CN104076266A

CN104076266A - Method for extracting subthreshold swing of MOSFET of double-material double-gate structure

Info

Publication number: CN104076266A
Application number: CN201410300741.1A
Authority: CN
Inventors: 胡光喜; 向平; 刘冉; 郑立荣
Original assignee: Fudan University
Current assignee: Fudan University
Priority date: 2014-06-27
Filing date: 2014-06-27
Publication date: 2014-10-01

Abstract

The invention belongs to the technical field of semiconductors, in particular to a method for extracting the subthreshold swing of a metal-oxide-semiconductor field-effect transistor with a double-material double-gate structure. The invention obtains the potential distribution of the dual-material double-gate structure MOSFET, and then obtains the analytical model of the sub-threshold swing by using the obtained potential according to the definition of the sub-threshold swing. The sub-threshold swing analytical model is simple in form and clear in physical concept, which provides a fast tool for circuit simulation software to study new dual-material double-gate structure devices.

Description

A Method for Extracting the Subthreshold Swing of MOSFET with Dual-Material and Double-Gate Structure

技术领域technical field

本发明属于半导体技术领域，具体涉及一种提取双材料双栅结构金属-氧化物-半导体场效应晶体管(MOSFET)亚阈值摆幅的方法。The invention belongs to the technical field of semiconductors, and in particular relates to a method for extracting a subthreshold swing of a metal-oxide-semiconductor field-effect transistor (MOSFET) with a double-material double-gate structure.

背景技术Background technique

随着集成电路芯片集成度不断提高，器件几何尺寸不断缩小，MOSFET器件已经逐步从平面结构向非平面立体结构发展。而在各类非传统平面器件结构中，双栅结构MOSFET的栅极控制能力强，能够更好抑制短沟道效应，降低器件的静待功耗。将双材料栅极MOSFET与双栅MOSFET结合,就能结合两者的优点,使得器件有更好的短沟道特性和性能。由于靠近漏端使用功函数较小的材料做栅，可以减小平带电压，增大有效栅压，可以减小漏端沿沟道方向电场，从而减小热载流子效应。由于以上优势，对这种双栅MOSFET结构创建解析模型变得尤为重要，并且其亚阈值摆幅提取模型日益受到工业界关注。传统体硅MOSFET亚阈值摆幅模型已经不再适合，这对于新型多栅纳米器件的建模与模拟带来了新的挑战。With the continuous improvement of integrated circuit chip integration and the continuous reduction of device geometry, MOSFET devices have gradually developed from a planar structure to a non-planar three-dimensional structure. Among all kinds of non-traditional planar device structures, the dual-gate structure MOSFET has strong gate control ability, which can better suppress the short-channel effect and reduce the static power consumption of the device. Combining dual-material gate MOSFET with dual-gate MOSFET can combine the advantages of both, making the device have better short-channel characteristics and performance. Since the gate is made of a material with a smaller work function near the drain end, the flat-band voltage can be reduced, the effective gate voltage can be increased, and the electric field along the channel direction at the drain end can be reduced, thereby reducing the hot carrier effect. Due to the above advantages, it is particularly important to create an analytical model for this dual-gate MOSFET structure, and its subthreshold swing extraction model is increasingly concerned by the industry. The traditional bulk silicon MOSFET subthreshold swing model is no longer suitable, which brings new challenges to the modeling and simulation of new multi-gate nanodevices.

亚阈值摆幅是MOSFET最为重要参数之一，它的定义为：亚阈值区域，电流每变化十倍，栅极偏压所需要变化量。为了使用电路模拟软件能够正确模拟电路特性，建立精确的亚阈值摆幅模型是非常重要的。Sub-threshold swing is one of the most important parameters of MOSFET. It is defined as: in the sub-threshold region, the gate bias voltage needs to change for every ten-fold change in current. In order to correctly simulate circuit characteristics using circuit simulation software, it is very important to establish an accurate subthreshold swing model.

发明内容Contents of the invention

本发明目的在于提供一种方便、正确提取双材料双栅MOSFET亚阈值摆幅的方法。The purpose of the present invention is to provide a convenient and correct method for extracting the sub-threshold swing of a dual-material dual-gate MOSFET.

本发明提供的提取双材料双栅MOSFET亚阈值摆幅的方法，关键是建立形式简洁、物理概念清晰，且精度高的双材料双栅MOSFET亚阈值摆幅解析模型。The key to the method for extracting the sub-threshold swing of the dual-material dual-gate MOSFET provided by the present invention is to establish an analytical model of the sub-threshold swing of the dual-material dual-gate MOSFET with simple form, clear physical concept and high precision.

