DFT calculations
Table of Contents
Method
Electronicstructure calculations are performed within the DFT framework using the projector augmentedwave method^{33} as implemented in the Vienna Ab initio Simulation Package (VASP)^{34,35}. The exchangecorrelation potential is described using the Perdew–Burke–Ernzerhof^{36} functional within the generalizedgradient approximation. Plane waves are used as a basis set with a kinetic energy cutoff of 450 eV. To sample the Brillouin zone, we use a 24 × 24 × 24 Γcentred kgrid. The SOC is included in our calculations as described in ref. ^{37}. We take into account the p and d semicore states for the Fe and Sn projector augmentedwave datasets used and the simplified onsite Hubbard U correction^{38} is added for the Fe d orbitals.
Flat band and Dirac/Weyl crossing in Fe_{3}Sn_{2}
The ARPES data from Fe_{3}Sn_{2} reported so far provide no evidence for the flat band predicted from the tightbinding calculations originally proposed to describe this material. We, along with other authors^{18}, attribute this to the breathing nature of the kagome layers as well as the hopping of the electrons across different kagome layers, that is, the electrons are not confined to the kagome layer only as assumed for the tightbinding calculations.
Dirac crossings at K points in triangular, honeycomb and kagome lattices are common in simple tightbinding calculations. However, these crossings can be gapped by SOC or small perturbations such as ‘breathing’ distortions of the kagome planes. Thus, our first approach to understanding this kagome material is to use DFT+U for the 3D material without making assumptions about 2D confinement of the electron movement. We adjust the U value to match the ARPES data.
Experimental detection of the Weyl nodes is difficult for ordinary vacuum ultraviolet ARPES because of their large number^{15} combined with ferromagnetic and crystallographic domain structures, resulting in a superposition of different band structures, as well as the presence of surface states. Theoretical guidance is also imperfect given that correlation effects can move the nodes.
‘Dirac points’, electron pockets and magnetization direction dependence in DFT calculations at various U values
We attempted to reproduce the ‘Dirac points’ predicted in the tightbinding model by changing the U value in the DFT calculations as shown in Extended Data Fig. 1 (also Brillouin zone labels). Our results show that the ‘Dirac points’ are not reproduced for U values between 0 and 1.3 eV. We can see that the closest resemblance of the band dispersion to the linear crossing is achieved at a U value of 0.5 eV or above, at which the gap is narrow but never fully closes.
We determined a Hubbard U value of 1.3 eV to reproduce the ARPES band structure around the Γ point at E_{F}. Band structures calculated using several different U values with magnetic moments pointing in the kagome plane are shown in Extended Data Fig. 1c. In Extended Data Fig. 2, the bandstructure evolution as a function of magnetic moment direction (M∥x, θ = 70° and M∥z) is also plotted with selected U values (0, 0.5 and 1.3 eV). Extended Data Fig. 3 plots the bands calculated in the ARPES energy window for the range k_{z} = (0–0.05)ΓZ (for more details about this, see the section ‘Laser μARPES’).
Sample growth and structural characterization
Fe_{3}Sn_{2} crystallizes in a rhombohedral structure with space group \(R\bar{3}m\), with crystal axis a,b = 5.34 Å, c = 19.80 Å and γ = 120°. Fe_{3}Sn_{2} single crystals were grown by a vapour transport method. Stoichiometric iron powder (Alfa Aesar, 99.9%) and tin powder (Alfa Aesar, 99.9%) were placed into an evacuated quartz tube. The tube was then annealed at 800 °C for 7 days before quenching in icy water. The crystal structure of the polycrystalline Fe_{3}Sn_{2} precursor is confirmed by the Xray diffraction data shown in Extended Data Fig. 4a. The obtained polycrystalline Fe_{3}Sn_{2} precursor was thoroughly ground and sealed with I_{2} (about 4 mg cm^{−3}) in a quartz tube 1 cm in diameter and 16 cm in length. Fe_{3}Sn_{2} single crystals were obtained under a temperature gradient of 650 °C (source) to 720 °C (sink) for two weeks.
