news & views TOPOLOGICAL INSULATORS
Topology and structural distortions Lattice distortions can be used to manipulate surface states in topological crystalline insulators. This discovery suggests new methods to control the motion of electrons in 2D electron systems.
Kai Sun
T
opological crystalline insulators are close relatives of topological insulators. Both materials are insulating in the bulk and have conductive states on their surface that are due to the special topological character of the bulk electron wavefunctions. However, in contrast to a topological insulator, whose key properties rely on time-reversal symmetry, the behaviour of a topological crystalline insulator depends on crystalline symmetries. Writing in Nature Materials, Vidya Madhavan and colleagues now provide insights into the role of crystalline symmetry and symmetry-breaking distortions in the determination of the properties of surface electrons in topological crystalline insulators1. In the bulk of topological crystalline insulators an energy bandgap separates the allowed electronic states, as expected for insulating materials. The surface of the material, however, has a distinct behaviour: points in momentum space exist at which the energy bandgap closes. The energymomentum dispersion of the electrons in the vicinity of these points assumes a conical shape called a Dirac cone. According to Dirac theory, electrons in this condition behave as massless particles free to move on the surface, thereby becoming conducting. In topological crystalline insulators, this surface behaviour relies on the mirror symmetry of these materials2. One example of such mirror symmetry is shown in Fig. 1a: if we introduce a mirror parallel to the red or green diagonal lines, the structure of the crystal coincides with its mirror image. Breaking such mirror symmetry, for example by distorting the lattice, can change the fate of the surface electrons giving them a finite mass — a Dirac mass — and can introduce an energy gap at the Dirac cone. Experimentally, the observation of the mass generation caused by the mirror symmetry breaking is a direct evidence for a topological crystalline insulator. Using scanning tunnelling microscopy (STM) Madhavan and co-workers studied the topological crystalline insulator Pb1–xSnxSe and observed on its surface two
coexisting types of Dirac cone; gapped (with electrons having a Dirac mass) and gapless (with massless electrons). This result was interpreted by assuming that a particular mirror symmetry of the lattice was broken, and a possible structure distortion pattern was proposed in which the Sn–Pb and Se sublattices move in opposite directions3 (Fig. 1b). In this study, the researchers were able to directly confirm this distortion using real-space imaging. In contrast to conventional low-bias STM imaging that detects a single atomic species per scan, Madhavan and co-workers optimized the scanning condition to simultaneously observe the relative positions of the Se, Sn and Pb atoms. Figure 1c shows an example of the structural distortion they observed in Pb1–xSnxSe, which is in agreement with that predicted in their previous work3. It is also critical to understand whether the whole 3D crystal or only the atoms near the 2D surface get distorted. In fact, although both types of distortion break the mirror symmetry and can lead to surface electrons with finite mass, they entail different fundamental properties such as, for example, the response to lattice defects or thermal fluctuations. Madhavan and co-workers report evidence supporting the idea of distortions taking place only on the 2D surface of the Pb1–xSnxSe crystal. The first proof comes from X-ray scattering, a a
b
technique sensitive to the bulk properties of Pb1–xSnxSe. The scattering spectra show the crystalline structure of Pb1–xSnxSe and suggest that distortions in the bulk are absent. Furthermore, the researchers varied the Sn content to push the system towards the quantum phase transition point — the point at which the material becomes a conventional insulator — and found that the lattice distortion remains unchanged but that the mass of the surface electrons reduces to zero. This observation is fully consistent with the picture of a surface distortion. In fact, because of their quantum nature the surface electrons partially penetrate in the bulk, as expected from the uncertainty principle. Near the quantum phase transition point, the electrons penetrate deeper into the bulk and become less sensitive to the surface condition. If the distortion is localized on the surface, its effect on the mass of the electrons becomes negligible at the transition point, which is in agreement with the researchers’ observations. This behaviour is not expected if the dominant contribution to the mass is from bulk distortions. For future studies, additional confirmation of the 2D nature of these lattice distortions could be obtained by studying the contribution of lattice defects or disorder. For a surface lattice distortion, infinitesimal amount of local c Pb Se
Pb Se
Figure 1 | Lattice distortions in topological crystalline insulators. a, Schematic of the PbxSn1–xSe crystal. Orange and yellow spheres show the position of Pb (or Sn) and Se atoms, respectively. The green and red lines represent crystallographic planes showing mirror symmetry. b, Distorted lattice obtained by a displacement of the atoms (red arrows). The mirror symmetry with respect to the red plane is broken. c, STM topography image (6.1 Å × 6.1 Å) of the distorted PbxSn1–xSe crystal. Figures adapted from ref. 1, Nature Publishing Group.
