Pressure-Driven Enhancement of Topological Insulating State in Tin

Mar 28, 2013 - College of Materials Science and Engineering & State Key Laboratory ... Department of Physics and HiPSEC, University of Nevada, Las Veg...
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Pressure-Driven Enhancement of Topological Insulating State in Tin Telluride Dan Zhou,†,‡ Quan Li,*,†,‡ Yanming Ma,† Qiliang Cui,† and Changfeng Chen*,‡ †

College of Materials Science and Engineering & State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012, China ‡ Department of Physics and HiPSEC, University of Nevada, Las Vegas, Nevada 89154, United States

ABSTRACT: Recent discovery of a distinct type of topological insulating state in tin telluride (SnTe) has reignited great interest in this narrow bandgap semiconductor. A pressing task is to understand the stability of this intriguing state of matter and the response of its electronic properties under the influence of external conditions that may alter the underlying fundamental physics. In this work, we examine by first-principles calculations the effect of pressure on the phonon dispersion, electronic band structure, Fermi surface, and charge distribution of SnTe. We show that pressure suppresses a soft optical phonon mode and enhances the stability of the cubic (B1, Fm3̅m) phase of SnTe, which is a key requirement for the observed topological insulating state. Pressure also drives an electronic topological transition that alters the shape and connectivity of the Fermi surface, and it also promotes a charge redistribution that results in an enhanced bonding interaction that strengthens the cubic crystalline structure. Our calculations further demonstrate that the electronic band gap of the cubic phase, another key parameter of the topological insulating state in SnTe, increases in size monotonically with increasing pressure. These results indicate that pressure makes the topological insulating state in the cubic phase of SnTe more stable and robust; pressure is also effective in tuning the electronic band structure, which is expected to have significant impact on a wide range of physical properties crucial to potential applications. insulator phase in SnTe.12 Most recently, the topological crystalline insulator phase and topological phase transition have also been observed in Pb1‑xSnxTe by ARPES measurements.13 Following these exciting discoveries, a pressing task is to explore the behavior of the recently identified topological insulating state and characterize its response to external physical conditions that may alter the underlying fundamental physics. Of particular interest are the possible changes in the stability of its cubic crystalline phase and the nature and size of its electronic band gap, which are essential ingredients for maintaining the topological insulating state in SnTe.10,11 Tin telluride is a direct narrow band gap semiconductor with a gap of 0.18 eV (0.30 eV) at room (4.2 K) temperature.14 It has been extensively studied over the past several decades, and

1. INTRODUCTION Topological insulators are intriguing states of quantum matter characterized by an insulating gap in the bulk and conducting gapless edges or surface states in the boundaries.1−3 Their discovery has generated great interest because of the scientific importance of the observed phenomena and promising potential of these materials for high-temperature spintronics applications.4−9 Search for additional members of this class of materials has been continuing. Recent theoretical work identified tin telluride (SnTe) as a distinct type of topological crystalline insulator, in which the metallic surface states are protected by the mirror symmetry of the crystal, in contrast to the time-reversal symmetry protection in the earlier identified Z2 topological insulators.10,11 Subsequently, angle-resolved photoemission spectra (ARPES) detected surface electronic states consisting of four Dirac cones (even number of so-called band-inversion points) in the first surface Brillouin zone, providing experimental evidence for the topological crystalline © 2013 American Chemical Society

Received: February 25, 2013 Revised: March 26, 2013 Published: March 28, 2013 8437

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Perdew−Burke−Ernzerh (PBE) generalized gradient approximation (GGA) functional28 and the Heyd−Scuseria−Ernzerhof (HSE) hybrid functional29,30 with spin−orbit interaction as the exchange-correlation potential were applied to investigate the electronic structures. The projector augmented wave (PAW) potentials31 with 5s25p2 and 5s25p4 electrons as valence were adopted for Sn and Te, respectively. An energy cutoff of 300 eV and 16 × 16 × 16 Monkhorst-Pack k meshes were chosen to ensure that enthalpy calculations were well converged to better than 1 meV/atom. The phonon frequencies were calculated using a direct supercell method (4 × 4 × 4, 128 atoms supercell), which uses the forces obtained by the Hellmann−Feynman theorem.32 Bader charge analysis33 was used to analyze the charge state. Note that the HSE calculation is only used to correct the band energies in assessment of the band gap, but it is not used in the calculation of electronic density of states, Fermi surfaces, phonon, and Bader charge analysis, which are less sensitive to the choice of the functionals.

