The GATE exam is used for recruitment into various research institutes. The students having bachelor's degree and masters degree can appear in Gate exam and it's mainly to test their subjects knowledge they have studied in their graduation or post graduation.
To know more about gate exam, you can visit the following link:-
Once you finish your post graduation/masters, after completion of post graduation, you can apply for higher study in the field of research, i.e. PHD ,so you should write GATE exam since after your mtech.
Yes,I'll suggest you to appear for GATE exam to getting admission for PHD.