本論文的主旨為利用 C 語言去模擬 1/4 球面網格之半導體元件，一開始利用
有限元素法去建立最基本的網格，進而將網格構成的單位元素等效成等效電路模
型以便將方程式(如帕松方程式和電子電洞連續方程式)帶入電路模型中進行運
算求所需之物理和電路特性。在架構建立完之後，可以透過網路上的動態幾何代
數軟體(GeoGebra)去塑造所期望的 1/4 球面網格，先在球座標上架構所需的點數
到運用體積比驗證我們的 1/4 球面網格建構是否精準，最後模擬我們想要的元件
特性曲線，可得知結果與理論值相差無幾，說明了我們此套開發的程式模擬架構
是能應用在 1/4 球面網格半導體元件的模擬。

The main purpose of this thesis is to use C language to simulate the semiconductor
element of the 1/4 spherical grid. In the beginning, the finite element method is used to
establish the most basic grid, and then the unit element grid can transform into an
equivalent circuit, which can convert the Equations (such as the Poisson equation and
the electron-hole continuity equation) to obtain the required physical and circuit
characteristics. After the architecture is established, we can use the dynamic geometric
algebra software (GeoGebra) to shape the 1/4 spherical grid. First, we set the nodes in
spherical coordinates and then used the volume ratio to verify the accuracy of the
1/4 spherical grid structure. Finally, we can simulate the characteristic curve we want
to show that the simulation results are almost the same as the theoretical values, which
shows that the program simulation framework we developed can be applied to the
simulation of 1/4 spherical grid semiconductor devices.

[1] L Baker, “C Mathematical Function Handbook”, Computing McGraw-Hill,
Oct.1991.
[2] T. D. Pauw and W. F. Pfeffer, “The Divergence Theorem for Unbounded Vector Fields”, Transactions of the American Mathematical Society, Vol. 359, No. 12, pp. 5915 - 5929, American Mathematical Society, Dec. 2007.
[3] R.A.Jabr, M. Hamad and Y.M. Mohanna, “Newton-Raphson Solution of Poisson’s Equation in a PN Diode”, International Journal of Electrical Engineering Education, Jan. 2007.
[4] N. Mohamed and M. Z. Sujod, “Finite Elements in Semiconductor Devices ”, 2009 International Conference on Information Management and Engineering, pp.108-110, Apr. 2009.
[5] Y.T.Liao, “3D Bridged Cube Element and Matrix Coefficient Verification and Its Applications to Semiconductor Device Simulation” National Central University, M.S.Thesis, June.2021.
[6] R.E.Bank, D.J.Rose, and W.Fichtner, “Numerical methods for semiconductor device”, IEEE Trans, Electron Devices, Sep.1983.
[7] Y.P.Chen, “3D grounded Cube Element and Matrix Coefficient Verification and Its Applications to Semiconductor Device Simulation” National Central University, M.S.Thesis, June.2021.
[8] J.F.Lee, “2D grounded Triangle Element and Matrix Coefficient Verification and Its Applications to Semiconductor Device Simulation” National Central University, M.S.Thesis, June. 2021.
[9] S. Micheletti, “Stabilized finite elements for semiconductor device simulation,” Compute & Visual Sci., vol. 3, pp. 177-183, 2001.
[10] D.A.Neamen, “Semiconductor physics and devices” 4th edition. McGraw-Hill, Jan.2011.
[11] M. Marrero-Martín, J. García, B. González and A. Hernández, "Circuit models for PN integrated varactors," IEEE Trans, Palma de Mallorca, pp. 1-4, 2011.
[12] M. Putti and C. Cordes, “Finite Element Approximation of the Diffusion Operator on Tetrahedra”, SIAM J. SCI. COMPUT., Vol. 19, No. 4, pp. 1154 – 1168, Society for Industrial and Applied Mathematics, July 1998.
[13] P. Feldmann and R.A. Rohrer, “Proof of the Number of independent Kirchhoff Equations in an Electrical Circuit”, IEEE Transactions on Circuits and Systems, vol. 38, No. 7, pp.681 - 684, IEEE, Jul. 1991.