Abstract
Recent theoretical studies employing densityfunctional theory have predicted BaBiO_{3} (when doped with electrons) and YBiO_{3} to become a topological insulator (TI) with a large topological gap (~0.7 eV). This, together with the natural stability against surface oxidation, makes the BismuthOxide family of special interest for possible applications in quantum information and spintronics. The central question, we study here, is whether the holedoped Bismuth Oxides, i.e. Ba_{1x}K_{x}BiO_{3} and BaPb_{1x}Bi_{x}O_{3}, which are “highTc” bulk superconducting near 30 K, additionally display in the further vicinity of their Fermi energy E_{F} a topological gap with a Diractype of topological surface state. Our electronic structure calculations predict the Kdoped family to emerge as a TI, with a topological gap above E_{F}. Thus, these compounds can become superconductors with holedoping and potential TIs with additional electron doping. Furthermore, we predict the BismuthOxide family to contain an additional Dirac cone below E_{F} for further hole doping, which manifests these systems to be candidates for both electron and holedoped topological insulators.
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Introduction
Topological insulators (TIs) are new quantum states of matter that are of fundamental interest for both condensedmatter physics studies and applications in spintronics, quantum information as well as thermoelectrics^{3,4}. The unique feature of the TIs is the existence of topologically protected and, therefore, robust conducting channels at the surfaces/edges of materials that are insulating in the interior. Due to the existence of bandinversion in their bulk electronic structure, this new type of insulators is topologically different from the conventional insulators in the sense that TIs cannot be adiabatically transformed into an atomic insulator without going through a phase transition. More precisely, charge conservation and time reversal symmetry around the Fermi level of band inversion establish the protecting symmetry of the nontrivial odd Dirac cone surface state which cannot be gapped out continuously.
Recently, a new direction in the search for topological insulators, with a substantial potential for the above applications, has emerged by identifying BaBiO_{3} as a TI in the electrondoped region^{1,2}. According to densityfunctional electronic structure calculations^{1}, this compound possesses the largest topological gap (~0.7 eV) among currently known TI materials and is naturally stable against surface oxidation and degradation, in contrast to other TIs. The large topological gap is induced by the strong spinorbit coupling (SOC) of Bismuth in cubic BaBiO_{3}, which causes an inversion between the Bis and Bip band at a timereversal invariant momenta (TRIM), i.e. the symmetry point R. Inside the corresponding topological gap a Diractype of topological surface state (TSS) then exists. So far, however, BaBiO_{3} has not yet experimentally been verified as a topological insulator.
The central question, which we want to address in this work, is to study to what extent the decisive role of the SOC of the s and pBismuth orbitals for the band inversion and the TI nature can be carried over to the large family of superconducting, i.e. doped, Bismuth Oxides. Here of particular interest are the potassium (K) and lead (Pb)doped “relatives” of BaBiO_{3}. As the most experimentally studied doped BaBiO_{3} compounds, these systems are naturally the first choice to understand the influence of doping on the topological nature of BaBiO_{3.} It has been known for a long time, that doping cubic BaBiO_{3} with K and Pb will convert this system to a superconductor^{5,6}(SC). Indeed, Ba_{1x}K_{x}BiO_{3} and BaPb_{1x}Bi_{x}O_{3} show the highest transition temperatures in copper and ironfree systems. Thus, the key question here is, whether the doping of BaBiO_{3} with K and Pb, i.e. extending the systems to the holedoped highTc bulk superconductors, will still preserve a “hidden” topological insulator phase achievable through additional electron or also holedoping. Our work answers this crucial question: for the experimentally known structures and phases of the K and Pbdoped BaBiO_{3} compounds, the topological nature is indeed found to be robust, providing the feasibility of doping, which still keeping the topological nontrivial band structure. This should encourage experimentalists to tune BaBiO_{3} by first achieving an appropriate doping and then performing electric gating on top of this.
