Higher T_c and Upper Critical Field in Novel Misfit Layered Compound Obtained by Indium-Addition Synthesis

Abstract

Indium-addition synthesis of a misfit layered compound (SnSe)_1.16(NbSe₂) was found to obtain a novel sample (In-sample) with another stacking structure in (SnSe)_1.16(NbSe₂), causing the increase in the superconducting transition temperature T_c and the in-plane upper critical field μ₀$H_{\mathrm{c}2}^{\mathrm{in}-\mathrm{plane}}$ (0). Crystal structure analysis using single crystals revealed that the In-sample has a significantly elongated lattice constant along the c axis due to thickening of the layer other than the NbSe₂, while retaining the original misfit layered structure. The T_c increased from 3.6 K to 5.4 K in the In-sample. Furthermore, the in-plane upper critical fields μ₀$H_{\mathrm{c}2}^{\mathrm{in}-\mathrm{plane}}$ (0) exceeded the Pauli limit μ₀H_p, reaching 29.4 T (μ₀H_p ~ 10 T). The coherence length along the c axis was reduced in the In-sample, indicating enhanced two-dimensionality. These results suggest that the In-sample not only has another stacking structure but also exhibits higher T_c and μ₀H_c2.

1. Introduction

Superconductivity in a material without spatial inversion symmetry has been extensively investigated due to the appearance of novel properties, such as the topological or parity-mixed superconducting state, the superconducting diode effect, and others [1,2,3]. Although these materials are rare, the range of candidate substances has been expanded to include materials without local spatial inversion symmetry [4]. One of the candidates is a misfit layered compound. Misfit layered compounds (MX)_m(TX)_n (m, n = layer numbers) have a unique crystal structure: insulating monochalcogenide MX layers (M = Sn, Pb and Bi) and conducting dichalcogenide TX₂ layers (T: transition metal, X = S, Se and Te) are alternately stacked through Van der Waals forces. The spatial inversion symmetry is broken locally at the boundary between the MX layer and the TX₂ layer because these layers have different lattices: MX and TX₂ layers have a NaCl-type square lattice and a triangular prism lattice [5,6,7,8,9]. Misfit layered compounds with metallic properties often show superconductivity [10,11,12,13,14,15]. The superconducting temperature T_c value varies by the number of MX and TX₂ layers [16]. Furthermore, this unique crystal structure gives rise to distinctive properties, such as a quite high upper critical magnetic field (μ₀H_c2). Here, we focus on NbSe₂-based misfit layered compounds (MSe)_m(NbSe₂)_n (M = Sn, Bi, Pb, La).

The (MSe)_m(NbSe₂)_n system features a structure in which superconducting 2H-NbSe₂ layers are alternately stacked with insulating MSe layers [17]. A T_c of (SnSe)_m(NbSe₂)_n is approximately 4 K for (m, n) = (1, 1), and approximately 5 K for (m, n) = (1, 2). The in-plane upper critical field (μ₀$H_{\mathrm{c}2}^{\mathrm{in}-\mathrm{plane}}$) reaches the Pauli limit (μ₀H_p) when (m, n) = (1, 1) but increases to approximately 1.5 times μ₀H_p when (m, n) = (1, 2) [18,19]. In the case of (LaSe)_m(NbSe₂)_n, the μ₀$H_{\mathrm{c}2}^{\mathrm{in}-\mathrm{plane}}$ reaches nearly 10 times μ₀H_p for (m, n) = (1, 1), and approximately twice μ₀H_p for (m, n) = (1, 2) [20]. (PbSe)_m(NbSe₂)_n for (m, n) = (1, 1) also shows a higher μ₀$H_{\mathrm{c}2}^{\mathrm{in}-\mathrm{plane}}$ exceeding μ₀H_p [21]. On the other hand, (BiSe)_1.10(NbSe₂) does not have a higher μ₀$H_{\mathrm{c}2}^{\mathrm{in}-\mathrm{plane}}$ but shows an anisotropic superconducting property with two-fold symmetry in the in-plane: the upper critical magnetic field along the b axis (μ₀$H_{\mathrm{c}2}^{\left|\right|\mathrm{b}}$) is about twice as large as that along the a axis (μ₀$H_{\mathrm{c}2}^{\left|\right|\mathrm{a}}$) [22,23]. These novel superconducting states appear in the conducting NbSe₂ layer. However, the bulk NbSe₂ does not show these novel properties: the NbSe₂ single crystal has a T_c of about 7.2 K, which is higher than that of the NbSe₂-based misfit layered compound. The μ₀H_c2(0) along the inter-plane and the in-plane of NbSe₂ are 4.1 T and 12.3 T (0.9 μ₀H_p) [24]. These results suggest that the insertion of the MX layer between the NbSe₂ layers enhances μ₀H_c2(0) and decreases T_c. However, the mechanism for the change in these superconducting properties in the misfit layered compound has not been revealed so far. An elucidation of the mechanism is essential because the novel superconducting state in the misfit compound can be controlled.

