How does temperature inversion complicate the assumption that hold is always worst at the fast/cold corner?

Short Answer

Cell delay normally shortens as temperature drops, because carrier mobility improves — this is why designers default to the coldest corner for hold analysis. At low supply voltage, though, delay can start increasing as temperature drops instead, because the threshold voltage's own temperature dependence takes over from mobility. This reversal is called temperature inversion, and it means the coldest corner is not automatically the fastest, or the worst, corner for hold.

Technical Explanation

• A transistor's delay depends on two competing temperature effects: carrier mobility (how fast electrons move through the channel) and threshold voltage Vt (the gate voltage needed to turn the device on).
• At nominal or high supply voltage, mobility dominates. Mobility improves as temperature drops, so the cell switches faster — this is the normal behavior designers rely on when they call the cold corner the fast corner.
• Vt also depends on temperature, and it moves the opposite way: Vt rises as temperature drops. At low supply voltage, the overdrive (supply voltage minus Vt) is already small, so a rising Vt eats a much bigger fraction of the available overdrive than it does at nominal voltage.
• Below a certain supply voltage, the Vt effect overtakes the mobility effect, and cell delay starts increasing as temperature drops instead of decreasing. This crossover is temperature inversion.
• The practical consequence: at a low-voltage operating point, the coldest temperature is not guaranteed to be the fastest corner anymore, so it is not guaranteed to be the worst corner for hold either. The tool has to be told to check the corner where inversion actually makes the device slowest, which can sit in the middle of the temperature range rather than at either extreme.
• Liberty (LIB) library characterization normally spans multiple temperature points per voltage so the timing tool can interpolate and expose this non-monotonic behavior; a library sparsely characterized only at temperature extremes can hide the inversion point entirely.
• What breaks: signing off hold only at the traditional fast/cold corner (for example, FFG at -40C) on a low-voltage design can miss the true worst hold corner, letting a hold violation slip through that only appears at an intermediate temperature.

Follow-up Question & Model Response

"How would you decide, for a specific low-voltage design, which temperature points actually need signoff corners once inversion is a possibility?"

Candidate Model Response: I would start from the library characterization points themselves — ask the library team at which supply voltages the vendor observed or modeled inversion, since that is a library-specific crossover, not a universal number. For any operating voltage at or below that threshold, I would add at least one mid-range temperature corner in addition to the two extremes, because the true delay maximum can sit between them. I would also cross-check with report_timing -derate (PT) at each candidate corner to confirm which one actually produces the smallest cell delay, rather than assuming the extreme temperature does.