Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
  • Open Access

Spin-orbital mechanisms for negative thermal expansion in Ca2RuO4

Wojciech Brzezicki, Filomena Forte, Canio Noce, Mario Cuoco, and Andrzej M. Oleś
Phys. Rev. B 107, 104403 – Published 6 March 2023
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • SINGLE TM−O−TM BOND ANALYSIS
  • PLAQUETTE CONSISTING OF d4 IONS…
  • PLAQUETTE WITH d3 IMPURITY
  • SUMMARY AND CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The phenomenon of negative thermal expansion (NTE) deals with the increase of the lattice parameters and the volume of the unit cell when the material is thermally cooled. The NTE is typically associated with thermal phonons and anomalous spin-lattice coupling at low temperatures. However, the underlying mechanisms in the presence of strong electron correlations in multiorbital systems are not yet fully established. Here, we investigate the role of Coulomb interaction in the presence of lattice distortions in setting out the NTE effect, by focusing on the physical case of layered Ca2RuO4 with the d4 configuration at each Ru ion site. We employ the Slater-Koster parametrization to describe the electron-lattice coupling through the dependence of the d-p hybridization on the Ru-O-Ru bond angle. The evaluation of the minimum of the free energy at finite temperature by fully solving the multiorbital many-body problem on a finite-size cluster allows us to identify the regime for which the system is prone to exhibit NTE effects. The analysis shows that the nature of the spin-orbital correlations is relevant to drive the reduction of the bond angle by cooling, and in turn the tendency toward a NTE. This is confirmed by the fact that a changeover of the electronic and orbital configuration from d4 to d3 by transition metal substitution is shown to favor the occurrence of a NTE in Ca2RuO4. This finding is in agreement with the experimental observations of a NTE effect which is significantly dependent on the transition metal substitution in the Ca2RuO4 compound.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    7 More
    • Received 23 September 2022
    • Revised 18 January 2023
    • Accepted 14 February 2023

    DOI:https://doi.org/10.1103/PhysRevB.107.104403

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Magnetic orderMagnetismThermal expansion
    1. Physical Systems
    2-dimensional systemsAntiferromagnetsMott insulators
    1. Techniques
    Hubbard model
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Wojciech Brzezicki1, Filomena Forte2,3, Canio Noce2,3, Mario Cuoco2,3, and Andrzej M. Oleś4,1

    • 1Institute of Theoretical Physics, Jagiellonian University, Prof. Stanisława Łojasiewicza 11, PL-30348 Kraków, Poland
    • 2Consiglio Nazionale delle Ricerche, CNR-SPIN, IT-84084 Fisciano (SA), Italy
    • 3Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno, Via Giovanni Paolo II 132, IT-84084 Fisciano (SA), Italy
    • 4Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

    Article Text

    Click to Expand

    References

    Click to Expand
    Issue

    Vol. 107, Iss. 10 — 1 March 2023

    Reuse & Permissions
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      Schematic representation of the TM1−O−TM2 bond for the case of (a) an undistorted configuration with θ=0, (b) a nonvanishing bond angle due to the misalignment between the TM1−O and TM1−TM2 directions. In (c) we provide a sketch to depict how a variation of the bond angle for a square geometry of the TM−O lattice can lead to a modification of the unit cell area. In particular, a thermal gradient ΔT that induces an increase of the TM−O−TM bond angle would lead to a reduction of the unit cell.

      Reuse & Permissions
    • Figure 2
      Figure 2

      (a) Contour map of the optimal bond angle for the TM(d4)−O−TM(d4) configuration at zero temperature as a function of the p-d hybridization parameters {Vpdσ,Vpdπ} for a representative value of the Coulomb strength and Hund coupling, i.e., U=2.3 eV and JH=0.5 eV. We notice that there are two distinct regimes of bond distortions that can be favored by the electronic correlations. For Vpdπ about larger than Vpdσ, the the tendency is to favor a small tilt of the bond angle which is close to zero. Otherwise, in the remaining portion of the phase diagram the trend is to stabilize a large bond angle of π/4. The transition between the two regimes is quite rapid and occurs in the proximity of the line Vpdσ≃Vpdπ. A variation of the electronic parameters does not alter significantly the phase diagram. (b) Evolution of the optimal angle as a function of Vpdσ for fixed values of Vpdπ, for the same choice of parameters as in (a).