These questions are answered via a theoretical study of the other two end compounds, i.e. KBiO_{3} and BaPbO_{3}, in their cubic phase. Additionally, we consider the doped compounds Ba_{0.5}K_{0.5}BiO_{3}, as well as BaPb_{0.7}Bi_{0.3}O_{3}, which facilitates an interpolation in our search for 3D TIs within the Ba_{1x}K_{x}BiO_{3} and BaPb_{1x}Bi_{x}O_{3} families.
We find that cubic KBiO_{3} has a very similar electronic structure as BaBiO_{3}, including again the band inversions at the Rpoint. However, compared to the band inversion in BaBiO_{3}, this happens at a higher energy in KBiO_{3}. Thus, cubic KBiO_{3} can also become a TI via electron doping, albeit experimentally this may harder to be realized. In BaPbO_{3}, the Bi sband moves to higher energy and while a direct energy gap is left at the Rpoint, preserving the topological nature^{1}, the indirect gap is zero. As a result, in the surface BZ the Dirac cone merges into the bulk bands. Under electron doping, BaPbO_{3} can then be tuned into a “topological metal”. As an important additional observation, we find that all three Bismuth Oxides, i.e. BaBiO_{3}, KBiO_{3} and BaPbO_{3} are found to contain another band inversion at the Γpoint below the Fermi level. For this reason, these systems can, in principle, also be tuned into TIs via hole doping. These results concerning the topological nature and the “hidden” Dirac cones are also carried over to arbitrary doping levels, as discussed in our work.
Results
Let us first briefly review the structural aspects of all three parent compounds and explain why we will focus in this paper on their cubic phase. One striking feature in BaBiO_{3} is the breathingmode distortions, induced by Bi ions. It gives rise to an ordered arrangement of Bi^{3+} and Bi^{5+} ions^{7}. Comprehensive crystallographic studies, by using neutron powder diffraction^{8,9}, found that BaBiO_{3} experiences a number of temperatureinduced phase transitions. They range from monoclinic in P2_{1}/n at low temperature and monoclinic in I2/m at room temperature to rhombohedral in R at ca 405 K and cubic in Fmm at ca 750800 K. In ref.1, the interesting topological phase was found to exist in the perovskite lattice (see crystal structure in Fig. 1(b)) of the parent compound BaBiO_{3} in the cubic phase, which is stable against the monoclinic lattice distortion. The perovskite structure of BaPbO_{3} is also quite stable^{10,11}. At temperatures above 673 K, it crystallizes in the simple perovskite structure in Pmm and transforms to tetragonal I4/mcm structure at temperatures below it. Further decreasing the temperature to 573 K, it changes to the orthorhombic Ibmm structure. KBiO_{3} does not form the perovskite structure but rather crystallizes in a cubic KSbO_{3}type tunnel structure with space group Im^{12}.
Despite the large variations in their crystal structures, as far as superconductivity is concerned, the Ba_{1x}K_{x}BiO_{3} compounds are found to confine to their cubic symmetry^{13,14,15}. For this reason, we restrict our study of the parent compound KBiO_{3} to the simple Pmm perovskite structure (see Fig. 1).
The strategy here is as follows: if the two end parent compounds are both TIs in the cubic structure, their random alloys with the same structure would have a large chance to be also TIs. Through firstprinciple calculations of a superstructure, we confirm this indeed to be the case for Ba_{1x}K_{x}BiO_{3}. Superconducting BaPb_{1x}Bi_{x}O_{3} is confined to the tetragonal distortion of the cubic phase^{16}. Similarly to Ba_{1x}K_{x}BiO_{3}, here we will also concentrate on the cubic phase of BaPbO_{3} and BaPb_{1x}Bi_{x}O_{3}. In addition, we also show that the tetragonal distortion does not change the topological nature of the cubic BaPbO_{3}.
