Introduction to Materials ScienceGraduate students (Applied Physics)

Prof. Michael Roth

Chapter 1Crystallography

Overview

Properties –Mechanical, Electrical, Thermal

Performance – in Engineering Components

Processing –Manufacturing/Materials Selection

Structure –Crystallographic Structure (contains information about atomic, molecular, and electron positions

Overview

Temperature dependent properties such as resistivity and heat capacityEnergy Band structure in metals, semiconductors, and insulatorsBinding Forces (Covalent, Ionic, Metallic, Van der Waals) and Mechanical Properties at room temperature

Molecular, Atomic, Electronic Periodic Arrangement

Free Electrons (conduction electrons) in periodic lattice

Crystallographic

Structure Properties

Atomic/Molecular Chemistry

Phonons (lattice vibrations)

Introduction

STATES OF MATTERPrimarily matter exists in three states which can be distinguished by their macroscopic and microscopicproperties. These are: solid, liquid and gas.Liquids and gases are fluid: unlike solids they flow and do not maintain their own shape.

Other states of matter can broadly be considered as special cases of one of these, or a combination, e.g.• Plasma (mixture of two gases: electrons and ions)• Glass (in some cases a like liquid, in others solid)• Gel (mixture of liquid and solid)• Sol (mixture of liquid and solid)• other colloids

The solid phase is the main source of an immense variety of mechanical, electronic, optical andelectro-optic devices, which emphasizes the importance of the field.

Introduction - continuedMACROSCOPIC PROPERTIES of SOLIDSA solid has definite dimensions. The variation of these with temperature or under a force gives amacroscopic description of the properties of the solid. Examples are:• Linear coefficient of thermal expansion• Hardness• Young’s modulus, Shear Modulus• StrengthA good working definition of a solid in macroscopic terms is:

A material which can sustain a shear forcewhich is actually another way of saying, “a material which does not flow”.

Other properties which describe all matter are:• Volume coefficient of thermal expansion• Density• Bulk modulus / compressibility• TemperatureAll of these are related to the microscopic structure of the material.

Introduction - continuedMICROSCOPIC DESCRIPTION OF MATTEROrder and DisorderOn a microscopic scale all matter consists of particles which interact with each other. These may beeither single atoms or molecules, which are made up of atoms which are chemically bound to eachother.In fluids, the particles are not bound and are free to move. The difference between liquids and gasesdepends on their separation and the strength of the interaction between them.In solids, the constituent particles are bound together and occupy definite positions relative to eachother. In crystalline materials these positions are regularly arranged on a lattice. This leadsautomatically to the description of matter in terms of two microscopically defined primary states:

Structurally all perfect crystals exhibit long range order. If the position of one particle is known, thepositions of all others can be determined using well defined rules.Gases, most alloys, and spin glasses are completely disordered. The position or type of a particle(magnetisation state in a spin glass) is independent of all other particles.Liquids, glasses, and other amorphous materials, have short range order. If the position or state of aparticle is known, some information about it’s (nearest or next-nearest) neighbors can be derived.

Crystals and Lattices

Crystal ≡ infinite periodic repetition of identical units (atoms, molecules, etc.) in space ⇒ an underlyinglatiice ≡ set of periodic points of identical environments.

A crystal structure is defined by crystal structure = lattice + basisThe lattice is a mathematical abstraction describing a regular periodic array of points (or positionvectors). It describes the symmetry and long range order of the crystal.

The basis is the building block which, when placed on the lattice, creates the physical structure of thecrystal. It may correspond to a single atom, ion or molecule, but usually consists of a group of them.

Primitive lattice ≡ one that includes all points with identical environments ≡ defines the smallest cell(otherwise it is a space lattice).

Translation Symmetry

We assume that there are 3 non-coplanar vectors a, b and cthat leave all the properties of the crystal unchanged afterthe shift as a whole by any of those vectors. As a result, anylattice point r' could be obtained from another point r as

r' = r +n1a + n2b + n3c,where ni are integers and r is the displacement operator.Such a lattice of building blocks is called the Bravais lattice.The crystal structure can be understood by the combinationof the properties of the building block (basis) and of theBravais lattice.

