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

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

Taichi Terashima    Hishiro T. Hirose National Institute for Materials Science, Tsukuba, Ibaraki 305-0003, Japan    David Graf National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA    Yonghui Ma    Gang Mu    Tao Hu State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China CAS Center for Excellence in Superconducting Electronics (CENSE), Shanghai 200050, China    Katsuhiro Suzuki Research Organization of Science and Technology, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan    Shinya Uji National Institute for Materials Science, Tsukuba, Ibaraki 305-0003, Japan    Hiroaki Ikeda Department of Physics, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
July 12, 2019

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 compehensively 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 . The carrier density is of the order of 10 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- magnetoresistance and an anomalous resistivity increase above about 30 T for , 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 ( 118 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.

preprint: Ver. 9


I introduction

Since the discovery of superconductivity at = 26 K in LaFeAs(OF) by Kamihara et al. Kamihara et al. (2008), iron-based superconductors have been studied extensively. By replacing La with smaller rare-earth elements, of up to 56 K was achieved in the same 1111-type structure immediately after the discovery Kito et al. (2008); Ren et al. (2008); Yang et al. (2008); Wang et al. (2008a). There are now various structure types: the 122- (e.g. BaFeAs Rotter et al. (2008); Sasmal et al. (2008)), 111- (e.g. LiFeAs Tapp et al. (2008); Pitcher et al. (2008); Wang et al. (2008b)), 11-type (e.g. FeSe Hsu et al. (2008)) structures and so on. Nonetheless, the above record has not yet been surpassed by other structure types as far as bulk materials are concerned. It is therefore crucial to unravel the normal-state electronic structure of the 1111 iron arsenides from which the highest among the iron-based superconductors emerges. Unfortunately, however, such studies suffer from difficulty in obtaining high-quality single crystals of the 1111-type iron arsenides.

There is a variant of the 1111 iron arsenides, i.e., CaFeAsF (or SrFeAsF) Matsuishi et al. (2008); Tegel et al. (2008); Han et al. (2008) that has the same ZrCuSiAs-type structure as LaFeAsO, but the LaO layers in LaFeAsO are replaced by the CaF layers (Fig. 1). The electronic band structure near the Fermi level is predicted to be similar to that of LaFeAsO Shein and Ivanovskii (2008). CaFeAsF exhibits a structural and an antiferromagnetic transition as LaFeAsO does Tegel et al. (2008); Xiao et al. (2009). The antiferromagnetic structure is the same stripe-type one as LaFeAsO Xiao et al. (2009). By partial substitution of Fe by Co, Ca(Fe, Co)AsF exhibits superconductivity with a maximum of 22 K Matsuishi et al. (2008). Further, of 56 K has been observed for rare-earth doped compounds Wu et al. (2009); Cheng et al. (2009). Following initial reports of single-crystal growth of (Ca, Na)FeAsF Shlyk et al. (2014) and CaFeAsF Tao et al. (2014), Ma et al. have succeeded in growing large high-quality single crystals of undoped CaFeAsF by self-flux method Ma et al. (2015).

In this article, we report resistivity and magnetic-torque measurements down to 0.03 K (0.4 K) up to 18 T (45 T) performed on high-quality single crystals of CaFeAsF. The temperature dependence of the resistivity is nonmetallic down to = 118 K and exhibits two clearly separated anomalies at and = 107 K, which are attributed to a structural and an antiferromagnetic transition, respectively. We observe clear quantum oscillations both in the resistivity and magnetic torque in the antiferromagnetic ground state. Two frequencies and are resolved, and their field-angle dependences indicate that they are from highly two-dimensional Fermi-surface cylinders. From detailed analyses of the oscillation phases, we find a nontrivial Berry phase associated with the oscillation, demonstrating that the cylinder is a Dirac-fermion cylinder. Our band-structure calculations indicate that the Fermi surface consists of a hole cylinder at the point of the Brillouin zone and a symmetry-related pair of electron cylinders originating from Dirac lines located away from . Thus the and cylinders are attributed to the electron and hole cylinders, respectively. The carrier density is extremely small, of the order of 10 per Fe. Additionally, we report a nearly linear-in- magnetoresistivity and an anomalous resistivity increase above about 30 T for . Among other things, our work opens up a new possibility to explore Dirac-fermion physics in a branch of the family of the iron-based superconductors with a simple band structure.

