The Nuts and Bolts of First-Principles Simulation

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

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

3. 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. 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. 23: Calculations using phonons

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

7. 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. 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. 23: Calculations using phonons

9. 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. 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. 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? 23: Calculations using phonons

12. 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. 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! 23: Calculations using phonons

14. 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: 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. 23: Calculations using phonons

16. 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. 23: Calculations using phonons

17. Sn Dispersion Curves 23: Calculations using phonons

18. Phase Diagram for Sn Theoretical phase transition temperature is slightly above the experimental value. The missing factor is anharmonicity. 23: Calculations using phonons

19. 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. 23: Calculations using phonons

20. 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: 23: Calculations using phonons

21. Elastic Constants • If V is the volume of the unit cell, then we can define the following brackets: The elastic constants are then: 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: 23: Calculations using phonons

23. 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. 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. 23: Calculations using phonons

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

28. 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 427-428, 58 (1999). 23: Calculations using phonons