi-Classical Transport Theory

2. Outline: • What is Computational Electronics? • Semi-Classical Transport Theory • Drift-Diffusion Simulations • Hydrodynamic Simulations • Particle-Based Device Simulations • Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical Simulators • Tunneling Effect: WKB Approximation and Transfer Matrix Approach • Quantum-Mechanical Size Quantization Effect • Drift-Diffusion and Hydrodynamics: Quantum Correction and Quantum Moment Methods • Particle-Based Device Simulations: Effective Potential Approach • Quantum Transport • Direct Solution of the Schrodinger Equation (Usuki Method) and Theoretical Basis of the Green’s Functions Approach (NEGF) • NEGF: Recursive Green’s Function Technique and CBR Approach • Atomistic Simulations – The Future • Prologue

3. Direct Solution of the Boltzmann Transport Equation • Particle-Based Approaches • Spherical Harmonics • Numerical Solution of the Boltzmann-Poisson Problem • In here we will focus on Particle-Based (Monte Carlo) approaches to solving the Boltzmann Transport Equation

4. Ways of Solving the BTE Using MCT • Single particle Monte Carlo Technique • Follow single particle for long enough time to collect sufficient statistics • Practical for characterization of bulk materials or inversion layers • Ensemble Monte Carlo Technique • MUST BE USED when modeling SEMICONDUCTOR DEVICES to have the complete self-consistency built in Carlo Jacoboni and Lino Reggiani, The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials, Rev. Mod. Phys. 55, 645 - 705 (1983).

5. Path-Integral Solution to the BTE • The path integral solution of the Boltzmann Transport Equation (BTE), where L=Nt and tn=nt, is of the form: K. K. Thornber and Richard P. Feynman, Phys. Rev. B 1, 4099 (1970).

6. The two-step procedure is then found by using N=1, which means that t=t, i.e.: Integration over a trajectory, i.e.probability that no scattering occurred within time integral t (FREE FLIGHT) Intermediate function that describes the occupancy of the state (p+eEt) at time t=0, which can be changed due to scattering events (SCATTER) +

7. Monte Carlo Approach to Solving the Boltzmann Transport Equation • Using path integral formulation to the BTE one can show that one can decompose the solution procedure into two components: • Carrier free-flights that are interrupted by scattering events • Memory-less scattering events that change the momentum and the energy of the particle instantaneously

8. Particle Trajectories in Phase Space

9. Carrier Free-Flights • The probability of an electron scattering in a small time interval dt is (k)dt, where (k) is the total transition rate per unit time. Time dependence originates from the change in k(t) during acceleration by external forces where v is the velocity of the particle. • The probability that an electron has not scattered after scattering at t = 0 is: • It is this (unnormalized) probability that we utilize as a non-uniform distribution of free flight times over a semi-infinite interval 0 to . We want to sample random flight times from this non-uniform distribution using uniformly distributed random numbers over the interval 0 to 1.

10. Generation of Random Flight Times Hence, we choose a random number Ith particle first random number We have a problem with this integral! We solve this by introducing a new, fictitious scattering process which does not change energy or momentum:

11. The sum runs over all the real scattering processes. To this we add the fictitious self-scattering which is chosen to have a nice property: Generation of Random Flight Times

12. Self-Scattering

13. Self-Scattering

14.                                                     Free-Flight Scatter Sequence for Ensemble Monte Carlo However, we need a second time scale, which provides the times at which the ensemble is “stopped” and averages are computed. Particle time scale  = collisions

15. Free-FlightScatterSequence R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, 1983.

16. Choice of Scattering Event Terminating Free Flight • At the end of the free flight ti, the type of scattering which ends the flight (either real or self-scattering) must be chosen according to the relative probabilities for each mechanism. • Assume that the total scattering rate for each scattering mechanism is a function only of the energy, E, of the particle at the end of the free flight • where the rates due to the real scattering mechanisms are typically stored in a lookup table. • A histogram is formed of the scattering rates, and a random number r is used as a pointer to select the right mechanism. This is schematically shown on the next slide.

17. Selection process for scattering Choice of Scattering Event Terminating Free-Flight We can make a table of the scattering processes at the energy of the particle at the scattering time:

18. Look-up table of scattering rates: Store the total scattering rates in a table for a grid in energy ………

19. Choice of the Final State After Scattering • Using a random number and probability distribution function • Using analytical expressions (slides that follow)

22. Representative Simulation Results From Bulk Simulations - GaAs Simulation Results Obtained by D. Vasileska’s Monte Carlo Code.

