Non-Trivial Topological Phase in the Sn_{1-x}In_xTe Superconductor

TL;DR

This study reveals a non-trivial topological phase in the Sn_{1-x}In_xTe superconductor, showing how In doping influences its electronic structure and topological surface states, with potential implications for topological superconductivity.

Contribution

The paper demonstrates the existence of a non-trivial topological phase in Sn_{1-x}In_xTe and details how In doping affects its topological surface states and electronic properties.

Findings

Band gap decreases with In content, becoming metallic for x>0.1.

Topological invariant indicates a non-trivial phase with mirror Chern number n_M = -2.

Topologically protected surface states are identified and affected by In doping.

Abstract

Whereas SnTe is a inverted band gap topological crystalline insulator, the topological phase of the alloy Sn_{1-x}In_xTe, a topological superconductor candidate, has not been clearly studied so far. Our calculations show that the Sn_{1-x}In_xTe band gap reduces by increasing the In content, becoming a metal for x>0.1. However, the band inversion at the fcc L point for both gapped and gapless phases has been maintained. Furthermore, the computed topological invariant shows a non-trivial phase with a mirror Chern number n_M = -2 for In concentrations of x=0.03125, x=0.125, and x=0.25. We also identify pairs of topologically protected states on the (001) surface of Sn_{1-x}In_xTe with +/- i mirror eigenvalues. The character of these topological states is affected by In dopant. As the In content x increases, the Dirac crossing point moves further away from the L point, and the Fermi velocity of the topological states increases significantly. Our results demonstrate a non-trivial topological phase for the superconductor Sn_{1-x}In_xTe, and provide a detailed description of the topological state properties.

Tables

Table 1. Table 1: Topological surface state parameters: k ( L ) D {}_{D}(L) is the Dirac crossing shift with respect to the pristine L point, Fermi wavenumber k F , electron velocity v e , hole velocity v h , and Fermi velocity v F . Wavenumbers in units of Å -1 , and the velocities are in units of m/s.

	Pristine	$x=0.03125$	$x=0.125$	Experimental reports
				($x$ in the range 0.23-0.41)
kD(L)	0.056	0.077	0.121
kF	0.000	0.101	0.131	0.04[29], 0.06-0.10[33]
ve	3.02	3.03	2.76
vh	1.85	1.82	3.86
vF	2.46	3.56	5.79	2.5[29], 5.8-6.0[33]

Equations

$$\mathcal{M}|s,\pm \sigma ⟩=\pm i|s,\pm \sigma ⟩;\mathcal{M}|a,\pm \sigma ⟩=\mp i|a,\pm \sigma ⟩.$$

Full text

K****eywords topological crystalline insulator, topological superconductor, protected surface states

1 Introduction

Topological phase of matter has received significant attention lately, starting from topological insulators (TIs) and more recently topological superconductors (TSCs) [1, 2, 3, 4, 5]. TIs are characterized by Dirac-like edge or surface states that can be protected by time reversal symmetry [6], or crystal lattice symmetry, the so called topological crystalline insulator (TCI) [7]. TSCs has special interest for potential quantum computing applications [5]. Topological superconductor phase has been proposed to be obtained throw proximity induced by attaching a TI onto a conventional superconductor [8, 9], or by doping a TI turning it into a superconductor phase, like Cu, Sr or Nb doped Bi2Se3 [10, 11, 12, 13, 14, 15, 16, 17, 18]. Recently, superconductivity has also been investigated by the doping of a TCI material, in particular In doped SnTe [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] or PbSnTe [30]. As SnTe is a TCI material [31, 32] and In doped SnTe is a superconductor, Sn1-xInxTe is expected to be a TSC material. Indeed, Sn1-xInxTe is a well characterized superconductor, whose critical temperature depends on the In doping rate, varying from T${}_{\rm{C}}=1.2~{}K$ in low In doping regime [21] to T${}_{\rm{C}}=4.8~{}K$ for higher In contents [24]. Although there is no doubt about the superconductivity phase, the preservation of its topological phase under In doping is a topic of current discussion. For low In concentration some experiments indicate a signature of topological surface states [21, 25], and recent experiments [33] show evidence of surface linear dispersion bands at high In doping rates. However, that does not confirm its topological order.

