Applicability of Field Model: An Asymptotic Parameters...

VLSI DESIGN1998, Vol. 8, Nos. (1-4), pp. 135-141Reprints available directly from the publisherPhotocopying permitted by license only

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Applicability of the High Field Model:An Analytical Study Via Asymptotic Parameters

Defining Domain Decomposition

CARLO CERCIGNANI a, IRENE M. GAMBA b, JOSEPH W. JEROME c,, and CHI-WANG SHU

Politecnico di Milano, 20133 Milano, Italy;b Courant Institute, New York University, New York, NY 10012,"

Department of Mathematics, Northwestern University, Evanston, IL 60208;a Division of Applied Mathematics, Brown University, Providence, RI 02912

In this paper, we present a mesoscopic-macroscopic model of self-consistent chargetransport. It is based upon an asymptotic expansion of solutions of the BoltzmannTransport Equation (BTE). We identify three dimensionless parameters from the BTE.These parameters are, respectively, the quotient of reference scales for drift and thermalvelocities, the scaled mean free path, and the scaled Debye length. Such parametersinduce domain dependent macroscopic approximations. Particular focus is placed uponthe so-called high field model, defined by the regime where drift velocity dominatesthermal velocity. This model incorporates kinetic transition layers, linking mesoscopicto macroscopic states. Reference scalings are defined by the background doping levelsand distinct, experimentally measured mobility expressions, as well as locallydetermined ranges for the electric fields. The mobilities reflect a coarse substitute forreference scales of scattering mechanisms. See [9] for elaboration.The high field approximation is a formally derived modification of the augmented

drift-diffusion model originally introduced by Thornber some fifteen years ago [25]. Weare able to compare our approach with the earlier kinetic approach of Baranger andWilkins [5] and the macroscopic approach of Kan, Ravaioli and Kerkhoven [20].

Keywords." Asymptotic parameters, mesoscopic-macroscopic model, augmented drift-diffusion,high field model, domain decomposition

1. INTRODUCTION

This presentation is motivated by the search formore reliable macroscopic models of high fieldtransport in submicron structures which are

computationally efficient. This is a research topicof major interest to the micro-electronics industry.

Transport in submicron structures differs fromtransport in bulk material in many ways. Itincludes far from equilibrium situations created

* Corresponding author.

135

136 C. CERCIGNANI et al.

by large electric fields in small structures; ballisticelectrons, when dimensions of the drift-forcingterm are of the same order as the mean free path;and new effects, introduced by the close proximityto the boundaries of regions where the densitygradients are (relatively) rapidly changing.Most of the previous work on transport in

submicron semiconducting structures used severalcorrections to drift-diffusion equations in whichthe current is a sum of drift and diffusion terms,where both have field-dependent mobility anddiffusion coefficients, respectively. References onmodels and computational methods for N+- NN+ structures of Si or GaAs can be found in theliterature of the last decades [24, 23, 16, 21, 25, 20].However, the computations have not yieldedresults which have been genuinely satisfactory,and the main reason has been the lack of under-stading of the accuracy of the modeling in manyheterogeneous structures. An alternative to thedrift-diffusion equations has been Monte Carlosimulation, a very costly computational procedure.An approach intermediate between drift-diffu-

sion and Monte Carlo simulation was generated bythe hydrodynamic like theory, using velocity mo-ments of the Boltzmann equation (see [6, 2, 3, 15])and realizing closure by imposing constitutiverelationships between the state macroscopic vari-ables. Extensive numerical modeling of N+ -N-N+ Si and GaAs structures under several geometricconstraints can be found in [12, 17, 18, 19]. For thederivation ofthese models, the distribution functionis often assumed to be a Maxwellian distribution.However, this assumption has been shown to bequestionable in lightly doped submicron structures,in part because of the presence of ballistic electrons.

