A. Structure refinement

All the observed XRD peaks were successfully indexed with an incommensurate composite crystal structure model similar to that for HMSs.^{Reference Miyazaki, Igarashi, Hayashi, Kajitani and Yubuta7}Figure 1 illustrates the adopted structure model. The Rh₁₇Ge₂₂ phase consists of two tetragonal subsystems of [Rh] and [Ge] with a common a-axis but c-axes of different length. The [Rh] subsystem has a β-Sn-type arrangement of Rh atoms and exhibits the three-dimensional (3D) space group I4₁/amd with cell dimensions of a ∼ 5.6 Å and c _Rh ∼ 4.6 Å [Fig. 1(a)]. The [Ge] subsystem originally had 3D unit cell dimensions of a _Ge = a/√2 ∼ 3.9 Å and c _Ge ∼ 3.6 Å, with Ge atoms at the origin (2a sites) of the P4/nnc space group. The unit vectors a_Ge and b_Ge in the [Ge] subsystem were suitably transformed into a = a_Ge + b_Ge and b = −a_Ge + b_Ge. Simultaneously, the origin in the [Ge] subsystem was shifted to (1/4, 1/4, 0) as shown in Fig. 1(b), to treat the composite crystal with a [Rh] subsystem-based superspace group notation. Using the incommensurate c-axis ratio γ = c _Rh/c _Ge, the compound can be expressed as RhGe_γ. The incommensurate structure model for the present compound showing the Rh and Ge atoms in the range of a × a × 4c _Rh is depicted in Fig. 1(c). The reason why the compound is called a NCL phase can easily be recognized, as the Rh atoms form a rigid rectangular chimney, and pairs of Ge atoms at the same height are located to build helically rotating ladders between the Rh atoms.

FIG. 1.

(a) Unit cells of the [Rh] subsystem and (b) the [Ge] subsystem. (c) Incommensurate crystal structure model of RhGe_γ showing the Rh and Ge atoms in the range a × a × 4c _Rh.

We adopted the (3 + 1)-dimensional superspace group I4₁/amd(00γ)00ss, in which the [Rh] subsystem has the 3D space group I4₁/amd, and the modulation vector k = (00γ) exists between the subsystems. The 00ss term indicates that the +1/2 translation should be added to the symmetry operations related to the terms 2/m and 2/d. A more detailed representation of the translation parts and symmetry operation can be found elsewhere.^{Reference Miyazaki and Khidirov12}

Figure 2 shows the observed, calculated, and difference profiles of the XRD data for RhGe_γ measured at 293 K. The short vertical lines below the patterns indicate the peak positions of possible Bragg reflections. The peaks denoted as hkl0 and hk0m are the fundamental reflections derived from the [Rh] and [Ge] subsystems, respectively, whereas the hklm peaks are satellite reflections. The final R factors were R _wp = 7.6% and R _p = 5.9%, with a goodness-of-fit factor of R _wp/R _E = 2.2. The lattice parameters were refined to be a = 5.6043(3) Å, c _Rh = 4.6161(2) Å and γ = 1.2927(1), viz., c _Ge = 3.5709(3) Å. The numbers in parentheses represent the estimated standard deviations of the last significant digit. The resulting c-axis ratio was slightly different from that of the commensurate Rh₁₇Ge₂₂ phase of 22/17 = 1.2941; the VEC value of the present sample is estimated to be 14.17.

FIG. 2.

Observed, calculated, and difference profiles of powder XRD patterns for RhGe_γ recorded at 293 K. The peaks denoted as hkl0 and hk0m are the fundamental reflections derived from the [Rh] and [Ge] subsystems, respectively, whereas the hklm peaks are satellite reflections.

