Introduction to Crystal Structure and Bonding. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

Transcription

1 Introduction to Crystal Structure and Bonding 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

2 Fundamental Properties of matter 2 Matter: - Has mass, occupies space Mass measure of inertia - from Newton s first law of motion. It is one of the fundamental physical properties. States of Matter 1. Solids Definite volume, definite shape. 2. Liquids Definite volume, no fixed shape. Flows. 3. Gases No definite volume, no definite shape. Takes the volume and shape of its container.

3 Element one type of atoms Compound Two or more different atoms chemically joined. Constituent atoms (fixed ratios) can be separated only by chemical means. Mixture - Two or more different atoms combined. Constituent atoms (variable ratios) can be separated by physical means. 3 Solid-State Physics branch of physics dealing with solids. Now replaced by a more general terminology - Condensed Matter Physics. To include fluids which in many cases share same concepts and analytical techniques.

4 Classification Of Solids 4 Can be classified under several criteria based on atomic arrangements, electrical properties, thermal properties, chemical bonds etc. Using electrical criterion: Conductors, Insulators, Semiconductors Using atomic arrangements: Amorphous, Polycrystalline, Crystalline.

5 Atoms And Bonding 5 The periodic table Ionic bonding Covalent bonding Metallic bonding van der Waals bonding

6 Atoms and bonding 6 In order to understand the physics of semiconductor devices, we should first learn how atoms bond together to form the solids. Atom is composed of a nucleus which contains protons and neutrons; surrounding the nucleus are the electrons. Atoms can combine with themselves or other atoms. The valence electrons, i.e. the outermost shell electrons govern the chemistry of atoms. Atoms come together and form gases, liquids or solids depending on the strength of the attractive forces between them. The atomic bonding can be classified as ionic, covalent, metallic, van der Waals,etc. In all types of bonding the electrostatic force acts between charged particles.

7 The Periodic Table 7 1A 2A 8A Li Be 3A 4A 5A 6A 7A He Na Mg B C N O F Ne K Ca 2B Al Si P S Cl Ar Rb Sr Zn Ga Ge As Se Br Kr Cs Ba Cd In Sn Sb Te I Xe Fr Rd Hg Ti Pb Bi Po At Rn Groups 3B,4B,5B,6B 7B,8B,1B lie in here A section of the periodic table

8 The Periodic Table 8 Ionic solids Group 1A (alkali metals) contains lithium (Li), sodium (Na), potassium (K),..and these combine easily with group 7A (halogens) of fluorine (F), chlorine (Cl), bromine (Br),.. and produce ionic solids of NaCl, KCl, KBr, etc. Rare (noble) gases Group 8A elements of noble gases of helium(he), neon (Ne), argon (Ar), have a full complement of valence electrons and so do not combine easily with other elements. Elemental semiconductors Silicon(Si) and germanium (Ge) belong to group 4A. Compound semiconductors 1) III-V compound s/c s; GaP, InAs, AlGaAs (group 3A-5A) 2) II-VI compound s/c s; ZnS, CdS, etc. (group 2B-6A)

9 Covalent bonding 9 Elemental semiconductors of Si, Ge and diamond are bonded by this mechanism and these are purely covalent. The bonding is due to the sharing of electrons. Covalently bonded solids are hard, high melting points, and insoluble in all ordinary solids. Compound semiconductors exhibit a mixture of both ionic and covalent bonding.

10 Ionic bonding 10 Ionic bonding is due to the electrostatic force of attraction between positively and negatively charged ions (between 1Å and 7Å). This process leads to electron transfer and formation of charged ions; a positively charged ion for the atom that has lost the electron and a negatively charged ion for the atom that has gained an electron. All ionic compounds are crystalline solids at room temperature. NaCl and CsCl are typical examples of ionic bonding. Ionic crystals are hard, high melting point, brittle and can be dissolved in ordinary liquids.