      Reuse & Permissions
    • Figure 3
      Figure 3

      Temperature dependencies of the bond angles for (a), (b) TM(d4)−O−TM(d4) and (c), (d) TM(d3)-O-TM(d4) bonds. Curves in (a) and (c) are for fixed JH and different values of U and in (b) and (d) are for fixed U and different JH; the values of U and JH are indicated. The other parameters are Vpdπ=1.3, Vpdσ=1.6, λ=0.075, δ=0.25, δort=0.09, all in eV.

      Reuse & Permissions
    • Figure 4
      Figure 4

      (a), (c) Temperature dependence of the scalar product of the spin moments at the TM sites for a TM(d4)-O-TM(d4) configuration, as a function of the bond angle at T=460 K and T=45 K for U=2.3 eV and U=2.1 eV, respectively. (b) and (d) provide the evolution of the vector product of the spin moments at the TM sites as a function of the bond angle at T=460 K and T=45 K for U=2.3 eV and U=2.1 eV, respectively.

      Reuse & Permissions
    • Figure 5
      Figure 5

      (a)–(c) Temperature dependence of the expectation value of the scalar product of the spin moments at the TM sites for a TM(d3)−O−TM(d4) configuration, as a function of the bond angle at two representative temperatures, T=460 K and T=45 K, for U=1.8 eV and U=2.0 eV, respectively. (b) and (d) provide the evolution of the vector product of the spin moments at the TM sites as a function of bond angle at T=460 K and T=45 K for U=1.8 eV and U=2 eV, respectively.

      Reuse & Permissions
    • Figure 6
      Figure 6

      Schematic view of a plaquette of four metallic sites (red dots) with surrounding oxygens (green dots) with cooperative rotational distortions by angle ±θ. The bond angles are explicitly highlighted in cyan.

      Reuse & Permissions
    • Figure 7
      Figure 7

      Results for the undoped d4 plaquette [inset in (d)]: (a) phase diagram as function of π and σ hybridization amplitudes; (b) typical curve of ground state energy versus bond angle θ in the intermediate phase; (c), (d) typical curve of the optimal bond angle and the ground state average of 〈S⃗1×S⃗2〉z versus δ(Vpdπ)≡Vpdπ−V0 for Vpdσ=1.5 eV in the intermediate phase marked by a dashed line in plot (a); (e), (f) thermal dependencies of the optimal bond angle and average (S⃗1×S⃗2)z in the intermediate phase for Vpdσ=1.5 and Vpdπ=1.2973 eV. The other parameters are U=8.0, JH=0.5, εp=−4.5, δ=0.35, δort=0.09, and λ=0.075, all in eV.

      Reuse & Permissions
    • Figure 8
      Figure 8

      Results for pure-d4 plaquette with C-AFM magnetic order (see the inset): (a), (b) typical curves of the optimal bond angle and the ground state averages of 〈S⃗1×S⃗2〉z and 〈S⃗1×S⃗3〉z versus δ(Vpdπ)=Vpdπ−V0 for Vpdσ=1.5 eV in the intermediate phase between θ=0 and θ=π/4 (see Fig. 6); (c), (d) thermal dependencies of the optimal bond angle and average 〈S⃗1×S⃗2〉z and 〈S⃗1×S⃗3〉z in the intermediate phase for Vpdσ=1.5 and Vpdπ=1.3034 eV. The other parameters are U=8.0, JH=0.5, εp=−4.5, δ=0.35, δort=0.09, λ=0.075, and h=0.2, all in eV.

