Prediction of flat bands in a ternary intermetallic electride LaCoSi

Abstract

Paramagnetic LaCoSi, a ternary intermetallic electride, consists of CoSi blocks separated by two layers of La atoms. Its structure is similar to that of the widely studied 111 system of iron-based superconductors. Utilizing angle-resolved photoemission spectroscopy and first-principles calculations, we demonstrate the existence of linear bands and flat bands mainly originating from the e_g orbitals of Co 3d states near the Fermi energy. The anomalous scattering rate of the linear bands varies linearly with the binding energy. The flat band above the Fermi energy indicated by the calculations could be modulated by substitutions and pressure to induce new ordered quantum phases, such as magnetism and superconductivity. Our findings reveal flat-band physics in electrides.

Graphical abstract

Keywords

electronic structure / flat bands / ARPES / electride

1 Introduction

Flat electronic bands at the Fermi energy (E_{\mathrm{F}}) can lead to interaction-driven instabilities, resulting in a wide range of new quantum phases [1-4]. Exploring these bands in new materials and manipulating them to study novel physical properties and effects is one of the most important topics in condensed matter physics and materials science. However, it remains challenging to find flat bands near E_{\mathrm{F}}, except in f-electron systems [5] and special geometric lattices such as kagome [6-14], Lieb [15-17], and twisted bilayer lattices [18-20]. Furthermore, flat bands in kagome materials located far from E_{\mathrm{F}} are difficult to tune properly to induce novel quantum phases. Constructing real materials with flat bands in the Lieb lattice poses a challenge while achieving large-scale and stable artificial materials with specific angles in moiré lattices is also quite difficult.

The flat bottom and top of the bands located at E_{\mathrm{F}} offer opportunities to uncover partially flat bands to induce new physical properties. 122 system of iron-based superconductors such as A(Fe{}_{1-x}Co{}_x){}_2As{}_2 (A = Ba, Sr, and Ca) exhibits strong doping- and orbital-dependent physics [21-27]. In the superconducting regime, the low-lying energy bands in the vicinity of E_{\mathrm{F}} are dominated by the t_{2g} orbitals which are more correlated than the e_g ones. Along with the increased filling of the electronic 3d shell, the chemical potential is shifted up and the strength of the electronic correlation is almost equal among the t_{2g} orbitals upon the half substituted AFeCoAs{}_2 [21, 22]. In fully doped ACo{}_2As{}_2 [23-25], the e_g orbitals fall on E_{\mathrm{F}} and begin to play a dominant role in the system. The flat band bottom with e_g orbital characters, spanning 2/3 of the momentum along the \mathrm{\Gamma }−M direction, is thought to be responsible for magnetic instabilities that might lead to quantum critical phenomena induced by doping and magnetic fields. In the isoelectronic Ba(Fe{}_{1/3}Co{}_{1/3}Ni{}_{1/3}){}_2As{}_2 [26, 27], the flat band may be related to its quantum critical behaviors.

LaCoSi is a ternary intermetallic electride composed of CoSi blocks separated by two layers of La atoms [28-30]. Its structure is similar to the 111 system of iron-based superconductors, such as LiFeAs and NaFeAs [31-33]. The presence of intercalated La atoms provides more free electrons in LaCoSi, causing the chemical potential to shift substantially compared to LiFeAs and NaFeAs. Investigations of iron-based superconductors have shown that the chemical potentials in this system can be easily adjusted by substituting 3d electrons while maintaining the main band structure unchanged. Considering the comparable band structures between the 111 and 122 systems and drawing inspiration from the production of the flat band bottom in A(Fe{}_{1-x}Co{}_x){}_2As{}_2 [21-27], it prompts the question of whether a flat band bottom exists in LaCoSi and introduces the possibility of some novel physics properties.

In this work, we investigate the low-energy electronic structure of LaCoSi using angle-resolved photoelectron spectroscopy (ARPES) and first-principles band structure calculations. We observe linear bands with anomalous scattering rates in the vicinity of E_{\mathrm{F}}. The calculations reveal the presence of a flat band bottom with e_g orbital characters above E_{\mathrm{F}}, similar to those found in ACo{}_2As{}_2. The out-of-plane orbitals (d_{z^2}) of La and Co electrons contribute to the k_z dispersions near E_{\mathrm{F}}, while the in-plane orbitals, La d_{x^2-y^2} and Co d_{xy} orbitals also do participate possibly due to the larger size of La{}^{3+} covalent atomic radius enhancing the orbital hybridization and interlayer coupling, unlike in the 111 system of ferropnictides.

