Temperature Inversion

Temperature inversion is a phenomenon which occurs in lower nodes,which makes the delay of a cell decreases when there is a rise in temperature contradictory the delay in higher nodes.
Lets unfold this,
If you look at the MOSFET drive current equation,

So, ID varies linearly as u (mobility) and (VGS-Vt)^2 or the overdrive voltage. We can conclude that Delay of a cell depend upon two factors mobility & threshold voltage( Vt)of a transistor.

How mobility & Vt  depend upon temperature

Due to rise in temperature, metal ions going to vibrate more, so mobility of charge carriers will decreases such that delay of a cell going to increase.

Threshold voltage is also going to decrease,with rise in temperature as number of minority carriers in the substrate going to increase, which makes the less Vt than usual required to form a channel.

To Summarise, increase in temperature, makes the delay of a cell

• Decreases due to Decrease in threshold voltage,
• Increases due to Decrease in the mobility.

So delay of a cell may increase or decrease depend upon which factor going to dominate either mobility or threshold voltage on final current.

When the VGS- Vt or overdrive voltage is large(in higher node-> high VGS), then decrease in threshold voltage due to variation in temperature is negligible because overall overdrive voltage has very less impact, so mobility factor is going to dominate here, results delay of a cell going to increase with rise in temperature .

When the overdrive voltage(VGS-Vt) is less (smaller node -> less VGS), then decrease in threshold voltage due to rise in temp going to dominate the overall overdrive voltage, results delay of a cell going to decrease with rise in temperature.

One thing should be noted that temperature inversion is come into picture at lower nodes (lower voltage) with more prominent effects on HVT cells.