Comparison of Monte Carlo and deterministic simulations of a silicon diode Jose A. Carrillo Λ, Irene M. Gamba y, Orazio Muscato z and Chi-Wang Shu x N

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1 Comparison of Monte Carlo and deterministic simlations of a silicon diode Jose A. Carrillo Λ, Irene M. Gamba y, Orazio Mscato z and Chi-ang Sh November 7, Abstract lectron transport models in Si transistors with channel length of :4 microns and nanometers are eamined and compared between classical Direct Monte Carlo Simlations and deterministic NO solvers for a self-consistent kinetic field-relaation Poisson model. This model is a well accepted low density redction of the fll non-eqilibrim transport phenomena. In this comparison we control the calibration of the field dependent, satrated mobility. Or comptations show that, at channel length of order :4 microns, the relaation model captres the the first two moments of the particle distribtion fnction inside the channel. In particlar a domain decomposition techniqe that implements classical drift diffsion in the high density regions and agmented drift diffsion inside the channel region gives a correction to the classical drift diffsion simlations, and prodces similar qalitative reslts to the Monte Carlo simlations with a :2 CPU time redction factor. However, we show that in the case of a nanometer channel, the kinetic field-relaation model fails to approimate well even the first moment, and in particlar it does not approimate well the crrent voltage crve otpt from Monte Carlo simlations, making it necessary to incorporate high energy effects into the collision operator. Λ Departamento de Matematica Aplicada, Universidad de Granada, 87 Granada, Spain. -mail: carrillo@gr.es y Department of Mathematics, University of Teas at Astin, Astin, TX 7872, USA. -mail: gamba@math.teas.ed z Dipartimento di Matematica, Universitá di Catania, Italy. -mail: mscato@dmi.nict.it Division of Applied Mathematics, Brown University, Providence, RI 292, USA. -mail: sh@cfm.brown.ed

2 Introdction e report a comptational comparison between a fll one space-three velocity dimensional Boltzmann Transport qation (BT) solved by a Direct Simlation Monte Carlo (DSMC) method and recent deterministic comptations corresponding to a one dimensional kinetic field-relaation model sing eighted ssentially Non-Oscillatory (NO) schemes, with a field dependent relaation time approimation for electron transport, both for a :4 microns and a nanometers channel of a n + -n-n + Si diode. This comparison is motivated in order to show the validation of this kinetic relaation model as a good approimation for moderate qasi-ballistic regimes [, 2] in the channel region p to orders of.3 microns for relative high energies. In addition, we show that comptations corresponding to a shorter Si nanodevice with a channel length of.5 micron, are not well modeled by the one dimensional field-dependent kinetic relaation system, for two-parameter, field dependent mobilities, when compared with DSMC for the fll BT. This is also considered to be qasi-ballistic []. The longer channel asymptotics correspond to a length scale that balances the strength of collisions with the drift strength. As a reslt, the density and crrents are well captred by the deterministic comptation at 2 fraction of the comptational time when sing a domain decomposition techniqe that incorporates a mlti-scale analysis. As a conseqence, for these qasi-ballistic regimes, we are able to obtain a mlti-scale comptational algorithm for crrent-voltage characteristics (IV-crves) which agree with Monte Carlo IV crves at a higher order of the relative error. This mlti-scale comptational algorithm has been presented in [4] and compared with the same one dimensional field dependent kinetic relaation model for which we perform the comparisons in this paper. In fact, when comparing the relative errors between both drift-diffsion and the mlti-scale algorithm otpt for IVcrves with Monte Carlo, the latter improves in a factor of 2 with respect to the former one. e see that for shorter (or faster") devices the relaation model fails to captre the high energy effects coming from stronger scattering rates. This means that the fll acostic and non-polar optical phonon scattering collision operators mst be modeled. As a conseqence, most classical hydrodynamical closres based on collisional models that fail to incorporate high energy effects will not reprodce a good fitting of Monte Carlo simlations, even for the lower moments. For completeness of this paper, we recall the models that are compted, with the eception of the classical Drift Diffsion Poisson (DDP) system with field dependent mobility. The classical kinetic relaation model for charged transport is derived from a low density approimation of the semi classical Boltzmann-Poisson system, taking into accont only collisions with backgrond imprities. Thogh this system is posed in the three dimensional velocity space, a one dimensional model recovers the important featres of the charge transport that are given in the direction parallel to 2

