Asalamalaykum, I was wondering if Tahajjud is permissible without first sleeping. As a student, who usually is busy studying during the night, can I still pray Tahajjud even if I have not slept to wake up? Also, when I make the intention of my Tahajjud namaz, are they followed as sunnat or nafl intention? Please let me know.
Alaikumassalam wa rahmatullah,
Yes, that is permissible. People use “sunnat” to refer to voluntary good deeds and acts of worship. Nafl means voluntary act of worship. So the two in their common usage have generally the same meaning when they refer to acts of worship like prayer or fasting. There is no need to worry about intention; if your intention is to pray tahajjud, then that is sufficient for it.
Technically whether something is obligatory or voluntary, it is sunna because it is part of the Prophet’s tradition 
. But in common usage, people only use sunna to refer to voluntary deeds. When they use it like this, a sunna act of worship has the same meaning as a nafl act of worship. All nafl acts of worship are sunna unless they have been invented later, such as the communal tarāwīh prayers on Ramadan nights, which are nafl prayers but since the Prophet did not perform such prayers communally, the communal part is not sunna.
For more on tahajjud please see my essay: Mysticism without Sufism: A Guide to Tahajjud, Islam’s Meditation Practice