Time to move into a type of controller that hardcore gamers love: physical ones.
There are a few devices that have a controller included as a part of their hardware. Some notable examples are XPeria Play—one of the pioneer phones that had a sliding gamepad—and Nvidia Shield, the latest in this category.
XPeria Play was one of the first devices with an integrated gamepad:
Nvidia Shield is one of the most powerful Android devices with a gamepad:
On the other hand, there are many brands that build game controllers for smartphones and they have been fairly popular among traditional gamers. All of them are Bluetooth controllers that can be connected to your phone or tablet. Some of them are designed to make your phone fit into it like Gametel (another pioneer) and most of the MOGA models.
There are also a few Android-powered devices that use controllers as the main input source. Here, we are talking about microconsoles...