The cleaved surface for laser ARPES is confirmed to be (001), the ab plane, from lowenergy electron diffraction measurements on a freshly cleaved surface, as shown in Extended Data Fig. 4b. The typical cleaved surface is shown in Extended Data Fig. 4c, indicating a flat surface suitable for ARPES measurements.
Magnetic domain and surface termination probed by XPEEM
We first consider the magnetic domain configuration and different surface terminations, which typical ARPES averages over, by using Xray photoemission electron microscopy (XPEEM) to obtain spatially resolved Xray absorption spectroscopy (XAS) and Xray photoelectron spectroscopy (XPS) maps of the sample using the SPELEEM III instrument (Elmitec GmbH) at the Surfaces/Interfaces: Microscopy (SIM) beamline of the Swiss Light Source. The aim is to establish whether there are variabilities in the sample on the length scale of the laser μARPES experiments. In our case, the XAS is monitored using photoelectrons with kinetic energy below approximately 1–2 eV (lowenergy secondary electron, ‘bulk’ sensitive to a depth of 3–5 nm from the surface), whereas the XPS photoelectrons have a kinetic energy of 96 eV (surface sensitive). The sample was cleaved in vacuum and investigated at 80 K, the base temperature of the cryostat. Extended Data Fig. 5a shows the magnetic domains of the system, which are obtained from the pixelwise ratio of XAS collected for left circular (CL) and right circular (CR) polarized photons at the Fe L_{3} edge. Extended Data Fig. 5b shows the map of the photoelectron yield summed over both polarizations (CL + CR), confirming that the iron content in the ‘bulk’ of the sample is homogeneous.
The surface termination of the Fe_{3}Sn_{2} samples is probed by mapping the Sn 3d_{3/2} and Fe 2p_{3/2} local photoemitted electron intensities. The photon excitation energy was varied such that the kinetic energy of the detected electrons was fixed at 96.8 eV to achieve a high surface sensitivity (about 2 Å according to the universal curve^{39}). The photoemission peaks are determined by recording XPEEM image sequences with varying incident photon energy with Xrays linearly polarized in the plane of the sample. Extended Data Fig. 5c shows the XPS map for the same region as in panels a and b, obtained for the Sn 3d levels and indicating variations in the Sn intensity. Similar results are observed for a different cleaved surface, shown in Extended Data Fig. 5f. Panel d shows the intensity histogram from the red square in panel c: it is bimodal (in this case, well characterized as the sum of two Gaussians), which indicates that there are two possible Sn populations for the surfaces formed after cleaving Fe_{3}Sn_{2}. The ratio between the mean intensities for the two Gaussians is I_{1}/I_{2} = 1.8 ± 0.6, in which I_{1} is the brighter intensity and I_{2} is the darker intensity mean value (the derivation is described below). The bimodal distribution of the Sn intensity, also visible in the sharp step in intensity of the line scan shown in panel d, arises because the material consists of (kagome) Fe bilayers alternating with stanene monolayers. In the topmost two layers, one expects either one (one stanene layer) or two Fe layers (no stanene layer), as shown in panel h. Thus, the clear contrast in Sn distribution (panel c) arises from stanene termination for the brighter areas and kagome termination for the smaller, lower intensity islands. Furthermore, given the much bigger area associated with the larger Sn photoemission intensity, we conclude that most of the sample is stanene terminated, agreeing with the suggestion made in ref. ^{15}. Comparison of panels c and a also demonstrates that the magnetic domain pattern is uncorrelated with the surface termination.
A second XPS map is shown in Extended Data Fig. 5e–g, showing a clear cleaved surface and an area with glue residue and a crack. The XPS intensity distributions for the Sn 3d_{3/2} and Fe 2p_{3/2} peaks are shown in panels f and g, respectively. Electrons emitted from the Sn 3d_{3/2} orbitals have two distinct intensities in the spatially resolved map in panel f, revealing a large region of higher intensity together with smaller islands of lower intensity. By comparison, the iron emission map shown in panel g is more homogeneous and has a lower signaltonoise ratio owing to the lower photoemission yield of the Fe 2p transition. We assign the bright area in panel f as stanene terminated and the darker area as kagome terminated. However, we cannot determine the exact kagome termination, that is, whether it is one or two Fe layers (panel h i or iii) within the intensity resolution of our measurements. A comparison between these two sets of data shows that the area attributed to kagome termination can vary in size and is not limited to the small areas shown in panel c. The XPS spectrum shown in panel g is obtained by integrating over an area larger than that for the Sn XPS peak shown in panel f.
