Electrons in bismuth

Édelman, V. S.

doi:10.1080/00018737600101452

Cited by 205 publications

(65 citation statements)

References 53 publications

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“…What is important here is that the ratio M is an intrinsic value, which is not affected by impurities or vacancies, and it can be accurately determined by the quantum oscillation measurements. So far, the ratio M has been determined repeatedly by experiments in strongly spin-orbit coupled materials, such as Bi [6,7,8,9,10] and Bi 2 Se 3 [11,12,13]. On the other hand, however, the clear theoretical understandings of M has been pushed aside except for the two-band Dirac electron systems [3,14,5].…”

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confidence: 99%

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Crystalline spin–orbit interaction and the Zeeman splitting in Pb_1−xSn_xTe

Hayasaka

Fuseya

2016

J. Phys.: Condens. Matter

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Abstract. The ratio of the Zeeman splitting to the cyclotron energy (M = ∆E Z / ω c ), which characterizes the relative strength of the spin-orbit interaction in crystals, is examined for the narrow gap IV-VI semiconductors PbTe, SnTe, and their alloy Pb 1−x Sn x Te on the basis of the multiband k · p theory. The inverse mass α, the g-factor g, and M are calculated numerically by employing the relativistic empirical tight-binding band calculation. On the other hand, a simple but exact formula of M is obtained for the six-band model based on the group theoretical analysis. It is shown that M < 1 for PbTe and M > 1 for SnTe, which are interpreted in terms of the relevance of the interband couplings due to the crystalline spin-orbit interaction. It is clarified both analytically and numerically that M = 1 just at the band inversion point, where the transition from trivial to nontrivial topological crystalline insulator occurs. By using this property, one can detect the transition point only with the bulk measurements. It is also proposed that M is useful to evaluate quantitatively a degree of the Dirac electrons in solids.The spin-orbit interaction (SOI) affects the eigenstate of electrons in solids in a variety of ways. It strongly depends on the crystal structure and the momentum of the carrier. One of the most fundamental such effects is the modification of the band structure. For example, in semiconductors of the diamond and zincblende structures, the band modification can be characterized by the spin-orbit splitting energy. But this is not the whole information of the crystalline SOI. Another important information can be obtained under a magnetic field, where we cannot attain to only with the band calculations. The one-body Hamiltonian under the magnetic field can be separated into two part in general: the symmetric and the antisymmetric part with respect to the commutation of the kinematical momentum operator [1,2,3,4,5]. The eigenenergy of the symmetric part is given in terms of the cyclotron energy as ω c (n + 1/2) with an anisotropic cyclotron mass. The eigenenergy of the antisymmetric part is given by the Zeeman energy with an anisotropic g-factor. This Zeeman energy does not originate from the bare electron spins, but originates from the orbital motion of electrons. The antisymmetric part is relevant only in the case with the sizable crystalline SOI. Therefore, the effect of the crystalline SOI is clearly reflected by the antisymmetric part, whose relative strength is characterized by the ratio of the Zeeman splitting to

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“…On the other hand, however, the clear theoretical understandings of M has been pushed aside except for the two-band Dirac electron systems [3,14,5]. Actually, the anisotropic and large M in Bi has been a puzzle for more than half a century [6,7,8,9,10]. Very recently, this long-standing puzzle was eventually solved based on the multiband k · p theory [15].…”

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Crystalline spin–orbit interaction and the Zeeman splitting in Pb_1−xSn_xTe

Hayasaka

Fuseya

2016

J. Phys.: Condens. Matter

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“…This effect, reminiscent of the Coulomb pseudo-gap in localized electronic states, affects only electrons in the immediate vicinity of the Fermi level. It presents the most striking departure from the non-interacting picture of electrons in bulk bismuth.A naturally occurring element known since ancient times [3,4], the semi-metal bismuth has played an important role in the history of solid state physics [5]. Over the last five years, bismuth has attracted new attention focused on its unusual properties in the presence of a strong magnetic field.…”

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confidence: 99%

“…A naturally occurring element known since ancient times [3,4], the semi-metal bismuth has played an important role in the history of solid state physics [5]. Over the last five years, bismuth has attracted new attention focused on its unusual properties in the presence of a strong magnetic field.…”

