此量子校正的能量運輸模型是由七條自身伴隨的非線性的偏微分方
程式所組成，它所探討的是奈米導體裝置中，電子流和電洞流的穩定性，能量轉換的穩定性，再加上古典電位能和量子電位能的穩定性。我們引用Kopteva [14] 二維空間期後誤差估計的方法在無因次的量子校正的能量運輸模型上，這個誤差估計的方法能幫助我們在網格點適應性加切時，能當作為誤差指標。對於QCET偏微分方程模型，我們在一維二極管模型問題的期後誤差估計數值試驗，已經取得七個很好的誤差指標。根據一維模型數據經驗，在二維模型中加入七個條方程式做為網格點適應性加切的誤差指標。在二維量子校正的能量運輸模型問題中，在相同收斂條件下做網格點適應性加切，數值呈現，新的誤差估計方法比舊的誤差估計方法，總網格數減少了11%。

The quantum-corrected energy transport (QCET) model consisting of seven self-adjoint nonlinear PDEs describes the steady state of electron and hole flows, their energy transport, and classical and quantum potentials within a nano-scale semiconductor device. We develop a second-order maximum norm a posteriori error estimate proposed by Kopteva [14] for the QCET which after scaling involves the scaled Debye length, intrinsic carrier density, Planck constant, and thermal conductivity as the singular perturbation parameters. This estimate can be used as an error indicator for the refinement process in an adaptive algorithm. We present explicit formulas for computing the error indicators which are indispensable for the adaptive computations of the semiconductor device simulation for advanced nano-devices. Our numerical experiments on the a posteriori error estimation for the 1D QCET model problem have shown good results of the proposed error indicators for all seven PDEs of the QCET model. With the 1D QCET Model's numerical results, we take the seven PDEs to make adaptive finite element mesh in the 2D QCET model. For the 2D QCET problem, it is shown that the total number of nodes of the final adaptive mesh using the new estimation method has been decreased to 11% from that of the old method under the same stopping criteria of the adptive algorithm.

[1] M. G. Ancona, G. J. Iafrate, Quantum correction to the equation of state of an electron gas in a semiconductor, Phys. Rev. B 39 (1989)
9536-9540.
[2] M. G. Ancona and H. F. Tiersten, Macroscopic physics of the silicon inversion layer, Phys. Rev. B 35 (1987) 7959-7965.
[3] M. G. Ancona, Z. Yu, R.W. Dutton, P.J.V. Voorde, M. Cao, D. Vook, Density-gradient analysis of MOS tunneling, IEEE Trans. Electron. Dev.47 (2000) 2310.
[4] B. A. Biegel, M. G. Ancona, C. S. Ra¤erty, Z. Yu, E¢cient multi- dimensional simulation of quantum con…nement e¤ects in advanced MOS devices, NAS Tech. Report NAS-04-008, 2004.
[5] R.-C. Chen, J.-L. Liu, A quantum corrected energy transport model for nanoscale semiconductor devices, J. Comput. Phys. 204 (2005) 1347 (2000) 2310.
[6] R.-C. Chen, J.-L. Liu, An iterative method for adaptive …nite element so- lutions of an energy transport model of semiconductor devices, J. Com- put. Phys. 189 (2003) 579-606.
[7] R.-C. Chen, J.-L. Liu, Monotone iterative methods for the adaptive finite element solution of semiconductor equations, J. Comput. Applied Math. 159 (2003) 341-364.
[8] R.-C. Chen, J.-L. Liu, An accelerated monotone iterative method for the quantum-corrected energy transport model, J. Comp. Phys. 227 (2008)6266-6240.
[9] D. Connelly, Z. Yu, D. Yergeau, Macroscopic simulation of quantum mechanical e¤ects in 2-D MOS devices via the density gradient method, IEEE Trans. Electron Devices 49 (2002) 619-626.
[10] D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables I and II, Phys. Rev., 85 (1952) 166-179 and 180-93.
[11] C. de Falco, E. Gatti, A. L. Lacaita, R. Sacco, Quantum-corrected drift- di¤usion models for transport in semiconductor devices, J. Comput. Phys. 204 (2005) 533-561.
[12] C. de Falco, J. W. Jerome, and R. Sacco, Quantum corrected drift- di¤usion models: Solution …xed point map and …nite element approxi- mation, J.Comput. Phys., 228 (2009), pp. 1770-1789.
[13] P. Degond, S. Gallego, F. Méhats, An entropic quantum drift-di¤usion model for electron transport in resonant tunneling diodes, J. Comput. Phys. 221 (2007) 226-249.
[14] N. Kopteva, Maximum norm a posteriori error estimate for a 2d singu- larly perturbed reaction-di¤usion problem, SIAM J. Numer. Anal., 46 (2008), 1602-1618.
[15] D. Vasileska, K. Raleva, and S.M. Goodnick, Modeling heating e¤ects in nanoscale devices: the present and the future, J. Comput. Electron,7 (2008) pp. 179—182.
[16] C. Y. Lin, Maximum Norm A Posteriori Error Estimate For the Quantum-Coreceted Energy Transport Model Part I: Theory, Master thesis, National University of Kaohsiung 2009
[17] N. Ben Abdallah, A. Unterreiter, On the stationary quantum drift- di¤usion model, Z. Angew. Math. Phys. 49 (1998) 251—275.
[18] J.-L. Liu R.-C. Chen C.-T. Lee. A singular perturbed formulation of the quantum-corrected energy transport model, preprint, 2009.
[19] S. Odanaka, Multidimensional discretization of the stationary quantum drift-di¤usion model for ultrasmall MOSFET structures, IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 23 (2004) 837–842.
[20] R. Pinnau, Uniform convergence of an exponentially …tted scheme for the quantum drift di¤usion model, SIAM J. Numer. Anal. 42 (2004)
1648-1668.
[21] E. Pop, S. Sinha, K. E. Goodson, Heat generation and transport in nanometer-scale transistors, Proc. IEEE 94 (2006) 1587-1601.
[22] C. S. Ra¤erty, B. Biegel, Z. Yu, M. G. Ancona, J. Bude, R. W. Dutton, Multi-dimensional quantum e¤ect simulation using a density-gradient model and script-level programming techniques, Proc. SISPAD (1998)
137-140.
[23] S. Carl and J. Jerome, Drift-di¤usion in electrochemistry: thresholds for boundary ‡ux and discontinuous optical generation, Applicable Anal. 83 (2004) 915—931.