本发明建立的双材料双栅结构MOSFET亚阈值摆幅解析模型，为电路模拟软件提供一种快速精确解析双栅结构模型。The dual-material double-gate structure MOSFET sub-threshold swing analysis model established by the invention provides a fast and accurate analysis double-gate structure model for circuit simulation software.

具体步骤如下：Specific steps are as follows:

(1)建立双材料双栅MOSFET(1) Build a dual-material dual-gate MOSFET

双材料双栅MOSFET与双栅MOSFET类似，中间是硅。沟道采用p型掺杂，源漏则采用n型重掺杂。栅极采用不对称的结构，其中一个栅极采用两种功函数不同的材料制备。为了得到阶梯型的沟道电势分布，材料M2采用功函数较小的n型重掺杂多晶硅(功函数为4.17eV)，材料M1则采用功函数较高的p型多晶硅(功函数为5.25eV)。两端栅极外接同样的偏压。A dual-material dual-gate MOSFET is similar to a dual-gate MOSFET with silicon in the middle. The channel is doped with p-type, and the source and drain are heavily doped with n-type. The gate adopts an asymmetric structure, and one of the gates is made of two materials with different work functions. In order to obtain a stepped channel potential distribution, material M2 uses n-type heavily doped polysilicon with a small work function (work function is 4.17eV), and material M1 uses p-type polysilicon with a high work function (work function is 5.25eV ). The gates at both ends are externally connected with the same bias voltage.

沟道长度为L一端的栅极被划分为两个部分，分别对应两种功函数不同的栅极材料。材料M1对应的长度为L₁，材料M2对应的长度为L₂，L＝L₁+L₂。材料M1的功函数为5.25eV，材料M2的功函数为4.17eV。t_ox1，t_ox2为前栅和背栅氧化层厚度，t_si为沟道厚度。The gate at one end of which the channel length is L is divided into two parts corresponding to two gate materials with different work functions. The length corresponding to the material M1 is L ₁ , the length corresponding to the material M2 is L ₂ , and L=L ₁ +L ₂ . The work function of the material M1 is 5.25 eV, and the work function of the material M2 is 4.17 eV. t _ox1 and t _ox2 are the thickness of the front gate and back gate oxide layer, and t _si is the thickness of the channel.

(2)求解沟道电势的泊松方程，得到沟道电势(2) Solve the Poisson equation of the channel potential to obtain the channel potential

沟道电势的泊松方程可以表示为：The Poisson equation for the channel potential can be expressed as:

$\frac{{&PartialD; &PartialD;}^{22} φ φ ((x x,, y the y))}{{&PartialD; &PartialD; x x}^{22}} + + \frac{{&PartialD; &PartialD;}^{22} φ φ ((x x,, y the y))}{&PartialD; &PartialD; {y the y}^{22}} = = \frac{q q {N N}_{A A}}{{ϵ ϵ}_{si the si}} - - - - - - ((11))$

其中q为电子电荷，N_A为沟道的掺杂浓度，ε_si为硅的介电常数，φ(x,y)为沟道电势。Where q is the electron charge, N _A is the doping concentration of the channel, ε _si is the dielectric constant of silicon, and φ(x,y) is the channel potential.

根据电场在沟道、氧化层交界面连续以及源、漏两端的电压，边界条件可表示为：According to the continuity of the electric field at the interface between the channel and the oxide layer and the voltage across the source and drain, the boundary conditions can be expressed as:

φ(x＝0,y)＝V_S (2)φ(x=0,y)=V _S (2)

φ(x＝L,y)＝V_S+V_DS (3)φ(x=L,y)=V _S +V _DS (3)

$φ φ ((x x,, y the y = = 00)) = = {V V}_{GFF GFF} + + \frac{{ϵ ϵ}_{si the si}}{{C C}_{ox ox 11}} \frac{&PartialD; &PartialD; φ φ}{&PartialD; &PartialD; y the y} {| |}_{y the y = = 00} - - - - - - ((44))$

$φ φ ((x x,, y the y = = {t t}_{si the si})) = = {V V}_{GFB GFB} + + \frac{{ϵ ϵ}_{si the si}}{{C C}_{ox ox 22}} \frac{&PartialD; &PartialD; φ φ}{&PartialD; &PartialD; y the y} {| |}_{y the y = = {t t}_{si the si}} - - - - - - ((55))$

其中，VS为内建电势，VDS为漏源电压，COX1、COX2分别为上栅和下栅氧化层单位面积电容，V_GFF、V_GFB分别为上栅和下栅有效栅压。Among them, VS is the built-in potential, VDS is the drain-source voltage, COX1 and COX2 are the capacitance per unit area of the oxide layer of the upper gate and the lower gate respectively, and V _GFF and V _GFB are the effective gate voltages of the upper gate and the lower gate respectively.