      Reuse & Permissions
    • Figure 9
      Figure 9

      Results for the d3-doped d4 plaquette (insets; dopant at i=4): (a) phase diagram as function of π and σ hybridization amplitudes; (b) typical curve of ground state energy versus bond angle θ close to the phase boundary; (c) optimal angle versus temperature for Vpdσ=1.5 and Vpdπ=1.2848 eV, no magnetic order imposed; (d) optimal angle versus temperature for Vpdσ=1.5 and Vpdπ=1.2745 eV, C-AFM magnetic order is imposed as shown by red (blue) color of the nearest-neighbor bonds; (e), (f) averages of (S⃗i×S⃗j)z along the nearest-neighbor bonds versus temperature for the same parameters as in (d). The other parameters are U=Ũ=8.0, JH=J̃H=0.5, Ie=1.0, εp=−4.5, δ=0.35, δort=0.09, and λ=0.075, all in eV.

      Reuse & Permissions
    • Figure 10
      Figure 10

      Spin-spin correlations for the d3-doped d4 plaquette (insets; dopant at i=4) as functions of π hybridization amplitude across the phase transition between θ=0 and θ=π/4 phases at zero temperature. (a)–(d) Correlations 〈SiαSjα〉 for the nearest neighbors. (e), (f) Correlations 〈S⃗i×S⃗j〉z along the nearest-neighbor bonds. The other parameters are U=Ũ=8.0, JH=J̃H=0.5, Ie=1.0, εp=−4.5, δ=0.35, δort=0.09, λ=0.075, and Vpdπ=1.5, all in eV.

      Reuse & Permissions
    • Figure 11
      Figure 11

      Results for the undoped d4 plaquette with finite JH and λ increased by a factor of χ=3.2 compared to Sec. 3: (a), (b) typical curve of the optimal bond angle and the ground state average of 〈S⃗1×S⃗2〉z versus δ(Vpdπ)≡Vpdπ−V0 for Vpdσ=1.5 eV in the intermediate-angle phase; (c), (d) thermal dependencies of the optimal bond angle and average (S⃗1×S⃗2)z in the intermediate phase for Vpdσ=1.5 eV and Vpdπ=1.2962 eV. The other parameters are U=8.0, JH=1.6, εp=−4.5, δ=0.35, δort=0.09, and λ=0.24, all in eV.

      Reuse & Permissions
    • Figure 12
      Figure 12

      Results for the undoped d4 plaquette with C-AFM magnetic order (see the inset) and with JH and λ increased by a factor of χ=3.2 compared to Sec. 3: (a), (b) typical curves of the optimal bond angle and the ground state averages of 〈S⃗1×S⃗2〉z and 〈S⃗1×S⃗3〉z versus δ(Vpdπ)=Vpdπ−V0 for Vpdσ=1.5 eV in the intermediate phase between θ=0 and θ=π/4; (c), (d) thermal dependencies of the optimal bond angle and average 〈S⃗1×S⃗2〉z and 〈S⃗1×S⃗3〉z in the intermediate phase for Vpdσ=1.5 and Vpdπ=1.2928 eV. The other parameters are U=8.0, JH=1.6, εp=−4.5, δ=0.35, δort=0.09, λ=0.24, and h=0.2, all in eV.

      Reuse & Permissions
    • Figure 13
      Figure 13

      Optimal angle versus temperature for a single (a) d4−d4 bond and (b) d3−d4 bond, for different values of JH (renormalized by a factor of χ=3.2). The other parameters are Vpdσ=1.6, Vpdπ=1.3, U=8.0, εp=−4.5, δ=0.25, δort=0.09, λ=0.24, and h=0.2, all in eV.

      Reuse & Permissions
    • Figure 14
      Figure 14

      Results for a pure d4 plaquette as obtained within the effective low-energy spin-orbital exchange approach: (a) electronic free energy versus bond angle for a given Vpdσ=1.5 eV and different values of Vpdπ=V0+δ(Vpdπ) chosen in the intermediate-angle phase; (b) optimal angle versus Vpdπ for a purely electronic free energy (red curve) and for the phenomenological electronic-phononic free energy [Eq. (E1)] assuming a representative value for the coupling constant, i.e., k=6 meV. The inset of (b) shows the optimal angle versus temperature dependence for Vpdπ=1.2902 eV. The other electronic parameters are U=8.0, εp=−4.5, δ=0.35, δort=0.09, and λ=0.075; all values are in eV.

      Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review B

    Reuse & Permissions

    It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

    ×

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×