Enhanced Seebeck coefficient by a filling-induced Lifshitz transition in KxRhO2

TL;DR

This study reveals that in K$_{x}$RhO$_{2}$, a filling-induced Lifshitz transition at a critical composition enhances the Seebeck coefficient, indicating a topological change in the Fermi surface that affects thermoelectric properties.

Contribution

It provides experimental evidence of a Lifshitz transition in K$_{x}$RhO$_{2}$ and links it to anomalous thermoelectric behavior, a novel insight in layered rhodium oxides.

Findings

Seebeck coefficient is anomalously enhanced below x=0.64

A critical composition x* ≈ 0.65 marks a change in transport behavior

Filling-induced Lifshitz transition causes topological change in Fermi surface

Abstract

We have systematically measured the transport properties in the layered rhodium oxide K$_{x}$RhO$_{2}$ single crystals ($0.5\lesssim x \lesssim 0.67$), which is isostructural to the thermoelectric oxide Na$_{x}$CoO$_{2}$. We find that below $x = 0.64$ the Seebeck coefficient is anomalously enhanced at low temperatures with increasing $x$, while it is proportional to the temperature like a conventional metal above $x=0.65$, suggesting an existence of a critical content $x^{*} \simeq 0.65$. For the origin of this anomalous behavior, we discuss a filling-induced Lifshitz transition, which is characterized by a sudden topological change in the cylindrical hole Fermi surfaces at the critical content $x^*$.

Full text

The layered cobalt oxide NaxCoO2 has attracted a great deal of attention for its rich physical properties since the discovery of the large Seebeck coefficient Terasaki1997 . This compound consists of alternately-stacked CoO2 and Na layers, and the conductive CoO2 layer is formed by edge-shared CoO6 octahedra, resulting in a two-dimensional (2D) triangular lattice of Co ions. The uniqueness of this compound is the complex electronic properties drastically changing with Na content $x$ Foo2004 . For rich Na content region, the large Seebeck coefficient is observed with metallic resistivity Motohashi2001 ; Lee2006 . The magnetic property varies from Pauli-paramagnetic to Curie-Weiss (CW) behavior across a critical Na content $x^{*}_{\rm Na}\simeq 0.62$ Yokoi2005 ; Yoshizumi2007 ; Okamoto2010 ; Lang2008 . A spin-density-wave and a charge-ordered insulating state have also been suggested for $0.75\leq x$ and at $x\sim$ 0.5, respectively Motohashi2003 ; Sales2004 ; Huang2004 ; Sugiyama2004 ; Ning2008 . Moreover, hydration induces unconventional superconductivity at around $x=0.3$ Takada2003 ; Jin2003 ; Mazin2005 ; Yang2005PRB ; Ogata2007 .

The crucial points for understanding various emergent phenomena in NaxCoO2 are the characteristic electronic band structure and the topology of the Fermi surface. In this compound, the $t_{{\rm 2}g}$ bands from Co 3$d$ orbitals are composed of an $a_{{\rm 1}g}$ band and doubly degenerate $e_{g}^{\prime}$ bands, and the Fermi level lies across the $a_{1g}$ band to form a cylindrical Fermi surface reflecting the quasi-2D crystal structure Singh2000 ; Korshunov2007 ; Kuroki2007PRL ; Hasan2004 ; Yang2005 ; Yang2004 ; Qian2006 ; Geck2007 ; Arakane2011 . Now the shape of the $a_{1g}$ band is peculiar, including a somewhat flat portion around a local minimum at the $\Gamma$ point, which is called pudding-mold band shape Kuroki2007JPSJ . With increasing Na content, the Fermi level rises and then touches the local minimum to create a small electron pocket as is resolved by the angle resolved photoemission spectroscopy (ARPES) Arakane2011 . Detailed thermodynamic study has revealed that this topological change in the Fermi surface, known as the Lifshitz transition Lifshitz1960 , occurs at the critical content $x^{*}_{\rm Na}\simeq 0.62$ Okamoto2010 . The Fermi level then locates around the flat region for $x>x^{*}_{\rm Na}$, resulting in the large Seebeck coefficient due to a significant difference in the velocities of electrons and holes Kuroki2007JPSJ . The flat band with large density of states may also contribute to the CW-like magnetic behavior observed above $x^{*}_{\rm Na}$.

The issue to be addressed is whether the richness of these intriguing properties universally emerges in the related systems. KxCoO2 ($x\sim 0.5$) shows two phase transitions at low temperatures Yokoi2008 ; Watanabe2006 , reminiscent of Na0.5CoO2, and similar electronic structures in these materials are suggested both experimentally and theoretically Qian2006 ; Lee2007 . The $4d$ system NaxRhO2 is also isostructural to NaxCoO2 Varela2005 ; Krockenberger2007 and exhibits a metal-insulator transition which is explained by an Ioffe-Regel criterion Zhang2018 , but the detailed electronic properties are still unexplored.

