Negative-mass electronic transport in Gallium Nitride using analytic approximations in Monte-Carlo Simulations

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 20th January 2012

2. Outline • Introduction • Cosine Band-structure approximation • Algorithm • Implementation of approximation • Use of parallelisation • Results for GaN/GaAsxN1-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 where EB 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 GaNxAs1-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 Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 4

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

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] [1] – K. Tomizawa, Numerical Simulation of Submircon Semiconductor Devices,Artech House, London, 1993 Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 6

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 • 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. • ~32 minutesSingle core of an Intel Core 2 Duo Processor – 3.0GHz, Windows 7 64-bit, Intel Fortran Compiler using full optimisation. (64-bit binary) • ~25 minutesTwo 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 minutesFour 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

9. Validation Average Electron Velocity (x107 cm/s) [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 Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 9

10. Results - GaN 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] [1] – D. R. Naylor, et. al, Solid State Communications (2012), ArticleIn 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 Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 10

11. Results - GaN - Negative Effective Mass •Г valley – +ive effective mass •Г valley – -ive effective mass • Upper valley for the Cosine approximation for the parabolic approximation 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. Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 11

12. Results – GaN – Transient properties GaN with a 1.2eV valley separation using the cosine band-structure approximation Average Electron Velocity (x107 cm/s) Applied Field (kV/cm) Time Elapsed (ps) Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 12

13. Results – GaN0.01As0.99 [1] – D. R. Naylor, et. al, Submitted to Journal of Applied Physics Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 13

14. Results – GaN0.01As0.99 [1] – D. R. Naylor, et. al, Submitted to Journal of Applied Physics Negative-mass electronic transport in GaN using analytic approximations | 20 January 2012 | 14

15. Conclusions • Our Cosine band-structure implementation gives comparable results to full band MC codes for GaN and analytic results for GaNxAs1-xusing 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