Your idea is very innovative, since it turns the display into a solar energy concentrator. The inventors of the quantum-dot window are concentrating on scaling them up so that windows in homes and buildings can also be concentrated solar energy harvesters. Scaling down to mobile device size would require photovoltaic cells that were very thin to fit around the edge of a display, but the concentration factor should be impressive since the entire face of the display would be funneling incident light to the PVs. Only time will tell if your idea is workable, but I see promise.
What would be the application of the Quantum dot enabled solar energy harvesting? The mobile devices would be able to harvest solar energy to generate power sufficient for charging the battery of the mobile devices? Recently while driving to my office I kept my phone on the seat next to mine. Sunlight was falling on its display and I was thinking (or rather wishing) that, there should have been a technology such that the display of the mobile device (or a transparent screen on it...the screen protector itself) to be able to harvest electrical energy from the solar energy to charge the battery of the device, so that my mobile will get charged on the way to my office... :)