Measurements of Electronic Band Structure in CeCoGe₃ by Angle-Resolved Photoemission Spectroscopy

Abstract

In this paper, we present a comprehensive study of the electronic structure of CeCoGe₃ throughout the entire Brillouin zone in the non-magnetic regime using angle-resolved photoemission spectroscopy (ARPES). The electronic structure agrees in large part with first principles calculations, including predicted topological nodal lines. Two new features in the band structure are also observed, namely a surface state and folded bands, the latter of which is argued to originate from a unit cell reconstruction.

1. Introduction

The field of unconventional superconductivity was invigorated by the discovery of superconducting phases in materials lacking inversion symmetry. The lack of parity as a good quantum number permits the existence of anti-symmetric spin–orbit coupling interactions that can lead to unexpected behavior in the superconducting state. These interactions result in superconducting pairings that are a mixture of singlet and triplet states, as well as complex structures of the superconducting gap that can include point nodes and line nodes [1,2,3].

Electronic correlations and band topology add further richness to the potential pairing mechanisms and phenomenologies in superconductors that lack inversion symmetry. It was recently demonstrated [4] that these constituents coexist in non-centrosymmetric heavy fermion materials with chemical formula CeTX₃ (T = transition metal, X = Si or Ge) and a tetragonal BaNiSn₃ crystal structure. Collectively, these compounds span the full Doniach phase diagram from localized magnetism to heavy fermion physics without magnetic order [5,6]. CeCoGe₃, the compound of interest in this manuscript, is in the intermediate regime, featuring enhanced carrier mass and magnetic ordering [7,8,9]. This competition between magnetism and Kondo physics was demonstrated via the temperature dependence of spectral weight measured by ARPES [10]. Recent studies have also investigated predicted Weyl nodes near the Fermi energy ($E_F$) [4,11,12] large field-induced anomalous Hall conductivity reported [13].

CeCoGe₃ is susceptible to superconductivity under hydrostatic pressure [14] and can be driven to the magnetic quantum phase transition via chemical substitution [7,15]. In the superconducting state, CeCoGe₃ and related compounds are distinguished by extremely high upper critical magnetic fields along the c-axis [16,17]. This, together with the broken inversion symmetry, topological band structure, and the appearance of superconductivity at the endpoint of magnetic order, has been cited in the proposal of unconventional, possibly triplet, superconductivity in this and related compounds [18]. Experimental fermiology is one of the ingredients for evaluating unconventional superconducting mechanisms.

Here we report a comprehensive ARPES study of the non-magnetic electronic structure of CeCoGe₃ throughout the three-dimensional (3D) Brillouin zone. There is overall agreement with first-principles calculations that assume localized f electrons. However, two additional features are seen experimentally: a two-dimensional (2D) surface-like band and band folding that appears to be most consistent with a unit cell doubling.

2. Materials and Methods

Single crystals of CeCoGe₃ were synthesized using the solution growth method [19,20,21]. First, the stoichiometric composition of CeCoGe₃ was arc-melted, flipped upside-down and arc-melted again multiple times to ensure a homogeneous mixture. It was then combined with bismuth in a ratio of Ce₈Co₈Ge₂₄Bi₆₀. The entire mixture was initially heated to 1150 °C within 6 h, followed by a dwell time of 72 h. The temperature was then slowly decreased to 750 °C over 90 h, after which the remaining molten bismuth flux was removed by centrifugation [20,21,22,23].

ARPES and XPS measurements were performed at the MERLIN endstation (Beamline 4.0.3) at the Advanced Lightsource and Beamline 5.2 at the Stanford Synchrotron Radiation Lightsource (SSRL) using photon energies between 26 and 150 eV. Beam spot size at SSRL was ≈15 × 8 $\mathrm{\mu }$$\mathrm{m}$, and the analyzer energy resolution was ≈18 meV. MERLIN data were collected with spot size ≈50 × 70 $\mathrm{\mu }$$\mathrm{m}$ and energy resolution ≈25 meV. All photoemission spectra were collected at 30 K, above any of the magnetic ordering transitions [19].

