Calibration of a Dimensional Hydrodynamic with Monte Carlo...

VLSI DESIGN1998, Vol. 8, Nos. (1-4), pp. 515-520Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach SciencePublishers imprint.

Printed in India.

Calibration of a One Dimensional HydrodynamicSimulator with Monte Carlo Data

O. MUSCATO a,,, S. RINAUDOb and P. FALSAPERLA

Dipartimento di Matematica, Viale Andrea Doria, 95125 Catania (Italy)"b SGS-THOMSON Microelectronics, Stradale Primo Sole 50, 95121 Catania (ltaly)

In this paper we use the code Exemplar for matching a hydrodynamic 1D, time-dependent simulator and the transport coefficients obtained by the Monte Carlosimulator Damocles. This code is based on the Least Square method and it does notrequire any a priori knowledge about the simulator (analytical form of the equationsetc.). The stationary electron flow in a one dimensional n+-n-n+ submicron silicondiode is simulated.

Keywords." TCAD, VLSI, BTE, transport theory, fluid mechanics, electronic devices

1. INTRODUCTION

Electronic transport in semiconductors can bedescribed by hydrodynamic models (hereafterHM), obtained by taking the moments of theBoltzmann transport equation (hereafter BTE):the resulting mathematical model consists of aninfinite hierarchy of Partial Differential Equationsexpressing balance laws for the particle number n,velocity , total energy E, deviatoric stress tensor

(0"), energy flux (or heat flux h) and so on,coupled with the Poisson equation.

Recently Anile and Muscato [1] presented anExtended Hydrodynamic model where the closureof the moment hierarchy is obtained by exploitingthe entropy principle: this system, which is

hyperbolic, consists of 13 scalar equations in the13 unknowns (n, , E, ( ij), ) and completelydetermines the description of the stress and ofthe heat flux, at variance with the other models.Such a model has been tested successfully withMonte Carlo (hereafter MC) simulations in Silicon[2]. The Left H and Side of the balance equations,called production terms, represent the average rateof change of carrier total energy Qw, momentum

Qp, energy-flux Qs and stress Q</j> due to thescattering of carriers with the lattice. Usually theyare approximated with ad hoe empirical formula[3] or as relaxation terms [4]: this last approxima-tion leads to a serious inconsistency with one ofthe fundamental principles of Linear IrreversibleThermodynamics, the Onsager Reciprocity Princi-

*Corresponding author.

515

5!6 O. MUSCATO et al.

pie. In order to tackle this problem, we expandedthe electron distribution function in Hermitepolynomials around the state of global thermalequilibrium (limiting ourselves to the first twopolynomials), where the electrons lie close to thebottom of the conduction band. By using a wellknown procedure due to Grad [5] we obtained thefollowing equations:

Qw (1)"rw

Qp a Poo + a Pl + a P2 nff

+ bP + bP--oo+ bl-o kBTo

(2)

Qs ag + asl + kB Tonff

+ + +(3)

where To is the room temperature, E0 the latticetotal energy, no a reference impurity concentration(1018cm-3). Since in this approximation thedistribution function is quasi-isotropic [2] thedeviatoric stress tensor /is negligible and by thismethod we cannot extract the correspondingproduction term (unless we consider higher orderHermite polynomials).For the sake of simplicity we modeled the stress

production as a relaxation term:

Q<ij>-<ij>

(4)

We should emphasize that:

the coefficients appearing in Eqs. (1-4) are notfitting parameters but rather will be extracted byMC data obtained by the Damocles code [6], inthe case of parabolic spherical band approxima-tion, in order to be consistent with the Anile andMuscato model, obtained under these restric-tions;

these coefficients are not functions of thepositions in the device as some authors claim.

With such coefficients we obtained a closed set ofhydro equations which has been solved by anadequate numerical scheme. Since the MonteCarlo method gives a stochastic solution of theBTE, the results are noisy: how does the hydrosolution change for small variations of the para-meters Tw, T, a0, al, a2, b0, bl, b29. In order toanswer this question the simulator Exemplar,developed by one of us [7], has been used.The plan of the paper is the following: in section

2 we discuss the optimization problem; in section 3we simulate the usual n+-n-n + diode with theDamocles code and with our Extended hydrody-namic model. We discuss the range of validity ofthe optimization procedure for various biases andconclusion are drawn.

