Abstract
ZrTe_{5} and HfTe_{5} have attracted increasingly attention recently since the theoretical prediction of being topological insulators (TIs). However, subsequent works show many contradictions about their topological nature. Three possible phases, i.e. strong TI, weak TI, and Dirac semimetal, have been observed in different experiments until now. Essentially whether ZrTe_{5} or HfTe_{5} has a band gap or not is still a question. Here, we present detailed firstprinciples calculations on the electronic and topological properties of ZrTe_{5} and HfTe_{5} on variant volumes and clearly demonstrate the topological phase transition from a strong TI, going through an intermediate Dirac semimetal state, then to a weak TI when the crystal expands. Our work might give a unified explain about the divergent experimental results and propose the crucial clue to further experiments to elucidate the topological nature of these materials.
Introduction
Topological insulator (TI) is a new class of material which is an insulator in its bulk, while having time reversal symmetry protected conducting states on the edge or surface^{1,2,3}. A large number of realistic materials have been theoretically proposed and experimentally confirmed, such as Bi_{2}Se_{3} and Bi_{2}Te_{3}^{4,5}. However, the layered transitionmetal pentatelluride ZrTe_{5} and HfTe_{5} is a particular example. ZrTe_{5} and HfTe_{5} were studied more than 30 years ago due to the large thermoelectric power^{6} and mysterious resistivity anomaly^{7,8}. Recently, Weng et al. predicted that monolayer ZrTe_{5} and HfTe_{5} are good quantum spin Hall insulators with relatively large bulk band gap (about 0.1 eV) by first principles calculations^{9}. The threedimensional (3D) bulk phase of ZrTe_{5} and HfTe_{5} are also predicted to be TIs, which are located at the vicinity of a transition between strong and weak TI, but without detailed description^{9}.
Nevertheless, subsequent experiments show many contradictions about the topological nature of ZrTe_{5} or HfTe_{5}. Several experimental works suggested that ZrTe_{5} is a Dirac semimetal without a finite band gap by different characterization methods, such as Shubnikovde Haas oscillations, angleresolved photoemission spectroscopy (ARPES), and infrared reflectivity measurements^{10,11,12,13,14,15}. Of course there are also other experimental works holding opposite point of view. For example, in two recent scanning tunneling microscopy (STM) experiments, they unambiguously observed a large bulk band gap about 80 or 100 meV in ZrTe_{5}^{16,17}, implying that there is no surface state on the top surface and therefore ZrTe_{5} should be a weak TI. Another APRES work also favored a weak TI for ZrTe_{5}^{18}. However, there are two other ARPES works which believed that ZrTe_{5} is a strong TI^{19,20}. For instance, by using the comprehensive ARPES, STM, and first principles calculations, Manzoni et al. found a metallic density of state (DOS) at Fermi energy, which arises from the twodimensional surface state and thus indicates ZrTe_{5} is a strong TI^{19}.
The divergence of these experiments make ZrTe_{5} (HfTe_{5}) being a very puzzling but interesting material, which needs more further experimental and theoretical studies. Therefore, in order to figure out the physical mechanism behind those contradictory experimental results, we revisited the band structures of ZrTe_{5} and HfTe_{5}, and carefully studied their relationship with the volume expansion. We find a clear topological transition between a strong and weak TI in ZrTe_{5} and HfTe_{5}, accompanied by an intermediate Dirac semimetal state between them. This work could shed more light on a unified explain about the different experimental results, and propose the crucial clue to further experiments to elucidate the topological nature of ZrTe_{5} and HfTe_{5}.
Results
As shown in Fig. 1(a), ZrTe_{5} and HfTe_{5} share the same basecentered orthorhombic crystal structure with Cmcm (No. 63) space group symmetry. Trigonal prismatic ZrTe_{3} chains oriented along the a axis make up the ZrTe_{5} natural cleavage plane. Each chain consists of one Zr atom and two different kinds of Te atoms. ZrTe_{3} chains are connected by zigzag Te chains along the c axis, building a two dimensional structure of ZrTe_{5} in the ac plane. One crystal unit cell contains two ZrTe_{5} planes piled along the b axis, the stacking orientation of ZrTe_{5}. The Brillouin zone and high symmetry kpoints of ZrTe_{5} (HfTe_{5}) are shown in Fig. 1(b), in which a^{*}, b^{*}, and c^{*} are the reciprocal lattice vectors.