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2015 Macmillan Publishers Limited. All rights reserved
1
news & views inhomogeneity or randomness will destroy the long-range coherence of the ordering and will lead to the formation of coherent domains with finite size, analogous to 2D Ising ferromagnetism in the presence of a random magnetic field4. This phenomenon has been seen in the study of different (but closely related) distortion patterns that break the rotational symmetry in 2D or quasi-2D materials such as 2D electron gases and high-temperature superconductors5. For distortions in the 3D bulk, however, the long-range coherence is expected to remain stable unless the randomness reaches a critical threshold. As a result, a careful study of the dependence of long-range coherence on the amount of lattice defects in these topological crystalline insulators may offer additional clues on whether these distortions are 2D or 3D in nature. As a further development, more insight on the dispersion
2
relation of the surface electrons could be achieved by using other techniques such as angle-resolved photoemission spectroscopy (ARPES) and quasiparticle scattering. In fact, although the energy scale of the Dirac mass (~20 meV) and the momentum splitting between two Dirac points are both small, they are within the resolution of the state-of-the-art ARPES tools. Topological states of matter present exotic properties that are impossible in conventional materials. Madhavan and colleagues show, on the one hand, a way to control the Dirac mass generation in surface topological states. On the other, they suggest new ways to control the energy gap of a 2D electronic system by tuning lattice distortions, for example by using strain or pressure as a control parameter. The capability of tuning the fundamental properties of topological states of matter
could be a step towards future technologies such as topological transistors and even topological quantum computation6. ❐ Kai Sun is in the Department of Physics, University of Michigan, 450 Church Street, Ann Arbor, Michigan 48109, USA. e-mail: [email protected] References
1. Zeljkovic, I. et al. Nature Mater. http://dx.doi.org/10.1038/ nmat4215 (2015). 2. Fu, L. Phys. Rev. Lett. 106, 106802 (2011). 3. Okada, Y. et al. Science 341, 1496–1499 (2013). 4. Cardy, J. Scaling and Renormalization in Statistical Physics (Cambridge Univ. Press, 1996). 5. Fradkin, E., Kivelson, S. A., Lawler, M. J., Eisenstein, J. P. & Mackenzie, A. P. Annu. Rev. Condens. Matter Phys. 1, 153–178 (2010). 6. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Rev. Mod. Phys. 80, 1083–1159 (2008).
Published online: 16 February 2015
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2015 Macmillan Publishers Limited. All rights reserved
Topology and structural distortions Lattice distortions can be used to manipulate surface states in topological crystalline insulators. This discovery suggests new methods to control the motion of electrons in 2D electron systems.
Kai Sun
T
opological crystalline insulators are close relatives of topological insulators. Both materials are insulating in the bulk and have conductive states on their surface that are due to the special topological character of the bulk electron wavefunctions. However, in contrast to a topological insulator, whose key properties rely on time-reversal symmetry, the behaviour of a topological crystalline insulator depends on crystalline symmetries. Writing in Nature Materials, Vidya Madhavan and colleagues now provide insights into the role of crystalline symmetry and symmetry-breaking distortions in the determination of the properties of surface electrons in topological crystalline insulators1. In the bulk of topological crystalline insulators an energy bandgap separates the allowed electronic states, as expected for insulating materials. The surface of the material, however, has a distinct behaviour: points in momentum space exist at which the energy bandgap closes. The energymomentum dispersion of the electrons in the vicinity of these points assumes a conical shape called a Dirac cone. According to Dirac theory, electrons in this condition behave as massless particles free to move on the surface, thereby becoming conducting. In topological crystalline insulators, this surface behaviour relies on the mirror symmetry of these materials2. One example of such mirror symmetry is shown in Fig. 1a: if we introduce a mirror parallel to the red or green diagonal lines, the structure of the crystal coincides with its mirror image. Breaking such mirror symmetry, for example by distorting the lattice, can change the fate of the surface electrons giving them a finite mass — a Dirac mass — and can introduce an energy gap at the Dirac cone. Experimentally, the observation of the mass generation caused by the mirror symmetry breaking is a direct evidence for a topological crystalline insulator. Using scanning tunnelling microscopy (STM) Madhavan and co-workers studied the topological crystalline insulator Pb1–xSnxSe and observed on its surface two
