Module 2 : Normal metals

Lecture 1 : Electrical conductivity and heat capacity followed by problem solving

Basic Properties Of Metals In The Normal State

Before delving into the properties of superconductors, it is important to give a brief account of the salient features of normal metals. In a non-interacting electron picture, the contribution of the elecrons to various properties can be calculated. This is worked out in standard texts on solid state physics and the summary of the results is given below.

Electrical conductivity

We recall that in the non-interacting-electrons description of metals, one considers electrons moving in a periodic potential of the lattice. In this ``Bloch'' picture, a perfect crystal is expected to have an infinite electrical conductivity. In real materials, a finite conductivity appears due to the inherent imperfections and defects. Additionally, at non-zero temperatures, lattice vibrations lead to a deviation from periodicity and contribute to electron scattering. In summary, one gets resistivity T for T where is the Debye temperature. On the other hand, for T one obtains T 5. The residual resistivity as 0 decreases with decreasing amounts of impurities. A typical value of resistivity of a good metal at room temperature is of the order of ohm cm. As an example, the variation of the resistivity with temperature for silver is shown in the figure below

The resistivity variation with temperature is shown for a normal metal. At high temperature, a linear variation of resistivity with temperature is seen while at low temperatures a residual resistivity is present which is linked to the presence of defects and impurities.

For ideal metals, the thermal conductivity due to electrons is given by where is the mean square electronic speed, is the relaxation time, and is the specific heat capacity of the electrons. The ratio of the thermal conductivity to the electrical conductivity times temperature ( T) of an electron gas is a universal constant (Lorentz number) and this is called Wiedmann-Franz law. The value of the Lorentz number is of the order of in typical metals.

Heat capacity

The temperature dependence of the specific heat capacity of an electron gas is given by where is the Boltzmann constant and is the density-of-states at the Fermi level. The value of in typical metals is of the order of 1 . In the Debye model, the phonon contribution to the heat capacity per atom is given by where is the Debye temperature. The typical variation is shown in figure below

The heat capacity variation for a typical metal is shown in the figure. At high temperatures, it tends to a constant value while at low temperatures the heat capacity is a combination of a linear term (due to electrons) and a cubic term (due to lattice vibrations or phonons). When C/T is plotted as function of T2, a straight line is obtained. The intercept on the y-axis gives information about the electron density of states while the slope provides information about characteristic energy associated with lattice vibrations. In the low-temperature limit T , the phonon contribution reduces to . In a typical metal, the low-temperature heat capacity has, therefore, a combination of a linear and a cubic term in temperature. The low-temperature heat capacity measurement serves as an important probe of the Fermi surface properties of the electron gas.