SrAuSi3: A Noncentrosymmetric Superconductor - Chemistry of

Feb 14, 2014 - National Institute for Materials Science (NIMS), 1-2-1 Sengen, ... φ) with an azimuthal quantum number l of 0, 1, 2, 3, ..., for the s...
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SrAuSi3: A Noncentrosymmetric Superconductor Masaaki Isobe,*,† Hiroyuki Yoshida,† Koji Kimoto,‡ Masao Arai,§ and Eiji Takayama-Muromachi∥ †

Superconducting Properties Unit, National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan Surface Physics and Structure Unit, National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan § Computational Materials Science Unit, National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan ∥ National Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan ‡

ABSTRACT: A novel strontium−gold silicide (SrAuSi3) was successfully synthesized for the first time using a high-pressure technique, at ∼6 GPa. X-ray Rietveld analysis and scanning transmission electron microscopy studies clearly revealed that SrAuSi3 crystallizes in a BaNiSn3-type structure, in space group I4mm with the following lattice parameters: a = 4.4090(1) Å, and c = 9.9475(3) Å. The structure was found not to have inversion symmetry in real space. From direct measurements of polycrystalline SrAuSi3, the electrical resistivity was zero, the static magnetic susceptibility was almost a full volume Meissner diamagnetic signal, and the specific heat exhibited a sharp jump, all of which were accompanied by a phase transition due to bulk superconductivity at a critical temperature (TC) of 1.54 K. The SrAuSi3 compound is the first noncentrosymmetric superconductor that includes Au as a principal constituent element.

1. INTRODUCTION Inversion symmetry is an important key in understanding the superconducting state. In the case in which the superconducting system possesses inversion symmetry in the crystal structure, the wave function of Cooper pairs Ψ(r1, σ1; r2, σ2) can be described as a multiplication of a spin part χ(σ1, σ2) and an orbital part ψ(r1, r2), resulting in Ψ(r1, σ1; r2, σ2) = χ(σ1, σ2)·ψ(r1, r2). The spin part is either spin-singlet (1/√2)(|↑↓⟩ − |↓↑⟩) or spin-triplet |↑↑⟩ (for Sz = 1), (1/√2)(|↑↓⟩ + |↓↑⟩) (for Sz = 0), or |↓↓⟩ (for Sz = −1). The orbital part can be written as a kind of spherical harmonics ψ(r2 − r1) ∝ Ylm(θ, φ) with an azimuthal quantum number l of 0, 1, 2, 3, ..., for the s-, p-, d-, f-, ..., states, respectively. The wave function has to be kept in odd symmetry for an exchange of counter pair fermions; Ψ(r2, σ2; r1, σ1) = −Ψ(r1, σ1; r2, σ2). Therefore, if the spin part is a singlet (antisymmetric), the orbital part must be an even function, i.e., s- or d-wave state. Reversely, if the spin part is a triplet (symmetric), the orbital part must be an odd function, i.e., p- or f-wave state. In either case, parity of the wave function can be definitely determined for the superconducting state. For an s-wave state, the orbital part of the wave function is isotropic in real and momentum space and the superconducting gap uniformly opens anywhere on the Fermi surface. In contrast, for a p- or d-wave state, the orbital part is anisotropic and the gap partly closes on the Fermi surface with nodes. The gap structure directly affects the superconducting properties. Parity is, therefore, the most basic key factor describing the superconducting state. On the other hand, in structures lacking inversion symmetry, however, the parity is then no longer a meaningful label for © 2014 American Chemical Society

defining the symmetry, and the pairing states cannot be classified as singlet or triplet. Missing inversion symmetry allows parity-violated superconductivity in which the admixture of spin-singlet and spin-triplet pairing states within the same orbital channel can occur.1−15 The noncentrosymmetry introduces a local electrical field into the crystal and thereby creates asymmetric spin−orbit coupling (SOC) between electron’s momentum and its spin, given by HSO = (ℏ2/ 4m*2c2)(∇V × k)·σ, where ∇V is the potential gradient generated at asymmetric crystallographic sites in the unit structure, k is the momentum of an electron, and σ is the Pauli matrix. This SO interaction breaks symmetry for an operation of space inversion k → −k or spin inversion σ → −σ. The SOC can be regarded as interaction between the spin σ and the effective Zeeman field (Heff = ∇V × k), which splits degenerated bands and thus the Fermi surfaces into two subparts with different spin helicities of up or down spin. The spin vector of electrons points in the direction perpendicular to the Fermi momentum kF and ∇V. For example, in the case of a Rashba-type asymmetric field [∇V ∝ nz = (001)], the spin component can be given by σ ∝ (−ky, kx, 0). In the superconducting state, where the Fermi surface splitting energy is by some amount larger than the superconducting gap, the Cooper pair (k, −k) is expected to form within each Fermi surface. Inter Fermi surface Cooper pairing is unlikely, because it may result in a loss of energy. It should be Received: January 6, 2014 Revised: February 12, 2014 Published: February 14, 2014 2155

