Equilibrium and non-equilibrium Mott transitions at finite temperatures

Transcription

1 Equilibrium and non-equilibrium Mott transitions at finite temperatures Stefanos Papanikolaou Department of Physics, Cornell University

2 Collaborators A. Shekhawat (Cornell), S. Zapperi (U. Milan), J. P. Sethna (Cornell) R. M. Fernandes (Iowa), R. Sknepnek (Iowa), J. Schmalian (Iowa) E. Fradkin (U. Illinois), P. Phillips (U. Illinois)

3 Outline Equilibrium Mott transitions and Mott-erials Ising Universality at equilibrium finite-t critical points and conductivity complexity Non-Equilibrium Mott transitions and resistance jumps Novel avalanche critical behavior at non-equilibrium V

4 Equilibrium Mott transitions Direct conducting-insulating transitions for systems with odd number of electrons per unit cell. Simple viewpoint based on two energy scales: U (local Coulomb repulsion) and W (predicted bandwidth) Direct transitions between two aledlimiting by a series cases realized in many materials: (Cr 1 x V x ) 2 O 3, VO 2, NiS 2 x Se x, NiS, κ 2 -ET organics

5 T (K) Mott Transition from a Spin Liquid to a Fermi Liquid in the Spin-Frustrated Organic Conductor - ET 2 Cu 2 CN 3 Equilibrium Mott transitions NiS 2! ("cm) Y. Kurosaki, 1 Y. Shimizu, 1,2, * K. Miyagawa, 1,3 K. Kanoda, 1,3 and G. Saito 2 1 Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo, , Japan 2 Division of Chemistry, Kyoto University, Sakyo-ku, Kyoto, , Japan 3 CREST, Japan Science and Technology Corporation, Kawaguchi , Japan (Received 15 October 2004; revised manuscript received 6 April 2005; published 18 October 2005) The pressure-temperature 80 phase diagram of the organic Mott insulator - ET 2 Cu 2 CN 3, a model 4.0 GPa P GPa 6.2 GPa 7.5 GPa T N 40 system of the spin liquid on triangular lattice, has been investigated by 1 H NMR and resistivity measurements. The spin-liquid phase is persistent before the Mott transition to the metal or superconducting phase under pressure. At the Mott transition, the spin fluctuations are rapidly suppressed and the Fermi-liquid features Tare N observed in the temperature dependence of the spin-lattice relaxation rate 20 and resistivity. The characteristic T N curvature of the Mott boundary in the phase diagram highlights a crucial effect of the spin frustration on the Mott transition. P 0, 2.6, 3.0, 3.2, 3.3, 3.36, 3.4 GPa T (K) 300! (µ"cm) T (K) 100 DOI: /PhysRevLett PACS numbers: Nf, a, Kn, k V 2 O 3 κ (ET) 2 X T MI T N NiS 2 Insulator Magnetic interaction on the verge of the Mott transition is one of the chief subjectspmin the physics of strongly correlated electrons, because striking phenomena such as unconventional AFM superconductivity emerge from the mother Mott 0 0 insulator 2 with 4 antiferromagnetic 6 8 (AFM) 10 order. Examples are transition P (GPa) metal oxides such as V 2 O 3 and La 2 CuO 4, in which localized paramagnetic spins undergo the AFM transition at low temperatures [1]. The ground state of the Mott insulator is, however, no more trivial when the spin frustration works between the localized spins. Realization of the spin liquid has attracted much attention since a proposal of the possibility in a triangularlattice Heisenberg antiferromagnet [2]. Owing to the extensive materials research, some examples of the possible spin liquid have been found in systems with triangular and kagomé lattices, such as the solid 3 He layer [3], Cs 2 CuCl 4 [4], and - ET 2 Cu 2 CN 3 [5]. Mott transitions between metallic and insulating spin-liquid phases are an interesting new area of research. The layered organic conductor - ET 2 Cu 2 CN 3 is the only spin-liquid system to exhibit the Mott transition, to the authors knowledge [5]. The conduction layer in - ET 2 Cu 2 CN 3 consists of strongly dimerized ET [bis(ethlylenedithio)-tetrathiafulvalene] molecules with one hole per dimer site, so that the on-site Coulomb repulsion inhibits the hole transfer [6]. In fact, it is a Mott insulator at ambient pressure and becomes a metal ambient pressure. Then the Mott transition in - ET 2 Cu 2 CN 3 under pressure may be the unprecedented one without symmetry breaking, if the magnetic order does not emerge under pressure up to the Mott boundary. In this Letter, we report on the NMR and resistance studies of the Mott transition in - ET 2 Cu 2 CN 3 under pressure. The result is summarized by the pressuretemperature (P-T) phase diagram in Fig. 1. The Mott Mott insulator (Spin liquid) (dr/dt) max onset T C R = R 0 + AT 2 Superconductor (1/T 1 T) max Pressure (10-1 GPa) κ (ET) 2 Cu) 2 (CN) 3 Crossover T 1 T = const. Metal (Fermi liquid)