Due to the weak the van der Waals (vdw) interaction in the layer ZrTe_{5} (HfTe_{5})^{9}, the vdw corrected correlation functional is necessary in order to obtain good theoretical lattice constants. In Table 1, we present the optimized lattice constants of ZrTe_{5} and HfTe_{5} based on the optB86bvdw functional, as well as the experimental ones^{21}. We find that the theoretical and experimental lattice constants are well consistent with each other and the maximum difference between them is less than 1%. For comparison, we also optimized the structures of ZrTe_{5} and HfTe_{5} by using the standard PerdewBurkeErnzerhof (PBE) exchange correlation^{22}. It is obvious that there is a large error (about 9%) in the lattice b (in the stacking direction), which indicates that standard PBE failed to describe the structures of ZrTe_{5} andHfTe_{5}, and the vdw correction is necessary.
In order to explore the possible topological phase transition in ZrTe_{5} (HfTe_{5}), we then study their electronic properties under different volumes. Based on the above optimized structure, we change volume of the unit cell by hand and then optimize the atom positions and lattice constants under each variant volumes. This process can simulate the hydrostatic pressure experiments or the thermal expansion effect due to finite temperatures. It is noted that we did not change the volume drastically and the system is far away from the region of superconductivity phase under high pressure found in ZrTe_{5} and HfTe_{5}^{23,24,25}. In Fig. 2(a), we present the change of lattice constants a, b, and c under different volume expansion ratios defined as (V − V_{0})/V_{0} × 100%, where V_{0} is the unit cell volume at theoretical ground state listed in the Table 1. It is found that all the lattice constants have similar linear dependence on the volume of the unit cell. But the inplane lattice constants a and c changes much slower with the volume than that of the lattice constant b, which indicates the weak interlayer binding energy along the b direction in ZrTe_{5}. The paraboliclike relationship between total energy of unit cell and volume is expected and given in Fig. 2(b). The blue vertical dotted line represent the experimental volume at low temperature (10 K), which is nearly 1.9% smaller than our calculated value.
Then we have calculated the band structures and the DOSs with spinorbit coupling (SOC) under variant volumes, and three of them are shown in Fig. 3. The calculated band structure (Fig. 3(a)) at the ground state volume (ΔV = 0) is similar as previous theoretical computation^{9}, although we use a different high symmetry kpath. A clear direct band gap about 94.6 meV at Γ point is found in the band structure. Of course it is also found that the valence band maxima is between the Γ and Y point and conduction band minima is between the A_{1} and T point. Therefore the indirect band gap is much smaller than the direct one at Γ point, which is about 41.7 meV in Fig. 3(a). The present of a clear band gap is confirmed in its corresponding DOS (Fig. 3(d)). Our calculated band gap is comparable with the values observed in the previous experiments, which are 80 or 100 meV^{16,17}. We also calculated the 3D isoenergy surface (not shown here) of ZrTe_{5} in the whole Brillouin zone by the wannier functions and confirmed again that there is a global band gap in ZrTe_{5} when ΔV = 0%.
