Electronic structure, magnetic and transport properties of antiferromagnetic Weyl semimetal GdAlSi

GdAlSi single crystals were grown by the standard self flux method with excess Al as a flux Bobev et al. (2005); Puphal et al. (2019). Gd ingot (99.9$\%$), Al ingot (99.999$\%$), and Si chips (99.999$\%$) in a molar ratio of 1:10:1 were mixed in an alumina crucible. The crucible was then sealed into a quartz tube under a partial pressure of argon gas. The content was heated to 1050^∘C, kept for 24 hours at that temperature, and then cooled to 700^∘C at a rate of 3^∘C/hour. The excess flux could not be completely removed by centrifuging. Therefore, the remaining Al-flux was further removed by dissolution in a NaOH-H₂O solution. The very clean plate-like single crystals were filtered from the solution. The typical size of the crystal is 4mm$\times$3mm$\times$0.3mm as shown in the inset of Fig. 1(c). The crystal structure and phase purity were determined by x-ray diffraction and energy dispersive x-ray spectroscopy methods. The powder of crushed single crystals was characterized at 28-ID-1 beamline of the National Synchrotron Light Source II at Brookhaven National Laboratory with an x-ray wavelength of 0.1665Å. The plane orientation of the single crystal was confirmed by Cu-K_α radiation in the Rigaku Smartlab diffractometer. Magnetotransport measurements were carried out in a physical property measurement system (PPMS, Quantum Design) via the standard six-probe method and magnetic measurements were executed in a magnetic property measurement system (MPMS, Quantum Design). In-situ strain study was performed in PPMS using Razorbill high-force strain cell.

ARPES experiments were performed at the Electron Spectro Microscopy (ESM) 21-ID-1 beamline of the National Synchrotron Light Source II, USA. The beamline is equipped with a Scienta DA30 electron analyzer, with base pressure better than $\sim$ 1$\times$10^-11 mbar. Prior to the ARPES experiments, samples were cleaved using a post inside an ultra-high vacuum chamber (UHV) at $\sim$ 15 K. The total energy and angular resolution were $\sim$ 15 meV and $\sim$ 0.1^∘, respectively. All measurements were performed using linearly-horizontal (LH) polarized light. The incident angle of light was 55 degrees with respect to the sample normal. The analyzer slits were oriented along the vertical direction.

The density-functional-theory (DFT) calculations were done using VASP DFT package Kresse and Furthmüller (1996); Kresse and Joubert (1999); Blöchl (1994). The primitive Brillouin zone was sampled with a regular $11\times 11\times 11$ mesh containing 126 irreducible $k$ points. Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional Perdew et al. (1996) within the generalized gradient approximation (GGA) was used in all the calculations. The GGA + $U_{\textrm{eff}}$ method  Dudarev et al. (1998) was used to handle the Gd-4f orbitals. $U_{\textrm{eff}}$ of 6 eV was chosen in our calculations Gaudet et al. (2021); however, we have also verified that the results presented here remain robust for a large range of $U_{\textrm{eff}}$ values. The calculations for the non-magnetic phase were done within open-core approximation by freezing the Gd-4f states. The spin-orbit coupling (SOC) was treated in the second variation method. The slab and Weyl points calculations were done by using Wannier90+WannierTools software Mostofi et al. (2014); Wu et al. (2018) by taking 56$\times$56 Wannierised Hamiltonian. Gd 5$p$, Gd 4$d$ orbitals, Al $p$, and Si $p$ orbitals were used in the Wannierisation procedure to accurately reproduce the DFT bands in the energy window from $-2$ to 2 eV.

