Fermi surface and Berry phase analysis for Dirac nodal line semimetals: cautionary tale to SrGa$_2$ and BaGa$_2$

Abstract

A Berry phase of odd multiples of $π$ inferred from quantum oscillations (QOs) has often been treated as evidence for nontrivial reciprocal space topology. However, disentangling the Berry phase values from the Zeeman effect and the orbital magnetic moment is often challenging. In centrosymmetric compounds, the case is simpler as the orbital magnetic moment contribution is negligible. Although the Zeeman effect can be significant, it is usually overlooked in most studies of QOs in centrosymmetric compounds. Here, we present a detailed study on the non-magnetic centrosymmetric $\mathrm{SrGa_2}$ and $\mathrm{BaGa_2}$, which are predicted to be Dirac nodal line semimetals (DNLSs) based on density functional theory (DFT) calculations. Evidence of the nontrivial topology is found in magnetotransport measurements. The Fermi surface topology and band structure are carefully studied through a combination of angle-dependent QOs, angle-resolved photoemission spectroscopy (ARPES), and DFT calculations, where the nodal line is observed in the vicinity of the Fermi level. Strong de Haas-van Alphen fundamental oscillations associated with higher harmonics are observed in both compounds, which are well-fitted by the Lifshitz-Kosevich (LK) formula. However, even with the inclusion of higher harmonics in the fitting, we found that the Berry phases cannot be unambiguously determined when the Zeeman effect is included. We revisit the LK formula and analyze the phenomena and outcomes that were associated with the Zeeman effect in previous studies. Our experimental results confirm that $\mathrm{SrGa_2}$ and $\mathrm{BaGa_2}$ are Dirac nodal line semimetals. Additionally, we highlight the often overlooked role of spin-damping terms in Berry phase analysis.

Figures (13)

• Figure 1: Calculated band structures and Fermi surfaces of $\mathrm{SrGa_2}$ and $\mathrm{BaGa_2}$. (a,c) Left: Band structure calculations with SOC (solid lines) and without SOC (dashed lines) of $\mathrm{SrGa_2}$ (a) and $\mathrm{BaGa_2}$ (c) along high-symmetry paths. The bands that cross the Fermi energy are highlighted in different colors. The inset of (c) shows the Brillouin zone with labels of high-symmetry points. The magenta boxes highlight the Dirac nodal lines along the K-H direction. Right: Zoomed-in view of the Dirac nodal lines along the K-H direction. A small gap develops between the two bands that form the Dirac nodal lines because of SOC. (b,d) Calculated Fermi surfaces for $\mathrm{SrGa_2}$ (b) and $\mathrm{BaGa_2}$ (d) with SOC included. The colors of the Fermi pockets are consistent with those in (a,c). The labels represent the QO frequencies as shown in Fig. \ref{['fig:3']} and Fig. \ref{['fig:5']}.
• Figure 2: Powder X-ray diffraction pattern of $\mathrm{SrGa_2}$. The upper inset is the crystal structure projection along the c-axis showing honeycomb layers of Ga atoms and triangular layers of alkalines. The lower inset shows one piece of single crystal, with each grid equal to 1mm.
• Figure 3: In-plane resistivity and Hall resistivity of $\mathrm{SrGa_2}$ and $\mathrm{BaGa_2}$. (a) In-plane zero field resistivity $\rho_{xx}$ of $\mathrm{SrGa_2}$ and $\mathrm{BaGa_2}$. (b,c) In-plane MR and Hall resistivity of $\mathrm{SrGa_2}$ at different temperatures .
• Figure 4: Angle-dependent dHvA oscillations of $\mathrm{SrGa_2}$ and $\mathrm{BaGa_2}$. (a,d) Angle-dependent dHvA oscillations of $\mathrm{SrGa_2}$ (a) and $\mathrm{BaGa_2}$ (d). The inset of (a) is a sketch of the measurement. (b,e) FFT spectra of $\Delta M$ of $\mathrm{SrGa_2}$(b) and $\mathrm{BaGa_2}$(e). The FFT spectra at $\theta$ = 79° and 90° for $\mathrm{BaGa_2}$ are amplified by 40 and 60 times, respectively. An offset to the FFT amplitude has been applied for better visualization. Dashed lines are guides to the eye for the frequency changes. (c,f) QO frequencies from fundamental oscillations as a function of angle for $\mathrm{SrGa_2}$ (c) and $\mathrm{BaGa_2}$ (f). Symbols are extracted values from FFTs in (b,e), dotted lines are values from DFT, and dashed lines are estimated values of a cylindrical Fermi surface.
• Figure 5: ARPES measurements of the electronic structure of (a)-(f) SrGa$_2$ and (g)-(k) BaGa$_2$. (a)-(b) Fermi surface of SrGa$_2$ near $k_z$ = 0 (102 eV) and $\pi$ (81 eV) measured by linear horizontal (LH) photons. Fermi pockets calculated from density functional theory (DFT) are overlaid on top, the colors of which correspond to those assigned in Fig. \ref{['fig:4']}(b). (c)-(d) High symmetry band dispersions along $\Gamma$-$K$-$M$ and $A$-$H$-$L$ with DFT bands superimposed for comparison. The Dirac point (DP) at $K$ is circled in red. (e) Constant energy contour at $E-E_{\rm F}=-0.85$ eV in the $k_z-k_x$ plane through varying the photon energy, where the photon energies used in (a) and (b) are marked by the dashed blue curves. (f) Band dispersions along the out-of-plane $H$-$K$-$H$ direction showing spin-orbit-coupling(SOC)-split nodal lines of SrGa$_2$. The cut direction is marked by the vertical red arrow in (e). (g) Fermi surface of BaGa$_2$ measured with 86.5 eV LH polarized photons showing both $k_z$ = 0 (solid contours) and $k_z$ = $\pi$ pockets (dashed contours) predicted by DFT. (h)-(i) Equivalent cuts as in (c) and (d) but for BaGa$_2$. The $\overline{K}$ and $\overline{L}$ notation in (h) and (i) is due to the large curvature of the cut shown by the dashed curve in (j), so the second "$K$" or "$L$" deviates from the high-symmetry plane significantly. (j) $k_x$-$k_z$ constant energy contour of BaGa$_2$ at $E-E_{\rm F}=-0.4$ eV. From the blue double arrow along $H$-$K$-$H$, the band dispersions containing the SOC-split Dirac nodal line are extracted and shown in (k). SS: surface states. LH: linear horizontal. LV: linear vertical. CR: circular right. These light polarizations are denoted for different momentum segments of the displayed band dispersions in (c), (d), (h), and (i) to showcase a more complete band structure.