3 the force field. Sch a configration is a correct description of a channel-like flow, as in the modeling of diodes, de to the simple geometry of the device. Therefore this eqation + vf e m (t; )f v = fi (M o (f) f) (.) f = f(t; ; v) is the density fnction for an electron at position 2 [;L] and velocity v 2 IR at time t, where L is the device channel length. The constants e and m represent the nit charge and effective electron mass, respectively. The electric field = (t; ) is self-consistently prodced by the electrons moving in a fied ion backgrond with density C(), called doping profile. is determined by the Poisson eqation " o = e((f) C()) (.2) (t; ) = where " o is the permittivity of the material and Z Z (t; ) = f(t; ; v)dv; j(t; ) = vf(t; ; v)dv; 2 [;L]; t (.3) IR IR are, respectively, the charge and crrent densities of the electrons. M o is the absolte Mawellian given by ψ! M o =(2ß o ) =2 ep v2 2 o ; where o is the lattice temperatre, that is, o = k B m T o with the Boltzmann constant k B and the lattice temperatre T o in Kelvin. Here, the relaation time fi depends on the absolte vale of the force field in sch away that the mobility μ = e mfi is linear for small vales of jj with slope μ and has a horizontal asymptote v d as jj becomes large. e have chosen the following formla for describing the mobility, μ() = e m fi() = 2μ r (.4) μ v d where the parameters μ (low-field mobility) and v d (satration speed) are adjsted for calibrating the drift speed μ in the blk Si with respect to the DSMC data obtained by thedamocles code [8]. In this way, wemake sre that both models in or comparison refer to the same physical device. The calibration can be performed by means of different procedres. Here, the calibration has been made by sing the tool emplar [9], where the minimm of the residal fnction (i.e. the sqare of the difference of the Monte Carlo data and the data obtained by Formla (.4) is obtained by the simple algorithm. Accordingly, we take μ =799 cm2 and Vsec v d=.57 7 cm. In Figre we have sec 3

4 plotted the drift speed of the Monte Carlo rn compared to the plot of the fnction in (.4) given by the above procedre. There are different choices to calibrate the relaation model, for instance, to find nmerically, by sing the mobility reslts from Damocles, an approimation to the limit at infinity v d and the initial slope μ. Another possible choice wold be to se an interpolation of the data from Damocles as inpt for the mobility in the relaation code. These other two different calibrations have been tested nmerically and they give very similar reslts to the emplar calibration and it seems the reslts qalitatively are not very sensitive to the calibration procedre. e shall see that the transport regime corresponding to the larger channel device (:4μm) is well modeled by the field dependent relaation-poisson system (.)-(.2)-(.4). As it was worked ot in [4], sch a device model can, in fact, be modeled by a hybrid domain decomposition techniqe consisting of the comptation of classical drift-diffsion-poisson (DDP) models in the n + regions which correspond to dominant collision processes, and agmented-drift-diffsion-poisson (ADDP) models in the n-channel region which corresponds to drift-collision balance processes that take over the channel for this qasi-ballistic regime. The (ADDP) system in its dimensionalized form reads [6] t (J) = (.5) e J = μ + fiμ ( μ +!) fi[( o +2μ 2 2 )] + fiμ(μ) (.6) " o The constant! is fied as = ; and " o = e((f) C()): (.7)! =(μ)j =! ; (.8) where! is some point inthecomptational region. The total crrent j(t; ), as defined in (.3), is approimated by J(t; ) soltion of (.5)-(.8). e refer to [6] and the references therein for a deeper discssion of this system and the role of!. The density (t; ) and crrent J(t; ) that solve the hybrid DDP and ADDP system, with the latter given by (.5)-(.8), are Chapman-nskog asymptotic approimations of the first and second moments of f(t; ; v) soltion of the kinetic system (.)-(.2)-(.4) in the n+ and n-channel regions respectively. 2 Nmerical comparisons e solve nmerically the fll BT by sing Damocles, the Monte Carlo code developed in IBM [8]: the spherical parabolic band approimation is sed and acostic and non-polar optical phonon scattering mechanisms are considered. e compared the reslts with respect to the deterministic simlation of the one dimensional relaation-poisson system (.)-(.2)-(.4). e shall call the nmerical 4

