Solid state physics 02-crystal structure

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Solid state physics 02-crystal structure

  1. 1. Solid State Physics UNIST, Jungwoo Yoo 1. What holds atoms together - interatomic forces (Ch. 1.6) 2. Arrangement of atoms in solid - crystal structure (Ch. 1.1-4) - Elementary crystallography - Typical crystal structures - X-ray Crystallography 3. Atomic vibration in solid - lattice vibration (Ch. 2) - Sound waves - Lattice vibrations - Heat capacity from lattice vibration - Thermal conductivity 4. Free electron gas - an early look at metals (Ch. 3) - The free electron model, Transport properties of the conduction electrons ---------------------------------------------------------------------------------------------------------(Midterm I) 5. Free electron in crystal - the effect of periodic potential (Ch. 4) - Nearly free electron theory - Block's theorem (Ch. 11.3) - The tight binding approach - Insulator, semiconductor, or metal - Band structure and optical properties 6. Waves in crystal (Ch. 11) - Elastic scattering of waves by a crystal - Wavelike normal modes - Block's theorem - Normal modes, reciprocal lattice, brillouin zone 7. Semiconductors (Ch. 5) - Electrons and holes - Methods of providing electrons and holes - Transport properties - Non-equilibrium carrier densities 8. Semiconductor devices (Ch. 6) - The p-n junction - Other devices based on p-n junction - Metal-oxide-semiconductor field-effect transistor (MOSFET) ---------------------------------------------------------------------------------------------------------------(Final) All about atoms backstage All about electrons Main character Main applications
  2. 2. Solid State Physics UNIST, Jungwoo Yoo Types of Bonding Ionic Bonding Van Der Waals Bonding Metallic Bonding Covalent Bonding Hydrogen Bonding High Melting Point Hard and Brittle Non conducting solid NaCl, CsCl, ZnS Low Melting Points Soft and Brittle Non-Conducting Ne, Ar, Kr and Xe Variable Melting Point Variable Hardness Conducting Fe, Cu, Ag Very High Melting Point Very Hard Usually not Conducting Diamond, Graphite Low Melting Points Soft and Brittle Usually Non-Conducting İce, organic solids
  3. 3. Solid State Physics UNIST, Jungwoo Yoo Crystal Structure 1. Crystal lattice, basis, unit cell, crystal planes and direction 2. Closed packed structure (cubic, hexagonal) 3. The body-centered cubic structure 4. Structure of ionic solids 5. The diamond and zincblende structures 6. X-ray crystallography A solid is said to be crystal if atoms are arranged in such a way that their positions are exactly periodic.
  4. 4. Solid State Physics UNIST, Jungwoo Yoo Solid Mateirals Crystalline Polycrystalline Amorphous Single Crystal
  5. 5. Solid State Physics UNIST, Jungwoo Yoo 5 Single Crystal Single Pyrite Crystal Amorphous Solid Single crystal has an atomic structure that repeats periodically across its whole volume. Crystalline solid Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetry
  6. 6. Solid State Physics UNIST, Jungwoo Yoo 6 Polycrystalline Pyrite form (Grain) Polycrystalline solid 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. These ordered regions, or single crytal regions, vary in size and orientation wrt one another. These regions are called as grains (domain) and 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 - 100 microns in diameter. Polycrystals with grains that are <10 nm in diameter are called nanocrystalline
  7. 7. Solid State Physics UNIST, Jungwoo Yoo 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 Amorphous silicon can be used in solar cells and thin film transistors.
