A High-Throughput Approach to Determining Energy-Dependent Impact Ionization Probabilities

<b>Introduction</b><br/>Impact Ionization plays a critical role in a wide range of devices, including avalanche photodiodes, LED devices, power transistors, and MOSFETS. Carrier generation by Impact ionization is also used in the contacts of emerging wide bandgap optoelectronic devices [1] to efficiently inject carriers, mitigating the doping challenges associated with conventional wide bandgap material systems.<br/><br/>The impact ionization probability is a critical parameter in the search for novel wide-bandgap materials. However, current impact ionization models can be poorly predictive, often assuming a parabolic band dispersion relation or failing to take advantage of recent advances in ab-initio electronic structure calculations. On the other hand, advanced models require the calculation of the transition matrix elements which comes at an immense computational cost, precluding high-throughput studies for material screening and prediction.<br/><br/>For high-throughput material screening, there is a critical need for a fast and accurate model for the impact ionization threshold. In this study, we bridge this critical gap and develop a high-throughput computational method to calculate the impact ionization threshold as a function of the primary carrier momentum and from this, the energy-dependent probability of impact ionization for ultra-wide bandgap semiconductors.<br/><br/><b>Methodology</b><br/>The threshold for impact ionization was determined by minimizing the energy of the primary carrier subject to energy and momentum conservation. The band structures were acquired from the Materials Project database, enabling a high-throughput computational study of up to 76,000 materials. These band structures were computed via density functional theory (DFT) using the generalized gradient approximation (GGA). A scissor correction was first applied to the Materials Project band-structure data, utilizing experimental bandgaps where available, and by linear regression of the GGA error elsewhere. The band-structures were then interpolated in k-space, applying the symmetry operations of the associated Brillouin zone (BZ), before implementing interpolation by radial basis functions (RBF), using a basis set of cubic splines. The minimization problem comprises nine continuous variables (the k-point coordinates for each of the non-primary carriers) and three discrete variables; their band indices. Minimization was achieved through brute force over the band combinations and by basin-hopping with sequential least squares programming (SLSQP) for the continuous variables. The impact ionization threshold was computed for primary carriers along high symmetry k-space paths and additionally for some number of random primary momenta (k-points) within the first BZ for k-space interpolation. An energy-dependent impact ionization density of states was computed by integrating the states where the threshold is less than the given carrier energy over the constant carrier energy surface of the band structure. Dividing this by the total DOS gives the energy-dependent impact ionization probability.<br/><br/><b>Results</b><br/>In this work, we developed a new model to rapidly calculate the impact ionization threshold of wide bandgap semiconductors based on DFT-calculated band structures. We validate the impact ionization thresholds against measurements in the literature showing that the impact ionization probability is a reliable estimator for the impact ionization scattering rate for high-throughput material screening and prediction. We then use our model to calculate impact ionization probability curves for several conventional ultrawide bandgap material systems including GaN, AlN, β-Ga2O3, and MgO, and show a range of interesting asymmetry in electron/hole impact ionization rates that could be exploited to make more efficient optoelectronic devices.<br/><br/>[1] Impact Ionization Light-emitting Diodes, by N. Krause. (2021, Feb. 18). <i>US20210050474A. </i>[Online]. Available: https://patents.google.com/patent/US20210050474A1/en?oq=20210050474