1. Carbon Nanomaterials and Technology
EE 6909
Dr. Md. Sherajul Islam
Assistant Professor
Department of Electrical and Electronics Engineering Khulna University of Engineering & Technology
Khulna, Bangladesh

2. Basics: Crystal Structure
LECTURE - 2
Basics: Crystal Structure
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

3. The (Common) Phases of Matter
Gases
Liquids & Liquid Crystals
Solids
This doesn’t include Plasmas, but these are the “common” phases!!
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).
“Condensed Matter” includes both of these. We’ll focus on Solids!

4. Solids
Solids consist of atoms or molecules undergoing thermal motion about their equilibrium positions, which are at fixed points in space.
Solids can be crystalline, polycrystalline, or amorphous.
Solids (at a given temperature, pressure, volume) have stronger interatomic bonds than liquids.
So, Solids require more energy to break the interatomic bonds than liquids.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

5. Crystal Structure Topics 1. Periodic Arrays of Atoms
2. Fundamental Types of Lattices
3. Index System for Crystal Planes
4. Simple Crystal Structures
5. Direct Imaging of Crystal Structure
6. Non-ideal Crystal Structures
7. Crystal Structure Data
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

6. Periodic Arrays of Atoms
Experimental Evidence of periodic structures.
(See Kittel, Fig. 1.)
The external appearance of crystals gives some clues to this. Fig. 1 shows that when a crystal is cleaved, we can see that it is built up of identical “building blocks”. Further, the early crystallographers noted that the index numbers that define plane orientations are exact integers.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).
Cleaving a Crystal

7. Elementary Crystallography
Solid Material Types
Crystalline
Polycrystalline
Amorphous
Single Crystals
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

8. The Three General Types of Solids
Single Crystal
Polycrystalline
Amorphous
Each type is characterized by the size of the ordered region within the material.
An ordered region is a spatial volume in which atoms or molecules have a regular geometric arrangement or periodicity.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

9. Crystalline Solids
A Crystalline Solid is the solid form of a substance in which the atoms or molecules are arranged in a definite, repeating pattern in three dimensions.
Single Crystals, ideally have a high degree of order, or regular geometric periodicity, throughout the entire volume of the material.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

10. A 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.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).
Single Pyrite Crystal
Amorphous Solid
Single Crystals

11. Polycrystalline Solids
A Polycrystalline Solid is made up of an aggregate of many small single crystals (crystallites or grains). Polycrystalline materials have a high degree of order over many atomic or molecular dimensions. These ordered regions, or single crystal regions, vary in size & orientation with respect to one another. These regions are called grains (or domains) & are separated from one another by grain boundaries.
The atomic order can vary from one domain to the next. The grains are usually 100 nm microns in diameter. Polycrystals with grains that are < 10 nm in diameter are called nanocrystallites.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).
Polycrystalline
Pyrite Grain

12. Amorphous Solids
Amorphous (Non-crystalline) Solids are 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. They do not have any long-range order, but they have varying degrees of short-range order. Examples of amorphous material include amorphous silicon, plastics, & glasses.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

13. Departures From the “Perfect Crystal”
A “Perfect Crystal” is an idealization that does not exist in nature. In some ways, even a crystal surface is an imperfection, because the periodicity is interrupted there.
Each atom undergoes thermal vibrations around their equilibrium positions for temperatures T > 0K. These can also be viewed as “imperfections”.
Real Crystals always have foreign atoms (impurities), missing atoms (vacancies), & atoms in between lattice sites (interstitials) where they should not be. Each of these spoils the perfect crystal structure.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

14. Crystallography
Crystallography ≡ The branch of science that deals with the geometric description of crystals & their internal arrangements. It is the science of crystals & the math used to describe them. It is a VERY OLD field which pre-dates Solid State Physics by about a century! So (unfortunately, in some ways) much of the terminology (& theory notation) of Solid State Physics originated in crystallography.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

15. Crystallography A Basic Knowledge of Elementary
Crystallography is Essential
for Solid State Physicists!!!
A crystal’s symmetry has a profound influence on many of its properties.
A crystal structure should be specified completely, concisely & unambiguously.
Structures are classified into different types according to the symmetries they possess.
In this course, we only consider solids with “simple” structures.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

16. (Scanning Tunneling Microscope)
Crystal Lattice
Crystallography focuses on the geometric properties of crystals. So, we imagine each atom replaced by a mathematical point at the equilibrium position of that atom. A Crystal Lattice (or a Crystal) ≡ An idealized description of the geometry of a crystalline material. A Crystal ≡ A 3- dimensional periodic array of atoms. Usually, we’ll only consider ideal crystals. “Ideal” means one with no defects, as already mentioned. That is, no missing atoms, no atoms off of the lattice sites where we expect them to be, no impurities,…Clearly, such an ideal crystal never occurs in nature. Yet, 85-90% of experimental observations on crystalline materials is accounted for by considering only ideal crystals!
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).
Platinum Surface
(Scanning Tunneling Microscope)
Crystal Lattice
Structure of Platinum
Platinum