11 Ionic bonding 11 The metallic elements have only up to the valence electrons in their outer shell will lose their electrons and become positive ions, whereas electronegative elements tend to acquire additional electrons to complete their octed and become negative ions, or anions. Na Cl

12 Comparison of Ionic and Covalent Bonding 12

13 Potential energy diagram for molecules This typical curve has a minimum at equilibrium distance R 0 R > R 0 ; the potential increases gradually, approaching 0 as R the force is attractive V(R) 0 R0 Repulsive 13 R R < R 0 ; the potential increases very rapidly, approaching at small radius. the force is repulsive r R Attractive

14 Metallic bonding 14 Valance electrons are relatively bound to the nucleus and therefore they move freely through the metal and they are spread out among the atoms in the form of a lowdensity electron cloud. A metallic bond result from the sharing of a variable number of electrons by a variable number of atoms. A metal may be described as a cloud of free electrons Therefore, metals have high electrical and thermal conductivity

15 Metallic bonding 15 All valence electrons in a metal combine to form a sea of electrons that move freely between the atom cores. The more electrons, the stronger the attraction. This means the melting and boiling points are higher, and the metal is stronger and harder. The positively charged cores are held together by these negatively charged electrons. The free electrons act as the bond (or as a glue ) between the positively charged ions. This type of bonding is nondirectional and is rather insensitive to structure. As a result we have a high ductility of metals - the bonds do not break when atoms are rearranged metals can experience a significant degree of plastic deformation.

16 van der Waals bonding 16 It is the weakest bonding mechanism. It occurs between neutral atoms and molecules. The explanation of these weak forces of attraction is that there are natural fluctuation in the electron density of all molecules and these cause small temporary dipoles within the molecules. It is these temporary dipoles that attract one molecule to another. They are as called van der Waals' forces. Such a weak bonding results low melting and boiling points and little mechanical strength.

17 van der Waals bonding 17 The dipoles can be formed as a result of unbalanced distribution of electrons in asymmetrical molecules. This is caused by the instantaneous location of a few more electrons on one side of the nucleus than on the other. symmetric asymmetric Therefore atoms or molecules containing dipoles are attracted to each other by electrostatic forces.

18 Classification of solids 18 SOLID MATERIALS CRYSTALLINE POLYCRYSTALLINE AMORPHOUS (Non-crystalline) Single Crystal

19 Crystalline Solids 19 Atoms arranged in a 3-D long range order. Single crystals emphasizes one type of crystal order that exists as opposed to polycrystals.

20 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.

21 Crystalline Solid 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 Single Pyrite Crystal Single Crystal Amorphous Solid

22 Polycrystalline Solids 22 Atomic order present in sections (grains) of the solid. Different order of arrangement from grain to grain. Grain sizes = hundreds of m. An aggregate of a large number of small crystals or grains in which the structure is regular, but the crystals or grains are arranged in a random fashion.

23 Polycrystalline Solids 23

24 Polycrystalline Solid Polycrystal is a material made up of an aggregate of many small single crystals (also called crystallites or grains). The grains are usually 100 nm microns in diameter. Polycrystals with grains that are <10 nm in diameter are called nanocrystalline Polycrystal Polycrystalline Pyrite form (Grain) 24

25 Amorphous Solids 25 No regular long range order of arrangement in the atoms. Eg. Polymers, cotton candy, common window glass, ceramic. Can be prepared by rapidly cooling molten material. Rapid minimizes time for atoms to pack into a more thermodynamically favorable crystalline state. Two sub-states of amorphous solids: Rubbery and Glassy states. Glass transition temperature Tg = temperature above which the solid transforms from glassy to rubbery state, becoming more viscous.

