PTAT, CTAT and temperature

A voltage that rises with temperature

The thermal voltage U_T = kT/q (k Boltzmann's constant, q the electron charge, T in kelvin) is 25.87 mV at 27 °C and grows by k/q = 86.17 µV per kelvin. A quantity proportional to absolute temperature is called PTAT.

In weak inversion, below threshold, a MOS current is exponential in V_GS: each factor e of current costs n·U_T, with the slope factor n between 1 and 2 (E2.6). Feed two identical devices currents in a ratio K and their gate voltages differ by ΔV_GS = n·U_T·ln K. The threshold cancels, so ΔV_GS is PTAT.

Compare the slope with the formula. ΔV_GS = n·U_T·ln 8 grows by n · 86.17 µV · ln 8 = n · 179.2 µV per kelvin. The measured 230.4 µV/K gives n = 1.29.

The value fits less well: with n = 1.29 the formula gives 69.3 mV at 27 °C, the run 102.3 mV. The formula needs both devices deep in weak inversion. The 1 µA device is, with g_m/I_D = 22.49 V⁻¹ (E2.5: above 20). The 8 µA device, at 17.18 V⁻¹, is in moderate inversion, where V_GS grows faster than the logarithm of the current, and that extra part is not PTAT.

A voltage that falls with temperature

The lower panel, V_GS at a constant 1 µA, falls 750.6 µV/K. A quantity that falls as temperature rises is called CTAT (complementary to absolute temperature).

It falls because the threshold falls with temperature (E2.7). Mobility falls too, which pushes V_GS up, but at a small current the threshold wins; at large currents the two cancel, and above that V_GS rises.

Why the beta multiplier's current climbs

In the beta multiplier the voltage across R_S is V_GS1 − V_GS2, a ΔV_GS of the same kind, so it rises with temperature. With a fixed resistor, so does I = ΔV_GS/R_S. An ideal SPICE resistor has no temperature coefficient unless you give it one.

A real resistor has a temperature coefficient. SKY130's res_high_po poly resistor (W = 0.35 µm, L = 9.08 µm) measures 9.733, 9.997 and 10.57 kΩ at −40, 27 and 125 °C. In place of the ideal 10 kΩ it cancels part of the rise.

RS s 0 10k

VR s2 r2 0
XRS r2 0 0 sky130_fd_pr__res_high_po W=0.35 L=9.08

Still far outside ±5 %. But the beta multiplier is also called a constant-g_m bias. Its current moves so that the transconductance g_m of its devices stays nearly still, and an amplifier biased from it keeps its gain and bandwidth (E5).

The poly resistor flattens the current but makes g_m move more, because g_m is set mainly by R_S: in the square law g_m1 = 2(1 − 1/√K)/R_S. Pick the resistor for the quantity you need constant.

Adding PTAT to CTAT

One voltage rises 230.4 µV/K, the other falls 750.6 µV/K. Scale the rising one by w = 750.6/230.4 = 3.258 and add: the slopes cancel at 27 °C.

The sum is a shallow bowl: both slopes change with temperature, and one weight cancels only their first-order parts. The rest is called curvature.

A bandgap reference builds this in silicon, usually from bipolar transistors: the base-emitter voltage is the CTAT term and the difference of two of them the PTAT term. A bipolar junction follows its equations closely and spreads less than a MOS threshold. SKY130 has a parasitic PNP, but this track's library has no bipolar models, so the lesson stops at the principle.

Try it

Run it and read the lines: ΔV_GS and V_GS at three temperatures and their slopes at 27 °C, the g_m/I_D of the two devices, and the weighted sum. Open Plots to see all the node voltages against temperature.

Now change things and predict first. Set I8 0 n8 8u to 0.8u and I1 to 0.1u: both devices go deeper into weak inversion. Now n from the slope (1.29) and n from the 27 °C value (1.90) should move toward each other; check with dv_slope and dv_27. Change w to 3.0 and to 3.5 and watch the bottom of the bowl move to the hot or the cold end. Replace .lib sky130 tt with ss: the CTAT voltage moves with the corner, the PTAT one hardly does.

Check yourself

* An ideal 10 k resistor carrying 1 uA, swept over temperature
I1 0 r 1u
R1 r 0 10k
.control
dc TEMP -40 125 5
meas dc v125 FIND v(r) AT=125
.endc
.end