Also the superconductivity mechanism in Sn1-xInxTe is currently in discussion. Some previous experiments indicate that the superconductivity should be odd-parity pairing [19, 22], but more recently it has been attributed to a s-wave BCS model [23, 24, 27, 28]. Another intriguing question with respect to In doped SnTe TCI is its doping character. While low In concentrations generate hole-like doping, for high In doses it turns into electron-like carriers [26, 29]. The hole-like can be expected, as In is a group III element substituted at a group IV atom, but no electron-like behaviour would be expected.

In this work we show that Sn1-xInxTe for In concentrations of $x=0.03125$, $x=0.125$ and $x=0.25$ the valence and conduction bands are inverted, giving a non-null Chern number. We have identified pairs of topological states protected by the mirror symmetry on the (001) surface of Sn1-xInxTe with opposite mirror eigenvalues $\pm i$. Our results show that the experimentally observed hole to electron carrier-like transition as a function of increased In concentration can be explained in terms of the depopulation of the In-5s orbital, which is pinned at the Fermi level. Furthermore, our results also indicate that Sn1-xInxTe is a $s-$wave superconductor.

2 Method

The mirror Chern number has been computed using the hybrid Wannier charge center scheme [34, 35], where the Wannier functions are constructed from first-principles calculations. The band structure and the projection of the mirror eigenvalues on the topological surface states have been computed using the density functional theory and generalized gradient approximation for the exchange and correlation functional [36]. Fully relativistic pseudopotentials within the projector augmented wave (PAW) scheme [37] have been used self-consistently within plane wave basis set with the kinetic energy cut-off of 340 eV. We used the Vienna Ab initio Simulation Package (VASP) for band structure and mirror projections [38, 39], and Quantum ESPRESSO [40] to compute the Wannier functions. Both SnTe and Sn1-xInxTe were modelled in the rock-salt structure. We used the experimental lattice parameter for SnTe, and following the lattice variation observed experimentally [22, 26] making a linear reduction of the lattice parameter for the In content compounds. The Brillouin zone (BZ) was sampled by using the Monkhorst-Pack special ${\bm{k}}$-points scheme [41], with grid-sizes as discussed later.

Calculations for bulk Sn1-xInxTe systems were performed by using a 64-atom cubic unit cell containing 32 cations (Sn/In) and 32 anions (Te) at the rock-salt basis sites. For $x=0,0.03125,0.125$ and 0.25 the number of In dopants per cell was 0, 1, 4 and 8, respectively. The BZ sampling was done with the (10$\times$10$\times$10) mesh of ${\bm{k}}$-points.

To compute surface states we modelled the Sn1-xInxTe system using two types of cells containing an atomic slab and a minimum of 15 Å of vacuum region. For $x=0$ (pristine SnTe) and $x=0.125$ we used a tetragonal cell with the atomic slab containing 128 and 64 atoms, respectively. For $x=0.125$ each slab in a tetragonal unit cell contained 4 In, 28 Sn and 32 Te atoms. For $x=0.03125$ we considered a cubic slab with 256 atoms (4 In, 124 Sn and 128 Te) in a unit cell. The BZ sampling for the cubic and tetragonal cells was done with (10$\times$10$\times$1) and (7$\times$7$\times$1) ${\bm{k}}$-point meshes, respectively.

In order to discuss symmetry-related features in the band structure results for both bulk and slab geometries, we have presented a schematic sketch of a sample cubic cell of size $a$ and of a sample tetragonal cell of base $(a/\sqrt{2}\times a/\sqrt{2})$ on the left hand side of Fig. 1. On the right hand side of Fig. 1 we have shown the correspondence between the important BZ symmetry points for bulk (fcc) and their projection on the BZs for the cubic and tetragonal structures used in our modelling. In particular, we note that the bulk L points map onto the ${\bar{M}}_{\rm c}$ point of the cubic structure and onto the ${\bar{X}}_{\rm t}$ point of the tetragonal structure.

3 Results and Discussions

3.1 Topological Phase for In Doped Bulk SnTe

At room temperature SnTe has the simple rocksalt crystal structure. This IV-VI compound is a narrow-gap (Eg=0.18 eV) semiconductor, with the conduction and valence band edges located at the four equivalent L points in the fcc BZ (two of which are marked in Fig. 1). In our modelling we chose to use the experimental lattice constant of 6.321 Å in order to facilitate comparison of electronic states in the gap region. In Fig. 2(a) we have plotted the electronic band structure of bulk SnTe using a simple cubic (2$\times$2$\times$2) unit cell (8 times the volume of the cubic cell in Fig 1) as mentioned in the Method section. The L point of the fcc BZ is folded onto the zone centre $\bar{\Gamma}$ for this periodic structure. As shown in the bottom panel of Fig. 2(a) we obtain a band gap of 0.17 eV for bulk SnTe when the SO interaction is included.

