Vibrational Normal Modes
or “Phonon” Dispersion Relations
in Crystalline Materials
“Phonon” Dispersion Relations
in Crystalline Materials
• So far, we’ve discussed results for the “Phonon” Dispersion
Relations ω(k) (or ω(q)) only in model, 1-dimensional lattices.
• Now, we’ll have a Brief Overview of the Phonon
Dispersion Relations ω(k) in real materials.
• Both experimental results & some of the past theoretical
approaches to obtaining predictions of ω(k) will be discussed.
– As we’ll see, some past “theories” were quite complicated in
the sense that they contained N (N >> 1) parameters which were
adjusted to fit experimental data. So, (my opinion)
They were really models & NOT true theories.
– As already mentioned, the modern approach is to solve the
electronic problem first, then calculate the force constants for the
lattice vibrational predictions by taking 2nd derivatives of the total
electronic ground state energy with respect to the atomic positions.
Two Part Discussion
Part I
• This will be a general discussion of ω(k) in crystalline
solids, followed by the presentation of some representative
experimental results for ω(k) (obtained mainly in neutron
scattering experiments) for several materials.
Part II
• This will be a brief survey of various Lattice Dynamics
models, which were used in the past to try to understand the
experimental results.
– As we’ll see, some of these models were quite complicated in
the sense that they contained LARGE NUMBERS of adjustable
parameters which were fit to experimental data.
– The modern method is to first solve the electronic problem.
Then, the force constants which for the vibrational problem are
calculated by taking various 2nd derivatives of the electronic
ground state energy with respect to various atomic displacements.
The Classical Vibrational Normal Mode Problem
(in the Harmonic Approximation)
ALWAYS reduces to solving:
Here, D(q) ≡ The Dynamical Matrix
D(q) ≡ The spatial Fourier Transform of the
“Force Constant” Matrix Φ
q ≡ wave vector,
I ≡ identity matrix
ω2 ≡ ω2(q) ≡ vibrational mode eigenvalue
NOTE!
• There are, in general, 2 distinct types of vibrational
waves (2 possible wave polarizations) in solids:
Longitudinal
• Compressional: The vibrational amplitude is
parallel to the wave propagation direction.
and
Transverse
• Shear: The vibrational amplitude is
perpendicular to the wave propagation direction.
• For each wave vector k, these 2 vibrational
polarizations will give
2 different solutions for ω(k).
• We also know that there are, at least, 2 distinct
branches of ω(k) (2 different functions ω(k) for each k)
The Acoustic Branch
• This branch received it’s name because it
contains long wavelength vibrations of the
form ω = vsk, where vs is the velocity of
sound. Thus, at long wavelengths, it’s ω vs. k
relationship is identical to that for ordinary
acoustic (sound) waves in a medium like air.
The Optic Branch
Discussed on the next page:
The Optic Branch
• This branch is always at much higher frequencies than
the acoustic branch. So, in real materials, a probe at
optical frequencies is needed to excite these modes.
• Historically, the term “Optic” came from how these
modes were discovered. Consider an ionic crystal in
which atom 1 has a positive charge & atom 2 has a
negative charge. As we’ve seen, in those modes, these
atoms are moving in opposite directions. (So, each unit cell
contains an oscillating dipole.) These modes can be excited
with optical frequency range electromagnetic radiation.
• We’ve already seen that the 2 branches have
very different vibrational frequencies ω(k).
So, when discussing the vibrational frequencies ω(k),
it is necessary to distinguish between
Longitudinal & Transverse Modes (Polarizations)
&
At the same time to distinguish between
Acoustic & Optic Modes.
• So, there are four distinct kinds of modes for ω(k).
