Phenomenological classification of metals based on resistivity

Qikai Guo, César Magén, Marcelo J. Rozenberg, and Beatriz Noheda

Phys. Rev. B 106, 085141 – Published 29 August 2022

Abstract

Efforts to understand metallic behavior have led to important concepts such as those of strange metals, bad metals, or Planckian metals. However, a unified description of metallic resistivity is still missing. An empirical analysis of a large variety of metals shows that the parallel resistor formalism used in the cuprates, which includes $T$ -linear and $T$ -quadratic dependence of the electron scattering rates, can be used to provide a phenomenological description of the electrical resistivity in all metals, where these two contributions are shown to correspond to the first two terms of a Taylor expansion of the resistivity, detached from their physics origin, and thus valid for any metal. Here, we show that the different metallic classes are then determined by the relative magnitude of these two components and the magnitude of the extrapolated residual resistivity. These two parameters allow us to categorize a few systems that are notoriously hard to ascribe to one of the currently accepted metallic classes. This approach also reveals that the $T$ -linear term has a common origin in all cases, strengthening the arguments that propose the universal character of the Planckian dissipation bound.

Received 14 October 2021
Revised 7 August 2022
Accepted 12 August 2022

DOI:https://doi.org/10.1103/PhysRevB.106.085141

Physics Subject Headings (PhySH)

Electrical conductivity Electrical properties

Metals

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Qikai Guo ^1,2,*, César Magén ^3,4, Marcelo J. Rozenberg ⁵, and Beatriz Noheda ^1,6,†

¹Zernike Institute for Advanced Materials, University of Groningen, 9747AG Groningen, The Netherlands
²School of Microelectronics, Shandong University, Jinan 250100, China
³Instituto de Nanociencia y Materiales de Aragón (INMA), CSIC-Universidad de Zaragoza, 50009 Zaragoza, Spain
⁴Laboratorio de Microscopías Avanzadas (LMA), Universidad de Zaragoza, 50018 Zaragoza, Spain
⁵Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France
⁶CogniGron Center, University of Groningen, 9747AG Groningen, The Netherlands

^*Corresponding author: qikaiguo@sdu.edu.cn
^†Corresponding author: b.noheda@rug.nl

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Issue

Vol. 106, Iss. 8 — 15 August 2022

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Images

Figure 1
Resistivity of various metallic systems. (a) Compilation plot showing the resistivity of various metallic systems. (b) Blow-up of the intermediate region in (a). The dashed line indicates the linear- $T$ -resistivity slope of 1 $μ Ω$ cm; the arrow shows the MIR limit (0.7 $m Ω$ cm) of ${La}_{2 - x} {Sr}_{x} {CuO}_{4}$ . [Data source: Cu, Nb [6, 10, 11, 12]; Al, Co, and Pd [13]; Pb [14]; ${La}_{2 - x} {Sr}_{x} {CuO}_{4}$ with $x = 0.17$ –0.33 [15], with $x = 0.04$ and 0.07 [16, 17]; ${Bi}_{2} {Sr}_{2} {Ca}_{0.89} Y_{0.11} {Cu}_{2} O_{y}$ [18]; ${Rb}_{3} C_{60}$ [19, 20]; ${YBa}_{2} {Cu}_{3} O_{6.45}$ [21]; ${Sr}_{2} {RuO}_{4}$ [22]; ${CeRu}_{2} {Si}_{2}$ [23, 24]; ${CrO}_{2}$ [25, 26, 27, 28]; ${VO}_{2}$ [29]; ${Nb}_{3} Sb$ [30]; ${SmNiO}_{3}$ [31]; ${Nd}_{0.825} {Sr}_{0.175} {NiO}_{2}$ [32]; Nd-LSCO ( $p = 0.24$ ) and Bi2212 ( $p = 0.23$ ) [33]; $Ba {({Fe}_{1 / 3} {Co}_{1 / 3} {Ni}_{1 / 3})}_{2} {As}_{2}$ [34]; ${Sr}_{3} {Ru}_{2} O_{7}$ [9].]
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Figure 2
Fitting of the electrical resistivity of various metallic systems using Eq. (1). (a) Underdoped cuprate. (b) Ruthenates. (c) Simple metals. (d) Nickelate ( ${NdNiO}_{3}$ ). The triangle in (d) indicates the temperature above which the data deviate from a $T$ -linear dependence. (Data sources: The resistivity of ${NdNiO}_{3}$ is measured in the present work, while the data of other systems are extracted from Refs. [9, 13, 16, 22], respectively.)
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Figure 3
Observed trends in the DC-PRF coefficients. From the fit of $ρ (T)$ in various metals to the DC-PRF model of Eqs. (1) and (2), the coefficients (a) $A_{1}$ and (b) $A_{2}$ are obtained and plotted as a function of the corresponding room-temperature resistivity ( $ρ_{300 K}$ ). The blue dashed line in (a) indicates the maximum value obtained for $A_{1} \sim 1 μ Ω$ cm/K. Error bars are also obtained from the fitting results. Encircled in yellow are the strange metals. Shadows are a guide to the eye showing the general evolution of the coefficients. (c) The $m^{*} / n$ ratio is plotted as a function of $ρ_{300 K}$ , showing a trend similar to $A_{1}$ . In the inset, $A_{1} n / m^{*}$ is shown to remain at, or slightly below, $k_{B} / (e^{2} ℏ)$ , which corresponds to the Planckian dissipation limit (PDL), also for the normal metals and bad metals. (d) Saturation resistivity ( $ρ_{sat}$ ) as a function of residual resistivity ( $ρ_{0}$ ). Encircled in blue are the bad metals.
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Figure 4
Key parameters for the general classification of metals. (a) Residual resistivity ( $ρ_{0}$ ) vs $T^{*} = A_{1} / A_{2}$ . Bad metals, recognized by their large $ρ_{0}$ , are shown to display the largest $A_{2}$ and the lowest $T^{*}$ (the quadratic term dominates the scattering in the widest temperature range), while in strange metals (encircled in orange in all the figures) it is the linear term that dominates at most temperatures. (b) $ρ_{0}$ vs $A_{2}$ , representing the magnitude of coherent contributions to the electron scattering rate.
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