coexisting types of Dirac cone; gapped (with electrons having a Dirac mass) and gapless (with massless electrons). This result was interpreted by assuming that a particular mirror symmetry of the lattice was broken, and a possible structure distortion pattern was proposed in which the Sn–Pb and Se sublattices move in opposite directions3 (Fig. 1b). In this study, the researchers were able to directly confirm this distortion using real-space imaging. In contrast to conventional low-bias STM imaging that detects a single atomic species per scan, Madhavan and co-workers optimized the scanning condition to simultaneously observe the relative positions of the Se, Sn and Pb atoms. Figure 1c shows an example of the structural distortion they observed in Pb1–xSnxSe, which is in agreement with that predicted in their previous work3. It is also critical to understand whether the whole 3D crystal or only the atoms near the 2D surface get distorted. In fact, although both types of distortion break the mirror symmetry and can lead to surface electrons with finite mass, they entail different fundamental properties such as, for example, the response to lattice defects or thermal fluctuations. Madhavan and co-workers report evidence supporting the idea of distortions taking place only on the 2D surface of the Pb1–xSnxSe crystal. The first proof comes from X-ray scattering, a a
b
technique sensitive to the bulk properties of Pb1–xSnxSe. The scattering spectra show the crystalline structure of Pb1–xSnxSe and suggest that distortions in the bulk are absent. Furthermore, the researchers varied the Sn content to push the system towards the quantum phase transition point — the point at which the material becomes a conventional insulator — and found that the lattice distortion remains unchanged but that the mass of the surface electrons reduces to zero. This observation is fully consistent with the picture of a surface distortion. In fact, because of their quantum nature the surface electrons partially penetrate in the bulk, as expected from the uncertainty principle. Near the quantum phase transition point, the electrons penetrate deeper into the bulk and become less sensitive to the surface condition. If the distortion is localized on the surface, its effect on the mass of the electrons becomes negligible at the transition point, which is in agreement with the researchers’ observations. This behaviour is not expected if the dominant contribution to the mass is from bulk distortions. For future studies, additional confirmation of the 2D nature of these lattice distortions could be obtained by studying the contribution of lattice defects or disorder. For a surface lattice distortion, infinitesimal amount of local c Pb Se
Pb Se
Figure 1 | Lattice distortions in topological crystalline insulators. a, Schematic of the PbxSn1–xSe crystal. Orange and yellow spheres show the position of Pb (or Sn) and Se atoms, respectively. The green and red lines represent crystallographic planes showing mirror symmetry. b, Distorted lattice obtained by a displacement of the atoms (red arrows). The mirror symmetry with respect to the red plane is broken. c, STM topography image (6.1 Å × 6.1 Å) of the distorted PbxSn1–xSe crystal. Figures adapted from ref. 1, Nature Publishing Group.
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2015 Macmillan Publishers Limited. All rights reserved
1
news & views inhomogeneity or randomness will destroy the long-range coherence of the ordering and will lead to the formation of coherent domains with finite size, analogous to 2D Ising ferromagnetism in the presence of a random magnetic field4. This phenomenon has been seen in the study of different (but closely related) distortion patterns that break the rotational symmetry in 2D or quasi-2D materials such as 2D electron gases and high-temperature superconductors5. For distortions in the 3D bulk, however, the long-range coherence is expected to remain stable unless the randomness reaches a critical threshold. As a result, a careful study of the dependence of long-range coherence on the amount of lattice defects in these topological crystalline insulators may offer additional clues on whether these distortions are 2D or 3D in nature. As a further development, more insight on the dispersion
2
relation of the surface electrons could be achieved by using other techniques such as angle-resolved photoemission spectroscopy (ARPES) and quasiparticle scattering. In fact, although the energy scale of the Dirac mass (~20 meV) and the momentum splitting between two Dirac points are both small, they are within the resolution of the state-of-the-art ARPES tools. Topological states of matter present exotic properties that are impossible in conventional materials. Madhavan and colleagues show, on the one hand, a way to control the Dirac mass generation in surface topological states. On the other, they suggest new ways to control the energy gap of a 2D electronic system by tuning lattice distortions, for example by using strain or pressure as a control parameter. The capability of tuning the fundamental properties of topological states of matter
could be a step towards future technologies such as topological transistors and even topological quantum computation6. ❐ Kai Sun is in the Department of Physics, University of Michigan, 450 Church Street, Ann Arbor, Michigan 48109, USA. e-mail: [email protected] References
1. Zeljkovic, I. et al. Nature Mater. http://dx.doi.org/10.1038/ nmat4215 (2015). 2. Fu, L. Phys. Rev. Lett. 106, 106802 (2011). 3. Okada, Y. et al. Science 341, 1496–1499 (2013). 4. Cardy, J. Scaling and Renormalization in Statistical Physics (Cambridge Univ. Press, 1996). 5. Fradkin, E., Kivelson, S. A., Lawler, M. J., Eisenstein, J. P. & Mackenzie, A. P. Annu. Rev. Condens. Matter Phys. 1, 153–178 (2010). 6. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Rev. Mod. Phys. 80, 1083–1159 (2008).
Published online: 16 February 2015
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2015 Macmillan Publishers Limited. All rights reserved
