Positive-bias gate-controlled metal–insulator transition in ultrathin VO2 channels with TiO2 gate dielectrics

The next generation of electronics is likely to incorporate various functional materials, including those exhibiting ferroelectricity, ferromagnetism and metal–insulator transitions. Metal–insulator transitions can be controlled by electron doping, and so incorporating such a material in transistor channels will enable us to significantly modulate transistor current. However, such gate-controlled metal–insulator transitions have been challenging because of the limited number of electrons accumulated by gate dielectrics, or possible electrochemical reaction in ionic liquid gate. Here we achieve a positive-bias gate-controlled metal–insulator transition near the transition temperature. A significant number of electrons were accumulated via a high-permittivity TiO2 gate dielectric with subnanometre equivalent oxide thickness in the inverse-Schottky-gate geometry. An abrupt transition in the VO2 channel is further exploited, leading to a significant current modulation far beyond the capacitive coupling. This solid-state operation enables us to discuss the electrostatic mechanism as well as the collective nature of gate-controlled metal–insulator transitions, paving the pathway for developing functional field effect transistors.

M etal-insulator transitions (MITs) in transition metal oxides often introduce nontrivial changes in a wide range of electrical, magnetic and optical properties 1 . Therefore, the MITs can incorporate various functionalities in materials themselves rather than in the device structures 2 , further boosting the development of electronics in conjunction with silicon (Si) technology. The MIT originates from electrons in narrow d-orbit bands, which are often localized by electronelectron and electron-phonon interactions 1 . These interactions can be weakened by external stimuli such as temperature change and electron doping. Then, all the localized electrons are delocalized, making a sudden change from low to high electrical conductivity, namely the MIT.
The dynamic control of the doping-induced MIT has been attempted via electrostatic gating in cuprates and manganites [3][4][5][6] for the purpose of gate-controlled MITs. Although the gatecontrolled MITs have partially been successful, the expected large conductivity modulation has been hindered by the gradual transitions during the MIT due to the coexisting metallic and insulating domains 4,6 . On the other hand, a larger conductivity modulation is expected by MITs in vanadium dioxides (VO 2 ; ref. 7) because VO 2 shows a relatively abrupt transition in thin films [8][9][10] , and even a completely discontinuous transition in free-standing nanobeams 11 . In addition, VO 2 has several industrial advantages: a simple composition, a relatively low temperature of 300 o C for growing a single-crystalline film on TiO 2 (refs 8-10) and a transition temperature higher than the room temperature 12 .
The gate-controlled MIT in VO 2 has been studied for more than a decade. At first, the three-terminal devices with VO 2 channels were fabricated with the conventional gate dielectrics such as SiO 2 , HfO 2 and parylene [13][14][15][16][17] . However, these devices did not show properties simply expected for the gate-controlled MIT. For example, in a three-terminal device using SiO 2 , the influence of the negative gate voltage (V G ) was observed, but not in a monotonic way 14 . Another example showed current modulation by both positive and negative V G , where only several % conductivity modulation took more than a few tens of seconds indicating the influence of interface traps 15 . In a HfO 2 gate device, the current modulation was accompanied by an unexpected hysteresis, which was attributed to the field-induced strain effect 16 . Although the parylene gate showed a clear modulation of infrared ray absorption by negative V G and a negligible modulation by positive V G , the quantitative information was limited 17 . One of the challenges in these previous studies is the limited number of electronic charges accumulated by the conventional gate dielectrics, in the order of 1 Â 10 13 cm À 2 . In this sense, the gate-controlled MITs in VO 2 have made a conspicuous development with an emergence of the ionic liquid gate [18][19][20][21][22][23][24] . The ionic liquid gate utilizes the electrical double layer at the solid-liquid interface to accumulate electrons in the transistor channel by an order of magnitude larger density than those achieved by the conventional gate dielectrics. Furthermore, the ionic liquid gate has reported a novel mechanism of gate-controlled MITs in the VO 2 channel: while the V G accumulates electrons only within the electrostatic screening length from the interface, the MIT is induced far beyond this screening length 19 . However, this mechanism has rather been controversial because of the possible electrochemical reaction in the ionic liquid gate, which may induce MITs by forming oxygen vacancy or interstitial hydrogen throughout the VO 2 channel 20,21,24 . Besides, the ionic liquid gate has a critical disadvantage of a slow response 22 , severely limiting its device application. To elucidate the mechanism of gate-controlled MITs and to validate their practical interest, the solid-state gate control of the MIT is strongly desired.
