Correlation in transport coefficients of hole-doped CuRhO$_2$ single crystals

TL;DR

This study systematically investigates the transport properties of single-crystalline CuRhO$_2$, revealing phonon-drag effects and correlations between resistivity and thermopower that clarify its thermoelectric behavior.

Contribution

It provides the first detailed transport measurements on single-crystalline CuRhO$_2$, highlighting the relationship between resistivity and Seebeck coefficient in this material.

Findings

Phonon-drag effect causes a peak in Seebeck coefficient.

Resistivity and Seebeck coefficient decrease with Mg substitution.

Low-temperature transport follows simple metallic relationships.

Abstract

To clarify the origin of the nontrivial thermoelectric properties observed in the delafossite oxide CuRhO$_2$ polycrystals, we have performed the systematic transport measurements on the single-crystalline CuRhO$_2$ samples. In the parent compound, we find a pronounced peak structure due to a phonon-drag effect in the temperature dependence of the Seebeck coefficient, which is also confirmed by the size effect experiments. In the Mg-substituted crystals, in contrast to the results of the polycrystals, both the resistivity and the Seebeck coefficient decrease with increasing Mg content $y$. In particular, the coefficient $A$ for the $T^2$ term of the resistivity and the $T$-linear coefficient for the Seebeck coefficient at low temperatures are well described within a simple relationship expected for metals, which is also applicable to the correlated materials with low carrier densities.

Equations

$$l_b$$

$$l_{\mathrm{ph}}^{-1}=l_T^{-1}+l_b^{-1}.$$

$$S_{\mathrm{ph}}=\frac{\beta v{l}_{\mathrm{ph}}}{{\mu }_{e}T},$$

$$S\simeq S_d+\frac{\beta v{l}_T}{{\mu }_{e}T}\left(1-\frac{l_T}{l_b}\right),$$

$$A=\frac{\hslash }{e^2}{\left(\frac{k_B}{{\epsilon }_F}\right)}^2{l}_{\mathrm{quad}},$$

$$\frac{S}{T}=-\frac{{\pi }^2}{2}\frac{k_B}{e}\frac{k_B}{{\epsilon }_F}.$$

$$\frac{A}{(S/T{)}^2}=\frac{4}{{\pi }^4}\frac{\hslash }{k_B^2}{l}_{\mathrm{quad}},$$

Full text

I Introduction

The search for the guiding principles toward the efficient thermoelectric materials is of great importance both in fundamental and applied physics. The oxides have attracted a keen attention as a key class of thermoelectric materials with a certain merit of their high stability at high temperature in air Maignan2002 ; Koumoto2006 ; Snyder2008 ; Terasaki2016 . In particular, since the discovery of the good thermoelectric properties in NaxCoO2 Terasaki1997 , various cobalt oxides are investigated, and interesting mechanisms to enhance the Seebeck coefficient, $S$, including the spin-orbital entropy of correlated $d$ electrons Koshibae2000 ; Koshibae2001 and the peculiar shape of the electronic band structure Kuroki2007 , are proposed and examined.

The rhodium oxides are the isovalent $4d$ counterpart for the cobalt oxides, essential to the thorough understanding of the origin of the thermoelectric transport properties in this system Okada2005 ; Klein2006 ; Okamoto2006 ; Kobayashi2007 ; Kobayashi2009 ; Shibasaki2010 ; Okazaki2011 . In this respect, the delafossite CuRhO2 is a suitable material Kuriyama2006 ; Shibasaki2006 ; Guilmeau2009 ; Maignan2009 ; Le2009 ; Vilmercati2013 ; Daou2017 , since the two-dimensional (2D) RhO2 layer is consisting of the edge-shared RhO6 octahedra to form a triangular lattice as is similar to the 2D triangular lattice of Co ions in NaxCoO2. The parent compound is insulating and becomes conducting with the hole doping by the Mg2+ substitution to the Rh sites. Indeed, polycrystalline CuRh0.9Mg0.1O2 exhibits a large Seebeck coefficient with a metallic resistivity Guilmeau2009 ; Maignan2009 , which may originate from a “pudding mold” type band structure suggested by Usui et al. Usui2009 , as is similar to the case of NaxCoO2. Although there exists a contribution of the Cu layer Le2009 , combined spectroscopic experiments have shown that the Rh $4d$ electron is dominant for the conduction phenomena Vilmercati2013 , as is similar to other hole-doped delafossite compounds Yokobori2013 .

