Long range Coulomb interactions at low densities of polarons

Transcription

1 Long range Coulomb interactions at low densities of polarons S. Fratini - Grenoble Subtitle: on the role played by the long-range polarization (large polarons) in metal-insulator transitions P. Quémerais (LEPES-CNRS Grenoble) G. Rastelli, S. Ciuchi (Università dell Aquila, Italy) G. Chuev (Institute of theoretical & experimental biophysics Russian Academy of Science, Pushchino, Russia) J.-L. Raimbault (Ecole Polytechnique, France)

2 Outline - metal-insulator transition scenarios - what happens in a polarizable medium? Wigner crystal of polarons: melting & instability - 2 examples, 1 analogy: cuprates and metal-amonia solutions

3 Metal-Insulator Transitions (a selection) Anderson localization (monoelectronic model) Disorder can localize a free particle (no interaction between carriers) Peierls transition (2kF susceptibility in 1D) formation of a CDW, a gap opens at the Fermi level Mott-Hubbard transition (short range interaction) half-filled band --> should be metallic, often insulating (ex: transition metal oxides, organic salts...) Mott transition in semiconductors (long-range Coulomb interactions & screening) instability of the metallic phase against the formation of localized states around charged dopants (ex: nonpolar doped semiconductors) Polarization catastrophe (long-range Coulomb interactions) polarizability of individual units diverges due to collective effects (local field, cf. Clausius-Mossotti) Wigner crystallization (long-range Coulomb interactions) minimization of electrostatic energy at low concentrations (ex: electrons on liquid He, semiconductor heterostroctures)

4 The Mott Transition in semiconductors (1949) Long-range Coulomb interactions / screening conduction band Donor atoms in usual semiconductors (Si,Ge) Non-polar: κ= ε s= ε Bound state around the donor 2 e r Free electron screening No more bound state when Mott Criterion nc1/3a0*=0.26 Impurity levels Impurity band - Instability of the metallic phase - carrier density jumps at the transition - works in a number of doped nonpolar semiconductors (later understood as Anderson localization)

5 [Thomas, J Phys Chem 88, 3749 (84)]

6 Polarization catastrophe (1927) (predicts if a solid is a metal or an insulator from the polarizability of the individual atoms, before Bloch theory...)

7 Dilute gas of polarizable units, each with characteristic frequency ω 0 responds to external field Dense system (solid, liquid): units respond to external field macroscopic polarization (transverse) Polarizability ω0 α = e2 m ω02 p = dipole Transverse frequency reduced by Lorenz local field factor (dipole-dipole interactions) ωcoll2 = ω02 - ωp2/3 The restoring force for the electrons vanishes at critical nc ---> metal

8 Metals: R>M/d Insulators: R<M/d ~1/n

9 Wigner crystallization (1934) At low enough densities, electrons in a compensating jellium of positive charge crystallize --> insulating phase (at T=0K) n : density of electrons Dimensionless density parameter r =Rs /a0 s H=HkinHint Kinetic energy per particle behaves as: 1/rs2 Coulomb interaction ~ 1/rs At low densities, the minimization of energy yields a crystallized state 3D --> BCC 2D --> Hexagonal

10 electron jellium sphere Wigner decomposition The system is separated into neutral spheres of radius Rs containing one electron -e and a compensating charge e {Ri} Bravais lattice sites ri = Ri ui Quadratic expansion of

11 Mean Field (Wigner) Neutral spheres do not interact, E=0 ui Beyond mean field Dipole-dipole interactions uj ui Warning!! : not all crystallized structures are MECHANICALLY stable. Ex: Simple Cubic <0

12 Melting towards electron liquid: Eliq<EWC Mean field (Wigner 1933) Dipole-dipole interactions (bcc) (Carr 1961) BUT: Comparison of energies is not accurate enough (the liquid is strongly interacting at such low densities) => Phenomenological Lindeman criterion, originally for thermal melting of solids The crystal melts when the oscillations of the localized particles attain a fraction of the interparticle distance [Pines & Nozières 1958] 2D QMC: 3D

13 On the formation of (large) polarons and their crystallization

14 Polarizable medium Dipolar Liquid Ionic Crystal Two sources of polarization 1) Electronic polarizability (core electrons) fast dynamics (follows instantaneously the motion of added charge) 2) Ionic distortion, molecular orientation slow dynamics (leads to polaron formation) Dielectric constant depends on frequency NH3 La2CuO4 Ionic crystal εs ε (isotropic) Chen & al. 91

15 Formation of Fröhlich polarons Ionic distortion LO phonons create a polarization to screen the electron charge localizes the electronic wave function self-consistent problem α 2= Hydrogenic bound state phonon energy

16 Single polaron: already a nontrivial problem Weak coupling: perturbation theory Strong coupling: static polarization field Electron localized in a static potential well All coupling theory: path integrals [R. Feynman 1958]

17 From weak to strong coupling: Feynman Path-Integral approach NOT quadratic! What choice for the trial model?

18 Trial model the polarization is replaced by a fictitious particle - not fixed, does not break translational invariance! - K, M are variational parameters - many-body --> two-body - after minimization, the trial model itself gives a good representation of the polaron

19 Polaronic Wigner crystal (PWC) Take a system of polarons at low density and let them interact through the Coulomb repulsion OR Take a Wigner Crystal of electrons and put it in a host polar material Polarizable host electron jellium η= ε /ε s Polarizability ratio (no bipolaron formation if η>0.1) Two competing effects arise from the ionic polarization Static screening of LR repulsion: reduces interaction energy Polaron formation: Decrease of kinetic energy MP>>m tends to DESTABILIZE the WC tends to STABILIZE the WC Who wins?