2 Methods

High-quality single crystals of LaCoSi were synthesized by the self-flux method [34]. La, Co, and Si with an atomic ratio of 15:5:1 were loaded into a MgO crucible and then sealed in an evacuated quartz tube. The tube was heated to 1000 °C, kept for 12 hours, and then cooled to 650 °C with a rate of 5 °C/hour. The crystals were separated from the flux by a centrifuge. ARPES measurements within a wide range of photon energies were performed at the 03U and 09U endstations at the Shanghai Synchrotron Radiation Facility (SSRF). In our measurements, we have employed both linearly horizontal and vertical polarized lights. Due to orbitals hybridization, there is minimal difference in the data obtained from the two polarizations. Consequently, the data presented here is taken by horizontally polarized light, which typically offers better flux and resolution compared to vertically polarized light. The energy and angular resolutions were better than 15 meV and 0.2°, respectively. Samples of size smaller than 1 mm × 1 mm were cleaved in situ, producing flat mirror-shaped (001) surfaces. During the measurements, the temperature was maintained at T = 20 K and the pressure was maintained at less than 5\times {10}^{-11} Torr.

First-principles calculations based on density functional theory (DFT) were performed based on the generalized gradient approximation of the Perdew−Burke−Ernzerhof type [35], and used the projector-augmented wave method, as implemented in the Vienna ab initio simulation package [36]. The plane-wave cut-off energy was set to 500 eV. The first Brillouin zone (BZ) was sampled using a 16 × 16 × 9 k-point mesh. The energy convergence criteria were defined as 10{}^{-8} eV. Spin−orbit coupling (SOC) was taken into account self-consistently. The experimental lattice constants a = b = 4.08283 Å and c = 7.19361 Å were used in the calculations [30].

3 Results and discussion

The crystal structure of LaCoSi is shown in Fig.1(a). It crystallizes in a tetragonal structure with the space group P4/nmm (No. 129). Each Co atom is coordinated by four Si atoms, and La atoms are positioned between the CoSi layers, forming a layered structure. In comparison with isostructural LiFeAs and NaFeAs in the 111 system of iron-based superconductors, the ratio c/a of LaCoSi is approximately 1.762 [30], which falls between the values of LiFeAs (1.683) and NaFeAs (1.782) [31]. The CoSi layer is significantly compressed along the c-axis partially due to the larger size of the covalent bond radius of La{}^{3+} compared to that of Li{}^{+} and Na{}^{+} [37]. Consequently, the La intercalated layer serves not only as a charge reservoir, similar to the Li and Na layers in the 111 system, but also enhances interlayer interactions and contributes states near E_{\mathrm{F}}. Fig.1(b) shows that the density of states (DOS) around E_{\mathrm{F}} is mainly derived from the Co 3d and La 5d states, accompanied by fewer Si 4p states. A sharp increasing peak marked by the arrow indicates that some bands with high DOS at E_{\mathrm{F}} could be dispersionless.

We have performed high-resolution ARPES measurements along the high-symmetry directions. The corresponding three-dimensional (3D) BZ with high-symmetry points and Fermi surfaces (FSs) of LaCoSi are illustrated in Fig.1(c) and (d). A four-angle star-shaped and a squared FS surround the center of the BZ, the \overline{\mathrm{\Gamma }} point. The shadowy FS along the \overline{\mathrm{\Gamma }}−\overline{M} lines should correspond to the calculated high DOS at E_{\mathrm{F}}. Due to the effects of matrix elements, the shadowy FSs exhibit asymmetry in two perpendicular directions within the first BZ. However, they demonstrate C4 symmetrical intensity in the second BZ. \overline{\mathrm{\Gamma }}−\overline{M}−\overline{X} represents two-dimensional (2D) BZ projected on the (001) plane. To investigate k_z dispersions in detail, we have measured along the \overline{\mathrm{\Gamma }}−\overline{X} direction using photon energies (hv) from 20 to 130 eV. Fig.2(a) shows the intensity as a function of the photon energy and k_x at E_{\mathrm{F}} and E_{\mathrm{F}}−0.5 eV, respectively. According to the periodicity of intensity and free-electron final state model [38], with an empirical value of the inner potential of \sim 11 eV [39] and c = 7.19361 Å [30], we found that hv = 70 and 100 eV are close to the \mathrm{\Gamma } points, and 60 and 85 eV are close to the Z points, consistent with that in 111 system of ferropnictides. Corresponding to the two FSs in Fig.1(d), two bands cross E_{\mathrm{F}} on the \overline{\mathrm{\Gamma }}−\overline{X} direction, \alpha and \beta, as marked in Fig.2(a). The \alpha band does not show distinguishable k_z dispersions and the intensity of the \beta band changes clearly with the photon energies.