The low-temperature transport nature of CuRh1-yMgyO2 is, on the other hand, still controversial: while the resistivity certainly varies from insulating to conducting behavior with the hole doping, the temperature variations of the Seebeck coefficient for $0\leq y\leq 0.2$ fall into a single curve below 150 K Shibasaki2006 , highly incompatible with the conventional behavior where the Seebeck coefficient decreases in magnitude with increasing carrier density Snyder2008 . Since the Seebeck coefficient measures the temperature derivative of the chemical potential, $\partial\mu/\partial T$ Peterson2010 ; Yamamoto2017 , the observed $y$-independent Seebeck coefficient indicates that the compressibility, $\partial n/\partial\mu$ ($n$ being the carrier density), is divergently large Shibasaki2006 . This implies an occurrence of a phase separation, in which a percolation network for the doped holes is away from the substituted Mg ions. Thus it represents an intriguing doping effect to be further addressed where the substituted ions act as the carrier donors/acceptors with weak scattering, although the earlier studies are performed by using polycrystalline samples.

The aim of this paper is to clarify the low-temperature thermoelectric properties of CuRh1-yMgyO2 by using single-crystalline samples. We find that the parent compound shows a peak structure in the temperature dependence of the Seebeck coefficient at around 120 K. We also observe a size effect of the peak structure, indicating an existence of the phonon-drag effect. In the Mg-substituted crystals, the Seebeck coefficient systematically varies with the hole doping unlike the polycrystalline data. Moreover, we show that the relation between the $T$-linear coefficient for the Seebeck coefficient, $S/T$, and the coefficient $A$ for the $T^{2}$ term in the Fermi-liquid resistivity is described within a simple model for metals. In contrast to other universal relations, such as the Kadowaki-Woods ratio $A/\gamma^{2}$ ($\gamma$ being the electronic specific heat coefficient) Kadowaki1986 and $S/\gamma T$ ratio suggested by Behnia et al. Behnia2004 , the present relation among the transport coefficients is widely applicable to the correlated materials even with low carrier density.

II Experiments

The experiments were performed using CuRh1-yMgyO2 single crystals ($y=0,0.03,0.06,0.1$). Powders of Rh2O3 (99.9%), MgO (99.99%), and CuO (99.9%) were mixed in a stoichiometric ratio and calcined in air at 1203 K for 36 h. The powders were then ground and calcined again in air at 1203 K for 72 h Shibasaki2006 . The single crystals were then grown by a self-flux method as is similar to the crystal growth of CuAlO2 Koumoto2001 . CuO (99.9%) powder was added with the calcined powder as a flux. The concentration of CuRh1-yMgyO2 powder was set to be 10% in molar ratio. The mixture was put in a platinum crucible and heated up to 1423 K in air with a heating rate of 200 K/h. After keeping 1423 K for 10 h, it was slowly cooled down with a rate of 0.5 K/h, and at 1323 K, the power of the furnace was switched off. As-grown samples were rinsed in 1 M HNO3 to remove the flux, and then annealed at 673 K for 20 h in air to obtain the homogeneous samples. Typical crystal dimension was $1\times 0.3\times 0.01$ mm3.

X-ray diffraction (XRD) measurements were performed at room temperature by using an x-ray diffractometer (Rigaku UltimaIV) with Cu K$\alpha$ radiation in a $\theta$-2$\theta$ scan mode. For the powder XRD experiments, we ground the single crystals grown by the self-flux method. In the case of the single-crystalline experiments, the scattering vector was normal to the surface of the sample. The in-plane resistivity was measured with a conventional dc four-probe method. The excitation current of $I=20~{}\mu$A was provided by a Keithley 6221 current source and the sample voltage was measured with a synchronized Keithley 2182 nanovoltmeter. These two instruments were operated in a built-in Delta mode to cancel unwanted thermoelectric voltage. The Seebeck coefficient was measured by using a steady-state method with a typical temperature gradient of 0.5 K/mm made by a resistive heater. The thermoelectric voltage of the sample was measured with Keithley 2182A nanovoltmeter. The temperature gradient was measured with a differential thermocouple made of copper and constantan in a liquid He cryostat. The thermoelectric voltage from the wire leads was subtracted.

