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A 2015 perspective on the nature of the steady-state and transient electron transport within the wurtzite phases of gallium nitride, aluminum nitride, indium nitride, and zinc oxide: a critical and retrospective review

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Abstract

Wide energy gap semiconductors are broadly recognized as promising materials for novel electronic and optoelectronic device applications. As informed device design requires a firm grasp of the material properties of the underlying electronic materials, the electron transport that occurs within the wide energy gap semiconductors has been the focus of considerable study over the years. In an effort to provide some perspective on this rapidly evolving and burgeoning field of research, we review analyzes of the electron transport within some wide energy gap semiconductors of current interest in this paper. In order to narrow the scope of this review, we will primarily focus on the electron transport that occurs within the wurtzite phases of gallium nitride, aluminum nitride, indium nitride, and zinc oxide in this review, these materials being of great current interest to the wide energy gap semiconductor community; indium nitride, while not a wide energy gap semiconductor in of itself, is included as it is often alloyed with other wide energy gap semiconductors, the resultant alloys being wide energy gap semiconductors themselves. The electron transport that occurs within zinc-blende gallium arsenide is also considered, albeit primarily for bench-marking purposes. Most of our discussion will focus on results obtained from our ensemble semi-classical three-valley Monte Carlo simulations of the electron transport within these materials, our results conforming with state-of-the-art wide energy gap semiconductor orthodoxy. A brief tutorial on the Monte Carlo electron transport simulation approach, this approach being used to generate the results presented herein, is also provided. Steady-state and transient electron transport results are presented. The evolution of the field, and a survey of the current literature, are also featured. We conclude our review by presenting some recent developments on the electron transport within these materials.

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Notes

  1. Yoder [1] defines a wide energy gap semiconductor as being that possessing an energy gap equal to 2.2 eV or wider.

  2. In principle, SiC can crystallize in the form of an infinite number of polytypes. Thus far, over 250 polytypes of SiC have actually been experimentally observed [39].

  3. The more common polytypes of SiC possess wide and indirect energy gaps that range between 2.2 and 3 eV [51, 64, 65]. SiC is also found to exhibit a high breakdown field [66, 67], an elevated thermal conductivity [68, 69], and favorable electron transport characteristics [70]. This constellation of material properties associated with the various polytypes of SiC, and the recognition of the device opportunities thus engendered, was, in large measure, responsible for igniting interest into this material in the first place.

  4. InN, while not a wide energy gap semiconductor in of itself, its room temperature energy gap only being around 0.7 eV [71], is often alloyed with the other III–V nitride semiconductors, and thus, is often considered an honorary member of the wide energy gap semiconductor family [72–75].

  5. Initial interest in the III–V nitride semiconductors focused on GaN, the wurtzite phase of this material exhibiting a wide and direct energy gap of around 3.39 eV [31]. Wurtzite GaN also exhibits a high breakdown field [103, 104], elevated thermal conductivity [105, 106], and superb electron transport characteristics [107–110]. These attributes make GaN ideally suited for both electronic and optoelectronic device applications [111–126].

  6. ZnO, while currently finding applications as a material for low-field thin-film transistor electron device structures [132] and as a potential material for transparent conducting electrodes [133], also possesses a direct energy gap [134, 135] with a magnitude that is very similar to that exhibited by GaN [136]. Thus, it might be expected that, with some further improvements in its material quality, ZnO may also be employed for some of the device roles currently implemented or envisaged for GaN.

  7. The wurtzite phases of GaN, AlN, InN, and ZnO are the most common forms of these materials, other crystalline forms also being available.

  8. All of the materials considered in this analysis possess direct energy gaps.

  9. Structurally, these reviews are quite similar, different materials and conditions being considered.

  10. This requires that the electron ensemble has settled on a new equilibrium state. By an equilibrium state, however, we are not necessarily referring to thermal equilibrium, thermal equilibrium only being achieved in the absence of an applied electric field.

