NEWS & VIEWS RESEARCH also occur through changes in gene regulation, without a change in the protein-coding sequence, and Brawand et al. also addressed this. The cichlid genomes show evidence for enhanced rates of evolution in putative regulatory elements, and high evolutionary turnover in microRNAs — a class of RNA molecule that regulates gene expression. Furthermore, the genomes reveal 40 new microRNA-encoding genes that, intriguingly, show complementary patterns of expression relative to the genes they are hypothesized to regulate. This suggests that they are involved in suppressing gene expression, perhaps to stabilize and refine expression patterns that have been acquired during the radiation. The authors attribute the great diversity of changes seen across these genomes to a period of relaxed selection that occurred early in the radiation. During this time, the selective pressures that maintained the stability of the genome were reduced, thereby allowing genetic variation to accumulate and produce subsequent diversification into the lineages we observe today. However, accelerated evolution can result either from neutral evolution due to relaxed selection, or from positive natural selection acting through new selective pressures. Most of the genomic signatures in the paper do not strongly distinguish between these two possibilities. Indeed, it seems most likely that the retention of gene duplicates and rapid genetic divergence were primarily driven by positive natural selection, as species adapted to the great diversity of ecological niches available in the lakes. Subsequent extinction of early lineages could have led to an apparent burst of rapid change on the branch leading to the extant species. There may be no need to invoke a genetic revolution when plain old natural selection can explain the observed patterns. Although the five genomes offer some impressive insights into cichlid biology, I believe that the most exciting advances will come from analysis of more-closely related genomes within each radiation. The cichlids offer in abundance two of the characteristics that have facilitated analysis of adaptive traits in other taxa: there are many closely related species that show highly divergent morphology, and there is repeated evolution of similar traits in parallel. Whole-genome sequencing of multiple individual fishes with both divergent and convergent ecological traits will provide rich pickings for understanding how genetic changes are associated with specific ecological characteristics. Brawand et al. have scratched the surface of this task by reanalysing sequence data from samples of six species found in Lake Victoria6; these suggest that even very closely related species show quite high levels of divergence across the genome. These genomes will facilitate further studies that will undoubtedly enhance our understanding of cichlid biology. It may be rash, but I will make one prediction. Work in organisms
ranging from sticklebacks7 to butterflies8 has shown that recent adaptive events can make use of ancient genetic variants. This may be surprising, but can occur because gene flow within a species, or sometimes even between species8, can provide ‘pre-adapted’ variants that permit populations to adapt rapidly to new challenges. So I predict that similarities between cichlids in different lakes that are currently considered to have evolved independently will in fact turn out to have resulted in part from ancient shared variation that may have arisen early in the radiation9. ■ Chris D. Jiggins is in the Department of Zoology, University of Cambridge,
Cambridge CB2 3EJ, UK. e-mail: [email protected] 1. Wagner, C. E., Harmon, L. J. & Seehausen, O. Nature 487, 366–369 (2012). 2. Genner, M. J. et al. Mol. Biol. Evol. 24, 1269–1282 (2007). 3. Brawand, D. et al. Nature 513, 375–381 (2014). 4. Lee, H. J., Kusche, H. & Meyer, A. PLoS ONE 7, e44670 (2012). 5. Ohno, S. Evolution by Gene Duplication (Springer, 2014). 6. Wagner, C. E. et al. Mol. Ecol. 22, 787–798 (2013). 7. Colosimo, P. F. et al. Science 307, 1928–1933 (2005). 8. The Heliconius Genome Consortium. Nature 487, 94–98 (2012). 9. Loh, Y.-H. E. et al. Mol. Biol. Evol. 30, 906–917 (2013). This article was published online on 3 September 2014.
