Solid State Physics 460- Lecture 2a Structure of Crystals (Kittel - - PowerPoint PPT Presentation

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Solid State Physics 460- Lecture 2a Structure of Crystals (Kittel - - PowerPoint PPT Presentation

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Solid State Physics 460- Lecture 2a Structure of Crystals (Kittel Ch. 1) See many great sites like Bobs rock shop with pictures and crystallography info: http://www.rockhounds.com/rockshop/xtal/index.html Physics 460 F 2006 Lect 2a 1

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Physics 460 F 2006 Lect 2a 1

Solid State Physics 460- Lecture 2a Structure of Crystals (Kittel Ch. 1)

See many great sites like “Bob’s rock shop” with pictures and crystallography info: http://www.rockhounds.com/rockshop/xtal/index.html

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Physics 460 F 2006 Lect 2a 2

Crystals

From Last Time

  • A crystal is a repeated array of atoms
  • Crystal

¤ Lattice + Basis Crystal Lattice of points (Bravais Lattice) Basis of atoms

  • Crystals can be classified into a small number of

types – See text for more details

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Physics 460 F 2006 Lect 2a 3

Two Dimensional Crystals

From Last Time

a1 a2

φ

Basis Lattice

  • Infinite number of possible lattices and crystals
  • Finite number of possible lattice types and crystal

types

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Physics 460 F 2006 Lect 2a 4

Possible Two Dimensional Lattices

From Last Time

a1 a2

φ General oblique

a1 a2

Hexagonal Φ = 60, a1 = a2 6-fold rotation , reflections

a1 a2

Square 4-fold rot., reflect.

a1 a2

Rectangular 2-fold rot., reflect. Centered Rectangular 2-fold rot., reflect.

a1 a2

  • These are the only possible special crystal

types in two dimensions

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Physics 460 F 2006 Lect 2a 5

Three Dimensional Lattices

  • Every point on the Bravais lattice is a multiple of

3 primitive lattice vectors T(n1,n2,n3) = n1 a1 + n2 a2 + n3 a3 where the n’s are integers

a1 a2 a3

X y z

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Physics 460 F 2006 Lect 2a 6

Three Dimensional Lattices Simplest examples

  • Orthorhombic: angles 90 degrees, 3 lengths different

Tetragonal: 2 lengths same Cubic: 3 lengths same

  • Hexagonal: a3 different from a1, a2 by symmetry

Simple Orthorhombic Bravais Lattice

a1 a2 a3

Hexagonal Bravais Lattice

a1 a2 a3

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Physics 460 F 2006 Lect 2a 7

Cubic Lattices

Length of each side - a a3 a2 a1 a a a

Simple Cubic Body Centered Cubic (BCC)

Primitive lattice vectors a1 = (1,0,0) a a2 = (0,1,0) a a3 = (0,0,1) a One atom per cell at position (0,0,0)

Conventional Cell with 2 atoms at positions (000), (1/2,1/2,1/2) a

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Physics 460 F 2006 Lect 2a 8

Cubic Lattices

Length of each side - a a3 a2 a1 a3 a2 a1

Face Centered Cubic (FCC) Simple Cubic

Primitive lattice vectors a1 = (1,0,0) a a2 = (0,1,0) a a3 = (0,0,1) a One atom per cell at position (0,0,0)

Conventional Cell with 4 atoms at positions (000 ), (0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0)a

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Physics 460 F 2006 Lect 2a 9

Face Centered Cubic Two views - Conventional Cubic Cell

X y z X

Conventional Cell of Face Centered Cubic Lattice 4 times the volume of a primitive cell

y z

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Physics 460 F 2006 Lect 2a 10

Face Centered Cubic (fcc) Also called cubic closed packed (ccp)

X y z X

Each atom has 12 equal neighbors We will see later that this is a “close packed” lattice

y z

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Physics 460 F 2006 Lect 2a 11

Face Centered Cubic

X y z

a1 a3 a2

Wigner-Seitz Cell One Primitive Cell

Primitive lattice vectors a1 = (1/2,1/2,0) a a2 = (1/2,0,1/2) a a3 = (0,1/2,1/2) a One atom per cell at position (0,0,0)

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Physics 460 F 2006 Lect 2a 12

Body Centered Cubic

X y z

a3 a1 a2

Wigner-Seitz Cell One Primitive Cell

Primitive lattice vectors a1 = (1/2,1/2,-1/2) a a2 = (1/2, -1/2,1/2) a a3 = (-1/2,1/2,1/2) a One atom per cell at position (0,0,0)

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Physics 460 F 2006 Lect 2a 13

Lattice Planes - Index System

s1 a1 Plane through the points s1 a1 , s2 a2, s3a3 Each s can be an integer or a rational fractions s2 a2 s3 a3

  • Define the plane by the reciprocals 1/s1, 1/s2, 1/s3
  • Reduce to three integers with same ratio h,k,l
  • Plane is defined by (h,k,l)
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Physics 460 F 2006 Lect 2a 14

Schematic illustrations of lattice planes Lines in 2d crystals

Lattice

  • Infinite number of possible planes
  • Can be through lattice points or between

lattice points

a1 a2

φ

Basis

(22) (11) (01) (14) (02)

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Physics 460 F 2006 Lect 2a 15

Schematic illustrations of lattice planes Lines in 2d crystals

Lattice

(01)

a1 a2

φ (14)

  • Equivalent parallel planes
  • Low index planes: more dense, more widely

spaced

  • High index planes: less dense, more closely

spaced Basis

(02)

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Physics 460 F 2006 Lect 2a 16

Schematic illustrations of lattice planes Lines in 2d crystals

a1 a2

(01) (02) (14)

