1. Computational Electronics Generalized Monte Carlo Tool for Investigating Low-Field and High Field Properties of Materials Using Non-parabolic Band Structure Model Raghuraj Hathwar Advisor : Dr. Dragica Vasileska

2. Computational Electronics Outline Motivation of modeling different materials - Strained Silicon - III-V and II-VI materials - Silicon Carbide The generalized Monte Carlo code - Free-Flight and drift velocity calculation Rappture interfacing Results Conclusions and future work.

3. Computational Electronics Technology Trends

4. Computational Electronics Strained Silicon The four minima of the conduction band in directions parallel to the plane of strain are raised. This results in higher electron mobility. There is also a splitting of the light and heavy hole bands leading to increased hole mobility.

5. Computational Electronics III-V and II-VI Materials High electron mobility of compared to silicon. AlGaAs/GaAs are lattice matched. AlGaN/GaN interfaces have spontaneous polarization.

6. Computational Electronics Silicon Carbide (SiC) Very useful in high voltage devices because of its thermal conductivity, high band gap and high breakdown field. In fact the thermal conductivity of 4H-SiC is greater than that of copper at room temperature.

7. Computational Electronics The Boltzmann Transport Equation The Chamber-Rees Path Integral The Monte Carlo Method

8. Computational Electronics The Generalized Monte Carlo Flow Chart

9. Computational Electronics Types of Scattering Acoustic Phonon Scattering Zeroth order Intervalley Scattering First order Intervalley Scattering Piezoelectric Scattering Polar Optical Phonon Scattering Ionized Impurity Scattering

10. Computational Electronics Fermi’s Golden Rule and Scattering Rates Calculation Calculate the Matrix Element Use Fermi’s Golden Rule Sum over all k’ states

11. Computational Electronics Band Structure Model e.g. GaAs Full Band Structure (equilibrium) 3 Valley Approximation

12. Computational Electronics E-k relation for a General Valley Here k 1, k 2 and k 3 are the wave vectors along the three mutually perpendicular directions that define the valley and m 1, m 2 and m 3 are the effective masses of the electrons along those directions

13. Computational Electronics Conversion from Anisotropic Bands to Isotropic Bands In order to make the conversion between energy and momentum easy all anisotropic bands are converted to isotropic bands using Which gives the following E-k relation where

14. Computational Electronics Carrier Free-Flight From Newton’s 2 nd law and Q.M.

15. Computational Electronics For simplicity the wave vectors of all electrons are only stored in the x,y and z coordinate system. Therefore before drifting, the wave vectors are transformed from the x,y,z coordinate system to the 1,2,3 coordinate system using, where [a 1 b 1 c 1 ], [a 2 b 2 c 2 ] and [a 3 b 3 c 3 ] are the three mutually perpendicular directions that define the valley.

16. Computational Electronics The electric fields must also be transformed to the directions along the wave vectors The electrons are then drifted and transformed back into the x,y,z coordinate system.

17. Computational Electronics Drift Velocity Calculation

18. Computational Electronics The drift velocities must then be transformed to the x,y,z coordinate system so that an average can be taken over all electrons.

19. Computational Electronics Rappture Integration The Rappture toolkit provides the basic infrastructure for a large class of scientific applications, letting scientists focus on their core algorithm when developing new simulators. Instead of inventing your own input/output, you declare the parameters associated with your tool by describing Rappture objects in the Extensible Markup Language (XML). Create an xml file describing the input structure. Integrate the source code with Rappture to read input values and to output results to the Rappture GUI.

20. Computational Electronics Material Parameters and Simulation Parameters

21. Computational Electronics Valley Parameters

22. Computational Electronics Scattering Parameters

23. Computational Electronics Drift Velocity vs Electric Field Electron Energy vs Electric Field Silicon

24. Computational Electronics Gallium Arsenide (GaAs) Drift Velocity vs Electric Field Electron Energy vs Electric Field

25. Computational Electronics Fraction of electrons in the L valley vs Electric Field

26. Computational Electronics Drift Velocity vs Electric FieldElectron Energy vs Electric Field Germanium (Ge)

27. Computational Electronics Rappture GUI Results

28. Computational Electronics

29. Conclusions and Future Work Uses non-parabolic band structure making it as accurate as possible for an analytic representation of the band structure. Interfacing the tool with Rappture enables easy handling of the parameters and reduces the complexity of using the tool. Existing materials band structures can be easily modified to improve existing results. New materials can easily be added to the code. The tool can be extended to include impact ionization scattering to better model high field properties. Full band simulation for holes.