Overview Solid = collection of atoms (~1023, a huge number) Crystalline: single crystals, polycrystalline materials Amorphous, glassy materials Others: liquid crystals, quasicrystals, partially ordered polymers, . Theory of solids to explain and predict p
Crystallographic restriction theorem See: http:/en.wikipedia.org/wiki/Crystallographic_restriction_theorem Problem: Consider rotational symmetry operations of a 2D periodic lattice with the rotation axis perpendicular to the lattice plane. Show that only
Ferromagnetism A simple picture: ferromagnetism arises from "atomic magnets" aligned in the same direction. What is the interaction to align them? Each atomic magnetic has a magnetic moment ~ g0 B . Dipolar interaction energy between neighboring atomic ma
Magnetism B H 4 M H M = magnetization
Bi Bo Bournday conditions: B 0 4 1 D H j 0 , assuming steady state and no free current c c t
H i H o
Ferromagnetic Paramagnetic Diamagnetic
M 0 for B a 0 (zero applied field); generally nonlinear BH ~ constant ~ 1 + O
Infrared Properties (of Ionic Crystals) Assume the system is nonmagnetic, 1 . Optical properties determined by the dielectric function, D E. Assume cubic symmetry. D E Assume no free charges. D 0 DEP ik E 0
E ik E 4 ( = bound charge density) Longitudinal
Neutron Scattering Counts E E' E Spectrum at a fixed scattering geometry: elastic peak caused by defect scattering (or Bragg diffraction) one-phonon emission peaks at lower energies (Stokes peaks) one-phonon absorption peaks at higher energies (anti-Stoke
Anharmonic Effects Assume one atom per unit cell. Potential energy U r R U
R u R
U U 0 harmonic potential u 2 O u 3 O u 4
Terms O u 3 , O u 4 anharmonic effects or phonon-phonon interactions uR 1 N
ks
2 M s (k )
a
ks
+ a-ks e s (k )eik R
O u 3 . a a a
Lattice Specific Heat Classical: Equi-partition theorem:
1 1 1 kx x2 mvx 2 kT 2 2 2
6 degrees of freedom each atom 3 each from kinetic energy and potential energy. Internal energy/volume u Volume specific heat CV 3
3NkT ; N = total number of atoms (includ
Lattice Waves Thus far, static lattice model. In reality, atoms vibrate even at T 0 because of zero-point vibration. Monatomic Crystals Basis = 1 atom.
r R R u R;t
Lattice: R ni ai
i
i 1,2,3
Actual atomic position = lattice position + vibration
u 0
r
ave
Electron Dynamics (of Semiconductors) Basic assumption: independent electron approximation (single-particle picture) Question: what happens to an electron under an external field (transport properties)? Bloch state nk e ik r uk r
p nk i nk e ik r p k uk
Screening (Electron-Electron Interaction) Classical picture
Eext
Classical metal Ein 0
The internal field is zero. Electrons are free to move; a layer of charge (screening charge) accumulates at the surface. The external field is screened. Quantum picture
Density of States: g Definition: g d = number of states between and d (per unit volume) For a single band k , g d
1 V
1 8 d
3 k ,s d
d
2
3
k
1 4 3
dS dk
d
dk
dS = area element on constant energy surface k dk = change in k, along k k (perpendicular to d
Band Structure Calculations using a plane wave basis set
2 2 k r U r k r k k r 2m
Fourier transform: U r UK e iK r
K
k r e ik r uk r e ik r ck-K e iK r e ik r ck+K e iK r
K K
2 2m
c k K
k-K K
2
e
i k K r
UK e iK r ck-K e
K K
i k K r
k
c
K
k-K
e
i k
Electronic States in a Crystal ~1023 interacting electrons and ion cores a big computational problem.
p2 1 p2 Z Z e2 1 e2 n n n HT i i 2m 2 i , j ri r j n 2Mn 2 n,n ' R n R n Hspin-orbit-interaction . Hel H core H el core Hsoi . Z e2 n n,i ri R n
Frozen
X-Ray Diffraction A method to determine crystal structure. Assumption: ignoring thermal vibrations of lattice (including zero-point vibrations) for now. A charged particle of charge q in an EM field:
1 q H p A V r 2m c
2
1 2 q q q2 p A p p A 2 A2 V r 2m
Reciprocal Lattice Direct lattice given by R ni ai
i
Definition: reciprocal lattice is a Bravais lattice given by
K = reciprocal lattice vectors i bi , where i = integers, and
a2 a3 a 3 a1 a1 a 2 b1 2 ; b2 2 ; b3 2 a1 a 2 a 3 a1 a 2 a 3 a1 a 2 a 3
Useful
Review Drude model (free electron approximation) + (independent electron approximation) + (Maxwell Boltzmann statistics) Sommerfeld model (free electron approximation) + (independent electron approximation) + (Fermi Dirac statistics) Next: Band structure
Sommerfeld Theory of Metals a quantum theory of independent free electrons Physical picture: Focus on one electron. Assume all other charges in the system are smeared out into a static neutral background (atomic details, fluctuations, & correlations ignor
Physics 560 Homework problem set 1 1. A&M Problem 1, Chapter 1. 2. A&M Problem 2, Chapter 1. 3. A&M Problem 3, Chapter 1. 4. Recall 1
i 4
, where is the conductivity derived from the valence electrons.
This expression was derived assuming that the posit