  • Planes “slice through” the basis of physical

atoms

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Physics 460 F 2006 Lect 2a 17

Lattice planes in cubic crystals

X y z X z y

(100) and (110) planes in a cubic lattice (illustrated for the fcc lattice)

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Physics 460 F 2006 Lect 2a 18

(111) lattice planes in cubic crystals

X y z

Face Centered Cubic Lattice Lattice planes perpendicular to [111] direction Each plane is hexagonal close packed array of points

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Physics 460 F 2006 Lect 2a 19

Stacking hexagonal 2d layers to make close packed 3-d crystal

  • Each sphere has 12 equal neighbors
  • 6 in plane, 3 above, 3 below
  • Close packing for spheres
  • Can stack each layer in one of two ways, B
  • r C above A
  • Also see figure in Kittel

A B C

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Physics 460 F 2006 Lect 2a 20

Stacking hexagonal 2d layers to make hexagonal close packed (hcp) 3-d crystal

  • Each sphere has 12 equal neighbors
  • Close packing for spheres
  • See figure in Kittel for stacking sequence
  • HCP is ABABAB….. Stacking
  • Basis of 2 atoms
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Physics 460 F 2006 Lect 2a 21

Hexagonal close packed

a1 a2 a3

A B A Hexagonal Bravais Lattice Two atoms per cell

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Physics 460 F 2006 Lect 2a 22

Stacking hexagonal 2d layers to make cubic close packed (ccp) 3-d crystal

Cube ABC stacking

  • Each sphere has 12 equal neighbors
  • Close packing for spheres
  • See figure in Kittel for stacking sequence
  • CCP is ABCABCABC….. Stacking
  • Basis of 1 atom
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Physics 460 F 2006 Lect 2a 23

Stacking hexagonal 2d layers to make cubic close packed (ccp) 3-d crystal

Cube [111] direction in cube

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Physics 460 F 2006 Lect 2a 24

Face Centered Cubic (fcc) Also called cubic closed packed (ccp)

X y z

Recall from before [111] direction in cube Each atom has 12 equal neighbors The figure at the right shows the face centered character

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Physics 460 F 2006 Lect 2a 25

(111) planes in an fcc crystal

X y z

C B A C B A ABCABC... stacking of hexagonal planes fi fcc crystal fcc is a close packed crtsal – cubic close packed - ccp [111] direction in cube [111] direction in cube

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Physics 460 F 2006 Lect 2a 26

More on stacking hexagonal 2d layers

A B C B A B A B A C A C B B B A A A CCP Other polytype HCP

  • Infinite number of ways to stack planes
  • Polytypes occur in some metals, some

compounds like silicon carbide (SiC)

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Physics 460 F 2006 Lect 2a 27

Cubic crystals with a basis

NaCl Structure with Face Centered Cubic Bravais Lattice

X y z

ZnS Structure with Face Centered Cubic Bravais Lattice C, Si, Ge form diamond structure with

  • nly one type of atom
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Physics 460 F 2006 Lect 2a 28

NaCl Structure with Face Centered Cubic Bravais Lattice

X y

NaCl Structure

z

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Physics 460 F 2006 Lect 2a 29

CsCl Structure

CsCl Structure Simple Cubic Bravais Lattice

X y z

a3 a2 a1

From http://www.ilpi.com/inorganic/structures/cscl/index.html

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Physics 460 F 2006 Lect 2a 30

Atomic planes in NaCl and ZnS crystals

(110) planes in NaCl crystal rows of the Na and Cl atoms

X y z

(110) plane in ZnS crystal zig-zag Zn-S chains of atoms

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Physics 460 F 2006 Lect 2a 31

(110) plane in diamond structure crystal

(100) plane in ZnS crystal zig-zag Zn-S chains of atoms (diamond if the two atoms are the same)

X y z

Calculated valence electron density in a (110) plane in a Si crystal (Cover of Physics Today, 1970)

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Physics 460 F 2006 Lect 2a 32

(111) planes in ZnS crystals

X y z

(111) planes in cubic ZnS crystal C B A C B A CCP

Note: ABAB... stacking gives hexagonal ZnS

Zn S [111] direction in cube Zn S Zn S Zn S Zn Zn S S

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Physics 460 F 2006 Lect 2a 33

A

Simple Cubic Bravais Lattice

B B B B B B B B O

Perovskite Structure ABO3

A atoms have 12 O neighbors B atoms have 6 closer O neighbors Many compounds form the perovskite structure, SrTiO3, BaTiO3, LaMnO3, . . .

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Physics 460 F 2006 Lect 2a 34

Symmetries of crystals in 3 dimensions

  • All Crystals can be classified by:
  • 7 Crystal systems (triclinic, monoclinic,
  • rthorhombic, tetragonal, cubic, hexagonal,

trigonal)

  • 14 Bravais Lattices (primitive, face-centered or

body-centered for each system – 14 of the 7x3 possibilities describe all Bravais lattices )

  • 32 Points groups (rotations, inversion, reflection)
  • See references in Kittel Ch 1, G. Burns, “Solid

State Physics”

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Physics 460 F 2006 Lect 2a 35

Is a crystal really different from a liquid?

Crystal Liquid

Yes – the crystal has “order” – different directions are different Other crucial differences? Yes – dislocations

Crystal with a “dislocation”

Example of a dislocation

  • a crystal with an extra plane
  • f atoms on the left
  • The dislocation can move but

it cannot disappear! Important for deformations, … See Kittel Ch. 20

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Physics 460 F 2006 Lect 2a 36

Next Time

  • Diffraction from crystals
  • Reciprocal lattice
  • Read Kittel Ch 2