1. Theoretical approach to physical properties of atom-inserted C 60 crystals 原子を挿入されたフラーレン結晶の 物性への理論的アプローチ Kusakabe Lab Kawashima Kei

2. Contents Introduction – Crystal structures of atom-inserted C 60 crystals (Objects of my study) – Cs 3 C 60 crystal (The main object to study from now on) Mott insulator-superconductor transition of Cs 3 C 60 Way to study ― Theoretical approach to physical properties by computational simulations ― First principles calculation in DFT within LDA Current studies ― Computational simulations for C 60 Crystal Future works ― Computational simulations for Cs 3 C 60 crystal Summary

3. Crystal structures of atom-inserted C 60 crystals Conventional unit cell of a FCC C 60 Crystal Superconductivity found in 1990s. Insulator (Band gap ≒ 1.2ev) SC Insulator Metal Insulator

4. The main object to study from now on － Cs 3 C 60 crystal Cs 3 C 60 Crystal (A15 structure) Cs 3 C 60 Crystal (A15 structure) In 2008, superconductivity in Cs 3 C 60 crystal was found by Takabayashi group. Cs atom

5. Pressure dependence of T c of Cs 3 C 60 crystal Low pressure region Ref: ALEXEY Y. GANIN et al. Nature Mat., Vol. 7(2008) Superconductors have perfect anti-magnetism( 完全反磁性 ).

6. Below about 47K, Cs 3 C 60 is Mott insulator. Under more than 3kbar, Cs 3 C 60 is superconductor. Anti-ferro magnetism Electron pair Mott insulator – Superconductor transition

7. Ref: Metal AFI : Anti-ferro insulator (Mott insulator ) SC : Superconductor T N is the temperature at which the zero-field magnetization begins to increase. T c is the temperature at which the zero-field magnetization begins to decrease. A copper-oxide crystal Phase diagram of Cs 3 C 60 Hole density per Cu atom

8. Way to study ― Theoretical approach to physical properties( 物性 ) by computational simulations Numerical calculations of the physical properties using computers (Parallel calculation) Experimental facts Input data of a material Resulting output data Comparison Calculations by other groups

9. Advantages and disadvantages of computational simulations Advantages – You can estimate physical properties of materials easily using only computers. – You can analyze unknown materials. – You can perform accurate calculations of elastic properties( 弾性 ) and phonon dispersion etc. Disadvantages – Sometimes estimated physical properties of materials do not agree with experimental facts. – It is not so easy to analyze correctly systems such as strongly correlated electron systems( 強相関電子系 ) and high-temperature superconductors( 高温超伝導体 ).

10. First principles method In DFT( 密度汎関数理論 ) within LDA( 局所密度近似 ) In first principles method, you begin with Schrödinger eigen equation, and analyze physical properties of materials theoretically. Schrödinger eigen equation in a crystal at r.

11. Band structures of C 60 -based crystals C 60 (FCC) - Insulator K 3 C 60 (FCC) - Metal Ba 6 C 60 (BCC) - Semimetal Unoccupied states Occupied states Fermi energy Ref: O. Gunnarsson, Reviews of Modern Physics, Vol. 68, No. 3, 575-606(1996) ・ Steven C. Erwin, Phys. Rev. B, Vol. 47 No.21, 14657- 14660(1993) Wave vector space Band gap

12. Current study ― Theoretical simulations for C 60 Crystal 1.Optimize the atomic positions (60 C atoms in a unit cell) 2.Obtain the optimum lattice constant (length of the one edge of FCC conventional unit cell) 3.Band structure 4.Density of states (DOS)

13. 1. Optimize the atomic positions Parts of an input data Initial values &control calculation='relax' &system ibrav=2 celldm(1)=26.79 nat=60 ntyp=1 ATOMIC_POSITIONS (angstrom) C -0.707 0.000 3.455 C -1.425 1.164 3.005 ・ C 2.285 -2.579 0.728 To obtain the optimized atomic positions, you set the values of the initial lattice constant and the initial atomic potions to the experimental values. Optimized atomic positions

14. 2. Get the optimum lattice constant Parts of input data Total energy vs lattice constant lista=’26.55 26.60 26.65 26.70.....' for a in $lista do &control calculation=‘scf' &system ibrav=2 celldm(1)=$a nat=60 ntyp=1 ATOMIC_POSITIONS (angstrom) C -0.713 0.000 3.485 C -1.437 1.174 3.031 ・ C 2.303 -2.601 0.734 done Experimental value 26.79 Bohr Experimental value 26.79 Bohr 26.63 Bohr 誤差約 0.6%

15. 3. Band structure Ref: O. Gunnarsson, Reviews of Modern Physics, Vol. 68, No. 3, 575-606(1996) Band gap By O.Gunnarsson groupBy me Experimental band gap of C 60 crystal is about 1.2 ev.

