Towards a unified description of metallic transport

Qikai Guo,1 Cesar Magen,2, 3 Marcelo J. Rozenberg,4 and Beatriz Noheda1, 5, ∗

1Zernike Institute for Advanced Materials, University of Groningen, The Netherlands2Instituto de Nanociencia y Materiales de Aragon (INMA),

CSIC-Universidad de Zaragoza, 50009 Zaragoza, Spain3Laboratorio de Microscopıas Avanzadas (LMA),Universidad de Zaragoza, 50018 Zaragoza, Spain

4Universite Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France5CogniGron center, University of Groningen, The Netherlands

Understanding metallic behaviour is still oneof the central tasks in Condensed Matter Physics.Recent developments have energized the inter-est in several modern concepts, such as strangemetal [1, 2], bad metal [3, 4], and Planckian metal[5–7]. However, a unified description of metal-lic resistivity applicable to the existing diversityof materials is still missing. Herein we presentan empirical analysis of a large variety of metals,from normal metals to strongly correlated met-als, using the same phenomenological approach.The electrical resistivity in all the cases follows aparallel resistor formalism [8], which takes bothT -linear and T -quadratic dependence of the scat-tering rates into account. The results reveal thesignificance of the model by showing that the dif-ferent metallic classes are determined by the rela-tive magnitude of these two components. Impor-tantly, our analysis shows that the T -linear termarises from the Planckian dissipation limit andit is present in all considered systems. This for-malism extends previous reports on strange andnormal metals [6, 9], facilitating the classificationof materials with non-linear resistivity curves, animportant step towards the experimental confir-mation of the universal character of the Planckiandissipation bound.

Interactions of electrons with other (quasi-)particles(e.g. phonons, magnons or electrons themselves) areresponsible for the electrical transport of metallic sys-tems. In simple metals, electron-electron interactionslead to a Fermi liquid description [10] of the resistivityat low-temperature (T ) as a T 2 dependence; while a lin-ear increase of resistivity is usually observed at high-Tbecause of the boosted scattering strength between elec-trons and phonons. However, this well-defined regimemeets with problems in strongly correlated metals, ei-ther when the metallic system is driven close to a Quan-tum Critical Point, which gives rise to a ”strange metal”with a T -dependence of resistivity of the type T n, with1≤ n < 2) at low-T [11]; or when the increased scatter-ing drives the mean-free path (`) to approach the Mott-Ioffe-Regel (MIR) limit [12], which compels the resistiv-

∗ b.noheda@rug.nl

ity to show saturation at high-T [13]. However, in someso-called incoherent or bad metals, the resistivity over-comes this upper limit, implying such large scatteringrates that, according to Heisenberg’s principle, the un-certainty in the quasi-particles energy prevents their co-herence, thus, disqualifying the quasi-particle descriptionaltogether [4, 14].

Different conduction mechanisms become dominant atdifferent temperatures and, thus, an overall descriptionof metallic resistivity over a wide temperature range re-quires considering the combined effect of the various con-tributions in a phenomenological manner. T -linear resis-tivity has been typically associated to electron-phononscattering and, thus, such a dependence was not expectedat low temperatures. However, it is now well establishedthat T -linear resistivity can emerge well below the Debyetemperature in various systems, from simple metals tostrongly correlated metals, as long as the scattering rate(1/τ) per Kelvin of the charge carriers reaches a univer-sal bound, kB/~. This so-called ”Planckian dissipation”limit (PDL) is independent of the distinct behaviour andconduction mechanism [6]. These findings challenge thesignificance of a specific scattering mechanism in deter-mining ”strange” metallic transport and motivates thesearch for a phenomenological description that applies toa large variety of metals.

In Fig. 1a, we put together the ρ(T) curves of awide diversity of metallic systems. These systems in-clude cuprates with different doping levels, ruthenates,heavy fermions, alkali-doped C60, iron pnictides, tran-sition metals and monovalent metals. These materialshave been classified as simple metals, correlated metals,strange metals, bad metals or Planckian metals. Com-pared with the slowly increasing resistivity of simple met-als (in the yellow region), the slope of ρ(T) at around300 K of the systems shown in the blue region (whichin most of cases is the maximum slope), is large, as ex-pected from strong electron scattering. Among them,under-doped cuprates and Rb3C60 are well-establishedbad metals [13]. The resistivity in these systems cancross the ρMIR limit at relatively low temperatures andapproach a value far beyond it at high temperatures, vi-olating the quasi-particle scenario.

