[Evoryboypoto]

I had asked Mufti AbuHajirah this same question once. He said no.
One can pray tahajjud anytime after isha till fajr. But certainly the are times when tahajjud has more reward.
I think its in the earlier part of the last one-third of the night. Someone please confirm.

The last third is more rewarding than the earlier parts.

Even though its not a condition to sleep to pray tahajjud, its better to do so because you will get more reward for giving up your sleep.