Indium substitution effect on the topological crystalline insulator family (Pb$_{1-x}$Sn$_{x}$)$_{1-y}$In$_{y}$Te: Topological and superconducting properties

Topological crystalline insulators (TCIs) have been of great interest in the area of condensed matter physics. We investigated the effect of indium substitution on the crystal structure and transport properties in the TCI system (Pb$_{1-x}$Sn$_{x}$)$_{1-y}$In$_{y}$Te. For samples with a tin concentration $x\le50\%$, the low-temperature resisitivities show a dramatic variation as a function of indium concentration: with up to ~2% indium doping the samples show weak-metallic behavior, similar to their parent compounds; with ~6% indium doping, samples have true bulk-insulating resistivity and present evidence for nontrivial topological surface states; with higher indium doping levels, superconductivity was observed, with a transition temperature, Tc, positively correlated to the indium concentration and reaching as high as 4.7 K. We address this issue from the view of bulk electronic structure modified by the indium-induced impurity level that pins the Fermi level. The current work summarizes the indium substitution effect on (Pb,Sn)Te, and discusses the topological and superconducting aspects, which can be provide guidance for future studies on this and related systems.


I. INTRODUCTION
Topological insulators (TIs) are a class of materials that are currently the focus of considerable attention since they represent a new state of matter in which the bulk is an insulator with an "inverted" energy gap induced by a strong spin-orbit coupling (SOC), which leads to the emergence of unusual gapless edge or surface states protected by time-reversalsymmetry [1][2][3][4]. First discovered in two-dimensional systems, one of the simplest topological insulators is the quantum spin Hall state, in which the SOC plays the same role as the magnetic field in the quantum Hall effect. In the quantum Hall effect, the bulk conductance is zero while the edge states are conducting with current flowing in one direction around the edge of the system. Similarly, in the quantum-spin-Hall state, the bulk is still insulating while edge-state electrons with opposite spins propagate in opposite directions, consistent with time-reversal symmetry. Theoretical concepts were soon generalized to three dimensions and shown experimentally in materials such as Bi 1−x Sb x [5]. As in the 2D case, the direction of an electron's motion along the surface of a 3D topological insulator is locked to the spin direction, which now changes continuously as a function of propagation direction, resulting in an unusual "planar metal". In the bulk of a TI, the electronic band structure resembles that of an ordinary band insulator, with the Fermi level falling between the conduction and valence bands. On the surface of a TI there are special states that fall within the bulk energy gap and allow surface metallic conduction. Although ordinary band insulators can also support conductive surface states: the locking of the spin and propagation directions eliminates the possibility of backscattering from nonmagnetic impurities.
The first key experiment in this field was the observation of the 2D quantum-spin-Hall effect in a quantum-well structure made by sandwiching a thin layer of mercury telluride (HgTe) between layers of mercury cadmium telluride (Hg x Cd 1−x Te), following a theoretical prediction [6]. The first 3D TI to be probed using angle-resolved photoemission spectroscopy (ARPES) was the semiconducting alloy Bi 1−x Sb x [7]. Simpler versions of the 3D TI were theoretically predicted in Bi 2 Te 3 , Sb 2 Te 3 [8] and Bi 2 Se 3 [8,9] compounds with a large bulk gap and a gapless surface state consisting of a single Dirac cone. Later ARPES experiments indeed observed the linear dispersion relation of these surface states [9,10]. These discoveries confirmed the ubiquitous existence in nature of this new topological state.
In 2011, the notion of "topological crystalline insulators (TCIs)" was introduced to ex-tend the topological classification of band structures to include certain crystal point group symmetries [11]. This new state of matter features metallic surface states with quadratic band dispersion on high symmetry crystal surfaces, and it was shown that such a situation is realized in an insulating crystal having rocksalt structure. It has caused quite a sensation since the first example, SnTe, has been theoretically [12] and experimentally [13] confirmed to exhibit topological surface states in ¡001¿, ¡110¿ and ¡111¿ surfaces. Soon after this discovery, the topological surface state in the Pb-doped Pb 1−x Sn x Te and Pb 1−x Sn x Se have been verified by ARPES and Landau level spectroscopy using scanning tunneling microscopy and spectroscopy [14][15][16], thus expanding the range of relevant materials. Alongside SnTe and the related alloys Pb 1−x Sn x Se/Te, other chalcogenides such as SnS and SnSe that incorporate lighter elements have also been predicted to be TCIs even without the SOC [17]. In theory, by applying external pressure, normal IV-VI rocksalt chalcogenides can be tuned into TCIs [18]. Besides the materials with rocksalt crystal structure, Hsieh et.al. predicted that the antiperovskite family are also promising materials for exploring the topological and other related properties [19]. More recently, a new phase of Bi is stablized by strain, has been found to be a TCI based on mirror symmetry, similar to SnTe [20].