However, it has been mathematically provedthat hydrodynamic models relax to energy trans-port models. The latter have been derived asmacroscopic limits associated with a particularchoice of space-time scale that makes somecollision mechanisms dominant (see [10] for a

rigorous proof in the case of the relaxation of thefull hydrodynamic model). These models, how-ever, cannot describe situations where the drift-

forcing term and the resulting scaled mean freepath are of the same order, but they are

asymptotically correct in regions where the pre-defined background is very slowly varying (i.e.,almost constant doping). A new approach at-

tempting to fill in the middle ground among drift-diffusion, hydrodynamic energy transport, andMonte Carlo simulation (see [4, 5, 26]) was theuse of direct, numerical solution of the Boltzmannequation within the relaxation time approxima-tion. The use of the Boltzmann equation fornon-electron transport is justified in the abovereferences, as well as in [7, 13]. The relaxation timeapproximation within the Boltzmann-Poisson sys-tem approach, positing that all scattering pro-cesses can be characterized by a few scatteringrates setting up the scale of relaxation timeconstants, may not be strictly valid for GaAs [11].Our goal, however, is to present a full solvable

model for an N+-N-N+ structure, as analternative to all previous calculations, which willproduce comparable results with greater efficiencythan attained by [4, 5, 20]. In particular, we

incorporate into the modeling the macroscopiclimit associated with the choice of space, time,potential drop, and local electric field scales,combined with the relaxation time scale, whichmakes the drift-forcing term and collisions bothdominant, and of the same scale set up by the localscaled mean free path.As a consequence of this approach, we incorpo-

rate a level of modeling that takes into accountmesoscopic-macroscopic multiscales, and corre-

sponding limit derivations, according to differentscales becoming dominant in different regions ofthe N+ -N-N+ device, depending on the spatialinhomogeneity of the lightly doped region.

2. PRELIMINARIES OF THE HIGHFIELD MODEL

We identify three dimensionless parameters, viz.,

The ratio r/of drift and free velocity (the latterusually taking on the thermal velocity);

HIGH FIELD MODEL: ANALYTICAL STUDY 137

.. The scaled mean free path e andThe scaled Debye length -,/from the electrostaticpotential equation of a self-consistent model.

The dimensionless parameters induce domaindependent macroscopic approximations, which arevalid within regimes that do not change the scaleof the Debye length. This is of paramountimportance in the macroscopic derivation asso-ciated with the BTE under a self-consistent field:the scaling of the forcing term depends on thescaled external and internal fields, and must notoverride the limiting regime of the macroscopicderivation. Therefore, this model incorporateskinetic transition layers linking mesoscopic andmacroscopic states.

Reference scalings are defined by the back-ground doping levels and distinct, experimentallymeasured mobilities which serve to set up the scaleof the relaxation time approximation of thescattering mechanisms. In addition, scales takeinto account locally determined ranges for theelectric fields. Numerical experiments of thisproposed model are also presented by the authorsin [8]. Although this is a first approach to domaindecomposition models, based on a mesoscopic-macroscopic linking, our results seem to give an

improvement (see also [13]) of computationalefficiency, relative to the kinetic computations ofBaranger and Wilkins [5]. Moreover, the modelitself seems to be an improvement over theaugmented drift-diffusion model used by Kan,Ravaioli and Kerkhoven [20]. This model, basedon a model introduced empirically by Thornber[25], employs necessary corrections to the usualmacroscopic approach to high field models.

3. BOLTZMANN-POISSON SYSTEMSCALES

Finally, we concentrate on the derivation of themacroscopic models for semiconductors, based on

asymptotic expansions of multiscale dimensionlessBTE-Poisson systems. For the sake of simplicity

we shall consider only electrons. The semiclassicalBoltzmann-Poisson system, within a parabolicband-relaxation time approximation, may bewritten:

