A general purpose device simulator coupling Poisson and Monte Carlo transport with applications to deep submicron MOSFETs

A general purpose device simulator coupling Poisson and Monte Carlo transport with applications to deep submicron MOSFETs
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  360 zyxwvutsrqponml EEE TRANSACTIONS ON COMPUTER-AIDED DESIGN. VOL. 8. NO. 4 APRIL 1989 A General Purpose Device Simulator Coupling Poisson and Monte Carlo Transport with Applications to Deep Submicron MOSFET’s FRANCO VENTURI, R. KENT SMITH, ENRICO C. SANGIORGI, MEMBER, IEEE, MARK R. PINTO, MEMBER, IEEE, AND BRUNO RICCO, MEMBER, IEEE Abstract-An efficient, self-consistent device simulator coupling Poisson equation and Monte Carlo (MC) transport has been developed, suitable for general silicon devices, including those with regions of high dopingkhrrier densities. Key features include an srcinal iteration scheme between MC and Poisson equation and an almost complete vec- torization of the program. The simulator has been used to characterize non-equilibrium effects in deep submicron nMOSFET’s. Substantial overshoot effects are noticeable at gate lengths of 0.25 pm at room tem- peratures. I. INTRODUCTION EVICE simulation tools used in semiconductor tech- D ology development have to date been based primar- ily on the drift-diffusion (DD) model for carrier transport, a simplification of the more general Boltzmann transport equation zyxwvutsrqp   11. Recently reported experimental work on sil- icon MOSFET’s [2], [3] however, indicates that non- equilibrium transport effects, not directly treatable using DD , become important in determining intrinsic device characteristics at small geometries. A simulator which could treat the Boltzmann equation directly would be able to correctly model these effects, the extent of which still remains unclear. Furthermore, the analysis zyxwvut f hot carrier induced phenomena could be addressed more rigorously The numerical method generally proposed to solve the Boltzmann equation is the Monte Carlo (MC) technique, first applied to the charge transport in semiconductors by Kurosawa zyxwvutsrqp 5]. The MC method consists of following in space and time the history of each individual particle and does not rely on the relaxation time approximation. The major drawback of the MC approach comes from the intensive computational requirements that become [41. Manuscript received July 27, 1988; revised October 17, 1988 and No- vember 4, 1988. The review of this paper was arranged by Guest Editor J. Fossum. F. Venturi and B. Ricco are with the Department of Electronics, Uni- versity of Bologna, Bologna, Italy. R. K. Smith and M. R. Smith are with AT&T Bell Laboratories, Murray Hill, NJ 07974. E. C. Sangiorgi is with the Department of Physics, University of Udine, Udine, Italy. IEEE Log Number 8825906. particularly severe if the carrier distributions obtained from the MC calculation must be computed self-con- sistently with the electrostatic potential, given by Poisson equation. As a result, in the past the MC method has been felt to be impractical for use in studying devices such as MOSFET’s with very heavily doped source and drain re- gions zyxwvu 6]. The purpose of this paper is to describe a new, com- putationally efficient, self-consistent MC simulator which can analyze devices with completely general geometries and impurity profiles. As an example of the applications and usefulness of such a tool, an analysis of non-equilib- rium effects in deep submicron MOSFET’s will be pre- sented showing that at gate mask lengths of 0.25 pm these can account for an additional 20 percent increase in per- formance at room temperature. 11. THE MONTE CARLO METHOD The MC method applied to charge transport in semi- conductors [7] relies on the simulation of a large number of carriers subject to external forces (electric field) and given scattering mechanisms. A carrier trajectory consists of a sequence of ballistic flights (free flights), terminated by collisions (scattering events). The duration of each free flight and the type of terminating scattering event are se- lected stochastically, according to given probabilities de- scribing the microscopic processes. Because each particle is followed individually along its entire trajectory, a com- plete set of informations, including carrier position, mo- mentum and energy can be collected; these data can be suitably averaged to obtain local carrier densities and dis- tribution functions in a form similar to that provided by macroscopic models (e.g., DD). Due to its intrinsic nature, the MC method provides a microscopic description of the physical transport phenom- ena; in addition it is structured in such a way that new and/or more sophisticated scattering models can be easily added to the simulator. Most