Serguei Brazovskii and Natasha Kirova Natal 2012 Physics of synthetic conductors as low dimensional correlated electro

1. SergueiBrazovskii and Natasha Kirova Natal 2012 Physics of synthetic conductors as low dimensional correlated electronic systems. Lecture 3, part 1 Spontaneously deformed states

2. OVERVIEW • Polarons • Lattice relaxation effects in polymers • Jahn-Teller effect • Peierls transition

3. MOLECULAR PHYSICS MEETS CONDENSED MATTER PHYSICSOVERCOMING THE JARGON BARRIER!!! • PHYSICS - SOLID STATE BAND MOLECULAR PHYSICS - MOLECULAR ORBITAL • Valence band, VB, continuous HOMO, discrete • Conduction band, CB, continuous LUMO, discrete • Fermi energy, EF (Electro)chemical potential • Bloch orbital, localized/delocalized Molecular orbital, localized/delocalized • Tight binding band calculation Molecular orbital calculation • Band gap, Eg HOMO-LUMO gap • Direct band gap Dipole allowed • Indirect band gap Dipole forbidden • Phonon, lattice vibration/libration Molecular vibration/rotation • Peierls distortion, CDW Jahn Teller distortion • Polarons, magnons, plasmons No analogues in molecules • ExcitonsExcitons = Molecule excited states

4. www.britannica.com: Polaron…. An electron accompanied by this kind of electrical displacement of neighbouring charges constitutes a polaron. What’s a polaron? www.wikipedia.com : A polaron is aquasiparticle composed of an electron and its accompanying polarization field. Formation of polarons: • A conduction electron in an ionic crystal or a polar semiconductor repels the negative ions and attracts the positive ions • A self - induced potential arises which acts back on the electron and modifies its physical properties Energy gain U must exceed the cost of deformation + kinetic energy of confined electron Origin of word “polaron” – selpftrapping in polar crystals (NaCl, LiF, etc) Today used of all types of selftrapping or autolocalization Polaron effect is strong in ionic, but weak in covalent crystals.

7. At D>1 long range Coulomb interactions are necessary

8. 1D polaron: Spontaneous deformation forms the potential well U for an electron l U0 Energy functional: K- elasticity contant Ψ – electron wave function m*- electron effective mass ~1/2m*l2 ~ -U0 ~KU02l/2 E=1/2m*l2 -U0+KU02l/2 Min over U0 : U0~1/Kl E l=∞ - always unstable 1/l Minimum at l~K/m* Mp>m* ( polaron = e + strain field ). At D>1 long range Coulomb interactions are necessary

9. Polaroniceffects– ground state • - electron interaction (flat arrangement of phenyl rings) • competes with Van der Waals • hydrogenerepulsion (perpendicular arrangement of phenyl rings) Band gap can be tuned by changing the mutual orientation of the phenyl rings (we change the band dispersion) PPV: 20oEg 2.4 e V PPP: 30oEg 3.2 e V

10. Polaronic effects, electronic excitations • For excited states dominant lattice relaxation modes are phenyl ring librations: • Emptying destabilizing level results in increased molecular rigidity • polymer chain becomes more flat • Gap decreases • Delocalized excitations enhance this effect • Same effect should be seen with increasing pressure or under doping • Opposite effect with increasing temperature • Chain chirality diminishes under pumping The process of the exciton relaxation canbeseen in femtosecondoptics - red shift of the luminescence during the first 50 ps. Relaxed exciton has lowerenergywith respect to created pure Coulomb exciton for about 0.1-0.2 eV, hence – higherbindingenergy -3 -2 -1 0 1 2 3 Momentum p

11. Stimulated Emission red-shift in mLPPP films Polaronic effects: SE and PL Lanzanni et al, 2002 Hayes et al, PRB, 52 (1995) 11569

