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Computer Physics Communications 169 (2005) 309–312

www.elsevier.com/locate/cp

A numerical iterative method for solving Schrödinger and Poisequations in nanoscale single, double and surrounding gat

metal-oxide-semiconductor structures

Yiming Li a,b,∗, Shao-Ming Yuc

a Department of Computational Nanoelectronics, National Nano Device Laboratories, Hsinchu 300, Taiwanb Microelectronics and Information Systems Research Center, National Chiao Tung University, Hsinchu 300, Taiwan

c Department of Computer and Information Science, National Chiao Tung University Hsinchu 300, Taiwan

Available online 8 April 2005

Abstract

Numerical solution of the Schrödinger and Poisson equations (SPEs) plays an important role in semiconductor siWe in this paper present a robust iterative method to compute the self-consistent solution of the SPEs in nanoscaoxide-semiconductor (MOS) structures. Based on the global convergence of the monotone iterative (MI) method inthe quantum corrected nonlinear Poisson equation (PE), this iterative method is successfully implemented and tessingle-, double-, and surrounding-gate (SG, DG, and AG) MOS structures. Compared with other approaches, varioussimulations are demonstrated to show the accuracy and efficiency of the method. 2005 Elsevier B.V. All rights reserved.

PACS: 72.15.Rn; 73.40.Ty; 73.40.Qv; 02.70.Fj; 02.70.-c

Keywords: Schrödinger and Poisson equations; Quantum corrected Poisson equation; Numerical iterative method; Monotone iterativNanoscale MOS structures

toingDG,er-

178,

on-

ntopro-ialsectsed

rob-

1. Introduction

Advanced process technology has allowed usfabricate diverse semiconductor devices includnanoscale MOS structures. For nanoscale SG,and AG MOS structures, the displacement of the inv

* Corresponding author. Postal address: P.O. Box 25-Hsinchu 300, Taiwan.

E-mail address: ymli@mail.nctu.edu.tw(Y. Li).

0010-4655/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.cpc.2005.03.069

sion carrier density away from the interface of silicand silicon dioxide (Si/SiO2) due to the quantum mechanical (QM) effect cannot be neglected[1–3]. Thusany accurate analysis must take the QM effect iconsideration. A set of the SPEs subject to an appriate boundary condition at the interface of materplays an accurate way of incorporating the QM eff[1–3]. Various solution techniques have been propofor solving only the SG MOS structures[1–3]. How-ever, these methods may encounter divergence p

.

310 Y. Li, S.-M. Yu / Computer Physics Communications 169 (2005) 309–312

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Fig. 1. (a) The computed electron density and (b) the potential othree MOS structures. The inset figures, shown inFig. 1(a), are thecross-sectional views of the MOS structures.

lem when applied to simulate the DG and AG MOstructures.

In this paper, we propose a robust numericalerative method for solving the SPEs in the SG, Dand AG MOS structures shown in the inset ofFig. 1.This unified iterative scheme successfully integrathe MI method[4] and the potential relaxation update method[1], where the effect of strong quantucorrection charge density[2] is taken into consideration simultaneously. First of all the nonlinear PEsolved with the MI method instead of the Newtoniterative method[4]. The calculated potential(φ) en-ergy, after a relaxation update, is used in the sotion of the Schrödinger equation (SE). The calculaelectron wavefunctions(ζjk) and eigenenergies(Ejk)

are applied to estimate the electron density and

derivatives. The calculated quantum corrected etron density is feedback into the PE and we solvePE with the MI method. Iteration is terminated wha self-consistent solution is obtained. Compared wother well known approaches[1,2], application of thismethod to simulate the nanoscale SG, DG, andMOS structures under various testing conditionsconfirmed the robustness of the method. In Sectio2,we state the SPEs for the SG, DG, and AG MOS strtures and discuss the iterative method. In Section3, wediscuss the results. Finally, we draw conclusions.

