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  • Open Access

Oxidation States, Thouless’ Pumps, and Nontrivial Ionic Transport in Nonstoichiometric Electrolytes

Paolo Pegolo, Federico Grasselli, and Stefano Baroni
Phys. Rev. X 10, 041031 – Published 12 November 2020
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  • INTRODUCTION
  • THEORY
  • NONTRIVIAL TRANSPORT
  • CONCLUSIONS
  • DATA AVAILABILITY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • Supplemental Material
    References

    Abstract

    Thouless’ quantization of adiabatic particle transport permits one to associate an integer topological charge with each atom of an electronically gapped material. If these charges are additive and independent of atomic positions, they provide a rigorous definition of atomic oxidation states and atoms can be identified as integer-charge carriers in ionic conductors. Whenever these conditions are met, charge transport is necessarily convective; i.e., it cannot occur without substantial ionic flow, a transport regime that we dub trivial. We show that the topological requirements that allow these conditions to be broken are the same that would determine a Thouless’ pump mechanism if the system were subject to a suitably defined time-periodic Hamiltonian. The occurrence of these requirements determines a nontrivial transport regime whereby charge can flow without any ionic convection, even in electronic insulators. These results are first demonstrated with a couple of simple molecular models that display a quantum-pump mechanism upon introduction of a fictitious time dependence of the atomic positions along a closed loop in configuration space. We finally examine the impact of our findings on the transport properties of nonstoichiometric alkali-halide melts, where the same topological conditions that would induce a quantum-pump mechanism along certain closed loops in configuration space also determine a nontrivial transport regime such that most of the total charge current results to be uncorrelated from the ionic ones.

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    • Received 28 June 2020
    • Revised 25 August 2020
    • Accepted 29 September 2020

    DOI:https://doi.org/10.1103/PhysRevX.10.041031

    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.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Electric polarizationElectrical conductivitySolvationTransport phenomena
    1. Physical Systems
    InsulatorsLiquids
    1. Techniques
    Density functional theoryMolecular dynamicsWannier function methods
    Condensed Matter, Materials & Applied PhysicsInterdisciplinary PhysicsAtomic, Molecular & Optical

    Authors & Affiliations

    Paolo Pegolo1, Federico Grasselli1,*, and Stefano Baroni1,2,†

    • 1SISSA-Scuola Internazionale Superiore di Studi Avanzati, 34136 Trieste, Italy
    • 2CNR-Istituto Officina dei Materiali, SISSA, 34136 Trieste, Italy

    • *Present address: COSMO—Laboratory of Computational Science and Modelling, IMX, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.
    • †baroni@sissa.it

    Popular Summary

    Electric conductors are usually classified into two families: metals and ionic conductors. In metals, electrons move almost freely, whereas in normal ionic conductors (such as stoichiometric molten salts), electrons are transported by ions to which they are bound. However, it is possible to design simple ionic systems that straddle these two regimes, with electrons that are still localized, but whose position cannot be uniquely ascribed to any particular ion. This leads to anomalies resulting in good conductors that are transparent like nonmetals, but where charge transport is not accompanied by any sizable mass transport, as it is in ordinary ionic conductors. Here, we investigate this regime in terms of topological quantum numbers and show that the anomalous charge transport in these intermediate systems relates to nontrivial topological features of their electronic ground state.

    We find that these topological features can induce a quantized charge-pump mechanism and charge transport with no net ionic displacement while retaining the insulating character of the ground state. We first demonstrate these results with simple molecular models where massless charge transport occurs. We then examine the impact of our findings on the transport properties of certain alkali-halide melts, where most of the total charge current is uncorrelated with the ionic current.

    Our work paves the way to a deeper understanding of charge transport in liquid electrolytes, where the anomalous transport regime lies across the transition between an electronically insulating phase and a metallic one. Furthermore, anomalous charge transport may produce a high electric conductivity in electrolytes that is not accompanied by a large heat conductivity, boosting the quest for unconventional materials for thermoelectric applications.

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    Vol. 10, Iss. 4 — October - December 2020

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    Images

    • Figure 1
      Figure 1

      (a) Two-dimensional atomic configuration space with periodic boundary conditions. (b) Representation of the ACS on a 2-torus. Open paths in the plane whose end points are one the periodic image of the other map onto closed paths in the torus; closed paths in the plane map onto trivial loops in the torus (i.e., closed paths that can be continuously shrunk to a point). The concatenation of the two green paths with the reversed blue one is a trivial loop on the torus. When strong adiabaticity holds, the total transported charge in a trivial path is zero.

