Introduction

The interaction of an electron with a phonon plays a key role in emergent phenomena such as the polaron, charge density wave, and superconductivity. The electron-phonon interaction manifests itself, among others, in the ac-response of the material, including optical reflectance and dielectric functions1. In HTSC, the ab-plane optical conductivity exhibits an electronic continuum at the mid-IR range. Interestingly, for most HTSC compounds such as YBa2Cu3O7−δ2, Bi2Sr2CaCu2O83, and others4,5, a particular type of spectral feature, i.e., narrow dips or minima appear on top of the broad mid-IR band at multiple photon energies. In 1992, Reedyk and Timusk discovered that the minima are associated with optical phonons propagating along the c-axis of the lattice, specifically, longitudinal optical phonons. The unusual activation of the c-axis phonons in the ab-plane reflectivity, normally forbidden due to the momentum selection rule, results from the coupling of the in-plane electron with the c-axis LO phonons6. This electron-phonon interaction has drawn attention from the perspective of possible superconductivity pairing mechanisms. On the other hand, there has been a question as to whether a similar kind of interaction occurs in other layered metallic oxides as well. To the best of our knowledge, such material has not yet been reported.

The delafossite PdCoO2 consists of triangular Pd-planes that alternate with the CoO6 planes and are stacked along the c-axis. The in-plane electrical conduction occurs predominantly in the Pd-sheet7,8,9,10,11,12,13,14,15,16, giving rise to dc-conductivity σ = 3.8 × 105Ω−1cm−1 at room temperature17, which is, remarkably, higher than noble metals such as Au or Ag18. The mean free path of electrons is as long as 20 μm at low temperatures, making this material a promising candidate for hydrodynamic and other non-local transport studies19,20. An optical study by Homes et al. suggested that, importantly, the ab-plane electrons may couple with c-axis LO phonons in PdCoO221. This claim was based on two phonon-like peaks that are expected to be silent in the ab-plane reflectivity yet appear in the actual measurements. This interesting suggestion, however, was not supported by compelling experimental evidence.

In this work, we directly address this issue by performing optical measurements using a distinct approach from ref. 21: First, we probe both the ab-plane and the c-axis. For the ab-plane study, we employ a thin film, PdCoO2, instead of a single crystal. The latter has an extremely high reflection in the infrared range, which, as mentioned in ref. 21, poses difficulty in carrying out a quantitative analysis. Such a problem can be largely alleviated by using a PdCoO2 thin film for which reflectivity is significantly reduced. Additionally, a thin film allows for transmission measurements, which, when combined with the reflection, leads to precise optical dielectric functions. Second, for the c-axis study, we take advantage of a single crystal PdCoO2 in combination with a focused beam of microscopic Fourier transform infrared (FTIR) spectroscopy, which makes the optical measurement possible despite the limited sample dimension along the c-axis. Through the complementary studies on a thin film (for the ab-plane) and a single crystal (for the c-axis), we firmly establish that the ab-plane electrons of PdCoO2 couple with a longitudinal c-axis optical phonon. While the coupling in HTSC occurred between the (non-Drude) mid-infrared band and c-LO, it is the Drude carrier that couples in PdCoO2.

Results

Extended Drude analysis

Figure 1a shows the reflectance R(ω) and transmittance T(ω) of the PdCoO2 thin film for ℏω < 0.1 eV. We fit R(ω) and T(ω) simultaneously using the multilayer (film+substrate) analysis algorithm of the Kramers-Kronig (KK) constrained RefFit program22. The dielectric functions of the bare Al2O3-substrate were characterized separately and fed into the analysis. Figure 1b displays σ1(ω) and σ2(ω), the real and imaginary optical conductivity of PdCoO2, respectively, obtained from the fit at T = 10 K. They consist of an intra-band (Drude) response in the low-energy range and interband transitions at high energy ℏω > 0.8 eV. As PdCoO2 possesses only one conduction band as evidenced by experiments and theory9,10,11,13,17,23,24, the Drude conductivity represents the response of the single-band electron. When compared to the previous optical study on a single crystal PdCoO221, the interband transitions of our film are almost identical, whereas the Drude peak is notably broader. The latter is attributed to additional scatterings of the carrier at the twin boundary and the top / bottom surfaces of the film25. In the inset of Fig. 1a, we highlight that there is a distinct peak-like feature at ℏω = 90 meV in R(ω). We label it conveniently as Peak-Ω and will revisit it frequently later for data analysis. To add, \({\Omega }^{{\prime} }\) refers to a dip at a lower energy.

