Transport properties of binary phosphide AgP2 denoting high Hall mobility and low lattice thermal conductivity

This study found that polycrystalline AgP2 shows intrinsic semiconducting electrical conductivity with Hall mobility of 51 cm2 V−1 s−1, which is as high as that of Mg2Si, and lattice thermal conductivity of 1.2 W K−1 m−1, which is as low as that of Bi2Te3. First-principles calculations theoretically indicate AgP2 as an intrinsic semiconductor, and indicate the estimated carrier relaxation time τ as 3.3 fs, which is long for a polycrystalline material. Moreover, the effective mass of hole m* is approximately 0.11 times that of free electrons. These results indicate that long τ and light m* of the carrier are the origins of the high experimentally obtained Hall mobility. Phonon calculations indicate that the Ag atoms in AgP2 exhibit highly anharmonic phonon modes with mode Grüneisen parameters of more than 2 in the 50–100 cm−1 low-frequency range. The large anharmonic vibrations of the Ag atoms reduce the phonon mean free path. Moreover, the lattice thermal conductivity was found, experimentally and theoretically, to be as low as approx. 1.2 W K−1 m−1 at room temperature by phonon–phonon and grain-boundary scattering.


Introduction
Thermoelectric conversion technology can directly convert thermal energy into electrical energy. Thus, because it is difficult to recover energy from low-temperature waste heat in a steam turbine, thermoelectric conversion is attracting attention as a technology to recover energy from low-temperature waste heat. The most widely used thermoelectric materials are Te compounds, such as Bi 2 Te 3 [1,2] and PbTe [3,4]. However, because of the low earth abundance of Te -approximately the same as that of Pt [5]-alternative materials are needed to replace Te compounds for mass production. In recent years, sulfides [6][7][8], silicides [9,10], and half-Heusler [11] compounds have been proposed as alternatives to Te compounds, and are being actively studied worldwide. Moreover, phosphides are gaining increasing attention as a candidate group for new thermoelectric materials. Formation of AgP 2 at Ag-InP junction interfaces has been reported [12]. Moreover, transition metal phosphides are well known as catalysts for various hydrotreatment processes such as hydrogen desulphurization and hydrogen denitrogenation. Actually, Ni, Co, Fe, Mo, and W phosphides have shown promise for use as catalysts [13].
Theoretical calculations have also been conducted for phosphides. First-principles calculations using GGA and HSE06 functional have been performed to elucidate electronic and phonon properties of PdPSeX (X=O, S, Te) and graphene-like boron phosphide monolayers [14,15]. First-principles calculations have been performed for AlP nanocrystals. Reportedly, the electronic properties approach those of bulk AlP as the nanocrystal size is increased [16].
However, because of their relatively low atomic weight and high speed of sound, many semiconductor phosphides, such as GaP [17] and InP [18], display high lattice thermal conductivity, which limits their application as thermoelectric materials. Recently, phosphides with low lattice thermal conductivity, such as the complex crystal-structured Ag 6 Ge 10 P 12 [19] and chain-structured Ag 3 SnP 7 [20], have been discovered. In this Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. study, we focused on the binary phosphide AgP 2 , which comprises a simpler composition and relatively complex crystal structure. Figure 1 shows the crystal structure of AgP 2 [21,22], which displays a P2 1 /c space group and monoclinic crystal system. The structure comprises three P(1) and one P(2) atoms that coordinate around Ag to form a tetrahedron (grey tetrahedra in figure 1). Although AgP 2 is a relatively simple binary system, it contains a relatively large number of atoms, namely four Ag (4e sites) and eight P (4e sites) atoms for a total of 12 atoms per primitive cell. Li et al chemically synthesised AgP 2 nanocrystals [23]. They reported that AgP 2 has a three-fold lower overpotential than that of Ag and is an excellent electrocatalyst for the reduction of CO from CO 2 . The synthesis of single crystals and polycrystals of AgP 2 using iodine has also been reported [24]; however, to the best of our knowledge, the physical properties of AgP 2 polycrystals have not been reported to date. In theoretical calculations, J. H. Pöhls et al reported the phonon bands of AgP 2 in a high-throughput screening study of density functional theory (DFT) calculations for metal phosphides [25]. However, this study focused on NiP 2 and did not discuss the detailed electronic and phonon structures of AgP 2 . In this study, we established a method to synthesise polycrystalline samples of AgP 2 and fabricate high-density sintered samples. We also investigated the electronic and phonon properties from both experimental and theoretical aspects.

