Transcript Solid state physics - accueil - nanomat

Solid state physics
N. Witkowski
Introduction
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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
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6 laboratory courses (6x3h): 1 extended report + 4 limited reports
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house 4, level 0, office 60111,
e-mail:[email protected]
Semiconductor physics
Specific heat
Superconductivity
Magnetic susceptibility
X-ray diffraction
Band structure calculation
Given between 23rd feb-6th march
Registration : from 9th feb on board F and Q
House 4 ground level
Info comes later
Home work
Evaluation : written examination 13 march (to be confirmed)
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5 hours, 6 problems
document authorized « Physics handbook for science and engineering» Carl
Nordling, Jonny Osterman
 Calculator authorized
 Second chance in june
What is solid state ?
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Single crystals
Long range order and 3D
translational periodicity
graphite 1.2 mm
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Polycristalline
crystals
Single crystals assembly
diamond
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Quasicrystals
Long range order no no 3D
translational periodicity
Al72Ni20Co8
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Amorphous
materials
Disordered or random atomic
structure
silicon
4 nmx4nm
Outline
Corresponding
chapter in Kittel book
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[1] Crystal structure
[2] Reciprocal lattice
[3] Diffraction
[4] Crystal binding
no lecture
[5] Lattice vibrations
[6] Thermal properties
[7] Free electron model
[8] Energy band
[9] Electron movement in crystals
Metals and Fermi surfaces
[10] Semiconductors
[11] Superconductivity
[12] Magnetism
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7,9
8
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11
Chap.1
Crystal structure
Introduction
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Aim :
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A : defining concepts and definitions
 B : describing the lattice types
 C : giving a description of crystal structures
A. Concepts, definitions
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A1. Definitions
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Crystal : 3 dimensional periodic
arrangments of atomes in
space. Description using a
mathematical abstraction : the
lattice
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Lattice : infinite periodic array
of points in space, invariant
under translation symmetry.
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Basis : atoms or group of
atoms attached to every lattice
point
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Crystal = basis+lattice
A. Concepts, definitions
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Translation vector :
arrangement of atoms looks
the same from r or r’ point
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r’=r+u1a1+u2a2+u3a3 : u1, u2
and u3 integers = lattice
constant
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a1, a2, a3 primitive
translation vectors
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T=u1a1+u2a2+u3a3
translation vector
r = a1+2a2
r’= 2a1- a2
T=r’-r=a1-3a2
A. Concepts, definitions
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A2.Primitive cell
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Standard model
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volume associated with one
lattice point
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Parallelepiped with lattice
points in the corner
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Each lattice point shared
among 8 cells
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Number of lattice
point/cell=8x1/8=1
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Vc= |a1.(a2xa3)|
A. Concepts, definitions
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Wigner-Seitz cell
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planes bisecting the lines
drawn from a lattice point to
its neighbors
A. Concepts, definitions
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A3.Crystallographic unit
cell
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larger cell used to display
the symmetries of the cristal
Not primitive
B. Lattice types
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B1. Symmetries :
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Translations
Rotation : 1,2,3,4 and 6
(no 5 or 7)
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three 4-fold axes
of a cube
four 3-fold
axes of a cube
six 2-fold
axes of a cube
Mirror reflection : reflection
about a plane through a
lattice point
Inversion operation (r -> -r)
planes of symmetry parallel in a cube
B. Lattice types
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B2. Bravais lattices in 2D
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5 types
general case :
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oblique lattice |a1|≠|a2| , (a1,a2)=φ
special cases :
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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
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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
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
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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
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 centered
monoclinic
B. Lattice types
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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
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 centered
orthorhombic
Body centered
orthorhombic
Face centered
orthorhombic
B. Lattice types
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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
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
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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
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
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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
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
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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
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
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B4. Examples : bcc
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Bcc cell : a, 90°, 2 atoms/cell
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Primitive cell : ai vectors from the
origin to lattice point at body
centers
z
a3
a2
y
x
a1
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Rhombohedron : a1= ½ a(x+y-z),
a2= ½ a(-x+y+z), a3= ½ a(x-y+z),
edge ½ a 3
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Wigner-Seitz cell
B. Lattice types
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B5. Examples : fcc
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fcc cell : a, 90°, 4 atoms/cell
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Primitive cell : ai vectors from the
origin to lattice point at face centers
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Rhombohedron : a1= ½ a(x+y), a2= ½
a(y+z), a3= ½ a(x+z), edge ½ 2a
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Wigner-Seitz cell
z
x
y
B. Lattice types
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B6. Examples : fcc - hcp
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different way of stacking the closepacked planes
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Spheres touching each other about
74% of the space occupied
fcc : 3 planes A B C
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B7. Example : diamond structure
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fcc structure
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4 atoms in tetraedric position
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Diamond, silicon
hcp : 2 planes A B
C. Crystal structures
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C1. Miller index
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lattice described by set of parallel planes
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usefull for cristallographic interpretation
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In 2D, 3 sets of planes
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Miller index
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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
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C2. Miller index : example
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plane intercepts axis :
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3a1 , 2a2, 2a3
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inverses : 1/3 , 1/2 , 1/2
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integers : 2, 3, 3
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h=2 , k=3 , l=3
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Index of planes : (2,3,3)