5 otpt corresponding to these comptations (-D BT). e sed NO finitedifference method [7] to solve in a deterministic way the (-D BT) as introdced in [5]. NO methods are high order accrate yet non-linearly stable (essentially non-oscillatory) in the presence of sharp gradients or shocks in the soltion. It is important to notice that we are comparing 24 hor rnning time for a DSMC simlation verss a 45 minte comptation for a NO simlation of the (-D BT) and a 3.5 minte comptation of the hybrid domain decomposition method (DDP-ADDP) sed in the longer channel case. Net, we make the comparison for two different devices. The main difference between them is the length scale. The second device is mch shorter than the first one. e refer to these two devices as longer and shorter channel in this seqel. The shorter device has been stdied also in [3]. Units are =cm 3 for density, 7 cm=sec for mean velocity, V for potential, V=cm for force field, ev for energy and 2 A=cm 2 for crrent. The vales of the parameters, common to both Si devices, are given by m =:26 :99 ( 3 kg), e = :62 ( 8 C), k b = : ( 8 J=K), " o = :7 8:8548 ( 8 F=μm), T o =3K and backgrond density n i =:45 =cm Longer channel Here, we consider a one dimensional Si n + -n-n + strctre of length :6μm with a channel length of :4μm. Ths, the domain of the device model is Ω=[; :6], and the doping profile, given by C(), is a sharp step fnction with density vales 5 7 =cm 3 in»» : and in :5»» :6 corresponding to the n + regions; and 2 5 =cm 3 in : < < :4 corresponding to the n-channel region. The nmerical reslts are shown in Figre 2 with the emplar approimation of the mobility. e observe a remarkable coincidence of the reslts of the -D velocity relaationtime model with respect to the fll BT when comparing density, force field and potential. For comparisons between (-D BT) and the hybrid (DDP)-(ADDP) comptations the first three moments, potential and forced field see [4]. e also show the comparison of electron energies for both simlations. In order to compare the energies for the -D velocity model we need to assme that the corresponding pdf in 3-D corresponds to the -D pdf mltiplied by a Mawellian with zero mean velocity and backgrond temperatre in the orthogonal directions. As observed, the comparison of the energy is mch worse than the one for the mean velocity. As a conseqence the -D relaation-time model fails in captring the energy of the fll BT. Despite of the discrepancy with the mean velocity and energy (Figre 2), we see that the -D model with the emplar approimation is fairly accrate in captring the IV-crves (Figre 2, lower right corner). e have also plotted there theivcrves of the drift-diffsion-poisson (DDP) system and the hybrid domain decomposition sing drift-diffsion-poisson and agmented-drift-diffsion-poisson (DDP-ADDP) system with field-dependent mobility given by (.4). Recall that it reqires only approimately 3 mintes to nmerically solve the (DDP-ADDP) 5

6 system, compared with a 24-hor DSMC simlation. Shorter channel Now, we consider a shorter one dimensional Si n + -n-n + strctre of length nm with a n-channel of nm. Here, the doping profile given by C() is a sharp step fnction with density vales 5 8 =cm 3 in»» and in»» ; and 5 =cm 3 in <<. The rest of the parameters and nits remain the same. The nmerical reslts are shown in Figre 3. Here, the deterministic comptations of the kinetic relaation system clearly nderestimate the mean velocity and overestimate the density with respect to the DSMC comptations of the fll BT. As a conseqence the IV-crves (Figre 3, lower right corner) are not well captred by the kinetic comptation of the one dimensional field dependent relaation (D-BT). Notice that the compted IVcrve of the hybrid (DDP-ADDP), being an asymptotic approimation to the field dependent relaation (D-BT), can only approimate well that of the (D-BT), not that of the DSMC comptations of the fll BT. In addition the (D-BT) comptation of the energy grossly nderestimates that compted with the DSMC for the fll BT, making this kinetic approimating model nreliable. 3 Conclsions e have compared two Si diode models, a longer one with a :4μm channel length, and a shorter nano-device with a :5μm channel length, sing the DSMC Damocles for the fll BT and the deterministic NO scheme for the -D field dependent relaation approimation to the fll BT. e show that the density, IV-crves, force field and potential corresponding to the longer device is well modeled by the -D kinetic relaation model and its corresponding hybrid DDP-ADDP macroscopic simlation, both compted with a deterministic NO scheme, redcing the comptational time from 24 hors to abot 3 mintes for the similar IV-crves. In particlar the IV-crve prodced by the hybrid DDP-ADDP model approaches the one with Monte Carlo simlation by a factor of 2 when compared with the corresponding DDP solver. This factor does not depend sensitively on the mobility approimation or the calibration procedre. The comptational cost of the Drift- Diffsion-Poisson solver and the hybrid DDP-ADDP one are of the same order. However the energy is not well modeled and the mean velocity is fairly approimated by the D kinetic relaation model when compared with the Monte Carlo simlation of the fll BT. For the nano-device, the the D-kinetic field dependent relaation model fails to prodce a good approimation for any of the moments. The comparisons for the IV-crves show the need of better approimations and asymptotics for the fll BT for high energetic processes in order to improve the agreement with Monte Carlo simlations. 6