  11. By electron drift velocity, we are referring to the average electron velocity, determined by statistically averaging over the entire electron ensemble.

  12. The results depicted in this review that are plotted as a function of the applied electric field strength are determined through an averaging process, over time and the electron ensemble, corresponding to each applied electric field strength selection. That is, for each selection of the applied electric field strength, a data point is determined, this data point corresponding to a time average over the ensemble of electrons for which the applied electric field strength has been held constant. The number of electrons in each simulation is such that the statistical fluctuations that occur, inherent to all Monte Carlo simulations, are sufficiently small that seemingly continuous plots arise, i.e., the Monte Carlo simulation points are sequentially connected in order to produce these plots. A similar approach is employed for the determination of the transient electron transport results, wherein a sequence of Monte Carlo simulation results are sequentially connected in order to produce these plots. The continuous nature of these plots arises as a consequence of the number of electrons employed in the simulation, i.e., sufficient in order to make the error sufficiently small. More on this matter is discussed in footnote 21.

  13. The Monte Carlo approach to simulating the electron transport within semiconductors has been employed by many authors. A Monte Carlo electron transport simulation resource, with source code included, may be found at https://nanohub.org/resources/moca. Further information about the Monte Carlo approach, beyond the electron transport context, may also be found at http://www.codeproject.com/Articles/767997/Parallelised-Monte-Carlo-Algorithms-sharp and http://www.codeproject.com/Articles/32654/Monte-Carlo-Simulation?q=Monte+Carlo+code.

  14. Albrecht et al. [135] generalize this relationship to include a second-order non-parabolicity coefficient that reduces to the traditional Kane model in the limit that this second-order non-parabolicity coefficient is set to zero. No dramatic impact on the results is observed.

  15. The longitudinal and transverse sound velocities are equal to

    $$\begin{aligned} \sqrt{\frac{C_{l}}{\rho }} \text{ and } \sqrt{\frac{C_{t}}{\rho }}, \end{aligned}$$

    respectively, where \(C_{l}\) and \(C_{t}\) denote the respective elastic constants and \(\rho \) represents the mass density.

  16. Piezoelectric scattering is treated using the well established zinc-blende scattering rates, and thus, a suitably transformed piezoelectric constant, \(\hbox {e}_{14}\), must be selected. This may be achieved through the transformation suggested by Bykhovski et al. [222, 223]. The \(\hbox {e}_{14}\) value selected for wurtzite GaN is that suggested by Chin et al. [147]. The \(\hbox {e}_{14}\) values selected for wurtzite InN and wurtzite ZnO are that corresponding to wurtzite GaN. AlN has a lager \(\hbox {e}_{14}\) value, as it has more pronounced piezoelectric properties; see O’Leary et al. [160] for a justification.

  17. All inter-valley deformation potentials are set to \(10^{9}\) eV/cm, following the approach of Gelmont et al. [146].

  18. We follow the approach of Bhapkar and Shur [151], and set the inter-valley phonon energies equal to the optical phonon energy, a relationship which holds approximately for the case of GaAs [224].

  19. Each conduction band structure is modeled as possessing three distinct “valleys,” each of the valley minima corresponding to a minima in the corresponding actual conduction band structure. These valleys are specified according to their locations in the band structures, the degeneracy of each valley, the effective mass of the electrons at each valley minimum, and the non-parabolicity coefficient corresponding to each valley.

  20. For the case of direct-gap semiconductors, the \(E_{o}\) energy gap corresponds with the regular energy gap, \(E_{g}\). For the case of indirect-gap semiconductors, the \(E_{o}\) energy gap exceeds \(E_{g}\). Adachi [15] refers to the \(E_{o}\) energy gap as the lowest direct-gap energy gap.

