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SOLID STATE PHYSICS Pre-requisite: Quantum Mechanics , Electricity & Magnetism I & II, Heat and Thermodynamics, Statistical Physics. Objectives: 1. To develop a basic knowledge of crystallography 2. To understand the x-ray diffraction in crystal investigation 3. To understand the binding forces in crystalline material 4. To develop the understanding of lattice dynamics and its uses in derivation of theories of specific heat 5. To understand the behavior of free electrons in metals and Fermi Energy. Course out lines Structure of Solids Lattices and basis, Symmetry operations, Fundamental types of lattice, Position and orientation of planes in crystals, Simple crystal structures, Atomic potential, space groups and binding forces. Crystal diffraction and reciprocal lattice: Diffraction of X-rays, Neutrons and electrons from crystals, Bragg’s law, Reciprocal lattice, Reciprocal lattice to sc, bcc, fcc, orthorhombic and hexagonal crystals, Laue method, rotating crystal method, Powder methods, Scattered wave amplitude, Ewald construction and Brillouin zone, Fourier analysis of the basis.

Lecture 1 SSP

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Page 1: Lecture 1 SSP

SOLID STATE PHYSICS

Pre-requisite:

Quantum Mechanics , Electricity & Magnetism I & II, Heat and Thermodynamics, Statistical Physics. Objectives:

1. To develop a basic knowledge of crystallography

2. To understand the x-ray diffraction in crystal investigation

3. To understand the binding forces in crystalline material

4. To develop the understanding of lattice dynamics and its uses in derivation of theories of specific heat

5. To understand the behavior of free electrons in metals and Fermi Energy.  Course out lines

Structure of Solids

Lattices and basis, Symmetry operations, Fundamental types of lattice, Position and orientation of planes in crystals, Simple crystal structures, Atomic potential, space groups and binding forces.

Crystal diffraction and reciprocal lattice:

Diffraction of X-rays, Neutrons and electrons from crystals, Bragg’s law, Reciprocal lattice, Reciprocal lattice to sc, bcc, fcc, orthorhombic and hexagonal crystals, Laue method, rotating crystal method, Powder methods, Scattered wave amplitude, Ewald construction and Brillouin zone, Fourier analysis of the basis.

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Phonons and Lattice Vibrations: Lattice heat capacity, classical model, Einstein model, Enumeration of normal models, Density of state in one, two and three dimensions, Debye model of heat capacity, Comparison with experimental results, Thermal conductivity and resistivity, Umklapp processes.

Recommended Books:

1. C. Kittle, Introduction to Solid State Physics, 7th Ed. By, Kohn Wiley, 1996.

2. N. M. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart & Winston, 1976,

3. S. R. Elliott, The Physics and Chemistry of Solids, Wiley, 1998.

4. M.A. Omar, Elementary Solid State Physics, Pearson Education 2000.

5. H.M. Rosenberg, The Solid State, 3rd Edition, Oxford Science Publications 1990.

6. M.A. Wahab, Solid State Physics, Narosa Publishing House, 1999.

7. G. Burns, High Temperature Superconductivity, An Introduction, Academic Press 1992.

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Lecture 1

• Solid State Physics

• Crystal, Lattice and Basis

• Position of lattice sites

• Weigner Seitz cell

Dr. Abdul Majid Sandhu, Department of Physics, University of Gujrat.

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Aim of Solid State Physics

Study of crystalline structures (Strctual, optical, electronic properties).

Study of electrons within crystals.

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Scope of Solid State Physics

Understanding the electrical properties of solids is right at the heart of modern society and technology.

The entire computer and electronics industry relies on tuning of a special class of material, the semiconductors, which lies right at the metal-insulator boundary. Solid state physics provide a background to understand what goes on in semiconductors.

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Types of Types of mattermatter

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Gases

Gases have atoms or molecules that do not bond to one another in a range of pressure, temperature and volume.

These molecules haven’t any particular order and move freely within a container.

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Liquids and Liquid Crystals

Similar to gases, liquids haven’t any atomic/molecular order and they assume the shape of the containers.

Applying low levels of thermal energy can easily break the existing weak bonds.

+

-

+

-

+

-

+

-

+

-

+

-

+

-

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Crytals

Solids consist of atoms or molecules executing thermal motion about an equilibrium position fixed at a point in space.

Solids can take the form of crystalline, polycrstalline, or amorphous materials.

Solids (at a given temperature, pressure, and volume) have stronger bonds between molecules and atoms than liquids.

Solids require more energy to break the bonds.

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Crystal Structure 10

SOLID MATERIALS

CRYSTALLINE POLYCRYSTALLINE AMORPHOUS(Non-crystalline)

Single Crystal

ELEMENTARY CRYSTALLOGRAPHYELEMENTARY CRYSTALLOGRAPHY

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Crystalline Solid

Crystalline Solid is the solid form of a substance in which the atoms or molecules are arranged in a definite, repeating pattern in three dimension.

Single crystals, ideally have a high degree of order, or regular geometric periodicity, throughout the entire volume of the material.

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Crystal Structure 12

Crystalline Solid

Single Crystal

Single Pyrite Crystal

AmorphousSolid

Single crystal has an atomic structure that repeats periodically across its whole volume. Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetry

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Polycrystalline Solid

PolycrystallinePyrite form

(Grain)

Polycrystal is a material made up of an aggregate of many small single crystals (also called crystallites or grains).