The discovery of TIs and TCIs has also stimulated the search for topological superconductors (TSCs), whose surfaces should exhibit Majorana fermions [2]. Superconductors derived from TIs by doping have been considered as TSC candidates, such as Cu x Bi 2 Se 3 [21,22].
Since the topological surface states are protected from backscattering by disorder, it should be safe to tune the chemical potential through chemical substitution. The ARPES studies performed on Sn 1−x In x Te (SIT) at x = 0.045 confirmed that the topological surface states remain intact after In doping [23]. In fact, SnTe becomes a superconductor upon substituting 2% or more of Sn with In, which introduces hole carriers [24][25][26][27]. Similarly, doping the TCI Pb 0.5 Sn 0.5 Te with ¿ 10% indium also induces superconductivity [28]. This spurs interest in searching for the superconducting analogue, a time-reversal-invariant topological superconductor (TSC), in this system. Point-contact spectroscopy experiments performed on SIT with varies In concentrations found that a zero-bias conductance peak is observed only in the cleanest samples with x ≈ 0.04, suggesting that there is competition between topological and non-topological superconducting states, and that disorder may determine the outcome [27].
A challenge for characterizing the transport properties of surface states in TI/TCI mate-rials such as Bi 2 Se 3 and SnTe is the dominance of a pronounced bulk conductance [29,30].
Despite considerable efforts to reduce the bulk carrier density, such as modifying crystal growth method [31], reducing sample thickness [32] and chemical counterdoping [33][34][35], the bulk conduction has proved difficult to suppress. Inspired by the goal of finding truly bulk-insulating topological materials, we have found that indium doping the TCI materials (Pb,Sn)Te can yield a huge bulk resistivity while maintaining topological surface states [36].
In this article, we present a review of the effects of indium substitution on the crystal structure, resistivity behavior, and electronic band structure in the TCI family (Pb 1−x Sn x ) 1−y In y Te (PSIT). By varying the indium concentration, samples show an extreme range of lowtemperature resistivities: with a few percent indium doping, the samples show weak-metallic behavior; with ∼6% indium doping, samples have true bulk-insulating resistivity and present evidence for nontrivial topological surface states; with higher indium doping levels, superconductivity with a transition temperature T c positively correlated to the indium concentration was observed. We consider this behavior from the standpoint of the localized impurity states associated with the indium dopants.

A. Crystal structure
SnTe is a IV-VI semiconductor that crystallizes in the cubic rocksalt structure at room temperature, and maintains this structure after a certain degree of substitution of Sn with Pb and/or In (Figure 1a). Due to the unchanged crystal structure, the crystal point group symmetries that are essential to maintain the topological surface states remain the same.
Because of the difference in lattice constants of the end members (PbTe ¿ SnTe ¿ InTe) [37], the lattice parameters of (Pb 1−x Sn x ) 1−y In y Te compounds vary with x and y.   Common tools to characterize the surface states include angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM). To apply these techiques, one typically needs atomically-flat and well-oriented surfaces. For the topological insulator Bi 2 Se 3 , this is not be a problem, since it can be easily cleaved due to weak coupling between its layers. In the case of SnTe related compounds, however, the situation is more challenging due to their isotropic cubic structures. To illustrate, Figures 1c-1e show microscope photos of pristine surfaces. The flat, shiny planes are cleaved surfaces, and they become smaller with increasing Pb/In substitution. Thus, it appears that substitution of Pb or In atoms introduces lattice distortion and leads to smaller cleaved surfaces for surface-sensitive studies.
STM studies of SIT single crystal samples have been successfully performed [23], as discussed in Sec. 2.5. Direct ARPES studies of PSIT single crystals are few, and it has proved more practical to perform measurements on thin films evaporated from previously characterized bulk samples [36,39].