6 Liquid-Gas critical point: Ising universality DMFT calculations towards a density order parameter in 2D (Kotliar et al. 00, DiCastro et al. 79) Thus, expected in the Ising universality class, but identification non-trivial: order parameter: scaling fields: h P P c, predictions: Σ Σ(T, P) Σ(T c, P c ) Σ (a) t β σ Σ (b) h 1/δ σ Σ/ h (c) t γ σ β σ + γ σ = β σ δ σ t T T c

7 Liquid-Gas critical point: Ising universality Experiment Material Findings Comments Limelette et al. (Science 2003) aled by a series (Cr 1 x V x ) 2 O 3 (β, γ, δ) = (1/2, 1, 3) 3D Ising Mean-Field Ising, Narrow critical region Kagawa et al. (Nature 2005) κ-et organic salts under pressure (β, γ, δ) = (1, 1, 2) 2D Ising...if conductivity = energy density (SP et al. PRL 2007) Takeshita et al. (private comm. 2007) NiS 2 under pressure (β, γ, δ) = (1/2, 1, 3) 3D Ising Mean-Field Ising, Narrow critical region M. de Souza et al. (PRL 2007) Kagawa et al. (Nature 2010) κ-et organic salts under pressure κ-et organic salts under pressure l 1 l/ T t 0.85 NMR signature of energy coupling to the conductivity 2D Ising...if phase diagram axes are rotated (SP et al. PRL 2007) 2D Ising scenario signatures

8 Ising critical points and resistor networks σ insul σ conduct S i = ±1 label grains of high-low carrier density. grain size: decoherence length Effective Ising Hamiltonian: l φ h = σ ins σ cond βh = 1 T S i S j + h T S i ij i

9 Ising critical points and resistor networks σ insul σ conduct S i = ±1 label grains of high-low carrier density. grain size: decoherence length Effective Ising Hamiltonian: l φ h = σ ins σ cond βh = 1 T S i S j + h T S i ij i

10 Infinite contrast limit: Looking at the fractal σ insul = 0: percolating clusters conductivity diffusion of an ant in the incipient infinite Ising cluster. critical exponents controlled by fractal properties No known relation between the conductivity exponent t and thermodynamic exponents 2.5 Σ Tc 2 g m = g ε = Σ Tc =3.9 L L

11 Infinite contrast limit: Looking at the fractal σ insul = 0: percolating clusters conductivity diffusion of an ant in the incipient infinite Ising cluster. critical exponents controlled by fractal properties No known relation between the conductivity exponent t and thermodynamic exponents 2.5 Σ Tc 2 g m = g ε = Σ Tc =3.9 L L