When ZrTe_{5} expands from its ground state, the band gap decreases gradually until the valence and conduction bands touch each other at a critical volume expansion ratio about 2.72%. Then, a Dirac point is formed at Γ point, which can be clearly seen in Fig. 3(b). This behavior is also confirmed in its corresponding Vshaped DOS near the Fermi energy, as shown in Fig. 3(e), which is the feature of Dirac point in band structure. It is noted that this Dirac point is 4fold degenerate since ZrTe_{5} has both the space inversion and time reversal symmetry. As the crystal continues to expand, the band gap of ZrTe_{5} opens again, and ultimately reaches a value of about 102.6 (direct) or 27.7 meV (indirect) under a volume expansion 6.12%. (see Fig. 3 (c) and (f)). This band gap is also confirmed by the 3D isoenergy surface of ZrTe_{5} in the Brillouin zone. Therefore from Fig. 3, we can clearly see a transition from a semiconductor to a semimetal and then to a semiconductor again in ZrTe_{5} when it expands. In order to check whether such a transition is topological or not, we have calculated the Z_{2} indices under each volume^{26}. It is found that the Z_{2} indices are all (1;110) when the volume expansion is less than 2.72%, while it is (0;110) when the volume expansion is larger than 2.72%. This definitively confirms that ZrTe_{5} undergoes a topological phase transition from a strong TI, to an intermediate Dirac semimetal state, and finally turns to a weak TI when its unit cell expands from 0 to 6.12% in our calculation. We noted that our calculated weak indices (110) are different from Weng’s calculation^{9} but same as Manzoni’s^{19} since the weak indices of Z_{2} depend on the choice of the unit cell^{26}.
The surface states of ZrTe_{5} in the strong and weak TI phase have also been calculated based on the wannier functions, shown in Fig. 4. The surface band structures are very similar as the ones presented in Weng’s work^{9}, since we use the similar high symmetry kpath in the surface Brillouin zone. From Fig. 4, we can see that there is a Dirac point at Γ point in top surface’s band structure for the ZrTe_{5} of ΔV = 0% while it does not for the case of ΔV = 6.12%. This key difference confirms again that ZrTe_{5} is a strong TI when ΔV = 0% and it becomes a weak TI when ΔV = 6.12%.
The detailed phase diagrams of such a topological phase transition of ZrTe_{5} and HfTe_{5} are given in Fig. 5, in which all the calculated absolute value of direct band gaps at Γ point under different volumes are plotted. In Fig. 5(a) we can find that the band gap of ZrTe_{5} decreases linearly as the volume increases from a negative volume expansion ratio about −6%, with a rate around −33 meV per 1% change of volume, where the negative value means a decrease of the band gap when the crystal expands. The band gap disappears at ΔV = 2.72%. Then it raises linearly with volume in a similar rate of 28 meV per 1% change of volume. Therefore ZrTe_{5} undergoes a topological transition from a strong TI to a weak TI due to volume expansion. Such a transition must need a zerogap intermediate state, which is the Dirac semimetal state found at about ΔV = 2.72% in our calculation. Similar phase diagram is also found recently by Manzoni et al.^{19}, in which they present the band gap at Γ point as a function of the interlayer distance, but not the volume of the unit cell. It is known that the monolayer ZrTe_{5} is a quantum spin Hall insulator^{9}. When we stack many monolayers of ZrTe_{5} into a 3D bulk ZrTe_{5} crystal, it would be a 3D strong or weak TIs which depends on the strength of coupling between the adjacent layers. From Manzoni’s^{19} and our calculation, it is obvious that the interlayer distance is the key factor that causes the transition between the strong and weak TI phases in ZrTe_{5}. In Fig. 5(b), we also show that HfTe_{5} has the very similar topological phase transition, with almost the same transition critical volume expansion ratio at about 2.72%. The band gap of HfTe_{5} also changes linearly as the volume increases with a rate about −31 and 26 meV per 1% change of volume in the strong and weak TI region respectively.