• The terminologies used, with their abbreviations are:
Longitudinal Acoustic Modes  LA Modes
Transverse Acoustic Modes  TA Modes
Longitudinal Optic Modes  LO Modes
Transverse Optic Modes  TO Modes
A Transverse Acoustic Mode for the Diatomic Chain
The type of relative motion illustrated here carries over
qualitatively to real three-dimensional crystals.
The vibrational amplitude is highly exaggerated!
This figure illustrates the case in which the lattice has
some ionic character, with + & - charges alternating:
A Transverse Optic Mode for the Diatomic Chain
The type of relative motion illustrated here carries over
qualitatively to real three-dimensional crystals.
The vibrational amplitude is highly exaggerated!
This figure illustrates the case in which the lattice has
some ionic character, with + & - charges alternating:
Polarization & Group Velocity
A crystal with 2 atoms or more per unit cell
will ALWAYS have BOTH Acoustic & Optic Modes.
If there are n atoms per unit cell in 3 dimensions,
there will ALWAYS be 3 Acoustic Modes & 3n -3 Optic Modes.
Vibrational Group
Velocity:
Frequency, 
Acoustic Modes
0
d
vg 
dK
Speed of Sound:
Wave vector, K
(p/a)
d
vs  lim
K 0 dK
Polarization
Lattice Constant, a
TA & TO
LA & LO
xn
For 2 atoms per unit
cell in 3 d, there are a
total of 6 polarizations
The transverse modes
(TA & TO) are often
doubly degenerate, as
has been assumed in
this illustration.
yn
Frequency, 
yn-1
xn+1
Optic Modes
LO
TO
Acoustic
Modes
LA
TA
0
Wave vector, K p/a
1st Brillouin Zones: For the FCC, BCC, & HCP Lattices
Direct: FCC
Reciprocal: BCC
Direct: BCC
Reciprocal: FCC
Direct: HCP
Reciprocal: HCP
(rotated)
1st Brillouin Zone of FCC Lattice
Direct Lattice
Reciprocal Lattice
Measured Phonon Dispersion Relations in Si
(Inelastic, “Cold” Neutron Scattering)
Normal Mode Frequencies (k)
Plotted for k along high symmetry
directions in the 1st BZ.
ω
1st BZ for the
Si Lattice
(diamond; FCC,
2 atoms/unit cell)
k
Normal Modes of Silicon
L = Longitudinal, T = Transverse
O = Optic, A = Acoustic
Theoretical (?) Phonon Dispersion
Relations in GaAs
Normal Mode Frequencies (k)
Plotted for k along high symmetry
directions in the 1st BZ.
ω
1st BZ for the
GaAs Lattice
(zincblende; FCC, 2
atoms/unit cell)
k
• For Diamond Structure materials, such as Si, & Zincblende
Structure materials, such as GaAs, for each wavevector q, there are
6 branches (modes) to the
“Phonon Dispersion Relations” ω(q)
• For Diamond Structure materials, such as Si, & Zincblende
Structure materials, such as GaAs, for each wavevector q, there are
6 branches (modes) to the
“Phonon Dispersion Relations” ω(q)
• These are:
3 Acoustic Branches
1 Longitudinal mode: LA branch or LA mode
+ 2 Transverse modes: TA branches or TA modes
In the acoustic modes, the atoms vibrate
in phase with their neighbors.
• For Diamond Structure materials, such as Si, & Zincblende
Structure materials, such as GaAs, for each wavevector q, there are
6 branches (modes) to the
“Phonon Dispersion Relations” ω(q)
• These are:
3 Acoustic Branches
1 Longitudinal mode: LA branch or LA mode
+ 2 Transverse modes: TA branches or TA modes
In the acoustic modes, the atoms vibrate
in phase with their neighbors.
and
3 Optic Branches
1 Longitudinal mode: LO branch or LO mode
+ 2 Transverse modes: TO branches or TO modes
In the optic modes, the atoms vibrate
out of phase with their neighbors.
Measured Phonon Dispersion Relations in FCC Metals
(Inelastic, “Cold” Neutron Scattering)
Pb
1st BZ for the
FCC Lattice
Cu
Measured Phonon Dispersion Relations in FCC Metals
(Inelastic X-Ray Scattering)
Al
1st
BZ for the
FCC Lattice
Unit Cell for
the FCC
Lattice
Measured Phonon Dispersion Relations for C in the
Diamond Structure (Inelastic X-Ray Scattering)
1st BZ for the
Diamond
Lattice
Measured Phonon Dispersion Relations for Ge in the
Diamond Structure (Inelastic “Cold” Neutron Scattering)
1st BZ for the
Diamond
Lattice

