Quasi-linear magnetoresistance and paramagnetic singularity in hypervalent bismuthide

Abstract

Materials featuring hypervalent bismuth motifs have generated immense interest due to their extraordinary electronic structure and exotic quantum transport. In this study, we synthesized high-quality single crystals of La₃ScBi₅ characterized by one-dimensional hypervalent bismuth chains and performed a systematic investigation of the magnetoresistive behavior and quantum oscillations. The metallic La₃ScBi₅ exhibits a low-temperature plateau of electrical resistivity and quasi-linear positive magnetoresistance, with anisotropic magnetoresistive behaviors suggesting the presence of anisotropic Fermi surfaces. This distinctive transport phenomenon is perfectly elucidated by first-principles calculations utilizing the semiclassical Boltzmann transport theory. Furthermore, the nonlinear Hall resistivity pointed towards a multiband electronic structure, characterized by the coexistence of electron and hole carriers, which is further supported by our first-principles calculations. Angle-dependent de Haas-van Alphen oscillations are crucial for further elucidating its Fermiology and topological characteristics. Intriguingly, magnetization measurements unveiled a notable paramagnetic singularity at low fields, which might suggest the nontrivial nature of the surface states. Our findings underscore the interplay between transport phenomena and the unique electronic structure of hypervalent bismuthide La₃ScBi₅, opening avenues for exploring novel electronic applications.

Introduction

Magnetoresistance (MR), defined as the variation in resistance under an applied magnetic field, serves as a crucial probe into Fermi surface properties and holds promise for applications in magnetic sensors and hard disk drives^1,2. In the realm of extreme quantum limits, where the magnetic field quantizes orbital motion, the dominance of the lowest Landau level (LL) typically leads to the expectation of pronounced quasi-linear MR at high fields^3,4. The exotic MR behavior in Dirac and Weyl materials has led to a flurry of extensive research, including significant negative linear MR in Dirac semimetals Cd₃As₂⁵ and Na₃Bi⁴, and titanic MR in Type II Weyl semimetal WTe₂⁶. These Dirac and Weyl fermions in those topological materials also manifest various exotic quantum phenomena, including chiral anomaly and nontrivial quantum oscillations^7,8. In addition, several non-magnetic materials with zero-gap properties show large unsaturated positive MR, the possible mechanism is based on the classical theory of long transport mean free path caused by mobility distribution or the quantum effect with linear dispersive band structure^9,10. Consequently, MR will exhibit unconventional behaviors across diverse novel materials^11,12, thereby fostering the development of potential electronic devices.

The “heavy” element bismuth and its compounds are predestined for topological properties induced by spin-orbit coupling (SOC)¹³. The AMnBi₂ (A = Sr, Ca, Eu, or Yb) systems^14,15 exhibit fascinating properties that derive from their building unit, a two-dimensional Bi square net containing relativistic fermions. In parallel, the Ln₃MX₅ system (Ln = rare earth; M = transition metals; X = As, Sb, Bi) is predicted to be a topological candidate owing to its linear Bi chains. The quasi-one-dimensional chains in Sm₃ZrBi₅ give rise to dispersive linear and flat bands, while Bi-Bi interaction results in topological band crossing^16,17. Experimental verification of nontrivial topological bands in the Ln₃MX₅ family has been successfully demonstrated via de Haas van Alphen (dHvA) oscillation analysis in La₃MgBi₅^18,19. These compounds are intriguing due to their narrow-gap topological bands influenced by SOC, potentially exhibiting novel electronic transport properties when subjected to a magnetic field. Notably, research on the MR of the low-dimensional Ln₃MX₅ system remains limited.