Polycrystalline material have a high degree of order over many atomic or molecular dimensions, called, grains which are separated from one another by grain boundaries.

The grains are usually 100 nm - 100 microns in diameter. Polycrystals with grains that are <10 nm in diameter are called nanocrystalline

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Amorphous Solid

Amorphous (Non-crystalline) Solid is composed of randomly orientated atoms , ions, or molecules that do not form defined patterns or lattice structures.

Amorphous materials have order only within a few atomic or molecular dimensions.

Amorphous materials do not have any long-range order, but they have varying degrees of short-range order.

Examples to amorphous materials include amorphous silicon, plastics, and glasses.

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Departure From Perfect Crystal

Strictly speaking, one cannot prepare a perfect crystal. For example, even the surface of a crystal is a kind of imperfection because the periodicity is interrupted there.

Another example concerns the thermal vibrations of the atoms around their equilibrium positions for any temperature T>0°K.

As a third example, actual crystal always contains some foreign atoms, i.e., impurities. These impurities spoils the perfect crystal structure.

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CRYSTALLOGRAPHY

What is crystallography?

The branch of science that deals with the geometric description of crystals and their internal arrangement.

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CRYSTAL LATTICE

In crystallography, only the geometrical properties of the crystal are of interest, therefore one replaces each atom by a geometrical point located at the equilibrium position of that atom.

Platinum Platinum surface Crystal lattice and structure of Platinum

(scanning tunneling microscope)

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An infinite array of points in space,

Each point has identical surroundings to all others.

Arrays are arranged exactly in a periodic manner.

Crystal Lattice

α

a

bCB ED

O A

y

x

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Crystal Structure 19

Crystal Structure

Crystal structure can be obtained by attaching atoms, groups of atoms or molecules which are called basis (motif) to the lattice sides of the lattice point.

Basis + Crystal Lattice = Crystal Structure

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Recipe of Crystal

20

Crystal structure = lattice + basis

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Crystal Structure 21

Crystal Lattice

Bravais Lattice (BL) Non-Bravais Lattice (non-BL)

All atoms are of the same kind All lattice points are equivalent

Atoms can be of different kind Some lattice points are not equivalentA combination of two or more BL

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A two-dimensional Bravais lattice with different choices for the basis

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Crystal structure

Don't mix up atoms with lattice points

Lattice points are infinitesimal points in space

Lattice points do not necessarily lie at the centre of atoms

Crystal Structure = Crystal Lattice + Basis

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Translational Lattice Vectors – 2D

A space lattice is a set of points such that a translation from any point in the lattice by a vector;

Rn = n1 a + n2 b

locates an exactly equivalent point, i.e. a point with the same environment as P . This is translational symmetry. The vectors a, b are known as lattice vectors.

P

Point D(n1, n2) = (0,2)

Point F (n1, n2) = (0,-1)

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Lattice Vectors – 3D

An ideal three dimensional crystal is described by 3 fundamental translation vectors a, b and c. If there is a lattice point represented by the position vector r, there is then also a lattice point represented by the position vector where u, v and w are arbitrary integers.

  r’ = r + u a + v b + w c      (1)

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Unit Cell in 2D

The smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal.

Sa

b

S

S

S

S

S

S

S

S

S

S

S

S

S

S

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Unit Cell in 2D

The smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal.

S

S

The choice of unit cell

is not unique.

a

Sb

S

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2D Unit Cell example -(NaCl)

We define lattice points ; these are points with identical environments

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Crystal Structure 29

Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same.

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Crystal Structure 30

This is also a unit cell - it doesn’t matter if you start from Na or Cl

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Crystal Structure 31

- or if you don’t start from an atom

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Crystal Structure 32

This is NOT a unit cell even though they are all the same - empty space is not allowed!

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Crystal Structure 33

In 2D, this IS a unit cellIn 3D, it is NOT

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Crystal Structure 34

Why can't the blue triangle be a unit cell?

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Crystal Structure 35

Unit Cell in 3D

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Crystal Structure 36

Unit Cell in 3D

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Crystal Structure 37

Three common Unit Cell in 3D

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Crystal Structure 38

UNIT CELL

Primitive Conventional & Non-primitive

Single lattice point per cell Smallest area in 2D, orSmallest volume in 3D

More than one lattice point per cell Integral multibles of the area of primitive cell

Body centered cubic(bcc)Body centered cubic(bcc)

Conventional Conventional ≠ Primitive cell≠ Primitive cellSimple cubic(sc)Simple cubic(sc)

ConventionalConventional = Primitive cell = Primitive cell

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Crystal Structure 39

Wigner-Seitz Method

A simply way to find the primitivecell which is called Wigner-Seitzcell can be done as follows;

1. Choose a lattice point.2. Draw lines to connect these

lattice point to its neighbours.3. At the mid-point and normal to

these lines draw new lines.

The volume enclosed is called as a Wigner-Seitz cell.

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Crystal Structure 40

Wigner-Seitz Cell - 3D

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Crystal Structure 41

Lattice Sites in Cubic Unit Cell