The isostructural KxRhO2 shows moderately large Seebeck coefficient with half the magnitude of NaxCoO2 Yubuta2009 ; Shibasaki2010 ; Yao2012 . The optical study suggests that the bandwidth of KxRhO2 is twice broader compared with NaxCoO2 owing to a difference of the orbital sizes between Co 3$d$ and Rh 4$d$ electrons Okazaki2011 . Recent ARPES measurement has clarified that the band shape of KxRhO2 is closely similar to that of NaxCoO2 Chen2017 , indicating a possible emergence of rich electronic phase diagram. Furthermore, interesting phenomena, including a significant enhancement of thermoelectric efficiency by hydration Saeed2012 and a realization of topological quantum Hall effect Zhou2016 , have been theoretically predicted in KxRhO2. However, the physical properties of KxRhO2 have been little known due to the difficulty of the synthesis of well controlled K content, single-phase samples Zhang2013 ; Zhang2016 .

In this paper, we present the potassium composition dependence of the transport properties in KxRhO2 for $0.5\lesssim x\lesssim 0.67$ using pure single-crystalline samples, which are systematically prepared by a self-flux method and a K de-intercalation process. We find that the Seebeck coefficient is highly sensitive to the K content $x$ and enhanced only below a critical content $x^{*}\simeq 0.65$. This result is attributed to a filling-induced Lifshitz transition at $x^{*}$, at which the topology of the cylindrical hole surface dramatically changes. Moreover, near $x=0.5$, distinct anomaly is observed at $T\simeq 75$ K, implying that this rhodate also offers a fascinating platform for various electronic phases to be investigated.

Single-crystalline samples of KxRhO2 were grown by a self-flux method Shibasaki2010 ; Yubuta2009 . A mixture of K2CO3 (99.999%) and Rh2O3 (99.9%) with a molar ratio of $50:1$ was put in an alumina crucible and kept at 1373 K for 1 h, and then cooled down to 1123 K with a rate of 5 K/h. After washing with distilled water, we obtained thin hexagonal crystals as shown in the inset of Fig. 1(a). X-ray diffraction (XRD) measurements were performed by an X-ray diffractometer (Rigaku UltimaIV) with Cu K$\alpha$ radiation in a $\theta$-2$\theta$ scan mode. In the case of the single crystalline samples, the scattering vector was normal to the surface of the sample. The K content of as-grown samples is $x\sim 0.67$. To get the lower potassium content samples, we dipped the samples in I2-acetonitrile solutions Zhang2016 . The potassium content was controlled by both the concentration of the solution and the length of time to dip. This process was conducted at room temperature. Subsequently, we annealed these dipped samples at 473 K for about 2 days in air to obtain single phase samples. To elucidate the relation between the $c$-axis lattice parameter and potassium content, we measured the K content $x$ using an electron probe microanalyzer (JEOL JXA-8100) after we determined the $c$-axis lattice parameter by XRD measurement. The in-plane resistivity and the Seebeck coefficient were simultaneously measured by using a conventional dc four-probe method and a steady-state method. The thermoelectric voltage contribution from the wire leads was subtracted.

We first show the XRD patterns of a single crystal in each process of the K de-intercalation. The XRD patterns of as-grown and after-de-intercalation crystals are depicted in Figs. 1(a) and 1(b), respectively. As reported by Zhang $et~{}al$ Zhang2013 ; Zhang2016 , several sets of the peaks are observed in after-de-intercalation crystals. On the other hand, after annealing, we find that a single-phase pattern is recovered as shown in Fig. 1(c). In the isostructural NaxCoO2, it is well known that the low Na-content sample is easily hydrated and a heating process removes H2O molecules from the hydrated sample Chen2004 . Thus the present result indicates that a hydrated phase is also formed in KxRhO2 after de-intercalation but can be recovered to the non-hydrate phase by the successive annealing. The existence of the hydrated KxRhO2 is also suggested by an ion exchange experiment Mendiboure1987 . Note that no superconductivity is observed in the hydrated KxRhO2 at present.

The powder XRD pattern of non-hydrate sample for $x=0.651$ is represented in Fig. 1(d). Note that the measured powder is not ground and just collected as small single crystals with the size of $\sim 50$ $\mu$m, because the grinding induces the hydration significantly. As seen in Fig. 1(d), all the peaks are indexed using the $\gamma$-NaxCoO2-type structure (space group $P6_{3}/mmc$) with the lattice parameters of $a=3.076(2)~{}\AA$ and $c=12.233(5)~{}\AA$. In the earlier reports, the $c$-axis lattice parameter is found to be 13.6 $\AA$ Yubuta2009 ; Yao2012 , much larger than the present result. This discrepancy indicates that the powder XRD patterns in the earlier reports refer to the hydrated phase, as is also pointed out by means of the optimization of the lattice constants Saeed2012 . On the other hand, the space group is preserved among the hydrated and non-hydrate samples, similar to the case of NaxCoO2 Takada2003 .

