Charge modulation in two-dimensional compounds. Pierre Monceau. University Grenoble Alpes and CNRS Institut Néel Grenoble

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1 Charge modulation in two-dimensional compounds Pierre Monceau University Grenoble Alpes and CNRS Institut Néel Grenoble Pierre Monceau ECRYS 2014

2 Summary - Charge order and superconductivity (a short historical survey) - MX 2 - other 2D systems (BEDT compounds and cuprates) - RTe 3 Pierre Monceau ECRYS 2014

3 Acknowledgments Pierre Monceau ECRYS 2014

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5 Fröhlich conductivity from moving lattice waves 5 Contrary to a semiconductor for which the energy gap at the Fermi level is due to the ionic potential,and therefore bound to the crystal frame, Fröhlich model: there can be states with current flow if the energy gap is displaced with the electrons and remains attached to the FS. If the lattice wave moves with the electrons with velocity vs, the order parameter will vary as e2ikf(x vs t). Figure (b): Fermi distribution of the same chain as in Figure (a) when the modulation is displaced with a uniform velocity, vs. The two planes of the FS are at ( π/k F ) + q and (+π/k F ) + q with m v s =h/2πq. In the Galilean frame in motion with v s, the system is again unstable with regard to a distortion which opens an energy gap at the new FS with the wave vector Q = 2k F. All the electrons are below the gap and in this frame the electronic current is zero. If the velocity, v s, is small, i.e. if the kinetic energy of the electrons in their translation, k F v s, is small compared with the Peierls gap, in the static reference frame, the current in the sample will be J = nev s Allender et al., PRB9 (1974) 119 Pierre Monceau ECRYS 2014

6 Sliding CDW Pierre Monceau ECRYS 2014

7 Long range CDW coherence Pierre Monceau ECRYS 2014

8 Normal-CDW current conversion Brazovskii et al. PRB61 (2000) Pierre Monceau ECRYS 2014

9 Phase slip in narrow superconducting strips Dolan and Jackel PRL39 (1977) 1628 Pierre Monceau ECRYS 2014

10 Electronic crystals: an experimental overview 61 (2012) 325 Pierre Monceau ECRYS 2014

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12 Magnetic Superlattice formation in chromium W.M.Lomer, Proc. Phys. Soc. 80 (1962) 489 Pierre Monceau ECRYS 2014

13 Superconductivity in intercalation compounds of graphite with alkali metals: C 8 K T c 0.15K but no superconductivity in C 8 Li now C 6 Yb T c = 6.5K, C 6 Ca T c = 11.5K Pierre Monceau ECRYS 2014

14 Model of Lawrence and Doniach- LT Pierre Monceau ECRYS 2014

15 Pierre Monceau ECRYS 2014 Geballe et al.,phys. Rev. Lett. 27 (1971) 314

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18 Pierre Monceau ECRYS 2014 Renker et al., Phys. Rev. Lett. 30 (1973) 1144

19 Coleman et al., Solid St. Com. 12 (1973) 1125 Pierre Monceau ECRYS 2014

20 Se-Se distances Chain III: 2.37 A ChainI : A Chain II: 2.91 A Pierre Monceau ECRYS 2014

CDW/PLD Static CDW order: periodic modulation of the electronic density refecting an enhancement of the dielectric response of the conduction electrons at the CDW wave vector q CDW, but stabilized by

21 CDW/PLD Static CDW order: periodic modulation of the electronic density refecting an enhancement of the dielectric response of the conduction electrons at the CDW wave vector q CDW, but stabilized by a coupling to the crystal lattice -common assumption: singularity in the dielectric function at q CDW = 2k F -alternative possibility: wave vector dependence of the electron-phonon coupling η Pierre Monceau ECRYS 2014 Chan and Heine, J.Phys.F 3 (1973) 795

