Fermi-level flat band in a kagome magnet

Abstract

The band structure in a kagome lattice can naturally exhibit flat band, Dirac cones, and van Hove singularity, enabling rich interplays between correlation and topology. However, the flat band is rarely detected just at the Fermi level in kagome materials, which would be crucial to realize emergent flat band physics. Here, combining angle-resolved photoemission spectroscopy, transport measurements and first-principles calculation, we observe a striking Fermi-level flat band in paramagnetic YCr₆Ge₆ as a typical signature of electronic kagome lattice. We explicitly unveil that orbital character plays an essential role to realize electronic kagome lattice in crystals with transition-metal kagome layers. We further engineer this material with magnetic rare earth elements to break the time-reversal symmetry of the Fermi-level kagome flat band. Our work establishes a Fermi-level flat band in a kagome magnet as an exciting quantum platform.

1 Introduction

Due to the unique triangular-based corner sharing geometry, the kagome lattice (Fig. 1a) provides an exciting platform [1] for the realization of a series of unconventional quantum phases, including spin liquids [2], noncolinear magnetism [3], Chern energy gap [4], and unconventional charge ordering [5]. Particularly, the destructive quantum interference in the kagome lattice can lead to a flat band [6], which are proposed to host many emergent physics, such as the potential realization of high temperature fractional quantum Hall effect when a combination of spin-orbital coupling, time-reversal symmetry breaking and partial electron filling of the flat band is considered [7]. Along this direction, flat bands in several kagome materials have been successfully identified [8–17]. However, in those materials, the flat band is fairly far away from the Fermi level (usually on the order of 100 meV away from Fermi level), limiting its potential for further fundamental research and its application value in quantum devices.

Figure 1

Crystal and electronic structures of YCr₆Ge₆. (a) Crystal structure of YCr₆Ge₆, with Cr-kagome planes in a non-distorted stacking form. (b) Bulk Brillouin zone with the high-symmetry points indicated. (c) Photoemission intensity plot at $E_{F}$ in the $k_{z} = 0$ plane. (d)–(e) Photoemission intensity plot along the Γ-M and Γ-K directions, with the momentum path indicated as cut1 and cut2 in (c), respectively. (f)–(g) Photoemission intensity along the cut3 and cut4 directions. (h)–(i) Corresponding EDC plots along the cut3 and cut4 direction in (c), with the flat band peak marked

In this paper, using angle-resolved photoemission spectroscopy (ARPES), we report the direct observations of typical kagome band structure with a Fermi-level flat band in paramagnet YCr₆Ge₆ (Fig. 1a), consisting with previous transport and first-principle calculation results [18]. We unveil that orbital character plays an important role in formation of flat band in kagome materials, with band flatness decomposed for $d_{x^{2} - y^{2}} / d_{xy}$ states. We further introduce ferromagnetism in the system by substitution Y with magnetic Gd, in which the Fermi-level flat band is preserved. Beside of systems with transition metals V [5, 16, 17, 19], Mn [3, 4, 15, 20–23], Fe [8, 10, 24, 25] and Co [9, 11–14, 26–28] kagome planes, Cr-based kagome magnet (Y,Gd)Cr₆Ge₆ provides a good platform to study the intrinsic properties of Fermi-level flat band, and its interplay with spin-orbit coupling (SOC) and time reversal symmetry breaking.

2 Results

2.1 Fermi-level flat band in YCr₆Ge₆

Firstly, we map the in-plane band structure with $h\nu = 55$ eV, corresponding to the $k_{z} = 0$ plane in the bulk Brillouin zone (BZ) (Fig. 1b). The Fermi surface (FS) consists of a big pocket (named as α) centered at the Γ point, and a hot spot (labeled as β) at the K point (Fig. 1c). The α band crosses Fermi energy ($E_{F}$) along the Γ-M (Fig. 1d) and Γ-K (Fig. 1e) directions, with Fermi momentum indicated by arrows. Interestingly, additional flat band, named as γ, is clearly observed near $E_{F}$. We note spectra weight of the α and flat γ bands in the first BZ are much less intense than that from higher BZs (Fig. 1c-e), due to the matrix element effect of photoemission. To eliminate the strong intensity variation and focus on the band dispersion, we plot the experimental results along the Γ-M direction in the 2nd BZ, with the ARPES intensity and energy distribution curve (EDC) plotted in Fig. 1f and h, respectively. The flat band γ clearly appears with peaks near $E_{F}$ in EDC plot along the Γ-M direction between two 2nd Γ points. Similarly, the flat band γ is explicitly observed near $E_{F}$ along the Γ-K-M direction in higher BZs as seen from results shown in Fig. 1g and i. In addition to the high-symmetry directions, the γ band shows no dispersion at $E_{F}$ in whole the $k_{z} = 0$ plane (see Additional file 1 Fig. S3), indicating an electronic kagome lattice origin of the Fermi-level flat band γ.

