Giant Peltier Conductivity in an Uncompensated Semimetal Ta2PdSe6

Thermoelectric properties of single crystal Ta2PdSe6 is investigated by means of transport measurements, and a density functional calculation. We found a giant Peltier conductivity of 100 Acm-1K-1 at 10 K and successfully explained it by means of conventional semiconductor theory. We concluded that an uncompensated semimetal, high mobility, and heavy effective mass are responsible for the giant Peltier conductivity. Our finding opens a new ground in the field of thermoelectrics to explore much better semimetals for a new possible application such as an electric current generator for a superconducting magnet.


Introduction
Itinerant electrons in solids in a thermal equilibrium are driven not only by an external electric field, but also by a temperature gradient ∇T, and form a steady flow, i.e., an electrical current density j. While the former case is known as Ohm's law, the latter is rather recognized as the origin of the Seebeck effect that an electric field is generated by ∇T in an open circuit. In fact, Ohm used j generated ∇T to find Ohm's law. The proportionality constant between j and ∇T has been called the Peltier conductivity P as j = P(−∇T). Despite many studies for thermoelectrics, a theoretical upper limit of P has yet to be explored, and accordingly an unexpectedly large P has a great potential to revolutionize modern electronics.
Here, we report the observation of a giant P of 100 Acm -1 K -1 at 10 K in a single crystal of the layered semimetal Ta2PdSe6 [1]. This value is 200 times larger than the maximum P of the commercially available thermoelectric material Bi2Te3 [2] [3], and, to the best of our knowledge, it is the largest ever reported for a bulk material. This value may open a novel heat-to-electricity conversion such as a current generator, in which a 1 cc sample generates an electric current of 100 A with a temperature difference of 1 K. This is applicable to an isolated current source set in a cryogenic space. Using a two-carrier model for a perfectly uncompensated semimetal, we clarified why and how such a giant value is realized in the title compound.

Method
High-quality single crystals of Ta2PdSe6 were grown by means of I2 vapor transport. Powders of tantalum (99.9%), palladium (99.9%), and selenium (99.9% or 99.999%) were loaded into an evacuated quartz tube with a I2 concentration of ~3 mg/cm 3 . Then, a temperature difference of 145 ℃ between 875 ℃ and 730 ℃ in a three-zone furnace was used for crystal growth for 4 days. Single phase polycrystalline sample was synthesized using the same starting powder. Once the raw powder was heated up to 550 ℃, then the regrinded powder was heated at 730 ℃ for 48 h in a tube furnace.
Ta2PdSe6 crystals were chemically characterized by scanning electron microscopy with energy dispersive X-ray spectroscopy (JEOL JSM-7500F). The ratio of Ta:Pd:Se was evaluated to be 1.91 : 0.93 : 6.14, which agrees with the stoichiometric ratio of Ta2PdSe6 (Fig. S1). We also conducted a synchrotron single crystal X-ray diffraction (XRD) measurement at BL02B1 in SPring-8. We employed a wavelength of 0.30963 Å to obtain a high-resolution data. We used a gas-blowing device for sample temperature control, and a Pilatus3 X 1M CdTe detector [4] for measuring two-dimensional (2D) diffraction patterns (Fig. S2).
Diffraction intensity averaging was performed using SORTAV [5], and crystal structure refinement was performed by means of the SHELXL least squares program [6]. The refined ratio of Ta and Se against Pd was 1.997 and 5.998 respectively, indicating the atomic deficiency is less than 1%. A summary of the structural analysis is provided in Table 1 and S1. Transport properties, including electrical resistivity, thermopower, and Hall resistivity, were measured using a PPMS (Quantum Design). The electrical resistivity was measured along the b-axis by a four-probe method using gold wires with 20 μm diameters and the silver paste. The thermopower along the b-axis was measured with a steady state and the two-probe technique. The sample bridged two separated copper heat baths, and the resistance heater (KYOWA KFLB-02-120-C1-11) created a temperature difference between the two heat baths, which was monitored through a copper-constantan differential thermocouple.
The contribution of the voltage leads was carefully subtracted. The Hall resistivity with the four-probe technique was achieved by sweeping an out-of-plane magnetic field from -4 to 4 T with a steady current along the b-axis. The typical setup for the transport measurements is shown in Fig. S3. The resistivity at each magnetic field were collected using ΔR mode of a nano-ohmmeter LR-700 (LINEAR RESEARCH INC). The Hall resistivity ρyx was obtained by calculating (ρyx(+H) -ρyx(-H))/2. The specific heat measurements were performed by relaxation method with a commercial measurement system (Quantum Design PPMS Dynacool) by using polycrystalline Ta2PdSe6. The heat capacity was measured from 3 K to 300 K.
The band structure calculations were performed using the pseudopotential method based on the projector augmented wave (PAW) formalism [7] with plane-wave basis sets implemented in Quantum Espresso (version 6.6) [8]. The cut-off energies for plane waves and charge densities were set to 44 and 448 Ry in the full relativistic calculations. We conducted structure optimization using the structural parameter obtained from the single crystal XRD at 100 K as an initial input. We used a 14 × 14 × 4 uniform k-point mesh with the cold smearing method during self-consistent loops and 30 × 30 × 10 points with the "tetrahedra_opt" method for density-of-states and Fermi-surface calculation. Figure S4 shows the orbitalresolved Fermi surface.

