Calculation of magnetic properties of metals by means of the magnetic-field-containing relativistic tight-binding approximation method

Magnetic properties of metals are investigated through electronic structure calculations based on the recently proposed magnetic-field-containing relativistic tight-binding approximation (MFRTB) method [Phys. Rev. B 91, 075122 (2015)]. It is found that electronic energy bands for a metal immersed in a uniform magnetic field have a cluster structure in which multiple energy bands lie within a small energy width. Each cluster corresponds to an energy level that is derived on the basis of the semiclassical approximation. While the cluster is responsible for the de Haas–van Alphen (dHvA) oscillations, constituent energy bands of the cluster cause additional oscillation peaks of the magnetization that are not explained by the conventional Lifshitz-Kosevich formula. Also, the energy width of the cluster leads to the reduction of the amplitude of the dHvA oscillations, which can be observed as the pseudo Dingle temperature and/or the overestimation of the curvature of the Fermi surface.

Figure 1

Dependences of the total energy (blue) and magnetization (red) on the inverse of the magnitude of the magnetic field in the range $p/q=0.0594\text{–}0.178$. Symbols (2a), (2b), (2c), and (2d) indicate the magnetic fields at which we calculate the DOSs [see Figs. 2–2].

Figure 2

Magnetic field dependence of the DOSs for systems immersed in high magnetic fields. (a) DOS for the system immersed in the magnetic fields (2a) of Fig. 1. (b) DOS for the system immersed in the magnetic fields (2b) of Fig. 1. (c) DOS for the system immersed in the magnetic fields (2c) of Fig. 1. (d) DOS for the system immersed in the magnetic fields (2d) of Fig. 1.

Figure 3

Dependences of the total energy (blue) and magnetization (red) on the inverse of the magnitude of the magnetic field in the range $p/q=0.130\text{–}0.132$. Symbols (5a)–(5d) indicate the magnetic fields at which we calculate the DOSs [see Figs. 5–5].

Figure 4

Energy band structure for the case of the magnetic field (5d) that is indicated in Fig. 3. This energy band structure corresponds to the DOS of Fig. 2. Symbols $Z$, $R$, $M$, and $\mathrm{\Gamma }$ in the horizontal axis denote special $\mathit{k}$ points in the magnetic first Brillouin zone [12]. Coordinates of special $\mathit{k}$ points $Z$, $R$, $M$, and $\mathrm{\Gamma }$ are given by $(0,0,\pi /a)$, $(\pi /a,0,\pi /a)$, $(\pi /a,\pi /qa,0)$, and $(0,0,0)$, respectively.

Figure 5

Magnetic field dependence of the energy band structure for the system immersed in a magnetic field. (a) In the case of the magnetic field (5a) of Fig. 3. (b) In the case of the magnetic field (5b) of Fig. 3. (c) In the case of the magnetic field (5c) of Fig. 3. (d) In the case of the magnetic field (5d) of Fig. 3.

Figure 6

Energy dependences of $m_c^{\mathrm{MFRTB}\left(\text{A}\right)}(E,k_z^{\mathrm{ext}})/m$, $m_c^{\mathrm{MFRTB}\left(\text{B}\right)}(E,k_z^{\mathrm{ext}})/m$, $m_c^{\mathrm{MFRTB}\left(\text{C}\right)}(E,k_z^{\mathrm{ext}})/m$ and $m_c(E,k_z^{\mathrm{ext}})/m$ for the electron orbital ($k_z^{\mathrm{ext}}=\pi /a$) in the case of $p/q=1/1427\approx 6.998\times {10}^{-4}(B=9.82\text{T}$).

Figure 7

Magnetic field dependences of $\mathrm{\Delta }{m}_c^{\mathrm{MFRTB}\left(\text{X}\right)}(E_F,k_z^{\mathrm{ext}})$ (X = A, B, and C) for (a) the electron orbital ($k_z^{\mathrm{ext}}=\pi /a$) and (b) the hole orbital ($k_z^{\mathrm{ext}}=0$).

Figure 8

Magnetic field dependences of the total energy (filled circle) in the $p/q$ range (c), $5.069\times {10}^{-3}–5.666\times {10}^{-3}$. The solid line and plots denote resultant fitted curves of the LK formula and calculation results of the MFRTB method, respectively.

Figure 9

Magnetic field dependence of the total energy for the cases of a magnetic field ranging from $p/q=2.878\times {10}^{-2}$ to $5.703\times {10}^{-2}$. The solid line and plots denote calculation results of the LK formula with rigorous values of parameters (Table 2) and those of the MFRTB method, respectively. The inset is the magnified view of the dependence for the high magnetic field region.