Probing the electronic properties of ternary A n M3n-1B2n (n = 1: A = Ca, Sr; M = Rh, Ir and n = 3: A = Ca, Sr; M = Rh) phases: observation of superconductivity.

We follow the evolution of the electronic properties of the titled homologous series when n as well as the atomic type of A and M are varied where for n = 1, A = Ca, Sr and M = Rh, Ir while for n = 3, A = Ca, Sr and M = Rh. The crystal structure of n = 1 members is known to be CaRh2B2-type (Fddd), while that of n = 3 is Ca3Rh8B6-type (Fmmm); the latter can be visualized as a stacking of structural fragments from AM3B2 (P6/mmm) and AM2B2. The metallic properties of the n = 1 and 3 members are distinctly different: on the one hand, the n = 1 members are characterized by a linear coefficient of the electronic specific heat γ ≈ 3 mJ mol-1 K-2, a Debye temperature θD ≈ 300 K, a normal conductivity down to 2 K and a relatively strong linear magnetoresistivity for fields up to 150 kOe. The n = 3 family, on the other hand, exhibits γ ≈ 18 mJ mol-1 K-2, θD ≈ 330 K, a weak linear magnetoresistivity and an onset of superconductivity (for Ca3Rh8B6, Tc = 4.0 K and Hc2 = 14.5 kOe, while for Sr3Rh8 B6, Tc = 3.4 K and Hc2 ≈ 4.0 kOe). These remarkable differences are consistent with the findings of the electronic band structures and density of state (DOS) calculations. In particular, satisfactory agreement between the measured and calculated γ was obtained. Furthermore, the Fermi level, EF, of Ca3Rh8B6 lies at almost the top of a pronounced local DOS peak, while that of CaRh2B2 lies at a local valley: this is the main reason behind the differences between the, e.g., superconducting properties. Finally, although all atoms contribute to the DOS at EF, the contribution of the Rh atoms is the strongest.

Introduction

Recently, we observed superconductivity in a number of new non-centrosymmetric ternary Li–Rh–B phases with Tc ranging from 2 to 3 K [1]. In an attempt to raise such Tc, we investigated the A–M–B system wherein Li was replaced by alkaline earth atoms, Rh by another noble platinium atom while B was maintained since such a light-mass metalloid may be conducive to higher frequency modes. Traces of superconductivity were observed in compositions wherein the majority phase belongs to the AM3B2 series (A = Ca, Sr; M = Rh,Ir; n is an integer) [2].

Following this lead, we carried out a systematic magnetoresistivity characterization of this series [2]: the obtained results revealed distinct differences among the electronic properties of the n = 1 (AM2B2 where A = Ca, Sr; M = Rh, Ir) and the n = 3 (A3Rh8B6; A = Ca,Sr) phases. In the present work, further investigations were carried out to study their electronic properties (in particular, the superconductivity, the main concern of this work). For this purpose, various preparation routes and differing heat treatments as well as different starting compositions were applied. The structural, elemental, magnetic, thermal and electric transport properties were extensively investigated. The results confirm the strong difference in electronic properties of these two series: as an example, all of the studied n = 1 phases are normal conductors, while the studied n = 3 ones are superconductors. The experimental results are satisfactorily interpreted in terms of the theoretically calculated band structures and density of state (DOS) curves.

The crystal structures of the n = 1 and 3 families, together with those of the related AM3B2, are shown in figure 1 [3]. All these structures can be visualized as a sequence of stacked layers: taking AM3B2 as a reference, the M-layers of AM2B2 contain only 2/3 of the M atoms and are shifted by b/2 along the y-direction. On the other hand, the structure of A3M8B6, (AM3−B2, δ = 1/3) can be visualized as a stacking of the structural fragments of AM3B2 (AM3−B2, δ = 0) and AM2B2(AM3−B2, δ = 1); there are six Rh layers in the unit cell of A3M8B6 but only two of these exhibit 1/3 vacant Rh-sites (for further structure details see [3, 4]).

Figure 1.

The unit cells of (a) AM3B2 (P6/mmm) [3], (b) AM2B2 (Fddd ) [3] and (c) A3M8B6 (Fmmm) (A = Ca, Sr; M = Rh, Ir) [3]. The structure of A3M8B6 can be visualized as a stacking of structural fragments of AM3B2 of panel (a), labeled A132, and AM2B2 of panel (b), labeled A122, with the ratio 2:1 [3, 4]. Some relevant structural parameters of the studied AM3B2 (n = 1, 3) compounds are provided in table 2.

Table 2.

