Reconstruction of the Fermi surface induced by high magnetic field in the quasi-two-dimensional charge density wave conductor $\eta \text{−}\mathrm{M}{\mathrm{o}}_4{\mathrm{O}}_{11}$

Low-dimensional charge density wave (CDW) systems, which undergo successive so-called Peierls phase transitions at characteristic temperatures associated with corresponding changes in electronic structure, are of great importance for exploring novel quantum physics. Here, we report systematic research of the magnetotransport properties for the quasi-two-dimensional CDW conductor $\eta \text{−}\mathrm{M}{\mathrm{o}}_4{\mathrm{O}}_{11}$ under high magnetic fields up to 59 T. Measurements of in-plane and out-of-plane magnetoresistance and Hall resistance have been carried out to determine the Fermi surface (FS) structure at low temperatures and high magnetic fields. Beyond the quantum limit (QL), fast quantum oscillations (QOs) with complex periods have been found in both MR and Hall resistance, representing a direct measurement of the area of the FS. In addition, we observe a conspicuous phase transition at 45 T. The QO frequency and carrier density successively increase above the QL and phase transition fields. Consequently, we propose that such remarkable feature of the magnetic field dependence of the fast QOs for $\eta \text{−}\mathrm{M}{\mathrm{o}}_4{\mathrm{O}}_{11}$ arises from the field-induced density wave transition and FS reconstruction. We have further discussed the mechanism and provided the evolution on the geometry of the FS with magnetic field in the second CDW state based on the field-dependent FS model.

Figure 1

(a) Temperature dependence of the resistivity ρ for $\eta \text{−}\mathrm{M}{\mathrm{o}}_4{\mathrm{O}}_{11}$ with decreasing temperature. The current directions are parallel to the $b$ axis for ${\rho }_{xx}$ (black line) and the $a$ axis for ${\rho }_{zz}$ (red line). Black arrows indicate the CDW transition temperatures $T_{c1}=105$ K and $T_{c2}=30$ K, respectively. The inset of (a) shows the typical crystal that exhibits a characteristic morphology with natural (100) facet. The red arrows in the inset mark the corresponding crystallographic directions of the grown crystal in the $bc$ plane. (b) The $B-T$ phase diagram with several characteristic fields at different temperatures. “FIPT” denotes the field-induced phase transition.

Figure 2

Magnetic field dependence of (a) ${\rho }_{xx}$ (blue line), ${\rho }_{xy}$ (yellow line), and ${\rho }_{zz}$ (red line) at $T=1.4$ K and with magnetic field up to 59 T, respectively. Note that ${\rho }_{xy}$ has been amplified 5 times and ${\rho }_{zz}$ has been reduced by 8000 times for clarity. The thick line is for the ascending field and the thin line is for the descending field. All the curves shown below are up-sweep ones. (b) Detailed ${\rho }_{xx}$ at various temperatures from 1.4 to 50 K. The red arrow indicates the maximum MR. (c) ${\rho }_{xy}$ at 1.4, 4.2, and 10 K. (d) ${\rho }_{zz}$ from 1.4 to 8 K. The inset of (d) displays the low-field results of ${\rho }_{zz}$ as $B<25\mathrm{T}$. The red arrows mark two characteristic fields of the QL and FIPT, respectively.

Figure 3

(a), (b) The high-field oscillation component $\mathrm{\Delta }{\rho }_{xx}$ and $\mathrm{\Delta }{\rho }_{zz}$ plotted against inverse field 1/$B$ above the $B_{\mathrm{QL}}$ and $B_{\mathrm{PT}}$ after subtracting a smoothed background, respectively. (c), (d) The FFT spectra of ${\rho }_{xx}$ in two field ranges of $19\mathrm{T}<B<45\mathrm{T}$ and $45\mathrm{T}<B<59\mathrm{T}$, respectively. The black dotted curves in (c), (d) indicate the FFT results of ${\rho }_{xy}$ at $T=1.4$ K over different field ranges. (e) The FFT results of ${\rho }_{zz}$ as $45\mathrm{T}<B<59\mathrm{T}$.

Figure 4

(a)–(e) The Lifshitz-Kosevich (LK) fitting of the temperature dependence of the FFT amplitudes for different frequencies, respectively. The red dots are the experimental results, and the black solid lines are the fitted results according to the LK formula shown in the main text.

Figure 5

The schematic pictures of the FS evolution with magnetic field for the quasi-2D CDW conductor $\eta \text{−}\mathrm{M}{\mathrm{o}}_4{\mathrm{O}}_{11}$. (a) FS image as $B<19\mathrm{T}$ according to the band structure calculations [25]. (b) The estimated FS image as $19\mathrm{T}<B<45\mathrm{T}$. (c) The proposed FS picture as $45\mathrm{T}<B<59\mathrm{T}$. The open and red shaded areas in (a)–(c) denote the electron and hole FSs, respectively. The black arrows in (c) indicate the two wave vectors $Q_1$ and $Q_2$, respectively.