A Hydrodynamical Model for Carriers and Phonons With Generation-Recombination , Including Auger Effect

The asymptotic procedure proposed allows to derive closed hydrodynamical equations from the kinetic equations of carriers and phonons (treated as a partecipating species) in a photon background. The direct generation-recombination processes are accounted for. The fluid-dynamical equations constructed for the chemical potentials of carriers, temperature, and drift velocity, are related to the extended thermodynamical (ET) ones for the chemical potentials of carriers, temperature, and drift velocity. In the drift-diffusion approximation the constitutive laws are derived and the Onsager relation recovered.


Introduction
In bipolar devices an interacting population of positively charged carriers (holes) must be taken into account, besides electrons and phonons.The hole-phonon interactions, similarly to electron-phonon, are emission/absobption phenomena.
Several generation-recombination (GR) events occur in semiconductors.In the presence of a photon background the most important ones are the radiative GR events.In the direct GR events a photon (pt) interact with a valence band electron and a couple electron-hole (e-h)is created: If the intensity of the photon field is low we must consider the Auger effect [1], which consists of two different processes and their inverse (a) electron capture: an electron fills an hole.The resulting energy is absorbed by another electron , and vice versa e + e + h e * , where * means "more energetic".(b) hole capture: an hole combine with an electron.The resulting energy is absorbed by another hole, and vice versa h + h + e h * We start with the Bloch-Boltzmann-Peierls coupled kinetic equations for the distribution functions of carriers and phonons.A small parameter is introduced to account for the umklapp processes and both the interaction kernels and the distribution functions are expanded with respect to .The lowest order equations show that the drifted maxwellian approximation is justified.
A hydrodynamical model, whose equations are similar to the ET ones [2] , is then constructed for the temperature T and the drift velocity V of the system, in addition to the chemical potentials µ e , µ h of electrons and holes.Here such a model is not based on the Maximum Entropy Principle, like in ET, but strictly on kinetic theory.
The calculation of the source terms due to GR events takes advantage from the smallness of the GR collision frequencies In the drift-diffusion approximation the constitutive laws are derived and the Onsager symmetry relationships verified.
We stress that in the present model (1) The displaced Maxwellians approximation is not an ad hoc assumption but is justified by the expansion we apply (2) Phonons are treated as a partecipating species , which brings energy and momentum (3) The correct Phonon-phonon, carrier-phonon, and GR interaction kernels are utilized: we avoid the use of the relaxation time approximation.
The most qualifying point is (2).In fact, the usual assumption that the phonon field can be treated as a fixed background is dropped here, since "any thermal gradient give rise to transport of heat by the phonons, whilst an electric current, thought carried by electrons, cannot fail to transfer some of its momentum to the lattice vibrations, and drag them along with it" [3].The present model can be seen as a generalization of a previous one [4] (with generation-recombination, more mathematical oriented), by means of the explicit treatment of phonons.

Kinetic Equations
Consider three interacting populations: electrons (e), holes (h), and phonons (pn where Observe that, being At the right hand sides of the equations for phonons we have where (b is an appropriate vector belonging to the reciprocal lattice), which account for three-phonon processes: , is the difference between the number of phonons k emitted by electrons with any quasimomenta p and the number of phonons absorbed by electrons with any p , where b is a vector of the reciprocal lattice appropriate to the present interaction.
For carriers we have The first term corresponds to to processes with emission of a phonon having quasimomentum k by an electron having a given quasimomentum p and reverse processes.The second term corresponds to processes with absorption of a phonon by an electron with quasimomentum p and reverse processes.The w's are transition probabilities which account for energy conservation and satisfy the following symmetry relations: Consider now the carrier-phonon system in contact with a photon medium.Let N pt be a Planck's distribution function at the temperature T pt : where ω pt = ck pt (c is the speed of light).The collision integrals for the GR interactions are given by where γ = h, e.
The transition probabilities account for energy conservation and satisfy the following symmetry relations: The Auger GR contributions can be written [1] as where where W e and U h are the Waldmann kernels of Auger processes, for electrons and holes, respectively.We consider now a system, exact but not closed, of four balance equations, to be utilized later.By projecting the equation for particles α over 1 we have : where the carrier number density (N α ) and the electric currents J α are given by , where we separated the radiative (R) and Auger (A) effects.By projecting the equations for particles α over 2p and the phonon ones on k, summation gives the following balance equation for momentum: where Finally, by projecting the equations for particles α over 2E α and the phonon ones over ω g pn , summation gives the following balance equation for energy : where the energy density W and the energy flux F W are given by and due to radiative GR events are given by Moreover, the contribution of Auger effect reads S A = (∂f e /∂t) A (p e + p h )dp e ,

