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 M. Calandra et al. Phys. Rev. B 82, (2010) M. Casula et al. to appear on Phys. Rev. Lett.

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

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

Outline Wannier interpolation scheme to calculate adiabatic and non-adiabatic phonon dispersions on ultradense electron and phonon momentum grids. Phonon dispersion and electron-phonon coupling in MgB 2 Adiabatic and non adiabatic phonon dispersion in CaC 6 Theory: Applications: Electron-phonon coupling in K-doped Picene (78 atoms per cell)

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

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

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

The functional has the form: whereis 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. Baroni et al., Rev. Mod. Phys. 73, 515 (2001) Gonze and Lee, PRB 55, (1997) M. Calandra et al. Phys. Rev. B 82, (2010) Force-Constants functional

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

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

The ω dependence of Difficulties in calculating dynamical force-constants in metals In the force-constants definition, T=T 0 =300K= Ryd is the physical temperature which, in metallic systems, requires an enormous number of k-points to be evaluated. should be calculated self-consistently and it is thus very expensive.

Stationary condition for F I J The following condition holds: A linear error inaffects 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 and a symmetric one on ρ(r’).

Approximated force constants functional We then define an approximate force constant functional: Where: is not anymore evaluated self-consistently at the physical temperature T=T 0 The ω dependence ofis neglected and the static limit is considered. SELF-CONSISTENCY at finite ω and at T=T 0 is not needed. but at a much hotter one T=T ph ≈0.03 Ryd at which the phonon calculation is carried out. Converging at T ph requires much less k-points, N k (T ph ) <<N k (T 0 ) The error in the phonon frequencies and on the functional is of order 2

From theory to a practical calculation scheme. Passing in Fourier space, summing and subtracting the standard adiabatic force constants calculated from first principles at a temperature T ph, namely - where: And the deformation potential matrix element (electron-phonon coupling) is: with

From theory to a practical calculation scheme. The DYNAMICAL force constants on an ULTRADENSE k-point grid N k (T 0 ) at very low temperature T 0 are obtained from the calculation of the STATIC force constants on a COARSE grid N k (T ph ) and a hot temperature T ph 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 implementing the method proposed in Giustino et al. PRB 76, (2007) N. Marzari and D. Vanderbilt, PRB 56, (1997) I. Souza et al., PRB 65, (2002) Mostofi et al. Comput. Phys. Comm. 178,685 (2008)

APPLICATIONS MgB 2

Adiabatic phonon dispersion in MgB 2 Substantial enhancement of the in-plane E 2g Kohn anomaly related to inter-cylinders nesting. Kortus et al. PRL 86, 4656 (2001) M. Calandra et al. Phys. Rev. B 82, (2010) A. Shukla et al., PRL 90, (2003)

Adiabatic phonon dispersion in MgB 2 Substantial enhancement of the in-plane E 2g Kohn anomaly related to inter-cylinders nesting. A Kohn-anomaly appears on E 2g and B 1g branches along ΓA The ultradense k-point sampling leads to phonon frequencies in better agreement with experiments Kortus et al. PRL 86, 4656 (2001) M. Calandra et al. Phys. Rev. B 82, (2010) A. Shukla et al., PRL 90, (2003)

Accuracy of Wannier interpolation Linear response Wannier interpolation

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

Effect on Eliashberg function Significant discrepancy in the main peak position of the Eliashberg function (E 2g mode)

Effect on Eliashberg function Reduction of the Energy position of the E 2g mode respect to previous works with improved sampling on phonon frequencies. Accurate k-point sampling on λ only is not sufficient, phonon frequencies need to be accurately converged! Much lower value of ω log

APPLICATIONS CaC 6

Adiabatic phonon dispersion in CaC 6 Many Kohn anomalies occur at all energy scales in the phonon spectrum. (see black arrows) The low energy anomaly on Ca xy 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, (2006) M. Calandra and F. Mauri, PRL 95, (2005) J. S. Kim et al., PRB 74, (2006)

Adiabatic phonon dispersion in CaC 6 Many Kohn anomalies occur at all energy scales in the phonon spectrum. (see black arrows) The low energy anomaly on Ca xy 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)

Non adiabatic (NA) phonon dispersion in CaC 6 Giant NA effects predicted at zone center seen in Raman scattering. Saitta et al., PRL 100, (2008) Dean et al., PRB 81, (2010) It is unclear to what extent NA effects extend from zone center. Can NA effects be relevant for superconductivity ?

