# Surface Potential Function

In Section 3.4.2.2, we solved coupled Equation 3.105 and Equation 3.106 for surface potential ф, and center potential ф_{0}. In this section, we will derive a surface potential function using a transform variable defined as (3 = /(ф,) using the generalized solution of Poisson’s equation derived in Section 3.4.2.2.

In order to derive an expression for (3, we use the generalized solution of Poisson’s equation given by Equations 3.95 and 3.96 to obtain the relation between the gate voltage *V _{g}* and surface potential ф

_{( }

Thus, to calculate the surface potential through a single continuous equation, we define the transformation variable (3 as the argument of the cosine function in ф(л' = *tJ2)* in Equation 3.124 so that [21]

Now, we define [18]

Then, from Equation 3.126, we can show Then, we get from Equation 3.127 as

Now, substituting for *a* and *b* from Equation 3.127 and ф_{()} from Equation 3.128 into Equation 3.123 we can write for ф(х) as

Then, we can further simplify Equation 3.129 as follows

Now, substituting the expressions fora from Equation 3.126 into Equation 3.130, we get Then rearranging Equation 3.131, we can show

From Equation 3.132, we can show that the surface potential, at x = *±tJ2* (Si/Si0_{2 }interface)

And, (at *x=t _{sj}l*2)

In order to derive an expression for (f relating to the gate voltage *V _{g}* and surface potential ф„ we use the boundary condition given by Equation 3.100 as

Then substituting for ф(х = *±tj*2) from Equation 3.133 and с/ф(х = *±tJ2)/dx* from Equation 3.134 into Equation 3.135, we get

After simplification of Equation 3.136, we can show

Thus, through a change of variable, the expression for the surface potential ф, can be represented by the function

Equation 3.138 can be solved for p at a given value of *V* to determine the surface potential and determine the characteristics of DG-MOS capacitors. We have shown that for an undoped or lightly doped silicon substrate *nj* / *N _{h}* «

*n*therefore, for an undoped or lightly doped substrate, Equation 3.138 can be expressed as

_{h}

Equation 3.138 and Equation 3.139 are used to determine the FinFET device performance with appropriate modifications for the lateral electric field due to applied drain voltage as discussed in Chapter 10 [8].

# Unified Expression for Inversion Charge Density

In Section 3.4.2.1, we derived the expression [Equation 3.63] for the total induced charge in а /Муре semiconductor substrate of multigate MOS capacitors given by

As discussed in Section 3.4.2.1, on the right-hand side of Equation 3.140, the first term within the square brackets is due to the minority carrier electrons in the /Муре substrate, whereas the second term is due to the majority carrier bulk doping concentration *N _{a}.* For a

*lightly doped body,*the bulk charge

*Q*therefore, neglecting the bulk charge term in Equation 3.140, we can express the inversion charge density in а /Муре substrate of multigate MOS capacitor as

_{h}« Q-,Equation 3.141 can be further simplified as

/ Ф<>-

In *strong inversion* ф_{5} » ф_{0}, therefore, **l-e *** ^{m}* approaches 1.

/ Ф()~Фх

In *weak inversion,* we can simplify the term *l -e ^{m}* , assuming linear potential profile from .v = 0 (center point) to x =

*±t*(surface). Then if

_{s}/2*E*is the average electric field in the region between

*x*=

*tJ2*to the center point at

*x*= 0, then using Gauss’ law [Equation 3.68], we can write

If we assume that potential varies linearly from center potential ф_{0} to the surface potential ф_{(}, then Equation 3.143 can be expressed as

Thus, the inversion charge is given by where:

C„ = is the depletion capacitance of silicon body

Now, substituting Equation 3.145 into Equation 3.142, we get

We can further simplify Equation 3.146 by Taylor series expansion of the term exp[-G */(2C _{si}v_{kT})^* to keep the first term and neglect the higher order terms since

*Q/(2C*1 in the weak inversion regime. Then, we get

_{si}v_{kT}) <

Therefore, using Equation 3.147 we can write the expression for the inversion charge density *Qi(LD)* for a lightly doped substrate as

Equation 3.148 is an implicit equation in *Q _{{}* and is solved iteratively with coupled Equations 3.96 and 3.101 to compute

*V*dependence of inversion charge. It is reported that the numerical device simulation data show that a modified form of Equation 3.148 fits the real data on MOS capacitors, as given below [23]

_{g}

Now, let us derive an expression for *O,* in *heavily doped* DG-MOS capacitor systems. We know that the second term on the right-hand side of Equation 3.140 represents the bulk charge *Q _{b}* and is given by Equation 3.108 as

In the *strong inversion* regime of heavily doped DG-MOS capacitors, ф, term is large and *Q _{t} » Q_{h}.* However, in the

*weak inversion*regime

*O, « Q*and therefore, (Ф.5

_{h}^{-}Фо) is a small perturbation in the total surface potential ф

_{(}and ф,» ф

_{0}. Using this assumption, we can simplify Equation 3.140 for a heavily doped body as

And

In the second expression of Equation 3.152, we have used Equation 3.43 for

/ i

*N _{a}* = n, exp — . Then we can express Equation 3.151 as

V * ^{V}kT* ,

We know that the total charge in the semiconductor is given by *Q _{s}* =

*Q, + Q*therefore, Equation 3.153 can be expressed as

_{b};After taking the square to both sides of Equation 3.154, we get

Now, multiplying both sides of Equation 3.155 by *O,,* we get the inversion charge density after simplification as

From the similarities of charge expressions in Equation 3.149 for *Qj(LD)* and 3.156 for *Qi(HD),* a unified expression can be used to calculate the inversion charge density for a wide range of devices as a function of *Q _{h}* and is given by

where: *Q _{a}* = 2

*Q*+ 5C„v

_{h}_{№}with

*C*=

_{si}*K*sj; Q is the fixed depletion charge and for an ultrathin-body is given by

_{sj}e_{h}

*qN*It is reported that the unified charge density model agrees very well with inversion charge density calculated using exact equation for a wide range of body doping concentration [22,23].

_{b}t_{si}.