Discussion
The changing rate of our calculated band gap is quite significant especially in a small band gap semiconductor material. Therefore, we can conclude that the electronic properties of ZrTe_{5} (HfTe_{5}) are indeed very sensitive to the change of the volume and they are indeed located very close to the boundary between the strong and weak TI. Although our optimized and the experimentally measured volume of ZrTe_{5} (HfTe_{5}) both indicate that they should be within the strong TI region, we think it still has the possibility that ZrTe_{5} (HfTe_{5}) can locate in a weak TI region due to different growth methods and characterization techniques in experiments. According to Fig. 5, it is even possible that ZrTe_{5} (HfTe_{5}) can be very close to the intermediate Dirac semimetal state if it happens to have a proper volume expansion ratio, which, however, is quite challenging in experiment. Another more possible reason which can explain the semimetal behavior found in experiments is due to the defect and doping, which make the ZrTe_{5} (HfTe_{5}) being a degenerate semiconductor. In a degenerate semiconductor, the Fermi energy is located within the conduction or valence band due to the doping effect, and the crystal will behave like a metal. But in this case, the energy gap still exist just below or above the Fermi energy and the Dirac point is not needed in the energy gap. This possibility is verified in a recent experimental work by Shahi et al. They found that the resistance anomaly of ZrTe_{5}, which was observed in many existing experiments, is due to the Te deficiency, while the nearly stoichiometric ZrTe_{5} single crystal shows the normal semiconducting transport behavior^{27}. In order to avoid the possible artificial effect induced by the cleavage in both STM and ARPES experiments, we suggest that nondestructive optical measurements for the existence of a direct band gap at Γ point, and its change under different temperatures, in the high quality and stoichiometric single crystals are probably useful to elucidate the topological nature in ZrTe_{5} and HfTe_{5}.
Finally we show the importance of our calculated change rate of band gap. First we can roughly estimate the bulk thermal expansion coefficient from experimental lattice constants of ZrTe_{5} (Table 1) to be about 3.4 × 10^{−5} K^{−1}, which means that the volume will change about 1% when the temperature changes from 0 to 300 K, equivalently the band gap of ZrTe_{5} at Γ point will change about −33 meV for strong TI phase or 28 meV for weak TI phase, according to our calculation. In a recent highresolution ARPES work^{28}, Zhang et al. found a clear and dramatic temperature dependent band gap in ZrTe_{5}, from which we then can estimate that change rate of observed band gap is about 26 or 37 meV from 0 to 300 K depending on the methods used in their experiment. These two values are both well consistent with our calculated result. Moreover, the positive change rate found in the experiment^{28} implies that the ZrTe_{5} crystal used in their experiment is probably a weak TI according to our calculated phase diagram in Fig. 4.
In summary, we have studied the band structures of ZrTe_{5} and HfTe_{5} at variant volumes by first principles calculations. A clearly volume dependent strong and weak topological phase transition is found, accompanied by an intermediate Dirac semimetal state at the boundary between the transition. The direct band gap of ZrTe_{5} at Γ point changes linearly with the volume, which is −33 meV and 28 meV in a strong and weak TI phase respectively, if the volume of ZrTe_{5} increases 1%, or equivalently if the temperature increases from 0 to 300 K. The results for HfTe_{5} is very similar to those of ZrTe_{5}. Our calculated results indicate that the electronic properties and topological nature of ZrTe_{5} and HfTe_{5} are indeed very sensitive to the lattice constants of crystals, which is probably the reason for the divergent experimental results at present. We suggest that high quality and stoichiometric single crystal with accurate structure refinement at different temperatures would be helpful to resolve the divergent experimental results in ZrTe_{5} and HfTe_{5}.
Methods
The geometric and electronic properties of ZrTe_{5} and HfTe_{5} are calculated by the density functional theory in the generalized gradient approximation implemented in the Vienna Abinitio Simulation Package (VASP) code^{29,30}. The projected augmented wave method^{31,32} and the van der Waals (vdw) corrected optB86bvdw functional^{33,34} are used. The planewave cutoff energy is 300 eV and the kpoint mesh is 8 × 8 × 4 in the calculations. And a denser kpoint mesh of 24 × 24 × 12 is used in the density of state (DOS) calculation. Spinorbit coupling (SOC) is included in the calculation except for the structural optimization.
The theoretical ground states of ZrTe_{5} and HfTe_{5} are obtained by fully optimization of the atom positions and lattice constants, until the maximal residual force is less than 0.01 V/Å. Then we vary and fix the volumes of unit cell and still optimize the atom positions and lattice constants to study the possible topological transition in ZrTe_{5} and HfTe_{5}.
