1 Free Electron Model for Metals Metals are very good at conducting both heat and electricity. A lattice of in a “sea of electrons” shared between all nuclei (moving freely between them): This is referred to as the free electron model for metals. This model explains many of the properties of metals: a) Electrical Conductivity: The mobile electrons carry current. b) Thermal Conductivity: The mobile electrons can also carry heat. c) Malleability and Ductility: Deforming the metal still leaves each cation surrounded by a “sea of electrons”, so little energy is required to either stretch or flatten the metal. d) Opacity and Reflectance (Shininess): The electrons will have a wide range of energies, so can absorb and re-emit many different wavelengths of light.

2 Band Theory for Metals (and Other Solids) How do we describe electrons in a metal? These solids can be treated in a way similar to molecular orbital theory; As there are no distinct molecules to orbitals are delocalized covering the entire dimensions of the lattice, called states. Same basic approach: a) Combine atomic orbitals from every to make states giving a large number of a very large molecular orbitals. the number of states produced = the number of atomic orbitals b) The Pauli exclusion: each state hold two electrons. c) Electrical conductivity requires that electrons must be able to gain enough energy to achieve an excited states. The highest energy state when no such excitation has occurred (i.e. in the ground state metal) is called the Fermi level.

3 Band Theory for Lithium MOs produced by linear combination of the 2s orbitals in Li 2, Li 3 and Li 4. Note that, for every atom added: An additional MO is formed The energies of the MOs get closer together When a sample contains a very large number of Li atoms (e.g ×10 23 atoms in g), The states formed are so close in energy that they form a band of energy levels. A band is named for the AOs from which it was made (e.g. 2s band)

4 Band Theory for Metals For alkali metals, the valence s band is only half full. e.g. sodium: 1s 2 2s 2 2p 6 3s 1 For N sodium atoms there are N 3s electrons. Hence N states are formed each holding up to 2 electrons. Therefore N /2 states in the 3s band will be full and N /2 states will be empty in the ground state. Alkali metals are good conductors as the valence band is half full. This means that it is easy for electrons in the valence band to be excited into empty higher energy states. These empty higher energy states are in the same band, which means that the valence band for sodium is also the conduction band. Valence band Conduction Band

5 Band Theory for Metals In an alkaline earth metal, the valence s band is full. e.g. Beryllium 1s 2 2s 2 2p 0 For N atoms of Be there are 2N electrons in the 2s orbitals. These make up N states, each able to hold two electrons. Hence all the states of the 2s band are full in the ground state. So, why are alkaline earth metals conductors? The 2s band in Be overlaps with the 2p band. The energy of the 2s and 2p AOs in metals are close. Hence electrons in the valence band can easily be excited into the conduction band. In Be, the conduction band is the 2p band. Conduction Band Valence Band

6 Band Theory for Non-metals Insulators do not conduct electricity. e.g. Diamond C: 1s 2 2s 2 2p z 2 2p x 0 2p y 0 For N atoms of Carbon, there will be 4N valence electrons. The 4N valence AOs combine to make two bands each with 2N states. The lower energy band is the valence band with 4 N electrons and hence full in the ground state. The higher energy band is the conduction band Which is empty in in ground state. The energy gap between the valence band and the conduction band is large so that it is difficult for an electron in the valence band to absorb enough energy to be excited into the conduction band.

How big does the band gap have to be for a material to be an insulator? Depends on how much energy is available to the average electron. Thermal energy: RT = for a mole of gas k B ·T= for in individual particle k B = × J/K = R/A Boltzmans Constant T =temperature (Kelvin) For an insulator the band gap is much larger than k B ·T, e.g. Diamond: ~200×k B ·T For a conductor the band gap is smaller than or similar to k B ·T e.g. Sodium: 0×k B ·T, tin: 3×k B ·T For a semiconductor the band gap is about ten times larger than k B ·T e.g. Silicon: ~50×k B ·T Band gaps are measured by absorption spectroscopy, where the lowest energy of light absorbed corresponds to the band gap energy. 7 Band Theory For Insulators

8 Band Theory for Semiconductors Intrinsic Semiconductors - Have a moderate band gap. A small fraction of the electrons in the valence band can be excited into the conduction band. The holes these electrons leave in the valence band can also carry current as other electrons in the valence band can be excited into them. Extrinsic Semiconductors –Have impurities (Dopants) added to them to increase the current they can conduct by either providing extra electrons or extra holes: 1) extra electrons gives an n-type semiconductor. (n – negative) 2) extra holes gives a p-type semiconductor. (p - positive)

9 Band Theory for n-Type Semiconductors How does an n-type semiconductor work? e.g. silicon ([Ne]3s 2 3p 2 ) is doped with phosphorus ([Ne]3s 2 3p 3 ) Si : valence band is full and the conduction band is empty. P : provides an additional band (Donor band) full of electrons that is higher in energy than the valence band of silicon. Electrons of the donor band are more easily excited into the conduction band. (compared to electrons in the valence band of silicon).

10 Band Theory for p-Type Semiconductors How does a p-type semiconductor work? e.g. silicon ([Ne]3s 2 3p 2 ) is doped with aluminium ([Ne]3s 2 3p 1 ) Si : valence band is full and the conduction band is empty. Al : provides an additional empty band that is lower in energy than the conduction band of silicon. Electrons of the valence band of silicon are more easily excited into this acceptor band. (compared to the conduction band of silicon).

11 Diodes By changing the dopant and its concentration, the conductivity of a semiconductor can be fine-tuned. e.g. Diodes An n-type and a p-type semiconductor are connected. The acceptor band in the p-type semiconductor gets filled with the extra electrons from the n-type semiconductor. The extra holes from the p-type semiconductor thus move to the n-type semiconductor. Electron depletion at the n-type semiconductor Electron accumulation at the p-type semiconductor If a diode is connected to a circuit so that electrons flow into the n-type semiconductor, current can flow. If a diode is connected to a circuit so that the electrons flow into the p-type semiconductor, electrons will pile up there and the current will stop. acceptor donor Current

12 Photodiodes & Solar Cells In a photodiode, the p-type semiconductor is exposed to light. This excites electrons from the acceptor band into the conduction band. They are attracted to the neighbouring n-type semiconductor, which has built up a slight positive charge. This causes current to flow, and is how many solar cells work. acceptor donor Photocurrent