1. 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

2. 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, 224513 (2015) W. Sano, T. Koretsune, T. Tadano, R. Akashi, RA, arXiv:1512.07365

3. 3 High T c SC in sulfur hydrides A. P. Drozdov et al., Nature (2015) T c =203K@ 150GPa Isotope effect  ~0.3 → Phonon mechanism

4. 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. 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. 6 McMillan’s formula Logarithmic averaged frequency Ele-ph coupling Coulomb pseudopotential McMillan PR1968 Allen-Dynes PRB 1975

7. 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. 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. 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. 10 H 3 S: Eliashberg function Large >2 ! Duan et al., Sci. Rep. 2015 ~200meV !

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

12. 12 H 3 S: experiment A. P. Drozdov et al., Nature (2015) M. Einaga et al., arXiv:1509.0315 H3SH3S D3SD3S Isotope effect  ~0.3 203K

13. 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. 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. 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. 16 Beyond Migdal-Eliashberg Theory (3) Zero point motion with ZPM w/o ZPM E-E F [eV] DOS [states/eV/atom]

17. 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. 18 Beyond Migdal-Eliashberg Theory (4) Anharmonicity Frequency [cm -1 ] anharmonic harmonic Electron-phonon coupling: weaker Phonon frequency: higher

19. 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:1512.07365 R. Akashi, M. Kawamura, S. Tsuneyuki, Y. Nomura, and RA, PRB 91, 224513 (2015)

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

21. 21 Non-empirical calculation of retardation effect T c calc =225 K Converged ! The energy range of W can be covered by m~1000-5000 # of Matsubara freq. Const DOS approx. overestimate T c exp ~160K@250GPa energy DOS 

22. 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. 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. 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. 25 Anharmonicity Electron-phonon coupling: weaker Phonon frequency: higher T c : 202 K → 181 K (-21 K) Frequency [cm -1 ] anharmonic harmonic

26. 26 Vertex correction : Einstein Phonon Simplifying... At the lowest Matsubara frequency... ( ω n = π/β ) 〈 Γ q (1) 〉 q → ~ -0.22 (H 3 S) T c decreases by 30K

27. 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:1509.0315 T c exp ~160K Significant impact on T c Result

28. 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:1512.07365 R. Akashi, M. Kawamura, S. Tsuneyuki, Y. Nomura, and RA, PRB 91, 224513 (2015)

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

30. 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, 024545 (2005) M. Marques et al, PRB 72, 024546 (2005)

31. 31 Oliveira et al., PRL 60, 2430 (1988) Kreibich & Gross PRL 86, 2984 (2001) M. Lüders et al, PRB 72, 024545 (2005) M. Marques et al, PRB 72, 024546 (2005) SCDFT: Kohn-Sham BdG equation

32. 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, 024545 (2005) M. Marques et al, PRB 72, 024546 (2005) SCDFT: Gap equation

33. 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, 024545 (2005) M. Marques et al, PRB 72, 024546 (2005)

34. 34 SCDFT: benchmark calculation

35. 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 111 057006 (2013), JPSJ 061016 (2014)

36. 36 Li under high pressures: Plasmon effect R. Akashi & RA, PRL 111 057006 (2013), JPSJ 061016 (2014)

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

38. 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:1509.0315 T c exp ~160K T c exp is reproduced by non-empirical calculation Significant impact on T c Result

39. 39 H 3 S vs H 2 S @140GPa@250GPa 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. 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. 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. 42 Conventional ME vs Post ME + W. Sano, T. Koretsune, T. Tadano, R. Akashi, RA, arXiv:1512.07365

43. 43 Retardation effect in SCDFT

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

45. 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, 167001 (2003) High T c ~20K under P~30GPa

46. 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. 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