Introduction to Materials Science Graduate students (Applied Physics)

Transcription

1 Introduction to Materials Science Graduate students (Applied Physics) Prof. Michael Roth Chapter 1 Crystallography

2 Overview Performance in Engineering Components Properties Mechanical, Electrical, Thermal Processing Manufacturing/ Materials Selection Structure Crystallographic Structure (contains information about atomic, molecular, and electron positions

3 Overview Phonons (lattice vibrations) Crystallographic Structure Atomic/Molecular Chemistry Properties Molecular, Atomic, Electronic Periodic Arrangement Free Electrons (conduction electrons) in periodic lattice Temperature dependent properties such as resistivity and heat capacity Energy Band structure in metals, semiconductors, and insulators Binding Forces (Covalent, Ionic, Metallic, Van der Waals) and Mechanical Properties at room temperature

4 Introduction STATES OF MATTER Primarily matter exists in three states which can be distinguished by their macroscopic and microscopic properties. 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 and electro-optic devices, which emphasizes the importance of the field.

5 Introduction - continued MACROSCOPIC PROPERTIES of SOLIDS A solid has definite dimensions. The variation of these with temperature or under a force gives a macroscopic description of the properties of the solid. Examples are: Linear coefficient of thermal expansion Hardness Young s modulus, Shear Modulus Strength A good working definition of a solid in macroscopic terms is: A material which can sustain a shear force which 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 Temperature All of these are related to the microscopic structure of the material.

6 Introduction - continued MICROSCOPIC DESCRIPTION OF MATTER Order and Disorder On a microscopic scale all matter consists of particles which interact with each other. These may be either single atoms or molecules, which are made up of atoms which are chemically bound to each other. In fluids, the particles are not bound and are free to move. The difference between liquids and gases depends 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 each other. In crystalline materials these positions are regularly arranged on a lattice. This leads automatically 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, the positions 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 a particle is known, some information about it s (nearest or next-nearest) neighbors can be derived.

7 Crystals and Lattices Crystal infinite periodic repetition of identical units (atoms, molecules, etc.) in space an underlying latiice set of periodic points of identical environments. A crystal structure is defined by crystal structure = lattice + basis The lattice is a mathematical abstraction describing a regular periodic array of points (or position vectors). 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 the crystal. 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).

8 Translation Symmetry Crystalline structure means spatial periodicity or translation symmetry. We assume that there are 3 non-coplanar vectors a, b and c that leave all the properties of the crystal unchanged after the shift as a whole by any of those vectors. As a result, any lattice point r' could be obtained from another point r as r' = r +n 1 a + n 2 b + n 3 c, where n i 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 combination of the properties of the building block (basis) and of the Bravais lattice. Note that There is no unique way to choose a, b and c. 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. Vectors a, b and c chosen in such a way are called primitive. The volume cell enclosed by the primitive translation vectors is called the primitive unit cell. The volume of the primitive cell is: V = a (b c) Primitive unit cell 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.)

9 1D and 2D Lattices One-dimensional lattices chains a) 1D chains are shown in the right Figure. We have only 1 translation vector 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 lattices Bravais lattice. Full Non-Bravais lattice. A and A 1 There are 5 basic 2D lattices translational symmetry. sites are not equivalent. Not invariant under AA 1 translation. The choice of the unit cell is not unique. Wigner-Seitz unit cell reflects the symmetry. It may be chosen as in the Figure: (i) draw lines to connect a given lattice point to all nearby lattice points. (ii) at the midpoint and normal to these lines, draw new lines (planes in 3D). The smallest volume enclosed is the Wigner-Seitz primitive cell. All the space of the crystal may be filled by these primitive cells, by translating the unit cell by the lattice vectors.