12. The Jahn-Teller effect Jahn-Teller theorem: “there cannot be unequal occupation of orbitals with identical energy” Molecules will distort to eliminate the degeneracy a nonlinear symmetrical molecule with a partially filled set of degenerate orbitals will be unstable with respect to distortion and thus it will distort to lift a degeneracy to a lower-symmetry geometry and thereby remove the electronic degeneracy a case of self trapping – extended version of Jahn-Teller effect. Lattice deformations create a dip in the potential energy U of electron. Exit in transition metal complexes, manganites etc, where d or f orbitals are partially filled

13. Cyclobutane C4H4 4 carbon atoms – 4 π-electrons Wave functions: Ψ1=cos 0•n=cnst Ψ2=cos(πn/2) Ψ2=sin(πn/2) Ψ4=cosπn=(-1)n Half filled doubly degenerate level Instability square - rectangle Jahn-Teller effect

14. H H C C H C CH C C H H Isolated benzene ring Degenerate level is completely filled – no gain from reducing the hexagon symmetry. We only loose the deformation energy N=8 – again the instability, like for N=4

15. Fullerene Crystal • Fullerene (C60) • Cage diameter - 0.71 nm • Optical gap – 1.9 eV • Crystalline Fullerene • fcc structure with lattice constant of 1.417 nm • C60-C60 distance – 1.002 nm

16. Application to C60 t1uLUMO can adopt up to 6 electrons

17. Charge transfer crystals : • n=0, dielectrics • n=1, metal at high temperature quasi 1D polymeric chain at room temperature quenched dimer lattice at low temperature. • n=2 1/3 filled band, should be a metal, but a semiconductor • n=3 half filled band metal, superconductor • n=4 2/3 filled band, should be a metal; but a semiconductor, semimetal under pressure, bct • n=5 does not exist ? • n=6 completely filled t1u, dielectric 1D polymeric phase under illumination, 2D polymeric phase under pressure

18. Uniaxial cage distortion: Splitting of the t1u level • Two-fold degenerate (x,y) level E1=-Δ/3 • Non degenerate z-level E2= 2Δ/3 Rem.: Center of gravity of the levels does not change E2,n2 2Δ/3 -Δ/3 E1,n1 The Jahn-Teller effect on the isolated molecule

19. We need bimodal distortion

20. Excited states of the molecule,antipolaronic effect Δ Δ/4 Δ/2 Triplet exciton 1 Triplet exciton 2 Ground state Excited state

21. Jahn-Teller effect in A4C60 crystal E E p p

22. Electrons on deformable lattice - instabilities Interaction between two single-electron atoms (generally different) Very large distances E1=E0; E2=-E0 1 2 Finite distances: overlap of single-atomic wave functions, the hopping integral t~exp(-x/a) E0 -E0 Hamiltonian for the eigenstates: Anti-bonding totally excited state - repulsion Ground state -attraction → larger t First excited state: Eb+Eb=0, no interaction

23. Usually we pay the energy Total energy If E0<<t Always non trivial minimum at t≈2/K If E0>>t Instability threshold at E0K=2

24. H H H C C C C C H H E E F -kF kF -kF kF Dream – a metal Reality - insulator Peierls effect - one dimentional chain of equidistant atoms is unstable with respect to the dimerisation: Spontaneus symmetry braking results in the dielectric state, the gap is open. Wexp(-

26. We substract the energy of the parent metal For the ground state Δ=Δ0 shouldbefoundselfconsistently

27. Non degenerate ground state W -0 0 

28. Combined Peiels effect in diatomic linear chain polymer =π/2 Joint effect of extrinsic ∆e and intrinsic ∆i contributions to dimerization gap ∆. ∆e comes from the build-in site dimerization – non-equivalence of sites A and B. ∆i comes from spontaneous dimerization of bonds, the Peierls effect. R` R` R R E E F F Threshold effect : ∆i WILL NOT be spontaneously generated – it is a - if ∆e already exceeds the wanted optimal Peierls gap. Chemistry precaution: make a small difference of ligands R and R’ -kF kF -kF kF

29. Q.So what? A. Electrons added to conduction band are absolutely unstable forming selflocalized states with the levels inside the Peierls gap (next lecture)