B. Detailed description of the modulated structure

For most incommensurate composite crystals, positional (displacive) modulation of the atomic sites is commonly observed, owing to the potential interactions between atoms arising from the different periodicity of the two subsystems. In the present study, we introduced up to fourth-order cosine and sine components of the Fourier terms, A _ij and B _ij (i = 1–4, j = x, y, z). The positional modulation function u _j(ν) of an atom then becomes:

(1)$${u_j}\left( \nu \right) = \mathop\sum\limits_{i = 1}^4 {\left[ {{A_{ij}}\cos \left( {2\pi i\nu } \right) + {B_{ij}}\cos \left( {2\pi i\nu } \right)} \right]} \quad ,$$

where ν represents the fourth superspace coordinate without modulation in the [Rh] or [Ge] subsystem. For the [Rh] subsystem, ν is defined as γz _Rh, whereas ν equals (1/γ)(z _Ge + t) for the [Ge] subsystem; the notation z stands for the z-coordinate without modulation. The parameter t is defined as the distance from 3D space and, therefore, corresponds to another expression of the (3 + 1)-dimensional superspace coordinate.

Table I summarizes the refined Fourier amplitudes for the positional modulations and the isotropic displacement parameters, U _iso, for each atom. Owing to the superspace group symmetry, the number of refinable parameters for the Fourier terms is limited via A _1x = A _1y = B _1x = −B _1y and A _3x = A _3y = −B _3x = B _3y. Only the even terms of the sine wave along the z direction are allowed for the Rh atoms. In contrast, the odd terms of the sine and cosine waves are allowed for both the x and y components for the Ge atoms, along with the B _4z term (see details, Refs. Reference Miyazaki, Igarashi, Hayashi, Kajitani and Yubuta7 and Reference Miyazaki and Khidirov12).

TABLE I.

Refined positional modulation wave components and isotropic displacement parameters, U _iso, for RhGe_γ at 293 K.

Figure 3 shows the positional modulations obtained for the x, y, and z coordinates of each atom, plotted against the fourth superspace coordinate, ν. The right vertical axes are rescaled to represent the displacement (in angstroms) for each atom. All the displacements are periodic in the interval 0 ≤ ν ≤ 1.0. For Rh atoms, positional modulation is only allowed in the z direction, and the maximum displacement (∼0.20 Å) from z = 0 is observed at ν ∼ 0.14, 0.36, 0.64, and 0.86. The displacement of the Ge atoms along z is slightly smaller than that of the Rh atoms, and a maximum displacement of (∼0.13 Å) from z = 1/4 is seen at ν ∼ 0.06, 0.19, 0.31, 0.44, etc. On the basis of the large sine and cosine components, the rotational modulation in the x and y directions is significant for the Ge atoms. The modulation waves in the x and y directions are identical to the phase-shift Δν = 1/4. A maximum displacement of ∼0.56 Å from x = 1/4, equivalent to Δx ∼ 0.1, occurs at ν ∼ 0.13 and 0.63, and from the y = 1/4 position at ν ∼ 0.37 and 0.87. A similar rotational modulation, i.e., Δx ∼ 0.1, is reported for most incommensurate NCLs, such as (Mo_1−xRh_x)Ge_γ,^{Reference Rohrer, Lind, Eriksson, Larsson and Lidin13} (Cr_1−xMo_x)Ge_γ,^{Reference Rohrer, Lind, Eriksson, Larsson and Lidin14} and HMSs.^{Reference Miyazaki, Igarashi, Hayashi, Kajitani and Yubuta7}

FIG. 3.

Positional modulations obtained for the Rh and Ge atoms plotted as a function of the fourth superspace coordinate, ν. The right vertical axes are rescaled to represent the displacement for each atom.