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noted that in this case, the Cooper pair |k, +σ⟩ |−k, −σ⟩ is not a spin-singlet state, because the proper counterpart of this state, | k, −σ⟩ |−k, +σ⟩, is formed on another Fermi surface. Thus, superimposition is not possible. This is attributed to the fact that the spin orientation is tightly restricted by the Fermi momentum through the asymmetric SOC. Many theories have predicted that the pairing state should be a mixture of spinsinglet and -triplet states.1−11 Parity mixing is the most striking feature of superconductivity in noncentrosymmetric systems. It is also theoretically predicted that the superconducting-gap function (order parameter) is the admixture of what would normally be expected for singlet and triplet states, forming the anisotropic gap structure, including line nodes on the Fermi surfaces even in the absence of strong electronic correlations.1−11 This outcome is of particular interest to all in the field of superconductivity research and should be investigated through experiments to confirm the theoretical hypothesis. Following the pioneering work on the CePt3Si16 compound by Bauer et al., a wide variety of noncentrosymmetric superconductors have been reported, e.g., CeRhSi3,17,18 CeIrSi3,19,20 CeCoGe3,21 BaPtSi3,22 CaIrSi3,23−25 CaPtSi3,24,25 UIr,26 BiPd,27 Li2Pd3B,28−33 Li2Pt3B,30,31,33,34 Ru7B3,35,36 Mg 10 Ir 19 B 16 , 37,38 Mo 3 Al 2 C, 39,40 Rh 2 Ga 9 , 41,42 Ir 2 Ga 9 , 41,42 LaNiC2,43,44 Y2C3,45,46 La2C3,46 etc. In particular, the Cebased heavy-fermion systems have attracted attention, because they exhibit an unusually large upper critical field HC2 far beyond the Pauli limit.18,19 However, the participation of the felectrons may make the issue more complicated, because the electronic correlation screens out the not fully understood nature of noncentrosymmetric superconductivity. Therefore, we have chosen to study nonmagnetic f-electron-free noncentrosymmetric superconductors. Thus far, unconventional phenomena with the spin-triplet Cooper pairing, which is expected to arise from the lack of inversion symmetry, have been reported for only Li2Pt3B,33,34 Mo3Al2C,39,40 and LaNiC2.43 Most of the other noncentrosymmetric systems reported are, however, conventional s-wave full-gap superconductors with dominant spin-singlet Cooper pairs.22,24 It is, therefore, important to explore novel types of unconventional noncentrosymmetric superconductors to improve our understanding of noncentrosymmetric physics. During the course of our study of new materials, we focused our search on BaNiSn3-type crystal structure compounds. This is because a large number of BaNiSn3-type compounds have been reported, as described with the general chemical formula AMX3 (A = Ce, La, Ca, Sr, or Ba; M = Co, Ni, Ru, Rh, Pd, Os, Ir, or Pt; X = Si, Ge, Sn, etc.). After many trials, we recently succeeded in synthesizing a novel 1-1-3 strontium−gold silicide (SrAuSi3) by utilizing a high-pressure technique (gigapascal pressures). We found that SrAuSi3 exhibits bulk superconductivity with a critical temperature (TC) of ∼1.6 K. To the best of our knowledge, this SrAuSi3 is the first noncentrosymmetric superconductor that includes Au as a principal constituent element. Because Au is a heavier element than Ir and Pt, stronger SOC would be expected, if the 5d or 6p nonspherical (anisotropic) orbitals effectively contribute to the bands crossing the Fermi level. In this paper, we report the synthesis, crystal structure, electrical resistivity, specific heat, and magnetic properties of the novel noncentrosymmetric superconductor SrAuSi3. It was found that SrAuSi3 exhibits an unusual temperature dependence of the specific heat and a puzzling TC suppression in spite of its moderately strong coupling Cooper pairing; this may be

an indication of an unconventional superconducting state. We discuss the origin of these unusual properties in the superconducting state from the viewpoint of the electronic structure based on the results of relativistic density functional theory (DFT) band calculation of SrAuSi3.

2. EXPERIMENTAL SECTION Polycrystalline samples of SrAuSi3 were prepared using a high-pressure synthesis technique. Starting reagents, SrSi2 (3 N), Au (4 N), and Si (4 N) powders, were mixed in an agate mortar in a molar ratio of 1:1:1. The mixture was pressed into a disk pellet (6.9 mmϕ × ∼3 mmt, weight of ∼500 mg). The processing steps, i.e., weighing, mixing, and producing the pellet, were conducted in a glovebox filled with dry argon gas. The pellets were put in a high-pressure cell with a pressure medium of hexagonal boron nitride (h-BN) powder, heated, and maintained at 1500 °C for ∼1 h under 6 GPa using a flat-belt-type high-pressure apparatus. The temperature and pressure measurements were calibrated with a W5%Re−W26%Re thermocouple and pressure standard materials (Bi, Tl, and Ba), respectively. During the heat treatment, the chemical reaction SrSi2 + Au + Si → SrAuSi3 was expected to dominate. After being heated, the pellets were quenched to room temperature within a few seconds before the pressure was released. The final product was dense and metallic shiny black in color and retained the pellet shape. The phase purity and crystal structure of the products were studied using conventional powder X-ray diffraction (XRD). The XRD data were collected at room temperature using a diffractometer (Rigaku, RINT-Ultima III) equipped with a Cu Kα radiation source and a conventional slit system. The atomic composition of grains in the ceramic sample was determined using a field emission electron probe microanalyzer (EPMA; JEOL, JXA-8500F) operated at 15 kV; SrTiO3, Au, and Si were used as the standard materials. The crystal structure of the samples was observed using a high-resolution scanning transmission electron microscope (STEM; FEI, Titan3) operating at 300 kV equipped with a spherical aberration (Cs-) corrector (CEOS GmbH, DCOR).47 The structural parameters were refined using the X-ray Rietveld method with the analysis software RIETAN-2000.48 The crystal structure was visualized using VESTA.49 The electrical resistivity was measured with the standard dc fourprobe method with a gauge current of 1 mA using a commercial apparatus (Quantum Design, PPMS) equipped with a 3He refrigerator. The specific heat was measured for a small bulk specimen using the PPMS system and the time relaxation method. Static magnetic data were collected for a pulverized sample (weight of ∼120 mg) using a SQUID magnetometer (Quantum Design, MPMS) equipped with a 3 He cryo-system (IQUANTUM, iHelium3). The ab initio calculation was performed by means of the fullpotential linearized augmented plane-wave (FLAPW) method using the WIEN2k software package.50 The generalized gradient approximation (GGA)51 based on the DFT52 was employed as the exchangecorrelation energy functional. The spin−orbit interaction is included as a perturbation to the scalar-relativistic equations. Experimental lattice parameters and atomic coordinates were used for the calculation. The muffin-tin sphere radii (R) were chosen as 2.5 au for Sr, 2.4 au for Au, and 2.0 au for Si. The self-consistent calculations converged well with the wavenumber cutoff parameter K, which satisfies the relationship RminK = 8, where Rmin is the smallest muffin radius of 2.0 au The Brillouin zone integration was approximated by a tetrahedron method with 159 k points in the irreducible Brillouin zone (IBZ). For the Fermi surface plot, eigen energies were computed using a finer k mesh that consists of 2393 k points in the IBZ.