In the asbence of SO coupling, the VBM and CBM states at the L point in the BZ for IV-VI compounds are contributed by (s-cation, p-anion) and (s-anion, p-cation), respectively [42]. This is the expected normal ordering of the band edges in III-V and II-VI compunds [43, 44]. Due to relativistic interactions the cation/anion characters are inverted in SnTe, and the projected orbital contributions are $\Phi_{-}=[\phi^{Sn}_{P}+\phi^{Te}_{S}]$ (in orange) for VBM and $\Phi_{+}=[\phi^{Te}_{P}+\phi^{Sn}_{S}]$ (in purple) for CBM. The Bloch eigenstates on the $\Gamma~{}L_{1}~{}L_{2}$ plane in the fcc BZ are invariant under the mirror symmetry of rocksalt structure with respect to the family of {110} planes. The Bloch eigenstates for such mirror planes are also eigenstates of the mirror symmetry operator ${\cal M}$, with eigenvalues $\pm i$. For each class of Bloch eigenstates we compute the mirror Chern number using the Wannier charge center technique [45] obtaining $n_{M}=-2$ in agreement with previous results [31, 32].

For Sn1-xInxTe the bulk conduction and valence bands edges (CBM and VBM) for $x=0.03125$, $x=0.125$ and $x=0.25$ are still inverted at the $\bar{\Gamma}$(L) point, as we can see in Fig. 2-(f)-(h). However for low In content $x=0.03125$ there is a clear total band gap, for higher In concentration it becomes a gapless system. Without the inclusion of spin-orbit interaction the conduction band dips inside the valence band, making Sn1-xInxTe metallic (see Fig. 2-(b)-(d)). This is in contrast to the pristine case ($x=0$), where there is still a very small gap between the VBM and CBM (Fig. 2-(a)) without the inclusion of the spin-orbit interaction.

For low In doping, $x=0.03125$, we find an In state inside the bulk band gap and the Fermi level lies slightly below the valence band maximum (VBM) (see Fig. 2-(f)). This makes the system a gapped p-type doped semiconductor, in agreement with a recent study [26]. In order to understand the In effects we have highlighted (in pink) the In-5s contribution to the band structure (identified within the range -0.6 eV up to +0.5 eV) in Fig. 2(j). Some of these features are close to the $\bar{\Gamma}$ point and some away from $\bar{\Gamma}$ but close to $E_{F}$. However, close to the $\bar{\Gamma}$ point there is a negligible In-5s contribution in the SnTe band gap region.

By increasing the In concentration to 12.5%, Fig. 2-(k), the In-derived levels go inside the valence band at the $\bar{\Gamma}$ point. There is a clear development of In-5s related bands inside and across the band gap. More importantly, there are more significant (cf. larger symbols) In-5s contributions just below $E_{F}$ across the BZ. Similarly for the high $x$ regime of 25% of In, the metallic character is enhanced with more In-5s levels connecting valence and conduction bands as shown in Fig. 2-(l). It is interesting to note that there is a flat band originating from In-5s orbitals quite close to the Fermi level (around $E_{f}-0.054$ eV). For both $x=0.125$ and $x=0.25$ several In-5s bands are empty around $E_{F}$, with larger contributions for $x=0.25$. This result can explain the transition from hole-like carriers in low In doping to electron-like carriers for higher In concentrations observed recently in experiments [29]. This transition can occur when the In-5s impurity level is partially occupied, turns it into In*+3*, instead of the expected In*+1* state. This $p-$ to $n-$type transition has been found to occur even for a lower In concentration [26], which can be attributed to the presence of other defects like Sn vacancies that lower the Fermi level. We will return to discuss this point by examining the electronic density of states later.

The p-orbitals of cation and anion has been used as a basis to compute the mirror Chern number for pristine SnTe [31], as well as IV-VI monolayers [46]. The p-orbitals of cations contributing to VBM and the p-orbitals of anions contributing to CBM, are all aligned along the [111] direction. This inverted orbital ordering produces a negative band gap. In Fig. 3 we show schematically the evolution of the VBM and CBM orbital characters when In ions are incorporated at Sn sites. We can still identify the $\Phi_{-}=[\phi^{Sn}_{P}+\phi^{Te}_{S}]$ and $\Phi_{+}=[\phi^{Te}_{P}+\phi^{Sn}_{S}]$ for the VBM and CBM, respectively, but we have an additional In state which is always around the band gap region. The In contribution of this state is quite low at the L (${\bar{\Gamma}}_{t}$) point but increases away from this point. Importantly, the In state keeps the same character as the VBM, wheather it is inside the band gap or above the CBM.