The maximallylocalized Wannier functions of ZrTe_{5} are fitted based on the Zr’s d and Te’s p orbitals by the Wannier90 code^{35} and then the surface states are calculated by the WannierTools^{36}.
Additional Information
How to cite this article: Fan, Z. et al. Transition between strong and weak topological insulator in ZrTe_{5} and HfTe_{5}. Sci. Rep. 7, 45667; doi: 10.1038/srep45667 (2017).
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.
Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2011).
 2.
Qi, X. L. & Zhang, S. C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2010).
 3.
Ando, Y. Topological insulator materials. J. Phys. Soc. Jap. 82, 102001–1–32 (2013).
 4.
Zhang, H. et al. Topological insulators in Bi_{2}Se_{3}, Bi_{2}Te_{3} and Sb_{2}Te_{3} with a single Dirac cone on the surface. Nat. Phys. 5, 438–442 (2009).
 5.
Chen, Y. L. et al. Experimental realization of a threedimensional topological insulator, Bi_{2}Te_{3}. Science 325, 178–181 (2009).
 6.
Jones, T. E., Fuller, W. W., Wieting, T. J. & Levy, F. Thermoelectric power of HfTe_{5} and ZrTe_{5}. Solid State Commun. 42, 793–798 (1982).
 7.
Okada, S., Sambongi, T. & Ido, M. Giant resistivity anomaly in ZrTe_{5}. J. Phys. Soc. Jap. 49, 839–840 (1980).
 8.
DiSalvo, F. J., Fleming, R. M. & Waszczak, J. V. Possible phase transition in the quasionedimensional materials ZrTe^{5} or HfTe_{5}. Phys. Rev. B 24, 2935–2939 (1981).
 9.
Weng, H., Dai, X. & Fang, Z. Transitionmetal pentatelluride ZrTe_{5} and HfTe_{5}: A paradigm for largegap quantum spin Hall insulators. Phys. Rev. X 4, 011002 (2014).
 10.
Chen, R. Y. et al. Magnetoinfrared spectroscopy of Landau levels and Zeeman splitting of threedimensional massless Dirac fermions in ZrTe_{5}. Phys. Rev. Lett. 115, 176404 (2015).
 11.
Chen, R. Y. et al. Optical spectroscopy study of the threedimensional Dirac semimetal ZrTe_{5}. Phys. Rev. B 92, 075107 (2015).
 12.
Li, Q. et al. Chiral magnetic effect in ZrTe_{5}. Nat. Phys. 12, 550–555 (2016).
 13.
Liu, Y. W. et al. Zeeman splitting and dynamical mass generation in Dirac semimetal ZrTe_{5}. Nat. Commun. 7, 12516 (2016).
 14.
Yuan, X. et al. Observation of quasitwodimensional Dirac fermions in ZrTe_{5}. arXiv 1510.00907 (2015).
 15.
Zheng, G. et al. Transport evidence for the threedimensional Dirac semimetal phase in ZrTe_{5}. Phys. Rev. B 93, 115414 (2016).
 16.
Li, X. B. et al. Experimental Observation of Topological Edge States at the Surface Step Edge of the Topological Insulator ZrTe_{5}. Phys. Rev. Lett. 116, 176803 (2016).
 17.
Wu, R. et al. Evidence for Topological Edge States in a Large Energy Gap near the Step Edges on the Surface of ZrTe_{5}. Phys. Rev. X 6, 021017 (2016).
 18.
Moreschini, L. et al. Nature and topology of the lowenergy states in ZrTe_{5}. Phys. Rev. B 94, 081101 (2016).
 19.
Manzoni, G. et al. Evidence for a Strong Topological Insulator Phase in ZrTe_{5}. Phys. Rev. Lett. 117, 237601 (2016).
 20.
Manzoni, G. et al. Temperature dependent nonmonotonic bands shift in ZrTe_{5}. J. Electron Spectrosc. Relat. Phenom. in press (2016).
 21.
Fjellvag, H. & Kjekshus, A. Structural properties of ZrTe_{5} and HfTe_{5} as seen by powder diffraction. Solid State Commun. 60, 91–93 (1986).
 22.