Ii Methods

High-quality single crystals of CaFeAsF were prepared by a CaAs self-flux method in Shanghai Ma et al. (2015). The crystals were platelike with an -plane area of up to 1 mm and a thickness of 50 m or less.

A 20-T superconducting magnet with a dilution refrigerator at the National Institute for Materials Science and the 45-T hybrid magnet with a He refrigerator at the National High Magnetic Field Laboratory were used. In either magnet, samples can be rotated in situ, and the field angle is measured from the axis. Electrical resistivity measurements were performed with a four-contact method both for current and (see Fig. 2 for the contact configuration for ). The contacts were spot-welded or made with conducting silver paste. A total of six samples were prepared: two -plane and one -axis samples were used for SdH measurements, while the remaining one -plane and two -axis samples were used only for zero-field temperature-dependence measurements. Microcantilever torque measurements were performed on another two crystals. Consistent quantum-oscillation results were obtained for the three SdH and two torque crystals.

Fully relativistic electronic band-structure calculations with spin-orbit coupling were performed by using the WIEN2k code with the PBE-GGA potential Blaha et al. (2001); Perdew et al. (1996). We used the experimental lattice parameters of the orthorhombic crystal structure (space group Cmma) at = 2 K: = 5.514594 Å, = 5.476802 Å, = 8.576845 Å, = 0.66329 Xiao et al. (2009)111Y. Xiao (private communication).. The antiferromagnetic ordering with = (1,0, 1/2) (w.r.t Cmma) Xiao et al. (2009) was incorporated by using space group Ibam (#72) with a doubled dimension (Fig. 1). In the self-consistent calculations, the muffin-tin radii of Ca, Fe, As and F atom were set to 2.15, 2.31, 2.19, and 2.26 in units of the Bohr radius, respectively. The maximum modulus of reciprocal vectors was chosen such that . The corresponding Brillouin zone was sampled using a -mesh for the self-consistent calculations, and -mesh for calculations in the density of states.

Figure 1: Crystal and magnetic structure of CaFeAsF. The , , and directions of the orthorhombic Cmma cell are indicated. The black frame indicates the magnetic cell with the doubled 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: (Color online). Temperature dependence of resistivity in CaFeAsF. The -axis and -plane resistivities were measured using two different samples. The inset explains the contact configuration for the -axis measurements. The upper panel shows the anisotropy .

Iii Temperature dependence of resistivity

Figure 2 shows the in-plane and -axis resistivities as a function of temperature . The resistivities increase initially as is lowered from room temperature as reported in Ma et al. (2015). These measurements were made on the very same high-quality single crystals that exhibit quantum oscillations at low temperatures. This indicates that the observed non-metallic behavior is intrinsic nature of CaFeAsF.

The resistivities decrease sharply below = 118 K and exhibit a kink at = 107 K as reported in Ma et al. (2015). These anomalies at and most likely correspond to a structural and an antiferromagnetic phase transition, respectively, as observed in a neutron powder-diffraction experiment by Xiao et al Xiao et al. (2009). The -axis resistivity of this sample exhibits a sudden faint drop at = 20 K, which is not observed in the other samples and hence is probably due to a crack. The resistivity anisotropy is large and basically increases with decreasing temperature.

The room-temperature in-plane and -axis resistivities, averaged out over the three in-plane and three -axis samples, are 1.9(8) mcm and 2.3(6) mcm, respectively. The large errors can be ascribed to errors in the sample dimensions, irregular sample shapes and inhomogeneous current distribution. The large anisotropy of the order of 10 is broadly consistent with the large upper-critical-field anisotropy of 9 observed for CaFeCoAsF Ma et al. (2017) as the two anisotropies are approximated by and , respectively.

