Inverted Temperature Dependence.

It is known that with increase in temperate, the resistivity of a metal wire(conductor) increases. The reason for this phenomenon is that with increase in temperature, thermal vibrations in lattice increase. This gives rise to increased electron scattering. One can visualize this as electrons colliding with each other more and hence contributing less to the streamline flow needed for the flow of electric current.

There is similar effect that happens in semiconductor and the mobility of primary carrier decreases with increase in temperature. This applies to holes  equally as well as electrons.

But in semiconductors, when the supply voltage of a MOS transistor is reduced, and interesting effect is observed. At lower voltages the delay through the MOS device decreases with increasing temperature, rather than increasing. After all common wisdom is that with increasing temperature the mobility decreases and hence one would have expected reduced current and  subsequently reduced delay. This effect is also referred to as low voltage Inverted Temperature Dependence.
Lets first see, what does the delay of a MOS transistor depend upon, in a simplified model.

Delay = ( Cout * Vdd )/ Id [ approx ]

Cout = Drain Cap
Vdd = Supply voltage
Id = Drain current.

Now lets see what drain current depends upon.

Id = µ(T) * (Vdd – Vth(T))α

µ = mobility
Vth = threshold voltage
α = positive constant ( small number )

One can see that Id is dependent upon both mobility µ and threshold voltage Vth. Let examine the dependence of mobility and threshold voltage upon temperature.

μ(T) = μ(300) ( 300/T )m
Vth(T) = Vth(300) − κ(T − 300)
here ‘300’ is room temperature in kelvin.

Mobility and threshold voltage both decreases with temperature. But decrease in mobility means less drain current and slower device, whereas decrease in threshold voltage means increase in drain current and faster device.

The final drain current is determined by which trend dominates the drain current at a given voltage and temperature pair. At high voltage mobility determines the drain current where as at lower voltages threshold voltage dominates the darin current.

This is the reason, at higher voltages device delay increase with temperature but at lower voltages, device delay increases with temperature.


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