分子結晶の電子構造（バンド構造） - Kyoto U

1 NC Electronic structure (band structure) of molecular crystals Saturated hydrocarbons (molecular orbital composed of -bonds) Crystallization Overlap of MOs between molecules is small (Probability of electron transfer between molecules is small) Electronic structure of the crystal is not so much different from the isolated molecule electron clouds have to be extended outer side to increase the intermolecular MO overlap lone-pair electrons -electrons In most cases, MO overlap is not extended to the whole crystal due to localized character of the electron cloud For the -conjugated systems, MO overlap can be extended to the whole crystal LUMO HOMO - Consider the case where planar -conjugated molecules are stacked one-dimensionally with overlapping the - electron clouds Electronic structure of the isolated molecule Electronic structure when two molecules are stacked with - overlap long interplanar distance short interplanar distance CC p 2p 2p CCCC When two p-orbitals form a bond When the inter-atomic distance is long When the inter-atomic distance is short (When overlap between the orbitals is small) (When overlap between the orbitals is large) LUMO HOMO 2t 4t t.

2 Conduction band Valence band one molecule two molecules three molecules four molecules Infinite number of molecules transfer integral (proportional to the overlap integral) 4t van der Waals (EG) HOMO-LUMO EG HOMO-LUMO 2 eV Neutral and closed-shell molecules There is no attractive interaction between the molecules that compresses the van der Waals radii The band width is rather narrow The energy gap is not so much different from the HOMO-LUMO gap in the isolated molecule In general, HOMO-LUMO gap in the -conjugated molecules is rather large (> 2 eV) Most of the crystals are semiconductors with low electrical conductivity (insulators) EG How can the electrical conductivity be increased?

3 Make EG small (i) HOMO-LUMO Make HOMO-LUMO gap small (it is difficult for ordinary hydrocarbons) (ii) Make the band width wider (pressurization) make vacancies in the valence band put electrons in the conduction band Make the component molecules open-shell structure neutral radicals cation radical salts anion radical salts charge-transfer complexes At present, most of the conducting crystals are obtained from these compounds + NNOO- . nitronyl nitroxide radical open-shell -conjugation is localized bulky the radical group is protected by the bulky groups due to high reactivity path NNNNNNNNLiLi(phtnalocyaninate) Li(Pc) Pc2- : Li+(Pc1-) : Pc open-shell -conjugation is delocalized molecule is planar path Neutral radical crystals examples These features are disadvantageous for MO overlapping to produce conduction paths The molecule cannot be a component of conductors but can be a component of molecular magnets Closed-shell -conjugated system Pc -conjugated system becomes open-shell structure Formation of conduction paths by stacking of the planar molecules Consider the band structure Li(Pc) 4t LUMO SOMO Li(Pc) Li(Pc) 2t Li(Pc) Li(Pc)

4 T Since molecules stack with overlapping MOs accommodaAng unpaired electron , only the bonding interacAon operates between the molecules t becomes larger Only one-half of the band is occupied Similar to metallic bands Wider band width Advantageous for electrical conduction One-dimensional column of Li(Pc) 4t Li(Pc) 10-3 S cm-1 eV Au, Ag, Cu; 106 S cm-1 Si; 10-5 S cm-1 Half-filled band On-site Coulomb repulsion energy: U) Only one-half of the band is occupied Similar to metallic bands Wider band width Advantageous for electrical conduction However, the electrical conductivity is only 10-3 S cm-1 at room temperature, and the temperature dependence is semiconducting (activation energy.

5 EV) Room temperature conductivity Each molecule has one electron Transfer of electron accommodating two electrons accommodating two holes electron transfer costs electrostatic repulsion energy Band structure of Li(Pc) 4t Half-filled band 4t U MoC Li(Pc) U = eV, t = eV half-filled band >> U In this state, two electrons occupy one energy level If electrostatic repulsion between the charges is large, it is required extra energy to put the second electron in the same energy level Hypothetical band structure The band accommodates the second electron Although each level accommodates only one electron , the band is completely filled As a result, the electronic structure resembles that of intrinsic semiconductors (Mott insulator) By the way, following energy values are estimated for Li(Pc) U = eV, t = eV The values are consistent with its electrical properties The metallic state is maintained for alkali metals despite their half-filled bands, since their band width is sufficiently larger than U Donor D D - e- X- (D+ )X- [(Dy)+ ]X- [ D + )(X-)]

6 , < 1] Fully oxidized salt Partially oxidized salt (y > 1) Fully oxidized salt (D+ )X- MoC spin-singlet pair Easily oxidized molecules The electronic structure is practically similar to that in neutral radical crystals Mott insulators However, dimerization occurs frequently (stabilized by formation of spin-singlet pair of two unpaired electrons) insulators Cation radical salts [(Dy)+ ]X- Partially oxidized salt (y > 1) One-dimensional system D1/2+)(X-)1/2 (D2)X 1/2+ 1/2+ 1/2+ 1/2+ 1/2+ 1/2+ 1/2+ a tight-binding approximation R Electrons are tightly bound to the molecular orbital Potential energy of the isolated molecule (the lattice point is defined by R) Assuming the solution is V Arrange the lattice points one-dimensionally The difference is defined by V As the first order approximation, we assume that the wave function of the one-dimensional array (molecular orbital of the conduction band) can be expressed by linear combination of HOMO (a form satisfies the periodic condition) H = !

