1. Quest for Higher Tc or RTS
3 possible routes:
1. (conventional) phonon SC
2. Unconventional cuprates SC
3. Fe-based SC
1. H3S under Extremely high pressures
2. YBCO under pulsed Laser pumping
3. Monolayer FeSe on an STO substrate
Under extreme conditions:

2. Sulfur Hydride @200 GPa : Phonon-SC (?) Theoretical prediction:
A.P. Drozdov, M.I. Eremets, I.A. Troyan, arXiv:
A.P. Drozdov, M.I. Eremets, I.A. Troyan, V. Ksenofontv & S.I. Shylin, Nature 525, 73 (2015).
(Max Planck, Mainz)
Tc = 203 K
Im3m: H3S (?)
Theoretical prediction:
“P-induced metallization of dense (H2S)2H2 with high-Tc superconductivity”
D. Duan et al., Sci. Reports 4, (2014). (Jilin Univ., Changchun)

3. S atoms form a bcc sublattice.
D. Duan et al., Sci. Reports 4, (2014).
arXiv:
Structural analysis cannot distinguish between R3m and Im-3m.

4. Rules of Matthias for Discovering New Superconductors
#1: Find cubic crystals, high symmetry is best.
#2: Find d-electron metals and/or metals with the average number of valence electrons, preferably odd numbers 3, 5, and 7, peaks in DOS are good.
#3: Stay away from magnetism,
exclude metals showing or in close vicinity of magnetism.
#4: Stay away from oxygen and insulators,
exclude metals near a metal-insulator transition such as oxide
materials.
#5: Stay away from theorists

5. A15 Superconductors
Tc (K)
Liq. N２
Liq. H２
1973
Tc = 23 K
Liq. He
year

6. Strong electron-phonon interaction in A15 Resistivity saturation
Strong phonon scattering at elevated T’s  Very short mean free path, l ~ a (or kFl ≤ 2p)
Resistivity saturation Cu

7. BCS Tc Limit/Tc Ceiling ?
Tc (K)
Liq. N２
BCS limit ?
Liq. H２
1973
Tc = 23 K
Liq. He
year

8. History of Phonon High-Tc
200
History of Phonon High-Tc
Tc (K)
100
~ 30 K
MgB2
39 K
BaKBiO3
C60
Nb3Ge
23 K
2001
2014
1988, 1991
1973
1970
1980
1990
2000
2010
year

9. Tc bound (ceiling) of “phonon” SC
kBTc ~ ħW0 e-1/l
(BCS-Migdal)
(l = N(EF) <I 2> / MW02 )
kBTc ≪ ħW0 ≪ EF
For very large W0 and moderate l, no Tc ceiling as long as ħW0 ≪ EF
“Metallic Hydrogen: A High-Temperature Superconductor?”
N.W. Ashcroft, Phys. Rev. Lett. 21, 1748 (1968).

10. Tc bound (ceiling) of “phonon” SC
Migdal-Eliashberg (McMillan-Allen-Dynes)
ħ<w>
kBTc
Weak to moderate el-ph coupling:
kBTc ~ ħW0 e-1/l
For very large W0 and moderate l, no Tc ceiling as long as ħW0 ≪ EF
Migdal
Strong el-ph coupling:
For large l; Tc ~ l1/2
Some may wonder if there is an upper bound of Tc for BCS phonon superconductivity. This is not correct. The widely used McMillan-Allen-Dynes formula for Tc basically has no Tc bound. Even in the case of not strong electron-phonon coupling Tc can be high when relevant phonon frequency is very high. This is the case with MgB2 and probably with hydrogen dominant metallic alloys including H3S. For large electron-phonon coupling Tc in the McMillan-Allen-Dynes formula asymptotically increases as proportional to square root of the electron-phonon coupling constant. However, in this case the crystalline lattice tends to be unstable and suppresses superconductivity.
No Tc ceiling as long as a lattice instability can be avoided.

11. Doped Bismuthate (BaBiO3)
Strong electron-phonon coupling
Strong electron-phonon coupling Bi Ba
Superconductivity (Tc max~ 12, 30 K) is unstable against CDW formation.
L.F. Mattheiss et al., Phys. Rev. B 37, 3745 (1988).
A.W. Sleight et al., Solid State Commun. 17, 27 (1975).
R.J. Cava et al., Nature 332, 814 (1988).

12. Parent Insulator BaBiO3
3D-CDW
Bi5+
Bi3+
Superconductor
CDW Insulator
Breathing-mode distortions
T.M. Rice & L. Sneddon, Phys. Rev. Lett. 47, 689 (1981).
S. Tajima, SU et al., Phys. Rev. B 32, 6302 (1985).
CDW gap
S. Uchida, K. Kitazawa, S. Tanaka, Phase Transitions 8: (1987).