Experiment details
In our first synthesis of AgP 2 , we attempted direct reaction synthesis in which the starting materials Ag and P are enclosed and heated in a quartz tube. However, in the direct reaction method, unreacted white phosphorus precipitated. The sample burned in air. We were unable to obtain the AgP 2 phase. Therefore, AgP 2 was synthesised by a chemical vapour phase transport method using iodine as the transport agent. The starting materials were Ag (99.9%, Kojundo Chemical Laboratory Co., Ltd.), P (99.9999%, Rare Metallic Co., Ltd.), and 2.56×10 -4 mol I 2 (99.9%, Fujifilm Wako Pure Chemicals Co., Ltd.). The mass of P was set at 0.4 g to prevent the quartz glass tube from exploding due to the vapour pressure of phosphorus. The mixture of starting materials was vacuum-sealed in a quartz tube (thickness, 1.8 mm; diameter, 8.5 mm; length, 10 cm) at a pressure of <5×10 -5 Torr. Polycrystalline AgP 2 was obtained by heating the quartz tube containing the sample at 600°C for 8 h. The polycrystalline AgP 2 pellets were sintered at 400°C for 15 min under 300 MPa pressure using the hot-press method.
The Seebeck coefficient S and thermal conductivity κ at 7-340 K were measured by the steady-state twoprobe method using a physical property measurement system (PPMS, Qantum Design Co., Ltd.). The electrical resistivity from 3 to 340 K and Hall coefficient R H from 50 to 300 K were measured by the four-probe method, using the AC transport option in PPMS. The R H was determined from the slope of the straight line attained by measuring the Hall voltage against the magnetic field in the range of ±5 T (see figure S1 in the Supporting Information (available online at stacks.iop.org/MRX/9/055901/mmedia)). A silver paste was used to adhere the electrode to the sample.

Computational methods
The E-k relationship, density of state, force on the cell, and total energy of AgP 2 were calculated using OpenMX [26]-a software package based on DFT, norm-conserving pseudopotentials, and optimised pseudoatomic basis functions. The exchange-correlation potential was calculated using the GGA-PBE functional [27]. The k-path of the first Brillouin zone was obtained using See-kpath [28]. The pseudo-atomic orbital basis functions were Ag7.0-s3p2d2f1 and P7.0-s3p2d2f1, each denoting the element, cutoff radius (Bohr units), and specification of optimised orbitals, respectively. The grid of k-points for the primitive cell was 16×18×14, and that for the supercell (2×2×2 primitive cell) was 4×5×3. The quasi-Newton method [29][30][31][32][33][34] was used for structural relaxation calculations, and the atomic positions and lattice parameters were optimised without applying any constraints until the force on the atoms was <10 -6 Ha Bohr −1 . The symmetry of the crystal before and after structural relaxation was analysed using Spglib [35], which confirmed that the space group did not change after structural relaxation. For all calculations, we set an energy cutoff of 500 Ryd for numerical integration.
The electronic properties were calculated using the electron transport code BoltzTraP [36], based on the Boltzmann transport theory. The electrical conductivity σ and Seebeck coefficient S can be expressed by equation (1): where e is the elementary charge, T is the absolute temperature, K n is the transport coefficient, η(E, T) is the spectrum conductivity at energy E and temperature T, μ is the chemical potential, and f FD is the Fermi-Dirac distribution function. The Fourier completion factor was set to 13, and the total number of k-points used for the transport coefficient approximated 50000. In the calculation of the Seebeck coefficient temperature dependence, the Fourier completion factor was set to 25 because numerous k-points are required when the chemical potential is located in the gap. The phonon properties were calculated using ALAMODE [37]. A 2×2×2 supercell of a structurally relaxed AgP 2 primitive cell was constructed, and the second-and third-order interatomic force constants (IFCs) were calculated using the supercell method. The atomic displacements used in the IFC calculations were 0.04 and 0.08 Å for the second-and third-order IFCs, respectively.