it has remained a topic of great interest because of its fundamental physics and potential applications in electronic devices. An unusual feature of the electronic structure of SnTe is that the ordering of its conduction and valence band near the Fermi energy is inverted relative to a normal semiconductor like PbTe whose electronic band structure smoothly connects to the atomic limit.10 The so-called “negative band gap” (or band inversion) in SnTe occurs between a valence band maximum near the L points in the Brillouin zone with L6− symmetry and the conduction band minimum with L6+ symmetry, which are opposite to those in PbTe.15−22 A transition from the normal band structure in PbTe to that of SnTe with the nontrivial inverted band gap has been illustrated by examining the band gap evolution in Pb1‑xSnxTe as a function of alloy composition,23 which shows that the band gap of the alloy initially decreases with increasing value of x, leading to its full closure, and then reopens and increases in the inverted direction with further rising x. Moreover, ARPES and ab initio calculations revealed complex Fermi surface structure near the L points, showing topological changes in the constantenergy surface from disconnected pockets, to open tubes, and then to cuboids as the binding energy (or hole-doping) increases.24 The narrow band gap of SnTe is very sensitive to changes of external conditions such as pressure, doping, or temperature. Given the essential role of the inverted band gap in the topological insulating state of SnTe,10,11 it is crucial that we establish an understanding of its behavior under changing external conditions. While the response of the electronic properties of SnTe to changing temperature and hole-doping has been extensively studied,10,14,24−26 the effect of pressure on the structural stability of the cubic phase of SnTe and its electronic properties remains largely unexplored. In this article, we present a systematic first-principles study of the influence of applied pressure on the structural and electronic properties of the topological insulating state that exists in the cubic phase of SnTe. We have calculated the phonon dispersion curve to examine the dynamic stability of the cubic SnTe structure under pressure, and our results indicate that pressure suppresses a soft optical phonon associated with a structural instability, thus strengthening the cubic phase. Our electronic band structure calculations reveal a significant pressure effect on the topological structure of the Fermi surface and its impact on the charge redistribution, which, in turn, strengthens the bonding and structural stability of the cubic phase. The band structure calculations also show that the electronic band gap of SnTe increases in size considerably with applied pressure, making the topological insulating state more robust under pressure. These results show that pressure stabilizes and enhances the topological crystalline insulator state of SnTe and effectively tunes its electronic structure and Fermi surface, which are expected to have significant impact on the transport and optical properties crucial for its applications. In the following sections, we will first provide some details of our computational methods and procedure, and then present the results and discussion on the calculated structural and electronic properties of SnTe under changing pressure. We summarize our main conclusions at the end of the article.

3. RESULTS AND DISCUSSION With the consideration of spin−orbit coupling, the theoretical equilibrium lattice parameters calculated by the standard PBEGGA functional and HSE hybrid functional for SnTe are 6.404 Å and 6.356 Å, which are 1.3% and 0.5% larger than the experimental data of 6.327 Å,34 respectively. To investigate the effect of pressure on the dynamic structural stability of SnTe, we have calculated phonon dispersion relationships at various pressure points. From our recent study of structural evolution of SnTe at high pressure using an integrated approach of angle dispersive synchrotron X-ray diffraction combined with a firstprinciples structural search method, SnTe undergoes pressuredriven phase transitions from the ambient-pressure Fm3̅m structure to three coexisting intermediate phases of Pnma, Cmcm, and a GeS type structure at 5.0 GPa.35 Therefore, the upper limit of pressure in this study was chosen as 5.0 GPa. The obtained results shown in Figure 1 indicate that the cubic (Fm3̅m) structure is unstable with imaginary transverse optical phonon frequencies near the zone center (Γ point). This result is in agreement with the previously reported Raman scattering experiment.25 This structural instability can be understood in terms of the strong coupling between the interband electronic excitation and the transverse optical phonon.36 Because of the softening phonon mode, the Fm3m ̅ structure transforms to a rhombohedral structure with a space group of R3m driven by a small dimerization in the unit cell. Our results show that rising pressure leads to monotonically increasing frequencies of the soft optical phonon mode (Figure 1a). The squared phonon frequency of the optical branch at the Γ point exhibits an almost linearly increasing relationship with pressure (Figure 1b), and the cubic phase is dynamically stable above 0.5 GPa. The current lattice dynamics results demonstrate that rising pressure is able to stabilize the cubic phase of SnTe, which is consistent with the experimental results.25,26 We now turn to the electronic band structure and examine the influence of pressure, especially its effect on the band gap, which is a crucial physical parameter for maintaining the topological insulating state in SnTe and for its applications. We have performed the band structure calculations using the standard PBE-GGA and the HSE hybrid functional. In heavy atoms, relativistic shifts of the electron energy levels lead to a strong spin−orbit coupling. The band structures of SnTe and PbTe have been previously examined by electronic band