12 Zero contrast limit: Perturbative limit σ ins σ cond : Possible to calculate the conductivity in a cluster perturbation theory, in powers of the contrast parameter g Consider generally: σ ij = σ 0 (1 + g m (S i + S j ) + g ɛ S i S j ) Solve the Kirchoff laws perturbatively (Blackman 1976) (...) Result:

13 Zero contrast limit: Perturbative limit σ ins σ cond : Possible to calculate the conductivity in a cluster perturbation theory, in powers of the contrast parameter g Consider generally: σ ij = σ 0 (1 + g m (S i + S j ) + g ɛ S i S j ) Solve the Kirchoff laws perturbatively (Blackman 1976) (...) Result: Σ σ 0 = 1 + g m S + (g + g 2 mγ αβ )(SS) α + g g m Γ αβ (SS) α S β + g 2 Γ αβ (SS) α (SS) β + O(g 3 )

14 Zero contrast limit: Perturbative limit σ ins σ cond : Possible to calculate the conductivity in a cluster perturbation theory, in powers of the contrast parameter g Consider generally: σ ij = σ 0 (1 + g m (S i + S j ) + g ɛ S i S j ) Solve the Kirchoff laws perturbatively (Blackman 1976) (...) Result: when Ising critical: m + Σ σ 0 = 1 + g m S + (g + g 2 mγ αβ )(SS) α + g g m Γ αβ (SS) α S β + g 2 Γ αβ (SS) α (SS) β + O(g 3 )

15 Zero contrast limit: Perturbative limit σ ins σ cond : Possible to calculate the conductivity in a cluster perturbation theory, in powers of the contrast parameter g Consider generally: σ ij = σ 0 (1 + g m (S i + S j ) + g ɛ S i S j ) Solve the Kirchoff laws perturbatively (Blackman 1976) (...) Result: Σ σ 0 = 1 + g m S + (g + g 2 mγ αβ )(SS) α + g g m Γ αβ (SS) α S β + g 2 Γ αβ (SS) α (SS) β + O(g 3 )

16 Zero contrast limit: Perturbative limit σ ins σ cond : Possible to calculate the conductivity in a cluster perturbation theory, in powers of the contrast parameter g Consider generally: σ ij = σ 0 (1 + g m (S i + S j ) + g ɛ S i S j ) Solve the Kirchoff laws perturbatively (Blackman 1976) (...) Result: when Ising critical: + Σ σ 0 = 1 + g m S + (g + g 2 mγ αβ )(SS) α + g g m Γ αβ (SS) α S β + g 2 Γ αβ (SS) α (SS) β + O(g 3 )

17 Zero contrast limit: Perturbative limit At the Ising critical point: Conductivity has odd and even parts which scale like the magnetization and energy density g m = g ε = 0.01 σ even T c = L g m = g ε = 0.01 σ odd T c = L g m = 0.001, g ε = 0.01 Σ Tc Σ 0 = 0.02 L Σ Tc Σ 0 = L σ even T c σ odd T c Σ Tc Σ L L L Crossover length scale where magnetization coupling becomes dominant

18 Experiments and Ising scenario Suggestion: g m g Then, energy-controlled true-critical exponents ( Σ ) (β σ, γ σ, δ σ ) = (1, 7/8, 15/8) very close to experimental finding ~ ( 1, 1, 2 ) Crossover between energy dominated regime (smaller length scales) and order parameter dominated regime (larger length scales). But, why g m so small: Strong grain-interface scattering

19 General Model: Disorder and quasi-2d Appropriate model for criticality observed in organic materials: Strongly Anisotropic Random-Field Ising model H = J xy {ij} xy S i S j J z Dimensional crossover for the critical point [Zachar et al. 2003] {kl} z S k S l + i h i S i, 3D Materials small critical region, RFIM universality disorder is relevant J xy F quasi-2d materials wide 2D Ising critical region, still RFIM universality 1/2 ξ z > 1 ξ xy > 1 ξ z, ξ xy < 1 P J 3D C 0 0 J z /J xy 1

20 Outline Equilibrium Mott transitions and Mott-erials Ising Universality at equilibrium finite-t critical points and conductivity complexity Non-Equilibrium Mott transitions and resistance jumps Novel avalanche critical behavior at non-equilibrium

21 Avalanches in resistance jumps in VO 2 Resistance decreases in jumps Power law distributed ( Sharoni et al. PRL 101, (2008) ) minimal model for Mott avalanches? Are the jumps quantum (Mott-driven) or classical (disorder driven)?

