典型MOSFET的次臨界導通電流I_D作為施加的柵極電壓V_GS的函數，並且次臨界擺幅表達式SS(T)=m*ln10*kT/q具有作為溫度T的函數的玻爾茲曼極限，由SS（T）給出= ln10 * kT / q。儘管如此，根據不同製程的低溫溫度研究皆指出，低於20K或更低的SS值至少比Boltzmann極限預測的SS值大上幾個數量級。又根據近年的研究，隨著溫度的降低，接近帶邊緣的界面陷阱密度逐漸增加的概念儼然成為用來理解次臨界擺幅飽和的主流觀念，然而從次臨界區域的SS實驗結果中於超低溫萃取的過高的界面陷阱濃度值仍然存在爭議而引發討論。因此部分研究者開始嘗試重新理解考慮能帶尾模型的次臨界擺幅數學理論，並假設I_D與臨界溫度以下的exp⁡((qV_GS)/W_t )成正比以解釋FET中SS的低溫飽和度（W_t是烏爾巴赫能量）。
在本論文中，我們也採用能態尾部的能態密度理論建構數學模型，來探索PN接面考慮尾態的元件電流其次臨界擺幅飽和度。通過導電（價）帶邊緣的狀態密度乘以費米-狄拉克分佈函數的數值積分來計算其在空乏區邊界上的粒子濃度，使其在中性區域擴散，以模擬PN接面中的帶尾電流。當臨界溫度T_0約等於35/70/105/140K時，其分別對應於W_t等於3/6/9/12meV，次臨界擺幅的飽和值SS(T <T_0 )=ln10*(kT_0)/q=ln10*W_t/q，其由超幾何函數的數學性質也獲得了驗證且巧妙地說明了本論文中PN接面從300K下降到20 K的SS（T）模擬圖形，其和在次臨界區域內匹配MOSFET的SS（T）有著極為相似的結果。我們的一系列分析亦點出當我們進行各種類型的低溫電路設計時，非規則性造成的能帶尾部延伸效應將影響低溫量子元件的開關等性質。

The drain current I_D of a typical MOSFET in subthreshold region is a function of the applied gate voltage V_GS, and the subthreshold swing expression SS(T)=(dV_GS)/(dlogI_D ) has a Boltzmann limit as a function of temperature T, which is determined by SS(T)=m*ln10*kT/q. Nonetheless, recent low-temperature temperature studies based on different processes have pointed out that SS values below 20K or lower are at least several orders of magnitude larger than that predicted by Boltzmann's limit. According to recent studies, as the temperature decreases, the concept that the interface trap density gradually increases close to the edge of the band has become the mainstream concept for understanding sub-threshold swing saturation. However, the excessively high interface trap concentration values extracted from the SS experimental results in the sub-threshold region at the ultra-low temperatures are still controversial and aroused discussion. Therefore, some researchers began to try to re-comprehend the mathematical application of subthreshold swing considering the band tail model, and assume that I_D is proportional to exp⁡((qV_GS)/W_t ) below the critical temperatures to explain the saturation of SS at low temperatures in FET (W_t is the Urbach energy).
In this dissertion, we also use the band-tail DOS theory to construct a mathematical model and explore the subthreshold swing saturation of IV curves considering the tail state of the PN junction. Via calculating the numerical integration of the particle concentration at the boundary of the depletion region by multiplying the state density at the edge of the conduction (valence) band with Fermi-Dirac distribution function and then let it diffuses in the quasi-neutral region to simulate the band-tail current in PN junction. When the critical temperature T_0 is approximately equal to 35/70/105/140K, which respectively correspond to W_t at 3/6/9/12meV, the saturation value of the sub-threshold swing SS(T <T_0 )=ln10*(kT_0)/q=ln10*W_t/q, which is also verified by the mathematical properties of the hypergeometric function and ingeniously illustrates the SS(T ) simulation graph of the PN junction falling from 300K to 20K in this work. Moreover, there is good match to the SS(T ) of the MOSFET in the subthreshold region. Our series of analyses also highlight that when we design various types of low-temperature circuits, the band tail extension effect caused by disorder would affect the switching properties of low-temperature quantum devices.

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