solid state physics n. witkowski. based on « introduction to solid state physics » 8th edition...
TRANSCRIPT
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:[email protected]
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)