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, Colchester20th January 2012

Outline

• Introduction

• Cosine Band-structure approximation

• Algorithm– Implementation of approximation– Use of parallelisation

• Results for GaN/GaAsxN1-x

• Conclusions

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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.

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Cosine Band-structure approximation

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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

Cosine Band-structure approximation

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parabolic

k.pcosine

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]

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[1] – K. Tomizawa, Numerical Simulation of Submircon Semiconductor Devices, Artech House, London, 1993

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.

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Algorithm – Ensemble Monte Carlo code

• ~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)

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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.

Validation

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100000 1000000 100000000

50000

100000

150000

200000

250000

300000

GaAs - theoryGaAs - experiment - [1]InP - theoryInP - experiment - [2]GaN k.pGaN cosine

Applied Electric Field (kV/cm)[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

Aver

age

Elec

tron

Vel

ocit

y (x

107

cm/s

)

Results - GaN

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[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]

Results - GaN - Negative Effective Mass

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• 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

Results – GaN – Transient properties

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Aver

age

Ele

ctro

n Ve

loci

ty (x

107

cm/s

)

Applied Field (kV/cm)Time Elapsed (ps)

GaN with a 1.2eV valley separation using the cosine band-structure approximation

Results – GaN0.01As0.99

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[1] – D. R. Naylor, et. al, Submitted to Journal of Applied Physics

Results – GaN0.01As0.99

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[1] – D. R. Naylor, et. al, Submitted to Journal of Applied Physics

Conclusions

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

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.)

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