Electron distribution function of the Boltzmann transport equation under high electric fields with optical phonon scattering

Abstract We address the Boltzmann transport equation for electrons in semiconductors subjected to high electric fields, with a focus on optical phonon scattering. By deriving a differential equation for the electron distribution function, we explore solutions corresponding to a density of states dependent on the electron kinetic energy raised to the power of p . For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:math> , we reproduce the well-established result that the average electron kinetic energy scales with the square of the electric field. However, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>7</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>8</mml:mn> </mml:mrow> </mml:mfrac> </mml:math> , we find that the average kinetic energy increases linearly with the electric field, a behavior that aligns more closely with Monte Carlo simulations in the high-field regime of silicon bulk. This new analytical approach provides deeper insight into the field dependence of the average electron kinetic energy. Additionally, we present simulation results that illustrate the evolution of the distribution function under different high-field conditions, where the average electron kinetic energy exhibits a linear dependence on the electric field.