Result and Discussion
The Peltier conductivity P can be understood from the Seebeck effect, in which a voltage difference of ΔV is generated across a sample subjected to a temperature difference of ΔT. The proportionality constant S is called the Seebeck coefficient, and ΔV = SΔT. In the presence of ΔT, materials can behave like a battery with an open circuit voltage of SΔT and an internal resistance of the sample R, as schematically shown in the inset of Fig. 1. In case the load resistance is much smaller than R, the maximum thermoelectric current is calculated to be (SΔT)/R, from which P is evaluated as S/ρ as a parameter intrinsic to materials, where ρ is the resistivity. Figure 1 (a) shows |P| of various materials [9] plotted as a function of conductivity σ = 1/ρ. We can find that Ta2PdSe6 locates at the top-level even at 300 K, and the highest at 10 K among the thermoelectric materials. Figure 1 (b) shows a comparison of the temperature dependence of |P| between Ta2PdSe6 and other representative low-and middle-temperature thermoelectric materials [10]- [14]. We find that |P| of others except for YbAgCu4 takes at most of the order of 1 Acm -1 K -1 , and gradually decreases as temperature decreases. Since optimized thermoelectric materials show a substantial residual resistivity accompanied by the T-linear S at low temperatures, their |P| is expected to be linear in T at low temperatures. On contrary, |P| of Ta2PdSe6 rapidly increases below 100 K to reach a giant value of 100 Acm -1 K -1 at 10 K, indicating the giant P results from an extremely low residual resistivity.  [10], Na0.88CoO2 [11], Ta4SiTe4 [12], YbAgCu4 [13], and CsBi4Te6 [14].  Fig. 2 (a). The resistivity ρ reaches a low value of 10 -7 Ωcm at 2 K with a residual resistance ratio (rrr) of 694. This rrr value is much better than that of other chalcogenide semimetals [15] [16]. S takes a relatively large value of 40 μV K -1 at 20 K. Consequently, the calculated P result in the giant value of 100 Acm -1 K -1 at 10 K as shown in Fig. 2 (c). Note that the power factor, which is a measure of electric power of the sample subjected to a temperature difference of 1 K, also becomes a huge value of 2.4 mWcm -1 K -2 at 15 K, although we previously reported a relatively large value of 13 μWcm -1 K -2 at 300 K [17]. We also pointed out this compound is semimetallic, for S shows a sign change near 100 K. These trends are well reproduced between different single crystals (see Figs S5 (a) and S5 (b)).
We should point out that the contribution of the phonon drag effect to S is negligible in Ta2PdSe6. As shown in Fig. S5, the single crystal prepared by using low-purity (99.9%) selenium powder shows worse conductivity than that prepared by high-purity (99.999%) selenium powder at the lowest temperature. This indicates that the carriers are scattered more frequently by the introduced impurities in the low-purity sample. Nevertheless, S of the low-purity sample is almost the same as the high-purity one. If the phonon drag effect effectively contributed to S, S would have to be affected by the impurity doping, since the mean free path of phonons is generally longer than that of electrons. This proves that the phonon drag effect is negligible and the diffusive part of electrons mainly contributes to S of Ta2PdSe6.  Figures 3 (a) and 3 (b) shows the crystal structure of layered selenide Ta2PdSe6, the layers of which consist of face-shared TaSe6 prisms and square-planar PdSe4. The crystal structure is visualized by VESTA [18].Ta2PdSe6 was first synthesized in 1985, and its electric resistivity was investigated [1].
Ta2PdSe6 has a structural similarity to the excitonic insulator candidate Ta2NiSe5 [19], a material in which we found unique structural [20][21] and transport properties [22] [23]. Thus, we focused on Ta2PdSe6 as a related material and then identified the giant Peltier conductivity.
We now take a closer look at the electronic states of Ta2PdSe6. Before a band structure calculation, we performed a structural optimization to obtain a lowest-temperature structure by starting from a crystal structure determined by a synchrotron single-crystal X-ray diffraction measurement at 100 K ( Table 1).
The relaxed lattice and atomic coordination parameters are listed in Table 2. Figures 3 (c) and 3 (e) shows the Fermi surface visualized by FermiSurfer [24] and band dispersion around the Fermi energy (EF) along the MCLC1 path [25]. There is an electron Fermi surface near the I, X, and N points, while a hole Fermi surface near the Γ, Y, L, and Z points, indicating a semimetallic ground state. Figure 3 (f) shows the total and partial density of states, demonstrating that Ta 5d and Se 4p components mainly contribute to the lowenergy electronic states. This is also shown by the orbital-resolved Fermi surface in Fig.S4. The calculated Fermi surface in Fig.3 (c) is shaped like a ragged, corrugated plane perpendicular to the b axis, indicating a pseudo one-dimensional (1D) electronic structure.   