Transport and ordering of strongly correlated electrons in kagome system at partial filling

Transcription

1 Transport and ordering of strongly correlated electrons in kagome system at partial filling Eduard TUTIŠ Institute of Physics, Zagreb, Croatia ECRYS 2017, Cargèse

2 in collaboration with & inspired by EXPERIMENTAL László Forró Jaćim Jaćimović Areta Olariu Dejan Đokić Cristian Vaju EPFL,Switzerland CRYSTALS Patrick Batail THEORY Ivo Batistić Andrea Kadović University of Angers, France Zagreb, Croatia

3 ElectronicCRYStals 2017

4 Physical system

5 Quasi-2D (layered system) (EDT-TTF-CONH 2 ) 6 [Re 6 Se 8 (CN) 6 ] Baudron, Batail, Coulon, Clérac, Canadell, Laukhin, Melzi, Wzietek, Jérome, Auban-Senzier, Ravy J. Am. Chem. Soc. 127(33), (2005)

6 (EDT-TTF-CONH 2 ) 6 [Re 6 Se 8 (CN) 6 ] Baudron et al, JACS (2005) Synthesis Structure NMR ESR DC resistivity El. parameters (ext. Huckel) Dimers form kagome lattice

7 (EDT-TTF-CONH 2 ) 6 [Re 6 Se 8 (CN) 6 ] 3 sites per unit cell 2 electrons per cell

8 Evolution in temperature at ambient pressure = = = or T [K] spin Heisenb erg chains (S=1/2) charge ordering metal

9 Two observed dimer configurations in the low temperature phase Baudron et al, JACS (2005) 410 mev 590 mev Electronphonon coupling

10 Model?

11 Model: Hubbard-Holstein Hamiltonian? t U,V g K, M

12 Experiment

13 Resistivity Acivated by 900 K! T(K) Metallic but high resistivity! k F l 10-2 Does not qualify for a metal! T(K)

14 Seebeck coefficient (EXP) Mixed (e, h) T(K) transport

15 Less conductive under PRESSURE!

16 Phase diagram Kagome bad metal Kagome semiconductor Triclinic semiconductor

17 Electronic bands?

18 (EDT-TTF-CONH 2 ) 6 [Re 6 Se 8 (CN) 6 ] 3 sites per unit cell 2 electrons per cell 1/3 of bands filled

19 Band model Fermi Energy at Dirac point? E F?

20 & Opening the gap by lifting the site degeneracy? B C A e B,C =0 e A =-gu 1

21 Magnetic susceptibility in high temperature (HT) phase 1/χ Curie-Weiss at high T AF correlations Θ CW ~175 K HT localised s=1/2 spins over 3 sites! 2/3 filling > Strong correlations! The 200action is not 260at 280 the 300 Dirac Point! Température (K) µ = 2g µ SS ( + 1) μ eff ~2.2 μ B per formula unit Metallic (from ρ) but it is not a metal! eff (SQUID, ESR) B el. per u. cell

22 Band model? E F?

23 Other model/picture?

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25 Dense system of small polarons?

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27 spin S=1/2, & no double occupancy S = k log 2(1 ) B n q n w.o. off-site interaction Chaikin and Beni, Phys. Rev B 13, 647 (1974) extensions of Hikes formula (Hikes, 1961) 2(1 n) S = 0 for = 1 i.e. TEP n n = 2 3

28 Seebeck coefficient (EXP) Thermopower shows correlated hopping regime! T(K) S = 0 Chaikin and Beni, Phys. Rev B 13, 647 (1974)

29 Thermopower: Charge flow attached to entropy flow

30 Ordinary metal

31 Charge flow is attached to configuration and spin entropy flow

32 Opposite flow of Configuration and Spin Entropy attached to Carrier n = 2 3 S = 0 i.e.

33 Extended lattice-gas model is Ising spins Blume-Emery-Griffiths model (1971)

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35 Hopping rate - Adiabatic small polaron W ω π E a 0 = exp 2 T e.g. Y.A. Firsov (2007) as in A.S. Alexandrov, Polarons in Adv. Mat.

36 MODEL CALCULATION: Magnetic susceptibility J = 0.15 V (Monte Carlo simulation)

37 How to address transport properties of dense lattice gases? Resistivity Thermolectric power

38 Thermoelectric power S Q J 1 Q 1 J E = = µ qt qt J n qt J n Q1 = E µ Q1=heat per particle

39 Thermoelectric power S TEP Q J 1 Q 1 J E = = qt qt J n qt J n µ Q = E µ Q=heat per particle Linear response (Kubo formulae): S TEP 1 JEJ n 1 ξ µ Tq JJ n n Tq σ µ = =

40 Kubo formulae Electrical conductivity σ = β q JJ n n = q lim dτexp ( iωτ) ( ) (0) 0 0 m Jn τ Jn ω iω Vd ω = iω+ δ n Thermopower ξ J J lim dτexp ( iωτ m ) J E( τ ) Jn(0) β = q E n q ω 0 iω Vd 0 ω = iω+ δ n