In this study, we demonstrated that Indium (In)-addition synthesis stabilized a novel sample with a different stacking structure (In-sample) compared with (SnSe)_1.16(NbSe₂), which exhibits an enhanced superconducting state. Single crystals of (SnSe)_1.16(NbSe₂) with or without Indium addition were successfully obtained by the molten salt flux method. The lattice constant along the c axis of the In-sample increased in comparison with (SnSe)_1.16(NbSe₂), which originates from the layer other than NbSe₂ thickening, while maintaining the misfit stacking structure. The T_c increased from 3.6 K to 5.6 K in the In-sample. The $H_{\mathrm{c}2}^{\left|\right|\mathrm{a}}$/H_p (0) and $H_{\mathrm{c}2}^{\left|\right|\mathrm{b}}$/H_p (0) increased twice in value compared to (SnSe)_1.16(NbSe₂). These results indicate that the In-sample exhibits an enhanced superconducting state relative to (SnSe)_1.16(NbSe₂).

2. Experimental/Methods

Single crystal samples were synthesized using the molten salt flux method. In the molten salt flux method, the target single crystals are obtained together with byproducts. Therefore, after synthesis, the target crystals were picked up from the obtained products. Since the composition of the obtained crystals often differs from the nominal starting composition, compositional analysis is necessary to determine the actual composition. We prepared two types of samples, with or without the addition of In. A sample without In addition (denoted as the ref. sample) was prepared to weigh to a stoichiometric ratio of Se: Nb: Sn = 5.16: 2: 1.16 mol% to a total of 0.8 g. The other sample with the In addition (denoted as the In-sample) was also prepared with a ratio of Se: Nb: In: Sn = 5.16: 2: 0.8: 0.2 mol% to a total of 0.8 g. The purities of the reagents are as follows: Se (99.9%; Kojundo Chemical Laboratory Co., Ltd., Sakado, Saitama, Japan), Nb (99.9%; Kojundo Chemical Laboratory Co., Ltd., Sakado, Saitama, Japan), In (99.99%; Soekawa Rikagaku Co., Ltd., Itabashi, Tokyo, Japan) and Sn (99.99%; Kojundo Chemical Laboratory Co., Ltd., Sakado, Saitama, Japan). A mixture of KCl (99.9%; Kojundo Chemical Laboratory Co., Ltd., Sakado, Saitama, Japan) and CsCl (99.9%; Iwatani Corporation, Osaka-shi, Osaka, Japan) was used as the flux, weighed at a 3: 5 mol% ratio to a total of 5 g. The metallic elements were first mixed in an agate mortar, followed by the addition of KCl and CsCl, and thoroughly mixed again. The mixture was then vacuum sealed in a quartz tube and sintered in an electric furnace. The thermal process is as follows: the temperature is increased to 700 °C, followed by slow cooling to 600 °C at a rate of v (°C/h). The rate of v is 0.5 for the ref. sample and 1.0 for the In-sample. After sintering, the mixture in the quartz tube was taken out and immersed in pure water to remove the flux. The obtained single crystals were characterized by X-ray diffraction (XRD; Ultima IV, Rigaku, Akishima, Tokyo, Japan), Scanning Electron Microscopy (SEM; S-2150, Hitachi High-Tech Corporation, Minato, Tokyo, Japan) with Energy Dispersive X-ray Spectroscopy (EDX; S-775X1, HORIBA, Ltd., Minami, Kyoto, Japan) for compositional analysis, and Transmission Electron Microscopy (TEM; Tecnai Osiris, Thermo Fisher Scientific, Hillsboro, OR, USA) measurements for structural confirmation. The XRD measurements were performed using Cu Kα radiation over the 2θ range of 10–70°. For compositional analysis, EDX spectra were measured on freshly cleaved crystal surfaces. The measurement was carried out over several regions of approximately 50 × 50 μm², and five different spots were measured in each region. A composition was determined from the average of these measurements. In addition, some samples were measured by synchrotron X-ray diffraction at BL02B1 in SPring-8 (2024B1917). Superconducting properties were evaluated using PPMS (PPMS; Quantum Design, Inc., San Diego, CA, USA) and MPMS (MPMS; Quantum Design, Inc., San Diego, CA, USA). The current was applied along the a axis for electrical resistivity measurements. In addition, the magnetic fields were applied along the a, b, and c axes when the upper critical field was evaluated. A previous report on (BiSe)_1.10(NbSe₂) showed the streak on the crystal surface along the b axis [23]. An analogous streak was observed on the surface of both the ref. and In-samples. We therefore identified the direction parallel to the streak as the b axis and another direction perpendicular to it as the a axis. For the magnetization measurements, a thin platelet-like single crystal with approximate dimensions of 1 mm × 1 mm × 10 μm was used. The magnetic field was applied perpendicular to the platelet plane, namely along the c axis.

