Introduction

The emergence of superconductivity (SC) in the vicinity of a competing electronic state has received great attention over the past few decades. Well-known examples include cuprates1, heavy fermions2, transition metal chalcogenides3, and iron pnictides4. Their parent compounds exhibit either an antiferromagnetic5,6,7 or a charge-density-wave (CDW) transition8, which is sometimes accompanied by a change from metallic to nonmetallic behavior9,10. Understanding the interplay between these transitions and SC not only sheds light on the pairing mechanism, but also provides an effective route for the search of new superconductors. Recently, the topological aspects of superconductors have become a focus of interest11,12,13. These topological superconductors (TSCs) have a bulk superconducting gap but host gapless edge states consisting of Majorana fermions. Many proposals to create TSCs have been put forward and a significant progress has been made. In particular, chemical doping of topological materials is shown to be one of the most promising ways for this purpose14,15,16,17,18,19,20. However, in all known cases, there exit no competing ground state in the parent compounds.

Re3Ge7 is the only binary phase in the Re-Ge system and crystallizes in the orthorhombic structure with the Cmcm space group21, which is sketched in Fig. 1a, b. Its structure can be viewed as consisting of isolated Re3B-type ReGe3 and double NbAs2-type Re2Ge4 prisms21. Although Re3Ge7 has been known to exist for nearly 40 years, its thermodynamic and transport properties are reported only very recently22. The results indicate that the compound is weakly diamagnetic while undergoes a second-order phase transition below 58.5 K as confirmed by specific-heat measurements. This transition is accompanied by a drop in the diamagnetic susceptibility, a metal-to-insulator-like transition in resistivity, and a strong reduction in electron carrier concentration. While it is suspected that the phase transition in Re3Ge7 has a structural origin, no low-temperature X-ray study has been performed. On the other hand, theoretical calculations suggest that Re3Ge7 is a high-symmetry point topological semimetal23. Since the combination of these properties in a single material is uncommon, it is of significant interest to see what would be the ground state once the phase transition is suppressed.

Fig. 1: Schematic crystal structure and x-ray diffraction results of Re3Ge7−xGax.
figure 1

a A three-dimensional view of the orthorhombic structure of Re3Ge7. b Structure of Re3Ge7 projected perpendicular to the a-axis. The circles indicate the ReGe3 and Re2Ge4 building blocks. c Room temperature powder XRD patterns for the series of Re3Ge7−xGax samples. d–g Variation of a-, b-, c-axis lattice parameters and unit-cell volume of Re3Ge7−xGax as a function of the Ga content x.

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Here we show that, upon hole doping by substituting Ge with Ga, Re3Ge7 turns into a bulk superconductor. The resulting temperature-doping phase diagram for Re3Ge7−xGax resembles closely those of the correlated electron systems. Especially, as the metal–insulator transition is suppressed continuously and disappears completely for x > 0.2, a dome-like superconducting phase is observed for x between 0.2 and 0.4 with a maximum Tc of 3.37 K at x = 0.25. The effect of Ga doping on the electronic band structure and topology in Re3Ge7 is investigated by first principles calculations, and the implication of these results is discussed.

Results and discussion

Structural characterization

The X-ray diffraction (XRD) results for the series of Re3Ge7−xGax samples at room temperature are displayed in Fig. 1c. The patterns are very similar and all the diffraction peaks can be well indexed on the basis of an orthorhombic unit-cell with the Cmcm space group. The refined lattice parameters as well as the unit-cell volume V are plotted as a function of the Ga content x in Fig. 1d–g. For undoped Re3Ge7 (x = 0), the a-, b-, and c-axis lattice constants are found to be 3.227(1) Å, 9.044(1) Å, and 21.960(1) Å, respectively, which are in excellent agreement with the previous report21. With increasing x, both a- and b-axis expand, which is as expected since since the atomic radius of Ga (1.388 Å) is larger than that of Ge (1.349 Å)24. Nevertheless, the c-axis shrinks more rapidly, which leads to a small contraction of the unit-cell volume (up to ~0.2%).

