Solid state physics

1. Solid state physics N. Witkowski

2. Introduction • 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 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 ? Long range order and 3D translational periodicity • Single crystals graphite 1.2 mm 4 nmx4nm • Polycristalline crystals Single crystals assembly diamond • Quasicrystals Long range order no no 3D translational periodicity Al72Ni20Co8 • Amorphous materials Disordered or random atomic structure silicon

4. Outline Corresponding chapter in Kittel book • [1] Crystal structure 1 • [2] Reciprocal lattice 2 • [3] Diffraction 2 • [4] Crystal binding no lecture 3 • [5] Lattice vibrations 4 • [6] Thermal properties 5 • [7] Free electron model 6 • [8] Energy band 7,9 • [9] Electron movement in crystals 8 Metals and Fermi surfaces 9 • [10] Semiconductors 8 • [11] Superconductivity 10 • [12] Magnetism 11

5. Chap.1Crystal structure

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

7. 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

8. A. Concepts, definitions • Translation vector : arrangement of atoms looks the same from r or r’ point • r’=r+u1a1+u2a2+u3a3 : u1, u2 and u3 integers = lattice constant • a1, a2, a3 primitive translation vectors • T=u1a1+u2a2+u3a3 translation vector r = a1+2a2 r’= 2a1- a2 T=r’-r=a1-3a2

9. 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= |a1.(a2xa3)|

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

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

12. 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

13. B. Lattice types • B2. Bravais lattices in 2D • 5 types • general case : • oblique lattice |a1|≠|a2| , (a1,a2)=φ • special cases : • square lattice: |a1|=|a2| , φ= 90° • hexagonal lattice: |a1|=|a2| , φ= 120° • rectangular lattice: |a1|≠|a2| , φ= 90° • centered rectangular lattice: |a1|≠|a2| , φ= 90°

14. B. Lattice types • B3. Bravais lattices in 3D: 14

15. B. Lattice types • B3. Bravais lattices in 3D: 14 Base centered monoclinic

16. B. Lattice types • B3. Bravais lattices in 3D: 14 Base centered orthorhombic Body centered orthorhombic Face centered orthorhombic

17. B. Lattice types • B3. Bravais lattices in 3D: 14 Body centered tetragonal

18. B. Lattice types • B3. Bravais lattices in 3D: 14 Simple cubic sc Body centered cubic bcc Face centered cubic fcc

19. B. Lattice types • B3. Bravais lattices in 3D: 14

20. B. Lattice types • B3. Bravais lattices in 3D: 14

21. B. Lattice types z a3 a2 • B4. Examples : bcc • Bcc cell : a, 90°, 2 atoms/cell • Primitive cell : ai vectors from the origin to lattice point at body centers • Rhombohedron : a1= ½ a(x+y-z), a2= ½ a(-x+y+z), a3= ½ a(x-y+z), edge ½ a • Wigner-Seitz cell y x a1

22. B. Lattice types z • B5. Examples : fcc • fcc cell : a, 90°, 4 atoms/cell • Primitive cell : ai vectors from the origin to lattice point at face centers • Rhombohedron : a1= ½ a(x+y), a2= ½ a(y+z), a3= ½ a(x+z), edge ½ a • Wigner-Seitz cell x y

23. 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 C hcp : 2 planes A B

24. 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 <u,v,w> direction http://www.doitpoms.ac.uk/tlplib/miller_indices/lattice_index.php

25. C. Crystal structures • C2. Miller index : example • plane intercepts axis : • 3a1 , 2a2, 2a3 • inverses : 1/3 , 1/2 , 1/2 • integers : 2, 3, 3 • h=2 , k=3 , l=3 • Index of planes : (2,3,3)