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Matteo Calandra , Gianni Profeta , Michele Casula and Francesco Mauri

Adiabatic and non-adiabatic phonon dispersion in a Wannier functions approach: applications to MgB 2 , CaC 6 and K-doped Picene. Matteo Calandra , Gianni Profeta , Michele Casula and Francesco Mauri. M. Calandra et al. Phys. Rev . B 82 , 165111 (2010)

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Matteo Calandra , Gianni Profeta , Michele Casula and Francesco Mauri

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  1. Adiabatic and non-adiabatic phonon dispersion in a Wannier functions approach: applications to MgB2 , CaC6 and K-doped Picene Matteo Calandra, Gianni Profeta, MicheleCasula and Francesco Mauri M. Calandra et al. Phys. Rev. B 82, 165111 (2010) M. Casulaet al. to appear on Phys. Rev. Lett.

  2. Motivation of the work Description of phonon dispersions in metals needs an ultradense sampling of the Fermi surface. 300 K Fermi Surface Impossible to sample a sharp Fermi surface with a coarse grid. An ultradense grid is needed.

  3. Motivation of the work Description of phonon dispersions in metals needs an ultradense sampling of the Fermi surface. 4000 K Fermi Surface A finite temperature Tph is introduced not sampling sampling Nk(Tph) is the number of k-points necessary to sample the Fermi surface a a given Tph. Typical values: Tph=0.03 Ryd ≈4700 K Nk(Tph)=123

  4. Outline Theory: Wannier interpolation scheme to calculate adiabatic and non-adiabatic phonon dispersions on ultradense electron and phonon momentum grids. Applications: Phonon dispersion and electron-phonon coupling in MgB2 Adiabatic and non adiabatic phonon dispersion in CaC6 Electron-phonon coupling in K-doped Picene (78 atoms per cell)

  5. Time-dependent force-constants matrix J displacement from equilibrium cell atom in the cell at equilibrium Displacing atom I at time t’ induces a force FJ(t) on atom J at time t. The force-constants matrix is then and its ω and Fourier transforms are defined as: WARNING Complex Quantity!

  6. From forces to phonon frequencies The force-constants matrix is complex, we define: If then the self-consistent equation Gives the phonon frequencies.

  7. Force-Constantsfunctional We write the force constants in a functional form. Using linear response we look for a functional such that: with is the time dependent charge density. and

  8. Force-Constantsfunctional The functional has the form: where is the Hartree and exchange correlation kernel and T is the temperature. is the number of k-points necessary to converge the sum at a temperature T. Baroniet al., Rev. Mod. Phys. 73, 515 (2001) Gonze and Lee, PRB 55, 10355 (1997) M. Calandra et al. Phys. Rev. B 82, 165111 (2010)

  9. Force-Constantsfunctional (details) The first term contains the product of the screened potential matrix elements: for for + …. This term depends on ω explicitly in the denominator but also implicitly in ρ and ρ’. The solution of this equation requires self-consistency in ω ! Baroniet al., Rev. Mod. Phys. 73, 515 (2001) Gonze and Lee, PRB 55, 10355 (1997) M. Calandra et al. Phys. Rev. B 82, 165111 (2010)

  10. Force-Constantsfunctional (details) The functional has the form: double-counting coulomb term Baroniet al., Rev. Mod. Phys. 73, 515 (2001) Gonze and Lee, PRB 55, 10355 (1997) M. Calandra et al. arXiv:1007.2098

  11. Difficulties in calculatingdynamicalforce-constants in metals The ω dependence of should be calculated self-consistently and it is thus very expensive. In the force-constants definition, T=T0=300K=0.0019 Ryd is the physical temperature which, in metallic systems, requires an enormous number of k-points to be evaluated.

  12. Stationary condition for FI J The following condition holds: and a symmetric one on ρ(r’). A linear error in affects the functional and the phonon frequencies at second order! This property can be used to efficiently calculate adiabatic and non-adiabatic phonon dispersion in a NON SELF-CONSISTENT WAY

  13. Approximated force constants functional We then define an approximate force constant functional: Where: The ω dependence of is neglected and the static limit is considered. SELF-CONSISTENCY at finite ω and at T=T0 is not needed. is not anymore evaluated self-consistently at the physical temperature T=T0 but at a much hotter one T=Tph≈0.03 Ryd at which the phonon calculation is carried out. Converging at Tph requires much less k-points, Nk(Tph) <<Nk(T0) The error in the phonon frequencies and on the functional is of order 2

  14. Fromtheory to a practicalcalculationscheme. Passing in Fourier space, summing and subtracting the standard adiabatic force constants calculated from first principles at a temperature Tph, namely - where: And the deformation potential matrix element (electron-phonon coupling) is: with

