### Research

*This page focusses on my PhD research. This will be updated with my current work once I have more time!*

Monte Carlo simulations have been used for many years in the simulation of many semiconductors. For most materials, such as Silicon and Gallium Arsenide, the use of the** **popular parabolic and **k.p** band-structure approximations provide an excellent analytic approximations to the true band structure, and allow for relatively quick simulation run times. However, these approximations do not provide a satisfactory approximation to the band-structure of GaN around the Gamma point, missing vital features of the structure that have been postulated to cause negative effective mass transport in GaN, and ultimately, a retardation in average electron velocity at high fields. Therefore, to accurately simulate electron transport in GaN, full-band structure simulations have had to be used, which are computationally expensive to run due to the numerical integrations involved throughout the simulation.

The cosine band-structure, originally proposed by Ridley, Schaff and Eastman in 2005 [2] was suggested as an analytic model that would be able to accurately simulate the negative mass transport at higher energies. It takes the form

E = (E_{B}/2)(1 – cos(**k** a)) (1)

where E_{B} is the width of the band, which is 2.7eV, **k** is the electron wave-vector (in reciprocal space), and c is the hexagonal lattice constant along the c-axis which is 5.186Å. It turns out that this is an excellent approximation to the band-structure of GaN around the Gamma point, as can be seen in a paper by *Dyson* and *Ridley* [2, fig 1.] It can be seen that this band-structure approximation contains the negative-mass states by simply looking at the velocity-k expression for the approximation,

V** _{k}**=1/ℏ dE

**/d**

_{k}**k**= E

_{B}sin(

**k**a)/2(2)

It is easy to see that after the electrons attain an energy of E_{B}/4, the electron velocity will decrease as the energy increases, suggesting that the “negative mass” states are accounted for in some form through the use of this approximation.

My PhD project focused on GaN and the cosine band structure with the aim of creating a Monte Carlo code for GaN based devices. It is hoped that the use of the cosine band-structure approximation in Monte Carlo codes will be significantly faster than using a numerical full-band simulation, yet still provide accurate results. Using equation 1, some scattering rates using this new band-structure have been derived (including polar-optical phonon scattering, which has previously been derived [3]), and are currently being implemented into Monte Carlo codes designed to simulation electron transport in bulk GaN, and in future, devices. Preliminary results from the code for bulk GaN are promising, the current output of the bulk GaN code appears to be in excellent agreement of other full band models and also in close agreement with experimental data. We also used our simulation to model the dilute nitride Gallium Nitrogen Arsenide with favourable results.

The results of my PhD project can be seen in my thesis, available on the University of Hull Hydra Document service, and in two papers, Steady-state and transient electron transport in bulk GaN employing an analytic bandstructure and Steady state and transient electron transport properties of bulk dilute GaN_{x}As_{1−x}.

During my Postdoctoral contract, I looked at implementing “Hot Phonon” effects, that is, how the evolution of phonon occupation in systems affects the carrier transport. While I have now left my post, work is still ongoing at my old institution to analyze the data I was able to generate.

[1] A. Dyson, B.K. Ridley et. al, *physica status solidi (c).* **4, **pp. 528-530, 2007. Link

[2] B.K. Ridley, W.J.Schaff and L.F. Eastman, *J. Appl. Phys.* **97,** pp. 094503-094509, 2005. Link

[3] A. Dyson and B.K. Ridley, *J. Appl. Phys.* **104**, pp. 113701-113709, 2008. Link