OtF+ v VxF-e

E(x, t) VvF- (F)M F,E -Vq, V. (eVq) e((F) Nd(x)), (1)

where F(x, v, t) is the density function for theelectron at position x, velocity v and time t. Theconstants e, m, - and e represent the electriccharge constant, its mass, the relaxation timereference scale, and permittivity of the material,respectively. The functions b and E of (x, t)represent the electrostatic potential and its field.The bracket (.) denotes the usual average of thedistribution function with respect to the velocityvariable. Finally M= M(v, x) denotes the Max-wellian centered at zero velocity with 0 O(x)representing the background temperature of thelattice, measured in units of specific energy.Derivations of mesoscopic limits associated withsystem (1) depend on a particular choice of space-time scales, collision mechanism scales, and driftvelocity scales induced by self-referencing fieldscales. These regimes, as we shall see, can makesome terms of the BTE dominant. The idea thatfollows then, is to expand the solution of the BTEabout the solution of the grouml equation corre-

sponding to the dominant terms. Of course, this willdepend on the form of scattering mechanisms thatset up the scales of the collision terms. Lately, avery nice survey of a hierarchy of macroscopicmodels for semiconductors under limit regimesthat correspond to dominant collision mechanismshas been presented in [1]. These macroscopicmodels correspond to classical drift-diffusion andso-called energy transport models. They arise fromexpansion about kernels of the collision operators,which are field independent, so they do notincorporate the scales of the external, or evenself-induced electric field. Diffusions are fieldindependent, and consequently they cannot ap-proximate the ballistic electron distribution func-


tions that have been observed by Barenger andWilkins [4, 5]. After the work initiated by [14]and [22] on strong forcing scaling for externalfields, where the dominant term in the scaled Eq.(1) is given by the balance of the force-drift (elm)E(x, t)X7vF and the collision terms, the following 3-scale dimensionless formulation has been formu-lated in [9] in g Euclidean dimensions:

rift + v. VxF+rlvxO VvF-- {(F) M- F}

zxO (e) Ue( ) (2)

7"01/2 is the scaled mean free path for awhere e Tlength scale L and relaxation time r, and 0 is thereference scale of the background energy. Thedimensionless constant r- U/O 1/2, where U, in

7"e, takes intounits of velocity, is given byaccount the reference scale of the local electric

eLfield. Finally, the dimensionless constant -y- ,.[4]represents the scaled Debye length, where scalesthe density of the fixed background Nd(x).Now, a strong force regime corresponds to

values of r/ O(1), -y O(1) and e < L. In thiscase, system (2) is obtained, where the distributionfunction F is scaled with the drift velocity U(instead of the low field scalings that use thethermal velocity scale for F). The time scale is thenfixed at LU-.

4. ASYMPTOTIC LIMITSAND THE HIGH FIELD MODEL

Clearly, any asymptotic expansion is justifiedunder the assumption that the scaling does notbreak down. Thus, any expansion of the BTE-Poisson system (2) requires that the scale of theinternal field defined by the Poisson equation (i.e.,depending on the value of 3’) does not override thescaling of the BTE.

is the reference scale of the predefined Na(x)and will change scale whenever Nd(x) does.Accordingly, so will the local field. Therefore, as

an immediate consequence of the fact that-y q,(#) can be easily checked for GaAs (as in[5, 20]), the dimensionless quantities vary in theN+regions from the N regions. Thus, in N+ re-

gions, r/ e o(1). Because of scaling times andthe distribution function F, accordingly, (2) yieldsthe Drift-Diffusion-Energy Transport regime asthe relative electric field remains moderate and thecollision mechanisms become dominant.However, in the N region it is computed by the

authors in [8] that e (1/7) and r/ O(1). Thus,while 3’ remains of order O(1), it has been shownthat densities and the electric field satisfy theequations,

pt-t- V J- O,J- --T[Vx(lg(O q- 2E @ E))] #oE

e(#,U- w),+’r#EpV. (#e) "r#p-

27. w O, curl w #E (R) 7p,E --VO, (3)

in their dimensionalized formulation. Here # is themobility. The new variable w can be interpreted asthe curl of a magnetic field, associated with thehigh electric current. This model is a correction ofthe one proposed by Thornber [25], where thediffusion and transport coefficients were givenempirically.We point out that in this relaxation time

approximation, the mobility coefficient is givenby # , so clearly the saturation of the mobilitydepends on how the scattering rates scale for r

changing with respect to the electric field. Thestandard assumption based on curve fitting is totake #- C/(1 + C21EI2) 1/2 (see [16]).We stress that this regime is valid only for the

strong force scaling. The transition regime fromweak to strong forcing scaling, and conversely,ought to be done by carefully considering thecorresponding kinetic transition layers. The im-plementation and solution of this transitionproblem require solution of boundary valueproblems at the kinetic and macroscopic level aswell.


