Topological isolator

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Idealized electronic band structure of a topological isolator. The Fermi energy lies in the band gap which is crossed by topologically protected surface states.

In physics, a topological insulator (in detail: an insulator with topologically protected surface conductivity ) is a solid that behaves like an electrical insulator inside , i.e. despite the presence of an external electrical field, it completely prevents any electrical current, which, however, at the same time on its surface (or at the outer edges) allows the movement of charge carriers (usually there is an almost resistance-free metallic conductivity here). A similar phenomenon is known as the marginal channel model .

The name combines physical aspects ("isolator behavior") with the mathematical discipline topology , which u. A. Terms like “the inside” and “the surface” reflect.

This unusual behavior is difficult to understand and the phenomenon is relatively new to be discovered. At low temperatures it leads to a large and almost resistance-free ("dissipation-free") electrical conductivity of the system. Otherwise, freedom from dissipation is known from superconductors . But there it concerns precisely the interior, although there too supercurrents occur on the surface.

Some topological insulators show a quantum spin Hall effect , for example the system of topological insulators in quantum wells, on which topological insulators were experimentally demonstrated for the first time in 2007 by Laurens Molenkamp's group .

In the meantime, improvements in materials have been achieved in the very young field of topological insulators. In 2016, for example, a monolayer of bismuth on silicon carbide was synthesized . Due to the resulting large energy gap of 0.8 eV, the use of the phenomenon of a topological insulator and quantum spin Hall material is conceivable at room temperature .

General

A system that has a large energy gap with regard to the volume effects between the valence band and conduction band , as in insulators, can for topological reasons have conductive , i.e. energy gap-free states on the surface that are topologically protected , e.g. B. due to time reversal invariance of the interactions. Topologically protected means: Any changes to the parameters of the system have no effect on the protected properties because (or if so) the topological relationships always remain unchanged during the measurement. The parameters of the system can change, but with a constant topological invariant - here with time-reversal variance - the new and old system belong to the same, i. W. by Fig. 1 characterized equivalence class .

The corresponding topological invariant herein refers to the symmetry against reversal of movement, the so-called. Time reversal symmetry , (and consequently reverse impulse and angular momentum vectors). It is always given if the changes in the interaction only affect potential and / or spin-orbit scattering , but it is violated if additional magnetic disturbances dominate. In the first two cases one has so-called Kramers degeneracy: states with opposite k-vectors and opposite spins have the same energy.

Proposal and implementation

Topological isolators were predicted by Charles L. Kane in 2005 and independently by Shoucheng Zhang in 2006 . Zhang also predicted a realization in tellurium-cadmium quantum wells . This was demonstrated in 2007 at low temperatures by a group led by Laurens W. Molenkamp at the University of Würzburg . At the end of 2013, Molenkamp received a Leibniz Prize from the German Research Foundation for his investigations into the phenomenon. After these first attempts had to be made at very low temperatures due to the very small volume band gap, progress has now been made in the research area. According to theoretical prediction, researchers in Würzburg led by Werner Hanke succeeded in producing bismuth on silicon carbide in 2017. Due to the arrangement of the Bi atoms in a honeycomb lattice, the system resembles graphs at first glance , but the large spin-orbit coupling of the Bi atoms and their interaction with the substrate creates a volume band gap of 0.8 eV what makes room temperature applications possible.

Theoretical interpretation

Inside a topological insulator, the electronic band structure is similar to that of an ordinary insulator with the Fermi energy between the conduction and valence bands. On the surface of the topological insulator, however, there are special states whose energies lie within the band gap, which enable measurable, ideally dissipation-free charge transport on the surface: For energies that lie in the actual band gap, there are on the surface, as in the graph marked by the green arrows, correlated pairs of such surface states with anti-parallel spin of the charge carriers (electrons) and opposite direction of movement. One model for the explanation is the edge channel model , which explains the quantum Hall effect that occurs, according to which there is only one of the two spin types of the electrons (spin-up or spin-down) on one side, since the spin of an electron unites "Twist" generated in one of the two corresponding directions, for example to the right or left side. The mechanism is also here analogous to the theory of superconductivity and is reminiscent of the singlet mechanism in the formation of the so-called Cooper pairs there , but here the spin is fixed perpendicular to the momentum, "spin-momentum locking". (So ​​there are not only analogies, but also subtle differences.) At the respective "edges" in the marginal canal model, the resulting landing level is bent upwards and the intersection of the Fermi level between two landing levels with the orbitals creates a leading "topologically protected" Area in which there is a spin degeneration.