20 Feynman approach to the PWC - Fröhlich e-ph interaction Coulomb e-e repulsion (long-range) - Integrate out phonons, expand for small oscillations Mean field: polaron in a harmonic potential: average effect of other localized polarons Beyond mean field: polaron-polaron correlations

21 Mean field: polaron in a harmonic potential Simple picture based on trial model: the polaron is a composite particle, with 2 independent d.o.f. Sketch of absorption spectrum (i) center of mass Electron-electron interactions Melting of PWC: 2 degrees of freedom => 2 Lindeman criteria (ii) relative motion

22 Melting of PWC: two scenarios (i) (ii)

23 Weak and intermediate coupling, melting driven by center of mass fluctuation Electrons are dressed by the phonons => The PWC melts as an ordinary WC, but rc is renormalized by Mp and ε s strong coupling, fluctuations of internal d.o.f. the polarons have a large mass and cannot delocalize => polaron dissociation ~ Mott criterion! Mott

24 Beyond mean field: collective modes - trial model: 2 particles/site, in 3D => 6 branches - dispersion caused by dipole-dipole interactions Polaron peak in σ(ω) Wigner crystal of electrons Wigner crystal of polarons

25 Remember polarization catastrophe? Increasing density Polaron peak Optcond, figura articolo Softening of transverse collective frequency is a precursor to the polarization catastrophe ωpol2=ω02-ωp2/3ε Characterist frequency of isolated polarons Softening due to collective effects Softening of polaronic collective mode transfer of spectral weight to low freq. collective mode ALL DUE TO DIPOLE-DIPOLE INTERACTIONS alternative to screening, cf. [Cataudella et al.] [Devreese et al.]

26 Dielectric properties: ε (k,w) Static limit: <0 It can overscreen the coulomb repulsion between 2 test charges [cf. talk by Takada] moves the ions away from equilibrium positions Dynamical response: Becomes generally negative when approaching the polarization catastrophe PWC attraction Could induce pairing of free electrons coexisting with a PWC The collective bosonic excitations of the PWC could mediate an attractive interaction BUT: hard to justify theoretically e-ph attraction

27 Can all this be of some relevance in real compounds?

28 Cuprates: softening of MIR peak [Lupi et al. PRL 99] [Lucarelli et al. IJMP 03]

29 STM: hole crystallization? Measure tunneling I vs. bias V with atomic resolution => conductance di/dv= g(r,v) ~ Local DOS FT of local DOS => spots in reciprocal space - q varies with bias => qp interference, g(q,v) ~ A(k,V) A(kq,V) - dispersionless q, 4a0x4a0 modulations, A corr. length => charge ordering Several interpretations (crystal of superconducting pairs, condensate with spatial modulations, stripes, Wigner crystal of holes...) A Wigner crystal with periodicity 4x4 can be stabilized provided that the holes are subject to an additional localizing effect whose characteristic energy scale is ~ 150meV, the same that is observed in σ(ω) candidates => polaron energy Ep, exchange energy [G.R., S.F. & P.Q.: Eur. Phys. Jour. B, 42, 305 (2004)]

30 Na-CCOC, non SC (low doping) [Hanaguri et al., Nature 430, 1001 (2004) Bi-2212, superconducting [McElroy, PRL (2005)] Inhomogeneous gap maps, from OD to UD NON-Dispersive features in PG regions

31 Metal-Ammonia solutions: Na-NH3 Solvated electrons=polarons σ(ω) Optical absorption peak at 0.8 ev

32 Phase diagram [JC. Wasse, website] [Edwards 2001] Polarons are blue! Bronze metallic phase (polarons have dissociated)

33 Softening of polaron peak [Schlauf et al]

34

35 A possible analogy? region with negative dielectric constant? A suggestion: If the counter charges are mobile as in liquid NH3, phase separation If counter charges are frozen as in the cuprates, superconductivity

36 Conclusion There is a lot of interesting physics in long range interactions, why restrict to short range?

37 References P.Q.: Mod. Phys. Lett., 9, 1665 (1995) P.Q & S. F: Mod. Phys. Lett., 11, 1303 (1997) P.Q. & S.F: Int. Jour. Mod. Phys B, 12, 3131 (1998) first prediction of the red shift S.F. &P.Q.: Mod. Phys. Lett. 12, 1003 (1998) S.F & P.Q.: Eur. Phys. Jour. B, 14, 99 (2000) P.Q. & S.F.: Physica C, , 225 (2000) S.F. & P.Q.: Eur. Phys. Jour. B, 29, 41 (2002) P.Q., J.L.R. & S.F,: J. Phys. IV France 12, pr9-227 (2002) on the analogy with metal-amonia solutions G.R., S.F. & P.Q.: Eur. Phys. Jour. B, 42, 305 (2004) on the modulations observed by STM G.R and S.C.: Phys. Rev. B 71, (2005) G.R. and S.F.: in preparation on the stabilization of a WC in low-dimensional solids