Asymmetric Cherenkov acoustic reverse in topological insulators

Sergey Smirnov

Phys. Rev. B 90, 125305 – Published 8 September 2014

Abstract

A general phenomenon of the Cherenkov radiation known in optics or acoustics of conventional materials is a formation of a forward cone of, respectively, photons or phonons emitted by a particle accelerated above the speed of light or sound in those materials. Here we suggest three-dimensional topological insulators as a unique platform to fundamentally explore and practically exploit the acoustic aspect of the Cherenkov effect. We demonstrate that by applying an in-plane magnetic field to a surface of a three-dimensional topological insulator one may suppress the forward Cherenkov sound up to zero at a critical magnetic field. Above the critical field the Cherenkov sound acquires pure backward nature with the polar distribution differing from the forward one generated below the critical field. Potential applications of this asymmetric Cherenkov reverse are in the design of low energy electronic devices such as acoustic ratchets or, in general, in low power design of electronic circuits with a magnetic field control of the direction and magnitude of the Cherenkov dissipation.

Received 9 June 2014
Revised 26 August 2014

DOI:https://doi.org/10.1103/PhysRevB.90.125305

Authors & Affiliations

Sergey Smirnov

Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany

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Issue

Vol. 90, Iss. 12 — 15 September 2014

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Images

Figure 1
The schematic picture of the Cherenkov acoustic reverse on a surface of a three-dimensional topological insulator subject to an in-plane magnetic field. The helical particle's momentum and energy before exciting the Cherenkov sound are $p$ (black thick horizontal arrows) and $ɛ$ , while after emitting a phonon with momentum $q$ (red wavy arrows) they become $p^{'}$ (black thick inclined arrows) and $ɛ^{'}$ . An in-plane magnetic field (blue vertical arrows) is applied in the direction opposite to the $y axis$ . The black circles represent low energy isotropic surfaces of constant energy given by Eq. (3). Their center $p_{H}$ (black dot) is the magnetic field dependent Dirac point. In the upper part $p_{H} < | p |$ resulting in pure forward Cherenkov sound. In the lower part $p_{H} > | p |$ . In this case the Cherenkov sound reverses and acquires pure backward nature. In both cases the Cherenkov sound distribution is shown as the shaded red area with phonon momenta $q$ (red wavy arrows) inside.
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Figure 2
The polar distributions of the Cherenkov sound intensity $W (ϕ)$ on a surface of ${Bi}_{2} {Te}_{3}$ for the case $| H | < H_{c}$ . The magnitude of the helical particle momentum $| p | = 4.05 \times 10^{- 27}$ is chosen to be small to stay within low energies well captured by the Dirac isotropic model. The polar distributions correspond to different magnitudes of the magnetic field, i.e., to different values of the parameter $h$ : $0.0$ (black), $50.0$ (red), $100.0$ (blue), $150.0$ (green). The critical value of $h$ corresponding to $H_{c}$ is $h_{c} = 272.0$ . The Cherenkov sound is purely forward for all $h < h_{c}$ . When $h$ increases, $W (ϕ)$ decreases and totally vanishes at $h = h_{c}$ .
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Figure 3
The polar distributions of the Cherenkov sound intensity $W (ϕ)$ on a surface of ${Bi}_{2} {Te}_{3}$ for the case $| H | > H_{c}$ . The helical particle momentum is the same as for Fig. 2. The polar distributions correspond to $h = 450.0$ (red), $500.0$ (blue), $600.0$ (green), $700.0$ (black). The Cherenkov sound is of pure backward nature for all $h > h_{c}$ .
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Figure 4
The polar distributions of the Cherenkov sound intensity $W (ϕ)$ on a surface of ${Bi}_{2} {Te}_{3}$ for the case $| H | < H_{c}$ . The helical particle momentum is the same as for Fig. 2. The polar distributions correspond to $h = 100.0$ and $ϕ_{H} = - π / 2$ (black), $ϕ_{H} = - π / 4$ (red) and $ϕ_{H} = - 3 π / 4$ (blue). The Cherenkov sound is mainly forward but narrow backward angular sectors appear for $ϕ_{H} = - π / 4$ and $ϕ_{H} = - 3 π / 4$ .
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Figure 5
The polar distributions of the Cherenkov sound intensity $W (ϕ)$ on a surface of ${Bi}_{2} {Te}_{3}$ for the case $| H | > H_{c}$ . The helical particle momentum is the same as for Fig. 2. The polar distributions correspond to $h = 700.0$ and $ϕ_{H} = - π / 2$ (black), $ϕ_{H} = - π / 4$ (red) and $ϕ_{H} = - 3 π / 4$ (blue). The Cherenkov sound is mainly backward, but narrow forward angular sectors appear for $ϕ_{H} = - π / 4$ and $ϕ_{H} = - 3 π / 4$ .
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