From the IV curve you attached one may conclude that the contacts to the nanowire are non-ohmic and that the properties of the metal-semiconductor interface determine the IV characteristic. There is not much more to be concluded from that data.
Since you are dealing with NW, cathodluminescence is a fruitful method to investigate band-band and excitonic transitions.
Jun-Sik is okay. I would like to add that in order to determine the band gap from the measured I-V curve, one has to determine precisely the current conduction mechanism. In the sense one has to get an expression relating the the current to voltage and the semiconductor parameters including the energy gap. It is so that the intrinsic concentration ni is the parameter which is related to energy gap. So, if the material is intrinsic one would directly measure ni as a function of temperature and from the curve of ni versus 1/T one can determine the energy gap as the activation energy for the thermal generation process. The other method is to to produce a pn junction and determines the reverse saturation current which is also activated thermally by the energy gap. So, i is required to measure the reverse saturation current as a function of temperature and from it one can determine the energy gap.
However as Jun-Sik hinted the energy gap can be measured directly from the optical absorption edge.
If I am understanding what you have done correctly, it sounds as if you have connected a single nano-wire as if it is a resistor, and measured the I-V characteristics. If this is the case, Then there is nothing within the nano-wire that is going to create an energy field and cause charge separation, allowing you to examine the band-gap. Your I-V curve is nearly symmetrical, and looks like a highly resistive semiconductor with weakly rectifying Schottky barriers on both sides. And if you connected as I believe you have, this is exactly what I would expect to see.
If you can grow enough, spectroscopic ellipsometry would be very useful, but hard to get reliable data from a single nano-wire. I believe what you need to do is a series of tests with different metal connections on one end of your nano-wire. Find a metal that makes an OHMIC contact with your nano-wire, and coat 1 end of your nano wire with that. Find a metal that makes a rectifying contact with your wire, and coat the other end with that, and attempt your measurement again. By studying the I-V curve, what you are really looking at is the energy barrier. If you choose a metal with a high enough work function, and a metal with a low enough work function (assuming your nano-wire has a band gap) you can create an electron only device, and a hole only device. From there, by examining the energy barriers your device encounters, you can infer your energy band gap.
A great resource for this measurement, and many others, is Semiconductor Material and Device Characterization by Dieter K. Schroder. I found a 3rd edition online for free, and it has been very useful.
It is very likely that the transport is space-charge-limited. This is especially the case if your wire is not nominally doped. You can check my project folder for mor information about this.