2. THE OPTIMIZATION PROBLEM

The optimization problem is well known: definingthe residual function

N

R() -wi(f(xi, ) yi)2 (5)i=1

where Y is the vector of the parameters, f(xi, ) isthe analytical description of the event computed in

xi (modeled by the vector parameter ), Yi the realevent value obtained in xi and wi an appropriateweight factor. In a mathematical language theproblem of the least squares method could beexpressed in the form:

min{R(7)fi7 e Rn} (6)

where n is the dimension of the vector 7 and 7 thesolution of the system, generally, non-linear

VR() -0 (7)

and R" the n dimensional space of the real R.If we know the analytical equation off(x, 7), we

could evaluate and resolve the system vR() 0,

CALIBRATION OF A ONE DIMENSIONAL SIMULATOR 517

using one of many methods proposed (e.g.Marquardt). However, if we do not know theanalytical form off(x, E), it is not possible to findthe solution of the problem (6) finding the solutionof the system (7). This happens when we want todetermine a set of parameters to model a generaldevice with any kind of simulator: we run thesimulator obtaining the macroscopic quantitiesbut not the derivatives of the Residual function. Inthis case we can resort numerical methods as thesimplex method and Powell’s method, whichminimize the Residual function, without havingto know its derivatives.The Exemplar code runs the simulator to

calculate the function f(xi, ) (i= 1,...,N) thencompute and optimize the Residual function withthe Simplex or Powell method (according the user

choice) and the procedure restarts until a toleranceof the Residual function is obtained.The proposed algorithm can be used with any

simulators which can read their input from filesand store their output in column-formatted ASCIIfiles. It was implemented in a software programwith a friendly user interface based on X-Window,which resolves the problem of the optimizationwell, but, since during the optimization it runs thesimulator, it needs a large computational time.

-50

MC Eq.(2)

"700 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 O. 0.5(micron)

FIGURE Average rate of change of carrier momentum Qpvs. distance: Monte Carlo data (with ***) and the fitting formulaEq. (2) (with ooo).

-25MC Eq.(3) Volt bias

.300 0.05 0’.1 0.’15 0.2 0.25 0’.3 0.35 0.4 0.45 0.5(micron)

3. SIMULATION OF A ONE DIMENSIONALn + -n-n + SILICON DIODE

FIGURE 2 Average rate of change of carrier energy flux Qsvs. distance: Monte Carlo data (with ***) and the fitting formulaEq. (3) (with ooo).

The n + -n-n + diode consists of two n + regions0.1 gm-long doped to a density ofN 1018 cm-3,while the central n region is 0.3 gm wide, with a

doping density of N 1016 cm-3. We considerOhmic boundary conditions, To 300 K, and Vof applied bias; simulations refer to the stationaryregime.

In Figures and 2 we plot the MC data for Qvand Qs and the fitting with eqs. (2) and (3) (withooo): we see that our functional form fits well thedata. We observe that the MC data are noisyespecially near the contacts (this run needed

month of CPU in a IBM Risc 6000 590) andconsequently the fitting coefficients are not unique.By using these coefficients the Extended Hydro-

dynamic model is solved by using a simulatorbased on the splitting method between relaxationand convection (Tadmor scheme) [8]. Then we runthe Exemplar code in order to find the bestcoefficients which fit well the MC data. In Figures3, 4 and 5 we compare the MC data (with ***), thehydro data without optimization (with ooo), thehydro data optimized with Exemplar (with xxx).

518 O. MUSCATO et al.

1.4

0.4

(micron)

Monte Carlo Hydro Exemplar Volt bias

x

0.4 0.45 0.5

FIGURE 3 Average drift velocity vs. distance computed byusing the hydrodynamic simulator without optimization (ooo),the hydro data optimized with the Exemplar code (xxx) andMonte Carlo data (***), with Volt bias.

0.15

-0.05

-0.1


(micron)

FIGURE 5 Heat-flux vs. distance computed by using thehydrodynamic simulator without optimization (ooo), the hydrodata optimized with the Exemplar code (xxx) and Monte Carlodata (***), with Volt bias.


0.25

0.2

0.1

0.05

oi5 o. o/15 0.2 o., ot o., o:, o:;. o.(micron)

FIGURE 4 Average electron energy vs. distance computed byusing the hydrodynamic simulator without optimization (ooo),the hydro data optimized with the Exemplar code (xxx) andMonte Carlo data (***), with Volt bias.