Besides flat band, YCr₆Ge₆ shows complete hallmarks of electronic kagome lattice including Dirac dispersions and saddle points. Figure 2a shows band structure evolution in the $k_{z} = 0$ plane. From $E_{F}$ (Fig. 1c) to higher $E_{B}$ (Fig. 2a-I), the hole-like α pocket at the Γ points also becomes bigger and the β band at K point evolves from a single hot spot to a circular pocket. Further to higher $E_{B}$ (Fig. 2a-II), the α and β pockets further expand and touch each other at $T_{\alpha -\beta }$ point along the Γ-K direction at $E_{B} \sim 0.26$ eV. Then the α and β pockets become even bigger at higher $E_{B}$, and the two α bands touch each other at the saddle point $S_{\alpha}$ along the Γ-M direction at $E_{B} \sim$ 0.4 eV (Fig. 2a-III). At $E_{B}$ higher than the $S_{\alpha}$ point, the α band evolves from big hole-like pockets around different Γ points to small electron-like pockets around the K point. Then two β bands touch each other at the saddle point $S_{\beta}$ along the M-K direction at $E_{B} = 0.5$ eV (Fig. 2a-IV). The β band evolves from six small hole-like pockets at different K points to single big electron-like pockets centered at the Γ point as $E_{B} > 0.5$ eV, that becomes smaller and moves towards to the center of BZ with higher $E_{B}$, as seen from Fig. 2a-IV-VI. Furthermore, the electron-like α pocket at the K point shrinks into a single point $D_{\alpha}$ at $E_{B} \sim 0.6$ eV (Fig. 2a-V), and become a hole-like pockets at higher $E_{B}$ (Fig. 2a-VI). The touching point $T_{\alpha -\beta }$, Dirac point $D_{\alpha}$ and saddle points $S_{\alpha}$ and $S_{\beta}$ are clearly resolved in the intensity and curvature plots along the Γ-K-M and Γ-M directions in Fig. 2b-e. The effective masses $m^{*}$ of the α and β bands change the sign along the Γ-K-M and Γ-M directions at the $S_{\alpha}$ and $S_{\beta}$ points, respectively.

Figure 2

Band structure evolution of YCr₆Ge₆ in the $k_{z} = 0$ plane. (a) Photoemission intensity plots at different constant $E_{B}$ in the $k_{x}-k_{y}$ plane. (b) Photoemission intensity plot along the Γ-K-M-K-Γ direction, with the extracted band dispersions (crosses) and guiding lines overlaid on top. (c) The corresponding curvature plot of the spectrum in (b). (d)–(e) Same as (b)–(c) but along the Γ-M-Γ direction. The touching points and saddle points are labelled in (a), (b) and (d)

2.2 Orbit-selectivity in formation of flat band

Our photon energy dependent ARPES measurements in Fig. 3b further indicate that inter-kagome-layers interaction is non-negligible in YCr₆Ge₆. The periodical variation along the $k_{z}$ direction is a direct evidence of the bulk electronic states origins for the ARPES signals. We note that the periodicity along $k_{z}$ is of $4\pi/c$, as twice as that in the bulk BZ along the $k_{z}$ direction ($2\pi/c$), because the two Cr kagome lattice in the unite cell are crystalline equivalence (Fig. 1a). Along the out of plane direction Γ-A-Γ (Fig. 3c), ARPES intensity near $E_{F}$ only appears near the Γ point, corresponding to the flat band γ closed to $E_{F}$ in the $k_{z} = 0$ plane. The photon energy dependent ARPES results demonstrate a planar flat band nature of γ, which is nondispersive feature in the $k_{z} = 0$ plane, indeed dispersive along the $k_{z}$ direction.