Now we explain how the giant P realized in terms of a two-carrier model [26]. The partial conductivities of the electron (σe) and hole (σh) can be written as where n (p), e, and μe(h) are the concentration of electrons (holes) per valley, the element charge, and the carrier mobility of the electron (hole), respectively. Note that the factor of two in σe represents the valley degeneracy. Then, the net Hall coefficient RH is described as Imposing a semimetallic condition of 2n = p, we rewrite equation (3) as where f = μh/(μh+μe). For pseudo 1D parabolic bands, the partial thermopowers of electron (Se) and hole (Sh) are given by (see supporting information) where kB, h, me(h) and A, are the Boltzmann constant, the Planck constant, the effective mass of electrons To find p, mh, and f, we measured the Hall resistivity ρyx at various temperatures, as shown in Fig. 4 (a).
Little deviation from linear-field dependence implies that RH is well defined by ρyx/μ0H at 1 T. RH clearly shows a rapid decrease from 10 -2 down to 10 -3 cm 3 C -1 around 100 K, as shown in Fig. 4 (b). Assuming a heavily uncompensated condition of f ~ 1 at 2 K, we obtain p ~7×10 20 cm -3 from equation (4), which is roughly consistent with the calculated carrier concentration of 7.5 × 10 20 cm -3 . We also evaluate μh at 2 K to be 9×10 4 cm 2 V -1 s -1 from RH/ρ. Furthermore, using equation (9), we get mh =2.9m0 (m0 is the bare electron mass) for S = 40 μV K -1 and T = 20 K. This value is roughly consistent with the effective mass of 3.5m0 estimated from the electron specific heat coefficient γ and n (see Table 3 and Fig. S6). Thus, we conclude that the lightly doped holes with high mobility and heavy mass are responsible for the giant Peltier conductivity in Ta2PdSe6. In the field of thermoelectrics, the so-called B-factor is a measure of good thermoelectric materials [27].
It is proportional to (m*) 3/2 μ/κL, where m*, μ, and κL are the effective mass, the mobility, and the lattice thermal conductivity of a material, respectively. The numerator (m*) 3/2 μ characterizes the power factor and is 4000 times larger for Ta2PdSe6 than for Bi2Te3 (m* = 0.2m0 and μ = 1200 cm 2 V -1 s -1 ). This is indicative of the large power factor in the present compound (2.4 mWcm -1 K -2 at 15 K). The 1D electronic structure plays a vital role to achieve such high (m*) 3/2 μ, since light holes along the 1D direction are responsible for the high mobility, whereas holes perpendicular to the 1D direction are responsible for the heavy mass [28]. We also note that Ta2PdS6, an isostructural compound of Ta2PdSe6, shows relatively high power factor ~ 30 μWcm -1 K -2 at 300 K [17].
At higher temperatures, the electron conduction begins to contribute. Since we have determined p and mh already, only me is left as an unknown parameter in equation (9). Setting me = 0.9m0 as an adjustable parameter, we obtain f from the measured S though equation (9). In Fig. 4 (b), we show the thus-obtained 2f-1, which reasonably matches with RH(f). In short, all the measured transport parameters are quantitatively and consistently understood in terms of low carrier concentration, heavy mass, high hole mobility, and a crossover from f = 1/2 (compensated) to f ~ 1 (heavily uncompensated).
The giant Peltier conductivity and huge power factor can be used as a current source for a superconducting solenoid isolated in a cryogenic space. For 100 Acm -1 K -1 , a cubic sample of 1 cc would supply a thermoelectric current of 100 A to the zero-resistance solenoid for a temperature difference of 1 K. An absence of external current leads can make the system compact and concise and reduce cooling costs for such a solenoid. This can be a novel application of heat-to-electricity conversion. We notice that the induction voltage can be larger than SΔT, so that the field-sweeping rate must be kept low.
Our finding suggests that uncompensated semimetals can generate substantial electricity at low temperatures. This type of application will break new ground in the field of thermoelectrics. Semimetals are of high mobility in general and show good electrical conduction without impurity doping [29]. The uncompensated condition of f ~ 1 partly comes from electron-hole asymmetry as shown in Fig. 3 (c), and ternary or quaternary compounds may satisfy this condition. Such materials have never been researched, and therefore we believe that much better semimetals exist but are to date unknown.

This PDF file includes:
Figs. S1 to S6 Table S1 Characterization of a single crystal Ta2PdSe6 Structural analysis of Ta2PdSe6   Table S1. Experimental setup for the transport measurements for Ta2PdSe6 Figure S3. Experimental setup for the transport measurements for Ta2PdSe6. Four probe configurations for the resistivity (a) and Hall resistivity (b) measurements. In the configuration of A, the four gold wires are attached to the side of the single crystal. The measured resistivity with this configuration corresponds to the data #7 in Fig. S3(a).