Brief History of Solid State Physics Along with astronomy, the oldest subfield of what we now refer to as Physics. Pre-scien>fic >mes: stones, bronzes, iron, jewelry...Lots of empirical knowledge but, prior to the end of the 19th century, almost no understanding. Crystals: periodic structures of atoms and molecules. A common no>on in crystallography and mineralogy well before the periodic structure was proven by X-rays (1912). Special branch of mathemacs: group theory. Early discoveries Mahiessen Rule Agustus Mahiesen (1864) ρ(T ) = ρ + ρin (T ) 0 purity-dependent material- but not purity-dependent ρin (T ) ∝ T (for T > 50 ÷ 70 K) Interpretaon ρ0 : impurities, defects... ρin :lattice vibrations (phonons) In general, all sources of scattering contribute: ρ= ρ ∑n n Wiedemann-Franz Law Gustav Wiedemann and Rudolph Franz (1853) thermal conductivity = const for a given T electrical conductivity Ludvig Lorentz (1872) thermal conductivity = const electrical conductivity iT 2 π 2 ⎛ k ⎞ "Lorentz number"= B ⎝⎜ ⎠⎟ 3 e −8 2 8 2 Ltheor = 2.45i10 WiOhm/K Lexp i10 WiOhm/K 0 C 100 C Ag 2.31 2.37 Au 2.35 2.40 Cd 2.42 2.43 Cu 2.23 2.33 Pb 2.47 2.56 Pt 2.51 2.60 W 3.04 3.20 Zn 2.31 2.33 Ir 2.49 2.49 Mo 2.61 2.79 Hall Effect Edwin Hall (1879, PhD) Drude model Paul Drude (1900) Drude model dp p = −eE − ev × B − dt τ j ne2τ dc conductivity: σ = = E m V 1 Hall constant: R = H = − H j i B en 2 1 ⎛ k ⎞ Lorentz number= B 3⎝⎜ e ⎠⎟ 2 π 2 ⎛ k ⎞ as compared to the correct value B ⎝⎜ ⎠⎟ 3 e Assump>ons of the Drude model 1 3 m v2 = k T Maxwell-Boltzmann staLsLcs 2 2 B m 2 Wrong. In metals, electrons obey the Fermi-Dirac stas>cs v ≈ const(T ) 2 Classical dynamics (second law) Quantum mechanics was not invented yet... Scaering mechanism: collisions between electrons and lace Wrong. QM bandstructure theory: electrons are not slowed down by a periodic array of ions; instead, they behave of par>cles of different mass Yet, σ =ne2τ / m does not contain the electron velocity The formula still works if τ is understood as phenomenological parameter Great predic>on of the Drude model j ne2τ dc conductivity: σ = = By measuring these two quanLLes E m one can separate the T dependences VH 1 of the relaxaon me and the electron Hall constant: RH = = − number density j i B en Metals and insulators ρ −RH n = −1/ eRH T Metals: number density is T independent Insulators: free carriers freeze out relaxaon >me is T dependendent as T goes down Sommerfeld theory of metals Arnold Sommerfeld (PhD, 1928) free electrons obeying Fermi-Dirac stas>cs .independence of n from T .linear dependence of the specific heat in metals at low temperatures $ .correct value of the Lorentz number #$ . below room T, the Lorentz number becomes T dependent ☐ .origin of scaering ☐ .posi>ve value of the Hall constants in certain metals ☐ . positive magnetoresistance (an increase of the resistivity with B) ☐ 2k 2 E = F F 2m f (E) 4 3 3 kBT π k F = (2π ) n 3 k F Metals: EF = 1÷10 eV E 4 5 F EF / kB = 10 ÷10 K Fermi sphere Quantum-mechanical theory electron dynamics Felix Bloch (1928, PhD) interference of electron waves scaered by ions%energy bands E Posi>on of the chemical poten>al is determined by the number of the electrons If a band is less than half ful l%effec>ve carriers are electrons RH<0 µ If a band is more than half full%effec>ve carriers are “holes” µ Holes=posi>vely charged