10 Reciprocal Lattice of a 2D Lattice Kittel pg. 38

11 3D Lattices 3D Bravais lattices In 3 D there are only 14 inequivalent lattices which satisfy translational symmetry. These are the Bravais lattices. If we allow non-primitive unit cells, the 14 lattices can be grouped in the seven crystal systems. To count lattice points: centre of cell = 1; face = 1/2; edge = 1/4; corner = 1/8. System Triclinic Monoclinic Orthorhombic Tetragonal Cubic Trigonal Hexagonal Conventional Unit cell a b c α β γ a b c α = β = 90 γ a b c α = β = γ = 90 a = b c α = β = γ = 90 a = b = c α = β = γ = 90 a = b = c 120 > α = β = γ 90 a = b c α = β = 90, γ = 60 Bravais Lattice P (primitive) P I P C I (body-centered) F (face-centered) P I F P I F R (rhombohedral primitive) P

12 3D Wigner-Seitz Cells In the above we have used one of the two most common ways of creating the primitive unit cell. By joining lattice points we find the primitive unit cell to be the volume enclosed by the three primitive translation vectors (more accurately - the planes defined by pairs of them). For the face centred cubic (fcc) lattice this unit cell is the rhombohedron 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 particular lattice point than any other. The surface of this unit cell is then defined by the planes which bisect all lattice vectors drawn from that point. For the same lattice as above, the Wigner-Seitz cell is the rhombic dodecahedron shown on the right. Both of these unit cells represent 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 is shared between the 8 corners.

13 Point Group Symmetry The natural way to describe a crystal structure is a set of point group operations which involve operations applied around a point of the lattice. We shall see that symmetry provides important restrictions upon vibration and electron properties (in particular, spectrum degeneracy). Usually are discussed: Reflection, σ: Reflection across a plane (mirror symmetry). Symbol m. e.g., H 2 O molecule, 2 mirror planes: plane of molecule and Inversion, I: Transformation r r, fixed point is selected as origin (lack of inversion symmetry may lead to piezoelectricity); symbol. E.g., homonuclear diatomic molecules: and m, bcc lattice (center of cube) 1 1 Rotation, C n : n-fold rotation axes: symbol 1, 2, 3, 4, 6. Rotation by an 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 only n = 1, 2, 3, 4, 6 are compatible with translational symmetry! Rotation-inversion axes: n-fold rotation with simultaneous inversion; symbol 2,3, 4,6 Improper rotation, S n : Rotation C n, followed by reflection in the plane normal to the rotation axis. 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.

14 32 Crystallographic Point Groups Isometric (cubic) Hexagonal Tetragonal Trigonal

15 32 Crystallographic Point Groups and Summary Triclinic (1), Mononoclinic (2) and Orthorhombic (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 groups 230 space groups also encompass the following symmetry elements: screw axis: rotation + translation glide mirror plane: reflection in a plane + translation

16 System (1) Class Name (2) AXES Hermann- 2-fold 3-fold 4-fold 6-fold Planes Center Maugin Symbols (3) Isometric Tetartoidal Diploidal yes 2/m 3 Hextetrahedral m Gyroidal Hexoctahedral yes 4/m 3 2/m Tetragonal Disphenoidal Pyramidal Dipyramidal yes 4/m Scalenohedral m Ditetragonal pyramidal mm Trapezohedral Ditetragonal-Dipyramidal yes 4/m 2/m 2/m Orthorhombic Pyramidal mm2 Disphenoidal Dipyramidal yes 2/m 2/m 2/m Hexagonal Trigonal Dipyramidal Pyramidal Dipyramidal yes 6/m Ditrigonal Dipyramidal m2 Dihexagonal Pyramidal mm Trapezohedral Dihexagonal Dipyramidal yes 6/m 2/m 2/m Trigonal Pyramidal Rhombohedral yes 3 Ditrigonal Pyramidal m Trapezohedral Hexagonal Scalenohedral yes 3 2/m Monoclinic Domatic m Sphenoidal Prismatic yes 2/m Triclinic Pedial Pinacoidal yes 1

17 Crystal Directions Any lattice vector can be written as that given by e.q. r = n 1 a + n 2 b + n 3 c. The direction is then specified by the three integers [n 1 n 2 n 3 ]. 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 equivalent due 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 and belong to the same family of directions. Two examples for the cubic lattice are given below: [ ] [ ] [ ] [ ] 100 = 100, 010, 001, 100, 010, = 111, 111, 111, 111, 111, 111, 111, 111 The angle (α) between any two vectors [n 1 n 2 n 3 ] and [m 1 m 2 m 3 ] n1m 1 + n2m2 + n3m3 Is Cosα = n + n + n m + m + m Hexagonal lattices usually employ 4 indexes.

