In some papers, I can see the drift velocity of electrons equaling thermal velocity. Can anyone tell me when both almost equal each other?
The drift velocity is the net velocity of electrons in a certain direction under an applied field. The thermal velocity is has no net direction because it is randomly distributed and occurs in any metal at finite temperatures. Since the two velocities are different, it does not make any sense to say they are qualitatively equal, even though you may equate them quantitatively as in the previous answer.
The drift velocity of electrons in a metal is given by the equation
where $I$ is the electric current in the metal wire, $n$ is the number of electron density, $A$ is the cross sectional area of the metal wire and $v_D$ is the drift velocity.
From this we get
The thermal velocity is given by
where $T$ is the temperature of the metal, $k_B$ is Boltzmann’s constant, $m_e$ is electron mass and $v_T$ is the thermal velocity. From the last equation we get
Equating $v_D$ and $v_T$ gives the following
Condition for the two speeds to be equal: