RIKEN Center for Emergent Matter ScienceCenter for Emergent Matter Science Ryotaro ARITARyotaro ARITA Non-empirical post-Eliashberg study on high T c superconductivity in H 3 S

RIKEN Wataru Sano Takashi Koretsune Yusuke Nomura Univ. Tokyo Ryosuke Akashi Terumasa Tadano Shinji Tsuneyuki 2 Collaborators R. Akashi, M. Kawamura, S. Tsuneyuki, Y. Nomura, RA, PRB 91, (2015) W. Sano, T. Koretsune, T. Tadano, R. Akashi, RA, arXiv:

3 High T c SC in sulfur hydrides A. P. Drozdov et al., Nature (2015) T c 150GPa Isotope effect  ~0.3 → Phonon mechanism

4 McMillan’s formula Logarithmic averaged frequency Ele-ph coupling Coulomb pseudopotential McMillan PR1968 Allen-Dynes PRB 1975 Empirical parameter Fully ab initio calculation of T c = big challenge

5 Outline P=250GPaTcTc TcTc const DOS225 K Energy dependent DOS168 K-57K Self consistency in  193 K+25K Zero point motion202 K+9K Anharmonicity181 K-21K Vertex correction151K-30K Plasmon effect171K+20K Significant impact on T c  Fully non-empirical calculation of T c free from  *   ln /E F, T c /E F not small → we have to consider effects neglected in ME T c exp (~160K at 250GPa) is reproduced from first principles

6 McMillan’s formula Logarithmic averaged frequency Ele-ph coupling Coulomb pseudopotential McMillan PR1968 Allen-Dynes PRB 1975

We need large and for large For covalent system, is usually large, but … 7 Doped diamond Diamond is a 3D insulator → we need to dope many carriers ~ 0.3 E - E F [eV] DOS [states/eV/atom] doping T c < 10K Hard system with very high 

8 MgB 2 graphite MgB 2 σ band is occupied σ band makes FS E 2g phonon couples with  electrons for  electrons = 1.2 2D DOS for  band Kortus et al., PRL2001 Yildirim et al., PRL2001 d BB =1.78 Å T c = 39K

9 H 3 S: Density of states Van Hove singularity Im-3m structure Large Strong  bond d HS < 1.5 Å Duan et al., Sci. Rep. 2015

10 H 3 S: Eliashberg function Large >2 ! Duan et al., Sci. Rep ~200meV !

11 High T c SC in metallic hydrogen H = lightest element → large

12 H 3 S: experiment A. P. Drozdov et al., Nature (2015) M. Einaga et al., arXiv: H3SH3S D3SD3S Isotope effect  ~ K

13 Self-consistent perturbation theory: lowest-order dressed-phonon and dressed Coulomb contribution to  retained (Nambu-Gor’kov formalism) Migdal-Eliashberg Theory We assume  D /E F (T c /E F ) is sufficiently small

14 Non-adiabatic SC  D /E F (T c /E F ) is not small ? “Open Pandora’s box” We need to consider effects neglected in ME

15 Beyond Migdal-Eliashberg Theory (1)Dynamical structure of V c : plasmon assisted SC (2)Vertex correction  D /E F is not small ? If small q is dominant in SC is enhanced

16 Beyond Migdal-Eliashberg Theory (3) Zero point motion with ZPM w/o ZPM E-E F [eV] DOS [states/eV/atom]

17 Beyond Migdal-Eliashberg Theory (4) Anharmonicity P1 Cccm R3mIm-3m Structural phase transition from R3m to Im-3m Duan et al., Sci. Rep. 2015

18 Beyond Migdal-Eliashberg Theory (4) Anharmonicity Frequency [cm -1 ] anharmonic harmonic Electron-phonon coupling: weaker Phonon frequency: higher

19 Non-empirical calculation of T c First- principles Extension of DFT (Superconducting DFT) DFT + (post) Migdal- Eliashberg W. Sano, T. Koretsune, T. Tadano, R. Akashi, RA, arXiv: R. Akashi, M. Kawamura, S. Tsuneyuki, Y. Nomura, and RA, PRB 91, (2015)

20 Conventional calculation based on ME ✔ Momentum average & constant DOS ✔ Coulomb pseudo potential W VcVc V ph ωDωD 0 empirical parameter

21 Non-empirical calculation of retardation effect T c calc =225 K Converged ! The energy range of W can be covered by m~ # of Matsubara freq. Const DOS approx. overestimate T c exp energy DOS 

22 Non-empirical calculation of retardation effect # of Matsubara freq. T c [K] 225 K (constant DOS approx.) → 168 K (energy dependent DOS) Converged ! Const DOS approx. overestimate T c by 57K energy DOS

23 Feedback effect in self-consistent calc. of  In self-consistent calculation, mass enhancement effect becomes weaker (feedback effect) T c : 168 K → 193 K (+25 K)

24 Zero point motion with ZPM w/o ZPM E-E F [eV] DOS [states/eV/atom] T c : 193 K → 202 K (+9 K)

25 Anharmonicity Electron-phonon coupling: weaker Phonon frequency: higher T c : 202 K → 181 K (-21 K) Frequency [cm -1 ] anharmonic harmonic

26 Vertex correction : Einstein Phonon Simplifying... At the lowest Matsubara frequency... ( ω n = π/β ) 〈 Γ q (1) 〉 q → ~ (H 3 S) T c decreases by 30K

27 P=250GPaTcTc TcTc const DOS225 K Energy dependent DOS168 K-57K Self consistency193 K+25K Zero point motion202 K+9K Anharmonicity181 K-21K Vertex correction151K-30K Plasmon effect?? A. P. Drozdov et al., Nature (2015) M. Einaga et al., arXiv: T c exp ~160K Significant impact on T c Result

28 Non-empirical calculation of T c First- principles Extension of DFT (Superconducting DFT) DFT + (post) Migdal- Eliashberg W. Sano, T. Koretsune, T. Tadano, R. Akashi, RA, arXiv: R. Akashi, M. Kawamura, S. Tsuneyuki, Y. Nomura, and RA, PRB 91, (2015)

29 DFT for normal states v   Hohenberg-Kohn theorem one-to-one correspondence Kohn-Sham equation

30 DFT for superconductors electron density anomalous density [v,  ]  [ ,  ] Hohenberg-Kohn theorem for superconductors Oliveira et al., PRL 60, 2430 (1988) Kreibich & Gross PRL 86, 2984 (2001) M. Lüders et al, PRB 72, (2005) M. Marques et al, PRB 72, (2005)

31 Oliveira et al., PRL 60, 2430 (1988) Kreibich & Gross PRL 86, 2984 (2001) M. Lüders et al, PRB 72, (2005) M. Marques et al, PRB 72, (2005) SCDFT: Kohn-Sham BdG equation

32 Once F xc is given, we can calculate T c without adjustable parameters Linearized gap equation Oliveira et al., PRL 60, 2430 (1988) Kreibich & Gross PRL 86, 2984 (2001) M. Lüders et al, PRB 72, (2005) M. Marques et al, PRB 72, (2005) SCDFT: Gap equation

33 SCDFT: exchange correlation functional F (anomalous Green fn.) F xc e-ph = F xc e-e = Static screened Coulomb V c F (anomalous Green fn.) Kohn-Sham perturbation theory ( F, D, V c are obtained from first-principles calc.) M. Lüders et al, PRB 72, (2005) M. Marques et al, PRB 72, (2005)

34 SCDFT: benchmark calculation

35 SCDFT for plasmon mechanism F (anomalous Green fn.) F xc e-ph = F xc e-e = F (anomalous Green fn.) Dynamical screened Coulomb V c (  ) with plasmon-pole approximation Kohn-Sham perturbation theory ( F, D, V c are obtained from first-principles calc.) R. Akashi & RA, PRL (2013), JPSJ (2014)

36 Li under high pressures: Plasmon effect R. Akashi & RA, PRL (2013), JPSJ (2014)

37 H 3 S: Plasmon effect  Temperature (K) (eV) Dynamical SCDFT (phonon+plasmon) Static SCDFT (phonon only) +20K

38 P=250GPaTcTc TcTc const DOS225 K Energy dependent DOS168 K-57K Self consistency193 K+25K Zero point motion202 K+9K Anharmonicity181 K-21K Vertex correction151K-30K Dynamical Coulomb171K+20K A. P. Drozdov et al., Nature (2015) M. Einaga et al., arXiv: T c exp ~160K T c exp is reproduced by non-empirical calculation Significant impact on T c Result

39 H 3 S vs H 2 S E-E F [eV] DOS [states/eV/f.u.] E-E F [eV] H 2 S, P-1H 3 S, Im-3m No van Hove singularity

40 P=140GPaTcTc TcTc const DOS56 K Energy dependent DOS66 K+10K Self consistency63 K-3K Zero point motion44 K-19K Anharmonicity33 K-11K Vertex correction24 K-9K Dynamical Coulomb44 K+20K A. P. Drozdov et al., Nature (2015) T c exp =30~60K T c exp is reproduced by non-empirical calculation Result

41 Conclusion  We performed a fully non-empirical calculation of T c free from  *  We found that effects neglected in ME on T c is significant  We succeeded in reproducing T c exp (~160K at for H 3 S, ~40K for H 2 S)

42 Conventional ME vs Post ME + W. Sano, T. Koretsune, T. Tadano, R. Akashi, RA, arXiv:

43 Retardation effect in SCDFT

44 Importance of dynamical Coulomb int. “Plasmon mechanism” Y. Takada, 1978

45 High T c in Li under high pressures Shimizu et al., Nature 419, 597 (2002) Struzhkin et al., Science 298, 1213 (2002) Deemyad and Schilling, PRL 91, (2003) High T c ~20K under P~30GPa

46 Discussion: can we enhance T c ? SC suppressed Non-adiabatic SC  D /E F is not small ? For H 3 S, Q c ~1, but … If forward scattering is dominant, the vertex correction P enhances T c Grimaldi et al., PRL 1995

47 Discussion: can we enhance T c ? C. Heil and L. Boeri, PRB 2015 Construct a set of “virtual” isovalent atom X, which interpolates between actual chalchogen atoms Mixing S and O → H-X bond becomes stronger → enhance T c