Doped Mott insulators and high temperature superconductivity T. Senthil (MIT) and M. Randeria (OSU) Earlier version: lectures by Patrick Lee and TS at MIT, Sept 09. Thanks to Patrick Lee for some of the slides. Wednesday, December 15, 2010 Plan Lecture 1: The problem of doping a Mott insulator Lecture 2: Cuprate phenomenology + minimal theory Lecture 3: More cuprate phenomenology + minimal theory Secondary goal: introduction to modern experimental probes of electronic solids Wednesday, December 15, 2010 Outline for Lecture 1 1. Cuprates as doped Mott insulators 2. Magnetism and Mott insulators 3. Doping a Mott insulator -(i) some general theoretical questions -(ii) experiments on a few materials. Wednesday, December 15, 2010 Outline for Lecture 1 1. Cuprates as doped Mott insulators 2. Magnetism and Mott insulators 3. Doping a Mott insulator -(i) some general theoretical questions -(ii) experiments on a few materials. Wednesday, December 15, 2010 Wednesday, December 15, 2010 What is a Mott insulator? Insulation due to jamming effect of Coulomb repulsion Coulomb cost of two electrons occupying same atomic orbital dominant ⇒Electrons can’t move if every possible atomic orbital site is already occupied by another electron. Odd number of electrons per unit cell: band theory predicts metal. Wednesday, December 15, 2010 A useful theoretical model Wednesday, December 15, 2010 Complications in many real Mott insulators 1. Orbital degeneracy: More than one atomic orbital may be available for the electron to occupy at each site. 2. Multi-band model may be more appropriate starting point (definitely so if there is orbital degeneracy) 3. Spin-orbit interactions 4. (Obviously) must include long range Coulomb +............................. In this lecture I will primarily consider situations in which many of these complications (mainly 1-3) are likely unimportant. Fortunately the cuprates likely fall in this class! Wednesday, December 15, 2010 When Mott insulator? Periodic Table of Elements 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 1 Atomic # 2 2 K 1 H Symbol He Hydrogen Name Solid Metals Nonmetals Helium 1.00794 Atomic Mass C 4.002602 Alkali metals Alkali earth metals Alkaline metals Transition metals Poor nonmetals Other gases Noble 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 K 1 2 Hg Liquid Lanthanoids 3 4 5 6 7 8 L 2 Li Be B C N O F Ne Lithium Beryllium Boron Carbon Nitrogen Oxygen Fluorine Neon 6.941 9.012182 H Gas 10.811 12.0107 14.0067 15.9994 18.9984032 20.1797 2 2 Actinoids 2 2 2 2 2 2 K 11 8 12 8 13 8 14 8 15 8 16 8 17 8 18 8 L 1 2 Unknown 3 4 5 6 7 8 M Rf 3 Na Mg Al Si P S Cl Ar Sodium Magnesium Aluminium Silicon Phosphorus Sulfur Chlorine Argon 22.98976928 24.3050 26.9815386 28.0855 30.973762 32.065 35.453 39.948 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 K 19 8 20 8 21 8 22 8 23 8 24 8 25 8 26 8 27 8 28 8 29 8 30 8 31 8 32 8 33 8 34 8 35 8 36 8 L 8 8 9 10 11 13 13 14 15 16 18 18 18 18 18 18 18 18 M 4 K 1 Ca 2 Sc 2 Ti 2 V 2 Cr 1 Mn 2 Fe 2 Co 2 Ni 2 Cu 1 Zn 2 Ga 3 Ge 4 As 5 Se 6 Br 7 Kr 8 N Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton 39.0983 40.078 44.955912 47.867 50.9415 51.9961 54.938045 55.845 58.933195 58.6934 63.546 65.38 69.723 72.64 74.92160 78.96 79.904 83.798 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 K 37 8 38 8 39 8 40 8 41 8 42 8 43 8 44 8 45 8 46 8 47 8 48 8 49 8 50 8 51 