KBiO_{3} and Ba_{0.5}K_{0.5}BiO_{3}
Figure 1 displays the electronic structure of simple cubic KBiO_{3} with the lattice constant 4.2886 Å^{13,17}. Its band structure is very similar to that of BaBiO_{3}. Thus, the discussions here also apply to BaBiO_{3}. Cubic KBiO_{3} is a metal, with a band carrying a large weight of the Bi sorbital crossing the Fermi level. The potassium states stay at higher binding energy and are not relevant to the bandinversions we discuss here. The bands displayed in Fig. 1(a) are mainly from the bismuth and oxygen states. The bandinversions of KBiO_{3} are shown in Fig. 1(a) as the interchange of the red and green colors at special TRIM (see, for example, the lightyellow area around the Rpoint). The red and green colors are used to label Bi s and porbitals, respectively. The interchange of the two colors inside the lightyellow area indicates that the order of these two bands is inverted. Due to this reason, the adiabatic connection to the atomic limit of the system is then lost, which makes this system topologically different from the conventional insulator. An important point for the more general TI physics in these compounds is that, the topological gap shown in the lightyellow area at the Rpoint is induced by the strong SOC of Bi, as discussed in ref.1 for BaBiO_{3}.
In addition, we found a band inversion at the Γpoint, however, below the Fermi level. Similar to the band inversion at R, the Bi sorbital and porbital weights interchange at Γ around 5.8 eV below E_{F}. It may experimentally not be easy to shift E_{F} into such heavily holedoped region. However, it should be feasible for ARPES to detect the corresponding Diractype surface states in the occupied bands.
To further validate the topological nature of cubic KBiO_{3} at both hole and electrondoped regimes, we calculate the surface states by constructing a slab geometry along the [001] direction of bulk KBiO_{3}. Fig. 2(a, b) displays the electronic structure of such a slab with thickness of 60 atomic layers. As one can clearly see from Fig. 2(a), there are two Dirac cones located at and . They stay at different sides of the Fermi level. Though, the energy gap below E_{F} is smaller than that at the Rpoint (above E_{F}), it is clear that there is also a Diraccone positioned inside it. Thus, both topological surface states can, in principle, be accessed via “chemical engineering”, see the schematic plot in Fig. 3.
By calculating the projected density of states, we, further, verified that both Dirac cones are stemming from surface states (results are not shown here). With the current slab setup displayed in Fig. 2, each branch of the surface states is doubly degenerate as this slab contains two surfaces with the same atomic layer, which do not interact with each other. This degeneracy can be removed by terminating one surface with a different atomic layer, as shown in Fig. 2(b). When one surface is terminated by the BiO layer, while keeping the other one unchanged (KO layer), the different potentials at these two surfaces separate the two Dirac cones in energy, which lifts the degeneracy. Furthermore, the Z_{2} topological invariant calculated in Table 1 also confirms KBiO_{3} to be a strong topological insulator.
Since both cubic BaBiO_{3} and KBiO_{3} can be TIs, we next show that their random alloys in the same cubic phase, in principle, can be also TI. Ba_{1x}K_{x}BiO_{3} is found to crystallize in the Pmm phase for 0.3 < x < 0.5^{13}. By taking the experimental lattice constant a = 4.2618 Å^{13} and setting c = 2a (relaxation on c does not change the following conclusion), we calculated the electronic structure of Ba_{0.5}K_{0.5}BiO_{3} by using a supercell with two Bi atoms (see Fig. 4 for the corresponding crystal structure and the BZ). Comparing with the cubic structure of BaBiO_{3} and KBiO_{3}, the length of the caxis is doubled. This leads to a folding of the BZ and to a mapping the Rpoint of the cubic BZ (see the BZ in Fig. 1) to the Mpoint. So does the bandinversion.
We found both bandinversions in cubic BaBiO_{3} and KBiO_{3} to exist in their corresponding superstructure. Now they are at the TRIM Mpoint and Γpoint. Here, we adopted the case x = 0.5 as an example to illustrate the surface states. This case allows us to work with a superlattice that is only twice the size of the cubic cell. As displayed in the right plot of Fig. 4, in a slab of Ba_{0.5}K_{0.5}BiO_{3} constructed with the superlattice bulk band structure along the [001] direction, these two bandinversions induce Dirac cones located at the and points at the surface BZ. Thus, substituting Ba with K does not change the topological nature of the system, see also Table 1 for the topological invariant. The band inversion, as well the Dirac cones are robust to this substitution, as long as the alloys crystallize in the cubic structure.