双栅材料的栅极采用两种功函数不同的材料来形成。由于有效栅压表达式为V_GFF＝V_GS-V_FBF,V_FBF为前栅平带电压，且V_FBF在两种材料中对应不同的值，因此(4)式中的V_GFF应当为一个分段函数。如果定义V_FBF1和V_FBF2分别为两种材料的平带电压的话，那么V_GFF就能够表示为V_GFF＝V_GS-V_FBF1(0<x<L₁),V_GFF＝V_GS-V_FBF2(L₁<x<L₁+L₂)。V_FBF1和V_FBF2可以通过材料的功函数来计算得到。定义r₁＝V_GFB/V_GFF,r₂＝C_ox2/C_ox1，并且代入式(5)，边界条件可以重新表示为：The gate of the dual gate material is formed using two materials with different work functions. Since the effective gate voltage expression is V _GFF =V _GS -V _FBF , V _FBF is the front gate flat band voltage, and V _FBF corresponds to different values in the two materials, so V _GFF in formula (4) should be one piecewise function. If V _FBF1 and V _FBF2 are defined as the flat band voltages of the two materials respectively, then V _GFF can be expressed as V _GFF =V _GS -V _FBF1 (0<x<L ₁ ), V _GFF =V _GS -V _FBF2 (L ₁ <x<L ₁ +L ₂ ). V _FBF1 and V _FBF2 can be calculated from the work function of the material. Define r ₁ =V _GFB /V _GFF , r ₂ =C _ox2 /C _ox1 , and substitute into formula (5), the boundary conditions can be re-expressed as:

$φ φ ((x x,, y the y = = {t t}_{si the si})) = = {r r}_{11} {V V}_{GFF GFF} - - \frac{{ϵ ϵ}_{si the si}}{{r r}_{22} {C C}_{ox ox 11}} \frac{&PartialD; &PartialD; φ φ}{&PartialD; &PartialD; y the y} {| |}_{y the y = = {t t}_{si the si} - - - - - - ((66))}$

解的形式可以表示为：The form of the solution can be expressed as:

$φ φ ((x x,, y the y)) = = {V V}_{S S} + + \frac{{V V}_{DS DS}}{L L} x x + + {Σ Σ}_{n no = = 11}^{\infty \infty} {A A}_{n no} ((y the y)) sin sin \frac{nπx nπx}{L L} - - - - - - ((77))$

其中n为整数，An为待定系数。Among them, n is an integer, and An is an undetermined coefficient.

将式(7)代入泊松方程(1)，可以得到：Substituting equation (7) into Poisson equation (1), we can get:

$\frac{{d d}^{22} {A A}_{n no} ((y the y))}{{dy dy}^{22}} - - {k k}_{n no}^{22} {A A}_{n no} ((y the y)) = = {f f}_{n no} - - - - - - ((88))$

其中k_n＝nπ/L，f_n＝(2qN_A/nπε_si)[1-(-1)ⁿ]。式(8)是一个常微分方程，解为：where k _n =nπ/L, f _n =(2qN _A /nπε _si )[1-(-1) ⁿ ]. Equation (8) is an ordinary differential equation, the solution is:

${A A}_{n no} ((y the y)) = = {C C}_{n no} {e e}^{{k k}_{n no} y the y} + + {D D.}_{n no} {e e}^{- - {k k}_{n no} y the y} - - {f f}_{n no} / / {k k}_{n no}^{22} - - - - - - ((99))$

将式(9)代入边界条件(4)与(6)，然后作傅里叶展开，可以进一步求出系数C_n与D_n分别为：Substituting equation (9) into boundary conditions (4) and (6), and then performing Fourier expansion, the coefficients C _n and D _n can be further obtained as follows:

