3.23 Threshold Voltage

A simple analytical threshold voltage $V_{th}$ definition for $GEOMOD = 0$ , $1$, and $2$ was derived and implemented as operating point info in BSIM-CMG. For a long channel device, $V_{th}$ is defined as the value of $V_g$ at which the drift and diffusion components of the source to drain current at the source side are equal. Based on this definition, it can be shown that at $V_g = V_{th}$, the charge at source side is given by [18],

$Q_{is} = C_{ox} \cdot \dfrac{kT}{q} \qquad (3.640)$

Next, the surface potential at the source is approximately calculated from the charges as follows ([4], Ch. 3, p. 66)

$\psi_s \approx \dfrac{kT}{q} \cdot ln \Bigg[ \dfrac{Q_{is} \cdot (Q_{is} + 2 Q_{bulk} + 5 C_{si} \frac{kT}{q}) }{2 q \cdot n_i \cdot \epsilon_{sub} \cdot \frac{kT}{q}} \Bigg] + \phi_B + \Delta V_{t,QM} \qquad (3.641)$

The Gauss law demands that at the source side

$V_g = V_{fb} + \psi_s + \dfrac{Q_{is} + Q_{bs}}{C_{ox}} \qquad (3.642)$

Substituting (3.640) and (3.641) in (3.642) results in the following expression for $V_{th}$ for a long-channel device:

$V_{th0} = V_{fb} + \dfrac{kT}{q} \cdot ln \Bigg[ \dfrac{C_{ox} \frac{kT}{q} \cdot \Big( C_{ox} \frac{kT}{q} + 2 Q_{bulk} + 5 C_{si} \frac{kT}{q} \Big) }{2 q \cdot n_i \cdot \epsilon_{sub} \cdot \frac{kT}{q}} \Bigg] + \phi_B + \Delta V_{t,QM} + \dfrac{kT}{q} + q_{bs}$

$(3.643)$

Corrections due to threshold voltage roll-off, DIBL, reverse short-channel effect, and temperature are added accordingly:

$V_{th} = V_{th0} + \Delta V_{th,all} \qquad (3.644)$

References

[4] M. V. Dunga, Ph.D. Dissertation: Nanoscale CMOS Modeling. UC Berkeley, 2007.

[18] C. Galup-Montoro, M. C. Schneider, A. I. A. Cunha, F. Rangel de Sousa, H. Klimach, and F. Siebel, "The Advanced Compact MOSFET (ACM) model for circuit analysis and design," in IEEE Custom Integrated Circuits Conference, 2007, pp. 519-526.