22 Fermi surface nesting? FS nesting does not play any role at all and can be definitively ruled out as a cause for CDW formation. Instead, we argue, the instability must be due to the electron-ion interaction (Johannes et al. PRB73 (2006) ) In the Peierls picture, lattice distortion is a secondary effect that arises in response to an electronically driven charge redistribution that would occur regardless of whether or not the ions subsequently shift from their high symmetry positions. In real systems, the electronic and ionic instabilities always occur simultaneously. Computational attempts to stabilize a CDW without allowing the ions to move have failed in all cases we are aware of, most particularly for a prototypical CDW metal NbSe 2. We will show that the concurrence of the two transitions is not a coincidence but arises from the fact that CDW formation relies on the lattice distortion as an essential element and not the reverse (Johannes and Mazin PRB77 (2008) ) Pierre Monceau ECRYS 2014

Peierls transition in the weak coupling limit The Peierls

mean-field theory of a 1D electron-lattice system in the

The coupled electron-lattice system is modeled by a

23 Peierls transition in the weak coupling limit The Peierls transition to a CDW/PLD ground state is described by a mean-field theory of a 1D electron-lattice system in the weak-coupling limit. The coupled electron-lattice system is modeled by a Fröhlich Hamiltonian: Rudolf Peierls ( ) Herbert Fröhlich ( ) Pierre Monceau ECRYS 2014

24 K. Rossnagel, J. Phys.:Condens.Matter 23(2011) Pierre Monceau ECRYS 2014

25 Soft phonon mode in NbSe 2 Weber et al. PRL107 (2011) Pierre Monceau ECRYS 2014

26 Lattice dynamical calculations based on density functional perturbation theory (DFPT) show that q CDW of the ordered phase is determined entirely by the wave vector dependence of the electron-phonon coupling Weber et al. PRL107 (2011) Pierre Monceau ECRYS 2014

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28 Visualizing the CDW transition in 2H-NbSe 2 The static CDW order is established in nanoscale regions in the vicinity of defects at temperatures that are several times the bulk transition temperature T CDW Chockalingam et al. arxiv (2013) Pierre Monceau ECRYS 2014

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35 ARPES Rahn et al. PRB85 (2012) Pierre Monceau ECRYS 2014

36 Triangular to stripe order in NbSe 2 induced by local strain The 1Q CDW regions may result from a local strain-induced phase q 1Q 2/7 Q 0 q 3Q 0.328Q 0 Soumyanarayanan et al. PNAS 110 (2013) 1623 Pierre Monceau ECRYS 2014

37 Transition by breaking of analycity in ICDW Pierre Monceau ECRYS 2014

38 Critical scattering in (TaSe 4 ) 2 I Inverse correlation lengths along three orthogonal directions a + b *, a * - b *, c* from elastic (ω = 0) scans through the critical scattering distribution at (2+η, 2-η, 4+δ). Very small anisotropy parameter, of the order of 1.8, at variance with that typically found for other 1D materials. This reflects the broad temperature extension above T P of the 3D-fluctuation regime J.E. Lorenzo et al., J. Phys.: Condens. Matter, 10 (1998) 5039 Pierre Monceau ECRYS 2014

39 Reflectivity in (TaSe 4 ) 2 I Polarized reflectance of (TaSe 4 ) 2 I at T=300K (T P =263K) Temperature dependence of the longitudinal conductivity and Peierls gap. The gap energy is nearly the same between 420K and 140K Longitudinal conductivity function and Peierls gap at 300 Steep decrease of the conductivity below 0.4eV formati of an energy gap well above the Peierls transition H.P. Geserich et al. Physica 143B (1986) 198 Pierre Monceau ECRYS 2014

40 Inelastic neutron scattering in (TaSe 4 ) 2 I 50K 100K 180K 230K 270K 285K 300K Constant-Q scans at the satellite position above and below T P = 253K. Temperature dependence of the TA phonon frequency and damping at the CDW satellite position in (TaSe 4 ) 2 I J.E. Lorenzo et al., J. Phys.: Condens. Matter, 10 (1998) 5039 Pierre Monceau ECRYS 2014