The consequence of this is that the scattering is strongly suppressed and the transport on the surface proceeds almost without dissipation. These states are identified by an index similar to the gender of a surface in the mathematical discipline of topology and are an example of a topologically ordered state.

Further examples

Topologically protected edge states (1D) were predicted in quantum wells (very thin layers) of mercury telluride between cadmium telluride ( Andrei Bernevig , Shoucheng Zhang, Taylor Hughes) and observed experimentally shortly afterwards by the group of Laurens Molenkamp . Later they were predicted in three-dimensional systems from binary compounds with bismuth . The first experimentally realized three-dimensional topological insulator was observed in bismuth-antimony. A short time later topologically protected surface states were also in pure antimony , bismuth selenide , bismuth telluride and antimony telluride by different groups using ARPES proven. Various other material systems are now believed to behave like a topological insulator. In some of these materials, the Fermi energy is in the valence or conduction band due to naturally occurring defects. In this case, it has to be pushed into the band gap by means of doping or a gate voltage.

Similar edge currents also occur in the quantum Hall effect . However, this requires strong magnetic fields, (mostly) low temperatures and two-dimensional systems.

A helical Dirac fermion, which behaves like a massless relativistic particle, was also observed in a topological insulator.

Topological isolators for light in optical waveguides were realized by Alexander Szameit and colleagues in 2013. They used twisted waveguide structures inscribed in quartz glass by means of a laser and were later able to prove experimentally the prediction that so-called topological Anderson insulators (see also Anderson localization , cf.) work. They showed that the transport of light on the surface of a regular topological insulator was prevented by a small variation in the structure, but nevertheless took place again when further, irregular disturbances were introduced.

Mathematical classification

Mathematically, the general theory of the topologically protected boundary states is described by cohomology groups .

Strictly speaking, a distinction is made between the somewhat more general term “topology-protected” and the somewhat weaker term “symmetry-protected”, which is authoritative here. “Symmetry-protected” does not mean that the affiliation of the “protected states” to the respective symmetry class results from the originally or last existing symmetry; rather, it is required that the original symmetry, e.g. B. the time reversal invariance, remains unchanged during the entire measurement process, which is not always the case. So only "protection for topological reasons that are kept constant (shorter: for reasons of symmetry , or more precisely: because of time-reversal symmetry )".

In the work Classification of symmetry-protected topological phases , the mathematical behavior of the systems is described in detail from a theoretical-physical point of view and the 10 classes of Altland and Zirnbauer are discussed: Ten classes result because on the one hand the symmetrized or antisymmetrized   time reversal - particle hole - or chiral symmetries or antisymmetries are decisive, but on the other hand also the so-called "trivial" transformation and the operator product

For some mathematically known classes no experimental realization has yet been found.