5. DOMAIN DECOMPOSITIONAND TRANSITION LAYERS

Appropriate boundary data at the kinetic level andthe corresponding limiting data for the mesoscopicapproximation are presented in [9]. For kineticlevel distribution functions there are prescribedexact neutral space charge conditions at contactboundaries and standard insulating boundaryconditions at insulating walls. Then, these condi-tions, together with the asymptotic approxima-tions, yield Dirichlet boundary conditions at themacroscopic level, with a correction term of theorder of the asymptotic parameter (i.e., the orderof the scaled mean free path e). The contactcondition, at the kinetic level, on a space boundarysection I" reads [9]:

P

on v. n < 0,(4)

where n denotes the outer normal with respect tothe boundary section F, and P denotes the lowestorder term in the expansion for either weak forcescaling (i.e., P is a Maxwellian) or strong forcescaling (i.e., P peaks asymmetrically and is electricfield dependent).The corresponding macroscopic condition for

the inner boundary can be computed explicitly [9]and yields

Here,

p(x) Na(x) + o(e)K(x),

K(x) --(vF)>. n(F_)<. F(_)> n

F___F() + eF(1) + O(ff2), F() <F)P,

f- dv./n<Of dv.

We finally remark that the Ohmic contacts onN+regions correspond to a local scaled mean freepath e of much smaller order than in the N

regions. Hence, at boundary ohmic contacts onN+ regions, condition (5) is almost a neutral spacecharge condition at the macroscopic level, ashistorically has been, and still is typically pre-scribed. However, at the transition layer from theN+ to the N regime, the transfer of the data entailsa kinetic computation using condition (4); then,one prescribes an exact background density, andthus delivers, at the next level, for the strong forcemodel, a condition for the density that is an O(e)derivation from the background density. This factcan actually be observed in the computations ofBarenger and Wilkins [4, 5].Numerical implementations of strong-weak

forcing decomposition are presented by theauthors [8] in this same issue. However, we usethere a matching with hydrodynamic computa-tions of well accepted performance. In Figure 3 of[8], we present a graph of r/, which clearlycorrelates with the suggested domain decomposi-tion. Theoretical justifications and alternativenumerical methods of these domain decomposi-tion ideas are underway and will be presented infuture work.

Acknowledgements

The second author is supported by the NationalScience Foundation under grant DMS-9623037.The third author is supported by the NationalScience Foundation under grant DMS-9424464.The fourth author is supported by the NationalScience Foundation under grants ECS-9214488and ECS-9627849, and the Army Research Officeunder grant DAAH04-94-G-0205.

References

[1] Ben Abdallah, N. and Degond, P. (1996). On a hierarchyof macroscopic models for semiconductors. J. Math.Phys., 37, 3306-3333.

[2] Anile, A. M. and Muscato, O. (1995). Improved hydro-dynamical model for carrier transport in semiconductors.Phys. Rev. B, 51, 16728-16740.

[3] Baccarani, G. and Wordeman, M. R. (1985). Aninvestigation of steady-state velocity overshoot effects inSi and GaAs devices. Solid State Eleetr., 28, 407-416.


[4] Barenger, H. U. and Wilkins, J. W. (1984). Ballisticelectrons in an inhomogeneous submicron structure:Thermal and contact effects. Phys. Rev. B, 30, 7349-7351.

[5] Barenger, H. U. and Wilkins, J. W. (1987). Ballisticstructure in the electron distribution function of smallsemiconducting structures: General features and specifictrends. Phys. Rev. B, 36, 1487-1502.

[6] Blotekjaer, K. (1970). Transport equations for electrons intwo-valley semiconductors. IEEE Trans. Electron Devices,17, 38-47.

[7] Cercignani, C. (1987). The Boltzmann Equation and itsApplication. Springer-Verlag, New York.