3. RESULTS AND DISCUSSION 3.1. Crystal Structure. Figure 1a is the powder XRD profile of the SrAuSi3 sample. Almost all the Bragg reflections in the XRD profile can be systematically indexed with a bodycentered tetragonal system with the following lattice parameters: a = 4.4090(1) Å, and c = 9.9475(3) Å. Extinctions of the 2156

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to the literature values obtained from ref 53. Figure 1b is the best refinement X-ray Rietveld profile. Resultant reliability factors were as follows: Rwp = 10.21%, RB = 2.53%, RF = 1.22%, and S = Rwp/Re = 1.5154. The difference curve (solid blue line at the bottom of the plot) clearly indicates that the observed Xray profile can be satisfactorily reproduced using the proposed structural model. The refined structure parameters and the crystallographic data of SrAuSi3 are summarized in Table 1. Table 1. Atomic Coordinates (x, y, z), Isotropic Displacement Parameters (B), and Crystallographic Data of SrAuSi3 atom

site

Sr 2a Au 2a Si1 2a Si2 4b formula molecular weight temperature wavelength (Å) space group lattice constants

Figure 1. (a) Powder X-ray diffraction (XRD) pattern of the SrAuSi3 sample. All the prominent Bragg peaks are systematically indexed with a body-centered tetragonal lattice with the following cell parameters: a = 4.4090(1) Å, and c = 9.9475(3) Å. This majority phase was identified as a novel phase SrAuSi3. For the remaining peaks, some small extra reflections marked with A and S are assigned to secondary phases SrAu2Si2 and Si, respectively. (b) X-ray Rietveld analysis pattern of the SrAuSi3 sample: observed data (small red crosses), calculated profile (solid green line), and the difference (solid blue line). The green vertical tick marks indicate 2θ positions of the Bragg reflections for SrAuSi3 (top), SrAu2Si2 (middle), and Si (bottom). The inset is a magnification of the pattern in the 2θ = 30−50° region. Reliability factors are as follows: Rwp = 10.21%, RB = 2.53%, and RF = 1.22%. Refined structural parameters are summarized in Table 1.

Z density (calcd) (g/cm3) R factors mass fractions refinement software

ga

x

y

B (Å2)

z b

1 0 0 0.0 1 0 0 0.6291(3) 1 0 0 0.3784(8) 1 0 1/2 0.2575(6) SrAuSi3 368.84 room temperature 1.540593 (Cu Kα) I4mm (No. 107) a = 4.40903(16) Å, c = 9.94755(32) Å, V = 193.3762(117) Å3 2 6.3346

0.42(4) 0.63(1) 0.12(6) 0.12c

Rwp = 10.21%, Rp = 7.48%, RB = 2.53%, RF = 1.22%, S = Rwp/Re = 1.5154 SrAuSi3 (71.77 wt %), SrAu2Si2 (22.57 wt %), Si (5.66 wt %) RIETAN-2000

Occupancies for all the atoms are fixed to 1 (g = 1). bThe fractional coordinate z of the Sr site is fixed to null [z(Sr) = 0.0]. cThermal parameters of the Si atoms were grouped and refined together [constraint on B(Si2) = B(Si1)]. a

reflections are as follows: h + k + l = 2n for hkl, h + k = 2n for hk0, h + l = 2n for h0l, l = 2n for hhl, and l = 2n for 00l. Possible space groups are, therefore, centrosymmetric I4/mmm (No. 139) and noncentrosymmetric I42̅ m (No. 121), I4m ̅ 2 (No. 119), I4mm (No. 107), and I422 (No. 97). The intensity pattern and the Bragg angles of this phase are obviously distinct from those of the already-known ThCr2Si2-type structure phase of SrAu2Si2 [I4/mmm; a = 4.37(1) Å, and c = 10.14(2) Å].53 Therefore, the main phase in the product can be assigned to the novel compound SrAuSi3. The SrAuSi3 sample looks almost monophasic. However, the XRD profile includes some extra Bragg reflections because of a small quantity of secondary phases. For example, small peaks around 2θ values of ∼35.5° and ∼47.4° were identified as SrAu2Si2 and Si, respectively. According to the phase rule, another secondary phase should be included in the sample to balance the chemical composition in the system. The undetectably small impurity is expected to be a Sr−Si binary phase. The crystal structure of SrAuSi3 was more precisely studied using the X-ray Rietveld analysis method. A structural model for SrAuSi3 was composed on the basis of the BaNiSn3-type structure in space group I4mm. The atomic coordinates and the thermal displacement parameters were refined for all the sites of SrAuSi3. The secondary phases, SrAu2Si2 and Si, were also analyzed simultaneously with the main phase, SrAuSi3. For the secondary phases, only the lattice constants and peak profile parameters were refined and the atomic coordinates were fixed

Table 2. Selected Interatomic Distances for the Structure of SrAuSi3a atom 1−atom 2

no. of equivalent distances

interatomic distance (Å)

Au(i)−Si1(i) Au(i)−Si2(ii) Si1(i)−Si2(i) Si2(i)−Si2(iii) Sr(iv)−Si2(i) Sr(iv)−Si1(i) Sr(iv)−Au(i) Sr(i)−Si2(i)

1 4 4 4 4 4 4 4

2.493(6) 2.548(2) 2.511(4) 3.117(1) 3.267(4) 3.343(3) 3.371(1) 3.379(4)

Symmetry operators: (i) x, y, z; (ii) x + 1/2, y − 1/2, z + 1/2; (iii) −y + /2, x, z; (iv) x + 1/2, y + 1/2, z + 1/2.

a

1

The selected interatomic distances are listed in Table 2. Mass fractions of the main phase and the secondary phases contained in the sample are 71.77 wt % for SrAuSi3, 22.57 wt % for SrAu2Si2, and 5.66 wt % for Si. The SrAuSi3:SrAu2Si2 molar ratio is 0.823:0.177, agreeing with the value determined from the EPMA measurements. It should be noted that in Table 1, the thermal displacement parameter B of the heavier element (Au) is larger than that of the lighter element (Si). This result may be due to slight atomic mixing between the Au and Si sites. 2157

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The large difference in atomic weight between Au and Si can cause the significant deviation from the proper B value even for slight atomic mixing. Figure 2a is an illustration of the crystal structure of SrAuSi3, using the parameters from Table 1. The crystal structure of

Figure 3. Typical SEM image of a cleavage surface of a SrAuSi3 ceramic sample. Figure 2. Crystal structures of (a) SrAuSi3 [BaNiSn3 type, I4mm (noncentrosymmetric), a = 4.4090(1) Å, and c = 9.9475(3) Å] and (b) SrAu2Si2 [ThCr2Si2 type, I4/mmm (centrosymmetric), a = 4.37(1) Å, and c = 10.14(2) Å].53 Two structure types are derivatives of the BaAl4-type structure phase, where they have the same lattice points but a different order of the sequence of the atoms along the c-axis: (a) ...Sr−(Au, Si, Si)−Sr−(Au, Si, Si)−Sr... for SrAuSi3 and (b) ...Sr−(Si, Au, Si)−Sr−(Si, Au, Si)−Sr... for SrAu2Si2. Panel a has no mirror plane perpendicular to the c-axis.

