Solid state physics N. Witkowski. Based on « Introduction to Solid State Physics » 8th edition...

Solid state physics

N. Witkowski

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 marchRegistration : 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

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

Al72Ni20Co8

silicon

Outline

[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

Corresponding chapter in Kittel book

Chap.1Crystal 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-foldaxes 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

systemNumber 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 2 |a1|=|a2|≠|a3| , α=β=γ=90°

Cubic 3 |a1|=|a2|=|a3| , α=β=γ=90°

Trigonal 1 |a1|=|a2|=|a3| , α=β=γ<120°≠90°

Hexagonal 1 |a1|=|a2|≠|a3| , α=β=90° γ=120°

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

systemNumber 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 2 |a1|=|a2|≠|a3| , α=β=γ=90°

Cubic 3 |a1|=|a2|=|a3| , α=β=γ=90°

Trigonal 1 |a1|=|a2|=|a3| , α=β=γ<120°≠90°

Hexagonal 1 |a1|=|a2|≠|a3| , α=β=90° γ=120° Base centeredmonoclinic

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

systemNumber 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 2 |a1|=|a2|≠|a3| , α=β=γ=90°

Cubic 3 |a1|=|a2|=|a3| , α=β=γ=90°

Trigonal 1 |a1|=|a2|=|a3| , α=β=γ<120°≠90°

Hexagonal 1 |a1|=|a2|≠|a3| , α=β=90° γ=120° Body centeredorthorhombic

Face centeredorthorhombic

Base centeredorthorhombic

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

systemNumber 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 2 |a1|=|a2|≠|a3| , α=β=γ=90°

Cubic 3 |a1|=|a2|=|a3| , α=β=γ=90°

Trigonal 1 |a1|=|a2|=|a3| , α=β=γ<120°≠90°

Hexagonal 1 |a1|=|a2|≠|a3| , α=β=90° γ=120°

Body centered tetragonal

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

systemNumber 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 2 |a1|=|a2|≠|a3| , α=β=γ=90°

Cubic 3 |a1|=|a2|=|a3| , α=β=γ=90°

Trigonal 1 |a1|=|a2|=|a3| , α=β=γ<120°≠90°

Hexagonal 1 |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

systemNumber 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 2 |a1|=|a2|≠|a3| , α=β=γ=90°

Cubic 3 |a1|=|a2|=|a3| , α=β=γ=90°

Trigonal 1 |a1|=|a2|=|a3| , α=β=γ<120°≠90°

Hexagonal 1 |a1|=|a2|≠|a3| , α=β=90° γ=120°

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

systemNumber 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 2 |a1|=|a2|≠|a3| , α=β=γ=90°

Cubic 3 |a1|=|a2|=|a3| , α=β=γ=90°

Trigonal 1 |a1|=|a2|=|a3| , α=β=γ<120°≠90°

Hexagonal 1 |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

xy

z

a1

a2a3

3

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

xy

z

2

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)