The increase in band gap of a semiconductor has drastic changes in electrical and optical properties. For example, if a semiconductor was transparent in infrared region, on increasing band gap it may be transparent in visible region. Electrical conductivity may decrease on increase in band gap. Use of a semiconductor in solar cells also depend on band gap. Optimum value for this purpose is 1.4 eV. It may not be showing photo luminescence at low band gap but may show photoluminescence when band gap becomes higher. In conclusion one can say that band gap is a critical parameter from application point of view.
Whether there will be any application for a higher band gap, will depend upon the current band gap and the relevant properties of the semiconductor for an application. Say the application is solar cell and the current band gap is less than the optimal band gap, which is in the range of 1.4 - 1.5 eV. But, if the absorption coefficient of the semiconductor is not attractive, then there is not much sense in increasing its ban gap, say, by alloying with other semiconductors.
Change in bandgap (Eg) with change in material composition for compound semiconductors is generally referred as Bandgap Enginnering and it has very useful applications in case of Electronic devices like HEMTs (AlGaAs/GaAs, AlGaN/GaN, InGaN/GaN), HBTs etc. Application also includes in opto-electronic devices like LASERs, LEDs, PDs, Solar cells etc. In hetero-junctions like AlGaAs/GaAs (AlGaN/GaN), higher bandgap of AlGaAs (AlGaN) than that of GaAs (GaN) will create a barrier and give rise to a high density sheet charge (2DEG) without/less doping. This high-mobility 2DEG is used as channel for HEMTs.
With increase in Eg, intrinsic carrier density (ni) decreases i.e. conductivity decreases 9at same temperature). This ni is responsible for leakage current in devices, so with increase in Eg, leakage current will reduce. As a result those devices can be reliably operated in harsh environments (like Space , high temperature applications).
With higher Eg barrier height will increase and making ohmic contact will be difficult.
However increase in barrier height for higher Eg will reduce the gate leakage current in case of MESFET, HEMTs.
For optical applications, increase in Eg can make a semiconductor from opaque to semi-transparent.
The increases of the energy gap for semiconductors lead to decreases the absorption region and the substance may be transparency and the absorption edge shifted toward low wavelengths. This may be useful for solar cell applications .
There is another application for increasing the bandgap value that has been forgotten by the other pals: high temperature electronics, nice for motor cars and space (to explore planets closer to the sun) sensors and/or microprocessors. In the first part of the 90's, there were some groups that have studied this problem, seeking for applications of some materials such as diamond, silicon carbide and boron phosphide as well. The discovery of blue ray emission, carbon nanotubes, photovoltaics, and other nice problems have ecclipsed this subject matter.