(Invited) A New Analysis of the Dependence of Critical Electric Field on Semiconductor Bandgap
Authors: Kaplar Robert J. , Slobodyan Oleksiy , Flicker Jack D. , Hollis Mark A. 
Published: February 2020

Recently there has been much interest in wide- and ultra-wide-bandgap (WBG/UWBG) semiconductors for power conversion, radio-frequency, and other applications. The benefits of these materials for high-power devices ultimately stems from the increase in critical electric field (EC) with increasing bandgap (EG). Most researchers cite the work of Hudgins et al. [IEEE Trans Power Elec. 18, 907 (2003)] which reported that EC ~ EG
2.0 for indirect-gap materials and EC ~ EG
2.5 for direct-gap materials. These dependencies are based on empirical power-law fits to reported experimental data. The exponent in the power-law is critical when predicting the performance of power devices composed of new UWBG materials, since the Unipolar Figure-of-Merit (UFOM) scales as EC
3. This work has re-analyzed old data and also analyzed new data, and has found EC ~ EG
1.8 with no difference between direct- and indirect-gap materials. A theory that explains this dependence has also been derived and shows good agreement with the experimental data.
A key point is that the critical electric field is not a constant for a given material. Rather, EC depends on temperature and doping. Further, it is defined for a triangular field distribution, i.e. a non-punch-through (NPT) drift region, not a punch-through (PT) drift region with a trapezoidal field distribution. Careful examination of the literature revealed that critical electric field values have been reported for various doping levels, and that in general no distinction between values derived from PT and NPT structures has been made. Fortunately, in most cases values have been reported at room temperature, which is taken to be standard. As such, we have derived a procedure that allows EC values to be normalized to a fixed doping (1×1016 cm-3) and a NPT field distribution. The method equates the ionization integrals for the non-standard and standard configurations and backs out the values of EC normalized to the standard configuration.
Using this algorithm, normalized values of EC for a variety of semiconductors were tabulated and plotted against bandgap. These data are shown in the attached Figure, where a power-law fit yielding an exponent of 1.8 for both direct- and indirect-gap semiconductors is also shown. The conventional semiconductors Ge, Si, InP, GaAs, and GaP are all included. The narrow-gap semiconductors InSb and InAs are not included, as it is believed that breakdown in these materials may have substantial contributions from tunneling, in addition to impact ionization. GaSb is also excluded due to experimental uncertainty. The wide-bandgap semiconductors SiC and GaN are included in the analysis, although considerable debate still exists concerning the critical field of GaN. For the UWBG semiconductors such as Ga2O3, AlN, and diamond, even more uncertainty exists, and only the critical electric field of diamond has been included in the analysis, which is nevertheless subject to large uncertainty.
To explain the observed experimental data, a model based on the ionization integral has been derived. In this model, the ionization integral is transformed from an integral over space into an integral over electric field. The ionization rate as a function of field is then inserted into the integral. For this, the lucky-drift model of Ridley is used [J. Phys. C 16, 3373 (1983)]. This model is an extension of the more-commonly-cited lucky electron model first proposed by Shockley. In the extended model of Ridley, elastic momentum relaxation is considered in addition to energy relaxation. The model depends on three input parameters, which are the threshold energy required for ionization (Eth), the mean carrier relaxation length (l), and the ratio of the average energy lost per collision to the threshold energy (r). These parameters all depend on bandgap, with Eth ~ EG, l ~ EG
-3/2, and r ~ EG
-1. Inserting these dependencies into the ionization integral and numerically extracting the electric field at which avalanche occurs for a given bandgap yields a curve of EC versus EG, which closely matches the experimental EC ~ EG dependence.
This work was partially supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly-owned subsidiary of Honeywell International Inc, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA-0003525. This material is also based upon work supported by the Assistant Secretary of Defense for Research and Engineering under Air Force Contract No. FA8702-15-D-0001. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Assistant Secretary of Defense for Research and Engineering.

Figure 1