Suppressed Magnetic Circular Dichroism and Valley-Selective Magnetoabsorption due to the Effective Mass Anisotropy in Bismuth

We measure the far-infrared reflectivity and Kerr angle spectra on a high-quality crystal of pure semimetallic bismuth as a function of magnetic field, from which we extract the conductivity for left- and right-handed circular polarizations. The high spectral resolution allows us to separate the intraband Landau level transitions for electrons and holes. The hole transition exhibits 100% magnetic circular dichroism; it appears only for one polarization as expected for a circular cyclotron orbit. However, the dichroism for electron transitions is reduced to only $13\pm 1\%$, which is quantitatively explained by the large effective mass anisotropy of the electron pockets of the Fermi surface. This observation is a signature of the mismatch between the metric experienced by the photons and the electrons. It allows for a contactless measurement of the effective mass anisotropy and provides a direction towards valley polarized magnetooptical pumping with elliptically polarized light.

Figure 1

(a) Schematic representation of the interaction of charge carriers in bismuth with light. The Fermi surface of bismuth is sketched in the interior of the sample, with a hole pocket (red) oriented along the trigonal ($z$) axis (inspired by Ref. [18]). Three electron pockets (green) are oriented almost perpendicularly to the trigonal axis (with a 6° tilt), rotated by 120° with respect to each other. A magnetic field is applied along the $z$ axis. Radiation with linear polarization ($E_{\mathrm{i}\mathrm{n}\mathrm{c}}$) is propagating almost parallel to the field. Close to the intraband resonances the light reflects ($E_{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}$) with an elliptical polarization and is rotated by the Kerr angle ${\theta }_K$. (b) Top view on the trigonal plane. The effective electric field $E_{\mathrm{tot}}$ in the crystal generates currents (both in real space) in the different pockets (in reciprocal space) according to $j=\sigma E_{\text{tot}}$. The current in the hole pocket results in complete magnetic circular dichroism. The sum of the currents in the electron pockets shows strongly reduced circular dichroism due to the large effective mass anisotropy (bottom).

Figure 2

(a), (b) The real part of the measured conductivity for right and left circularly polarized light, ${\sigma }_{+}$ and ${\sigma }_{-}$, as a function of the photon energy. The magnetic field steps are 0.5 T and the same color coding applies to both panels. The curves are displaced for clarity. The symbols at each peak correspond to Figs. 3 and 3.

Figure 3

Energies of the Landau levels for electrons (a) and holes (b), labeled with their $n$ and spin. The Fermi level (green dots) is taken from Ref. [18]. The relevant transitions for this experiment are indicated as arrows. For electrons, transitions for spin up from level $n$ have the same energy as transitions for spin down from level $n+1$. For holes all transitions have the same energy. (c), (d) Measured energies of the Landau level transitions as a function of magnetic field for ${\sigma }_{+}$ and ${\sigma }_{-}$. The different symbols mark experimental points of the different series of transitions. The lines represent the theoretical field dependence as given by Eq. (1) (holes) and Eq. (2) (electrons).

Figure 4

(a) Comparison of ${\sigma }_{+}$ and ${\sigma }_{-}$ as a function of energy measured at 5.5 T, showing the large dichroism for the hole transition together with the strongly reduced dichroism for the electron transitions. The tiny peak indicated with an arrow is discussed in the text. (b) Example of how the spectral weight for the hole transitions is derived for the ${\sigma }_{-}$ spectrum at 5.5 T. We use a Lorentzian at each transition and a Drude term to account for the low frequency spectral weight. (c) The spectral weight of the intraband transitions, integrated over energies 0–25 meV as a function of magnetic field. The symbols denote the total spectral weight in ${\sigma }_{+}+{\sigma }_{-}$ (diamonds), in ${\sigma }_{+}$ (circles), in ${\sigma }_{-}$ (crosses), in the hole transition only (squares), and in ${\sigma }_{-}$ when the hole spectral weight is subtracted (dots). (d) The magnetic circular dichroism ratio for the electron transitions at the magnetic fields where the hole peak can be subtracted. The line is the theoretical value based on the effective mass anisotropy of the electron pockets. The error bars are explained in the text.

Figure 5

(a) Relative power absorbed in each electron pocket for elliptically polarized light as a function of the azimuth angle of the polarization ellipse $\theta$. (b),(c) Schematics of the valley-polarized magneto-absorption for $\theta =0°$ and $\theta =90°$, respectively.