Research ArticleCONDENSED MATTER PHYSICS

Two-dimensional ground-state mapping of a Mott-Hubbard system in a flexible field-effect device

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Science Advances  10 May 2019:
Vol. 5, no. 5, eaav7282
DOI: 10.1126/sciadv.aav7282
  • Fig. 1 Bandfilling and bandwidth control of κ-Cl in the same sample.

    (A) Molecular arrangement of the BEDT-TTF layer in κ-Cl (top view). (B) Conceptual phase diagram based on the Hubbard model (37). The vertical axis denotes the strength of the electron correlation. κ-Cl is originally located near the tip of the insulating region and is shifted along both directions to investigate the superconducting region. (C) Schematic side view of the device structure. The doping concentration and effective pressure are controlled by an electric double-layer gating and bending of the substrate with a piezo nanopositioner, respectively. The resistivity is measured by the standard four-probe method. (D) Sheet resistivity versus temperature plots under hole doping at tensile strain S = 0.41%. The dashed line indicates the pair quantum resistance h/4e2. (E) Resistivity versus temperature plots under different tensile strains at gate voltage VG = 0 V.

  • Fig. 2 Electron-hole asymmetric phase diagram of κ-Cl.

    (A to G) Contour plots of the sheet resistivity ρ under tensile strains S = 0.35% (A), 0.39% (B), 0.41% (C), 0.44% (D), 0.46% (E), 0.50% (F), and 0.55% (G) as a function of temperature and gate voltage. (H) Contour plots of the sheet resistivity ρ at 5.5 K as a function of gate voltage and tensile strain. Black dots in all figures indicate the data points where the sheet resistivity was measured. The doping concentration estimated from the average density of charge accumulated in the charge displacement current measurement (fig. S2) is shown for reference on the upper horizontal axis in (H). AFI, antiferromagnetic insulator; h-SC, p-type SC; e-SC, n-type SC. In the region below the white dashed line at S = 0.35%, the surface resistivity under doping cannot be measured because the nondoped bulk is superconducting below 12 K.

  • Fig. 3 VCA calculations.

    (A) Antiferromagnetic and dx2y2 superconducting order parameters, M and D, respectively, versus doping concentration δ for several values of U/t. M and D for the metastable and unstable solutions (empty symbols) are also shown at U/t = 4 and 4.5 under electron doping (corresponding to positive δ). (B) Doping concentration δ versus chemical potential μ relative to that at half filling (μhalf) for several values of U/t. The results for the metastable and unstable solutions at U/t = 4 and 4.5 are indicated by dashed lines, while the results obtained by the Maxwell construction are denoted by solid vertical lines. This implies the presence of phase separation and a first-order phase transition. It is noteworthy that there is a steep (nearly vertical) increase in δ with increasing μ for larger values of U/t under electron doping, suggesting a strong tendency toward phase separation. The values of μhalf are μhalf = 1.3725t for U/t = 3.5, μhalf = 1.8375t for U/t = 4, and μhalf = U/2 for U/t 4.5. (C) Chemical potential μ versus doping concentration δ for U/t = 4 (see fig. S8 for more details). δ1 and δ2 are the doping concentrations of the two extreme states in the phase separation. All results in (A) to (C) are calculated using the VCA for the single-band Hubbard model on an anisotropic triangular lattice (t′/t = −0.44) with a 4 × 3 cluster. (D) Noninteracting tight-binding band structure and density of states (DOS) for t′/t = −0.44 with t = 65 meV. Here, Γ = (0,0), Z = (0,π/c), M = (π/a,π/c), and X = (π/a,0), with a and c being the lengths of the primitive translation vectors indicated in Fig. 1A. The Fermi level for half filling is set to zero and denoted by dashed lines. (E) Single-particle spectral functions and DOS of κ-Cl at half filling in an antiferromagnetic state at zero temperature, calculated by VCA. The Fermi level is denoted by a dashed line at zero energy. The flat features seen away from the Fermi level indicate incoherent continuous spectra due to the electron correlation. The reason why they appear rather discretized is because of the discrete many-body energy levels in the VCA calculation, for which a finite-size cluster is used to obtain the single-particle excitation energies.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/5/eaav7282/DC1

    Section S1. Estimation of injected charge density

    Section S2. Phase separation tendency of the VCA calculations

    Fig. S1. Optical images, surface profile, and contact resistance properties of the flexible EDLT based on κ-Cl.

    Fig. S2. Estimation of injected charge density.

    Fig. S3. Suppression of SC by applying a magnetic field.

    Fig. S4. Resistivity dip under small electron doping.

    Fig. S5. Resistance versus temperature plots under low electron doping at tensile strain S = 0.41%.

    Fig. S6. Comparison of temperature dependences of the resistivity between our device and bulk crystals.

    Fig. S7. Cluster size dependence of the phase separation tendency.

    Fig. S8. Swallowtail-shaped grand potential and Maxwell construction under electron doping with moderate correlation.

    Reference (50)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Estimation of injected charge density
    • Section S2. Phase separation tendency of the VCA calculations
    • Fig. S1. Optical images, surface profile, and contact resistance properties of the flexible EDLT based on κ-Cl.
    • Fig. S2. Estimation of injected charge density.
    • Fig. S3. Suppression of SC by applying a magnetic field.
    • Fig. S4. Resistivity dip under small electron doping.
    • Fig. S5. Resistance versus temperature plots under low electron doping at tensile strain S = 0.41%.
    • Fig. S6. Comparison of temperature dependences of the resistivity between our device and bulk crystals.
    • Fig. S7. Cluster size dependence of the phase separation tendency.
    • Fig. S8. Swallowtail-shaped grand potential and Maxwell construction under electron doping with moderate correlation.
    • Reference (50)

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