III Results and discussion

III.1 Structural and transport properties

Figures 1(a) and 1(b) represent the x-ray diffraction patterns of CuRh1-yMgyO2 powders and single-crystalline samples, respectively. No impurity phase is detected in the powder data, and only $(00l)$ reflections are observed in the single-crystalline samples, showing that the measured crystal surface is the $ab$ plane. Figure 1(c) depicts the Mg content $y$ dependence of the lattice parameters. The $a$-axis length decreases with increasing $y$, while the $c$-axis length is almost constant for $y$, as is consistent with the previous reports Shibasaki2006 . The volume reduction with increasing $y$ is explained by the comparison of the ionic radii for Mg2+ ($0.72$ $\AA$), Rh3+ ($0.665$ $\AA$), and Rh4+ ($0.60$ $\AA$) Shannon1976 : If we simply express the valence state of CuRh1-yMgyO2 as CuRh${}^{3+}_{1-2y}$Rh${}^{4+}_{y}$MgyO2 from the charge neutrality, the average ionic radius of the Rh site $R_{\rm avg}$ is calculated as $R_{\rm avg}=0.665\times(1-2y)+0.66\times 2y$. The reduction of $R_{\rm avg}$ at $y=0.1$ is then calculated as 0.15%. On the other hand, experimental reduction of the $a$-axis length at $y=0.1$ is 0.13%, which well agrees with the theoretical calculation.

Figure 2(a) shows the temperature variations of the in-plane resistivity, $\rho(T)$, for CuRh1-yMgyO2 crystals ($0\leq y\leq 0.1$). The resistivity systematically decreases with the Mg substitution, indicating the successful hole doping. Figure 2(b) depicts $\ln\rho$ as a function of $T^{-1}$ for the parent compound, and the dotted line represents a thermal activation fitting $\rho\propto\exp(\Delta/2k_{B}T)$, where $\Delta=0.17$ eV is the activation energy. Now this value is smaller than the energy gap value of 0.8 eV estimated from the band calculations Usui2009 ; Maignan2009 , indicating a thermal excitation from the valence band to an acceptor level. Below 100 K, the resistivity deviates from the activation formula since an impurity band conduction sets in. In the metallic samples for $y\geq 0.06$, as shown in Fig. 2(c), the low-temperature resistivity shows a Fermi-liquid behavior $\rho(T)=\rho_{0}+AT^{2}$ with a small upturn below 20 K, as is similar to the polycrystalline study Maignan2009 and to the results of other layered oxides Masset2000 ; Limelette2005 ; Hsieh2014 ; Ikeda2016 . The doping dependence of $A$ will be mentioned later.

The temperature variations of the Seebeck coefficient are displayed in Fig. 3. In sharp contrast to the polycrystalline results Shibasaki2006 , the Seebeck coefficient systematically varies with the Mg content $y$. In the parent compound, it exhibits a relatively large value of $S\sim 200$ $\mu$V/K at room temperature and shows a peak structure at $T\simeq 120$ K. Such a structure may originate from a phonon-drag effect, because it is not resolved in the polycrystalline sample with a considerable phonon scattering effect at the grain boundaries Masui2002 . Also note that the observation of the phonon-drag effect supports the thermal activation transport in the resistivity as mention above, rather than the variable range hopping conduction. For $y\geq 0.03$, the Seebeck coefficient shows a metallic behavior, and the overall temperature dependence of $S$ qualitatively agrees with the theoretical calculations Usui2009 . The low-temperature $T$-linear coefficient decreases with the doping, which will be argued with the results of the resistivity.