importantly, this approach avoids the assumption of the relaxation time approxima- tion which may not be valid in some conditions of inter- est. Furthermore, energy threshold phenomena like im- 0278-0070/89/0400-0360 01 OO zyxwvu   989 IEEE  VENTURI zyxwvusrqponm   zyxwvusrqponm / : SIMULATOR COUPLING POISSON zyxwvusrqp   MC TRANSPORT 36 1 pact ionization, gate current, and hot-carrier degradation can be most successfully treated [4], [SI because the en- tire energy distribution function is calculated, while other methods rely on average quantities of the distribution. zyxwvu s already mentioned, the computational constraints would seem to limit extensive use of this technique in technology development in particular if real, multidimen- sional devices are treated. 111. FEATURES F THE SIMULATOR In order to adequately deal with advanced devices op- erating under extreme conditions (for example, with high internal fields) a MC simulator should present the follow- ing characteristics: a) the carrier distribution function and the electrostatic potential (obtained via Poisson equation) must be computed self-consistently b) should be CPU time efficient to allow an extensive use in technology de- velopment; c) include accurate physical models for the dominant scattering mechanisms, though saving the com- putation efficiency; d) include terminating regions where safe (i.e., equilibrium) boundary conditions can be im- posed; e) should be able to deal with “rare events” such as low carrier concentrations compared to high ones and a few hot carriers in the presence of a multitude of colder ones. The first point, as will be shown later, is important in submicron devices where non-equilibrium phenomena cause the electrostatic potential to differ significantly from that obtained with conventional (DD) simulations. The desired consistency can be achieved by means of an iter- ative procedure coupling Poisson equation and the MC transport solution and, to this end, the potential obtained with DD calculations represents an obvious choice of ini- tial guess. Very serious problems, however, arise from the noise inevitably associated with the MC carrier con- centration, and they seriously threaten the convergence of the iterative procedure. As for the last point, the general problem is that rate events require the simulation of a vast number of common ones to achieve sufficient degree of confidence in their statistical description. This problem can be tackled by means of multiplication techniques expanding the statis- tical significance of occurring rare events; however pre- vious applications of the algorithm [12], [8] seem to be unsufficiently sophisticated to deal with complex devices as silicon MOSFET’s. In this context the present work advances the state of the art in two aspects. First it features an effective method to couple MC transport and Poisson equation providing a viable solution for the problem of the statistical noise mentioned above. Second it presents a new approach for a well-designed implementation of a MC code on a vector processor machine. The obtained improvements have made it possible to perform cost-effective MC simulations for such complex devices as advanced silicon sub-micron MOSFET’s. As for the other features mentioned above, our simu- lator makes use of sophisticated techniques available in the literature and already used elsewhere [7]-[9], [ll], [12]. In some of these cases, however, several improve- ments have been introduced to adapt such techniques to the needs of real device simulations. In particular an im- proved version of the multiplication method [SI, [ 121 used to deal with scarcely populated regions has been devel- oped to suit the multi-particle simulation case. In this case suitable weight factors W, are assigned to each geometric element so that the density of simulated “virtual” parti- cles remains constant across the device and so does the statistical noise. In fact, when a particle enters the geo- metric element zyxw   its weight is compared with Wi. f the former is greater than the latter, the particle is “split” into a set of smaller sub-particles; in the opposite case, the particle is gathered with other “light” ones possibly present in the same cell. The gathering algorithm does not affect the amount of total charge present in the device be- cause the new particle weights are chosen to be close but in general not to coincide with the reference one ( W, ) associated with the cell. The whole process of splitting and gathering particles keeps constant the amount of mo- bile charge present in the device during the simulation. This choice is justified even in a self-consistent scheme, because the mobile charge is dominated by that in the source and drain regions. In order to keep constant the statistical error in the re- gion of the