In Re3Ge7, Re atoms occupy two different crystallographic sites (0, 0.074, 0.559) and (0, 0.48, 0.25), and there are four distinct sites for Ge: (0, 0.061, 0.163), (0, 0.324, 0.137), (0, 0.353, 0.532), and (0, 0.761, 0.25). For the structural refinements of Re3Ge7−xGax, the Ga atoms are assumed to be distributed randomly on the four Ge sites (see Table 1). Representative refinement results in the 2θ region near the strongest peak for x = 0, 0.2, and 0.4 are shown in Fig. 2a–c (results for full patterns are shown in Supplementary Fig. 1). In all cases, the calculated XRD patterns match well with the observed ones, which is corroborated by the small Rwp (4.4–5.9%) and Rp (3.2–4.3%) factors. Hence all samples are free of discernible impurities, confirming their high quality. In passing, we have also performed low-temperature XRD measurements on Re3Ge7 down to 15 K. The data reveal that the lattice parameters vary smoothly with decreasing temperature and there is no evidence for a structural transition (see Supplementary Fig. 2).

Table 1 Atomic coordinates for Re3Ge7−xGax.
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Fig. 2: Structural refinement of Re3Ge7−xGax.
figure 2

a–c Structural refinement profiles in the 2θ range of 20 to 50∘ for the Re3Ge7−xGax samples with x = 0, 0.2, and 0.4, respectively.

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Suppression of the metal–insulator-like transition

A systematic change in the resistivity (ρ) of Re3Ge7−xGax is observed with increasing x, which is shown in Fig. 3a. On cooling below room temperature, the pristine Re3Ge7 exhibits a metallic behavior but undergoes a metal–insulator transition below TMI = 57.3 K. At 1.8 K, the residual resistivity ratio (RRR) is much smaller than unity. This overall behavior is well consistent with that observed in single-crystal samples22. When Ga is introduced into the system, the ρ at high temperature increases monotonically and the metal–insulator transition is gradually suppressed. Actually, the ρ value at 1.8 K for x = 0.1 is only about half that for x = 0. In addition, the application of magnetic field has little effect on the TMI while results in a positive magnetoresistance at low temperature (data not shown). As x increases above 0.2, TMI can no longer be resolved and RRR becomes significantly larger than one with a maximum value of 7.4 at x = 0.25, signifying a good metallic behavior.

Fig. 3: Evolution of resistivity, magnetic susceptibility, specific-heat, and Hall coefficient with Ga content in Re3Ge7−xGax.
figure 3

a–c Temperature dependence of resistivity, magnetic susceptibility, and specific heat below 300 K for the series of Re3Ge7−xGax samples, respectively. The vertical dashed line is a guide to the eyes. In (a), the arrow marks the increasing of Ga content x. In (c), the horizontal line is the Dulong–Petit limit of 3NR = 249.42 J mol−1 K−1, and the inset shows a zoom of the data between 50 and 62 K. d, e Magnetic field dependence of Hall resistivity at various temperatures for the samples with x = 0 and 0.2, respectively. The circles and arrows are a guide to the eyes. f Temperature dependence of Hall coefficient for the series of samples. The anomalies are marked by the arrows, and the data for both x = 0 and 0.1 are divided by a factor of 10 for better illustration.

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The doping evolution of temperature-dependent magnetic susceptibility χ(T) measured under a field of 7 T for Re3Ge7−xGax is displayed in Fig. 3b. All the χ(T) data are negative and nearly temperature independent, substantiating that the magnetic properties of these samples are dominated by the core diamagnetism. With increasing x, however, the χ(T) data become less negative, and hence the Pauli paramagnetic contribution seems to grow with the Ga content. This trend implies an enhancement of the density of states at the Fermi level [N(EF)] induced by Ga doping, consistent with the results shown below. For x ≤ 0.2, a drop in χ(T) is detected and its magnitude decreases as the increase of x. Note that the onset temperature of this anomaly agrees with TMI determined from resistivity measurements (see the dashed line). Across the metal–insulator-like transition, it is reasonable to speculate that a gap opens at the Fermi level and hence N(EF) decreases3. Since the Pauli paramagentic susceptibility is proportional to N(EF), this leads to a decrease in χ as observed experimentally. At low temperature, the χ(T) data exhibit a weak upturn, which is ascribed to a small amount of paramagnetic impurities.At low temperature, the χ(T) data exhibit a weak upturn, which is ascribed to a small amount of paramagnetic impurities.

