Efficient Solution Scheme Expansion Boltzmann Transport ...

VLSI DESIGN1998, Vol. 6, Nos. (1-4), pp. 239-242Reprints available directly from the publisherPhotocopying permitted by license only

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An Efficient Solution Scheme for the Spherical-HarmonicsExpansion of the Boltzmann Transport Equation Applied

to Two-Dimensional DevicesMARIA CRISTINA VECCHIa, JAN MOHRINGb and MASSIMO RUDANa

aDipartimento di Elettronica, InJbrmatica e Sistemistica, UniversiM di Bologna, Viale Risorgimento 2, 40136 Bologna, ItalybUniversitiit Kaiserslautern, FB Mathematik, Erwin Schr6dingerstraj3e, 67653 Kaiserslautern, Germany

This paper presents a novel numerical scheme applicable to the solution of the Boltzmanntransport equation by means of a spherical-harmonics expansion. This scheme improves thesolution at low energies, keeping the desired accuracy in the calculation of the mean quanti-ties while saving a significant amount of CPU time. This is important in view of the applica-tions of the method, since the typical number of nodes to be used in the combinedspace-energy domain is in the range of 104-105.

Keywords: Submicron Devices, Boltzmann Equation, Spherical-Harmonics Expansion, NumericalMethods, Semiconductor-Device Simulation

1. MATHEMATICAL MODEL

In recent years the spherical-harmonics expansion(SHE) of the Boltzmann transport equation (BTE) hassuccessfully been tested in a wide range of problemsin the field of electron transport simulation 1,2]. Themethod transforms the differential equation in energyand space into a differential equation in space and adifference equation in energy [3]. This allows one to

connect to each point in space only those nodes in

energy coupled via the scattering operator. To achievea detailed knowledge of the distribution function inenergy, a rather fine discretization of the energydomain is necessary. Thanks to the structure of theSHE equation, the energy discretization can bedecomposed in several grids, say L, displaced in theenergy domain where the solution has to be com-

puted. In order to correctly compute the averages

239

(namely concentration, mean energy, mean velocity,etc.), the calculation of the solution on several grids isespecially important in the low-energy range. We pro-pose here a new scheme that improves the solution at

low energies, keeping the desired accuracy in the cal-culation of the mean quantities while saving a signifi-cant amount of CPU-time.

The SHE method in steady state provides the sec-ond-order difference-differential equation [3]

-g- [N+opfo Nopfo } -0, (1)

in the unknown distribution function f0. The equationis solved in a three-dimensional domain (x, y, H) withboundaries defined in space by the boundaries of thedevice and in energy by Hmin =-qtp(x,y) and Hmax

Emax -qtp(x,y), where Emax is the maximum energy of

240 MARIA CRISTINA VECCHI et al.

the band system. A prismatic mesh with triangularelements in the (x,y) plane is adopted. The nodes at

constant total energy H are uniformly spaced in

energy by intervals AH hoop with n integer.Thanks to the absence in (1) of partial derivativeswith respect to H each node is connected along the Hdirection only to the nodes at (x,y, H + hO3op) via thephonon scattering operator. The resulting algebraicsystem is thus decomposed in L decoupled subsys-tems. In the solution scheme we present here, eqn. (1)is solved in the full energy domain [Hmin, Hmax] onlyonce, while several additional solutions (i.e. L 1) are

computed in the reduced energy domain [Hmin,rn Hop], where Hop hcoop q(p(x,y). The terms in

eqn. (1) which are external to [Hmin, rn Hop are

expressed by interpolating the values previously com-puted on the full domain. By letting rn 10 one can

express the values of the distribution function outsidethe domain by an interpolation scheme that maintainsthe linearity of the system; conversely, to compute thelocal solution with rn one has to resort to a more

sophisticated interpolation scheme. Both schemes arebased on a Lagrange polynomial interpolation per-formed on the logarithm of the distribution functionbut, while the first one only involves previously com-

puted values of the distribution function, the secondone involves also the unknowns of the local system. Itshould be added that, in addition to the loss of linear-

ity of the system, a local solution scheme with rndoes not ensure the conservation of the electron cur-

FIGURE Sketch of the fluxes relative to node in the case ofthe one-interval local solution

rent density because of the early truncation of thesolution domain. This can be understood by consider-

ing the fluxes in a simplified one-dimensional case

(Fig. 1). The x axis is in the horizontal direction,hence node and the two adjacent nodes i- 1, +have the same total energy, say Hi. The H axis is inthe vertical direction, hence the energy of node + is

Hi+ ho)op. In particular H is assumed here to repre-sent the lowest energy at location i, namely, there isno other node below node i. The incoming and outgo-ing contributions to the fluxes off0 for node are also

represented in the figure. The dotted arrows indicatesthe fact that the second term in brackets of (1) is miss-

ing from the system matrix when the scheme with

rn is used, because no equation such as (1) is writ-

ten for node +. Still considering the one-interval

scheme, it is worth adding that the value of fi usedin the calculation of the flux i not fully consistent

with foi. This leads to a non-conservation of the car-

rier distribution flux which yields a non-conservation

of the current density at the device contacts. In theory,this phenomenon also takes place when the schemewith rn 10 is adopted; however, it occurs at rela-

tively higher energies, hence its effect on the distribu-

tion function and its derivative is not relevant.

