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A numerical method to compute isotropic band models from anisotropic semiconductor band structures

A numerical method to compute isotropic band models from anisotropic semiconductor band structures
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  IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 12, NO. 9, SEPTEMBER 1993 zyx 327 A Numerical Method to Compute Isotropic Band Models from Anisotropic Semiconductor Band Structures Antonio Abramo, Franco Venturi, Enrico Sangiorgi, zyxwvut ember, IEEE, Jack M. Higman, and Bruno Riccb Abstract-A numerical method for the determination of iso- tropic band models has been developed and applied to silicon. The resulting model accurately approximates both density of states and group velocity of the corresponding anisotropic band structure, thus providing an excellent agreement to both the collision and non-homogeneous terms of the Boltzmann trans- port equation. The model, represented through a simple set of energy-wave vector tables, has been implemented into a Monte Carlo device simulator, but can also be extended to alternative methods for solving Boltzmann equation. Simulation of homo- geneous silicon shows a very good agreement with available ex- perimental data. Comparison with results obtained using the complete anisotropic band structure, both in homogeneous and non-homogeneous silicon devices, confirms the validity of the model. I. INTRODUCTION HE Monte Carlo (MC) method is widely recognized T s the most accurate tool to simulate submicrometer devices and, for this reason, a continuous effort is devoted to the development of suitable MC microscopic models achieving a good trade-off between accuracy and com- putational efficiency. As for the former aspect, models based on a single band [l] are no longer adequate to de- scribe high-field phenomena, in which hot carriers play a significant role; conversely, the use of the complete an- isotropic semiconductor band structure (ASBS) zyxwvut 2] [4] requires a huge CPU time, due eyentially to the an- isotropic dispersion relationship zyxwvutsrq ( k). In this context, a good alternative is provided by multiband isotropic models [5]-[7] which accurately describe transport phenomena in semiconductors, are significantly less CPU-time consum- ing, and can be easily implemented in alternative methods to solve Boltzmann transport equation [8]-[lo]. The models of [5], [6] are given in terms of analytical electron and hole bands that fit the ASBS density of states (DOS) as a function of energy. The DOS in fact, con- Manuscript received June 29, 1992; revised March zyxwvutsrq   1993. This paper was recommended by D. Scharfetter. A. Abramo and E. Sangiorgi are with the Department of Electronics, University of Bologna, Italy. F. Venturi is with the Department of Information Technology, Univer- sity of Parma, Italy. J. M. Higman was with the Beckman Institute, University of Illinois at Urbana-Champaign. He is now with Advanced Products Research and De- velopment Laboratory, Motorola, Inc., Austin. TX 78721. IEEE Log Number 9208765. tains integr ted information over the whole band structure and, in particular, the electron-phonon scattering rate at a given energy is proportional to the DOS of the final state. Therefore, the fitting of the DOS ensures that the simpli- fied models feature the same ver ge electron-phonon scattering rate of the more complex ASBS. The fitting of the DOS in itself, however, does not ensure that carrier dynamics, which depends on carrier velocity, is properly treated in the simplified model. In add ion, in the ASBS the carrier groupyelocity (1 h) V;E( k ) is a function of the wave vector k, while for any isotropic band-model it depends on the magnitude of the wave vector 11 k 11 only, thus on the energy E. This fact leads to the conclusion that a comparison between the carrier velocity of the ASBS and that of isotropic models can be done only through suitable averages. In this framework, this paper advances the state of the art, presenting a new isotropic band model for bipolar transport in silicon conceived to best fit the behavior in energy of both density of states and a mean group velocity calculated from the ASBS. The E(k) rela- tionship is given in terms of numerical isotropic bands, one for each of the main symmetry points in the conduc- tion and valence band. The band model is consistently used in the evaluation of the scattering probabilities, in- cluding electron-phonon and electron-impurity interac- tion. The reduction from the three-dimensional momen- tum space of the ASBS to the one-dimensional energy space lowers the memory and CPU time requirements to the level of the single band models. Extensive comparisons with results obtained using the complete ASBS demonstrate the validity of this approach. This paper is organized in the following way: in Sec- tion I1 the physical background of the problem is sketched; Section I11 discusses the validity of the approach by means of comparison with simulations performed using the ASBS model; in Section IV the numerical procedure for obtain- ing the isotropic model is described; the simulation results obtained with the new model are given in Section V; and finally, conclusions are drawn in Section VI. 11. PHYSICAL ACKGROUND This section introduces preliminary physical consider- ations as a theoretical basis of our work. Section I11 in- stead shows the results of some tests that have been per- 0278-0070/93 03.00 @ 1993 IEEE  formed in order to validate these theoretical assumptions. Therefore, all results shown in Figs. 1-6 refer to the ASBS and not to the isotropic model discussed in this paper. As already discussed in zyxwvutsrq 5], he zyxwvutsr OS(E) contains zyxwvu n- tegrated information over the ASBS. In addition, the elec- tron-phonon scattering rate is proportional to the DOS (E). Although the information contained in the DOS is neces- sary to correctly treat transport in semiconductors, carrier dynamics also plays a fundaqental role. In particular, the carrier group velocity zyxwvutsrq g k) is defined as the vector (1 /A)ViE( zyxwvutsrq   nd, therefore, carries the information about the detailecj shape of the band structure. Notice that, in general, Gg k) s not a function of energy only, and thus electrons at the same energy can experience large differences in velocity due to the anisotropicity of the band structure. Conversely, the group velocity in2n isotropic band model is a function of the magnitude of k and, there- fore, of energy only. Hence, an isotropic band model re- lated to the ASBS group velocity should feature, at each energy, a group velocity u,(E) corresponding to a suit- able average of all the group velocities of the ASBS at the same energy. Following this approach, we chose to define u,(E) as the magnitude of the group velocity Jg(k) averaged over surfaces+ of constant energy. In particular, we weighted all the k states contributing to u,(E) by their DOS, thus obtaining: where: Therefore, an isotropic model, reproducing both DOS(E) and u,(E), will feature the same average elec- tron-phonon scattering rate and group velocity that the carriers would have in the ASBS if they were uniformly distributed over each equi-energy surface. The leading idea was that the consistent treatment of carrier scattering and dynamics should lead to excellent approximation of both the+ collision and the non-horflogeneous terms ((df(?, k, t)/dt),,,, and Gg V,f( 3, k, t of the Boltz- mann transport equation, that depend on the semiconduc- tor DOS and carrier gFup velocity, respectively. From the ASBS E( k) data computed on a regular grid (spaced 1 /20 of ?r/a,, a, being the lattice constant) using the well-known empirical pseudo-potential method zyxwvut   1 11, the two functions of interest (namely DOS(E) and zyxwvut g E)) were obtained following the method of [12]. For pur- poses peculiar to the derivation of the present model (see Section IV for details), the irreducible wedge of the first Brillouin zone (1 48 of the whole zone) was divided into three parts Si, = 1, 3), each centered on a selected direction Fig. 1. ASBS along principal directions: different line types emphasize the different regions around the following principal points: (a) conduction band: 0.85X dotted); r (dashed); L (solid). (b) valence band: X (dotted); r (dashed); L (solid). ici nd made up of all the nearest_ ’ vectors (usually known as Voronoi polyhedra). The k,; vectors were cho- sen at the local energy minima (maxima) of the conduc- tion (valence) band: zyxw .85X ’, and L for electrons; X, I’’ and L for holes (see Fig. 1). For each region 6; n the k space, the corresponding DOSi (,E) and ugi E) were cal- culated integrating over all the k states in the Si egion so that: 6(E E( zy   d (3) and 1 6(E E(;)) dz 6i It immediately follows that the total DOS(E) and ug E) are related to the previous quantities as follows: c DOS;(E) = DOS(E) 1 c ugi E) DOSi (E) The total DOS(E) and u,(E) are plotted in Figs. 2  and 3. In addition, the individual contributions to the DOS of the three portions 6; n which the Brillouin zone has been divided, are also plotted in Fig. 2.  Looking at Fig. 2, it can be noticed that the peaks of the electron DOS, located at = 1.9 eV and = 3 eV, are mainly due to states in the L region (solid line); in Fig. 1 the DOS maximum at 3 eV appears as the highest minimum at the L point; corre-  ABRAMO zyxwvutsrqponmlk t al.: ISOTROPIC BAND MODELS zyxwvutsrqpon valence band conduction band 1329 20, zyxwvut   I , , , I - 4 2 zyxwvutsrqpon   zyxwvutsrqp rn2 n 0 -4.5 -3.0 1.5 0.0 1.5 3.0 4.5 energy [ev] Fig. 2. DOS(E) rom the ASBS (thick solid line); the remaining lines show the contributions of the zyxwvutsrq ’; o the total DOS (see Fig. 1). 