Body effect: threshold variation and its approximation

Figure 3.7: MOSFET chain with static voltages

[n-MOSFET chain] [p-MOSFET chain]

It is known that a MOS transistor with the source-body voltage different from zero has the threshold voltage modified by the body effect, that is if $V_{sb}\neq 0$, with $V_{sb}$ the source-body voltage (let's remember that for a n-MOSFET $V_b = V_{SS}$ and for a p-MOSFET $V_b = V_{DD}$), then $\vert V_{Th}\vert _{V_{sb} \neq 0} > \vert V_{Th}\vert _{V_{sb} = 0}$. The initial conditions of the chain nodes are set by the initial condition on the output. So if the output node is discharging, then one (and only one) n-MOSFET is switching from off to on. It means that all the other MOSFETs are already on, and while the starting voltage of the output node is $V_{DD}$, all the internal nodes have as a starting voltage $V_{DD} - V_{Tn}^{*}$.

With the notations of previous paragraphs, the N-th (topmost) n-MOS transistor has $V_{s_n}^N = V_{DD} - V_{Tn}^{*}$, with $V_s$ source potential and $V_{Tn}^*$ the threshold voltage modified by the body effect. All the internal transistors have $V_{d_n}^i=V_{s_n}^i=V_{DD}-V_{Tn}^*$, while the first one has $V_{d_n}^1=V_{DD}-V_{Tn}^*$ and $V_{s_n}^1=0$.

The threshold voltage variation as a function of $V_{sb}$ is given by:

$$\Delta V_{Tn}=\gamma(\sqrt{2\vert\Phi_p\vert+V_{sb}} - \sqrt{2\vert\Phi_p\vert})\; ,$$

with $\gamma= \frac{\sqrt{2\varepsilon_s q N_a}}{C_{ox}}$ and $\Phi_p = -\frac{KT}{q}\ln{(\frac{N_a}{n_i})}.$

The source potential of the top transistor is

$$V_s = V_{DD}-V_{Tn}^*\; ,$$

and, if $V_{Tn0}$ is the threshold voltage with $V_{sb}=0$, then $V_{Tn}^*=V_{Tn0}+\Delta V_{Tn}$ and we can solve for $V_{sb}$:

$$\begin{split}V_{sb}&= \pm \frac{\gamma\sqrt{4\gamma\sqrt{2\ve... ... + V_{DD} - V_{Tn0} + \frac{\gamma^2}{2} \quad (>0) \end{split}$$

We can find an analogue equation for p-MOSFETs: knowing that, for the p-MOS chain depicted in figure 3.7(b), the drain potential of transistor is $V_{d_p}^P = 0$, while $V_{s_p}^P = -V_{DD} - V_{Tp}^*$; for the middle transistors $V_{d_p}^j=V_{s_p}^j=-V_{DD}-V_{Tp}^*\,$; and for the first (top-MOSt) transistor $V_{d_p}^1 =-V_{DD}-V_{Tp}^*$ and $V_{s_p}^1=V_{DD}$.

The threshold voltage variation function of $V_{sb}$ again is:

$$\Delta V_{Tp}=-\gamma(\sqrt{2\vert\Phi_p\vert+V_{sb}} - \sqrt{2\vert\Phi_p\vert})$$

(for p-MOS transistors threshold voltage is negative).

Again, solving:

$$V_{sb} = - V_{DD} - V_{Tp}^* = - V_{DD} - V_{Tp0} + \gamma(\sqrt{2\vert\Phi_p\vert+V_{sb}} - \sqrt{2\vert\Phi_p\vert})$$

where $V_{Tp0}$ is the threshold voltage with $V_{sb}=V_{DD}$; thus we find:

$$\begin{split}V_{sb}& = \pm \frac{\gamma\sqrt{4\gamma\sqrt{2\v... ... - V_{DD} - V_{Tp0} - \frac{\gamma^2}{2} \quad (<0) \end{split}$$

Figure 3.8: Threshold variation with V$_{\text{sb}}$ (solid line) and its linear approximation (dashed line)

[n-MOSFET] [p-MOSFET]

The threshold variation is approximated in the model by a linear approximation given by:

$$V_{T_n} = \alpha_n V_{sb} + \beta_n$$

$$V_{T_p} = \alpha_p V_{sb} + \beta_p$$

with $\alpha_{n,p}$ and $\beta_{n,p}$ constants:

$$\alpha_n = \frac{V_{Tn}^*-V_{Tn0}}{V_{DD}-V_{Tn}^*} \beta_n = V_{Tn0}$$

$$\alpha_p = \frac{V_{T_p}^*-V_{T_p0}}{V_{DD}+V_{Tp}^*} \beta_p = \frac{V_{T_p}\left( V_{DD}^*+V_{Tp0}\right)}{V_{DD}+V_{Tp}^*}$$

In figure 3.8(a) and 3.8(b) the actual threshold variation (of a n-MOS transistor and a p-MOS transistor) when a $V_{sb}$ voltage is applied is compared with the linear approximation used in our model, for a $0.7\,\mathrm{\mu m}$ technology.

The max error due to the linear approximation is limited to $7\%$.