GdAlSi crystallizes in a LaPtSi-type body centered tetragonal structure with the polar space group $I4_{1}md$ (No. 109) as shown in Fig.  1(a). The measured lattice parameters from the XRD refinement fitting are $a=b$ = 4.12584(4)Å and $c$ = 14.42816(20)Å [Fig. 1(b)]. Fig. 1(c) shows the XRD diffraction pattern for $c$-axis which confirms that the plate-like surface is $ab$-plane. Fig. 1(d) represents the core level photoemission spectrum of GdAlSi crystal, which clearly manifests the characteristic peaks originating from Si 2$p$, Al 2$p$, and Gd 4$f$ orbitals.

Fig.  2(a) shows the temperature dependent zero-field cooled (ZFC) magnetic susceptibility ($\chi$) along the crystallographic $c$-axis ($B\parallel c$) and in $ab$-plane ($B\parallel ab$). An antiferromagnetic (AFM) transition is observed at $T_{\mathrm{N}}$= 32 K due to the ordering of Gd³⁺ moments. This compound shows a very weak magnetocrystalline anisotropy, $\chi_{c}/\chi_{ab}=0.95$, which is similar to that of a spiral magnet SmAlSi Yao et al. (2023). Such a weak magnetocrystalline anisotropy allows the spins to arrange in a spiral magnetic order. To determine the effective magnetic moment of the Gd ions, the inverse susceptibility ($1/\chi$) is fitted with the modified Curie-Weiss law, $\chi(T)=\chi_{0}+C/(T-\theta_{p})$, in the paramagnetic region (100 K - 300 K). Here, $\chi_{0}$, $C$, and $\theta_{P}$ are the temperature-independent susceptibility, Curie constant, and paramagnetic Curie temperature, respectively. The estimated effective magnetic moment of Gd³⁺ is 8.11$\mu_{B}$ for $B\parallel ab$ and 8.26$\mu_{B}$ for $B\parallel c$ which are close to the theoretical value of $g\sqrt{S(S+1)}\mu_{B}=7.94\mu_{B}$ for $S=7/2$. The estimated values of the $\theta_{P}$ from the fitting are $-$109 K for $B\parallel ab$ and $-$116 K for $B\parallel c$. The negative values of paramagnetic Curie temperature are consistent with AFM ordering in this compound. We found that the $\chi(T)$ curve below $T_{\mathrm{N}}$ is quite different from that observed in a typical AFM system [Fig. 2(b)]. A bifurcation between ZFC and FC curves is observed below $T_{\mathrm{N}}$ for $B\parallel ab$ and these curves cross each other at 15 K. Moreover, isothermal M(B) curves show an unusual hysteresis behavior for $B\parallel ab$ as shown in Fig. 2(d),(e),(f). The hysteresis around 1 T and $-$1 T are not symmetric [ Fig. 2(e),(f)]. The area of hysteresis loop around 1 T is larger than that of around $-$1 T. We confirmed this unusual behavior by performing measurements repeatedly with several crystals. Overall, the hysteresis becomes weaker with increasing temperature, and it disappears for $T\geq$ 15 K [ Fig. 2(f)]. No hysteresis is observed for $B\parallel c$ as shown in Fig. 2(c). The value of magnetization of GdAlSi at 7 T is 0.7 $\mu_{B}$, which is an order of magnitude smaller than that of Gd³⁺ moment. Similar behavior is also observed in its sister spiral magnetic compound SmAlSi Yao et al. (2023). Moreover, SmAlSi also shows magnetic hysteresis only for $B\parallel a$ around 4 T rather than zero field region. A schematic diagram is shown in Fig. 2(g) for visualization of spin orientation in a typical spiral magnetic structure. Another small kink in $\chi(T)$ is observed at $T^{*}$= 4.5 K for $B\parallel ab$ as shown in the inset of Fig. 2(c). Further studies are required to get an insight into the magnetic ordered state of this compound.