spread over the surface in a regular pattern. The grain size is typically 10−20 μm (width) by 30−100 μm (length). Atomic compositions of the grains were measured using the EPMA method. More than 10 grains having a relatively wide plane surface were randomly selected for the measurement; the results are summarized in Table 3. All the grains have a similar Table 3. Atomic Compositions of Randomly Selected Grains in the Polycrystalline SrAuSi3 Sample, Detected by the EPMA Method

SrAu2Si2 is also given in Figure 2b for the sake of comparison. Both the structure types [BaNiSn3 type (I4mm) and ThCr2Si2 type (I4/mmm)] are ternary derivatives of the BaAl4 type (I4/ mmm); they have the same lattice points but a different order sequence of the smaller atomic sites (Au and Si) along the caxis. The most striking difference in the structures between SrAuSi3 and SrAu2Si2 is on an inversion center for symmetry. SrAuSi3 has neither the mirror plane perpendicular to the c-axis nor the inversion center for the unit structure. At low temperatures, the noncentrosymmetry should be conserved in the SrAuSi3 structure. This is because the inversion symmetry operator (−x, −y, −z) is not included in any maximal nonisomorphic subgroups of space group I4mm, even if structural phase transition lowers the symmetry. The crystal structure of SrAuSi3 is essentially identical to those of the many other AMSi3-type family compounds, such as CeIrSi3,54 CeRhSi3,54 LaIrSi3,55,56 LaRhSi3,56 CaIrSi3,24 CaPtSi3,24 BaPtSi3,57 BaPdSi3,57 etc. Some characteristic features obtained from the structural analysis for SrAuSi3 are summarized as follows. (i) The Au atom coordinates with four basal Si2 atoms and one apical Si1 atom. The interatomic distances are 2.493(6) Å for the Au−Si1 bond and 2.548(2) Å for the Au−Si2 bond, close to the value of 2.45 Å expected from the summation of the two atomic radii: 1.34 Å for Au and 1.11 Å for Si. (ii) The Si1 atom coordinates to four Si2 atoms with a distance of 2.511(4) Å, within a permissible range of the typical Si1−Si2 distances of 2.50−2.63 Å reported for AMSi3type structures.24,54−57 The Si1−Si2 distance in SrAuSi3 is, if anything, a little shorter than those in most others of the AMSi3 type. (iii) The Sr atom is accommodated in the localized Si−Au cluster. The Sr−Si and Sr−Au distances are in the range of 3.26−3.38 Å, consistent with those of the other AMSi3-type compounds. Figure 3 is a typical SEM (scanning electron microscopy) image of a cleavage surface of the SrAuSi3 ceramic sample. It can be observed that many large rectangular crystal grains are

composition (atom %) grain number

Sr

Au

Si

1 2 3 4 5 6 7 8 9 10 11 12 13 14 average standard deviation ideal ratio SrAuSi3 SrAu2Si2

19.49 19.41 19.19 19.43 20.03 21.05 20.77 19.87 19.44 20.25 20.14 20.65 20.06 19.33 19.93 0.58

23.43 22.14 22.76 23.33 22.79 21.85 22.73 23.41 23.84 24.03 23.48 24.41 22.95 23.73 23.21 0.71

57.08 58.45 58.05 57.24 57.18 57.11 56.50 56.72 56.71 55.72 56.38 54.94 56.98 56.95 56.86 0.86

20.0 20.0

20.0 40.0

60.0 40.0

composition around the average composition of 19.9(5), 23.2(7), and 56.8(8) atom % for Sr, Au, and Si, respectively. The average composition is close to the ideal Sr:Au:Si ratio of 20.0:20.0:60.0 for SrAuSi 3 rather than the ratio of 20.0:40.0:40.0 for SrAu2Si2, indicating that these phases are essentially SrAuSi3. However, in a stricter sense, the observed composition is ∼3% rich in Au and poor in Si, as compared with the ideal value of SrAuSi3. This suggests that the minority SrAu2Si2 intrudes into the majority SrAuSi3 as a crystal grain, similar to an intergrowth structure. The average composition is Sr0.99(2)Au1.16(3)Si2.84(4) (∼0.84 × SrAuSi3 + 0.16 × SrAu2Si2), 2158

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indicating that ∼16 mol % SrAu2Si2 is included in the basal SrAuSi3 grain. This SrAuSi3:SrAu2Si2 molar ratio is consistent with the value of 0.823:0.177 determined from the Rietveld analysis. SrAu2Si2 domains should be of a size that allows X-ray coherent scattering for diffraction. The structure of the phases in a crystal grain was directly observed using a high-resolution scanning transmission electron microscope (STEM). Panels a and b of Figure 4 are annular

Figure 4c is an ABF image of SrAu2Si2 taken projected along the [111] direction. They were observed in small portions in a crystal grain. The large darkest spots, middle spots, and faint spots were assigned to Au, Sr, and Si, respectively. The arrangement of the atoms in this lattice image is obviously different from that of SrAuSi3 shown in Figure 4a. Two structures can clearly be distinguished for the projection pattern. Figure 4d is an annular dark-field (ADF) image of the structure around a domain boundary between the SrAuSi3 (top left corner) and SrAu2Si2 (bottom right corner) domains in a crystal grain. This image indicates that the secondary SrAu2Si2 domains partly coexist with the main SrAuSi3, similar to lattice imperfections in a crystal. Figure 4e is a zoom-out image of the crystal grain. The bright beltlike domains were identified as SrAu2Si2, intruding into the main SrAuSi3 grain. In contrast, Si is removed from the grain as an extrinsic phase. 3.2. Physical Properties. Figure 5 is a plot showing the temperature (T) dependence of the electrical resistivity (ρ) for

Figure 4. Lattice images of a SrAuSi3 sample, taken by using a highresolution scanning transmission electron microscope (STEM). Panels a and b are annular bright-field (ABF) images of SrAuSi3, projected along the [111] and [001] directions, respectively. Panel c is an ABF image of SrAu2Si2 for the [111] projection. SrAu2Si2 is the secondary phase that intrudes into the crystal grain of the majority SrAuSi3. Panel d is an annular dark-field (ADF) image of the structure around a domain boundary between SrAuSi3 (top left) and SrAu2Si2 (bottom right) in a crystal grain. Panel e is a zoom-out image of the sample grain.

