Raghuraj Hathwar Advisor : Dr. Dragica Vasileska

1. 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. 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. Technology Trends

4. 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. 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. 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. The Monte Carlo Method • The Boltzmann Transport Equation • The Chamber-Rees Path Integral

8. The Generalized Monte Carlo Flow Chart

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

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

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

12. E-k relation for a General Valley Here k1 , k2and k3are the wave vectors along the three mutually perpendicular directions that define the valley and m1 , m2and m3 are the effective masses of the electrons along those directions

13. 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. Carrier Free-Flight From Newton’s 2nd law and Q.M.

15. 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 [a1b1c1], [a2b2c2] and [a3b3c3] are the three mutually perpendicular directions that define the valley.

16. 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. Drift Velocity Calculation

18. 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. 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. Material Parameters and Simulation Parameters

21. Valley Parameters

22. Scattering Parameters

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

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

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

26. Germanium (Ge) Drift Velocity vs Electric Field Electron Energy vs Electric Field

27. Rappture GUI Results

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.