Thermistor (NTC)

A thermistor is a resistor whose resistance depends strongly on temperature. The negative-temperature-coefficient (NTC) type used in optics is a sintered ceramic of transition-metal oxides, which behaves as a semiconductor: warming it frees more carriers and its resistance falls roughly exponentially. The usual part for laser packages and optical mounts is specified as 10 kΩ at 25 °C with a β value near 3950 K, which gives 33.6 kΩ at 0 °C, 3.59 kΩ at 50 °C and a slope of −4.4%/K at room temperature. In a laser package it is the sensor that closes the loop of a thermoelectric cooler, usually mounted on the submount next to the chip.

Beta and Steinhart-Hart equations

The simplest description is the β (beta) equation,

$$R(T) = R_0 \exp\!\left[\beta\left(\frac{1}{T} - \frac{1}{T_0}\right)\right],$$

with $T$ in kelvin and $R_0$ the resistance at $T_0$ = 298.15 K. Its logarithmic derivative gives the temperature coefficient,

$$\alpha = \frac{1}{R}\frac{dR}{dT} = -\frac{\beta}{T^2}.$$

For β = 3950 K at 298.15 K, α = −4.44%/K, so a 10 kΩ part changes by 444 Ω/K and by 4.4 Ω for a 0.01 K step. A platinum RTD, at 0.385%/K, is about eleven times less sensitive in relative terms, which is the main reason thermistors are used where millikelvin resolution is needed with simple electronics.

The β equation is a fit over a stated range, usually 25–50 °C or 25–85 °C, and it drifts outside it. The three-parameter Steinhart-Hart equation,

$$\frac{1}{T} = A + B\ln R + C\,(\ln R)^3,$$

follows a real thermistor to about 0.01 K or better over a span of about 100 K when its coefficients are fitted to three calibration points. For a common 10 kΩ material, A = 1.129148 × 10⁻³, B = 2.34125 × 10⁻⁴ and C = 8.76741 × 10⁻⁸ (SI units, R in ohms) return 25.00 °C at 10 kΩ. A β equation fitted to the same part between 25 °C and 50 °C (β = 3936 K) reads 0.48 °C at a true 0 °C and 75.45 °C at a true 75 °C: harmless for a set point near room temperature, and a real error for a controller that steps the laser over a wide range.

Reading the resistance and self-heating

A controller either drives a small constant current through the thermistor and measures the voltage, or places it in a divider with a fixed resistor. With 100 µA through 10 kΩ the voltage is 1.0 V and the sensitivity 44 mV/K. A divider of the thermistor and a matched 10 kΩ resistor from a 2.5 V reference gives 28 mV/K at its midpoint, where the response is most linear, which is why the fixed resistor is chosen equal to the thermistor's resistance at the set point.

The measuring current heats the sensor. The dissipated power is $P = I^2 R$, and the temperature rise is $P$ divided by the dissipation constant, which is of order 1 mW/K for a small bead in still air and larger when the bead is bonded to a metal submount. At 100 µA the power is 0.1 mW and the error up to about 0.1 K in air; at 10 µA it is 1 µW and about 1 mK, at the cost of a 4.4 mV/K signal that needs a quieter amplifier.

Placement for laser temperature control

The controller holds the thermistor at its set point; the laser's active region runs above it by the thermal resistance between them times the dissipated power, about 7 K for the DFB laser example under junction temperature. A thermistor close to the chip keeps that offset small and the loop fast, because the thermal delay between heater and sensor, a thermal RC time constant, limits how much gain the loop can use before it oscillates.

The stability requirement follows from the wavelength temperature coefficient. A DFB laser tuning at 0.09 nm/K moves 0.9 pm, about 112 MHz at 1550 nm, for 0.01 K, which is the stability a good TEC loop holds at the thermistor. Holding the sensor constant does not hold the junction constant when the drive current or ambient temperature changes, which is one reason dense-WDM transmitters add a wavelength locker.

Pitfalls

• Using a β value from a different material or temperature range; controllers that accept Steinhart-Hart coefficients avoid the extrapolation error.
• Lead and contact resistance, which adds directly at low thermistor resistance; a 1 Ω lead error is 2 mK at 444 Ω/K.
• Calling a thermistor reading the junction temperature.

Common questions

What does NTC mean?

Negative temperature coefficient: the resistance decreases as temperature increases. PTC thermistors, whose resistance rises sharply above a switching temperature, are used as resettable fuses and over-temperature cut-outs, and are not used for precision sensing.

Why is a 10 kΩ thermistor so common?

At room temperature 10 kΩ gives a convenient voltage with microampere currents, keeps self-heating small, and makes lead resistance negligible. Higher values (100 kΩ and above) suit higher temperatures, where the resistance would otherwise become small.

References: J. S. Steinhart and S. R. Hart, "Calibration curves for thermistors," Deep-Sea Research 15, 497 (1968); J. Fraden, Handbook of Modern Sensors, 5th ed. (Springer, 2016).