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 mathema cs: group theory. Early discoveries Ma hiessen Rule
Agustus Ma hiesen (1864)
ρ(T ) = ρ + ρin (T ) 0 purity-dependent material- but not purity-dependent
ρin (T ) ∝ T (for T > 50 ÷ 70 K) Interpreta on
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 Assump ons of the Drude model
1 3 m v2 = k T Maxwell-Boltzmann sta s cs 2 2 B
m 2 Wrong. In metals, electrons obey the Fermi-Dirac sta s cs v ≈ const(T ) 2 Classical dynamics (second law)
Quantum mechanics was not invented yet... Sca ering mechanism: collisions between electrons and la ce
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 quan es E m one can separate the T dependences VH 1 of the relaxa on 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 relaxa on me is T dependendent as T goes down Sommerfeld theory of metals Arnold Sommerfeld (PhD, 1928) free electrons obeying Fermi-Dirac sta s 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 sca ering ☐ .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 sca ered by ionsenergy 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 leffec ve carriers are electrons RH<0 µ If a band is more than half fulleffec ve carriers are “holes” µ Holes=posi vely charged electronsRH>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 equa on 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 la ce 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 la ces 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 La ce dynamics
Classical thermodynamics: specific heat for a system of coupled oscillators (Dulong-Pe t law)
CV = 3kBn Experiment: marked devia ons from the Dulong-Pe t law
Albert Enstein: quantum monochroma c oscillators modern language: op cal phonons Paul Debye: quantum sound waves Dulong-Pe t CV modern language: acous c phonons “Black-body radia on” 3 CV ∝ T Max Born: modern theory of la ce dynamics Important consequence: electrons are not slowed down because of sca ering at sta onary ions. 3 room But they are slowed down by sca ering from T T T vibra ng ions. This is why relaxation time depends on T! X-ray sca ering from crystals: confirma on 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 transforma ons (1947-1948) He-4 atoms are bosons Bose-Einstein condensa on 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 a rac on
Normal metals: Coulomb repulsion dominates Superconductors: phonon-mediated a rac 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 Bra ain, 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 Rela ve standard uncertainty 3.2 x 10-10
Klaus von Klitzing (Nobel Prize 1985) Theore cal explana on: 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 excita on 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