Free Electron Model for Metals
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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: Delocalization of electrons explain how heat can move through the lattice. c) Malleability and Ductility: Deforming the metal still leaves each cation surrounded by a sea of electrons, 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. 1
2 Band Theory for Metals 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 as MO: 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 used b) 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. 2
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 P-band When a sample contains a very large number of Li atoms (e.g 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 S-band 3
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 Conduction Band unoccupied occupied Valence band 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. 4
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 Conduction Band unoccupied occupied For N atoms of Be there are 2N electrons in the 2s orbitals. Valence Band 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. 5
6 Band Theory for Non-metals Insulators do not conduct electricity. e.g. Diamond C: 1s 2 2s 2 2p 2 For N atoms of Carbon, there will be 4N valence electrons. The 4N valence AOs combine to make two bands Each with 2N states that have incompatible symmetry, g and u, and are separated by large energy gap. The lower energy band, 2s g is the valence band with 4 N electrons and thus is full in the ground state. The higher energy band, 2s u * is the conduction band which is empty in the 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. unoccupied occupied 2s u * 2s g 6
7 Band Theory For Insulators 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 T = temperature (Kelvin) Boltzmans Constant For an insulator the band gap is much larger than k B T, Ex) Diamond: ~200 k B T Effectively 0 probability For a conductor the band gap is smaller than or similar to k B T Ex) Sodium: 0 k B T High probability For a semiconductor the band gap is about ten times larger than k B T Ex) Silicon: ~50 k B T Low probability Band gaps are measured by absorption spectroscopy, where the lowest energy of light absorbed corresponds to the band gap energy. 7
8 The Size of Band Gaps 8
9 Intrinsic Semiconductors Have a moderate band gap. Band Theory for Semiconductors 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, known as dopants, added in order to increase the current they can conduct. The dopants can either provide extra electrons or provide extra holes: A semiconductor doped to have extra electrons is an n-type semiconductor ( n is for negative ) A semiconductor doped to have extra holes is a p-type semiconductor ( p is for positive ) 9
10 Band Theory for n-type Semiconductors How does an n-type semiconductor work? e.g. Si ([Ne]3s 2 3p 2 ) is doped with P ([Ne]3s 2 3p 3 ) In silicon, the valence band is completely full and the conduction band is completely empty. The phosphorus provides an additional band full of electrons that is higher in energy than the valence band of silicon. Electrons in this donor band are more easily excited into the conduction band (compared to electrons in the valence band of silicon). 10
11 Band Theory for p-type Semiconductors How does a p-type semiconductor work? e.g. Si ([Ne]3s 2 3p 2 ) is doped with Al ([Ne]3s 2 3p 1 ) In silicon, the valence band is completely full and the conduction band is completely empty. The aluminium provides an additional empty band that is lower in energy than the conduction band of silicon. Electrons in the valence band of silicon are more easily excited into this acceptor band (compared to the conduction band of silicon). 11
12 Devices Through careful choice of both dopant and concentration, the conductivity of a Semiconductor can be fine-tuned. There are many applications of semiconductors and doping in electronics. Ex) Diodes An n-type and a p-type SC are connected. The acceptor band in the p-type SC is filled with the extra electrons from the n-type SC. The extra holes from the p-type SC move to the n-type SC. Negative and positive charges move in opposite directions, thus a charge separation builds up which stops the flow of both electrons and holes from moving unless the diode is connected to a circuit.. 12
13 Diodes If a diode is connected to a circuit such that the electrons flow into the p-type semiconductor, electrons are replenished and current can flow. Initially with -q accumulation Initially with +q accumulation +ve terminal -ve terminal Charge accumulation reduced If a diode is connected to a circuit such that the electrons flow into the n-type semiconductor, electrons will pile up there and the current will stop. Initially with -q accumulation Initially with +q accumulation -ve terminal +ve terminal Charge accumulation increased 13
14 LED s LEDs are semiconductor (p-n junction diode) light sources. When an LED is switched on, electrons are able to recombine with holes. This causes the release of energy in the form of a photon. The wavelength of the light emitted (photon), and thus its color, depends on the band gap energy of the materials forming the p-n junction. LEDs display some key advantages over traditional light sources and have many diverse applications nm (GaP, AlGaInP) nm (ZnS, InGaN) nm (AlGaAs, GaAsP) 14
15 Solar Cells In a photodiode, the p-type semiconductor is exposed to light. This can excite electrons from the former acceptor band into the conduction band. They are then attracted to the neighboring n-type semiconductor (which has built up a slight positive charge). This causes current to flow, and is how many solar cells work. hn 15
16 Solar Cells Solar cells are devices that convert light into electricity. There are lots of methods/materials re: solar cell development. A large percentage of solar cells on the market still use p-n junction silicon 16
17 Band Theory for Non-metals Non metal conductors?. e.g. graphene C: 1s 2 2s 2 2p 2 For N atoms of Carbon, there will be 4N valence electrons. A total of 3N 2s and 2p xy AO s combine to form 2s g and 2s u * bands each containing 3N/2 states. These are separated by a large energy gap. The remaining N 2p z AO s combine to make 2p u and 2p g * bands each containing N/2 states. These bands coalesce to form one continuous band with a total of N states. The lower energy band, 2s g, is the valence band with 3 N electrons and hence full in the ground state. The middle band, made up from is the 2p u and 2p g * bands, is the conduction band containing N and hence is half occupied in ground state. Hence making it is a good conductor, just like the 3s band does for Sodium 2s u * 2p g * 2p u 2s g unoccupied Conduction Valence 3N e s fully occupied 3N/2 N Valence N e s half occupied 3N/2 The highest energy band, 2s u *, is empty in in ground state. 17
18 Carbon Nanotubes SEM Image TEM Image Added to conventional materials: i) to increase tensile strength ii) to make them conductive iii) Organic solar cells 18
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