3. Results and Discussions

XRD results for each sample are shown in Figure 1. The peaks corresponding to the (00l) planes were observed because the X-ray was irradiated onto the ab-plane of the single crystals. The period of the (00l) planes was estimated to be 12.32(2) Å for the ref. sample and 14.73(4) Å for the In-sample. This result indicates that the interlayer distance along the c axis is expanded in the In-sample.

Compositional analysis results using SEM-EDX measurements are depicted in Figure 2. To examine the crystal homogeneity, SEM-EDX measurements were performed at multiple points on the sample surface. Composition ratios were calculated from this multi-point analysis. The atomic ratio for the ref. sample is Se: Nb: Sn = 2.92(5): 1: 1.1(1), while that for the In-sample is Se: Nb: In: Sn = 3.54(5): 1: 0.53(5): 0.98(9). These results indicate that single crystals of (SnSe)_1.16(NbSe₂) and the In-sample with the above composition were obtained under the present synthesis conditions. However, this compositional change cannot be explained by a simple intercalation or a partial substitution of In for Sn in the ref. sample, as discussed later. Furthermore, the composition is independent of the measurement locations, indicating that a crystal with little inhomogeneity is obtained.

Crystal structure analysis was performed using TEM and synchrotron XRD measurements of the In-sample to obtain more information about the crystal structure. SAED patterns obtained by TEM measurements are depicted in Figure 3. Figure 3a shows the diffraction patterns of the a*c*-plane. The stacking period along the c axis is approximately 14.77 Å, consistent with the XRD results. The diffraction pattern of the a*b*-plane in Figure 3b shows distinct patterns composed of triangular lattice spots (red circles) and square lattice spots (blue squares). These patterns are analogous to a previous TEM study of a misfit compound (SbS)_1+δ(NbS₂) [25]. Its analogous patterns indicate that the In-sample remains a misfit stacking structure consisting of a layer with a triangular lattice and a square lattice. Lattice constants of the In-sample are calculated from Figure 3 and summarized in

Table 1. Summary of the lattice parameters for the ref. sample and the In-sample. The lattice parameters for the In-sample were independently calculated from the SAED patterns and from the single-crystal X-ray diffraction (SC-XRD) measurements. For the ref. sample, the lattice parameters were taken from powder XRD, as reported in previous work [18].

In-Sample					(SnSe)1.16(NbSe2) [18]
Subsystem	a (Å)	b (Å)	c (Å)	method	a (Å)	b (Å)	c (Å)	method
Square	5.76(8)	5.88(6)	14.8(1)	SAED	5.940	5.968	12.340	P-XRD
Triangular	3.38(4)	5.88(6)			3.444	5.968	24.68
Square	5.628	5.935	14.398	SC-XRD
Triangular	3.384	5.919

. The lattice constants along the a and b axes are close to those of previous results [18,19]. The lattice mismatch factor δ = 2 ∙ (${\mathrm{a}}_{{\mathrm{NbSe}}_2}$/${\mathrm{a}}_{\mathrm{SnSe}}$) – 1 is 0.17, which is an analogous value of 0.16 in the ref. sample.

