Metals and Insulators


Metals have "free" electrons able to carry current. How electrons in metals can be "free" was a mystery in classical physics and requires quantum mechanics for its explanation. The electrical conductivity approaches a finite lower limit as the temperature is reduced toward absolute zero. The limit depends on purity and defect structure. In contrast, for nonmetals, including semiconductors, the conductivity tends to zero with decreasing temperature. We include metallic oxides like CoO(2) and the high-temperature superconductors like La(2-x)Sr(x)CuO(4) and heavily doped semiconductors as "metals" because they all have a finite low temperature electrical conductivity.

Electrons in both metals and insulators are described by "Bloch wave functions" (BWF). The BWF is a plane wave e^ik.r in space modulated by a function Uk(x,y,z) having the periodicity of the crystal lattice in which the electrons move. These electrons have an infinite mean free path in a perfect lattice at absolute zero.

The finite electrical conductivity or reciprocal resistance is caused by scattering from delocalized quantized thermalized sound vibrations of the crystal called phonons, and by localized defects in the crystal structure.

Why can we ignore the direct Coulomb force between the electrons in the metal? First look at why an insulator is an insulator. The BWF get "Bragg reflected" which causes energy gaps in the energies associated with the BWF. Furthermore the identical electrons obey the Pauli exclusion principle in which no more than one electron can be in the same BWF. The important structure is the density of BWF states N(E) with respect to the energy E of these states. N(E)dE is the number of BWF states in the range of energies from E to E + dE.

Consider a monatomic lattice.The outer low energy levels of an electron in the atom are delocalized by the atom-atom interactions and broaden into a band of energies because of the periodicity of the lattice arrangement of the atoms.

Plot N(E) vs E. For a simple monovalent metal like silver, for example, N(E) rises monotonically from zero at E = 0 with decreasing slope, and the lowest energy band of BWF states is only half-filled. The top of the occupied set of BWFs is called the Fermi energy Ef determined by the Pauli exclusion principle.

The Pauli exclusion principle is caused by repulsive quantum exchange forces between the electrons in addition to the classical Coulomb forces. Therefore, there are empty BWFs energetically available for the electrons to quantum jump into when an electric field is applied to the metal. Remember the repulsive quantum force prevents an electron from moving into a BWF already occupied by another identical twin electron.

Bear in mind that these BWFs are delocalized, i.e., they are spread over the entire crystal. They are each giant or "macroscopic" quantum states. Quantum states do not have to be small. For example, the quantum state of a low energy neutron in an interferometer can be many centimeters in size.

Next consider a simple insulator. The density of states function N(E) rises from zero at E = 0 and then drops down to zero again. N(E) stays zero for the width of the energy gap before rising again for the second band. In an insulator, all of the BWFs in the lowest energy band are filled with electrons.

Now apply an electric field. Unless the field is extremely strong, the virtual work that would be done on an electron over the mean free path between collisions is smaller than the energy gap .Any collisions of the electrons with phonons and defects would randomize the ordered motion disrupting the current flow induced by the external electric field. Therefore, there is no net response to the external weak field. The field cannot suppy enough energy to overcome the barrier of the gap. If the field is made stronger, the insulator will break down when the supplied energy per electron is greater than the gap energy.

A semimetal like calcium has overlapping N(E) curves with the Fermi energy Ef in the overlap region.


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