2. RESULTS

We present here, for the case rn 10, the comparisonof the results of this solution scheme (consisting offull solution and 9 local solutions) with those of a fullsolution computed on 10 grids in [Hmin, Hmax]. Thetest device is a 2D-MOS transistor with gate length0.33 lttm. The device has a bulk doping concentrationof 1017 cm-3, a Gaussian p-type channel implant with

a 2 1017 cm-3 peak located at the surface and a

0.04 m standard deviation. The source and drain

junction depth is 0.11 m and the oxide thickness is 7nm. The simulation has been performed with

VGS VDS--3 V, using a non-uniform mesh with tri-

angular elements. Figs. 2 and 3 present the compari-son of the electron concentrations and the normalizedelectron mean energies along a section of the device

parallel to the Si-SiO2 interface, and Fig. 4 compares

AN EFFICIENT SOLUTION SCHEME FOR THE SPHERICAL-HARMONICS EXPANSION 241

10

I0

10

10

10,,

0

SHE-10 full gridsSHE-! full grid and 9 local solutions

0.0 0.2 0.4 0.6 0.8Position tm)

FIGURE 2 Electron concentration along a section parallel to theSi-SiO interface computed with the SHE full-domain and thelocal schemes

the drain currents. For comparison the current com-

puted by HYDRO-HFIELDS [4], using a mobilitymodel consistent with SHE, is also reported in Fig. 4.

In conclusion, the scheme based on the solution of Lsubsystems displaced in the full-energy domain can

be replaced without loss of accuracy by a schemewhich considers a single complete solution and (L- 1)

30

1o

o

SHE-10 full gridsSHE-1 full grid and 9 local solutions

0.0 0.2 0.4 0.6 0.8Position (lain)

FIGURE 3 Normalized electron energy along a section parallel tothe Si-SiO2 interface computed with the SHE full-domain and thelocal schemes

local solutions in the interval [Hmin, rn Hop]. Thanksto its structure, the new method is intrinsically moreefficient: in the example shown here, a reduction fac-tor of about 5 in the CPU-time has been achieved.

FIGURE 4 Output characteristic computedHYDRO-HFIELDS and two solution schemes of the SHE

with

Acknowledgements

This work was part of ADEQUAT (JESSI BTI1) andwas funded as ESPRIT Project 8002. The activity ofM. C. Vecchi has been supported by a fellowship pro-vided by the Integrated Systems Laboratory (IIS) ofthe ETH Z0rich.

References[1] N. Goldsman, L. Henrickson, and J. Frey. A Physics-based

Analytical/Numerical Solution to the Boltzmann TransportEquation for Use in Device Simulation. Solid-state electron-ics, 34(4):389-396, 1991.

[2] A. Gnudi, D. Ventura, G. Baccarani, and E Odeh.Two-dimensional MOSFET Simulation by Means of a Multi-dimensional Spherical Harmonics Expansion of the Boltz-mann Transport Equation. Solid-State Electronics,36(4):575-581, 1993.

[3] D. Ventura, A. Gnudi, and G. Baccarani. MultidimensionalSpherical Harmonics Expansion of Boltzmann Equation forTransport in Semiconductors. Applied Mathematics Letters,5(3):85-90, 1992.

[4] A. Forghieri, R. Guerrieri, P. Ciampolini, A. Gnudi, M.Rudan, and G. Baccarani. A new discretization strategy ofthe semiconductor equations comprising momentum andenergy balance. IEEE Trans. on CAD of ICAS,CAD-7(2):231-242, 1988.

242 MARIA CRISTINA VECCHI et al.

Biographies

Maria Cristina Vecchi received the degree and thePhD in Electrical Engineering from the University ofBologna, Italy, in 1990 and 1995, respectively. Since

1990 she has been working at the Department of Elec-tronics (DEIS) of the same University in the field ofthe numerical simulation of semiconductor devices,with emphasis on the optimization techniques forprocess design and on the simulation of optical sen-

sors. In 1992-93 she visited the IBM T.J. WatsonResearch Center of Yorktown Heights, NY, workingon higher-order transport models in semiconductors.In 1994 she visited the Integrated System Laborato-ries of the ETH, Ziirich, working on the deterministicsolution of the Boltzmann transport equation.Jan Mohring received the degree in industrial

mathematics in 1992 from the University of Kaisers-lautern, Germany. Since then he has been workingtowards the Ph.D. degree in mathematics at the Uni-

versity of Kaiserslautern and, for four months, at theDepartment of Electronics at the University of Bolo-gna, Italy. His thesis will deal with numerical

improvements and a mathematical consolidation of a

deterministic method to solve the Boltzmann equationin device simulation.

Massimo Rudan was born in Bologna, Italy, in

1949. He received a degree in Electrical Engineeringin 1973 and a degree in Physics in 1976, both fromthe University of Bologna, Italy. He joined the

Department of Electronics (DEIS) of the Universityof Bologna in 1975, where he investigated the physi-cal properties of the MOS structures and the problemsof analytical modelling of semiconductor devices.From 1978 he has been teaching an annual course ofQuantum Electronics in the Faculty of Engineering ofthe same University. Since 1983 he has been workingin a group involved in numerical analysis of semicon-ductor devices, acting as Task leader in a number ofEEC-supported Projects in the area of CAD for VLSI.In 1986 he has been a visiting scientist, on a one-yearassignment, at the IBM Thomas J. Watson ResearchCenter at Yorktown Heights, NY. In 1990, he was

appointed Full Professor of Microelectronics at the

University of Bologna.

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