The crystal structure of the In-sample was also evaluated by the synchrotron XRD measurement at Spring-8. Figure 4 shows representative X-ray diffraction images. Diffraction spots attributed to the triangular and square lattices are clearly identified. The lattice constants obtained from these X-ray diffraction patterns are summarized in Table 1. These values are close to those obtained from the SAED patterns. These findings demonstrate that the In-sample not only keeps the misfit stacking structure but also has an enlarged lattice constant along the c axis. It is noted that the spots highlighted in yellow in Figure 4 could not be observed in the TEM images. These spots correspond to the reflection from a diagonal direction of the square lattice.

In order to determine which of the triangular and square lattice layers is thicker, Bright-Field (BF) and High-Angle Annular Dark-Field (HAADF) images were obtained, as shown in Figure 5. Both images indicate that two layers with different thicknesses are stacked on top of each other. The thickness of each layer is estimated to be approximately 5 Å and 9 Å. The 5 Å layer is brighter in the BF image but darker in the HAADF image, indicating that it consists of lighter elements such as Nb and Se. On the other hand, the 9 Å layer is composed of heavier elements such as Sn and In, as indicated by the same comparison of images. This contrast difference between layers is also confirmed by the line-cut intensity profiles shown in Figure 5c. This BF (HAADF) image and the lattice constants of the triangular lattice in Table 1 indicate that the 5 Å layer is the NbSe₂ layer. A previous report supports this result: both the SnSe and NbSe₂ layers have a thickness of approximately 6 Å [26]. On the other hand, the 9 Å layer is not determined to be the SnSe layer, although the lattice constants along the a and b axes are analogous to those of (SnSe)_1.16(NbSe₂) [26]. The type of the 9 Å layer is discussed later. These results suggest that the enlarged layer in the In-sample is not the NbSe₂ layer but rather the other layer (9 Å layer), resulting in an increased c axis lattice constant.

Figure 6a shows a high-resolution (HR) image of the ac-plane of the In-sample. Brighter stripes are observed along the c axis. A line profile along the c axis is shown in Figure 6b, which corresponds to the orange square in the inset of Figure 6a. Based on a previous TEM measurement of (SnSe)_1.16(NbSe₂), the Nb atoms in the NbSe₂ layers are brighter than the other atoms in the HR image [27]. Thus, two peaks, located on both sides and of higher intensity, are attributed to Nb atoms in the NbSe₂ layer. The distance of these peaks is approximately 14 Å, which is consistent with the TEM, XRD and SC-XRD results. Furthermore, the four intermediate peaks between the Nb-related peaks are considered to originate from the structure of the 9 Å layers.

Here, we discuss the crystal structure of the In-sample. The In-sample has an enlarged lattice constant along the c axis compared with the ref. sample, which is attributed to the expansion of the 9 Å layers consisting of Sn, In and Se ions. The possibilities for the origins of the expansion are as follows: (1) a substitution of In for Sn in SnSe layers or (2) an intercalation of In between layers in the ref. sample. However, this is not the case, as shown below. (1, 2-1) The composition of Se/Nb deviates from three, which should remain approximately three when the In ion is substituted or intercalated into the ref. sample. (1, 2-2) The lattice expansion is too large to be explained by intercalation. (1, 2-3) The TEM measurements and SAED patterns revealed no intercalated atoms or layers. These results indicate that the In-samples are neither the In-substituted nor the In-intercalated ref. sample (SnSe)_1.16(NbSe₂).

The case (2)′, where (InSe₂) layers are inserted into (SnSe)_1.16(NbSe₂), i.e., (SnSe)_1.16(NbSe₂)(InSe₂)_x, is not considered as follows [28,29]. (2′-1) The (SnSe)_1.16(NbSe₂)(InSe₂)_x should have three types of layers in an alternating stacking structure. However, in the out-of-plane TEM images and in the BF and HAADF images obtained in this study, only two types of stacked layers were observed. Furthermore, the line-cut profiles of the TEM images showed four peaks between the NbSe₂ layers, which cannot be explained by a simple stacking of SnSe and InSe₂ layers, as in the previous study [28]. (2′-2) The electron diffraction patterns taken with the incident beam perpendicular to the layers showed that only the spots originating from the NbSe₂ and the square lattice layers were present. If InSe₂ layers were present, additional diffraction spots corresponding to InSe₂ should appear, but such spots are not detected. (2′-3) The composition analysis also does not support the insertion of InSe₂ layers. The EDX composition of the In-sample is Se: Nb: In: Sn = 3.54(5): 1: 0.53(5): 0.98(9). The ratio of Se to In ion becomes Se: In = 0.56: 0.53, where the contribution from SnSe and NbSe₂ layers (Se: 0.98 + 2.00 = 2.98) is subtracted in the Se composition. This ratio is not consistent with InSe₂. For this reason, the composition analysis also does not support the formation of additional InSe₂ layers. From the considerations of (2′-1, 2, 3), we concluded that the insertion of discrete InSe₂ layers is unlikely.

