Strong-Coupling Superconductivity in BaPd2As2 Induced by Soft Phonons in the ThCr2Si2-type Polymorph

TL;DR

This study investigates two polymorphs of BaPd2As2, revealing that the ThCr2Si2-type exhibits strong-coupling superconductivity likely driven by soft phonons, unlike the non-superconducting CeMg2Si2-type.

Contribution

It demonstrates that soft phonons in the ThCr2Si2-type polymorph induce strong-coupling superconductivity, providing insight into phonon-mediated pairing mechanisms.

Findings

ThCr2Si2-type has low Debye temperature indicating soft phonons

Superconductivity in ThCr2Si2-type is strong-coupling

CeMg2Si2-type remains a normal metal down to 1.8 K

Abstract

The specific heat of two polymorphs of BaPd2As2 was measured. The ThCr2Si2-type polymorph (space group I4/mmm, $D_{4h}^{17}$, No. 139) is a previously reported superconductor with a transition temperature Tc $\simeq$ 3.5 K, while the CeMg2Si2-type polymorph (P4/mmm, $D_{4h}^{1}$, No. 123) is a normal metal and does not exhibit superconductivity down to 1.8 K. Our results revealed that the ThCr2Si2-type has an anomalously low Debye temperature, indicative of soft phonons, compared to the CeMg2Si2-type. Moreover, a large specific-heat jump at Tc indicated that the superconductivity of ThCr2Si2-type is a strong-coupling type, which is likely derived from soft phonons.

Tables

Table 1. Table 1: The superconducting transition temperature T c subscript 𝑇 c T_{\rm c} , electronic specific-heat coefficient γ 𝛾 \gamma , Debye temperature Θ D subscript Θ D \Theta_{\rm D} , and normalized specific-heat jump Δ ​ C e / γ ​ T c Δ subscript 𝐶 e 𝛾 subscript 𝑇 c \Delta C_{\rm e}/\gamma T_{\rm c} for AEPd 2 As 2 (AE = Ca, Sr, and Ba) with the ThCr 2 Si 2 -type structure (space group I ​ 4 / m ​ m ​ m 𝐼 4 𝑚 𝑚 𝑚 I4/mmm , D 4 ​ h 17 superscript subscript 𝐷 4 ℎ 17 D_{4h}^{17} , No. 139) and for BaPd 2 As 2 with the CeMg 2 Si 2 -type structure ( P ​ 4 / m ​ m ​ m 𝑃 4 𝑚 𝑚 𝑚 P4/mmm , D 4 ​ h 1 superscript subscript 𝐷 4 ℎ 1 D_{4h}^{1} , No. 123).

Compound	Structure type	$T_c$	$\gamma$	${\Theta }_D$	$\Delta C_e/\gamma T_c$
		(K)	(mJ/K2mol)	(K)
CaPd2As2*	ThCr2Si2	1.27	6.52	276	1.1
SrPd2As2*	ThCr2Si2	0.92	6.43	298	1.0
BaPd2As2	ThCr2Si2	3.5	6.1	144	2.3
BaPd2As2	CeMg2Si2	–	5.4	259	–
* Ref. \citenanand.

Full text

Energetically low-lying phonons often result in strong-coupling superconductivity with an enhanced superconducting transition temperature $T_{\rm c}$ [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Thus, engineering materials to produce low-lying phonons has become an important issue in superconductivity. A promising route to producing low-lying phonons is a local anharmonic vibration of the ion that is weakly bound in a cage-like structure. Such a vibration is called a rattling phonon and characterized by a low-frequency Einstein mode. A remarkable example is the $\beta$-pyrochlore oxide KOs2O6 with $T_{\rm c}$ = 9.6 K [1, 2, 3, 4], as well as Ba3Ir4Ge16 with $T_{\rm c}$ $\simeq$ 6 K [5, 6]. Another route to producing low-lying phonons is structural instability that becomes evident from the occurrence of a structural phase transition due to applied pressure or chemical doping. Prominent examples of such superconductors include the 122-type pnictides BaNi2(As1-xPx)2 with $T_{\rm c}$ = 3.3 K [7], BaNi2(Ge1-xPx)2 with $T_{\rm c}$ = 2.9 K [8], and Ba(Ni1-xCux)2As2 with $T_{\rm c}$ = 3.2 K [9], all of which exhibit strong-coupling superconductivity because of a structural phase transition and the subsequent phonon softening characterized by an anomalously low Debye frequency. The MnP-type compound IrGe with $T_{\rm c}$ = 4.7 K is a rare example that exhibits strong-coupling superconductivity with low-lying phonons but does not exhibit a structural phase transition [10]. Such superconductors suggest how low-lying phonons may be produced for enhancing superconductivity.

