Modelling Impact-Ionization in the Framework of the Spherical-Harmonics Expansion of the Boltzmann Transport Equation with Full-Band Structure Effects

IJalld-st.ruct urc, ollect s ha.ve been incorporated in the framework of {.he SphericalIIarnlol~ics k;xpansion ( S H E ) of the Boltz~nann Transport Equat.ion ( B T E ) for ~lcctrons ill silicon [ I ] , using t.he densit,y of st.at.es ( D O S ) and the group velocity ( C V ) obt.ained from t,he full-band syst.em [2]. In t,his paper an impactionizat.ion model is present,ed along wit,h the numerical results. The model is consistent wit.11 t,he full-band system mentioned above and is a.ble t,o fit the irrlpac1.-iooizatio~r coefficient., t.he impact-ionization quant,urr~ yield, and t,he dat,a from soft, x-ray pllot,oerrlission spettroscopy available i l l recent, 1iterat.ure (e.g., 131 1. 1. Physical nlot l t .1 The SH b; 01' t .11~ H'I'E: has becr~ t.ested successfully in a wide range of problerns in the field of c l c c t r o ~ ~ transpori sirr~ulat.ion [$, 51. T h e main advantage of this rnethod is t,he large tlynar~ric rallgr of i1.s det.errninistic solution and the abilit,y of predicting t,he electron ~1ist.ribrlt.io11, i r l bot.ll t.hc spal.ia1ly honrogcneous and non-honiogcneous cases, without t.ht, hra.vy c.o~~ll)ut.at ionill bi~rdcn t.ypical of stochast,ic metl-~ods. Full-band st,ructure c,ffc~.ts il.rct i ~ ~ ( . o r l ) ~ ) r i ~ l t ~ t l t,hrol~gh t.he DOS and GV independent,ly calculated from the f ~ ~ l l l ~ n ~ ~ t l s y s t c ~ ~ ~ [2] I,y rllt-ans of il, suit,a.ble averaging procedure. The framework of t IIP S I I I; ~ric~t.hod in stt:a.tly st.ai,e provides the different,ia.l equation [4] Thr \vurbol\ t~accx the following rncaning: g ( E ) is the DOS, u, (E) the modulus of t Ile c : ~ . T the tot a1 4cattcling ratc, F the electric field, cop a constant proportional M. C. Vecchi et al.: Modelling Impact-Ionization in the Framework 417 to the optical-phonon coupling constant, No, the optical-phonon occupation number, N,f , = N,, + I , g*(E) = g ( E & hw,,), where hw,, is the optical-phonon energy, and sirilililrly [or f : (E) . Impact ionization is also considered: ciig(E) is the t,ot.al in~pact,-ionizat,ion scatt,ering rate and A(E1, E) is a suitable kernel [4] . The non-linear opt.iniization code PROFILE [6] has been used to obtain the best set of scattering pa.ramet,ers by fitting suitable average quantities (mean velocity, energy, impact-ioniza.t.ion coefficient) provided by the Monte Carlo code DAMOCLES [2] and experimental rr~easurernents in spatially-homogeneous conditions. The fitting procedure based on t.he full-band structure, but still using the impact-ionization model of [4], provitles t.he impact,-ionization scattering rate shown in Fig. 1. It is seen that the adoption of a full-bai~tl s1,ructure brings the result of SHE closer to that of Monte Carlo analysis, also showri in the figure. Although the agreement between SHE and Monte Carlo dat,a of Fig. 1 is fair. it, can considerably be irnproved by a sounder description of the impa.ct.-ionization mechanism, as shown below. 2. Impact-Ioilization model -4 three-th~esllold ~ r ~ o d c l is worked out. In order to avoid the simulation of the electrons in tbc kalerice band, the latter is assumed flat and full of electrons. The 5cat terlng illat tix, deliked in the Born approximation [ i ] , is: ' 3 ~ i ; ( ( k , kt, k") = x ~ : ( k , kt, k") = j = 1 7 = C bj? [a: $ (k' k ) 2 ] 2 S ( E El E" EGj ) , (2) j =1 where EGJ is t,hc ionization threshold, a, the inverse screening length, and bi? a normalizing constant,. The values of the parameters have been determined in order to reproduce t.he sca.tt.ering rat,e presented in [3 ] . Such scattering rate, in turn, is consist,ent wit,h the experimental data of [ S ] and [9 ] . The scattering rate of [3] is shown in Fig. 2 along wit.h t,he scat,tering rate obtained by SHE using 2, while the values of t,he adopted paranlrt,ers a.re reported in Table I. 1 Table I: impact-ionization model parameters It is worth adding that this calculation dealt only with impact ionization, namely, the other parameters mentioned in the previous section have been left unchanged. The impact-ionization coefficient is shown in Fig. 3 and compared with experimental da ta 110, I I ] in a large interval of electric fields: the good agreement in the low-field region 41 8 M. C. Vecchi et al.: Modelling Impact-Ionization in the Framework is related t.o Lhe presence of a soft threshold in (2). Fig. 4 shows the effect of the impa.ct-ionization model on the electron energy-distribution function at 200 kV/cm: the high-energy tail conipui.ed with the new model (2) is a few orders of magnitude lower than t.lle on<) obl,ained wit,h t,he old model, due to the higher scattering rate provided by (.he new model at high energies (compare Figs. 1 and 2). These results emphasize t,he importance of a correct description of the band structure and impact ionization especially in the analysis of carrier transport at high energies. On the other hand, t,heg a.lso show the ability of the SHE scheme to efficiently incorporate the feat,ures 01 the transport mechanisms to a rather general extent, and reproduce the result,s of sta.t#e-of-t,he-art, st,ochastic methods.