Note that

n There is no unique way to choose a, b and c.n We choose a as shortest period of the lattice, b as the

shortest period not parallel to a, c as the shortest period not coplanar to a and b.

n Vectors a, b and c chosen in such a way are called primitive.

The volume cell enclosed by the primitive translation vectorsis called the primitive unit cell.The volume of the primitive cell is: V = a ⋅ (b × c)

Crystalline structure means spatial periodicity or translation symmetry.

a, b, c : lattice parameters, or lattice constants, and their complementary angles are α, β and γ .(Note: α is the angle between b and c, etc.)

Primitive

unit cell

1D and 2D LatticesOne-dimensional lattices – chains a)1D chains are shown in the right Figure. We have only 1 translationvector |a| = a, V = a. White and black circles are the atoms of b)different kind. a) is a primitive lattice with one atom in a primitive cell;b) and c) are composite lattices with two atoms in a cell. c)

Two-dimensional latticesBravais lattice. Full Non-Bravais lattice. A and A1 There are 5 basic 2D latticestranslational symmetry. sites are not equivalent. Not

invariant under AA1 translation.

The choice of the unit cell is not unique. Wigner-Seitz unit cell reflectsthe symmetry. It may be chosen as in the Figure: (i) draw lines toconnect a given lattice point to all nearby lattice points. (ii) at the midpointand normal to these lines, draw new lines(planes in 3D). The smallest volume enclosedis the Wigner-Seitz primitive cell. All the spaceof the crystal may be filled by these primitivecells, by translating the unit cell by the latticevectors.

Kittel pg. 38

Reciprocal Lattice of a 2D Lattice

3D Lattices3D Bravais latticesIn 3 D there are only 14 inequivalent lattices which satisfy translational symmetry. These are the Bravais lattices. Ifwe allow non-primitive unit cells, the 14 lattices can be grouped in the seven crystal systems. To count latticepoints: centre of cell = 1; face = 1/2; edge = 1/4; corner = 1/8.

Bravais LatticeConventional Unit cell

System

P (primitive)a ≠ b ≠ cα ≠ β ≠ γ

Triclinic

PI

a ≠ b ≠ cα = β = 90° ≠ γ

Monoclinic

PC

I (body-centered)F (face-centered)

a ≠ b ≠ cα = β = γ = 90°

Orthorhombic

PIF

a = b ≠ cα = β = γ = 90°

Tetragonal

PIF

a = b = cα = β = γ = 90°

Cubic

R (rhombohedralprimitive)

a = b = c120° > α = β = γ ≠ 90°

Trigonal

Pa = b ≠ cα = β = 90°, γ = 60°

Hexagonal

3D Wigner-Seitz CellsIn the above we have used one of the two most common ways ofcreating the primitive unit cell. By joining lattice points we find theprimitive unit cell to be the volume enclosed by the three primitivetranslation vectors (more accurately - the planes defined by pairsof them). For the face centred cubic (fcc) lattice this unit cell is therhombohedron on the left.

Another, often more useful, primitive unit cell is the Wigner-Seitz cell.This is defined as the volume of space nearest to a particularlattice point than any other. The surface of this unit cell is thendefined by the planes which bisect all lattice vectors drawn from thatpoint. For the same lattice as above, the Wigner-Seitz cell is therhombic dodecahedron shown on the right. Both of these unit cellsrepresent the same lattice, translate with the same lattice vectors,and have the same volume. The only difference is that in Wigner-Seitz cell the lattice point is at the centre, and in the first example it isshared between the 8 corners.