  21. The number of electrons employed in each simulation is determined through a consideration of the magnitude of the root-mean square error associated with the Monte Carlo simulation results. If the number of electrons is too small, then the root-mean square error, and the statistical fluctuations that arise as a consequence, become so large that the results obtained become difficult to discern from the background noise. Unfortunately, the use of large numbers of electrons dramatically increases the running time. In fact, in some cases, the computations demanded exceeded the capability of the hardware employed. For our purposes, the selection of the number of electrons to employ corresponds to a trade-off between the error, as determined by the root-mean square, and the running time. We find that for the steady-state electron transport simulations, the simulations are performed over longer periods of time, and thus, a smaller number of electrons are required for the error to be sufficiently low; for our specific case, three-thousand electrons is observed to work well for the steady-state electron transport simulations. In contrast, as the transient electron simulations occur over a shorter period of time, a larger number of electrons are required in order for the error to be sufficiently low; for our specific case, ten-thousand electrons is observed to work well for the transient electron transport simulations. Further details, related to this issue, are presented in the analysis of Jensen et al. [221].

  22. Intense interest into the material properties of the III–V nitride semiconductors, GaN, AlN, and InN, and into their corresponding device applications, began in earnest in the early 1990s [76].

  23. Interest in the material properties of ZnO began in earnest in the early-2000s [84].

  24. Interest in the material properties of ZnO was ignited later than that associated with GaN, primarily on account of material quality considerations, i.e., high-quality GaN was prepared earlier, and a lack of familiarity with means of effectively handling II–VI compound semiconductors, many GaN processing techniques being borrowed directly from the GaAs case.

  25. While every effort was made to provide a reasonable sampling of the electron transport literature corresponding to each material, some key references may have been neglected. We apologize to authors for these potential oversights.

  26. The results presented by O’Leary et al. [155] were built upon preliminary results presented previously, also by O’Leary et al. [232].

  27. In 1986, Tansley and Foley [279] measured the spectral dependence of the optical absorption coefficient associated with wurtzite InN and determined that the 300 K energy gap associated with this material is around 1.89 eV. This value became the defacto standard for the field until 2002, when Wu et al. [71] demonstrated, using higher quality forms of wurtzite InN, that the 300 K energy gap associated with this material is actually around 0.7 eV. Other experimental measurements confirmed the narrower energy gap value suggested by Wu et al. [71, 280, 281]. This revised value for the wurtzite InN energy gap is now widely accepted by the semiconductor materials community.

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Acknowledgments

Financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. The work performed at Rensselaer Polytechnic Institute was supported by the Army Research Laboratory under the auspices of the ARL MSME Alliance program.

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Authors and Affiliations

  1. School of Engineering, The University of British Columbia, 3333 University Way, Kelowna, BC, V1V 1V7, Canada

    Poppy Siddiqua, Walid A. Hadi & Stephen K. O’Leary

  2. Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY, 12180-3590, USA

    Michael S. Shur

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  1. Poppy Siddiqua
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  2. Walid A. Hadi
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  3. Michael S. Shur
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  4. Stephen K. O’Leary
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Correspondence to Stephen K. O’Leary.

Additional information

Many of the results presented herein, and portions of the text, are borrowed from our previous review articles, i.e., “Steady-state and transient electron transport within the III–V nitride semiconductors, GaN, AlN, and InN: a review,” which was published in the Journal of Materials Science: Materials in Electronics in 2006 [110] and “Steady-state and transient electron transport within the wide energy gap compound semiconductors gallium nitride and zinc oxide: an updated and critical review,” which was published in the Journal of Materials Science: Materials in Electronics in 2014 [204]. Copyright permission, for the figures, tables, and text, was obtained from Springer.

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Siddiqua, P., Hadi, W.A., Shur, M.S. et al. A 2015 perspective on the nature of the steady-state and transient electron transport within the wurtzite phases of gallium nitride, aluminum nitride, indium nitride, and zinc oxide: a critical and retrospective review. J Mater Sci: Mater Electron 26, 4475–4512 (2015). https://doi.org/10.1007/s10854-015-3055-7

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