Investigation of Planckian behavior in a high-conductivity oxide: PdCrO2

Abstract

The layered delafossite metal PdCrO[Formula: see text] is a natural heterostructure of highly conductive Pd layers Kondo coupled to localized spins in the adjacent Mott insulating CrO[Formula: see text] layers. At high temperatures, T, it has a T-linear resistivity which is not seen in the isostructural but nonmagnetic PdCoO[Formula: see text]. The strength of the Kondo coupling is known, as-grown crystals are extremely high purity and the Fermi surface is both very simple and experimentally known. It is therefore an ideal material platform in which to investigate "Planckian metal" physics. We do this by means of controlled introduction of point disorder, measurement of the thermal conductivity and Lorenz ratio, and studying the sources of its high-temperature entropy. The T-linear resistivity is seen to be due mainly to elastic scattering and to arise from a sum of several scattering mechanisms. Remarkably, this sum leads to a scattering rate within 10[Formula: see text] of the Planckian value of k[Formula: see text]T/[Formula: see text].

Keywords: Kondo coupling; Planckian metal; T-linear resistivity; disorder.

Fig. 1.

(A) SEM image of a microstructured PdCoO${}_2$ sample mounted by the “epoxy-free” method. (B) SEM image of the PdCrO${}_2$ device mounted using the “free-standing” method. The measured regions of two devices appear to be darker in the SEM image because the gold is removed from them. The rest of the devices, Pt contacts, and substrate are covered with sputtered gold. The Pt contacts are slotted to decrease their spring constant and give the overall device enough mechanical flexibility to avoid the PdCrO${}_2$ fracturing during cool-downs.

Fig. 2.

In-plane resistivity data for the PdCrO${}_2$ microstructure shown in Fig. 1 (green) compared with data from previous work (36) on two single crystals (gray). The difference between the results from the two crystals is due to inevitable uncertainty in the geometrical factors used to convert resistance to resistivity when working with traditional hand-mounted silver epoxy contacts. Geometrical errors can be made much smaller with microstructures. Overall, the agreement between the three measurements is excellent. A T-linear resistivity is observed between approximately 150 K and the highest measurement temperature of 500 K.

Fig. 3.

(A) Increase of resistivity as a function of electron dose for two delafossite metals: PdCoO${}_2$ and PdCrO${}_2$. The samples were in a liquid hydrogen bath at a temperature of approximately 22 K. (B) Change to the in-plane resistivity of the PdCrO${}_2$ microstructure shown in Fig. 1, in its as-prepared state and then after two successive exposures to high-energy electron radiation with defect concentrations of 0.16$\%$ and 0.29$\%$, obtained using the dose calibration established in detail in ref. . The gradient of the $T$-linear resistivity is unchanged, within small experimental errors, although the residual resistivity is increased by over a factor of twenty.

Fig. 4.

(A) Thermal conductivity and (B) Lorenz ratio of PdCrO${}_2$ as a function of temperature, measured on a bulk single crystal of PdCrO${}_2$. For the data in panel B, the resistivity was measured using the same contacts as those used for the thermal conductivity, to eliminate geometrical uncertainties in $L$.

Fig. 5.

(A) The total heat capacity of magnetic PdCrO${}_2$ and nonmagnetic PdCoO${}_2$ as a function of temperature up to 390 K. (B) Density functional calculation for the vibrational heat capacity for nonmagnetic PdCoO${}_2$ and spin-polarized PdCrO${}_2$. (C) The entropy difference between the two compounds over the same range of temperatures (green), split into two components: the phonon part (purple) deduced from the calculated data of panel B (SI Appendix, Fig. S3) and the remainder, which we attribute to magnetic entropy not included in the density functional calculations. The latter comes very close to the expected value of Rln4 by 390 K.

Fig. 6.

Resistivity of magnetic PdCrO${}_2$ and nonmagnetic PdCoO${}_2$ from the microstructures studied in this work (green and purple lines) and the bulk crystals studied in ref. (gray lines), and the difference between the two (orange).