The specific heats Cp of Re3Ge7−xGax are also measured and shown in Fig. 3c. The data for all x values almost overlap with each other. In addition, the Cp values at high temperatures are close to the Dulong–Petit limit of 3NR = 249.92 J mol−1 K−1, where N = 10 and R = 8.314 J mol−1 K−1 is the molar gas constant. On close examination, a Cp anomaly is found near 57 K for x = 0 and 0.1 (see the inset), confirming that the metal–insulator transition is of second order. At higher x values, no such anomaly is discernible, suggesting that the transition is either too weak to be detected or completely suppressed.

The effect of Ga doping on the carrier concentration of Re3Ge7−xGax is further investigated by the Hall effect measurements. Figure 3d, e shows the magnetic field dependence of Hall resistivity ρH for two cases of x = 0 and 0.25, which show clear contrast. In the former case, the ρH data are negative above 50 K, but become positive at lower temperatures. This indicate the presence of both hole and electron carriers in the system, which naturally explains the nonlinear ρH as a function of magnetic field. In the latter case, however, ρH is positive and depends linearly on the field. The temperature dependence of Hall coefficient RH for the series of samples shown in Fig. 3f. Here RH is determined as RH = ρH/B in the low-field region. With decreasing temperature, RH for x = 0 and 0.1 undergoes a sign reversal from negative to positive, and then rises steeply before reaching a plateau. Note that the sign-reversal temperature is close to TMI, suggesting that the two phenomena are intimately related. At higher x values, RH is always positive and no sign-reversal occurs. Nevertheless, a rise in RH is still observable below a temperature around TMI for x = 0.2. Remarkably, the low-temperature RH values for x ≤ 0.1 are about two-orders magnitude higher than those for x ≥ 0.25. This implies that the latter has a much higher hole concentration, which is estimated to be ~1.2–2 × 1021 cm−3 assuming a one-band model. These results demonstrate that Ga doping introduces holes in Re3Ge7. Nonetheless, given the orthorhombic structure and multiband nature of Re3Ge7−xGax, Hall measurements on single crystals are needed to determine more precisely the evolution of carrier concentration with varying Ga content.

Emergence of superconductivity

As a consequence of this doping, SC is induced in Re3Ge7−xGax over an x range of 0.2–0.4. This is demonstrated by the ρ(T) and χ(T) data below 4 K in Fig. 4a, b. As can be seen in Fig. 4a, a drop to zero resistivity is observed for all x values in this range. With increasing x, the resistive transition first shifts to higher temperatures and then to lower temperatures, displaying a non-monotonic behavior. The Tc values, determined from the midpoints of the ρ drops, are 1.81, 3.37, 2.56, 2.08, and 1.49 K for x = 0.2, 0.25, 0.3, 0.35, and 0.4, respectively. Meanwhile, the zero-field cooling susceptibility χZFC of these samples measured under 1 mT exhibits a strong diamagnetic response, whose onset temperature coincides with Tc. In addition, their shielding factions are estimated to exceed ~150% without demagnetization correction.