In this work, at temperatures close to the MIT, we achieved the gate-controlled MIT by accumulating a significant number of electrons via high-permittivity titanium dioxide (TiO 2 ) gate dielectric, realizing current modulation far beyond capacitive coupling. The solid-state device operation not only enables us to discuss the electrostatic mechanism and the collective nature of gate-controlled MITs but also implies the critical role of interfacial structural compatibility: the gate-controlled MIT is possible only for such an interface that all the interfacial atoms can be involved in the structural transformation during the MIT. These results underscore the interaction of the accumulated electrons at the interface with the three-dimensional phase transition, paving the pathway for realizing functional transistors based on the MITs.

Results
Inverse-Schottky gate. We focused on TiO 2 gate dielectric, which has a significantly high relative permittivity above 100 (ref. 25). Given this permittivity, even a relatively thick TiO 2 gate dielectric (several tens nanometres) can achieve a subnanometre equivalent oxide thickness. In Si technology, the TiO 2 gate dielectric has not been used because of its insufficient band offset against Si conduction band edge. On the other hand, it can have a sufficient band offset against VO 2 that has larger electron affinity 26 . A single-crystalline niobium-doped TiO 2 (Nb:TiO 2 ), an N-type semiconductor, is used as a back-gate electrode as shown in Fig. 1a,b to realize the inverse-Schottky-gate geometry as shown in Fig. 1c,d. In the terminology of 'inverse-Schottky gate', the VO 2 channel is regarded as a metal electrode because of the nanoscale electrostatic screening length of VO 2 in both the metallic and the insulating phases 21 . In this inverse-Schottky-gate geometry, the depleted single-crystalline Nb:TiO 2 at the interface can be exploited as a gate dielectric, enabling a high permittivity as well as high breakdown voltage. In other words, the inverse-Schottkygate geometry achieves electrostatic gating with the aid of strong interfacial electric field created by semiconductor space charge, which was also exploited in other oxide devices 5,27,28 . For the device operation, the Nb:TiO 2 gate dielectric accumulates electrons in the initially insulating VO 2 channel (Fig. 1c) and stabilizes the metallic phase when the accumulated electron density exceeds the threshold as indicated by the black shadow in Fig. 1d. Furthermore, Nb:TiO 2 has the same crystalline structure of rutile as VO 2 , and, hence, an epitaxial interface is formed in between 8,9 . The advantage of this epitaxial interface can be seen in the ideal characteristics of this inverse-Schottky-gate junction as shown in Fig. 2a. The junctions with (101)-and (001)-oriented Nb:TiO 2 had current-voltage characteristics with a clear rectification at room temperature. The exponential relationship for the negative V G (forward bias of the Schottky junctions) followed the thermionic emission model with the ideality factor below 1.1. No dielectric breakdown was observed up to Fig. 2b also showed linear relationships between 1/C 2 and V (Mott-Schottky plot) for both (101) and (001) orientations. The relative permittivities are estimated to be 111 and 146 for (101)-and (001)-oriented Nb:TiO 2 , respectively, from the slopes in Fig. 2b, which are approximately consistent with the past literatures 25 . The built-in potentials of 0.7 eV were extracted from the intercepts of the extrapolated lines at the voltage axis in both cases, which approximately equals the value estimated from the work functions 26,29 . The calculated depletion widths with no bias are 25 and 29 nm for (101)-and (001)-oriented Nb:TiO 2 , respectively, corresponding to equivalent oxide thicknesses of 0.9 and 0.8 nm in the case of metal-insulator-semiconductor structures. Although the decrease in TiO 2 permittivity under high electric field has been reported 30 , the plots of 1/C 2 are almost linear to V G in Fig. 2b, indicating that the decrease in permittivity is insignificant at the electric field of 2.1 Â 10 6 V cm À 1 for (101) orientation and 1.8 Â 10 6 V cm À 1 for (001) orientation (both corresponding to V G ¼ 10 V). This high permittivity up to high electric field enables electrons more than 1 Â 10 14 cm 2 to accumulate at the VO 2 /Nb:TiO 2 interface.