Measured and calculated crystallographic parameters of the studied AM2B2 and A3Rh8B6 (A = Ca, Sr, Ba; M = Rh, Ir) samples. The occupied Wyckoff positions of AM2B2 are B (16f), M (16g) and A (8a), while those of A3Rh8B6 are B (8h,16m), M (8f,8i,16j) and A (4a,8i)—all in standard settings. For comparison, CaRh3B2 crystalizes in the CeCo3B2 structure with P6/mmm; B (2d), Rh (3f), Ca (1b); a = 5.551 Å, c = 2.919 Å [3]. For all compositions, the atomic occupations and thermal parameters are taken from the cited references (for bond lengths and angles, see the same references). The lattice parameters of the different (S1,S2,S3) samples of A3Rh8B6 are equal (within the quoted statistical errors). fu represents the number of formula units per unit cell.

			Parameters from experiments					Parameters from theory			Parameters from reference
Compound	Space group	fu	c/a	a (Å)	b (Å)	c (Å)	b/a	a (Å)	b (Å)	c (Å)	a (Å)	b (Å)	c (Å)
CaRh2B2	Fddd	8	5.8396(21)	9.2535(24)	10.6071(18)	1.58	1.82	5.8185	9.2225	10.5674	5.832	9.24	10.606	[11]
CaIr2B2	Fddd	8	5.8741(12)	9.2544(17)	10.7267(13)	1.58	1.83	5.8550	9.2240	10.6906	5.877	9.257	10.727	[11]
SrRh2B2	Fddd	8	5.9826(12)	9.3906(9)	10.6682(9)	1.57	1.78	5.9626	9.3592	10.6325	5.989	9.399	10.672	[12]
SrIr2B2	Fddd	8	6.0221(7)	9.3983(13)	10.7123(76)	1.56	1.78	6.0020	9.3669	10.6765	5.989	9.404	10.723	[12]
Ca3Rh8B6	Fmmm	4	5.4872(46)	9.7035(56)	16.9683(24)	1.77	3.09	5.4816	9.6656	16.9900	5.500	9.698	17.047	[3]
Sr3Rh8B6	Fmmm	4	5.5655(40)	9.9042(40)	17.1186(20)	1.78	3.08	5.5534	9.8878	17.0607	5.572	9.921	17.118	[3]

Experiment

Instruments

Powder diffractograms were collected on an x-ray diffractometer equipped with a Si detector and employing Cu K radiation. Structural analyses (using the Rietveld method and the structural models shown in figure 1) were carried out on all synthesized samples to evaluate the sample homogeneity, phase content and structural parameters. Elemental analyses were carried out with the help of an energy-dispersive x-ray (EDAX) analyzer. Magnetizations, M(T,H), were measured on a superconducting quantum interference device magnetometer, while magnetoresistivities, ρH(T), were measured on a conventional collinear four-point technique. Specific heats were measured on a relaxation-type calorimeter.

Calculation method and procedure

To solve the scalar-relativistic Kohn–Sham equations, calculations based on the density-functional theory were performed. We used the augmented plane wave plus local orbital method [5-7] as embodied in the WIEN2K code [8]: here, the wave functions are expanded in terms of spherical harmonics inside non-overlapping atomic spheres of radius R, while in the remaining space of the unit cell (the interstitial region), plane waves were used. The exchange and correlation effects were treated within the recently proposed generalized gradient approximation of Perdew et al [9].

Reasonable values were used for the atomic spherical radius (ratom) as well as the smallest muffin tin radius (R) and the largest wave number of the basis set (KMAX). As an example, for CaRh2B2, rCa is 1.32 Å and both rRh and rB are fixed at 0.95 Å, while for Ca3Rh8B6, rCa is 1.32 Å, rRh is 1.11 Å and rB is 0.95 Å. On the other hand, the product RKMAX (which controls the size of the basis set) is fixed at 7.0 for both CaRh2B2 and Ca3Rh8B6 phases.

Integration in the reciprocal space was performed using the tetrahedron method taking up to 512 k-points in the first irreducible Brillouin zone. Once self-consistency of the potential was achieved, quantum mechanically derived forces were obtained and the ions were displaced according to a rank-one multisecant scheme, and then the relaxed atomic positions were obtained. The tolerance for ionic forces was 1 mRyd au−1. For all the studied phases, the unit cell dimensions were fully relaxed (see below in table 2).

Synthesis of AM3B2 (n = 1,3)

It was found difficult to synthesize stable, single-phase samples of these seemingly incongruent AM3B2 phases directly from stoichiometric starting compositions (such a difficulty might have been the barrier that hampered the search for superconductivity in this series). Instead, these compounds were precipitated, via peritectic reactions, from A-rich compositions. Dozens of samples were synthesized by two preparation routes: (i) the conventional argon arc-melt procedure, which was found to produce heavy losses in the easily evaporating alkaline earth elements, and (ii) the standard solid-state method in which pure elements were reacted in BN or Ta crucibles: this route (used throughout this work) was found to produce the best stoichiometric composition. For all compositions, a significant improvement in sample quality was obtained when as-prepared A–M–B samples were annealed in a Ta crucible for 18–20 h at 850 °C (used for all samples in this work).