The results from inelastic neuton scattering experiments on (TaSe 4 ) 2 II have shown: -absence of a clear soft-mode behaviour at the phase transition -presence of a central peak whose intensity

41 The results from inelastic neuton scattering experiments on (TaSe 4 ) 2 II have shown: -absence of a clear soft-mode behaviour at the phase transition -presence of a central peak whose intensity diverges at T P This indicates that the phase transition in (TaSe 4 ) 2 I falls in the class of order-disorder transitions, as predicted by strong-coupling theories. The strong-coupling model accounts for: Pierre Monceau ECRYS 2014

42 NbSe 3 The lattice dynamics of NbSe 3 obtained by IXS indicate the absence of a Kohn anomaly and of a distinct phason dispersion. These results are inconsistent with models on «weak»electron-phonon coupling Pierre Monceau ECRYS 2014 H. Requardt et al., Phys. Rev. B66 (2002)

43 Aubry model (1) Aubry et al. J. Stat. Phys. 67 (1992) 675 Raimbault and Aubry JPCM 7 (1995) 8287 Pierre Monceau ECRYS 2014

Aubry model (2) CDW can be viewed as a simple order-disorder mechanism model of pseudo-spin Strong e-ph interactions bring electrons and ions close together (polaronic effect).

44 Aubry model (2) CDW can be viewed as a simple order-disorder mechanism model of pseudo-spin Strong e-ph interactions bring electrons and ions close together (polaronic effect). The potential in which the ion moves becomes anharmonic. At a strong enough coupling te potential possesses two minima, so that the system can be described in terms of Ising spins Pierre Monceau ECRYS 2014 (Nunez-Regueiro et al. PRL55 (1985) 1931) Gor kov PRB 85 (2012) )

The helical charge density distribution arises from a rotation of the dominant CDW component as

45 CDW chirality There were two states in the 3 intensity peaks of the CDW decrease: clockwise and anticlockwise 1T-TiSe 2 Chirality in CDW results in the threefold symmetry breaking. The helical charge density distribution arises from a rotation of the dominant CDW component as one progresses through consecutive layers Ishioka et al. PRL 105 (2010) Pierre Monceau ECRYS 2014

Chirality: twofold symmetry in optical reflectivity The transient variation in optical reflectance was measured by changing the polarization angle E pol of the incident probe laser pulse Twofold

46 Chirality: twofold symmetry in optical reflectivity The transient variation in optical reflectance was measured by changing the polarization angle E pol of the incident probe laser pulse Twofold chiral symmetry although 1T-TiSe 2 has a trigonal lattice structure observed in optical polarimetry experiment with the emitted light to be in one of only two preferred polarization states Ishioka et al. PRL 105 (2010) Pierre Monceau ECRYS 2014

47 Chirality in 2H-TaS 2 Guillamon et al. New Journal of Physics 13 (2011) Pierre Monceau ECRYS 2014

48 Chirality in 2H-TaS 2 Structure of chiral domains Guillamon et al. New Journal of Physics 13 (2011) Pierre Monceau ECRYS 2014

Chirality and orbital order van Wezel EPL 96 (2011) 67011 The periodic displacement wave follows the redistribution of electronic charge in order to minimize Coulomb energy.

and Se leads to an anisotropic electron-phonon coupling for each orbital sector which results in a partly transversal polarization of the displacement wave The transition in TiSe2 is

49 Chirality and orbital order van Wezel EPL 96 (2011) The periodic displacement wave follows the redistribution of electronic charge in order to minimize Coulomb energy. The ability of the lattice distortions to align with the charge modulation is limited by the anisotropy of the electron-phonon coupling In 1T-TiSe 2, the anisotropy of the p orbitals in Te and Se leads to an anisotropic electron-phonon coupling for each orbital sector which results in a partly transversal polarization of the displacement wave The transition in TiSe2 is accompanied by a transfer of electronic spectral weight from the maximum of the Se 4p band at the center of the first Brillouin zone to the 3 minima of the Ti 3d bands at the zone boundaries Pierre Monceau ECRYS 2014