Fig. 4: Superconducting transition and upper critical field of Re3Ge7−xGax.
figure 4

a, b Low-temperature resistivity and magnetic susceptibility for Re3Ge7−xGax with 0.2 ≤ x ≤ 0.4, respectively. The vertical dashed line is a guide to the eyes. c Low-temperature specific-heat Cp data for all the Re3Ge7−xGax samples plotted as Cp/T versus T2. The solid lines are fits by the Debye model. d Temperature dependence of normalized electronic specific-heat for the samples with 0.25 ≤ x ≤ 0.35. The back lines are energy conserving constructions to estimate the specific-heat jump and the red line denotes the theoretical BCS curve. e Temperature dependence of resistivity under various fields up to 0.56 T for the sample with x = 0.2, and the field increment is 0.08 T. The horizontal line and arrow indicate the 50% drop of normal-state resistivity and field increasing direction, respectively. f Upper critical field versus temperature phase diagram for the samples with 0.25 ≤ x ≤ 0.35. The solid lines are fits to the data by the Ginzburg–Landau model.

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The bulk nature of SC is confirmed by the plots of low-temperature Cp/T versus T2 in Fig. 4c. A clear Cp jump is observed for the x value of 0.25, 0.3 and 0.35. As for x = 0.2 and 0.4, the absence of such an anomaly is due to that their Tc values are below the lowest measurement temperature (1.8 K). On the other hand, the normal-state Cp data for both undoped and Ga-doped Re3Ge7 are well fitted by the Debye model

$${C}_{{{{\rm{p}}}}}/T\,=\,\gamma \,+\,\beta {T}^{2}\,+\,\delta {T}^{4},$$
(1)

where γ and β(δ) are the electronic and phonon specific-heat coefficients, respectively. From β, the Debye temperature ΘD is calculated as

$${{{\Theta }}}_{{{{\rm{D}}}}}\,=\,{\left(\frac{12{\pi }^{4}NR}{5\beta }\right)}^{1/3}.$$
(2)

The obtained γ and ΘD are listed in Table 2. Intriguingly, while no systematics in ΘD are observed, γ exhibits a nonmonotontic x dependence with a maximum value of 13.6 mJ mol−1 K−2 at x = 0.25. This value is larger than those of the conventional metals25 and comparable to those of some iron-based superconductors with a similar carrier concentration26.

Table 2 Structural refinement results and physical parameters of Re3Ge7−xGax.
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It is prudent to note that the γ value (=2.5 mJ mol−1 K−2) of our polycrystalline Re3Ge7 sample is much larger than that (~4 μJ mol−1 K−2) of the single crystalline one22. Moreover, the signs of their RH data at low temperature are opposite. These contrasts are most probably due to slight difference in stoichiometry of the two samples. As a matter of fact, we have also grown Re3Ge7 crystals with (00l) orientation using a flux method different from that in ref. 22, where crystals with (0l0) orientation were obtained. The Hall measurements indicate that its RH is positive in the whole temperature range (data not shown). It is thus possible that the sign of RH depends on the crystal orientations. Since our polycrystalline samples consist of many small crystals with random orientations, it is no wonder that the sign of RH is different from that measured on single crystals.

The normalized electronic specific-heat Cel/γT for 0.25 ≤ x ≤ 0.35, obtained by subtraction of the phonon contribution, is shown in Fig. 4d. It turns out that the Cel/γT jump decreases with increasing x and hence increasing Tc. Using an entropy conserving construction, ΔCel/γT is determined to be 0.83 and 1.08 for x = 0.25 and 0.3, respectively, which are significantly smaller than the BCS value of 1.4327. Indeed, the temperature dependence of their Cel/γT data show a clear deviation from the weak coupling BCS theory27, hinting at the presence of multiple superconducting gaps or even gap nodes. Hence, to better understand the gap structure, Cp measurements at temperatures much below Tc are needed. Nevertheless, since inhomogeneity could be present is our polycrystalline samples, such investigation is best performed on single crystals and thus left for future studies.

