1. The Nuts and Bolts of First-Principles Simulation
23: Calculating Phonon Dispersion Relations and other matters!
Durham, 6th-13th December 2001
CASTEP Developers’ Group with support from the ESF k Network

2. 23: Calculations using phonons
Outline
Introduction phonon dispersion calculations.
Some examples for a range of materials.
Some surfaces.
Density of states.
Free energies.
Phase transitions.
Nuts and Bolts 2001
23: Calculations using phonons

3. 23: Calculations using phonons
Introduction
We will use a linear response method to calculate phonon properties of a crystalline material.
This avoids the need to create a supercell as in the frozen phonon method.
LR theory allows us to calculate phonon properties for any q-vector. Arbitrary q-vector leads to very large supercells (in principle) using frozen phonons.
Nuts and Bolts 2001
23: Calculations using phonons

4. Examples of LRT Phonons
First we will look at GaAs.
The dispersion curve has been calculated by repeated calculation of w(q) for a range of q-vectors.
As with band structures, use primitive cell.
Nuts and Bolts 2001
23: Calculations using phonons

5. Dispersion Relation for GaAs
GaAs phonon dispersion relation along some lines of high symmetry.
Dots indication neutron-scattering data.
Nuts and Bolts 2001
23: Calculations using phonons

6. Phonons in Carbon-diamond
For many nearly all group-V and III-V materials, the maximum phonon frequency occurs at the G point.
Anomaly in diamond – G point frequency is a local minimum.
Experiment: Inelastic x-ray and neutron scattering data.
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7. 23: Calculations using phonons
Anomaly in Diamond
Examine q-vectors in all directions around G and we see that there is a local minimum.
Shown here in a (100) plane. A similar results is found is other directions.
Nuts and Bolts 2001
23: Calculations using phonons

8. Explanation for Diamond
The dispersion of the uppermost branch has a strong over-binding in any direction.
Examination of the phonon eigenvectors indicate direction of atom motion for each q-vector – examine binding for yourself!
This results in a minimum frequency at the zone centre.
This is at variance with other tetrahedral semiconductors.
Nuts and Bolts 2001
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9. 23: Calculations using phonons
Wurtzite Structures
Wide band gap wurtzite semiconductors (eg. GaN) are of interest for blue/UV LED’s.
Behaviour of carriers (electrons or holes) are influenced by interaction with phonons.
Hence lattice-dynamical properties are very important input into various phenomenological models.
Experimentally difficult to measure phonon dispersion since decent crystals are hard to grow, hence experimental information is limited.
Nuts and Bolts 2001
23: Calculations using phonons

10. GaN – The Wurtzite Structure
Semiconducting GaN in the hexagonal Wurtzite structure.
The Brillouin Zone constructs and a path of high symmetry.
Nuts and Bolts 2001
23: Calculations using phonons

11. Dispersion Relation in Wurtzite Structure
Agreement with experiment is reasonably good.
Deviations from experiment exist (eg GM line).
What’s better: experiment of theory?
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12. 23: Calculations using phonons
Density of States
Calculating the density of states for phonons is performed in a similar manner to that of the electronic density of states.
Sample q-vectors over the Brillouin zone (accounting for symmetry) and integrate up.
Interpolation of frequencies between the calculated values at the given q-vectors can be used to increase integration accuracy.
Nuts and Bolts 2001
23: Calculations using phonons

13. Dispersion and DOS of Si
Note: Singularities in DOS is difficult to achieve, but important to get right for free energy calculations!
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14. 23: Calculations using phonons
Free Energy
An important application of ab initio phonon calculations is the ability to evaluate free energies.
This is important in the study of phase diagrams.
Whichever phase has the lowest free energy will (should!) be the one that’s observed.
Free energy of a temperature (T)-volume (V) system is given as follows:
Nuts and Bolts 2001
23: Calculations using phonons

15. Example of Phase Transitions
Tin is commonly found in one of two allotropic forms at ambient pressure.
The low temperature phase (a-Sn) is a zero gap semiconductor in the diamond structure.
Above 13oC the crystal transforms into the b phase (white tin which is metallic): body-centred tetragonal.
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16. 23: Calculations using phonons
a-b Sn Transition
The a-b transition is a simple example of an entropy driven structural phase transition transition.
It can be examined using the vibrational properties of the two phases of the material.
Examining the free energy equation, each term is easily accessible to the LR-DFPT calculations presented in the previous lecture.
Nuts and Bolts 2001
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17. 23: Calculations using phonons
Sn Dispersion Curves
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18. 23: Calculations using phonons
Phase Diagram for Sn
Theoretical phase transition temperature is slightly above the experimental value.
The missing factor is anharmonicity.
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19. 23: Calculations using phonons
Surface Phonons
As with the electronic density of states of a surface, we can perform a similar calculation for phonons:
This is the surface phonon dispersion curve for InP (100) surface.
Solid lines indicate the phonons that are localised at the surface.
Shaded area are bulk phonons projected onto surface BZ.
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20. 23: Calculations using phonons
Elastic Constants
A full derivation of elastic constants from the Dynamics matrix can be found in A. A. Maradudin, et. al. “Theory of lattice dynamics in the harmonic approximation”, Academic Press, 1971.
I will skip a few steps, but basic results are as follows:
Allow a,b,g,l,m,n to run over the co-ordinate axes and define:
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21. 23: Calculations using phonons
Elastic Constants
If V is the volume of the unit cell, then we can define the following brackets:
The elastic constants are then:
Nuts and Bolts 2001
23: Calculations using phonons

22. Definition of Elastic Constants
The elastic constants can then be expressed in their more familiar form by pairing indices by the equivalences:
This leads to the standard form, eg. c11, c12, c44, etc
These can be used to calculate various modulus, e.g. for a cubic crystal, the bulk modulus is:
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23. 23: Calculations using phonons
Structure of MgO
MgO (along with FeO) is hypothesised to be a major constituent of the lower mantle.
Experiments at lower mantle pressures are not possible.
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23: Calculations using phonons

24. Elastic Constants of MgO
Elastic constants of Minerals at high pressure is of physical and geological interest, e.g. for interpretation of seismological data
Solid Line: Theory.
Dashed Lines: Interpolated experimental data using two models.
Nuts and Bolts 2001
23: Calculations using phonons

25. Bulk/Shear Modulus for MgO
The shear modulus, G, is calculated as follows:
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23: Calculations using phonons

26. Energy Propagation in MgO
The longitudinal and shear-wave velocities, VP and VS of an isotropic aggregate can be calculated using:
Nuts and Bolts 2001
23: Calculations using phonons

27. Propagation Direction in MgO
Dependence of the longitudinal and two shear-wave velocities of MgO on propagation direction at pressures of 0 GPa (solid) and 150 GPa (dashed).
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28. 23: Calculations using phonons
References
P. Pavone, J,. Phys. Condens. Matter 13, 7593 (2001).
B. B. Karki, S. J. Clark, et. al., American Mineralogist 82, 52 (1997).
J. Frirsch, et. al., Surface Science , 58 (1999).
Nuts and Bolts 2001
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