For low In concentration, $x=0.03125$, the In state is located inside the band gap and its contribution is pure p-orbital ($\Phi_{-}^{In}$) aligned along the [111] direction, the same character as that for the VBM cation Sn ion. As the gap is still inverted, the non-trivial topology is maintained. For high In concentration the In state is resonant inside the conduction band, and both VBM and CBM characters are the same as for the pristine one. Using the same procedure as described above the computed mirror Chern number for each of $x=0.03125$ and 0.25 is $n_{M}=-2$.

3.2 Topological Surface States

In the previous sub-section we have clearly verified the band inversion and non-trivial topological characteristics of pristine SnTe and Sn1-xInxTe. In a recent experimental study, using ARPES measurements, Sato et al [21] reported the existence of topological surface states, leading to the possibility of topological superconductivity in Sn1-xInxTe. In order to clarify the topological nature, we also performed calculations with a repeated slab geometry for pristine ($x=0$) and In doped system containing a low ($x=0.03125$) and a higher In doped concentration ($x=0.125$). Details of the slab makeup, including number of atomic layers and vacuum region, have been presented earlier. The mirror symmetry has been maintained for each slab geometry. However, substitutional In impurity breaks the top and bottom surface symmetries, yielding a reduction in the hybridization between the topological states of opposite surfaces.

Expressing, with respect to the mirror plane, symmetric and antisymmetric real space parts of wavefunction as $|s\rangle$ and $|a\rangle$, respectively, and the spinor as $|\sigma\rangle$, the mirror eigenvalue equations can be stated as

[TABLE]

This allows for two sets of topological surface states to be realized: the state $(|s,\sigma\rangle+|a,-\sigma\rangle)$ corresponding to the eigenvalue $+i$ and the state $(|s,-\sigma\rangle+|a,\sigma\rangle)$ corresponding to the eigenvalue $-i$. Panel (a) in Fig. 4 shows the band structure of the pristine ($x=0$) SnTe slab geometry. The bulk band structure projected on to the tetragonal slab BZ is shown as shaded region. Consistent with the mirror symmetry and band inversion, a pair of surface bands corresponding to the mirror eigenvalues $\pm i$ inside the bulk band gap region are identified (and shown in blue and red colors) extending along the ${\bar{\Gamma}}_{t}$-${\bar{X}}_{t}$ direction of the tetragonal BZ. The tiny band gap (around 36 meV) between the two surface bands in the middle of the bulk band gap, instead of Dirac crossing, is due to the interaction between the top and bottom surfaces of the slab. We also clearly identify a surface state (non topological origin) lying at around $E_{F}-0.17$ eV for part of the ${\bar{\Gamma}}_{t}$-${\bar{X}}_{t}$ direction.

As we can see in panels (b) and (c) of Fig. 4, topological surface states are present for both the low ($x=0.03125$) and high ($x=0.125$) In concentration regimes. The mirror symmetry is preserved for both In concentrations, as required for the presence of symmetry protected surface states. It is interesting to note that the topological state branch (blue color) coming from the valence band has contributions from In orbitals (pink color), while the topological state coming from the conduction band (red color) has pure SnTe character, similar to the pristine case in panel (a).

In a previous work in TCI [47] it was found that, as the slab thickness increases the interaction between top and bottom topological surface states reduces, leading to a reduction in the surface states band gap. We find that the surface state gaps for both $x=0.03125$ and $x=0.125$ are smaller than that for $x=0$. Since the pristine SnTe slab, and the slab for $x=0.03125$ have the same thickness, we conclude that the presence of In impurity further reduces the surface band gap. The presence of substitutional In breaks the top and bottom surface symmetries, yielding a reduction of the hybridization between the topological states of opposite surfaces. Our work reveals that Indium doping of SnTe shifts the Dirac crossing energy position and also affects the Fermi velocities of the topological surface states. The main changes occur for topological hole states. As the doping moves the Fermi level downwards, the Fermi velocity increases with respect to the pristine system.

