**eV**and here I will explain the following topic:

# Drude Model and the Classical Charge Transport in Metals

## Formula

## What do the formula symbols mean?

## Current Density

`$$ \class{red}{\boldsymbol j} $$`Unit

`$$ \frac{ \mathrm A }{ \mathrm{m}^2 } $$`

## Electric field (E field)

`$$ \class{purple}{\boldsymbol E} $$`Unit

`$$ \frac{\mathrm{V}}{\mathrm{m}} = \frac{\mathrm{N}}{\mathrm{C}} = \frac{\mathrm{kg} \, \mathrm{m}}{\mathrm{A} \, \mathrm{s}^3} $$`

## Mean free time

`$$ \tau $$`Unit

`$$ \mathrm{s} $$`

## Electron density

`$$ n $$`Unit

`$$ \frac{1}{\mathrm{m}^3} $$`

## Mass

`$$ m_{\text e} $$`Unit

`$$ \mathrm{kg} $$`

## Elementary charge

`$$ e $$`Unit

`$$ \mathrm{C} = \mathrm{As} $$`

*exact*value:

`$$ e ~=~ 1.602 \, 176 \, 634 ~\cdot~ 10^{-19} \, \mathrm{C} $$`

**Explanation**

## Video

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

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:

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:

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

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

:

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

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:

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

):

Or expressed with mobility 6

:

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:

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**.

## Was ist die dynamische Leitfähigkeit?

Dynamische Leitfähigkeit \( \sigma(\omega) \) ist die *frequenzabhängige* elektrische Leitfähigkeit und zwar aufgrund eines externen zeitabhängigen elektrischen Feldes \( E(t) \).

Schau dir die folgende Differentialgleichung an:

Mit \( E(t) = E_0 \, e^{i\omega \, t} \) und \( v(t) = v_0 \, e^{-i\omega \, t} \), folgt für die dynamische Leitfähigkeit:

Das Interessante ist: Der Realteil von Gl. 2

ist frequenzunabhängig, aber abhängig von der Streuzeit \( \tau \). Beim Imaginärteil ist es genau andersherum.