${C C}_{n no} = = \frac{{C C}_{ox ox 11} [[(({ϵ ϵ}_{si the si} {k k}_{n no} - - {r r}_{22} {C C}_{ox ox 11})) (({f f}_{n no} - - {G G}_{n no} {k k}_{n no}^{22})) + + {r r}_{22} (({C C}_{ox ox 11} + + {ϵ ϵ}_{si the si} {k k}_{n no})) (({f f}_{n no} - - {H h}_{n no} {k k}_{n no}^{22})) {e e}^{{k k}_{n no} {t t}_{si the si}}]]}{{k k}_{n no}^{22} [[{C C}_{ox ox 11} {ϵ ϵ}_{si the si} {k k}_{n no} ((11 + + {r r}_{22})) ((11 + + {e e}^{{22 k k}_{n no} {t t}_{Si Si}})) - - (({r r}_{22} {C C}_{ox ox 11}^{22} + + {ϵ ϵ}_{si the si}^{22} {k k}_{n no}^{22})) ((11 - - {e e}^{{22 k k}_{n no} {t t}_{Si Si}}))]]} - - - - - - ((1010))$

以及as well as

${C C}_{n no} = = {C C}_{ox ox 11} {e e}^{{k k}_{n no} {t t}_{si the si}} \frac{[[(({ϵ ϵ}_{si the si} {k k}_{n no} + + {r r}_{22} {C C}_{ox ox 11})) (({f f}_{n no} - - {G G}_{n no} {k k}_{n no}^{22})) {e e}^{{k k}_{n no} {t t}_{si the si}} - - {r r}_{22} (({C C}_{ox ox 11} - - {ϵ ϵ}_{si the si} {k k}_{n no})) (({f f}_{n no} - - {H h}_{n no} {k k}_{n no}^{22}))]]}{{k k}_{n no}^{22} [[{C C}_{ox ox 11} {ϵ ϵ}_{si the si} {k k}_{n no} ((11 + + {r r}_{22})) ((11 + + {e e}^{{22 k k}_{n no} {t t}_{Si Si}})) - - (({r r}_{22} {C C}_{ox ox 11}^{22} + + {ϵ ϵ}_{si the si}^{22} {k k}_{n no}^{22})) ((11 - - {e e}^{{22 k k}_{n no} {t t}_{Si Si}}))]]} - - - - - - ((1111))$

其中G_n、H_n分别为Among them, G _n and H _n are respectively

${G G}_{n no} = = \frac{22}{nπ nπ} [[{V V}_{S S} ((11 - - {((- - 11))}^{n no})) - - {V V}_{GFF GFF 11} ((11 - - cos cos \frac{nπ nπ {L L}_{11}}{L L})) - - {V V}_{GFF GFF 22} ((cos cos \frac{nπ nπ {L L}_{11}}{L L} - - {((- - 11))}^{n no})) - - {((- - 11))}^{n no} {V V}_{DS DS}]] - - - - - - ((1212))$

${H h}_{n no} = = \frac{22}{nπ nπ} [[(({V V}_{S S} - - {r r}_{11} {V V}_{GFF GFF})) ((11 - - {((- - 11))}^{n no})) - - {((- - 11))}^{n no} {V V}_{DS DS}]] - - - - - - ((1313))$

这样就可以得到双材料双栅的沟道电势表达式。如果引入更多的不同功函数的材料，那么可以用类似的方法求解。In this way, the channel potential expression of the double material double gate can be obtained. If more materials with different work functions are introduced, then similar methods can be used to solve them.

(3)建立双材料双栅MOSFET的亚阀值摆幅解析表达式(3) Establish an analytical expression for the subthreshold swing of a dual-material dual-gate MOSFET

亚阈值摆幅定义为：Subthreshold swing is defined as:

$S S = = \frac{&PartialD; &PartialD; {V V}_{GS GS}}{&PartialD; &PartialD; {log log}_{1010} {I I}_{DS DS}} - - - - - - ((1414))$

在亚阈值区域，双材料双栅MOSFET处于弱反型条件下，源漏电流压可以近似表示成：In the subthreshold region, the dual-material dual-gate MOSFET is under weak inversion conditions, and the source-drain current voltage can be approximately expressed as:

I_ds∝n_min(y) (15)I _ds ∝ n _min (y) (15)

n_min(y)可以表示成 $n_{miin} (y) = (n_{i}^{2} / N_{A}) e^{q φ_{\min} (y) / kT} .$ φ_min为x方向的电势最小值。代入式(15)之后，可以将亚阈值摆幅的表达式重新改写为：n _min (y) can be expressed as ${no}_{miin} (the y) = ({no}_{i}^{2} / N_{A}) e^{q φ_{\min} (the y) / kT} .$ φ _min is the minimum value of the electric potential in the x direction. After substituting into equation (15), the expression of the subthreshold swing can be rewritten as:

$S S = = 2.3 2.3 {V V}_{t t} {[[\frac{&PartialD; &PartialD; φ φ (({x x}_{00},, y the y))}{&PartialD; &PartialD; {V V}_{GS GS}}]]}^{- - 11} - - - - - - ((1616))$

其中V_t＝k_BT/q，k_B为波尔兹曼常数，T为温度。Where V _t =k _B T/q, k _B is Boltzmann's constant, and T is temperature.