C O N D EN SE D- M AT T E R P H YS I CS
Catching relativistic electrons Low-energy electrons have been found to mimic relativistic high-energy particles in cadmium arsenide. This defines the first stable ‘3D Dirac semimetal’, which holds promise for fundamental-physics exploration and practical applications. ZHIHUAI ZHU & JENNIFER E. HOFFMAN
I
n classical Newtonian mechanics, an object’s energy varies as the square of its velocity or momentum (Fig. 1a) — a rule that car drivers should treat with respect. Photons, neutrinos and other light, fast-moving particles are governed instead by Einstein’s theory of relativity: their energy scales linearly with their momentum, with fixed velocity equal to the slope of the increase. Such relativistic high-energy particles hold the key to fundamental understanding of our Universe. But where do electrons — which determine the more practical properties of the materials immediately around us — fit into this picture? Electrons move very fast, but their motion is not primarily relativistic in conventional solids. However, in a paper published in Physical Review Letters, Borisenko et al.1 report the discovery of relativistic motion of low-energy electrons in cadmium arsenide (Cd3As2). Taken together with similar findings described in three independent papers, by Neupane et al.2, Liu et al.3 and Jeon et al.4, this result paves the way for future relativistic electronics. The realization that low-energy electrons can mimic high-energy relativistic particles occurred a decade ago with the isolation of two-dimensional (2D) carbon in the form of graphene5. This material has dual significance for the exploration of fundamental physics and for revolutionary applications; it has
prompted more than 100,000 publications, some 7,000 patent applications and a 2010 Nobel prize. Electrons in graphene are described as massless Dirac fermions because they have half-integer spin, which makes them fermions, and their linear energy–momentum relationship obeys Dirac’s famous wave equation, which first united quantum mechanics and special relativity almost a century ago. Graphene is also a semimetal, meaning that its Fermi energy (the dividing line between filled and empty electronic states) sits ideally at its ‘Dirac point’ — where its valence and conduction energy bands meet (Fig. 1b) — and may be easily tuned using an applied voltage. The resultant charge carriers may be either electrons or holes (the absence of electrons) and have high mobility: a measure of inverse electrical resistivity per carrier, which increases with carrier velocity but decreases with carrier scattering. Graphene’s moderately high carrier velocity of about 10 5 metres per second, combined with the reduced intrinsic scattering possibilities caused by the small carrier density inherent to a Dirac semimetal, can give a mobility up to 140 times that of silicon — the material of choice for most electronic applications. Therefore, graphene offers promise for making novel, high-efficiency electronic devices. However, graphene is challenging to fabricate and manipulate in large sheets, and its mobility is extremely susceptible to scattering from environmental
1 8 S E P T E M B E R 2 0 1 4 | VO L 5 1 3 | NAT U R E | 3 1 9
© 2014 Macmillan Publishers Limited. All rights reserved
RESEARCH NEWS & VIEWS a
b
E
c
E
E
Dirac point px
p Classical objects or electrons
2D Dirac fermions
px
pz
3D Dirac fermions
Figure 1 | Energy–momentum spectra of electrons. a, Classical objects and electrons exhibit a parabolic relationship between their energy (E) and momentum (p). b, Two-dimensional (2D) Dirac fermions, such as electrons in graphene, have valence (blue) and conduction (purple) energy bands with a linear energy–momentum relationship. These touch at a point called the Dirac point in the 3D parameter space formed by E, px and py. Shown here is a 2D slice of the 3D space. c, A 3D slice of the 4D (E, px, py, pz) energy–momentum relationship of 3D Dirac fermions such as those discovered1–4 in Cd3As2, with two Dirac points along a special high-symmetry axis (pz).
defects because graphene is all surface. A second kind of 2D Dirac semimetal arises from another relativistic effect of electrons called spin–orbit coupling — the interaction between an electron’s spin and the induced magnetic field from the electron’s orbital motion. Spin–orbit coupling is generally small for materials that are made up of light atoms such as carbon, but for materials containing heavy atoms such as bismuth and cadmium, the interaction can be significant; for example, it can invert the valence and conduction bands in the bulk of an insulator. This inversion can lead to surface Dirac fermions that are topologically protected — surface carriers that are robust against some local disorder and have their spin locked to their momentum (that is, the carrier’s momentum determines its spin). These ‘topological insulators’ 6,7 provoked tremendous excitement in recent years about possible applications such as low-energy-consumption spintronic devices, which manipulate the spin rather than the charge of electrons, for high-performance computing. But despite their name, existing topological insulators have excess conducting bulk electrons, which overwhelm the surface Dirac fermions and foil their use. Meanwhile, new ideas were brewing, suggesting that 3D Dirac semimetallic states could exist in the bulk of a solid material. It was known that such states could occur under finely tuned conditions, such as the exact concentration of bismuth at which spin–orbit coupling becomes strong enough to invert the bulk energy bands in antimony–bismuth alloys8 (Sb1–xBix). But more recent theoretical work predicted the robust occurrence of such states in pure materials that have certain crystalline symmetries: first, unstable BiO2 (ref. 9), then air-sensitive Na3Bi (ref. 10) and, finally, the stable compound Cd3As2 (ref. 11). Furthermore, when time-reversal or spatial-inversion symmetries are broken — for example, by application of a magnetic field or pressure — each Dirac point can split into two copies at which the electrons become Weyl fermions12, which have opposite chirality (spin orientation with
