1. Negative-mass electronic transport in Gallium Nitride using analytic approximations in Monte-Carlo Simulations Daniel R. Naylor*, Angela Dyson* & Brian K. Ridley† *Department of Physics, University of Hull †School of Computing Science and Electronic Engineering, University of Essex, Colchester 20 th January 2012

2. Outline Introduction Cosine Band-structure approximation Algorithm – Implementation of approximation – Use of parallelisation Results for GaN/GaAs x N 1-x Conclusions Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 2

3. Introduction There has been a lot of interest in negative effective mass states in materials such as Gallium Nitride – Potential for use in generation of Terahertz EM radiation A full band implementation would take negative mass states into account, however runtimes for such codes quickly become unmanageable. A novel analytic band-structure approximation has been developed that includes the NM states, whilst still retaining the advantages of an analytic approximation. Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 3

4. Cosine Band-structure approximation Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 4 where E B is the width of the band and a is the lattice constant along the c-axis. A very good fit for some highly non-parabolic materials, such as Gallium Nitride around the Γ point. [1] Also a good fit (with slight modification) for the E- band as predicted in the band anti-crossing model in GaN x As 1-x (x ~ 1%) Potential to study negative mass states at higher energies in the band using an analytic form, without reverting to a slow, numerical full-band model. [1] – A. Dyson, B. K. Ridley, Journal of Applied Physics, 104(11) 2008, p.113709. doi: 10.1063/1.3032272

5. Cosine Band-structure approximation Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 5 parabolic k.p cosine

6. Algorithm Based on algorithms by Tomizawa. [1] Rewritten to make use of FORTRAN 95 language features and to be based on the cosine band structure approximation. Different codes have been developed/are in development, in order of increasing complexity – Single Electron (SMC) – Ensemble (EMC) – 1D Device (Coupled EMC and Poisson solver) [in progress] Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 6 [1] – K. Tomizawa, Numerical Simulation of Submircon Semiconductor Devices, Artech House, London, 1993

7. Algorithm – Ensemble Monte Carlo code Scattering rates (based on the cosine form) are pre- calculated for a range of electron energies. Electrons are selected in turn, are drifted for a small increment of time and then are scattered. – We have parallelised this step, as we assume that there is no electron-electron interaction. Drift time and scattering mechanisms are selected through the use of a random number generator. Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 7

8. Algorithm – Ensemble Monte Carlo code ~32 minutes Single core of an Intel Core 2 Duo Processor – 3.0GHz, Windows 7 64-bit, Intel Fortran Compiler using full optimisation. (64-bit binary) ~25 minutes Two cores of an Intel Core 2 Duo Processor – 3.0GHz, Windows 7 64-bit, Intel Fortran Compiler, using OpenMP and full optimisation. (64-bit binary) ~15 minutes Four cores of an Intel Core 2 Quad Processor – 2.5GHz, Windows XP 32-bit, Intel Fortran Compiler, using OpenMP and full optimisation. (32-bit binary) ~5 minutes Four cores of two Intel Xeon Processors (eight cores total) – 2.67GHz, Ubuntu Linux 11.10 64-bit, GCC gfortran Compiler, using OpenMP and full optimisation. (64-bit binary) Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 8 Sample average run-times Using GaN parameters run over a range of 51 electric-field strengths, 0-500kV/cm in 10kV/cm steps, simulation time 4ps with 15000 particles.

9. Validation Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 9 [1] Blakemore, J. S., J. Appl. Phys. 53, 10 (1982) pp. R123-R181. doi: 10.1063/1.331665 [2] Maloney, T. J. and J. Prey, J. Appl. Phys. 48, 2 (1977) pp. 781-787. doi: 10.1063/1.323670 Average Electron Velocity (x10 7 cm/s)

10. Results - GaN Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 10 [1] – D. R. Naylor, et. al, Solid State Communications (2012), Article In Press, doi:10.1016/j.ssc.2011.12.029 [2] – J. Barker et. al, J. Appl. Phys. 97, 063705 (2005), doi:10.1063/1.1854724 EMC – cosine band-structure approximation [1] EMC – k.p band-structure approximation [1] Simple hydrodynamic-like model (using fitted parameters) Sample experimental data (Barker et. al) – [2]

11. Results - GaN - Negative Effective Mass Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 11 Г valley – +ive effective mass Г valley – -ive effective mass Upper valley Distribution of electron energies vs. their velocities in the direction of the applied field (of 200kV/cm). Black curve – expected velocity of electron as predicted by the cosine band structure if the electron was travelling solely parallel to the field in the Γ valley. Green curve – as predicted by the parabolic band structure. for the Cosine approximation for the parabolic approximation

12. Results – GaN – Transient properties Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 12 Average Electron Velocity (x10 7 cm/s) Applied Field (kV/cm) Time Elapsed (ps) GaN with a 1.2eV valley separation using the cosine band-structure approximation

13. Results – GaN 0.01 As 0.99 Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 13 [1] – D. R. Naylor, et. al, Submitted to Journal of Applied Physics

14. Results – GaN 0.01 As 0.99 Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 14 [1] – D. R. Naylor, et. al, Submitted to Journal of Applied Physics

15. Conclusions Our Cosine band-structure implementation gives comparable results to full band MC codes for GaN and analytic results for GaN x As 1-x using BAC Occupation of negative mass states can be comparable to the occupation of satellite valley states Proper parallelisation significantly improves runtimes Our code provides an excellent foundation for further development without major escalation in runtimes Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 15

16. Acknowledgements Dr. Jianzhong Zhang DRN acknowledges EPSRC for financial support AD & BKR acknowledge ONR for financial support (sponsored by Dr. Paul Maki under grant nos. N00014-09-1-0777 & N00014-06- 1-0267.) Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 16