Matter, Materials, Crystal Structure

and Bonding

Chris J. Pickard

Why should a theorist care?

Where the atoms are determines what they do

Where the atoms can be determines what we can do

Overview of Structure and Bonding

Atoms and Elements

Non-crystalline materials

Crystalline materials

Bonding and cohesion

Atoms and Elements

Molecules

Naphalene

Molecules are finite collections of covalently bonded atoms

They vary from diatomic molecules, to massive proteins

Polymers

Polyethylene (or polythene)

Polymers are long chain molecules,often with a carbon backbone(sometimes silicon)

The repeat unit, or monomer, isrepeated and joined together manytimes

The chemical formula for polyethyleneis (CH2)N

Many properties depend simply on N

Nanoparticles

Silver Nanoparticle

Nanoparticles range in size from 1 100 nm and can contain from 10 to100,000 atoms

Their properties are not necessarilythe same as the bulk material

The physical, mechanical, electronic,magnetic, optical and chemicalproperties can all differ

Nanostructures

(9,9) armchair nanotube

More complex structure thannanoparticles

Might be designed, and often selfassembled

Nanotechnology is a very hot topic,with strong technological promises fornanoelectronics etc.

Amorphous Solids

Amorphous Carbon

Amorphous solids have no long rangeorder

The are disordered, and the kind ofdisorder is structural as opposed tocompositional

In this example both sp3 and sp2

bonding is found

Can have a variety of electronic andmechanical properties

Order in randomness

A silica glass

The atomic positions are rarelyentirely random

The structural constraints imposesome order on the randomness

The result is short range order,without long range order

This order can be probedexperimentally

The radial distribution function

0 2 4 6r (Angstrom)

0

5

10

15

Rad

ial d

istr

ibut

ion

func

tion

The RDF is the probability density offinding particle i at a given distancefrom particle j:

Gi,j =N

i6=j (r |ri rj|)

Can be measured in scatteringexperiments (X-ray, neutron)

For a crystalline solid this would justbe a set of delta spikes . . .

Crystalline solids

A crystal is a homogeneous solid with a regularly repeating atomic arrangement

The majority of the solid state physicists theoretical tools have been developedfor dealing with crystals

Most current scientific questions concern non ideal crystals (or structures thatare simply not crystalline)

Our tools are still useful . . .

Ionic crystals

Halite (NaCl)

The Alkali Halides are typical ioniccrystals

Can only exist for compounds(different elements)

Are electronic wide band gapinsulators

Are of technological importance fortheir optical properties

Pure Metal

Aluminium

The pure form of the majority ofthe elements in the periodic table aremetallic

Metals conduct electricity freely, andreflect light

Many pure metals are soft, buttransition metals are harder

Alloys

Ordered metallic alloy

Alloys are mixtures of elements, notexclusively metallic and can be eitherordered or disordered

The properties depend dramaticallyon the composition, and history ofthe alloy

Understanding alloying is ofimmense economic and technologicalimportance

Covalent Crystals

Diamond

The strong directional covalent bondslead to mechanically strong crystals

Covalent crystals are less wellinsulating than ionic crystal

Carbon can form a range of differentcovalent bonds, in this case sp3

Graphite

Bernal Graphite

The graphene sheets are purely sp2

covalently bonded

Graphite can exist in different phases,according to the stacking

Bernal graphite (structure found byJ. D. Bernal in 1924) has an. . . ABABABA. . . stacking

Graphite is a semi-metal (a non-zeroDOS at the Fermi level)

Semiconductors

Gallium Arsenide (GaAs)

Semiconductors are still less insulatingthan ionic crystals

Band-gaps range from a fraction of anelectron volt, to several

They can be doped to control theelectronic properties

They are the basis of thesemiconductor industry

Zeolites

Ferrierite

Built up from SiO4 tetrahedral units

Like the flexible hybridisation ofcarbon, these units allow manystructural possibilities

Zeolites are open silicates

They are industrially important ascatalysts

Minerals

Amphibole

Minerals are materials found naturallyin the earth

Many are silicate based (again, on theSiO4 tetrahedra)

The structures can be very complex,contain many atomic species, anddepend on the geological history ofthe sample

A wide range of properties

High Temperature Superconductors

YBa2Cu3O7

YBCO is a complex layered metaloxide

It is in the Perovskite structure

YBCO superconducts at 92K

Oxidation state of CuO planesand chains control superconductingproperties

Molecular crystals

Solid cubane

Molecules play the role of atoms toform molecular crystals

At non-zero temperature, themolecules will be in often complexmotion

Molecular crystals can be organicnon-linear optical materials, andother electronic properties are oftechnological interest

Cubic Ice

Cubic water ice (H2O)

Water ice is another form of molecularcrystal

Held together by directional, butweak, hydrogen bonds

The H2O molecule can vibrate, buthave less freedom to rotate

Molecular crystal with hydrogen bonds

Indigo

Many organic molecules (pigmentslike Indigo, or drug molecules) are heldtogether by both hydrogen bonds, andeven weaker van der Waals bonds

There can be very many possiblepolymorphs

The physical properties of the crystalwill depend on the exact polymorphpresent

Bonding

The energy of a collection of atoms is often lower if they are close

The atoms stick together

In specific ways, depending on the nature of the atoms and their neighbours

This bonding can be classified as Covalent, Ionic, Mixed, Metallic, Molecularand Hydrogen Bonding

Of course, in reality there is a continuum

Covalent Bonding

Diamond

The atoms involved in a covalent bondcan be said to be sharing valenceelectrons so as to attain a magicaloctet

Visually, this expresses itself as asignificant charge density between theatoms

Covalent bonds are strong, gettingstronger (and shorter) the moreelectrons that are shared

Ionic Bonding

Halite (NaCl)

The electrons are not shared and thereis a concentration of charge density onthe more electronegative ion

The ion that donates electronsbecomes positively charged (cations),and the receiving electron negativelycharged (anions)

The bond is due to the electrostaticinteraction between the oppositelycharged ions, and non-directional

Mixed Ionic Covalent Bonding

III-V Semiconductor (GaAs)

The ionicity is weaker, due to thecloser electro-negativity of the atoms,but still present

There is an increase in charge densitybetween bonded atoms

The bond has both covalent andionic character, and the bonding isdirectional

Metallic Bonding

Aluminium

The electrons are shared across theentire solid, and are delocalised : anextreme version of covalent bonding

The more electrons shared, thestronger the metallic bonding: Mghas a higher melting point than Na

The bonding is non directional, canhave a covalent character, and resultsin close packed solids

Hydrogen Bonding

Cubic water ice (H2O)

Weak directional bonds due tothe electrostatic attraction of polarmolecules (or permanent dipoleswithin molecules)

These permanent dipoles are due todifferences in electronegativity

Hydrogen is unusual: it cannot fullyionise (first ionisation potential toohigh), but also can form only onecovalent bond

Van der Waals Bonding

Solid cubane

Weak non-directional bonds due tothe electrostatic attraction betweenfluctuating dipoles

Always present, but only dominatesif all other bonds are saturated orimpossible

The charge density between van derWaals bonded elements is verysmall

Cohesive Energy

2 4 6 8 10 12 14 16 18 20Lattice Parameter (Angstrom)

-1250

-1200

-1150

-1100

-1050

Tot

al E

nerg

y (e

V)

The total energy of diamond withlattice parameter

See Chapter 20 of Ashcroft andMermin for a more (slightly more ...)quantitative approach

To do a proper job, you really needto use a first principles QM basedapproach

See the lecture courses of Drs.Molteni and Towler next term