respect to their direction of motion). These Weyl fermions could enable robust spintronics in three dimensions. Cd3As2 has been known for more than 50 years13 for its extraordinary carrier mobility, which is larger than that of suspended graphene and among the highest of any bulk semiconductor. Thanks to the recent studies by Borisenko et al.1, Neupane et al.2 and Liu et al.3 — who all conducted experiments on Cd3As2 using a technique called angle-resolved photo emission spectroscopy (ARPES) — we now understand that the high mobility arises from high-velocity 3D Dirac semimetallic states. During ARPES experiments, mono chromatic light is incident on a sample and electrons can absorb a photon and escape from the material. To unveil the full 3D energy– momentum relationship of electrons within Cd3As2, a challenging but crucial step was to precisely measure the energy and momentum of emitted electrons while tuning the photon energy through a wide range. The data1–3 clearly show a linear energy–momentum relationship, with two Dirac points along a crystal axis of four-fold rotational symmetry (Fig. 1c). This result proves that electrons in this material are 3D massless Dirac fermions as predicted11. Measurement of the energy– momentum slope gives electron velocity as high as about 106 m s–1 (ref. 2), but with a tenfold discrepancy between the three studies1–3, which could be due to differences in sample quality or the angle of the exposed surface. Liu et al. additionally demonstrated that the carrier concentration in Cd3As2 could be finely tuned by ‘doping’ the surface of the material with potassium atoms3, making it a flexible platform for future studies. Most recently, Jeon et al.4 used a scanning tunnelling microscope to confirm Cd3As2 as a 3D Dirac semimetal down to atomic length scales, and to visualize how dopant atoms scatter carriers primarily in the valence band, preserving the mobility of carriers in the high-velocity conduction band. Furthermore, Jeon and colleagues applied a magnetic field, which is not possible in an ARPES experiment.
3 2 0 | NAT U R E | VO L 5 1 3 | 1 8 S E P T E M B E R 2 0 1 4
© 2014 Macmillan Publishers Limited. All rights reserved
Although the field breaks the time-reversal symmetry that would be necessary to split the Dirac fermions into the more exotic chiral Weyl fermions, its orientation in this experiment also breaks the four-fold rotational symmetry of the crystal that was necessary to realize the Dirac fermions in the first place. This means that the first glimpse of Weyl fermions will need to wait for a follow-up experiment in which the magnetic field has a different orientation. The work on Cd3As2 (refs 1–4), together with the lower-mobility Na3Bi reported earlier this year14, confirms the existence of motion of Dirac fermions inside 3D materials. Despite its exciting new physics, the application potential of Cd3As2 is limited by its small band-inversion energy — the relativistic nature is not robust at room temperature4. Furthermore, Cd3As2 is not exactly something you want in your drinking water. Nevertheless, given the new understanding that robust Dirac fermions can arise in solids from general crystalline symmetries and strong spin–orbit coupling, there are probably numerous 3D Dirac semimetals yet to be discovered9. Immediate research priorities include magnetic-field and pressure control to isolate chiral Weyl fermions in existing materials, realization of these materials as thin films to access a phenomenon known as the quantum spin Hall effect to visualize the spatial flow of surface Dirac fermions15, and computational modelling to predict new materials and heterostructures with larger band-inversion energies16. Then exotic applications, such as a ‘chiral battery’ or a ‘quantum amplifier’ of magnetic field, may be on the horizon17. ■ Zhihuai Zhu and Jennifer E. Hoffman are in the Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA. e-mails: [email protected]; [email protected] 1. Borisenko, S. et al. Phys. Rev. Lett. 113, 027603 (2014). 2. Neupane, M. et al. Nature Commun. 5, 3786; http://dx.doi.org/10.1038/ncomms4786 (2014). 3. Liu, Z. K. et al. Nature Mater. 13, 677–681 (2014). 4. Jeon, S. et al. Nature Mater. 13, 851–856 (2014). 5. Geim, A. K. & Novoselov, K. S. Nature Mater. 6, 183–191 (2007). 6. Hasan, M. Z. & Kane, C. L. Rev. Mod. Phys. 82, 3045 (2010). 7. Qi, X.-L. & Zhang, S.-C. Rev. Mod. Phys. 83, 1057 (2011). 8. Hsieh, D. et al. Nature 452, 970–974 (2008). 9. Young, S. M. et al. Phys. Rev. Lett. 108, 140405 (2012). 10. Wang, Z. et al. Phys. Rev. B 85, 195320 (2012). 11. Wang, Z., Weng, H., Wu, Q., Dai, X. & Fang, Z. Phys. Rev. B 88, 125427 (2013). 12. Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Phys. Rev. B 83, 205101 (2011). 13. Rosenberg, A. J. & Harman, T. C. J. Appl. Phys. 30, 1621 (1959). 14. Liu, Z. K. et al. Science 343, 864–867 (2014). 15. Kane, C. L. & Mele, E. J. Phys. Rev. Lett. 95, 226801 (2005). 16. Burkov, A. A. & Balents, L. Phys. Rev. Lett. 107, 127205 (2011). 17. Kharzeev, D. E. & Yee, H.-U. Phys. Rev. B 88, 115119 (2013).
