Magnetic oscillations of in-plane conductivity in quasi-two-dimensional metals

We develop the theory of transverse magnetoresistance in layered quasi-two-dimensional (Q2D) metals. Using the Kubo formula and harmonic expansion, we calculate intralayer conductivity in a magnetic field perpendicular to conducting layers. The analytical expressions for the amplitudes and phases of magnetic quantum oscillations (MQO) and of the so-called slow oscillations (SO) are derived and applied to analyze their behavior as a function of several parameters: magnetic field strength, interlayer transfer integral, and Landau-level width. Both the MQO and SO of intralayer and interlayer conductivities have approximately opposite phase in weak magnetic field and the same phase in strong field. The amplitude of SO of intralayer conductivity changes sign at ${\omega }_c{\tau }_0=\sqrt{3}$. There are several other qualitative differences between magnetic oscillations of in-plane and out-of-plane conductivity. The results obtained are useful to analyze experimental data on magnetoresistance oscillations in various strongly anisotropic Q2D metals.

Figure 1

The amplitude of quantum oscillations of in-plane diagonal conductivity for three different ratios $t_z/E_F$. (a) The amplitude of quantum oscillations of normalized in-plane diagonal conductivity $A_{xx}^{\text{QO}}d/G_0$ given by Eq. (41) as a function of $1/\lambda \sim B_z$ [here $G_0=e^2/\left(\pi \hslash \right)$ is the quantum of conductance] for three different values of Fermi energy $E_F$. The positions of local minima of MQO amplitude (beat nodes) do not depend on Fermi energy $E_F$, but the amplitude of MQO amplitude does according to Eq. (41) and discussion after Eq. (42). The taken parameters are $m_{*}=0.04m_e$, ${\mathrm{\Gamma }}_0=14.5K$, $t_z=10\text{meV}$, $E_F=\{5,10,20\}{t}_z$. (b) The same as in panel (a) at fixed $E_F$ but for three different values of $t_z$. The values at local minima of MQO depend on $t_z$. The taken parameters are $m_{*}=0.04m_e$, ${\mathrm{\Gamma }}_0=14.5K$, $t_z=\{1/5,1/15,1/20\}{E}_F$, $E_F=200\text{meV}$.

Figure 2

Quantum oscillations of in-plane diagonal conductivity as compared to $R_{D}cos\left(\overline{\alpha }\right)$ and to the oscillating part ${\rho }_{\text{osc}}$ of the DOS, where ${\rho }_{\text{osc}}\equiv -2{\rho }_{0}cos\left(\overline{\alpha }\right)R_D{J}_0\left(\lambda \right)$. The phases of DOS and of ${\sigma }_{xx}$ oscillations coincide everywhere except the proximity of beat nodes. The taken parameters are $m_{*}=0.04m_e$, ${\mathrm{\Gamma }}_0=14.5K$, $t_z=20\text{meV}$, $E_F=200\text{meV}$.

Figure 3

Quantum oscillations of intralayer diagonal conductivity ${\sigma }_{xx}^{\text{QO}}$, of interlayer conductivity ${\sigma }_{zz}^{\text{QO}}$, and of magnetization $\tilde{M}$ as a function $1/\lambda =B_z/\left(2\pi \mathrm{\Delta }F\right)$. One can see that the oscillations of ${\sigma }_{zz}^{\text{QO}}$ are shifted from $\tilde{M}$ by a quarter of a period, but at $1/\lambda \approx 0.01036$ the phase shift is close to $\pi$. Except the beat nodes, the oscillations of ${\sigma }_{xx}$ have the same phase as those of the density of states, but are in antiphase with the oscillations of ${\sigma }_{zz}$. The parameters for numerical calculations are $m_{*}=0.04m_e$, ${\mathrm{\Gamma }}_0=14.5K$, $t_z=10\text{meV}$, $E_F=200\text{meV}$.

Figure 4

The amplitude (a) and magnitude (b) of slow oscillations of ${\sigma }_{xx}$ and ${\sigma }_{zz}$. The parameters are the same as in Fig. 3. (a) Amplitude $A_{xx}^{\text{SO}}$ of slow oscillations of normalized intralayer diagonal conductivity ${\sigma }_{xx}^{\text{SO}}d/G_0$ as a function of $1/{\gamma }_0=\hslash {\omega }_c/\left(2\pi {\mathrm{\Gamma }}_0\right)\propto B_z$. This plot demonstrates that the amplitude $A_{xx}^{\text{SO}}$ changes its sign at ${\gamma }_0=\pi /\sqrt{3}$. (b) Slow oscillations of intralayer ${\sigma }_{xx}$ and interlayer ${\sigma }_{zz}$ conductivity as a function of $1/\lambda =\hslash {\omega }_c/\left(4\pi t_z\right)\propto B_z$. The slow oscillations of ${\sigma }_{xx}$ and ${\sigma }_{zz}$ are in antiphase at low magnetic field and have the same phase at high field.