We examine the relation between the $c$-axis lattice parameter and the K content $x$ using non-hydrate crystals as shown in the inset of Fig. 1(d). The $c$-axis lattice parameter decreases almost linearly with increasing K content $x$. In this system, when the K content increases, the Coulomb force holding the RhO2 layers is enhanced to decrease the $c$-axis lattice parameter, which is also seen in NaxCoO2 Foo2004 . The present data are fitted using a linear function $c=c_{0}+c_{1}x$ with $c_{0}=13.13(4)$ $\AA$ and $c_{1}=-1.37(7)$ $\AA$ as shown by the dashed line. The K content of crystals for the transport measurements is calculated from the $c$-axis lattice parameter using this relation in a similar manner to that of NaxCoO2 Yokoi2005 . The $x$ values have a common error which is estimated at up to 0.04 due to the large error of $c_{0}$ and $c_{1}$. However, we can obtain relative amount of K content accurately by the comparison of the $c$-axis lattice parameter. From the K content, the hole concentration is approximately determined as $1-x$ holes/Rh Chen2017 .

We then discuss the transport properties of KxRhO2. Figure 2 shows the temperature dependence of the Seebeck coefficient and the resistivity for $x=0.50$ and 0.51. The resistivity is normalized by the room-temperature resistivity $\rho_{\rm RT}\sim 1$ m$\Omega$cm, which has a large error because the sample thickness is too small to be measured correctly. We also note that the value of $\rho_{\rm RT}$ does not change significantly for all the crystals in the present study. A distinct anomaly is observed in both the Seebeck coefficient and resistivity at around 75 K, as marked by the dashed line. An origin of this anomaly is not clear at present. In NaxCoO2, the two distinct phase transitions were observed, and the ground state is a charge-ordered insulator at around $x=0.5$ Foo2004 ; Yokoi2005 . Moreover, an interesting topological state has been suggested theoretically for $x=0.5$ Zhou2016 .

We next show the temperature dependence of the transport properties for $0.57\lesssim x\lesssim 0.67$ in Figs. 3(a) and (b). Around room temperature, the Seebeck coefficient almost linearly decreases with decreasing temperature for all the samples. On the other hand, the temperature dependence does not systematically vary with the K content. For $x\gtrsim 0.65$, the Seebeck coefficient is roughly proportional to temperature as shown in Fig. 3(b). In contrast, as shown in Fig. 3(a), the temperature dependence for $0.57\lesssim x\lesssim 0.64$ is qualitatively different from that for $x\gtrsim 0.65$: At low temperatures, the Seebeck coefficient is enhanced with a broad peak structure. Note that the phonon drag effect is unlikely since we have observed similar temperature variation even in the polycrystalline samples (not shown).

The anomaly is clearly demonstrated in the Seebeck coefficient divided by temperature, $S/T$, shown in Figs. 3(c) and (d). For $0.57\lesssim x\lesssim 0.64$, $S/T$ gradually increases with lowering temperature. It also increases with increasing $x$ but the behavior drastically changes in the narrow range of $0.64<x<0.65$ into almost no temperature dependence, as expected in a conventional metal, suggesting that there is a critical content $x^{*}$ in this narrow range. Figure 4(a) represents $S/T$ as a function of $x$ at constant temperatures and the contour plot is depicted in Fig. 4(b). At low temperature, $S/T$ shows a divergent behavior at $x^{*}$ only from the lower $x$ side. This is distinct from the divergence of $S/T$ from both sides of the critical point of the Landau-type phase transition associated with broken symmetry Sakai2016 . Also, no structural transition is observed at $x^{*}$. It is a key point that the enhancement is observed for only low temperature, while $S/T$ continuously increases with $x$ near room temperature, which is readily explained by the change in the carrier concentration.

Now we discuss the origin of the enhancement of the Seebeck coefficient. According to the observed band structure of K0.62RhO2, the Fermi level lies across the $a_{1g}$ band Chen2017 . The Fermi level is elevated as $x$ increases, and as is the case in NaxCoO2, the topological change in the Fermi surface would also occur at a certain content in KxRhO2. Note that the present critical content $x^{*}$ is close to $x^{*}_{\rm Na}\simeq 0.62$, at which the Lifshitz transition occurs in NaxCoO2 Okamoto2010 . Moreover, near the Lifshitz transition, the low-temperature Seebeck coefficient is predicted to increase steeply when the Fermi level approaches a critical point from only one side, the one for which the number of Fermi surfaces is larger, owing to the divergence in the energy derivative of the density of states Varlamov1989 ; Abrikosovbook , as is also explored in several alloys experimentally Egorov1982 ; Egorov1984 . Thus the observed enhancement indicates the existence of a Lifshitz transition in KxRhO2. We also note that the resistivity has no singularity around the Lifshitz transition Varlamov1989 . In the insets of Figs. 3(a) and (b), we show the temperature dependence of the normalized resistivity, which exhibits no significant change across $x^{*}$, consistent with above picture.