Because of its high content in these A-rich starting compositions, one needs to drain away any excess of A. However, in some cases, an A inclusion was found to precipitate within the surroundings of the grain boundaries.

After various adjustments and optimizations of the above-mentioned procedures, we managed to prepare various almost single-phase samples. In general, it is relatively easier to prepare a single-phase AM2B2 sample than that of A3Rh8B6 (cf figures 2 and 3): the latter are fragile, deteriorate easily if left exposed to air and are often contaminated with weak impurity phases (such a contamination was also reported in other A3M8B6 isomorphs such as La3Ru8B6 and Y3Os8B6 [4]). For most A3M8B6 samples, the following impurity traces (<8%) were identified: AM2B2, AM3B2 or AM2. Some of these contaminants (<6%) are superconductors: for example, ARh2 (e.g. CaRh2, Tc = 6.4 K; CaIr2, Tc = 4–6.15 K; SrRh2, Tc = 6.2 K; SrIr2, Tc = 5.7 K) [10] or an unidentified superconducting contaminant with Tc ≈ 9 K. The superconductivity of these contaminants should not be confused with the superconductivity of the main A3M8B6 phase: the latter is manifested at a similar Tc, independent of the type of the contaminations, and is well correlated with the pronounced peak in DOS calculations (see below). Accordingly, it is emphasized that the conclusions reached in this work are independent of the presence of these contaminations since their features can be easily identified.

Figure 2.

X-ray diffractograms of AM2B2 (A = Ca, Sr; M = Rh, Ir) samples. Symbols: the measured intensities; solid lines: Rietveld fits; short bars: Bragg positions. Vertical arrows indicate traces of contaminating phases. Fit parameters are given in table 2.

Figure 3.

X-ray diffractograms of A3Rh8B6 (A = Ca, Sr) samples. Symbols: the measured intensities; solid lines: Rietveld fits (see table 2); short bars: Bragg positions. The second, minority phase, denoted by the lower row of the Bragg bars, in panel (a) is Sr2Rh5B4, while in panel (b) is CaRh2B2. Vertical arrows indicate unidentified contaminating peaks, the presence of which leads to much higher goodness of fit R-factors (e.g. R ∼ 10). Inset: representative diffractograms of some Ca3Rh8B6 samples that were prepared with different starting compositions: Ca:Rh:B = 14:8:6 (S1), Ca:Rh:B = 10:8:6 (S2) and Ca:Rh:B = 6:8:6 (S3). Note the differences in intensities of the main and contaminating phases.

For the purpose of improving the phase purity of these A3Rh8B6 samples, various starting compositions were tested. As an illustration, we discuss below the structural and magnetic properties of three different Ca-rich starting compositions: namely Ca:Rh:B = 14:8:6, 10:8:6 and 6:8:6 (identified, respectively, as S1, S2 and S3—see the inset of figure 3 and table 1). In all cases, a majority Ca3Rh8B6 phase was formed. For Sr-based samples, the starting compositions Sr:Rh:B = 6:8:6 and 6:8:12 were tested (see table 1).

Table 1.

Representative EDAX analysis of A3Rh8B6 (A = Ca, Sr) (for comparison, data of CaRh2B2 are also included). These results (given in weight and atomic percentage) compare favorably with the expected values and are consistent with the x-ray diffraction analysis (see the text and figures 2 and 3). The following atomic weights were used: Ca (40.078), Sr (87.62), Rh (102.9055) and B (10.811). For each composition, the values are normalized to that of Rh (since its determination is the most reliable); in contrast, the value of the light-mass B is obtained by subtraction since its direct determination by EDAX is not reliable. Due to experimental difficulty, we were not able to measure EDAX of the S3 sample.