Transistor. By exploiting these ideal characteristics of the inverse-Schottky-gate geometry, the gate-controlled MIT was investigated on the VO 2 channel with 6-nm thickness. Figure 3a shows the electrical resistivity versus temperature of the VO 2 channel deposited on the (101)-oriented Nb:TiO 2 gate (the lattice mismatch B0.8% in the [010] direction and B1.6% in the [10-1] direction) as a function of V G . The data show the MIT with the resistivity jump by more than two orders of magnitude around 312 K, accompanied by a hysteresis reflecting the first-order phase transition of VO 2 (ref. 7). Remarkably, the MIT was shifted to a lower temperature by as much as 1 K with the application of V G ¼ 9 V. This magnitude of modulation is approximately comparable to the magnitude reported in the ionic liquid gating 18 (0.5 K for accumulated electron density B2.5 Â 10 14 cm À 2 ), which seems to avoid the electrochemical reaction by strictly limiting the magnitude of gate voltage. In the case of the (001)-oriented Nb:TiO 2 (the lattice mismatch B0.8%), on the other hand, the MIT was not clearly shifted by the application of V G ¼ 5 V (Fig. 3c). In the (001) orientation, the MIT was observed around 290 K, which was different from the transition temperature (T MIT ) for the (101)-oriented interface because of the orientation-dependent strain in the VO 2 channel by the Nb:TiO 2 substrate 31 . T MIT was summarized as a function of the accumulated electron density (N 2D ) in Fig. 3d. Here T MIT was calculated by averaging the transition temperatures during heating and cooling in Fig. 3a,c. T MIT decreased almost linearly to N 2D , and the slope was much larger for the (101) orientation than the (001) orientation.
Next, the gate-controlled MIT in the VO 2 channel on (101)oriented Nb:TiO 2 was evaluated at three different temperatures: 306.3 K (in the insulating phase), 315.0 K (around the transition temperature) and 325.3 K (in the metallic phase), in the transfer characteristics of Fig. 4a. The channel current (I D ) and the gate leakage current (I G ) were plotted as a function of V G . I G was more than two or three orders of magnitude smaller than I D throughout the measurement (except for V G o0.2 V where the Schottky diode in the gate is forward biased near the drain electrode). The current modulation by V G was small at 306.3 and     Fig. 4b, demonstrating the current modulation far beyond capacitive coupling. It should be noted that the actual electron mobility could deviate from 0.5 cm 2 V À 1 s À 1 , especially when the VO 2 becomes metallic, given the uncertainty of the Hall measurement in the metallic phase 32 . However, if we can assume that the Hall measurement is relatively valid in the insulating phase, the calculation in Fig. 4a,b may still hold because it basically calculates the electron accumulation in the insulating phase.
Origin of gate-controlled MITs. We checked the time response of the channel resistance to V G to confirm that the slow process such as electronic trapping and ionic diffusion is not dominant.
To investigate the possible influence of these slow processes independently of the MIT in VO 2 , the time response was measured far below T MIT , where VO 2 remains in the insulating phase and the application of V G ¼ 5 V decreases the resistivity only by B10% as shown in Fig. 3b. When V G was abruptly increased from 0.2 to 5 V as shown in Fig. 5a, the resistivity change was faster than the time resolution of this measurement (100 ms). These results indicate that there is neither electronic trapping nor ionic diffusion slower than 100 ms. The fast response was also confirmed by the frequency dependence of capacitance at VO 2 /Nb:TiO 2 (101) Schottky junction far below T MIT , where the VO 2 was in the insulating phase (Fig. 5c). The capacitance was approximately constant below 10 kHz, indicating that the capacitance measurement was free from the process slower than 100 ms. Above 10 kHz, the measured capacitance rapidly decreases because of the effect of series resistance in the electrical circuit. The absence of ionic diffusion in our devices was also confirmed in Fig. 3d. The transition temperature recovered when V G was changed from 0.2 to 9 V (red filled diamonds from left to right in Fig. 3d), and finally set to 0.2 V again (red empty diamond). This volatile effect of V G indicates the absence of electrochemical reactions, in contrast to the ionic liquid gate where the recovery of the initial state requires an additional process of heating 24 or applying the opposite V G even outside the VO 2 hysteresis 19 . It should also be noted that the switching power and speed in our device are physically consistent as shown in Supplementary Fig. 1 and Supplementary Note 1, which maps various limits on the power-speed relationship in the VO 2 channel field effect transistors.