If we, further, compare the location of the energy gap at the Rpoint in three different systems, i.e. BaBiO_{3}, KBiO_{3} and the superlattice BaK[BiO_{3}]_{2}, the potassium doping effect becomes obvious: K contains less valence electrons compared to Ba. Thus, substituting Ba with K effectively dopes holes into the system. As a result, the electronic structure of KBiO_{3} can be essentially understood as that of BaBiO_{3} with a global shift to a larger positive value in energy.
BaPbO_{3} and BaPb_{0.7}Bi_{0.3}O_{3}
The Bi sp band inversions in BaBiO_{3} are left unaffected with the substitution of Ba with K, as the Ba and K states all stay at binding energies outside of the relevant energy regime for this band order reversal. Thus, it is not surprising that their alloys Ba_{1x}K_{x}BiO_{3}, in principle, also stay as TIs in the cubic phase. However, this is not the case in the BaPb_{x}Bi_{1x}O_{3} systems. Chemically, Pb is adjacent to Bi in the periodic table, with also a large atomic number and, thus, a strong SOC (but weaker than that of Bi). A similar topological phase can then be expected.
From the following discussions, we will see that the relatively stronger SOC of Bi is crucial for realizing the topological phase in BaBiO_{3}. By substituting Bi with Pb, the reduced SOC leads to a strong shift of the Pb stype band, which, in the end, pushes the spinfiltered surface states into the bulk states.
At temperatures above 673 K, BaPbO_{3} crystallizes in the same cubic structure as BaBiO_{3}. Fig. 5(a) displays the electronic structures for the bulk and an [001] slab of cubic BaPbO_{3}. The lattice constant is 4.2997 Å, taken from ref.11. Replacing Bi with Pb strongly modifies the bandinversions at R and Γ points. On the one hand, the bandinversion in cubic BaPbO_{3} moves to an energy of ~2 eV higher than that in BaBiO_{3}. This is similar to what happens in KBiO_{3}, i.e. Pb contains less valence electrons than Bi. Thus, replacing Bi with Pb effectively dopes the system with holes, which shifts the Fermi level downwards.
On the other hand, compared to BaBiO_{3} and KBiO_{3} (see Fig. 1), the details of the bulk electronic structure are also strongly modified. The stype band moves to higher binding energy. At R around 5 eV, the ptype bands shows stronger dispersion along RM and RZ directions and the indirect energy gap here becomes negative. Thus, it is of no surprise to see that, when cubic BaPbO_{3} is terminated with a surface, the surface states originating from this bandinversion merge into the bulk states (see the bluecolored plot of Fig. 5(a)). With the tetragonal distortion (Fig. 5(c)), the bandinversion around 5 eV is still preserved (see also Table 1 for the topological invariant), which now appears at Γ. The separation of the Pb stype and ptype bands in energy is also similar to that in the cubic phase (see Fig. 5(a)). For the same reason, the Dirac cone of the tetragonal BaPbO_{3} and BaPb_{0.7}Bi_{0.3}O_{3} also merge into the bulk bands but with their topological nature preserved.
In the righthand plot of Fig. 5(c), the electronic structure of the tetragonal alloy BaPb_{0.7}Bi_{0.3}O_{3} is studied by using the virtual cluster approximation. This is exactly the phase where superconductivity is observed in experiments. The band inversion survives the tetragonal distortion and the chemical substitution. Thus, one may be tempted to speculate about topological superconductivity emerging from the coexistence of the superconductivity at T < T_{c} and the topological insulating nature. It is of crucial importance, however, to note that while the topological insulating phase can indeed be achieved on the basis of our electronic structure calculations, one is actually comparing two different systems (see also Fig. 3): one stands for the doped “highT_{c}” superconductor with the chemical potential E_{F} as given by our calculations. The other, i.e. the nominal TI system, requires an additional shift in the chemical potential to the energies, where the band inversion takes place. However, we find that, in both K and Pbdoped BaBiO_{3}, the topological gaps below the Fermi level are present. Particularly, in the Pbdoped BaBiO3, the Pb substitution of Bi displays clearly different influences on the topological gaps at above and below the Fermi level. The bandinversion and the energy gap below the Fermi level turns out to be more robust, which gives rise to surface states displayed in the righthand plots in Fig. 5(a) and (b), respectively. If the two surfaces of the slab are terminated with different atomiclayers, one of the Dirac cones will merges into the bulk states, while the other one still is positioned inside the energy gap (see again the surface states plots in Fig. 5 (a) and (b)).
