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Solid state physics N. Witkowski

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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 Introduction Given between 23rd feb-6th march Registration : from 9th feb on board F and Q House 4 ground level Info comes later Home work

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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 Al 72 Ni 20 Co 8 silicon

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Subject of study Phenomena 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 ( to Pa) Magnetic field (-50 T) Electric field ( - 1 GV/m) Material metals semiconductors insulators

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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 …

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Outline [1] Crystal structure1 [2] Reciprocal lattice2 [3] Diffraction2 [4] Crystal binding no lecture3 [5] Lattice vibrations4 [6] Thermal properties5 [7] Free electron model6 [8] Energy band7,9 [9] Electron movement in crystals8 Metals and Fermi surfaces9 [10] Semiconductors8 [11] Superconductivity10 [12] Magnetism11 Corresponding chapter in Kittel book

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Chap.1 Crystal structure

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

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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

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A. Concepts, definitions Translation vector : arrangement of atoms looks the same from r or r’ point r’=r+u 1 a 1 +u 2 a 2 +u 3 a 3 : u 1, u 2 and u 3 integers = lattice constant a 1, a 2, a 3 primitive translation vectors T=u 1 a 1 +u 2 a 2 +u 3 a 3 translation vector r = a 1 +2a 2 r’= 2a 1 - a 2 T=r’-r=a 1 -3a 2

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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= |a 1.(a 2 xa 3 )|

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

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

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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

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B. Lattice types B2. Bravais lattices in 2D 5 types general case : oblique lattice |a 1 |≠|a 2 |, (a 1,a 2 )=φ special cases : square lattice: |a 1 |=|a 2 |, φ= 90° hexagonal lattice: |a 1 |=|a 2 |, φ= 120° rectangular lattice: |a 1 |≠|a 2 |, φ= 90° centered rectangular lattice: |a 1 |≠|a 2 |, φ= 90°

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B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120°

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B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120° Base centered monoclinic

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B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120° Body centered orthorhombic Face centered orthorhombic Base centered orthorhombic

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B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120° Body centered tetragonal

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B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120° Simple cubic sc Body centered cubic bcc Face centered cubic fcc

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B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120°

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B. Lattice types B3. Bravais lattices in 3D : 14 system Number of lattices Cell axes and angles Triclinic1|a 1 |≠|a 2 |≠|a 3 |, α≠β≠γ Monoclinic2|a 1 |≠|a 2 |≠|a 3 |, α=γ=90°≠β Orthorhombic4|a 1 |≠|a 2 |≠|a 3 |, α=β=γ=90° Tetragonal2|a 1 |=|a 2 |≠|a 3 |, α=β=γ=90° Cubic3|a 1 |=|a 2 |=|a 3 |, α=β=γ=90° Trigonal1|a 1 |=|a 2 |=|a 3 |, α=β=γ<120°≠90° Hexagonal1|a 1 |=|a 2 |≠|a 3 |, α=β=90° γ=120°

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B. Lattice types B4. Examples : bcc Bcc cell : a, 90°, 2 atoms/cell Primitive cell : a i vectors from the origin to lattice point at body centers Rhombohedron : a 1 = ½ a(x+y-z), a 2 = ½ a(-x+y+z), a 3 = ½ a(x-y+z), edge ½ a Wigner-Seitz cell x y z a1a1 a2a2 a3a3

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B. Lattice types B5. Examples : fcc fcc cell : a, 90°, 4 atoms/cell Primitive cell : a i vectors from the origin to lattice point at face centers Rhombohedron : a 1 = ½ a(x+y), a 2 = ½ a(y+z), a 3 = ½ a(x+z), edge ½ a Wigner-Seitz cell x y z

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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 Chcp : 2 planes A B

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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 direction

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C. Crystal structures C2. Miller index : example plane intercepts axis : 3a 1, 2a 2, 2a 3 inverses : 1/3, 1/2, 1/2 integers : 2, 3, 3 h=2, k=3, l=3 Index of planes : (2,3,3)

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