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PH 409: Introduction to Condensed Matter Physics

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PH 409: Introduction to Condensed Matter Physics i PH 409: Introduction to Condensed Matter Physics Course instructor : Kantimay Das Gupta Lectures are held: Mon (10.35-11.30), Tue (11.35-12.30), Thu (8.30-9.25) [IC 1] Course Contents: 1. Describing condensed matter microscopically structure factor • lattice and fourier transforming of densities. • Unit cell concept • Xray scattering • 2. The free electron gas in metals Drude theory of transport • Thermodynamic quantities • response to electromagnetic fields • 3. Bloch’s theorem Gap at zone boundaries and reduced zone scheme. Draw it with free electrons, show that in • the second zone, filled states are not near the zone center. Band structure as a perturbation problem. • Classification of metals/insulators/semiconductors • Kronig-Penny model of a solid. • 4. Tight binding method Linear Combination of Atomic Orbitals as the way to build a molecule from atoms. • ID example, 2D Graphene, 3D, some lattices like cubic/BCC/FCC. • What are the shortcomings. APW and then e-e interaction. • Mott insulators (NiO) if possible. • Revision of Brillouin zone. Drawing the free electron Fermi surface. The Fermi level in Alkali • metals, divalent/trivalent metals. Why some metals can give hole-like Hall voltage. • 5. semiclassical dynamics of electrons in a band. Introduce the equations, write Bloch oscillation condition, group velocity near zone boundary, • why k k-G flipping does not give unphysical results? → Concept of holes: what is the correct k vector of a hole, equation of motion, group velocity etc. • Concept of orbits in a magnetic field (electron like, hole like and open) • Concept of holes • de-Haas van Alphen oscillation mechanism. • Optical absorption and dHvA as two key methods of band structure determination • 6. Phonons specific heat (lattice + free electron) , • thermal conductivity Bloch-Gruniessen if possible. • ii One example of inelastic neutron scattering (perhaps from liq. Helium) • 7. Chapter on bonding: electrovalent, covalent solids 8. Magnetism Pauli paramagnetism in the electron gas • writing the free energy of an electron gas in magnetic field • diamagnetism, • ferromagnetism • exchange interaction , ferro-antiferro, hysteresis, • spin wave magnon, • 9. Special topics (may be modified...) Conservation laws in a crystal • fluctuation, dimensionality and order • Collective modes/excitations • 10. disordered material glass, colloids etc. • Introduce g(r) and its significance • BOOKS: There are a (very!) large number of books on introductory Condensed matter physics. Two of the most commonly used ones at this level are probably 1. C. Kittel, Introduction to Solid State Physics 2. N.W. Ashcroft and N.D. Mermin, Solid State Physics There are many more texts like: 1. J. M. Ziman, Principles of the Theory of Solids 2. P. M. Chaikin and T. Lubensky, Principles of the Condensed Matter Physics. (This has a focus on statistical and ”soft” matter) 3. N.F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys (The early development of the subject with very close attention to interpreting experimental data.) From time to time I will suggest other references during the lectures. Evaluation : 1. Class Quiz : 15% 2. Mid sem : 30% 3. Class Quiz : 15% 4. End sem : 40% iii This material is not a book or the complete course. It gives you some guidelines only. You should always refer to the recommended books to see the standard development and data relating to a chapter. iv Contents 1 Structure factor and lattice 3 1.1 Fourier transform and Structure factor . ............... 3 1.1.1 Whatisacrystal? ............................... ..... 5 1.1.2 Fourier transform of the lattice points . ............. 7 1.2 Thereciprocallattice. ........... 8 1.2.1 Specifying directions and planes in a lattice . ............... 9 2 Diffraction and basics of crystal structure 11 2.1 Braggdiffraction .................................. ....... 11 2.1.1 Powder diffraction : Debye Scherer ”camera” . ........... 13 2.1.2 Handling the basis in the scattering problem . ............. 16 2.2 Afewcommonlatticetypes . ......... 17 2.2.1 SimpleCubic ................................... 18 2.2.2 Body-centered-cubic lattice . ........... 18 2.2.3 Face-centered-cubic lattice . ............ 19 2.2.4 Diamondlattice ................................ ..... 20 2.3 A Geometrical excursion: Wigner Seitz cell, Brilloiun zonesetc ............... 25 2.3.1 WignerSeitzcell ............................... ...... 25 2.3.2 In3D.......................................... 26 3 Free electrons in metals: basic thermodynamics, transport and electrodynamics 33 3.1 Thebasicmodel ................................... ...... 33 3.1.1 How does the Fermi distribution look? . ........... 