Band theory of metals

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Band theory is also known as the molecular orbital theory (MOT) of metals or the zone theory. It is based on the quantum mechanical treatment of metallic bonding in a crystal lattice. Metallic bonding is simply defined as the attractive force that develops between positively charged metal atoms embedded in a sea of delocalized electrons and negatively charged electrons. A large number (N) of metal atoms held together make a giant structure called a metallic crystal (NN).

 The band theory proposes that metal atomic orbitals combine to form molecular orbitals that extend throughout the metallic structure. Continue reading to learn more about the band theory and how it explains the superior electrical conductivity of metals.

What is band theory of metals – Definition

The foundation for the molecular orbital theory or the band theory of metals was laid back in 1927 by Walter Heitler and Fritz London. As per the band theory, N number of atomic orbitals combine to form N molecular orbitals when a large number of metal atoms interact in a metallic crystal. The molecular orbitals belong to the crystal as a whole and not to any individual metal atom. It is due to the minute energy difference between these molecular orbitals that they appear to be a continuum, known as a quasi-continuous energy band. 

MOT in metallic bonding

Example:  Metallic bonding in sodium (Na) metal explained through the lens of the band theory.

The electronic configuration of a 11Na atom is 1s2 2s2 2p6 3s1

Each Na atom has only 1 electron in its valence shell. When two Na atoms (Na1 and Na2) interact, the electron wave of the 3s atomic orbital of Na1 overlaps with the valence electron wave of Na2. Two 3s atomic orbitals combine to form two molecular orbitals, i.e., a low-energy bonding molecular orbital and a higher-energy antibonding molecular orbital. As per the Aufbau principle of filling electrons, both the valence electrons of combining atoms are placed in bonding molecular orbital while the higher-energy antibonding molecular orbital stays empty.

On a larger scale, in the sodium crystal, a large number, such as 1 mole (6.02 x 1023) 3s atomic orbitals of the Na atoms interact to form 6.02 x 1023 molecular orbitals (MOs). Due to the combination of a large number of atomic orbitals at once, the bonding and anti-bonding MOs are closely spaced in energy. This results in a continuous energy band belonging to the sodium crystal as a whole.

The interesting fact is that as per the band theory, not only valence shell atomic orbitals of 1 mole Na-atoms combine to form MOs. Rather, the 1s, 2s, 2p and 3p orbitals also overlap with the respective orbitals of adjacent atoms to form MOs. This results in different types of energy bands in the crystal structure.

How does band theory explain the electrical conductivity of metals?

The presence of delocalized electrons is an essential requirement for electrical conduction in metals. In the example above, the 1s, 2s and 2p bands of sodium crystal are completely filled. These are known as non-conduction bands.

The 3s band is only half-filled, while the 3p band is empty. The half-filled valence band (3s) of sodium crystal can overlap with its empty 3p band due to a small energy gap or band gap between the two. The 3s electrons can thus easily jump to the 3p band in the overlapping zone. This explains the good electrical conductivity of metals such as sodium. The 3p band in which the valence electrons are free to move is known as the conduction band.

All the images in this article are by Ammara W.

Smaller the energy difference (band gap) between the valence and conduction bands of a metal crystal, the higher the electrical conductivity of the metallic structure.

Related terms in the MO diagram of metals

  • The electronic energy bands that allow electrons to jump in between them are known as the permitted bands or a Brillouin zone.
  • The width of a Brillouin zone depends on the overlapping between respective electron clouds.
  • The 3s and 3p bands in sodium crystal are together called permitted bands that form a Brillouin zone.
  • The 3p band itself is known as the conduction band as it allows free movement of the incoming electrons from 3s, the valence band.
  • In NaN crystal, there is a large energy difference between 1s, 2s and 2p bands. Therefore, electrons cannot jump in between these bands. These are thus known as non-conduction bands.
  • The energy difference between non-conduction bands, that the electrons cannot cross is called forbidden zone.
  • The energy level below which all bands are completely filled is known as the Fermi level.

What do you think is the reason behind the partial electrical conductivity of semiconductors, and why are insulators non-conductors of electricity according to the band theory?  Let’s uncover that in the next section of the article.

Why do insulators not conduct electricity as per the band theory?

Non-metals such as diamond and phosphorus are poor conductors of heat and electricity, also known as insulators. The band theory explains the reason for this non-conductivity as a high energy gap between the partially filled band and the empty band of the crystalline structure. Electrons cannot jump from the low energy partially filled 3p band of phosphorus to its high-energy empty 3d band due to the large energy gap. The energy difference between these two bands is also known as the forbidden zone.

An insulator usually has an energy difference of 5eV between its valence band and the conduction band.

Electrical conductivity of semiconductors as per band theory

Solids such as silicon and germanium are intrinsic semiconductors. A semiconductor is defined as a material whose electrical conductivity lies between that of a conductor and an insulator. These are commonly used in the electronic industry. For instance, silicon (Si) chips are used in watches, computer systems, and also for making transistors and solar cells etc., owing to its semiconduction property.

As per band theory, a comparatively smaller energy gap exists between the partially filled and empty bands of silicon crystal. At room temperature, the electrons cannot jump from the partially filled 3p band to its empty 3d band, so Si is a poor conductor at r.t.p. However, when provided with some external energy, the electrons get excited, able to jump in between the two energy bands and thus conduct electricity. As electrons jump from the partially filled valence band to the empty conduction band, positively charged holes are created in the valence band. Also, increasing temperature decreases the forbidden energy gap in the semiconductor structure.

This concept explains why increasing temperature increases the electrical conductivity of semiconductors.

Contrarily, the electrical conductivity of metals decreases by increasing temperature, do you know why? Find out here.

Revise your concepts on the band theory of metals by practicing this worksheet.

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