Domain of validity for semiconductor equations If I understand correctly, the distinction between semiconductors and insulators is a matter of convention? A semiconductor is basically an insulator with an (enough) narrow band gap, usually around $3$ $\rm eV$.
I have a few equations used for semiconductors, will they still work for insulators? Or are they gradually starting to breakdown for larger band gaps? Which effects remain and which fade away?
In this specific case I'm working on $\rm TiO_2$ which is (if I'm not mistaken?) a wide-band semiconductor, which classifies between semiconductor and insulator.
Example of semiconductor equations in use:

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*Conductivity: $\sigma = ne\mu_e+pe\mu_h$


*Inner voltage: $$V_{bi}=\frac{k_BT}{q}\ln{\frac{N_AN_D}{n_0p_0}}$$


*Intrinsic carrier concentration: $$n_i=p_i=2\left(\frac{k_BT}{2\pi h^2}\right)^{3/2}(m_em_h)^{3/2}\exp\left(-\frac{E_g}{2k_BT}\right)$$


*Doping:
Baseline, is it okay to use semiconductor formula's on $\rm TiO_2$? Any other wide-band semiconductor? Any insulator? What is the general rule?
 A: Anything will conduct electricity with a large enough electric field. What distinguishes a semiconductor from a non-semiconductor in practice is how it conducts.

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*Silicon, $\rm TiO_2$, and many other materials conduct electricity by something resembling the free flow of electrons and/or holes (a.k.a. single-particle approximation). We can treat all these materials in a similar way, using similar equations, and we can call them all "semiconductors".


*Other materials conduct electricity in other different ways, and therefore are not called semiconductors. For example, if you apply a small electric field to glass, current will flow via electron tunneling from one bound state to the next (Poole-Frenkel effect), which does not at all resemble the free flow of electrons and/or holes. If you apply a large electric field to glass, current will flow by the process we call "dielectric breakdown", involving motion of ions and defects etc., and which, again, has no resemblance whatsoever to the free flow of electrons and/or holes.
I like this operational definition because it explains otherwise-inexplicable terms like how doped diamond is called "semiconducting diamond". You might think, diamond is either a semiconductor or it's not, so how can you describe a certain type of diamond as being "semiconducting diamond"? Well, because when it's appropriately doped, it conducts current by something resembling the motion of free electrons or holes, and when it's not appropriately doped, it conducts current by Poole-Frenkel effect, dielectric breakdown, or whatever.
It's a general rule that as the bandgap increases, the material becomes more and more ionic, and the electrons and holes look less and less like free particles and more and more like polarons. It's also a general rule that as the bandgap increases, would-be dopants are less and less likely to be able to ionize to create "free" carriers (they tend to create levels deep in the bandgap). Therefore, as bandgap increases, it's less and less likely that you'll find anything that should be called a semiconductor.
But there's no hard boundary. Even aluminum nitride $(6.2\ \rm eV)$ and diamond $(5.5\ \rm eV)$ can behave as semiconductors---in special, carefully-controlled situations.
$\rm TiO_2$ is very much a typical semiconductor, as far as I know. It differs from silicon, $\rm GaAs$, etc., mainly in that it's impossible to make it intrinsic $\rm TiO_2$ or p-type $\rm TiO_2$ (well, some would argue it's merely "almost impossible"). But if you keep that in mind, the equations you cite should still apply.
A: As you mentioned, the difference between insulators and semiconductors is a matter of convention. There is no absolute way of judging whether a material is a semiconductor or insulator. Insulators have "large" band gap while semiconductors have a relatively "narrow" band gap. There are no clear thresholds to decide which is which. However, one can roughly say that a material having a band gap energy less that its melting temperature is a semiconductor. Have a look at the second paragraph of "In semiconductor physics" of this page. The inverse of that statement is not true.
Speaking of Titanium dioxide in particular. It was shown by experiments to be a semiconductor, have a look here.  It is also known to be strongly N-type intrinsic semiconductor, have a look here.
With respect to equations, I suppose the answer is yes you can use them. In this paper the authors study the mobilities of holes and electron holes. The paper also indicates that Titanium dioxide can be doped as any other semiconductor. There are many papers on the internet studying the behavior of doped Titanium dioxide with nitrogen and chromium as an example. So it can be doped and you can compute its conductivity using expression above.
As a second indicator of the possibility of treating Titanium dioxide as a regular semiconductor, have a look at this PhD thesis. In it, the author treats Titanium dioxide as a typical semiconductor. He even sites many of the standard formula used in semiconductor physics in the introductory chapters of his work.
I am not an expert on Titanium dioxide in particular. But literature suggests it can be treated as semiconductor because it is.
A: The equations you mention are derivable by assuming the existence of a band structure with a band gap surrounding the Fermi level, and taking the limit of low electron/hole concentrations in conduction/valence bands. They work just fine in insulators, although they over-complicate things when it's not possible to dope the insulator.
In contrast, you can't use those equations in: Highly doped semiconductors or small-gap semiconductors, semimetals, metals. In these cases the Fermi level is too close to a conductive band, or is even inside a band. 