Another scenario is (3) the insertion of two SnSe layers, where In is partially substituted for Sn, between the NbSe₂ layers. Based on the above discussion of the square lattice layer, it is highly likely that the thick layer corresponds to two {(Sn,In)Se} layers. Previous reports have shown that the thickness of a single SnSe layer was approximately 6 Å. Therefore, a simple double-SnSe-layer model would give a total thickness of about 12 Å, which is larger than the experimentally observed value of 9 Å. However, partial substitution of In for Sn may reduce the layer thickness because the ionic radius of In³⁺ is smaller than that of Sn²⁺. In fact, the presence of In is confirmed by both EDX and WDS measurements. These results support the interpretation that a partially In-substituted double {(Sn,In)Se} layer is present in our compound. By subtracting the NbSe₂ component from the EDX composition of the In-sample (Se: Nb: In: Sn = 3.54: 1: 0.53: 0.98), we obtained Se: (In + Sn) = 1.54: 1.51. This composition is comparable to {(Sn, In)Se}_1.5(NbSe₂), indicating that the stacking structure consists of two (Sn, In)Se layers and one NbSe₂ layer. In this case, the expansion of SnSe layers in the In-sample is well explained. This structure aligns with observations from BF, HAADF, and SAED images; it consists of a thick SnSe layer and thin NbSe₂ layers. Therefore, we finally suggest that the In-sample is (SnSe)₂(NbSe₂), where the In ion partially substitutes for the Sn site. On the other hand, an ideal value of 1 + δ is 1.16 estimated from the misfit layered compounds (SnSe)_1.16(NbSe₂). Thus, the composition with two insulating layers is expected to be {(Sn, In)Se}_2.32(NbSe₂). This result suggests that the defect is present in either the (In, Sn)Se or NbSe₂ layer. Therefore, further investigation of the crystal structure of the defects and related aspects is necessary to determine the detailed structure.

Next, we will compare the main experimental results between the In-sample and ref. sample of (SnSe)_1.16(NbSe₂). This comparison is not intended to evaluate a simple doping effect, because the In-sample most likely has a different stacking structure and composition from the ref. sample. As discussed above, the In-sample is most likely an In-substituted (SnSe)₂(NbSe₂). The reason for this comparison is that the superconducting properties, such as the upper critical fields, have not been studied in (SnSe)₂(NbSe₂), although superconductivity with T_c ~ 1.9 K has been reported for (SnSe)₂(NbSe₂) [27]. (SnSe)_1.16(NbSe₂) was therefore chosen as a reference to compare with the In-sample, in which the superconducting NbSe₂ layer is effectively a single layer in the unit cell, allowing us to discuss these superconducting properties quantitatively.

Figure 7 shows the temperature dependence of the resistivity measurements for the ref. and In-samples. The current is applied along the a axis. Metallic behavior and superconducting transition are observed in both samples. The resistivity of the In-sample is lower than that of the ref. sample by more than one order of magnitude for the entire temperature range. This decrease in resistivity may be attributed to a change in carrier density induced by In-substitution. Furthermore, the higher residual resistivity ratio (RRR) of the In-sample may be attributed to improved crystallinity, although the underlying mechanism remains unclear.

The T_c is defined as the temperature at which the electrical resistivity decreases to 50% of its normal state value. The T_c is 3.6 K for the ref. samples, which is consistent with the previous report [18]. On the other hand, the T_c of the In-sample increases to 5.4 K.