ranging from sticklebacks7 to butterflies8 has shown that recent adaptive events can make use of ancient genetic variants. This may be surprising, but can occur because gene flow within a species, or sometimes even between species8, can provide ‘pre-adapted’ variants that permit populations to adapt rapidly to new challenges. So I predict that similarities between cichlids in different lakes that are currently considered to have evolved independently will in fact turn out to have resulted in part from ancient shared variation that may have arisen early in the radiation9. ■ Chris D. Jiggins is in the Department of Zoology, University of Cambridge,
Cambridge CB2 3EJ, UK. e-mail: [email protected] 1. Wagner, C. E., Harmon, L. J. & Seehausen, O. Nature 487, 366–369 (2012). 2. Genner, M. J. et al. Mol. Biol. Evol. 24, 1269–1282 (2007). 3. Brawand, D. et al. Nature 513, 375–381 (2014). 4. Lee, H. J., Kusche, H. & Meyer, A. PLoS ONE 7, e44670 (2012). 5. Ohno, S. Evolution by Gene Duplication (Springer, 2014). 6. Wagner, C. E. et al. Mol. Ecol. 22, 787–798 (2013). 7. Colosimo, P. F. et al. Science 307, 1928–1933 (2005). 8. The Heliconius Genome Consortium. Nature 487, 94–98 (2012). 9. Loh, Y.-H. E. et al. Mol. Biol. Evol. 30, 906–917 (2013). This article was published online on 3 September 2014.
C O N D EN SE D- M AT T E R P H YS I CS
Catching relativistic electrons Low-energy electrons have been found to mimic relativistic high-energy particles in cadmium arsenide. This defines the first stable ‘3D Dirac semimetal’, which holds promise for fundamental-physics exploration and practical applications. ZHIHUAI ZHU & JENNIFER E. HOFFMAN
I
n classical Newtonian mechanics, an object’s energy varies as the square of its velocity or momentum (Fig. 1a) — a rule that car drivers should treat with respect. Photons, neutrinos and other light, fast-moving particles are governed instead by Einstein’s theory of relativity: their energy scales linearly with their momentum, with fixed velocity equal to the slope of the increase. Such relativistic high-energy particles hold the key to fundamental understanding of our Universe. But where do electrons — which determine the more practical properties of the materials immediately around us — fit into this picture? Electrons move very fast, but their motion is not primarily relativistic in conventional solids. However, in a paper published in Physical Review Letters, Borisenko et al.1 report the discovery of relativistic motion of low-energy electrons in cadmium arsenide (Cd3As2). Taken together with similar findings described in three independent papers, by Neupane et al.2, Liu et al.3 and Jeon et al.4, this result paves the way for future relativistic electronics. The realization that low-energy electrons can mimic high-energy relativistic particles occurred a decade ago with the isolation of two-dimensional (2D) carbon in the form of graphene5. This material has dual significance for the exploration of fundamental physics and for revolutionary applications; it has
prompted more than 100,000 publications, some 7,000 patent applications and a 2010 Nobel prize. Electrons in graphene are described as massless Dirac fermions because they have half-integer spin, which makes them fermions, and their linear energy–momentum relationship obeys Dirac’s famous wave equation, which first united quantum mechanics and special relativity almost a century ago. Graphene is also a semimetal, meaning that its Fermi energy (the dividing line between filled and empty electronic states) sits ideally at its ‘Dirac point’ — where its valence and conduction energy bands meet (Fig. 1b) — and may be easily tuned using an applied voltage. The resultant charge carriers may be either electrons or holes (the absence of electrons) and have high mobility: a measure of inverse electrical resistivity per carrier, which increases with carrier velocity but decreases with carrier scattering. Graphene’s moderately high carrier velocity of about 10 5 metres per second, combined with the reduced intrinsic scattering possibilities caused by the small carrier density inherent to a Dirac semimetal, can give a mobility up to 140 times that of silicon — the material of choice for most electronic applications. Therefore, graphene offers promise for making novel, high-efficiency electronic devices. However, graphene is challenging to fabricate and manipulate in large sheets, and its mobility is extremely susceptible to scattering from environmental
1 8 S E P T E M B E R 2 0 1 4 | VO L 5 1 3 | NAT U R E | 3 1 9
© 2014 Macmillan Publishers Limited. All rights reserved
RESEARCH NEWS & VIEWS a
b
E
c
E
E
Dirac point px
p Classical objects or electrons
2D Dirac fermions
px
pz
3D Dirac fermions