In Table I we summarise the topological surface state parameters for Sn1-xInxTe. For pristine SnTe, the Dirac crossing point is identified at a location with wavenumber $k_{D}(L)$ away from the bulk L point. For Sn1-xInxTe the Dirac crossing point moves further way from the L point as $x$ increases. The Fermi wavenumber $k_{F}$ also increases as $x$ increases. Our computed values of $k_{F}$ are similar to the values reported in experimental works [29, 33], albeit for different $x$ values and different surfaces. The electron and hole velocities have been computed in a range of 0.2 eV above and below the Dirac crossing point. The electron velocity, $v_{e}$, is almost independent of the $x$ values. It is found that the hole velocity, $v_{h}$, has increased significantly for $x=0.125$. In contrast, the Fermi velocity of the topological states, $v_{F}$, increases steadily with increase in $x$. Our computed Fermi velocities are in reasonable agreement with those obtained from quantum oscillation and ARPES measurements [29, 33]. In particular, the increase in $v_{F}$ with increase in $x$ has also been revealed from the ARPES measurements in [33] for $x=0.23$ and $x=0.41$ for the Sn1-xInxTe(111) surface.

We have estimated the Dirac crossing point to lie at 220 meV and 380 meV above the Fermi level for $x=0.03125$ and $x=0.125$, respectively. The relative increase in $E_{D}-E_{F}$ with increase in $x$ has also been reported in [33] from a quantitative analysis of the ARPES spectrum for the (111) surface with $x=0.23$ and $x=0.41$. Bulk states are always inside the topological surface bands, i.e. with k(bulk)$<$kD(surf). This is consistent with the analysis of the ARPES study in [33] for their $x=0.23$ and $x=0.41$ samples.

3.3 Superconductivity Character

Transport, magnetization, and heat capacity measurements show that In substitution at Sn sites turns the TCI SnTe into a superconductor [22]. The superconductivity character of bulk SnxIn1-xTe can also be qualitatively analysed from our first principles calculations. The electronic density of states distribution near the Fermi level, $DOS(E_{F})$, plays a dominant role in forming the BCS type superconducting state. According to the BCS theory, the formation energy of a Cooper pair increases linearly with $DOS(E_{F})$ [48]. With this in mind, in Fig. 5 we project the total and In contribution to the density of states. Comparing the results for $x=0$ (pristine SnTe) and the low In concentration of $x=0.03125$ we find that there is a clear development of $DOS(E_{F})$ for the latter case. There is a progressively larger development of $DOS(E_{F})$ for the larger values of $x=0.125$ and $x=0.25$. Our computed density of states per unit cell at the Fermi level for the three doping concentrations are 0.19, 0.73 and 0.78 states/eV for $x=0.03125$, $x=0.125$ and $x=0.25$, respectively. These values are in quite good agreement with the estimated density of states based on magnetic susceptibility data [26] of 0.44 and 1.17 states/eV for $x=0.05$ and $x=0.40$, respectively. For each of the three $x$ values In-5s orbitals make almost the total contribution towards $DOS(E_{F})$. Our results thus support a s-wave superconductivity in SnxIn1-xTe, in agreement with recent experimental observations [23, 24, 25, 27, 28]. We observe also from Fig. 5 that the In-5s orbital becomes less occupied for higher In doping regime, which prompts the $p-$ to $n-$type transition as a function of increasing In concentration, providing support to the analysis of experimental measurements in [26, 29].

4 Conclusion

We report on the non-trivial topological properties of the superconductor Sn1-xInxTe. The band inversion characteristic, viz. swapping of the normal contributions from p-anion and s-cation orbitals for the valence and conduction band edges (VBM and CBM), has been identified to arise in the presence of spin-orbit interaction. This inversion is preserved for In content systems with $x=0.03125,0.125$ and 0.25. The non-trivial phase has been confirmed by computing the mirror Chern number which is non-null for all those In concentrations. Topological states with opposite $\pm i$ mirror eigenvalues has been identified on the (001) surface of the Sn1-xInxTe. With increase in In concentration, the location of the surface Dirac crossing point shifts progressively towards larger wavenumber. Also, the doping moves the Fermi level downwards and the Fermi velocity progressively and significantly increases as $x$ increases. We also verified that the density of states at the Fermi level for finite $x$ values has been found to come mostly from In 5s orbitals. This confirms the s-wave nature of the superconducting state in agreement with some experimental studies. The results presented in this work demonstrate a non-trivial topological phase for the superconductor Sn1-xInxTe, and provide a detailed description of the topological state properties.

Acknowledgements

One of the authors (TMS) acknowledge the financial support from the Brazilian Agencies INCT in Carbon Nanomaterials, CNPq, CAPES, FAPEMIG, and the computational facilities of LNCC and Cenapad.