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

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.

3. They were really models
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.

4. A 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.

5. A Two Part Discussion 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.

6. D(q) ≡ Spatial Fourier Transform of
Classical Vibrational Normal Mode Problem
(in the Harmonic Approximation)
ALWAYS reduces to solving:
Here, D(q) ≡ The “Dynamical Matrix”
D(q) ≡ Spatial Fourier Transform of
the “Force Constant” Matrix Φ
q ≡ wave vector, I ≡ identity matrix
ω2 ≡ ω2(q) ≡ vibrational mode eigenvalue

7. NOTE! There are, in general, 2 distinct types of vibrational waves (2 possible wave polarizations) in solids:

8. 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.

9. 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 (or q), these 2 vibrational polarizations will give
2 different solutions for ω(k).

10. We also know that there are, at least, 2 distinct branches of ω(k) (2 different functions ω(k) for each k)

11. 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.

12. 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:

13. 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).

14. it is necessary to distinguish between Longitudinal & Transverse Modes
So, when discussing the vibrational frequencies ω(k),
it is necessary to distinguish between
Longitudinal & Transverse Modes
(Polarizations) &

15. it is necessary to distinguish between Longitudinal & Transverse Modes
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.

16. it is necessary to distinguish between Longitudinal & Transverse Modes
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 4 distinct kinds of modes for ω(k).

17. Longitudinal Acoustic Modes  LA Modes
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 4 distinct kinds of modes for ω(k).
The terminologies used, with their abbreviations are:
Longitudinal Acoustic Modes  LA Modes

18. it is necessary to distinguish between Longitudinal & Transverse Modes
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 4 distinct kinds of modes for ω(k).
The terminologies used, with their abbreviations are:
Longitudinal Acoustic Modes  LA Modes
Transverse Acoustic Modes  TA Modes

19. it is necessary to distinguish between Longitudinal & Transverse Modes
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 4 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

20. it is necessary to distinguish between Longitudinal & Transverse Modes
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 4 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

21. Transverse Acoustic Mode for 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:

22. Transverse Optic Mode for 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:

23. 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:

24. 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

25. 1st Brillouin Zones: FCC, BCC, & HCP Lattices
Direct: FCC Reciprocal: BCC
Direct: BCC
Reciprocal: FCC
Direct: HCP
Reciprocal: HCP
(rotated)

26. 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

27. 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)

28. 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.

29. 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.

30. Vibrational Normal Modes in Silicon
L = Longitudinal, T = Transverse
O = Optic, A = Acoustic

31. Measured Phonon Dispersion Relations in Si
(From 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

32. Measured Phonon Dispersion Relations in Si
3THz ~ 100 cm-1 ; 1meV ~ 8 cm-1
THz
G. Nilsson and G. Nelin, PRB 6, 3777 (1972)
W. Weber, PRB 15, 4789 (1977)

33. Measured Phonon Dispersion
Relations in 3C SiC
J. Serrano et al., APL 80, 23 (2002)
cm-1 LO TO
J. Serrano et al., APL 80, 23 (2002)

34. 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

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

36. 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

37. 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

38. 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

39. Measured Phonon Dispersion Relations for KBr
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
FCC Lattice L 

40. Measured & Calculated Phonon Dispersion Relations 1st BZ for the
for Zr in the BCC Structure (Inelastic, “Cold” Neutron Scattering)
Data Points, 2 Different Models: 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

41. Measured Phonon Dispersion Relations in Sn
-Sn (diamond structure, “gray tin”)
-Sn (body centered tetragonal , “white tin”)

42. Measured & Calculated Phonon Dispersion in MgSiO3
Calc Exp
Calc Exp
0 GPa
Calc:
Karki, Wentzcovitch,
de Gironcoli, Baroni
PRB 62, 14750, 2000
Exp:
Raman [Durben &
Wolf 1992] Infrared
[Lu et al. 1994] -
100 GPa

43. Measured Phonon Dispersion in GaN
T. Ruf et al., PRL 86, 906 (2001)