Carbon Nanomaterials and Technology

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

2. LECTURE - 2 Basics: Crystal Structure

3. The(Common) Phases of Matter This doesn’t include Plasmas, but these are the “common” phases!! “Condensed Matter”includesboth of these.We’ll focus onSolids!

4. Solids • Solids consist of atoms or moleculesundergoing thermal motionabout their equilibrium positions, which are at fixed pointsin space. • Solids can be crystalline, polycrystalline,or amorphous. • Solids(at a given temperature, pressure, volume) have stronger interatomic bondsthan liquids. • So,Solidsrequire more energy to break the interatomic bondsthan liquids.

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

6. Periodic Arrays of Atoms Experimental Evidenceof 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. Cleaving a Crystal

7. Elementary Crystallography

8. The Three General Types of Solids • Single Crystal • Polycrystalline • Amorphous • Each type is characterized by the size of the orderedregion within the material. • An ordered region is a spatial volume in which atoms or molecules have a regular geometric arrangement or periodicity.

9. Crystalline Solids • A Crystalline Solidis the solid form of a substance in which the atoms or moleculesare 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.

10. A Single Crystalhas 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. SinglePyriteCrystal AmorphousSolid Single Crystals

11. Polycrystalline Solids • A Polycrystalline Solidis made up of an aggregate of many small single crystals(crystallites or grains). Polycrystalline materialshave a high degree of order over many atomic or moleculardimensions.These ordered regions, or single crystal regions, vary in size & orientationwith respect to one another.These regions are called grains(or domains)& are separated from one another by grain boundaries. • The atomic ordercan vary from one domain to the next.The grains are usually 100 nm - 100 microns in diameter. Polycrystals with grains that are < 10 nm in diameter are called nanocrystallites. Polycrystalline PyriteGrain

12. Amorphous Solids • Amorphous (Non-crystalline) Solidsare composed of randomly orientated atoms, ions, or molecules that do not form defined patterns or lattice structures.Amorphous materialshave 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 materialinclude amorphous silicon, plastics, & glasses.

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.

14. Crystallography Crystallography ≡The branch of science that deals with the geometric descriptionof 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.

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.

16. Crystal Lattice Crystallography focuses on the geometric properties of crystals. So, we imagine each atom replaced by a mathematicalpoint 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! Platinum Surface (Scanning Tunneling Microscope) Crystal Lattice Structure of Platinum Platinum

17. y B C D E α b O x a A 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  

18. 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

19. The structure of an Ideal Crystalcan 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

20. Crystalline Periodicity • In a crystalline material, the equilibrium positions of all the atoms form a crystal Crystal Structure ≡ Lattice + Basis Lattice 2 Atom Basis Crystal Structure

21. Crystalline Periodicity Crystal Structure ≡ Lattice + Basis For another example, see the figure. Crystal Structure  Lattice  Basis 

22. Crystalline Periodicity Crystal Structure ≡ Lattice + Basis For another example, see the figure. Basis  Crystal Structure  Lattice

23. A Two-Dimensional (Bravais) Lattice with Different Choices for the Basis

24. 2 Dimensional Lattice y y B C D B C D E α b F G b x O x O a A a A Lattice with atoms at the corners of regularhexagons E H

25. Crystal Structure = Lattice + Basis The atoms do not necessarily lie at lattice points!! Basis   Crystal Structure

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.

28. Honeycomb net: Bravais lattice with two point basis

29. Translation Vector T Translation(a1,a2), Nontranslation Vectors(a1’’’,a2’’’)

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.

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) • 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. Will contain lattice points only Lattice Unit cell of a Will contain entities which decorate the lattice Crystal * 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. 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 Contributions to the unit cell: Left point = 0.5, Middle point = 1, Right point = 0.5. Total = 2 • 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 Doubly Non-primitive UC Triply Non-primitive UC

39. 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 b  a • Unit cell in 2D is described by 3 lattice parameters: a, b,  • 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. 2D UC-1 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. 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