First-principles calculations were carried out using the full potential linear muffin tin orbital method including spin–orbit coupling [24]. The effects of Ce $4f—$orbital electronic correlations are captured within the Local Density Approximation + Gutzwiller (LDA+G) formalism [25,26]. The local crystal field effects on the Ce $4f_{7/2}^1$ and Ce $4f_{5/2}^1$ multiplets were captured using a band-dependent double-counting scheme [4]. Topological features within the Brillouin zone (BZ) were identified recursively using a Berry curvature link-variable approach [27].

3. Results

CeCoGe₃ crystallizes in a non-centrosymmetric tetragonal crystal structure with the space group I4mm (No. 107); the structural (conventional) unit cell is shown in Figure 1a, but the primitive cell is used to define the BZ and for performing calculations. The conventional unit cell can be viewed as a layered structure with Ce, Co, and two adjacent non-equivalent Ge layers. The 3D BZ is shown in Figure 1b, with high-symmetry points marked. Throughout the text, energy vs. momentum cuts are taken either parallel to $\Gamma -\Sigma$ or parallel to $\Gamma -X$ at different values of $k_z$. The $\Gamma$ plane, Z plane, and $N-P$ plane refer to planes through these high symmetry points and parallel to $k_x-k_y$. $\overline{\Gamma Z}$ refers to the line between $\Gamma$ and Z along the $k_z$ axis. In Figure 1b, the colored planes indicate planes of corresponding color in later figures.

Figure 2a shows a survey XPS spectrum. All elements of the compound are present, and no extraneous peaks are observed, which indicates a cleave without contamination from Bi flux. An example spectrum for a sample with residual flux is included in the Supplementary Materials; samples such as these tend to yield poor-quality ARPES spectra. Figure 2b highlights the valence bands, with an emphasis on Ce levels. In Figure 2c, we focus on the Ge-$3d$ states, which clearly show at least four peaks. These spectra were measured in the 50–150 eV photon energy range and fitted using a background plus two Voigt-profile spin–orbit-split doublets from which the integrated area of each component was extracted. Both linear and Shirley background functions were tested in the fitting (See Supplementary Materials). The results of this fit are summarized in Figure 2d, which plots the area ratio of the two doublets. Though the results are background-dependent at low photon energy, the doublet at higher binding energy (doublet 1) is strongest at a photon energy of ≈90 eV, where the inelastic mean free path (IMFP) for Ge-$3d$ electrons is the smallest (Figure 2e). For the Ge-$3d$ core levels, IMFPs were calculated with SESSA v2.2.2 [28], yielding the two core-level curves (3d_5/2 and 3d_3/2), which differ slightly due to their different kinetic energies at a given photon energy. For valence band photoelectrons (Figure 2e), a reference IMFP was computed using the semi-empirical Gries G1 expression [29]. In this model, the effective electron number Z* represents the number of electrons that most strongly contribute to inelastic scattering. The envelope spans Gries-G1 results using $Z^{\ast }=Z_{\mathrm{eff}}(Z_{\mathrm{eff}}={({\sum }_i{f}_i{Z}_i^{2.94})}^{1/2.94}=43.44)$ [30] and Z* = 30 (valence-electron count [31]), while the central curve uses the direct atomic-number average. IMFP values for Ge-$3d$ levels and valence bands are calculated only over the measurement range of these quantities in this manuscript.

