Answer to Question #155920 in Molecular Physics | Thermodynamics for Vincent

Question

Discuss the free electron model and its failures

Answer 1

In the free-electron model the conduction electrons are assumed to be completely free, except for a potential at the surface, which has the effect of confining the electrons to the interior of the specimen. According to this model, the conduction electrons move about inside the specimen without any collisions, except for an occasional reflection from the surface, much like the molecules in an ideal gas. Because of this, we speak of a free-electron gas.

Let us look at the model a little more closely. It is suprising that it should be valid at all, because, at first sight, one expects the conduction electrons to interact with the ions in the background, and also with each other. These interactions are strong, and hence the electrons ought to suffer frequent collisions; a picture of a highly nonideal gas should therefore emerge.

The failures of the free electron model can be briefly summarized as follows:

a. The temperature and the directional dependence of the DC electrical conductivity could not be adequately explained by the Sommerfeld theory.

b. The free electron model explains the contribution of the linear term in T to the specific heat of metals, but the magnitude of this contribution is inadequate for many metals in the sense that it is either too large or too small. In addition, there is a cubic term (T³) that contributes to the specific heat of metals but cannot be explained by the free electron model.

c. The major failure of the free electron model is that it does not explain the significant difference between metals, insulators, and semiconductors. To be specific, it fails to explain why some elements

crystallize as good conductors (metals) of which the conductivity decreases with the increase of temperature, some others crystallize as insulators that are very poor conductors, and the rest crystallize as semiconductors of which the conductivity is very poor at low temperatures but increases with increase of temperature unlike that of metals. In fact, it has been commented that the difference of electrical resistivity at room temperature between that of good conductors (10⁻¹⁰ −10⁻⁶) ohm-cm and of good insulators (10¹⁴ −10²² ) ohm-cm is one of the widest range of any physical property ever found in nature. In contrast, the electrical resistivity of semiconductors at room temperature is in the range (10⁻² −10⁹) ohm-cm but is strongly dependent on temperature.