[8] Cercignani, C., Gamba, I. M., Jerome, J. W. and Shu,C.-W., Applicability of the high field model: A prelimi-nary numerical study. This volume.

[9] Cercignani, C., Gamba, I. M. and Levermore, C. D.,(1997). High field approximations to Boltzmann-Poissonsystem boundary conditions in a semiconductor. Appl.Math. Lett., 10, 111 117.

[10] Chen, G.-Q., Jerome, J. and Zhang, B., Existence and thesingular relaxation limit for the inviscid hydrodynamicenergy model. Proceedings, Workshop on Modeling andComputation for Applications in Mathematics, Science andEngineering, Oxford University Press, to appear.

[11] Conwell, E. (1982). Handbook of Semiconductors, 1(W. Paul, ed.), p. 513, North-Holland.

[12] Fatemi, E., Jerome, J. and Osher, S. (1991). Solution ofthe hydrodynamic device model using high-order non-oscillatory shock capturing algorithms. IEEE Trans.Computer-Aided Design ofIntegrated Circuits and Systems,10, 232- 244.

[13] Ferry, D. K. and Grondin, R. O. (1991). Physics ofSubmicron Devices. Plenum Press, New York.

[14] Frosali, G., van der Mee, C. V. M. and Paveri Fontana, S.L. (1989). Conditions for run-away phenomena in thekinetic theory of particle swarms. J. Math. Phys., 3, 1177-1186.

[15] Gnudi, A., Odeh, F. and Rudan, M. (1988). An efficientdiscretization scheme for the energy continuity equation insemiconductors. In Proceedings of SISDP, 387-390.

[16] Jerome, J. W. (1996). Analysis of Charge Transport; AMathematical Theory of Semiconductor Devices. Springer.

[17] Jerome, J. W. and Shu, C.-W. (1994). Energy models forone-carrier transport in semiconductor devices. In Semi-conductors, Part II, 59 of IMA Volumes in Mathematicsand its Applications (W. M. Coughran, J. Cole, P. Lloydand J. K. White, eds.), pp. 185-207. Springer-Verlag,New York.

[18] Jerome, J. W. and Shu, C.-W. (1995). The response of thehydrodynamic model to heat conduction, mobility, andrelaxation expressions. VLSI DESIGN, 3, 131-143.

[19] Jerome, J. W. and Shu, C.-W. (1995). Transport effectsand characteristic modes in the modeling and simulationof submicron devices. IEEE Trans. Computer-AidedDesign of Integrated Circuits and Systems, 14, 917-923.

[20] Kan, E. C., Ravaioli, U. and Kerkhoven, T. (1991).Calculation of velocity overshoot in submicron devicesusing an augmented drift-diffusion model. Solid-StateElectr., 34, 995- 999.

[21] Markowich, P. A., Ringhofer, C. A. and Schmeiser, C.(1990). Semiconductor Equations. Springer-Verlag, Vienna.

[22] Poupaud, F. (1992). Runaway phenomena and fluidapproximation under high fields in semiconductor kinetictheory. Z. Angew. Math. Mech., 72, 359-372.

[23] Selberherr, S. (1984). Analysis and Simulation of Semi-conductor Devices. Springer-Verlag, New York.

[24] Sze, S. M. (1981). Physics of Semiconductor Devices. JohnWiley, New York.

[25] Thornber, K. K. (1983). Current equations for velocityovershoot. IEEE Electron Device Lett., 3, 69-71.

[26] Trugman, S. A. and Taylor, A. J. (1986). Analytic solutionof the Boltzmann equation with applications to electrontransport in inhomogeneous semiconductors. Phys. Rev.B, 33, 5575- 5584.

