Page 257 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

THERMAL SENSORS 237

the voltage sensitivity S T is a constant depending on the drive current:

k ET //o \ dV out & B . (k , A to im
Vout = - In — -f 1 and S T = -rtr- = — In — + 1 ) (8.10)
q \h J dT q \I S J

The overall temperature sensitivity of the diode depends on the relative size of the drive
current and saturation. When the drive current is set to a value well above the saturation
current, Equation (8.10) becomes

r, ^ IQ % &B /o , , ^ j /0 , , ,
y out % - In — and ST — In — when /o ^> 7 S (8.11)
q I* q I s
Let us suppose that the forward current is 0.1 uA and about 100 times the diode saturation
current of approximately 1 nA; then, the expected temperature sensitivity is +0.2 mV/K.
However, in practice, the temperature-dependence of a diode depends on the strong
temperature-dependence of the saturation current itself. The actual value can be obtained
experimentally from the temperature-dependence of the forward junction voltage of a
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silicon diode , that is, -2 mV/°C, and therefore V f0 oc T.
In a similar way, a bipolar transistor can be used as a temperature sensor. For example,
Figure 8.9(b) shows an n-p-n transistor in a common-emitter configuration and constant
current circuit. From our basic theory, the base-emitter voltage V BE is proportional to the
absolute temperature and simply related to the collector current Ic by
k^T / /c \
VBE = — In ( TH where 7 co = ^E-/S (8.12)
q Vco/
where A E is the area of the emitter, J s is the saturation current density, and I C0 is the
reverse saturation current. More accurate models can be developed from, for example,
those discussed previously for a bipolar transistor (Equation (4.20)), but the base-emitter
current approximates well in practice to

V BE ^ V BEO + AT (8.13)

where A. is an empirical constant that depends on the current density and process param-
eters and the offset voltage V BEO has a typical value of 1.3 V when the base-collector
voltage V BC is set to zero.
To make a truly PTAT sensor, it is necessary to fabricate two transistors - one with an
emitter area A EI and the other with A E2. Then the difference in their base-emitter voltages
is directly proportional to the absolute temperature and is given by

A i/ ^7 ^r \ i . /A E2 \
A VBE = (VBEI - ^BE2) = - In - — — — ^ - In -— (8. 14)
q \Ic2J si AEI/ q \AEI/
When the two transistors are identical, the collector currents and saturation current densi-
ties are equal, and the ratio of the emitter areas only determines the sensor's response.

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Typically 0.7 V at 25 °C for silicon (and 0.25 V for germanium).