Magnetoconductivity of Dirac electrons in bismuth

The magnetoconductivity of bismuth is theoretically investigated on the basis of the Kubo formula to interpret the longstanding mystery of linear magnetoresistance in bismuth. First, the magnetoconductivity for the isotropic Dirac model is studied. It is found that the inverse magnetoconductivity increases quadratically at low magnetic fields and is saturated at high fields. This high field property is in contrast to that obtained by the semiclassical theory, where the inverse magnetoconductivity keeps quadratic increase. Next, the magnetoconductivity for the extended Dirac model, which is a realistic model for bismuth, is studied. The inverse magnetoconductivity so obtained is not saturated, but is reduced at high fields. Implications of present results to the linear magnetoresistivity of bismuth are discussed.


Introduction
In 1928, Kapitza reported that the magnetoresistance of bismuth increases linearly with respect to the external magnetic field [1]. Next year, he further reported such a linear dependence can be observed in various materials, which is known as "Kapitza's linear law" [2][3][4][5]. The linear magnetoresistance was a mystery at that time since the all existing theory predicted a quadratic dependence at small fields and a saturating behavior at high fields. This problem was partly solved by Lifshits and Peschanskii for the case with open Fermi surface [6]. Much later, it was shown by Abrikosov that the magnetoresistance becomes linear at extremely high magnetic fields, in the so-called quantum limit, where only the lowest Landau level is occupied, by taking into account the infinite order of impurity scattering beyond the Born approximation [5,7]. The theory of Abrikosov seems to succeed to give a reasonable explanation for the linear magnetoresistance on bismuth. However, it is well known that the effective Hamiltonian of electrons in bismuth is given by the form equivalent to the Dirac Hamiltonian and the carrier density greatly varies as a function of the magnetic field due to the charge neutrality condition of compensated metals [8][9][10]. These specific features of bismuth are indispensable to explain various properties of bismuth, such as the angle resolved quantum oscillation [8,9] and diamagnetism [11], but they were not taken into account in the Abrikosov's theory.
In this work, we investigate the magnetoconductivity of bismuth by taking into account the quantum effect of whole Landau levels on the basis of the Kubo formula [12], the Dirac dispersion, the spatial anisotropy, and the field dependence of the Fermi energy, aiming at giving a possible interpretation to the Kapitza's linear law.

Theory
We consider the isotropic Dirac Hamiltonian under an external magnetic field [10,[13][14][15]: where 2∆ is the energy gap, γ is the isotropic velocity, and σ is the Pauli matrix. π = −i∇ + (e/c)A, where A is the vector potential. The eigenenergy of this Hamiltonian is given by where the cyclotron frequency is defined as Even if we take into account the anisotropy of the velocity, the above isotropic model is essentially valid only with a replacement m c → m h in Eq. (2) [9,15,16]. It is a remarkable characteristic of Dirac electron system that the lowest Landau level, E LLL = E n=0,σ=−1 , is fixed at the initial energy even at high magnetic fields [10]. Note that the cyclotron mass m c = ∆/γ 2 is much less than the bare electron mass, m, in the case of bismuth, e. g., m c /m 0.0019 for the field parallel to the bisectrix axis [9]. This is the reason why we can reach quantum limit at relatively low magnetic field. The complete and normalized wave function has the form where χ u is a wave function of free electrons under a magnetic field [13]. The matrix elements of the velocity operator, which is given by v i = ∂H /∂k i , are calculated as After some straightforward calculations, we obtain Then the matrix elements for v x is given by The operator π ± changes the orbital quantum number n → n ± 1, while σ ± changes the spin quantum number σ → σ ± 1. The only good quantum number is j = n + 1/2 + σ, and the operator v x changes j as j → j ± 1, namely, orbital and spin quantum number cannot be changed simultaneously.
The diagonal conductivity σ xx is given on the basis of the Kubo formula as [12,17] where ε n = (2n + 1)πT , ω λ = 2πλT (n, λ: integer). Here we introduced the effect of impurity scattering, Γ, as iε n = iε n + iΓε n /|ε n |. For the Green function part, the summation with respect to ε n can be carried out as . (11) (ω λ was analytically continued as iω λ → ω.) The contributions from the second and third terms of Eq. (11) correspond to the "Fermi surface term", and that from the first and fourth terms correspond to the "Fermi sea term".
For the interband transition, we have where ∑ n,kz,σ denotes summations with respect to n, k z and σ except for n = 0, k z = 0, σ = −1 of the valence band, and we omitted the k z dependences in energy. N L is the degrees of the Landau-level degeneracy, N L = eB/2πc = ω c ∆/2πγ 2 . A cv n+1σ,nσ means A(E n+1,σ , −E n,σ ). ("c" and "v" abbreviate conduction and valence bands, respectively.) The first and second term corresponds to the orbital transition, and the third term to the spin transition, which appear also in the Wolff's theory [13]. The fourth term is a new term only appears in the interband transition (not appear in the Wolff's theory). The total conductivity is calculated both from the intra-and inter-band contributions.