L
Measured Phonon Dispersion Relations for KBr
in the NaCl Structure (FCC, 1 Na & 1 Cl in each unit cell)
(Inelastic, “Cold” Neutron Scattering)
1st BZ for the
Diamond
Lattice

L
Measured & Calculated Phonon Dispersion Relations
for Zr in the BCC Structure (Inelastic, “Cold” Neutron Scattering)
Data Points, 2 Different Theories: Solid & Dashed Curves)
1st BZ for the
BCC Lattice
Models for Normal Modes ω(k) in 3 Dimensions
Outline of Calculations with
Newton’s 2nd Law Equations of Motion
Assuming the Harmonic Approximation
(r)  Interatomic Potential
s  Displacements from Equilibrium
nth unit cell

In the harmonic approximation, expand  in a
Taylor’s series of displacements s about the
equilibrium positions. Cut off the series at the
term that is quadratic in the displacements.
The following illustrates this procedure:
0
 (rn i  sn i )
 2 (rn i  sn i )
1
 (rn i  sn i )   (rn i  sn i )  
sn i 
sn i sm j

rn i
2 n i ,m j rn i rm j
n i
0
N unit cells, each with n atoms means that there are 
3Nn Coupled Newton’s 2nd Law Equations of Motion
Lattice Dynamics in 3 Dimensions - Outline Calculations
of ω(k) in the Harmonic Approximation
(r)  Interatomic Potential
s  Displacements from Equilibrium
nth unit cell

Expand  in a Taylor’s series in displacements s about
equilibrium. Keep only up to quadratic terms:
 (rn i  sn i )
 2 (rn i  sn i )
1
 (rn i  sn i )   (rn i  sn i )  
sn i  
sn i sm j
rn i
2 n i ,m j rn i rm j
n i
  “Force Constant” Matrix   mni j
Hamiltonian in the
 2 (rn i  sn i )

Harmonic Approx.
rn i rm j
Analogous to 1 d
F = -(d/dx)  Fn i    (rn i  sn i )  H  1  M  sn2 i  1
2nd
sn i
Resulting Newton’s
Law
Equation of Motion
2 n i

2  
 nmi j sn i sm j
n i ,m j
M  sn i     nmi j sm j
m j
N unit cells, each with n atoms means that there are 
3Nn Coupled Newton’s 2nd Law Equations of Motion
Force Constant Matrix Properties
M  sn i     mni j sm j  mx  kx
Analogous to the 1d
Harmonic Oscillator
 mni j are analogous to elastic coefficients k
Analogous to the 1 d
m j
Spring Constant
Schematic view
of the lattice. 

m j
n i

n j
m i
Various symmetries of the
Force Constant Matrix
 mni j   (0m i n )  j from translational invariance
m j

 n i  0
m
Formally Solve the Equations of Motion –
Use a Spatial Fourier Series Approach
sn i 
1
u i (q)ei ( qrn t )  Ta sn i (q)  e  iqa sn i (q)
M
1
2
 u i (q)  
 nmi j eiq ( r r )u j (q)
n
m j
Di j 
1
M M 
m
M M 
  nmi j eiq (rn rm ) 
m
1
M M 
  0pij e
iq ( r p )
p
Di j  dynamical matrix (does not depend on rn )
After some work, the equations of motion become:
 2u i (q)   Di j u j (q)   ( Di j   2i j  u j (q)  0
j
(
j


