Classical simulation of optically excited phonon dynamics in Bismuth

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Classical simulation of optically excitedphonon
dynamics in Bismuth

• Donal ODonoghue
• Prof. Stephen Fahy

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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

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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

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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

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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

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Pump probe experiments
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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

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Bismuth crystal structure

• Rhombohedral unit cell
• Two unit atoms per unit cell
• Peierls Distortion of simple cubic structure

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Bismuth A1g phonon mode

• A1g mode consists of atoms beating against each
  other along trigonal axis
• Phonon characterised by variation of x

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Bismuth Exciting A1g mode
Equilibrium shift depends on of electrons
excited to conduction band
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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
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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

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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

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Vibration modes in crystal lattices

• System of 3rN coupled differential equations
  describe the atomic dynamics fully

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Traveling waves in periodic structures

• Assume plane wave-like solutions for atomic
  displacements

• redtc

(wavevectors in the Brillouin Zone)
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Dynamical Matrices

• Plane-wave ansatz reduces problem to linear
  homogenous system with 3r equations

• Introduce dynamical matrices

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Dynamical Matrices

• System of equations of form

• Eigenvalues of D(q) give allowed phonon
  frequencies, eigenvectors give phonon amplitudes

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Harmonic approximation

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

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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.

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Classical simulation what we know

• Potential energy of A1g mode (per unit cell), as
  a function of relative displacement x and
  electron hole plasma density n.

x is the relative displacement of atoms in a unit
cell
Equilibrium value of x depends on of electrons
in conduction band
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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
  202020 grid of q-points in BZ, at several
  values of (x0,n)
• Need to interpolate between these values

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Polynomial interpolation - 1

• Third order series expansion
• Negative values introduces unstable modes
  (artefact of numerics not physically realistic)

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Polynomial interpolation 2

• Linear expansions in x0 and n
• Guarantees positive eigenvalues, in rectangular
  grid, no unphysical instabilities

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Classical simulation dynamics
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Computational scheme
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Classical simulation sample output
f2.6 Thz
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Preliminary results - summary
A1g Amplitude Decay Time
plasma density n0 n1
large ampl. simul. 5.5 ps 2.9 ps
small ampl. simul. 5.5 ps 3.3 ps
Pert. theory 3.6 ps 2.4 ps
Experiment 4.0 ps 0.8 ps

• Minor variation (increase) of damping rate with
  amplitude
• Disagreement with DFT calculations needs further
  investigation

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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.