The word topology refers to the contours of a surface or the shape of an object. In mathematics, topology classifies objects by the number of holes they have. A ball is a sphere with no hole, whereas a doughnut, with its one hole, is topologically different. The ball is topologically equivalent to an apple, and a doughnut to a cup, but not to a ball or a pretzel, since going from one topology to another would require a dramatic change, like ripping a hole. For this reason, the topological states discovered in some materials are robust and resist disruptions, unless they are as dramatic as the one mentioned previously.
Topological materials provide certain electronic states that persist despite a modification to their physical shape. What’s important isn’t the shape itself but the structure of its electronic bands; regions of electronic energy distribution particular to each material. The electronic band structure is the digital print of a solid or crystalline structure. The energy region at which electrons can flow without resistance is called the conduction band -typically within metallic materials- whereas if the material was in absolute zero, electrons would be confined in the lower energetic valence band, where energy needs to be injected in order for these electrons to jump into the conduction band and flow. So, the conduction band if the lowest closest range of energies to the valence band. The largest the gap between conduction and valence bands, more difficult it is to jump the gap, and materials in this situation are called insulators.
In metals, these energy ranges overlap, so electrons move easily into the conduction band, allowing current to flow. Insulators have a wide band gap, so electrons cannot jump from the valence to the conduction band. Semiconductors have a smaller band gap, so current can flow if the electrons absorb the right amount of energy.
In topological materials, there can be the case where the bulk of the material is insulator, whereas the surface of the material is conductive; in this case the energetic band is also physically located in a particular region of space, such that one can combine electric properties of different materials in a single material and control the location of these properties. Bismuth Telluride is an example, and its atomic structure is shown below in Figure 1.
Figure 1: The topological material trisodium bismuthide. Black = Na, and purple = Bi. Credit: James Collins/Arc Centre Of Excellence In Future Low-Energy Electronics Technologies/Monash University
These weird effects are not disrupted by small defects in a crystal; if there’s some damage in the surface, the current simply flows around it, because the surface state is very stable. Not even a major disruption such as a high increase in temperature is enough to modify topological properties.
This ruggedness results from certain stable electronic states within the materials, which typically contain heavy metals. When electrons in a current hit a defect in the material, they simply flow around it, instead of being scattered or experiencing resistance as in traditional conductors.
Topological insulators have additional interesting properties, for example, current could flow only in one direction on a surface, or even at edges of a crystal structure, as the one achieved by physicists at the University of Zurich, with a new class of materials: higher-order topological insulators. Electric current flows without dissipation on the edges of these crystalline solids, while the rest of the crystal remains insulating.
Schematic of a higher-order topological insulator in the shape of a nanowire, with conducting channels on its edges. Credit: UZH
Another characteristic of the topological properties, is that if the material was perfectly malleable, one could reshape it into different equivalent topological shapes, without changing that basic topological property. Similarly, a doughnut could be remolded into a coffee cup, with the doughnut’s hole becoming the opening in the cup’s handle.
All of the above opens a route to a different classification of materials following their topological properties, and revealing new properties of materials that were thought to be well understood; this last translates into an infinity of applications in electronics, catalysis and quantum computing… And, maybe even redefining the periodic table. The periodic table classifies elements with respect to their bulk properties, so if new properties arrive concerning topological criteria, maybe a new topological periodic table will arise.
In order to know before hand if a material is topological, researchers are developing a method based on its constituent elements, crystal structure and positions of atoms.
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