Point Group SymmetryThe natural way to describe a crystal structure is a set of point group operations which involve operationsapplied around a point of the lattice. We shall see that symmetry provides important restrictions upon vibration andelectron properties (in particular, spectrum degeneracy). Usually are discussed:

Reflection, σ: Reflection across a plane (mirror symmetry). Symbol m.e.g., H2O molecule, 2 mirror planes: plane of molecule and ----------

Inversion, I: Transformation r → –r, fixed point is selected as origin (lack of inversion symmetry maylead to piezoelectricity); symbol . E.g., homonuclear diatomic molecules: and m, bcc lattice (center of cube)

Rotation, Cn: n-fold rotation axes: symbol 1, 2, 3, 4, 6. Rotation byan angle 2π /n about the specified axis. There are restrictions for n.Indeed, if a is the lattice constant, the quantity b = a +2aCosφ(see Fig. below). Consequently, Cosφ = i/2 (i – integer), and onlyn = 1, 2, 3, 4, 6 are compatible with translational symmetry!

Rotation-inversion axes: n-fold rotation with simultaneous inversion;symbol

Improper rotation, Sn: Rotation Cn, followed by reflection in the planenormal to the rotation axis.

1 1

2,3, 4,6

A 5-fold axis of symmetry cannot exist in a periodic lattice because it is not possible to fill the area of a plane with a concentrated array of pentagons. We can, however, fill all the area of a plane with just two distinct designs of ”tiles” or elementary polygons. In 3D this building principle leads to so called quasicrystals.

32 Crystallographic Point GroupsIsometric (cubic) Hexagonal

Tetragonal Trigonal

32 Crystallographic Point Groups and Summary

Triclinic (1), Mononoclinic (2) andOrthorhombic (3) Symmetries

Summary

Crystals may be described by combination of symmetry elements* that carry the crystal into itself.

Combination of point symmetry elements ⇒ 7 point groups (crystal systems)

§ Lattice + basis (has its own symmetry) ⇒ 32 crystallographic point groups§ Crystal systems + translation operation ⇒ 14 ”Bravais lattices”

(space groups) (translation ⇒ centered lattices)

§ Bravais lattices + basis ⇒ 230 space groups230 space groups also encompass the following symmetry elements:screw axis: rotation + translationglide mirror plane: reflection in a plane + translation

AXES System (1) Class Name (2)

2-fold 3-fold 4-fold 6-fold Planes Center

Hermann- Maugin Symbols (3)

Tetartoidal 3 4 - - - - 23 Diploidal 3 4 - - 3 yes 2/m 3 Hextetrahedral 3 4 - - 6 - 4 3m Gyroidal 6 4 3 - - - 432

Isometric

Hexoctahedral 6 4 3 - 9 yes 4/m 3 2/m Disphenoidal 1 - - - - - 4 Pyramidal - - 1 - - - 4 Dipyramidal - - 1 - 1 yes 4/m Scalenohedral 3 - - - 2 - 4 2m Ditetragonal pyramidal - - - - 4 - 4mm Trapezohedral 4 - 1 - - - 422

Tetragonal

Ditetragonal-Dipyramidal 4 - 1 - 5 yes 4/m 2/m 2/mPyramidal 1 - - - 2 - mm2 Disphenoidal 3 - - - - - 222

Orthorhombic

Dipyramidal 3 - - - 3 yes 2/m 2/m 2/mTrigonal Dipyramidal - 1 - - 1 - 6 Pyramidal - - - 1 - - 6 Dipyramidal - - - 1 1 yes 6/m Ditrigonal Dipyramidal 3 1 - - 4 - 6m2 Dihexagonal Pyramidal - - - 1 6 - 6mm Trapezohedral 6 - - 1 - - 622

Hexagonal

Dihexagonal Dipyramidal 6 - - 1 7 yes 6/m 2/m 2/mPyramidal - 1 - - - - 3 Rhombohedral - 1 - - - yes 3 Ditrigonal Pyramidal - 1 - - 3 - 3m Trapezohedral 3 1 - - - - 32

Trigonal

Hexagonal Scalenohedral 3 1 - - 3 yes 3 2/m Domatic - - - - 1 - m Sphenoidal 1 - - - - - 2

