Brief History of Solid State Physics

Brief History of Solid State Physics

 Along with astronomy, the oldest subfield of what we now refer to as Physics.  Pre-scienfic 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 noon 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 L = 2.45i10−8 WiOhm/K2 L i108 WiOhm/K2 theor exp

  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 Assumpons of the Drude model

1 3 m v2 = k T Maxwell-Boltzmann stascs 2 2 B

m 2 Wrong. In metals, electrons obey the Fermi-Dirac stascs 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 parcles 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 predicon of the Drude model

j ne2τ dc conductivity: σ = = By measuring these two quanes 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 stascs .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 ☐ .posive 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 Posion of the chemical potenal is determined by the number of the electrons

If a band is less than half ful leffecve carriers are electrons RH<0 µ If a band is more than half fulleffecve carriers are “holes” µ Holes=posively charged electronsRH>0 insulator µ metal phase shi between incoming and reflected waves 2ka

allowed forbidden 2π a a 2ka = π N ⇒ λ = = N k 2 Shroedinger equaon with a periodic potenal 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 properes 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-Pet law)

CV = 3kBn Experiment: marked deviaons from the Dulong-Pet law

Albert Enstein: quantum monochromac oscillators modern language: opcal phonons Paul Debye: quantum sound waves Dulong-Pet CV modern language: acousc 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 staonary 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 superconducvity -1911

Kamerlingh Onnes

Co. Scienfic American Meissner-Ochsenfeld effect (1933)

Walther Meissner Superfluidity (moon without fricon) in He-4

Pyotr Kapitsa (1937) John F. Allen and Don Misener (1937)

T < Tλ = 4.2 K @1 atm Richard Feynman: verces (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 interacon among electrons in metals: i) Coulomb repulsion ii) Phonon-mediated aracon

Normal metals: Coulomb repulsion dominates Superconductors: phonon-mediated aracon dominates Herbert Froelich below the crical temperature

Leon Cooper Cooper pairs Bardeen-Cooper-Schrieffer Theory of Superconducvity (1957)

Leon Cooper John Bardeen Robert Schrieffer High-temperature superconducvity 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) Theorecal explanaon: Robert Laughlin Fraconal 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 fraconal electric charge! Solid statenanoscience

2D: electron gases, graphene

Konstann Andre Novoselov Geim Nobel Prize 2010 1D: carbon nanotubes and quantum wires