Atom	1st analysis		2nd analysis		3rd analysis		4th analysis		Average		Expected
	wt%	at.%	wt%	at.%	wt%	at.%	wt%	at.%	wt%	at.%	wt%	at.%
Sr3Rh8B6 (from the starting composition Sr:Rh:B=6:8:6)
Sr	22.62	17.45	24.99	19.30	24.12	18.65	24.44	18.90	24.04	18.58	22.84	17.65
Rh	71.53	47.06	71.53	47.06	71.53	47.06	71.53	47.06	71.53	47.06	71.53	47.06
B	5.85	35.49	3.48	33.64	4.35	34.29	4.03	34.04	4.43	34.36	5.63	35.29
Sr3Rh8B6 (from the starting composition Sr:Rh:B=6:8:12)
Sr	29.85	22.99	28.48	21.58	29.50	22.63	28.92	22.02	29.19	22.31	22.84	17.65
Rh	64.60	42.35	65.65	42.36	64.86	42.35	65.31	42.35	65.10	42.35	71.53	47.06
B	5.55	34.66	5.87	36.06	5.64	35.02	5.77	35.62	5.71	35.34	5.64	35.29
Ca3Rh8B6 (from the starting composition Ca:Rh:B=10:8:6, labeled as S2 batch)
Ca	14.31	21.16	10.87	16.09	10.75	15.90	10.77	15.94	11.68	17.28	11.93	17.65
Rh	81.64	47.06	81.64	47.06	81.64	47.06	81.64	47.06	81.64	47.06	81.64	47.06
B	4.05	31.78	7.49	36.85	7.61	37.04	7.59	37.00	6.68	35.66	6.43	35.29
Ca3Rh8B6 (from the starting composition Ca:Rh:B=14:8:6, labeled as S1 batch)
Ca	11.33	16.75	10.67	15.77	10.05	14.87	10.67	15.79	10.68	15.80	11.93	17.65
Rh	81.64	47.06	81.64	47.06	81.64	47.06	81.64	47.06	81.64	47.06	81.64	47.06
B	7.03	36.19	7.69	37.17	8.31	38.07	7.69	37.15	7.68	37.14	6.43	35.29
CaRh2B2 (from the starting composition Ca:Rh:B=2:2:2)
Ca	13.11	17.51	13.19	17.64	13.16	17.56	13.14	17.54	13.15	17.56	14.98	20.00
Rh	76.94	40.00	76.94	40.00	76.94	40.00	76.94	40.00	76.94	40.00	76.94	40.00
B	9.95	42.49	9.87	42.36	9.90	42.44	9.92	42.46	9.91	42.44	8.08	40.00

Results

Structural and EDAX characterization

Representative diffractograms of AM2B2 samples are shown in figure 2 and those of A3Rh8B6 samples in figure 3. Both XRD and EDAX analyses (table 1) indicate a single-phase character for the AM2B2 samples and a major-phase character for the A3M8B6 samples.

The crystal structures of the studied AM3B2 samples are shown in figure 1. The experimentally determined structural parameters are given in table 2: evidently, the lattice parameters of both series are in excellent agreement with the reported values as well as with those obtained from our theoretical calculations (see below). The ratios of these orthorhombic lattice parameters (see table 2) emphasize the similarity of their bonding character: it is remarkable that the b/a ratios of the ARh2B2 (similarly A3Rh8B6) family are equal even though A and M are widely varied. Similar arguments hold for the c/a ratios; however, due to the difference in their stacking arrangements, the evolution of the c/a ratio with n is much stronger than that of the b/a ratio.

Magnetic, thermal and magnetoresistive characterization

SrM2B2 and Sr3Rh8B6(M = Rh,Ir).

Magnetization (figure 4), magnetoresistivity [2], and specific heat (figure 5) curves of SrRh2B2 and SrIr2B2 show no superconducting signal down to 1.8 K. By contrast, magnetization (figure 4), magnetoresistivity (inset of figure 4) [2] and specific heat (figures 5 and 6) of Sr3Rh8B6 show a relatively large, bulk superconducting signal with an onset at Tc = 3.4(2) K. In addition to such a difference in their superconducting properties, SrM2B2 and Sr3Rh8B6 exhibit two other differences: (i) based on figures 5 and 6 and table 3, the Sommerfeld coefficient, γL, of SrIr2B2 is six times lower than that of Sr3Rh8B6 but, on the other hand, θD of the former is only 10% lower than that of the latter. (ii) Based on earlier results [2], the SrM2B2 members exhibit a strong linear magnetoresistivity but, by contrast, Sr3Rh8B6 shows an extremely weak magnetoresistive effect. These differences are expected to be valid for all pairs of SrM2B2 and Sr3M8B6.

Figure 4.

Thermal evolution of the mass susceptibility (20 Oe) of polycrystalline SrRh2B2, SrIr2B2 and Sr3Rh8B6 samples. Inset: isofield magnetoresistivity of Sr3Rh8B6 for various fields. The critical points (taken at the onset) are indicated by the vertical arrows.

Figure 5.

Zero-field C/T versus T2 curves of A3Rh8B6, CaRh2B2 and AIr2B2 (A = Ca, Sr). Two different Ca3Rh8B6 samples (S2 and S3; not S1 due to accidental deterioration) were measured: a strong reminder of the influence of sample history on the measured properties (see also figures 3 and 8). Symbols represent the measurements while the solid lines represent the high-temperature (10 < T < 25 K) fits to γH + βHT2 (subscript H indicates a high-temperature fit parameter, see table 3). Inset: a low-temperature expansion of the same curves but with solid lines that represent low-temperature (2 < T < 10 K) fits to γL + βLT2 (see table 3).

