1. Semiconductor Device Modeling
Sts. Cyril and Methodius University - Skopje FACULTY OF ELECTRICAL ENGINEERING AND INFORMATION TECHNOLOGIIES
Semiconductor Device Modeling
Lecture 7. Particle-Based Device Simulation Methods
1. Free-flight Generation
2. Final State After Scattering
3. Ensemble Monte-Carlo Simulation
4. Device Simulation Using Particles – Monte Carlo Device Simulation
Knowledge Alliance EPP BG-EPPKA2-KA
Micro-Electronics Cloud Alliance (MECA)

2. Electronic Structure, Lattice Dynamics Electromagnetic Fields
Device Simulation
Before we explain particle based device methods…
Electronic Structure, Lattice Dynamics
Transport Equations
Electromagnetic Fields
Device Simulation
J, r
E, B

3. Hierarchy of Semiconductor Simulation Models
Schrödinger Equation
Boltzmann Transport Equation
Monte-Carlo Particle-Based App.
Hydrodynamic and Energy Balance
Transport Approaches
Drift-Diffusion
Approaches
Compact Approaches

4. Monte Carlo Method for the Solution of the Boltzmann Transport Equation
Outline
Path Integral Solution of the Boltzmann Transport Equation (BTE)
Ways of Solving BTE
The Monte Carlo Method Described
Scattering Tables Construction
Carrier Initialization in Momentum Space
Free-flight Sequence
Choice of Scattering Events
Choice of Final State After Scattering
Free-flight-scatter Flow-Chart
Ensemble Monte Carlo
Ensemble Averages
Representative Simulation Results
Inclusion of the Pauli Exclusion Principle

5. Path Integral Solution of 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. + Path Integral Solution of the BTE
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. Path Integral Solution of the BTE
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. Path Integral Solution of the BTE - Free-Flight-Scatter Sequence -

9. Ways of solving the BTE 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, (1983).

10. The Monte Carlo Method - Scattering Tables Construction -
First, bandstructure of the material is chosen
For consideration of electron transport in silicon material system, 6 equivalent conduction band valleys are considered
For the case of GaAs material, for example, 3 non-equivalent valleys must be accounted for for proper treatment of high field transport in this material system. Namely, the gamma, X and L valleys must be taken into consideration if high field transport has to be accounted for properly.
Next, relevant scattering mechanisms are identified
For silicon material system relevant scattering mechanisms are acoustic phonon scattering, Coulomb scattering and intervalley g- and f-type of phonon scattering
For the case of GaAs, relevant scattering mechanisms are acoustic phonon scattering, polar optical phonon scattering, Coulomb scattering and non-equivalent intervalley phonon scattering

11. The Monte Carlo Method - Scattering Table Construction -
Continuing on the intervalley scattering in silicon, it can be due to g- and f-phonons.
G-phonons move the carriers on the opposite end of the same axis. There is only one final valley into which the phonons can scatter due to g-phonons and the valley degeneracy of this scattering mechanism is therefore 1.
F-phonons move the electrons from the, for example, (100) valley into a (010) valley, or (001) valley, just to name a few. Indeed there are 4 final valleys into which the electrons can scatter into and the valley degeneracy factor for f-phonons is 4.
A graphical illustration of the g- and the f-phonon processes in silicon is shown in the following slide,

12. The Monte Carlo Method - Scattering Table Construction -
Graphical representation of the g- and f- phonon processes in silicon.

13. The Monte Carlo Method - Non-parabolicity and Full-Band -

14. The Monte Carlo Method - Non-parabolicity and Full-Band -
Electron distribution in momentum space, for an electric field of
50 kV/cm in the (111) direction, at 300 K. Higher energy electrons are
shown with lighter shades of gray.
E. Pop, R. W. Dutton and K. E. Goodson, JOURNAL OF APPLIED PHYSICS VOLUME 96, NUMBER 9 1 NOVEMBER 2004

15. The Monte Carlo Method - Non-parabolicity and Full-Band -
Phonon dispersion in silicon along the (100) direction, from neutron
scattering data (symbols). The lines represent quadratic
approximation. The f and g phonons participate in the intervalley scattering
of electrons
E. Pop, R. W. Dutton and K. E. Goodson, JOURNAL OF APPLIED PHYSICS VOLUME 96, NUMBER 9 1 NOVEMBER 2004

16. The Monte Carlo Method - Scattering Table Construction
For the case of GaAs we have three non-equivalent valleys. Namely, we gave the gamma valley (single), the three X valleys and the four L valleys. Then, a scattering table is needed for each of these three nonequivalent valleys. The possible intervalley transittions are:
For gamma valley: gamma  X (3) , gamma  L(4)
For X valley: Xgamma (1), XX(2), XL(4)
For the L-valley, we have: L gamma(1), L X(3), L  L(3)
The graphical representation of these possible transitions is shown in the next slide.