7 22m 2+V(x)= !22m 2+U(x R)+ V= H 0+ V k(x+R')=eikR (x+R' R)R =eikR'eik(R R') x R R'() R R =eikR x R()= k(x)R x x + R k: k=l L L: l = 1, 2, 3, Wave number vector Whole length of the one-dimensional array M kF=M 2L kF=La 2L= 2a = k* H kdx k* kdx = k* H 0 kdx +e ikRR *(x R) VeikR'R' (x R')dx k* kdx k* kdx =N k* H 0 kdx =N 0 eik(R R')R * (x R) V (x R')dx=eikR''R'' * (x+R'') V (x)dx R R -R = R = 0+1 Neik(R' R)R * (x R) V (x R')dx R' R N R R D 1/2 3/4 4t (x) R 0 R0 R0 = a (x) is localized near the lattice point It is enough for calculation to consider only 0 and R0 (nearest neighbors) as R Formal charge of D is + Each MO that composes the band accommodates up to two electrons The band is -filled If the distance between the lattice points is equivalent, the system becomes a metal eikR0R0 =e ika+eika=coska isinka+coska+isinka=2coska + + + + + + + + + a a a Peierls instability 4a However.

8 The one-dimensional metallic state is unstable Lattice distortion Four molecules accommodate two holes Stabilization is achieved by lowering the energy of electrons near EF + + + + + + + + + 2a U 2a When dimerization occurs in the initial state In the case where the short-range Coulomb repulsion is small In the case where the short-range Coulomb repulsion is large Hypothetical band structure without dimerization Dimerized state Metal Insulator + + + + + + + + + a 2a Wigner 1+ 1+ 1+ 1+ When the long-range Coulomb repulsion is large Charges arrange equidistantly Two molecules share one charge Charge ordered phase Charge is localized on one molecule The band structure is similar to that for large short-range Coulomb repulsion Unable to describe the band structure Becomes insulator - SSSSSSSSSSSSTTF BEDT-TTF (ET) HOMO HOMO Two-dimensional electronic systems -conjugated molecules with certain molecular frameworks can form two-dimensional - interactions ta tb a b A B n = an A + bn B H11 = H22 = 0 + 2tacos(kxa), H12 = H21* = tb(1 + e-ikxa)(1 + e-ikyb) = 0 + 2tacos(kxa) 4tbcos(kxa/2)cos(kyb/2) 0 = 0, a = 5 , b = 10 , ta = eV, tb = eV 1/2+ Formal charge of each molecule is + assuming = 0 + 2tacos(kxa) 4tbcos(kxa/2)cos(kyb/2)

9 Ky = 0 kx = 0 Fermi kx ky kz kx ky kz Band structure Fermi surfaces One-dimensional system Peierls MoC MoC Two-dimensional electronic systems One-dimensional electronic systems Stable metallic state Unstable metallic state Transition into a superconducting ground state Peierls instability Mott insulators Charge ordered phase However, in some cases, the electronic structure is suffered by perturbations Mott insulators Charge ordered phase Band width Correlation energy of electrons Strongly correlated electronic systems Acceptor A A e- Cation+ Cation+(A- ) Cation+[(Ay)- ] [(Cation+) A - ) , < 1] Fully reduced salt Partially reduced salt (y > 1) Fully reduced salt Easily reduced molecules The electronic structures are practically similar to those in cation radical salts Partially reduced salt Electrons occupy the levels from the bottom of the band Cation A2 Anion radical salts -filled (Cation)3C60, cation = K, Rb C60 K C (K+)3(C60)3- LUMO K+ Half-filled band U Mott Due to screening of U, the system does not become a Mott insulator Each levels in the conduction band accommodates up to 6 electrons Exception: Six-step reversible reduction is possible Electrons occupy triply degenerated LUMO Two-dimensional systems are rather rare for the anion radical salts D A D+ A- D + A - D0 A0 Full charge transfer: ionic ground state Partial charge transfer (ionic ground state) No charge transfer.

10 Neutral ground state (0 < < 1) Donor HOMO Vacuum level Acceptor LUMO ( ) ( ) Ionization potential electron affinity D0 A0 HOMO LUMO V. L . D+ A- HOMO LUMO V. L . >> << How is the ground state of charge-transfer complex dominated? Relation between and is important Charge-transfer complexes Since and should be measured for the isolated molecule in vacuum, the data are not always available 0 V vs. SCE TTF0/TTF+ ET0/ET+ Per0/Per+ Chl-/Chl0 TCNQ-/TCNQ0 F4 TCNQ-/F4 TCNQ0 SSSSSSSSSSSSOOClClClClCNCNNCNCCNCNNCNCFF FFS trong donor Strong acceptor Complex with neutral ground state Complex with ionic ground state Highly conducting complex with segregated stacking structure (partial charge transfer) Complex that shows neutral-to-ionic transition For convenience, redox potential can be used Difference comes from the solvation effect D0 A0 HOMO LUMO V.