Figure 6.

C/T versus T2 curves of one Sr3Rh8B6 sample and two different Ca3Rh8B6 samples (identified as S2 and S3, see figures 3 and 8 and table 3). Each sample was measured at zero (open symbols) and 20 kOe (filled symbols). The broadening in the superconducting transition in Sr3Rh8B6 as well as the absence of signature at Tc of Ca3Rh8B6 are attributed to sample-dependent influences. Inset: an enlarged scale of C versus T curves of the same measurements given in the main panel. Symbols: measurements; solid lines: low-temperature fits to the sum γLT + βLT3. The obtained values of γL and θL,D are given in table 3.

Table 3.

Normal-state γ and θD of five representative samples. γbarecal was calculated from equation (1) and λ from equation (2). The measured parameters were obtained from the least-squares fit of the expression γT + β T3 to the data of figure 5. It is noted that there are two temperatures regions wherein the above expression gives a satisfactory fit [15]: (i) the high-temperature range (10 < T < 25 K, the upper measuring limit), the obtained parameters are denoted by the subscript H; and (ii) the low-temperature range (2 < T < 10 K), the obtained parameters are denoted by the subscript L.

	θL,D	θH,D	γL	γH	γbarecal
Compound	K		mJ mol−1 K−2				λλNt(EF)
Sr3Rh8B6	325(5)	309(6)	19(1)	8(1)	11.3	0.68	3.31
Ca3Rh8B6:S2	325(5)	313(6)	14(1)	7.8(7)	11.7	0.19	0.97
Ca3Rh8B6:S3	358(4)	340(7)	18(1)	7.5(2)	11.7	0.53	2.72
SrIr2B2	282(1)	277(1)	3.1(1)	3.5(4)	2.1	0.46	0.40
CaRh2B2	350(2)	315(1)	3.4(1)	1.0(1)	1.93	0.77	0.64
CaIr2B2	298(1)	288(1)	3.1(1)	2.0(1)	1.8	0.74	0.53

Figure 8.

(a) Mass dc susceptibilities of CaRh2B2,CaIr2B2 and Ca3Rh8B6. Inset: an expansion of the χdc(T) curves of S1, S2 and S3 samples of Ca3Rh8B6 (see figures 3 and 6). The vertical arrows mark the onset of Tc. One notes two very weak additional superconducting signals: one at Tc = 6.4(2) K (related to CaRh2) and another, unidentified, at Tc = 8.7(2) (currently under further investigation); the thermal evolutions of Hc2 of both contaminants are shown in figure 10.

On analyzing the superconducting transitions manifest in the ρH(T) curves of Sr3Rh8B6 (inset of figure 4 and [2]), one obtains the H–T phase diagram shown in figure 7. Evidently, for H ≽ Hc2(0), the superconducting state is quenched: this is confirmed, independently, in the specific heat curves of figure 6.

Figure 7.

Thermal evolution of Hc2(T) of Sr3Rh8B6 as obtained from the isothermal longitudinal magnetoresistivity (see the inset of figure 4). The lines indicate the calculated Hc2(T) using the WHH [13, 14] expression (solid line: α = 0.08 and λso → ∞) and the quadratic expression (dashed line: Hc2(t) = Hc2[(1 − t2)/(1 + t2)], Hc2(0) = 3.85 kOe and Tc = 3.4 K). A dotted line expressing the WHH calculation for α = 0.08 and small finite λso (≈0) lies on the top of the solid line (see text).

Two procedures were used for the analysis of Hc2(T): (i) the quadratic expression (dashed line in figure 7) fits the data quite reasonably near Tc = 3.4 K and predicts Hc2(0) = 3.85 kOe, which is higher than the calculated Hc2(0) = −0.693Tc(∂Hc2/∂T) = 2.7 kOe. (ii) The Werthamer–Helfand–Hohenberg (WHH) expression [13, 14], which is usually parameterized in terms of α (a measure of the Pauli spin paramagnetism) and λso (a measure of the spin–orbit scattering). As usual [13, 14], α was calculated as follows: αcal = 5.33 × 10−5(∂Hc2/∂T) = 0.06 and αcal = 2.3ργ = 0.08 (ρ = low-T normal resistivity); among these, αcal was found to give a better description of the measured Hc2(T) curve. Furthermore, it was found that, in spite of the higher-Z value of Rh atom, the calculated WHH curve with finite λso does not differ from that with λso → ∞ (see figure 7): this coupled with the lower value of Hc2(0) suggests that spin effects in Sr3Rh8B6 have no strong influence on the evolution of Hc2.

CaM2B2 and Ca3Rh8B6(M = Rh, Ir).