Character of gate-controlled MITs. One of the issues on the gate-controlled MIT is whether it is different from the temperature-induced MIT in its character. First, we consider the possibility that the gate-controlled MIT is different from the temperature-induced MIT and is determined only by electron density as assumed in the past literature 13,14,20,22 . In this case, the accumulated electron density in our gate-controlled MIT can be compared with critical electron densities for MITs in several other experiments. In our device, N 2D to decrease T MIT by 1 K is B9.0 Â 10 13 cm À 2 . On the other hand, the electron density to decrease T MIT by 1 K in W-doped VO 2 is B6.8 Â 10 20 cm À 3 (ref. 33), corresponding to B1.4 Â 10 13 cm À 2 in a 6-nm VO 2 film. Considering that W doping can also affect crystal structure and hinder a fair comparison, we also focused on the Hall measurement in non-doped VO 2 films 34 , assuming that the transition is induced when the electron density reaches B3.2 Â 10 19 cm À 3 in the insulating phase. On the basis of this assumption, the temperature dependence of the Hall electron density indicates that B5.0 Â 10 18 cm À 3 is needed to decrease T MIT by 1 K, which corresponds to B3.0 Â 10 12 cm À 2 in a 6-nm VO 2 film. In either case, the electron density that is needed for the gate-controlled MIT is estimated to be only 3-16% of the N 2D in our device. This discrepancy seems to be not compatible with the assumption that the gate-controlled MIT is driven only by carrier density. In reality, however, it is also possible that not  Second, we summarize another possibility that the gatecontrolled MIT is similar to the temperature-induced MIT, which is known to be a first-order transition mainly balancing the phonon entropy and the orbital energy 36 . In this case, V G only modifies the transition temperature of the temperature-induced MIT, and when the modified transition temperature crosses the measurement temperature, the gate-controlled MIT is observed. This possibility was hinted by the slow time response (B10 s) around T MIT (Fig. 5b), where V G induces transition from the insulating phase to metallic phase. A possible interpretation of this slow response would be the nucleation of metallic phase domains, which is usually observed in the temperature-induced MIT 37,38 . Actually, the nucleation of the metallic phase was also implied by the small but abrupt steps in resistivity in Supplementary Fig. 2 and Supplementary Note 2. Besides, the first-order nature of the gate-controlled MIT was also suggested by the fact that the critical N 2D (or V G ) for the MIT increases as the temperature decreases from T MIT (Fig. 3d). This strong temperature dependence is reasonable if the metallic phase has a higher entropy than the insulating phase, and hence less favoured at lower temperature, just as reported in temperature-induced MIT 36 . At this moment, it is not feasible to determine the character of the gate-controlled MIT, and is to be further investigated via structural or optical analyses, which are beyond the scope of this paper.
Territory of gate-controlled MITs. In our devices, the current modulation is taking place not only in the vicinity of the interface, but in the whole 6-nm VO 2 film. This is clearly evinced by the fact that the resistivity versus temperature curve in Fig. 3a was shifted by V G almost in parallel without any change in its shape. Given the screening length of VO 2 is at most a few nanometres 22 , it is implied that the current modulation occurs beyond the electrostatic screening length. The current modulation beyond the electrostatic screening length was also previously reported in the VO 2 film with ionic liquid gate 19 . However, this current modulation with ionic liquid gate is controversial because of the possible ionic diffusion 20,21,24 , leaving uncertainty about its origin. In our devices, the solid-state current modulation using the epitaxial VO 2 /Nb:TiO 2 interface avoids the ionic diffusion as mentioned previously, providing better insights into the collective nature of MITs.