Iv quantum oscillations

iv.1 Theoretical background

Quantum oscillation of the resistivity can be described by Richards (1973); Shoenberg (1984)


where is the background resistivity, a positive coefficient, and we have neglected harmonics. We have tacitly assumed that since the Hall resistivity is much smaller than Cheng et al. (2009). The frequency is related to a Fermi-surface cross sectional area as . The temperature and Dingle reduction factors are given by and , where , , and the coefficient is 14.69 T/K. The Dingle temperature is inversely proportional to the carrier scattering time : . The spin reduction factor describes the interference of oscillations from up- and down-spin electrons and is given by , where is the spin factor. is 0 for a two-dimensional (2D) Fermi-surface (FS) cylinder while it is + or when the oscillation is from a minimum or maximum cross section of a three-dimensional (3D) FS pocket. is the Berry phase, which is 0 for normal electrons but for Dirac fermions Mikitik and Sharlai (1999, 1998). The sign of is essential to determine the Berry phase in systems where electronic bands are doubly degenerate at each due to spin degrees of freedom as is the case with the present one *[][.Evenifasystemunderconsiderationisantiferromagneticallyordered; thedoubledegeneracyremainsaslongastheproductoftheinversion(andatranslationifnecessary)andthetimereversalisasymmetryofthesystem.]Cvetkovic13PRB. If and is neglected, resistivity maxima occur for normal fermions when takes integer values, while minima for Dirac fermions. If , the opposite is true.

iv.2 Frequency and effective mass

Figure 3: (a) In-plane resistivity in CaFeAsF and its magnetic-field derivative as a function of applied parallel to the axis. (b) Fourier transforms of d/d vs over the field range 5–17.8 T for various field directions . Note that the horizontal axis is . The spectra are vertically shifted so that the baseline of a spectrum for an angle is placed at of the vertical axis. (c) Temperature dependence of the and oscillation amplitudes for . The curves are fits to the temperature reduction factor . (d) Dingle plot for the frequency for . The vertical axis is logarithm of a reduced amplitude. The line is a linear fit.

Figure 3(a) shows the in-plane resistivity and field derivative d/d as a function of up to 17.8 T applied along the axis. The field derivative shows an initial rise up to 0.6 T, above which it stays approximately constant except oscillations. Namely, the magnetoresistance is nearly linear in .

Quantum oscillations are discernible above 5 T. Figure 3(b) shows Fourier transforms of oscillations as a function of for various field directions . Two frequencies and are resolved: = 19.6 T and = 46.7 T for , which correspond to 0.14 and 0.34% of the cross section of the antiferromagnetic Brillouin zone. For both frequencies, stays constant for the investigated range, indicating that they are from highly 2D FS cylinders. The effective masses associated with the and orbits for are estimated from the temperature dependences of the oscillation amplitudes to be 0.383(4) and 0.92(1), respectively, in units of the free electron mass [Fig. 3(c)]. Figure 3(d) shows the field dependence of the oscillation amplitude in the form of a Dingle plot. The amplitudes were estimated from the extremums of the pure oscillation obtained below in Sec. IV.3. From the linear fit to the data, = 2.7 K, corresponding to a carrier mean free path of 21 nm ( , see Table I for ).

In order to search for higher frequencies, i.e., larger FS pockets, we extended measurements up to = 45 T by using the hybrid magnet. The inset of Fig. 4(a) shows semilogarithmic plots of in-plane resistivity vs field. For , the resistivity at = 0.4 K shows a remarkable increase above 30 T. The increase is suppressed as the temperature is raised, and almost disappears at = 20 K. For nearly ( = 91), no such an increase is observed. A similar behavior is observed for the -axis resistivity (data not shown). Although the origin of the anomalous resistivity increase is not clear at present, it may be related to the quantum limit (i.e., the limit where only the lowest Landau level is occupied) of the orbit ( = 19.6 T). Apart from the anomalous increase, the undulation of the = 0.4 K curve for is dominated by the oscillation: the maxima at 12, 16, and 23 T are ascribable to it, and no faster oscillation is seen.

The main panel of Fig. 4(a) shows Fourier transform analysis of resistivity data measured in the hybrid magnet. We have analyzed the logarithmic derivative instead of d/d to suppress huge background due to the anomalous resistivity increase near . The spectra show peaks corresponding to , , and their harmonics: peaks around 100 T are ascribable to 2 and/or 6. No higher fundamental frequency than exists.

We also measured magnetic torque [Fig.4(b)]. The undulation of the torque curves is again basically explicable with the oscillation, and no faster oscillation is seen. The Fourier spectra confirm that there is no frequency higher than . To summarize, our experimental results indicate that the Fermi surface is composed of only two types of extremely small cylinders and .

