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 =∑ n ρ 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 i T
π 2 ⎛ kB ⎞
"Lorentz number"=
⎜
⎟
3 ⎝ e ⎠
2
Ltheor = 2.45i10 −8 WiOhm/K 2
Lexp i10 8 WiOhm/K 2
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
VH
1
Hall constant: RH =
=−
jiB
en
1 ⎛ kB ⎞
Lorentz number= ⎜ ⎟
3⎝ e ⎠
2
π 2 ⎛ kB ⎞
as compared to the correct value
⎜ ⎟
3 ⎝ e⎠
2
Assump>ons of the Drude model Maxwell-­‐Boltzmann staLsLcs 1
3
m v 2 = kBT
2
2
m 2
v ≈ const(T )
Wrong. In metals, electrons obey the Fermi-­‐Dirac sta>s>cs 2
Classical dynamics (second law) Quantum mechanics was not invented yet... 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 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 = =
E
m
VH
1
Hall constant: RH =
=−
jiB
en
By measuring these two quanLLes one can separate the T dependences of the relaxaLon Lme and the electron number density Metals and insulators ρ
−RH
n = −1 / eRH
T
Metals: number density is T independent relaxa>on >me is T dependendent Insulators: free carriers freeze out 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 scaOering ☐  posi>ve value of the Hall constants in certain metals ☐  positive magnetoresistance (an increase of the resistivity with B) ☐
f (E)
 2 k 2F
EF =
2m
4 3
3
π k F = ( 2π ) n
3
Metals: EF = 1 ÷ 10 eV
kBT
kF
EF
Fermi sphere EF / kB = 10 4 ÷ 10 5 K
Quantum-­‐mechanical theory electron dynamics Felix Bloch (1928, PhD) interference of electron waves scaOered by ionsenergy bands Posi>on of the chemical poten>al is determined by the number of the electrons µ
µ
µ
insulator metal allowed forbidden E If a band is less than half ful leffec>ve carriers are electrons RH<0 If a band is more than half fulleffec>ve carriers are “holes” Holes=posi>vely charged electronsRH>0 phase shic between incoming and reflected waves 2ka
a 2π
a
2ka = π N ⇒ λ =
=N
k
2
Shroedinger equa>on with a periodic poten>al energy ⎡ 2 2
⎤
⎢ − 2m ∇ + U ( r ) ⎥ψ = Eψ
⎣
⎦
U ( r + n1a1 + n2 a 2 + n3a 3 ) = U ( r ); n1,2,3 = 0, ±1, ±2...
a1
a2
Symmetries of lafce determine proper>es of the eigenstates Bloch Theorem ψ k ( r ) = eikir uk ( r )
uk ( r + a ) = uk ( r )
E (k) = E (k + b)
b i = ( 2π )
3
a j × ak
V
a3
pseudo (crystal momentum) k and k + b are equivalent
a1
a2
Bravais lafces in 3D: 14 types, 7 classes Ag,Au,Al,Cu,Fe,Cr,Ni,Mb… 1.
2.
3.
4.
5.
6.
7.
Cubic ✖3 Tetragonal✖2 Hexagonal✖1 Orthorhombic✖4 Rhombohedral✖1 Monoclinic✖2 Triclinic✖1 Ba,Cs,Fe,Cr,Li,Na,K,U,V… α − Po
He,Sc,Zn,Se,Cd… Auguste Bravais (1850) S,Cl,Br F Sb,Bi,Hg 17 Lafce dynamics Classical thermodynamics: specific heat for a system of coupled oscillators (Dulong-­‐Pe>t law) CV = 3kB n
Experiment: marked devia>ons from the Dulong-­‐Pe>t law CV
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” 3 Dulong-­‐Pe>t CV ∝T
T
3
room T
T
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! X-­‐ray scaOering 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 Meissner-­‐Ochsenfeld effect (1933) Walther Meissner Co. Scien>fic American 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 aOrac>on Herbert Froelich Leon Cooper Normal metals: Coulomb repulsion dominates Superconductors: phonon-­‐mediated aOrac>on dominates below the cri>cal temperature Cooper pairs Bardeen-­‐Cooper-­‐Schrieffer Theory of Superconduc>vity (1957) John Bardeen Robert Schrieffer Leon Cooper 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) 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 quantization of ρ xy 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 statenanoscience 2D: electron gases, graphene Nobel Prize 2010 Konstan>n Novoselov Andre Geim 1D: carbon nanotubes and quantum wires