We examine the detailed mechanism of the Lifshitz transition. In Fig. 5, we show the schematic band structure and the Fermi surface of NaxCoO2 and KxRhO2. Since the $\gamma$-type structure has two Co/Rh ions in the unit cell, we consider two $a_{1g}$ bands labelled as higher and lower bands in Figs. 5(a) and 5(e) Singh2000 , although these are almost degenerate and difficult to be resolved separately by ARPES measurements. In NaxCoO2, there are two hole cylinders for $x<x^{*}_{\rm Na}$ as shown in Fig. 5(b), and with increasing $x$, the Fermi level rises and then touches the local minimum of the lower band at $x=x^{*}_{\rm Na}$ [Fig. 5(c)] to create a small electron pocket at the $\Gamma$ point for $x>x^{*}_{\rm Na}$ [Fig. 5(d)]. In this case, the low-temperature Seebeck coefficient would be critically enhanced owing to the disappearance of the electron pocket only when $x$ approaches $x^{*}_{\rm Na}$ from the high $x$ side. Indeed, negatively enhanced Seebeck coefficient is seen for $x>x^{*}_{\rm Na}$, although the experiments have been done in polycrystalline samples Okamoto2010 .

On the other hand, the enhancement of the Seebeck coefficient is observed for $x<x^{*}$ in KxRhO2, indicating that the topological change in the Fermi surfaces differs from that of NaxCoO2. According to recent ARPES measurement Chen2017 , the band structure of KxRhO2 is closely similar to that of NaxCoO2, and thus we consider a local minimum structure at the $\Gamma$ point as well. In NaxCoO2, this minimum structure is induced by a third nearest hopping $t_{3}$, and the local minimum sinks more deeply as the ratio $|t_{3}/t_{1}|$ becomes larger ($t_{1}$ being the first nearest hopping) Kuroki2007JPSJ . Now, the 4$d$ orbital is broader than 3$d$ one, but the ionic radius of Rh is also larger than Co, resulting in an accidental cancelation of the effect of the broader $4d$ orbital in some materials Okada2005 . In KxRhO2, the broader bandwidth is indeed observed Okazaki2011 ; Chen2017 , but $|t_{3}/t_{1}|$ may be smaller than that of NaxCoO2 because oxygen $2p$ contribution in $t_{3}$ would be small due to the larger Rh-Rh distance. In such a case, the local minimum structure of the band top would be shallow in KxRhO2 as illustrated in Fig. 5(e), also implied by the DFT calculations Chen2017 .

This scenario is schematically depicted in Figs. 5(e-h). In KxRhO2, the shallow local minimum structure may be undetectable because of thermal fluctuations even at low temperatures. For increasing $x$, the Fermi level is elevated and then the lower $a_{1{\rm g}}$ band is fully occupied at $x^{*}$. In this case, the inner hole cylinder gradually diminishes and then completely disappears for $x>x^{*}$, which is also supported by a highly 2D character of the Fermi surface Chen2017 . Thus, in KxRhO2, the low-temperature enhancement of $S/T$ occurs only for $x<x^{*}$, where the number of hole cylinders is larger than that for $x>x^{*}$. It should be noted that the disappearance of the electron pocket in NaxCoO2 contributes to the Seebeck coefficient negatively but the disappearance of the hole cylinder in KxRhO2 contributes positively, enhancing the total Seebeck coefficient that also includes the contribution from the outer hole cylinder. Therefore, our results suggest the Lifshitz transition as a fundamental means to increase the Seebeck coefficient at low temperatures in this system, in addition to the previously reported pudding-mold band structure Kuroki2007JPSJ and the large entropy flow of the $d$ electrons Koshibae2000 .

To summarize, we have synthesized single crystal KxRhO2 for $0.50\lesssim x\lesssim 0.67$ and find that, while the Seebeck coefficient monotonically increases with increasing $x$ near room temperature, it exhibits an enhancement at low temperatures only below $x^{*}\simeq 0.65$. As an origin, we propose a Lifshitz transition associated with disappearance of the inner hole cylinder due to the flat band structure around the $\Gamma$ point. Moreover, we have discovered a distinct phase transition for $x\sim$ 0.5, suggesting that KxRhO2 also offers a fascinating platform for various electronic phases, similar to NaxCoO2.