13. IR & Raman phonons in BaBiO3
IR-active
Raman-active
BaBiO3 \\
breathing mode
BaBiO3

14. Resonant Raman scattering from the breathing-mode phonon in BaBiO3
CDW gap
2-phonon
tuning incident light energy
l = 5145 Å
3-phonon
4-phonon

15. Strong covalent bond and strong electron-phonon coupling in BaBiO3
s-bonding Bi6s-O2p bands spread over ~ 16 eV
Extremely large deformation potential for breathing-mode lattice distortions
breathing mode
L.F. Mattheiss & D.R. Hamann, Phys. Rev. B 28, 4227 (1983).

16. Relativistic Effect on Atomic Energy Levels
P3/2
P3/2
S1/2
s, p
P1/2
P1/2
j = l + s
S1/2
“non-relativistic”
“spin-orbit　interaction”
“mass-correction”
Hso = (Z/137)2 l·s　
m = m0/(1 – v2/c2)1/2　

17. Charge Disproportionation (Valence Skipper)
“ionic picture” (too simplified)
Ba2+Bi4+O2-3
2Bi4+(6s)1  Bi3+(6s) Bi5+(6s)0 6p 6p 6p
relativistic effect: “mass-correction”
“closed shell”
~ (Z/137)2 Ry 6s 6s 6s
Bi: Z = 83
“closed shell”
5s, 5p, 5d, …

18. Strong Covalent Bond in BaBiO3
Antibonding
BaBi1-xPbxO3
CDW Gap
Doping
Ba1-xKxBiO3
Bi6s：Bi4+
O2ps
Bonding
Bi4+(6s)1  Bi3+(6s)2 + Bi5+(6s)0

19. Strong Covalent Bond in MgB2
sp2-Antibonding (s *)
B2px, 2py
B2px, 2py
Nonbonding Mg
B2s
B2s
Charge Transfer
s -band
sp2-Bonding (s)

20. Superconductivity next to CDW
SC often emerges from CDW phase, but Tc is not high compared to SC competing with AF (SDW) phase: i) phonon vs electronic pairing ? Ii) conventional s-wave vs unconventional pairs ? Iii) charge (e) vs spin (sx, sy, sz); 1:3 (Aoki) ?
E. Morosan, H.W. Zandbergen, N.P. Ong, R.J. Cava et al., Nature Phys. 2, 544 (2006).
K.E. Wagner, R.J. Cava et al., Phys. Rev. B 78, (2008).

21. Superconductivity next to CDW Hole-doped cuprate two-leg ladders
CuxTiSe2 P
Tc max~ 12 K 12 8
Tc(K)
No spin order 4 2 4 6 8
P (GPa)
3cL
K.M. Kojima, N. Motoyama, H. Eisaki, SU, J. Electron Spectroscopy and Related Phenomena , 237 (2001).
x ~ 11 (n=1/3)
N. Motoyama, SU et al., Europhys. Lett. 58, 758 (2002).

22. Charge and SC orders are intertwined.
Tc max~ 100 K
400
300 PG T*
Temperature T (K)
200
TSCon
TCDWon
100 Tc AF FL
d-SC
0.3
0.1
0.2
Hole doping p
QCP ?

23. Optimized Phonon Mechanism in MgB2
1) Strong sp2 bonding: Layer structure
2) Light element : Combined with the strong sp2 bond makes W0 very high.
3) A simple metal: ħW0 ≪ EF (Migdal)
4) Strong electron-phonon coupling without instability: Cylindrical FS (s-FS) avoids a structural instability.
These seem to conspire to optimize the phonon-mediated superconductivity in MgB2.
W. E. Pickett, Physica C 468, 126 (2007).

24. “Covalent Bond Driven Metallic”
Tc ~ 40 K
MgB2: Mg2+(B2)2-
ħW0 ~ 80 meV
l ~ 1
s-band
Bsp2 hybridization s-bonding band
J.M. An and W. E. Pickett, Phys. Rev. Lett. 86 (2001).
s-FS

25. Doped Fullerides (C60)-Cs3C60
Basically isotropic s-wave pairing, but the highest Tc is realized in the vicinity of AF phase.
A15-Cs3C60
P = 7 kbar P
A.Y. Ganin et al., Nature Mater. 7, 367 (2008).

26. History of Phonon High-Tc
200
History of Phonon High-Tc
203 K
GPa
Tc (K)
100
30 K
MgB2
39 K
BaKBiO3
Nb3Ge
23 K
2001
1988
1973
1970
1980
1990
2000
2010
year

27. Tc = 203 K Superconductivity in Sulfur Hydride (H3S) @ P=200 GPa
Resistance
Magnetization/ Meissner
A.P. Drozdov, M.I. Eremets, I.A. Troyan, V. Ksenofontv & S.I. Shylin, Nature 525, 73 (2015).

