Fermi Surface with Dirac Fermions in CaFeAsF Determined via Quantum Oscillation Measurements

Despite the fact that 1111-type iron arsenides hold the record transition temperature of iron-based superconductors, their electronic structures have not been studied much because of the lack of high-quality single crystals. In this study, we comprehensively determine the Fermi surface in the antiferromagnetic state of CaFeAsF, a 1111 iron-arsenide parent compound, by performing quantum oscillation measurements and band-structure calculations. The determined Fermi surface consists of a symmetry-related pair of Dirac electron cylinders and a normal hole cylinder. From analyses of quantum-oscillation phases, we demonstrate that the electron cylinders carry a nontrivial Berry phase $\pi$. The carrier density is of the order of ${10}^{-3}$ per Fe. This unusual metallic state with the extremely small carrier density is a consequence of the previously discussed topological feature of the band structure which prevents the antiferromagnetic gap from being a full gap. We also report a nearly linear-in-$B$ magnetoresistance and an anomalous resistivity increase above about 30 T for $B\parallel c$, the latter of which is likely related to the quantum limit of the electron orbit. Intriguingly, the electrical resistivity exhibits a nonmetallic temperature dependence in the paramagnetic tetragonal phase ($T>118\mathrm{K}$), which may suggest an incoherent state. Our study provides a detailed knowledge of the Fermi surface in the antiferromagnetic state of 1111 parent compounds and moreover opens up a new possibility to explore Dirac-fermion physics in those compounds.

Popular Summary

Superconductors carry electrical current without energy loss when they are cooled below a certain temperature (known as the transition temperature). At the heart of this phenomenon is the Fermi surface, a boundary that appears when one plots the three-dimensional momentum of electrons in the system. The Fermi surface separates occupied quantum-mechanical states from unoccupied ones in the momentum space; electrons on the Fermi surface form pairs that allow them to superconduct. Although it sounds imaginary, one can see the Fermi surface via various experimental probes. Little is known about the Fermi surface of the parent compounds of iron-based superconductors known as the “1111-type iron arsenides.” These materials hold the record for the highest transition temperature ($-217^{\circ }C$). Here, we characterize the Fermi surface in one of the parent compounds using experimental measurements and theoretical calculations.

We measure quantum oscillations in CaFeAsF, which provide information about the size and shape of the Fermi surface and the effective mass of the electrons. With the aid of band-structure calculations, we conclude that the Fermi surface consists of a cylinder enclosing holelike carriers and a pair of cylinders enclosing electronlike carriers. The carrier density is extremely small, of the order of 0.001 carriers per iron atom. Furthermore, we found that the electronlike carriers carry a nontrivial Berry phase, indicating that they are so-called Dirac fermions, particles that are expected to produce novel quantum-mechanical phenomena and hence are currently searched for in various materials.

Our results greatly increase our knowledge of the electronic structure of 1111 compounds and open a new opportunity to explore Dirac-fermion physics in compounds related to iron-based superconductors.

Figure 1

Crystal and magnetic structure of CaFeAsF. The $a$, $b$, and $c$ directions of the orthorhombic Cmma cell are indicated. The black frame indicates the magnetic cell with the doubled $c$ dimension, and the arrows indicate the Fe moment direction. Fe1 and Fe2 label magnetic sublattices used to calculate partial density of states in Fig. 7.

Figure 2

Temperature dependence of resistivity in CaFeAsF. The $c$-axis and $ab$-plane resistivities were measured using two different samples. The inset explains the contact configuration for the $c$-axis measurements. The upper panel shows the anisotropy ${\rho }_c/{\rho }_{ab}$.

Figure 3

(a) In-plane resistivity in CaFeAsF and its magnetic-field derivative as a function of $B$ applied parallel to the $c$ axis. (b) Fourier transforms of $d\rho /dB$ vs $1/B$ over the field range 5–17.8 T for various field directions $\theta$. Note that the horizontal axis is $F\cos \theta$. The spectra are vertically shifted so that the baseline of a spectrum for an angle $\theta$ is placed at $\theta$ of the vertical axis. (c) Temperature dependence of the $\alpha$ and $\beta$ oscillation amplitudes for $B\parallel c$. The curves are fits to the temperature reduction factor $R_T$. (d) Dingle plot for the $\beta$ frequency for $B\parallel c$. The vertical axis is a logarithm of a reduced amplitude. The line is a linear fit.

Figure 4

(a, inset) In-plane resistivity in CaFeAsF as a function of $B$ up to 45 T for $B\parallel c$ and nearly $B\parallel ab$ ($\theta =91°$). (a, main) Fourier transforms of $d\ln \rho /dB$ vs $1/B$ over the field range 11.4–44.7 T for various field directions $\theta$. Note that the horizontal axis is $F\cos \theta$. The spectra are vertically shifted as in Fig. 3. (b) Magnetic torque $\tau$ in CaFeAsF (upper right) and corresponding Fourier transforms of $d\tau /\mathrm{dB}$ vs $1/B$ plotted against $F\cos \theta$.

Figure 5

(a) Oscillatory part of the resistivity as a function of $1/(B\cos \theta )$ for field directions near $\theta =52.6°$. All the curves are from the hybrid measurements. (b) Oscillatory part of the resistivity as a function of $1/(B\cos \theta )$ after box smoothing. The box (window) size is the same as one oscillation period of the $\beta$ frequency, and hence the $\beta$ oscillation is effectively removed. The curves for $\theta =0$, 28.3, and 48.3° are from the 20-T measurements, while the others from the hybrid measurements.

Figure 6

(a) Oscillatory part of the resistivity for $B\parallel c$ as a function of $1/B$ (solid curve). The original data are the same as those of Fig. 3. The crosses are zeros of the $\beta$ oscillation determined from $d^2\rho /d{B}^2$, and the connecting curve (dashed curve) is obtained by cubic spline interpolation. The $\beta$ oscillation (lower curve, vertically shifted) is obtained by subtracting the interpolated curve from the original. (b) Fourier transforms of the original, interpolation, and $\beta$-oscillation curves. (c) Landau-index plot for the $\beta$ frequency. Integer and half integer indices are assigned to resistivity maxima (solid circles) and minima (open circles), respectively.

Figure 7

Results of the band-structure calculations with spin-orbit coupling. (a) The total and partial densities of states. Up spin is shown above the axis, and down spin below. Fe1 and Fe2 refer to iron sites belonging to different magnetic sublattices (see Fig. 1). (b) Band structure near the Fermi level. The vertical arrow indicates the Dirac point. The color coding is based on orbital weights as explained by the inset, where $e_g$ refers to $d_{Z^2}+d_{X^2-Y^2}$, while $t_{2g}$ $d_{XZ}+d_{YZ}+d_{XY}$. (c) Fermi surface in the antiferromagnetic Brillouin zone.