The temperature dependence of magnetic susceptibility is shown in Figure 8. A superconducting transition is observed at T_c ~ 4.4 K. The zero-resistivity temperature, T_c^zero, is approximately 4.2 K, which is close to the onset temperature of the diamagnetic signal in the M(T) measurement, approximately 4.4 K. A similar relationship between the zero-resistivity temperature in ρ(T) and the magnetic onset temperature in M(T) has also been reported for the ref. compound, (SnSe)_1.16(NbSe₂) [18]. Therefore, the ρ(T) and M(T) results are reasonably consistent. It is noted that the ZFC signal is not saturated until 2.5 K. It is possible that a lower temperature is required for saturation to be observed, or that the In distribution in the In-sample broadened the transition. Further M(T) measurements at lower temperatures are needed to confirm this discussion.

The apparent shielding fraction assuming the (SnSe)₂(NbSe₂) structure is calculated to be about 320% at 2.5 K. This value exceeds 100% because of the large demagnetization effect arising from the thin platelet-like sample shape. Using the approximation formula proposed by Prozorov and Kogan [30], the demagnetization factor was estimated to be N ~ 0.985 for the present measurement geometry. After correcting the apparent shielding fraction using this demagnetization factor, the shielding fraction is estimated to be approximately 77%. The superconducting transition of the In-sample is broader than that of the ref. sample. This broadening may be attributed to a slight inhomogeneity in the In distribution in the In-sample, as the additional phase is not detected by laboratory and synchrotron XRD or TEM measurements.

Figure 9 shows the temperature dependence of the electrical resistivity under magnetic fields for the ref. (a–c) and In-samples (d–f). Superconductivity for both samples is robust to the magnetic field along the in-plane (Figure 9a,b,d,e) compared to that along the c axis (Figure 9c,f). This trend originated from the two-dimensional electronic structure due to the layered structure.

The temperature dependence of the upper critical field along the a, b, and c axes calculated from Figure 9 is presented in Figure 10. The solid lines in Figure 10 are the fitting lines estimated for an anisotropic two-band superconductor in the Ginzburg–Landau (GL) scenario, expressed as follows [31]:

$${\mathit{\mu }}_0{H}_{\mathrm{c}2}\left(T\right)={{\mathit{\mu }}_{0}H}_{\mathrm{c}2}\left(0\right)\left[1-{\left(T/T_{\mathrm{c}}\right)}^2\right]/\left[1+{\left(T/T_{\mathrm{c}}\right)}^2\right]$$

(1)

The reason for adopting the two-band model is that NbSe₂ has been reported to exhibit multiband superconductivity [32,33,34]. Furthermore, the upper critical fields of NbSe₂-based misfit superconductors have also been estimated using this model [18,19]. Therefore, this model was also used to estimate the μ₀H_c2 in the In-sample.

In this paper, μ₀H_c2(T) values along the a, b and c axes are denoted as μ₀$H_{\mathrm{c}2}^{\left|\right|a}$, μ₀$H_{\mathrm{c}2}^{\left|\right|b}$ and μ₀$H_{\mathrm{c}2}^{\left|\right|\mathrm{c}}$, respectively. The upper critical field values at 0 K are estimated from the fitting lines. (SnSe)_1.16(NbSe₂) has μ₀$H_{\mathrm{c}2}^{\left|\right|a}$(0) = 8.6 T, μ₀$H_{\mathrm{c}2}^{\left|\right|b}$(0) = 9.5 T and μ₀$H_{\mathrm{c}2}^{\left|\right|\mathrm{c}}$(0) = 1.7 T. The In-sample has μ₀$H_{\mathrm{c}2}^{\left|\right|a}$(0) = 23.0 T, μ₀$H_{\mathrm{c}2}^{\left|\right|b}$(0) = 29.4 T, and μ₀$H_{\mathrm{c}2}^{\left|\right|\mathrm{c}}$(0) = 2.4 T. The In-sample exhibited higher μ₀H_c2 values than the ref. sample in all directions. The μ₀$H_{\mathrm{c}2}^{\left|\right|a}$ and μ₀$H_{\mathrm{c}2}^{\left|\right|b}$ of the ref. sample show an analogous temperature dependence. On the other hand, the μ₀$H_{\mathrm{c}2}^{\left|\right|b}$ in the In-sample has a higher value in comparison with that of μ₀$H_{\mathrm{c}2}^{\left|\right|a}$, indicating the in-plane anisotropy between μ₀$H_{\mathrm{c}2}^{\left|\right|a}$ and μ₀$H_{\mathrm{c}2}^{\left|\right|b}$ is enhanced slightly in the In-sample. These results demonstrate that the In-sample exhibits higher T_c, μ₀H_c2 and in-plane anisotropy of μ₀H_c2 than the ref. sample. A previous study of (BiSe)_1.10(NbSe₂) showed that μ₀$H_{\mathrm{c}2}^{\left|\right|b}$ was larger than μ₀$H_{\mathrm{c}2}^{\left|\right|a}$ for all temperature ranges, consistent with the current study [22]. However, the in-plane anisotropy of (BiSe)_1.10(NbSe₂) is larger than the In-sample. This difference may be attributed to differences in Bi and Sn in the MX layers. The origin of this difference is an open question so far.