The Xray magnetic circular dichroism map, collected at T = 80 K, reveals a magnetic domain pattern with a hierarchy of characteristic features ranging in size from 50 to <1 µm, indicating the need for μARPES to avoid averaging of momentum and energyresolved states associated with different magnetization directions. The lengthscale distribution is similar to that for previous magnetic force microscopy (MFM) studies^{14}, which also showed that, at the lower temperatures of our μARPES experiments described below (about 6 K), the magnetic domain pattern was characterized by larger domains and that, although they are smaller at 80 K, they are still predominantly polarized parallel to the kagome planes (domains with perpendicular polarization are a minority and would produce subtly different band structures; see Fig. 1b and ref. ^{15}). At the same time, the different surface terminations as established by XPS are characterized by length scales in the same range, implying that μARPES could also detect differences in surface states should they matter for the photon energy chosen.
Intensity ratio of the XPS signal
The photoemission electron micrographs showing different terminations in Fe_{3}Sn_{2} can be numerically analysed as follows:

1.
We assume that the photon penetration depth is much deeper than the escape depth of the most energetic photoelectron, that is, an electron at E_{F}. Thus, we ignore the exponential decay component of the photon intensity.

2.
Assumption (1) leads to an expression for the photoelectron intensity from a given atom called a, on layer called b, with distance d from the surface as
$${I}_{ab}={C}_{ab}{n}_{ab}\exp \left(\frac{d}{\lambda }\right)$$
in which λ is the escape depth of the electron, C_{ab} is the proportionality constant that contains the photon crosssection of atom a at layer of type b and n_{ab} is the atom a density at layer of type b.

3.
Generalizing (2), we can express the total intensity from atoms a in all layers of type b as a summation
$${I}_{ab{\rm{total}}}={C}_{ab}{n}_{ab}\mathop{\sum }\limits_{i=1}^{\infty }\exp \left(\frac{{d}_{i}}{\lambda }\right)$$
in which we assume the proportionality constant C_{ab} to be the same for all similar layers at different depths.
With these three assumptions, we can obtain the total XPS intensity from both Sn and Fe. First, we notice that there are three possible terminations and call them double kagome termination (S_{kk}), single kagome termination (S_{k}) and stanene termination (S_{s}), as shown in Extended Data Fig. 5h.
Double kagome termination (S
_{kk}) case
The total Sn and Fe intensity from this termination can be expressed as
$$\begin{array}{l}{I}_{{\rm{Sn}},{S}_{{\rm{kk}}}}=\mathop{\sum }\limits_{m=0}^{\infty }\left(\exp \,\left(m\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right)\exp \,\left(\frac{{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{s}}}{n}_{{\rm{Sn}},{\rm{s}}}\right)\\ \,\,\,+\,\mathop{\sum }\limits_{m=0}^{\infty }\left(\exp \,\left(m\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}\right)\\ \,\,\,+\,\mathop{\sum }\limits_{m=0}^{\infty }\left(\exp \,\left(m\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right)\exp \,\left(\frac{{z}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}\right)\\ \,\,\,\,=\,\frac{\exp \,\left(\frac{{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{s}}}{n}_{{\rm{Sn}},{\rm{s}}}+{C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}+\exp \,\left(\frac{{z}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}}{1\exp \,\left(\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right)}\\ {I}_{{\rm{Fe}},{S}_{{\rm{kk}}}}=\mathop{\sum }\limits_{m=0}^{\infty }\left(\exp \,\left(m\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}\right)\\ \,\,\,+\,\mathop{\sum }\limits_{m=0}^{\infty }\left(\exp \,\left(m\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right)\exp \,\left(\frac{{h}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}\right)\\ \,\,\,\,=\,\frac{{C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}+\exp \,\left(\frac{{h}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}}{1\exp \,\left(\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right)}\end{array}$$
