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Drude Model and the Classical Charge Transport in Metals

Important Formula

Formula: Drude Model

$$\class{red}{j} ~=~ \frac{n \, e^2 \, \tau}{m_{\text e}} \, E$$ $$\class{red}{j} ~=~ \frac{n \, e^2 \, \tau}{m_{\text e}} \, E$$ $$E ~=~ \frac{m_{\text e}}{ n \, e^2 \, \tau } \, \class{red}{j}$$ $$\tau ~=~ \frac{ m_{\text e} }{ n\,e^2 } \, \frac{\class{red}{j}}{E}$$ $$n ~=~ \frac{ m_{\text e} }{ \tau \, e^2 } \, \frac{\class{red}{j}}{E}$$ $$m_{\text e} ~=~ n\,\tau \, e^2 \, \frac{E}{\class{red}{j}}$$

Rearrange formula

What do the formula symbols mean?

Current Density

$$ \class{red}{\boldsymbol j} $$ Unit $$ \frac{ \mathrm A }{ \mathrm{m}^2 } $$

Electric current density represents the current passing through a cross-sectional area.

Electric field (E field)

$$ \class{purple}{\boldsymbol E} $$ Unit $$ \frac{\mathrm{V}}{\mathrm{m}} = \frac{\mathrm{N}}{\mathrm{C}} = \frac{\mathrm{kg} \, \mathrm{m}}{\mathrm{A} \, \mathrm{s}^3} $$

Applied external electric field causing a current density $j$. As the E-field increases, the current density also increases.

Mean free time

$$ \tau $$ Unit $$ \mathrm{s} $$

Mean free time is the time that elapses between two collisions. During this time, the electron is accelerated collision-free by the electric field to the drift velocity.

Electron density

$$ n $$ Unit $$ \frac{1}{\mathrm{m}^3} $$

Electron density specifies the number of electrons per volume.

Mass

$$ m_{\text e} $$ Unit $$ \mathrm{kg} $$

Mass of the electron. The rest mass of the electron is: $ m ~\approx~ 9.109 \,\cdot\, 10^{-31} \, \mathrm{kg} $.

Elementary charge

$$ e $$ Unit $$ \mathrm{C} = \mathrm{As} $$

The elementary charge is a physical constant and is the smallest, freely existing electric charge in our universe. It has the exact value: $$ e ~=~ 1.602 \, 176 \, 634 ~\cdot~ 10^{-19} \, \mathrm{C} $$

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In general, the electrical conductivity $ \sigma $ is defined as a constant of proportionality between the electric field $ \boldsymbol{E} $ and the current density $ \class{red}{\boldsymbol{j}} $:

$$ \begin{align} \class{red}{\boldsymbol{j}} ~=~ \sigma \, \class{purple}{\boldsymbol{E}} \end{align} $$

Scattering of an electron according to the Drude model.

In the Drude model, the charge carrier gas (mostly electron gas in metals) with particles of charge $ q $ and mass $ m $ is assumed to be a classical gas. The individual charge carriers move with the thermal velocity $ \boldsymbol{v}_{\text{th}} $ in a chaotic way.

If now an external electric field $ \boldsymbol{E} $ is switched on, the charge carriers are accelerated along the electric field. In the process, they collide with the atoms and are thus decelerated. A small period of time passes until the charge carrier collides with the next atom. This time is called mean free time $ \tau $. During this time period, the charge carrier is accelerated by the E-field to the velocity $ \boldsymbol{v}_{\text d} $, which is called drift velocity. It is the average velocity $ \boldsymbol{v} $ of the charge carrier minus the thermal velocity: $ \boldsymbol{v}_{\text d} = \boldsymbol{v} - \boldsymbol{v}_{\text{th}} $. Without external E-fields, the drift velocity is obviously $ \boldsymbol{v}_{\text d} = 0 $ and the average velocity corresponds to the undirected thermal velocity.