Figure 4: (a, inset) In-plane resistivity in CaFeAsF as a function of up to 45 T for and nearly ( = 91). (a, main) Fourier transforms of d/d vs over the field range 11.4–44.7 T for various field directions . Note that the horizontal axis is . The spectra are vertically shifted as in Fig. 3(b). (b) Magnetic torque in CaFeAsF (upper right) and corresponding Fourier transforms of d/dB vs plotted against .

iv.3 Oscillation phase

Figure 5: (a) Oscillatory part of the resistivity as a function of 1/() for field directions near = 52.6. All the curves are from the hybrid measurements. (b) Oscillatory part of the resistivity as a function of 1/() after box smoothing. The box (window) size is the same as one oscillation period of the frequency, and hence the oscillation is effectively removed. The curves for = 0, 28.3, and 48.3 are from the 20-T measurements, while the others from the hybrid measurements.

We now consider the spin reduction factors and oscillation phases by analyzing the waveform of the observed oscillations. To avoid ambiguity due to background subtraction, we subtract the same linear background with = 260 cm and = 89.9 cm/T from the raw data. The parameters were determined by fitting a line to the curve of Fig. 3(a) in the field range 5–17.8 T.

For a 2D FS cylinder, the effective mass varies as 1/cos as the field direction is varied. varies accordingly and may cross zero to change sign. For such a “spin-zero” (), where is an integer.

We begin with for the oscillation. The Fourier transforms in Fig. 4(a) show that the amplitude almost vanishes at = 52.6 but revives at 57.4, suggesting a spin-zero near 52.6. Figure 5(a) shows the oscillatory part of the resistivity for field directions around = 52.6. The curves at = 38.1, 42.9, and 47.7 show a minimum near , while those at 52.6, 57.4, and 62.2 a maximum: that is, the spin-zero exists between = 47.7 and 52.6. Since the fundamental oscillation is weak at these angles ( = 47.7 and 52.6) because of , the waveform is strongly influenced by the harmonics and the oscillation, which explains the fact that the peak positions at these angles slightly differ from the other angles. Similarly, looking at , we notice the second spin-zero between = 62.2 and 67.0 and the third between 67.0 and 71.8.

Let us assume that the first spin-zero is at , where = 1.43 and hence . If = 2 ( = 3.49), the second and third spin-zeros occur at = 62.6 and 69.1, which are consistent with the experimental observations. No other choice of is consistent with the experiment: cannot explain the second spin-zero while produces extra spin-zeros. Since = 0.92 at = 0, and hence . The conclusion at = 0 is confirmed by a more general analysis which shows that the experimental observations can be explained with a value between 3.30 and 3.55, corresponding to the first spin-zero between 49.2 and 52.6. Since shows a maxima at at low angles where [Figs. 5(a) and 6(a)], the carriers are normal fermions ( = 0).

Figure 6: (a) Oscillatory part of the resistivity for as a function of (solid curve). The original data is the same as that of Fig. 3(a). The crosses are zeros of the oscillation determined from d/d, and the connecting curve (dashed curve) is obtained by cubic spline interpolation. The oscillation (lower curve, vertically shifted) is obtained by subtracting the interpolated curve from the original. (b) Fourier transforms of the original, interpolation, and -oscillation curves. (c) Landau-index plot for the frequency. Integer and half integer indices are assigned to resistivity maxima (solid circles) and minima (open circles), respectively.

We now consider for the orbit. Figure 6 shows the oscillatory part of the resistivity for as a function of . We decompose this data into the oscillation and the rest, the latter of which contains the oscillation, as follows: We determine the positions of the zeros of the oscillation (crosses in the figure) by interpolating the maximum and minimum positions of d/d, which are dominated by the oscillation. We apply cubic-spline interpolation to them, subtract the interpolated curve (dashed curve) from the data, and obtain the pure oscillation (lower curve). The Fourier transforms in Fig. 6(b) confirms the success of the decomposition. Figure 6(c) shows a Landau-index plot based on the resistivity maxima and minima of the thus obtained oscillation. Integer and half integer indices are assigned to resistivity maxima and minima, respectively. A linear fit (solid line) gives a slope of 0.0213 T, which corresponds to = 47.0 T, in excellent agreement with the Fourier analysis. The extrapolation of the line intercepts the = 0 line at = 0.02(2). The obtained intercept is much closer to the 2D value (0) than the 3D values () and supports the two dimensionality of the cylinder. Note that the intercept also confirms that = 0