The value of μ₀H_c2 can be expressed using the anisotropic Ginzburg–Landau relation:

$${\mathit{\mu }}_0{H}_{\mathrm{c}2}^i\left(0\right)= {\displaystyle \frac{{\Phi }_0}{2\mathsf{\pi }{\xi }_j\left(0\right){\xi }_k\left(0\right)}}$$

(2)

where Φ₀ is the flux quantum and ξ represents the coherence length. The indices of i, j and k represent the cyclic permutation of the directions a, b, and c. The coherence lengths ξ_i (i = a, b, c) along each axis were calculated using this equation and the experimental values of μ₀H_c2. All values are summarized in

Table 2. Lattice constants along the c axis, T_c, μ₀H_c2(0) for each axis, H_c2/H_p, and coherence lengths for each sample.

	(SnSe)1.16(NbSe2)	In-Sample
c	12.32	14.77
Tc	3.6	5.4
μ0$H_{\mathrm{c}2}^{\left|\right|a}$(0)	8.6	23.0
μ0$H_{\mathrm{c}2}^{\left|\right|b}$(0)	9.5	29.4
μ0$H_{\mathrm{c}2}^{\left|\right|c}$(0)	1.7	2.4
$H_{\mathrm{c}2}^{\left|\right|a}$(0)/Hp	1.3	2.3
$H_{\mathrm{c}2}^{\left|\right|b}$(0)/Hp	1.4	3.0
$H_{\mathrm{c}2}^{\left|\right|c}$(0)/Hp	0.3	0.2
ξa (Å)	13.4	10.3
ξb (Å)	14.8	13.2
ξc (Å)	2.6	1.1
γab	1.1	1.3
γac	5.2	9.6
γbc	5.7	12

. The $H_{\mathrm{c}2}^{\left|\right|a}\left(0\right)$/H_p and $H_{\mathrm{c}2}^{\left|\right|b}\left(0\right)$/H_p of the In-sample are higher than those of the ref. sample. The coherence length along the c axis, ξ_c, is significantly smaller than the thickness of the SnSe layers separating the superconducting NbSe₂ layers in both ref. and In-samples. In particular, the ξ_c of the In-sample is lower than that of the ref. sample. This fact indicates that the two-dimensional electronic structure is enhanced in the In-samples. Since the enhanced two-dimensionality can increase the orbital upper critical field for magnetic fields applied parallel to the layers, the μ₀H_p restricts the in-plane μ₀H_c2. However, the in-plane μ₀H_c2 in our sample exceeds the μ₀H_p. The NbSe₂ thin film also shows higher in-plane μ₀H_c2 (~4 μ₀H_p in trilayers to ~10 μ₀H_p in a monolayer), which is discussed in Ising pairing in superconductivity [35]. Similar behavior has also been reported in NbSe₂-based misfit layered compounds. For instance, (PbSe)_1.14(NbSe₂)₃ showed higher in-plane μ₀H_c2 (~4 μ₀H_p). The angular dependence of μ₀H_c2, TDO (tunnel diode oscillator) measurements, and theoretical calculations suggest that Ising superconductivity and the FFLO state are realized, depending on the magnetic field [14]. (LaSe)_1.14(NbSe₂)_n (n = 1, 2) also has such a higher in-plane μ₀H_c2(~10 μ₀H_p and ~5 μ₀H_p) [36]. The possibility of Ising superconductivity is discussed based on strong spin–orbit coupling, as confirmed by ARPES, NMR, and first-principles calculations [36,37]. Therefore, the μ₀H_c2 value for exceeding the Pauli paramagnetic limit is suggestive of the existence of the Ising superconductivity. Based on these previous studies related to NbSe₂ and NbSe₂-based misfit layered compounds, a similar phenomenon may occur in the Pauli-limit-exceeding in-plane μ₀H_c2 observed in the present In-sample. However, direct experimental evidence for Ising superconductivity, such as angular-dependent μ₀H_c2 measurements or TDO measurements, was not obtained in this study. Therefore, the origin of the large in-plane μ₀H_c2 cannot be determined from the present results alone, and further experimental studies are required.