current flow, the weights Wi re automatically chosen so that the density of “virtual” particles is uni- form along the whole interface, including source, chan- nel, and drain regions. In this way the density of simu- lated particles in the vertical direction of the channel decreases sharply following the behavior of the actual car- rier density. For the study of the energy tails, instead, regions within the device can be specified where the multiplication al- gorithm is applied to the most energetic electrons (re- garded as “interesting” rare events) in a recursive way so that any range of the distribution function can be stud- ied. The MC package requires an initial guess for the solu- tion of the electrostatic potential. Solutions obtained from an advanced DD simulator, e.g., [lo], provide remark- ably good guesses for most problems, certainly much bet- ter than any zyxw d zyxwv oc guess, thus speeding up the MC-Pois- son convergence. Furthermore, the computation time spent in calculating the DD solution is an extremely small fraction of the MC solution time. The MC analysis is generally performed only in a small domain or window within the device as simulated by DD, thus avoiding as much computational effort as possible that would be expended in simply reproducing the DD results. The ideal domain would consist of only those re- gion(s) where the DD model is expected to be in error, e.g., the channel of a MOSFET. However, because only equilibrium guarantees a correct choice of boundary con- ditions, the MC domain must extend in regions where vir- tually no electric field is present, e.g. the MOSFET source and drain.  362 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 8. NO. 4 APRIL zyx 989 zy In device simulation a grid is used to discretize Pois- son’s equation. Furthermore the grid is also used to dis- cretize the electric field during a carrier’s flight as it is essential to account for the local electric field on each trajectory, particularly in areas such as that near the drain of a MOSFET where the field is very nonuniform. To this purpose we have used a general two-dimensional trian- gular grid to discretize the simulation domain. This choice has two major advantages over other grids (e.g. rectan- gular). First, as with state-of-the-art DD simulators, e.g. [lo], triangular grids permit the treatment of devices with nonplanar geometries which are increasingly prevalent in VLSI technologies. Second, the computation of carrier trajectories becomes vastly simplified with triangular ele- ments as the electrostatic potential is assumed linear and hence the field is constant therein. In the calculation of the free flight time duration, it is most convenient to have the total scattering probability per unit time constant and in particular independent on the electron momentum and energy. An artificial scattering mechanism, the so-called self-scattering zyxwvuts   111, is, there- fore, introduced such that the sum of the true scattering rate and of the self-scattering rate is always constant. However, because of the boundary condition constraints the MC domain will always include low field regions where the energy dependent scattering rate is low, to- gether with high-field regions with very high scattering rate. This causes the number of self-scatterings to be un- acceptably high because in the low field regions the con- stant scattering rate is made up mostly by the self-scatter- ing rate. To avoid this problem we developed an algorithm which suitably discretizes the total scattering rate in order to minimize the number of self-scatterings while saving the convenience of the method. This scheme, described in details in [8], limits the number of self-scatterings to less than 10 percent of the total with a speed-up factor of ten, even in the most challenging applications. Finally, the physical models and scattering mechanisms included in this study are those discussed in [8]. In par- ticular the acoustic and optical phonons, collision with ionized impurities and impact ionization are described in [7], while the surface scattering model adopted comes from [9]. IV. SELF-CONSISTENT OUPLING CHEMES When a significant fraction of the electron population inside a device is under non-equilibrium conditions, the carrier-distributions obtained via MC must be computed self-consistently with the electrostatic potential. This point is demonstrated in Fig. 1 which shows a comparison of surface densities for a MOSFET with a gate mask length Lgate = 0.25 pm. Notice first that the DD model predicts a much higher carrier concentration at the surface essen- tially because, in contrast with MC calculations, it ne- glects field induced electron heating within the channel, thus confining the carrier position in a narrow portion of the potential energy well, close to the