Monoclinic

Prismatic 1 - - - 1 yes 2/m Pedial - - - - - - 1 Triclinic Pinacoidal - - - - - yes 1

Crystal Directions

Any lattice vector can be written as that given by e.q. r = n1a + n2b + n3c. The direction is then specified by the threeintegers [n1n2n3]. If the latter have a common factor, this factor is removed. For example, [111] is used rather than [222],or [100], rather than [400]. When we speak about directions, we mean a hole set of parallel lines, which are equivalentdue to transnational symmetry. Opposite orientation is denoted by the negative sign over a number.Similar notations are used for primitive cubic and noncubic lattices.Directions related by symmetry are structurally equivalent andbelong to the same family of directions. Two examples for thecubic lattice are given below:

The angle (α) between any two vectors [n1n2n3] and [m1m2m3]

Is

Hexagonal lattices usually employ 4 indexes.

[ ] [ ] [ ]100 100 , 010 , 001 , 100 , 010 , 001 =

[ ]111 111 , 111 , 111 , 111 , 111 , 111 , 111 , 111 =

1 1 2 2 3 32 2 2 2 2 21 2 3 1 2 3

Cos n m n m n mn n n m m m

α + +=

+ + + +

Crystallographic PlanesThe orientation of a plane in a lattice is specified by Miller indices. They are defined as follows. We find intercept of theplane with the axes along the primitive translation vectors a1, a2 and a3. Let these intercepts be x, y, and z, so that x isfractional multiple of a1, y is a fractional multiple of a2 and z is a fractional multiple of a3. Therefore we can measure x,y, and z in units a1, a2 and a3 respectively. We have then a triplet of integers (x y z). Then we invert it (1/x 1/y 1/z) andreduce this set to a similar one having the smallest integers by multiplying by a common factor. This set is called Millerindices of the plane (hkl). For example, if the plane intercepts x, y, and z in points 1, 3 and 1, the index of this plane willbe (313).The Miller indices specify not just one plane but an infinite set of equivalent planes.Note that for cubic crystals the direction [hkl] is perpendicular to a plane (hkl) havingthe same indices, but this is not generally true for other crystal systems. Examplesof the planes in a cubic system:

Indexing of the Hexagonal SystemThe lattice is defined by the vectors a1, a2, a3 and c, so the Miller-Bravais indicesare written as (hkil) where h+k= -i. For example, the side planes of the hexagonallattice are clearly shown as:

Crystal Structures - 1

Simple Cubic (sc)The lattice points at the vertices of a cube. The conventional unit cell therefore contains only one lattice point or one basis. This is the least dense of all structures. In a hard sphere model the packing density is only about 52%.

Body Centred Cubic (bcc)The conventional unit cell contains two lattice points - at the centre of the cube and at the vertices, i.e. at (0,0,0) and (½,½,½).The bcc unit cell is not primitive, and has twice the volume of the Wigner-Seitz cell.In a hard sphere model, the spheres only touch along the body diagonal of the cube, and not at the edges. The packing density is about 68%.

Only one element, Polonium (Po), crystallises in this structure

Examples of materials with a bcc structure are some elemental metals (Fe, Na, Li) and alloys.

Crystal Structures - 2Face Centred Cubic (fcc):some noble gases, Cu, Ag, Au, Ni, Pd, Pt, Al

(Left) The fcc structure with illustration of the close-packed planes perpendicular to the space diagonal ([111] direction).(Right) Illustration of the close packing of atoms and the nearest-neighbor distance.

The conventional unit cell contains four lattice points - at the center of the faces of cube and at the vertices,i.e. at (½,½,0), (0,½,½), (½,0,½) and (0,0,0).

The conventional fcc unit cell is also not primitive, and has a volumefour times that of the Wigner-Seitz cell. In a hard sphere model, the balls on the {111} planes touch their six neighbours. These are known as the close packed planes. The fcc structure is one of the two close packed structures. For this reason it is sometimes referred to as the cubic close packed structure. Ithas a packing density of ~ 74%.