7 ven thogh the correct collision mechanism shold inclde optical phonon collisions, at larger devices with relatively low voltage bias the levels of energy injection are low enogh, so that the the dominant collisions are acostical, eplaining the agreement of the relaation and Damocles for :4μ and voltage drops from.4 to V. Beyond that bias, the field-relaation model and Monte Carlo differ significantly. This regime can be identified as the qasi-ballistic one [3]. This fact brings p the need to compare the diode comptation with a deterministic acostical-optical phonon collision solver. This is part of a work in progress. Acknowledgments The athors wold like tothank M. Lndstrom and K. Banoo for helpfl discssions, and the Teas Institte for Comptational and Applied Mathematics (TICAM) and the Institte for Mathematics and its applications (IMA) where part of this research was done. The first athor is spported by the DGSIC MC-Spain Perfeccionamiento de Doctores en el tranjero fellowship and DGS (Spain) PB The second athor is spported by the National Science Fondation nder grant DMS and TARP The third athor is partially spported by the Italian Consiglio Nazionale delle Ricerche nder grant PS and the M.U.R.S.T. project Problems of Kinetic Theories 998. The forth athor is spported by the National Science Fondation nder grant CS and by the Army Research Office nder grants DAAG and DAAD The first and third athor also acknowledges partial spport by the grant RBFMRXCT9757 (TMR-Network) from the U. References [] F. Assad, Z. Ren, P. Beni and M.S. Lndstrom, Performance Limits of Si MOSFT's, IDM Tech. Digest, p.547, Dec [2] M.S. Lndstrom, lementary scattering theory of a MOSFT, I lectron Dev. Lett., 8, pp , 997. [3] K. Banoo, J.-H Rhew, M.S. Lndstrom, C.-. Sh and J.. Jerome, Simlating qasi-ballistic transport in Si nanotransistors, Proceedings IC. VLSI Design, to appear. [4] J. A. Carrillo, I.M. Gamba and C.-. Sh, Comptational macroscopic approimations to the -D relaation-time kinetic system for semicondctors, Physica D, to appear. [5] C. Cercignani, I.M. Gamba, J.. Jerome and C.-. Sh, Device benchmark comparisons via kinetic, hydrodynamic, and high-field models, Compter Methods in Applied Mechanics and ngineering, 8,, pp

8 [6] C. Cercignani, I.M. Gamba and C.D. Levermore, A drift-collision balance asymptotic to a Boltzmann-Poisson system in bonded domains, to appear in SIAM J. Appl. Math.. [7] G. Jiang and C.-. Sh, fficient implementation of weighted NO schemes, J. Compt. Phys., 26, 996, pp [8] S.. La, M. V. Fischetti, D. J. Frank, Monte Carlo analysis of semicondctor devices: The DAMOCLS program, IBMJ. Res. Develop., 34, 99, [9] S. Rinado, A general algorithm to calibrate any kind of simlators withot any knowledge abot analytical form of the implemented models, SGS- THOMSON Microelectronics, internal report. [] P.A. Markowich, C.A. Ringhofer and C. Schmeiser, Semicondctor qations, Springer, New York, 99. [] G. Timp, J. Bde, et al., The Ballistic Nano-transistor, IDM Tech. Digest, p.55, Dec

9 7 drift speed force field Figre : Drift speed for blk Si. Solid line: Formla (.4) with emplar vales; circle symbols: the BT system reslt by DSMC Damocles. 9

10 Frame 8 9 Ag crrent crrent Frame 8 9 Ag crrent Frame 8 9 Ag crrent Frame 8 9 Ag crrent Frame 8 9 Ag crrent Frame 8 Nov Figre 2: Nmerical comparison of density (top left), mean velocity (top right), force field (middle left), potential (middle right), energy (bottom left) and IV-crves (bottom right) for the Si n + -n-n + device with a :4μm channel at V bias = :5V. Relaation -D BT by NO (sqare symbols and dotted line); BT system by DSMC Damocles (solid line); DDP system (dashed line) and the hybrid DDP-ADDP system (dash-dotted line).

11 crrent crrent crrent crrent crrent crrent Figre 3: Nmerical comparison of density (top left), mean velocity (top right), force field (middle left), potential (middle right), energy (bottom left) and IV-crves (bottom right) for the Si n + -n-n + device with a :5μm channel at V bias =:6V. Relaation -D BT by NO (sqare symbols and dotted line); BT system by DSMC Damocles (solid line); DDP system (dashed line) and the hybrid DDP-ADDP system (dash-dotted line).