This solution is about home key problem in Samsung Galaxy Tab 3 T111 in most cases this problem happens because of water or rust damages. So disassemble the phone and check these parts that are shown in the diagram given bellow.

On you have make sure there is rust or carbon on the motherboard so make it clean with electronics cleaner and apply heat to make it dry.

In the diagram above you can see red and green line track that is given to show you the track and parts used in Samsung Galaxy Tab 3 T111 home key functions.