Classical simulation of optically excited phonon dynamics in Bismuth

1. Classical simulation of optically excitedphonon dynamics in Bismuth Donal O’Donoghue Prof. Stephen Fahy

2. Talk outline • Coherent phonons & optically excited Bismuth • Project motivation and goals • Theory: lattice vibrations, dynamical matrices, anharmonic phonon decay • Classical simulation of dynamics • Preliminary results

3. What is a phonon? • Vibration of a crystal lattice • Wave like stretching and deformation of inter-atomic bonds • ‘Acoustic’ and ‘Optic’ modes of longitudinal and transverse waves

4. Creating coherent phonons • At thermal equilibrium, coherent phonons are unlikely to exist • A femtosecond light pulse can induce coherent atomic motion over a macroscopic region

5. Decay of coherent phonons • Coherent phonons can decay due to coupling with electrons and other phonon modes, and due to scattering by crystal impurities and defects

6. Pump – probe experiments

7. Bismuth – general properties • Element 83, atomic mass 209 g/mol • Semi-metal, similar to Antimony and Arsenic • Potential applications in electronic, optoelectronic, and semiconductor devices

8. Bismuth – crystal structure • Rhombohedral unit cell • Two unit atoms per unit cell • ‘Peierls Distortion’ of simple cubic structure

9. Bismuth – A1g phonon mode • A1g mode consists of atoms beating against each other along trigonal axis • Phonon characterised by variation of x

10. Bismuth – Exciting A1g mode Equilibrium shift depends on % of electrons excited to conduction band

11. E. Murray et al. Project Goals • Calculate decay rate of A1g mode using full classical dynamical simulation • Investigate variation of decay rate with amplitude, e-h plasma density A. Hurley, ‘Decay of Photo-excited Vibrations in Bismuth’

12. Vibration modes in crystal lattices • Potential energy of system changes when atoms are displaced from equilibrium lattice sites • Using series expansion to approximate changes in F due to displacements from equilibrium

13. Vibration modes in crystal lattices • In analogy to the spring constant ‘k’ in 1-D, there are generalised spring constants in 3-D, called ‘coupling constants’

14. Vibration modes in crystal lattices • System of 3rN coupled differential equations describe the atomic dynamics fully

15. Traveling waves in periodic structures • Assume plane wave-like solutions for atomic displacements • redtc (wavevectors in the Brillouin Zone)

16. Dynamical Matrices • Plane-wave ansatz reduces problem to linear homogenous system with 3r equations • Introduce dynamical matrices

17. Dynamical Matrices • System of equations of form • Eigenvalues of D(q) give allowed phonon frequencies, eigenvectors give phonon amplitudes

18. Harmonic approximation • Linear coupling terms • Independent normal modes of vibration • No energy transfer, modes do not decay

19. Anharmonic coupling • Include third order terms in series expansion of potential energy F • This introduces non-linear coupling terms into equations of motion • Normal modes can no longer be decoupled • Transfer of energy between modes occurs, A1g mode will decay into two other phonons.

20. Potential energy of A1g mode (per unit cell), as a function of relative displacement xand electron hole plasma density n. E. Murray et al. Classical simulation – what we know x is the relative displacement of atoms in a unit cell Equilibrium value of x depends on % of electrons in conduction band

21. Classical simulation – what we need • Coupling of A1g mode to other modes in BZ, as a function of phonon co-ordinate x0 and electron hole plasma density n. • Have D(q) (including 3rd order terms) for 20×20×20 grid of ‘q-points’ in BZ, at several values of (x0,n) • Need to interpolate between these values

22. Polynomial interpolation - 1 • Third order series expansion • Negative values introduces unstable modes (artefact of numerics – not physically realistic)

23. Polynomial interpolation 2 • Linear expansions in x0 and n • Guarantees positive eigenvalues, in rectangular grid, no unphysical instabilities

24. Classical simulation – dynamics

25. Computational scheme

26. Classical simulation – sample output f=2.6 Thz

27. Preliminary results - summary • Minor variation (increase) of damping rate with amplitude • Disagreement with DFT calculations needs further investigation A1g Amplitude Decay Time

28. Acknowledgements / References • Prof Stephen Fahy • Phonon frequency shifts and damping in optically excited Bismuth – E. Murray, PhD Thesis. • Decay of Photo-excited Vibrations in Bismuth – A. Hurley, 4th year report • Solid-State Physics - An Introduction to Principles of Materials Science, Springer, 2nd edition, 1995. Ibach,H & Luth, H.