Resistivity of non-ohmic materials

Question

Is resistivity of non-ohmic materials constant with constant temperature, or it depends on some other factors as well? I'm aware that for non-ohmic devices, the ratio V/I is not a constant which means that the resistance (R) is variable, but from the relation R=rho*l/a (rho=resistivity),R should be constant for constant resistivity. So does resistivity vary with voltage, current(or any other factor) here or is this relation not always true for non-ohmic materials? (I know that this is a silly question, still I'm trying to seek help here.)

Answer 1

Fixed resistance is only idealization of what really happens. Resistors (conductors) usually have a complex dependence on temperature and other variables (frequency, voltage, current etc.), which might or might not be simplified depending on what you are trying to analyze. In general, conductor resistivity increases with increasing temperature and the reason for this phenomena lies in the field of quantum mechanics. The resistivity of a conductor can be approximated as:

$$\rho(T) = \rho_0 \cdot \bigl( 1 + \alpha \cdot (T-T_0) \bigr)$$

where $\rho_0$, $T_0$, and $\alpha$ are material-dependent parameters, and $T$ is temperature.

Some other elements have resistance which depends mainly on frequency (aka impedance). Frequency dependency is not a problem because we have tools (Laplace and Fourier transform) which handles frequencies naturally. But they also depend on other variables, most importantly temperature, voltage and current. Examples include inductors and capacitors. Devices with active control can sometimes be described with impedance, for example power converters etc.

Finally, there are devices and elements which are nonlinear and cannot be simplified to resistance or impedance. They are described by complex voltage-current relationship. Examples include semiconductors such as diodes and transistors etc. Semiconductors in general have strong dependency on temperature.

Figure below shows voltage-current relationship of a diode at a fixed temperature. Although deep inside the diode is composed of conductors which do obey Ohm's law, due to other effects it's voltage-current characteristics is nowhere near that of a simple resistor.

^{Fig. Diode voltage-current relationship (P. Horowitz, W. Hill, "The Art of Electronics", 3ed, 2015)}

I would also like to point out that there are devices with negative impedance. I know, mind-blowing! Examples include systems that are controlled to draw constant power - when voltage drops, current must increase to maintain the power, hence negative impedance. These are especially nasty for the power system.

To conclude, resistivity depends on temperature and geometry (length, cross-section area) with material-dependent parameters. Voltage and current do not directly affect resistivity (someone with background in quantum mechanics might prove this wrong). However, current flow does cause temperature variations in the conductor which affects resistivity, hence current indirectly does affect resistivity.