Fig. 1: Reflectance, transmittance, and optical conductivity of PdCoO2.
figure 1

a Reflectance and transmittance of a PdCoO2 thin film (thickness = 90 nm). The inset highlights that there is a peak around ~90 meV. b Real and imaginary parts of the optical conductivity. Inset depicts the wide-range σ1(ω) up to 4 eV, which consists of Drude component (red) and multiple interband transition peaks (colored).

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In Fig. 2, we show the scattering rate 1/τ(ω) and effective mass m*(ω) of the Drude carrier. They are calculated from the Drude σ1(ω) and σ2(ω) using the extended Drude analysis formula.

$$\frac{1}{\tau (\omega )}=\frac{{\omega }_{{{{\rm{p}}}}}^{2}}{4\pi }\frac{{\sigma }_{1}(\omega )}{{\sigma }_{1}^{2}(\omega )+{\sigma }_{2}^{2}(\omega )}$$
(1)
$$\frac{{m}^{* }(\omega )}{{m}_{{{{\rm{b}}}}}}=\frac{{\omega }_{{{{\rm{p}}}}}^{2}}{4\pi }\frac{{\sigma }_{2}(\omega )}{{\sigma }_{1}^{2}(\omega )+{\sigma }_{2}^{2}(\omega )}\frac{1}{\omega }$$
(2)
Fig. 2: Extended Drude analysis results.
figure 2

a Frequency-dependent scattering rate 1/τ(ω) and b mass enhancement m*(ω)/mb. They are calculated from the Drude conductivity through Eqs. (1) and (2), where ωp (= 28500 cm−1) was determined from \({\omega }_{{{{\rm{p}}}}}^{2}/8=\int\nolimits_{0}^{{\omega }_{{{{\rm{c}}}}}}{\sigma }_{{{{\rm{Drude}}}}}({\omega }^{{\prime} })d{\omega }^{{\prime} }\) with σDrude(ω) = σ(ω)-σInterband(ω) and the cutoff frequency ωc = 15000 cm−1. The notation Ω corresponds to the optical feature in the reflectivity, Peak-Ω in Fig. 1a. The spurious noise at ω = 450 cm−1 (dashed line) is caused by the substrate. (See Supplementary Fig. 2).

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The 1/τ(ω) increases markedly for ω > Ω and m*(ω) drops from the same frequency. At ℏω = 90 meV, there is a dispersive structure in 1/τ(ω) and m*(ω) that triggers the frequency-dependent changes. This structure originates from the Peak-Ω in R(ω). The frequency-dependent 1/τ(ω) and m*(ω) are the characteristic behavior of an electron-boson interaction: For a conducting material with an electron-boson interaction, 1/τ(ω) increases as ω exceeds the boson energy, and simultaneously, m*(ω)/mb begins to decrease from a dressed mass (>1) to bare band mass (=1)26,27,28,29,30,31. Figure 2 shows that an electron-boson coupling is occurring in PdCoO2 with a boson mode located at ω = Ω.

To ensure that the frequency-dependent 1/τ(ω) and m*(ω) are intrinsic properties of PdCoO2, we synthesized a pure Pd-thin film (thickness = 15 nm) using molecular beam epitaxy (MBE) on the same substrate (Al2O3) and performed the same optical measurements (R and T) and extended Drude analysis. The Pd-film is classified as a noble metal such as Au or Ag films. In contrast to PdCoO2, 1/τ(ω) and m*(ω) of the Pd-film are independent of frequency as expected for a simple Drude metal. (See Supplementary Fig. 1) This comparative study supports that 1/τ(ω) and m*(ω) in Fig. 2 are not artifacts caused by, for example, the Al2O3 substrate but are the genuine properties of PdCoO2. In addition, the Pd-film does not show Peak-Ω (and \({\Omega }^{{\prime} }\)) in R(ω), indicating that Peak-Ω (and \({\Omega }^{{\prime} }\)) is intrinsic to PdCoO2 as well. (See Supplementary Fig. 2). In Fig. 2, 1/τ(ω) and m*(ω) become considerably noisy for the range ω < Ω. The unwanted noises are caused mostly by the strong optical phonons of Al2O3, which we present in detail in Supplementary Fig. 2. Ideally, the substrate phonons should be isolated from the thin film during the data analysis (fitting). However, in practice, they are not perfectly removed, causing the noise. Importantly, Supplementary Fig. 2 demonstrates that, unambiguously, Peak-Ω arises not from the substrate but from PdCoO2. Given that Peak-Ω is driving the frequency-dependent 1/τ(ω) and m*(ω), it is crucial to unveil the origin of Peak-Ω in order to understand the electron-boson interaction of PdCoO2.