We next examine the oscillation. Figure 5(b) shows the oscillatory resistivities after box-smoothing. The box (window) size has been matched to the period of the oscillation so that the oscillation has been suppressed effectively. Looking at , we notice that the curves show a minimum for but a maximum for up to 71.8 (it is difficult to classify the curve for ). This indicates that the first spin-zero for the oscillation occurs between 48.3 and 57.4. Let us assume that it is at , where = 0.596 and hence . For , changes sign at least once between = 57.4 and 71.8, which is inconsistent with the experimental observation. Hence, , and at (). A more general analysis indicates that a value between 1.41 and 1.73 can explain the experimental observation and confirms at . Since the curve for shows a minimum near and maxima near and 2.5, for the oscillation. That is, the cylinder is a Fermi surface of Dirac fermions. (It is difficult to determine whether = 0 or for the oscillation: the number of the observed oscillation periods is too small and the resistivity maxima and minima cannot be determined accurately because the oscillation is weak.)

V comparison with band-structure calculations

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 refers to , while . (c) Fermi surface in the antiferromagnetic Brillouin zone.

We now consider results of our band-structure calculations. Figure 7 shows the calculated density of states, band structure, and Fermi surface. Those results are reminiscent of similar results for the antiferromagnetic state of LaFeAsO Yin et al. (2008). The density of states shows that the Fermi level is located at the bottom of a deep pseudo gap. The density of states at is of the order of 0.1 states/eV per spin per cell. The difference in the occupied states below between up- and down-spin at the Fe1 or Fe2 site gives an antiferromagnetic moment of 2.0 , which is much larger than the experimental value of 0.49(5) Xiao et al. (2009). We realize that the significant difference in the Fe1 (or Fe2) partial density of states between up- and down-spin is not restricted to near but exists even at 4 eV below . This is very different from a textbook picture of a spin-density wave where the spin polarization is expected to occur only near and indicates that the antiferromagnetic order involves states far below as was pointed out early on in e.g. Johannes and Mazin (2009).

The calculated band structure [Fig. 7(b)] shows that two bands cross , giving rise to a hole cylinder at the center of the Brillouin zone and two symmetry-related electron cylinders off toward the antiferromagnetic direction [Fig. 7(c)]. We note that the electron cylinder encloses a Dirac line: See the band structure along the line between and ‘’. An and a band cross immediately below without spin-orbit coupling. Spin-orbit coupling induces a gap of about 5 meV, which is too small to be visible in Fig. 7(b). We also note that the two -crossing bands are very flat along the line Z, which indicates a highly two-dimensional electronic state and is consistent with the large anisotropy of the resistivity. This contrasts with the case of antiferromagnetic state of BaFeAs, where the band structure is three dimensional, producing closed Fermi pockets Shimojima et al. (2010); Terashima et al. (2011).

We attribute the and frequencies to the electron and hole cylinders, respectively. The Dirac nature of the cylinder is thus confirmed theoretically as well. Based on this assignment and from the experimental frequencies and effective masses, we can estimate the carrier densities to be = 1.7 holes/Fe and = 1.4 electrons/Fe, and the Sommerfeld coefficient to be 1.3 mJ/(mol K). The slight imperfection of the carrier compensation can be ascribed to possible small corrugation of the cylinder, error in the frequency, and/or off-stoichiometry. See table I for other parameters such as Fermi velocity and energy.

FS cylinder
carrier type Dirac electron normal hole
(T) 19.6 46.7
0.383(4) 0.92(1)
(10 per Fe) 1.4 1.7
(10 cm) 2.2 2.6
) 0.024 0.038
(10 m/s) 7.4 4.7
(meV) 12 5.9
Table 1: Carrier type, quantum-oscillation frequency , effective mass , carrier density , Fermi momentum , Fermi velocity , and effective Fermi energy determined from the quantum-oscillation measurements. for and . for the cylinder refers to the sum of the carriers of the two cylinders occurring in the Brillouin zone. To derive , a linear dispersion () was used for the cylinder, while a quadratic one () for .