50 Polar charge and orbital order in 2H-TaS 2 van Wezel PRB85 (2012) Pierre Monceau ECRYS 2014

51 Two-dimensional (BEDT-TTF) 2 X Charge order two-dimensional (ET) 2 X Ferroelectricity θ- (BEDT-TTF) 2 RbZn(SCN) 4 Non-linear conductivity θ- (BEDT-TTF) 2 CsZn(SCN) 4 CO melting α-(bedt-ttf) 2 I 3 frustration Glassiness Pierre Monceau ECRYS 2014

52 (ET) 2 X compounds Pierre Monceau ECRYS 2014

¼ filled band Shibata et al. Tsuchiizu et al. 2001 Ground state phase diagram of the 1D extended Hubbard model at 1/4 filling on the plane of U/t and V/t S. Ejima et al. Europhys. Lett.

53 ¼ filled band Shibata et al. Tsuchiizu et al Ground state phase diagram of the 1D extended Hubbard model at 1/4 filling on the plane of U/t and V/t S. Ejima et al. Europhys. Lett. 70, , 2006 The mean field approximation of the 1D Hubbard model show that when V exceeds a critical value, V c,, charge disproportionation occurs among sites with alternating «charge rich» and «charge poor» sites (Seo and Fukuyama 1997). With D 0 numerical calculations on the plane U and V for a fixed D where the metallic phase at D = 0 is replaced by the Mott insulating phase, and a phase with Wigner crystal-type CD is still present in the large U and V region Pierre Monceau ECRYS 2014

center -Stack 2: B and C located on the inversion centers Y.

54 Structures of θ- (BEDT-TTF) 2 RbZn(SCN) 4 and α-(bedt-ttf) 2 I 3 4 (ET) molecules in the unit cell - Stack 1: A and A combined by an inversion center -Stack 2: B and C located on the inversion centers Y. Tanaka and K. Yonemitsu,J. Phys. Soc. Jpn 79, , 2010 Pierre Monceau ECRYS 2014

Stripe phases Different spatial patterns of stripe phases are stabilized depending on the anisotropy of the transfer integrals t c and t p and of the values of intersite Coulomb energies along the

55 Stripe phases Different spatial patterns of stripe phases are stabilized depending on the anisotropy of the transfer integrals t c and t p and of the values of intersite Coulomb energies along the stacking direction V c and along the bonds in the transverse direction V p -horizontal -vertical -diagonal -3fold θ- (BEDT-TTF) 2 RbZn(SCN) 4 q 1 =(1/3, k, ¼) q 2 =(0, k, ½) Seo: J. Phys. Soc. Japan 69 (2000) 805 neel.cnrs.fr Pierre Monceau ECRYS 2014

56 3 fold order Origin of the triply periodic order indexed by q 1? - Interplay between the Fermi surface nesting and the intersite Coulomb repulsion (Kuroki J. Phys. Soc. Jpn 75, ) - q 1 -type modulation comes from the intersite Coulomb repulsion and the q 2 -type from the Fermi surface nesting (Odagawa and Motome Phys. Rev. Lett. 98, ) Pin-ball neel.cnrs.fr liquid model (Hotta and Furukawa, Phys. Rev.B 74, ) 56 frustration with phase charge-ordered liquid under geometrical separation with a statistically ordered phase Pierre Monceau ECRYS 2014

Charge ordering from NMR in θ- (ET) 2 RbZn(SCN) 4 13 C-NMR with H=8T normal to the conducting layers With T decrease, the spectra become broadened indicating a continuous molecule to molecule

57 Charge ordering from NMR in θ- (ET) 2 RbZn(SCN) 4 13 C-NMR with H=8T normal to the conducting layers With T decrease, the spectra become broadened indicating a continuous molecule to molecule distribution of Knight shift. from Miyagawa et al. Phys. Rev. B62 (2000) R7679 Below195K, the spectrum consits of a broad line (A) (with a much larger relaxation rate)and a paket doublet (line B). Separation of the BEDT-TTF into two inequivalent molecules Also from vibrational spectroscopy K. Yamamoto et al. Phys. Rev. B65 (2002) Pierre Monceau ECRYS 2014