Single kagome termination (S
_{k}) case
$$\begin{array}{l}{I}_{{\rm{Sn}},{S}_{{\rm{k}}}}=\mathop{\sum }\limits_{m=0}^{\infty }\left(\exp \,\left(m\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right)\exp \,\left(\frac{{z}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{s}}}{n}_{{\rm{Sn}},{\rm{s}}}\right)\\ \,\,\,+\,\mathop{\sum }\limits_{m=0}^{\infty }\left(\exp \,\left(m\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right)\exp \,\left(\frac{2{z}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}\right)\\ \,\,\,+\,\mathop{\sum }\limits_{m=0}^{\infty }\left(\exp \,\left(m\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}\right)\\ \,\,\,=\,\frac{\exp \,\left(\frac{{z}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{s}}}{n}_{{\rm{Sn}},{\rm{s}}}+\exp \,\left(\frac{2{z}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}+{C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}}{1\exp \,\left(\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right)}\\ {I}_{{\rm{Fe}},{S}_{{\rm{k}}}}=\mathop{\sum }\limits_{m=0}^{\infty }\left(\exp \,\left(m\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right)\exp \,\left(\frac{2{h}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}\right)\\ \,\,\,+\,\mathop{\sum }\limits_{m=0}^{\infty }\left(\exp \,\left(m\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}\right)\\ \,\,\,=\,\frac{\exp \,\left(\frac{2{h}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}+{C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}}{1\exp \,\left(\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right)}\end{array}$$
Stanene termination (S
_{s}) case
$$\begin{array}{l}{I}_{{\rm{Sn}},{S}_{{\rm{s}}}}=\mathop{\sum }\limits_{n=0}^{\infty }\left(\exp \,\left(n\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{s}}}{n}_{{\rm{Sn}},{\rm{s}}}\right)\\ \,\,\,+\,\mathop{\sum }\limits_{n=0}^{\infty }\left(\exp \,\left(n\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right)\exp \,\left(\frac{{z}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}\right)\\ \,\,\,+\,\mathop{\sum }\limits_{n=0}^{\infty }\left(\exp \,\left(n\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right)\exp \,\left(\frac{{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}\right)\\ \,\,=\,\frac{{C}_{{\rm{Sn}},{\rm{s}}}{n}_{{\rm{Sn}},{\rm{s}}}+\exp \,\left(\frac{{z}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}+\exp \,\left(\frac{{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Sn}},{\rm{k}}}{n}_{{\rm{Sn}},{\rm{k}}}}{1\exp \,\left(\frac{2{z}_{{\rm{ks}}}+{z}_{{\rm{kk}}}}{\lambda }\right)}\\ {I}_{{\rm{Fe}},{S}_{{\rm{s}}}}=\mathop{\sum }\limits_{n=0}^{\infty }\left(\exp \,\left(n\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right)\exp \,\left(\frac{{h}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}\right)\\ \,\,+\,\mathop{\sum }\limits_{n=0}^{\infty }\left(\exp \,\left(n\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right)\exp \,\left(\frac{{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}\right)\\ \,\,=\,\frac{\exp \,\left(\frac{{h}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}+\exp \,\left(\frac{{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}}{1\exp \,\left(\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right)}\end{array}$$
in which C_{Sn,s} is the proportionality constant for Sn atoms in the stanene layer, n_{Sn,s} is the Sn density in the stanene layers, C_{Sn,k} and C_{Fe,k} are the proportionality constants for Sn and Fe atoms in the kagome layer, respectively, n_{Sn,k} and n_{Fe,k} are the Sn and Fe densities at the kagome layer, respectively, h_{kk} is the distance of Fe layers between two adjacent kagome layers (the kagome bilayer), h_{ks} is the distance of Fe layers in the kagome layer to the nearest stanene layer, z_{kk} is the distance of Sn atom layers between two adjacent kagome layers (the kagome bilayer) and z_{ks} is the distance of Sn atom layers from a kagome layer to the nearest stanene layer.