det D(q)   I  0  for each q : d  r eigenvalues  (q)
2
2
s
So, the mathematics of
All of FORMAL Lattice Dynamics can
be summarized as finding solutions to
• The remainder is the use of various models
& theories for the “force constants” which
enter the force constant matrix Ф & thus
the dynamical matrix D.
• There are many different models & theories
which were designed to determine the force
constants which enter the dynamical matrix D.
These can broadly be divided into 4 groups:
1. Force Constant Models
2. Shell Models
3. Bond Models
4. Bond Charge Models
• Within each group, there are MANY variations
on these models!
• Going down the list: The models get more complex &
(in my opinion) harder to understand in terms of the
physics behind them.
Common Features of All Models
(or Theories):
1. All model the ion-ion interactions with some
parameters in the force constant matrix .
2. All find these parameters by fitting to various
experimental quantities.
A few of the many quantities used to do the fitting are:
Bulk Modulus; Shear Modulus; BZ center LO, TO, LA,
& TA frequencies; BZ edge LO, TO, LA, TA frequencie
+ Many Others
Since the goal was to explain neutron scattering data, people
tried to use non-neutron scattering data to fit the parameters.
3. All used the fitted parameters in the matrix  to
compare to neutron scattering data & to predict
results of neutron scattering experiments.
Force Constant Models
• These models are the crudest approach taken & the closest
in spirit & actual calculations to the 1d models we discussed.
• They model the force constant matrix  with as few
parameters as possible & fit to data mentioned.
• Assumption: The atoms (the ion core + valence electrons) are
HARD SPHERES, coupled by “springs”, characterized
by spring constants (~ like the 1d models)
• They include short range forces only. But have no
Coulomb forces!
• There are various types of “springs”:
1st, 2nd, 3rd, 4th, 5th, … neighbor coupling!!
• The spring forces have directional dependences, with
different spring constants for
coupling in different directions.
• The “best” force constant models require 12 to
20 DIFFERENT force constants per material!
A Rhetorical Question:
Is this physically reasonable & satisfying?
• Such models give good (q) for the
Group IV covalent solids:
C (diamond), Si, Ge, α-Sn
• But, they FAIL for many covalent & ionic compounds, such as
The III-V & II-VI materials, GaAs, CdTe, etc.
• This happens because
Coulomb (ionic) forces are ignored!
• Also, the bonds in these compounds are
partially ionic (there is a charge separation).
A Rhetorical Question!!
Is a 15 to 20 adjustable parameter “theory”
REALLY A THEORY?
• A quote in several references:
“The parameters are not easily understood
from a physical point of view.”
(In my opinion, this is putting it mildly!)
• Often, these models need up to
th
5
&
th
6
neighbor (or higher)
force constants!
• A physically realistic qualitative expectation
for relative size of the force constants
connecting neighbors at various distances is:
The force constant size should decrease
as the distance increases.
• However, it’s been found that, in order to get a good
fit to data, some of these models require instead that
the size of some force constants must increase with
increasing distance!! For example:
Φ4nn > Φ1nn
& other, absurd, completely unphysical results!
• In addition, no matter how many force constants are
assumed, these models cannot explain a lot of data!
– For example, the flattening of TA near the BZ edges.
• Often, these models were found to work ok
for purely covalent solids like
C (diamond), Si, Ge,…
but to do a poor job on ionic compounds
in which Coulomb Effects are important!
• To deal with these problems, “better” theories or
models were introduced. One such group of
models is called
The Shell Models
Shell Models
• The force constant models all assume “hard sphere” atoms
(ion core + valence electrons). From our discussion of bonding &
from electronic properties studies, we know that this is a
Very BAD assumption for covalently bonded
solids as well as for many other solid types!
• Our knowledge of bonding & electronic properties tells that:
The valence electrons are NOT rigidly attached to the ions!
The Main Idea of the Shell Model:
Each atom is modeled as a rigid ion core plus an
“independent” valence electron shell.
Also, the valence electron shell AND the ion core can move.
That is, the Atoms are Deformable!
So, in the extensions of the force constant
models to the Shell Models,
the atoms are deformable!
That is,
The ions & valence electron shells
are all moving.
• Also, Coulomb Interactions are included by putting
charges on the shells & the ion cores.
• In these models, the atomic displacements induce dipole
moments on the atoms.
So, there are dipole-dipole interactions between
unit cells as well as force constants to couple
the cells.
Best Shell Model results for (q)
• Ge - A good fit to neutron data is found with only 5 parameters!
• GaAs & other Compounds A good fit to neutron data is found with
~ 10 - 12 parameters.
• That is, the combined force constant shell model doesn’t
do much better than pure force constant models!
Physics Criticisms
1. The valence electrons in covalent materials are NOT
in the shells around the ion cores!
2. The valence electrons in these materials ARE in the
covalent bonds between the cores!
3. The fitting parameters are ~unphysical & have limited
use for modeling properties other than (q).
Other Physics Criticisms
• These models make an artificial division
of valence electrons between atoms which