17. A Lattice is Defined as an Infinite Array of Points in Space
Crystal Lattice
Mathematically
A Lattice is Defined as an Infinite Array of Points in Space
in which each point has
identical surroundings
to all others. The points
are arranged exactly
in a periodic manner.
2 Dimensional Example  α a b C B E D O A y x
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

18. The simplest structural unit for a given solid is called the BASIS
Ideal Crystal ≡ An infinite periodic repetition of identical structural units in space.
The simplest structural unit we can imagine is a Single Atom. This corresponds to a solid made up of only one kind of atom ≡
An Elemental Solid.
However, this structural unit could also be a group of several atoms or even molecules.
The simplest structural unit for a given solid is called the BASIS
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

19. Crystal Structure ≡ Lattice + Basis
The structure of an Ideal Crystal can be described in terms of a mathematical construction called a Lattice.
A Lattice ≡
A 3-dimensional periodic array of points in space. For a particular solid, the smallest structural unit, which when repeated for every point in the lattice is called the Basis.
The Crystal Structure is defined once both the lattice & the basis are specified. That is
Crystal Structure ≡ Lattice + Basis
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

20. Crystalline Periodicity
In a crystalline material, the equilibrium positions of all the atoms form a crystal
Crystal Structure ≡ Lattice + Basis
Lattice 
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).
2 Atom Basis 
Crystal 
Structure

21. Crystalline Periodicity
Crystal Structure ≡ Lattice + Basis
For another example, see the figure.
Crystal Structure 
Lattice 
Basis 
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

22. Crystalline Periodicity
Crystal Structure ≡ Lattice + Basis
For another example, see the figure.
Basis 
Crystal Structure 
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).
Lattice 

23. A Two-Dimensional (Bravais) Lattice with Different Choices for the Basis
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

24. Lattice with atoms at the corners
2 Dimensional Lattice
Lattice with atoms at the corners
of regular hexagons α a b C B E D O A y x O A C B F b G D x y a E H
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

25. Crystal Structure = Lattice + Basis
The atoms do not necessarily lie at lattice points!!
Crystal Structure = Lattice + Basis
Basis  
Crystal
Structure
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

26. Bravais Lattice
An infinite array of discrete points with an arrangement and orientation that appears exactly the same, from any of the points the array is viewed from.
A three dimensional Bravais lattice consists of all points with position vectors R that can be written as a linear combination of primitive vectors. The expansion coefficients must be integers.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

27. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

28. Honeycomb net: Bravais lattice with two point basis
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

29. Translation(a1,a2), Nontranslation Vectors(a1’’’,a2’’’)
Translation Vector T
Translation(a1,a2), Nontranslation Vectors(a1’’’,a2’’’)
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

30. Primitive Unit Cell
A primitive cell or primitive unit cell is a volume of space that when translated through all the vectors in a Bravais lattice just fills all of space without either overlapping itself or leaving voids.
A primitive cell must contain precisely one lattice point.
The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines nanotechnology as “the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge (the nanoscale is the size range from approximately 1 nm to 100 nm).

31. Fundamental Types of Lattices
Crystal lattices can be mapped into themselves by the lattice translations T and by various other symmetry operations.
A typical symmetry operation is that of rotation about an axis that passes through a lattice point. Allowed rotations of : 2 π, 2π/2, 2π/3,2π/4, 2π/6
(Note: lattices do not have rotation axes for 1/5, 1/7 …) times 2π

32. Five fold axis of symmetry cannot exist

33. Two Dimensional Lattices
There is an unlimited number of possible lattices, since there is no restriction on the lengths of the lattice translation vectors or on the angle between them. An oblique lattice has arbitrary a1 and a2 and is invariant only under rotation of π and 2 π about any lattice point.