Non adiabatic (NA) phonon dispersion in CaC 6 Giant NA effects predicted at zone center seen in Raman scattering. Saitta et al., PRL 100, (2008) Dean et al., PRB 81, (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! Raman

C 22 H 14 Picene Phenantrene C 14 H 10 Coronnene C 24 H 12 SUPERCONDUCTING HYDROCARBONS T c = 7K or 18K T c = 5KT c = 3-18 K ??? two phases ?? Insulating molecular crystal becoming superconducting upon K intercalation. Mitsuhashi et al., Nature (2010)Wang et al., arXiv: Kubozono et al., unpublished M. Casula, M. Calandra, G. Profeta and F. Mauri, To appear on PRL

C 22 H 14 K 3 Picene - Structure MoleculeCrystal with K 3 intercalation K-intercalation changes the angle between the molecules from 61° (Picene) to 114° (K 3 Picene). Two molecules/cell = 78 atom/cell T. Kosugi et al., J. Phys. Soc. Japan 78, 11 (2009) Rigid doping of undoped Picene completely unjustified! T. Kosugi et al., J. Phys. Soc. Japan 78, 11 (2009) T. Kosugi et al. arXiv: H. Okazaki, PRB 82, (2010)

K 3 Picene – Electronic structure Very narrow Bandwith ≈0.3 eV Substantial mixing of the LUMO+1 with other electronic states. Substantial variation of the DOS on the phonon frequency energy scale! Ultradense k-point sampling needed (we need a smearing smaller than 0.1 eV at least). Difficulties with the electron-phonon calculation: The full K 3 Picene crystal structure needs to be taken into account (78 atoms/cell) A CHALLENGE FOR FIRST PRINCIPLES CALCULATIONS

K 3 Picene – Electron-phonon coupling Bandwith ≈0.3 eV Strong variation of the electron-phonon coupling detectable for σ < Bandwidth N k = atoms/cell All curves converged K 3 Picene is not a molecular crystal (for what concerns the EP coupling). Molecular Crystal Not a molecular crystal

K 3 Picene – Electron-phonon coupling DOS/spin Fermi functions EP matrix element Usually ω qν is neglected in the Dirac function and the Fermi functions differences is replaced with its derivative, leading to Is this justified in K 3 Picene where ω qν is a substantial part of the bandwith ?

K 3 Picene – Electron-phonon coupling DOS/spin Fermi functions EP matrix element Usually ω qν is neglected in the Dirac function and the Fermi functions differences is replaced with its derivative, leading to Is this justified in K 3 Picene where ω qν is a substantial part of the bandwith ? NO (17% reduction of the electron-phonon coupling) !!!! A. Subedi et al. PRB 84, (2011) Rigid band doping of undoped picene λ AD =0.78, ω log =126 meV ω log =18 meV

K 3 Picene – Electron-phonon coupling 40% of the electron-phonon coupling comes from K and intermolecular modes (~80% if we restrict to intramolecular electronic states!).

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. MgB 2 : occurrence of new Kohn anomalies. An accurate determination of phonon frequencies is necessary to have well converged Eliashberg functions and ω log. CaC 6 : non adiabatic effects are not localized at zone center but extend throughout the full BZ on C xy vibrations. Linear response: Quantum-Espresso code Wannierization : Wannier90 K 3 Picene : Intercalant and intermolecular phonon-modes contribute substantially (40%) to the EP coupling. 17% reduction of λby inclusion of ω. ω log =18 meV M. Casula,et al. To appear on PRL See also M. Casula poster on K 3 Picene

Electronic structure undoped versus doped Picene K 3 Picene *Picene EfEf PICENE K 3 PICENE Totally different electronic structure and Fermi surface. T. Kosugi et al., J. Phys. Soc. Japan 78, 11 (2009) K-induced FS

K 3 Picene – Role of K Energy (eV) K 3 Picene *Picene (compensating background and K 3 picene crystal structure) Additional Fermi Surface when adding K

Eliashberg function: Casula v.s. Subedi α 2 F(ω)/N(0) M. Casula,et al. To appear on PRL A. Subedi et al. PRB 84, (2011)

Low energy phonon spectrum

Wannierized band structures MgB 2 CaC 6

Adiabatic CaC 6 phonon dispersion

Eliashberg function in CaC 6

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

Baroni et al., Rev. Mod. Phys. 73, 515 (2001) Gonze and Lee, PRB 55, (1997) M. Calandra et al. arXiv: double-counting coulomb term In standard time-independent linear-response theory the same object is written as: 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! Force-Constants functional (details)

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