Longitudinal interlayer magnetoresistance in strongly anisotropic quasi-two-dimensional metals

In strongly anisotropic quasi-two-dimensional (2D) metals, where the interlayer band width is less than the Landau level separation, the role of impurity scattering is enhanced by a magnetic field perpendicular to the conducting layers. This leads to a longitudinal magnetoresistance (MR) in contrast to the prediction of classical theory based on the constant-$\tau$ approximation. The MR has a square-root dependence $R_{zz}\left(B_z\right)$ in a strong field, being linear in the intermediate region. The crossover field allows to estimate the interlayer transfer integral or electron mean-free time. Longitudinal interlayer MR, being robust to the increase of temperature or long-range disorder, is easy for measurements and provides a useful tool to investigate the electronic structure of quasi-2D compounds.

Figure 1

Average interlayer magnetoresistance ${\overline{R}}_{zz}\left(B_z\right)=1/{\overline{\sigma }}_{zz}$ as a function of magnetic field $B_z/{\Gamma }_0$, calculated at $4t_z<\hslash {\omega }_c$ using Eqs. (8) and (16) at four different values of $t_z/{\Gamma }_0=0.2$ (solid black line), $t_z/{\Gamma }_0=4$ (dashed red line), $t_z/{\Gamma }_0=8$ (dotted blue line), and $t_z/{\Gamma }_0=12$ (dash-dotted green line). At $t_z\gg {\Gamma }_0$ there is a wide interval $4t_z<\hslash {\omega }_c\ll {\left(4t_z\right)}^2/{\Gamma }_0$ of linear MR (see dash-dotted green line), while at smaller $t_z$ MR has a square-root field dependence ${\overline{R}}_{zz}\left(B_z\right)\propto \sqrt{B_z}$ (solid black and dashed red lines).

Figure 2

The same as in Fig. 1 but as function of the square root of the magnetic field. At $t_z\lesssim {\Gamma }_0$ MR ${\overline{R}}_{zz}\left(B_z\right)\propto \sqrt{B_z}$ (solid black and dashed red curves are linear). At larger $t_z\gg {\Gamma }_0$ this dependence transforms to linear MR ${\overline{R}}_{zz}\left(B_z\right)\propto B_z$ (see dotted blue and dash-dotted green lines, which look like parabolas).

Figure 3

Interlayer conductivity ${\sigma }_{zz}\left(\Delta \epsilon \right)$ calculated from Eqs. (8) and (16) in the strong-field limit $\hslash {\omega }_c/{\Gamma }_0=40$ as a function of energy counted from the nearest LL at four different values of $t_z/{\Gamma }_0=0.1$ (solid blue line), $t_z/{\Gamma }_0=2$ (dashed green line), $t_z/{\Gamma }_0=5$ (dotted red line), and $t_z/{\Gamma }_0=10$ (dash-dotted magenta line). In the limit $2t_z^2\gg {\Gamma }_0\hslash {\omega }_c$ the interlayer conductivity has approximately a semicircle shape of width $2t_z$. In the opposite limit ${\Gamma }_0\hslash {\omega }_c\gg t_z^2$ the “conducting band” of each LL has the width $\sim \sqrt{{\Gamma }_0\hslash {\omega }_c}$.