雙閘極暨絕緣層上矽場效應電晶體有效位勢量子修正模式的線性迴歸

Abstract

現今在系統晶片積體電路中的半導體元件尺寸已經縮小到奈米刻度的尺寸，隨著尺寸的縮小，因應各種特殊設計，半導體元件中氧化層的厚度也隨之變薄。因為氧化層厚度變薄的因素，在氧化層及通道的界面便產生了能量井，量子效應也就產生。在模擬上我們該如何考慮所謂的量子效應，是一個重要的議題。 傳統上，為了模擬量子效應會加入水丁格(Schrödinger equation)方程式在半導體方程式中。然而，水丁格方程式在數值計算上相當耗時及會有數值收斂上的麻煩，在二維度或三維度空間中邊界條件的設定也不容易。為了避免此方程式在模擬上的困難，許多替代的量子修正模型也陸續被提出，在這許多的模型中，大都還是存在著偏微分方程式。近年來被提出的有效位勢(effective potential)理論，是一個簡單的積分方程式。除此之外，在演算法中也大大的改善了耗時的缺點。不過在有效位勢模型中，存在著一個具有不確定性的變數(波包的標準差，standard deviation of wave packet)。隨著標準差的變化，所模擬得到的結果也會有所差異。為了得到正確的值，吾人利用波松-水丁格方程式的結果為基準，調整波包的標準差以達到兩者的結果最為接近。而另外一個問題隨之出現，隨著元件外加不同的條件(偏壓、氧化層厚度…等等)，標準差的值也會隨之變化。 在此論文中，所探討的元件結構為雙閘極以及絕緣層上矽金屬氧化物半導體場效電晶體為主，探討不同的條件對波包的標準差的影響為何。吾人在各種不同的外加條件下，以波松-水丁格方程式的結果為基準，求出各個不同的波包標準差值。接著利用統計的方法，建立出波包標準差以及各外加條件的模型。首先，我們以散佈圖觀察各外加條件對波包的標準差的關係圖，發現之間並沒有複雜的關係，所以我們建立一個二階的線性模型。經過變數轉換得到不錯的結果。 在此提出的模型在結構，外加條件上有所限制，可以將此模型的適用性擴展到更多結構、或是特性相似的半導體元件上。文章中所提出的統計方法可以廣為應用在其他的半導體元件特性分析上。

Within the next decade or so, it is expected that gate lengths will shrink to 45 nm or less in devices found in integrated circuits. Quantum effects are known to occur in the channel region of MOSFET devices, in which the carriers are confined in a triangular potential well at the semiconductor-oxide interface. How might we expect quantum mechanics to arise in the transport through these small devices? Typically, these effects are quantified by a simultaneous solution of the Schrödinger and Poisson equations, which can be a very time consuming procedure if it needs to be incorporated in realistic device simulations. Besides, different methods are proposed to include quantization effects in simulation of carrier transport in nanoscale devices. For instance, Hansch, MlDA, Van Dort, Density Gradient model … etc. Among these approaches, Density Gradient method are used generally. However, the quantum potential is defined in terms of the second derivative of the square root of local density. Such and approach is highly sensitive to noise in the determination of the local carrier density. Recently, Ferry propose an efficient method, effective potential, to include quantum effects. This approach avoids complex computation. Later, an more complicated effective potential is develop, but it is not included in our discussion. Effective potential method is quite convenient to calculate. However, one variable, standard deviation of wave packet, in the model influence the results quite significantly. Unfortunately, value of this parameter is not known exactly. How to determine the value is an interesting problem. In this thesis, we do some simulations with various conditions to calibrate value of the variable by Schrödinger equation. And try to establish a model of standard deviation of wave packet by using statistical methods. First, we draw the scattering plots and find that correlations between outer conditions and value of standard deviation of wave packet are simple. So we just establish a second order multiple linear model. We get results which are satisfied through power transformation. The model is established corresponding to double-gate and silicon-on-insulator (SOI) MOSFET structures. Though the model is not suitable for any structure, conditions of devices. This method can be expanded to establish other models more generally.