The electrical resistivity ($\rho_{xx}$) is measured as a function of temperature within the $ab$-plane as shown in Fig. 3(a). The $\rho_{xx}(T)$ shows metallic behavior and a kink is observed at $T_{\mathrm{N}}$= 32 K due to the influence of magnetic ordering. We applied in-situ strain to the crystals along the in-plane direction while measuring temperature dependence of $\rho_{xx}$. The $\rho_{xx}$ increases with the increase of applied compressive strain. No changes in $T_{\mathrm{N}}$ are detected under maximum force of 39.2 Newton which corresponds to 0.033$\%$ compressive strain, indicating that the onset of magnetism is robust [the inset of Fig. 3(a)]. The estimated residual resistivity ratio [$\rho_{xx}(300K)/\rho_{xx}(2K)$=3] is comparable to that observed in other materials of this family Lyu et al. (2020b). Fig. 3(b) displays the magnetic field dependence of transverse magnetoresistance (MR) for $B\parallel c$. The observed maximum MR is $53\%$ at 2 K and 14 T, which gradually decreases with increasing temperature.

The field dependence of Hall resistivity, after removing the MR contribution using the expression $\rho_{yx}=[\rho_{yx}(B)-\rho_{yx}(-B)]/2$, is shown in Fig. 3(c). The positive $\rho_{yx}$ indicates that hole carriers dominate in GdAlSi. The carrier density and Hall mobility are estimated from the linear fitting of the $\rho_{yx}(B)$ curve at low field region using the relation $n$=1/($eR_{\textrm{H}}$) and $\mu$=$R_{\textrm{H}}$/$\rho_{xx}{(0)}$, where $R_{\textrm{H}}$ is the slope of $\rho_{yx}(B)$ curve. The temperature dependence of $n$ and $\mu$ are shown in Fig. 3(d). The carrier density of GdAlSi ($\sim 3\times 10^{20}$ cm^-3) is much smaller than that of typical metals ($10^{22}-10^{23}$ cm^-3) indicating a semimetallic behavior, consistent with the small value of density of states at the Fermi level ($E_{F}$) [see Fig. 5]. The values of $n$ and $\mu$ of GdAlSi are of the same order of magnitude as those reported for other members of this family Yang et al. (2021); Lyu et al. (2020b). As shown in Fig. 3(d), above magnetic ordering temperature $T_{N}$, the carrier density is found to be constant as a function of temperature up to 300 K, suggesting nearly no changes in the electronic states near Fermi level, which is expected for a system like this with Fermi level substantially above band edge, and this is consistent with the DFT calculations discussed later. The Hall mobility decreases slowly with increasing temperature due to increased carrier scattering at higher temperature. Below $T_{N}$, magnetic order sets in and enhances as temperature decreases. Increased magnetic orders decreases magnetic scattering to the carriers that resulted in higher mobility and lower numbers of mobile carriers, i.e. lower value of $n$ at low temperatures.

The $\rho_{yx}(B)$ at 2 K exhibits an anomaly around 3 T and the curve strongly deviates from linear field dependence at higher magnetic field as shown in Fig. 3(e). The fitting with conventional two-band model fails to reproduce the Hall conductivity data. Rather, this behavior is reminiscent of the Berry curvature induced anomalous Hall effect (AHE) observed in AFM Weyl semimetals Suzuki et al. (2016); Singha et al. (2019). To gain further insight, the anomalous Hall conductivity ($\sigma_{xy}^{A}$) is separated from the observed Hall conductivity ($\sigma_{xy}$) using the relation $\sigma_{xy}^{A}=\sigma_{xy}-\sigma_{xy}^{N}$, where $\sigma_{xy}^{N}$ is the normal Hall conductivity estimated from linear extrapolation of the low field fitting curve ($\rho_{yx}^{N}$). Both $\sigma_{xy}$ and $\sigma_{xy}^{N}$ are obtained through tensor conversions from the measured $\rho_{xx}$ and $\rho_{yx}$, $\sigma_{xy}=\rho_{yx}/(\rho_{xx}^{2}+\rho_{yx}^{2})$ and $\sigma_{xy}^{N}=\rho_{yx}^{N}/(\rho_{xx}^{2}+\rho_{yx}^{N2})$. The field dependence of $\sigma_{xy}^{A}$ is plotted in Fig. 3(f) at various temperatures from 2 K to 300 K. The maximum value of $\sigma_{xy}^{A}$ reaches 1310 $\Omega^{-1}$cm^-1 around 7.5 T at 2 K. This value is quite large compared to those reported in RAlX family and other AFM Weyl semimetals [See Table 1]. The peak of $\sigma_{xy}^{A}$ becomes broader and shifts towards higher magnetic field with increasing temperature. Moreover, the value of $\sigma_{xy}^{A}$ decreases gradually with increasing temperature and reaches $\sim 155$ $\Omega^{-1}$cm^-1 at 300 K. We note that GdAlSi has the highest AHC, nearly two times higher than the second highest AHC material presented in table 1. Furthermore, the observed AHC in GdAlSi is present at room temperature, which is not observed in other materials compared in table 1.