B. Resistivity behaviors of In-doped Pb 1−x Sn x Te
The evolution of the electronic properties with composition have been investigated through transport measurements. Here we take (Pb 0.5 Sn 0.5 ) 1−y In y Te as an example to illustrate the effect of indium substitution. As shown in Figure 2, pure Pb 0.5 Sn 0.5 Te shows a metallic-like behavior with a p-type carrier density similar to SnTe. By introducing increasing amounts of indium into the Pb 0.5 Sn 0.5 Te system, single crystal samples show quite divergent, nonmonotonic variations in resistivity in the normal state. For the samples with one percent or less indium, the resistivity is weakly metallic, just like the resistivity be-6 havior of pure SnTe [26] or Pb 1−x Sn x Te without indium doping [40]. Increasing y to 0.06, we observe that the resistivity at 10 K rises by five orders of magnitude. With further increases of y, the resistivity drops, but remains semiconducting, consistent with earlier studies [28,[41][42][43]. This resistivity behavior in the normal state is quite different from the case of In doped SnTe [26], where all samples are weakly metallic in the normal sate. At low temperature, samples show true bulk-insulating resistivity and and present evidence for nontrivial topological surface states [36]. With higher indium doping levels, superconductivity with a transition temperature T c positively correlated to the indium concentration was observed, and the highest T c , ∼ 4.7 K, was achieved for 45% indium doped SnTe samples [25,26] and 30% indium doped Pb 0.5 Sn 0.5 Te samples [36].
The effect of indium substitution is similar for other (Pb,Sn)Te compositions. Nonmonotonic variation in the normal-state resistivity with y is also found in transport measurements of PSIT for many series with different x values. Specifically, x=0.5 is not the only system that shows large bulk resistance when doped with a low concentration of indium. In the whole family of (Pb 1−x Sn x ) 1−y In y Te, maximum resistivities that surpass 10 6 Ωcm are observed for x=0.25-0.30. Even for x=0.35, doping with 6% In results in a rise in resistivity by 6 orders of magnitude at low temperature. These phenomena can be well explained in a picture where the chemical potential is pinned within the band gap, which will be discussed in detail in a later section.
A common test of the topological character of surface states involves measurements of magnetoresistance (MR) at low temperature [44]. The symmetry-protected coupling of spin and momentum for surface states makes them immune to weak localization effects. Application of a transverse magnetic field breaks time-reversal symmetry [45], thus removing the topological protection and leading to a field-induced increase in resistance. Figure 3a shows  8 the x=0.35 sample clearly shows a WAL effect even with a few percent indium, which is consistent with the ARPES evidence that the topological surface states of In-doped SnTe are maintained when In doping concentration is roughly 4.5% [23]. To be more quantitative, we convert the data to conductance, G, and compare with the theoretical formula for WAL [48], where ψ is the digamma function and α is a number equal to 1/2 times the number of conduction channels; B φ = Φ 0 /(8πl 2 φ ), with Φ 0 = h/e and l φ being the electronic phase coherence length. For our system, one expects four Dirac cones crossing the Fermi surface [45,49], which would give α = 2. Figure 3b shows that we get a good fit to the 20-K data for the x = 0.35, y = 0.02 sample with α = 0.37 and l φ = 51 nm. Moving to T = 5 K in Fig. 3d, the low-field data can be described by keeping α fixed and increasing l φ to 58 nm; however, the data also exhibit a large oscillation about the calculated curve for |B| > 0.2 T. This may be due to a Landau level crossing the Fermi energy [45]. Turning to the x = 0.5, y = 0.06 sample, the 5-K data in Fig. 3d are well described by the WAL formula for |B| < 1 T, with α = 2.25 and l φ = 100 nm, but at larger |B| we need an additional component that varies as −B 2 . The latter contribution might come from bulk states.

C. Phase diagram
To summarize the effect of indium substitution on PSIT materials, we present in Fig. 4, a ternary phase diagram of the system to illustrate trends for several properties: the character of the low-temperature resistivity (metallic, insulating, superconducting) and the solubility limit. Here, the end members are SnTe, PbTe, and InTe. The closer to the end member, the higher concentration of that component. Each of the six dashed lines starting with the same end member InTe represent a series of PSIT with the fixed Sn:Pb ratios, as labeled by x.