26 Amorphous Solids 26 Continuous random network structure of atoms in an amorphous solid

27 Amorphous Solid Amorphous (non-crystalline) Solid is composed of randomly orientated atoms, ions, or molecules that do not form defined patterns or lattice structures. 27

28 Single- Vs Poly- Crystal 28 Properties of single crystalline materials vary with direction, ie anisotropic. Properties of polycrystalline materials may or may not vary with direction. If the polycrystal grains are randomly oriented, properties will not vary with direction i.e isotropic. If the polycrystal grains are textured, properties will vary with direction i.e anisotropic

29 Single- Vs Poly- Crystal 29

30 Single- Vs Poly- Crystal m -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic.

31 31 Solid state devices employ semiconductor materials in all of the above forms. Examples: Amorphous silicon (a-si) used to make thin film transistors (TFTs) used as switching elements in LCDs. Ploycrystalline Si Gate materials in MOSFETS. Active regions of most solid state devices are made of crystalline semiconductors.

32 Crystalline Solids: Short-range Order Long-range Order Solids Amorphous solids: ~Short-range Order No Long-range Order 32

33 Crystals The periodic array of atoms, ions, or molecules that form the solids is called Crystal Structure Crystal Structure = Space (Crystal) Lattice + Basis Space (Crystal) Lattice is a regular periodic arrangement of points in space, and is purely mathematical abstraction Crystal Structure is formed by putting the identical atoms (group of atoms) in the points of the space lattice This group of atoms is the Basis 33

34 Hard Sphere Model of Crystals 34 Assumes atoms are hard spheres with well defined diameters that touch. Atoms are arranged on periodic array or lattice Repetitive pattern unit cell defined by lattice parameters comprising lengths of the 3 sides (a, b, c) and angles between the sides (,, ).

35 35 Lattice Parameters a c b

36 Atoms in a Crystal 36

37 The Unit Cell Concept 37 The simplest repeating unit in a crystal is called a unit cell. Opposite faces of a unit cell are parallel. The edge of the unit cell connects equivalent points. Not unique. There can be several unit cells of a crystal. The smallest possible unit cell is called primitive unit cell of a particular crystal structure. A primitive unit cell whose symmetry matches the lattice symmetry is called Wigner-Seitz cell.

38 38 Each unit cell is defined in terms of lattice points. Lattice point not necessarily at an atomic site. For each crystal structure, a conventional unit cell, is chosen to make the lattice as symmetric as possible. However, the conventional unit cell is not always the primitive unit cell. A crystal's structure and symmetry play a role in determining many of its properties, such as cleavage (tendency to split along certain planes with smooth surfaces), electronic band structure and optical properties.

39 39 Unit cell

40 40 b a Unit cell: Simplest portion of the structure which is repeated and shows its full symmetry. Basis vectors a and b defines relationship between a unit cell and (Bravais) lattice points of a crystal. Equivalent points of the lattice is defined by translation vector. r = ha + kb where h and k are integers. This constructs the entire lattice.

41 41 By repeated duplication, a unit cell should reproduce the whole crystal. A Bravias lattice (unit cells) - a set of points constructed by translating a single point in discrete steps by a set of basis vectors. In 3-D, there are 14 unique Bravais lattices. All crystalline materials fit in one of these arrangements. In 3-D, the translation vector is r = ha + kb + lc

42 Crystal System 42 The crystal system: Set of rotation and reflection symmetries which leave a lattice point fixed. There are seven unique crystal systems: the cubic (isometric), hexagonal, tetragonal, rhombohedral (trigonal), orthorhombic, monoclinic and triclinic.

43 Bravais Lattice and Crystal System 43 Crystal structure: contains atoms at every lattice point. The symmetry of the crystal can be more complicated than the symmetry of the lattice. Bravais lattice points do not necessarily correspond to real atomic sites in a crystal. A Bravais lattice point may be used to represent a group of many atoms of a real crystal. This means more ways of arranging atoms in a crystal lattice.