ARPES spectra were collected throughout the BZ, and most of the detailed spectra focus near planes cutting through the $\Gamma -Z$ line ($\overline{\Gamma Z}$) at 5 roughly equidistant positions: $\Gamma$ plane ($k_z=0.0$$\overline{\Gamma Z}$, red), $k_z=0.25$$\overline{\Gamma Z}$ (orange), $0.5$$\overline{\Gamma Z}$ (also called $N-P$ plane, purple), $0.75$$\overline{\Gamma Z}$ (green), and the Z plane ($1.0$$\overline{\Gamma Z}$, cyan). Figure 3 shows high-symmetry cuts parallel to $\Gamma -\Sigma$ at these values of $k_z$. The spectra shown are a sum of spectra collected with left circularly polarized (LCP) and right circularly polarized (RCP) light at the photon energy indicated in each panel. These energy vs. momentum spectra are energy distribution curve (EDC) normalized by taking each momentum channel and dividing it by the integrated area of that channel. The spectra tend to have weaker intensity close to $E_F$, which is also seen in previously published data [10]. Qualitatively, our spectra show good agreement with LDA+G calculations in Ref. [4], albeit with several differences. First, experimental spectra show a rigid shift of $\approx 180$ meV relative to the calculation. All calculations shown in this manuscript employ this rigid shift relative to Ref. [4]. We note that a rigid shift does not give perfect agreement, especially in the NP plane (Figure 3c,h) (see Supplementary Materials for overlays of calculations on ARPES data). Additionally, there are measured bands that are absent in the calculation. When these additional bands cross the predicted bands, they form ‘x-shaped’ features at low binding energy, and some examples are marked by vertical arrows in Figure 3a–e (See Supplementary Materials for zoomed-in images of this feature). Additional extra features are marked by horizontal arrows in some panels. At most planes in the BZ, the position and dispersion of these addition features is captured by calculated bands half the $\overline{\Gamma Z}$ distance; for example, extra bands near the $\Gamma$ plane agree well with bands near the Z plane. Examples of these offset bands are plotted in dashed lines in Figure 3f,g,i,j. Panel (h) corresponds to the NP plane, and half a BZ away in $k_z$ is still the NP plane.

Figure 4 explores constant energy maps at $E_F$ (Fermi surface maps) both along $k_z$ and perpendicular to $k_z$. All maps are generated with a 60 meV integration window centered around $E_F$. Figure 4a,b were collected in the $\Gamma -X-Z$ plane, where photon energy was tuned between 26 and 118 eV to access different values of $k_z$. An inner potential value of 15.5 eV was used, as determined from the periodicity of the spectra and from matching Fermi surface maps to calculations [32]. Different polarizations were used in panels (a) (linear horizontal, LH) and (b) (RCP+LCP) to capture different band features. Figure 4a,b were taken on the same cleave. Panel (a) shows some features that agree with overlaid calculated Fermi surfaces, such as near the BZ boundary at Z, but other experimental features not captured in calculation, such as near $\Gamma$ where the measured Fermi surface is further away from the zone center than the calculation. Additionally, there are ‘vertical’ features prominent in the bottom half of Figure 4a, one of which is marked with pink arrow. As these features are very sharp and do not disperse as a function of $k_z$, they are presumed to be surface states. These features are not visible in panel (b). In those data, the Fermi surface in the $\Gamma -X-Z$ plane appears to have a periodicity twice that of the BZ in $k_z$.

In Figure 4c1–g3 we examine Fermi surface maps as a function of $k_x-k_y$ for different values of $k_z$ located at momentum planes roughly equivalent to those indicated in Figure 1b. All of these maps are collected with LH polarization. For each measured plane of the BZ, Fermi surface maps collected at different photon energies are shown in columns 1 and 2 in order to disambiguate effects of matrix elements and differing surface sensitivity. Calculated Fermi surfaces are shown in column 3. While the calculations are exactly at the planes indicated in the figure, data are collected along the trajectories indicated in (a) which are not planar and not located exactly equidistantly along the $k_z$ axis. The surface bands discussed in Figure 4a are seen most clearly as sharp quarter-circles furthest from the center of the zone in (c1), (d1), (e1), (e2), and (g1). Panels (c2), (d2), and (g2) agree reasonably well with their accompanying calculation. Meanwhile, some of the other measurements agree better with calculations elsewhere in the BZ: (c1), ignoring the surface band, has a diamond-shape surrounding a square similar to (g3); (d1) has a prominent diamond-shaped feature lacking in (d3); (e2) has bands additional to the predicted square; (f2) has bands other than the predicted circle.