We calculated the phonon-phonon scattering intensity Γ q and phonon lifetime τ q using equation (2): where N is the number of all phonon modes, ω q represents the harmonic phonon frequency at q, F 3 is the thirdorder IFC, and n is the Bose-Einstein distribution function. Finally, we calculated the lattice thermal conductivity κ lat by solving the Boltzmann transport equation, assuming a relaxation time approximation, using equation (3): where m and n are elements of the lattice thermal conductivity tensor, T is the absolute temperature, V is the unit cell volume, N q is the number of phonon modes at q, C q is the specific heat, and v q is the phonon group velocity.

Results and discussion
3.1. Synthesis of polycrystalline AgP 2 Figure 2 shows the XRD spectra of the powdered and hot-pressed AgP 2 samples. These spectra are in good agreement with the XRD spectra calculated from the crystal structure of AgP 2 , indicating that we successfully obtained a single-phase sample. The lattice constants calculated from the XRD spectra of the hot-pressed sample were a=6.227(6) Å, b=5.060(5) Å, and c=7.81(1) Å, all of which are within 0.2% of the literature values [21]. The peak intensity was different from that observed in the simulation, suggesting that the sample was slightly oriented. The SEM-EDS compositional analysis revealed a 1:1.9(1) Ag:P ratio, which was near-identical -within the error range-to the preparation composition. Figure 3(a) shows the temperature dependence of the electrical resistivity of hot-pressed AgP 2 . The heat flow and current direction were in-plane same direction of pellet for all measurements. The electrical resistivity ρ decreases exponentially with increasing temperature, indicating that AgP 2 is an Arrehenius-type semiconductor. The electrical resistivity at 300 K is 24.9 mΩm, which is relatively high. The electron thermal conductivity estimated from the Wiedemann-Franz law is almost zero, and thus, the contribution of electronic thermal conduction can be neglected. Figure 3(b) shows the lnρ-T −1 plot. The plot is linear at temperatures >80 K, indicating that AgP 2 is an Arrehenus-type semiconductor. At high temperatures >480 K, the slope increases, indicating that the excitation of carriers with higher activation energy occurs. The estimated activation energies (E a ) in the temperature range 80-480 K and at temperatures >480 K are 46 and 249 meV, respectively. This suggests that the activation energy corresponds to thermal excitation from the valence band to the impurity level and from the valence band to the conduction band. A possible impurity level is the loss of a small amount of P. Calculations of the electronic densities of state with a 1.6% loss of P show that a sharp impurity level-derived density of state occurs at the upper edge of the valence band. (See figure S2 in the Supporting Information). We evaluated thermal stability using high-temperature measurements of ρ and Seebeck coeeficient S, and SEM-EDS measurements, which revealed AgP 2 as destabilized around 680 K, resulting in P desorption. The section of Supporting Information describing thermal stability of AgP 2 presents important details. Figure 4 shows the temperature dependence of the Hall coefficient R H of AgP 2 . R H is positive across all temperature ranges, indicating that the major carrier of AgP 2 is a hole. As the temperature increases, the R H decreases exponentially. This qualitatively explains the electrical resistivity results. Assuming a single carrier model, the activation energy estimated from the temperature dependence of R H is ∼43 meV, which is in good agreement with the E a estimated from figure 3(b). This corresponds to an exponential increase in the carrier concentration due to thermal excitation from the upper valence band to the impurity level in the 50-300 K temperature range. The Hall mobility of AgP 2 , estimated from the Hall coefficient and electrical resistivity, is ∼51 cm 2 V −1 s −1 at 300 K. This value is comparable to that of polycrystalline Mg 2 Si [38]. Figure 5(b) shows the lattice thermal conductivity κ lat -T −1 plot. The κ lat of AgP 2 at 300 K is as low as 1.2 W K −1 m −1 . Surprisingly, this value is lower than that of Ag 6 Ge 10 P 12 [19], which has a complex crystal structure containing as many as 28 atoms in the basic unit cell, and is comparable to the lattice thermal conductivity of Bi 2 Te 3 [39]. At temperatures >152 K, the plot is linear, and phonon-phonon scattering is dominant at relatively low temperatures. This implies that the probability of the phonon-phonon scattering of AgP 2 is high at relatively low temperatures.