To identify the intensity spots on the maps, we have recorded the energy−momentum distributions of the photocurrents on the k_z \sim 0 and k_z \sim \pi planes, respectively, as shown in Fig.2(b). The \alpha and \beta bands evidently cross E_{\mathrm{F}} with the momentum of 0.30 and 0.61 Å{}^{-1} and the Fermi velocities of 1.37 \pm 0.05 and 1.53 \pm 0.04 eV·Å, respectively. The observed band structures are similar in the two k_z planes. However, without any renormalization and with an energy shift up of about 58 meV, the appended calculated bands show obviously k_z dispersions which are inconsistent with the experimental observations. This disagreement is mainly caused by the limitation of k_z resolution and the k_z broadening effect in ARPES experiments, which leads to the electronic states integrating over a certain k_z region of the bulk BZ and the states on the k_z \sim 0 and k_z \sim \pi planes having the main contributions in the ARPES spectra, as reported in LaSb, HfSiS, and TiB{}_2, etc. [40-42]. Explicitly, the observed experimental data match better with the calculated results from the superimposition of the electronic states on the k_z \sim 0 and k_z \sim \pi planes. Considering the correlation effect and interband scattering, the bands will be relatively shifted at different momentum points, which could explain the missing hole-like bands at the \mathrm{\Gamma } points.

The enlarged view of the \alpha band near E_{\mathrm{F}} on the right panel of Fig.2(b) shows the dispersion linearly changing with the binding energy up to about 90 meV. The linear band is also clearly displayed on the plot of momentum distribution curves (MDCs), which are fitted by using the Lorentz function, as indicated by the black solid curves in Fig.2(c). The scattering rates obtained from the fitted MDCs vary linearly with the binding energy up to 50 and 90 meV on the k_z \sim 0 and k_z \sim \pi planes, respectively. Despite the equal theoretical values of the half-width at half maximum (HWHM) on the right and left branches of the linear bands, some points deviate from the lines with opposite tendencies due to the effect of the intensity of near bands. The average values are even better for the linear relationship. At least, anomalous scattering rates are inconsistent with the 3D Fermi-liquid theory (1/\tau \propto {\omega }^2) or the 2D Fermi-liquid theory [1/\tau \propto ({\omega }^2/{ϵ}_F)\ln (4{ϵ}_{\mathrm{F}}/\omega )] [26, 27, 43].

We further measured the energy−momentum distributions of the photocurrents along the \overline{\mathrm{\Gamma }}−\overline{M} direction, as the intensity plots shown in Fig.3(a). The append calculated bands are still not renormalized and just shifted up of about 58 meV. The overall findings along \overline{\mathrm{\Gamma }}−\overline{X} in Fig.2 are almost reproduced here: the observed experimental data include the superimposition of the electronic states on the k_z\sim 0 and k_z\sim\pi planes and the hole-like bands at the \mathrm{\Gamma } points are missed possibly due to the bands relatively shift induced by the correlation effect and interband scattering. We also found that the scattering rates of the \alpha band obtained from the fitted MDCs vary linearly with the binding energy up to 40 meV in the planes k_z\sim 0 and k_z\sim\pi, deviating from Fermi-liquid behaviors [26, 27, 43].

Additionally, as indicated by the blue arrows in Fig.3(a), the calculations indicate the flat bands with binding energies of about 33 and 100 meV on the k_z\sim 0 and k_z\sim\pi planes, respectively. We have performed DFT calculations with orbital projections to study electronic structures in the vicinity of E_{\mathrm{F}} computationally. As mentioned in the orbital-resolved DOS in Fig.1(b), the Co 3d and La 5d electrons mainly contribute states around E_{\mathrm{F}} and a sharp peak appears at E_{\mathrm{F}}. We therefore displayed their orbital weights indicated by the colors and sizes of the bands as shown in Fig.4. Firstly, as we expected, t_{2g} orbitals of Co 3d falling deep below E_{\mathrm{F}} and the flat band bottom with e_g orbital characters spanning 2/3 of the momentum along the \mathrm{\Gamma }−M direction close to E_{\mathrm{F}}, that is similar to that in ACo{}_2As{}_2 [23-27]. However, as mentioned above, the calculated bands need to be shifted up to about 58 meV to match the main experimental data. On the one hand, the relative band shift can be induced by the electronic correlation effect and interband interaction. On the other hand, in the ARPES experiments, the components of the cleaved surfaces are not ideal due to vacancies, defects, reconstruction, etc., thus La{}^{3+} ions can be lost. Especially for the intercalated electride LaCoSi, the low work function indicates the loosely bound nature of surface electrons and favors donating surface electrons to adsorptive impurities [28, 29]. Furthermore, the corresponding linear dispersions are also mainly derived from e_g orbitals of Co 3d. k_z dispersions are not only contributed by the out-of-plane orbitals (d_{z^2}) but also the in-plane orbitals (La d_{x^2-y^2} and Co d_{xy}) possibly due to hybridization of La and CoSi orbitals enhancing interlayer interactions.

4 Conclusions

We have demonstrated the existence of flat bands and linear bands with their orbital characters in a ternary intermetallic electride LaCoSi. The flat band near E_{\mathrm{F}} can be tunned through substitutions and pressure, leading to the emergence of new ordered quantum phases, such as magnetism and superconductivity. Additionally, a linear band with an anomalous scattering rate has been observed before these adjustments, indicating a tendency towards non-Fermi liquid behavior possibly associated with the flat band. Our work broadens the search for flat bands and provides a feasible way to study new quantum phases by tuning partial flat bands.

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