The current modulation throughout the VO 2 film was further corroborated by the VO 2 thickness dependence. Figure 6a,b shows the V G dependence of resistivity for VO 2 thicknesses of 8.7 and 28 nm, respectively. In Fig. 6c, the change of T MIT (DT MIT ) is plotted as a function of N 2D for three different VO 2 thicknesses. As the VO 2 thickness increases, the magnitude of gate control becomes smaller. Here the slope (DT MIT /DN 2D ) is a good measure of the magnitude of gate control and was plotted as a function of VO 2 thickness (Fig. 6d) and its inverse (Fig. 6e). As shown in accumulated electrons interact not only with the interface region but also with the whole VO 2 film. The origin of this collective nature could be related to the domain boundary energy between metallic and insulating phases in VO 2 films. It is well established in the conventional phase transition that small nuclei of phase are unstable because of the energy cost of the phase domain boundary. If we could apply this concept to the MIT in our device, the nucleation of metallic phase only in the vicinity of the interface would be suppressed because it leads to a large area of phase domain boundary inside the film. Instead, the energetically favourable nucleation involving larger volume may occur. Then apparently, the electrostatic effect beyond the screening length could be observed.
Effect of Joule heating. Several device characteristics indicate that the Joule heating seems to have a negligible influence on the gate-controlled MIT. To check this, we considered the influence of Joule heating in two different ways. First, I D was increased (0.01, 0.1, 0.3 and 1 mA) with the gate floated, where the Joule heat was generated mainly by V D Â I D . As shown by blue diamonds in Fig. 7a, T MIT showed almost no change as a function of Joule heat that was calculated for the insulating VO 2 around the transition temperature. Second, V G was increased with constant V D , where Joule heat was mainly generated by the gate leakage and was calculated by V G Â I G for the insulating VO 2 around the transition temperature. In this case, T MIT monotonically decreased as shown in Fig. 3d and also by red squares in Fig. 7a. This discrepancy between two plots in Fig. 7a shows that the V G dependence of T MIT cannot simply be attributed to Joule heating. For the second case, we further compared T MIT and Joule heat (V G Â I G ) as functions of N 2D in Fig. 7b. While the decrease in T MIT is linear to N 2D , the Joule heat is strongly nonlinear, indicating that the decrease in T MIT seems not directly related to Joule heat.
While we have shown the Joule heating by homogeneous gate leakage is irrelevant of the gate-controlled MIT, it also has to be considered that the gate leakage current is inhomogeneous, causing local temperature rise (hotspot) in the VO 2 film. The formation of hotspots can thermally induce local transition from the insulating phase to the metallic phase. Although the total power of this local Joule heating was shown to be not enough to induce transition of the whole film, the metallic phase in the local hotspot can propagate to the rest of the film if it is in a superheated (metastable) state. Thus, the local Joule heating may also account for the observed gate-controlled MIT (Point 'A' in Supplementary Fig. 3a). The scenario of local Joule heating is also compatible with the slow time response in Fig. 5b as well as the collective transitions in Fig. 6.
Considering the device characteristics in details, however, we can possibly distinguish the influence of local Joule heating (Supplementary Note 3). While the local Joule heating may trigger the transition in the whole film on the high-temperature side of hysteresis (Point 'A' in Supplementary Fig. 3a) because of superheating ( Supplementary Fig. 3b), the same mechanism does not apply to the low-temperature side. On the low-temperature  side of hysteresis, the metallic phase in the hotspot cannot stabilize metallic phase in the rest of the film (Point 'B' in Supplementary Fig. 3c) because the rest of the film is not superheated (Supplementary Fig. 3d), in contrast to the high-temperature side of hysteresis ( Supplementary Fig. 3b).
In other words, because the insulating phase is more stable than the metallic phase on the low-temperature side of hysteresis, the propagation of metallic phase is energetically forbidden. As a result, the local Joule heating should lead to an asymmetric influence on hysteresis between the high-temperature side and the low-temperature side ( Supplementary Fig. 3c), which is inconsistent with the symmetric influence of V G in Fig. 3a or Supplementary Fig. 3a. Thus, our device characteristics seem to be not explained only by Joule heating in any way, although we have to keep an open mind considering the long-standing problem of Joule heating in voltage-induced MITs in two-terminal VO 2 devices 39,40 .

Discussion
While we experimentally showed that V G induces the MIT in the whole VO 2 channel through electrostatically accumulating electrons, the specific mechanisms of the interaction between the accumulated electrons and the whole-VO 2 channel are yet to be investigated. For example, while the accumulated electrons are expected to be in the conduction band of the insulating VO 2 , another possibility is that they are rather in the in-gap states and only some of them are thermally excited to the conduction band to induce the MIT. We can also consider the influence of inhomogeneity between the metallic and insulating phases during the MIT, which has been observed in polycrystalline 41,42 and single-crystalline VO 2 films 43,44 . The inhomogeneity of different phases is known to cause percolative conduction and sometimes enhance the magnitude of gate control in manganite-channel field effect transistors 4,6 .