surface. Second, the MC concentrations obtained using the DD potential 10’7~ zyxwvu   .1 zyxw .2 zyxw .3 0.4 0.5 zy .6 0.7 DISTANCE (pm) Fig. 1 Electron surface concentration for a MOSFET with a gate mask length Lgate = 0.25 zyxwv n at zyxwvu cs = 1.5 V, V,, = 2.0 V calculated with DD (solid), non-self-consistent MC (dashed-dotted) and self-consistent MC (dashed) approach. (i.e., non self-consistent MC) are unacceptably different from the fully self-consistent solution. Thus non-self-con- sistent approaches like that described in [8], though much simpler and faster than complete solutions, can only be used when most of the electron population is reasonably well described by the DD model. To obtain the self-consistent solution, our program starts from an initial guess generated by a DD simulator and then iteratively solves the transport problem (by MC) and Poisson equation until convergence is achieved. A flowchart illustrating this procedure, which can be likened to the well-known “Gummel” iteration procedure [ 131, is shown in Fig. 2.  It is worth noticing that the output of the MC is the updated carrier concentration n, but it can also be specified by an auxiliary function such as the elec- tron quasi-fermi level &. The standard way to implement the coupling scheme of Fig. 2  is to substitute the updated MC carrier density nMC n Poisson equation, e.g., * (eV$) = P = q(P ZMC -k NDA) (l) where NDA is the net doping concentration and E the di- electric constant. Assuming that the hole concentration p can be neglected throughout the entire simulation domain, (1) becomes linear in and can be easily solved. Unfor- tunately, we have found that this simple scheme cannot be applied to cases with high carrier concentrations (as a silicon MOSFET) because of stability problems arising from the unavoidable noise of the MC solution. To clarify this statement, an example is illustrated in Fig. 3  where a solution for a n+-n-n+ device was modified in the n+ re- gion by changing the electron concentration in a grid node from 2 X lo2’ to 2.0005 X 10’’ to estimate the effect of very small noise (0.025 percent) in the MC solution. The result is an unacceptable variation in the potential that cannot be recovered with successive iterations. One way to overcome this problem is that of solving Poisson equation very often in order to deal at each step only with extremely small changes in the carrier popula-  VENTURI zyxwvusrqponm r zyxwvusrqponm /.: SIMULATOR COUPLING POISSON zyxwvusrq   MC TRANSPORT 363 MONTE CARLO FLOWCHART zyxwvutsr   zyxwvutsrq ES zyxwvutsrqponmlkjihgfe n * ,yl ,@; zyxwvutsrqponmlkjihgfedcbaZ E’ ,I,* zyxwvutsrqpon If Fig. 2. Flowchart of the Monte Carlo program 3.0 3’5 I -i I I 0 0.5 1.0 1.5 2.0 2.5 DISTANCE ( pm) Fig. 3. Effect of a small localized charge variation (0.025 percent) on the potential of a n+-n-n+ device. Curves A and B refer to the linear and nonlinear Poisson case. respectively. tion unable to produce irreversible divergences in the it- eration scheme. To this purpose the electron plasma fre- quency ( = ( 1 fs)-’ in the case of MOSFET’s [14]) can provide a guideline for the time interval between succes- sive solutions of Poisson equation in the iteration scheme (i.e., At < 1 fs). Using such a value the calculations, though reasonably safe, implies very large computation time. In this work we use an alternative method that provides satisfactory convergence property and is much more cost- effective in terms of computing resources. Such a method, rather than the MC electron concentration, exploits the equivalent concept of the electron quasi-fermi level & de- fined as where TL is the lattice temperature, kB the Boltzmann con- stant, zyxw , he intrinsic concentration and i? denotes the de- pendence on position. In terms of q n the electron cencen- tration n can be written as n ?) = ni xp By using this expression Poisson equation becomes nonlinear in zyxw c while & is kept fixed. This scheme guar- antees stability because even large variations in the MC carrier densities lead to very small changes in the poten- tial due to their exponential relationship. In fact, curve B in Fig. 3  shows that for the same amount of charge vari- ation used for the linear case, the potential remains fixed when the nonlinear scheme is used. In Fig. 4  the spatially dependent convergence rate of this coupling scheme is examined for two points along the channel of a 0.25-pm gate length MOSFET. In particular Fig. 4(a) illustrates the electron energy plot along the MOSFET surface, showing an equilibrium energy point (40 meV)-labeled “A”-that lies in the source and a very high energy point (461 meV)-labeled “B”-in the chan- nel. Fig. 4(c) illustrates the convergence behavior of point B; the x and y axes represent the potential and the quasi- fermi level that define the