are covalently bonded together.
• Actually, these valence electrons are
shared in the covalent bonds!
• So, people introduced “better” theories
or models, such as the Bond Models.
Bond Models
• In covalent materials, the valence electrons are in
the covalent bonds between the atoms & along
the directions from an atom’s near neighbors.
• Bond models: Extend the “Valence Force
Field” Method to covalent solids.
• Valence Force Field Method (VFFM):
• Used in theoretical molecular chemistry to
explain vibrational properties of covalent
molecules.
• In this model, the vibrations are analyzed in terms
of “valence forces” for bond stretching &
bending.
VFFM Advantages:
• The force constants for bond stretching &
bending are ~ characteristic of particular
bonds & are transferable from one molecule
to another, which contains same bond (e.g.
the force constants for a C-C bond are ~ the
same no matter what solid it is in!
Bond Models:
• The force constants are ~ the same for an AB bond in a solid as they are in molecules.
• Extension of the VFFM to covalent solids,
2 atoms / unit cell.
• The bond potential energy V is expanded
about equilibrium positions for all possible
degrees of freedom of bending, stretching,
etc. of bond.
• Expansion stopped at 2nd order in
deviations from equilibrium.
Simple harmonic oscillators in all
degrees of vibrational freedom!
Disappointment!
• Despite the greater physical appeal &
(hopefully) the better physical realism of
such models, to get good fits to ω(q)
(neutron scattering data), the bond models
need ~ a similar number of parameters as
the shell models!
So, after all the work on the Bond
Models, it turns out that there is no real
advantage of them over the shell models!
The Keating Model ≡ The VFFM with 2 or
3 parameters + a charge parameter.
• Good for elastic properties at long wavelengths (later).
• BAD for frequencies!
Other models & extensions of the VFFM:
5 or 6 parameters
• Often do well for trends in frequencies & BAD
for other vibrational properties (like elastic properties)!
• So, people introduced “better” theories
(models) like the
Bond Charge Models
Bond Charge Models
• The most difficult part of modeling the force
constant matrix is accurately including the longrange (Coulomb) electron-ion interaction.
• The Shell Models + charges: Attempts
to simulate this. However, fails to account for
dielectric screening. Also, for covalent bonds,
charge is not on atoms, but between them!
• The Bond Models: Account for covalent
bonding, but neglects Coulomb screening.
Review of Screening:
• Look at a specific ion at origin.
• Let the Coulomb interaction with one electron
 Vo(r).
• But, the presence of other electrons reduces this:
• The presence of all other charges (ions & electrons) near
ion of interest causes effective interaction to be reduced.
• It is shown in EM courses that the true potential is of form
V(r)  Vo(r)exp(-r/ro)
• Usually, this is simulated in simpler way:
V(r)  Vo(r)/ε
Here, ε = dielectric constant
• Screening in classical E&M:
V(r)  Vo(r)/ε
ε = dielectric constant:
Really, ε = ε(q,ω) but neglect this.
• This is too simple to work well for vibrational
spectra! Reason: the implicit assumption is that
valence electrons are “free” , except for Coulomb
interactions with ion of interest.
• We could treat Coulomb effects (ion with charge Ze) by
V(r)  -(Ze2)/(εr),
but this is too crude.
• Instead, localize some of valence electrons on the
bonds, so that some screen in this way & others don’t.
Bond Charge Model
• A portion of valence electron charge is
localized on bonds (between atoms).
• Another portion is “free” & contributes to
screening.
• The portion which contributes to screening
is an empirical, adjustable parameter.
NOTE!
• This is a model! Don’t it take too seriously or
literally. It is designed to simulate actual
effects. It’s physical significance is questionable.
• In this model, the valence electrons are
divided into two parts:
1. “Free” charge which contributes to screening
2. Localized charge in the bonds between atoms
• The fraction of the valence electron charge which
is localized on bonds is an adjustable parameter,
• The bond charge fraction Zb is defined by
Zbe  -2e/ε
• This is the theory definition.
In practice, Zb is an adjustable parameter
• The bond charge fraction:
Zbe  -2e/ε
• When Zb is determined, often it is found that
Zbe < 1.0e !
• Don’t take this too seriously!! Remember that
It is a crude MODEL!
• In addition, there must be spring-like force
constants for coupling between the ions.
Bond Charge Model Results for Si
Zbe  0.35 e
(< 1.0e)
• This actually compares favorably to the
Si charge density calculated by state of
the art electronic structure
(pseudopotential) codes.
• It also compares favorably with X-Ray
experiments on the Si charge density.
• Both show a build-up of  0.4 e per
bond between Si atoms!
Bond Charge Model Results for Si
Zbe  0.35 e
(< 1.0e)
• The 4 valence electrons from each atom are divided into
1. Bond Charges localized on each bond
Zbe  0.35 e.
Point charges are assumed. In reality the bond
charge is spread out over a volume.
2. “Free” Charges which screen.
Each Si contributes Zbe/2 to the bond charge
and (4-2Zb)e to the free charge.
• In addition, this has to be combined with force
constants to couple ions.
• The Bond Charge Model combines the
“best” of force constant, shell, & bond models!
• A further refinement:
Adiabatic Bond Charge Model
(ABCM).
• Allows bond charges to follow motion
of ions so that they are not located
exactly in middle of bond.
• Gets good ω(q) for Si & other materials
with only 4 parameters!