What is the limiting value of band gap (in eV) that makes a material semiconductor? Or what is the value of band gap that separates insulator and semiconductor?
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3$\begingroup$ There is no commonly accepted definition. If you can make a standard semiconducting device in the material, it is a semiconductor. $\endgroup$– Jon CusterCommented Dec 10, 2015 at 13:35
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The transition of electron from the valence band to the conduction band is a probabilistic event. Electrons get this excess energy equivalent to the band gap ($\Delta E$) from the thermal energy ($kT$, where $k$ is the Boltzmann constant and $T$, temperature in Kelvin). Hence, the probability of such an event will be proportional to $e^{-\Delta E/kT}$.
Answering your question, although there is no standard defined value of band gap, you can see since it's an exponentially falling smooth curve that one can define a cutoff $\Delta E$, saying that for band gap $> \Delta E$ there are practically no electrons transiting to the conduction band, qualifying the material as an insulator (i.e., exhibiting insulator-like properties).
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3$\begingroup$ However, this completely ignores the dopants, which are close to the band edges and thus can get carriers in to the conduction (electrons) and valence (holes) bands at reasonable temperatures. Silicon carbide has a band gap in excess of 3eV, more than twice that of GaAs. Aluminium nitride, a so-called ultra-wide bandgap material, has a band gap of 6 eV, which is getting pretty close to silicon dioxide at ~8.9eV. Yet, aluminum nitride can be used to make devices because good dopants exist. So, the ability to dope is more important than a specific band gap. $\endgroup$ Commented Dec 10, 2015 at 15:06