Transport properties of hot-pressed AgP 2
The experimental results revealed that AgP 2 is a phosphide with both high Hall mobility and low lattice thermal conductivity. In this study, we investigated the electron and phonon transport properties of AgP 2 using first-principles calculations to theoretically clarify the origin of the high Hall mobility and low lattice thermal conductivity.    Figure 6(a) shows the E-k relationship for AgP 2 . The origin of the energy coincides with the chemical potential at 300 K, as calculated from first-principles calculations. The modified chemical potential μ Exp was determined from the chemical potential at 300 K, where the calculated and experimental Seebeck coefficients agree (see figure S3 in the Supporting Information). The slight shift in the position of the chemical potential, compared to the first-principles calculation, suggests the presence of some defects in the AgP 2 sample. This is possibily an impurity level caused by a defect in the P atom, based on the results of figure 3(b) and S2 (Supporting Information). Thus, μ Exp is located in the forbidden band, and AgP 2 is an intrinsic semiconductor. The band gap E g estimated from the E-k relationship is 498 meV, which is almost twice the activation energy (E a =249 meV) estimated from the slope of the lnρ-T −1 plot [figure 3(b)] at high temperatures (>480 K). At the upper end of the valence band, the band at the Y 2 point has the highest energy, and this band contributes the most to the electrical conduction of AgP 2 . Figure 6(b) shows the electrical conductivity στ −1 of AgP 2 obtained from the E-k relationship. The στ −1 decreases exponentially from the band edge to the forbidden band, with a στ −1 value of 1.22×10 16 Ω −1 m −1 s −1 at μ Exp . Compared to the electrical resistivity ρ at 300 K, the carrier relaxation time τ is estimated at ∼3.3 fs. This value is relatively long for a polycrystalline material [40].

Theoretical transport properties of AgP 2 3.3.1. Electronic properties
Based on the obtained carrier relaxation time τ and experimental Hall mobility μ H , we estimated the effective mass m * of holes in AgP 2 at 300 K using equation (4): where e is the elementary charge. The effective mass of holes m * in AgP 2 at 300 K was estimated to be 0.11 m 0 , while the effective mass of free electrons is m 0 . This indicates that the effective mass of carriers in AgP 2 is much lighter than that of free electrons. Thus, the large Hall mobility of AgP 2 was attributed to the relatively long carrier relaxation time and small effective mass of AgP 2 . Figure 7 shows the highest occupied molecular orbital (HOMO) at the Y 2 point. At this point, the Ag 4d orbitals and P(1) 3p orbitals are dominant. Because P (2) has no orbitals at the Y 2 point, the conduction hole at this point is understood to propagate on the Ag-P(1) network.
