A Closed-Form Expression for Minimum Operating Voltage of CMOS D Flip-Flop

In this paper, a closed-form expression for estimating the minimum operating voltage (
$V_{\mathrm{ DDmin}}$
) of D flip-flops (FFs) is proposed.
$V_{\mathrm{ DDmin}}$
is defined as the minimum supply voltage at which the FFs are functional without errors. The proposed expression indicates that
$V_{\mathrm{ DDmin}}$
of FFs is a linear function of the square root of logarithm of the number of FFs, and its slope depends on the within-die variation of the threshold voltage (
$V_{\mathrm{ TH}}$
) and its intercept depends on the balance between nMOS and pMOS, which is mainly due to the die-to-die
$V_{\mathrm{ TH}}$
variation. The proposed expression of
$V_{\mathrm{ DDmin}}$
is validated by the simulation results as well as the silicon measurements. Finally, we discuss the dependence of
$V_{\mathrm{ DDmin}}$
on the device parameters.