  8. 8. Solid State Physics UNIST, Jungwoo Yoo What is crystallography? The branch of science that deals with the geometric description of crystals and their internal arrangement. Crystallography is essential for solid state physics • Symmetry of a crystal can have a profound influence on its properties. • Any crystal structure should be specified completely, concisely and unambiguously • Structures should be classified into different types according to the symmetries they possess. Crystallography
  9. 9. Solid State Physics UNIST, Jungwoo Yoo Crystal Lattice CB ED O A G F Graphene
  10. 10. Solid State Physics UNIST, Jungwoo Yoo Crystal Lattice B O A • An infinite array of points in space, • Each point has identical surroundings to all others. • Arrays are arranged exactly in a periodic manner. Lattice ? a  b  Lattice vector: ba  ,  bvauT   vu, are integer Translational invariance Bravais Lattice Lattice is completely defined by lal and lbl and  For graphene a=b=2.46 Å,  = 120º
  11. 11. Solid State Physics UNIST, Jungwoo Yoo Crystal Lattice CB ED O A G F Graphene
  12. 12. Solid State Physics UNIST, Jungwoo Yoo Crystal Lattice
  13. 13. Solid State Physics UNIST, Jungwoo Yoo Crystal Lattice
  14. 14. Solid State Physics UNIST, Jungwoo Yoo Basis a b O A group of atoms associated with each lattice point to represent crystal structure  Lattice vector: ba  , Basis vector: bar  3 1 3 2  Basis: C(0,0), C(2/3, 1/3) Crystal structure = Lattice Basis Basis gives identity to the lattice to from crystal
  15. 15. Solid State Physics UNIST, Jungwoo Yoo Crystal structure = Lattice Basis
  16. 16. Solid State Physics UNIST, Jungwoo Yoo 5 Bravais lattice in 2D
  17. 17. Solid State Physics UNIST, Jungwoo Yoo Lattice in 3D cwbvauT   Translational invariance Lattice vector: cba  ,, wvu ,, are integer Lattice vector
  18. 18. Solid State Physics UNIST, Jungwoo Yoo S a b S S S S S S S S S S S S S S 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. 2D crystal
  19. 19. Solid State Physics UNIST, Jungwoo Yoo 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. The choice of unit cell is not unique. 2D crystal S S a Sb S
  20. 20. Solid State Physics UNIST, Jungwoo Yoo We define lattice points ; these are points with identical environments 2D Unit Cell example – (NaCl)
  21. 21. Solid State Physics UNIST, Jungwoo Yoo Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same.
  22. 22. Solid State Physics UNIST, Jungwoo Yoo This is also a unit cell - it doesn’t matter if you start from Na or Cl
  23. 23. Solid State Physics UNIST, Jungwoo Yoo - or if you don’t start from an atom
  24. 24. Solid State Physics UNIST, Jungwoo Yoo This is not a unit cell
  25. 25. Solid State Physics UNIST, Jungwoo Yoo Are they unit cells ?
  26. 26. Solid State Physics UNIST, Jungwoo Yoo Unit Cell in 3D
  27. 27. Solid State Physics UNIST, Jungwoo Yoo simple cubic (sc) body-centered cubic (bcc) face-centered cubic (fcc) 3 Common Unit Cell in 3D
  28. 28. Solid State Physics UNIST, Jungwoo Yoo Unit Cell Primitive Conventional & Non-Primitive Single lattice point per cell Smallest area in 2D, or Smallest volume in 3D More than one lattice point per cell Integral multiples of the area of primitive cell Simple cubic(sc) Conventional = Primitive cell Body centered cubic(bcc) Conventional ≠ Primitive cell
  29. 29. Solid State Physics UNIST, Jungwoo Yoo Primitive and conventional cells of FCC primitive unit cell conventional unit cell Primitive vectors      .ˆˆ 2 ,ˆˆ 2 ,ˆˆ 2 3 2 1 xz a a zy a a yx a a       Q: How many lattice point ?
  30. 30. Solid State Physics UNIST, Jungwoo Yoo Primitive and conventional cells of bcc primitive unit cell conventional unit cell Primitive vectors      .ˆˆˆ 2 ,ˆˆˆ 2 ,ˆˆˆ 2 3 2 1 zyx a a zyx a a zyx a a       Q: How many lattice point ?