The upper critical fields Bc2 for Re3Ge7−xGax with 0.25 ≤ x ≤ 0.35 are determined by resistivity measurements under magnetic fields. An example for x = 0.25 is shown in Fig. 4e. As expected, the resistive superconducting transition shifts toward lower temperatures and becomes broadened as the field increases. At each field, the Tc value is determined using the same criterion as above. The resulting Bc2 versus temperature phase diagrams are displayed in Fig. 4f. All the Bc2(T) data are well described by the Ginzburg–Landau (GL) model

$${B}_{{{{\rm{c}}}}2}(T)\,=\,{B}_{{{{\rm{c}}}}2}(0)\frac{1\,-\,{t}^{2}}{1\,+\,{t}^{2}},$$
(3)

where Bc2(0) is the zero-temperature upper critical field and t = T/Tc. The obtained Bc2(0) is 1.15 T for x = 0.25, 0.68 T for x = 0.3, and 0.58 T for x = 0.35. Once Bc2(0) is known, the GL coherence length ξGL can be calculated by the equation

$${\xi }_{{{{\rm{GL}}}}}\,=\,\sqrt{\frac{{{{\Phi }}}_{0}}{2\pi {B}_{{{{\rm{c}}}}2}(0)}},$$
(4)

where Φ0 = 2.07 × 10−15 Wb is the flux quantum. This gives ξGL values of 16.9, 22.0, and 23.8 nm for x = 0.25, 0.3, and 0.35, respectively.

Electronic phase diagram

The above results, which are also summarized in Table 2, allow us to construct the electronic phase diagram of Re3Ge7−xGax presented in Fig. 5a. Upon Ga doping, TMI in Re3Ge7 is gradually suppressed and disappears abruptly at x > 0.2. Note that, below TMI, the carrier concentration decreases by more than one order of magnitude as indicated by the above Hall measurements. This drastic variation is closely related to the second-order phase transition and hence differs from the case of a semimetal with a small pocket. Instead, the negative temperature coefficient of resistivity (dρ/dT < 0) below TMI is reminiscent of an insulating-like behavior, as we labeled in the phase diagram. On the other hand, SC emerges for x ≥ 0.2, and Tc shows a dome-like dependence on x with a maximum of 3.37 K observed at x = 0.25. The contrasting behavior of TMI and Tc suggests that there is a competition between the insulating-like and superconducting phases, though the two phases might coexist in a narrow x range between 0.2 and 0.25. Moreover, the maximum in Tc coincides well with that of γ, suggesting that the enhanced density of states (DOS) may play an important role in the superconducting pairing. It should be pointed out, for each x value, we have carried out measurements on multiple samples. An example of x = 0.25 and data summary are shown in Supplementary Fig. 3 and Table 1, respectively. These results indicate that the error bar in characteristic temperatures is within the symbol size, and hence the overall phase diagram is well reproducible.

Fig. 5: Electronic phase diagram of Re3Ge7−xGax and its comparison with the variation of electronic specific heat.
figure 5

a T-x electronic phase diagram of Re3Ge7−xGax. Note that the Tc data are multiplied by a factor of 2 for clarity. b x dependence of electronic specific-heat coefficient, respectively. The vertical dashed line is a guide to the eyes.

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Calculated band structure

To gain insight into the effect of Ga doping in Re3Ge7, we performed first principles calculations on the electronic structure of Re3Ge7−xGax with x up to 0.4. The calculated band dispersions with and without spin–orbit coupling (SOC) are shown in Fig. 6a, and the Brillouin zone is sketched in Fig. 6b. One can see that there are several bands crossing EF and the Ga doping mainly leads to a rigid band shift of the Fermi level (EF) (see Supplementary Fig. 4). Without considering SOC, there are three fourfold degenerate points about 0.3 eV above EF along the X-R line. Nevertheless, when turning on SOC, only the one located at R point remains ungapped. Group theory analysis of the k ⋅ p perturbation matrix elements shows that the energy bands around this fourfold point have linear dispersions28, which is protected by the nonsymmorphic space group Cmcm. Since R is a time-reversal invariant momentum, the linear dispersions for Kramers partners must be reversed. Thus despite that the band dispersion is tilted, the fourfold point at R is a type-I Dirac point29, in agreement with the previous report23.