4. Discussion

We discussed the origin of the In-sample properties compared with (SnSe)_1.16(NbSe₂). First, we recall the stacking structure of the In-sample. The c axis of the In-sample expands, which is due to the thickening of the layer other than NbSe₂. This expansion is expected to separate the conductive NbSe₂ layers, thereby enhancing the two-dimensionality of the NbSe₂ layers. This enhancement results in an increase in μ₀H_c2 along the in-plane directions (a and b axes), as NbSe₂ thin films and (LaSe)_1.14(NbSe₂)_n. The In-sample also exhibits a higher T_c than the ref. sample. One possibility leading to a higher T_c is (1) a change in the number of SnSe layers. However, previous studies on (SnSe)_m(NbSe₂) have reported that T_c decreases with increasing m [27,38]. Therefore, this scenario cannot account for the observed T_c enhancement. The other possibility is (2) a change in the carrier density of the NbSe₂ layer. It has been reported that T_c decreases with electron doping and increases with hole doping in NbSe₂ by an ionic liquid gate [39,40]. Furthermore, in misfit layered compounds (MSe)(NbSe₂) (M = Sn, Bi, Pb, La), the MSe layer has been shown to act as an electron donor to the NbSe₂ layer, suggesting that the insertion of a SnSe layer may reduce T_c compared to pristine NbSe₂ [41,42]. Indeed, the T_c of (MSe)(NbSe₂) is lower than that of pristine NbSe₂ [18,20,21,22,23,24]. In addition, the reported relationship between T_c and the number of MSe layers, described by (1), is consistent with this result: T_c decreases as the number of MSe layers increases [27,38]. Furthermore, previous studies on (SnSe)_1.16(NbSe₂)_n (n = 1, 2) reported that the hole carrier density decreases with increasing the number of SnSe layers, accompanied by a decrease in T_c [19]. In (LaSe)_1.14(NbSe₂)_n (n = 1, 2), the n = 2 sample has a higher hole carrier density than the n = 1 sample [36]. Based on these previous studies, changes in carrier density affect T_c in (MSe)(NbSe₂). Therefore, the In-sample may also alter the carrier state of the NbSe₂ layer due to differences in valence between the In and Sn ions, contributing to the change in T_c. This suggestion is consistent with the decrease in resistivity relative to the ref. sample, as shown in Figure 7. However, Hall-effect measurements, carrier concentration analysis, and valence-state measurements have not been performed in the present study. Therefore, the detailed mechanism of the T_c enhancement cannot be determined from the present results alone. To confirm the suggestion, an evaluation of the carrier concentration and valence of the In ion is needed.

5. Conclusions

We investigated the novel sample (In-sample) obtained by the In-addition synthesis in (SnSe)_1.16(NbSe₂). In the In-sample, the layer other than the NbSe₂ layer is thicker, as confirmed by XRD and TEM measurements. On the other hand, the misfit stacking structure was maintained after the In addition. The T_c of the In-sample increased to 5.4 K in comparison to 3.6 K for the ref. sample. The $H_{\mathrm{c}2}^{\left|\right|a}\left(0\right)$/H_p and $H_{\mathrm{c}2}^{\left|\right|b}\left(0\right)$/H_p were 1.3 and 1.4 for the ref. sample, while they were 2.3 and 3.0 for the In-sample. The calculated ξ_a(0) and ξ_b(0) were almost the same value between the ref. and In-samples. However, the ξ_c(0) in the In-sample is smaller than that of the ref. sample, which is related to the increase in the thickness of the layer other than NbSe₂. This small ξ_c(0) causes a substantial rise in μ₀$H_{\mathrm{c}2}^{\left|\right|a}\left(0\right)$ and μ₀$H_{\mathrm{c}2}^{\left|\right|b}\left(0\right)$. Therefore, the enhanced superconducting properties of the In-sample are likely associated with both In incorporation and the accompanying structural modification. These results are essential for controlling the novel superconducting state in misfit layered compounds with higher μ₀H_c2.