Figure 5. Temperature (T) dependence of the electrical resistivity (ρ) for a SrAuSi3 polycrystalline sample: (a) data taken in the wide temperature range of 0.6−300 K under zero magnetic field (H = 0) and (b) data below 2 K, showing the behavior of the superconducting transition under the various magnetic fields (H = 0−2500 Oe).

a SrAuSi3 polycrystalline sample. Figure 5a is a plot over a wide temperature range of 0.6−300 K under zero magnetic field. A superconducting transition occurs around ∼1.6 K. In the normal state, the T dependence of the resistivity exhibits typical metallic behavior of a Bloch−Grüneisen type. The residual resistivity (ρ0) is ∼0.1 mΩ cm, and the residual resistivity ratio (RRR; ρ300K/ρ0) is ∼1.8. Figure 5b is a plot of the T dependence of the resistivity (ρ) below 2 K, taken under various magnetic fields (H = 0−2500 Oe). At H = 0, an abrupt resistivity drop and zero resistivity due to the superconducting transition were clearly observed at a TC of ∼1.6 K. The transition temperature TC gradually decreases as the magnetic field increases. The superconducting transition is completely invisible when H = 2500 Oe at the lowest temperature of 0.6 K, suggesting that the upper critical field of SrAuSi3 is not large.

bright-field (ABF) lattice images of SrAuSi3, taken along the [111] and [001] directions, respectively. In ABF images, darker spots can be identified as heavier atoms. In Figure 4a, each kind of atom is lined up along the [111] direction in the crystal structure, looking like spots overlapping on the same lattice point in the STEM image. As a result, all atom types can be distinguished as different lattice point spots; the large darkest spots were assigned to Au, the middle dark spots to Sr, and the faint spots to Si. In Figure 4b, a square lattice with a cell parameter of 0.44 nm can be observed. The large darkest spots can be assigned to a projection of an array of overlapping Sr, Au, and Si1 atoms while the faint spots to Si2 atoms. These experimental results clearly confirm that the novel BaNiSn3type structural phase of SrAuSi3 is formed in the crystal grain. 2159

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Figure 6 is a plot of the magnetic data for SrAuSi3. Figure 6a is the temperature dependence of the magnetic susceptibility χ

Figure 7. Upper critical field HC2 as a function of T. The data points are given by the transition temperatures TConset (red circles), TCmid (green circles), TC0 (blue circles), and TCmag (black circles) under applied magnetic fields (H). The solid line indicates linear extrapolation of the TCmid plot to 0 K, and Horb indicates the orbital limit at 0 K given by the Werthamer−Helfand−Hohenberg (WHH) theory.

temperature data points near ∼0 K and the broad transition probably due to a superconducting fluctuation under the magnetic field. However, the upper critical field can be estimated from the HC2 versus T plot, approximately 2−3 kOe. The tentative linear extrapolation of the TCmid line gives a rough estimate of the upper critical field HC2(0) ∼ 2.2 kOe for SrAuSi3. Thus, HC2(0) is much lower than the Pauli limit of the upper critical field expected from Bardeen−Cooper−Schrieffer (BCS) theory, i.e., μ0HPauli(0) = 1.86TC, ∼3 T for SrAuSi3,60,61 suggesting that orbital pair breaking is the essential mechanism that limits the upper critical field for SrAuSi3. For conventional weak coupling BCS-type superconductors, the orbital limit of the upper critical field can be described by the Werthamer−Helfand−Hohenberg (WHH) theory, in which T ∼ 0 K, HC2 approaches the limit value Horb(0) = αHC2′TC, where HC2′ = −[dHC2(T)/dT]|H=0, and α is a purity factor taking a value between 0.73 (clean limit) and 0.693 (dirty limit).62,63 For SrAuSi3, the orbital limit value can be estimated to be ∼1.5 kOe. However, the observed experimental data of HC2(T) do not fit the WHH theory; its T dependence rather looks like a T linear behavior at least within the measurable temperature range down to ∼0.6 K (∼0.4TC). The origin of the deviation of HC2(T) from the WHH theory is not clear. However, it seems to be impossible to explain such a deviation only within the framework of the electron−phonon model with weak or intermediate coupling.64 Presumably, it might be due to a nonspherical Fermi surface65−67 or gap anisotropy.68 It has been reported by theoretical studies that a cylindrical twodimensional (2D) Fermi surface in a solid can enhance HC2 under a magnetic field parallel to the 2D plane.67 This effect might be related to the experimental results for SrAuSi3. Specific heat data under zero magnetic field for the SrAuSi3 sample are shown in Figure 8. Figure 8a is the temperature dependence of the specific heat (raw) data (Cp). The Cp values for a molar unit were calculated assuming the molar weight of target phase SrAuSi3. A sharp specific heat jump was observed around 1.6 K, indicating the superconducting transition to be bulk in nature. Figure 8b is the Cp/T versus T2 plot. By numerically fitting the normal (nonsuperconducting) state portion of the data within the temperature range of 2−5 K with the formula Cp ∼ Cn = γnT + βT3 + β5T5, we obtain the solid line shown in the figure. The linear term γnT corresponds to the electronic contribution in the normal state, for which we obtain an electronic specific heat coefficient (γn) of ∼6.0 mJ

Figure 6. (a) Temperature dependence of the magnetic susceptibility of a SrAuSi3 powder sample. The data were collected for the powder sample at an H of 10 Oe in a heating process, after zero-field cooling (ZFC) or field cooling (FC). The right vertical axis indicates the volume fraction of the superconducting phase, −4πχ. (b) M−H curve at 0.46 K. The upper critical field at 0.46 K is around 2000 Oe.