From the expressions above, we can conclude that \({I}_{{\rm{Fe}},{S}_{{\rm{s}}}} < {I}_{{\rm{Fe}},{S}_{{\rm{k}}}} < {I}_{{\rm{Fe}},{S}_{{\rm{kk}}}}\) based on the following relation
$$\begin{array}{c}\frac{\exp \left(\frac{{h}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}+\exp \left(\frac{{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}}{1\exp \left(\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right)}\\ < \frac{\exp \left(\frac{2{h}_{{\rm{ks}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}+{C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}}{1\exp \left(\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right)} < \frac{{C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}+\exp \left(\frac{{h}_{{\rm{kk}}}}{\lambda }\right){C}_{{\rm{Fe}},{\rm{k}}}{n}_{{\rm{Fe}},{\rm{k}}}}{1\exp \left(\frac{2{h}_{{\rm{ks}}}+{h}_{{\rm{kk}}}}{\lambda }\right)}\end{array}$$
According to this model, we should be able to see the Fe peak contrast between different terminations, which we do not identify in our data owing to insufficient signaltonoise ratio.
Meanwhile, for Sn intensity, we can posit that
$${n}_{{\rm{Sn}},{\rm{s}}}\approx 2{n}_{{\rm{Sn}},{\rm{k}}}$$
on account of the twice higher density of Sn in the stanene rather than in the kagome layer. It is also reasonable to assume
$${C}_{{\rm{Sn}},{\rm{s}}}\approx {C}_{{\rm{Sn}},{\rm{k}}}$$
to obtain the double kagome termination (S_{kk}) case
$${I}_{{\rm{S}}{\rm{n}},{S}_{{\rm{k}}{\rm{k}}}}\approx {C}_{{\rm{S}}{\rm{n}},{\rm{k}}}{n}_{{\rm{S}}{\rm{n}},{\rm{k}}}\frac{2\exp \left(\frac{{z}_{{\rm{k}}{\rm{s}}}+{z}_{{\rm{k}}{\rm{k}}}}{\lambda }\right)+1+\exp \left(\frac{{z}_{{\rm{k}}{\rm{k}}}}{\lambda }\right)}{1\exp \left(\frac{2{z}_{{\rm{k}}{\rm{s}}}+{z}_{{\rm{k}}{\rm{k}}}}{\lambda }\right)},$$
the single kagome termination (S_{k}) case
$${I}_{{\rm{S}}{\rm{n}},{S}_{{\rm{k}}}}\approx {C}_{{\rm{S}}{\rm{n}},{\rm{k}}}{n}_{{\rm{S}}{\rm{n}},{\rm{k}}}\frac{2\exp \left(\frac{{z}_{{\rm{k}}{\rm{s}}}}{\lambda }\right)+\exp \left(\frac{2{z}_{{\rm{k}}{\rm{s}}}}{\lambda }\right)+1}{1\exp \left(\frac{2{z}_{{\rm{k}}{\rm{s}}}+{z}_{{\rm{k}}{\rm{k}}}}{\lambda }\right)},$$
and the stanene termination (S_{s}) case
$${I}_{{\rm{S}}{\rm{n}},{S}_{{\rm{s}}}}\approx {C}_{{\rm{S}}{\rm{n}},{\rm{k}}}{n}_{{\rm{S}}{\rm{n}},{\rm{k}}}\frac{2+\exp \left(\frac{{z}_{{\rm{k}}{\rm{s}}}}{\lambda }\right)+\exp \left(\frac{{z}_{{\rm{k}}{\rm{s}}}+{z}_{{\rm{k}}{\rm{k}}}}{\lambda }\right)}{1\exp \left(\frac{2{z}_{{\rm{k}}{\rm{s}}}+{z}_{{\rm{k}}{\rm{k}}}}{\lambda }\right)}.$$
With these assumptions, we have the Sn intensity relation for different terminations as
$${I}_{{\rm{Sn}},{S}_{{\rm{kk}}}} < {I}_{{\rm{Sn}},{S}_{{\rm{k}}}} < {I}_{{\rm{Sn}},{S}_{{\rm{s}}}}$$
From the intensity relation of both Sn and Fe, we can conclude that they are inversely related to each other.