literature

References and footnotes

  1. ↑ a b Felix Reis, Gang Li, Lenart Dudy, Maximilian Bauernfeind, Stefan Glass, Werner Hanke, Ronny Thomale, Jörg Schäfer and Ralph Claessen: Bismuthene on a SiC substrate: A candidate for a high-temperature quantum spin Hall material . In: Science , July 21, 2017 Vol. 357 No. 6348 pp. 287-290.
  2. ↑ Werner Hanke, Research Topics , University of Würzburg, accessed August 3, 2019
  3. ↑ Thomas Guhr, Axel Müller-Groening, Hans-Arwed Weidenmüller : Random Matrix Theories in Quantum Physics: Common Concepts . In: Physics Reports, Volume 299, 1998, pp. 189-425, arxiv : cond-mat / 9707301 .
  4. ↑ a b Markus König, Steffen Wiedmann, Christoph Brune, Andreas Roth, Hartmut Buhmann, Laurens W. Molenkamp, ​​Xiao-Liang Qi, Shou-Cheng Zhang: Quantum Spin Hall Insulator State in HgTe Quantum Wells . In: Science . 318, No. 5851, November 2, 2007, pp. 766-770. doi : 10.1126 / science.1148047 . Retrieved March 25, 2010.
  5. ↑ Würzburg: Leibniz Prize for Würzburg researchers ( Memento of the original from July 5, 2015 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / www.uni-wuerzburg.de
  6. ↑ Chia-Hsiu Hsu, Zhi-Quan Huang, Feng-Chuan Chuang, Chien-Cheng Kuo, Yu-Tzu Liu, Hsin Lin and Arun Bansil: The nontrivial electronic structure of Bi / Sb honeycombs on SiC (0001) . In: New Journal of Physics , February 10, 2015, Vol. 17, No. 2, p. 025005.
  7. ^ Charles L. Kane, Eugene J. Mele: PHYSICS: A New Spin on the Insulating State . In: Science . 314, No. 5806, December 15, 2006, pp. 1692-1693. doi : 10.1126 / science.1136573 . Retrieved March 25, 2010.
  8. ^ CL Kane, EJ Mele: Z 2 Topological Order and the Quantum Spin Hall Effect . In: Physical Review Letters . 95, No. 14, September 30, 2005, p. 146802. doi : 10.1103 / PhysRevLett.95.146802 .
  9. ↑ B. Andrei Bernevig, Taylor L. Hughes, Shou-Cheng Zhang: Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells . In: Science . 314, No. 5806, December 15, 2006, pp. 1757-1761. doi : 10.1126 / science.1133734 . Retrieved March 25, 2010.
  10. ^ Liang Fu, CL Kane: Topological insulators with inversion symmetry . In: Physical Review B . 76, No. 4, July 2, 2007, p. 045302. doi : 10.1103 / PhysRevB.76.045302 . Shuichi Murakami: Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase . In: New Journal of Physics . 9, No. 9, 2007, pp. 356-356. ISSN  1367-2630 . doi : 10.1088 / 1367-2630 / 9/9/356 . Retrieved March 26, 2010.
  11. ↑ D. Hsieh, D. Qian, L. Wray, Y. Xia, YS Hor, RJ Cava & MZ Hasan: A Topological Dirac insulator in a 3D quantum spin Hall phase . In: Nature . 452, No. 9, 2008, pp. 970-974. Accessed in 2010.
  12. ↑ M. Z Hasan, CL Kane: Topological Insulators . In: Rev. Mod. Phys. . 82, 2010, p. 3045. arxiv : 1002.3895 .
  13. ↑ Hsin Lin, L. Andrew Wray, Yuqi Xia, Suyang Xu, Shuang Jia, Robert J. Cava, Arun Bansil, M. Zahid Hasan: Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena . In: Nat Mater . 9, No. 7, July 2010, pp. 546-549. ISSN  1476-1122 . doi : 10.1038 / nmat2771 .
  14. ↑ D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, JH Dil, J. Osterwalder, L. Patthey, AV Fedorov, H. Lin, A. Bansil, D. Grauer, YS Hor, RJ Cava, MZ Hasan: Observation of Time-Reversal-Protected Single-Dirac-Cone Topological-Insulator States in Bi 2 Te 3 and Sb 2 Te 3 . In: Physical Review Letters . 103, No. 14, 2009, p. 146401. doi : 10.1103 / PhysRevLett.103.146401 .
  15. ↑ H.-J. Noh, H. Koh, S.-J. Oh, J.-H. Park, H.-D. Kim, JD Rameau, T. Valla, TE Kidd, PD Johnson, Y. Hu and Q. Li: Spin-orbit interaction effect in the electronic structure of Bi2Te3 observed by angle-resolved photoemission spectroscopy . In: EPL Europhysics Letters . 81, No. 5, 2008, p. 57006. doi : 10.1209 / 0295-5075 / 81/57006 . Retrieved April 25, 2010.
  16. ↑ D. Hsieh, Y. Xia et al. a .: A tunable topological insulator in the spin helical Dirac transport regime . In: Nature , 460, 2009, p. 1101, doi: 10.1038 / nature08234 .
  17. ^ MC Rechtsman, JM Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit: Photonic Floquet Topological Insulatorse . In: Nature , Volume 496, 2013, pp. 196-200
  18. ↑ openaccess.leidenuniv.nl Contribution from the University of Leiden : Theory of the topological Anderson insulator
  19. ↑ Simon Stützer, Yonatan Plotnik, Yaakov Lumer, Paraj Titum, Netanel H. Lindner, Mordechai Segev, Mikael C. Rechtsman, Alexander Szameit: Photonic topological Anderson insulators . In: Nature , No. 560, pp. 461–465 (August 22, 2018), accessed on August 24, 2018
  20. ↑ Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen: Symmetry protected topological orders and the group cohomology of their symmetry group . Review (2011), arxiv : 1106.4772 .
  21. ^ Frank Pollmann, Andreas Schnyder: Classification of symmetry-protected topological phases . In: Physik-Journal , 14 (8/9), 2015, pp. 65–69 ( online (PDF) )
  22. ↑ The time-reverse operator, for example, is represented in quantum mechanics not by a unitary but by an anti-unit operator, because complex numbers are converted into conjugate complexes.
  23. ^ R. Winkler, U. Zülicke: Discrete Symmetries of low-dimensional Dirac models: A selective review with a focus on condensed-matter realizations , ANZIAM J 0 (2014) 1–15 arxiv : 1206.0355 .