Polarization-dependent measurements and origin of Peak-Ω

In Fig. 3a, we measured the c-axis reflectance Rc(ω) of the PdCoO2 single crystal. To measure Rc(ω), a focused IR beam from a microscopic FTIR polarized along the c-axis (E ∥c) was illuminated on the side facet of the d ~ 100 μm-thick single crystal. In this manner, reproducible data were obtained for ω > ~500 cm−1. The Rc(ω) shows a prominent structure at ℏω = 90 meV and a minor one at 70 meV. The wide-range Rc(ω) (inset) reveals an insulating behavior of the c-axis, which contrasts sharply with the metallic R(ω) of the ab-plane. We fit the Rc(ω) using the KK-constrained RefFit and calculated the complex c-axis optical conductivity σc(ω) and dielectric constant εc(ω). In the fit, we constrained σc(ω) to match the c-axis dc-conductivity at ω = 032. In Fig. 3b, we show the real part of σc(ω). σc1(ω) is peaked at ℏω = 730 cm−1 and 570 cm−1, which represents two transverse optical phonons (TO) of the c-axis. These c-TO phonons propagate along the ab-plane, q ∥ab. We also calculate the dielectric loss function Im[−1/εc(ω)], which shows two peaks Ωc and \({\Omega }_{{{{\rm{c}}}}}^{{\prime} }\) representing the c-axis LO phonons. They propagate along q ∥c. Note that, remarkably, Ωc is very close to Peak-Ω, suggesting that it is a possible source of Peak-Ω. The c-TO phonon (q ∥ab), which is also close to Peak-Ω, is not excited in the normal-incidence thin film measurements (q ∥c) in Fig. 1, thus cannot create Peak-Ω. As for \({\Omega }_{{{{\rm{c}}}}}^{{\prime} }\), its energy is close to the dip \({\Omega }^{{\prime} }\) of Fig. 1.

Fig. 3: Optical conductivity and dielectric loss function of the c-axis.
figure 3

a Reflectance Rc(ω) measured with the light polarized as E ∥c on the PdCoO2 single crystal. Inset shows the wide-range Rc(ω) up to 1.8 eV. b The c-axis optical conductivity and dielectric loss function. Inset shows that the light propagates along the ab-plane, q ∥ab, and E-field is polarized along the c-axis, E ∥c. Here Ωc and \({\Omega }_{{{{\rm{c}}}}}^{{\prime} }\) denote the two peaks of the Im[−1/εc(ω)].

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To confirm the presumption that Peak-Ω originates from Ωc, we perform further polarized reflectance measurements. In Fig. 4a, incident light propagates along the c-axis while the electric field is parallel to the ab-plane. This optical configuration (q ∥c, E ∥ab) can activate the ab-plane TO and the c-axis LO phonon. In Fig. 4b, we employed a different optical configuration q ∥a and E ∥b, which activates the b-TO but not the c-LO. (To note, we use ‘a’ and ‘b’ to represent two orthogonal axes of the ab-plane but they do not indicate any specific crystallographic directions. The terms a-TO and b-TO are equivalent to ab-TO.) The R(ω) in Fig. 4d, e show that Peak-Ω is activated in Fig. 4a but is absent in Fig. 4b, demonstrating that ab-TO is excluded from the source of Peak-Ω, thus leaving the c-LO (Ωc) the only remaining candidate. The ab-TO phonons are another possible source of Peak-Ω but, according to ref. 21, they are far from Peak-Ω. In general, a c-axis optical phonon of a layered material does not appear in the ab-plane reflectivity due to forbidden symmetry. In PdCoO2, however, Ωc manifests itself in the ab-plane reflectivity as a result of coupling with the ab-plane Drude carrier. This coupling leads to the frequency-dependent 1/τ(ω) and m*(ω) of Fig. 2.