status of a grid node. Each ver- tical line corresponds to a Poisson solution-where the potential is changed and the quasi-fermi level is kept fixed-while each horizontal one represents a MC solu- tion. The staircase-like behavior of the electrical param- eters ( and +,J at point B requires a large number of iterations to reach convergence because the actual solu- tion is far from the initial guess (i.e., the DD solution). On the other hand at point A, where the electrons are cold, the electrical parameters remain very close to the initial guess and convergence is reached in very few steps (Fig. 403)). To this regard it must be noticed that the scale used to plot Fig. 4(b) is expanded greatly compared to Fig. 4(c). The example of Fig. 4 shows that the coupling with nonlinear Poisson equation guarantees stability but fea- tures slow convergence rate in the regions where the elec- trons are hot. To increase the convergence speed, we have developed a method based on the use of the electron tem- perature T, instead of the lattice temperature TL in (3). The thought here would be that the greater the tempera- ture used in (2) and (3), the larger is the change in poten- tial caused by the Poisson solver. In this way, in fact, we are exploiting additional information, i.e., the average electron energy, from the MC simulator to adapt (2) to the case of strong non-equilibrium conditions. More in details, starting from the average electron en- ergy E,, given by the MC calculation, the electron tem- perature T, is computed as Therefore, (2) and (3) become (4)  364 zyxwvusrqponm EEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, zyxwv OL. 8, NO. zyx . APRIL 1989 zy 0 zyxwvutsr 2 04 0 DISTANCE pm) (a) 0.570 zyxwvutsrqpon   .565 - > I 9 0.560 Y 0.555 0.550 I I I 0 0.005 0 01 0.015 0.02 QUASI-FERMI LEVEL (V) (b) 68 0 70 0 72 0 74 zyxwvutsrqponml .76 0 78 080 0.82 QUASI-FERMI LEVEL zyxwvutsrqponmlkj V) (C) Fig. 4. Spatially dependent convergence rate for the nonlinear Poisson coupling. (a) Average electron energy at the surface of MOSFET with Lgate = 0.25 pm. (b) Convergence rate of point A in the domain defined by the potential and the quasi-fermi level. (c) Convergence rate of point B in the domain defined by the potential and the quasi-fermi level. and Because T, = TL for the points such as A inside the source and drain regions, the stability of the solution in 1.30 - > J f 1.26- z W - 1.22- 1181 1 I I I 068 070 072 074 076 078 080 08 PUASI-FERMI LEVEL VI Fig. 5. Convergence rate of point B in Fig. 4(a) when using the lattice (dashed line) or the electronic temperature (solid line) in the nonlinear Poisson equation. the highly doped regions is still described by Fig. 4(b), while the convergence rate is much faster in the other parts of the device. In Fig. 5  the convergence rate at point B for the pure nonlinear coupling scheme (equations (2) and (3)) and that one which makes use of the electronic tem- perature (equations (5) and (6)) are compared. Notice that in the new scheme a Poisson solution is represented by a diagonal line because the variable in the x axis has been computed using TL to allow comparison between the two methods, It is clear from the presented data that the new approach is over four times faster, while of course yield- ing the same solution. Considering, as an example, the case of a quarter mi- cron MOSFET, Fig. 6 shows the L-2 norm of the relative error in the electrostatic potential as a function of the number of self-consistent iterations. The error first de- creases rapidly but eventually is limited by the statistical noise of the MC procedure. Within the accuracy of the MC calculations, we find that a reasonable stopping cri- teria is about 0.5 mV in this norm. For the examples we have tried, this tolerance is achieved after about ten iter- ations. Fig. 7 illustrates the distribution of the potential update (A*) during the convergence process in the case of a MOSFET with Lgate = 0.15 pm. Each line connects the points with the same A* after the Poisson solution while the difference between two contiguous lines is 5 mV for Fig. 7(a) and (b) and 0.5 mV for Fig. 7(c). The three plots refer to the situation after the first, fourth, and fourteenth iteration, respectively. At the beginning of the conver- gence process (Fig. 7(a)) the update is large across the whole device with a maximum (over 50 mV) at the end of the channel where electron heating, hence the differ- ence between DD and MC solutions, is stronger. How- ever, because of the fast convergence process the poten- tial update rapidly decreases in few iterations (see Fig. 7(b) where the maximum A* is between 5-10 mV) and eventually levels off (Fig. 7(c)) with a random distribu- tion across the whole device and a peak below 1 mV.
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