Notably, most correlated metals are located in anintermediate region between the good metals and thebad metals, as shown in the blow-up plot of Fig. 1b.

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FIG. 1: Resistivity of various metallic systems. a, Compilation plot showing the resistivity of various metallicsystems. b, Blow-up plot of the intermediate region in a. The dashed line indicates the linear-T -resistivity slope of 1µ Ω cm; the arrow shows the MIR limit (0.7 mΩ cm) of La2-xSrxCuO4. (Data source: Cu, Nb [15]; Al, Co, and Pd

[16]; La2-xSrxCuO4 with x=0.17-0.33 [8], with x=0.04 and 0.07 [17]; Bi2Sr2Ca0.89Y0.11Cu2Oy [18]; Rb3C60 [19];YBa2Cu3O6.45 [20]; Sr2RuO4 [21]; CeRu2Si2 [22]; CrO2 [23]; VO2 [24]; SmNiO3 [25]; Nd0.825Sr0.175NiO2 [26];

Nd-LSCO (p=0.24) and Bi2212 (p=0.23) [9]; Ba(Fe1/3Co1/3Ni1/3)2As2 [27]; Sr3Ru2O7 [6].)

Interestingly, a number of these intermediate systemsremain unclassified, such as Sr2RuO4 [21] and CrO2

[23], which have been reported to possess propertiesof both conventional and bad metals. Other systems,like Sr3Ru2O7, Nd-LSCO (p=0.24), Bi2212 (p=0.23),and Ba(Fe1/3Co1/3Ni1/3)2As2, have been discussed asPlanckian metals [6, 9, 27]. However, regardless of theirdifferent origin and classification, the resistivity of all themetallic systems in the intermediate region show manycomparable features. For instance, the maximum slopeof resistivity in most of materials is well below an up-per limit of 1 µΩ cm/K. The same limit has been re-ported in high-T c cuprates and was associated with themomentum-averaged scattering rate (~/τ ∼ πkBT) [28],which corresponds to the PDL. As mentioned earlier, thisconcept has been put forward as the common origin of thelinear-T -resistivity in systems with very different scat-tering mechanisms, including high-T c superconductors,other electron-correlated systems and even simple met-als [6].

In an effort to unify the behaviour of the differentmetallic systems, we follow the seminal work of N. E.Hussey et al. in high-T c cuprates [8], showing that theresistivity of La2-xSrxCuO4 at various doping levels canbe successfully described by a parallel resistor formalism[29]:

ρ(T )−1 = ρideal(T )−1 + ρsat−1 (1)

where ρideal is the resistivity in the absence of saturation,which is shunted by the large value of ρsat at high tem-peratures. An adequate definition of ρideal is then needed

in order to describe ρ(T) in a wide temperature range. Adual-component model, with linear and quadratic terms,has been used in the cuprates [8, 30] as:

ρideal(T ) = ρ0 +A1T +A2T2 (2)

where ρ0 represents the residual resistivity. Generalizingthis to other types of metals implies that A1 takes eitherelectron-phonon or the Planckian limit behaviour intoaccount and that A2 (approximately) describes higherorder contributions (Fermi liquid or others).

Here we show that this dual-component parallel-resistor formalism (DC-PRF) can describe the metallicbehaviour of very distinct systems independently of thedominant scattering mechanism. The DC-PRF has beenused to fit the remarkable variety of different resistivitydata shown in Fig. 1, from the bad metals to the goodmetals. As shown in Fig. 2a-c and Supplementary in-formation, the electrical resistivity of all these materialscan be well described by Eqs. 1-2.

This analysis provides us with a unified view of metallicbehaviour. As shown in Fig. 3a,b, the fitting of resistiv-ity to all those metallic systems, reveals a clear evolutionof coefficient A1 and A2 as a function of their room tem-perature resistivity (ρ300K). Interestingly, the data in-cludes NdNiO3 (NNO), which we consider in the presentwork as both, a test case and illustration of the utility ofthe formalism. In fact, this compound is attracting sig-nificant attention, since superconductivity was reportedin the infinite layer system Nd1−xSrxNiO2 [31], and is aremarkable example of a material whose metallic behav-ior has been particularly difficult to classify [25, 32, 33].