The electronic structure and electronic transport properties of AgP 2 were investigated using first-principles calculations. AgP 2 is an intrinsic semiconductor and exhibits a high Hall mobility comparable to that of Mg 2 Si. This is due to its relatively long relaxation time (3.3 fs) and small effective carrier mass (0.11 m 0 ). The carrier conduction is dominated by hole conduction through a network of Ag 4d orbitals and P(1) 3p orbitals. Figures 8(a) and (b) show the phonon dispersion relations and partial densities of state of the constituent atoms of AgP 2 . The phonon dispersion relation is colour-mapped for each phonon mode according to the value of the mode Grüneisen parameter. The optical phonon modes occur in the low-frequency region (∼60 cm −1 ), suggesting that the scattering probability of phonon-phonon scattering at the Brillouin zone boundary is high at room temperature. This qualitatively supports the experimental results for κ lat -T −1 . A relatively large anharmonic phonon mode with γ > 2 occurs in the 50-100 cm −1 frequency range. This mode has a small energy dispersion and a slight gap in the 90-100 cm −1 frequency region. From the partial densities of state of the phonons, the contribution of the Ag atom to the phonon mode is dominant in the frequency region below 100 cm −1 , indicating that the highly anharmonic phonon mode with small energy dispersion in the 50-100 cm −1 region originates from the Ag atom. The phonon mode of the Cu atom in CuP 2 , which has the same crystal  structure as AgP 2 , was reported by Ji et al [41] to display a large-mode Grüneisen parameter and high anharmonicity, and the phonon mode of the Ag atom in AgP 2 is similar.

Phonon properties
To clarify the origin of the anharmonic phonon mode of the Ag atoms in AgP 2 , we calculated the difference in electron density in real space and investigated the bonding state. The difference in electron density was calculated by subtracting the electron densities of the isolated Ag and P atoms from the electron density of the AgP 2 crystal. Figure 9 shows a two-dimensional colour map of the difference electron density distribution near the P(1)-Ag-P(1)-Ag and P(1)-Ag-P(2) bonds in AgP 2 . The Ag-P bonds are ionic because of the polarity from the difference electron density between the bonds. P(1)-Ag-P(1)-Ag is a quadrupole, while P(1)-Ag-P(2) comprises a dipole-like charge polarity. In addition, the polarity from the difference electron density between the Ag-P(2) bonds is larger than that between the Ag-P(1) bonds, suggesting that the bonding strength of the latter is weaker than that of the former. These results indicate that the bonding around the Ag atoms is strongly anisotropic. The difference electron density between the P-P bonds is concentrated at the centre of the two atoms, which is a typical covalent bond (Figure not shown). AgP 2 is a crystal with a mixture of ionic and covalent bonds, and the strong anisotropy of the bonds around the Ag atoms is suggested to be the origin of the large anharmonic vibration of the Ag atoms.  To compare the anharmonicity of the phonon structure of AgP 2 with that of other materials, the meansquare displacement (MSD) of the constituent atoms, relaxation time of phonon-phonon scattering, and phonon group velocity were compared with those of CuP 2 [42], which has the same crystal structure, and InP [43], which constituent atoms are close in atomic weight. The calculation conditions for the phonon properties of CuP 2 and InP are listed in table S1 in the Supporting Information. Figures 10(a)-(c) show the temperature dependence of the MSD of the constituent atoms in AgP 2 , CuP 2 , and InP, respectively. The MSD increases monotonically with increasing temperature, and the scattering crosssection in phonon-phonon scattering also increases. In AgP 2 , the MSD of the Ag atoms at 300 K is between 1.6and 1.7-fold larger than the that of the P atoms. Notably, the MSD of the Cu atoms in CuP 2 is approximately 1.5fold larger than that of the P atoms, but not as large as that of the Ag atoms in AgP 2 . This indicates that the scattering cross-section of the Ag atoms in AgP 2 is larger than that of the Cu atoms in CuP 2 . The MSD of In in InP is approximately 1.1-fold larger than that of P, and the In atoms in InP do not vibrate as much against P as do the Ag atoms in AgP 2 . Figures 11(a)-(c) respectively show the phonon relaxation time τ q , absolute value of the phonon group velocity |v q |, and mean free path (MFP) of phonon-phonon scattering in the three-phonon processes of AgP 2 , CuP 2 , and InP (see equation (3) for definitions of τ q and v q ). The temperature of the