2. COMPUTATIONAL DETAILS We have carried out first-principles energetic and electronic band structure calculations using the density functional theory as implemented in the VASP code.27 Both the standard 8438

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Figure 1. (a) Calculated phonon dispersion relationship of the cubic structure of SnTe at different pressures. The inset shows the first Brillouin zone with selected high-symmetry points indicated. (b) The calculated squared phonon frequency ω2 of the optical branch at the Γ point as a function of pressure.

Figure 2. Band structures of SnTe at (a) 0 GPa with the lattice parameter determined by using the PBE or HSE functionals, (b) no applied pressure, but using the experimental lattice parameter of a = 6.327 Å, and (c) at 5 GPa using the lattice parameter by using the PBE or HSE functionals. (d) The calculated band gap of SnTe as a function of pressure, where results at pressures corresponding to lattice parameters below the experimental value are represented by the dashed lines.

structure calculations with and without the consideration of spin−orbit coupling.15,22,24,37,38 These calculations identify significant effects of the spin−orbit coupling on the band structures for SnTe and PbTe, e.g., splitting in the degenerate bands along high-symmetry directions and modifying the band gaps.15,22,24,37,38 It is therefore necessary to consider the spin− orbit coupling explicitly to describe the electronic properties of SnTe. The calculated results are shown in Figure 2. The results of our calculations using the standard PBE-GGA functional without applied pressure (Figure 2a) are in good agreement with recently reported data.10,23 In SnTe, the L6+ state is mainly derived from Te and the L6− state from Sn. The occupation in the conduction band minimum (L6+ state) and valence band maximum (L6− state) in SnTe is switched (i.e., inverted) near the L points relative to those in PbTe, and this peculiar behavior stems from the different relativistic shifts of the valence states between Pb and Sn, e.g., 2.75 eV for the s state and 0.73 eV for the p state.16 The PBE-GGA calculations yield a small direct band gap of 0.112 eV at the L points with the optimized lattice parameters, indicating its narrow band gap semiconductor nature. This band gap by PEB-GGA is smaller than the experimental value of 0.18 eV measured at room temperature.14 It should be mentioned that the experimental band gap of 0.30 eV14 at the low temperature of 4.2 K corresponds to the rhombohedral phase and, therefore, cannot be directly compared to the calculated result obtained for the cubic phase. As the GGA usually underestimates electronic band gap, we have performed the calculations using a hybrid functional where 25% of the GGA exchange potential is replaced by screened Fock exchange. This type of HSE hybrid functional29,30 usually provides a better description of the electronic band structure, especially the band gap.39−41 Intriguingly, our calculations show that the band gap obtained using the HSE hybrid functional is reduced to 0.004 eV, which is smaller than that using the standard PBE-GGA. This downward shift of the HSE band gap is attributed to the

unusual inverted band gap of SnTe. In a normal semiconductor, HSE tends to move the bottom of the conduction band upward while moving the top of the valence band downward, thus further opening up the band gap and correcting the underestimation by the GGA calculations. In SnTe, the bottom of the conduction band and top of the valence band states are switched; the HSE calculations still have the same effect on these states, resulting in the lowering of the bottom of the conduction band and rising of the valence band. This reversed process leads to the significantly reduced band gap as obtained in our calculations. It should be noted, however, that the theoretical lattice parameters obtained in both the GGA and HSE calculations are larger than the experimental value obtained at room temperature. Using the experimental lattice parameter of 6.327 Å, our calculations produced the electronic band gap of 0.173 and 0.094 eV for SnTe using the PBE-GGA and HSE functional, respectively. While the GGA band gap is in better agreement with the experimental value of 0.18 eV, one should exercise caution in drawing conclusions about the accuracy of the GGA and HSE functionals applied to SnTe since the reported calculations do not explicitly consider the temperature effect (i.e., thermal vibration of atomic positions) that may have a significant influence on the band gap.42 It is important that both functionals produce the same trend for the response of the band gap to pressure as we discuss below. Our calculated results presented in Figure 2 show that the inverted band gap of SnTe is sensitive to the change of external pressure. As the pressure increases, both the GGA and HSE calculations predict that the size of the inverted band gap will have a positive pressure dependence. This indicates that the topological crystalline insulator state in SnTe will be enhanced by pressure. Note also that the results using the HSE and PBE8439