Authors’ Bigraphies

Carlo Cercignani received his Univ. Milano LaureaPhysics in 1961 and his Univ. Milano LaureaMathematics in 1963. From 1963 to 1966 he wasAssistant and Associate Professor of PlasmaPhysics at the University of Milano. He was a

Visiting Associate Professor of Applied Mathe-matics at M.I.T. during 1966-67. He came toPolitecnico di Milano in 1968 and has beenProfessor in Theoretical Mechanics since 1975.His scientific specialties include the kinetic theoryof gases and its applications to rarefied gasdynamics, neutron transport and semiconductors;nonequilibrium statistical mechanics, fluid dy-namics and singular integral equations. Hiscurrent research interests are analytical andnumerical treatment of kinetic and transportequations for gases and semiconductors. He is amember of Accademia Nazionale dei Lincei,Istituto Lombardo and Acad6mie des Sciences deParis. In addition he is a member of the NationalCommittees for Mathematics of the Italian Re-search Council (C.N.R., President), ScientificCommittee of the National Group for Mathema-tical Physics (G.N.F.M.) of C.N.R., a member ofGeneral Assembly of IUTAM, General Assemblyof IMU, Congress Committee of IUTAM, Scien-tific Committee of 1st European MathematicalCongress, and Advisory Committee for Symposiaon Rarefied Gas Dynamics. He serves on thefollowing editorial boards (1993): Fluid DynamicsResearch, Transport Theory and Statistical Phy-sics, MECHANICS Research Communications,Annali di Matematica Pura e Applicata, AppliedMathematics Letters, European Journal of Me-


chanics B (Ass. Ed.), Nonlinear Differential Case Western Reserve University during 1968-Equations and Applications, Surveys in Mathe- 70. He joined Northwestern University in 1970,matics for Industry, Continuum Mechanics and where he has been Professor of Mathematics andThermodynamics. He has authored or coauthored Applied Mathematics since 1976 He has heldseven scientific books and more than two hundred sabbatical positions at Oxford University, Eng-scientific publications. He is the recipient of the land, 1974-75, University of Texas, 1978-79Gold Medal for Mathematics of the Accademia and Bell Laboratories, Murray Hill, 1982-83. Hedei XL (1982), Prize "Citta’ di Cagliari" for was visiting scholar at the University of ChicagoApplied Mathematics (1992), Docteur Honoris in 1985. He received the Distinguished AlumnusCausa of the University Pierre et Marie Curie Award from Purdue University’s School of(Paris VI) (1992) and the Humboldt Prize (Aca- Science in 1996. His research interests includedemic Year 1995-1996). applied analysis, numerical analysis, computa-

Irene M. Gamba received the Ph.D. degree in tional electronics, and ion transport in biology.Mathematics from The University of Chicago in The most recent of his three books, Analysis of1989. From 1990 to 1991 she was at Purdue Charge Transport, was published by Springer inUniversity as a Visiting Assistant Professor and 1996.Research Associate. She was an Assistant Profes- Chi-Wang Shu received the B.S. degree insor at The College of New Jersey, Trenton, NJ, Mathematics from the University of Science andduring 1991-92. She came to the Courant Technology of China (USTC) in 1982 and the Ph.Institute of Mathematical Sciences, New York D. degree in Applied Mathematics from theUniversity, in 1992. There she was an NSF Post University of California at Los Angeles (UCLA)doctoral Fellow during 1992-94, an Assistant in 1986. He was a post doctoral fellow at theProfessor from 1994-96 and Associate Professor Institute for Mathematics and Its Applicationsduring 1996-97. She has accepted a position as (IMA), University of Minnesota, during 1986-Professor of Mathematics at the University of 1987 and came to Brown University in 1987, whereTexas at Austin beginning in August, 1997. Her he has been Professor of Applied Mathematicsresearch interests include nonlinear analysis ap- since 1996. He was a co-recipient of the Chineseplied to gas dynamics and charged-particle trans- Academy of Science Award in Numerical Analysisport systems in the mathematical modeling of and Scientific Computing in 1995. His researchmicroelectronic devices, interests include numerical analysis, scientific

Joseph W. Jerome received the Ph.D. degree in computing and computational physics. He cur-Mathematics from Purdue University in 1966. He rently serves on the editorial boards of Mathe-was visiting Assistant Professor at the Mathe- matics of Computation, SIAM Journal onmatics Research Center, University of Wisconsin, Numerical Analysis and Journal of Computationalduring 1966-68 and was Assistant Professor at Mathematics.

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