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FIG. 2: Fitting of the electrical resistivity of various metallic systems using Eq. 1. a, Underdopedcuprate; b, Ruthenates; c, Simple metals; d, Nickelate (NdNiO3). The triangle in d indicates the temperature abovewhich the data deviates from a T -linear dependence. Data source: resistivity of NdNiO3 is measured in the present

work; while data of other systems are extracted from Refs. [6, 16, 17, 21], respectively.)

For the present study we have used high quality epitax-ial NNO films grown on LaAlO3 substrates, which havebeen characterized in detail in our previous work [34, 35].High-angle annular dark field (HAADF) STEM imageshown in Supplementary Section 1 demonstrates the highcrystalline quality of the NNO films with an atomicallysharp interface with the substrate. In this 10 nm film,a first-order metal-to-insulator transition happens below150 K. Measurements of resistivity in an extended tem-perature range allow for clear determination of the T -dependence in the metallic state.

As shown in Fig. 2d, a cuprate-like linear-T -resistivityis observed in NNO in a ultra-wide temperature range(about 400 K). In our previous works, we showed thatthis T -linear behaviour of resistivity can be achieved inoptimized NNO films with low epitaxial strain and lowoxygen vacancy content [34] and, more interestingly, ithas signatures of Planckian dissipation [35]. With thefurther increase of T, the rise of ρ(T) shows an obviousdeviation from the linear dependence, which is caused bythe addition of a parallel saturation resistance that takesover the behaviour in the high temperature regime [4,13, 29]. Moreover, as in NdNiO3 strong electron-electroninteractions are expected, the combined effect of all thesecontributions should be considered. Indeed, we can show

that the metallic resistivity of the NdNiO3 film over atemperature range of 600 K can be well fit with the DC-PRF of Eqs. 1-2.

One of the interesting features unveiled in Fig. 3 isthat the increase of A1 saturates at a maximum value∼ 1 µΩ cm/K, which we have previously discussed inrelation to the definition of the intermediate region of Fig.1a. However, the DC-PRF allows to extract the linearcontribution to resistivity in a wider variety of metallicsystems and in a wider temperature range. In this way,we find that the upper limit is, actually, obeyed by allthe correlated systems, even in those well-established badmetals.

The same A1 ∼1 µΩ cm/K limit has been reported inhigh-T c cuprates [28] and is associated with the PDL.Indeed, we notice that the extracted A1 from thosestrange metals (inside the orange-encircled region) ap-proximately approaches this upper limit. Despite beingderived for simple and isotropic Fermi surfaces, one canuse the Drude formula of conduction to estimate that theuniversal Planckian bound on dissipation (1/τ=kBT/~)to estimate A1=(m∗/n)(kB/e2~), which includes the car-rier density (n) and carrier effective mass (m∗) and, thus,it is system-specific. Therefore, due to their lower m∗/nratio (see Fig. 1c and Supplementary Information section

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FIG. 3: Evolution of extracted coefficients from the fit of ρ(T) in various metals. From the fit of ρ(T) invarious metals to the DC-PRF model of Eqs. 1-2, the coefficients a, A1 and b, A2 are obtained and plotted as a

function of the corresponding room temperature resistivity (ρ300K). The blue dashed line in a, indicates themaximum value obtained for A1 ∼ 1 µΩ cm/K. Error bars are also obtained from the fitting results. shadows are aguide-to-the-eye showing the general evolution of the coefficients. c, the m*/n ratio is plotted as a function of ρ300K,

showing a trend similar to A1. In the inset A1n/m∗ is shown to remain at, or slightly below, kB/(e2~), whichcorresponds the Planckian dissipation limit, also for the normal metals and bad metals. d, (ρsat) versus A2,

representing the electron correlations. Bad metals, recognized by their large ρsat, are shown to display the largest A2

(the quadratic term dominates the scattering in the widest temperature range); while in strange metals (encircled inorange in all the figures) is the linear term the one dominating at most temperatures (see section 5 in

Supplementary Information).

4), the simple metals are expected to display smaller A1

values than the correlated metals. Indeed, as shown inthe inset of Fig. 1c, the product A1 n/m

∗, which char-acterizes the PDL, confirms that such a limit is generallyobeyed [6]. Thus, our analysis shows that the Planckianbound is a significant contribution to the ρ(T) in all in-vestigated metals. Until now, the relevance of this limitin the energy dissipation of carriers has not been reportedto exist in systems that show non-linear-T resistivity.