  15. Fromtheory to a practicalcalculationscheme. The DYNAMICAL force constants on an ULTRADENSE k-point grid Nk(T0) at very low temperature T0 are obtained from the calculation of the STATIC force constants on a COARSE grid Nk(Tph) and a hot temperature Tph If a fast calculation of the deformation potential in throughout the BZ is available. To interpolate the deformation potential matrix element we use Maximally localized Wannier functions N. Marzari and D. Vanderbilt, PRB 56, 12847 (1997) I. Souzaet al., PRB 65, 035109 (2002) Mostofiet al. Comput. Phys. Comm. 178,685 (2008) implementing the method proposed in Giustino et al. PRB 76, 165108 (2007)

  16. APPLICATIONS MgB2

  17. Adiabatic phonon dispersion in MgB2 Substantial enhancement of the in-plane E2g Kohn anomaly related to inter-cylinders nesting. Kortuset al. PRL 86, 4656 (2001) M. Calandra et al. Phys. Rev. B 82, 165111 (2010) A. Shuklaet al., PRL 90, 095506 (2003)

  18. Adiabatic phonon dispersion in MgB2 Substantial enhancement of the in-plane E2g Kohn anomaly related to inter-cylinders nesting. A Kohn-anomaly appears on E2g and B1g branches along ΓA Kortuset al. PRL 86, 4656 (2001) The ultradensek-point sampling leads to phonon frequencies in better agreement with experiments M. Calandra et al. Phys. Rev. B 82, 165111 (2010) A. Shuklaet al., PRL 90, 095506 (2003)

  19. Accuracy of Wannier interpolation Linear response Wannier interpolation

  20. Effect on EP coupling λ=0.74 (This Work) In agreement with other calculations on “sufficiently large grids” -> λ=0.73-0.77 Ahn and Pickett, PRL 86, 4366 (2001) Kong et al., PRB 64, 020501 (2001) Bohnenet al, PRL 86, 5771 (2001) Liu, MazinKortus, PRL 87, 087005 (2001) Choi et al. , Nature 418, 758 (2002) Eiguren and C. Ambrosch-Draxl, PRB 78, 045124 (2008) Electron-phonon coupling almost converged (with time) ? Does denser k-point sampling in the calculation of phonon frequencies have any effect on EP coupling ?

  21. Effect on Eliashbergfunction Significant discrepancy in the main peak position of the Eliashberg function (E2g mode)

  22. Effect on Eliashbergfunction Reduction of the Energy position of the E2g mode respect to previous works with improved sampling on phonon frequencies. Much lower value of ωlog Accurate k-point sampling on λ only is not sufficient, phonon frequencies need to be accurately converged!

  23. APPLICATIONS CaC6

  24. Adiabatic phonon dispersion in CaC6 Many Kohn anomalies occur at all energy scales in the phonon spectrum. (see black arrows) The low energy anomaly on Caxy phonon modes is not at X (as it was inferred on the basis of Fourier interpolated branches) but nearby. M. Calandra and F. Mauri, PRB 74, 094507 (2006) M. Calandra and F. Mauri, PRL 95, 237002 (2005) J. S. Kim et al., PRB 74, 214513 (2006)

  25. Adiabatic phonon dispersion in CaC6 Many Kohn anomalies occur at all energy scales in the phonon spectrum. (see black arrows) The low energy anomaly on Caxy phonon modes is not at X (as it was inferred on the basis of Fourier interpolated branches) but nearby. The anomaly is present at all energy scales (nesting)

  26. Non adiabatic (NA) phonon dispersion in CaC6 Giant NA effects predicted at zone center seen in Raman scattering. Saittaet al., PRL 100, 226401 (2008) Dean et al., PRB 81, 045405 (2010) It is unclear to what extent NA effects extend from zone center. Can NA effects be relevant for superconductivity ?

  27. Non adiabatic (NA) phonon dispersion in CaC6 Giant NA effects predicted at zone center seen in Raman scattering. Raman Saittaet al., PRL 100, 226401 (2008) Dean et al., PRB 81, 045405 (2010) It is unclear to what extent NA effects extend from zone center. Can NA effects be relevant for superconductivity ? NA effects are not localized at zone center but extend throughout the full Brillouin zone!