The susceptibility (figure 8), magnetoresistivity (figure 9) and specific heat (figure 5) indicate that CaIr2B2 is a normal conductor down to 1.8 K. Similar features (in particular, the non-superconductivity) were observed for CaRh2B2 [2]. On the other hand, the magnetization (figure 8) and magnetoresistivity (figure 9) of Ca3Rh8B6 reveal a superconducting state, Tc = 4.0(2) K, with its bulk character being reflected in: (i) a high degree of screening (figure 8), (ii) a small valued Hc2 which is well below the usual observed values of H and (iii) a satisfactory success of the WHH analysis in describing the thermal evolution of Hc2(T). Nevertheless, there are two anomalous features regarding the manifestation of this superconductivity: (i) the resistivity (figure 9) reaches its lowest value at Tc; however, it does not attain zero below Tc (this is evident also in Sr3Rh8B6—figure 4—and is attributed to contamination or loosely compacted superconducting grains, which in turn is reflected in the high fragility of these samples), (ii) the specific heat curve (figures 5 and 6) does not show any event at the onset of superconductivity. Such an absence of signature is well known in, e.g., the Fe-based high-Tc pnictides (namely Ba0.8K0.2Fe2As2 [16], Na1−FeAs [17] and Sr2VO3FeAs [18]) and is usually attributed to sample quality, a scenario which, for Ca3Rh8B6, is demonstrated in figure 8: the diamagnetic response of S1, S2 and S3 is sample dependent (see also the XRD analysis in the inset of figure 3). Similarly, the specific heat curves of S2 and S3, shown in figure 6 and table 3, are found to be sample dependent (due to accidental deterioration, not all of samples S1, S2 and S3 were measured by specific heat, EDAX or magnetoresistivity. Nonetheless, their features are expected to manifest the same trend that is evident from the magnetization (figure 8) and XRD (figure 3).

Figure 9.

Zero-field ρ(T) curve of Ca3Rh8B6: the lower-left expansion indicates that, due to the contaminating superconducting phases (see the text), the resistivity was gradually reduced but tends to its lowest value below Tc(=4 K) of the main phase. Lower-right inset: isothermal magnetoresistivity of Ca3Rh8B6 showing the transition at Hc2(T = 2.4 K). Upper-left inset: representative ΔρT(H)/ρT(0) isotherms (T = 4.8, 30, 50, 100 K) of CaIr2B2 exhibiting strong linear-in-H magnetoresistive features. The solid lines represent the linear relation: ΔρT(H)/ρT(0 =a0 + aT.H (for further details see [2]).

Figure 10.

Thermal evolution of Hc2(T) of Ca3Rh8 B6 as obtained from the magnetoresistivity curves. Inset: the thermal evolution of the two contaminating superconducting phases: namely CaRh2 (Tc = 6.4(3) K) and an unidentified phase (Tc = 8.7(3) K) (see figure 8). The lines indicate the calculated Hc2(T) using the WHH [13, 14] expression (solid line: α = 0.19 and λso → ∞) and the quadratic expression (dashed line: Hc2(t) = Hc2[(1 − t2)/(1 + t2)], Hc2(0) = 12 kOe and Tc = 4 K). A dotted line expressing the WHH calculation for α = 0.19 and λso ≈ 0 can hardly be separated from the solid line.

Figure 10 shows the thermal evolution of Hc2(T) of Ca3Rh8B6, as obtained from the analysis of the ρH(T) curves (such as in figure 9). Similar to the case of Sr3Rh8B6, the above-mentioned analysis was applied to Hc2(T) of Ca3Rh8B6: the quadratic expression (dashed line in figure 10) gave Tc = 4 K and Hc2(0) =12 kOe (>Hcalc2(0)), while the WHH analysis (solid line in figure 10) gave Tc = 4 K and Hcalc2(0) = 10.1 kOe. During the latter analysis, αcal = 0.19 was fixed by experiment while λso → ∞ (as the ρ of Ca3Rh8B6 is reduced by the onset of superconductivity in the contaminating phases, the calculated αcal = 0.015 is smaller than that calculated using the superconducting parameters). Evidently, this WHH analysis suggests, similar to the case of Sr3Rh8B6, that the paramagnetic and spin–orbit effects have no discernible influence on Hc2.

Band structure and DOS calculations

The band structure and DOS curves of all the studied AM2B2 and A3Rh8B6 compounds were calculated; below we show two representative examples, namely Ca3Rh8B6 and CaRh2B2: their band structures are given, respectively, in figures 11 and 12 while their corresponding DOS curves are shown in figures 13 and 14. The calculated band structure and DOS curves of the other AM2B2 and A3Rh8B6 phases show similar, corresponding features.

Figure 11.

The band structure of Ca3Rh8B6. Most of the Rh 4d bands are found between −1 and −6 eV. Fewer bands cross the Fermi surface but two of these have extrema that almost touch EF, leading to a local peak in the DOS curve (see figure 13) and an increased N(EF) value.