electrons%RH>0 insulator µ metal phase shic between incoming and reflected waves 2ka allowed forbidden 2π a a 2ka = π N ⇒ λ = = N k 2 Shroedinger equaon with a periodic poten>al energy ⎡ 2 ⎤ 2 U r E ⎢− ∇ + ( )⎥ψ = ψ a1 ⎣ 2m ⎦ U r + n a + n a + n a = U r ; n = 0,±1,±2... a ( 1 1 2 2 3 3 ) ( ) 1,2,3 2 Symmetries of lace determine proper>es of the eigenstates Bloch Theorem ikir ψ r = e u r a3 k ( ) k ( ) pseudo (crystal momentum) uk (r + a) = uk (r) k and k + b are equivalent E(k) = E(k + b) a1 a a 3 j × k a bi = (2π ) 2 V Bravais laces in 3D: 14 types, 7 classes Ag,Au,Al,Cu,Fe,Cr,Ni,Mb… 1. Cubic ✖3 2. Tetragonal✖2 3. Hexagonal✖1 Ba,Cs,Fe,Cr,Li,Na,K,U,V… α − Po 4. Orthorhombic✖4 5. Rhombohedral✖1 6. Monoclinic✖2 7. Triclinic✖1 He,Sc,Zn,Se,Cd… Auguste Bravais (1850) S,Cl,Br F Sb,Bi,Hg 17 Lace dynamics Classical thermodynamics: specific heat for a system of coupled oscillators (Dulong-Pe>t law) CV = 3kBn Experiment: marked deviaons from the Dulong-Pe>t law Albert Enstein: quantum monochromac oscillators modern language: op>cal phonons Paul Debye: quantum sound waves Dulong-Pe>t CV modern language: acous>c phonons “Black-body radiaon” 3 CV ∝ T Max Born: modern theory of lace dynamics Important consequence: electrons are not slowed down because of scaering at sta1onary ions. 3 room But they are slowed down by scaering from T T T vibrang ions. This is why relaxation time depends on T! X-ray scaering from crystals: confirmaon of periodicity Max von Laue (Nobel Prize 1914) Bragg’s law William Lawrence Bragg and William Henry Bragg ( 1913) Discovery of superconduc>vity -1911 Kamerlingh Onnes Co. Scien>fic American Meissner-Ochsenfeld effect (1933) Walther Meissner Superfluidity (mo>on without fric>on) in He-4 Pyotr Kapitsa (1937) John F. Allen and Don Misener (1937) T < Tλ = 4.2 K @1 atm Richard Feynman: ver>ces (1955) Lev Landau: phenomenological two-fluid model (1941) Nikolay Bogolyubov: canonical transformaons (1947-1948) He-4 atoms are bosons Bose-Einstein condensaon into the lowest energy state. T > Tλ T < Tλ Electrons are fermions. How to make bosons out of fermions? Pair them! Two types of interac>on among electrons in metals: i) Coulomb repulsion ii) Phonon-mediated arac>on Normal metals: Coulomb repulsion dominates Superconductors: phonon-mediated arac>on dominates Herbert Froelich below the cri>cal temperature Leon Cooper Cooper pairs Bardeen-Cooper-Schrieffer Theory of Superconduc>vity (1957) Leon Cooper John Bardeen Robert Schrieffer High-temperature superconduc>vity 1986 Alexander Müller Georg Bednorz non-phonon mechanism Field-effect transistor first patent: Lilienfeld (1925) working device: John Bardeen, Walter Braain, William Shockley (Nobel Prize 1956) Integer Quantum Hall Effect (1980) 2 von Klitzing constant R K = h / e Value 25 812.807 4434 Standard uncertainty 0.000 0084 Relave standard uncertainty 3.2 x 10-10 Klaus von Klitzing (Nobel Prize 1985) Theore>cal explanaon: Robert Laughlin Frac>onal quantum Hall effect (1982) Dan Tsui, Horst Stormer, Robert Laughlin: Nobel Prize, 1998 Robert Laughlin Dan Tsui Horst Stormer 2 quantization of ρxy in fractions of h / e Each plateau is a new elementary excitaon with 1/ 3,1/ 5,5 / 2... a frac>onal electric charge! Solid statenanoscience 2D: electron gases, graphene Konstan>n Andre Novoselov Geim Nobel Prize 2010 1D: carbon nanotubes and quantum wires .