Mott variable-range hopping based calibration procedure for NTC thermistors in wide cryogenic temperature range

In this paper we propose a procedure based on the Mott variable-range hopping model that describes the low-temperature conduction in strongly disordered systems such as negative temperature coefficient (NTC) thermistors. This procedure offers a close approximation to the physical model and provides an accurate R ( T ) fit for the resistance − temperature data over three orders of magnitude temperature range (0.04 K — 40 K) using only 4 fit coefficients. Eventually, the temperature corresponding to any resistance value of the thermistor T ( R ) can be determined with an approximation of an inverted R ( T ) function based on a Puiseux series expansion. This approach has been showed to be applicable to different NTC thermistors, including ruthenium oxide (RuO 2 ) and germanium (Ge) resistors with a range of resistance values. It provides a root mean square of the relative residual of the fit Δ T rel RMS ≤ 1 % and Δ T rel RMS ≤ 1 . 7 % using a n fit ∈ [ 4 , 10 ] and n fit = 4 fit coefficients for RuO 2 and Ge, respectively. Previously reported procedures for the calibration of resistance-temperature data in low-temperature thermistors do not reach this accuracy with a similar number of fit coefficients. • We report on a comprehensive calibration procedure to fit resistance-temperature data set providing the R ( T ) function and its inversion T ( R ) • The R ( T ) function is derived from Mott variable-range hopping which describes the low temperature carrier transport in disordered semiconductors in an extended temperature range. The T ( R ) function is approximated using Puiseux series expansion. • This calibration procedure has been tested on different NTC thermistors over three orders of magnitude of temperature range (40 mK to 40 K). • This calibration procedure reports a root mean square of the relative residual of the fit Δ T rel RMS ≤ 1 % and Δ T rel RMS ≤ 1 . 7 % using a number of fit coefficients n fit ∈ [ 4 , 10 ] and n fit = 4 for RuO 2 and Ge, respectively. None of the common calibration equations can reach such a level of accuracy over the specified temperature range with the same number of fit coefficients. • Calibration precision doesn’t degrade but remains stable when more fit coefficients are added, as it happens, for instance, to the expansion series of the Basic Equation.