8 52 8 53 8 54 8 L 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 M 8 8 9 10 12 13 14 15 16 18 18 18 18 18 18 18 18 18 N 5 Rb 1 Sr 2 Y 2 Zr 2 Nb 1 Mo 1 Tc 1 Ru 1 Rh 1 Pd 0 Ag 1 Cd 2 In 3 Sn 4 Sb 5 Te 6 I 7 Xe 8 O Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium Palladium Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon 85.4678 87.62 88.90585 91.224 92.90638 95.96 (97.9072) 101.07 102.90550 106.42 107.8682 112.411 114.818 118.710 121.760 127.60 126.90447 131.293 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 K 55 8 56 8 72 8 73 8 74 8 75 8 76 8 77 8 78 8 79 8 80 8 81 8 82 8 83 8 84 8 85 8 86 8 L 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 M 18 18 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 N 6 Cs 8 Ba 8 57–71 Hf 10 Ta 11 W 12 Re 13 Os 14 Ir 15 Pt 17 Au 18 Hg 18 Tl 18 Pb 18 Bi 18 Po 18 At 18 Rn 18 O Caesium 1 Barium 2 Hafnium 2 Tantalum 2 Tungsten 2 Rhenium 2 Osmium 2 Iridium 2 Platinum 1 Gold 1 Mercury 2 Thallium 3 Lead 4 Bismuth 5 Polonium 6 Astatine 7 Radon 8 P 132.9054519 137.327 178.49 180.94788 183.84 186.207 190.23 192.217 195.084 196.966569 200.59 204.3833 207.2 208.98040 (208.9824) (209.9871) (222.0176) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 K 87 8 88 8 104 8 105 8 106 8 107 8 108 8 109 8 110 8 111 8 112 8 113 8 114 8 115 8 116 8 117 118 8 L 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 M 32 32 89–103 32 32 32 32 32 32 32 32 32 32 32 32 32 32 N 7 Fr 18 Ra 18 Rf 32 Db 32 Sg 32 Bh 32 Hs 32 Mt 32 Ds 32 Rg 32 Uub 32 Uut 32 Uuq 32 Uup 32 Uuh 32 Uus Uuo 32 O Francium 8 Radium 8 Ru herfordium 10 Dubnium 11 Seaborgium 12 Bohrium 13 Hassium 14 Meitnerium 15 Darmstadtium 17 Roentgenium 18 Ununbium 18 Ununtrium 18 Ununquadium 18 Ununpentium 18 Ununhexium 18 Ununseptium Ununoctium 18 P (223) 1 (226) 2 (261) 2 (262) 2 (266) 2 (264) 2 (277) 2 (268) 2 (271) 1 (272) 1 (285) 2 (284) 3 (289) 4 (288) 5 (292) 6 (294) 8 Q For elements with no stable isotopes, the mass number of the isotope with the longest half-life is in parentheses. Periodic Table Design and Interface Copyright © 1997 Michael Dayah. http://www.ptable.com/ Last updated: May 27, 2008 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 57 8 58 8 59 8 60 8 61 8 62 8 63 8 64 8 65 8 66 8 67 8 68 8 69 8 70 8 71 8 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 19 21 22 23 24 25 25 27 28 29 30 31 32 32 La 9 Ce 9 Pr 8 Nd 8 Pm 8 Sm 8 Eu 8 Gd 9 Tb 8 Dy 8 Ho 8 Er 8 Tm 8 Yb 8 Lu 9 Lanthanum 2 Cerium 2 Praseodymium 2 Neodymium 2 Promethium 2 Samarium 2 Europium 2 Gadolinium 2 Terbium 2 Dysprosium 2 Holmium 2 Erbium 2 Thulium 2 Ytterbium 2 Lutetium 2 138.90547 140.116 140.90765 144.242 (145) 150.36 151.964 157.25 158.92535 162.500 164.93032 167.259 168.93421 173.054 174.9668 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 89 8 90 8 91 8 92 8 93 8 94 8 95 8 96 8 97 8 98 8 99 8 100 8 101 8 102 8 103 8 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 Ac 18 Th 18 Pa 20 U 21 Np 22 Pu 24 Am 25 Cm 25 Bk 27 Cf 28 Es 29 Fm 30 Md 31 No 32 Lr 32 Actinium 9 Thorium 10 Protactinium 9 Uranium 9 Neptunium 9 Plutonium 8 Americium 8 Curium 9 Berkelium 8 Californium 8 Einsteinium 8 Fermium 8 Mendelevium 8 Nobelium 8 Lawrencium 9 (227) 2 232.03806 2 231.03588 2 238.02891 2 (237) 2 (244) 2 (243) 2 (247) 2 (247) 2 (251) 2 (252) 2 (257) 2 (258) 2 (259) 2 (262) 2 Michael Dayah For a fully interactive experience, visit www.ptable.com.