The title compound BaPd2As2 has been known to crystallize in three forms: ThCr2Si2-type (space group $I4/mmm$, $D_{4h}^{17}$, No. 139) [11], CeMg2Si2-type ($P4/mmm$, $D_{4h}^{1}$, No. 123) [12], and the intergrowth structure of CaBe2Ge2-type and CeMg2Si2-type ($I4/mmm$) [12]. The ThCr2Si2-type structure consists of PdAs4 tetrahedra, while the CeMg2Si2-type consists of PdAs4 planar squares, as shown in Figs. 1(a) and 1(b), respectively. The intergrowth structure, shown in Fig. 1(c), consists of Pd4As tetrahedra, which are characteristic of the CaBe2Ge2-type structure such as BaPt2As2 [13, 14], as well as PdAs4 planar squares, which are characteristic of the CeMg2Si2-type structure. Guo et al. reported superconductivity at $T_{\rm c}$ = 3.85 K for the ThCr2Si2-type structure [15]. This $T_{\rm c}$ of BaPd2As2 is anomalously high compared with $T_{\rm c}$ = 1.27 and 0.92 K for CaPd2As2 and SrPd2As2, respectively [16], which are isoelectronic and isostructural with BaPd2As2 of the ThCr2Si2-type structure. On the other hand, no bulk superconductivity was observed for BaPd2As2 when the structure is the CeMg2Si2-type [16]. These observations suggest that a key factor in enhancing $T_{\rm c}$ may be present in BaPd2As2.

In this paper, we present the results of our synthesis of two polymorphs of BaPd2As2, namely, ThCr2Si2-type and CeMg2Si2-type, along with the results of specific-heat measurements. We found that when the structure was ThCr2Si2-type, BaPd2As2 exhibited superconductivity at $T_{\rm c}$ = 3.5 K with a large specific-heat jump at the superconducting transition $\Delta C/\gamma T_{\rm c}$ = 2.3, where $\gamma$ is the electronic specific-heat coefficient, which indicated that the superconductivity was a strong-coupling type. Further, low-lying phonons, which manifested themselves at an anomalously low Debye temperature $\Theta_{\rm D}$ = 144 K, was observed in the normal-state specific-heat data. On the other hand, when the structure was CeMg2Si2-type, BaPd2As2 did not exhibit superconductivity, and the Debye temperature of $\Theta_{\rm D}$ = 259 K was a typical value for 122-type pnictides.

Each polymorph of BaPd2As2, ThCr2Si2-type or CeMg2Si2-type, was selectively synthesized with different conditions. Single crystals were obtained for both polymorphs. The condition for obtaining the ThCr2Si2-type was strict. First, a mixture of Pd$:$As = 1$:$1.5 was sealed in an evacuated quartz tube, heated at 500 ∘C for 10 h, and then heated at 750 ∘C for 40 h, followed by furnace cooling. The ampule was heated repeatedly at 800 and 850 ∘C, following at 500 ∘C in each process, without intermediate grinding. The resultant lumps of PdAs2 with metallic luster were separated from the Pd2As powder and pulverized. Second, stoichiometric amounts of the Ba, Pd, and PdAs2 powders were placed in an alumina crucible, sealed in an evacuated quartz tube, and heated at 900 ∘C for 48 h, followed by a quenching in iced water. As a result, single crystals of ThCr2Si2-type were obtained with a typical dimension of 0.1–0.5 $\times$ 0.1–0.5 $\times$ 0.02 mm3. Note that a larger ratio of As in the starting composition resulted in As grains in the quartz tube during the PdAs2 synthesis. In this case, a mixture of ThCr2Si2-type and CeMg2Si2-type crystals was obtained, even if the As grains were removed and only PdAs2 lumps were used for the second step. On the other hand, the condition for obtaining the CeMg2Si2-type was less strict. We used a precursor PdAs2 synthesized from a mixture with Pd$:$As = 1$:$2. Stoichiometric amounts of Ba, Pd, and PdAs2 were placed in an alumina crucible, sealed in an evacuated quartz tube, and heated at 700–1150 ∘C, followed by either quenching in iced water or furnace cooling. Both conditions resulted in single crystals of CeMg2Si2-type with a typical dimension of 0.5 $\times$ 0.5 $\times$ 0.02 mm3. All of these observations suggest that the CeMg2Si2-type is a stable phase, while the ThCr2Si2-type is a metastable phase, consistent with total energies predicted by first-principles calculations[17].