We investigate the Sn intensity ratio between terminations to correlate with the experimental data we obtained from XPS. In this model, we use
$${z}_{{\rm{ks}}}\approx 2\,{\text{\AA }}\,{\rm{and}}\,{z}_{{\rm{kk}}}\approx 2.5\,{\text{\AA }}.$$
The summary of the intensity (proportional to C_{Sn,k}n_{Sn,k}) is given in Extended Data Table 1 for two different λ values.
Comparing with the experimental result of I_{bright}/I_{dark} = 1.84 ± 0.57 from Extended Data Fig. 5d, although we cannot tell if the darker region is coming from the single or the bilayer kagome, we can safely infer that the bright regions have stanene termination.
Laser µARPES
For the µARPES measurements, we use a 6.01 eV fourthharmonic generation continuous laser from LEOS as photon source, a custombuilt microfocusing lens to reduce the beamspot diameter to about 3 μm and an MB Scientific analyser equipped with a deflectionangle mode to map the dispersion relation while keeping the area of interest intact (that is, not changing owing to sample rotation). Typical energy and angular resolution (FWHM) were 3 meV/0.2°. The pressure during the measurement is kept at <10^{−10} mbar. The sample is mounted on a conventional sixaxis ARPES manipulator as described in ref. ^{40} and the sample position is scanned with an xyz stage of 100 nm resolution and better than 1 μm bidirectional reproducibility. A more detailed description can be found in ref. ^{41}. During the measurement, the temperature is first lowered to 6 K, at which the sample is cleaved in situ, and the sample drift at subsequent higher temperatures is tracked by using the edges of the sample as reference. The samples are prealigned to the highsymmetry cut with lowenergy electron diffraction after cleaving.
The perpendicular momentum k_{z} of the electrons measured by ARPES can be obtained from the expression
$${k}_{z}=\sqrt{\frac{2{m}_{{\rm{e}}}^{* }}{{\hbar }^{2}}\times \left({K}_{{\rm{out}}}+{V}_{{\rm{o}}}\right)\frac{2{m}_{{\rm{e}}}}{{\hbar }^{2}}{K}_{{\rm{out}}}{\sin }^{2}\phi }$$
in which \({m}_{{\rm{e}}}^{* }\) is the effective mass of the electron, K_{out} = hν − w − E_{b} is the kinetic energy of the electron, hν is the photon energy, E_{b} is the binding energy of the electron, w is the work function of the detector, V_{o} is the inner potential of the material and ϕ is the analyser slit angle (more details can be found in refs. ^{25,42}). The laser photon energy used, 6.01 eV, corresponds to a perpendicular momentum close to the centre of the Brillouin zone \({k}_{z}\approx i\times \frac{2{\rm{\pi }}}{c}\), in which i_{6.01eV} ≈ 5.93 ≈ 6, as shown in Extended Data Fig. 6a, whose inner potential used is taken from ref. ^{15} and assuming \({m}_{{\rm{e}}}^{* }={m}_{{\rm{e}}}\). The synchrotron photon energy used is 48 eV, which also lies roughly in the same position at the centre of the Brillouin zone, i_{48eV} ≈ 11.98 ≈ 12. From Extended Data Fig. 6a, we can see that these two photon energies differ by two Brillouin zones.
We can also estimate the uncertainty (δk_{z}) of k_{z} determination from the escape depth of the electron. In this case, we can focus on the electron close to E_{F} (K_{out} ≈ 1.64 eV), which has an escape depth around λ ≈ 60 Å according to the universal curve^{39}, giving us \(\delta {k}_{z}=\frac{1}{\lambda }\approx 0.016\,{{\text{\AA }}}^{1}\,\approx \)\(0.05\Gamma {\rm{Z}}\). In Fig. 2b, bottom, we show DFT calculations for k_{z} values between 0 and 0.05ΓZ, revealing that, at E_{F}, the dispersion along k_{z} could easily result in contributions to the inplane Δk of order 0.02 Å^{−1}. Furthermore, the γ band has a larger dispersion along ΓZ than the α band, accounting for the larger Δk seen in the experiments for the γ band.