  31. 31. Solid State Physics UNIST, Jungwoo Yoo
  32. 32. Solid State Physics UNIST, Jungwoo Yoo Crystal Structure 32 • A primitive unit cell is made of primitive translation vectors a1 ,a2, and a3 such that there is no cell of smaller volume that can be used as a building blo ck for crystal structures. • A primitive unit cell will fill space by repetition of s uitable crystal translation vectors. This defined by the parallelpiped a1, a2 and a3. The volume of a pr imitive unit cell can be found by • V = a1.(a2 x a3) (vector products) Cubic cell volume = a3 Primitive Unit Cell and Vectors
  33. 33. Solid State Physics UNIST, Jungwoo Yoo P = Primitive Unit Cell NP = Non-Primitive Unit Cell Primitive Unit Cell The primitive unit cell must have only one lattice point. There can be different choices for lattice vectors, but the volumes of these primitive cells are all the same
  34. 34. Solid State Physics UNIST, Jungwoo Yoo A simply way to find the primitive cell Wigner-Seitz cell 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 dr aw new lines. The volume(or area) enclosed is called a Wigner-Seitz cell. Wigner-Seitz Method
  35. 35. Solid State Physics UNIST, Jungwoo Yoo Wigner-Seitz Cell - 3D fcc wigner-seitz cell bcc wigner-seitz cell
  36. 36. Solid State Physics UNIST, Jungwoo Yoo Crystal Directions Any vector r which specify direction in a crystal can be written in terms of lattice vector cwbvauT   This direction in a crystal is expressed as  uvw If the numbers u v w have a common factor, this factor is removed. Ex. [111] rather than [222], or [100], rather than [400]. wvu ,, are smallest integer negative directions are indicated as wvu ,,
  37. 37. Solid State Physics UNIST, Jungwoo Yoo X = 1 , Y = ½ , Z = 0 [1 ½ 0] [2 1 0] [210] Ex. X = ½ , Y = ½ , Z = 1 [½ ½ 1] [1 1 2] [112]
  38. 38. Solid State Physics UNIST, Jungwoo Yoo Ex. We can move vector to the origin. X = -1 , Y = 1, Z = -1/6 [-1 1 -1/6] [6 6 1]
  39. 39. Solid State Physics UNIST, Jungwoo Yoo Crystal Planes • Within a crystal lattice it is possible to identify sets of equally spaced parallel planes. These are called lattice planes. • In the figure density of lattice points on each plane of a set is the same and all lattice points are contained on each set of planes. The set of planes in 2D lattice. b a b a
  40. 40. Solid State Physics UNIST, Jungwoo Yoo Miller Indices Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes. To determine Miller indices of a plane, take the following steps; 1) Determine the intercepts of the plane along each of the three crystallographic directions 2) Take the reciprocals of the intercepts 3) If fractions result, multiply each by the denominator of the smallest fraction  hkl lkh ,, are smallest integer (100)
  41. 41. Solid State Physics UNIST, Jungwoo Yoo Ex. (100) (110) (111) (200) (100) (102) (102) (233)
  42. 42. Solid State Physics UNIST, Jungwoo Yoo )111(),111(),111(),111(),111(),111(),111(),111(}111{ )001(),100(),010(),001(),010(),100(}100{   Indices of a Family or Form Indices {h,k,l} represent all the planes equivalent to the plane (hkl) through rotational symmetry. Sometimes when the unit cell has rotational symmetry, several nonparallel planes may be equivalent by virtue of this symmetry, in which case it is convenient to lump all these planes in the same Miller Indices, but with curly brackets.
  43. 43. Solid State Physics UNIST, Jungwoo Yoo Crystal Structure 43 Close packing Close-Packed Structure A A A A A A A A A AA A A A A A A A A A A A A A A A A BB B B B B B BB B B BB B BB BB B B BB B A A A A A A A A A AA A A A A A A A C C C C C C CC C C C C C CC C C C Two sequence: ABABABABAB…… ABCABCABC ……..
  44. 44. Solid State Physics UNIST, Jungwoo Yoo 44 A A A A A A A A A AA A A A A A A A BB B B B B B BB B B BB B BB BB B B BB B A A A A A A A A A AA A A A A A A A For ABABABAB For ABABABAB Hexagonal structure or hexagonal close-packed structure Lattice vector: a=b =120, c=1.633a, basis : (0,0,0) (2/3a ,1/3a,1/2c)
  45. 45. Solid State Physics UNIST, Jungwoo Yoo A A A A A A A A A AA A A A A A A A BB B B B B B BB B B BB B BB BB B B BB B C C C C C C CC C C C C C CC C C C For ABCABCABC For ABCABCABC Cubic close packed structure or face centered cubic structure [111] plane
  46. 46. Solid State Physics UNIST, Jungwoo Yoo How about square ? A A A A A A A A A B B BB A A A A A A A A A Body centered cubic structure
  47. 47. Solid State Physics UNIST, Jungwoo Yoo Coordination Number Coordinatıon Number (CN) : The Bravais lattice points closest to a given point are the nearest neighbours Because the Bravais lattice is periodic, all points have the same number of nearest neighbours or coordination number. It is a property of the lattice. For SC: BCC: FCC:6 8 12 Atomic packing factor APF = Volume of atoms in the unit cell Volume of unit cell For SC: BCC: FCC:
