Deterministic Solution of the Boltzmann Equation for Device Reliability Investigations

The geometry of metal-oxide-semiconductor field-effect transistors (MOSFETs) continues to shrink with each technology generation. As the supply voltages cannot be reduced at the same rate without jeopardizing reliable device operation, the electric fields within the devices increase. As a result, charge carriers can gain high energies, which are sufficient to break atomic bonds or ionize neutral atoms. To study such device degradation effects, the carrier distribution with respect to momentum is of utmost interest, since it essentially determines the defect creation probability. Within this project we plan to develop efficient and accurate methods for the computation of the carrier distribution function in order to improve the performance and reliability of future semiconductor devices. Established macroscopic models for carrier transport in semiconductor devices only yield rough estimates of the carrier density function at each point in the device. To obtain the required level of detail, the seven-dimensional Boltzmann transport equation (BTE) has to be solved. Unfortunately, even with modern computers, the numerical solution of the BTE is very challenging. The most prominent solution technique is the stochastic Monte Carlo method. However, the stochastic nature of the method leads to excessive execution times when low probability regions of the distribution function have to be accurately resolved. Since these low probability regions describe the carriers with highest energy, the Monte Carlo method is not satisfactory for the study of hot carrier degradation effects in semiconductor devices. As an alternative, the BTE can be solved using deterministic solution methods. In this project we focus on the deterministic spherical harmonics expansion (SHE) method, which has a sound mathematical and physical basis and has already provided excellent results in one- and two-dimensional scenarios at much shorter simulation times than Monte Carlo methods. Still, a limiting factor of the SHE method is the huge memory requirement. We plan to apply state-of-the art discretization schemes to the SHE method and incorporate additional details from a physical point of view in order to obtain a solid foundation for further reliability studies. To tackle the high complexity of the BTE, the numerical solution methods and the physical models involved, the expertise of both the Institute for Microelectronics and the Institute for Analysis and Scientific Computing are required and will be joined in this project.

The ever-increasing demands on the performance of semiconductor devices require the design of faster, smaller, and more energy-efficient devices. As devices approach fundamental physical limits, an increased insight into the underlying physical processes is mandatory. Such improved understanding can be obtained in an expensive and time- consuming trial-and-error approach in laboratories, or through computer simulations with much shorter turnaround time and much smaller cost. A combination of both two approaches is vital for overall progress. Within this project, new algorithms for the numerical solution of the Boltzmann transport equation have been developed and implemented. Typically, the Boltzmann transport equation is considered to be too difficult to solve on a computer for realistic device geometries, hence simpler approaches based on moments of the Boltzmann transport equation are used in productive environments. Still, solutions of the Boltzmann transport equation are considered to be the golden standard of semiconductor device simulations for which certain quantum-mechanical effects are considered which play only a secondary role for overall device operation (semiclassical regime). With the results of this research project, solutions of the Boltzmann transport equation can now be computed at a fraction of the computational cost of other methods, most prominently the so-called Monte Carlo method. These numerical improvements were obtained in the 'numerics' module. The second part of this project, the physics module, focused on incorporating better physical models and to apply the new capabilities to modeling questions faced in real devices. Our results in the numerics module enable highly detailed simulations of complicated device geometries such as recent three-dimensional tri-gate transistors, which are now state-of-the- art in modern processors. To do so, we developed a discretization scheme able to deal with general unstructured grids in three spatial dimensions. Moreover, we developed adaptive schemes to dramatically reduce memory requirements, so that we can now run simulations requiring only tens of Gigabytes of main memory, whereas former techniques required hundreds of Gigabytes for the same simulation. The results in the physics module enable the consideration of not only how charge carriers interact (scatter) with the crystal lattice, but also with each other. Also, we developed a model for considering the generation and recombination of electron-hole pairs, which, although generally considered important, has not been demonstrated for direct solutions of the Boltzmann transport equation before.