The optimized coefficients (shown in Tab. I)differ from the previous ones at maximum by 40%.We notice that the well known ’spurious’ peak inthe velocity curve Figure 3 is lowered and theenergy and heat flux curves Figures 4 and 5 arecloser to the MC data. However this ’spurious’peak cannot be eliminated by any small change inthe parameters and therefore its persistence callsfor a more radical examination of the basicassumptions of the model.

A crucial question to be addressed is what is therange of validity of the optimization procedure? Inorder to answer this question we simulate thedevice with the coefficients of Table I (obtainedwith Volt) for various biases. The result is that inthe range 0.8 / 1.2 Volt the hydro simulations fitwell the respective MC data (see Figs. 6, 7 and 8 incase of 1.2V): for higher biases the hydrosimulations diverge considerably with respect tothe MC data.

0,4

Monte Carlo Hydro 1.2 Volt bias

(micron)0.05 0.1 0.15 0.2 0.3 0.35 0.4 0.45 0.5

FIGURE 6 Average drift velocity vs. distance computed byusing the hydrodynamic simulator with transport coefficients ofTable (obtained with Volt) (ooo) and Monte Carlo data (***),with 1.2 Volt bias.

CALIBRATION OF A ONE DIMENSIONAL SIMULATOR 519

TABLE Transport coefficients (in psec-), channel 0.3 tm, Volt bias

a0 a a2 b0 b b2Qp -0.1376 -2.3042 1.3978 1.6078 0.1930 -0.00769Qs 5.7677 27.7312 24.1639 -21.7514 -2.6697 3.5269

7-w 0.4094 (in psec.)- 0.0712 (in psec.)

0.4-

0.35

0.25

0,2

0.15

0.05

Monte Carlo, Hydro 1.2 Volt bias

ooOooOo

o

O0 0.;5 0:1 0.’15 0:2 0.5(micron)

0.3 0.35 0.4 0.45 0.5

FIGURE 7 Average electron energy vs. distance computed byusing the hydrodynamic simulator with transport coefficients ofTable (obtained with Volt) (ooo) and Monte Carlo data(***), with 1.2 Volt bias.

0,15

0"05

-0.05

-0.1

Monte Carlo Hydro 1.2 Volt bias

0.05 0.1 0.15 0.2 0.25 0.35 0.4 0.45 0.5(micron)

FIGURE 8 Heat flux vs. distance computed by using thehydrodynamic simulator with transport coefficients of Table(obtained with Volt) (ooo) and Monte Carlo data (***), with1.2 Volt bias.

Finally we say that the modeling of theproduction terms and the closure of the hydromodel are crucial points. They should be obtainedby using the physics (e.g., an entropy principle,

expansions of the distribution function). Thetransport coefficients can be obtained either byexperiments or by MC simulations, and theirvalidity is restricted to a neighborhood.We are trying to improve our model and to

simulate 2D devices: work along this line is inprogress and will presented elsewhere.

Acknowledgements

This work has been supported by the C.N.R.Progetto Speciale Modelli Matematici per semi-

conduttori 1996, the MURST project 40% and60% 1996.

References

[1] Anile, A. M. and Muscato, O. (1995). Phys. Rev. B., 51,16728.

[2] Anile, A. M. and Muscato, O. (1996). Cont. Mech Therm.,1,1.

[3] Tang, T. W., Ramaswamy, S. and Nam, J. (1993). IEEETrans. Elec. Dev., 40, 1469.

[4] Baccarani, G. and Wordemann, M. E. (1982). Solid StateElec., 29, 970.

[5] Grad, H. (1958). In Thermodynamics of gases edited byS. Flugee, Handerbruch der Physik 12, Springer-Verlag,Berlin.

[6] Laux, S. E., Fischetti, M. V. and Frank, D. J. (1990). IBMJ. Res. Develop., 34.

[7] Rinaudo, S. (1995). A General algorithm to calibrate anykind of simultators without any knowledge about analyticalform of the implemented models, SGS-THOMSON Micro-electronics, internal report.

[8] Falsaperla, P. and Trovato, M. (1997). Preprint Universityof Catania.

Authors’ Biographies

Orazio Muscato is Assistant Professor of Theore-tical Mechanics at Catania University. His re-search interests include mathematical models forsemiconductors, Monte Carlo simulations, wave

propagation.

520 O. MUSCATO et al.

Salvatore Rinaudo is research staff member at Paolo Falsaperla is Graduade student at CataniaST. His research interests include process and University. His research interests include numer-devices numerical simulations and technology ical simulations and mathematical models forCAD. semiconductors.

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