34. UNIT CELLS (UC) Unit cell Lattice of a Crystal
A unit cell (also sometimes causally referred to as a cell) is a representative unit of the structure.  which when translationally repeated (by the basis vector(s)) gives the whole structure.
The term unit should not be confused with ‘having one’ lattice point or motif (The term primitive or sometimes simple is reserved for that).
If the structure is a lattice, the unit cell will be unit of that (hence will have points* only).
If the structure under considerations is a crystal, then the unit cell will also contain atoms (or ions or molecules etc.).  Note: Instead of full atoms (or other units) only a part of the entity may be present in the unit cell (a single unit cell)
The dimension of the unit cell will match the dimension of the structure**:  If the lattice is 1D  the unit cell will be 1D, if the crystal is 3D  then the unit cell will be 3D, if the lattice is nD  the unit cell will be nD.
Lattice
Will contain lattice points only
Unit cell
of a
Crystal
Will contain entities which decorate the lattice
* Strangely in crystallography often we even ‘split a point’ (and say that 1/8th belongs to the UC).
** One can envisage other possibilities– e.g. a 2D motif may be repeated only along one direction (i.e. the crystal is 3D but the repeat direction is along 1D)

36. Consider an infinite pattern made of squares
Why Unit Cells?
Instead of drawing the whole structure I can draw a representative part and specify the repetition pattern.
ADDITIONAL POINTS
A cell is a finite representation of the infinite lattice/crystal
A cell is a line segment (1D) or a parallelogram (2D) or a parallelopiped (3D) with lattice points at their corners  This is the convention
If the lattice points are only at the corners, the cell is primitive. Hence, a primitive unit cell is one, wherein the lattice points are only at the corners of the unit cell (or the ends of a line segment unit cell in 1D)
If there are lattice points in the cell other than the corners, the cell is non-primitive. 
Consider an infinite pattern made of squares  
This way the infinite information content of a crystal can be reduced to the information required to specify the contents of a unit cell (along with the lattice translation vectors). 
This can be thought of as a single square repeated in x and y directions

37. In general the following types of unit cells can be defined:
Primitive unit cell
Non-primitive unit cells
Voronoi cells
Wigner-Seitz cells

38. 1D
Unit cell of a 1D lattice is a line segment of length = the lattice parameter  this is the PRIMITIVE UNIT CELL (i.e. has one lattice point per cell).
Each of these lattice points contributes half a lattice point to the unit cell
Primitive UC
Contributions to the unit cell: Left point = 0.5, Middle point = 1, Right point = 0.5. Total = 2
Doubly Non-primitive UC
Triply Non-primitive UC

39. Though the whole lattice point is shown only half belongs to the UC
Unit cell of a 1D crystal will contain Motifs in addition to lattice points
NOTE:  The only kind of motifs possible in 1D are line segments  Hence in ‘reality’ 1D crystals are not possible as Motifs typically have a finite dimension (however we shall call them 1D crystals and use them for illustration of concepts)  We could have 2D or 3D motifs repeated along 1D (hence periodicity and ‘crystallinity’ is only along 1D)
Correct unit cell
Though the whole lattice point is shown only half belongs to the UC
Each of these atoms contributes ‘half-atom’ to the unit cell
Though this is the correct unit cell
Often unit cells will be drawn like this
Unit cell in 1D is described by 1 (one) lattice parameter: a

40. 2D Unit cell in 2D is described by 3 lattice parameters: a, b,  b  a
Special cases include: a = b;  = 90 or 120
Unit Cell shapes in 2D  Lattice parameters  Square  (a = b,  = 90)  Rectangle  (a, b,  = 90)  120 Rhombus  (a = b,  = 120)  Parallelogram (general)  (a, b, )

41. Note: these are the basis vectors (and included angle) for UC-1 above
Note: basis vectors (& included angle) will change based on the ‘unit cell’ chosen [which implies that lattice parameters will change as well !]
90
Rectangular
lattice
Note: these are the basis vectors (and included angle) for UC-1 above
90
Note: Symmetry of the Lattice or the crystal is not altered by our choice of unit cell!!

42. Different kinds of CELLS
Unit cell
A unit cell is a spatial arrangement of atoms which is tiled in three-dimensional space to describe the crystal.
Primitive unit cell
For each crystal structure there is a conventional unit cell, usually chosen to make the resulting lattice as symmetric as possible. However, the conventional unit cell is not always the smallest possible choice. A primitive unit cell of a particular crystal structure is the smallest possible unit cell one can construct such that, when tiled, it completely fills space.
Wigner-Seitz cell
A Wigner-Seitz cell is a particular kind of primitive cell, which has the same symmetry as the lattice.

43. 1D 2D Q & A How many ‘shapes’ of primitive unit cells are possible?
1D → one.
2D, 3D → Infinite (few examples in 2D given below). 1D 2D

44. Wigner-Seitz Cell
Is a primitive unit cell with the symmetry of the lattice
Created by Voronoi tessellation of space
The region enclosed by the Wigner-Seitz cell is closer to a given lattice point than to any other lattice point
Square lattice
Centred Rectangular lattice
Wigner-Seitz cells