Figure 12.

The band structure of CaRh2B2. In contrast to the case of Ca3Rh8B6 and within the displayed k-region, there are fewer bands located around EF and, furthermore, only two of these bands cross EF. This leads to the situation where EF is positioned at a valley of the N(EF) curve (see figure 14).

Figure 13.

The calculated total and partial DOS of Ca3Rh8B6 (in units of states per eV per primitive unit cell). Different atoms (as well as different sites of the same type of atom) contribute differently to N(EF). The Rh atoms are the major contributors to N(EF); the Rh atomic partial contributions are such that N(16j) > N(8i) > N(8f). Similarly, the partial contribution of B is such that N(16m) > N(8h), while for Ca one finds that N(8i) > N(4b).

Figure 14.

The calculated total and partial DOS curves of CaRh2B2 (in units of states per eV per primitive unit cell). Different atoms contribute differently to N(EF): N(16g) of Rh > N(16f) of B > N(8a) of Ca.

These curves as well as the results shown in table 4 indicate that Nt(EF) of each composition is substantial and that there is no band gap: these findings are consistent with the observed good metallic character. Furthermore, each Nt(EF) receives contributions from all atoms but the dominant contribution is from the Rh atoms (this is best illustrated in figure 13 of Ca3Rh8B6). Decomposing this Rh-atoms contribution, one notes that the main contribution is from the d orbitals (although there is a weak contribution from both s- and p-states). Furthermore, the d contribution at, say, the symmetry points Γ and X is dominated by the d orbitals (l = 2, m = ± 2) of the Rh1 (8i) atoms. On the other hand, the contribution of the Rh2 (16j) and Rh3 (8f) atoms is dominantly from the x2 − y2 (l = 2, m = ± 2) and z2 − 1 (l = 2, m = 0) orbitals. The above arguments are valid also for the case of Sr3Rh8B6. Similar features are also observed in CaRh2B2 (see figure 14): e.g. the stronger contributors are the Rh atoms, followed by B and then Ca atoms.

Table 4.

Total and partial DOS (in units of states per eV per primitive cell) of AM2B2 and A3Rh8B6 (A = Ca, Sr, Ba; M = Rh, Ir). See figures 13 and 14. The contributions of the s, p and d bands of each atom in Ca3Rh8B6 and CaRh2B2 are also given (in units of states per eV per formula unit).

ARh2B2	Partial/Total	NB(EF)		NM(EF)			NA(EF)		Nt(EF)
		16f		16g			8a
CaIr2B2	Total	0.09		0.24			0.06		0.72
SrRh2B2	Total	0.13		0.29			0.05		0.89
SrIr2B2	Total	0.11		0.29			0.06		0.86
CaRh2B2	s band	0.007		0.003			0.002
	p band	0.046		0.010			0.008
	d band	0.003		0.124			0.054
	Total	0.11		0.27			0.07		0.83
A3Rh8B6		8h	16m	8f	8i	16j	4b	8i
Sr3Rh8B6	Total	0.12	0.19	0.20	0.36	0.66	0.03	0.07	4.87
Ca3Rh8B6	s band	0.003	0.007	0.008	0.009	0.004	0.011	0.011
	p band	0.110	0.080	0.032	0.029	0.029	0.011	0.015
	d band	0.004	0.004	0.157	0.354	0.309	0.030	0.054
	Total	0.12	0.19	0.20	0.40	0.69	0.03	0.09	5.13

It is noted that the Rh 8i and 16j sites (located within the structural fragment of CaRh3B2) contribute almost twice as much as that from the 8f site (associated with the fragments of CaRh2B2). This may suggest that the observed superconductivity is related to the fragments of the CaRh3B2 phase; contrary to such an expectation, no superconductivity was observed in preliminary magnetization curves of CaRh3B2, measured down to 2 K [19].

Figures 11 and 12 indicate that the Ca3Rh8B6 bands seem to form both electron and hole pockets near EF, whereas for CaRh2B2 only electron pockets are noticeable; furthermore, for Ca3Rh8B6 (in contrast to CaRh2B2) there are more levels crossing EF; these features lead to the situation wherein EF of Ca3Rh8B6 is at almost the top of a local peak while EF of CaRh2B2 is positioned at a DOS local valley. As a consequence, Nt(EF) of Ca3Rh8B6 is almost fourfold higher than that of CaRh2B2: these features are taken to be the major reasons behind the observed differences among the transport and thermal properties of AM2B2 and A3Rh8B6 compounds. As an example, (i) the pronounced peak in the DOS of A3Rh8B6 (together with a relatively larger value of λNt(EF)—see below) is considered to be the main reason behind the surge of superconductivity in these compounds. (ii) These are also the reasons behind the difference in their measured γ: based on the theoretically calculated Nt(EF) (see e.g. figures 13 and 14), the bare Sommerfeld coefficient gives values (shown in table 3) which are surprisingly close to the measured ones (denoted, in the same table, as γL); the small discrepancy is attributed to normalization effects and these are characterized by the (1 + λ) factor where λ is the interaction (electron–phonon or electron–electron) parameter. Apparently, the calculated λ (table 3) for the superconducting A3Rh8B6 are smaller than those of the normal conducting AM2B2. This apparent contradiction is removed if we take into consideration the product λNt(EF) (the last column of table 3): indeed this product is much higher for A3Rh8B6 than for AM2B2.