22 Conducting / Insulating cluster properties in VO 2 zero voltage M.M. Qazilbash et al. (Science 2007) finite voltage Kim et al. (PRB 2007)

23 Non-equilibrium resistor networks Tc decreases linearly with voltage [Sharoni et al. 2008] Dielectric breakdown model[duxbury et al ] relevant for experiments [Sharoni et al. 2008, S. Ganapathy 2010] Fixed V, increase T slowly, we take avalanches (V=0 percolation)

24 Description of the model Square lattice anti-fuse network random thresholds uniform in [0,1] R i V Bond with RINS becomes RCOND when V i + at R i Temperature T driven across the transition Relevant parameters: contrast ratio: h = R COND [0, 1], V R INS

25 V Phase diagram

26 Phase diagram Two phases: a) bolt phase b) percolative phase V

27 V Phase diagram

28 Phase diagram V V=0, percolation; h is a relevant perturbation [Tremblay 84]

29 V Phase diagram

30 Phase diagram V h=0, infinite contrast; similar to fracture/ superconductor problems

31 V Phase diagram

32 Phase diagram V h=1, zero contrast; perturbation theory is applicable;

33 V Phase diagram

34 Phase diagram V Newly discovered phase transition! Universality controlled by the large-h regime Current experiments in the small-v, small-h regime

35 V Phase diagram

36 Phase structure: a) Percolative phase (h,v 0) R T Avalanche structure Resistance.vs. temperature Cluster structure small avalanche sizes -- almost absent hysteresis

37 Phase structure: b) Bolt Phase (h 0,V ) R Avalanche structure T Resistance.vs. temperature Cluster structure lightning-bolt -like structures emerge -- large hysteresis

38 <S 2 > <S> 2 Existence of a phase transition size distributions show cutoff scaling second moment has peak at the transition V

39 Phase transitions: a) High contrast (h 0) R T

40 Phase transitions: a) High contrast (h 0) R T Exponent relevant for experiments

41 Phase transitions: a) High contrast (h 0) R T

42 Phase transitions: b) Low contrast (h 1) R Avalanche structure T Resistance.vs. temperature Cluster structure

43 Phase transitions: Distributions at low contrast scaling distributions for both sizes and resistance jumps novel universality class, perturbation expansion suggests in a 2D longrange universality class

44 Perturbation theory at low contrast Universality class controlled by the h 1 regime. For h 1, perturbation theory in (1-h) V α = V 0 α +(1 h)γ αβ S β V 0 β +(1 h) 2 Γ αβ S β Γ βδ S δ V 0 δ Γ αβ = G il + G jk G ik G jl k β l G ij = Ω 1 d 2 q eiq (R(i) R(j)) i cos(q i) j i α For large r: Γ αβ sin2 θ r 2 Dipolar Long-Range Ising coupling to first-order

45 Voltage jump dependence (Sharoni et al. PRL (2008) ) experimental verification of the size dependence (below the critical point) indicator for the location of the critical point

46 Conclusions Finite temperature Mott transitions gradually in our grasp... Resistance avalanches, no depinning interface Non-equilibrium critical behavior controlled by a low contrast - high voltage fixed point Predictions for future experiments Refs: 1) S. Papanikolaou et al., Phys. Rev. Lett. 100, (2008) and 2) A. Shekhawat, S. Papanikolaou, S. Zapperi and J. P. Sethna (to be submitted)