Figure 3

The orbital-selective Fermi-level flat band in YCr₆Ge₆. (a) The curvature plot along the Γ-K-M-Γ direction. The extracted band dispersions and the guiding lines are overlaid on top. The red and yellow lines indicate the $d_{z^{2}}$ and $d_{x^{2} - y^{2}}/d_{xy}$ orbits. (b) Photoemission intensity plot at $E_{F}$ in the $k_{x}-k_{z}$ plane with $k_{y} = 0$. (c) Photoemission intensity along the Γ-A direction. (d) Band structure projection of Cr $3d_{z^{2}}$ orbital by first-principal calculations. Inset shows the contrast between calculated band without/with SOC (blue/green line). (e)–(f) Schematic of the Cr kagome lattice and the $3 d_{z^{2}}$ orbitals near the Fermi level. (g)–(i) Same as (d)–(f), but for the $d_{x^{2} - y^{2}}/d_{xy}$ orbits. (j) The density of states plot of Cr $d_{z^{2}}$ and $d_{x^{2} - y^{2}}/d_{xy}$ orbits

Our first-principal calculations projected on Cr-$3 d_{z^{2}}$ orbit (Fig. 3d) show a good agreement with the observed α and γ bands (Fig. 3a). Our calculations further suggest that the observed β band corresponds the lower branches of the Dirac dispersion for the $\text{Cr-}3d_{x^{2} - y^{2}}$ and $3d_{xy}$ states (Fig. 3g). As seen from the insets of Fig. 3d and g, SOC opens gaps around 10 and 30 meV for the Dirac points at the K point for the $d_{z^{2}}$ and $d_{x^{2} - y^{2}} / d_{xy}$ states, respectively. We note that a renormalization fact of 1.6 is needed for the calculation results to fit the overall features of experimental results (Additional file 1 Fig. S4), indicating a moderate electronic correlation effect in YCr₆Ge₆.

The unique behavior of the planar Fermi-level flat band γ is related to the intrinsic orbital character for the layered kagome lattice. As seen from Fig. 3e, intra-kagome-plane hopping process of the $d_{z^{2}}$ electrons is dominated by $t_{1}$ and other in-plane hopping terms ($t_{2}$ and $t_{3}$) are negligibly weak. Therefore, both the $d_{z^{2}}$ bands in the $k_{z} = 0$ plane (Fig. 3a and d) show a typical 2D electronic kagome lattice with Fermi-level flat band, von Hove singularity and Dirac dispersion. The hopping of $d_{z^{2}}$ electrons between the adjacent kagome planes ($t_{4}$ and $t_{5}$ in Fig. 3f) are considerable, as intermediated by the spacing layers (Fig. 1a). The inter layer hoping terms lead to promising $k_{z}$ dispersion of the planar Fermi-level flat band. The abnormal transport anisotropy in YCr₆Ge₆ ([18] and Additional file 1 Fig. S7), with the in-plane resistivity more than twice larger than the out-of-plane one, provides transport signatures of the planar flat band. The planar Fermi-level flat band also leads to a peak in density of states (DOS) at $E_{F}$ (Fig. 3j). The relatively big Sommerfeld coefficient (80.5 mJ K⁻² mol⁻¹) experimentally determined by heat capacity measurements serves as another transport property supporting the planar flat band.

Distinct from the $d_{z^{2}}$ electrons, the inter-layer interaction of $d_{x^{2} - y^{2}}$ and $d_{xy}$ orbits are expected to be weak (Fig. 3i), which results in less dispersive feature along the Γ-A direction (Fig. 3g). In the meantime, the $d_{x^{2} - y^{2}}$ and $d_{xy}$ orbits lay in the kagome plane and can induce additional in-plane hopping between the second and third nearest-neighbour sites (Fig. 3h), respectively. The additional terms $t_{2}$ and $t_{3}$ terms destroy the flatness of the $d_{x^{2} - y^{2}}$ and $d_{xy}$ orbits locating at about 0.7 eV above $E_{F}$.