Asymptotic Expansion and Hydrodynamical Equations
By following ref.[5], we expand both the interaction kernels and the distribution function with respect to a small parameter which accounts for umklapp (U) processes (which do not conserve momentum) in addition to normal (N) processes (which conserve momentum).
We start with carriers (the extension to phonons is trivial).The singular expansion for w αpn reads The sought expansions for n α and N g pn read According to [3], at low temperatures decreases exponentially as exp(−CT D /T ), where T is a characteristic temperature of the system, T D is the Debye temperature, and C is of the order of unity.Therefore we can say that the present expansion is valid for T << T D .
We can write now ∂n α ∂t αpn = ∂n α ∂t By taking into account momentum (it is a N − process) and energy conservation, the equations of order −1 are solved (see Appendix) by where that is the drifted Fermi-Dirac (FD) and Bose-Einstein (BE) distribution functions.The meaning of V is simple.Let vα the most probable velocity of carriers α, given by It is easily seen that vα = V.Analogously, for phonons, ûg pn = V.The distribution functions are usually expanded as follows This simplification is valid when the drift energy is small compared to thermal energy.Under this assumption, after some calculations we find and Starting from the equations of order 0 , a hydrodynamical model can be constructed now, related to ET one [2], for the temperature T and the drift velocity V of the system, in addition to the chemical potentials µ e , µ h .
By projecting the equation for carrier α over 1, the balance equation for particles α reads Hereinafter the subscript N in the source terms means that in their definition we utilize n N α for the integration.By projecting the equations for carriers over 2p and the phonon ones on k, summation gives the following balance equation for the total momentum: where we took advantage of due to momentum conservation for N-processes.Finally, by projecting the carrier equations over 2E α and the phonon ones over ω g pn , summation gives the following energy balance equation:

Source Terms
The source terms are small quantities since the relaxation time τ RG of the RG processes is much larger than the one (τ αpn ) of the α − pn interactions [6].Hence we shall utilize for their calculation an approximation which properly accounts for this smallness.
The equations of order 0 can be written, after a suitable adimensionalization, as follows which, due to (1,2,3), shows that in the present approximation V = 0, so that Q GR 0 , Q GR 1 , are simply approximated by setting n α = n 0 α and N g pn = N 0 g .Moreover, symmetry arguments lead to S GR = 0.In the low density approximation [6] (1 − n α 1) we can write where N α is the number density of particles α and Observe that Q 0 and Q 1 depend linearly on N e N h , while where are cubic with respect the number densities.

Revised Drift-Diffusion Approximation and Constitutive Laws
In the drift-diffusion approximation [7,8] (and, in particular for bipolar devices, [9]), we assume that the total momentum of the mixture does not vary appreciably over the momentum relaxation time.From the momentum balance equation we get The tensor IB, can be written in the following symmetric form By utilizing the drifted FD or BE distribution functions, the electrical (J α ) and thermal (U α , U p ) currents are given by where IK lm = IR l • IB −1 • IR m (∼ means transpose).Since IB = IB, the following Onsager symmetry relation is in order: IK lm = IK ml .
The cross effects of E h on J e and of E e on J h in a drift-diffusion model are discussed in [10].Moreover, we can calculate the energy flux by its very definition: The system of the drift-diffusion equations is obtained by inserting J e and J h into the carrier balance equations and F W into the energy balance one.