Variable-range hopping conduction in NTCR thermistors R. Schmidt, A.W. Brinkman, A. Basu Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK Introduction Results Conclusion Acknowledgements References The electrical conduction mechanism in ceramic NiMn 2 O 4+  NTCR thermistor material has been the subject of some controversy. Thermally activated hopping of electrons occurs between Mn 3+ and Mn 4+ cations on neighbouring octahedral lattice interstices of the cubic spinel lattice. The resistance decreases exponentially with increasing temperature (NTCR). The simplest way to describe this kind of electrical conduction in NTCR thermistors is in terms of an Arrhenius model R  ~ exp (  /k B T ).In this study it is shown that the conduction mechanism is better described by a variable-range hopping model of the sort: R ~ C (T 2x ) exp (T 0 /T ) x. D.c. resistance - temperature characteristics were measured for NiMn 2 O 4+  pellets, electron-beam evaporated films, rf magnetron sputtered films and screen-printed films. General formula for hopping mechanisms : (1) If the terms in the parentheses in (1) are comparable, variable-range hopping occurs [1] : (2) C is a constant depending on the effective Bohr’s radius and several other constants and the parameters N 0 and p which are connected to the density of states g(  ) : (3) Taking the natural logarithm of equation (3) gives : (4) x = (p+1)/(p+4). W is defined as : (5) From the results it becomes evident, that the following assumption can be made in a good approximation : (6) By neglecting |2x | according to rel.(7) and taking the natural logarithm of W one can obtain [2]: (7) C* is independent of temperature T, so by plotting ln W vs. ln T, the slope of the graph equals -x. PelletsElectron-beam evaporated films Rf magnetron sputtered films Screen-printed films ln W vs. ln T : x = 0.66 ln W vs. ln T : x = 0.43 ln W vs. ln T : x = 1.12 ln W vs. ln T : x = 0.49 ln(Resist.*Temp. -1 ) vs. 1/(T 0.5 ) : T 0 = K ln(Resist.*Temp. -1 ) vs. 1/(T 0.5 ) : T 0 = K ln(Resist.*Temp. -1 ) vs. 1/(T 0.5 ) : T 0 = K ln(Resist.*Temp. -1 ) vs. 1/(T 0.5 ) : T 0 = K The parameter x {x = (p+1)/(p+4)} were all close to 0.5 except for the sputtered film and it was concluded that the variable-range hopping model in eq.(3) is valid for all NiMn 2 O 4+  films and pellets. It is believed that the data collected from the sputtered film displays a high statistical error and no interpretation of the results is possible, as the value of x > 1 does not correspond to any hopping conduction mechanism. The resistance - temperature characteristics of an electron-beam evaporated film shows a high statistical error in the ln W vs. ln T plots as well. It may be therefore concluded that in thin sputtered and e-beam evaporated films the resistance measurements are subject to higher statistical errors leading to a more pronounced scattering of the resistance values around the trend line. In pellets and thick screen-printed films the ln W vs. ln T plots show a much better linearity. It is believed that the parameter x = 0.5. This would correspond to a parabolic shape of the density of states according to eq.(4). Ln(Resist. * Temp. -1) was plotted vs. 1/(Temp. 0.5 ). The slope of the graph equals the characteristic temperature T 0, which was in a similar range for all types of films suggesting that the assumption of x = 0.5 for all types of film is justifiable. It can be concluded that electrical transport in NiMn 2 O 4+  materials is well described by a variable-range hopping model R = C T exp(T 0 /T) 0.5 assuming a parabolic shape of the density of states. In pellets and thick films the mechanism was clearer and the linearity of the corresponding graphs better compared to thin films. The authors wish to thank Prof.Dr. A.Roosen and Dipl.Ing. A.Stiegelschmitt (University of Erlangen, Germany, Dept. of Material Science III) for guidance with developing screen-printing procedures [1] Mansfield, R., Hopping conduction in III-V compounds, in “Hopping transport in Solids”, Pollak, M. and Shklovskii, B., Editors. 1991, Elsevier Science: Amsterdam [2] Shklovskii, B.I. and Efros, A.L., Electronic properties of doped semiconductors. Solid State Sciences , Berlin: Springer. Theory Experimental Data analysis In order to examine the d.c. electrical conduction, pellets and films of NiMn 2 O 4+  were produced. Pellets were produced by pressing NiMn 2 O 4+  powder into disc form of 2-3 mm thickness and 35 mm diameter and sintering. Thin films of 1-2  m thickness were deposited on glass substrates by electron-beam evaporation and the crystallinity improved by an annealing process in air at 850°C for 10 min nm thick films were deposited by rf magnetron sputtering and again the crystallinity improved by an annealing process in air at 800 °C for 30 min. Screen-printed films of 20  m thickness were printed on thick film quality alumina substrates and sintered at 850°C for densification. Aluminium contacts were attached to the pellets and films and covered with silver paint to prevent oxidation. The resistance - temperature characteristics were measured in a standard cryostat system and in a purpose built heat calorimeter.