Diminishing topological Faraday effect in thin layer samples

A striking feature of three-dimensional (3D) topological insulators (TIs) is the theoretically expected topological magnetoelectric (TME) effect, which gives rise to additional terms in Maxwell's laws of electromagnetism with an universal quantized coefficient proportional to half-integer multiples of the fine-structure constant $\alpha$. In an ideal scenario one therefore expects also quantized contributions in the magnetooptical response of TIs. We review this premise by taking into account the trivial dielectric background of the TI bulk and potential host substrates, and the often present contribution of itinerant bulk carriers. We show (i) that one obtains a nonuniversal magnetooptical response whenever there is impedance mismatch between different layers and (ii) that the detectable signals due to the TME rapidly approach vanishingly small values as the impedance mismatch is detuned from zero. We demonstrate that it is methodologically impossible to deduce the existence of a TME exclusively from an optical experiment in the thin film limit of 3D TIs at high magnetic fields.

Figure 1

Kerr and Faraday rotation after interaction with a single interface in dependence of the permittivity of the second material (while ${\varepsilon }_{r,1}=1$). The dotted lines indicate the region of parameter space that cannot be accessed with physically reasonable material values.

Figure 2

Top: General layer-stack for two interfaces. The contribution from TME is indicated by the magnetoelectric polarization $P_3$, which can only be 0 (no TME contribution) or $1/2$ (TME contribution) [10]. Bottom: Faraday rotation through two interfaces in slab geometry for different combinations of permittivities of all three layers. Layers one and three are taken to be topologically trivial ($P_3=0$), layer two hosts the TME ($P_3=\frac{1}{2}$) with its permittivity set to 10.

Figure 3

Normalized transmission through the first interface of the three-layer systems dependent from multiple orders of magnitude of carrier concentrations in the second layer and the frequency of the incident electromagnetic wave. The normalization is relative to the incident power density in the first layer for each data point.

Figure 4

Cyclotron resonance properties for linearly polarized incident EM wave. (a), (b) Real part of dielectric function in circular basis ${\varepsilon }_{\pm }={\varepsilon }_{xx}\pm \text{i}{\varepsilon }_{xy}$. (c) Normalized transmission through the first interface. (d) Faraday rotation after transmission through the first interface. External magnetic field is along positive $z$ axis. Simulation performed without TME contribution.

Figure 5

Comparison of Faraday rotation with and without TME contribution. Rotation (a) without [and (b) with] TME contribution after the first interface. For noticeable difference the resonances are excluded. (c) Difference in rotation between simulation with and without TME contribution over wide frequency range, including resonances, after the first interface. As a guide to the eye the value for $\alpha /2$ is included. (d) Rotation with TME contribution after both interfaces. (e) Difference in rotation between simulation with and without TME contribution over wide frequency range, including resonances, after both interfaces.

Figure 6

Incident, reflected and transmitted EM wave at an interface, represented by the $x\text{−}y$ plane, between two materials $a$ and $b$. Only wave vector $\mathit{k}$ and the electric field $\mathit{E}$ of the EM wave are shown here. The magnetic field $\mathit{H}$ can be described by $\mathit{k}$ and $\mathit{E}$, see main text.

Figure 7

(a) Rotation angle for individual higher-order reflexes inside a substrate layer of finite thickness after full transmission. (b) Rotation angle for the superposition up to the $n\mathrm{th}$ multiple reflex inside a substrate layer of finite thickness after full transmission. The generalized layer stack is shown as inlay as well as the definition of the reflex index $n$ (reflexes offset for clarity). The dielectric constant of the TI layer is given by ${\varepsilon }_{r,TI}=25$, the substrate ${\varepsilon }_{r,\text{sub}}=10$ and the remaining layers are set to the value of vacuum ${\varepsilon }_{r,\text{vac}}=1$.