17. The Monte Carlo Method - Scattering Table Construction -
Intervalley scattering mechanisms in GaAs bulk material system

18. The Monte Carlo Method - Scattering Table Construction -
Scattering tables construction in the initialization phase of the Monte Carlo Algorithm

19. Scattering rates in the Gamma valley for GaAs

20. Scattering rates in the L-valley for GaAs

21. Scattering rates for the X-valley in GaAs

22. Cumulative scattering rates for the creation of the scattering table

23. The Monte Carlo Method Described - Carrier Initialization in Momentum space -

24. The Monte Carlo Method Described - Carrier Free-Flights -
In general the carrier free-flights are obtained by inverting the following equation
Rees introduced the concept of self-energy which allows for the analytical solution of the integral:

25. The Monte Carlo Method Described - Carrier Free-Flights -

26. The Monte Carlo Method Described - Carrier Free-Flights -

27. The Monte Carlo Method - Choice of scattering mechanisms
After a free-flight is completed one has to choose what type of scattering mechanism will take place.
This is accomplished with a selection of a random number uniformly distributed between 0 and 1.
It is assumed in this process that the cumulative scattering tables are normalized to unity.
A bin in the scattering table is then selected that corresponds to the particle energy.
The random number is compared against the values of various cells that denote the probability of a given scattering event to occur at the carrier energy.

28. The Monte Carlo Method - Choice of scattering mechanisms -

29. The Monte Carlo Method - Choice of Final State After Scattering -
Using a random number and probability distribution function, the final angle after scattering is given by:
Using analytical expressions (slides that follow)

30. The Monte Carlo Method - Choice of Final State After Scattering -
The analytical model is based on the following expressions for the calculation of the final angle after scattering:

31. The Monte Carlo Method - Choice of Final State After Scattering -

32. The Monte Carlo Method - Choice of Final State After Scattering -

33. The Monte Carlo Method - Free-Flight Scatter Sequence -
Synhronos ensemble

34. dte2
dtp B A
dte2
dt3 A
dte2
dt3 B
dte2

35. General Ensemble
Monte Carlo Code
Flowchart
Free-Flight
Scatter Routine

36. The Monte Carlo Method - Ensemble Averages -

37. Representative Simulation Results - Transient simulation results for Si -

38. Representative Simulation Results - Steady-state simulation results for Si bulk -

39. Representative Simulation Results - Transient Data: GaAs -

40. Representative Simulation Results
Steady-State Results
Gunn Effect

41. Initial and final wavevector along the field
Shift in the distribution due to application
of electric field.

42. Initial and final carrier energy
Shift in the distribution due to application
of electric field.

43. Inclusion of the Pauli Exclusion Principle
The influence of the final state on the scattering rate is important at low temperatures and high carrier densities.
This effect may be included via a self-scattering rejection method (Bosi and Jacoboni, J. Phys. C9, 315 (1976); Lugli and Ferry, IEEE Trans. Elec. Dev. 32, 2431 (1985)).
The electron (hole) distribution function f(kx,ky,kz) is updated in k-space (on a 2D or a 3D grid).
Once the final state has been selected, a new random number is generated:
If , then self-scattering is assumed to occur with no change of momentum or energy
If , then accept the scattering event

44. Two-Dimensional System Example
Simulation size:
Two-Dimensional System Example
The area per state in k-space is
cell size
Upon scattering, we increase or decrease the population in the cell, and the latter gives f(k)
Some important notes:
- The size of the grid in k-space determines how many electrons may occupy the grid.
- The accuracy improves as the number of particles increases.

45. Number of states that can be occupied
Within one cell is:

46. Mobility vs. Experimental Data

47. Modeling of SOI Devices Monte Carlo Particle Based Device Simulator
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
COMPUTEL

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

49. FD-SOI Devices: Monte Carlo vs. Hydrodynamic vs. Drift-Diffusion
25 nm

50. FD-SOI Devices: Monte Carlo vs. Hydrodynamic vs. Drift-Diffusion
14 nm

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

52. Conductivity of Thin Silicon Films

53. Particle-Based Device Simulator That Includes Heating
2.Average and smooth: electron density, drift velocity and electron energy at each mesh point
End of MCPS phase?
End of simulation?
End no
yes
Define device structure
t = 0
t = t + Dt
Transport Kernel (MC phase)
Field Kernel (Poisson Solver)
t = n Dt?
Start
3.Acoustic and Optical Phonon Energy Balance Equations Solver
1.Generate phonon temperature dependent scattering tables
Outer Gummel Loop
Initial potential, fields, positions and velocities of carriers

54. Current Degradation due to a Self-Heating for Different Technology Generation
Channel Length
(FD-SOI MOSFET)
VGS = VDS
(V)
TGATE=300K
Current Decrease
(%)
TGATE=400K
Current Decrease
(%)
25 nm
1.2
5.78
9.45
45 nm
1.2
5.93
10.79
60 nm
1.2
4.73
10.83
80 nm
1.5
11.0
17.96
90 nm
1.5
11.19
17.95
100 nm
1.5
11.20
18.48
120 nm
1.8
16.34
22.16
140 nm
1.8
15.6
21.85
180 nm
1.8
15.22
21.52

55. Heating vs. Different Technology Generation