Angular dependence of the Fermi surface cross-section area and magnetoresistance in quasi-two-dimensional metals

The analytical and numerical study of the angular dependence of magnetoresistance in layered quasi-two-dimensional (Q2D) metals is performed. The harmonic expansion analytical formulas for the angular dependence of Fermi-surface cross-section area in external magnetic field are obtained for various typical crystal symmetries. These formulas correct the previous results and allow the simple and effective interpretation of the magnetic quantum oscillations data in cuprate high-temperature superconducting materials, in organic metals and in other Q2D metals. Another analytical result for the azimuth-angle dependence of the Yamaji angles is obtained for the elliptic in-plane Fermi surface. The relation between the angular dependence of magnetoresistance and of Fermi-surface cross-section area is derived. The applicability region of all results obtained is investigated using the numerical calculations.

Figure 1

The illustration of the FS shape, given by Eq. (32), with the explanation of the angle notations, used in the text. The parameters $m=4$, $\beta =0.05$ and $2C_1{t}_c/E_F=0.2$.

Figure 2

(Color online) The in-plane FS shape (a) and the first Yamaji angle as function of the azimuth angle $\phi$ (b) for the elongated FS with monoclinic symmetry and the Fermi momentum given by $k_0\left(\varphi \right)={\sum }_{j=0}^4{k}_{4j0}\text{cos}\left(4j\varphi \right)$ with $\beta =k_{40}/k_{00}=0.5$. The higher harmonics are added to smooth the FS [see (a)]. The blue solid curve [see (b)] is the numerical result for the first Yamaji angle, obtained using Eq. (27); the red dotted curve is the result of Eq. (14); the green dashed curve is the result from Eq. (34) and magenta dash-dotted curve is the result from Eq. (24) and used in Refs. 8, 22. One can see that Eq. (14) gives the best result for Yamaji angles when the FS is strongly elongated. The harmonic expansion still gives a reasonable result, though the coefficient $\beta =0.5\sim 1$ is not small. The result of Eq. (24) gives much weaker $\phi$-dependence than the correct one.

Figure 3

(Color online) The in-plane FS shape (a) and the first Yamaji angle as function of the polar angle $\varphi$ (b) for the FS with tetragonal symmetry and the Fermi momentum given by Eq. (32) with $\beta =-0.07$. (a) shows that even for very small parameter $\beta$ the in-plane FS strongly differs from the circle. It is almost quadratic. Hence, the harmonic expansion of the $\varphi$ dependence of the FS shape is applicable for most compounds with tetragonal symmetry. In (b) the blue solid curve is the numerical result for the first Yamaji angle, obtained using Eq. (27). The red dashed curve is the result of Eq. (14); it gives too strong and opposite $\varphi$ dependence, which is completely incorrect. The green dotted curve is the result from Eq. (34); it gives very good agreement with the numerical result. The magenta dash-dotted curve illustrates Eq. (24), used by Bergemann et al. (Refs. 8, 22); this result gives too weak $\varphi$ dependence in agreement with the discussion in the end of Secs. 3A, 6.

Figure 4

The illustration of the FS shape given by Eq. (35). The parameters $m=2$, $\beta =0.05$, and $2C_1{t}_c/E_F=0.2$.

Figure 5

(Color online) The amplitude of the $\phi$ dependence of the extremal cross-section area as function of the polar angle $\theta$ for tetragonal (blue solid line) and hexagonal (green dashed line) symmetries in the case of straight interlayer electron hopping.

Figure 6

(Color online) The normalized conductivity ${\sigma }_{zz}$, calculated from Eq. (39) for the dispersion Eq. (49) as function of the polar angle $\theta$ for two different azimuthal angles $\varphi$ at several values of ${\omega }_c\tau =1$ (black dashed line), ${\omega }_c\tau =2$ (red dash-dotted line), ${\omega }_c\tau =4$ (blue short-dashed line) and ${\omega }_c\tau =8$ (green dotted line). The solid magenta line gives the result of Eq. (48).