Monte Carlo Modeling of Heat Generation in Electronic Nanostructures

This work develops a Monte Carlo (MC) simulation method for calculating the heat generation rate in electronic nanostructures. Electrons accelerated by the electric field scatter strongly with optical phonons, yet heat transport in silicon occurs via the faster acoustic modes. The MC method incorporates the appropriate energy transfer rates from electrons to each phonon branch. This accounts for the non-equilibrium energy exchange between the electrons and phonon branches. Using the MC method with an electron energy-dependent scattering rate intrinsically accounts for the non-locality of the heat transfer near a strongly peaked electric field. This approach provides more information about electronically generated heat at nanoscale dimensions compared to traditional macroscopic field-dependent methods. The method has applications in any region of high spatial or temporal non-equilibrium between electrons and phonons, and particularly facilitates careful microscopic analysis of heating in a nanoscale transistor. NOMENCLATURE Λ phonon mean free path Cs heat capacity per unit volume ks thermal conductivity T lattice temperature Q′′′ heat generation rate per unit volume τph phonon-phonon scattering time u phonon energy density per unit volume and unit solid angle uo equilibrium phonon energy density v phonon velocity J electron current density E electric field LA longitudinal acoustic TA transverse acoustic LO longitudinal optical TO transverse optical h̄ Planck’s constant divided by 2π k electron wave vector Γ(k) electron-phonon scattering rate M(k) electron-phonon matrix element Ek electron energy g(Ek) electron density of states ωq phonon frequency m∗ electron effective mass p′ electron momentum after a scattering event