Bose-Einstein distribution is 300 K, while the maximum τ q of AgP 2 is ∼500 ps, which is relatively short. The overall decrease in τ q compared to that of CuP 2 , which has the same crystal structure, indicates that the scattering probability in phononphonon scattering is higher than that of CuP 2 . This is qualitatively consistent with the MSD results in figure 10, in which the Ag atoms in AgP 2 present a larger phonon scattering cross-section than do the Cu atoms in CuP 2 . The overall relaxation time is shorter than that of InP, indicating that AgP 2 exhibits a large phonon-phonon scattering probability. The phonon group velocity in AgP 2 is shifted to a lower speed than that in CuP 2 , attributed to the difference in the atomic weights of Ag and Cu. The highest phonon group velocity in InP is relatively close to that of AgP 2 , indicating that the phonon group velocity in AgP 2 is not particularly slow. The frequency distribution of the phonon MFP in AgP 2 is shifted to a much shorter MFP than that in CuP 2 , which is caused by the high phonon-phonon scattering probability of the Ag atoms in AgP 2 and the lower phonon group velocity due to the relatively heavy atomic weight of the Ag atoms. Figure 12 shows the temperature dependence of the lattice thermal conductivity of AgP 2 . Following Matthiessen's rule, the average values of the lattice thermal conductivities in the a-, b-, and c-axes were calculated using equation (5): where κ lat_ave is the average lattice thermal conductivity along the a, b, and c axes , κ lat_a , κ lat_b , and κ lat_c are the respective lattice thermal conductivities along the a, b, and c axes. The lattice thermal conductivity of single-crystal AgP 2 obtained from theoretical calculations is 1.80 W K −1 m −1 at 300 K. This is an extremely low value comparable to that of single-crystal Bi 2 Te 3 [39]. The theoretical temperature dependence of the lattice thermal conductivity, assuming grain-boundary and phonon-phonon scattering for a particle size of 300 nm, well-reproduces the experimental value of κ lat for AgP 2 . This indicates that the average size of the effective grain boundary for phonons is ∼300 nm. Both experimental and theoretical results indicate that the large phonon scattering due to the anharmonic vibrations of the Ag atoms and the decrease in the phonnon group velocity associated with the relatively large atomic weight are the origins of the low lattice thermal conductivity of AgP 2 .

Conclusions
In this study, a polycrystalline sample of binary phosphide AgP 2 was synthesised by the chemical vapour transport method to obtain single-phase polycrystalline AgP 2 , which was then hot-pressed into a high-density sintered compact. The as-synthesised polycrystalline AgP 2 has a high Hall mobility μ H of 51 cm 2 V −1 s −1 at 300 K, which is comparable to that of Mg 2 Si, and a low lattice thermal conductivity κ lat of ∼1.2 W K −1 m −1 , which is comparable to that of Bi 2 Te 3 .
To clarify the origin of the high μ H and low κ lat , the electron and phonon transport properties were investigated in detail by first-principles calculations. The results revealed that AgP 2 has a semiconducting electronic structure in which the respective 4d and 3d orbitals of Ag and P dominate the electrical conduction at the Y 2 point in the upper valence band. The carrier relaxation time of AgP 2 is 3.3 fs at 300 K, which is relatively long for a polycrystalline material, and its effective mass hole is 0.11-fold lighter than that of free electrons. The high μ H of AgP 2 is attributed to the relatively long carrier relaxation time and small effective mass.
Phonon calculations revealed that AgP 2 exhibits highly anharmonic phonon modes in the low-frequency region (50-100 cm −1 ), with small energy dispersion from the Ag atoms and a mode Grüneisen parameter >2. Compared to CuP 2 , which has the same crystal structure, and InP, which has the same atomic weight of the constituent elements, AgP 2 exhibits a shorter phonon relaxation time and lower group velocity. The κ lat obtained from the theoretical calculations well-reproduced the experimental values, assuming phonon-phonon scattering and grain-boundary scattering by 300 nm grains. It is theoretically clarified that the high scattering probability of phonon-phonon scattering due to the large anharmonic vibration of Ag atoms is the origin of the low κ lat of AgP 2 . AgP 2 shows high potential as a thermoelectric material, but because it is an intrinsic semiconductor, its carrier concentration is low. To use AgP 2 as a thermoelectric material, it is necessary to dope the carriers by elemental substitution or introduction of defects.