We apply the Monte Carlo, stochastic Galerkin, and stochastic collocation methods to solving the drift-diffusion equations coupled with the Poisson equation arising in semiconductor devices with random rough surfaces. Instead of dividing the rough surface into slices, we use stochastic mapping to transform the original deterministic equations in a random domain into stochastic equations in the corresponding deterministic domain. A finite element discretization with the help of AFEPack is applied to the physical space, and the equations obtained are solved by the approximate Newton iterative method. Comparison of the three stochastic methods through numerical experiment on different PN junctions are given. The numerical results show that, for such a complicated nonlinear problem, the stochastic Galerkin method has no obvious advantages on efficiency except accuracy over the other two methods, and the stochastic collocation method combines the accuracy of the stochastic Galerkin method and the easy implementation of the Monte Carlo method.
AskeyR, WilsonJ. Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials. Mem Amer Math Soc, No 319. Providence: Amer Math Soc, 1985
2
BabuškaI, TemponeR, ZourarisG E. Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J Numer Anal, 2004, 42(2): 800-825 doi: 10.1137/S0036142902418680
3
Bäck,J, NobileF, TamelliniL, TemponeR. Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison. In: Proceedings of the International Conference on Spectral and High Order Methods (ICOSAHOM 09). Berlin: Springer-Verlag, 2010
4
BankR E, RoseD J. Global approximate Newton methods. Numer Math, 1981, 37: 279-295 doi: 10.1007/BF01398257
5
BankR E, RoseD J, FichtnerW. Numerical methods for semiconductor device simulation. SIAM J Sci Stat Comput, 1983, 4: 416-435 doi: 10.1137/0904032
6
BejanA. Shape and Structure, from Engineering to Nature. New York: Cambridge Univ Press, 2000
7
BürglerJ F, BankR E, FichtnerW, SmithR K. A new discretization for the semiconductor current continuity equations. IEEE Trans Comput-Aided Design Integrat Circuits Sys, 1989, 8: 479-489 doi: 10.1109/43.24876
8
BürglerJ F, ConghranW M, FichtnerJr W. An adaptive grid refinement strategy for the drift-diffusion equations. IEEE Trans Comput-Aided Design Integrat Circuits Sys, 1991, 10: 1251-1258 doi: 10.1109/43.88921
9
DebM K, BabuškaI M, OdenJ T. Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput Meth Appl Mech Engrg, 2001, 190: 6359-6372 doi: 10.1016/S0045-7825(01)00237-7
10
ElmanH C, MillerC W, PhippsE T, TuminaroR S. Assessment of collocation and Galerkin approaches to linear diffusion equations with random data. Int J Uncertainty Quant, 2011, 1(1): 19-33 doi: 10.1615/Int.J.UncertaintyQuantification.v1.i1.20
11
FishmanG. Monte Carlo: Concepts, Algorithms, and Applications. New York: Springer-Verlag, 1996 doi: 10.1007/978-1-4757-2553-7
LoèveM. Probability Theory. 4th ed. Berlin: Springer-Verlag, 1977
16
NovakE, RitterK. High dimensional integration of smooth functions over cubes. Numer Math, 1996, 75: 79-97 doi: 10.1007/s002110050231
17
NovakE, RitterK. Simple cubature formulas with high polynomial exactness. Constructive Approx, 1999, 15: 499-522 doi: 10.1007/s003659900119
18
SmolyakS A. Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl Akad Nauk SSSR, 1963, 148: 1042-1045(in Russian); Soviet Math Dokl, 1963, 4: 240-243
19
TaurY, NingH. Fundamentals of Modern VLSI Devices. 2nd ed. Cambridge: Cambridge Univ Press, 2009
20
Van TreesH L. Detection, Estimation, and Modulation Theory, Part I. New York: Wiley, 1968
21
WasilkowskiG, WozniakowskiH. Explicit cost bounds of algorithms for multivariate tensor product problems. J Complexity, 1995, 11: 1-56 doi: 10.1006/jcom.1995.1001
XiuD, HesthavenJ. High-order collocation methods for differential equations with random inputs. SIAM J Sci Comput, 2005, 27: 1118-1139 doi: 10.1137/040615201
24
XiuD, KarniadakisG E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput, 2002, 24(2): 619-644 doi: 10.1137/S1064827501387826
25
XiuD, KarniadakisG E. Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput Methods Appl Math Engrg, 2002, 191: 4927-4948 doi: 10.1016/S0045-7825(02)00421-8
26
XiuD, KarniadakisG E. Modeling uncertainty in flow simulations via generalized polynomial chaos. J Comput Phys, 2003, 187: 137-167 doi: 10.1016/S0021-9991(03)00092-5
27
XiuD, TartakovskyD M. Numerical methods for differential equations in random domain. SIAM J Sci Comput, 2006, 28: 1167-1185 doi: 10.1137/040613160
28
YuS, ZhaoY, ZengL, DuG, KangJ, HanR, LiuX. Impact of line-edge roughness on double-gate Schottky-barrier filed-effect transistors. IEEE Trans Electron Devices, 2009, 56(6): 1211-1219 doi: 10.1109/TED.2009.2017644