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
 23.
Zhou, Y. et al. Pressureinduced superconductivity in a threedimensional topological material ZrTe_{5}. PNAS 113, 2904–2909 (2016).
 24.
Qi, Y. et al. Pressuredriven Superconductivity in Transitionmetal Pentatelluride HfTe_{5}. Phys. Rev. B 94, 054517 (2016).
 25.
Liu, Y. et al. Superconductivity in HfTe_{5} Induced via Pressures. arXiv 1603.00514 (2016).
 26.
Soluyanov, A. A. & Vanderbilt, D. Computing topological invariants without inversion symmetry. Phys. Rev. B 83, 235401 (2011).
 27.
Shahi, P. et al. Bipolar Conduction is the Origin of the Electronic Transition in Pentatellurides: Metallic vs. Semiconducting Behavior. arXiv 1611.06370 (2016).
 28.
Zhang, Y. et al. Electronic Evidence of TemperatureInduced Lifshitz Transition and Topological Nature in ZrTe_{5}. arXiv 1602.03576 (2016).
 29.
Kresse, G. & Hafner, J. Ab initio molecular dynamics for openshell transition metals. Phys. Rev. B 48, 13115–13118 (1993).
 30.
Kresse, G. & Furthmüller, J. Efficiency of abinitio total energy calculations for metals and semiconductors using a planewave basis set. Comput. Mater. Sci. 6, 15–50 (1996).
 31.
Blöchl, P. Projector augmentedwave method. Phys. Rev. B 50, 17953–17979 (1994).
 32.
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 59, 1758–1775 (1999).
 33.
Klimes, J., Bowler, D. R. & Michaelides, A. Chemical accuracy for the van der Waals density functional. J. Phys.: Condens. Matter. 22, 022201 (2009).
 34.
Klimes, J., Bowler, D. R. & Michaelides, A. Van der Waals density functionals applied to solids. Phys. Rev. B 83, 195131 (2011).
 35.
Mostofi, A. A. et al. An updated version of wannier90: A tool for obtaining maximallylocalised Wannier functions. Comput. Phys. Commun. 185, 2309 (2014).
 36.
Wu, Q. S. & Zhang, S. N. WannierTools: An opensource software package for novel topological materials. https://github.com/quanshengwu/wannier_tools.
Acknowledgements
This work is supported by the National Key R&D Program of China (2016YFA0201104), the State Key Program for Basic Research (No. 2015CB659400), and the National Science Foundation of China (Nos 91622122, 11474150 and 11574215). The use of the computational resources in the High Performance Computing Center of Nanjing University for this work is also acknowledged.
Author information
Affiliations
National Laboratory of Solid State Microstructures and Department of Materials Science and Engineering, Nanjing University, Nanjing 210093, China
 Zongjian Fan
 , ShuHua Yao
 & Jian Zhou
Department of Physics, Shaoxing University, Shaoxing 312000, China
 QiFeng Liang
National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
 Y. B. Chen
Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China
 Jian Zhou
Authors
Search for Zongjian Fan in:
Search for QiFeng Liang in:
Search for Y. B. Chen in:
Search for ShuHua Yao in:
Search for Jian Zhou in:
Contributions
Y.B.C., S.H.Y., and J.Z. proposed the idea. Z.J.F., Q.F.L. and J.Z. carried out the calculations, analysed the results, and plotted the figures. Z.J.F. and J.Z. wrote the mannuscript. All authors reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Jian Zhou.
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Further reading

Magnetotransport Properties of Layered Topological Material ZrTe2 Thin Film
ACS Nano (2019)

Turning ZrTe5 into a semiconductor through atom intercalation
Science China Physics, Mechanics & Astronomy (2019)

TwoDimensional Conical Dispersion in ZrTe5 Evidenced by Optical Spectroscopy
Physical Review Letters (2019)

From Dirac Semimetals to Topological Phases in Three Dimensions: A CoupledWire Construction
Physical Review X (2019)

Evidence for a straintuned topological phase transition in ZrTe5
Science Advances (2019)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.