25. Transient Data

26. Steady-State Results Gunn Effect

27. Particle-Based Device Simulations • In a particle-based device simulation approach the Poisson equation is decoupled from the BTE over a short time period dt smaller than the dielectric relaxation time • Poisson and BTE are solved in a time-marching manner • During each time step dt the electric field is assumed to be constant (kept frozen)

28. Particle-Mesh Coupling • The particle-mesh coupling scheme consists of the following steps: • - Assign charge to the Poisson solver mesh • - Solve Poisson’s equation for V(r) • Calculate the force and interpolate it to the • particle locations • - Solve the equations of motion: Laux, S.E., On particle-mesh coupling in Monte Carlo semiconductor device simulation, Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on, Volume 15, Issue 10, Oct 1996 Page(s):1266 - 1277

29. Assign Charge to the Poisson Mesh 1. Nearest grid point scheme 2. Nearest element cell scheme 3. Cloud in cell scheme

30. Force interpolation • The SAME METHOD that is used for the charge assignment has to be used for the FORCE INTERPOLATION: xp-1 xp xp+1

31. Treatment of the Contacts From the aspect of device physics, one can distinguish between the following types of contacts: (1) Contacts, which allow a current flow in and out of the device - Ohmic contacts: purely voltage or purely current controlled - Schottky contacts (2) Contacts where only voltages can be applied

32. Calculation of the Current • The current in steady-state conditions is calculated in two ways: • By counting the total number of particles that enter/exit particular contact • By using the Ramo-Shockley theorem according to which, in the channel, the current is calculated using

33. Current Calculated by Counting the Net Number of Particles Exiting/Entering a Contact

34. Device Simulation Results for MOSFETs: Current Conservation VG=1.4 V, VD=1 V Drain contact Source contact Cumulative net number of particles Entering/exiting a contact for a 50 nm Channel length device Current calculated using Ramo-Schockley formula X. He, MS, ASU, 2000.

35. Simulation Results for MOSFETs: Velocity and Enery Along the Channel Velocity overshoot effect observed throughout the whole channel length of the device – non-stationary transport. For the bias conditions used average electron energy is smaller that 0.6 eV which justifies the use of non-parabolic band model.

36. Simulation Results for MOSFETs: IV Characteristics • The differences between the Monte Carlo and the Silvaco simulations are due to the following reasons: • Different transport models used (non-stationary transport is taking place in this device structure). • Surface-roughness and Coulomb scattering are not included in the theoretical model used in the 2D-MCPS. X. He, MS, ASU, 2000.

37. Simulation Results For SOI MESFET Devices – Where are the Carriers? SOI MOSFET SOI MESFET Applications: circuits based Micropower on weakly inverted MOSFETs Digital Watch Pacemaker Implantable cochlea and retina Low-power RF electronics. T.J. Thornton, IEEE Electron Dev. Lett., 8171 (1985).

38. Proper Modeling of SOI MESFET Device Gate current calculation: • WKB Approximation • Transfer Matrix Approach for piece-wise linear potentials Interface-Roughness: • K-space treatment • Real-space treatment Goodnick et al., Phys. Rev. B 32, 8171 (1985)

39. Output Characteristics and Cut-off Frequency of a Si MESFET Device Tarik Khan, PhD, ASU, 2004.

40. Output Characteristics and Cut-off Frequency of a Si MESFET Device Tarik Khan, PhD, ASU, 2004.

41. Modeling of SOI Devices • When modeling SOI devices lattice heating effects has to be accounted for • In what follows we discuss the following: • Comparison of the Monte Carlo, Hydrodynamic and Drift-Diffusion results of Fully-Depleted SOI Device Structures* • Impact of self-heating effects on the operation of the same generations of Fully-Depleted SOI Devices *D. Vasileska. K. Raleva and S. M. Goodnick, IEEE Trans. Electron Dev., in press.

42. FD-SOI Devices:Monte Carlo vs. Hydrodynamic vs. Drift-Diffusion 90 nm Silvaco ATLAS simulations performed by Prof. Vasileska.

43. FD-SOI Devices:Monte Carlo vs. Hydrodynamic vs. Drift-Diffusion 25 nm Silvaco ATLAS simulations performed by Prof. Vasileska.

44. FD-SOI Devices:Monte Carlo vs. Hydrodynamic vs. Drift-Diffusion 14 nm Silvaco ATLAS simulations performed by Prof. Vasileska.

45. FD-SOI Devices:Why Self-Heating Effect is Important? 1. Alternative materials (SiGe) 2. Alternative device designs (FD SOI, DG, TG, MG, Fin-FET transistors

46. FD-SOI Devices:Why Self-Heating Effect is Important? L ~ 300nm dS A. Majumdar, “Microscale Heat Conduction in Dielectric Thin Films,” Journal of Heat Transfer, Vol. 115, pp. 7-16, 1993.

47. Conductivity of Thin Silicon Films – Vasileska Empirical Formula

48. Particle-Based Device Simulator That Includes Heating

49. Heating vs. Different Technology Generation

50. Higher Order Effects Inclusion in Particle-Based Simulators • Degeneracy – Pauli Exclusion Principle • Short-Range Coulomb Interactions • Fast Multipole Method (FMM) V. Rokhlin and L. Greengard, J. Comp. Phys., 73, pp. 325-348 (1987). • Corrected Coulomb Approach W. J. Gross, D. Vasileska and D. K. Ferry, IEEE Electron Device Lett. 20, No. 9, pp. 463-465 (1999). • P3M Method Hockney and Eastwood, Computer Simulation Using Particles.