Fig. 4: Polarization-dependent reflectance of PdCoO2.
figure 4

Polarization-dependent reflectance of PdCoO2. a Light propagates along q ∥c and E-field is unpolarized. b Light propagates along q ∥a(b) and E-field is polarized along E ∥b(a). c s- and p-polarized lights are incident at an incidence angle θ = 10°. The reflectance data of (a–c) are shown in (d–f), respectively. The thin film was used for (a) and (c), and the single crystal was used for (b).

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In our near-normal (θ = 10°) reflectance measurement, incident light contains a small E ∥c component, which may cause the c-axis phonons to leak into the ab-plane reflectivity. In this case, Peak-Ω may appear in R(ω) even if the electron-phonon coupling was absent. To test if this is the case for Fig. 4d, we measured R(ω) using the s- and p-polarization as shown in Fig. 4c: In the s-polarization, the light has no E ∥c component, whereas the p-polarization does have a finite E ∥c component. The R(ω) in Fig. 4f shows that Peak-Ω is activated in the s-polarization with similar strength as in the p-polarization. This result rules out the leakage scenario of Peak-Ω. To reinforce our conclusion, we theoretically calculated the grazing-incidence R(ω) at incidence angles θ = 10° and 20°. For this, we used the ab-plane and c-axis dielectric functions measured in Figs. 1 and 3, respectively. The calculation results, shown in Supplementary Fig. 3, reveal that at θ = 20°, the c-LO leaks into the ab-plane reflectivity in the p-polarization, giving rise to a peak with 5 × 10−4 in height. However, this peak height is far weaker than the actual height of Peak-Ω in Fig. 4f, 0.01. Furthermore, at the experimental angle θ = 10°, the calculated leakage becomes even smaller, and the peak is too weak to be detected. This observation supports again that the E-field leakage cannot account for Peak-Ω in R(ω). We thus conclude that the c-LO does couple with the ab-plane Drude carrier manifesting itself as Peak-Ω in the ab-plane reflectance.

Fano analysis of the a b-plane and the c-axis

We performed the Fano fit of the ab-plane optical conductivity employing the Fano expression33,

$${\sigma }_{F}(\omega )=i{\sigma }_{0}\frac{{(q-i)}^{2}}{i+\epsilon }$$
(3)

where \(\epsilon =({\omega }^{2}-{\omega }_{0}^{2})/\gamma \omega\), σ0, ω0, γ and q are the strength, frequency, width, and asymmetry of the phonon, respectively. In Eq. (3), the Fano asymmetry increases as ∣q∣ becomes smaller33,34,35. For the ab-plane, Fano fit to the Peak-Ω is shown in Fig. 5b. The fit yields q = −1. Here the fit is good at ω > 715 cm−1 but becomes poor ω < 715 cm−1 due to the strong phonon peaks of the substrate. The two symbols * in Fig. 5 indicate the positions of the substrate optical phonons (Supplementary Fig. 5), which coincide with the deviations from the fit. The c-axis LO phonon emerges as a peak in the loss function Im[−1/εc], as we showed in Fig. 3b and explained in the main text. We fit Im[−1/εc] using \(\varepsilon (\omega )={\varepsilon }_{\infty }+\frac{4\pi i}{\omega }\)σ(ω) and Eq. (3). Fitting result shows q = 8 as shown in Fig. 5b. This q is larger than the ∣q∣ = 1 of the Peak-Ω of the ab-plane, indicating that the electron-phonon interaction is weaker in the c-axis than in the ab-plane. To look for the origin of this result, we note that the electron-phonon coupling strength is determined from

