Question 9.3: Charge Sharing Consider the idealized cross section of a sh...
Charge Sharing
Consider the idealized cross section of a short-channel transistor shown in the accompanying figure. The maximum depletion width away from the source and drain junctions is x_{dmax}, r_j is the junction radius, and r_2 is the radial distance to the corner of the trapezoid shown in the figure. W is the width of the channel and N_a is the dopant density.
Considering the reduced depletion-region charge that needs to be induced by the gate electrode, derive an approximate expression for the threshold voltage for small values of V_{DS} .
For small values of V_{DS} , we consider the charge induced by V_G to be approximately contained in a volume whose cross section is the trapezoid of width x_{dmax} and length varying from L at the surface to L_1 at the substrate side of the depletion region. The cross-sectional area is shown cross-hatched on the figure. If this charge is called Q_{d1} then
Q_{d1}=qx_{dmax}WN_a\frac{L+L_1}{2} (1)
The charge Q_{d1} in Equation 1 is approximately the depletion-layer charge that must be induced by the gate to bring the channel to the threshold condition. If the channel is long so that the space-charge regions at the source and drain are much smaller than L_1 , then L_1 approaches L. In that case, from Equation 1, Q_{d1} equals Q_d = q x_{dmax} N_a WL , as was assumed in the first-order theory of Sec. 9.1. For shorter channels, L_1 becomes appreciably less than L, and Q_{d1} is therefore less than Q_d as expccted from our qualitative arguments.
For a useful theory, L_1 must be related to the geometry of the MOSFET. This can be done approximately by assuming that when V_G = V_T , the depletion layer is x_{dmax} units wide both in the x-direction (perpendicular to the Si-SiO_2 interface) and along the radius of the diffused source and drain junctions. With this approximation r_2 = r_j + x_{dmax} From the geometry of the structure, we obtain
f\equiv \frac{Q_{d1}}{Q_d} =1-\frac{r_j}{L} \left(\sqrt{1+\frac{2x_{dmax}}{r_j} }-1 \right) (2)
The parameter f is therefore a function of the MOSFET geometry. The expression for the threshold voltage is written directly from Equation 8.3.18:
V_T=V_{FB}+V_C+2\left|\phi _p\right| +\frac{1}{C_{ox}} \sqrt{2\epsilon _sqN_a(2\left|\phi _p\right| +V_C-V_B)} (8.3.18)
V_T=V_{FB}+2\left|\phi _p\right| +V_S-\frac{fQ_d}{C_{ox}} \\ =V_{FB}+2\left|\phi _p\right| +V_S+\frac{f}{C_{ox}} \sqrt{2\epsilon _sqN_a(2\left|\phi _p\right| +V_S-V_B)} (3)
Despite the approximate nature of its derivation, Equation 3 is useful in predicting trends in the behavior of V_T for MOSFETs with short channel lengths.
The analysis leading to Equation 3 did not consider the difference between the space-charge dimensions at the source and at the drain and, therefore, represents V_T for small values of V_{DS} .
Because V_{DS} is typically much larger than the source-substrate bias V_{SB} , V_T is sensitive to V_D in short-channel MOSFETs. More elaborate geometric analyses include this effect.