∼ M/H. The data were collected for a powder sample at an H of 10 Oe in a heating process, after zero-field cooling (ZFC) or field cooling (FC). The magnetic susceptibility shows diamagnetic signals due to a superconducting transition around 1.5 K. The diamagnetic signal for the ZFC shielding effect at 0.46 K is approximately −9.5 × 10−3 emu/g, which corresponds to ∼75% of the full-volume Meissner signal of SrAuSi3. This ratio (∼75%) agrees with the mass fraction (∼72 wt %) of the SrAuSi3 phase included in the sample. The large diamagnetic signal confirms that the observed superconductivity is a bulk phenomenon originating from SrAuSi3 and not from any other minority phases in the sample. Figure 6b is an M−H curve at 0.46 K. As the magnetic field increases, the superconducting diamagnetic signal decreases and then approaches zero around an H of ∼2 kOe. This indicates that the upper critical field at 0.46 K is around 2 kOe. Figure 7 is a plot of the upper critical fields (HC2) as a function of temperature, determined from the electrical resistivity and magnetic susceptibility data for the polycrystalline sample. For this plot, four kinds of critical temperatures are defined: (i) TConset, the temperature at which ρ starts to drop from the normal resistivity (ρ = ρn ∼ ρ0); (ii) TCmid, the temperature at which ρ reaches half of the normal resistivity value (ρ = ρn/2); (iii) TC0, the temperature at which ρ reaches zero (ρ = 0); and (iv) TCmag, the temperature at which the Meissner signal appears. TCmag corresponds to the maximal critical field HC2max (=HC2ab or HC2c) of the single-crystal sample.58,59 As shown in Figure 7, HC2 increases almost linearly with a decrease in temperature. The exact value of the upper critical field HC2(0) at 0 K cannot be determined because of few 2160

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formula β = (12π4/5)rNAkB(1/ΘD3), where kB, NA, and r denote the Boltzmann constant, Avogadro’s number, and the number of atoms per formula unit, respectively. The Debye temperature observed for SrAuSi3 (∼410 K) is somewhat higher than those for CaIrSi3 (∼360 K),24 CaPtSi3 (∼370 K),24 and BaPtSi3 (∼345 K).22 In Figure 8a, the lattice specific heat (Cl), the normal state electronic specific heat (γnT), and their summation (Cn) are shown with two broken lines and a solid line, respectively. The contribution of the lattice part is almost negligibly small in the low-temperature range below 1.5 K. The electronic part of specific heat Cel in the superconducting state was obtained by subtracting the lattice part from the raw data: Cel(T) = Cp(T) − βT3 − β5T5. Figure 8c (main panel) is a Cel/T versus T plot. By idealizing the specific heat jump to ensure entropy balance around the transition, we determined the bulk thermodynamic critical temperature TC to be 1.54 K. The magnitude of the idealized specific heat jump at TC is ΔCel/TC ∼ 11.6 mJ mol−1 K−2, giving the normalized specific heat jump ΔCel/(γnTC) ∼ 1.92. This value is evidently larger than the BCS value of 1.43 and almost equal to the value for a niobium metal (1.87).70,71 This suggests that SrAuSi3 is in the category of moderately strong-coupling (=intermediatecoupling) superconductors for Cooper pairing. In Figure 8c, the electronic specific heat is rapidly attenuated below TC with a decreasing temperature, typical of strongcoupling superconductors.72 However, it was found that a kink exists around T = T* ∼ 1.3 K, below which Cel/T decreases almost linearly with a decreasing T; that is, the electronic specific heat Cel is proportional to quadratic temperature T2. This behavior is in contrast to the usual exponential decay of the specific heat for conventional full-gap s-wave superconductors and reminds us of a possibility that the superconducting energy gap might close partly on the Fermi surface with line nodes, as observed in CePt3Si.73,74 The existence of such a nodal gap is theoretically expected for noncentrosymmetric superconductors.7−10 However, a more likely possibility is that any secondary component of the specific heat may overlap with the TC ∼ 1.54 K superconducting electronic specific heat in the temperature range below T*. It may be due to the secondary phase in a crystal, because the entropy balance is not kept below TC, or for example, multigap superconductivity also might give such an unusual T dependence of the specific heat below TC. At present, we do not have a conclusive explanation for this unusual behavior. More precise measurements using a purer sample are necessary for a definitive answer. SrAuSi3 is a moderately strong-coupling superconductor. The electronic specific heat coefficient γn should be enhanced by electron−phonon coupling. The strength of electron−phonon coupling in SrAuSi3 was evaluated using the formula γn = (π2/ 3)N(EF)(1 + λep)kB2, where N(EF) is density of states (DOS) at the Fermi level and λep is the electron−phonon coupling constant.75 γn (=6.0 mJ mol−1 K−2) is already obtained from the present specific heat measurements. N(EF) is also estimated to be 1.3 states eV−1 cell−1 from the ab initio band calculation in this study. Substituting these values into the equation, we obtain an electron−phonon coupling constant (λep) of 0.97 for SrAuSi3. This λep value is close to that for a niobium metal (0.82) appropriate for a moderately strong-coupling superconductor.70−72,76 A generalized relationship between the electron−phonon coupling constant (λep) and the normalized specific heat jump [ΔCel/(γnTC)] for weak- to strong-coupling superconductors has been reported by Dalrymple et al.77 The

Figure 8. Specific heat data under zero magnetic field for a SrAuSi3 sample. (a) Plot of raw data for Cp vs T. Cp (solid red circles) indicates data points. Cn (solid blue line) and γnT (broken green line) are the specific heat for the normal state and its electronic part, respectively. Cl (broken black line) is the lattice part of the specific heat. (b) Plot of Cp/T vs T2. The solid line (Cn) indicates the numerical fitting curve of the normal (nonsuperconducting) state portion of the data within the temperature range of 2−5 K with Cp ∼ Cn = γnT + βT3 + β5T5, where the electronic specific heat coefficient γn = 6.0 mJ mol−1 K−2 and Debye temperature ΘD = [(12π4/5)rNAkB/β]1/3 ∼ 410 K. (c) Electronic specific heat data: Cel/T vs T (main panel) and Cel vs T (inset). For the main panel, the vertical line gives a bulk thermodynamic critical temperature (TC) of 1.54 K, and the horizontal line indicates the electronic specific heat coefficient γn. The normalized specific heat jump is ΔCel/(γnTC) ∼ 1.92. T* (∼1.3 K) indicates a bending point of the T dependence of Cel in the superconducting state. Below T*, Cel/T is proportional to T (see the text for a detailed explanation). For the inset, the solid curve indicates a T2 fit of the data points.

