The article isn't about energy storage, it's about graphene superconductivity. When it becomes superconductive, it allows the free flow of electrons without resistance which produces heat. One of the limiting factor of today's chips is heat dissipation; if no heat was generated, we could make them higher clocked and faster.

You might be thinking of supercapacitors, which store energy. http://physicsworld.com/cws/article/news/2010/nov/26/graphen...

I think this is not linked with battery storage. Superconductivity allows a material to conduct electrons without any resistance, thus losing less energy and generating less heat. I could be wrong though, I am no expert.

You can use superconductors for energy storage. An inductor stores energy, but can not be used for long-term energy storage due to resistance. A superconducting inductor can be used to store energy, as there is no loss due to resistance.

https://en.wikipedia.org/wiki/Superconducting_magnetic_energ...

Just to be sure, first, are you clear that your question has nothing to do with the article in question? Calcium-doped graphene superconducting at 4K means nothing for your phone battery.