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    • Figure 2
      Figure 2

      Linear H3+ in periodic boundary conditions. (a) I: Equilibrium configuration: The black circles and continuous line indicate atoms in the primitive supercell; gray circles and the dashed line indicate their periodic images. II and III are the two intermediate steps of the closed path described in the text. (b) Closed path in the 2D projection of the atomic configuration space relative to the atoms that participate in the loop; the yellowish areas indicate regions where the ground state is degenerate. (c) Dipole displaced along the closed path. Notice that the total displaced dipole is finite and an integer multiple of eL. (d) Closed path in the 3D projection of the ACS where also the x coordinate of atom A is shown. The metallic region encircled by the path extends for all values of xA, and the loop cannot be shrunk to a point without crossing it: The loop is, thus, nontrivial. 3D projections onto subspaces where xA is substituted with any other atomic coordinate have a similar appearance. An animation illustrating the trajectory of the atoms participating in the loop can be found in the file S1.mp4 in Supplemental Material [18]. In the animation, the green dot indicates the position of the Wannier center of the two electrons in the system.

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    • Figure 3
      Figure 3

      A planar configuration of the K3Cl system undergoing a loop in atomic configuration space. (a) Initial and final configurations. K and Cl atoms are indicated by pink and blue circles, respectively. The colored curved arrows indicate the 1D trajectories of the two K atoms participating in the loop. The color encodes the fictitious time parametrizing the loop (red→blue). (b) Closed path in the 2D projection of the atomic configuration space relative to the atoms that participate in the loop; the yellowish areas indicate regions where the ground state is degenerate. (c) Dipole displaced along the closed path. (d) Closed path in the 3D projection of the ACS where also the x coordinate of atom A is shown. An animation illustrating the trajectory of the atoms participating in the loop can be found in the file S2.mp4 in Supplemental Material [18]. The green dots indicate the positions of the Wannier centers of the electrons in the system.

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    • Figure 4
      Figure 4

      Time series of the HOMO-LUMO (blue) and HOMO–1-LUMO (orange) energy gaps. The horizontal red line indicates the thermal energy kBT. The horizontal green line is the average HOMO-LUMO gap for the stoichiometric K32Cl32 system.

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    • Figure 5
      Figure 5

      Overlay of several consecutive snapshots from a 435 fs-long sample of AIMD trajectory of our K33Cl31 model. K+ ions are depicted in pink and Cl− ions in blue, while the Wannier center associated to the lone bipolaronic pair is displayed in green.

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    • Figure 6
      Figure 6

      (a),(c) Different loops in the K33Cl31 atomic configuration space, whose initial and final configurations are the same. Cl atoms are depicted in blue. All K atoms but one are depicted in pink. One selected K atom, depicted in red, is moved from its initial position to its periodic image along the x direction, thus featuring a winding number nix=+1. All the other atoms feature zero winding numbers. The position of the lone pair is depicted in green. (b),(d) Dipoles displaced along the closed paths depicted on their left. The charge displaced along the two paths differ, in spite of the displacement of the same K atom and the same winding numbers. This difference indicates that no oxidation state can be uniquely associated to the (arbitrarily) chosen K atom, and transport anomalies have to be expected.

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    • Figure 7
      Figure 7

      Upper: Time series of the distances from the electron lone pair in K33Cl31 of the five nearest atoms. The horizontal lines are guides for the eye: dotted, distance equal to zero; dashed, maximum distance allowed in PBCs, i.e., 3L/2. Lower: Average diagonal elements of the Born effective-charge tensor of the five atoms described above; the horizontal dotted line marks the zero value.

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    • Figure 8
      Figure 8

      Adiabatic charge transport in molten K0.06/(KCl)0.94. Time series of the mean-square displaced dipole from definitions (2) (blue) and (4) (orange). The contribution due to the ionic cores and the tightly bound electrons is shown in green. The cross-correlation term is depicted in red. According to Eq. (3), the slope of the straight lines is a measure of the electric conductivity, whose actual value is estimated from cepstral analysis, as explained in Appendix pp2.

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    • Figure 9
      Figure 9

      Low-frequency portion of the PSD of the displaced dipole computed according to Eqs. (2) (blue) and (4) (orange). The noisy lines are the window-filtered PSDs (with a window of 0.1 THz), while the smooth lines are the cepstral estimates.

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