mol−1 K−2.69 This value is comparable to those in other Ce-free noncentrosymmetric superconductors CaIrSi3 (4.0 mJ mol−1 K−2),24 CaPtSi3 (2.1 mJ mol−1 K−2),24 BaPtSi3 (5.7 mJ mol−1 K−2),22 and LaPt3Si (9 mJ mol−1 K−2)16 but quite smaller than that in the Ce-based heavy fermion noncentrosymmetric superconductor CePt3Si (∼390 mJ mol−1 K−2).16 This result suggests that for SrAuSi3, mass enhancement due to electronic correlation is not significant. In specific heat Cp, the terms βT3 and β5T5 correspond to the lattice specific heat Cl, for which we obtain a Debye temperature (ΘD) of ∼410 K from the regular 2161

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results [λep = 0.97, and ΔCel/(γnTC) = 1.92] for SrAuSi3 satisfy the Dalrymple model. We evaluated the superconducting transition temperature TC using McMillan’s formula: TC = (ΘD/1.45) × exp{−1.04(1 + λep)/[λep − μ*(1 + 0.62λep)]}, where μ* is the Coulomb repulsion parameter.76 Using the standard value of 0.13 for μ*78 and the obtained values of λep (0.97) and ΘD (410 K), we obtain an ideal TC of ∼19 K for SrAuSi3. However, this result conflicts with the experimental result that the actual TC is 1.54 K. The observed TC is too low for a conventional superconductor with such a high Debye temperature (ΘD ∼ 410 K) and strong electron−phonon coupling (λep ∼ 0.97). This inconsistency is the most puzzling issue of the superconductivity observed in SrAuSi3. At the very least, within the usual ranges of the parameters, the McMillan model is not satisfactory for describing the superconducting state in SrAuSi3. Presumably, this phenomenon (TC suppression) might be due to the missing inversion symmetry. We suggest that parity mixing or gap anisotropy might suppress the TC for noncentrosymmetric superconductivity. The superconducting parameters of SrAuSi3 were estimated using the experimental data. From the HC2(T) curve in Figure 7, the upper critical field at 0 K is approximately μ0HC2(0) ∼ 0.22 T. The thermodynamic critical field HC(0) can be calculated from the free energy difference between the superconducting and normal states, given by the expression (1/2)μ0HC(0)2 = Fn − Fs = −(1/2)γnTC2 + ∫ T0 C Cel(T) dT. Numerical integration of the specific heat data gives a μ0HC value of ∼0.014 T. These values were put into the formula HC2/HC = √2κGL to obtain the Gintzburg−Landau parameter (κGL ∼ 11). This suggests that SrAuSi3 is a type II superconductor. Superconducting coherence length ξ(0) and penetration depth λ(0) were estimated to be ∼39 and ∼440 nm, respectively, using the formulas μ0HC2 = Φ0/[2πξ(0)2] and κGL = λ(0)/ξ(0), respectively, where Φ0 is fluxoid quantum (=2.0678 × 10−7 G cm−2). These parameters are summarized in Table 4 with other physical properties determined in this

Figure 9. Electronic band structure of SrAuSi3: bottom panel, SOC included; top panel, scalar relativistic only (without SOC); Brillouin zone, see Figure 12. Three hole bands (band 1, band 2, and band 3) cross the Fermi level to form Fermi surfaces (see the text). The Γi (i = 1−7) codes at the bottom of the figure indicate an irreducible representation of the symmetry for each colored band.80

Fermi level EF, for relativistic calculations including SOC (bottom panel) and scalar-relativistic calculations without SOC (top panel). The overall features of the band dispersion are similar to those in CaIrSi3,79 CaPtSi3,79 and BaPtSi3.22 Differences are mainly based on electron filling in SrAuSi3, in which the Au atom lifts up the Fermi level and changes the shape of the Fermi surfaces more than the Ir and Pt atoms. Focusing on the electronic bands near Fermi level EF, we found several bands cross the Fermi energy to form Fermi surfaces. One band (band 1) crosses the Fermi level in the directions of X−Γ and X−Z. Band splitting due to SOC is not significant for band 1. In contrast, two bands (band 2 and band 3) cross the Fermi level in the directions of Z−X and Z−Γ. The band splitting due to SOC is effective for band 2 and band 3. In particular, for band 2, a sizable energy splitting of ∼0.4 eV can be observed between two sub-bands in the Z−Γ direction. All the bands crossing the Fermi level are hole bands, suggesting that the type of carrier is a hole. To further investigate the character of electronic states, the weights of wave functions projected onto the angular momentum channels in each muffin-tin sphere are plotted in Figure 10. In this figure, the size of the circles represents projected weights (in Figure 10, a sequence of large circles appears to form a thick solid line). The Au d orbitals contribute mainly to the deep energy bands between −3 and −9 eV below the Fermi level. They may be assigned as Au 5d bands. Small

Table 4. Physical Properties of Polycrystalline SrAuSi3a

TC (K)b μ0HC2(0) (T) μ0HC(0) (T)c κGL ξ(0) (nm) λ(0) (nm) ΔCel(TC)/γnTC λep γn (mJ mol−1 K−2) ΘD (K)

SrAuSi3 (this work)

CaPtSi3 (ref 24)

CaIrSi3 (ref 24)

1.54 0.22 0.014 11 39 440 1.92 0.97 6.0 410

2.3 0.15 0.0094 11 47 520 − − 2.1 370

3.6 0.27 0.023 8.3 34 280 − − 4.0 360

a

The values for CaPtSi3 and CaIrSi3 are also given for the sake of comparison. bBulk TC determined from specific heat. cThermodynamic critical field at 0 K.

work. It is important to note that these results were obtained from data of polycrystalline samples of anisotropic materials. Nevertheless, these data are an essential starting point. The values should be verified using single-crystal, or purer, samples in the future. 3.3. DFT Electronic Structure. Figure 9 is the electronic band structure of SrAuSi3 over the range of −3 to 3 eV around 2162

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and without SOC (left panel). The partial DOS of Au and Si are widely distributed in the energy range below and above Fermi level EF. In particular, at E values from −9 to −3 eV, a large DOS that arises mainly from the Au 5d states exists. In contrast, the partial DOS of Sr is distributed almost only above EF. The finite states at EF are, therefore, mainly composed of Au and Si states, whereas the contribution from the Sr state is small. The intensity patterns of the partial DOS of the Au and Si states are similar, indicating that the Au and Si orbitals hybridize over the wide energy range around the Fermi level. A strong effect of the SOC can be observed in the Au 5d state dominant energy region from −8 to −4 eV, in which prominent DOS peaks for the SOC-free calculation are further divided into multiple peaks for the relativistic calculation, including the SOC. The value of the total DOS at the Fermi level is N(EF) ∼ 1.3 states eV−1 cel−1, almost the same for the calculation with or without SOC. Figure 12 is a plot of the Fermi surfaces of SrAuSi3. The overall view of the Fermi surfaces with a body-centered