Synchrotron versus laser ARPES
Crystallographic twins exist in Fe_{3}Sn_{2} and both will contribute to the ARPES data if the beam spot is bigger than the size of the domains. For example, we show in Extended Data Fig. 6b an example of ARPES data measured with photon energy 48 eV, temperature 17 K and a spot size of roughly 35 μm collected at the Surface/Interface Spectroscopy (SIS) beamline, Swiss Light Source, equipped with a Scienta R4000 hemispherical analyser. The overall feature shows a sixfold pattern with no indication of a threefold pattern. A closer look around the centre also shows a rather blurred circular shape, which can be attributed to a combination of all twinned domains that it may cover. This central feature can in fact be reconstructed from the laser ARPES results by combining data from both twinned domains and different polarizations (LH + LV) to give us the picture shown in Extended Data Fig. 6c. Indeed, we reproduce the synchrotron ARPES data but with higher momentum resolution.
Fitting MDCs
The MDCs are fitted to obtain the halfpeak width at halfmaximum (Δk), which—after correction for the δk_{z} effects described above—is the inverse of the average quasiparticle scattering length. We obtain Δk by fitting the MDC peak with a Voigt line shape, that is, a Lorentzian convolved with a Gaussian, for which the Gaussian simulates the instrument response. For momentum scans (MDC), the latter combines angular and energy resolutions to yield a FWHM
$$\delta k\approx \sqrt{{(\delta {k}_{x})}^{2}+{(\delta {k}_{y})}^{2}}=\sqrt{{(\delta {k}_{x})}^{2}+{\left(\delta E{\left(\frac{\partial E}{\partial k}\right)}^{1}\right)}^{2}}$$
in which \({v}_{{\rm{F}}}=\frac{1}{\hbar }\frac{\partial E}{\partial k}{ }_{{\rm{F}}}\). In this case, we set δk_{x} ≈ 0.001 Å^{−1} and δE ≈ 4.7 meV for the data in Fig. 3, δE ≈ 1.9 meV for the data in Fig. 4 and \(\frac{\partial E}{\partial k}\) as listed in Extended Data Table 2. We can therefore see that, typically, the broadening in the MDC is owing to the broadening in the energy distribution curve (EDC) because \(\delta {k}_{x} < \delta E{\left(\frac{\partial E}{\partial k}\right)}^{1}\).
Fitting sharp quasiparticle peak (β) EDC
Extended Data Fig. 7 shows the sharp quasiparticle peak of β at E_{F} at 6 K, fitted with a Voigt function in which the Gaussian FWHM was 4.7 meV to represent the detector response (dashed line). We obtain the resolutioncorrected Lorentzian FWHM of 2.5 meV.
To fit the 25 K data, we constrain the γ peak position by using the result of the 6 K peak position and let the β peak position at 25 K relax from the maximum separation obtained from the 6 K fitting while keeping the slope of E(k) versus k similar to the γ band.
In Extended Data Table 2, we summarize for the three pockets the Fermi velocity \({v}_{{\rm{F}}}=\frac{1}{\hbar }\frac{\partial E}{\partial k}{ }_{{\rm{F}}}\), the area of the electron pocket converted into the dHvA frequency as \(f(T)=\frac{\hbar }{2{\rm{\pi }}e}\times {\rm{area}}\) and the scattering lengths of the quasiparticle bounded from below (on account of k_{z} broadening^{25}) by \(\lambda =\frac{1}{\Delta k}\), in which Δk is the half width of the MDC peaks (at 6 K and at E_{F}). The area of the electron pockets is calculated by tracing the visible peaks of the corresponding band at E_{F} and assuming a circular extrapolation to make closed contours.