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 xdmax, rj is the junction radius, and r2 is the radial distance to the corner of the trapezoid shown in the

Question

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 [latex] x_{dmax}, r_j [/latex] is the junction radius, and [latex] r_2 [/latex] is the radial distance to the corner of the trapezoid shown in the figure. W is the width of the channel and [latex] N_a [/latex] 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 [latex] V_{DS} [/latex].

Accepted Answer

For small values of [latex] V_{DS} [/latex], we consider the charge induced by [latex] V_G [/latex] to be approximately contained in a volume whose cross section is the trapezoid of width [latex] x_{dmax} [/latex] and length varying from L at the surface to [latex] L_1 [/latex] at the substrate side of the depletion region. The cross-sectional area is shown cross-hatched on the figure. If this charge is called [latex] Q_{d1} [/latex] then

[latex] Q_{d1}=qx_{dmax}WN_a\frac{L+L_1}{2} [/latex] (1)

The charge [latex] Q_{d1} [/latex] 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 [latex] L_1 [/latex], then [latex] L_1 [/latex] approaches L. In that case, from Equation 1, [latex] Q_{d1} [/latex] equals [latex] Q_d = q x_{dmax} N_a WL [/latex], as was assumed in the first-order theory of Sec. 9.1. For shorter channels, [latex] L_1 [/latex] becomes appreciably less than L, and [latex] Q_{d1} [/latex] is therefore less than [latex] Q_d [/latex] as expccted from our qualitative arguments.

For a useful theory, [latex] L_1 [/latex] must be related to the geometry of the MOSFET. This can be done approximately by assuming that when [latex] V_G = V_T [/latex], the depletion layer is [latex] x_{dmax} [/latex] units wide both in the x-direction (perpendicular to the [latex] Si-SiO_2 [/latex] interface) and along the radius of the diffused source and drain junctions. With this approximation [latex] r_2 = r_j + x_{dmax} [/latex] From the geometry of the structure, we obtain

[latex] f\equiv \frac{Q_{d1}}{Q_d} =1-\frac{r_j}{L} \left(\sqrt{1+\frac{2x_{dmax}}{r_j} }-1 ight) [/latex] (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:

[latex] V_T=V_{FB}+V_C+2\left|\phi _p ight| +\frac{1}{C_{ox}} \sqrt{2\epsilon _sqN_a(2\left|\phi _p ight| +V_C-V_B)} [/latex] (8.3.18)

[latex] V_T=V_{FB}+2\left|\phi _p ight| +V_S-\frac{fQ_d}{C_{ox}} \ =V_{FB}+2\left|\phi _p ight| +V_S+\frac{f}{C_{ox}} \sqrt{2\epsilon _sqN_a(2\left|\phi _p ight| +V_S-V_B)} [/latex] (3)

Despite the approximate nature of its derivation, Equation 3 is useful in predicting trends in the behavior of [latex] V_T [/latex] 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 [latex] V_T [/latex] for small values of [latex] V_{DS} [/latex].
Because [latex] V_{DS} [/latex] is typically much larger than the source-substrate bias [latex] V_{SB} , V_T [/latex] is sensitive to [latex] V_D [/latex] in short-channel MOSFETs. More elaborate geometric analyses include this effect.

Question 9.3: Charge Sharing Consider the idealized cross section of a sh...

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