It's not about batteries
As others have noted, the actual claims talk about diodes, not batteries. However, this should not be taken to mean that the authors are talking about diodes used in batteries, and the possibility of reducing parasitic losses and the like. This research has nothing whatsoever to do with batteries at all. When batteries are mentioned, it's purely an example of why this research could be useful - a device using this technology in diodes could have greatly reduced power consumption compared to one with traditional diodes, and hence the battery used in such a device would last longer. They're talking entirely about reducing electronics power use, not about increasing battery capacity.
Unfortunately this has been mangled in the usual way by media attempts at science reporting to instead make grand claims about new battery technology, when in fact it is nothing to do with batteries at all. Here are some actual quotes from the press release:
"The material also has significantly less dissipative power compared to a semiconducting diode"
"new magnetic transistors and amplifiers that dissipate very little power"
"Less dissipative power in computer processors could also reduce the heat generated in laptop or desktop CPUs"
Note that there is nothing there about increasing battery capacity or efficiency; it's all about increasing the efficiency of electronics so that you can get more use out of existing batteries.