  28. SUPERCONDUCTING HYDROCARBONS M. Casula, M. Calandra, G. Profeta and F. Mauri, To appear on PRL Picene Coronnene Phenantrene C14H10 C22H14 C24H12 Insulatingmolecularcrystalbecomingsuperconductingupon K intercalation. Tc= 5K Tc= 3-18 K ??? Tc= 7K or 18K two phases ?? Mitsuhashiet al., Nature 464 76 (2010) Wang et al., arXiv:1102.4075 Kubozonoet al., unpublished

  29. K3Picene - Structure Molecule Crystal with K3 intercalation C22H14 Twomolecules/cell = 78 atom/cell T. Kosugiet al., J. Phys. Soc. Japan78, 11 (2009) K-intercalation changes the angle between the moleculesfrom 61° (Picene) to 114° (K3Picene). Rigid doping of undopedPicenecompletelyunjustified! T. Kosugiet al., J. Phys. Soc. Japan78, 11 (2009) T. Kosugiet al. arXiv:1109.2059 H. Okazaki, PRB 82, 195114 (2010)

  30. K3Picene – Electronic structure Substantialmixing of the LUMO+1 withotherelectronic states. Verynarrow Bandwith ≈0.3 eV Substantial variation of the DOS on the phonon frequencyenergyscale! Difficultieswith the electron-phononcalculation: Ultradensek-pointsamplingneeded (weneed a smearingsmallerthan 0.1 eV at least). The full K3Picenecrystal structure needs to betakenintoaccount (78 atoms/cell) A CHALLENGE FOR FIRST PRINCIPLES CALCULATIONS

  31. K3Picene – Electron-phononcoupling Molecular Crystal Bandwith ≈0.3 eV Not a molecular crystal Nk=120 3 78 atoms/cell All curvesconverged Strong variation of the electron-phononcouplingdetectable for σ < Bandwidth K3Piceneis not a molecularcrystal (for whatconcerns the EP coupling).

  32. K3Picene – Electron-phononcoupling EP matrixelement Fermi functions DOS/spin Usuallyωqνisneglected in the Dirac function and the Fermi functionsdifferences isreplacedwithitsderivative, leading to Is thisjustified in K3Picenewhereωqνis a substantial part of the bandwith ?

  33. K3Picene – Electron-phononcoupling EP matrixelement Fermi functions DOS/spin Usuallyωqνisneglected in the Dirac function and the Fermi functionsdifferences isreplacedwithitsderivative, leading to Is thisjustified in K3Picenewhereωqνis a substantial part of the bandwith ? NO (17% reduction of the electron-phononcoupling) !!!! ωlog=18 meV Rigid band doping of undopedpiceneλAD=0.78, ωlog=126 meV A. Subediet al. PRB 84, 020508 (2011)

  34. K3Picene – Electron-phononcoupling 40% of the electron-phononcouplingcomesfrom K and intermolecular modes (~80% if werestrict to intramolecularelectronic states!).

  35. Conclusion We develop a method to calculate adiabatic and non-adiabatic (clean limit) phonon dispersion in metals. The dispersions are calculated in a non self-consistent way using a Wannier-based interpolation scheme that allows integration over ultra dense phonon and electron momentum grids. MgB2 : occurrence of new Kohn anomalies. An accurate determination of phonon frequencies is necessary to have well converged Eliashberg functions and ωlog. CaC6 :non adiabatic effects are not localized at zone center but extend throughout the full BZ on Cxy vibrations. K3Picene : Intercalant and intermolecular phonon-modes contribute substantially (40%) to the EP coupling. 17% reduction of λby inclusion of ω. ωlog=18 meV M. Casula,et al. To appear on PRL Seealso M. Casula poster on K3Picene Linear response: Quantum-Espresso code Wannierization : Wannier90

  36. Electronic structure undoped versus dopedPicene K3 PICENE PICENE *Picene Ef K3Picene K-induced FS T. Kosugiet al., J. Phys. Soc. Japan78, 11 (2009) Totallydifferentelectronic structure and Fermi surface.

  37. K3Picene – Role of K Energy (eV) Additional Fermi Surface when adding K K3Picene *Picene (compensating background and K3picenecrystal structure)

  38. Eliashbergfunction: Casulav.s. Subedi A. Subediet al. PRB 84, 020508 (2011) α2F(ω)/N(0) M. Casula,et al. To appear on PRL

  39. Lowenergy phonon spectrum

  40. Wannierized band structures MgB2 CaC6

  41. Adiabatic CaC6 phonon dispersion

  42. Eliashbergfunction in CaC6

  43. From forces to phonon frequencies The force-constants matrix is complex, we define If then the relation Gives Allen formula.

  44. Force-Constantsfunctional (details) In standard time-independent linear-response theory the same object is written as: double-counting coulomb term Product of matrix elements involving the derivative of the external potential and the screened potential is present. No double-counting term is present. At convergence of the self-consistent process they must lead to the same result! Baroniet al., Rev. Mod. Phys. 73, 515 (2001) Gonze and Lee, PRB 55, 10355 (1997) M. Calandra et al. arXiv:1007.2098

  45. Force-Constantsfunctional (details) We define the following functional: What is the advantage of introducing this functional formulation ?

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