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Chemical Principles of Topological Semimetals
In the midst of the White House generated horror of the last days, I had the guilty pleasure of attending my favorite kind of lecture: A lecture that was not only on a subject about which I know nothing, but on a subject about which I never even heard, topological semimetals.
One of my goals in life is to feel as often as is possible like I'm the dumbest person in the room, and I definitely succeeded in this case.
The lecture was given by Dr. Leslie M. Schoop, the newest faculty member of the Princeton University Department of Chemistry.
I immediately went home after the lecture and began to look into the topic and was pleased to see that I recently downloaded (but clearly didn't read) a review article written by Dr. Schoop and her colleagues.
The article, from which the total of this post is taken is here: Chemical Principles of Topological Semimetals (Leslie M. Schoop,*,† Florian Pielnhofer,‡ and Bettina V. Lotsch, Chem. Mater., 2018, 30 (10), pp 3155–3176)
It's a relatively new, if rapidly expanding field, so I guess I can be excused for knowing nothing at all about it, but it apparently involves some novel particle physics apparently predicted by the mathematical physicist Hermann Weyl during the scientifically transcendent 20th century.
Since it involves the structure of matter, I plan to share this with my son when he returns from Europe, I believe he'll find it cool.
Much of the topic remains over my head, but I thought it might be interesting to post brief excerpts of the paper along with some of the beautiful graphics from it.
The practical application, should it ever develop, would be computers so fast as to revolutionize computation as much as the original digital computer did in the 20th century, the elusive quantum computer: At least this is what Dr. Shoop claimed.
The recent rapid development in the field of topological materials (see Figure 1) raises expectations that these materials might allow solving a large variety of current challenges in condensed matter science, ranging from applications in quantum computing, to infrared sensors or heterogeneous catalysis.(1?8) In addition, exciting predictions of completely new physical phenomena that could arise in topological materials drive the interest in these compounds.(9,10) For example, charge carriers might behave completely different from what we expect from the current laws of physics if they travel through topologically non-trivial systems.(11,12) This happens because charge carriers in topological materials can be different from the normal type of Fermions we know, which in turn affects the transport properties of the material. It has also been proposed that we could even find “new Fermions”, i.e., Fermions that are different from the types we currently know in condensed matter systems as well as in particle physics.(10) Such proposals connect the fields of high-energy or particle physics, whose goal it is to understand the universe and all the particles of which it is composed, with condensed matter physics, where the same type, or even additional types, of particles can be found as so-called quasi-particles, meaning that the charge carriers behave in a similar way as it would be expected from a certain particle existing in free space...
The caption:
Figure 1. Timeline of recent developments in the field of topologically non-trivial materials.
The intro continues:
...
he field of topology evolved from the idea that there can be insulators whose band structure is fundamentally different (i.e., has a different topology) from that of the common insulators we know. If two insulators with different topologies are brought into contact, electrons that have no mass and cannot be back scattered are supposed to appear at the interface. These edge states also appear if a topological insulator (TI) is in contact with air, a trivial insulator. 2D TIs have conducting edge states, whereas 3D TIs, which were discovered later, have conducting surface states. TIs have already been reviewed multiple times,(13?16) which is why we focus here on the newer kind of topological materials, namely topological semimetals (TSMs)...
The review then discusses the remarkable properties of graphene which Dr. Schoop remarked with some amusement can be made by peeling a single layer of carbon atoms off of graphite with masking tape.
...But let us first take a step back to look with a chemist’s eyes at graphene and try to understand why it is so special. As chemists we would think of graphene as an sp2-hybridized network of carbon atoms. Thus, three out of C’s four electrons are used to form the ?-bonds of the in-plane sp2-hybridized carbon backbone (Figure 2a). The remaining electron will occupy the pz-orbital, and since all C–C bonds in graphene have the same length, we know that these electrons are delocalized over the complete graphene sheet. Since graphene is an extended crystalline solid, the pz-orbitals are better described as a pz-band (Figure 2b). Since there is one electron per C available, this pz-band is exactly half-filled (Figure 2c).
