Quantum oscillations in a lead chalcogenide three-dimensional Dirac system

The de Haas–van Alphen (dHvA) and the Shubnikov–de Haas (SdH) oscillations were used to probe the properties of the Fermi surface in single crystals of ${\mathrm{Pb}}_{0.83}{\mathrm{Sn}}_{0.17}\mathrm{Se}$ with reduced charge concentration. Pronounced low-frequency oscillations of $\sim 8$ T in the [001] direction were observed, confirming the single Fermi surface cross section. Due to the low effective charge concentration, the ultraquantum limit is reached already at $\sim 10$ T. We observe $\pi$-Berry phase shift in the phase of both dHvA and SdH oscillations, which confirms the 3D Dirac nature of the energy band dispersion. In the construction of the Landau level diagram we use a combined indexing method for conductivity, magnetic susceptibility, and magnetization, which is based on the indexing used in the topological insulators. Moreover, the reliability of the indexing method is increased because we use, beside minima and maxima, zeros of the oscillations as well. Different microscopic parameters were calculated from the quantum oscillations in the magnetization, conductivity, and resistivity.

Figure 1

(a) $M$ of samples A1 and B1 at 5 K showing a linear dependence on $B$ with the dHvA oscillations. In the case of the sample A1, the oscillations are considerably stronger in amplitude than for the sample B1. (b) Strong linear-like magnetoresistance $\mathrm{MR}=[{\rho }_{xx}\left(B\right)-{\rho }_{xx}]/{\rho }_{xx}$ of samples A2 and B2 with pronounced SdH oscillations. For $B>10$ T both samples are in the ultraquantum limit. The inset to Fig. 1 shows ${\rho }_{xx}$ vs $T$ for the samples A2 and B2.

Figure 2

Analysis of the oscillatory component of the magnetization $\mathrm{\Delta }M$ for samples A1 and B1. Panels (a) and (b) show the temperature dependence of the dHvA oscillations. The oscillations start at a field as low as 1 T. In panel (a) Eq. (1) with fixed parameters [33] of $F, m_{\mathrm{c}}, T_{\mathrm{D}}$, and ${\varphi }_{\mathrm{p}}$ obtained by the indirect method as described in the text is fitted to the experimental data recorded at 5 K for sample A1. The inset shows the FFT analysis which reveals the single-frequency nature of the oscillations with $F_{\mathrm{FFT}}=8.5\left(4\right)$ T and $F_{\mathrm{FFT}}=7.4\left(5\right)$ T for samples A1 and B1, respectively. Using the LK theory the effective mass coefficient $\eta =m_{\mathrm{c}}/m_{\mathrm{e}}$ (c) and the Dingle temperature ($T_{\mathrm{D}}$) (d) are obtained. (c) Temperature dependence of normalized oscillation amplitude for the peaks positioned at $B=3.5$ T and 3.7 T, for samples A1 and B1, respectively, is used for retrieving the effective mass. The effective mass is 0.04(1) $m_{\mathrm{e}}$ for both samples, whereas sample B1 has a larger $T_{\mathrm{D}}$, which is in accordance with a smaller oscillation amplitude for the sample B1. The vertical axis on panel (d) is $F(\mathrm{\Delta }{\rho }_{xx},B,T)=ln(\mathrm{\Delta }M{B}^{1/2}sinh[2{\pi }^2{k}_{\mathrm{B}}T/\hslash {\omega }_{\mathrm{c}}])$ with $T$ set to 5 K.

Figure 3

Analysis of the SdH oscillations in resistivity $\mathrm{\Delta }{\rho }_{xx}$, for the samples A2 (a) and B2 (b). Both panels shows the $T$ dependence of the SdH oscillations after a linear background was subtracted. The peaks show the spin splitting at high magnetic fields. The inset shows the FFT analysis, which gives the peaks corresponding to the base frequency $F$ and the higher harmonics ($2F$ and $3F$) that originate from the spin splitting. The LK analysis of $T$ (c) and $B$ (d) amplitude attenuation, for the fields where the spin split effects are neglectable, is used for retrieving $m_{\mathrm{c}}$ and $T_{\mathrm{D}}$. The $m_{\mathrm{c}}$ values extracted from the amplitudes at $B=2.38$ T and 2.5 T, for sample A2 and B2, respectively, agree with the effective mass obtained from the dHvA oscillations, whereas the SdH oscillations give a higher $T_{\mathrm{D}}$ than the dHvA oscillations. The vertical axis on panel (d) is $F(\mathrm{\Delta }{\rho }_{xx},B,T)=ln(\mathrm{\Delta }{\rho }_{xx}{B}^{1/2}sinh[2{\pi }^2{k}_{\mathrm{B}}T/\hslash {\omega }_{\mathrm{c}}])$ with $T$ set to 1.8 K.

Figure 4

LL diagrams for the oscillatory part of the magnetization $\mathrm{\Delta }M$, resistivity $\mathrm{\Delta }{\rho }_{xx}$, and conductivity $\mathrm{\Delta }{\sigma }_{xx}$. The intercept, indicated by an arrow, directly recovers the ($\delta +\beta$) value from Eq. (2), and provides information on the Berry phase. For the Dirac case, $\beta =1/2$, and $\beta =0$ for the Schrödinger case, $\delta =\pm 1/8$. (a) Indexing of the dHvA magnetization data for samples A1 (red) and B1 (blue) gives the intercepts close to 3/8, which corresponds to the $\pi$-Berry phase shift. (b) The SdH resistivity indexing data. The intercept for the sample A2 (red) points to the existence of the $\pi$-Berry phase, whereas for the sample B2 (blue) it points to a vanishing Berry phase, which is in contradiction with the results obtained from the dHvA oscillations. (c) After transformation to the conductivity the intercepts of both samples, A2 (red) and B2 (blue), point to a $\pi$-Berry phase. For indexing of minima, maxima, and zeros for all three quantities we follow the rules given in Table 1.