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Quasiparticles in Solids: Electrons, Phonons, Plasmons, …
Ground state properties of a solid
(can be measured by elastic scattering, E=0)
Property
Measurement
charge density

X-ray diffraction
spin density
 
neutron diffraction
total energy
U
heat of formation
magnetic moment
force constant
U/B
f = 2U/r2
torque in B-field
force vs. displacement
Excited state properties of a solid (require excitation by inelastic scattering, E0)
Quasiparticle
Wave, oscillating quantity
electron, hole
wave function e ,h
photon
electromagnetic field A
phonon
vibration amplitude r
plasmon
charge density 
magnon
magnetization 
polariton = photon + phonon
electric polarization P
polaron
= electron + phonon
eandr
exciton
= electron + hole
eandh
All quasiparticles (= excitations in a solid) are fully characterized by quantum numbers:
Energy E = ћ, momentum p = ћk and angular symmetry (given by the point group).
These can be displayed by the band dispersion E(k) with labels for spin and symmetry.
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Band Dispersion E(k):
Electrons in Silicon
Conduction
Band
Band Gap EG
px,y
pz
Valence
Band
s
[111]
[100]
[110]
k
Phonons in Silicon
66 meV
L

[111]


[100]
X
K

[110]

k
For the k-axis labels see the fcc Brillouin zone in Lecture 10, Slide 3.
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Phonon Bands in the Diamond Structure:
The diamond structure has two atoms per unit cell, which have 2  3 = 6 degrees of freedom.
Therefore, there are 6 branches: 2TA, 2TO, 1LA, 1LO.
E=ћ
2 TO
1 LO
1 LA
2 TA
0
0
k
½Ghkl (zone boundary)
The general properties of various phonon modes are:

For transverse phonons the atoms move perpendicular to the k-vector (= propagation
direction)

For acoustic phonons two atoms in the same unit cell have the same phase, for optical
phonons the two are out of phase.

Atoms in adjacent unit cells have the same phase at the Brillouin zone center k=0.
They are out of phase at the zone boundary ½Ghkl and back in phase at Ghkl .
At the zone center:
TA phonons:
TO phonons:
k
r
At the zone boundary:
TA phonons (heavy atom moves):
TO phonons (light atom moves):
r
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Calculation of (k) for Phonons:
Consider the simplest case, a linear chain of equal atoms with only nearest neighbors
interacting with each other. Start with one unit cell and then connect the unit cells using a
plane wave ansatz. Each atom is displaced from the lattice, plus the atoms are displaced
relative to each other.
n1
n
n+1
a
a
urn
(1) Hooke’s law: Fn = f  (un+1 un) + f  (un1 un)
f = force constant,
un = δr = displacement
(2) Newton :
F=M a
Fn = M  d2un /dt2
F = force, M = mass
a = acceleration
(3) Plane wave ansatz:
ei ( k r   t ) , r = n a
un = ue i ( k n a   t )
u = amplitude
k = wave vector
a = lattice constant
(1) Fn = f  (un+1 + un1 2un) = f  (e i k a un + e  i k a un 2un) = f  [2cos(ka) 2]un
(3)
(2) Fn = M  d2un /dt2 = M  2 un
(3)
  = (2f /M)½  [1 cos(ka)]½ = 2(f /M)½  |sin(ka)|
This describes an acoustic phonon:

Classical physics gets the shape
of the phonon bands right, but
to calculate the force constant f
and the lattice constant a one
needs quantum physics. With
density functional theory (DFT)
one can obtain both f and a by
calculating the total energy U as
a function of the bond length.
0
k
/a
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General Properties of Waves:
Particle-wave duality in quantum physics connects each particle with energy E = ћ and
momentum p = ћk to a plane wave (r,t) = A  exp[i (krt)]
Velocity and Effective Mass
vph = E/|p| = /|k|
Phase velocity, for the propagation of an infinite plane wave.
vg = E/p = /k
Group velocity, for the propagation of a wave packet.
= slope of E(p)
It determines the speed of signals and of energy transport (vg c).
m* =
1
2E/p2
Effective mass, modeled after the kinetic energy E = p2/2m.
= inverse curvature of E(p)
A wave packet spreads over
time, if the phase velocity
depends on the frequency 
(“dispersion”). Waves with
higher frequency tend to move
faster due to their higher
energy E=ћ. The center of a
wave packet moves with the
group velocity vg . That determines how fast a signal pulse
propagates.
Solitons
In a non-linear medium, the phase velocity depends on the amplitude. The spread of a
wave packet due to dispersion can be compensated by an opposite spread due to nonlinearity. In the figure above the low frequency components with high amplitude are able
to catch up with the weaker high frequency components if the phase velocity increases
with amplitude. The result is a wave packet that does not change over time. Solitons
are used in long distance communications via fiber optics. A tsunami is a soliton.
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