Figure 1 | Energy–momentum spectra of electrons. a, Classical objects and electrons exhibit a parabolic relationship between their energy (E) and momentum (p). b, Two-dimensional (2D) Dirac fermions, such as electrons in graphene, have valence (blue) and conduction (purple) energy bands with a linear energy–momentum relationship. These touch at a point called the Dirac point in the 3D parameter space formed by E, px and py. Shown here is a 2D slice of the 3D space. c, A 3D slice of the 4D (E, px, py, pz) energy–momentum relationship of 3D Dirac fermions such as those discovered1–4 in Cd3As2, with two Dirac points along a special high-symmetry axis (pz).
defects because graphene is all surface. A second kind of 2D Dirac semimetal arises from another relativistic effect of electrons called spin–orbit coupling — the interaction between an electron’s spin and the induced magnetic field from the electron’s orbital motion. Spin–orbit coupling is generally small for materials that are made up of light atoms such as carbon, but for materials containing heavy atoms such as bismuth and cadmium, the interaction can be significant; for example, it can invert the valence and conduction bands in the bulk of an insulator. This inversion can lead to surface Dirac fermions that are topologically protected — surface carriers that are robust against some local disorder and have their spin locked to their momentum (that is, the carrier’s momentum determines its spin). These ‘topological insulators’ 6,7 provoked tremendous excitement in recent years about possible applications such as low-energy-consumption spintronic devices, which manipulate the spin rather than the charge of electrons, for high-performance computing. But despite their name, existing topological insulators have excess conducting bulk electrons, which overwhelm the surface Dirac fermions and foil their use. Meanwhile, new ideas were brewing, suggesting that 3D Dirac semimetallic states could exist in the bulk of a solid material. It was known that such states could occur under finely tuned conditions, such as the exact concentration of bismuth at which spin–orbit coupling becomes strong enough to invert the bulk energy bands in antimony–bismuth alloys8 (Sb1–xBix). But more recent theoretical work predicted the robust occurrence of such states in pure materials that have certain crystalline symmetries: first, unstable BiO2 (ref. 9), then air-sensitive Na3Bi (ref. 10) and, finally, the stable compound Cd3As2 (ref. 11). Furthermore, when time-reversal or spatial-inversion symmetries are broken — for example, by application of a magnetic field or pressure — each Dirac point can split into two copies at which the electrons become Weyl fermions12, which have opposite chirality (spin orientation with
respect to their direction of motion). These Weyl fermions could enable robust spintronics in three dimensions. Cd3As2 has been known for more than 50 years13 for its extraordinary carrier mobility, which is larger than that of suspended graphene and among the highest of any bulk semiconductor. Thanks to the recent studies by Borisenko et al.1, Neupane et al.2 and Liu et al.3 — who all conducted experiments on Cd3As2 using a technique called angle-resolved photo emission spectroscopy (ARPES) — we now understand that the high mobility arises from high-velocity 3D Dirac semimetallic states. During ARPES experiments, mono chromatic light is incident on a sample and electrons can absorb a photon and escape from the material. To unveil the full 3D energy– momentum relationship of electrons within Cd3As2, a challenging but crucial step was to precisely measure the energy and momentum of emitted electrons while tuning the photon energy through a wide range. The data1–3 clearly show a linear energy–momentum relationship, with two Dirac points along a crystal axis of four-fold rotational symmetry (Fig. 1c). This result proves that electrons in this material are 3D massless Dirac fermions as predicted11. Measurement of the energy– momentum slope gives electron velocity as high as about 106 m s–1 (ref. 2), but with a tenfold discrepancy between the three studies1–3, which could be due to differences in sample quality or the angle of the exposed surface. Liu et al. additionally demonstrated that the carrier concentration in Cd3As2 could be finely tuned by ‘doping’ the surface of the material with potassium atoms3, making it a flexible platform for future studies. Most recently, Jeon et al.4 used a scanning tunnelling microscope to confirm Cd3As2 as a 3D Dirac semimetal down to atomic length scales, and to visualize how dopant atoms scatter carriers primarily in the valence band, preserving the mobility of carriers in the high-velocity conduction band. Furthermore, Jeon and colleagues applied a magnetic field, which is not possible in an ARPES experiment.
