1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 23Electron Transport EquationsIn this lecture you will learn:• Position dependent non-equilibrium distribution functions• The Liouville equation• The Boltzmann equation• Relaxation time approximation• Transport equationsWilliam Schockley(1910-1989)ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityNote on NotationIn this handout, unless states otherwise, we will assume a conduction band with a dispersion given by:kMkEkETc..212kMkv.1In the presence of an electric field: EerEkMkrErkEcTc..2,12where:EcEvEf2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityPosition Dependent Non-Equilibrium Distribution FunctionWe generalize the concept of non-equilibrium distribution functions to situations where electron distributions could also be a function of position (as is the case in almost all electronic/optoelectronic devices):The local electron density is obtained upon integration over k-space: trkf ,,trkfkdtrndd,,22,FBZLocal Equilibrium Distribution Function:Electrons at a given location are likely to reach thermal equilibrium among themselves much faster than with electrons in other locations. The local equilibrium distribution function is defined by a local Fermi-level in the following way: KTtrErkEofetrkf,,11,,with the condition that the local Fermi level must be chosen such that:trkfkdtrkfkdtrnddodd,,22,,22,FBZFBZtrkf ,,krECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityCase of No Scattering: Liouville EquationQuestion: How does the non-equilibrium distribution function behave in time in the absence of scattering?trkf ,,In time interval “t ” each electron would have moved in k-space according to the dynamical equation:Eedttkdttrkf ,,value momentum finalvalue momentum initialttktkConsider an initial non-equilibrium distribution 2d dimensions at time “t ”, as shownThere is also an applied electric field, as shownkrBut in the same time interval “t ” each electron would have moved in real-space according to the equation:tkvdttrdvalue position finalvalue position initialttrtr3ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityThe distribution at time “t+t ” must obey the equation:ttrtkfttttrttkf ,,,,This is because in time “t “ the electron with initial momentum and position would have gone over to the state with momentum and position tkttk trkf ,,ttrkf ,,krCase of No Scattering: Liouville EquationEedttkdtkvdttrdtrttr   trkftttrkftdttrdtrkftdttkdtrkftrkfttrkftttdttrdrtdttkdkfttrtkfttttrttkfrk,,,,.,,.,,,,,,,,,,,,ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityCase of No Scattering: Liouville Equationtrkf ,,ttrkf ,,krEedttkdtkvdttrdWe have: trkftttrkftdttrdtrkftdttkdtrkftrkfrk,,,,.,,.,,,,The above equation implies that the underlined term must be zero:Liouville equationDescribes the deterministic evolution of electron distribution in k-space and real-space0.,,.,,,,dttrdtrkfdttkdtrkfttrkfrk0.,,.,,,,kvtrkfdttkdtrkfttrkfrk4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBoltzmann Equation: Liouville Equation with ScatteringNo Scattering With ScatteringNow we have:Boltzmann’s equationDeterministic evolution Non-deterministic evolutiontrkf ,,ttrkf ,,rtrkf ,,ttrkf ,,rkktttrtkfttttrttkf  scattering to due changes ,,,,scattering to due changes .,,.,,,, kvtrkfdttkdtrkfttrkfrkECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBoltzmann Equation: Relaxation Time Approximationscattering to due changes .,,.,,,,kvtrkfdttkdtrkfttrkfrkLocal Equilibrium:• Scattering is local in space – i.e. electrons at one location do not scatter from impurities, defects, phonons, and other electrons that are present at another location • Scattering restores local equilibrium – i.e. it drives the distribution function at any location to the local equilibrium distribution function at that locationtrkftrkfo,,,,scattering to due changes Note that:trkfkdtrkfkdtrnodddd,,22,,22,FBZFBZtrkftrkfkvtrkfdttkdtrkfttrkfork,,,,.,,.,,,,Boltzmann equation in the relaxation time approximation5ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityTransport Equations: Continuity Equationtrkftrkfkvtrkfdttkdtrkfttrkfork,,,,.,,.,,,,Boltzmann equation can be manipulated to give simpler transport equationsIntegrate LHS and RHS over k-space, multiply by two, and