Vi discussion

First of all, the standard band-structure calculations explain the Fermi surface in the antiferromagnetic state of CaFeAsF reasonably well despite the fact that they significantly overestimate the antiferromagnetic moment. This is very similar to what we saw in the case of the antiferromagnetic state of BaFeAs Terashima et al. (2011). Since the antiferromagnetic ordering involves deep states (at least) down to 4 eV below , band-energy adjustments of order of 100 meV, which are often employed to improve the agreement between the experimental and calculated Fermi surfaces, cannot resolve the disagreement about the moment.

It is interesting to note here that the Fermi surface in the low-temperature nonmagnetic state of FeSe strikingly differs from that expected from band-structure calculations Terashima et al. (2014); Tan et al. (2013); Maletz et al. (2014); Nakayama et al. (2014); Shimojima et al. (2014); Audouard et al. (2015); Watson et al. (2015a, b): notably, the carrier density is more than one order-of-magnitude smaller than calculated Terashima et al. (2014). In the case of FeSe, only the structural (nematic) transition occurs without antiferromagnetic ordering McQueen et al. (2009). The stark contrast between the above antiferromagnetic compounds and FeSe seems to suggest that interactions that make the Fermi surface significantly deviate from band-structure calculations are largely suppressed by antiferromagnetic ordering.

The observed antiferromagnetic metallic state of CaFeAsF is unusual in a sense that the carrier density is as low as 10 per Fe and that one type of the carriers is Dirac fermions. One may ask why CaFeAsF does not become an insulator after the antiferromagnetic ordering despite the strong antiferromagnetic interaction involving states far below . The unusual metallic state is actually a decisive demonstration of an early theoretical idea by Ran et al. Ran et al. (2009) (see also Morinari et al. (2010)). According to Ran et al., CaFeAsF is not allowed to become an insulator. Because of the topological feature of the iron-pnictide band structures, the antiferromagnetic gap has to have two Dirac nodes (in a 2D approximation). The nodes are close to but offset from , giving rise to FS pockets.

The theory by Ran et al. is also applicable to the antiferromagnetic state of FeAsO ( = rare earth) and FeAs ( = Ba, Sr, or Ca). The existence of Dirac fermions in those compounds has been suggested by magnetotransport Huynh et al. (2011); Tanabe et al. (2011); Pallecchi et al. (2011), angle-resolved photoemission spectroscopy Richard et al. (2010); Shimojima et al. (2010); Yi et al. (2011), and optical measurements Nakajima et al. (2011); Chen et al. (2017). The Fermi velocity ( m/s) of the Dirac cone in BaFeAs estimated from angle-resolved photoemission data Richard et al. (2010) is close to that of the present cylinder (Table I). However, quantum oscillation measurements so far have not confirmed a nontrivial Berry phase in those compounds Sebastian et al. (2008); Harrison et al. (2009); Analytis et al. (2009); Terashima et al. (2011); Sutherland et al. (2011); Graf et al. (2012); Rosa et al. (2014); Caglieris et al. (2017). Thus this is the first observation of a nontrivial Berry phase in iron-based superconductor parent compounds.

It would be interesting if one could eliminate the hole cylinder by slightly electron-doping CaFeAsF. One could investigate transport properties of Dirac fermions. Further, one might be able to induce topological superconductivity of Dirac fermions. In the case of BaFeAs, in addition to the Dirac electron pockets (the pockets in Terashima et al. (2011)), there are hole () and electron () pockets, and it is difficult to simultaneously remove both of the latter pockets by doping. Therefore CaFeAsF is best suited to testing those possibilities of Dirac-fermion physics.

We have observed nearly -linear magnetoresistance [Fig. 3(a)]. Similar -linear magnetoresistance has been reported for other Dirac-fermion systems such as Bi Kapitza (1928); Fuseya et al. (2015), CdAs Liang et al. (2015); He et al. (2014); Feng et al. (2015), NaBi Kushwaha et al. (2015), TlBiSSe Novak et al. (2015), and ZrSiS Singha et al. (2017), and further it has been reported for FeAsO Pallecchi et al. (2011); Pallecchi, I. et al. (2013) and BaFeAs Huynh et al. (2011) as well. It is sometimes analyzed with a model of quantum linear magnetoresistance proposed by Abrikosov Abrikosov (1998). However, this model is applicable only in the quantum limit. The present linear magnetoresistance starts at about 0.6 T, which is far smaller than a field to reach the quantum limit. This observation is similar to those reported in Liang et al. (2015); He et al. (2014); Feng et al. (2015); Kushwaha et al. (2015); Novak et al. (2015); Singha et al. (2017). Since the linear magnetoresistance appears fairly common among Dirac-fermion systems, it might be rooted in characteristic scattering mechanisms of Dirac fermions in magnetic fields as discussed in Liang et al. (2015); Feng et al. (2015).