Powder X-ray diffraction (XRD) studies confirmed that the resulting samples were a single phase of either ThCr2Si2-type or CeMg2As2-type. The magnetization $M$ and specific heat $C$ were measured using the Magnetic Property Measurement System (MPMS) and the Physical Property Measurement System (PPMS) by Quantum Design, respectively. The specific-heat measurement of a standard Cu sample yielded the electronic specific-heat coefficient $\gamma$ = 0.694(3) mJ/K2mol and Debye temperature $\Theta_{\rm D}$ = 337.3(9) K, which agreed reasonably well with literature values [18].

The superconducting transition in BaPd2As2 was examined by magnetization measurements, as shown in Fig. 2(a). In a manner consistent with previous reports [15, 16], BaPd2As2 exhibited a superconducting transition at 3.85 K with the ThCr2Si2-type structure, while superconductivity was not observed down to 1.8 K (the lowest temperature measured) for the CeMg2Si2-type structure.

Figure 3 shows the specific heat divided by the temperature $C/T$ as a function of the squared temperature $T^{2}$. The normal-state data could be fitted by $C/T=\gamma+\beta T^{2}+\delta T^{4}$, where $\gamma$ is an electronic specific-heat coefficient, $\beta$ is the coefficient of phonon contributions from which the Debye temperature $\Theta_{\rm D}$ is estimated, and $\delta$ is a constant related to phonon contributions. As shown in Fig. 3, the slope of the $C/T$ vs. $T^{2}$ curve of the ThCr2Si2-type was significantly steeper than that of the CeMg2Si2-type, suggesting the presence of significantly soft phonons in the ThCr2Si2-type structure. We estimated the Debye temperature $\Theta_{\rm D}$ = 144 and 259 K for the ThCr2Si2-type and CeMg2Si2-type, respectively. In contrast to the remarkable difference in the Debye temperature, the electronic specific-heat coefficient $\gamma$ showed similar values for the two structures, as can be seen from the almost identical intercepts of the $C/T$ vs. $T^{2}$ curves along the $C/T$ axis in Fig. 3. The $\gamma$ values are summarized in Table I. The observed $\gamma$ values were comparable to the first-principles calculation values, $\gamma$ = 5.664 and 4.639 mJ/K2mol for the ThCr2Si2-type and CeMg2Si2-type structures, respectively [17].

The superconducting transition temperature $T_{\rm c}$ = 3.5 K, determined by specific-heat measurements, as shown in Fig. 2(b), for BaPd2As2 was markedly higher than $T_{\rm c}$ = 1.27 and 0.92 K for CaPd2As2 and SrPd2As2, respectively [16]. Correspondingly, the Debye temperature $\Theta_{\rm D}$ = 144 K for BaPd2As2 was anomalously low compared with $\Theta_{\rm D}$ = 276 and 298 K for CaPd2As2 and SrPd2As2, respectively [16]. In contrast to the remarkable differences in both $T_{\rm c}$ and $\Theta_{\rm D}$ between BaPd2As2 and its isostructural compounds, CaPd2As2 and SrPd2As2, the electronic specific-heat coefficient $\gamma$ showed similar values, as summarized in Table I. These observations suggested that the enhanced superconductivity in BaPd2As2 originated not from the electronic properties but from phonons, specifically, from enhanced electron–phonon coupling due to soft phonons. Indeed, the normalized specific-heat jump at the superconducting transition $\Delta C_{\rm e}/\gamma T_{\rm c}$ = 2.3, which was determined based on the $\alpha$-model [19], as shown in Fig. 2(b), indicated strong-coupling superconductivity due to the enhanced electron–phonon coupling in BaPd2As2. On the other hand, the small value of $\Delta C_{\rm e}/\gamma T_{\rm c}$ $\simeq$ 1 indicated weak-coupling superconductivity in CaPd2As2 and SrPd2As2 [16].

