Solid state physics N. Witkowski

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

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

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

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 …

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

Chap.1 Crystal structure

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

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

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

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)|

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

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

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

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°

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°

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

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

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

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

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°

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°

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

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

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

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

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)