The presence of high-Z atoms (Rh and Ir with their higher partial contribution to N(EF)) may introduce a spin–orbit interaction which, in the absence of inversion symmetry [20, 21], would have a strong influence on the electronic properties: e.g. a split of the energy bands, a reduction of N(EF)—a lowering of γ—or acting as a pair-breaking potential. To investigate this scenario, spin–orbit interactions were introduced into the band structure calculations. As far as Nt(EF) is concerned, no perceptible changes were observed. This is attributed to the fact that the space groups of all compounds, in contradiction to our earlier claim [2], are centrosymmetric.

Discussion and conclusions

Both experiments and theory indicate that the properties of the A3M8B6 compounds are strongly different from those of the AM2B2 ones. Such differences include: (i) the structure of A3M8B6, in contrast to that of AM2B2, is a mixture of structural fragments of AM3B2 and AM2B2; as the space group of AM2B2 is Fddd while that of A3M8B6 is Fmmm [3], then all symmetry-related physical properties are expected to be different, (ii) γ of the n = 3 phases are relatively larger than those of the n = 1 ones (see table 3), (iii) the superconductivity is present in A3Rh8B6 but not in the AM2B2 compounds and (iv) the magnetoresistivity of the latter is relatively stronger than that of the former [2]. As mentioned above, the differences in the electronic properties can be understood in terms of the band structure and DOS calculations (see figures 11–14). In particular, these calculations give support to the surge of superconductivity in Ca3Rh8B6 (see above).

A comparison of figures 7 and 10 indicates that while Tc of Ca3Rh8B6 is only 15% higher than that of Sr3Rh8B6, Hc2(0) of the former is almost four times higher than that of the latter. The similarity in Tc is consistent with the observation that their γ (also θD and λNt(EF)) are not strongly different. This is also consistent with the features of figure 13: as the Ca contributes very weakly to Nt(EF) of Ca3Rh8B6, then a substitution of Ca by an isovalent Sr would not lead to a very different Nt(EF). On the other hand, the observation that Hc2(0) of Ca3Rh8B6 is higher than that of Sr3Rh8B6 is related to the fact that both Tc and (∂Hc2/∂T) of the former are higher than those of the latter (recall that Hc2(0)∝Tc(∂Hc2/∂T)).

The superconductivity in the studied A3Rh8B6 samples (as well as that of Y3Os8B6: Tc = 5.8 K, Hc2(0) ≈ 20 kOe [22]) is associated with the Fmmm Ca3Rh8B6-type phase, the very same one shown in figure 1(c). It is difficult, at this stage of investigation, to discuss the nature of this superconducting state; however, the observed superconducting features of A3Rh8B6 and Y3Os8B6 [22] suggest a BCS-type superconductivity with a singlet character, a low Tc, a weak Hc2(0) and a relatively smaller Meissner effect (all are strongly sensitive to sample history and impurities).

In summary, the application of a variety of preparation and heat treatment procedures enabled us to synthesize various AM3B2 (n = 1, 3) samples. Extensive structural, elemental and physical characterizations indicate that among the various stabilized compounds, superconductivity is detected only in A3Rh8B6 (A = Ca, Sr) samples; all studied AM2B2 samples are found to be normal conductors. There are additional striking differences among the AM2B2 and A3M8B6 phases: as compared to the latter, the former exhibits stronger magnetoresistivities and lower γ. Such strong differences are also evident in the calculated electronic band structure and DOS curves: as an example, EF of AM2B2 is positioned at a local valley of the DOS curve, while that of an A3M8B6 phase stands at almost the top of a local DOS peak; accordingly, their N(EF) and γ are different. Assuringly, the calculated γ‘s’ are found to approximate quite satisfactory the experimentally determined values.

Finally, these low-Tc superconducting A3Rh8B6 (A = Ca, Sr) phases are members of the homologous AM3B2 series. It is of interest to search for superconductivity in the other members of this series as well as to further explore how the metallic properties of this series are influenced by a variation in the pressure, in the substitution or in the stacking of the structural fragments of AM3B2 and AM2B2 [19].