34 3.1.2 What fixes the chemical potential (at T =0)?..................... 35 3.1.3 At finite T ........................................ 39 3.2 Transport in the free electron gas: Drude picture . ................. 44 3.2.1 Hall effect & Drude electrons in a magnetic field . ........... 46 3.2.2 Electrodynamics in the electron gas . ........... 47 3.2.3 Spatial variation of the potential, ǫ(q) ......................... 54 3.3 The basics of Boltzmann transport : shift in the distribution function . 57 3.3.1 Effect of E and B onthedistributionfunction . 58 4 Electrons in a periodic potential: Bloch’s theorem 63 4.1 Derivationofthetheorem . .......... 63 4.1.1 Translation invariance and Bloch’s theorem . .............. 64 4.1.2 Significance of k ..................................... 65 4.1.3 Origin of the band gap: solving the matrix equation . .............. 66 4.1.4 Band structure as a perturbation problem . ............ 68 4.1.5 TheKronig-Pennymodel . ...... 71 4.1.6 The Band gap: classification of conducting, non-conducting and semiconducting substances......................................... 77 4.2 Motion of an electron in a band: Bloch oscillation, group velocity and effective mass . 77 v vi CONTENTS 5 Tight binding or Linear Combination of Atomic Orbitals (LCAO) 81 5.1 Diatomic molecule and Linear chain of atoms . .............. 81 5.1.1 Diatomicmolecule .............................. ...... 81 5.1.2 Linear chain of atoms with nearest neighbour interaction .............. 83 5.1.3 Morethan1orbitalpersite . ........ 89 5.2 Measuring the effective mass: Cyclotron resonance . ................ 92 5.3 Orthogonalised Plane Waves (OPW) and Pseudopotential . ................. 92 5.3.1 The pseudopotential and the pseudo-wavefunction . ............... 94 5.3.2 Whathavewestillleftout? . ........ 94 6 Lattice vibrations 95 6.1 1-dimensionalmass&springchain . ............ 96 6.1.1 How many solutions to expect . ....... 97 6.1.2 1 dimensional chain with a two-atom basis . ............ 99 6.2 Writing the lattice vibrations in 3D . .............. 102 6.3 The lattice as a collection of harmonic oscillators . ................... 103 6.3.1 The harmonic oscillator revisited . ............ 103 6.3.2 Usingtheladderoperators . ........ 105 6.4 Internal energy and specific heat of the lattice . ................. 107 6.4.1 Low and high temperature limits of the Debye formula . .............. 108 6.5 Electrodynamics in the polar crystal with no free electrons.................. 110 6.5.1 Forced vibrations of a polar lattice due to an incident electromagnetic wave . 110 6.6 Electrons and phonons together: How an electron ”sees” a phonon?. 113 6.6.1 Using the Boltzmann Transport equation . ........... 116 7 Magnetism 121 7.1 Quantum mechanical effect of the magnetic field . ............. 121 7.1.1 Forces on a magnetic dipole . ........ 122 7.2 Magnetic moment of an isolated atom: Lande g-factor & Hund’srules . .. .. .. 123 7.2.1 Landeg-factor ................................. 124 7.2.2 Hund’srules ................................... 126 7.3 The electron gas in a magnetic field: oscillatory phenomena................. 129 7.4 Exchange interaction and ferromagnetism . ............... 129 7.4.1 Origin of the exchange interaction between two electrons............... 131 7.5 The ferromagnetic state from Coulomb interaction . ................. 134 7.6 The state with all spins pointing in the same direction . .................. 135 7.7 Astatewithonespinflipped . ......... 138 7.8 Magnitude of J : Why is Iron ferromagnetic? . ............. 140 7.8.1 The Heisenberg spin hamiltonian finally . ............ 141 7.8.2 Decrease of magnetisation of the excited states . ............... 142 A Fermi Surface, electrons and holes 145 A.1 Thefreeelectron”squarium” . ........... 145 A.1.1 1electronperatom .............................. 147 A.1.2 2electronsperatom ............................. 148 A.1.3 3electronsperatom ............................. 150 A.2 4electronsperatom ............................... ........ 151 A.3 Doingthesamein3D ................................ 154 A.3.1 1electronperatom .............................. 154 A.3.2 2electronsperatom ............................. 155 A.3.3 3electronsperatom ............................. 158 A.4 Electronsandholes............................... ......... 160 A.4.1 The equation of motion of a particle in a band . ........... 160 A.4.2 Signoftheeffectivemass . 160 A.4.3 Direction of the gradient . ......... 160 CONTENTS 1 A.4.4 Currentcarriedbyaband. ....... 161 A.4.5 The semiclassical evolution of a k-state ........................ 161 2 CONTENTS Chapter 1 Describing a collection of atoms: Structure factor and lattice Lecture notes for PH409 (Jul-Dec 2013): K. Das Gupta In elementary quantum mechanics, we usually work with a single particle. In condensed matter physics - we need to describe lots of particles together.
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