III.2 Phonon-drag effect

To discuss the phonon-drag effect in the CuRhO2 crystal in more details, we have examined the size effect on the Seebeck coefficient. It is well known that the phonon-drag contribution to the Seebeck effect can be enhanced by sizing up the crystal owing to the enhancement of the mean free path of the ballistic phonon. Figure 4 represents the temperature variations of the Seebeck coefficient of several CuRhO2 crystals with different dimensions. Here, a mean free path dominated by the crystal boundary scattering, $l_{b}$, can be estimated by using the sample dimensions as McCurdy1970

[TABLE]

where $D\times xD$ is the cross section of the crystal with a rectangular shape. As seen in Fig. 4, the peak structure in $S(T)$ is enhanced with increasing $l_{b}$, implying the phonon-drag effect.

It should be noted that, however, other scattering processes such as the phonon-phonon scattering should be dominant in the present temperature range, in contrast to the phonon-drag effect at low temperature Takahashi2016 . Let $l_{T}$ be a temperature-dependent mean free path, which is determined by the other scatterings, and then the phonon mean free path, $l_{\rm ph}$, is generally given as

[TABLE]

Here, $l_{T}\ll l_{b}$ holds around $T=120$ K, since the phonon transport should be diffusive. Now, the phonon-drag contribution to the Seebeck coefficient, $S_{\rm ph}$, is given as Herring1954

[TABLE]

where $\beta$, $v$, and $\mu_{e}$ are an electron-phonon coupling constant ($0<\beta<1$), the phonon velocity, and the carrier mobility, respectively. Then the total Seebeck coefficient is approximately expressed as

[TABLE]

where $S_{d}$ is the diffusive term of the Seebeck coefficient. The inset of Fig. 4 depicts the maximum Seebeck coefficient at the peak structure, $S_{\rm peak}$, as a function of $1/l_{b}$. According to Eq. 4, the experimental data are roughly fitted by a linear function of $1/l_{b}$ as shown by the dashed line, supporting the present phonon-drag model.

III.3 Relation in the transport coefficients

Now let us discuss the doping dependence of the resistivity and the Seebeck coefficient. As shown in Fig. 2(c) and Fig. 3, both the coefficient $A$ for the $T^{2}$ term of the resistivity and the $T$-linear coefficient for the Seebeck coefficient at low temperatures are reduced with increasing Mg content $y$, which is readily understood with the increase in the carrier density $n$ AM .

Here, let us compare the present results with other systems. It is well known that both $A$ and $S/T$ values are correlated with the electronic specific heat coefficient, $\gamma$, as is represented by Kadowaki-Woods Kadowaki1986 and Behnia plots Behnia2004 . Figures 5(a) and 5(b) represent the $A$ vs. $\gamma^{2}$ and $|S/T|$ vs. $\gamma$ plots, respectively, for various materials Jaccard1985 ; Aliev1984 ; Bredl1984 ; Sparn1985 ; Sato1987 ; Loneysen1994 ; Hiess1994 ; Izawa1999 ; Terashima2002 ; Amato1988 ; Lacerda1989 ; Sakurai1988 ; Maple1986 ; Jaccard1982 ; Havinga1973 ; AlamiYadri1999 ; Sales1976 ; Casanova1990 ; Bauer1991 ; Jaccard1985Yb ; Pott1981 ; Besnus1986 ; Konstantinovic2002 ; Mayama1998 ; Saito2017 ; Elizarova2000 ; Nakamae2003 ; Lee2006 ; Yoshino1996 ; Maeno1997 ; Okuda2001 ; Farrell1970 ; White1959 ; Fletcher1970 ; Mizutani1977 ; Fletcher1968 ; Trodahl1973 ; Andres1975 ; Nakamoto1995 ; Kambe1996 ; Palstra1986 ; Ebihara2000 ; Thompson1987 ; Tsujii1997 ; Mirzaei2013 ; Li2004 ; White1959 ; Rice1968 ; Budworth1960 ; Waite1956 . Note that we refer the literature data, but the analysis for the Fermi-liquid behavior is a remaining issue Tateiwa2012 ; Jaoui2018 . Here, although the Kadowaki-Woods plot can be corrected by involving the degeneracies Tsujii2003 ; Kontani2004 ; Tsujii2005 , both plots well explain the universal relations among those quantities for the correlated materials with a metallic carrier density. On the other hand, since both $A/\gamma^{2}$ and $S/\gamma T$ ratios include the carrier density $n$ Behnia2004 ; Takimoto1996 ; Continentino2000 , the data of correlated materials with low carrier density, such as a Kondo insulator CeNiSn Terashima2002 and slightly doped SrTiO3 Okuda2001 , significantly deviate from the universal relation in these plots. Note that there are several attempts to correct this deviation due to the carrier density Hussey2005 ; Jacko2009 .