The caption:
Figure 2. Intuitive approach for describing the electronic structure of graphene. (a) Real-space structure of graphene, highlighting the delocalized ?-system. (b) Orbital structure and band filling in graphene. (c) Corresponding electronic structure in k-space; only one atom per unit cell is considered. (d) Unit cell of graphene, containing two atoms. (e) Brillouin zone of graphene. (f) Folded band structure of panel (c), in accordance with the doubling of the unit cell. (g) Hypothetical version of distorted bands with localized double bonds. This type of distorted honeycomb can be found in oxide materials such as Na3Cu2SbO6.(58)
Some remarks on graphene as a prototype of the "Dirac Semimetal"
Most TSMs have in common that their unusual band topology arises from a band inversion. Unlike TIs, they do not have a band gap in their electronic structure. There are several classes of TSMs: Dirac semimetals (DSMs), Weyl semimetals (WSMs), and nodal line semimetals (NLSMs). All these kinds exist as “conventional” types; i.e., they are based on a band inversion. In addition, they can also be protected by non-symmorphic symmetry. The latter ones have to be viewed differently, and we will discuss them after introducing the conventional ones.
Dirac Semimetals
The prototype of a DSM is graphene. The “perfect” DSM has the same electronic structure of graphene; i.e., it should consist of two sets of linearly dispersed bands that cross each other at a single point. Ideally, no other states should interfere at the Fermi level. Note that in a DSM, the bands that cross are spin degenerate, meaning that we would call them two-fold degenerate, and thus the Dirac point is four-fold degenerate. When discussing degeneracies within this Review, we will always refer to spin orbitals. In any crystal that is inversion symmetric and non-magnetic (i.e., time reversal symmetry is present), all bands will always be two-fold degenerate. Time reversal symmetry (T-symmetry) means that a system’s properties do not change if a clock runs backward. A requirement for T-symmetry is that electrons at momentum points k and ?k have opposite spin, which means that the spin has to rotate with k around the Fermi surface since backscattering between k and ?k is forbidden. Introducing a perturbation, e.g., an external magnetic field, lifts the spin degeneracy and violates T-symmetry.
Dirac Semimetals
The prototype of a DSM is graphene. The “perfect” DSM has the same electronic structure of graphene; i.e., it should consist of two sets of linearly dispersed bands that cross each other at a single point. Ideally, no other states should interfere at the Fermi level. Note that in a DSM, the bands that cross are spin degenerate, meaning that we would call them two-fold degenerate, and thus the Dirac point is four-fold degenerate. When discussing degeneracies within this Review, we will always refer to spin orbitals. In any crystal that is inversion symmetric and non-magnetic (i.e., time reversal symmetry is present), all bands will always be two-fold degenerate. Time reversal symmetry (T-symmetry) means that a system’s properties do not change if a clock runs backward. A requirement for T-symmetry is that electrons at momentum points k and ?k have opposite spin, which means that the spin has to rotate with k around the Fermi surface since backscattering between k and ?k is forbidden. Introducing a perturbation, e.g., an external magnetic field, lifts the spin degeneracy and violates T-symmetry.
The caption:
Figure 3. Explanation of band inversions. (a) Rough density of states (DOS) of transition metals. Band inversions are possible between the different orbitals within one shell, but the material is likely to be metallic. (b) Band inversion between an s-band and a p-band. (c) Molecular orbital diagram of water. (d) Bands that cross and have the same irreducible representation (irrep) gap. (e) If the irreducible representations are different, the crossing is protected, but SOC might still create a gap.
"SOC" is spin orbit coupling.
Weyl Semimetals:
Weyl Semimetals
The difference between a DSM and a WSM is that, in the latter, the crossing point is only two-fold degenerate.(28,93,94) This is because in WSMs the bands are spin split; thus each band is only singly degenerate. If a spin-up band and a spin-down band cross, this results in a Weyl monopole, meaning that there is a chirality assigned to this crossing. Since there cannot be a net chirality in the crystals, Weyl cones always come in pairs. The resulting Weyl Fermions are chiral in nature and thus will behave physically different from “regular” Fermions. One example of this manifestation is the chiral anomaly, which we will discuss in the Properties and Applications of TSMs section below. Here, we will focus on the requirements necessary to realize a WSM.
