本文關(guān)鍵詞： 輸運過程 玻爾茲曼方程 流體力學(xué)方程 魏格納方程 矩展開 累積展開 Maple仿真　出處：《廣東工業(yè)大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

【摘要】：本課題對納米半導(dǎo)體器件仿真輸運理論進(jìn)行研究,以傳統(tǒng)、成熟的擴散-漂移為基礎(chǔ)的輸運理論已經(jīng)無法滿足目前納米半導(dǎo)體器件開發(fā)和仿真,急需通用、準(zhǔn)確、計算效率高的包含非平衡載流子輸運理論。流體力學(xué)模型仿真方法比傳統(tǒng)的擴散一漂移方法增加了高階偏微分方程,可以獨立描述器件小型化所帶來的載流子溫度、熱電子效應(yīng)和部分非平衡態(tài)效應(yīng)。本文從波爾茲曼傳輸方程入手,對玻爾茲曼傳輸方程進(jìn)行矩展開到前三階,將其轉(zhuǎn)化為粒子流守恒、動量守恒和能量守恒三個偏微分方程組,從而推導(dǎo)出流體力學(xué)方程,該模型適用于非拋物線能帶結(jié)構(gòu)。并詳細(xì)推導(dǎo)出通用流體力學(xué)模型(托馬斯模型)。引入魏格納方程,結(jié)合量子力學(xué)薛定諤方程,對魏格納方程進(jìn)行矩展開,得出量子流體力學(xué)模型。該模型完全考慮量子機械效應(yīng),彌補了半經(jīng)典玻爾茲曼方程所忽視的散射元散射所帶來的量子效應(yīng)。同時,對量子能量傳輸模型進(jìn)行簡化,并利用中心有限差分方法對該模型進(jìn)行數(shù)值離散,確定了電子密度和溫度的關(guān)系。接著引入累積展開,以帶漂移的麥克斯韋分布為例,檢查特征函數(shù)累積分布的有效性。在闡明累積展開相對于矩展開的優(yōu)勢后,得出累與矩的關(guān)系表達(dá)式,同時對波爾茲曼方程進(jìn)行傅里葉變換并提取特征函數(shù),得到關(guān)于累的偏微分方程組。納米器件中的波爾茲曼方程的碰撞項主要由聲子(晶格)散射、界面散射和電離雜質(zhì)散射組成。本文利用量子力學(xué)費米方法結(jié)合適用的載流子能帶結(jié)構(gòu)模型,研究光學(xué)聲子(Optic Phonon)、聲學(xué)聲子(Acoustic Phonon)、界面和電離雜質(zhì)微觀散射率,將微觀量子散射模型代替維象的遷移率模型和弛豫時間近似。利用累積展開,對碰撞項進(jìn)行傅里葉變換并提取特征函數(shù)后得到前三階碰撞累。接著,借助量子力學(xué)微擾知識,以光學(xué)聲子為例,得出碰撞項具體表達(dá)式,擺脫了目前非平衡輸運理論中的不準(zhǔn)確的弛豫時間近似。最后,利用Maple數(shù)學(xué)仿真軟件,分別對分布函數(shù)接近高斯分布時的系統(tǒng)特征函數(shù)、矩展開量與累積展開量關(guān)系表達(dá)式以及對玻爾茲曼方程進(jìn)行傅里葉變換后提取特征函數(shù)進(jìn)行模擬,驗證結(jié)果的準(zhǔn)確性。
[Abstract]:In this paper, the simulation transport theory of nanoscale semiconductor devices is studied. The transport theory based on traditional and mature diffusion-drift can not meet the current development and simulation of nanoscale semiconductor devices, so it is urgently needed to be universal and accurate. The computational efficiency includes the theory of non-equilibrium carrier transport. Compared with the traditional diffusion-drift method, the numerical simulation of fluid dynamics model increases the higher order partial differential equation, and can independently describe the carrier temperature brought by the miniaturization of the device. Hot electron effect and partial nonequilibrium state effect. In this paper, the Boltzmann transport equation is expanded to the first three order from the Boltzmann transport equation, which is transformed into three partial differential equations: particle flow conservation, momentum conservation and energy conservation. Thus, the hydrodynamic equation is derived, which is suitable for the non-parabolic band structure. A general hydrodynamic model (Thomas model) is derived in detail. The Wigner equation is introduced and the Schrodinger equation of quantum mechanics is combined. The quantum hydrodynamics model is obtained by moment expansion of the Wigner equation, which fully considers the quantum mechanical effect, which makes up for the quantum effect caused by scattering of scattering elements neglected by the semi-classical Boltzmann equation. At the same time, The quantum energy transfer model is simplified, the central finite difference method is used to discretize the model, and the relationship between electron density and temperature is determined. Then, the cumulative expansion is introduced, and the Maxwell distribution with drift is taken as an example. Check the validity of cumulative distribution of eigenfunction. After clarifying the advantage of cumulative expansion over moment expansion, the relation expression of cumulant and moment is obtained. At the same time, the Fourier transform of Boltzmann equation is carried out and the characteristic function is extracted. The collision term of the Boltzmann equation in nanoscale devices is mainly scattered by phonons (lattice). The composition of interface scattering and ionizing impurity scattering. In this paper, using the Fermi method of quantum mechanics combined with the applicable carrier band structure model, we study the optical phonon Optic Phonon, acoustic phonon acoustic phonon, the microscopic scattering rate of interface and ionization impurity. The microscopic quantum scattering model is used instead of the dimensional mobility model and relaxation time approximation. By using the cumulative expansion, the collision term is transformed by Fourier transform and the characteristic function is extracted to obtain the first three order collision tireds. Then, with the help of quantum mechanical perturbation knowledge, Taking the optical phonon as an example, the concrete expression of the collision term is obtained, which gets rid of the inaccurate relaxation time approximation in the current nonequilibrium transport theory. Finally, the Maple mathematical simulation software is used. The characteristic functions of the distribution function close to Gao Si's distribution, the expression of the relationship between the moment expansion and the cumulative expansion, and the feature function extracted from Boltzmann equation after Fourier transform are simulated, respectively, to verify the accuracy of the results.
【學(xué)位授予單位】：廣東工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：TN303

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