Figures 4(a) and 4(b) show interatomic distances plotted against the internal coordinate, t. The two periodic curves around 3.0 Å in Fig. 4(a) represent the four nearest-neighbor Rh–Rh distances, where each curve is actually made up of two overlapping curves associated with bonds of equal length. The Rh–Rh distances of 2.92 and 3.19 Å are longer than would be expected on the basis of the atomic radius of Rh, r _Rh = 1.345 Å.^{Reference Inoue, Chikazumi, Nagasaki and Tanuma15} However, such long distances are characteristics of the β-Sn-type coordination of the M atoms in incommensurate NCLs. In contrast to the eight coordination of the Mn atoms with Si in HMSs, each Rh atom is usually coordinated to six Ge atoms in the present germanide. The eight coordination of Rh–Ge bonds can be exceptionally observed at t ∼ 0 and 1/2, around which a pair of bonds with lengths of ∼3.0 Å, and another pair with lengths of ∼2.4 Å are formed. The nearest-neighbor Ge–Ge distances show a periodic variation as a function of t, as shown in Fig. 4(b). The minimum distance of 2.76 Å is slightly longer than that expected from the 2 × r _Ge value of 2.45 Å. The longer observed interatomic distances than those determined from a simple summation of atomic radii may be due to the large number of electron carriers introduced into the anti-bonding bands, as described in Sec. III. C.

FIG. 4.

Interatomic distances for (a) Rh–Rh and Rh–Ge, and (b) Ge–Ge as a function of the fourth superspace coordinate t for RhGe_γ at 293 K.

C. Electronic structure

To determine the electronic structures of incommensurate compounds having a dimensional symmetry higher than 3, a commensurate 3D structure must be approximated. Through the superspace approach, any approximated structure can be constructed as long as the relationship mc _Rh ≈ nc _Ge is satisfied. In the present study, we adopted an approximate unit cell of a × a × 10c _Rh, in which 10c _Rh = 46.161 Å roughly equals 13c _Ge = 46.422 Å. The space group and fractional coordinates of the Rh and Ge atoms, derived from the StructGen™ and Initialize calc processes in the calculation,^{Reference Blaha, Schwarz, Madsen, Kvasnicka and Luitz10} are summarized in Table II. The approximated structure contains 40 Rh sites (1a–1d) and 26 Ge sites (2e), with the monoclinic P2 space group (α = β = γ = 90°) whose unique axis passes through (1/2, 1/2, 0) and is parallel to the c-axis.

TABLE II.

Commensurate structure model used to calculate the electronic structure for RhGe_γ. Space group: P2 (No. 2, unique axis c), a = b = 5.604 Å, c = 10c _Rh = 46.161 Å, and α = β = γ = 90°.

In Fig. 5, we show the energy dependence of the total DOS and the contributions of the Rh-d and Ge-s + p bands of RhGe_γ with γ = 1.2927(1), on the basis of the commensurate approximation of the incommensurate structure. The overall features of these results are quite similar to those obtained from the structural data for Rh₁₇Ge₂₂ having a body-centered tetragonal unit cell with lattice parameters a = 5.604 Å and c = 78.45 Å, and comprising 9 Rh and 6 Ge sites.^{Reference Jeitschko and Parthé6} The broad VB ranging from −5 to −1 eV has a predominantly Rh-d character, while the p and s-character of Ge dominates the band between −12 and −8 eV. As expected, the compound can be regarded as a metal or more precisely a heavily electron-doped degenerated semiconductor. For such a case, the Seebeck coefficient can be represented as the well-known Mott equation mentioned below:

(2)$$S \approx {{{\pi ^2}} \over 3}{{k_{\rm{B}}^2} \over e}{1 \over {N\left( E \right)}}{\left( {{{\partial N\left( E \right)} \over {\partial E}}} \right)_{E = {E_{\rm{F}}}}}\quad ,$$

where k _B, e, and N(E) are the Boltzmann's constant, elementary charge, and DOS at a particular energy E, respectively. Hence, negative S values would be expected for the present sample, as the Fermi level E _F (E = 0) is located near the bottom of the conduction band (CB), only slightly above the narrow pseudo gap of ∼0.5 eV between it and the VB. However, the absolute value of |S| would be smaller because the energy derivative of the DOS at E _F is not steep. Instead, high σ values should be observed as the electrical conductivity is roughly proportional to N(E). A partial substitution of Rh with group 6–8 elements, such as Fe, would be expected to shift E _F toward the CB bottom, and the solid solution would show a larger negative |S|. If the E _F level is further shifted toward the VB top with VEC < 14, one may obtain p-type samples as well in the RhGe_γ-based solid solution.

