Quantum oscillations near the metamagnetic transition in ${\text{Sr}}_3{\text{Ru}}_2{\text{O}}_7$

We report a detailed investigation of quantum oscillations in ${\text{Sr}}_3{\text{Ru}}_2{\text{O}}_7$, observed inductively (the de Haas–van Alphen effect) and thermally (the magnetocaloric effect). Working at fields from 3 to 18 T allowed us to straddle the metamagnetic transition region and probe the low- and high-field Fermi liquids. The observed frequencies are strongly field dependent in the vicinity of the metamagnetic transition, and there is evidence for magnetic breakdown. We also present the results of a comprehensive rotation study. The most surprising result concerns the field dependence of the measured quasiparticle masses. Contrary to conclusions previously drawn by some of us as a result of a study performed with a much poorer signal-to-noise ratio, none of the five Fermi-surface branches for which we have good field-dependent data gives evidence for a strong-field dependence of the mass. The implications of these experimental findings are discussed.

Figure 1

(Color online) Schematic construction of the Fermi surface of ${\text{Sr}}_2{\text{RuO}}_4$, $a$ and $b$, and of ${\text{Sr}}_3{\text{Ru}}_2{\text{O}}_7$, $c$, $d$, $e$, and $f$, where in the last two, the $\sqrt{2}\times \sqrt{2}$ reconstruction of the BZ accompanied by backfolding was included, and the new BZ is represented by the dashed diamond. All axes were scaled by the parameters of the undistorted lattice. The red dots represent the location of a putative Van Hove singularity.

Figure 2

(Color online) Top panel: typical second-harmonic dHvA oscillations measured with $B\parallel c$-axis, at a temperature of 55 mK [see Mercure et al. for a discussion of the thermometry in this system (Ref. 11)]. Bottom panel: Fourier transform of the data in the top panel between 5 and 6.5 T (shown as a red square). Five frequencies were observed, at 0.15, 0.43, 0.90, 1.78, and 4.13 kT.

Figure 3

(Color online) (a) The change in sample temperature $T_S$ during a sweep of the magnetic field at 0.04 T/min (blue) and $-0.04\text{T}/\text{min}$ (red) as a function of magnetic field ${\mu }_{0}H$. The average sample temperature is 160 mK. (b) Fourier transform in $1/{\mu }_{0}H$ of the blue trace in panel (a). In green is given the Fourier transform of a dHvA measurement using first harmonic detection over a comparable magnetic field region at base temperature. The label (A) indicates the new frequency found at 110 T. (c) Temperature dependence of the amplitude of the 3 kT frequency of ${\text{Sr}}_2{\text{RuO}}_4$ as measured by magnetothermal oscillations. The red curve is a fit of the temperature derivative of the Lifshitz-Kosevich formula to the data. (d) Amplitude of the 110 T peak in the Fourier transform of the data in panel (a) as a function of sample temperature $T_S$. The red curve is a fit of the temperature derivative of the Lifshitz-Kosevich function to the data. This fit gives a mass $m_{110\text{T}}$ for the newly observed frequency of $8\left(1\right)m_e$.

Figure 4

(Color online) Field dependence of the dHvA spectra observed with the field aligned with the direction of the $c$-axis. The color scale corresponds to the amplitude of the spectrum.

Figure 5

(Color online) Top panel: angular dependence of the dHvA spectra observed in the low-field side of the metamagnetic transition, where the color represents the spectral intensity while the horizontal and vertical axes represent the angle and the dHvA frequency multiplied by $\text{cos}\left(\theta \right)$, respectively. Fourier transforms were taken between 5 and 6.5 T. Bottom panel: angular dependence of the spectra in the high-field side of the metamagnetic transition. Fourier transforms were taken between 10 and 18 T.

Figure 6

(Color online) Quasiparticle mass of five out of six frequencies identified in the low-field side as a function of applied magnetic field. These correspond, respectively, to the bands ${\alpha }_1$, ${\alpha }_2$, $\delta$, ${\gamma }_1$, and $\beta$. The anomalous phase region, situated between 7.9 and 8.1 T, is indicated with vertical red lines. The fits were calculated over field ranges of $0.01{\text{T}}^{-1}$ except for the 0.15 kT frequency, for which a width of $0.02{\text{T}}^{-1}$ was necessary in order to properly resolve the corresponding peaks in the spectra. Gaps in the data are due to the amplitudes becoming too small to fit because of beat patterns in the data.

Figure 7

(Color online) LK fits performed in various field ranges for three of the six detected frequencies, 0.15, 1.78, and 4.13 kT, which correspond to the $\beta$, ${\alpha }_1$, and ${\alpha }_2$ bands. As in the data of Fig. 6, for the case of 0.15 kT, inverse field windows of a width of $0.02{\text{T}}^{-1}$ were used while for the other two, the width was of $0.01{\text{T}}^{-1}$. The field regions are indicated in color boxes in the respective insets, and the same color schemes are respected for the data. Integrated peaks in the Fourier spectra are represented with various symbols while fitted LK curves are shown with solid lines. The values in the legends correspond to the extracted quasiparticle masses.

Figure 8

(Color online) Quasiparticle mass of five simulated dHvA frequencies similar to those detected in our experiments, calculated using the same method as that used in Fig. 6. The data was modeled using quasiparticle masses which were strongly enhanced near 8 T. The oscillations were calculated using the masses and mean-free path extracted from the experiment, as well as amplitudes, field-dependent frequencies, and beat patterns similar to those in the data and uncorrelated Gaussian distributed noise was added to the oscillations.

Figure 9

(Color online) Quasiparticle mass of five simulated dHvA frequencies similar to those detected in our experiments, calculated using the same method as that used in Fig. 6. The data was modeled using a constant quasiparticle mass and a scattering rate which was strongly enhanced near 7.9 T, driving the amplitude to zero at this field value, similar to the experimental observations.

Figure 10

Quasiparticle mass extracted from the merged peak composed of the frequencies at 0.11 and 0.15 kT, originating, respectively, from the ${\gamma }_2$ and $\beta$ bands, where in the simulation, the mass of ${\gamma }_2$ band is strongly enhanced while that of the $\beta$ band is not. The mass enhancement is washed out by the $\beta$ band, which possesses a mass of $5.5m_e$.

Figure 11

(Color online) Example of the two types of nonlinear LK fits to the data. The absolute value of the sum of the LK function (blue curve) and random Gaussian noise is shown in black stars, using a quasiparticle mass of $12m_e$. To this data was fitted the LK expression using a two parameter fit (red curve), which yielded a small underestimate of $11.4m_e\pm 0.6m_e$. A three parameter fit was also performed, featuring a free offset (green curve), which yielded an overestimate of the mass of $14m_e\pm 2m_e$.

Figure 12

(Color online) Simulations of dHvA data featuring a field-dependent signal-to-noise ratio, and its mass analysis using a three parameter fit featuring a free offset. The top panel presents noiseless simulated dHvA data, featuring two frequencies of 4.20 and 4.25 kT, leading to beat patterns, a mean-free path of $1\mu \text{m}$, and a quasiparticle mass of $12m_e$. The following two panels show the same data with two different added noise levels. These sets of data were analyzed with three parameter fits, the results of which are presented in the bottom three panels.