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Lorenz ratio of an impure compensated metal in the degenerate Fermi liquid regime

Woo-Ram Lee, Karen Michaeli, and Georg Schwiete
Phys. Rev. B 103, 115140 – Published 22 March 2021
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  • INTRODUCTION
  • LINEARIZED BOLTZMANN EQUATION FOR THE…
  • RELAXATION-TIME APPROXIMATION
  • FERMI-LIQUID COLLISION INTEGRAL
  • DISCUSSION
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The Lorenz ratio serves as a measure to compare thermal and electric conductivities of metals. Recent experiments observed small Lorenz ratios in the compensated metal WP2, indicating that charge flow is strongly favored over heat conduction. Motivated by these findings, we study transport properties of compensated metals in the presence of electron-electron collisions and electron-impurity scattering. We focus on intermediate temperatures, where the phonon contributions to transport are weak and elastic and inelastic scattering rates are comparable. Our exact solution for the kinetic equation in the presence of general Fermi-liquid interactions is used to extract the Lorenz ratio for short- and long-range interactions. We find that the Lorenz ratio develops a temperature dependence and gets enhanced as a consequence of disorder scattering. For collisions mediated by the Coulomb interaction, impurities give rise to a nonmonotonic dependence of the Lorenz ratio on the screening wave number with a minimum for intermediate screening strength. To help future experimental efforts, we establish a scheme to connect the exact results with the solution of the Boltzmann equation under the relaxation time approximation for all collision integrals. Our recipe provides simple phenomenological expressions for the transport coefficients and it allows for a physically transparent interpretation of the results.

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    • Received 30 September 2020
    • Revised 18 January 2021
    • Accepted 11 March 2021

    DOI:https://doi.org/10.1103/PhysRevB.103.115140

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Carrier dynamicsElectrical conductivityElectron thermal conductivityTransport phenomena
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Woo-Ram Lee1,2, Karen Michaeli3, and Georg Schwiete1

    • 1Department of Physics and Astronomy, The University of Alabama, Tuscaloosa, Alabama 35487, USA
    • 2Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA
    • 3Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel

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    Vol. 103, Iss. 11 — 15 March 2021

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    Images

    • Figure 1
      Figure 1

      The electron and hole bands for the compensated metal studied in this paper.

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    • Figure 2
      Figure 2

      Transport coefficients in the presence of Hubbard-like interactions as a function of temperature for different strengths of impurity scattering: The electric conductivity in units of σ0=Ne2(τei,1/m1+τei,2/m2) is shown in (a) and the thermal conductivity in units of κ0=(π2NT/3)(τei,1/m1+τei,2/m2) is plotted in (b). We used the following values for the calculation of the exact expressions for the transport coefficients: u11=1.29, W̃11=W̃22=1, W̃12/W̃11=W̃21/W̃22=2, m2/m1=1.5, and τei,2/τei,1=10. The red, blue, and black curves correspond to 1/εFτei,1 = 0.1, 0.01, 0.001, respectively. The dotted lines indicate the solutions to the phenomenological RTA approach, Eqs. (20), (24), and (25), by matching the relaxation times as described in the main text. The poor fit to κ illustrates the different effect that interband scattering has on the electric and thermal transport (see discussion in the main text).

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    • Figure 3
      Figure 3

      Transport coefficients in the presence of Hubbard-like interactions as a function of temperature for different strengths of impurity scattering: The parameters used for the plots are identical to those in Fig. 2. Here, however, the dotted lines indicate the solutions to the phenomenological RTA approach with two independent interband-scattering times as given by Eqs. (55) and (56). The ambipolar contribution has been dropped for comparison. The Lorenz ratio L(T) in units of L0=π2/(3e2) is also displayed.

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    • Figure 4
      Figure 4

      Transport coefficients for Hubbard-type interactions as a function of temperature for different strengths of interband scattering. The parameters used for the exact solutions are γei,1/εF=0.01, γei,2/εF=0.001, u11=1.29, and W̃11=W̃22. The black, blue, and red curves correspond to W̃12/W̃11=W̃21/W̃22=0.1,1,10, respectively. We consider bands of equal masses (solid line) and bands with different masses m2/m1=1.5 (dashed lines). The dotted lines illustrate the RTA expressions for the transport coefficient (with m1=m2).

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    • Figure 5
      Figure 5

      Transport coefficients in the presence of Hubbard-like interactions as a function of temperature for different strengths of impurity scattering. Parameters are chosen as in Fig. 3. In contrast to Fig. 3, the results of the phenomenological RTA approach (dotted lines) include the ambipolar contribution.

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    • Figure 6
      Figure 6

      Transport coefficients for the screened Coulomb interaction as a function of temperature for different screening wave numbers. Here, we set γei,1/εF=0.01, γei,2/εF=0.001 and m1=m2. The black, blue, and red colored curves correspond to kTF/pF=0.1,0.4,4, respectively. The dotted lines are the RTA expressions that have been obtained by matching scattering times in the clean forward scattering limit as explained in the main text.

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    • Figure 7
      Figure 7

      Transport coefficients for the screened Coulomb interaction as a function of kTF for different temperatures. We use here the same parameters as in Fig. 6. In all panels, black, blue, and red curves correspond to T/εF=0.05,0.1,0.2, respectively. The solid lines indicate the exact solution, while the RTA results are given by the dotted lines. We found the phenomenological parameters entering the RTA calculation by taking the limit of small kTF. Thus, the dotted curves start to strongly deviate from the solid lines as kTF/pF→1.

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    • Figure 8
      Figure 8

      Lorenz ratio in the presence of screened Coulomb interaction as a function of kTF for varying disorder scattering strengths. For all curves m1=m2, T/εF=0.05, and γei,1=γei,2. The black, gray, blue, purple, pink, red, and magenta lines correspond to γei,c/εF={1,4,8,12,16,20,24}×10−4, respectively. The dotted lines represent the RTA solutions in the limit of small k̃TF.

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    • Figure 9
      Figure 9

      Lorenz ratio in the presence of screened Coulomb interaction as a function of temperature for varying disorder scattering strengths. Here, we set m1=m2, kTF/pF=0.5, and γei,1=γei,2. The black, gray, blue, purple, pink, red, and magenta curves correspond to γei,c/εF=(1,4,8,12,16,20,24)×10−4, respectively.

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