The anomalously low Debye temperature $\Theta_{\rm D}$ and enhanced superconducting transition temperature $T_{\rm c}$ in BaPd2As2 was visualized using a $T_{\rm c}$ vs. $\Theta_{\rm D}$ plot for various 122-type compounds AETM2X2, where AE = Ca, Sr, and Ba; TM = Rh, Ir, Ni, and Pd; and X = P and As, with the tetragonal ThCr2Si2-type and distorted ThCr2Si2-type structures, as shown in Fig. 4(a). From the figure, lower values of $\Theta_{\rm D}$ tend to have higher values of $T_{\rm c}$; note that BaPd2As2 is placed at the highest value of $T_{\rm c}$ and lowest value of $\Theta_{\rm D}$ among the family of 122-type compounds with the ThCr2Si2-type structure, except for iron-based superconductors (not shown) such as BaFe2As2, in which the mechanism of superconductivity is different [24, 25]. This trend was evident in the series of CaPd2As2–SrPd2As2–BaPd2As2. However, this trend was less apparent in SrNi2P2–BaNi2P2 and SrNi2As2–BaNi2As2 likely as a result of the structural phase transition and resultant structural distortion of SrNi2P2 [23, 26] and BaNi2As2 [7, 27]. However, the substitution of tiny amount of P for As in BaNi2As2 resulted in an enhanced $T_{\rm c}$ with lowered $\Theta_{\rm D}$ [7], which was accompanied by the suppression of the structural phase transition [7], and the $T_{\rm c}$–$\Theta_{\rm D}$ relation became evident in SrNi2As2–BaNi2(As1-xPx)2 in Fig. 4(a). On the other hand, in the $T_{\rm c}$ vs. $\gamma$ plot, shown in Fig. 4(b), $T_{\rm c}$ is not obviously dependent on the electronic specific-heat coefficient $\gamma$. Moreover, CaPd2As2–SrPd2As2–BaPd2As2 exhibited different $T_{\rm c}$ values, even though they exhibited similar $\gamma$-values (approximately 6 mJ/K2mol). Similarly, the Ni-based compounds also exhibited quite different $T_{\rm c}$ values, even though their $\gamma$ values were similar (approximately 14 mJ/K2mol).

All of these observations indicated that the determining factor of $T_{\rm c}$ was the soft phonons, or energetically low-lying phonons, of the ThCr2Si2-type structure. For BaNi2As2, the in-plane vibrations of Ni and As have been predicted to produce low-lying phonons [28], which eventually resulted in a structural phase transition that distorted the Ni square network into zigzag chains [27], thus suggesting freezing of the in-plane Ni motion. On the other hand, BaPd2As2 did not exhibit a structural phase transition in spite of the enhanced electron–phonon coupling. The lack of a structural phase transition could explain why BaPd2As2 exhibited the highest $T_{\rm c}$ value among AETM2X2. The phonon mode responsible for the lowered Debye temperature $\Theta_{\rm D}$ of BaPd2As2 was not clear from this study. We speculate that the in-plane Pd and As vibrations have importance, according to the analogy with BaNi2As2, as well as the first-principles calculations [17, 29, 30]. We suggest the in-plane As vibration has particular importance to low-lying phonons because the ThCr2Si2-type structure could be transformed to the CeMg2Si2-type structure by the displacement of As, and vice versa, as can be seen from Figs. 1(a) and 1(b). Moreover, the ThCr2Si2-type is a metastable phase, while the CeMg2Si2-type is a stable phase, as we mentioned above. Thus, we expect that the ThCr2Si2-type polymorph possesses low-lying phonons due to the structural instability of metastable phase. If this is the case, polymorphism could provide another route to producing low-lying phonons and hence enhanced superconductivity [31, 32].

In conclusion, we have measured the specific heat of BaPd2As2 with two different forms of BaPd2As2: the superconducting ThCr2Si2-type (space group $I4/mmm$, $D_{4h}^{17}$, No. 139) and non-superconducting CeMg2Si2-type ($P4/mmm$, $D_{4h}^{1}$, No. 123). The very large specific-heat jump at $T_{\rm c}$ indicated that the ThCr2Si2-type was a strong-coupling superconductor. Compared with the CeMg2Si2-type and other ThCr2Si2-type pnictides, an anomalously low Debye temperature was observed in ThCr2Si2-type BaPd2As2, originating from low-lying phonons. The key factor that enhanced $T_{\rm c}$ in BaPd2As2 was low-lying phonons with a Debye temperature of $\Theta_{\rm D}$ = 144 K of the ThCr2Si2-type polymorph.

\acknowledgment

The authors are grateful to M. Akoshima and H. Abe for helping to calibrate the instrument for specific-heat measurements, and to K. Fujimura, N. Nishimoto, T. Mizukami, and H. Ishii for their technical assistance in the experiments and data analyses for this work. A part of this work was performed at the Advanced Science Research Center, Okayama University. This work was partially supported by Grants-in-Aid for Scientific Research (No. 26287082, 15H01047, 15H05886, and 16K05451) provided by the Japan Society for the Promotion of Science (JSPS) and the Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers from JSPS.