For low indium doping (blue region), samples show weak metallic resistivity, as in SnTe.
A few percent indium doping turns the Pb 1−x Sn x Te samples into true insulators (orange region). By increasing the In content further, superconductivity may be achieved (green region). When the indium content exceeds the solubility limit in the system (marked with white crosses), where additional In is no longer simply substituting for Pb/Sn, an impurity   Generally, it is believed that each In dopant will provide one less valence electron than Sn 2+ and Pb 2+ , so that indium should be considered as a p-type dopant. In the case of SnTe, one begins with a p-type semiconductor due to Sn vacancies. With In doping, the number of cation vacancies decreases, which partially compensates the expected impact of the In; nevertheless, the p-type carrier density initially grows with increasing In concentration [27,57]. The situation becomes more complicates for an Indium concentration above 10%, where the sign of the Hall resistivity changes [38], suggesting the possibility that two types of carriers are simultaneously present. In fact, in indium doped PbTe and Pb 1−x Sn x Te, In-doping results in far less than 1 electron per impurity atom, which suggests In doping also introduces an impurity band that causes the effect of pinning the Fermi level [58,59].
Evidence for the quasi-localized character of indium-induced states has been provided by a recent nuclear magnetic resonance study on Sn 0.9 In 0.1 Te [60].
In this scenario, the large bulk resistivity in series with x = 0.25 − 0.35 is a consequence of indium sites introducing localized impurity states that pin the chemical potential [51,52]; The magnitude of the resistivity will then depend largely on the size of the band gap, which is determined by the Sn content, x, instead of the indium content, y [51]. Figure 6 gives a summary of the variation in resistivity as a function of x in PSIT compounds with either 3% or 6% indium doping. The same trends are found as a function of x, although the low-temperature resistivities tend to be higher for y = 0.06.
It is worth mentioning that a long relaxation time was observed in the bulk resistance for several samples, especially those that are truly bulk insulators, i.e. x = 0.25, 0.30, The first plausible example of TSC (associated with TI or TCI compounds) was Cu x Bi 2 Se 3 [21].
Experimental evidence from point-contact spectroscopy [63,64] showing zero-bias conduc-tance peaks coexisting with a superconducting gap may be indicative of the unconventional superconductivity, which is necessary (but not sufficient) for TSC in inversion symmetric, time-reversal-invariant superconductors. Similarly, results for In-doped SnTe from both point-contact spectroscopy and high-resolution ARPES studies have been interpreted as evidence for odd-parity pairing and topological superconductivity [23,65].
A markedly different conclusion was drawn, however, in an STM study on Cu 0.2 Bi 2 Se 3 [66], which reported a superconducting gap without any zero bias anomalies. Later studies on the optimally doped TCI system SIT using thermal conductivity [67], magnetization and muon-spin rotation (µSR) measurements [68] also supported the conclusion that SIT has a ful superconducting gap in the bulk, and is more likely to be a conventional s-wave superconductor. Similarly, STM measurements [39] of the superconducting state as well as the superconducting energy gap in (Pb 0.5 Sn 0.5 ) 0.7 In 0.3 Te on the high-symmetry (001) surface lead to the same conclusion, that the superconducting sample seems to be fully gapped without any in-gap states, contrary to the expectations for a topological superconductor.
These controversies may be due to the complexity of the junctions in point contact measurements, since the spectra that are indicative of an unconventional superconductor can also be interpreted by other mechanisms [39,64]. On the other hand, the observed fully-gapped tunneling spectra in STM measurements on Cu x Bi 2 Se 3 and SIT can be also explained by the results of exotic pairing states with additional parameters [66]. In addition, in TCI compounds where the exotic surface states only exist on certain high-symmetry planes guaranteed by the mirror symmetry, the possibility of topological superconductivity feature cannot be ruled out from studies of the (001) plane alone [39]. Besides, due to the poor cleavability cubic (Pb 1−x Sn x ) 1−y In y Te, it might be tricky to expose the desired surface for surface-sensitive measurements.
