Physics Bootcamp

Subsection 56.3.1 Microscopic View of Ohm's Law

Consider metal as a box of electron gas that behaves as an ideal gas. According to the ideal gas kinetic theory, the root-mean square speed of electrons will be given by equating their kinetic energy to thermal energy for three degrees of freedom.

where \(m\) is the mass of the electron, \(v_{\textrm{rms}}\) is the root mean square (rms) speed, \(k_B\) the Boltzmann constant, and \(T\) the temperature in degrees Kelvin. Therefore, \(v_{\textrm{rms}}\) of electrons is

Let's calculate a numerical value to see how large the rms speed would be at room temperature, \(T = 300K\text{.}\)

This says that electron in a gas of electrons have very large speeds at room temperature. However, without an external electric field, the velocity of the electron would be in all possible directions, and so there is no net current flow. When an electric field \(\vec E\) is applied, the electron experiences a force \(\vec F\) in the direction opposite to the direction of the electric field.

Equating the magnitude of the force on the electron to \(m a\) of the electron gives the following for the acceleration of the electron.

In the Drude model the electron collides with the iron core. In between the collisions, the electron picks up speed, while right after a collision, it assumes a random speed again. If the average time each electron has been picking up speed since last collision is \(\tau\text{,}\) then the average speed of all electrons in the sample at any one instant will equal to \(a\tau\text{,}\) where \(\tau\) is the mean free time. Note that this average speed is the average of the speed of all the electrons in the metal at one instant since their last collisions, and not of any one electron. This speed \(v_d\) due to the acceleration of electrons as a result of the applied electric field is called the drift speed.

The drift of electrons will generate a current in the metal. Let \(n\) be number of conduction electrons per unit volume, then, the current density \(J\) will be

This is the expression of Ohm's law in the form \(J = \sigma E\) with \(\sigma\) the conductivity. Therefore, the Drude model gives us the following expression of conductivity in terms of microscpic properties.

Since electrons on average take time \(\tau\) between collisions, and move a distance of mean free length \(\lambda\) between collisions, according to the kinetic theory of gases we will also have the following for the rms speed of the electrons.

Replacing \(\tau\) here in terms of the mean free length and drift speed we get

Now, we can use the other result of the kineic theory of gases to relate this to the temperature of metal.

How good is this prediction? Let us calculate the conductivity of copper and compare the value to the experimental value of \(\sigma\text{.}\)

Subsection 56.3.2 Wiedemann-Franz Law

Metals are good electrical conductors and good thermal conductors. Recall that thermal conductivity \(K\) of a material is defined by the relation of rate at which heat flows per unit cross-sectional area and the gradient of temperature. In one-dimension flow along the \(x\)-axis we define \(K\) for the following equation.

In 1853 Wiedemann and Franz reported that the ratio of thermal conductivity \(K\) to electrical conductivity \(\sigma\) of metals were proportional to each other.

This is called Wiedemann-Franz Law. The temperature dependence of \(K/\sigma\) was investigated by Lorenz who concluded that \(K/\sigma\) was directly proportional to absolute temperature \(T\text{.}\)

The proportionality constant \(L\) is called Lorenz number. Equation (56.3.2) is sometimes known as the Wiedemann-Franz law, although, it should be most appropriately called the Wiedemann-Franz-Lorenz law. Experimentally, the Lorenz number depends slightly on temperature and on the metal.

We will not go through the steps of derivation of the \(K/\sigma\) based on the classical electron gas model of metals, but we state the conclusion and discuss its implications. The model predicts the following for \(K/\sigma\) of a metal.

The right side of this equation is independent of any property of the metal. Hence, \(K/\sigma\) will be same for all metals at a given temperature, in agreement with the original Wiedemann-Franz law. This prediction is an impressive achievement of the Drude model and led scientists to believe that there was some merit to the free electron picture of metals.

Comparing the theoretical relation in Eq. (56.3.3) to the Wiedemann-Franz-Lorenz law in Eq. (56.3.2) we see that the therory predicts that the Lorenz number will be a universal constant with the following value:

Table 56.3.2 contains experimental values of \(K/\sigma T\) of several metals at two different temperatures. The ratio \(K/\sigma T\) for most of the metals are near \(2.5\times 10^{-8}\:\textrm{W}.\Omega/\textrm{K}^{2}\text{,}\) although there are some exceptions, as in tungsten (W), whose \(K/\sigma T\) is around \(3.0\times 10^{-8}\:\textrm{W}.\Omega/\textrm{K}^{2}\text{.}\) The experimental values do lend credence to the fact that \(K/\sigma T\) is approximately universal. The numerical values themselves are around two to three times the predicted value. However, considering the crudeness of the model, these are very impressive agreements with the experiment, which leaves very little doubt that the motion of electrons in metal are responsible for both the electrical conduction and the thermal conduction.

Subsection 56.3.3 Temperature-dependence of Resistivity

The resistivity of a chemically pure metal with no defects is called the intrinsic resisitivity, \(\rho_i\text{.}\) In classical Drude model, electrons scatter from the nuclei of the atoms on lattice sites, which is responsible for the resistivity. According to Matthiessen's rule, the resistivity \(\rho_0\) due to electron's scattering from impurities and lattice defects add to the intrinsic resistivity to make up the net resistivity. The intrinsic resisitivity varies with temperature and goes to zero at zero degree Kelvin as some power of \(T\text{.}\)

Experimentally, if the temperature is not too low, the resistivity varies linearly with temperature. If the resisitivity is \(\rho_{\textrm{ref}}\) at some reference temperature \(T_{\textrm{ref}}\text{,}\) then the resisitivity at some other temperature \(T\) follows the following simple linear relation.

where \(\alpha\) is the temperature coefficient, which is around \(3\times 10^{-3}\) per degree Kelvin for most metals. Now, let us see how is this experiment addressed by the classical theory. We found above that the theory predicts the electrical conductivity \(\sigma\) in terms of the mean free path of the electrons, \(\lambda\text{,}\) the number density of electrons, \(n\text{,}\) and absolute temperature \(T\) in Eq. (56.3.1)

The resisitivity \(\rho\) is inverse of conductivity. Therefore, the resistivity should be

Thus, the prediction for resistivity with respect to the variation in temperature is that resistivity will vary as a square root of the absolute temperature.

This prediction is not in agreement with experiment since the resisitivity of most metals vary linearly with temperature as indicated in Eq. (56.3.1).