Figure 10. Band structures of SrAuSi3 for scalar-relativistic calculation, in which size of the circles on each band represents projected weights to angular momentum channels in each muffin-tin sphere. A sequence of large circles appears to form a thick solid line.

contributions from the Au s orbital are visible in the whole energy region in this figure. The Au p orbitals contribute mainly to the bands near the Fermi level, in particular to band 2 and band 3. The Si orbitals are widely distributed over all energies. In particular, the Si2 orbital contributes to band 1. The Sr orbitals contribute only to energies far above the Fermi level. These results clearly indicate that band 1 mainly originates from the Si orbitals, while band 2 and band 3 include large contributions from the Au- and Si-hybridized orbitals. Figure 11 is a plot of the partial and total density of states (DOS) of SrAuSi3 from calculations with SOC (right panel)

Figure 12. Fermi surfaces for SrAuSi3. (a) Overall view of the Fermi surfaces with a body-centered tetragonal (bct) lattice (c > a) Brillouin zone. (b) Cross sections of the irreducible Brillouin zone at kz = 0 and ky = 0, for band 1 and band 3 (left) and band 2 (right). The blue lines indicate cross sections of the Fermi surfaces for the scalar-relativistic calculation without SOC, while the red lines indicate those for the relativistic calculation with SOC.

tetragonal Brillouin zone is illustrated in Figure 12a. This result includes the SOC effect. A large cylindrical Fermi surface and two complicated-shape Fermi surfaces can be observed around the X point and the Z points, respectively. Figure 12b shows the cross sections of the irreducible Brillouin zone at kz = 0 and ky =

Figure 11. Total and partial density of states (DOS) for SrAuSi3 around Fermi energy EF. The right panel indicates DOS with SOC, while the left panel is for scalar-relativistic calculation without SOC. The energy scale is defined relative to EF. 2163

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conductivity in SrAuSi3. Specific heat and thermal conductivity measurements are important for obtaining thermodynamic information. ARPES (angle-resolved photoemission spectroscopy), STS (scanning tunneling spectroscopy), and NMR (nuclear magnetic resonance) experiments are much more informative with respect to the superconducting-gap state. To this end, synthesis of high-quality single-crystal samples is most strongly desired.

0. The blue lines indicate cross sections of the Fermi surfaces without SOC, while the red lines indicate those with SOC. Three kinds of Fermi surfaces exist in the Brillouin zone. One is around the X point, while the others are around the Z point. The cylindrical 2D-like Fermi surface around the X point is from band 1. This Fermi surface does not split into two subparts by the SOC, because it mainly originates from the Si orbitals. The wide surface with the cylindrical shape suggests that many holes are mobile in the double-Si layers of the crystal structure. The two Fermi surfaces around the Z point are from band 2 and band 3, respectively, and are complicated threedimensional (3D) shapes. These Fermi surfaces are partly divided by the SOC, because they include a significant contribution from the Au 5d and 6p orbitals. This suggests that the holes, in part, on these Fermi surfaces are possibly affected by the missing inversion symmetry in the crystal structure. The band calculation revealed that two types of carriers exist in SrAuSi3. One consists of the carriers conducted in the Si layers without the SOC effect, while the other consists of the carriers conducted toward the 3D directions in the crystal. A part of the latter type of carriers can feel the SOC effect due to the missing inversion symmetry. This type of carrier may be a key to understanding the origin of the unusual superconducting properties in SrAuSi3, i.e., unusual T dependence of the electronic specific heat and TC suppression. If these carriers participate in Cooper pairing, it may lead to an unconventional superconducting state for the observed phenomena in SrAuSi3.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We sincerely thank Dr. T. Taniguchi of the National Institute for Materials Science (NIMS) for much helpful advice on highpressure synthesis experiments, Mr. K. Kosuda of NIMS for EPMA measurements, and Dr. N. Shirakawa of the National Institute of Advanced Industrial Science and Technology (AIST) for magnetic measurements. M.I. thanks Ms. C. Azechi and Ms. E. Hashimoto of NIMS for technical assistance with synthesis experiments. This work was conducted on the Advanced Superconducting Materials Project administered by NIMS and partly supported by the Japan Society for the Promotion of Science (JSPS) through its “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program)”.

4. CONCLUSIONS SrAuSi3 is a novel high-pressure stable phase that has a BaNiSn3-type noncentrosymmetric crystal structure with a possible space group of I4mm and the following lattice parameters: a = 4.4090(1) Å, and c = 9.9475(3) Å. This compound shows bulk superconductivity with a critical temperature TC of 1.54 K and an upper critical field μ0HC2(0) of ∼0.22 T. The electron−phonon coupling constant (λep ∼ 0.97) and the specific heat jump [ΔCel/(γnTC) ∼ 1.92] at the superconducting transition in SrAuSi3 are comparable to those in a niobium metal, indicating that SrAuSi3 is a moderately strong coupling superconductor. However, curiously, the observed TC is too low to understand the superconductivity process only within the McMillan theory. This discrepancy might be due to the missing inversion symmetry of the crystal structure. It was also found that in the superconducting state, the electronic specific heat is proportional to T2 below a specific temperature (T*) of ∼1.3 K. This may be due to the secondary phase. However, there might be another possibility that an unconventional superconducting state, such as a nodal-gap or multigap state, might be realized in SrAuSi3. The DFT band calculation revealed that multiple Fermi surfaces exist with two types of carriers in SrAuSi3. The major carriers conduct in the Si layers without the SOC effect, while other types of carriers conduct toward the 3D directions in the crystal. A part of the latter type of carriers can be affected by the missing inversion symmetry through an asymmetric SOC. This type of carrier may be a key to understanding the origin of the unusual superconducting properties of SrAuSi3. The results of the calculation suggest the possibility of a multigap or anisotropic-gap superconductivity in SrAuSi3. More precise experiments using single-crystal samples are indispensable for improving our understanding of the super-

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REFERENCES

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Chemistry of Materials

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