44 44

45 1. Cubic (Isometric) System 45 3 Bravais lattices Symmetry elements: Four 3-fold rotation axes along cube diagonals a = b = c = = = 90 o a c b

46 By convention, the edge of a unit cell always connects equivalent points. Each of the eight corners of the unit cell therefore must contain an identical particle. 46

47 (1-a): Simple Cubic Structure (SC) 47 Rare due to poor packing (only Po has this structure) Close-packed directions are cube edges. Coordination # = 6 (# nearest neighbors) 1 atom/unit cell

48 48 Coordination Number = Number of nearest neighbors

49 49 One atom per unit cell 1/8 x 8 = 1

50 Atomic Packing Factor 50

51 (1-b): Face Centered Cubic Structure (FCC) 51 Exhibited by Al, Cu, Au, Ag, Ni, Pt Close packed directions are face diagonals. Coordination number = 12 4 atoms/unit cell All atoms are identical Adapted from Fig. 3.1(a), Callister 6e. 6 x (1/2 face) + 8 x 1/8 (corner) = 4 atoms/unit cell

52 52 FCC Coordination number = 12 3 mutually perpendicular planes. 4 nearest neighbors on each of the three planes.

53 53 How is a and R related for an FCC? [a= unit cell dimension, R = atomic radius]. All atoms are identical

54 (1-c): Body Centered Cubic Structure (BCC) 54 Exhibited by Cr, Fe, Mo, Ta, W Close packed directions are cube diagonals. Coordination number = 8 All atoms are identical 2 atoms/unit cell

55 55 How is a and R related for an BCC? [a= unit cell dimension, R = atomic radius]. All atoms are identical 2 atoms/unit cell

56 56 Which one has most packing?

57 57 Which one has most packing? For that reason, FCC is also referred to as cubic closed packed (CCP)

58 2. Hexagonal System 58 Only one Bravais lattice Symmetry element: One 6-fold rotation axis a = b c = 120 o = = 90 o

59 Hexagonal Closed Packed Structure (HCP) 59 Exhibited by. ABAB... Stacking Sequence Coordination # = 12 2D Projection 3D Projection A sites B sites A sites Adapted from Fig. 3.3, Callister 6e.

60 3. Tetragonal System 60 Two Bravais lattices Symmetry element: One 4-fold rotation axis a = b c = = = 90 o

61 4. Trigonal (Rhombohedral) System 61 One Bravais lattice Symmetry element: One 3-fold rotation axis a = b c = 120 o = = 90 o

62 5. Orthorhombic System 62 Four Bravais lattices Symmetry element: Three mutually perpendicular 2- fold rotation axes a b c = = = 90o

63 6. Monoclinic System 63 Two Bravais lattices Symmetry element: One 2-fold rotation axis a b c = = 90 o, 90 o

64 7. Triclinic System 64 One Bravais lattice Symmetry element: None a b c 90 o

65 65 The crystal system: Set of symmetries which leave a lattice point fixed. There are seven unique crystal systems. Some symmetries are identified by special name such as zincblende, wurtzite, zinc sulfide etc.

66 Layer Stacking Sequence 66 HCP A sites B sites A sites = ABAB FCC = ABCABC..

67 FCC: Coordination number 67 FCC Coordination number = 12 3 mutually perpendicular planes. 4 nearest neighbors on each of the three planes.

68 Diamond Lattice Structure 68 Exhibited by Carbon (C), Silicon (Si) and Germanium (Ge). Consists of two interpenetrating FCC lattices, displaced along the body diagonal of the cubic cell by 1/4 the length of the diagonal. Also regarded as an FCC lattice with two atoms per lattice site: one centered on the lattice site, and the other at a distance of a/4 along all axes, ie an FCC lattice with the twopoint basis.