16. 4. Density of states (DOS) D(ε) [states/ev ・ cell] ε [ev] Band gap D( ε ) shows the number of electronic quantum states per unit cell existing between ε and ε + Δ ε.

17. Numerical applications of DOS Some physical properties of electron system can be estimated from one electron energy and DOS. Total energy of electronic system Low-temperature Specific heat of electronic system Fermi distribution function Superconductive transition temperature by McMillan’s formula Electron-Phonon Coupling Constant Electron-Electron Coulomb Interaction ( μ =D( ε F )V c )

18. Future works ― Calculations for Cs 3 C 60 under higher pressures(1Gpa, 10Gpa, 100Gpa etc.) Electron-phonon coupling ( 電子 - フォノン結合 ) → important in Superconductivity based on BCS theory. Electron-phonon coupling ( 電子 - フォノン結合 ) → important in Superconductivity based on BCS theory. Very stable crystal structure is needed for phonon calculations! ・ Band structure ・ Density of states ・ Fermi surface ・ Band structure ・ Density of states ・ Fermi surface ・ Atomic positions ・ lattice constant ・ Atomic positions ・ lattice constant

19. Summary The main studying object from now on ― Cs 3 C 60 crystal Below about 47K under ambient pressure, it is an insulator called Mott insulator. By applying pressure, it transfers to a superconductor at low temperatures. I’ll try to study superconductive mechanism of Cs 3 C 60 under higher pressure by calculating electronic structure and electron-phonon coupling. Theoretical simulations based on first principles method You can estimate various physical properties of crystals using only computers. ― Crystal structure optimization, band structure, density of states, and phonon structure etc. What I learned from my studies up to now  I’ve got familiar with parallel calculation for many-electrons system.  I’ve learned that DFT within LDA has good calculation accuracy for some C 60 -based crystals.  I’ve got prepared for future works by calculating physical properties of C 60 crystal.

20. LDA vs LDA+DMFT for Cs 3 C 60 What can be done with LDA method – Calculation of Band structure, DOS, Charge density, Fermi surface, wave function etc. – Evaluation of T c based on BCS theory. What can be done with LDA+DMFT method – Calculation of Band structure, DOS, Charge density, Fermi surface, wave function etc. – Theoretical treatment of Mott insulator.

21. What is superconductivity? Perfect diamagnetism Perfect conductivity Meissner effect 2 main Characters of superconductivity Paramagnetism(Metal) Perfect diamagnetism(Superconductor) Magnetic field

22. Electron-doped C 60 superconductors Valence (The number of doped- electrons per C 60 molecule) BCS (Electron pair by phonon- electron interaction) Non-BCS (Electron pair by electron- electron interaction etc.) 3 K 3 C 60,RbK 2 C 60, Rb 2 KC 60 Rb 3 C 60, Rb 2 CsC 60,RbCs 2 C 60 (All FCC) Cs 3 C 60 (A15) →Future works 8 Ba4C60(BCO) 9 A 3 Ba 3 C 60 (BCC) (A=Alkali earth) 10 A 2 Ba 4 C 60 (BCC), Ca 5 C 60 12 Sr 6 C 60 (BCC), Ba 6 C 60 (BCC)

23. T c of BCS-type superconductors(valence=3) Material Lattice constant Calculated TcExperimental Tc K 3 C 60 14.253 Å 16.7K 19.28K RbK 2 C 60 14.299 Å 21.7K 21.80K Rb 2 KC 60 14.364 Å 24.6K 25.40K Rb 3 C 60 14.436 Å 28.6K 29.40K Rb 2 CsC 60 14.493 Å 34.3K 31.30K Ref : Ming-Zhu Huang et al. Physical Review B, Volume 46, Number 10, 6572-6577 (1992)

24. Pressure depencences Ref: ALEXEY Y. GANIN et al. Nature Mat., Vol. 7(2008) T c vs pressure (low pressure region) T c vs pressure (high pressure region) Superconducting fruction vs pressure (low pressure region)

25. 5. Space distribution of electron density

26. Example of theoretical simulations to explore high-pressure physics fccbcc sc (simple cubic) Ca-IV Ca-V Pressure [GPa] 2032113139 Strong-coupling theory of superconductivity tells T c =20K. Ca-IV structure by theoretical simulations

27. Parallel calculation system Queue QUEUE PRIO STATUS MAX JL/U JL/P NJOBS PEND RUN SUSP P1 230 Open:Active 128 - - - 856 728 128 0 B1 210 Open:Active 256 - - - 0 0 0 0 F4 120 Open:Active 4096 - - - 192 128 64 0 F8 120 Open:Active 4096 - - - 1792 1600 192 0 F16 120 Open:Inact 3072 - - - 5312 5312 0 0 L16 120 Open:Inact 256 - - - 384 384 0 0 F32 120 Open:Inact 3072 - - - 1664 1664 0 0 L32 120 Open:Inact 256 - - - 128 128 0 0 Current status of the calculation system