Magic-angle effects in the interlayer magnetoresistance of quasi-one-dimensional metals due to interchain incoherence

The dependence of the magnetoresistance of quasi-one-dimensional metals on the direction of the magnetic field show dips when the field is tilted at the so-called magic angles determined by the structural dimensions of the materials. There is currently no accepted explanation for these magic-angle effects. We present a possible explanation. Our model is based on the assumption that, the intralayer transport in the second most conducting direction has a small contribution from incoherent electrons. This incoherence is modeled by a small uncertainty in momentum perpendicular to the most conducting (chain) direction. Our model predicts the magic angles seen in interlayer transport measurements for different orientations of the field. We compare our results to predictions by other models and to experiment.

Figure 1

Interlayer magnetoresistance as a function of tilt angle $\theta$ in the $y\text{−}z$ plane. Even for a very small incoherent hopping between the chains of molecules the magic-angle effect is clearly seen. The parameter $k_{0}b$ is a dimensionless parameter describing spread in the distribution of momentum as the particle tunnels between the chains. $k_{0}b=0$ means full coherence, i.e., a delta-function distribution of $k_y$ values. The magnetic frequency ${\omega }_0=eb{v}_{F}B$ is the frequency at which the electrons traverse the open sheets of the Fermi surface. ${\rho }^0$ is the resistivity in zero field, we used $b=c$ and ${\omega }_0∕\Gamma =10$. We also included, as a comparison, the result when no incoherence is present $(k_{0}b=0)$, given by Eq. (16) in the text.

Figure 2

Interlayer magnetoresistance as a function of the magnetic field direction in the $x\text{−}z$ plane. $\theta$ is the angle between the field and the $z$ axis. The upper panel shows a numerical calculation of the so-called Danner-Kang-Chaikin oscillations,[40] from Eq. (19) in the text. The theoretical curve can be compared with Fig. 1 from Ref. 40 shown in the lower panel, with experiments done on ${\left(\mathrm{TMTSF}\right)}_2\mathrm{Cl}{\mathrm{O}}_4$ at ambient pressure and $T=0.5\mathrm{K}$. The dip around zero degree below $3\mathrm{T}$ is due to the sample becoming superconducting. Note that $\theta$ denotes the angle between the magnetic field and the $x$ axis. We used $b=c$, $k_{0}b=0.001$, with ${\omega }_0∕\Gamma =10$ and ${\omega }_0∕t_y=0.1$ at $B=7\mathrm{T}$.

Figure 3

Interlayer magnetoresistivity versus $y\text{−}z$ plane angle, $\alpha$, defined via $\tan \alpha =\sin \varphi ∕\tan \theta$ (see top figure). The middle panel shows the result from our numerical calculation of conductivity using Eq. (22). Modulations appear at the magic angles as the angle $\alpha$ is increased. We used $b=7.581\mathrm{\AA }$ and $c=13.264\mathrm{\AA }$. The other parameters used are $k_{0}b=0.1$, ${\omega }_0∕\Gamma =10$ and ${\omega }_0∕t_y=0.1$. The theoretical curve can be compared with Fig. 4 from Ref. 41 shown in the lower panel. This is an experiment done at $0.32\mathrm{K}$ on ${\left(\mathrm{TMTSF}\right)}_2\mathrm{P}{\mathrm{F}}_6$ with an applied hydrostatic pressure of $8.3\mathrm{kbar}$ to suppress the spin-density-wave state.