2.3 Ferromagnetism and Fermi-level flat band in GdCr₆Ge₆

We further systematically engineer this kagome metal with magnetic rare earth element Gd to break time-reversal symmetry of the Fermi-level flat band. The temperature dependent magnetic susceptibility measurements on GdCr₆Ge₆ reveals the emergence of ferromagnetic ordering below a Curie temperature $T_{C} \sim 11.2$ K, which is consistent to the Weiss temperature $\theta _{\mathrm{CW}} \sim 11.5$ K extracted from the data of paramagnetic phase (Fig. 4a). The magnetization M shows saturation and hysteresis below $T_{C}$ (Fig. 4b), further confirmed the ferromagnetism. Since Gd 4f shell is half filled, there is a large local moment in Gd. Therefore, the ferromagnetism could mainly arise from the localized 4f electrons in Gd, which is away from Fermi-level. The core-level spectra of YCr₆Ge₆ and GdCr₆Ge₆ are shown in Fig. 4c, with Gd-4f peaks at $E_{B} \sim 8$ eV and Y-3d one at $E_{B} \sim 155$ eV, respectively. The band structure of GdCr₆Ge₆ (Fig. 4d) is very similar as that of YCr₆Ge₆ (Fig. 2d), with the Fermi-level flat band preserved in GdCr₆Ge₆ above $T_{C}$. Time reversal symmetry breaking and SOC are expected to split the spin degeneracy of the flat band γ and separate it from the α band by opening a gap at the Γ point, which can induce a topological Chern band in ferromagnetic phase of GdCr₆Ge₆. In future, we hope to further resolve the magnetic splitting of the flat band through high resolution and ultra-low temperature ARPES, and explore the tantalizing unusual anomalous transverse transports as well as the quantization properties of the magnetic flat band [7, 29]. For partially substituted compounds, the susceptibilities and ARPES results also present similar magnetic properties and the Fermi-level flat band in electronic structure, as shown in Fig. S8 of Additional file 1.

Figure 4

The Fermi-level flat band in normal state of ferromagnetic GdCr₆Ge₆. (a) The temperature dependence of magnetic susceptibility. (b) The filed dependence of magnetization. The inset shows the hysteresis at $T= 2$ K. (c) Core-level spectra of YCr₆Ge₆ and GdCr₆Ge₆. (d) Photoemission intensity plot of GdCr₆Ge₆ along the Γ-M direction. (e) The corresponding EDC plots with the flat band peak marked

3 Conclusion and discussion

We directly observed a Fermi-level flat band from Cr-$d_{z^{2}}$ electrons in paramagnetic YCr₆Ge₆ and normal state of ferromagnetic GdCr₆Ge₆, together with other signatures of typical kagome lattice band structure e.g. von Hove singularity and Dirac dispersions. The Fermi-level flat band can explain the abnormal behaviours in transport measurements. Our results demonstrate that orbital character plays an essential role for realization of flat band in transition metal kagome layers. Topological Chern band are expected in the ferromagnetic state of GdCr₆Ge₆. The Cr-based kagome magnets (Y,Gd)Cr₆Ge₆ provide a novel platform for investigating Fermi-level flat band, and its interplay with time-reversal symmetry breaking and SOC.

4 Method

High-quality (Y,Gd)Cr₆Ge₆ single crystals were grown by the flux method using tin as a flux. The starting elements mixture was heated to 1100°C over 8 h, kept at 1100°C for 10 h and then slowly cooled to 600°C at the rate of 3°C/h, and finally decanted in a centrifuge. ARPES measurements were performed on the “surface and interface” beamline of the Swiss Light Source and “Dreamline” beamline of the Shanghai Synchrotron Radiation Facility, with an overall energy resolution of the order of 20 meV, angular resolution of 0.1°, at $T= 20$ K. The electronic structure of YCr₆Ge₆ was calculated based on the density functional theory and the local density approximation for the exchange correlation potential, as implemented in the plane-wave pseudopotential based Vienna ab initio simulation package. The wave functions were expanded in plane waves with a cutoff energy of 470 eV and Monkhorst–Pack k points were 9 × 9 × 5. The residual forces are less than 0.01 eV/Å and SOC is included by using the second-order variational procedure.