将前一部分沟道电势的结果代入(16)式,可以得到亚阈值摆幅的解析表达式为：Substituting the result of the channel potential in the previous part into Equation (16), the analytical expression of the subthreshold swing can be obtained as:

$\begin{matrix} S S = = 2.3 2.3 {V V}_{t t} {{{Σ Σ}_{n no = = 11}^{\infty \infty} \frac{22}{nπ nπ} sin sin ((\frac{nπ nπ {x x}_{00}}{L L})) ((11 - - {((- - 11))}^{n no})) \\ \frac{{C C}_{ox ox 11} [[{e e}^{{k k}_{n no} y the y} (({ϵ ϵ}_{si the si} {k k}_{n no} - - {r r}_{22} {C C}_{ox ox 11})) + + {e e}^{{k k}_{n no} ((22 {t t}_{si the si} - - y the y))} (({ϵ ϵ}_{si the si} {k k}_{n no} + + {r r}_{22} {C C}_{ox ox 11}))}{{C C}_{ox ox 11} {ϵ ϵ}_{si the si} {k k}_{n no} ((11 + + {r r}_{22})) ((11 + + {e e}^{22 {k k}_{n no} {t t}_{Si Si}})) - - (({r r}_{22} {C C}_{ox ox 11}^{22} + + {ϵ ϵ}_{si the si}^{22} {k k}_{n no}^{22})) ((11 - - {e e}^{{22 k k}_{n no} {t t}_{Si Si}}))} {}}}^{- - 11} \end{matrix} - - - - - - ((1717))$

计算时，y可用沟道的有效导电路径y_eff替代，也就是电荷的等效质心，When calculating, y can be replaced by the effective conduction path y _eff of the channel, which is the equivalent centroid of the charge,

${y the y}_{eff eff} = = {&Integral; &Integral;}_{00}^{{t t}_{si the si}} yn yn (({x x}_{00},, y the y)) dy dy / / {&Integral; &Integral;}_{00}^{{t t}_{si the si}} n no (({x x}_{00},, y the y)) dy dy - - - - - - ((1818))$

其中，n_i为本征载流子浓度。y_eff的表达式可改写为in, _ni is the intrinsic carrier concentration. The expression of y _eff can be rewritten as

${y the y}_{eff eff} = = {&Integral; &Integral;}_{00}^{{t t}_{si the si}} {ye yes}^{φ φ (({x x}_{00},, y the y)) / / {V V}_{t t}} dy dy / / {&Integral; &Integral;}_{00}^{{t t}_{si the si}} {e e}^{φ φ (({x x}_{00},, y the y)) / / {V V}_{t t}} dy dy - - - - - - ((1919))$

这样，就得到了双材料双栅MOSFET的亚阈值摆幅解析模型。In this way, the sub-threshold swing analytical model of dual-material dual-gate MOSFET is obtained.

附图说明Description of drawings

图1双材料双栅MOSFET二维结构示意图。Figure 1 Schematic diagram of the two-dimensional structure of a dual-material dual-gate MOSFET.

图2电势在沟道表面的分布情况。Figure 2 The distribution of the potential on the channel surface.

图3在不同的体硅厚度时，双材料双栅MOSFET亚阈值摆幅与沟道长度的变化关系。Figure 3 shows the relationship between the subthreshold swing and the channel length of a dual-material dual-gate MOSFET with different bulk silicon thicknesses.

图4在不同的氧化层厚度时，双材料双栅MOSFET亚阈值摆幅与沟道长度的变化关系。Figure 4 shows the relationship between the subthreshold swing and the channel length of a dual-material dual-gate MOSFET with different oxide layer thicknesses.

图5亚阈值摆幅建模流程示意图。Fig. 5 Schematic diagram of the modeling process of subthreshold swing.