Fig. 6: Calculated electronic structure and Fermi-surface nesting functions of Re3Ge7−xGax.
figure 6

a Bulk band structure for Re3Ge7−xGax without and with SOC. The yellow shaded region corresponds to the band filling of x = 1. b Brillouin zone of Re3Ge7−xGax with high-symmetry points and lines. c–g The Fermi-surface for Re3Ge7−xGax with x = 0, 0.05, 0.1, 0.2, and 0.4, respectively. h The calculated Fermi-surface nesting functions at high-symmetry lines. i A diagrammatic sketch of nesting vectors q1 and q2.

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Figure 6c–h shows the calculated Fermi-surface and its nesting function along the high-symmetry lines for Re3Ge7−xGax. For undoped Re3Ge7 (x = 0), its Fermi surface consists of two large hole pockets along the ky direction and two sets of small electron pockets crossing the G-X and Y-X1 lines. Upon Ga doping, the hole pockets remain nearly unchanged while the electron pockets become smaller and vanish for x ≥ 0.1. These results are well consistent with the evolution of RH data and confirm Ga as a hole dopant. Nevertheless, since the metal–insulator-like transition is not considered in the calculation, a direct comparison between the evolutions of theoretical DOS and experimental γ values with Ga content is not meaningful. On the other hand, the nesting function exhibits peaks around X, X1, Z points, and along G-X, G-Y-X1 lines, in addition to the universal one at Γ point. Nevertheless, only the peaks along G-X and Y-X1 lines (marked by q1 and q2 in Fig. 6h) are significantly suppressed with Ga doping, while the other peaks survive at higher doping level. The corresponding wave vectors for these peaks are q1 = (±0.06, 0, 0) and q2 = (±0.02, 1, 0) in the unit of (π/a, π/b, π/c). By careful inspection of the Fermi-surface structure, we find that the nesting peaks at q1 and q2 are closely related to the electron pockets, as shown in Fig. 6i. The q1 peak should be mainly contributed by the nesting between the hole and electron pockets. Nevertheless, since trace of q1 is still present at higher Ga doping levels, it is reasonable to infer that this peak is also contributed by the nesting between the hole pockets. For the peak of q2, it disappears as soon as the electron pockets vanish and thus should result from the nesting between the ellipsoid-shaped electron pockets around X point. This is corroborated by the 2D nesting functions shown in Supplementary Fig. 5. It is noted that the metal–insulator-like transition and the above-mentioned nesting peaks are suppressed at almost the same Ga doping level. It is therefore reasonable to speculate that the phase transition in pristine Re3Ge7 is associated with a nesting driven Fermi-surface instability30,31.

Nontrivial band topology

Now let’s examine the band topology of Re3Ge7−xGax. As illustrated in Fig. 6a, a moderate doping of x = 1 would lower the electron filling level into the yellow full-gapped region. Given that Re3Ge7 holds the spatial inversion symmetry, one can obtain the Z2 invariants by calculating the inversion parities32,33. This gives Z2 invariants of (1;110), which corresponds to a strong topological insulator (TI) phase. Since the Dirac point is protected by the nonsymmorphic space group, it is of interest to investigate the configuration that breaks the nonsymmorphic group symmetry. In Fig. 7a, we present a hypothetical structure of Re3Ge6Ga, which is derived by replacing 2 Ge atoms of Ge(1) site with Ga (per primitive cell). Such a configuration lowers the symmetry group to C2/m, which has only 2-dimensional double valued irreducible representations. Thus one expect that the bands are doubly degenerated at any high-symmetry points, which split the Dirac point into two doubly degenerated points. The calculated band structure for Re3Ge6Ga is shown in Fig. 7b. Indeed, the Dirac point at R point splits into a full-gapped region (the cyan shaded region), while the yellow shaded region remains gapped. The Z2 invariants are calculated to be (0;001) and (1;110) for the cyan and yellow regions, respectively. This not only demonstrates the robustness of the strong TI phase in Ga-doped Re3Ge7, but also indicates that Re3Ge7 will become a weak TI by breaking the nonsymmorphic group symmetry.

Fig. 7: Hypothetical crystal structure and corresponding electronic structure of Re3Ge6Ga.
figure 7

a The primitive cell for Re3Ge7 and a possible C2/m configuration of Re3Ge6Ga. b Bulk band structure for Re3Ge6Ga with space group C2/m. The cyan and yellow regions correspond to the band filling of x = 0 and 1, respectively.