FIG. 5.

Total DOS and the contribution of Rh-d and Ge-s + p bands of RhGe_γ with γ = 1.2927(1) on the basis of the commensurate approximation of the incommensurate structure.

D. TE properties

Figure 6 plots the temperature dependence of S, σ, and the power factor P = S ²σ for a polycrystalline sample of RhGe_γ. For the simultaneous measurements of S and σ, an SPS-processed pellet with a thickness of 1 mm was used. The pellet was cut into a rectangular shape, and both the electrical current and temperature gradient were applied perpendicular to the pressing direction. Samples for Hall measurement were cut from the remaining portion of the pellet. The sintered density was above 98% of the theoretical density of 9.01 g/cm³. Since samples as thick as 10 mm could not be obtained with a reasonable sintered density, we prepared another pellet with a thickness of 1 mm. The temperature gradient was applied parallel to the pressing direction to measure κ. As expected from the previous sections, a negative Seebeck coefficient of S = −25 μV was observed at 333 K. The absolute value of S was not as high as that for HMSs (∼130 μV) at around room temperature, but it gradually increased with T, reaching S = −53 μV at 984 K. The temperature dependence of the electrical conductivity also exhibited metallic behavior; σ was 7.3 × 10³ S/cm at 333 K, and decreased steadily to 4.8 × 10³ S/cm at 984 K. A lower |S| and higher σ imply that the RhGe_γ phase has a higher carrier concentration than HMSs. The carrier (electron) concentration, n _e, of RhGe_γ was 6.8 × 10²¹ cm⁻³ at 300 K, which is more than twice the hole concentration, n _h, for MnSi_1.7361, i.e., 2.5 × 10²¹ cm⁻³ at room temperature. The measured carrier mobility of μ = 6.7 cm²/Vs at room temperature, compared to the value of 1.7 cm²/Vs for MnSi_1.7361, should also lead to a higher electrical conductivity. Although the |S| values were smaller than 100 μV over the whole temperature range, the TE power factor was determined to be a reasonable value (0.47 mW/K² m at 333 K) and increased steadily exceeding 1 mW/K² m above 600 K; the highest value of 1.4 mW/K² m was recorded at 984 K.

FIG. 6.

Temperature dependence of the Seebeck coefficient S, electrical conductivity σ, and power factor P = S ²σ for a polycrystalline sample of RhGe_γ.

Reflecting the higher σ values, κ shows even higher values among NCLs. The total thermal conductivity was κ = 5.1 W/K m at 302 K and increased at a constant rate to 11.3 W/K m at 1000 K. As the measurement directions for σ and κ were different, pellets, to some extent, necessarily exhibited a preferred orientation. Nevertheless, the XRD pattern recorded from the pellet surface showed negligibly little difference to that for randomly oriented powder samples. Owing to the difference in measurement direction, we refrain from describing precise ZT values. However, a qualitative discussion is possible, and the maximum ZT of ∼0.1 at 900 K is presumably the best performance for the present germanide system. The compatibility factor^{Reference Snyder and Ursell16} of RhGe_γ, $\Sigma = {{\left( {\sqrt {1 + ZT} - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt {1 + ZT} - 1} \right)} {ST}}} \right. \kern-\nulldelimiterspace} {ST}}$, is evaluated to be Σ = 6.87 at 333 K and Σ = 0.28 at 984 K, based on the present ZT values, revealing that the germanide is not a good partner of the p-type analog HMS, Σ = 1.50 at 290 K and Σ = 0.62 at 1000 K.^{Reference Miyazaki, Kikuchi, Koumoto and Mori5}