1. Solid state physics
N. Witkowski

2. Introduction
Based on « Introduction to Solid State Physics » 8th edition Charles Kittel
Lecture notes from Gunnar Niklasson
40h Lessons with N. Witkowski
house 4, level 0, office 60111,
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 ? Single crystals Polycristalline crystals
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. Subject of study Phenomena Material Variables 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 1010 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 Corresponding chapter in Kittel book [1] Crystal structure 1
[2] Reciprocal lattice 2
[3] Diffraction
[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
[12] Magnetism

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+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

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= |a1.(a2xa3)|

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 |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°

16. B. Lattice types B3. Bravais lattices in 3D : 14 system
Number of lattices
Cell axes and angles
Triclinic 1
|a1|≠|a2|≠|a3| , α≠β≠γ
Monoclinic 2
|a1|≠|a2|≠|a3| , α=γ=90°≠β
Orthorhombic 4
|a1|≠|a2|≠|a3| , α=β=γ=90°
Tetragonal
|a1|=|a2|≠|a3| , α=β=γ=90°
Cubic 3
|a1|=|a2|=|a3| , α=β=γ=90°
Trigonal
|a1|=|a2|=|a3| , α=β=γ<120°≠90°
Hexagonal
|a1|=|a2|≠|a3| , α=β=90° γ=120°

17. B. Lattice types B3. Bravais lattices in 3D : 14 system
Number of lattices
Cell axes and angles
Triclinic 1
|a1|≠|a2|≠|a3| , α≠β≠γ
Monoclinic 2
|a1|≠|a2|≠|a3| , α=γ=90°≠β
Orthorhombic 4
|a1|≠|a2|≠|a3| , α=β=γ=90°
Tetragonal
|a1|=|a2|≠|a3| , α=β=γ=90°
Cubic 3
|a1|=|a2|=|a3| , α=β=γ=90°
Trigonal
|a1|=|a2|=|a3| , α=β=γ<120°≠90°
Hexagonal
|a1|=|a2|≠|a3| , 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
Triclinic 1
|a1|≠|a2|≠|a3| , α≠β≠γ
Monoclinic 2
|a1|≠|a2|≠|a3| , α=γ=90°≠β
Orthorhombic 4
|a1|≠|a2|≠|a3| , α=β=γ=90°
Tetragonal
|a1|=|a2|≠|a3| , α=β=γ=90°
Cubic 3
|a1|=|a2|=|a3| , α=β=γ=90°
Trigonal
|a1|=|a2|=|a3| , α=β=γ<120°≠90°
Hexagonal
|a1|=|a2|≠|a3| , α=β=90° γ=120°
Base centered
orthorhombic
Body centered
orthorhombic
Face centered
orthorhombic

19. B. Lattice types B3. Bravais lattices in 3D : 14 system
Number of lattices
Cell axes and angles
Triclinic 1
|a1|≠|a2|≠|a3| , α≠β≠γ
Monoclinic 2
|a1|≠|a2|≠|a3| , α=γ=90°≠β
Orthorhombic 4
|a1|≠|a2|≠|a3| , α=β=γ=90°
Tetragonal
|a1|=|a2|≠|a3| , α=β=γ=90°
Cubic 3
|a1|=|a2|=|a3| , α=β=γ=90°
Trigonal
|a1|=|a2|=|a3| , α=β=γ<120°≠90°
Hexagonal
|a1|=|a2|≠|a3| , 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
Triclinic 1
|a1|≠|a2|≠|a3| , α≠β≠γ
Monoclinic 2
|a1|≠|a2|≠|a3| , α=γ=90°≠β
Orthorhombic 4
|a1|≠|a2|≠|a3| , α=β=γ=90°
Tetragonal
|a1|=|a2|≠|a3| , α=β=γ=90°
Cubic 3
|a1|=|a2|=|a3| , α=β=γ=90°
Trigonal
|a1|=|a2|=|a3| , α=β=γ<120°≠90°
Hexagonal
|a1|=|a2|≠|a3| , 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
Triclinic 1
|a1|≠|a2|≠|a3| , α≠β≠γ
Monoclinic 2
|a1|≠|a2|≠|a3| , α=γ=90°≠β
Orthorhombic 4
|a1|≠|a2|≠|a3| , α=β=γ=90°
Tetragonal
|a1|=|a2|≠|a3| , α=β=γ=90°
Cubic 3
|a1|=|a2|=|a3| , α=β=γ=90°
Trigonal
|a1|=|a2|=|a3| , α=β=γ<120°≠90°
Hexagonal
|a1|=|a2|≠|a3| , α=β=90° γ=120°

22. B. Lattice types B3. Bravais lattices in 3D : 14 system
Number of lattices
Cell axes and angles
Triclinic 1
|a1|≠|a2|≠|a3| , α≠β≠γ
Monoclinic 2
|a1|≠|a2|≠|a3| , α=γ=90°≠β
Orthorhombic 4
|a1|≠|a2|≠|a3| , α=β=γ=90°
Tetragonal
|a1|=|a2|≠|a3| , α=β=γ=90°
Cubic 3
|a1|=|a2|=|a3| , α=β=γ=90°
Trigonal
|a1|=|a2|=|a3| , α=β=γ<120°≠90°
Hexagonal
|a1|=|a2|≠|a3| , α=β=90° γ=120°

23. B. Lattice types 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 z a3 a2 y x a1

24. B. Lattice types 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 z x y

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 C
hcp : 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 <u,v,w> direction

27. 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)