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Perspective and summary

Finally, we discuss the implications of our results and suggest some directions for future research. First, since the overall properties of Re3Ge7−xGax are similar to those of CuxTiSe23, the phase transition in Re3Ge7 could be due to the CDW formation, which calls for verification by temperature-dependent electron diffraction studies. Second, the emergence of SC by only ~3% Ga doping implies that Re3Ge7 lies on the verge of a superconducting instability. It is therefore of interest to see whether SC can be induced by doping with other elements or the application of high pressure. Third, the combination of SC and nontrivial band topology renders Re3Ge7−xGax a potential candidate for TSC11,12,13. In this respect, single-crystal growth is highly desirable for further spectroscopy measurements on the superconducting gap symmetry and possible in-gap states.

In summary, we have discovered SC in the orthorhombic Re3Ge7−xGax system. The pristine Re3Ge7 exhibits a second order metal–insulator-like phase transition below 57.3 K, which is suppressed upon Ga doping and disappears above x = 0.2. At this doping level, SC starts to be observed and Tc displays a dome-like dependence with a maximum value of 3.37 K at x = 0.25. The Hall effect measurements indicate that substitution of Ga for Ge introduces holes, consistent with the band structure calculations. The theoretical results further suggest that the phase transition in Re3Ge7 is likely driven by Fermi-surface nesting and superconducting Re3Ge7−xGax compositions exhibit nontrivial band topology characterized by strong Z2 invariants. Our results indicate that Ga-doped Re3Ge7 is a rare system that combines collective phase transition, nontrivial band topology and SC, which lays a foundation for further exploring the competition and interplay between these properties.

Methods

Sample synthesis

Polycrystalline Re3Ge7−xGax samples with x = 0, 0.1, 0.2, 0.25, 0.3, 0.35, and 0.4 were prepared by the solid-state reaction method. High-purity Re (99.99%), Ge (99.99%) powders, and Ga (99.999%) shots were weighed according to the stoichiometric ratio, mixed thoroughly and pressed into pellets in an argon-filled glove box. The pellets were then sealed in evacuated silica tubes and heated at 850 °C for several days, followed by slow cooling to room temperature. This process was repeated several times with intermediate grindings to ensure homogeneity.

Structural characterizations

The phase purity of resulting samples was examined by powder XRD using a Bruker D8 Advance X-ray diffractometer with Cu Kα radiation. The data were collected with a step-scan mode in the 2θ range from 5° to 120° and the structural refinements were performed using the program JANA200634.

Physical property measurements

Measurements of resistivity, Hall coefficient and specific heat were done on regular-shaped samples in a Quantum Design Physical Property Measurement System (PPMS-9 Dynacool). The resistivity was measured by the four-probe method and down to 150 mK at zero field using an adiabatic dilution refrigerator option. The Hall resistivity was measured by sweeping the field from −9 to 9 T, and the data were antisymmetrized to remove the magnetoresistance contribution. The zero-field cooling and field cooling magnetic susceptibility measurements down to 0.4 K were carried out using a Quantum Design Magnetic Property Measurement System (MPMS3).

Theoretical calculations

Our first principles calculations were performed within density functional theory, as implemented in the Vienna Ab-initio Simulation Package35. The Kohn–Sham equations were constructed on a projector augmented wave basis36. The exchange-correlation energy was calculated with a Perdew–Burke–Ernzerhof type functional37. For all calculations, we adopted a monoclinic primitive cell with experimental lattice parameters. The plane-wave energy cutoff for wavefunctions was set to 600 eV. The k-mesh was set 12 × 12 × 3 for self-consistent calculations and 24 × 24 × 6 for DOS calculations. To calculate the band dispersion over the whole Brillouin zone more efficiently, we constructed a tight-binding Hamiltonian with maximally localized Wannier functions38. The Fermi surfaces and nesting functions were then calculated with this tight-binding Hamiltonian.