Phonons In The Elk Code

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1 Phonons In The Elk Code Kay Dewhurst, Sangeeta Sharma, Antonio Sanna, Hardy Gross Max-Planck-Institut für Mikrostrukturphysik, Halle

2 If you are allowed to measure only one property of a material, make sure it s the A. Einstein

3 If you are allowed to measure only one property of a material, make sure it s the heat capacity A. Einstein

4 Why phonons? Measured directly with inelastic neutron scattering Thermodynamic quantities at low temperature: heat capacity, entropy, free energy, zero-point Phase transitions from the Gibb s free energy: G(P, T ) = E + P V T S Imaginary phonon frequencies indicate lattice instabilities Elastic tensor from q 0 limit of phonon dispersion Thermal expansion coefficient (from mode Grüneisen parameters) Temperature dependence of the band gap Static polarization ɛ(ω = 0) Electron-phonon coupling Superconductivity

5 The equilibrium positions of out crystal are the minimum of the Born-oppenheimer potential energy surface: ( ˆTe + ˆV e n (R) + ˆV e e + E n n (R)) Φ R (r) = E(R)Φ R (r) This is an artificial construct and in some cases may be a poor approximation to the average nuclear positions obtained from the correlated electron-nuclear wavefunction Ψ(r, R) Recent work entitled Exact Factorization of the Time-Dependent Electron-Nuclear Wave Function [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, (2010)] may make it easier to handle these cases

6 Let s assume that the potential energy surface is harmonic: E(R) = U t CU + E(R 0 ), where U R R 0

11 Elastic matrix We have to determine the spring constants between atoms: C αi,βj (R R 2 E ) = u αi (R) u βj (R ) = F αi(r) u βj (R ) for atoms α, β, polarisations i, j and primitive vectors R, R

12 Thus E = E (no linear term) C αi,βj(r R )u αi (R)u βj (R ) + αir βjr

13 Equation of motion: Use the anzatz M α 2 t 2 u αi(r) = βjr C αi,βj (R R )u βj (R ) u αi (R) = 1 ] [e αi (q) exp(i(q R ω(q)t)) + c.c. 2Mα to get where D αi,βj (q)e αi (q) = ω 2 (q)e αi (q) 1 D αi,βj (q) = exp(iq R)C αi,βj (R) Mα M β R

14 Ab-initio approaches for computing the dynamical matrix Supercell Parlinski-Li-Kawazoe method: random displacements and least-squares fitting (Wien2k, VASP) Supercell with molecular dynamics (Fourier transform of trajectories) Supercell method used in Elk Density functional perturbation theory (DFPT)

15 Density functional perturbation theory (DFPT) Start with the Hellmann-Feynman theorem Let the Hamiltonian depend on some parameter λ, therefore E λ = Ψ λ Ĥλ Ψ λ, where Ψ λ is the ground-state of Ĥλ

16 Density functional perturbation theory (DFPT) Start with the Hellmann-Feynman theorem Let the Hamiltonian depend on some parameter λ, therefore E λ = Ψ λ Ĥλ Ψ λ, where Ψ λ is the ground-state of Ĥλ Take the derivative E λ λ = Ψ λ λ Ĥ λ Ψ λ + λ Ψ λ Ĥλ Ψ λ + Ψ λ Ĥλ λ Ψ λ = Ψ λ λ Ĥ λ Ψ λ + λ Ψ λ Ĥλ Ψ λ λ =λ

17 Density functional perturbation theory (DFPT) Start with the Hellmann-Feynman theorem Let the Hamiltonian depend on some parameter λ, therefore E λ = Ψ λ Ĥλ Ψ λ, where Ψ λ is the ground-state of Ĥλ Take the derivative E λ λ = Ψ λ λ Ĥ λ Ψ λ + λ Ψ λ Ĥλ Ψ λ + Ψ λ Ĥλ λ Ψ λ = Ψ λ λ Ĥ λ Ψ λ + λ Ψ λ Ĥλ Ψ λ λ =λ }{{} = 0

18 Density functional perturbation theory (DFPT) Only the external potential depends on λ Therefore Ĥ λ = ˆT + ˆV λ ext + ˆV e e Thus finally = λĥλ λ ˆV ext λ E λ λ = Ψ λ λ ˆV λ ext Ψ λ E λ λ = d 3 rρ λ (r) λ V λ ext(r) for all λ

19 Density functional perturbation theory (DFPT) Taking another derivative: 2 E λ λ 2 = d 3 r λ ρ λ(r) λ V ext(r) λ + d 3 r ρ λ (r) 2 λ 2 V λ ext(r) Difficult part: ρ λ (r) λ

20 Density functional perturbation theory (DFPT) Recall the Kohn-Sham density is equal to the exact density ρ(r) = i φ i (r)φ i (r) So we get the linearised Kohn-Sham equations: e i = φ i V s φ i (H ε i ) φ i = ( H ε i ) φ i ρ(r) = φ i (r)φ i (r) + φ i (r) φ i (r) i V s (r) = V ext (r) + V H (r) + d 3 r δe xc δρ(r)δρ(r ) ρ(r)