The nonmetallic conductivity above is enigmatic. Band-structure calculations for the paramagnetic state predict a large Fermi surface similar to that in the paramagnetic state of other 1111 or 122 compounds Ma et al. (2017). On the other hand, it has theoretically been suggested that as the As height, the vertical distance of As from the Fe plane, increases, the Fe 3 band width decreases, leading to enhanced electronic fluctuations and stronger electronic correlations Vildosola et al. (2008); Kuroki et al. (2009). The As height in CaFeAsF is 1.41 ÅMa et al. (2015), which is appreciably larger than 1.32 Å in LaFeAsO de la Cruz et al. (2008). Therefore, because of enhanced fluctuations, CaFeAsF above may be in an incoherent state where the Fermi surface is not well-defined.

We also note that the conductivity becomes metallic below the structural (nematic) transition at , not the antiferromagnetic one at . Since no band folding occurs at and the orthorhombic distortion is tiny (0.3%), almost no change is expected for the calculated Fermi surface at . This may indicate that the metallic conductivity below is owing to the suppression of nematic fluctuations, which in turn may indicate the importance of those fluctuations in driving CaFeAsF into an incoherent state.

It is interesting to note that the conductivity near room temperature becomes metallic with Co doping: the resistivity of Ca(FeCo)AsF decreases from room temperature down to about 100 K, where it exhibits a broad and shallow minimum Xiao et al. (2009); Ma et al. (2016). This behavior is very similar to that observed in doped FeAsO, e.g., LaFeAs(OF) with = 0.03 or 0.05 and Sm(FeCo)AsO Dong et al. (2008); Kohama et al. (2008); Zhigadlo et al. (2012).

Finally, we mention a very recent report of quantum oscillation measurements on SmFeAsO Caglieris et al. (2017). Only one frequency was observed for , and its angular dependence was not determined. The reported frequency = 65(5) T is close to the frequency, but the effective mass 0.5(1) is about half of the mass.

Vii summary

We have compehensively determined the Fermi surface of antiferromagnetic CaFeAsF: it consists of the hole cylinder at the point and a pair of the Dirac electron cylinders symmetrically located at positions off toward the antiferromagnetic direction. We have proved that a nontrivial Berry phase is associated with the Dirac cylinders. The carrier density is extremely small, of the order of 10 per Fe. This unusual metallic state is a consequence of the previously studied topological nature of the band structure Ran et al. (2009). A nearly linear magnetoresistance is observed as in other Dirac-fermion systems. The anomalous resistivity increase observed above 30 T for is likely related to the quantum limit of the orbit. The band-structure calculations describe the Fermi surface reasonably well. The same calculations however largely overestimate the antiferromagnetic moment. Electron interactions that preserve the Fermi surface have to be sought to resolve this problem. The nonmetallic conduction observed above is intriguing given that the low-temperature antiferromagnetic state is metallic with a well-defined Fermi surface. It may suggest that CaFeAsF is in an incoherent state above . Finally, slight electron-doping can make CaFeAsF a system composed solely of Dirac fermions. Investigations of transport and other properties there would be fruitful, and moreover Dirac-fermion superconductivity might be induced.

We thank Yinguo Xiao for providing us with the structural parameters for the antiferromagnetic orthorhombic phase of CaFeAsF. We thank Satoru Matsuishi, Hiroyuki Yamase, Tomoaki Agatsuma, and Yuki Fuseya for valuable discussions. This work was supported by JSPS KAKENHI Grants No. JP17K05556, No. JP17J06088, No. JP16H04021, No. JP16H01081, the Youth Innovation Promotion Association of the Chinese Academy of Sciences (No. 2015187), and the National Natural Science Foundation of China (No. 11574338). A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1157490 and the State of Florida.


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