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Thermodynamic evidence for valley-dependent density of states in bulk bismuth

et al. 2014

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Electron-like carriers in bismuth are described by the Dirac Hamiltonian, with a band mass becoming a thousandth of the bare electron mass along one crystalline axis [1]. The existence of three anisotropic valleys offers electrons an additional degree of freedom, a subject of recent attention [2]. Here, we map the Landau spectrum by angle-resolved magnetostriction, and quantify the carrier number in each valley: while the electron valleys keep identical spectra, they substantially differ in their density of states at the Fermi level. Thus, the electron fluid does not keep the rotational symmetry of the lattice at low temperature and high magnetic field, even in the absence of internal strain. This effect, reminiscent of the Coulomb pseudo-gap in localized electronic states, affects only electrons in the immediate vicinity of the Fermi level. It presents the most striking departure from the non-interacting picture of electrons in bulk bismuth.A naturally occurring element known since ancient times [3,4], the semi-metal bismuth has played an important role in the history of solid state physics [5]. Over the last five years, bismuth has attracted new attention focused on its unusual properties in the presence of a strong magnetic field. More than 120 years after the first discovery of the Nernst-Ettingshausen effect (NE) in the very same material [6], giant NE oscillations were observed in the vicinity of the quantum limit [7]. This limit is attained when the magnetic field is strong enough to confine carriers to their lowest Landau level(s). With currently available magnetic fields, it is only accessible in solids with a carrier concentration as low as in bismuth. Theory expects a prominent role for electron interactions beyond this limit [8]. NE measurements extended to 33 T, well above the quantum limit [9], found features unexpected in the single-particle picture. Torque magnetometry studies by Li et al. [10] up to 31 T showed a structure in magnetization with hysteresis suggesting a correlation-induced phase transition.These results inspired new theoretical studies of the Landau spectrum in bismuth [11,12], which is remarkably complex because the electron dispersion is Diraclike, with an additional anisotropic Zeeman term [13,14]. In particular, in the vicinity of the high-symmetry axis known as the trigonal axis, the magnetic field at which an electron Landau level is depleted rapidly changes with the field orientation. Recently, the angle-resolved Landau spectrum was determined by NE measurements for the whole solid angle [15,16], and found to be in good agreement with theoretical calculations by Fuseya [15], based on an earlier model by Vecchi and co-workers [13]. Thus the non-interacting picture can successfully explain most features of the high-field phase diagram, including those previously attributed to electron interaction [9,10]. Still the unexpected hysteresis seen in the torque measurements [10], reminiscent of a first-order phase transition, remained unexplained. However, subsequent torque magneto...

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“…The Fermi energy determined from the carrier density at these pockets is very small, around 30 meV, and the effective masses of the carriers is also small in certain directions. 1,2 This peculiar electronic structure is closely related to the rhombohedral ͑A-7͒ geometric structure. For Bi in other configurations than the bulk A-7 crystal, very different properties can be expected for two reasons.…”

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confidence: 99%

Electronic structure and Fermi surface of Bi(100)

et al. 2005

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The surface electronic structure of Bi͑100͒ was studied by angle-resolved photoemission and the fullpotential linearized-augmented plane-wave film method. Experimentally, several electronic surface states were identified in the gaps of the projected-bulk band structure close to the Fermi level. Theory shows that these states belong to a spin-orbit split-surface band that extends through the whole Brillouin zone, and that some surface states penetrate very deeply into the bulk. In the experiment, the surface Fermi surface was found to consist of three features: an electron pocket at the ⌫ point, a hole pocket in the ⌫ -M direction ͑i.e., in the direction of the surface-mirror line͒, and a small Fermi-surface element close to the M Ј points.

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Electrons in bismuth

Cited by 205 publications

References 53 publications

Crystalline spin–orbit interaction and the Zeeman splitting in Pb_1−xSn_xTe

Crystalline spin–orbit interaction and the Zeeman splitting in Pb_1−xSn_xTe

Thermodynamic evidence for valley-dependent density of states in bulk bismuth

Electronic structure and Fermi surface of Bi(100)

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Electrons in bismuth

Cited by 205 publications

References 53 publications

Crystalline spin–orbit interaction and the Zeeman splitting in Pb1−xSnxTe

Crystalline spin–orbit interaction and the Zeeman splitting in Pb1−xSnxTe

Thermodynamic evidence for valley-dependent density of states in bulk bismuth

Electronic structure and Fermi surface of Bi(100)

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Crystalline spin–orbit interaction and the Zeeman splitting in Pb_1−xSn_xTe

Crystalline spin–orbit interaction and the Zeeman splitting in Pb_1−xSn_xTe