69 Diamond Lattice Structure 69 a = lattice constant

70 Diamond Lattice Structure 70

71 Two merged FCC cells offset by a/4 in x, y and z. 71 Basic FCC Cell Merged FCC Cells Omit atoms outside Cell Bonding of Atoms

72 72 8 atoms at each corner, 6 atoms on each face, 4 atoms entirely inside the cell

73 73 Zinc Blende Similar to the diamond cubic structure except that the two atoms at each lattice site are different. Exhibited by many semiconductors including ZnS, GaAs, ZnTe and CdTe. GaN and SiC can also crystallize in this structure.

74 74 Zinc Blende Each Zn bonded to 4 Sulfur - tetrahedral Equivalent if Zn and S are reversed Bonding often highly covalent

75 75 Zinc sulfide crystallizes in two different forms: Wurtzite and Zinc Blende.

76 GaAs 76 Green = Ga-atoms, Blue = As-atoms Equal numbers of Ga and As ions distributed on a diamond lattice. Each atom has 4 of the opposite kind as nearest neighbors.

77 Wurtzite (Hexagonal) Structure 77 This is the hexagonal analog of the zinc-blende lattice. Can be considered as two interpenetrating close-packed lattices with half of the tetrahedral sites occupied by another kind of atoms. Four equidistant nearest neighbors, similar to a zincblende structure. Certain compound semiconductors (ZnS, CdS, SiC) can crystallize in both zinc-blende (cubic) and wurtzite (hexagonal) structure.

78 WURTZITE 78 A sites B sites A sites

79 Wurtzite Gallium Nitride (GaN) 79

80 Miller Index For Cubic Structures 80 Miller index is used to describe directions and planes in a crystal. Directions - written as [u v w] where the integers u, v, w represent coordinates of the vector in real space. A family of directions which are equivalent due to symmetry operations is written as <u v w> Planes: Written as (h k l). Integers h, k, and l represent the intercept of the plane with x-, y-, and z- axes, respectively. Equivalent planes represented by {h k l}.

81 Miller Indices: Directions z 81 y x y z [1] Draw a vector and take components 0 2a 2a [2] Reduce to simplest integers [3] Enclose the number in square brackets [0 1 1] x

82 Negative Directions z 82 y x y z [1] Draw a vector and take components 0 -a 2a [2] Reduce to simplest integers [3] Enclose the number in square brackets 01 2 x

83 Miller Indices: Equivalent Directions 83 Equivalent directions due to crystal symmetry: z 1: [100] 2: [010] 3: [001] 3 y x 1 2 Notation <100> used to denote all directions equivalent to [100]

84 84 Directions

85 Miller Index Step 1 : Identify the intercepts on the x-, y- and z- axes. Step 2 : Specify the intercepts in fractional co-ordinates Step 3 : Take the reciprocals of the fractional intercepts 85 (i) in some instances the Miller indices are best multiplied or divided through by a common number in order to simplify them by, for example, removing a common factor. This operation of multiplication simply generates a parallel plane which is at a different distance from the origin of the particular unit cell being considered. e.g. (200) is transformed to (100) by dividing through by 2. (ii) if any of the intercepts are at negative values on the axes then the negative sign will carry through into the Miller indices; in such cases the negative sign is actually denoted by overstriking the relevant number. e.g. (00-1) is instead denoted by 00 1

86 The intercepts of a crystal plane with the axis defined by a set of unit vectors are at 2a, -3b and 4c. Find the corresponding Miller indices of this and all other crystal planes parallel to this plane. 86 The Miller indices are obtained in the following three steps: 1. Identify the intersections with the axis, namely 2, -3 and Calculate the inverse of each of those intercepts, resulting in 1/2, -1/3 and 1/4. 3. Find the smallest integers proportional to the inverse of the intercepts. Multiplying each fraction with the product of each of the intercepts (24 = 2 x 3 x 4) does result in integers, but not always the smallest integers. 4. These are obtained in this case by multiplying each fraction by Resulting Miller indices is Negative index indicated by a bar on top.