In Figure 5 we provide more details about the surface-like bands prominent for measurements with LH polarization. The dispersing bands closer to the zone center agree well with calculated bands. However, the additional band marked by the pink arrow differs substantially, especially at deeper binding energy. These extra features have a Fermi crossing around $k_{\left|\right|}=0.44$ Å, and the Fermi velocity is ≈2 eV$·$Å. At some photon energies, these surface bands have weak intensity at $E_F$ and stronger intensity at deeper binding energy (e.g., Figure 5c), while others show strong intensity at $E_F$ (e.g., Figure 5a,b), contributing to the sharp ’vertical’ features in Figure 4a.

In the non-magnetic state where the present measurements are performed, calculations have predicted that CeCoGe₃ is a topological metal, featuring multiple Weyl points and Weyl nodal lines throughout the BZ [4]. Figure 6 explores some support for these features, which are challenging to observe because the band splitting in the vicinity of the nodes tends to be small. Different momentum distribution curves (MDCs) are examined with LCP and RCP light, and the peak position is identified as the band position. Figure 6b shows the splitting and re-joining of bands between two predicted topological crossings, where each band is highlighted with a different polarization.

4. Discussion

The agreement between LDA+G in Ref. [4] and ARPES data is quite good throughout most of the BZ, as long as only weak hybridization with 4-f electrons is considered. This localized Ce-$4f$ electron character is supported by the Ce $4f^0$ peak appearing at binding energies deeper than 2 eV as well as a relatively high intensity of $4f_{7/2}^1$ as compared to $4f_{5/2}^1$ [33]. The strongest indicator in the present data is the overall energy shift between calculations involving hybridized $4f$ electrons (Ref. [4]) and measured band structure (Figure 3). This shift is well captured by performing calculations absent hybridized Ce electrons (Supplementary Materials and Ref. [10]). Weak hybridization between f electrons and conduction elections was also reported in studies on polycrystalline specimens [34]. The measured fermiology in Figure 4 is also qualitatively consistent with reported quantum oscillations measurements [9], except for the extra features from surface bands and reconstruction discussed below.

This compound is predicted to have topological band features in the non-magnetic state that are interesting in the context of this material’s moderate electronic correlations and tendency towards superconductivity under pressure. There is support for these nodal features via slight differences in dispersion for spectra taken with different polarization, in a manner that is consistent with predicted energy-dependent splitting and re-joining of bands (Figure 6). The example shown in that figure has a predicted Weyl crossing near $E_F$, with relevance to low energy phenomena like transport and superconducting pairing. When evaluating which Weyl crossings are relevant to transport, the rigid shift between measured band structure and predictions in Ref. [4] needs to be considered.

The crystal structure of CeCoGe₃ has two in-equivalent Ge positions [35], and it is thought that this compound cleaves with a Ge termination [10]. Both of these statements could be consistent with the two sets of Ge-$3d$ doublets that are shown in Figure 2c, and the relative intensity of the two doublets as a function of photon energy helps disambiguate between these scenarios. When measured at around 90 eV photon energy (Figure 2d), the relative intensity of the higher-binding-energy doublet (doublet 1) is highest. Although the detailed energy dependence is influenced by the choice of background at low photon energy, the general photon energy dependence is robust. This energy is also near the value where the IMFP is minimum for Ge-$3d$ electrons. Together this suggests, assuming the observation is primarily from initial-state effects, that the doublet at higher binding energy originates from the surface, and the relatively large (≈0.5 eV) chemical shift of the surface Ge species can be concomitant with the appearance of surface electronic states or surface structural reconstruction.

