Interaction of in-plane Drude carrier with c-axis phonon in $\rm PdCoO_2$

We performed polarized reflection and transmission measurements on the layered conducting oxide $\rm PdCoO_2$ thin films. For the ab-plane, an optical peak near $\Omega$ $\approx$ 750 cm$^{-1}$ drives the scattering rate $\gamma^{*}(\omega)$ and effective mass $m^{*}(\omega)$ of the Drude carrier to increase and decrease respectively for $\omega$ $\geqq$ $\Omega$. For the c-axis, a longitudinal optical phonon (LO) is present at $\Omega$ as evidenced by a peak in the loss function Im[$-1/\varepsilon_{c}(\omega)$]. Further polarized measurements in different light propagation (q) and electric field (E) configurations indicate that the Peak at $\Omega$ results from an electron-phonon coupling of the ab-plane carrier with the c-LO phonon, which leads to the frequency-dependent $\gamma^{*}(\omega)$ and $m^{*}(\omega)$. This unusual interaction was previously reported in high-temperature superconductors (HTSC) between a non-Drude, mid-infrared band and a c-LO. On the contrary, it is the Drude carrier that couples in $\rm PdCoO_2$. The coupling between the ab-plane Drude carrier and c-LO suggests that the c-LO phonon may play a significant role in the characteristic ab-plane electronic properties of $\rm PdCoO_2$ including the ultra-high dc-conductivity, phonon-drag, and hydrodynamic electron transport.

Introduction.-Theinteraction 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 functions [1].In HTSC, the ab-plane optical conductivity exhibits an electronic continuum at the mid-IR range.Interestingly, for most HTSC compounds such as YBa 2 Cu 3 O 7−δ [2], Bi 2 Sr 2 CaCu 2 O 8 [3], and others [4,5], 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 phonons [6].This electronphonon 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 to date.
The delafossite PdCoO 2 consists of triangular Pd-planes that alternate with the CoO 6 planes and are stacked along the c-axis.The in-plane electrical conduction occurs predominantly in the Pd-sheet [7][8][9][10][11][12][13][14][15][16], giving rise to dc-conductivity σ = 3.8 × 10 5 Ω −1 cm −1 at room temperature [17] which is, remarkably, higher than noble metals such as Au or Ag [18].The mean free path of electrons is as long as 20 µm at low-T, making this material a promising candidate for the hydrodynamic and other non-local transport studies [19].An optical study by Homes et al. suggested that, importantly, the ab-plane electrons may couple with c-axis LO phonons in PdCoO 2 [20].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. [20]: Firstly, we probe both the ab-plane and the c-axis.For the ab-plane study, we employ a thin film PdCoO 2 instead of a single crystal.The latter has an extremely high reflection in the infrared range, which, as mentioned in Ref. [20], poses difficulty in carrying out a quantitative analysis.Such a problem can be largely alleviated by using a PdCoO 2 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.Secondly, for the c-axis study, we take advantage of a single crystal PdCoO 2 in combination with a focused beam of microscopic FTIR, 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 PdCoO 2 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 PdCoO 2 .
Experiment.-Epitaxial PdCoO 2 thin films (thickness = 90 nm) were grown on an Al 2 O 3 substrate using the molecular beam epitaxy (MBE) technique and were thoroughly characterized through various methods such as XRD, RHEED, TEM, etc. [21].The ab-plane optical transmittance and reflectance in the infrared range were measured on the thin films using FTIR (Bruker Vertex 70v) in combination with the in-situ gold evaporation technique [22].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 3 on a 100 µm-thick high-quality single crystal grown using the flux method [23,24], 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.
Results.-Fig. 1 (a) shows the reflectance R(ω) and transmittance T(ω) of the PdCoO 2 thin film for ω < 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 program [25,26].The dielectric functions of the bare Al 2 O 3 -substrate were characterized separately and fed into the analysis.Fig. 1 (b) displays σ 1 (ω) and σ 2 (ω), the real and imaginary optical conductivity of PdCoO 2 , respectively, obtained from the fit at T = 10 K.They consist of an intra-band (Drude) response in the low-energy range and inter-band transitions at high energy ω > 0.8 eV.When compared to the previous optical study on a single crystal PdCoO 2 [20], the inter-band 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 film [21].In the inset of Fig. 1 (a) we highlight that there is a distinct peak-like feature at ω = 90 meV in R(ω).We label it conveniently as Peak-Ω and will revisit it frequently later for data analysis.To add, Ω refers to a dip at a lower energy.
At ω = 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 Note that, remarkably, Ω c 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 Ω c , its energy is close to the dip Ω of Fig. 1.
To confirm the presumption that Peak-Ω originates from Ω c , we perform further polarized reflectance measurements.In Fig. 4 (a) 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. 4 (b) we employed a different optical The R(ω) in Fig. 4 (a ) and (b ) show that Peak-Ω is activated in Fig. 4 (a), but is absent in  of Peak-Ω but, according to Ref. [20], 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 PdCoO 2 , however, Ω c 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.
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 were absent.
To test if this is the case for Fig. 4 (a ), we measured R(ω) using the s-and p-polarization as shown in Fig. 4 (c): 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. 4 (c ) 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 the calculated 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 Fig. 1 and Fig. 3, respectively.
The calculation results, shown in the Supplementary Fig. S3, 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. 4 (c ), 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.
Discussion.-To compare PdCoO 2 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 PdCoO 2 whereas it is the mid-IR band in HTSC [6].Therefore, in the latter, c-LO does not influence the dc-transport.On the contrary, the c-LO of PdCoO 2 may play a significant role in the ab-plane transport such as the hydrodynamic charge flow.We emphasize that PdCoO 2 is the first layered material in which the c-LO couples with the ab-plane Drude carriers.
To discuss Ω , we examine if it arises from Ω C like Ω did from Ω C .For this, we compare the ab-plane σ 1 (ω) with the c-axis Im[−1/ε c (ω)] in Fig. S4 following a similar approach as in Ref. [6].The Peak-Ω occurs at the same energy as Ω C but with a significantly narrower width.

FIG. 1 .
FIG. 1.(a) Reflectance and transmittance of a PdCoO 2 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 σ 1 (ω) up to 4 eV.

FIG. 3 .
FIG. 3. (a) Reflectance R c (ω) measured with the light polarized as E c on the PdCoO 2 single crystal.Inset shows the wide-range R c (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 Ω C and Ω C denote the two peaks of the Im[−1/ε c (ω)].

FIG. 4 .
FIG. 4. Polarization-dependent reflectance of PdCoO 2 .(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), (b), and (c) are shown in (a ), (b ), and (c ), respectively.The thin film was used for (a) and (c), and the single crystal was used for (b).

Fig. 4 (
Fig. 4 (b), demonstrating that ab-TO is excluded from the source of Peak-Ω, thus leaving the c-LO (Ω c ) the only remaining candidate.The ab-TO phonons are another possible sources