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GGA are in fair agreement above the pressures corresponding to the experimental lattice parameter. With increasing hydrostatic pressure (decreasing volume), the band structures, especially those near the Fermi energy, expand due to the expansion of the bandwidth by pressure. Because of the symmetry requirements, a crossing of the inverted conduction minimum L6+c and valence maximum L6−v is forbidden, and consequently, they will more strongly repel each other under pressure. The repulsion between the states of the same symmetry, i.e., the conduction minimum L6+c and the valence maximum L6+v will increase with increasing pressure, while the conduction band states L6+c and L6−c have different symmetries and can cross at high enough pressure. Our calculations show that the L6+c states at the L points are significantly pushed up under pressure (results at 5 GPa are shown in Figure 2c). Because of the effect of increased repulsion under high pressure, the shape of the conduction band minimum near the L points changes to that of an inverted saddle, and the conduction band maximum is moved slightly away from the L points. Recent ARPES experiments and density functional theory calculations revealed complex Fermi surface evolution in SnTe with increasing hole doping.24 A key finding is an electronic topological transition, also known as Lifshitz transition, as the binding energy increases, which corresponds to a rising level of hole doping. This result has important implications for understanding the transport and optical properties of SnTe, and there is also empirical evidence26 that doping tends to stabilize the cubic phase of SnTe. Here, we examine the evolution of the electronic structure of SnTe near the Fermi energy under pressure. In Figure 3, we plot constant energy surfaces of the cubic (Fm3̅m) SnTe at different energies by shifting the chemical potential downward, which correspond to Fermi surfaces at different hole (p-)doped concentrations, and compare the results taken at 0 and 5 GPa. With increasing downward energy shift (or higher doping concentration), the

calculated energy surfaces evolve from pockets (at Ef −0.3 eV) to interconnected quasicubic tubes (at Ef −0.5 eV) with cube corners at the L points of the first Brillouin Zone, signifying the occurrence a doping-induced electronic topological transition at zero pressure. This result is in agreement with the previous experimental photoemission spectroscopy and ab initio studies24 on the band structure of SnTe. Our calculations reveal that pressure driven evolution of the band structure promotes the electronic topological transition in SnTe. Results in Figure 3 clearly show that at higher pressure (5 GPa) the transition of the constant energy surfaces from disconnected pockets near the L points to interconnected quasicubic tubes occurs at energies closer to the Fermi energy, which corresponds to a lower level of hole doping. This indicates a stronger tendency of the system toward the electronic topological transition under pressure, which accelerates the process at an energy (e.g., Ef −0.3 eV) where doping alone would not have caused the transition. Pressure has an advantage in that it can be applied and released via external control, allowing a reversible modification of the electronic structure and the topology (i.e., connectivity) of the constant-energy surface, which, in turn, can lead to precise and reversible tuning of a wide range of physical properties, such as resistivity, thermoelectric power, Hall coefficient, phonon dispersion, and electron−phonon coupling.43−48 For further insights into the role of pressure in stabilizing the cubic structure of SnTe, we examine the pressure driven charge redistribution in real space associated with the electronic topological transition that occurs in the momentum space. We show in Figure 4 the band decomposed charge density projected onto the (100) plane of the crystal structure. The results indicate a peculiar reverse electron donation from tellurium to tin atoms. We employ the Bader charge analysis, which provides a description of electron transfer, to quantify the amount of charge belonging to each atom at different pressure points. The calculated charge amounts are 6.6310

Figure 3. Evolution of the three-dimensional constant-energy surfaces at different binding energies, corresponding to the Fermi surfaces at different hole(p-) doped levels, at 0 GPa (upper panels) and 5 GPa (lower panels).