20 ? 15 2 4 10 U - P zyxwvutsrqponml   E5 M 0 valence band conduction band zyxwvuts   -4.5 -3.0 1.5 0.0 1.5 3.0 4.5 energy [eV] Fig. 3. U, ,?) from the ASBS, as defined in I), for valence and conduc- tion band. spondingly in Fig. 3, the electron ug shows two pro- nounced drops, because the states at these energies ex- hibit low group velocity, proportional to the first derivative of the bands. In the energy range 1.9-3 eV, instead, ug exhibits a peak corresponding to the drop in DOS. As for holes, the continuous increase of the DOS is given by the contribution of bands around the points r and L (up to -1.2 eV) andX(up to -3 eV). 111. VALIDATION F THEORETICAL SSUMPTIONS To validate our approach, we now investigate if such zyxwv g E) s representative of the real situation of the electron population under the electric field action in the ASBS. In other words, if the carrier distribution function heated by strong and non-homogeneous fields still produces the same ug E) of equilibrium conditions. This investigation has two aims: first to check whether the carrier dynamics can be reduced to a spherically symmetrical problem; second if the particular expression (1) used for the average group velocity is meaningful. 0 1 2 3 4 energy [ev] Fig. 4. Electron U, ,?) from the ASBS (thick dashed line); electron U,@) computed with MC in homogeneous cases using the ASBS [I31 (thin solid lines). The simulations refer to field strengths of 250 and 500 kV/cm and directions ( 11 1 ) and ( 100). As can be seen, the different ug (E) are very close to each other, regardless of the electric field strength or lattice ori- entation. This strongly suggests that, for silicon homogeneous cases, the anisotropic effects are negligible. As a first step, we performed MC electron simulations of uniformly doped silicon slabs under different electric field strengths and directions using a simulator featuring the ASBS 1131. This is a recent version of the srcinal simulator described in 131. The main differences are the finer k-space grid for the E( k ) dispersion relationship and the use of a different impact ionization model 1141. We compared the a priori definition of ug (E) (1) with the av- erage electron velocity at each energy computed during the simulation. The aim was to determine to which extent the group velocity, averaged over the electron distribution heated by the field, depends on the field direction and magnitude because of anisotropic effects. The results, shown in Fig. 4 or field strengths of 250 and 500 kV/cm and for the two directions ( 11 1 ) and ( loo), prove that, for silicon, the average magnitude of electron velocity versus electron energy is marginally af- fected by anisotropic effects and resembles the u,(E) computed from the ASBS. This strongly suggests that the electron distribution function is evenly distributed over each equi-energy surface. In addition, we performed non-homogeneous simula- tions to check if the assumption previously discussed would hold also in this more realistic case. We simulated two non uniform, one dimensional test structures whose electric field profiles resemble the longitudinal one of a MOSFET biased with high drain voltage (Fig. 5). In par- ticular, the profiles consist of two constant plateaus con- nected by an exponentially increasing and a linearly de- creasing section. The profile of the shorter device corresponds to a total voltage drop of 2.88 V and is ob- tained from the other one (total voltage drop 5.76 V) by scaling the longitudinal direction by a factor of two. In Fig. 6 the MC group velocities computed at points in space indicated by bullets in Fig. 5 are compared with the  1330 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS. zyxwv OL. 12, NO. 9, SEPTEMBER 1993 zy 400 - zyxwvutsrqponmlk 00 2 00 00 .Y zyxwvut G U zyxwvutsrqponml   zyxwvutsrqpon   zyxwvutsrqponml A zyxwvutsrqponmlkjihgfedcb 0 0.0 0.2 0.4 0.6 0.8 distance [km] Fig. 5 Electric field profiles used for the one-dimensional non-homoge- neous simulations. The two profiles show the same maximum field but dif- ferent derivatives. 20 0 0 1 2 3 energy [ev] Fig. zyxwvutsrqponmlk . Electron ug E) rom the ASBS (dashed line); electron ug E) com- puted with MC in non-homogeneous cases using the ASBS 1131 (thin solid lines). The simulations refer to the electric field profiles of Fig. 5. The labels refer to the ones of Fig. 5. Thus, from Fig. 5  the electric field regime in which each curve was computed can be found. a priori definition of u,(E): again the good agreement confirms the validity of our assumption. IV. THE BAND MODEL The isotropic band model was obtained using the fol- lowing procedure. As already discussed, the irreducible wedge of the first Brillouin zone was divided into the three portions Pi, = 1, 3. This allows us to determine a single isotropic band for _each symmetry point, with the same effective mass at kci and