Figure 1

(Color online) Schematic of the Faraday and Kerr optical system capable of operating on various laser lines in the terahertz spectral region. The figure, adapted from Ref. 17 which details the Faraday experiment, depicts additional optics within the red-dashed trapezoid allowing Kerr measurements by capturing reflected light from the sample.Reuse & Permissions

Figure 2

(Color online) Calibration of the Faraday and Kerr angle measurement system. The Faraday angle, Kerr angle, and relative transmission and reflection from a GaAs 2DEG heterostructure were measured at an optical frequency of 5.24 meV and temperature of 80 K as a function of magnetic field. The Faraday and Kerr angles are simultaneously fit with effectively two parameters, the number density and mobility, found to be $n=2.0\times {10}^{11}{\text{cm}}^{-2}$ and $\mu =-1.19\times {10}^5{\text{cm}}^2/\text{V}\text{s}$. The fits (red line) are overlayed on top of the data (black dots) for comparison. The same two parameters are used to calculate reflection and transmission (normalized to the $B=0$ value) and overlayed on the data.Reuse & Permissions

Figure 3

(Color online) Kerr angle and reflection of linearly polarized light normalized to the $B=0\text{T}$ value were measured at 10 K and an optical frequency of 5.24 meV. Reflection of nearly circularly polarized light normalized to the $B=0\text{T}$ value was a separate measurement performed only on sample 1. [(a)–(d)] Sample 1 data (black dots) are depicted with several different fits superimposed. A homogeneous bulk response cannot describe the data (blue dashed-dotted). A homogeneous bulk response with surface states (red solid) fit the data. The weighted optical response of two crystallites with different carrier density (green dashed-dotted) can reproduce the salient features of the data. Slightly elliptically polarized light, characterized by parameter $\beta =0.13$, is used in calculations for Fig. 3d. [(e)–(h)] Sample 2 data (black dots) show a very different response but was only 2 mm away from sample 1 in the as-grown crystal. A homogeneous bulk response (blue dashed-dotted) does not quite describe the data. Addition of a surface-state response does not improve the fits. A weighted response of two crystallites with different carrier density (green dashed-dotted) more closely resembles the data. No reflectivity data using circular polarized light was taken for sample 2. $\beta$ is assumed zero in calculations in Fig. 3h.Reuse & Permissions

Figure 4

(Color online) Calculated optical response produced by the bulk properties and the expected surface-state response as measured by ARPES of ${\text{Bi}}_2{\text{Se}}_3$ as a function of magnetic field for series of surface-state Fermi energies measured above the Dirac point as given in the legend: $E_F=0$, 10, 20, 30, 40, 50, 75, 100, 150, 200, 250, 300, 350, and 400 meV. The Kerr angle as well as the reflectance (normalized to zero field) in a pure circular or linear state are calculated. Each row of graphs represents a different set of Drude parameters describing the bulk and surface-state response. The second and third row graphs, (e)–(h) and (i)–(l), are offset for clarity such that the Kerr angle is zero and $R/R\left[0\text{T}\right]$ is unity in zero magnetic field. The graphs with the Fermi energy $E_F=0$ correspond to the Dirac point where there is no contribution from the surface state, only bulk response. All graphs are calculated with a bulk mass $m_{cb}=0.15m_e$ and a bulk scattering rate ${\gamma }_b=10{\text{cm}}^{-1}$ and using the bulk phonon parameters as measured in Ref. 19. [(a)–(d)] Very low bulk density to demonstrate the response expected from the surface states: ${\omega }_{pb}=10{\text{cm}}^{-1}$, ${\gamma }_s=10{\text{cm}}^{-1}$. (e)–(h) ${\omega }_{pb}=400{\text{cm}}^{-1}$, ${\gamma }_s=3{\text{cm}}^{-1}$ (i)–(l) ${\omega }_{pb}=400{\text{cm}}^{-1}$, ${\gamma }_s=10{\text{cm}}^{-1}$. (m)–(p) ${\omega }_{pb}=400{\text{cm}}^{-1}$, ${\gamma }_s=25{\text{cm}}^{-1}$.Reuse & Permissions

Figure 5

(Color online) Calculated optical response produced by a hypothetical gate voltage which shifts the Fermi energy of the surface state by 30 meV. The change in Kerr angle produced by this shift at various Fermi energies for a specific surface scattering rate is shown for each column. All graphs are calculated using the bulk phonon parameters measured in Ref. 19 as well as the Drude bulk parameters ${\omega }_{pb}=400{\text{cm}}^{-1}$, ${\gamma }_b=10{\text{cm}}^{-1}$, and $m_{cb}=0.15m_e$. The surface-state 2D plasma frequency and mass are calculated based on the Fermi level and the measured ARPES dispersion relation (Ref. 11). The surface scattering rates are assumed to be (a) ${\gamma }_s=10{\text{cm}}^{-1}$, (b) ${\gamma }_s=25{\text{cm}}^{-1}$, (c) ${\gamma }_s=50{\text{cm}}^{-1}$, and (d) ${\gamma }_s=100{\text{cm}}^{-1}$.Reuse & Permissions