The specific mechanism of device operation could also relate to the larger current modulation for the (101)-oriented interface than the (001)-oriented interface. One of the possible origins for this interfacial orientation dependence can be the structural difference between the (101) and (001) interfaces. The (101) and (001) interfacial structures are schematically illustrated in Fig. 8a,b, where the rutile structure is indicated by the dashed lines. The rutile structure of the metallic phase VO 2 is distorted in the insulating phase in a monoclinic symmetry with the dimerization of vanadium atoms 7 as indicated by the empty circles. This dimerization leads to a critical difference between the (101)-and (001)-oriented interfaces; while half of the vanadium atoms cannot dimerize at the (001)-oriented interface as indicated by dashed blue circles in Fig. 8b, all of them can dimerize at the (101)-oriented interface because the dimers can be aligned parallel to the interface as shown in Fig. 8a. This dimerization in the (101) plane was also corroborated by the X-ray diffraction of a 71-nm VO 2 film on the (101)-oriented Nb:TiO 2 substrate in Fig. 8c, which shows the (0.5, 0, 0.5) superlattice peak of the VO 2 film corresponding to the doubling of the VO 2 unit cell in the plane-normal direction. Thus, the dimerization of interfacial vanadium atoms, which strongly depends on the geometric compatibility of the VO 2 structural distortion against the TiO 2 surface orientation, may possibly be responsible for the orientation-dependent current modulation, highlighting that MITs are not purely electronic but are often accompanied by structural transformation. On the other hand, this critical role of interfacial structural compatibility in gate-controlled MITs is to be further studied because the MIT and the structural transformation sometimes take place independently 14,45-48 . Indeed, it is also possible that the MIT by V G in our device is not accompanied by structural transformation, and the observed difference between the two interfacial orientations originates from the difference in structural imperfections.
In summary, the solid-state gate-controlled MIT in the ultrathin VO 2 channel was successfully demonstrated by the high-permittivity TiO 2 gate dielectric in the inverse-Schottky-gate geometry. This TiO 2 gate dielectric has a potential advantage compared with the ionic liquid gate suffering from the electrochemical effect and slow response, or the ferroelectric gate 4,49 impaired by a large hysteresis. This solid-state gate dielectric enabled us to discuss the electrostatic mechanism and the collective nature of gate-controlled MITs, which has caused much controversy in the past literature [13][14][15][16][17][18][19][20][21][22][23][24] . Besides, these results hinted the specific interaction mechanism between the accumulated electrons at the interface and the three-dimensional phase transition, providing the basis for novel device physics incorporating MITs and facilitating the extraction of a variety of functionalities.

Methods
Device fabrication. The VO 2 films were grown by pulsed laser deposition on the (101)-and (001)-oriented Nb:TiO 2 substrates with 0.05 wt% Nb concentration. The substrate temperature, the oxygen pressure and the laser fluence were set to 300°C, 1 Pa and 1 J cm À 2 , respectively. All the VO 2 films had atomically flat surfaces with the root mean square of B0.2 nm, except for the film with 28-nm thickness, which showed cracks because of the VO 2 -TiO 2 lattice mismatch 50 . The VO 2 films were patterned by diluted aqua regia, and Ag paste was used for the Ohmic contacts to the VO 2 films. Aluminium wires were bonded for the contact to the Nb:TiO 2 substrate. Electrical properties of the inverse-Schottky-gate junctions were measured for the VO 2 film of 9-nm thickness, which is below the critical thickness for crack formation 9,44 . For transistors, the thickness was decreased to 6 nm to enhance the gate control, and the source and the drain electrodes were formed with the channel length of 155 mm and the width of 50 mm.
Electrical measurement. The capacitance was measured by applying an AC voltage of 0.05 V in 500 Hz while sweeping the DC V G applied on the Nb:TiO 2 substrates. The transistor characteristics were measured with the source grounded, and V D ¼ 0.2 V. The temperature was swept from low temperature to high temperature for the heating sweep, reversed to low temperature for the cooling sweep and then V G was changed for the next set of sweeps. The transition temperature of the VO 2 film was defined by the largest negative slope of the logarithmic resistivity against the temperature.