Comparison of Non-Parabolic Hydrodynamic Models Based On Different Band Structure Models

This paper presents two non-parabolic hydrodynamic model formulations suitable for the simulation of inhomogeneous semiconductor devices. The first formulation uses the Kane dispersion relationship, (hk)2/2m W( +aW). The second formulation makes use of a power law, (hk)2/2m xWY, for the dispersion relation. The non-parabolicity and energy range of the hydrodynamic model based on the Kane dispersion relation is limited. The power law formulation produces closed form coefficients similar to those under the parabolic band approximation but the carrier concentration can deviate. An extended power law dispersion relation is proposed to account for band structure effects, (hk)2/2m xWl+yw. This dispersion relation closely matches the calculated band structure over a wide energy range and may lead to closed form coefficients for the hydrodynamic model.


INTRODUCTION
Current hydrodynamic models consist of a set of con- servation equations derived by taking moments of the Boltzmann transport equation.During the derivation of the conservation equations the parabolic band approximation is used to obtain rather simple coeffi- cients on the forcing terms in the flux equations.By relying on the parabolic band approximation higher order energy transport effects due to variations in the band structure are neglected.Accounting for band structure effects in hydrodynamic device simulation is important because parabolic models can not ade- quately account for high energy effects in semicon- ductors with non-parabolic band structures.
Non-parabolic hydrodynamic models have been reported for homogeneous material systems [1][2][3][4] using the Kane dispersion relationship [5].The gen- eral functional form obtained is similar to parabolic * Now with lIT Electro-Optical Products Division, Roanoke, VA hydrodynamic models with first order corrections on the diffusion term.However, the non-parabolic coeffi- cient in the field term and the forcing terms due to non-uniform band structure are neglected in the moment equations.Cassi and Ricc6 [6] introduced an alternative to the Kane relation in the form of a power law for the dispersion relationship.Instead of using a classical Kane dispersion law relating the energy and momentum, the band was fit over a specified energy range using two adjustable parameters.The approxi- mations and assumptions implied by assuming the power law formulation were absent.It was shown in [7] that the power law dispersion relation leads to a more simplistic and compact formulation than the classical Kane expression.A third non-parabolic dis- persion relation is also proposed.It is shown that this new relation more closely matches the k*P or pseudopotential band structure calculations over a very wide energy range in several material systems.

DISPERSION RELATIONS, CONCENTRATIONS, FLUX EQUATIONS
The dispersion relations considered in the derivation of the hydrodynamic conservation equations are; parabolic, Kane dispersion, power law, and extended power law 2me 2me where c is the non-parabolicity factor and x, y are fit- ting parameters over a specified energy range.If the power law is fit over the energy range 1.5 < W < 3.0 eV as suggested in [6] the deviation in carrier concentra- tion from the parabolic case and the Kane formulation is greater than 80% at most reduced energy values.However, when fit over the energy range 0 < W < 0.2 eV the deviation is only 2% [7].This is due to the distribution function weighing more heav- ily to the lower energy.Therefore, to accurately account for the carrier concentration the dispersion relation must match at lower energies and to simulate high energy effects it must match the band structure at higher energies.The insert in Figures 1-3 show the fit at lower ener- gies and displays that the extended power law is a good fit.In all cases the extended power law relation also provides a very good fit up to very high energy ranges, note that both of the other non-parabolic for- mulations are poor fits at the higher energy ranges.
Preliminary derivations indicate the extended power law relation may produce closed form coefficients for specific distribution functions.Note that there is a substantial difference in the definition of the y term in the power law case as compared to the extended power law formulation.In the power law case y approaches as non-parabolicity is decreased, on the other hand y approaches 0 as non-parabolicity is decreased in the extended power law formulation.

FLUX CONSERVATION EQUATIONS
The derivation and comparison of models based on the Kane formulation and power law are given in ref- erences [7][8].Similarly the electron energy flux equations using these dispersion relations can also be formulated [7].In reference [7] it was shown that as y approaches 2 in the power law formulation the field contribution to current is diminished.If the field pref- shows the low energy range fit actor is not included, as in previous models, then the flux would be over estimated.The resulting flux equations for the dispersion relations in equation ( 1) Energy (oV) FIGURE 3 Comparison of the three non-parabolic dispersion rela- tions to the k*P band structure calculations for GaN, the material parameters are listed in the material table.The insert shows the low energy range fit V( fI/VI-Y (1 + yW (1 + ln(W))) Vme + -I- me x fw'l-y 3meX(1 + yW (1 + In(W))) )- I 1 4(1-yW) ] (1 +yW(1 + ln(W))) 2 mex(1 + yW (1 + In(W))) (2(1-yW)-I)Vec (1 +yW( + n(W))) me( +yW( + n(W))) 3 41n(W)(1-yW) 3( +yW( + n(W))) +ln(W) (1 +yW(1 +ln(W))) + 5 (1 + ln(W)) (f fo) Vy- (5) Each of these equations must be integrated over all k space or equivalently over energy after some form of the distribution function or a closing relation for the distribution function is assumed.In reference [8] the distribution function was assumed to be a heated Fermi-Dirac to provide a valid basis of comparison between the different models simulated.
3. EXTENDED POWER LAW DISPERSIONCurrent work focuses on the development of a hydrodynamic formulation which uses an extended power law dispersion relation, (hk)2/2m xW I+yw.Figures1, 2, and 3 display comparison of this new formula- tion to band structure calculations and to the other non-parabolic formulations for S i, GaAs and GaN.The following values and fitting parameters were used in the comparison of the dispersion relations.
Comparison of the three non-parabolic dispersion relations to the k*P band structure calculations for Si, the material parameters are listed in the material table.The insert shows the Comparison of the three non-parabolic dispersion relations to the k*P band structure calculations for GaAs, the material parameters are listed in the material table.The insert FIGURE