具体实施方式Detailed ways

通过我们的解析模型计算，如图3所示，本发明对双材料双栅MOSFET亚阈值摆幅在不同体硅厚度下随沟道长度变化的解析结果。图4为双材料双栅MOSFET亚阈值摆幅在不同栅氧化层厚度下随沟道长度变化的解析结果。同普通的双栅MOSFET一样，双材料双栅MOSFET的亚阈值摆幅同样随沟道长度减少而减少。同时，栅氧化层的增加以及沟道厚度的增加，都会使得沟道的亚阈值特性变差，亚阈值摆幅增加。Calculated by our analytical model, as shown in FIG. 3 , the present invention shows the analytical results of the variation of the subthreshold swing of the dual-material dual-gate MOSFET with the channel length under different bulk silicon thicknesses. Figure 4 shows the analytical results of the subthreshold swing of the dual-material dual-gate MOSFET as a function of channel length under different gate oxide thicknesses. Like ordinary dual-gate MOSFETs, the subthreshold swing of dual-material dual-gate MOSFETs also decreases with decreasing channel length. At the same time, the increase of the gate oxide layer and the increase of the thickness of the channel will make the sub-threshold characteristics of the channel worse, and the sub-threshold swing will increase.

根据我们的解析模型，可以非常方便、准确提取双材料双栅MOSFET亚阈值摆幅。According to our analytical model, the subthreshold swing of dual-material dual-gate MOSFET can be extracted very conveniently and accurately.

Claims

1. a method of extracting two material double-gate structure MOSFET subthreshold swings, is characterized in that concrete steps are as follows:

(1) set up two material double grids MOSFETs

Two material double grids MOSFETs, centre is silicon, and raceway groove adopts p-type doping, and source is leaked and is adopted N-shaped heavy doping; Grid adopts asymmetric structure, and one of them grid adopts two kinds of material preparations that work function is different; The second material M2 adopts the less N-shaped heavily doped polysilicon of work function, and the first material M1 adopts the p-type polysilicon that work function is higher; The external same bias voltage of two ends grid;

Channel length is lthe grid of one end is divided into two parts, respectively corresponding two kinds of grid materials that work function is different; The length that the first material M1 is corresponding is l ₁, the length that the second material M2 is corresponding is l ₂ , L=L ₁ + L ₂; The work function of the first material M1 is 5.25eV, and the work function of the second material M2 is 4.17eV; t _ox1 , t _ox2for front grid and back of the body gate oxide thickness, t _sifor channel thickness;

(2) solve the Poisson equation of groove potential, obtain groove potential

The Poisson equation of groove potential is expressed as:

（1）

Wherein, for electron charge, for the doping content of raceway groove, for the specific inductive capacity of silicon, for groove potential;

According to electric field at raceway groove, oxide layer interface continuously and the voltage at source, leakage two ends, boundary condition is expressed as:

（2）

（3）

（4）

（5）

Wherein, vSfor Built-in potential, vDSfor drain-source voltage, cOX1, cOX2be respectively grid and lower gate oxide unit-area capacitance, v _gFF, v _gFBbe respectively grid and the effective grid voltage of lower grid; The grid of double-gate materials adopts two kinds of different materials of work function to form, in (4) formula v _gFFit is a piecewise function; Definition , , and substitution formula (5), boundary condition is expressed as again:

（6）

The form of separating is expressed as:

（7）

Wherein nfor integer, anfor undetermined coefficient;

By formula (7) substitution Poisson equation (1), obtain:

（8）

Wherein, , ;

The solution of formula (8) is:

（9）

By formula (9) substitution boundary condition (4) and (6), then make Fourier expansion, obtain coefficient with for:

（10）

And

（11）

Wherein, , be respectively

（12）

（13）

Like this, obtain the groove potential expression formula of two material double grids;

(3) the sub-threshold values amplitude of oscillation analytical expression of the two material double grids MOSFETs of foundation

Subthreshold swing is defined as:

（14）

In subthreshold value region, two material double grids MOSFETs are under weak transoid condition, and source-drain current is pressed and is approximated by:

（15）

be expressed as , for xthe electromotive force minimum value of direction; Substitution formula (15), is rewritten as the expression formula of subthreshold swing:

（16）

Wherein, , for Boltzmann constant, for temperature;

By result substitution (16) formula of groove potential, the analytical expression that obtains subthreshold swing is:

（17）

When calculating, effective conductive path of available raceway groove substitute the namely equivalent barycenter of electric charge:

（18）

Wherein, , for intrinsic carrier concentration, expression formula be rewritten as:

（19）。