Change of Gate-Source Voltage of a NMOS Transistor due to Temperature

Question

Consider the circuit below. I am performing a corner simulation in Cadence with temperatures of -40°C, 27°C and 85°C, to see how the gate-source voltage changes with temperature. Before simulating I calculated that by hand.

Since the transistor is in Saturation, the following formula can be applied to calculate the drain current (Sah / Shichman & Hodges Equation).

$$ I_D=\frac{\beta }{2}(V_{GS}-V_T)^2 $$

The formula can be rearranged to calculate \$V_{GS}\$ (since \$ I_D \$ is given by the current source).

$$ V_{GS}=\pm \sqrt{\frac{2}{\beta }I_D} + V_T $$

Furthermore the temperature dependence of the simulated transistor is the following.

$$ V_T=V(T_0)+TCV(T-T_0) $$

We can assume that \$TCV=-1.1mV/°C\$, \$ T_0 = 27°C \$, \$ \beta =\frac{75}{2} \mu A/V^2 \$, \$ I_D=10\mu A \$. Therefore the calculated gate-source voltages for the three different temperatures are

$$ V_{GS}(-40°C)=\pm \sqrt \frac{20\mu A}{\frac{75}{2} \frac{\mu A}{V^2}} + 0.6V - 1.1\frac{mV}{°C}(-40°C-27°C)=1.404V \\ V_{GS}(27°C) = 1.33V \\ V_{GS}(85°C) = 1.266V $$

We can see that with increasing temperature \$ V_T \$ goes down and so does \$ V_{GS} \$. But strangely the simulation shows me the opposite. Only the calculated \$ V_{GS} \$ at \$ 27°C \$ is correct.

Why is that the case? Did I do something wrong in my calculations?

Answer 1

The gate-source voltage of a MOS transistor changes because of two things. An increase of the threshold voltage and a decrease of mobility with increasing temperature.

Which effect dominates depends on the bias point, so Vgs can either go up or down with temperature.

The mobility \$\mu\$ is contained within $$ \beta = \mu C_{ox} \frac WL $$ and it is a function of temperature given by $$ \mu \propto T^{BEX} $$ where BEX depends on the technology (a typical value is -3/2).

For \$\beta\$ we find that $$ \beta(T) = \beta(T_r) \left(\frac{T}{T_r}\right)^\text{BEX} $$ wherer Tr is the reference temperature where \$\beta(T_r)\$ was determined.

Including this dependence should give you the right results, don't forget to use the absolute temperature in your calculations (Kelvin).