CHE-30005 Solids, Surfaces and Catalysis :Solid State Chemistry lecture 2

Rob JacksonLJ1.16, 01782 733042

r.a.jackson@keele.ac.ukwww.facebook.com/robjteaching

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Lecture plan

• Molecular orbitals in solids• Metals• Insulators• Band gaps• Semiconductors

– Intrinsic semiconductors– Extrinsic semiconductors

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Molecular orbitals in solids - 1

• The basic idea is to extend the idea of bonding in a diatomic molecule to a line of N atoms.

• We will consider atoms each with one outer electron in an s-orbital (e.g. Li).

• A diatomic molecule will have 2 atoms, and 2 orbitals which overlap to give a bonding orbital & an antibonding orbital.

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Molecular orbitals in solids - 2

• N=3 will give 3 orbitals, 1 bonding, 1 antibonding, and 1 intermediate in character.

• As N increases, more orbitals are added, and for a line of N atoms there will be N molecular orbitals ranging from fully bonding to fully antibonding – this is called a band of orbitals.

• This is illustrated on the next slide.

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Formation of bands

http://www.everyscience.com/Chemistry/Inorganic/Solid_State/a.1232.php

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Metals

• Metals adopt this form of bonding.• The band formed from overlap of

orbitals in, e.g., sodium, is called an s-band; bands involving s, p and d orbitals also occur in other metals.

• Sodium will be given as a detailed example at this point (also see next slide).

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Na band structure

• The addition of each Na atom adds a new Na orbital.

• Once the number of orbitals is large, a band is formed, with the valence electrons occupying the lower half of the band.

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Summary for Metals• Bands are not fully filled,

allowing electrons to be promoted to unoccupied levels, making them free to move through the metal, resulting in conduction.

• How does this model explain the effect of temperature on conduction?

http://www.chembio.uoguelph.ca/educmat/chm729/band/metal.htm

The Fermi Level is the highest occupied energy level at 0 K.

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The Fermi Level and occupation of bands

• The population of an energy level is related to its energy E by:

P = 1/[exp (E-EF)/kT)) +1]

(EF is the Fermi energy)

• If E >> EF, the expression becomes:

P exp [- (E-EF)/kT]

• This means that the population decays exponentially with increasing energy.

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Insulators - 1

• As examples of insulators we consider ionic compounds, e.g. NaCl.

• In ionic compounds there is in general a transfer of electrons from metal to non-metal, e.g. Na denotes its 3s electron to complete the Cl 3s-3p levels.

• In the band picture, this corresponds to a full Cl 3s-3p band, and an empty Na 3s band.

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Insulators - 2

• Na is [1s22s22p6]3s1, Cl is [1s22s22p6]3s23p5

• The valence electrons form 2 bands, a Cl 3s-3p band and a Na 3s band.

• The process of electron transfer corresponds to filling the Cl 3s-3p band.

• Cl is more electronegative than Na so the band is lower in energy.

• A band diagram is shown on the next slide:

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Band structure diagram for an insulator

• The valence band is fully occupied by the valence electrons.

• There is a gap between the valence band and conduction band which the electrons must cross for conduction to occur.

http://en.wikipedia.org/wiki/Image:Electronic_band_diagram.svg

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Or a simpler view:

http://ns.ph.liv.ac.uk/~ajb/radiometrics/glossary/insulator.html

The band gap for NaCl is ~8.97 eV at 77 K.

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Key Points

• The 3s-3p band contains 4N levels (each atom contributes 3s, 3px, 3py and 3pz orbitals).

• Each level (orbital) accommodates 2 electrons.

• NaCl has 8N valence electrons, so these occupy and fill the Cl 3s-3p band.

• MgO will be given as another example.

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MgO band structure

• Write down the outer electron configurations for Mg and O.

• What bands are formed?• Where are the valence electrons

located?

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Band Gaps

• In NaCl at 298K there is a gap of ~7 eV (~700 kJ mol-1) between the filled Cl 3s-3p band and the empty Na 3s band*.

• For conduction, electrons must be able to cross this gap (very unlikely at room temperature).

• Hence NaCl is an insulator at room temperature – what about the molten salt?

• * Note difference with 77K value – why the difference?

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Conclusions on Insulators

• Insulating properties are due to wide band gaps which prevent electrons accessing unoccupied levels in the next available band.

• Semiconductors are special cases of insulators with smaller band gaps.

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A definition of semiconductors

• Semiconductors are similar to insulators in that they have a valence band containing the valence electrons, separated by a band gap from the conduction band.

• The difference is the size of the band gap, which is much less (1.1 eV for Si, 0.7 eV for Ge).

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Example - silicon

• Si has valence electrons in 3s23p2

• Using hybridisation, the 4 valence electrons are placed in 4 equivalent sp3 orbitals: (sp3)1 (sp3)1 (sp3)1 (sp3)1

• Each atom supplies 4 levels, and 4 valence electrons, or 4N levels and 4N valence electrons in a crystal.

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Silicon – continued

• The levels are equally divided between bonding and antibonding levels, and the levels split into two bands.

• There will be 2N levels in each band, and the 4N valence electrons occupy the 2N levels in the valence band.

• Conduction can occur if an electron is promoted to the conduction band.

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Band structure of silicon

• The diagram shows the band in a horizontal form. In the valence band, each level is fully occupied so that all the electrons from sp3 levels are involved in bonding.

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Intrinsic semiconductors defined

• Silicon is an intrinsic semiconductor, in that it works well as a semiconductor in its pure form.

• Germanium is another example – obtain the band diagram using the same procedure as used for silicon.

• Extrinsic semiconductors have their conductivity enhanced by doping with other atoms.

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Extrinsic semiconductors - 1

(i) n-type semiconductorsThe material is doped with an atom with more valence electrons than the host material.e.g. for Si (3s23p2), dope with As (4s24p3)

• the additional electron that results from each substitution will go into a new band, the donor band.

• Note – doping levels low – 1 atom in 109

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Extrinsic semiconductors - 2

• The presence of the donor band reduces the band gap, since electrons can more easily move from the donor band to the conduction band.

• The charge carriers are electrons – hence this type of doping is called n-type.

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Extrinsic semiconductors - 3

(ii) p-type semiconductorsThe converse of n-type, where the material is doped with an atom with less valence electrons than the host material.

e.g. for Si (3s23p2), dope with Ga (4s24p1)• this effectively introduces holes into the

solid, represented by an empty band called the acceptor band.

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Extrinsic semiconductors - 4

• The mechanism of conduction is that electrons are promoted from the valence band to the acceptor band, leaving holes in the valence band.

• this gives mobility to the electrons in the valence band, and conduction occurs. Because it is caused by the presence of positive ‘holes’, it is called p-type conduction.

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Summary: the relative band structures of metals, insulators and semiconductors

http://www.physics.udel.edu/~watson/scen103/colloq2000/band-semi.html

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Summary

• The electrical conductivity of metals, insulators and semiconductors has been explained in terms of their electronic structure.

• Band structure diagrams have been introduced to represent this.