In Fig. 6, we instead plot $A$ as a function of $(S/T)^{2}$ for the materials shown in Figs. 5(a) and 5(b), in addition to the present data for CuRh1-yMgyO2 ($y=0.06,0.1$). All the data, including the low-density materials mentioned above, seem to fall into a single line. Now, the $A$ term in the resistivity is expressed as MottMI ; Lin2015

[TABLE]

where $\hbar$ is the reduced Planck constant, $e$ the elementary charge, and $\varepsilon_{F}$ the Fermi energy. Here $l_{\rm quad}$ is a characteristic length for the scattering events, which is given as $l_{\rm quad}=k_{F}\sigma_{\rm cs}$ ($k_{F}$ and $\sigma_{\rm cs}$ are the Fermi wavenumber and the collision cross section between two electrons, respectively). On the other hand, in a free-electron model, the Seebeck coefficient divided by temperature is given as

[TABLE]

Therefore, we express the ratio as

[TABLE]

in which the Fermi energy is cancelled out. The present ratio $A/(S/T)^{2}$ is, in other words, a direct measure for $l_{\rm quad}$. This ratio can also be expressed as $(2/3\pi^{3})(l_{\rm quad}/G_{\rm en})$, using the thermal conductance quantum, $G_{\rm th}$, divided by temperature, $G_{\rm th}/T=G_{\rm en}=(\pi/6)(k_{B}^{2}/\hbar)$. Since $G_{\rm en}$ ($=G_{\rm th}/T$) may be related to an entropy conductance, $A/(S/T)^{2}\simeq l_{\rm quad}/G_{\rm en}$ may imply a resistivity for an entropy flow, in analogy to the Fermi-liquid resistivity $AT^{2}\simeq(l_{\rm quad}/G_{0})(k_{B}T/\varepsilon_{F})^{2}$ ($G_{0}$ being the conductance quantum), which expresses a resistivity for the charge flow. Here, in Eq. 7, $l_{\rm quad}$ is indeed material-dependent but mostly lies in a narrow range between 1 and 40 nm Lin2015 , consistent with the plot shown in Fig. 6, in which various experimental data are on a single line calculated with a constant $l_{\rm quad}$ of 4 nm. It is an interesting issue to clarify the origin of the weak material dependence of $l_{\rm quad}$ for various compounds with very different size and shape of Fermi surfaces.

Noted that, in the present compound CuRh1-yMgyO2, a pudding-mold type band structure is suggested to enhance the Seebeck coefficient Usui2009 . The value of $S/T$ in Fig. 6 is, however, evaluated at low temperature, at which a simple metallic model can be applicable. Also, it is unlikely to apply the high-temperature spin-orbital entropy model Nishina2017 ; Takahashi2018 . For the present ratio $A/(S/T)^{2}$, an interesting exception may be a material around the Lifshitz transition, at which the Seebeck coefficient becomes divergently large while the resistivity remains intact Varlamov1989 ; Ito2019 .

IV Summary

In this study, we have carried out the transport measurements on the delafossite oxides CuRh1-yMgyO2 single crystals, and find the systematic doping variations of the resistivity and the Seebeck coefficient. Particularly, we find a correlation between the coefficient $A$ for the $T^{2}$ term of the Fermi-liquid resistivity and the $T$-linear coefficient for the Seebeck coefficient at low temperature, which can be described within a simple relationship expected for metals. This relation may also be applicable to the correlated materials with low carrier densities.

Acknowledgements.

This work was supported by JSPS KAKENHI Grants No. JP17H06136, No. JP18K03503, and No. JP18K13504.