  48. 48. Solid State Physics UNIST, Jungwoo Yoo Structure of Ionic Solids Sodium Chloride Both Cl- and Na+ represent fcc lattice and they are displaced by half unit in [100] direction Can be viewed as a fcc lattice with a basis of Na+(0,0,0), Cl-(½, ½, ½), or ? Cesium Chloride Both Cl- and Cs+ represent sc lattice and they are displaced by half unit in [111] direction Can be viewed as a sc lattice with a basis of Cs+(0,0,0), Cl-(½, ½, ½)
  49. 49. Solid State Physics UNIST, Jungwoo Yoo Structure of Covalent Bond Solids Height of atoms Diamonds Q: what type of hybridzation? sp3 Diamonds structure consists of two Interpenetrating fcc lattice Structure is far from being close packed structure Why so ? Nature of covalent bond restrict # of bonding i.e. nearest neighbor Nature of covalent bond very different from metallic, ionic, VDW Q: What other crystalline materials have diamond structure? Si, Ge Made of Carbon Can be viewed as a fcc lattice with a basis of C(0,0,0), C(1/4, 1/4, 1/4)
  50. 50. Solid State Physics UNIST, Jungwoo Yoo
  51. 51. Solid State Physics UNIST, Jungwoo Yoo Structure of Covalent Bond Solids For group III- group V GaAs (gallium arsenide) InSb (indium antimonide) Q: what type of bonding? covalent The structure of these materials closely related to ZnS (zincblende) Height of atoms Can be viewed as a fcc lattice with a basis of Zn(0,0,0), S(1/4, 1/4, 1/4)
  52. 52. Solid State Physics UNIST, Jungwoo Yoo X-ray Crystallography X-rays were discovered in 1895 by the German physicist Wilhelm Conrad Röntgen X ray, invisible, highly penetrating electromagnetic radiation of much shorter wavelength (higher frequency) than visible light. The wavelength range for X rays is from about 10-8 m to about 10-11 m, the corresponding frequency range is from about 3 × 1016 Hz to about 3 × 1019 Hz.  /hcE   For  = 1Å, X-ray E ~ 104 eV h = 6.626ⅹ10-34 J∙s = 4.136ⅹ10-15 eV
  53. 53. Solid State Physics UNIST, Jungwoo Yoo Production of x-ray Evacuated glass bulb Anode Cathode X rays can be produced in a highly evacuated glass bulb, called an X-ray tube, that contains essentially two electrodes—an anode made of platinum, tungsten, or another heavy metal of high melting point, and a cathode. When a high voltage is applied between the electrodes, streams of electrons (cathode rays) are accelerated from the cathode to the anode and produce X rays as they strike the anode K, L, M lines for emitted photon from electron Transition to n = 1, 2, 3 orbitals
  54. 54. Solid State Physics UNIST, Jungwoo Yoo Interference of wave Constructive interference is the result of synchronized light waves that add together to increase the light intensity. Destructive İnterference results when two out-of-phase light waves cancel each other out, resulting in darkness. Young’s double slit exp. Constructive interference Path difference: Dx = Destractive interference Path difference: Dx = n n = 0, 1, 2, 3 … n + /2 n = 0, 1, 2, 3 … To see interference: d ~  d L s Principle of superposition
  55. 55. Solid State Physics UNIST, Jungwoo Yoo X-ray crystallography Consider the solid as a diffraction grating with a spacing of ~1Å  /hcE   For  = 1Å, E ~ 104 eV Similar to diffraction grating, measurement of the x-ray diffraction maxima from a crystal allows us to determine the size of the unit cell Lattice point is the analogue of the line on an optical diffraction grating Basis represents the structure of the line
  56. 56. Solid State Physics UNIST, Jungwoo Yoo  A B CD 2 Bragg’s law W.L. Bragg considered crystals to be made up of parallel planes of atoms. Incident waves are reflected specularly from parallel planes of atoms in the crystal, with each plane is reflecting only a very small fraction of the radiation, like a lightly silvered mirror. In mirrorlike reflection the angle of incidence is equal to the angle of reflection. Therefore AC = DB for any angle. No interference pattern! The diffracted wave looks as if it has been reflected from the plane. Coherent scattering from a single plane is not sufficient to obtain a diffraction maximum. It is also necessary that successive planes should scatter in phase
  57. 57. Solid State Physics UNIST, Jungwoo Yoo Path difference: sin2d For constructive interference sin2d n Consider successive planes There is path difference between the light scattered from successive planes. And it varies with incident angle Bragg’s law DE + EF =
  58. 58. Solid State Physics UNIST, Jungwoo Yoo A beam corresponding to a value of n>1 could be identified by a statement such as ‘the nth-order reflections from the (hkl) planes’. (nh nk nl) reflection Third-order reflection from (111) plane a (333) reflection Bragg’s law Rewriting the Bragg law which makes n-th order diffraction off (hkl) planes of spacing ‘d’ look like first-order diffraction off planes of spacing d/n. Planes of this reduced spacing would have Miller indices (nh nk nl)        sin2 n d
