Bonding in Solids: Metals, Insulators, and Semiconductors

• Properties of metals can be explained by bonding in metals.

• Metals are:– Malleable - worked into useful shapes or foils.– Ductile - pulled into wires– Good conductors of heat and electricity.

• The bonding in metals cannot be explained in terms of ionic or covalent bonding.– There is little electronegativity difference from element to element

so metals and alloys do not form ionic bonds.– The outermost subshells are not filled, so a large number of

covalent bonds would be needed to satisfy the octet rule.

Bonding Models for Metals

Electron Sea Model: A regular array of metals in a “sea” of electrons.

Band (Molecular Orbital) Model: Electrons assumed to travel around metal crystal in MOs formed from valence atomic orbitals of metal atoms.

Models of Metallic Bonding

• The sea of electrons model is the first and simplest qualitative explanation of metallic bonding.

– The valence electrons are delocalized and move freely throughout the solid.

• Allows for the conduction of electricity

– Explains malleability, ductility, and conductivity properties of metals.

Electron Sea Model.

The electron sea model for metals postulates a regular array of cations in a "sea" of valence electrons. (a) Representation of an alkali metal (Group 1A) with one valence electron. (b) Representation of an alkaline earth metal (Group 2A) with two valence electrons.

Models of Metallic Bonding

• When a force is applied to a metal, the positively charged cores respond to the stress, deforming the metal.

• The free flow of electrons maintains the bonding throughout the process.

Models of Metallic Bonding

• Band theory is a quantitative model of bonding in solids.– The wave functions of the valence electrons interact with each

other.– Bonding molecular orbitals result from constructive interference.– Antibonding molecular orbitals result from destructive

interference.

• The number of molecular orbitals formed equals the number of atomic orbitals involved.– For every bonding molecular orbital formed, one antibonding

molecular orbital is also formed.– For odd number of bonding atoms, a nonbonding molecular orbital

is formed.– The molecular orbitals formed belong to the entire set of atoms

Bonding in Metals –Molecular Orbitals of Solids

Band Theory• Extension of MO

theory– Valence orbitals on N

atoms combine to form N molecular orbitals

– for large N, the energies of the MO’s are closely spaced

– Each valence orbital will generate a band with different energy range

•N/2 are filled.•The valence band.

•N/2 are empty.•The conduction band.

Empty MO

Filled MO

Band Theory and Conductivity

• Electrons fill the lowest energy band first.• The energy difference between the filled and empty bands

determines the electrical properties of the bulk material.• The band populated by valence electrons is the valence band.• The empty band above the valence band is the conduction band.• The energy difference between the valence and conduction band

is the band gap.• Current flows when electrons move from the valence band to the

conduction band.– Conductors have small band gaps. A very small amount of energy is

required to move electrons to the conduction band.– Insulators have large band gaps. A large amount of energy is required to

move electrons to the conduction band.– Semiconductors have band gap intermediate of conductors and insulators.

Band Theory

• The valence e- fill in these MO’s• Depending on the substance, bands are partially or completely full• A partial filled band allows conduction• For a filled band to conduct, e- must be promoted from the highest occupied

MO to the lowest unoccupied MO• The amount of energy required to promote these will determine how well the

substance conducts

Material Conductivity TypeCopper 106 S/cm ConductorSilicon 10-6 S/cm SemiconductorSilicon dioxide 10-12 S/cm Insulator

Consider 1 mole of Be atoms each with a [He] 2s2 electron configuration.• The NA atoms with have NA 2s orbitals which will combine to give ½ NA

bonding orbitals (bonding band) and ½NA antibonding orbitals(antibonding band).

• The 2 NA valence electrons will NA orbitals• completely fill the bonding band orbitals• completely fill the antibonding band orbitals

What is wrong with this argument?• Same number of occupied bonding and antibonding orbitals gives a bond

order of 0 (i.e. Solid Be would not form) • This predicts a large band gap between the highest occupied orbital and

lowest unoccupied orbital. Therefore, Be would be a poor conductor of electricity.

Be Atoms to Solid Be with Considering Only the Valence s Orbitals

Be Atoms to Solid for Be with Considing Only the Valence s Orbitals

Consider 1 mole of Be atoms each with a [He] 2s2 electron configuration.– The NA atoms with have NA 2s orbitals which will combine to give

½ NA bonding orbitals and ½NA antibonding orbitals. – The NA atoms with have 3NA 2p orbitals which will combine to

give 3/2 NA bonding orbitals and 3/2 NA antibonding orbitals.– Together the 2s and 2p orbitals give 2NA bonding orbitals (bonding

band) and 2NA antibonding orbitals (antibonding band).

This results in:– The 2NA valence electrons will only partially fill the bonding band

orbitals (bond order = NA)– This predicts a very small band gap between the highest occupied

orbital and lowest unoccupied orbital. Therefore, Be would be a good conductor of electricity.

Be Atoms to Solid Be with Considing Both the Valence s Orbitals and p Orbitals

Be Atoms to Solid Be with Considing Both the Valence s Orbitals and p Orbitals

Magnesium Metal

A representation of the energy levels (bands) in a magnesium crystal.

Results in a completely filled bonding band. Therefore, the a lot of energy is needed to promote an electron into the next higher energy level. So diamond is a very poor conductor of electricity (it is an insulator)

Carbon Atoms to Diamond

Molecular Orbital Energies

Partial representation of the molecular orbital energies in (a) diamond and (b) a typical metal.

Band Gap in Group IVA Diamond-structure Solids

• (Left column) Two hydrogen atoms approaching with 1s orbitals in phase result in an enhanced amplitude in the internuclear space

• (Right column) Out-of-phase orbitals in the two approaching hydrogen atoms cancel each out in the internuclear space, resulting in diminished amplitude (a node) between the atoms

Overlap as a Function of Atomic Distance

Factors Affecting Band Gap Energy

• Band gap energy increases with overlap of the atomic orbitals– As the size of the atoms decrease they can get closer together so

their orbital interaction increases and the band gap energy increases

• The band gap energy increases as the electronegativity difference between the atoms increases– As the difference increases the electrons are more “confined”

• In the bonding orbital the electron is located preferentially on the more electronegative atom

• In the antibonding orbital the electron is located preferentially on the less electronegative atom

• This results in greater difference energy between the bonding and antibonding orbitals (greater band gap energy)

Atomic Radii for Selected Atoms