21 Density functional perturbation theory (DFPT) If the perturbing potential is of the form V ext (r) = Z α cos(q R) τ αi r + R τ α R then Bloch state φ ik is coupled to states φ ik+q and φ ik q the density derivative is ρ(r) = ρ c (r) cos(q r) + ρ s (r) sin(q r) and the same for sin(q R)

22 Density functional perturbation theory (DFPT) We can also use perturbations of the form V ext (r) = Z α exp(iq R) τ αi r + R τ α R in which case φ ik is coupled to φ ik+q only, and the density derivative is ρ(r) = ρ(r) exp(iq r) and the effective potential derivative is also V s (r) = Ṽs(r) exp(iq r) ρ and V s are lattice periodic

23 Density functional perturbation theory (DFPT) This is very efficient, in the same way that spin-spirals are efficient for incommensurate magnetic structures The APW basis set depends on the atomic positions Pulay corrections make implementation of phonons difficult Previous implementations Henry Krakauer, Rici Yu and Cheng-Zhang Wang Robert Kouba and Claudia Ambrosch-Draxl in Wien2k

24 Supercell Slow Linear response Fast 1 Can be used for thermally stabilizing unstable phases: P. Souvatzis et al J. Phys.: Condens. Matter

25 Supercell Slow Easy to implement Linear response Fast Hard to implement 1 Can be used for thermally stabilizing unstable phases: P. Souvatzis et al J. Phys.: Condens. Matter

26 Supercell Slow Easy to implement Works with all features Linear response Fast Hard to implement Works with fewer features 1 Can be used for thermally stabilizing unstable phases: P. Souvatzis et al J. Phys.: Condens. Matter

27 Supercell Slow Easy to implement Works with all features Does not require f xc Linear response Fast Hard to implement Works with fewer features Requires f xc 1 Can be used for thermally stabilizing unstable phases: P. Souvatzis et al J. Phys.: Condens. Matter

28 Supercell Slow Easy to implement Works with all features Does not require f xc Can include anharmonicity 1 Linear response Fast Hard to implement Works with fewer features Requires f xc Purely harmonic 1 Can be used for thermally stabilizing unstable phases: P. Souvatzis et al J. Phys.: Condens. Matter

29 Phonons in Elk Elk uses an efficient supercell method: 1. Finds the smallest supercell which contains the q-vector perturbation 2. then performs calculations with cos-like and sin-like finite displacements 3. computes forces on all atoms and Fourier transforms to generate dynamical matrix rows 4. gathers all the dynamical matrix rows together to produce a phonon dispersion

30 Phonons in Elk This can be run across hundreds of computers sharing a common file system Each computer checks for the presence of a particular DYN file For example: DYN Q S02 A002 P3.OUT (means q = ( 1 4, 1 4, 1 2 ), α = (1, 2), i = 3) If it does not exist then the computer runs this task

31 Phonons in Elk The phonon dispersion is obtained by Fourier transforming the dynamical matrices to real space: Dαi, βj(r) = q Dαi, βj(q) exp(iq R), evaluating the matrix at an arbitrary q-vector: Dαi, βj(q ) = R Dαi, βj(r) exp( iq R), and diagonalising The acoustic sum rule is always enforced (acoustic branches go to zero)

32 LO-TO splitting in polar semiconductors is not yet in Elk

33 Phonons in Elk First-variational Hamiltonian (bordered band diagonal):

34 Phonons in Elk First-variational Hamiltonian (bordered band diagonal):

35 Phonons in Elk First-variational Hamiltonian (bordered band diagonal):

36 Elk also calculates Phonon density of states Heat capacity Entropy Free energy Zero point energy

37 Electron-phonon coupling matrices g n ik+q,jk = 1 2Mωn (q) φ ik+q e n (q) V s (q) φ jk Linewidths γ n (q) = 2πω n (q) g n ik+q,jk 2 δ(ε ik+q ε F )δ(ε jk ε F ) (interpolating like the dispersion) The Eliashberg function α 2 F (ω) = 1 γ n (q) 2πN F ω n (q) δ(ω ω n(q)) qn Electron-phonon coupling parameter λ = 2 0 dω α2 F (ω) ω

38 McMillan-Allen-Dynes superconducting critical temperature Eliashberg equations and superconducting gap (Antonio Sanna)

39 Phonon dispersion of bcc Nb (4 4 4 q-point set) Phonon linewidths of bcc Nb ( k-point set)

40 Question Can you think of a situation where at least one of the acoustic phonon branches is quadratic for small q, and not linear?

Introduction to density functional perturbation theory for lattice dynamics

Introduction to density functional perturbation theory for lattice dynamics SISSA and DEMOCRITOS Trieste (Italy) Outline 1 Lattice dynamic of a solid: phonons Description of a solid Equations of motion