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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
MaOhiessen Rule Agustus MaOhiesen (1864)
ρ T( ) = ρ0
purity-dependent
+ ρin (T )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= ρnn∑
Wiedemann-‐Franz Law
Gustav Wiedemann and Rudolph Franz (1853)
thermal conductivityelectrical conductivity
= const for a given T
Ludvig Lorentz (1872)
thermal conductivityelectrical conductivity iT
= const
"Lorentz number"=π 2
3kBe
⎛⎝⎜
⎞⎠⎟
2
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
0 C 100 C
Lexp i108 WiOhm/K2 Ltheor = 2.45i10−8 WiOhm/K2
Hall Effect
Edwin Hall (1879, PhD)
Drude model Paul Drude (1900)
Drude model
dpdt
= −eE− ev ×B− pτ
dc conductivity: σ = jE= ne
2τm
Hall constant: RH = VHj i B
= − 1en
Lorentz number= 13
kBe
⎛⎝⎜
⎞⎠⎟
2
as compared to the correct value π2
3kBe
⎛⎝⎜
⎞⎠⎟
2
Assump>ons of the Drude model
Maxwell-‐Boltzmann staLsLcs 12m v2 = 3
2kBT
Wrong. In metals, electrons obey the Fermi-‐Dirac sta>s>cs m2v2 ≈ const(T )
ScaOering mechanism: collisions between electrons and laRce
Wrong. QM bandstructure theory: electrons are not slowed down by a periodic array of ions; instead, they behave of par>cles of different mass
Classical dynamics (second law)
Quantum mechanics was not invented yet...
Yet, σ =ne2τ /m does not contain the electron velocityThe formula still works if τ is understood as phenomenological parameter
Great predic>on of the Drude model
dc conductivity: σ = jE= ne
2τm
Hall constant: RH = VHj i B
= − 1en
By measuring these two quanLLes one can separate the T dependences of the relaxaLon Lme and the electron number density
Metals and insulators
T
ρ−RH
Metals: number density is T independent relaxa>on >me is T dependendent
n = −1/ eRH
Insulators: free carriers freeze out as T goes down
Sommerfeld theory of metals
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 scaOering ☐ posi>ve value of the Hall constants in certain metals ☐ positive magnetoresistance (an increase of the resistivity with B) ☐
f E( ) kBT
Fermi sphere
kFEF
EF =2k2
F
2m43π k 3
F = 2π( )3 n
Metals: EF = 1÷10 eV EF / kB = 104 ÷105 K
Arnold Sommerfeld (PhD, 1928)
Quantum-‐mechanical theory electron dynamics
Felix Bloch (1928, PhD)
interference of electron waves scaOered by ionsenergy bands
E
allowed
forbidde
n
metal
insulator
µ
µ
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 If a band is more than half fulleffec>ve carriers are “holes” Holes=posi>vely charged electronsRH>0
a
phase shic between incoming and reflected waves
µ
2ka
2ka = πN ⇒λ = 2πk
= N a2
Shroedinger equa>on with a periodic poten>al energy
− 2
2m∇2 +U r( )⎡
⎣⎢
⎤
⎦⎥ψ = Eψ
U r + n1a1 + n2a2 + n3a3( ) =U r( ); n1,2,3 = 0,±1,±2...
Symmetries of lafce determine proper>es of the eigenstates
Bloch Theorem
ψ k r( ) = eikiruk r( )uk r + a( ) = uk r( )E k( ) = E k + b( )
bi = 2π( )3 a j × akV
a1a2
a3pseudo (crystal momentum)
k and k + b are equivalent
a1
a2
17
Bravais lafces in 3D: 14 types, 7 classes
Poα −
Ag,Au,Al,Cu,Fe,Cr,Ni,Mb…
Ba,Cs,Fe,Cr,Li,Na,K,U,V…
Sb,Bi,Hg
He,Sc,Zn,Se,Cd…
S,Cl,Br
F
1. Cubic ✖3 2. Tetragonal✖2 3. Hexagonal✖1 4. Orthorhombic✖4 5. Rhombohedral✖1 6. Monoclinic✖2 7. Triclinic✖1
Auguste Bravais (1850)
Lafce 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
T
CVDulong-‐Pe>t
room
Albert Enstein: quantum monochroma>c oscillators modern language: op>cal phonons Paul Debye: quantum sound waves modern language: acous>c phonons “Black-‐body radia>on”
Max Born: modern theory of lafce dynamics Important consequence: electrons are not slowed down because of scaOering at sta1onary ions. But they are slowed down by scaOering from vibra>ng ions. This is why relaxation time depends on T! T
T 3
CV ∝T 3
Max von Laue (Nobel Prize 1914)
X-‐ray scaOering from crystals: confirma>on of periodicity
William Lawrence Bragg and William Henry Bragg ( 1913)
Bragg’s law
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)
Lev Landau: phenomenological two-‐fluid model (1941) Nikolay Bogolyubov:
canonical transforma>ons (1947-‐1948)
Richard Feynman: ver>ces (1955)
He-‐4 atoms are bosons
Bose-‐Einstein condensa>on into the lowest energy state.
T < Tλ = 4.2 K @1 atm
T > Tλ T < Tλ
Electrons are fermions. How to make bosons out of fermions? Pair them!
Herbert Froelich
Two types of interac>on among electrons in metals: i) Coulomb repulsion ii) Phonon-‐mediated aOrac>on
Normal metals: Coulomb repulsion dominates Superconductors: phonon-‐mediated aOrac>on dominates below the cri>cal temperature
Cooper pairs Leon Cooper
Bardeen-‐Cooper-‐Schrieffer Theory of Superconduc>vity (1957)
John Bardeen Leon Cooper
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 BraOain, William Shockley (Nobel Prize 1956)
Integer Quantum Hall Effect (1980)
Klaus von Klitzing (Nobel Prize 1985)
von Klitzing constant
Value 25 812.807 4434 Standard uncertainty 0.000 0084 Rela>ve standard uncertainty 3.2 x 10-‐10
Theore>cal explana>on: Robert Laughlin
RK = h / e2
Frac>onal quantum Hall effect (1982)
Dan Tsui, Horst Stormer, Robert Laughlin: Nobel Prize, 1998
Dan Tsui
Horst Stormer
Robert Laughlin
quantization of ρxy in fractions of h / e2
1 / 3,1 / 5,5 / 2...Each plateau is a new elementary excita>on with a frac>onal electric charge!
Solid statenanoscience
2D: electron gases, graphene
Konstan>n Novoselov
Andre Geim Nobel Prize 2010
1D: carbon nanotubes and quantum wires