28. Further Optimization Phonon Mechanism in H3S ?
1) Strong covalent bonding (H1s-S3p)
2) Light element : Combined with the strong bond makes W0 very high.
3) A simple metal: ħW0 ≪ EF (Migdal) ??
4) Strong electron-phonon coupling without instability: Non-nesting FS (s-FS) avoids a structural instability.
Isotropic HTS suitable for SC power application
N. Bernstein, I.I. Mazin et al., Phys. Rev. B 91, (2015).

29. Why does theory successfully predict high Tc in H3S?
W. Sano, T. Koretsune, T. Tadano, R. Akashi, R. Arita, arXiv:
● Quantum nature of H atom
- zero-point motion
- anharmonic vibrations
● Validity of Migdal theorem
- van Hove singularity near EF
The effective Fermi energy is small; ‘EF’ ~ ħW0
m* = m / [1 + m ln (EF/ħW0)]
Y. Quan, W.E. Pickett, arXiv:
L.P. Gor’kov, V.Z. Kresin, arXiv:

30. Quantum Hydrogen Bond in R3m
I. Errea et al., Nature 532, 81 (2016).
d1: Covalent bond
d2: Hydrogen bond

31. Revival of Matthias’ Rules (?)
#1: Find cubic crystals, high symmetry is best.
#2: Find d-electron metals and/or metals with the average number of valence electrons, preferably odd numbers 3, 5, and 7,
peaks in DOS are good.
#3: Stay away from magnetism,
exclude metals showing or in close vicinity of magnetism.
#4: Stay away from oxygen and insulators,
exclude metals near a metal-insulator transition such as oxide
materials.
#5: Stay away from theorists

32. Why does theory successfully predict high Tc in H3S?
arXiv:

33. No Tc bound for “phonon” SC (?)
McMillan-Allen-Dynes (assuming m*= )
A. Bianconi: “SC above the lowest Earth temperature: 184 K (-89.2℃)”
GPa, Tc ~ 200 K
kBTc = ħW0 e-1/l
W0 = (K/ M)1/2 ~ 200 meV, <w> ~ 1300 K, l ~ 2.2
2. 0 Pa, Tc ~ 40 K
kBTc = ħW0 e-1/l
W0 = (K/ M)1/2 ~ 80 meV, <w> ~ 600 K, l ~ 1
1. doped BaBiO3 , Tc ~ 30 K
Unstable against CDW

34. No Tc bound for “phonon” SC (?)
McMillan-Allen-Dynes (assuming m*= )
GPa, Tc ~ 750 K
kBTc = ħW0 e-1/l
W0 = (K/ M)1/2 ~ 400 meV, <w> ~ 2300 K, l ~ 3
J.M. McMahon & D.M. Ceperley, Phys. Rev. B 84, (2011).
GPa, Tc ~ 200 K
kBTc = ħW0 e-1/l
W0 = (K/ M)1/2 ~ 200 meV, <w> ~ 1300 K, l ~ 2.2
2. 0 Pa, Tc ~ 40 K
kBTc = ħW0 e-1/l
W0 = (K/ M)1/2 ~ 80 meV, <w> ~ 600 K, l ~ 1
1. doped BaBiO3 , Tc ~ 30 K
Unstable against CDW

35. Even Higher Tc in Atomic Metallic H @2TPa
Phys. Rev. B 84, (2011).
“Metallic Hydrogen: A High-Temperature Superconductor?”
N.W. Ashcroft, Phys. Rev. Lett. 21, 1748 (1968).
Tcmax ~ 750 K 2000 GPa
ħW0 ~ meV <w> = 2300 K
l ~ 3

36. Supplementary Informations

37. Valence Skippers La

38. Charge Disproportionation (Valence Skipper)
2Tl2+(6s)1  Tl1+(6s) Tl3+(6s)0
2Pb3+(6s)1  Pb2+(6s) Pb4+(6s)0 6p 6p 6p
relativistic effect: “mass-correction”
“closed shell”
~ (Z/137)2 Ry 6s 6s 6s
“closed shell”
5s, 5p, 5d, …

39. Relativistic Element Hg Why is Hg a ‘liquid’ metal at RT ?

40. relativistic effect: “mass-correction”
Relativistic effect in Hg: Weak bond due to “closed shell” electronic structure
Hg1+(6s)1  Hg0 (6s) Hg2+(6s)0 6p 6p 6p
relativistic effect: “mass-correction” Hg
“closed shell”
~ (Z/137)2 Ry 6s 6s 6s
“closed shell”
5s, 5p, 5d, …