1. Vibrational Normal Modes or “Phonon” Dispersion Relations in Crystalline Materials

2. “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.

3. Two Part Discussion Part I Part II
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.

4. ALWAYS reduces to solving:
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

5. 2 different solutions for ω(k).
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).

6. Discussed on the next page:
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:

7. 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).

8. it is necessary to distinguish between
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

9. The vibrational amplitude is highly exaggerated!
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:

10. The vibrational amplitude is highly exaggerated!
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:

11. 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, 
Wave vector, K
(p/a)
LA Modes
TA Modes
Acoustic Modes
Speed of Sound:

12. Polarization Lattice Constant, a Frequency,  Wave vector, K TA & TOxn yn
yn-1
xn+1
LA & LO
Optic Modes
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. LO TO
Acoustic
Modes LA
Frequency,  TA
Wave vector, K
p/a

13. 1st Brillouin Zones: For the FCC, BCC, & HCP Lattices
Direct: FCC Reciprocal: BCC
Direct: BCC
Reciprocal: FCC
Direct: HCP
Reciprocal: HCP
(rotated)

14. 1st Brillouin Zone of FCC Lattice
in 3D, remember that the reciprocal lattice of fcc is bcc
not that the point symmetries are conserved
so remember: this is important because if we know it in the 1st BZ, we basically know it everywhere.
Direct Lattice Reciprocal Lattice

15. 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.
1st BZ for the
Si Lattice
(diamond; FCC, 2 atoms/unit cell) ω k

16. Normal Modes of Silicon
L = Longitudinal, T = Transverse
O = Optic, A = Acoustic

17. Theoretical (?) Phonon Dispersion
Relations in GaAs
Normal Mode Frequencies (k)
Plotted for k along high symmetry directions in the 1st BZ.
1st BZ for the
GaAs Lattice
(zincblende; FCC, 2 atoms/unit cell) ω k

18. 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)

19. 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: 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.

20. 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: 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.

21. Measured Phonon Dispersion Relations in FCC Metals Pb Cu
(Inelastic, “Cold” Neutron Scattering) Pb
1st BZ for the
FCC Lattice Cu

22. Al Measured Phonon Dispersion Relations in FCC Metals Unit Cell for
(Inelastic X-Ray Scattering) Al
phonon dispersion curve in 3D
different directions in the three dimensional k
We see acoustic branches but no optical branches.
Here three acoustic branches but in some parts degenerate.
Easy to see why: in some direction the two transversal acoustic waves are completely equivalent
More branches. In fact 3 branches for every atom per unit cell. Here: only one atom, three branches. Sometimes degenerate, especially in directions of high symmetry.
Unit Cell for
the FCC
Lattice
1st BZ for the
FCC Lattice

23. Measured Phonon Dispersion Relations for C in the 1st BZ for the
Diamond Structure (Inelastic X-Ray Scattering)
We see acoustic and optical phonons. More branches. In fact 3 branches for any atom per unit cell. Here two atoms, six branches, three acc, three opt
the lower the symmetry of the direction, the more likely that the waves are not degenerate
1st BZ for the
Diamond Lattice

24. Measured Phonon Dispersion Relations for Ge in the 1st BZ for the
Diamond Structure (Inelastic “Cold” Neutron Scattering)
We see acoustic and optical phonons. More branches. In fact 3 branches for any atom per unit cell. Here two atoms, six branches, three acc, three opt
the lower the symmetry of the direction, the more likely that the waves are not degenerate
1st BZ for the
Diamond Lattice  L

25. Measured Phonon Dispersion Relations for KBr 1st BZ for the
in the NaCl Structure (FCC, 1 Na & 1 Cl in each unit cell)
(Inelastic, “Cold” Neutron Scattering)
We see acoustic and optical phonons. More branches. In fact 3 branches for any atom per unit cell. Here two atoms, six branches, three acc, three opt
the lower the symmetry of the direction, the more likely that the waves are not degenerate
1st BZ for the
Diamond Lattice L 

26. Measured & Calculated Phonon Dispersion Relations 1st BZ for the
for Zr in the BCC Structure (Inelastic, “Cold” Neutron Scattering)
Data Points, 2 Different Theories: Solid & Dashed Curves)
We see acoustic and optical phonons. More branches. In fact 3 branches for any atom per unit cell. Here two atoms, six branches, three acc, three opt
the lower the symmetry of the direction, the more likely that the waves are not degenerate
1st BZ for the
BCC Lattice

27. Assuming the Harmonic Approximation
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
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:
nth unit cell 
N unit cells, each with n atoms means that there are 
3Nn Coupled Newton’s 2nd Law Equations of Motion

28. N unit cells, each with n atoms means that there are 
Lattice Dynamics in 3 Dimensions - Outline Calculations of ω(k) in the Harmonic Approximation
(r)  Interatomic Potential
s  Displacements from Equilibrium
Expand  in a Taylor’s series in displacements s about equilibrium. Keep only up to quadratic terms:
nth unit cell 
Hamiltonian in the Harmonic Approx.
  “Force Constant” Matrix 
Analogous to 1 d
F = -(d/dx) 
Resulting Newton’s 2nd Law Equation of Motion
N unit cells, each with n atoms means that there are 
3Nn Coupled Newton’s 2nd Law Equations of Motion

29. Force Constant Matrix Properties
Analogous to the 1d
Harmonic Oscillator
Analogous to the 1 d
Spring Constant
Schematic view
of the lattice. 
Various symmetries of the
Force Constant Matrix

30. After some work, the equations of motion become:
Formally Solve the Equations of Motion – Use a Spatial Fourier Series Approach
After some work, the equations of motion become:

31. All of FORMAL Lattice Dynamics can
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.

32. 2. Shell Models 3. Bond Models 4. Bond Charge Models
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.

33. 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.

34. different spring constants for coupling in different directions.
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.

35. Is this physically reasonable & satisfying? Group IV covalent solids:
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).

36. 5th & 6th neighbor (or higher)
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
5th & 6th neighbor (or higher)
force constants!

37. The force constant size should decrease as the distance increases.
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!

38. The Shell Models C (diamond), Si, Ge,…
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

39. That is, the Atoms are Deformable!
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!

40. The ions & valence electron shells are all moving.
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.

41. 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
~ 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).

42. 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.

43. 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.

44. VFFM Advantages: Bond Models:
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 A- B bond in a solid as they are in molecules.

45. Simple harmonic oscillators in all degrees of vibrational freedom!
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!

46. 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!

47. Other models & extensions of the VFFM:
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

48. Bond Charge Models
The most difficult part of modeling the force constant matrix is accurately including the long- range (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.

49. V(r)  Vo(r)exp(-r/ro)
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

50. V(r)  Vo(r)/ε V(r)  -(Ze2)/(εr),
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.

51. 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.

52. In practice, Zb is an adjustable parameter
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

53. Zbe  -2e/ε It is a crude MODEL!
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.

54. 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!

55. 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.

56. Adiabatic Bond Charge Model
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!