87 Miller Indices of Planes z= z 87 y x=a x y= x y z [1] Determine intercept of plane with each axis a [2] Invert the intercept values 1/a 1/ 1/ [3] Convert to the smallest integers [4] Enclose the number in round brackets (1 0 0)

88 Miller Indices of Planes z 88 y x x y z [1] Determine intercept of plane with each axis 2a 2a 2a [2] Invert the intercept values 1/2a 1/2a 1/2a [3] Convert to the smallest integers [4] Enclose the number in round brackets (1 1 1)

89 Planes with Negative Indices z 89 y x x y z [1] Determine intercept of plane with each axis a -a a [2] Invert the intercept values 1/a -1/a 1/a [3] Convert to the smallest integers [4] Enclose the number in round brackets 111

90 Equivalent Planes z (001) plane 90 (010) plane (100) plane x y Planes (100), (010), (001), (100), (010), (001) are equivalent planes. Denoted by {1 0 0}. Atomic density and arrangement as well as electrical, optical, physical properties are also equivalent.

91 91

92 92 In the cubic system the (hkl) plane and the vector [hkl] are normal to one another. This characteristic is unique to the cubic crystal system and does not apply to crystal systems of lower symmetry.

93 93 The (110) surface Assignment Intercepts : a, a, Fractional intercepts : 1, 1, Miller Indices : (110) The (100), (110) and (111) surfaces considered are the so-called low index surfaces of a cubic crystal system (the "low" refers to the Miller indices being small numbers - 0 or 1 in this case).

94 Crystallographic Planes 94 Miller Indices (hkl) reciprocals

95 95 The (111) surface Assignment Intercepts : a, a, a Fractional intercepts : 1, 1, 1 Miller Indices : (111) The (210) surface Assignment Intercepts : ½ a, a, Fractional intercepts : ½, 1, Miller Indices : (210)

96 Symmetry-equivalent surfaces 96 The three highlighted surfaces are related by the symmetry elements of the cubic crystal - they are entirely equivalent. In fact there are a total of 6 faces related by the symmetry elements and equivalent to the (100) surface - any surface belonging to this set of symmetry related surfaces may be denoted by the more general notation {100} where the Miller indices of one of the surfaces is instead enclosed in curlybrackets.

97 Angle ( ) between directions [h1 k1 l1] and [h2 k2 l2] of a cubic crystal is: ) )( ( ) cos( l k h l k h l l k k h h Angle Between Crystal Directions 97

98 Miller Index for Hexagonal Crystal System 98 Four principal axes used, leading to four Miller Indices: Directions [h k i l]; Planes (h k i l), e.g. (0001) surface. First three axes/indices are related: h + k + i = 0 or i = -h-k. Indices h, k and l are identical to the Miller index. Rhombohedral crystal system can also be identified with four indices.

99 99 Miller Index for Hexagonal System

100 Miller indices 10 0 Referring to the origin of the reciprocal lattice s definition, i.e, Bragg refraction, a reciprocal lattice vector G actually represents a plane in the real space z y {001} x (100) (200) Easier way to get the indices: Reciprocals of the intercepts

101 Wigner-Seitz primitive unit cell and first Brillouin zone The Wigner Seitz cell around a lattice point is defined as the locus of points in space that are closer to that lattice point than to any of the other lattice points. 101 The cell may be chosen by first picking a lattice point. Then, lines are drawn to all nearby (closest) lattice points. At the midpoint of each line, another line (or a plane, in 3D) is drawn normal to each of the first set of lines. 1D case 2D case Important 3D case: BCC

102 The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice 1D 2D 102 Real space Reciprocal space

103 3D: Recall that the reciprocal lattice of FCC is BCC X =??? 4 /a Why is FCC so important?

104 Why is FCC so important? It s the lattice of Si and many III-V semiconductors. 104 Si: diamond, a = 5.4 Å GaAs: zincblende Crystal structure = lattice + basis Modern VLSI technology uses the (100) surface of Si. Which plane is (100)? Which is (111)?