Thus, there are basically two forces acting on the charge carrier: the electric force $ - q \, \boldsymbol{E} $ and the frictional force caused by collisions: $ - m \, \frac{\boldsymbol{v}_{\text d}}{\tau} $. Thus, the equation of motion to be solved (differential equation) is the following one:

$$ \begin{align} m \, \frac{\text{d}\boldsymbol{v}}{\text{d}t} ~=~ - q \, \boldsymbol{E} ~-~ m\, \frac{\boldsymbol{v}_{\text d}}{\tau} \end{align} $$

Why the minus signs, you ask? To take into account that the two forces counteract each other. In the stationary case the two forces are in equilibrium, that is, the average velocity $ \boldsymbol{v} $ of the charge carriers does not change any more, consequently the time derivative must be zero: $ \frac{\text{d}\boldsymbol{v}}{\text{d}t} = 0 $. Thus, the equation of motion 2 becomes after the frictional force is brought to the other side of the equation:

$$ \begin{align} m\, \frac{\boldsymbol{v}_{\text d}}{\tau} ~=~ - q \, \boldsymbol{E} \end{align} $$

If the mean free time $\tau$ is known, the drift velocity can be calculated:

$$ \begin{align} \boldsymbol{v}_{\text d} ~=~ - \frac{q \, \tau}{m} \, \boldsymbol{E} \end{align} $$

Since the Drude model is mostly used to describe the electron gas in metals, we insert the negative elementary charge $ q = -e $ of the electron into Eq. 4:

$$ \begin{align} \boldsymbol{v}_{\text d} ~=~ \frac{e \, \tau}{m} \, \boldsymbol{E} \end{align} $$

Here the prefactor of $ \boldsymbol{E} $ is defined as electron mobility $ \mu $:

$$ \begin{align} \mu ~:=~ \frac{e \, \tau}{m} ~=~ \frac{ |\boldsymbol{v}_{\text d}| }{ |\boldsymbol{E}| } \end{align} $$

Thus, mobility is nothing but the mean free time $ \tau $ weighted by the specific charge $ \frac{q}{m} $. Or equivalently, it indicates the drift velocity that arises when a corresponding electric field $ \boldsymbol{E} $ is applied.

Using the electron density $ n $ (number of electrons per volume), Eq. 1 becomes:

$$ \begin{align} \class{red}{\boldsymbol{j}} ~=~ n \, e \, \boldsymbol{v}_{\text d} ~=~ \frac{n \, e^2 \, \tau}{m} \, \boldsymbol{E} \end{align} $$

So the electrical conductivity $\sigma$ according to the Drude model is given by (compare eq. 1):

$$ \begin{align} \sigma ~=~ \frac{n \, e^2 \, \tau}{m} \end{align} $$

Or expressed with mobility 6:

$$ \begin{align} \sigma ~=~ n \, e \, \mu \end{align} $$

So you can find out the mean free time $ \tau $ by measuring the electrical conductivity $ \sigma $ of the metal and the electron density $ n $. Typical values for the mean free time are in the range $ \tau \approx 10^{-14} \, \mathrm{s} $. With this mean free time and at a thermal velocity $ \boldsymbol{v}_{\text{th}} = 10^5 \, \frac{\mathrm m}{\mathrm s} $ (which is much larger than the drift velocity), the mean free path (distance traveled between two collisions) is in the range of the atomic distance: $ l_{\text{th}} = v_{\text{th}} \, \tau \approx 10^{-9} \, \mathrm{m} $. Thus, according to the Drude model, the equation 1 becomes:

$$ \begin{align} \class{red}{\boldsymbol{j}} ~=~ \frac{n \, e^2 \, \tau}{m} \, \boldsymbol{E} \end{align} $$

Drude's model thus predicts a linear relationship between the electric current density and the electric field, which corresponds to Ohm's law. The Drude model also correctly predicts (but by two erroneous assumptions that cancel each other out) the relationship between thermal and electrical conductivity (see Wiedemann-Franz law).

Note, however, that the Drude model fails at low temperatures because the model does not account for the Fermi distribution of electrons, even though the Fermi distribution plays an important role at low temperatures. According to the Drude model, all electrons contribute to the electrical conductivity, which leads to a too low theoretical value of the conductivity. The remedy is the improved Sommerfeld model.

What is dynamic conductivity?

Dynamic Conductivity $ \sigma(\omega) $ is the frequency-dependent electrical conductivity due to an external time-dependent electric field $ E(t) $.

Take a look at the following differential equation:

$$ \begin{align} m \, \dot{v} + \frac{m \, v}{\tau} = - e \, E \end{align} $$

With $ E(t) = E_0 \, e^{i\omega \, t} $ and $ v(t) = v_0 \, e^{-i\omega \, t} $, the dynamic conductivity follows:

$$ \begin{align} \sigma(\omega) = \frac{\sigma_0}{1-i\,\omega \, \tau} \end{align} $$

The interesting thing is: The real part of Eq. 2 is frequency-independent, but dependent on the scattering time $ \tau $. The opposite is true for the imaginary part.