When drift velocity equals thermal velocity?

Question

In some papers, I can see the drift velocity of electrons equaling thermal velocity. Can anyone tell me when both almost equal each other?

Answer 1

The drift velocity of electrons in a metal is given by the equation

$I=enAv_D$

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

$v_D= \frac{I}{enA}$

The thermal velocity is given by

$\frac{1}{2}m_ev_T^2=\frac{3}{2}k_BT$

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

$v_T= \sqrt{\frac{3k_BT}{m_e}}$

Equating $v_D$ and $v_T$ gives the following

Condition for the two speeds to be equal:

$\frac{I}{enA }=\sqrt{\frac{3k_BT}{m_e}}$.

Answer 2

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.

Answer 3

The theory of the above two answers is correct, but the equations provided are not quite correct. The equation $${{1}\over{2}}m_e(v_T)^2 = {{3}\over{2}}k_BT$$ is based on the assumption that the electrons behave as an ideal gas (and is true for semi-conductors). However, in metals, electron velocity actually has a very small dependence on temperature, and a much larger dependence on its Fermi energy. This comes from the application of density of states theory and Fermi-Dirac statistics.

$$E(T) = {{3}\over{5}} \times E_{F0}\times \left[1+\left({{5}\over{12}}\times\pi^2\right)\times(k\times T/(E_{F0})^2\right]$$

Where $E_{F0}$ is the Fermi energy at $T = 0\text{ K}$. Since $E_{F0} \gg kT$, the average energy can usually be taken to be ${{3}\over{5}}E_{F0}$. Depending on the metal you are working with, this is usually readily available in tables on-ne. For example, copper has an $E_{F0} = 7.0\text{ eV}$, meaning that $E_{\text{ave}} = 4.2\text{ eV}$, and $V_{\text{thermal}} = 1.2*10^6 \text{ m/s}$ at $t = 300\text{ K}$.