$${\alpha }^{2}F(\omega )=N(0)\frac{{\sum }_{{{{\bf{k}}}},{{{{\bf{k}}}}}^{{\prime} }}w({{{\bf{k}}}},{{{{\bf{k}}}}}^{{\prime} })| {M}_{{{{\bf{k}}}},{{{{\bf{k}}}}}^{{\prime} }}{| }^{2}\delta (\hslash \omega -\hslash {\omega }_{{{{\bf{k}}}}-{{{{\bf{k}}}}}^{{\prime} }})\delta ({\xi }_{{{{\bf{k}}}}})\delta ({\xi }_{{{{{\bf{k}}}}}^{{\prime} }})}{{\sum }_{{{{\bf{k}}}},{{{{\bf{k}}}}}^{{\prime} }}w({{{\bf{k}}}},{{{{\bf{k}}}}}^{{\prime} })\delta ({\xi }_{{{{\bf{k}}}}})\delta ({\xi }_{{{{{\bf{k}}}}}^{{\prime} }})}$$
(4)

where \({M}_{{{{\bf{k}}}},{{{{\bf{k}}}}}^{{\prime} }}\) is the electron-phonon matrix element, N(0) the density of states at the Fermi level, \(w({{{\bf{k}}}},{{{{\bf{k}}}}}^{{\prime} })={({v}_{{{{\bf{k}}}},i}-{v}_{{{{{\bf{k}}}}}^{{\prime} },i})}^{2}\) (i referring to the i-th component) is the electron velocity weighting factor36,37. Given that PdCoO2 possesses the cylindrical Fermi surface that is open in the c-axis direction11, the coupling strength can be different for the ab-plane and the c-axis. When the frequency is close to the c-axis LO mode energy minimum, kz of the Drude carrier changes very little upon scattering due to the purely c-axis phonon dispersion. That this leads to the anisotropic electron-phonon coupling strength can be most easily shown for the simplified case of cylindrical Fermi surface38, where \({v}_{x}={v}_{F}\cos \phi\), \({v}_{y}={v}_{F}\sin \phi\), and \({v}_{z}={v}_{\parallel }\sin {k}_{z}\) with vF and v∥ independent of kz. If the c-axis LO dispersion is approximated as \({(\hslash {\omega }_{{{{\bf{q}}}}})}^{2}={E}_{1}^{2}(1-\cos {q}_{z})+{E}_{0}^{2}\) near its minimum energy E0 (=90 meV in our measurement), we have the inequality relation

$$\begin{array}{lll}{\alpha }_{\perp }^{2}F(\omega )&=&N(0)| M{| }^{2}\frac{\hslash \omega }{2\pi \sqrt{({\hslash }^{2}{\omega }^{2}-{E}_{0}^{2})(4{E}_{1}^{2}+{E}_{0}^{2}-{\hslash }^{2}{\omega }^{2})}}\\ > {\alpha }_{\parallel }^{2}F(\omega )&=&N(0)| M{| }^{2}\frac{\hslash \omega }{4\pi {E}_{1}^{2}}\sqrt{\frac{{\hslash }^{2}{\omega }^{2}-{E}_{0}^{2}}{4{E}_{1}^{2}+{E}_{0}^{2}-{\hslash }^{2}{\omega }^{2}}},\end{array}$$
(5)

where ‘⊥’ refers to the components perpendicular to the c-axis, i.e., the ab-plane, ‘∥’ to the component parallel to the c-axis. Here \({M}_{{{{\bf{k}}}},{{{{\bf{k}}}}}^{{\prime} }}\) is taken to be constant due to a strongly screened electron-phonon interaction of PdCoO2. This theoretical consideration supports the larger asymmetry (smaller q) of the c-LO phonon in the ab-plane than in the c-axis in Fig. 5.

Fig. 5: Fano asymmetry analysis.
figure 5

Fano fit to a the optical conductivity data of ab-plane and b loss function of the c-axis at 300 K. Fano asymmetric parameters are q = −1 for the ab-plane and q = 8 for the c-axis. The two symbols * in Fig. 5 indicate the positions of the substrate optical phonons, which coincide with the deviations from the fit. The green curve in (a) shows the Fano asymmetric optical phonon.

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Discussion

To compare PdCoO2 with HTSC, they are the two types of rare materials that exhibit the coupling of the ab-plane carrier with c-LO phonons. One major difference, however, is that the Drude carrier couples in PdCoO2, whereas it is the mid-IR band in HTSC6. Therefore, in the latter, c-LO does not influence the dc-transport. On the contrary, the c-LO of PdCoO2 may play a significant role in the ab-plane transport, such as the hydrodynamic charge flow. We emphasize that PdCoO2 is the first layered material in which the c-LO couples with the ab-plane Drude carriers.

