You basically need the ID of the network controller. You are right about
which entry corresponds to it, but you need to run sudo lspci -nnk. Then
you'll see the ID of the device, mine looks similar to this:
02:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd.
RTL8111/8168B PCI Express Gigabit Ethernet controller *[10ec:8168]* (rev 03)
The part between the stars is the device ID, that's what you need to look for
on the net. There should be how-tos somewhere. I hope it actually works with
fully free drivers, but there are a number of things that require a non-free
firmware and thus cannot be supported by a libre kernel.