58 AC conductivity in θ- (BEDT-TTF) 2 RbZn(SCN) 4 Nad et al., J. Phys.:Condens. Matter 18, L509 30/09/2014 Pierre Monceau neel.cnrs.fr ECRYS

59 -3 orders of magnitude jump of conductivity at T CO with hysteresis T cooling =199K, T heating =204.5K -charge gap below T CO = 1900K -ε (T) shows a smooth monotonic increase from room temperature, more sharp, closed to divergence near T CO -ε (T) jumps down to a small magnitude sharply below T MI -the same jump up of ε in heating The ε growth above T CO may indicate the polarizability of the charge disproportionation seen in NMR The jump below T CO is associated with the 2c superstructure and the large charge gap Pierre Monceau neel.cnrs.fr ECRYS

$Comm 77 (1991) 265] Relative change of the sample length along a, b and c * Measured by capacitive dilatometry et x-ray diffraction from Heidmann et al.: Solid St. Comm.$

60 Metal-insulating phase transition in α-(bedt-ttf) 2 I 3 Abrupt phase transition at T=135.1K of first order Transition slightly hysteretic in specific heat with latent heat [Fortune et al. Solid St. Comm 77 (1991) 265] Relative change of the sample length along a, b and c * Measured by capacitive dilatometry et x-ray diffraction from Heidmann et al.: Solid St. Comm. 84 (1992) 711 Change of intensity of Bragg peak (for reflections only with an odd index with the a-component) (Nogami et al. Synth. Metals 16 (1986) 367 and T. Kakiuchi et al. J. Phys. Soc. Jpn. 76, , 2007) Dimerization of stacks I along the a axis of Peierls type Pierre Monceau ECRYS 2014

Charge ordering from NMR in α-(bedt-ttf) 2 I 3 13 C-NMR spectra at different temperatures below the M-I transition, the spectra consist of two Pake doublets.

61 Charge ordering from NMR in α-(bedt-ttf) 2 I 3 13 C-NMR spectra at different temperatures below the M-I transition, the spectra consist of two Pake doublets. The positions of the two doublets have differentt dependences. That indicates two differently charged BEDT-TTF molecules below the transition: Also from Raman spectroscopy Wojciechowski etal. Phys. Rev. B67 (2003) from Takano et al. J. Phys. Chem. Solids 62 (2001) 393 Pierre Monceau ECRYS 2014

62 Charge disproportionation in α-(bedt-ttf) 2 I 3 Horizontal stripe structure CD already above CO transition From infrared spectroscopy, NMR, and x-ray At room temperature: A=A = 0.60 B=0.68 C=0.44 In the CO state: A=0.81 A = 0.26 B=0.74 C=0. 23 Role of anions in the CO by changing the strenght of the donor-anion hydrogen bondi T. Kakiuchi et al. J. Phys. Soc. Jpn 76, , 2997 Alemany et al. PRB85 (2012) Pierre Monceau ECRYS 2014

63 ac conductivity in α-(bedt-ttf) 2 I 3 Conductivity Drop of ε below T MI Structural transition (dimerization) Nad et al. Physica B Pierre Monceau ECRYS 2014

64 Ferroelectricity is defined by the appearance of a macroscopic electric polarization and its reversibility by applying an external field For some ferroelectric materials, electron degrees of freedom and/or electronic interactions directly give rise to a macroscopic electric polarization and a ferroelectric transition electronic ferroelectricity Pierre Monceau ECRYS 2014

65 Electronic ferroelectricity Pierre Monceau ECRYS 2014

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Verwey transition in magnetite (Fe3O4), unveiled?

Verwey transition in magnetite (Fe3O4), unveiled? J.E. Lorenzo Keywords: Charge, orbital orderings; lattice distortion; spin reorientation; resonant X ray scattering S. Grenier N. Jaouen Y. Joly D. Mannix