18 Crystallographic Planes The orientation of a plane in a lattice is specified by Miller indices. They are defined as follows. We find intercept of the plane with the axes along the primitive translation vectors a 1, a 2 and a 3. Let these intercepts be x, y, and z, so that x is fractional multiple of a 1, y is a fractional multiple of a 2 and z is a fractional multiple of a 3. Therefore we can measure x, y, and z in units a 1, a 2 and a 3 respectively. We have then a triplet of integers (x y z). Then we invert it (1/x 1/y 1/z) and reduce this set to a similar one having the smallest integers by multiplying by a common factor. This set is called Miller indices 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 will be (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) having the same indices, but this is not generally true for other crystal systems. Examples of the planes in a cubic system: Indexing of the Hexagonal System The lattice is defined by the vectors a 1, a 2, a 3 and c, so the Miller-Bravais indices are written as (hkil) where h+k= -i. For example, the side planes of the hexagonal lattice are clearly shown as:

19 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.

20 Crystal Structures - 2 Face Centred Cubic (fcc): some noble gases, Cu, Ag, Au, Ni, Pd, Pt, Al 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). (Left) The fcc structure with illustration of the closepacked planes perpendicular to the space diagonal ([111] direction).(right) Illustration of the close packing of atoms and the nearest-neighbor distance. The conventional fcc unit cell is also not primitive, and has a volume four 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.

21 Crystal Structures - 3 Hexagonal close packed structure Another way of stacking close packed layers of atoms is to have every second layer above each other, i.e. a stacking sequence AB. The lattice is (simple) hexagonal with a basis of two atoms at (0,0,0) and ( 2, 1, 1 ) Important hcp metals: Zn, Cd, Be, Mg, Re, Ru, Os In this case the close packed layer forms the (001) basal planes of the hexagonal close packed structure. This structure has the same packing 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. often (c/a) > or < 1.633, which indicates deviations from purely metallic bonding.

22 Crystal Structures - 4 Caesium Chloride Structure Simple cubic lattice + Basis: Cl - at (0,0,0); Cs + at (½,½,½) (or vice versa) Sodium Chloride Structure Chlorine ions occupy fcc sites and sodium ions occupy edge sites. 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 is not equivalent to the Cl ion at the corner. In the bcc structure, each atom, whether at the centre or at the corner, is surrounded by eight equivalent sites. The bcc lattice can also be thought of as two intersecting cubic lattices displaced by half of the body diagonal. Both sub-lattices are identical. Sodium Chloride Structure (red ions represent Na + and blue ions represent Cl - )

23 Crystal Structures - 5 The diamond structure: E.g.: diamond, Si, Ge, a-sn Interpenetrating FCC lattices The primitive basis has two identical atoms at 000 and ¼¼¼ Conventional unit cell: 8 atoms Nearest neighbours: 4 Next nearest neighbours: 12 Most of this lattice is empty: packing fraction is 0.34, which is 46% of the HCP or FCC structure 5.43 Angstroms Silicon

24 Crystal Structures - 6 Zinc blende (cubic) ZnS structure (sphalerite): Many III-V semiconductors: GaAs, GaP, InSb,... Wurzite (hexagonal) ZnS structure Stacking sequence ABABAB... along cube diagonal two interpenetrating hcp lattices (i.e., hcp + basis) Many II-VI semiconductors: ZnO, ZnSe, ZnTe, CdS, CdSe Mixed forms with random cubic or hexagonal stacking or long period repeats also possible (so-called polytypes), e.g., SiC

25 Crystal Structures - 7 Perovskite (ABO 3 ) e.g. BaTiO 3 KTa X Nb 1-X O 3

26 Crystal Structures - 8 Fullerene (bucky ball) Discovered in 1985 by Sir Harold Kroto et al. (001) Carbon (C) Carbon nanotube Discovered in 1991 by Sumio Iijima

27 Crystal Structures - 9 (111) (0001)

28 Crystal Structures - 10 YBCO high-t c superconductor (1987) MgB 2 the newest high-t c superconductor (2002)

29 Crystal Structures - 11 Potassium titanyl phosphate KTiOPO 4 (KTP) KTP crystal structure is made up of a three-dimensional covalent framework of titanyl phosphate (TiOPO 4 ) - groups with K + cations loosely fitting into the Z-oriented channels in a spiral fashion.