41 Kubo formulae Electrical conductivity σ = β q JJ n n = q lim dτexp ( iωτ) ( ) (0) 0 0 m Jn τ Jn ω iω Vd ω = iω+ δ n Thermopower ξ J J lim dτexp ( iωτ m ) J E( τ ) Jn(0) β = q E n q ω 0 iω Vd 0 NOT VERY USEFUL FOR LATTICE GASES! ω = iω+ δ n

42 Transport in lattice gases Adding: + spin-spin interaction + configurations averaging via Monte Carlo simulations of the lattice gas arbitrary gas concentration G. D. Mahan, Phys. Rev.B, (1976)

43 electrical conductivity σ ξ 1 1 = β 2 Vd 2 2 q x x n n W E ijσ thermopower.. ( ) ( 1 ) ( ) i j j i ij ijσ 1 1 = β 2 Vd ( x x ) ( ) ( ) iσ, j jσ, i j 1 i ij ijσ 2 q X X n n W E ijσ X x + x x + x V n + J S. x i m i m iσ, j im m im m m i, j 2 2

44 EXPERIMENT

45 MODEL CALCULATIONS dimensionless transport coef. : Monte Carlo simulation on the lattice with 64x64 unit cells (12288 sites).

46 P Simplified physical picture

47 Coulomb interaction V drives the chain formation Activation energy for charge transport

48 What drives the crossover? T c S 0rd =0, E 0rd S disord, E disord Concentration of defects in c.o. Minimize F=E[n]-TS[n], Rough estimate of T c E 0rd ~ E disord -T c S disord eff

49 Conductivity eff Low T High T eff eff Activated with: =V+ E a σ ~ 1/T -> metallic

50 Pressure effects?

51 Dimer overlap with pr Resistivity vs Pressure low-t high-t low-t low-t high-t Main effect of Pressure is in Polaron energy and e-p coupling Modest variation of Coulomb interaction

52 Summary of the physical picture Transition to homogenous state GAP in the low temperature phase GAP in the non-metallic high temperature phase (P>>) E a V+ E a Non-metallic -- Metallic CROSSOVER at for Pressure: Increase of intra-dimer hybridization and effective electron-phonon coupling

53 Full dependence on particle concentration

54 Defects propagating in ordered phase at low temperature hole electron

55 Reading mobilities of holes and electrons n = 2 / 3 + δ V = a 1, J = 0.2, E = 0.2

56 conductivity T = a 0.1, E = 0.2

57 Thermoelectric power T = a 0.1, E = 0.2

58 Conclusions Evidence for dense polaron gas Dense polaronic system ( both experimentally and theoretical) is not easily accessible! (Needs both strong e-e correlations + strong e-p interaction.) Pressure-tuned hopping low P - diffusive polarons (mimics metallic transport); high P - activated polaronic transport. S=0 at n =2/3 de facto textbook case for hopping transport of strongly correlated electrons (a lattice gas polaron) with where strong spin entropy flow

59 in collaboration with & inspired by EXPERIMENTAL László Forró Jaćim Jaćimović Areta Olariu Dejan Đokić Cristian Vaju EPFL,Switzerland CRYSTALS Patrick Batail THEORY Ivo Batistić Andrea Kadović University of Angers, France Zagreb, Croatia

60

61 THE END

62 V/t=1.5 Ground State Phase Diagram Bipolaron lattice AF Order Homogenous phase H-F Calc.by I.Batistic

63 Antiferromagnetic chains Charge density Spin density

64 Bipolaron lattices Triangular lattice Bathroom foor? Charge density Charge density

65 Bordering AF/BP phases bipolaron AF bi-chain Charge density Spin density

66 .. and ome several more Another mixed AF/BP phase bipolaron AF chain Charge density Spin density

67 Where are we with? Bipolaron lattice Homogenous phase AF Order

68 Conductivity in gas phase (high T phase) ρ 0.2 Ωcm RT metallic upon T~E a

69 Reading thermopower of holes and electrons n = 2 / 3 + δ V = a 1, J = 0.2, E = 0.2

70 Electrons: Regaining individuality (yet remaining mobile!) decrease Bandwidth (W) increase Coulomb efects (U/W) strongly correlated electron systems increase Electon-phonon interaction form Polarons!? go.. Organics!

71 Model Hamiltonian t U,V g K, M

72 H.H.H. - Ground State Phase Diagram Bipolaron lattice AF Order I.Batistić, E.T. Homogenous phase Within Hartree-Fock and mean-field approximation for deformation

73 Robustness against Dirac point destruction: B C A e B,C =0 e A =-gu 1

74 Robustness against Dirac point destruction: B C A e B,C =0 e A =-gu 1

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80 Charge-ordered phase Transition temperature Semiconductor Gap = V + LT E a