Variation of resistance with temperature in semiconductors I've found in various books that the mathematical expression for the variation of resistance with temperature is the following:
$R=R_0\cdot exp\bigg(\frac{E_g}{2K_BT}\bigg)$
where $K_B$ is Boltzmann constant and $E_g$ is a constant related to the bandgap of the semiconductor.
The value $R_0$ is supposed to be the resistance for a reference value $T_0$ of the temperature of the material. But I don't see $T_0$ anywhere in this expression, which means that if I change $T_0$ and therefore $R_0$ changes, I would get different values of $R$, which is illogical. So, my question is, how can this happen? Does $E_g$ vary with the temperature in some way? But then, should I always use the value of $E_g$ for the temperature $T_0$ taken as reference?
 A: $R_0$ is not the resistance measured at a temperature $T_0$, as you have found. It is merely a proportionality constant. To estimate both the proportionality constant $R_0$ and the energy gap $E_g$ you would need two measurements of resistance ($R_1$ and $R_2$) at different temperatures ($T_1$ and $T_2$) and solve for both unknowns ($R_0$ and $E_g$) from the resulting equations
$$
R_1 = R_0\exp(E_g/2k_BT_1) \\
R_2 = R_0\exp(E_g/2k_BT_2)
$$
with solutions
$$
E_g = 2k_B\left(\frac{1}{T_1} - \frac{1}{T_2}\right)^{-1}\ln\left(\frac{R_1}{R_2}\right) \\
R_0 = R_2\exp\left(\frac{T_2}{T_2 - T_1}\right).
$$
If you already know the bandgap $E_g$, then you obviously only need one measurement ($R_1$ at $T_1$) and $R_0$ is given by
$$
R_0 = R_1\exp(-E_g/2k_BT_1).
$$
The result you cite can be traced back to a consideration of how the carrier concentration (electrons in the conduction band and holes in the valence band) changes with temperature in thermal equilibrum. In general, the temperature dependence of conductivity (and hence resistance) is a consequence of the temperature dependence of carrier concentration and carrier mobility. However, carrier mobility changes as a power law with small exponent, while carrier concentration changes exponentially. This causes the change in carrier concentration to be the main mechanism behind the change in conductivity and leads to your result for an intrinsic semiconductor.
A complete and elementary treatment obtaining said result can be found in Kittel's book "Introduction to Solid State Physics - Eight Edition" Chapter 8: Semiconductor crystals.
A: If the equation is supposed to behave as you described, then you must be missing a term.  Plugging in $T=T_{0}$, you get
$$R(T=T_{0})=R_{0}exp\bigg(\frac{E_{g}}{2k_{B}T_{0}}\bigg)\neq R_{0}$$
The only way to retrieve $R_{0}$ for $T=T_{0}$ would be if $T_{0}=\infty$ for that $R_{0}$, which is unlikely.  I do not know the correct formula for this phenomenon offhand, but I think your formula as written is incomplete.