One new experimental feature in the present data is a surface state that appears around the corners of the BZ. This feature manifests as dispersionless states as a function of photon energy in Figure 4a and Figure 5, and these bands are extremely polarization dependent, being almost entirely absent with circularly polarized light, but very strong with LH light. We note that some LH spectra simultaneously show these surface states and reconstruction features from unit cell reconstruction (Figure 5b,c), indicating that they are not mutually exclusive. These surface states may be associated with topological band features, or they may be topologically trivial surface bands related to the strong surface effects exhibited by terminating Ge atoms (Figure 2).

The other new experimental observation is the additional band features shown in Figure 3 and Figure 4 which show strong similarity with bands that are half a BZ away in $k_z$ for most cuts where they are observed. These extra features are interpreted as originating from a reconstruction of the unit cell from some incipient order. The discussion below assumes a c-axis reconstruction because known magnetic orders in this material are along c [8,19,36]. The evidence for this ordering wavevector ($\mathbf{q}=(0,0,1/2)$) in our data is (1) strong mixing of $\Gamma$-plane and Z-plane spectra, (2) the $\overline{\Gamma Z}/2$ periodicity of FS maps in Figure 4b, as well as (3) large discrepancies between calculation and experiment in the $N-P$ plane, where bands would cross and hybridize under a doubled unit cell. The presumed unit cell doubling has the same periodicity as the ground state bulk magnetic order [36]. One might argue that the highest-temperature magnetic order may be most relevant at the measurement temperature of 30K. Below $T_{N1}=21K$, the magnetic ordering vector is $\mathbf{q}=(0,0,2/3)$, and we do not fully rule out this ordering vector. The aspect of the data that more favors a $\mathbf{q}=(0,0,2/3)$ order is the observation that many of the FS maps outside of the $\Gamma$ and Z planes in Figure 4c1–g3 show strong Z-plane features of a square inside of a diamond (c1, d1, e2). A $\mathbf{q}=(0,0,2/3)$ order would repeat Z-plane spectra more times along $k_z$ even before considering the effects of perpendicular momentum resolution. Perpendicular momentum uncertainty/broadening originates from a short IMFP, and this can both broaden spectra and oversample extremal momenta of a dispersing band [37,38]. In the present experiments, momentum uncertainty is estimated to be $\Delta k_{\perp }=1/\mathrm{IMFP}\approx 0.2 {\AA }^{-1}$, or ≈17% of the BZ height. This amount of perpendicular momentum broadening could partially convolve spectra comparatively nearby to each other in $k_z$, such as those originating from a $\mathbf{q}=(0,0,2/3)$ reconstruction.

We propose that the folded bands originate from an additional reconstruction in the non-magnetic state due to bond distortion, charge disproportionation, or surface-stabilized magnetism, though our results do not directly distinguish between these proposals. This reconstruction may be primarily near the surface, as found in polar materials existing near magnetic phases [39,40]. The present measurements focus on photon energies where ARPES is extremely surface sensitive, with an IMFP varying by $\approx 10\%$ around 4 Å; information depth is typically considered to be 2–3× the IMFP. A surface reconstruction would dominate our surface-sensitive measurements but be imperceptible to bulk probes which have dominated existing literature. However, our data do not rule out a weak bulk incipient order, and if such features are found, particularly under pressure, it may be relevant for the superconducting mechanism, both as a candidate wavevector for interactions related to pairing and as an origin of a significantly different fermiology.

5. Conclusions

We have mapped out the electronic structure of CeCoGe₃ over the entire 3D BZ in the non-magnetic state at 30 K. The measured electronic structure agrees in large part with band structure calculations. Two new features in the band structure—a surface state and folded bands tentatively attributed to a unit cell doubling stabilized by the surface—are also discussed.