If we look at the (111) plane in the unit cell, the three face diagonals form the sides of an equilateral triangle. These are the [110], [011] and [101] directions. The lattice point at the centre of the (001) face sits at the centre of a triangle of three other lattice points on the (111) plane. With its neighbours, this point makes up a parallel close packed plane. Similarly the lattice point on a (010) face sits in the center of the inverted triangle. If we think of the structure as a stacking of close packed lattice planes, this sequence of three layers is repeated indefinitely. We refer to the stacking sequence as ABC .

Crystal Structures - 3

The lattice is (simple) hexagonal with a basis of twoatoms at (0,0,0) and ( )Important hcp metals: Zn, Cd, Be, Mg, Re, Ru, Os

often (c/a) > or < 1.633 , which indicates deviationsfrom purely metallic bonding.

2 1 1, ,3 3 2

Hexagonal close packed structureAnother way of stacking close packed layers of atoms is to have every second layer above each other, i.e. a stacking sequence AB.

In this case the close packed layer forms the (001) basal planes of the hexagonal close packed structure. This structure has the samepacking density as the fcc structure, 74%.

The atoms in layer B do not occupy equivalent sites to those in layer A. Consequently the hcp structure is not a lattice, and it has no primitive unit cell.

Crystal Structures - 4Caesium Chloride Structure Sodium Chloride StructureSimple cubic lattice + Basis: Chlorine ions occupy fcc sites and sodiumCl- at (0,0,0); Cs+ at (½,½,½) ions occupy edge sites.

(or vice versa) Cl- at (0,0,0); Na+ at (½,0,0)

Note that CsCl does not have a bcc structure.The Cs ion at the centre of the CsCl cubic cell isnot equivalent to the Cl ion at the corner. In thebcc structure, each atom, whether at the centre orat the corner, is surrounded by eight equivalentsites. The bcc lattice can also be thought of astwo intersecting cubic lattices displaced by half ofthe body diagonal. Both sub-lattices are identical.

Sodium Chloride Structure (red ions represent Na+ and blue ions represent Cl-)

Crystal Structures - 5

5.43 Angstroms

The diamond structure:

Silicon

v E.g.: diamond, Si, Ge, a-Snv Interpenetrating FCC latticesvThe primitive basis has two identical atoms at 000 and ¼¼¼v Conventional unit cell: 8 atomsv Nearest neighbours: 4v Next nearest neighbours: 12v Most of this lattice is empty: packing fraction is 0.34, which is 46% of the HCP or FCC structure

Crystal Structures - 6Zinc blende (cubic) ZnS structure (sphalerite):Many III-V semiconductors: GaAs, GaP, InSb, ...

Wurzite (hexagonal) ZnS structureStacking sequence ABABAB... along cube diagonaltwo interpenetrating hcp lattices (i.e., hcp + basis)Many II-VI semiconductors: ZnO, ZnSe, ZnTe, CdS, CdSeMixed forms with random cubic or hexagonal stacking orlong period repeats also possible (so-called polytypes), e.g., SiC

Crystal Structures - 7

Perovskite (ABO3)

e.g. BaTiO3

KTaXNb1-XO3

Crystal Structures - 8

Fullerene (bucky ball) (001) Carbon (C)Discovered in 1985 by Sir Harold Kroto et al.

Carbon nanotubeDiscovered in 1991 by Sumio Iijima

Crystal Structures - 9

(111) ⊥ (0001)

Crystal Structures - 10YBCO high-Tc superconductor

(1987)

MgB2 – the newest high-Tc

superconductor (2002)

Crystal Structures - 11

Potassium titanyl phosphateKTiOPO4 (KTP)

KTP crystal structure is made up of aKTP crystal structure is made up of athreethree--dimensional covalent framework of dimensional covalent framework of titanyl phosphate (TiOPOtitanyl phosphate (TiOPO44))-- groups with Kgroups with K++

cations loosely fitting into the Zcations loosely fitting into the Z--oriented oriented channels in a spiral fashion.channels in a spiral fashion.