  59. 59. Solid State Physics UNIST, Jungwoo Yoo Details of the structure can also be deduced from diffraction pattern Ex. NaCl, KCl Both shows diffraction peak for (100) NaCl shows weak but clear (111) peaks But, absence of (111) peaks for KCl K+ and Cl- both have argon electron shell a nearly same strength of x-ray scattering But, Na+ and Cl- have different shell structure (111) plane: only Cl- (222) plane: only Na+ For KCl: reflection from (222) plane induces destructive interference
  60. 60. Solid State Physics UNIST, Jungwoo Yoo X-ray Diffraction Methods Laue Rotating Crystal Powder Lattice parameters Polycrystal (powdered) Monochromatic beam Variable angle Lattice constant Single crystal Monochromatic beam Variable angle Orientation Single crystal Polychromatic beam Fixed angle
  61. 61. Solid State Physics UNIST, Jungwoo Yoo Laue methods The Laue method is mainly used to determine the orientation of large single crystals while radiation is reflected from, or transmitted through a fixed crystal. The diffracted beams form arrays of spots, that lie on curves on the film The Bragg angle is fixed for every set of planes in the crystal. Each set of planes picks out and diffracts the particular wavelength from the white radiation that satisfies the Bragg law for the values of d and θ involved The symmetry of the spot pattern reflects the symmetry of the crystal when viewed along the direction of the incident beam. Laue method is often used to determine the orientation of single crystals by means of illuminating the crystal with a continuos spectrum of X-rays
  62. 62. Solid State Physics UNIST, Jungwoo Yoo X-Ray FilmSingle Crystal X-Ray Film Single Crystal Transmission laue method Back reflection laue method
  63. 63. Solid State Physics UNIST, Jungwoo Yoo Rotating crystal method In the rotating crystal method, a single crystal is mounted with an axis normal to a monochromatic x- ray beam. A cylindrical film is placed around it and the crystal is rotated about the chosen axis As the crystal rotates, sets of lattice planes will at some point make the correct Bragg angle for the monochromatic incident beam, and at that point a diffracted beam will be formed The reflected beams are located on the surface of imaginary cones. By recording the diffraction patterns (both angles and intensities) for various crystal orientations, one can determine the shape and size of unit cell as well as arrangement of atoms inside the cell.
  64. 64. Solid State Physics UNIST, Jungwoo Yoo
  65. 65. Solid State Physics UNIST, Jungwoo Yoo The powder methods If a powdered specimen is used, instead of a single crystal, then there is no need to rotate the specimen, because there will always be some crystals at an orientation for which diffraction is permitted. Here a monochromatic X-ray beam is incident on a powdered or polycrystalline sample. This method is useful for samples that are difficult to obtain in single crystal form. The powder method is used to determine the value of the lattice parameters accurately. Lattice parameters are the magnitudes of the unit vectors a, b and c which define the unit cell for the crystal. For every set of crystal planes, by chance, one or more crystals will be in the correct orientation to give the correct Bragg angle to satisfy Bragg's equation. Every crystal plane is thus capable of diffraction. Each diffraction line is made up of a large number of small spots, each from a separate crystal. Each spot is so small as to give the appearance of a continuous line.
  66. 66. Solid State Physics UNIST, Jungwoo Yoo  If the sample consists of some tens of randomly orientated single crystals, the diffracted beams are seen to lie on the surface of several cones. The cones may emerge in all directions, forwards and backwards.  A sample of some hundreds of crystals (i.e. a powdered sample) show that the diffracted beams form continuous cones. A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giving diffraction lines. The lines are seen as arcs on the film. 66 • If a monochromatic x-ray beam is directed at a single crystal, then only one or two diffracted beams may result. The powder methods
  67. 67. Solid State Physics UNIST, Jungwoo Yoo The specimen is placed in the Debye Scherrer camera and is accurately aligned to be in the centre of the camera. X-rays enter the camera through a collimator. The powder diffracts the x-rays in accordance with Braggs law to produce cones of diffracted beams. These cones intersect a strip of photographic film located in the cylindrical camera to produce a characteristic set of arcs on the film.

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