In order to have spin split bands, we cannot have inversion (I) and time-reversal (T) symmetry at the same time, since the combination of these two symmetries will always force all bands to be doubly degenerate. In I asymmetric, i.e., non-centrosymmetric crystals, this degeneracy can be lifted with the help of SOC; this is the so-called Dresselhaus effect.(95)
The difference between a DSM and a WSM is that, in the latter, the crossing point is only two-fold degenerate.(28,93,94) This is because in WSMs the bands are spin split; thus each band is only singly degenerate. If a spin-up band and a spin-down band cross, this results in a Weyl monopole, meaning that there is a chirality assigned to this crossing. Since there cannot be a net chirality in the crystals, Weyl cones always come in pairs. The resulting Weyl Fermions are chiral in nature and thus will behave physically different from “regular” Fermions. One example of this manifestation is the chiral anomaly, which we will discuss in the Properties and Applications of TSMs section below. Here, we will focus on the requirements necessary to realize a WSM.
In order to have spin split bands, we cannot have inversion (I) and time-reversal (T) symmetry at the same time, since the combination of these two symmetries will always force all bands to be doubly degenerate. In I asymmetric, i.e., non-centrosymmetric crystals, this degeneracy can be lifted with the help of SOC; this is the so-called Dresselhaus effect.(95)
The caption:
Figure 4. Different ways to achieve a Weyl semimetal. (a) Effect of T- and I-symmetry breaking on a single band. (b) The same scenario for a Dirac crossing. In the case of T breaking, two Weyl crossings will appear on the high-symmetry line at different energies. In the case of I breaking, they will appear away from the high-symmetry line. (c) Schematic drawing of a type I (left) and a type II (right) WSM.
Figures for a 3D Dirac Semimetal, trisodium bismuthide, a Zintl salt (at least I knew about Zintl salts for the lecture):
The caption:
Figure 7. (a) Crystal structure of Na3Bi. (b) First Brillouin zone with high-symmetry points and highlighted Dirac points. (c) Bulk band structure. (d) 3D intensity plot of the ARPES spectra at the Dirac point. Panels b and d reprinted with permission from ref (143). Copyright 2014 The American Association for the Advancement of Science. Panel c reprinted with permission from ref (168). Copyright 2017 Springer Nature.
A Weyl Semimetal:
The caption:
Figure 8. (a) Crystal structure of TaAs. (b) Brillouin zone. (c) Band structure without and (d) with SOC. (e) Photoemission spectrum with overlaid calculated band structure. (f) Calculated and measured Fermi surface, displaying the Fermi arcs, which are the signature to identify WSMs. Panels b–d reprinted with permission from ref (149). Copyright 2015 Springer Nature. Panels e and f reprinted with permission from ref (91). Copyright 2015 Springer Nature.
A "Non-symmorphic Topological Semimetal: "
The caption:
Figure 9. (a) Crystal structure of ZrSiS. (b) Brillouin zone in space group 129, with highlighted degeneracy enforced by non-symmorphic symmetry. (c) Bulk band structure of ZrSiS. The two degeneracies enforced by non-symmorphic symmetry at the X point are highlighted in blue (above EF) and orange (below EF). (d) Effect of the c/a ratio of isostructural and isoelectronic analogues of ZrSiS on the non-symmorphically induced degeneracies at X. While most compounds exhibit these crossings below and above the Fermi level, there are two exceptions: HfSiTe and ZrSiTe. (e) ARPES spectrum of ZrSiS near X along ?-X. Two bands cross at X due to the non-symmorphic symmetry. Above the crossing, a very intensive surface state(205) is visible. Panel b reprinted with permission from ref (207). Copyright 2017 Elsevier. Panel c adapted and panel e reprinted with permission from ref (24). Copyright 2016 Springer Nature. Panel d reprinted with permission from ref (132). Copyright 2016 IOP Publishing.
And now, to generate some interest in saving the world after Elon Musk is done saving the world, a possible application, the ever popular solar hydrogen:
Figure 11. Schematic diagram of a topological Weyl semimetal for catalyzing the dye-sensitized hydrogen evolution. Reprinted with permission from ref (7). Copyright 2017 John Wiley and Sons.
Well, at least the degeneracy here doesn't involve that awful excuse for a human being in the White House.
A little interesting if still obscure, at least to me, science is a great way to escape. It's a pleasure to be the dumbest guy in the room, really a pleasure.
I wish you a pleasant day tomorrow.
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