Fig. 4 shows different magnetic patterns studied in this work using DFT calculation. Table. 2 shows the energy difference between various magnetic configurations. Our calculations find that all antiferromagnetic configurations are lower in energy compared to the ferromagnetic phase which is consistent with the experimental observation. In particular, A-type AFM pattern with in-plane FM and out-of-plane AFM arrangement of spins (udud) [Fig. 4(a)] is found to be lower in energy compared to up-down-down-up (equivalently up-up-down-down) [Fig. 4(b)] and in-plane AFM [Fig. 4(c)] arrangement. We also studied in-plane ferrimagnetic pattern Gaudet et al. (2021) by fixing spins in the up-down-down (udd) direction along 110-axis [Fig. 4(d)] Similarly, $120^{\circ}$ rotated spiral spin arrangement along 110 direction was also investigated Yao et al. (2023). While both of these patterns were found to have slightly higher energy than the udud pattern, the numbers are well within the uncertainty of the DFT calculations. The magneto-crystalline anisotropy energy is found to be weak in this system ($<$1 meV per Gd atom). This is consistent with the experimentally measured magnetic anisotropy for in-plane and out-of-plane applied magnetic field [see Fig. 2(a)]. In addition to the GGA functional [Table. 2], we also used SCAN-GGA functional to perform total energy calculations between different magnetic configurations for consistency check Sun et al. (2015); Wang et al. (2023). SCAN functional also gave udud phase as the lowest energy state by similar order of magnitude.

In Fig. 5, we compare the DFT calculated electronic structure of GdAlSi between the non-magnetic (NM) and lowest AFM phase (udud) found from our fixed-spin DFT calculations. Since there are no significant differences in the electronic dispersion between the magnetic and NM phase [Fig. 5(a) and 5(b)], we will first describe the electronic dispersion of NM phase in detail. The calculation for the NM phase was done within the open-core approximation where the Gd-4f states were frozen. Fig.  5(a) shows atom-projected electronic bands and density of states which reveal that the states around the Fermi level are derived mainly from Gd-d with small contributions from p-states of Gd, Al, and Si. Linear bands exist in the vicinity of the Fermi level which is consistent with vanishing DOS. While there are SOC induced gaps along high symmetry lines, band crossings exist along non-high symmetry directions giving rise to many Weyl points. An exhaustive search of band crossings in the entire BZ finds 18 pairs of Weyl points with Chern number of $\pm 1$ in the primitive BZ lying at around $\sim$ 50 to 100 meV above $E_{F}$. Despite having many pairs of Weyl nodes, most of them are in close vicinity to the Fermi level. While this system is very far from the ideal regime with single pairs of Weyl nodes, it still presents a unique opportunity to study Weyl mediated interactions due to the proximity of Weyl nodes to the Fermi level Ding et al. (2023a); Wang et al. (2022); Ding et al. (2023b); Belopolski et al. (2017); Liu et al. (2019); Nie et al. (2022); Lu et al. (2013). The Fermi surface plot [Fig. 5(d)] shows “hammer”-like feature from the electron bands along the $\bar{\Gamma}-\bar{X}$ direction and small ellipsoids from hole Weyl bands slightly away from $\bar{\Gamma}-\bar{M}$ direction. The electronic dispersion and FS plot for udud AFM phase shown in Figs. 5(b) and  5(e) are almost identical to the NM phase. The Gd-4f states, which are at $-$6 eV below $E_{F}$ are pushed to $-$10 eV by the application of Hubbard $U$ of 6 eV. There is a weak hybridization between the Gd-5d dominated Weyl bands with the Gd-4f states which enhances the exchange splittings between the Kramer’s pairs as highlighted by the dotted blue circle in the inset plot of Fig. 5(b). However, it is remarkable that except for a slight increment in the size of both the electron and hole pockets, the overall shape and topology of the FS remain intact. Therefore, in the next section, we use the electronic dispersion of the NM phase to make a comparison with the ARPES experiment.

