Warping in topological insulators

Both figures display experimental data on the “Dirac cone".

Both figures display experimental data on the “Dirac cone".

The outward warping is pronounced as well as the broadening of the boundaries of the cone which is caused by collisions of the electrons with obstructions in the real, non-ideal, topological insulator bismuth telluride.

The outward warping is pronounced as well as the broadening of the boundaries of the cone which is caused by collisions of the electrons with obstructions in the real, non-ideal, topological insulator bismuth telluride.

Topological insulators are promising to develop into a material for lossless electricity and information transport. Now, Jaime Sánchez-Barriga and colleagues from HZB investigated for the first time whether the direction of motion of electrons in topological insulators affects their behavior. In their work they identified directions along which electrons are much more prone to scattering losses and cannot conduct as well. To explain their results, they included the spin of the electron and questioned an established theory. The results could push topological insulator research, especially when a future BESSY-VSR will be available to provide much shorter light pulses to study the dynamics of the electrons. Their investigation into the "warping" of topological insulators is published by Physical Review B and has been selected as "Editor's Suggestion". Only six percent of the articles  published there receive this sort of acclaim.

The surfaces of topological insulators conduct electricity in principle without losses, i. e., the electrons manage to avoid hitting obstructions. Nevertheless, experts worldwide are asking themselves why this avoidance does not show up pronounced in the experiment. The HZB team with Jaime Sánchez-Barriga were able to show by simulations that the experimental data on the energy distribution of electrons are better explained when the spin of the electrons is considered. 

Electrons behave in the same way as light
To this end, they investigated the so-called warping of the Fermi surface which describes the energy distribution of the conduction electrons. This is not too much related to warp drive of the science fiction progamme Star Trek since it is not space-time that is bended but the relationship between energy and momentum of electrons at the surface of topological insulators. However, like in space-time they deal with a light cone, the so called Dirac cone, because the electrons in topological insulators behave almost like light. 

Electron speed depends on the direction
In an ideal Dirac cone, the electrons move with the same speed in all directions, like a soccer ball, still air provided, will roll equally fast in all cardinal directions. The material bismuth telluride is, however, known for a strong direction dependence of the electron velocity. Sánchez-Barriga and coworkers discovered now that this dependence is a bit different from the expectation. This means that the Dirac cone is not bent inwards but outwards; and although electrons at the surface of topological insulators should not suffer any collisions with obstructions, these collisions do appear in experiments: the Dirac cone appears washed out and with thicker walls.

Also the losses are direction dependent
Sánchez-Barriga and coworkers discovered that this broadening in the warped Dirac cone of bismuth telluride also depends on the direction. At first, this dependence has to be explained before the limits of lossless transport can be understood. This required, however, much scrutiny. At first, it has to be exluded that the different velocities, i. e., the very warping, cause the different scattering behavior, corresponding to the expectation that a faster ball is more likely to miss its path.

The simplest explanation is the so-called nesting. The nesting of a warped Dirac cone is high if its outline consists of many sections that are parallel to each other on opposite sides. This is maximally the case for a square and a hexagon, hardly for a circle and absent for a triangle.
Sánchez-Barriga and coworkers could not reconcile their results with nesting. Therefore, they searched for factors neglected so far and considered the spin, the angular momentum, of the electrons. In the picture of the kicked ball: “For bismuth telluride, the ball does not only roll faster when I kick it northwards as compared to eastwards. Only eastwards it rolls normally, northwards one cannot call it rolling since its spin axis points a bit skywards like a cut ball in socker” Jaime Sánchez-Barriga explains. The scientists performed simulations and only considering this spin, they could reproduce their experimental results.

It’s all about dynamics
The losses due to collisions can also be viewed upon as lifetime of the electrons which apparently depends on the direction. These dynamical effects occur on an extremely short timescale and shall be investigated with time resolution next. This research will profit largely from the possibilities of a future BESSY-VSR light source.

The published experiments have been conducted in the framework of the Helmholtz-Russia Joint Research Group of Andrei Varykhalov and are part of the DFG Priority Program “Topological Insulators” which is coordinated by Oliver Rader.

The paper is to be found here:
Anisotropic effect of warping on the lifetime broadening of topological surface states in angle-resolved photoemission from Bi2Te3
Phys. Rev. B 90, 195413 – Published 10 November 2014
J. Sánchez-Barriga, M. R. Scholz, E. Golias, E. Rienks, D. Marchenko, A. Varykhalov, L. V. Yashina, and O. Rader