To discuss \({\Omega }^{{\prime} }\), we examine if it arises from \({\Omega }_{{{{\rm{c}}}}}^{{\prime} }\) like Ω did from Ωc. For this, we compare the ab-plane σ1(ω) with the c-axis Im[−1/εc(ω)] in Supplementary Fig. 4 following a similar approach as in ref. 6. The Peak-\({\Omega }^{{\prime} }\) occurs at the same energy as \({\Omega }_{{{{\rm{c}}}}}^{{\prime} }\) but with a significantly narrower width. On the contrary, for the single crystal PdCoO2, an optical peak occurs in the ab-plane at the same energy and with similar width as \({\Omega }_{{{{\rm{c}}}}}^{{\prime} }\), supporting that the electron-phonon interaction persists for \({\Omega }^{{\prime} }\)21. In the thin film, σ1(ω) is highly uncertain in the region of \({\Omega }^{{\prime} }\), hindering precise determination of the spectral shape. To definitely establish the correlation with \({\Omega }_{{{{\rm{c}}}}}^{{\prime} }\), improved measurements that overcome the noise are needed. To further compare the thin film and single crystal results, we note that Peak-Ω has an asymmetric, Fano-like shape in σ1(ω) in both cases. However, they have the opposite Fano-asymmetry signs and different strengths of asymmetry (See Supplementary Fig. 4 and ref. 21). Such differences suggest that substantial thickness-dependent effects exist for the electron-phonon coupling. Lastly, while we focused primarily on electron-phonon coupling in this paper, our data may suggest that another type of interaction, such as the electron-electron interaction, may apply to PdCoO2 as well (See Supplementary Fig. 5).

In conclusion, we performed polarized infrared transmission and reflection measurements on a PdCoO2 thin film. In the ab-plane, the scattering rate 1/τ(ω) and effective mass m*(ω) of the Drude carriers increased and decreased for ω > Ω, respectively, driven by Peak-Ω. In the c-axis measurement on a single crystal, a longitudinal optical phonon was found at Ωc as evidenced by a peak of the loss function Im[−1/εc(ω)]. Further optical measurements employing different (q, E) configurations revealed that Peak-Ω is activated due to the interaction of the ab-plane Drude carriers with the c-LO phonon. This electron-phonon interaction leads to the frequency-dependent 1/τ(ω) and m*(ω). Our conclusion was established through the extensive supporting measurements on the pure Pd-film, bare Al2O3 substrate, and the s- and p-polarized grazing-incidence reflection calculations. The coupling of the ab-plane Drude electron with the c-LO phonon implies that c-LO may play a significant role in the characteristic ab-plane carrier dynamics of PdCoO2, such as the ultra-high dc-conductivity, phonon-drag, and hydrodynamic charge flow, which is worthy for further studies.

Methods

Sample growth and characterizations

Epitaxial PdCoO2 thin films (thickness = 90 nm) were grown on an Al2O3 substrate using the molecular beam epitaxy (MBE) technique and were characterized through various methods such as X-ray diffraction (XRD), reflection high energy electron diffraction (RHEED), transmission electron microscopy (TEM), etc.25. A 100 μm-thick high-quality single crystal was grown using the flux method and thoroughly characterized39,40.

Optical measurements

The ab-plane optical transmittance and reflectance in the infrared range were measured on the thin film samples (PdCoO2 + substrate) and the bare Al2O3 substrate using FTIR (Bruker Vertex 70v). The transmission and reflection power spectra were normalized by the blank and gold (Au), respectively, where the latter Au reference was coated on the sample using the in situ evaporation technique41. The results of these measurements are presented in Supplementary Fig. 7. A Spectroscopic Ellipsometer (J.A. Woollam VASE) was used to obtain the optical dielectric functions from 0.7 eV to 4 eV. The optical reflection of the c-axis was measured on a single crystal in combination with microscopic FTIR (Hyperion 2000). The a and b directions refer to a set of two orthogonal directions in the hexagonal plane, which is not aligned with respect to the crystal structure. For the c-axis reflectivity measurements, we carefully polished the side facet of the single crystal and then confirmed the surface quality by performing polarized reflectivity measurements, as presented in Supplementary Fig. 7.