To experimentally probe the electronic structure of GdAlSi, we have performed ARPES measurements. Fig. 6(a) represents the Fermi surface (FS) of GdAlSi measured in the AFM phase. We used a photon energy of 130 eV that covers two successive surface Brillouin zones (SBZs). One of the most notable features of the FS is the presence of high-intensity spots (enclosed by the circles) located near the $\bar{\Gamma}$ point, slightly offset along the $\bar{\Gamma}$-$\bar{M}$. A closed electron pocket is also observed around the $\bar{\Gamma}$. The overall features of the FS appear slightly different between the first and second zones, likely due to the photoemission matrix element effect. To understand the origin of the FS features, we have plotted the theoretical constant energy contour (at $E-E_{F}=-$120 meV) along with the Weyl nodes (magenta color dots) in the $k_{x}-k_{y}$ plane for $k_{z}=$ 0 and $k_{z}$ integrated from 0 ($\Gamma$) to 0.5 ($Z$) in Figs. 6(b) and 6(c), respectively. Comparison of experimental and theoretical results suggests that the high-intensity spots and arc-like features [the inset of Fig. 6(a)] observed in the ARPES Fermi surface possibly originate from the projection of the Weyl cones. Similar Weyl arcs were previously found in PrAlSi and SmAlSi Zhang et al. (2023); Lou et al. (2023); Sanchez et al. (2020). ARPES spectra along the $\bar{\Gamma}$-$\bar{M}$ and $\bar{\Gamma}$-$\bar{X}$ directions of the SBZ are shown in Fig. 6(d) and 6(e), respectively. The theoretical bands from the $k_{z}=$ 0 plane are overlayed on top of the ARPES spectra for comparison. A relatively good agreement between theory and experiment is obtained when the theoretical bands are shifted upward by $\sim$ 120 meV. This energy shift is most likely related to the surface effects. Overall, the ARPES spectra show some broadening due to the finite $k_{z}$ broadening. This can be better understood by comparing ARPES spectra to the projected DFT band structure results [Fig. 6(g) and (h)]. Although the theoretical band dispersions agree well with the ARPES along the $\bar{\Gamma}$-$\bar{M}$ direction, some discrepancies can be seen along the $\bar{\Gamma}$-$\bar{X}$ direction, especially in the first SBZ. For example, the M-shaped band (marked by an asterisk symbol) observed in ARPES at around $-$0.8 eV is not present in the theoretical calculations, even in the $k_{z}$ projected spectra [Fig. 6(h)]. Thus, this M-shaped band is possibly originating from the surface states. In Fig. 6(f), the ARPES spectrum cutting through the Weyl nodes [cut 1 in Fig. 6(a)] is displayed, with DFT bands superimposed. The comparison indicates that the material exhibits Weyl-like band dispersion near the Fermi energy. However, we could not resolve the Weyl points as they are expected to be above the Fermi energy from our DFT calculations.