Conclusions
In summary, it is a fascinating observation to explore topological band structure features “hidden” in the family of highTc superconductors, surrounding the BaBiO_{3} compounds. We found (1) the existence of an additional topological gap and the Dirac cone below the Fermi level, which was not discovered in the previous work^{1}. This largely enriches the possibility of observing the Dirac cone in experiment. It should be possible to detect this lower Dirac cone in ARPES; (2) the topological nature of BaBiO_{3} is robust with respect to K and Pbdoping. We believe our work represents one crucial step forward towards the final experimental realization of the topological phase of BaBiO_{3}. The robustness of the topological nature of BaBiO_{3} can make the electric gating much easier in electron dopedBaBiO_{3} than in the parent compound.
The topological insulator properties and the corresponding Dirac cones embedded within a band gap appear at a different chemical potential than the superconducting state. However, they coexist in the same material class. The SC and the TI states may be related via chemical doping, or other means (see Fig. 3). Our results demonstrate in particular, that, apart from the chemical potential shift, the overall electronic structure remains essentially unchanged under doping, like in a “rigid band structure” description. This observation substantially enhances the possibility of switching from one unconventional state to the other. In that sense, our findings may open up new avenues for the abovementioned possible applications, such as the fabrication of controlled SC/TI interfaces^{31}.
Methods
The calculations were mainly carried out within the fullpotential linearized augmented planewave (FPLAPW) method^{18}, implemented in the package WIEN2k^{19}. The package ELK (http://elk.sourceforge.net) and the Vienna Ab Initio Simulation Package (VASP)^{20,21,22,23} with PAW potentials^{24,25} are also employed for comparison. K_{max}R_{MT }= 9.0 and a 10 × 10 × 10 kmesh were used for the groundstate calculations in WIEN2k. R_{MT} represents the smallest muffintin radius and K_{max} is the maximum size of reciprocallattice vectors. The spinorbit coupling is included by a second variational procedure. The generalized gradient approximation (GGA) potential^{26,27} is used in all calculations. The Z_{2} invariant for all parent compounds are calculated within WIEN2k.
The surface electronic structures are further calculated using the maximally localized Wannier functions (MLWFs)^{28}, employing the WIEN2WANNIER^{29} and VASP2WANNIER90^{30} interfaces. The MLWFs are constructed in a nonselfconsistent calculation with an 8 × 8 × 8 kmesh.
Additional Information
How to cite this article: Li, G. et al. Topological nature and the multiple Dirac cones hidden in Bismuth highTc superconductors. Sci. Rep. 5, 10435; doi: 10.1038/srep10435 (2015).
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Acknowledgements
G. Li wants to acknowledge fruitful discussions with J.P., Hu., G. Li and W. Hanke acknowledge the DFG Grant Unit FOR1162 and SPP Ha 1537/242. R. Thomale is supported by ERCTOPOLECTRICSStG336012.
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G.L. designed and led the research, carried out calculations and analysis, documented the findings and prepared the figures; G.L. and W.H. wrote the main manuscript text. B.Y. and R.T. participated in scientific discussions. All authors contributed to the interpretation of results and to the finalization of the submitted manuscript.
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Li, G., Yan, B., Thomale, R. et al. Topological nature and the multiple Dirac cones hidden in Bismuth highTc superconductors. Sci Rep 5, 10435 (2015). https://doi.org/10.1038/srep10435
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DOI: https://doi.org/10.1038/srep10435
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