Angle-resolved Landau spectrum of electrons and holes in bismuth

In elemental bismuth, emptying the low-index Landau levels is accompanied by giant Nernst quantum oscillations. The Nernst response sharply peaks each time a Landau level intersects the chemical potential. By studying the evolution of these peaks when the field rotates in three perpendicular planes defined by three high-symmetry axes, we have mapped the angle-resolved Landau spectrum of the system up to 12 T. A theoretical model treating electrons at the $L$ point with an extended Dirac Hamiltonian is confronted with the experimentally resolved spectrum. We obtain a set of theoretical parameters yielding a good but imperfect agreement between theory and experiment for all orientations of the magnetic field in space. The results confirm the relevance of the Dirac spectrum to the electron pockets and settle the long-standing uncertainty about the magnitude of the $g$ factor for holes. According to our analysis, a magnetic field exceeding 2.5 T applied along the bisectrix axis puts all carriers of the three electron pockets in their lowest ($j=0$) spin-polarized Landau level. On top of this complex angle-dependent spectrum, experiment detects additional and unexpected Nernst peaks of unidentified origin.

Figure 1

(a) Sketch of the Brillouin zone and the Fermi surface of bismuth. The size of the Fermi surface is enhanced to make it visible. It consists of one hole pocket at the $T$ point and three electron pockets at the $L$ points of the Brillouin zone. (b) In our experiment, $\theta$ is the angle between the magnetic field and the trigonal axis, and $\phi$ is the angle between the binary axis and the projection of the magnetic field in the equatorial plane (defined by the binary and bisectrix axes). The three electron pockets are noted $e_1$, $e_2$, and $e_3$ as seen in the figure according to their position to the angle $\phi$.

Figure 2

Evolution of the bismuth model. The horizontal lines indicate the position of Landau levels with opposite spins.

Figure 3

Landau levels ($k_z=0$) as a function of $B^{-1}$ for a magnetic field along (a) trigonal, (b) bisectrix, and (c) binary direction. The origin of the energy is taken at the bottom of the conduction band at zero field.

Figure 4

Carrier density as a function of $B^{-1}$ for a magnetic field along (a) trigonal, (b) bisectrix, and (c) binary direction.

Figure 5

Comparison between theory (vertical lines) and experiment (solid curves) for a magnetic field along (a) trigonal, (b) bisectrix, and (c) binary direction. For each configuration, the Nernst voltage is plotted as a function of $B^{-1}$.

Figure 6

Top: Nernst voltage as a function of $B^{-1}$ with the magnetic field along the bisectrix. Electron peaks occur with a periodicity of 0.405 T${}^{-1}$. This period is twice the cross section of an electron ellipsoid along the bisectrix axis. When $B^{-1}$ is an even (odd) number of times this value, a Nernst peak is sixfold (fourfold) degenerate. Bottom: The $B^{-1}$ position of Nernst peaks as a function of their Landau index for the three pockets. Dirac spectrum leads to a vanishing intercept.

Figure 7

(a) The evolution of the field dependence of the Nernst signal $S_{xy}$ with the orientation of the magnetic field as the field rotates in the (trigonal, binary) plane. Curves are shifted for clarity. (b) Color map of the data presented in the upper panel. Bright lines track the angular evolution of the Nernst peaks corresponding to the intersection of a Landau level and the chemical potential.

Figure 8

Color map of $S_{xy}$ when the magnetic field rotates in three perpendicular planes: (a) (trigonal, binary) plane (i.e., $\phi =0^{\circ }$), (b) (trigonal, bisectrix) plane (i.e., $\phi ={90}^{\circ }$), and (c) (binary, bisectrix) plane (i.e., $\theta ={90}^{\circ }$). On top of the color map, theoretical lines corresponding to the hole (purple) and the three electron pockets $e_1$ (dotted), $e_2$ (black solid), and $e_3$ (gray solid) are plotted. In each panel there are two parts corresponding to the data obtained with two different pairs of electrodes.

Figure 9

The Landau spectrum for a magnetic field rotating in the (trigonal, bisectrix) plane according to theory (lines) and experiment (symbols). Additional peaks refer to those which can be clearly identified as not associated with an electron Landau sublevel.

Figure 10

The Landau spectrum for a magnetic field rotating in the (trigonal, binary) plane according to theory (lines) and experiment (symbols). Additional peaks refer to those which can be clearly identified as not associated with an electron Landau sublevel.

Figure 11

The Landau spectrum for a magnetic field rotating in the (binary, bisectrix) plane according to theory (lines) and experiment (symbols). Additional peaks refer to those which can be clearly identified as not associated with an electron Landau sublevel.

Figure 12

Zoom on color map of $S_{xy}$ for a magnetic field near the trigonal axis. The magnetic field rotates either in the (trigonal, binary) plane ($\phi =0^{\circ }$) (a) or in the (trigonal, bisectrix) plane ($\phi ={90}^{\circ }$). On top of the this color map, the theoretical Landau spectrum for the three electron pockets $e_1$ (dotted line), $e_2$(solid line), and $e_3$ (dash-dot line) are shown.

Figure 13

Nernst voltage as a function of $B^{-1}$ for $\phi ={90}^{\circ }$ as $\theta$ is scanned in the vicinity of 90${}^{\circ }$, which corresponds to the bisectrix direction. The curves are shifted for clarity. The black arrows indicate the position of two hole peaks with opposite spins ($4^{+}$ and $4^{-}$). Near $\theta ={90}^{\circ }$, experiment cannot resolve two distinct peaks. In other words, the Zeeman splitting becomes smaller than the width of a peak.