2. The Schrödinger and Poisson equations and theiterative method

To calculate the potential distribution and electrodensity in the nanoscale SG, DG, and AG MOS strtures, shown in the inset ofFig. 1, we consider thefollowing SPEs[1–3] in a unified formation

(1)d2φ(r)

dr2+ m

r

dφ(r)

dr= − q

εsi(p − n + ND − NA)

and

− h̄2

2mrk

(d2

dr2+ m

r

d

dr

)ζjk(r) + Ec(r)ζjk(r)

(2)= Ejkζjk(r),

where Eq.(1) is the PE and Eq.(2) is the SE in the ef-fective mass approximation. The PE is solved in whMOS structure and the SE is only solved in the regof the silicon films.m = 0 is for the SG and DG MOSstructures in the Cartesian coordinate andm = 1 is forthe AG MOS structure in the cylindrical coordinatn = ncl + nqm, where forncl , we consider the Feri–Dirac distribution and a two-dimensional electron gapproximation is used to calculate the quantum etron densitynqm. Eq. (1) is with the Dirichlet typeboundary condition on the boundary of the simulastructures. For Eq.(2), the electron wavefunction is asumed to be vanished on the interface of Si/SiO2. Allnotations used in Eqs.(1) and (2)are having their conventional physical meaning[1–4]. We briefly outlinethe proposed iterative method as follows:

(1) Solve Eq.(1) using the MI method,(2) Solve Eq.(2),(3) Estimate the derivative ofn,

Y. Li, S.-M. Yu / Computer Physics Communications 169 (2005) 309–312 311

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(4) Solve Eq.(1) with the MI method,(5) The computed potential is relaxing updated, an(6) Return to the step (2).

Iteration is terminated when the solution convergWe note the MI method converges monotonicawhen solves the nonlinear PE[4].

3. Results and discussion

Fig. 1 shows the computed potential distributiand electron’s density from the interface of Si/SiO2 forthe three MOS structures, respectively. In our calcution, eight subbands are computed in solving Eq.(2).As shown in the figure, the oxide thicknessTox, thesubstrate dopingNA, temperatureT , and gate volt-ageVG are with the same settings for all structurThe thickness of the silicon filmTsi for SG is equalto 100 nm andTsi = 20 nm for both the DG andAG MOS structures.Fig. 1(a) shows the superiority of the AG MOS among the MOS structures. Thigher electron, is due to good channel controllaity in AG MOS structure. To verify the robustneof the iterative method, we have studied the convgence of our method and compare with other two wknown algorithms: the first algorithm (Algorithm 1updates the computed potential using an explicitlaxation method[1] and the second one (Algorithm 2calculates the derivative ofnqm by evaluating its previ-ous computed quantities[2]. Comparison is subject tthe same device configurations and settings. We hexamined the convergence behavior of these thregorithms in terms ofTox, NA, T , VG, andTsi. Shownin Fig. 2, it is found that our method converges fall cases of simulation. However, Algorithms 1 andhave suffered different divergent problems, shownFig. 2(a)–(d), when solving the SPEs in the SG, Dand AG MOS structures underT = 300 K andT =77 K, respectively.

4. Conclusions

We have present an effective iterative methodthe self-consistent solution of the SPEs in nanosSG, DG, and AG MOS structures. Our method scessfully integrates the MI method and the poten

Fig. 2. The maximum norm errors of the computed potential vethe number of iteration for the three algorithms applied to SG, Dand AG MOS structures, respectively.

relaxation update method, where the effect of strquantum correction charge density was taken into csideration simultaneously. The solution methodshown good numerical stability and robust convgence for different simulation cases.

Acknowledgements

This work is supported in part by the NationScience Council of Taiwan under contracts NSC-2215-E-429-008 and NSC 93-2752-E-009-002-PAthe grant of the Ministry of Economic Affairs, Tawan under contract No. 93-EC-17-A-07-S1-0011, athe grant from Taiwan Semiconductor Manufactur

Company, Hsinchu, Taiwan in 2004–2005.

312 Y. Li, S.-M. Yu / Computer Physics Communications 169 (2005) 309–312

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