In contrast, the quadratic A2 coefficient does not dis-play a bound and continues increasing to reach the largestvalues in bad metals (see Fig. 3b). The behaviour of A2

follows clear trends consistent with the general knowl-edge of the increasing correlations across different ma-terials classes, further supporting the significance of theapproach. The different behaviour of A1 and A2 indi-cates two independent scattering processes. Two distinctinelastic scattering channels have been reported in high-T c cuprates [8, 36], being one contribution conjectured to

arise from conventional transport theory, while the otherone is correlated with the Planckian dissipation [2]. Ouranalysis is consistent with that and suggests that it isstraightforward to generalize such a picture to a largevariety metallic systems.

The different behaviour of ρ(T) in various systems ismainly determined by the relative strength of these twoprocesses. The relative importance of these two termscan be assessed by the magnitude of T *= A1/A2, that isthe temperature at which the linear and quadratic termsin ρ(T) become equal [37]. Section 5 of the Supplemen-tary Information shows ρsat as a function of T * evidenc-ing that the bad metals show the smallest values of T *;while the strange metals show the largest T *. A low(high) T * corresponds to a highly enhanced (depressed)A2, since A1 is of the same order for correlated metals.

Interestingly, in most of the investigated bad metals,ρo (obtained from the DC-PRF fits) is also significantlylarger than in other metals (see Supplementary Informa-

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tion Section 6), confirming the widespread notion thatbad metals are dirty metals[38]. This fact, next to thelow T * values, is responsible for the increased ρsat val-ues that define bad metals, see Fig. 3d. On the otherhand, the strange metals have a significantly smaller A2,compared to bad metals, and a similar A1, which givesrise to a strong decrease in the leading to a ρ(T) actionof the quadratic contribution, appearing linear in a widetemperature range.

Fig. 3d shows that metals can be primarily sorted inthree main classes according to their A2 and ρsat val-ues (low, intermediate and large) corresponding to nor-mal, correlated and bad metals, respectively. The strangemetals belong to the intermediate A2 regime, which hasthis parameter in the narrow range of [5×10−7 - 5×10−6]mΩ cm/K2. We have discussed the difficulties in theclassification of some system such as Sr2RuO4, CrO2 orNdNiO3. Our analysis shows that these materials be-long to the intermediate A2 regime of correlated mate-rials and are quite far from the values of bad and goodmetals. Interestingly, these three materials happen to beunexpectedly clean, according to the ρo obtained fromthe DC-PRF analysis, which are comparable to those ofthe simple metals, as shown in section 5 of the Supple-mentary information.

To conclude, the customary classification of metals asnormal, bad or strange, runs short to describe the com-plexity of electron correlated systems, often leading tocontroversial conclusions. The Hussey formalism consist-ing of T -linear (A1) and T - quadratic (A2) componentsadded to the residual resistivity, in parallel with a satura-tion term, can phenomenologically describe all observedbehaviours and provide a general framework to classifymetals in accordance to the relative magnitude of A2 andρsat. Generally, the saturation term represents the MIRlimit, while A2 describes the electron interactions. Theexception are the bad metals, for which the saturationterm overshoots the MIR limit, due to the combined ef-fect of a large ρo and an largely enhanced A2. However,regardless of their relative differences in these parame-ters, A1 is found to reach an upper bound, for sufficientlylarge (m∗/n) ratios. The clear link of this bound withthe Planckian dissipation limit supports its proposed uni-versality [6, 9], extending its scope to a larger number ofmetals and evidencing that all metals obey the Planckianconstrain.

I. ACKNOWLEDGEMENTS

We are indebted to Nigel Hussey, Jan Zaanen, ThomPalstra and Francisco Rivadulla for insightful discussions.We are grateful to Jacob Baas, Arjun Joshua and HenkBonder for their invaluable technical support. Qikai Guoacknowledges financial support from a China ScholarshipCouncil (CSC) grant and we both acknowledge finan-cial support from the Ubbo Emmius Funds (Universityof Groningen).

II. SUPPLEMENTARY MATERIAL

A. Methods

High quality epitaxial NdNiO3 thin films were de-posited on single-crystal LaAlO3 (LAO) substrates bypulsed laser ablation of a single-phase target (ToshimaManufacturing Co., Ltd.). Before deposition, the LAOsubstrates were thermally annealed at 1050 C underflowing O2 and etched by DI water to obtain an atom-ically flat surface with single-terminated terraces. Dur-ing the thin film deposition, the substrates were heatedto a temperature of 700 C and then the ablated ma-terial from the plume was uniformly deposited on thesubstrates. The oxygen pressure in the chamber duringdeposition was 0.2 mbar and the laser fluence on the tar-get was 2 J/cm2 with a laser frequency of 1 Hz. Thedistance between target and substrate was kept at 52.5mm during the deposition. After deposition, the sam-ples were cooled down to room temperature at 5 C/minunder a high oxygen pressure (900 mbar) to avoid theformation of oxygen vacancies in the lattice.

Following thin film growth, the structural and trans-port properties of all films were studied in detail. X-ray diffraction (XRD) have been performed by meansof a Panalytical Xpert MRD-Thin film diffractometer.For high-T measurement of XRD, a domed hot stagewith thermal control unit (DHS1100) was employed.Cross-sectional specimens of the films were prepared andstudied by scanning transmission electron microscopy(STEM) on a probe corrected FEI Titan 60–300 micro-scope equipped with a high-brightness field emission gun(X-FEG) and a CEOS aberration corrector for the con-denser system. This microscope was operated at 300 kV.Z contrast images were collected by High angle annulardark field (HAADF) STEM. A convergence angle of 24mrad was used to provide a probe size below 1 A. An-nular bright field (ABF) STEM images were acquired tovisualize the oxygen lattice of the nickelate film. For thispurpose, the same convergence angle of 25 mrad and anannular detector collected the scattered electrons in anapproximate angular range of 12-25 mrad.

The temperature dependence of resistivity for temper-atures below room temperature were carried out in avan der Pauw geometry using a Quantum Design Physi-cal Property Measurement System (PPMS), while thosemeasurements above room temperature were performedon a probe station with an Instec, Inc. heating stage. ACimpedance measurements were performed in the sameprobe station using a LCR Agilent E4980.

B. Characterization of NdNiO3 films

The crystalline structure of the NdNiO3 film was stud-ied by atomic resolution scanning transmission electronmicroscopy (STEM). A high-angle annular dark field(HAADF) STEM image shown in Fig. S1a indicates a

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high crystalline quality in the NNO films with atomi-cally sharp interface with the substrate. No misfit dis-locations or other common defects such as Ruddlesden-Popper (RP) faults, which are known to form in nicke-lates in the presence of excess A-cations or oxygen va-cancies, are observed in the lattice.

The annular bright field (ABF) image displayed in Fig.S1b provides further evidence of the octahedral oxygenrotations of the NNO orthorhombic phase by direct visu-alization of the oxygen lattice along the (110) orientation.

C. Fit to various metals

In the present work, a parallel resistor formalism [8]:1/ρ(T )=1/(ρ0+AT+BT 2)+1/ρsat, was employed to de-scribe the electrical resistivity of NdNiO3 film. For com-parison, the resistivity data of various metallic systemsfrom simple metals to strongly correlated metals wereextracted from previous works. In the following, we willdiscuss in detail the fits to all the materials using thesame approach.

1. La2-xSrxCuO4

A systematical analysis had been performed on theelectrical resistivity of La2-xSrxCuO4 by Cooper et al.[8]. Herein, we extracted the resistivity data from ref.[8] and performed the fit in the same way as describedin the main text for the NdNiO3 film. It is worth toemphasize that the ρsat in our fit is kept as a free pa-rameter; while a fixed value (900 µΩ cm) of ρsat is em-ployed in Cooper’s work. As shown in the figures below,the resistivity curves of all the reported doping levels byCooper et al. are well reproduced by our analysis. Theparameters used in the fit are shown in the inset of eachfigure. Interestingly, we found that the extracted ρsatfrom the fits show a decrease with increasing hole dopinglevels. For those with overdoping (p < 0.18), the valuesof ρsat are almost comparable with the fixed value usedby Cooper et al. However, the fits corresponding to thesamples with optimized doping (p=0.17 and 0.18), whichshow strange metal-like linear-T resistivity, give rise toa significantly larger ρsat. The measured resistivity atthese doping levels is far below the predicted ρsat fromthe fit, which is well consistent with the behaviour ofstrange metals.

The extracted A1 and A2 as a function of hole dopinglevels is shown in Figure S9. Notably, both marked kinks(solid dashed line) in A1 and A2 (corresponding to the α1

and α2 in their work, respectively) and also the evolutionof these two coefficients with hole doping, reported byCooper et al., are well reproduced by our fit. We believethat all these features demonstrate a reliability of ourapproach.

2. Bad metals

Bad metals are characterized by their large resistivity,which can increase across the predicted Mott-Ioffe-Regel(MIR) limit even at low temperatures. In the presentwork, resistivity data of four different systems, suchas underdoped La2-xSrxCuO4, Bi2Sr2Ca0.89Y0.11Cu2Oy,Rb3C60, YBa2Cu3O6.45, were extracted from therefs.[17–20] respectively. Among them, the re-sistivity of La1.96Sr0.04CuO4, La1.93Sr0.07CuO4, andBi2Sr2Ca0.89Y0.11Cu2Oy show obvious saturation withvalues well above the MIR limit. However, both Rb3C60

and YBa2Cu3O6.45 display a continuously increased re-sistivity in the whole investigated temperature rangewith no sign of saturation. Despite of this different per-formance, the fit with the parallel resistor model showsto describe the experimental data adequately over a widetemperature range.

3. Other correlated metals

• Sr2RuO4

An extended measurement of resistivity inSr2RuO4 had been performed by Tyler et al. [21].This material displays an interesting case amongstrongly correlated metals. Sr2RuO4 has beenproven to be a very good metal at low tempera-ture, following to a Fermi-liquid quasi-particle sce-nario. However, the increase of resistivity at high-Tshows no sign of saturation at the Mott-Ioffe-Regellimit, invaliding the quasi-particle description. In-deed, the fit of ρ(T) extracted from ref. [21] gave asaturation resistivity above 7.4 µΩ cm, which is farbeyond its calculated MIR limit (∼ 0.2 µΩ cm).

• CeRu2Si2

CeRu2Si2 is well known as a canonical heavyfermion compound. This material is characterizedby its rather large value of specific heat at low-Tand a metamagnetic-like transition [46]. The re-sistivity data of CeRu2Si2 analysed here were ob-tained from the work of Besnus et al. [22].

• CrO2

The resistivity data of CrO2 were extracted fromRef. [23]. CrO2 has also been discussed as badmetal. It shows signs of saturation but at highervalues than that predicted by the MIR limit. TheDC-PRF fit for this material is significantly worsethan for other systems. We have also found thatdifferent sources in the literature show different be-haviour [43–45] so the intrinsic temperature depen-dence of resisitivity in this material still needs to bedetermined.

• VO2

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FIG. S1: Characterization of NdNiO3 film. a Cross-sectional HAADF-STEM images of a 10-nm-thick NNOfilm grown on a LAO (001) substrate observed along a (100) crystal orientation. b Cross-sectional ABF-STEM

image with inverted contrast of the same film observed along the (110) crystal orientation where octahedra oxygenrotations of the orthorhombic NNO crystal structure can be observed.

FIG. S2: Fitting of ρ(T) of La1.83Sr0.17CuO4.

FIG. S3: Fitting of ρ(T) of La1.82Sr0.18CuO4.

FIG. S4: Fitting of ρ(T) of La1.79Sr0.21CuO4.

FIG. S5: Fitting of ρ(T) of La1.77Sr0.23CuO4.

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FIG. S6: Fitting of ρ(T) of La1.74Sr0.26CuO4.

FIG. S7: Fitting of ρ(T) of La1.71Sr0.29CuO4.

FIG. S8: Fitting of ρ(T) of La1.67Sr0.33CuO4.

An extended measurement of resistivity up to 840K was performed in VO2 by Philip et al. [24].Above the metal-insulator transition temperature(∼ 350 K), the temperature dependence of resis-tivity in the metallic phase of VO2 is linear. Thecalculated mean-free-path at 800 K is only 3.3 A,manifesting unconventional behaviour.

FIG. S9: Extracted coefficients a A1 and b A2 from thefit to the resistivity of La2-xSrxCuO4 as a function of

hole dopping.

FIG. S10: Fitting of ρ(T) of La1.96Sr0.04CuO4. Datafrom Ref.[17].

• SmNiO3

In comparison with NdNiO3 studied in this work,SmNiO3 displays a higher metal-insulator transi-tion temperature, above 400 K. An extended mea-surement of metallic resistivity has been reportedby Jaramillo et al. [25]. Moreover, they revealed abad-metallic behaviour of this material on the ba-sis of electrical and optical conductivity measure-ments. Indeed, the fit to the resistivity data ex-tracted from the same work gave a saturation re-sistivity of ∼ 1.077 µΩ cm, which is obviously largerthan the predicted MIR limit (0.5 µΩ cm) of this

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FIG. S11: Fitting of ρ(T) of La1.93Sr0.07CuO4. Datafrom Ref.[17].

FIG. S12: Fitting of ρ(T) of Bi2Sr2Ca0.89Y0.11Cu2Oy.Data from Ref. [18].

FIG. S13: Fitting of ρ(T) of Rb3C60. Data from Ref.[19].

material.

• Nd0.825Sr0.175NiO2

Rare-earth nickelates have attracted renewed inter-ests since the discovery of superconductivity in the

FIG. S14: Fitting of ρ(T) of YBa2Cu3O6.45. Data fromRef.[20].

FIG. S15: Fitting of ρ(T) of Sr2RuO4. Data fromRef.[21].

FIG. S16: Fitting of ρ(T) of CeRu2Si2. Data from Ref.[22].

related infinite-layer compound (Nd1-xSrxNiO2) byLi et al. [31]. The data studied in the present workare extracted from the subsequent work from thesame authors [26]. In this work, Li et al. reportedthe phase diagram of Nd1-xSrxNiO2 infinite layerthin films grown on SrTiO3. In our present work,

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FIG. S17: Fitting of ρ(T) of CrO2. Data from Ref. [23].

FIG. S18: Fitting of ρ(T) of VO2. Data from Ref. [24].

FIG. S19: Fitting of ρ(T) of SmNiO3. Data fromRef.[25].

resistivity of Nd0.825Sr0.175NiO2 was studied as itdisplays the best superconductivity and lowest re-sistivity among different doping levels.

FIG. S20: Fitting of ρ(T) of Nd0.825Sr0.175NiO2. Datafrom Ref.[26].

4. Planckian metals

• Nd-LSCO and Bi2212

The linear-T resistivity in high-Tc superconductorswith optimized doping has been a major puzzlein condensed matter physics. In a systematicallystudy by Legros et al. [9], they revealed that theorigin of the T -linear resistivity in many differ-ent cuprates is associated with a universal Planck-ian dissipation. Herein, the resistivity data oftwo systems: Nd-doped La2-xSrxCuO4 (Nd-LSCO)with p=0.24 and Bi2Sr2CaCu2O8+δ (Bi2212) withp=0.23, measured under high magnetic field werealso extracted and fit with the parallel resistormodel. By applying a high magnetic field, thesuperconductivity transition is suppressed and theT -linear resistivity towards absolute zero kelvin isobtained. Notably, the comparable parameters ob-tained from our fit in these two systems manifesta similar origin of T -linear resistivity in differentcuprates, which is well consistent with the conclu-sion of Legros et al.

FIG. S21: Fitting of ρ(T) of Nd-doped La2-xSrxCuO4

with p=0.24 (H=16 T). Data from Ref. [9].

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FIG. S22: Fitting of ρ(T) of Bi2Sr2CaCu2O8+δ

(Bi2212) with p=0.23 (H= 55 T). Data from Ref. [9].

• Sr3Ru2O7

As a magnetic-field-tuned quantum system,Sr3Ru2O7 can approach the quantum critical pointunder a critical field and, therefore, displays alow-T linear-dependence of resistivity [6]. Bruinet al.[6] demonstrated that the linear-T -resistivityof Sr3Ru2O7 under such conditions also arisesfrom the approach of the scattering rate to thePlanckian limit.

FIG. S23: Fitting of ρ(T) of Sr3Ru2O7. Data fromRef.[6].

• Ba(Fe1/3Co1/3Ni1/3)2As2

The resistivity data of Ba(Fe1/3Co1/3Ni1/3)2As2were obtained from the work of ref. [27].The linear dependence of resistivity in this non-superconducting iron pnictide system has also beenfound to obey a universal scaling relation betweentemperature and applied magnetic fields down tothe lowest energy scales [27].

• NdNiO3

FIG. S24: Fitting of ρ(T) of Ba(Fe1/3Co1/3Ni1/3)2As2.Data from Ref. [27].

FIG. S25: Fitting of ρ(T) of NdNiO3. Data frompresent work.

5. Simple metals

For comparison, the resistivity of several simple metalsis also included in this work. These simple metals hadalso been discussed as Planckian metals by Bruin et al.[6] The resistivity data of Cu and Nb was extracted fromthe work of Gunnarsson et al. [13]; while those of Al, Co,and Pd were obtained from Ref. [16]. Except for Nb, theresistivity of simple metals shows a linear-T dependenceat high-T . At low-T , a Fermi-liquid-like T 2 dependenceis observed.

However, these typical performance of resistivity is ab-sent in Nb. Among simple transition metals, Nb playsspecial role due to its high superconducting transitiontemperature. The resistivity of Nb shows a significantsaturation at ultra-high temperature. The fit to the ρ(T)of Nb gave a saturation resistivity about 100 µΩ cm,which is well consistent with that calculated by Gun-narsson et al. [13]. Moreover, as shown in Fig. 3c inmain text, the Nb shows a significantly larger values of α1

compared to α2. Interestingly, Nb has also been found toshow a more obvious deviation from the Planckian limit

12

in comparison with other simple metals [6]. We believethese anomalous behaviours of Nb are all attributed totheir large electron-phonon effects [42].

FIG. S26: Fitting of ρ(T) of Cu. Data from Ref. [13].

FIG. S27: Fitting of ρ(T) of Al. Data from Ref. [16].

FIG. S28: Fitting of ρ(T) of Co. Data from Ref. [16].

D. Effective mass and carrier density

Values of effective mass (m*) and carrier density (n)used in Fig. 3c and discussed in the main text are ob-

FIG. S29: Fitting of ρ(T) of Pd. Data from Ref. [16].

FIG. S30: Fitting of ρ(T) of Nb. Data from Ref. [13].

tained from refs. [9, 25, 39–41] and summarized in Table1. The last column shows A1n/m* in units of kB/~e2.For those metals with simple isotropic Fermi surfaces,A1n/m*= kB/~e2 at the Plackian dissipation limit [9].

E. Crossover temperature

We define T*= A1/A2 as the temperature at which thelinear term becomes equal to the quadratic term. Forsmall values of T* (bad metals), the quadratic term pre-vails for most temperatures, while for large T* (strangemetals), the linear term dominates up to (and beyond)room temperature. For the other metals, the crossovertemperature typically ranges between 70 K and 300 K.

F. Saturation resistivity and residual resistivity

13

TABLE I: Relevant parameters for different reported metals. Error bars for m* and n are obtained from therefs. [9, 25, 39–41]; while error bars in left two columns are calculated from the uncertainty of related parameters.

Material n(1027m-3) m*/m0 m*/n (m0/1027m-3) A1n/m* (kB/~e2)Bi2212 (p=0.23) 6.8 8.4 ± 1.6 1.2 ± 0.2 1.2 ± 0.3Nd-LSCO (p=0.24) 7.9 12 ± 4 1.5 ± 0.5 0.8 ± 0.3LSCO (p=0.26) 7.8 9.8 ± 1.7 1.3 ± 0.2 0.7 ± 0.2Rb3C60 3.9 ± 0.5 3.6 ± 0.5 0.9 ± 0.3 0.3 ± 0.2La1.96Sr0.04CuO4 3 ± 0.5 4 ± 0.5 1.3 ± 0.4 0.7 ± 0.2NdNiO3 10 ± 8 7 ± 1 0.7 ± 0.1 1.32 ± 1.1Cu 85 1.3 0.015 0.27 ± 0.1Nb 52 12 0.23 0.4 ± 0.2Al 60 1.4 0.023 0.6 ± 0.2

FIG. S31: Saturation resistivity (ρsat) versus T*= A1/A2. Bad metals, recognized by their large ρsat, areshown to display the lowest T* (largest A2, representing the largest efficiency of electron scattering). The same

symbols as in figure 3 of the main paper have been used.

14

FIG. S32: Saturation resistivity (ρSAT) as a function of residual resistivity (ρ0). Both parameters areextracted from the fit to resistivity data of each material, as discussed in section 3. The bad metals (encircled) show,as it is well known, the largest ρsat, which seem to originate from large ρ0 values in combination with the lowest T*

(the temperature at which the quadratic term takes over), mentioned in the main text. Interestingly, Sr2RuO4,CrO2 and NdNiO3, the three systems that have been discussed in the paper as notoriously difficult to classify, all

display very low ρ0 values, of the order of those of simple metals. The same symbols as in figure 3 of the main paperhave been used.

15

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