Figure 4. Electronic charge density distribution projected onto the (100) plane for the states in an energy window of 0.30 eV below the Fermi energy at 0 GPa (upper panel) and 5 GPa (lower panel). The charge density is measured in electrons per Å3. 8440

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gap of the cubic phase increases in size with rising pressure. Pressure also induces an electronic topological transition characterized by a change in the connectivity of the constant energy surface in the momentum space, which directly impacts transport and optical properties. These results show that pressure has a strong influence on the fundamental crystal and electronic structure of SnTe, and as a result, it makes the topological insulating state in SnTe more stable and robust. The sensitive pressure tuning of the electronic band gap and Fermi surface topology offers an effective tool to modulate a wide range of physical properties of SnTe that are important to its potential applications.

(3.3690), 6.6231 (3.3769), 6.6156 (3.3844), 6.6087 (3.3913), 6.6021 (3.3979), and 6.5957 (3.4043) electron for tellurium (tin) at 0, 1, 2, 3, 4, and 5 GPa, respectively. This result reveals that the charge states of tellurium and tin atoms change with increasing pressure. The tellurium (tin) atoms lose (gain) electron charge at an average rate of 7.0 × 10−3 e/GPa. It is noted that while SnTe crystallizes naturally in the cubic phase at ambient conditions, it is known to be in a nonstoichiometric state with a deficiency of tin atoms, resulting in an excess of holes in the material.20 Experimental results suggest that the excessive holes in SnTe tend to stabilize its cubic structure.26 From our calculated charge density results, one can see that at ambient conditions the electrons near the valence band edge are mainly contributed from the tellurium atoms (see Figure 4, where valence charge density within 0.3 eV below the Fermi energy is shown). Under pressure, the charges near the valence band maximum form an interconnected quasicubic filament in the momentum space (see Figure 3); correspondingly, in real space, these charges redistribute among the cubic lattice sites, which is expected to impact the bonding interaction and structural stability. Our phonon calculations reveal a persistent hardening of all the phonon branches of the cubic SnTe phase under increasing pressure (see Figure 1a). This clearly indicates a strengthening of the bonding interaction in the cubic structure in response to the pressure driven charge redistribution. Previous studies indicate that a carrier density of about 5 × 1020 cm−3 or 3.2 × 10−2 holes per SnTe formula unit under atmospheric pressure can suppress the transition from cubic to rhombohedral structure at low temperature.26 From our phonon calculations, the pressure to stabilize the cubic (Fm3̅m) phase of SnTe is 0.5 GPa. At this pressure, our charge analysis indicates a loss of about 3.5 × 10−3 e from tellurium per formula unit. This suggests that pressure driven charge density change is very effective in stabilizing the cubic structure of SnTe. It should be noted, however, that the hole doping by tin deficiency is likely distributed locally and randomly in the crystal structure; in contrast, our calculations study charge redistribution that occurs in an ideal crystal lattice. Consequently, it is not meaningful to make a direct comparison of the experimentally determined hole doping by tin deficiency and the theoretically calculated charge transfer, which is a net loss of electrons that can be viewed as an effective hole doping on the tellurium sites. Nevertheless, it is important that the empirical experimental evidence and our calculated results consistently point to the same overall trend of structural (cubic phase) stabilization by charge redistribution in SnTe either introduced by structural defects or driven by applied pressure.

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AUTHOR INFORMATION

Corresponding Author

*(Q.L.) Tel: +1-702-895-1714. Fax: +1-702-895-0804. E-mail: [email protected]. (C.C.) Tel: +1-702-895-4230. Fax: +1-702-895-0804. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Department of Energy through Cooperative Agreement DE-FC52-06NA26274 at UNLV and by NSFC (No. 11025418, 91022029, 51202084, 11074089, and 51172087) at JLU.

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REFERENCES

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4. CONCLUSIONS In summary, we have performed first-principles electronic and phonon calculations to examine the phase stability and electronic structure evolution in a topological crystalline insulator SnTe under pressure. Our phonon calculations show that pressure suppresses a soft optical phonon mode and stabilizes the cubic (Fm3̅m) phase of SnTe, which is predicted to be dynamically stable above 0.5 GPa in the absence of any carrier (hole) doping. Increasing pressure promotes a peculiar charge transfer in the form of a backdonation of electron from the tellurium to tin sites, which further strengthens the bonding interaction of the cubic structure as evidenced by the hardening of all the phonon branches under pressure. The electronic band structure of SnTe also responds sensitively to applied pressure. The inverted band 8441

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