Isotropic Dirac model
The results of the inverse magnetoconductivity for the isotropic Dirac model is shown in Fig.  1 as a function of ω c /∆. Here we fixed the Fermi energy to be E F /∆ = 5.8, where E F is measured from the center of the band gap. There are clear quantum oscillations, which cannot be obtained by the semi-classical theory. When the Fermi energy E F touches the minimum of the sub bands, σ iso xx shows a peak structure (appears as a dip in the plot of 1/σ iso xx ). The peak width is widened by the impurity scattering Γ. 1/σ iso xx exhibits B 2 dependence with the exception of the quantum oscillation in the weak field region, ω c /∆ 15. This is consistent with the semi-classical transport theory for free electrons [18], where, τ = 1/2Γ. The system reaches the quantum limit for ω c /∆ 17, where only the lowest Landau level is occupied. Beyond the quantum limit, 1/σ iso xx is saturated by the field. This is in contrast to the semi-classical result, where 1/σ xx keeps quadratic increase.

Model for bismuth
Next, we consider the anisotropy and the change of the Fermi energy E F according to the "extended Dirac model", whose energy is given as [9,19] where m h is the effective mass along the magnetic field, and β 0 = e/mc. This model also takes into account the spin splitting and the shift of the lowest Landau level by introducing the g -term in Eq. (14) [9,20,21]. This model has succeeded in giving good agreements with experiments on bismuth [9,19]. The anisotropy is taken into account in ω c and m h . The field dependences of E F (shown in the left panel of Fig. 2) is calculated so as to satisfy the charge neutrality condition [9]. The result of the inverse magnetoconductivity so obtained is shown in the right panel of Fig.  2 for the magnetic field parallel to the trigonal axis. Since E F does not vary so much at low fields, the properties of σ ex xx is basically the same as that of σ iso xx . At high fields, on the other hand, E F changes drastically as E F ∝ B in bismuth due to the charge neutrality condition [8,9] and the effect of the g -term becomes more relevant. Thus the properties of σ ex xx becomes different from those of σ iso xx as shown in Fig. 3. 1/σ ex xx is not saturated but greatly reduced at high fields in contrast to the saturated 1/σ iso xx . This reduction in 1/σ ex xx is a precursor to the quantum limit. For the field parallel to the trigonal axis, the second lowest Landau level, E 0,+ , approaches E F , but cannot cross E F even at high fields since both E F and E 0,+ increases in almost the same way [9]. The system keeps the state just before entering the quantum limit, namely, 1/σ ex xx keeps the dip structure at high fields. Consequently, the quantum effect and the change of E F make the behavior of conductivity completely different from that of semi-classical one.
In order to understand the behavior of magnetoresistance, we need to calculate the magnetic field dependence of Hall conductivity, σ xy (B). Moreover, we also need to estimate the contributions not only from electrons, but also from holes for bismuth. According to the semiclassical theory, the total magnetoresistance exhibits unsaturated ρ semi xx ∝ B 2 behavior at high fields when the carrier numbers of electrons and holes are equal. This consequence is obtained from 1/σ semi xx ∝ B 2 at high fields. Our result of 1/σ ex xx exhibit the drastic reduction in 1/σ ex xx at high fields. Therefore, it is naively expected that the ρ Bi xx will be reduced from the value o ρ semi xx at high fields and become close to ρ xx ∝ B behavior.

Conclusion
We have investigated the magnetoconductivity of Dirac electrons in bismuth on the basis of the Kubo formula. For the isotropic Dirac model, 1/σ iso xx (B) ∝ B 2 at low fields, which is consistent with the semi-classical results for free electrons. In the quantum limit, on the other hand, 1/σ iso xx is saturated. This is contrast to the semi-classical result. We have also calculated the magnetoconductivity for electrons in bismuth based on the extended Dirac model. We have took into account the spacial anisotropy, the spin splitting, and the field dependences of the Fermi energy E F . 1/σ ex xx ∝ B 2 at low fields, whereas 1/σ ex xx is drastically reduced at high fields. This reduction in 1/σ ex xx is significantly contrast to the semi-classical result, which keeps quadratic  Figure 3. Inverse magnetoconductivity for the extended and isotopic Dirac models at high fields.
increase. Consequently, the magnetoconductivity of bismuth will be reduced from the semiclassical value ρ semi xx ∝ B 2 toward ρ xx ∝ B. We believe such a conclusion is useful to interpret the Kapitza's linear law in bismuth.