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I have seen in several textbooks that say that when we connect a conductor with a battery then electrons start moving (drifting) in the battery in random up and down motion in the opposite direction of the applied electric field, the average velocity of electrons during this motion is then said to be the drift velocity , but it is also claimed that this drift velocity is constant.
Why is it so? If a constant electric field is present then why wouldn't the drift velocity just increase?
The conduction band electrons in the metal, are called either(there are two main opinions on this site):
delocalized
loosely bound to the atoms
Now the electrons move in zig zag way through the metal, as they interact with the atoms in the metal lattice. This constant interaction as they move in a zig zag way will give them their drift velocity, which is very slow.
Now you are asking why is this drift velocity not increasing as there is an external electric field present?
This external electric field is what gives these electrons kinetic energy, so they can move from one atom to another. If you remove this electric field, the electrons do not have the extra kinetic energy to keep moving, there is not electricity anymore.
You have to keep the electric field to keep giving the extra kinetic energy to the electrons to keep them moving through the lattice of the metal, interacting with the atoms.
The kinetic energy they gain from the electric field balances with the energy they lose as they interact with each atom. This is why the drift velocity is constant.
The velocity of electrons in a metal is given by two factors: on one hand, you have the motion of charges due to the applied electric field, which is an accelerated motion; on the other, moving negative charges scatter with the fixed positive charges of the metallic lattice.
The two effects counterbalance and the drift velocity becomes simply directly proportional to the applied electric field: $\textbf{v} = \mu \textbf{E}$, being $\mu$ the electron mobility, which directly comes from Ohm's law $\textbf{J} = \sigma \textbf{E}$.
A more detailed answer was given in Why is the drift velocity directly proportional to the electric field?
