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Solid state physics N. Witkowski. Based on « Introduction to Solid State Physics » 8th edition Charles Kittel Lecture notes from Gunnar Niklasson

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Presentation on theme: "Solid state physics N. Witkowski. Based on « Introduction to Solid State Physics » 8th edition Charles Kittel Lecture notes from Gunnar Niklasson"— Presentation transcript:

1 Solid state physics N. Witkowski

2 Based on « Introduction to Solid State Physics » 8th edition Charles Kittel Lecture notes from Gunnar Niklasson http://www.teknik.uu.se/ftf/education/ftf1/FTFI_forsta_sidan.html 40h Lessons with N. Witkowski  house 4, level 0, office 60111,  e-mail:witkowski@insp.jussieu.fr 6 laboratory courses (6x3h): 1 extended report + 4 limited reports  Semiconductor physics  Specific heat  Superconductivity  Magnetic susceptibility  X-ray diffraction  Band structure calculation Evaluation : written examination 13 march (to be confirmed)  5 hours, 6 problems  document authorized « Physics handbook for science and engineering» Carl Nordling, Jonny Osterman  Calculator authorized  Second chance in june Introduction Given between 23rd feb-6th march Registration : from 9th feb on board F and Q House 4 ground level Info comes later Home work

3 What is solid state ? Single crystals Polycristalline crystals Amorphous materials Quasicrystals Long range order no no 3D translational periodicity Long range order and 3D translational periodicity Single crystals assembly Disordered or random atomic structure 4 nmx4nm1.2 mmgraphite diamond Al 72 Ni 20 Co 8 silicon

4 Subject of study Phenomena  Crystalline structure  Atomic vibration, thermal properties  Electronic structure, electrical – optical properties  Superconductivity  Magnetism Variables  Temperature (mK – 3000 K)  Energy (provided by provided by photons, neutrons, electrons or ions, meV- keV)  Pressure (10 -10 to 10 10 Pa)  Magnetic field (-50 T)  Electric field ( - 1 GV/m) Material  metals  semiconductors  insulators

5 Motivations Wide range of technological applications  Materials science (applications of mechanical, electrical, optical, magnetic…properties of solids)  Semiconductor technology and micro-electronics  Microstructure engineering, nano-technology  Inorganic chemistry  Biological materials, biomimetics  Pharmaceutical materials science  Medical technology  …

6 Outline [1] Crystal structure1 [2] Reciprocal lattice2 [3] Diffraction2 [4] Crystal binding no lecture3 [5] Lattice vibrations4 [6] Thermal properties5 [7] Free electron model6 [8] Energy band7,9 [9] Electron movement in crystals8 Metals and Fermi surfaces9 [10] Semiconductors8 [11] Superconductivity10 [12] Magnetism11 Corresponding chapter in Kittel book

7 Chap.1 Crystal structure

8 Introduction Aim :  A : defining concepts and definitions  B : describing the lattice types  C : giving a description of crystal structures

9 A. Concepts, definitions A1. Definitions  Crystal : 3 dimensional periodic arrangments of atomes in space. Description using a mathematical abstraction : the lattice  Lattice : infinite periodic array of points in space, invariant under translation symmetry.  Basis : atoms or group of atoms attached to every lattice point  Crystal = basis+lattice

10 A. Concepts, definitions  Translation vector : arrangement of atoms looks the same from r or r’ point  r’=r+u 1 a 1 +u 2 a 2 +u 3 a 3 : u 1, u 2 and u 3 integers = lattice constant  a 1, a 2, a 3 primitive translation vectors  T=u 1 a 1 +u 2 a 2 +u 3 a 3 translation vector r = a 1 +2a 2 r’= 2a 1 - a 2 T=r’-r=a 1 -3a 2

11 A. Concepts, definitions A2.Primitive cell  Standard model volume associated with one lattice point Parallelepiped with lattice points in the corner Each lattice point shared among 8 cells Number of lattice point/cell=8x1/8=1 Vc= |a 1.(a 2 xa 3 )|

12 A. Concepts, definitions  Wigner-Seitz cell planes bisecting the lines drawn from a lattice point to its neighbors

13 A. Concepts, definitions A3.Crystallographic unit cell  larger cell used to display the symmetries of the cristal  Not primitive

14 B. Lattice types B1. Symmetries :  Translations  Rotation : 1,2,3,4 and 6 (no 5 or 7)  Mirror reflection : reflection about a plane through a lattice point  Inversion operation (r -> -r) three 4-fold axes of a cube four 3-fold axes of a cube six 2-fold axes of a cube planes of symmetry parallel in a cube

15 B. Lattice types B2. Bravais lattices in 2D  5 types  general case : oblique lattice |a 1 |≠|a 2 |, (a 1,a 2 )=φ  special cases : square lattice: |a 1 |=|a 2 |, φ= 90° hexagonal lattice: |a 1 |=|a 2 |, φ= 120° rectangular lattice: |a 1 |≠|a 2 |, φ= 90° centered rectangular lattice: |a 1 |≠|a 2 |, φ= 90°

16 B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120°

17 B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120° Base centered monoclinic

18 B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120° Body centered orthorhombic Face centered orthorhombic Base centered orthorhombic

19 B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120° Body centered tetragonal

20 B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120° Simple cubic sc Body centered cubic bcc Face centered cubic fcc

21 B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120°

22 B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120°

23 B. Lattice types B4. Examples : bcc  Bcc cell : a, 90°, 2 atoms/cell  Primitive cell : a i vectors from the origin to lattice point at body centers  Rhombohedron : a 1 = ½ a(x+y-z), a 2 = ½ a(-x+y+z), a 3 = ½ a(x-y+z), edge ½ a  Wigner-Seitz cell x y z a1a1 a2a2 a3a3

24 B. Lattice types B5. Examples : fcc  fcc cell : a, 90°, 4 atoms/cell  Primitive cell : a i vectors from the origin to lattice point at face centers  Rhombohedron : a 1 = ½ a(x+y), a 2 = ½ a(y+z), a 3 = ½ a(x+z), edge ½ a  Wigner-Seitz cell x y z

25 B. Lattice types B6. Examples : fcc - hcp  different way of stacking the close- packed planes  Spheres touching each other about 74% of the space occupied B7. Example : diamond structure  fcc structure  4 atoms in tetraedric position  Diamond, silicon fcc : 3 planes A B Chcp : 2 planes A B

26 C. Crystal structures C1. Miller index  lattice described by set of parallel planes  usefull for cristallographic interpretation  In 2D, 3 sets of planes  Miller index Interception between plane and lattice axis a, b, c Reducing 1/a,1/b,1/c to obtain the smallest intergers labelled h,k,l (h,k,l) index of the plan, {h,k,l} serie of planes, [u,v,w] or direction http://www.doitpoms.ac.uk/tlplib/miller_indices/lattice_index.php

27 C. Crystal structures C2. Miller index : example  plane intercepts axis : 3a 1, 2a 2, 2a 3  inverses : 1/3, 1/2, 1/2  integers : 2, 3, 3  h=2, k=3, l=3  Index of planes : (2,3,3)


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