multiplicity (Z) of the corre- sponding symmetry point in the ASBS. In particular we compute a set of Ei(k) featuring isotropic DOSY and U: in good agreement with the corresponding DOSi and ugi given by (3) and 4), espectively. Since the functions Ei(k) are isotropic, we can write the two basic equations of our model as follows: 1- dk and -+ where Zi s the multiplicity at kci and A a known propor- tionality factor. Equation 6 represents a set of two differ- ential equations in one unknown function Ei(k), and therefore cannot be solved exactly. An optimum solution can be found, which minimizes the weighted sum of the relative errors A given by: DOSy(E) DOSi(E) I DOS;(E) A = (1 - W uF(E) u,;(E) I u,;(E) (7) where w is a weighting factor in the range [0, 13 allowing to choose whether the DOS or the ug is to be better ap- proximated. In particular, a low value of w means a smaller error for DOS(E) than for ug E). We shall now consider the function k(E) nstead of E(k) for the following reasons: first because 7), solved for k at a given E, is a first-order non-linear differential equa- tion; second because the k(E) s a true mathematical func- tion, while the E(k) is not. Since (7) yields two different solutions for the two cases dEi/dk > 0 and dEi/dk < 0 and needs boundary conditions, the following constraints were imposed: 1) k(Ey) = k(EF) = 0, where, ~y~~ = Fin E(;); ke6, E? = gax E(;). ke6, dk - o for [E:~ , Ei]; dk - 0 for [Ei ?]; dEi dE; therefore, E; = E(k?). 3) Ei(k) E in [Eyin, ?]. The above constraints impose that each band consists of two branches: one electron-like featuring dE,/dk > 0 and one hole-like featuring dEi/dk < 0. Under these conditions, (7) was transformed into a fi- nite-difference problem and then numerically solved through the bisection technique. The total DOS(E) and u,(E) computed from (5)-(7) for different values of w( Us corresponding to different  ABRAMO zyxwvutsrqponmlk t al.: ISOTROPIC BAND MODELS 1331 zyx sets of zyxwvutsr i k)) re plotted in Figs. 7 and 8. It is worth no- ticing that while w zyxwvutsrqp   0 corresponds to zero error on the total DOS(E) (i.e., the DOS(E) obtained from the set of Ei k) oincides with that of the ASBS), w = 1 produces zero error on each ugi E) nly. The total group velocity u,(E) of the isotropic band structure, instead, is affected by the error made over the DOSi@) that are used as weighting factors in the determination of u,(E) (see zyxwvu 5) and Fig. 8). Therefore, values of w close to unity are not of practical interest. In Figs. 9  and 10 the relative errors made over DOS (E) and u, E) have been integrated over energy in the range zyxwv -4 V and shown as a function of the w parameter. At w = 0.5, for example, the error is equally distributed over DOS(E) and u, E), and is = 15 for conduction and = 20 for valence band, respectively. The electron and hole isotropic band structure obtained with w = 0.5 is reported in Fig. 11  , where different lines refer to the symmetry points of Fig. 1 Notice that the isotropic band related to the L symmetry point for elec- trons (solid line) reaches a much higher energy than the corresponding branch in the ASBS in Fig. 1. This can be explained considering that each isotropic band has been determined integrating over all states in each 6, nd not only over the ones along the so called principal directions shown in Fig. 1. The development of this model required also the exten- sion-of th_e formulae to compute the scattering rates P 11 k 11, 11 k for acoustic, optical phonons and ionized impurities from the well-known analytic band cases to the case where E(k) is given by means of numerical tables, as i,n o;r model. This extension was-made integrating the P , k ') expressions [ zyxwvut 1 over the k ' state space gener- ated by our zyxwvutsrqpo @') functions. A similar procedure has been applied in a recent work [7]. Using the same notation of [l] for the sake of simplicity, the scattering rates in our case are given by the following. Elastid acoustic phonon scattering: Intervalley optical phonon scattering: (9) Ionized impurity scattering (Brooks and Hemng [211): -4.5 -3.0 -1.5 0.0 1.5 3.0 4.5 energy [ev] Fig. 7. DOS(E) of the isotropic model computed for different values of the weighting factor w. he curve w = 0 corresponds to the exact DOS(E). The w step is 0.2. - zy   E 15 Y h :: 10 ? E5 - I 1 I 0 1 1 -4.5 -3.0 -1.5 0.0 1.5 3.0 4.5 energy [ev] Fig. 8. ug E) f the isotropic model computed for different values of the weighting factor w. he dotted line is the exact U,@ and does not corre- spond to the curve w = 1 (see text). The w step is 0.2. 40 r . 6e 30 Y h U V 4 M 2 20 4 . lo 0 ~ 0.0 0.2 0.4 0.6 0.8 1.0 weighting factor w Fig. 9. Present algorithm percentage error integrated over energy as a function of the w parameter for conduction band: DOS@) (A) and u,(E) zy 0) rrors. I +4= PS
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