Theoretical approach to physical properties of atom-inserted C60 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 LabKawashima Kei

Contents• Introduction

– Crystal structures of atom-inserted C60 crystals(Objects of my study)

– Cs3C60 crystal (The main object to study from now on)• Mott insulator-superconductor transition of Cs3C60

• Way to study ― Theoretical approach to physical properties by 　　 computational simulations ― First principles calculation in DFT within LDA

• Current studies ― Computational simulations for C60 Crystal • Future works

― Computational simulations for Cs3C60 crystal• Summary

Crystal structures of atom-inserted C60 crystalsConventional unit cell of a FCC C60 Crystal

Superconductivity found in 1990s.

Insulator (Band gap 1.2ev)≒

SC

SC

Insulator

Insulator

MetalMetal

Insulator

The main object to study from now on　－ Cs3C60 crystal

Cs3C60 Crystal(A15 structure) Interesting points

・ Transition from Mott insulator ( モット絶縁体 ) to metal, andsuperconducting transition ( 超伝導転移 ) at low temperatures under appropriate pressure.The phase diagram is similar to that of cupper oxide high-temperature superconductors( 銅酸化物高温超伝導体 ). ・ The maximum Tc is about 38K, that is the highest Tc among atom-inserted C60 crystals.

In 2008, superconductivity in Cs3C60 crystal was found by Takabayashi group.

Cs atom

Pressure dependence of Tc of Cs3C60 crystalLow pressure region

Ref: ALEXEY Y. GANIN et al. Nature Mat., Vol. 7(2008)

Superconductors have perfect anti-magnetism( 完全反磁性 ).

Below about 47K, Cs3C60 is Mott insulator.

Under more than 3kbar, Cs3C60 is superconductor.

Anti-ferro magnetism

Electron pair

Mott insulator – Superconductor transition

Ref:

Metal

AFI : Anti-ferro insulator (Mott insulator )SC : Superconductor

TN is the temperature at which the zero-field magnetization begins to increase.Tc is the temperature at which the zero-field magnetization begins to decrease.

A copper-oxide crystal

Phase diagram of Cs3C60

Hole density per Cu atom

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

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 ).

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.

Band structures of C60-based crystalsC60(FCC) - Insulator K3C60(FCC) - Metal Ba6C60(BCC) - Semimetal

Unoccupiedstates

Occupiedstates

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

Current study ― Theoretical simulations for C60 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 structure4. Density of states (DOS)

1. Optimize the atomic positions

Parts of an input data Initial values

&controlcalculation='relax'

&systemibrav=2 celldm(1)=26.79 nat=60 ntyp=1

ATOMIC_POSITIONS (angstrom)C -0.707 0.000 3.455C -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

2. Get the optimum lattice constantParts of input data Total energy vs lattice constantlista=’26.55 26.60 26.65 26.70 .....'for a in $listado&control

calculation=‘scf'

&systemibrav=2 celldm(1)=$a nat=60 ntyp=1

ATOMIC_POSITIONS (angstrom)C -0.713 0.000 3.485C -1.437 1.174 3.031

・・

C 2.303 -2.601 0.734

done

Experimental value26.79 Bohr

26.63 Bohr

誤差約 0.6%

3. Band structure

Ref: O. Gunnarsson, Reviews of Modern Physics, Vol. 68, No. 3, 575-606(1996)

Band gapBand gap

By O.Gunnarsson group By me

Experimental band gap of C60 crystal is about 1.2 ev.

4. Density of states (DOS)

D(ε) [states/ev ・cell]

ε [ev]

Band gap

Band gap

D(ε) shows the number of electronic quantum states per unit cell existing between ε and ε+Δε.

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(εF)Vc)

Future works ― Calculations for Cs3C60 under higher pressures(1Gpa, 10Gpa, 100Gpa etc.)

Electronic structure

Crystal structure

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

・ Atomic positions・ lattice constant

Summary• The main studying object from now on ― Cs3C60 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 Cs3C60 under higher pressure by calculating electronic structure and electron-phonon coupling.

• Theoretical simulations based on first principles methodYou 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 C60-based crystals. I’ve got prepared for future works by calculating physical

properties of C60 crystal.