Thermionic energy conversion, a process that allows direct transformation of thermal to electrical energy, presents a means of efficient electrical power generation as the hot and cold side of the corresponding heat engine are separated by a vacuum gap. Conversion efficiencies approaching those of the Carnot cycle are possible if material parameters of the active elements at the converter, i.e., electron emitter or cathode and collector or anode, are optimized for operation in the desired temperature range.
These parameters can be defined through the law of Richardson–Dushman that quantifies the ability of a material to release an electron current at a certain temperature as a function of the emission barrier or work function and the emission or Richardson constant. Engineering materials to defined parameter values presents the key challenge in constructing practical thermionic converters. The elevated temperature regime of operation presents a constraint that eliminates most semiconductors and identifies diamond, a wide band-gap semiconductor, as a suitable thermionic material through its unique material properties. For its surface, a configuration can be established, the negative electron affinity, that shifts the vacuum level below the conduction band minimum eliminating the surface barrier for electron emission.
In addition, its ability to accept impurities as donor states allows materials engineering to control the work function and the emission constant. Single-crystal diamond electrodes with nitrogen levels at 1.7 eV and phosphorus levels at 0.6 eV were prepared by plasma-enhanced chemical vapor deposition where the work function was controlled from 2.88 to 0.67 eV, one of the lowest thermionic work functions reported. This work function range was achieved through control of the doping concentration where a relation to the amount of band bending emerged. Upward band bending that contributed to the work function was attributed to surface states where lower doped homoepitaxial films exhibited a surface state density of ∼3 × 10[superscript 11] cm[superscript −2]. With these optimized doped diamond electrodes, highly efficient thermionic converters are feasible with a Schottky barrier at the diamond collector contact mitigated through operation at elevated temperatures.
We report on a new numerical approach for multi-band drift within the context of full band Monte Carlo (FBMC) simulation and apply this to Si and InAs nanowires. The approach is based on the solution of the Krieger and Iafrate (KI) equations [J. B. Krieger and G. J. Iafrate, Phys. Rev. B 33, 5494 (1986)], which gives the probability of carriers undergoing interband transitions subject to an applied electric field. The KI equations are based on the solution of the time-dependent Schrödinger equation, and previous solutions of these equations have used Runge-Kutta (RK) methods to numerically solve the KI equations. This approach made the solution of the KI equations numerically expensive and was therefore only applied to a small part of the Brillouin zone (BZ). Here we discuss an alternate approach to the solution of the KI equations using the Magnus expansion (also known as “exponential perturbation theory”). This method is more accurate than the RK method as the solution lies on the exponential map and shares important qualitative properties with the exact solution such as the preservation of the unitary character of the time evolution operator. The solution of the KI equations is then incorporated through a modified FBMC free-flight drift routine and applied throughout the nanowire BZ. The importance of the multi-band drift model is then demonstrated for the case of Si and InAs nanowires by simulating a uniform field FBMC and analyzing the average carrier energies and carrier populations under high electric fields. Numerical simulations show that the average energy of the carriers under high electric field is significantly higher when multi-band drift is taken into consideration, due to the interband transitions allowing carriers to achieve higher energies.