3 2 0 | NAT U R E | VO L 5 1 3 | 1 8 S E P T E M B E R 2 0 1 4
© 2014 Macmillan Publishers Limited. All rights reserved
Although the field breaks the time-reversal symmetry that would be necessary to split the Dirac fermions into the more exotic chiral Weyl fermions, its orientation in this experiment also breaks the four-fold rotational symmetry of the crystal that was necessary to realize the Dirac fermions in the first place. This means that the first glimpse of Weyl fermions will need to wait for a follow-up experiment in which the magnetic field has a different orientation. The work on Cd3As2 (refs 1–4), together with the lower-mobility Na3Bi reported earlier this year14, confirms the existence of motion of Dirac fermions inside 3D materials. Despite its exciting new physics, the application potential of Cd3As2 is limited by its small band-inversion energy — the relativistic nature is not robust at room temperature4. Furthermore, Cd3As2 is not exactly something you want in your drinking water. Nevertheless, given the new understanding that robust Dirac fermions can arise in solids from general crystalline symmetries and strong spin–orbit coupling, there are probably numerous 3D Dirac semimetals yet to be discovered9. Immediate research priorities include magnetic-field and pressure control to isolate chiral Weyl fermions in existing materials, realization of these materials as thin films to access a phenomenon known as the quantum spin Hall effect to visualize the spatial flow of surface Dirac fermions15, and computational modelling to predict new materials and heterostructures with larger band-inversion energies16. Then exotic applications, such as a ‘chiral battery’ or a ‘quantum amplifier’ of magnetic field, may be on the horizon17. ■ Zhihuai Zhu and Jennifer E. Hoffman are in the Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA. e-mails: [email protected]; [email protected] 1. Borisenko, S. et al. Phys. Rev. Lett. 113, 027603 (2014). 2. Neupane, M. et al. Nature Commun. 5, 3786; http://dx.doi.org/10.1038/ncomms4786 (2014). 3. Liu, Z. K. et al. Nature Mater. 13, 677–681 (2014). 4. Jeon, S. et al. Nature Mater. 13, 851–856 (2014). 5. Geim, A. K. & Novoselov, K. S. Nature Mater. 6, 183–191 (2007). 6. Hasan, M. Z. & Kane, C. L. Rev. Mod. Phys. 82, 3045 (2010). 7. Qi, X.-L. & Zhang, S.-C. Rev. Mod. Phys. 83, 1057 (2011). 8. Hsieh, D. et al. Nature 452, 970–974 (2008). 9. Young, S. M. et al. Phys. Rev. Lett. 108, 140405 (2012). 10. Wang, Z. et al. Phys. Rev. B 85, 195320 (2012). 11. Wang, Z., Weng, H., Wu, Q., Dai, X. & Fang, Z. Phys. Rev. B 88, 125427 (2013). 12. Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Phys. Rev. B 83, 205101 (2011). 13. Rosenberg, A. J. & Harman, T. C. J. Appl. Phys. 30, 1621 (1959). 14. Liu, Z. K. et al. Science 343, 864–867 (2014). 15. Kane, C. L. & Mele, E. J. Phys. Rev. Lett. 95, 226801 (2005). 16. Burkov, A. A. & Balents, L. Phys. Rev. Lett. 107, 127205 (2011). 17. Kharzeev, D. E. & Yee, H.-U. Phys. Rev. B 88, 115119 (2013).
