A Quasi-Two-Dimensional Analytical Model Of MOSFET's Threshold Voltage Base On Calculus Of Variations

With the rapid development of semiconductor technology, integrated circuits become more integrated, faster and greater information storage capacity. In order to meet these growing needs, MOS devices, the basic unit of integrated circuits, must continually shrink their size and channel length. The electrical properties of these devices cannot be simply analyzed by traditional one-dimensional methods now, and hence lots of two-dimensional methods have emerged.In this paper, we first established a normal MOS model, and then use the interaction of carriers on semiconductor surface and electric field to established energy integral of carriers. We proved that differential equations about quasi-two-dimensional threshold voltage are the approximate case of the energy integral. After that, we use the methods of direct variation to solve the expression of surface potential, and then we get a simply threshold voltage expression of ultra deep submicron MOSFET. In addition, we use the method mentioned above to establish integrations of LDMOS and solve it to get a threshold voltage expression. So that we theoretically proved that the quasi-two-dimensional variation methods in this paper can applicable to different MOSFET models.We use different parameters to simulate the same model by the software Medici. Then we use these parameters to theoretically calculate the expression and compared the outcome with the simulation results, and then we get a highly consistent result. We extract parameters which come from the particular MOS in the ADS software and use it to simulate in the Medici. The simulation results are also highly consistent with the result what we calculate. Meanwhile, we have also proved that quasi-two-dimensional threshold voltage differential equation is the special case of the energy integral variational result, which shows the variational energy integral solution is a high precision.