The debate on topological superconductivity has recently been reinvigorated by a nuclear magnetic resonance study of Cu 0.3 Bi 2 Se 3 [69]. There the authors find clear evidence for a breaking of the spin-rotation symmetry in the superconducting state, consistent with spintriplet pairing. This will surely motivated further investigations.

III. DISCUSSION
In recent years, In-doped SnTe and Pb 1−x Sn x Te have been studied extensively both as examples of topological crystalline insulators and potentially-interesting superconductors.
Our study of a broad range of compositions in the PSIT system shows that indium doping has a nonmonotonic effect on the electronic properties, which can be explained from the standpoint of the relative location of the indium-induced impurity band and the bulk band structure. In this article we have presented a summary which recaps our findings and conclusions, which can be instructive for future work on this system.
In the effort of looking for a new topological superconductor, Tl 5 Te 3 has been found to be tunable between superconducting and topological surface states by Sn-substitution [70,71], which is quite similar to the In-substitution effect on the Pb 1−x Sn x Te system. These facts may imply that the topological surface states and the bulk superconductivity are two competing parameters. The goal of mixing the superconducting and topological characters remains a challenge.
A plausible strategy of looking for Majorana fermions is to artificially construct topological insulator/conventional superconductor heterostructures and make of use the superconducting proximity effect [72][73][74][75][76][77]. Both SIT and PSIT would be perfect platforms for this purpose, since these systems undergo a continuous change from a TCI to a (likely conventional) superconductor. More specifically, the large bulk resistivity shows up in (Pb 1−x Sn x ) 1−y In y Te with x = 0.25-0.5, a p-type matrix can be realized in series the with x = 0.35-1.0, and superconductivity can also be realized in the latter compounds, which makes this system quite promising for exploitation in heterostructures.

IV. EXPERIMENTAL SECTION
A. Sample preparation.
In the Pb-Sn-In-Te alloy system, several elemental metals or compounds are found to be superconducting near liquid-helium temperatures, as indicated in Fig. 7 were set in a vertical position, heated at 950 • C in a three-zone vertical box furnace, with rocking to achieve homogeneous mixing of the ingredients. The crystal growth took place via slow cooling from 950 to 760 • C at the rate of 1.5 • C/hr, and then the samples were gradually cooled down to room temperature over another 3 days [26]. Figure 8a shows examples of the resulting crystals (before removal from the quartz tubes).
For a few compositions, where we needed large crystals for ARPES measurements, we used the modified floating-zone method. In the normal traveling-solvent floating-zone (TSFZ) method, a polycrystalline rod is prepared as feed rod and seed rod, fixed at the upper or lower shaft of the furnace. During the growth, the molten zone is not in contact with any container. However, elements such as indium and tin easily evaporate and are reactive when heated to high temperature. Therefore, we used the modified TSFZ method, as explained in detail in our earlier publication [28], where the starting material, prepared in the vertical Bridgman method, was sealed in a long quartz tube (∼15 cm) and mounted at the bottom shaft in the floating zone furnace. The space inside the large glass chamber surrounding the quartz tube was filled with high-purity Ar at 1 bar to avoid oxygen diffusion through the quartz. During the growth the rotating shaft kept moving downwards at a velocity of 0.5-1 mm/hr, so that the new crystal gradually grew from the bottom of the starting ingot, resulting in a sample such as that shown in Fig. 8b.
For each sample used in the magnetization, transport and other measurements, the chemical composition was characterized using energy-dispersive X-ray spectroscopy (EDS). In this article, chemical composition values x and y correspond to the measured concentrations.

B. Sample characterizations.
To identify the room-temperature crystal structure, each sample was characterized by X- To study the effect of indium substitution on the magnetic properties, dc magnetic susceptibility measurements were performed using a commercial superconducting quantum interference device (SQUID) magnetometer (MPMS, Quantum Design), for temperatures down to 1.75 K. The sample pieces were cut into an approximately cubic shape, typically weighing 0.1 g.
For transport measurements, thin bar-like samples with typical dimensions of 4×1.5×0.5 mm 3 were cut from the bulk crystal and then polished. Electrical resistance was measured in the standard four-probe configuration, using gold wires and room-temperature-cured, fastdrying silver paint for the ohmic contacts on top side, and performed with a Keithley digital multimeter (model 2001), using the MPMS for temperature control. Measurement errors due to the contact geometry are estimated to be less than 10%.