In this work, we report details of single-crystal growth of hypervalent bismuthide La₃ScBi₅ with one-dimensional Bi chains from the Ln₃MX₅ family. Density functional theory calculations indicate that SOC significantly affects La₃ScBi₅, as the band structure near the Fermi energy is predominantly comprised of Bi p orbitals. The electrical transport and magnetic properties were comprehensively investigated, leading to a study of the electron band structure in both the bulk and surface regions. The bulk crystal shows that the magnetoresistance exhibits a crossover between a conventional quadratic law and an unusual linear dependence, along with highly anisotropic electronic transport properties. Analysis of the Hall effect reveals that the Fermi surface of La₃ScBi₅ consists of multiple pockets arising from a multiband electronic structure. Even La₃ScBi₅, which has quasi-one-dimensional features and pronounced anisotropy, has a distinctly three-dimensional electronic structure, as evidenced by its angle-dependent dHvA oscillations. Notably, the presence of a prominent paramagnetic singularity suggests the potential nontrivial nature of its surface states. Our study employs La₃ScBi₅ as a model system to investigate hypervalent bismuthides, focusing on the quasi-linear MR and paramagnetic singularities observed in electron-rich motifs.

Results

Quasi-one-dimensional patterns of crystal structure

La₃ScBi₅ crystallizes in a rod-like, hexagonal structure, characteristic of other members in the Ln₃MX₅ family, with a space group of P6₃/mcm (No. 193). Analysis of the single-crystal X-ray diffraction (XRD) data reveals that La₃ScBi₅ exhibits lattice parameters of a = 9.746(3) Å and c = 6.561(3) Å, matching closely with the previous single-crystalline study²⁰. The single-crystal XRD data collection and refinement parameters are gathered in Tables SI-III (Supporting Information). The (h00) reflections in Fig. 1b demonstrate the excellent crystal quality.

Fig. 1: Crystal structure of La₃ScBi₅.

a Schematic diagram of crystal structure with a view along the c-axis projection. b Single-face X-ray diffraction pattern for a flux-grown single crystal. The inset presents an optical photograph of a typical single crystal. c Quasi-one-dimensional patterns in La₃ScBi₅: The separate one-dimensional (1D) ScBi₆ face-sharing octahedral chains, 1D Bi chains, and La zig-zag chains. d Electronic band structure of La₃ScBi₅ with spin-orbit coupling (SOC), which emphasizes the contribution of band projection of Bi chain p orbitals. e Partial density of states calculations with SOC, highlighting the contribution of Bi p states at the Fermi level.

Figure 1c showcases quasi-one-dimensional patterns within La₃ScBi₅, composed of the separate one-dimensional (1D) ScBi₆ face-sharing octahedral chains, 1D Bi chains, and La zig-zag chains aligned along the c axis, rather than forming a perfect 1D system. A notable feature is the Bi-Bi bond length within the chain, measuring approximately 3.28 Å, significantly longer than expected for a localized and covalent Bi-Bi single bond. This elongated yet symmetric bonding can be comprehended by “hypervalent” bonding, wherein a − 2 charge on Bi is stabilized by the delocalization of electrons within the chain²¹. The presence of extended hypervalent bonds necessitates an electron count of one electron per bond, resulting in half-filled bonds crucial for stabilizing symmetry-protected band crossings^22,23,24. Moreover, hypervalent bonds exhibit greater orbital overlap compared to metallic bonds, resulting in steep and widely dispersed bands^16,23. The double layers of the La atoms are stacked by 6₃ spiral axes parallel to the c direction, giving rise to La-based zig-zag chains with the nearest La-La distance of 4.01 Å.

To gain a deeper understanding of the electronic structure of La₃ScBi₅, the projection of Bi p orbitals onto the density functional theory band structure was analyzed according to the 4 d Wyckoff position, as shown in Fig. 1d. It is observed that the band near the Fermi level exhibits a significant contribution from 1D Bi chains. It is worth noting that the band projection between Γ and A reveals a steep slope, indicating a high Fermi velocity and suggesting excellent potential for thermal conductivity. This is primarily attributed to the 1D hypervalent Bi chains characteristics along the c-axis, where electron hopping is significantly greater than other directions, leading to more pronounced band dispersion. The partial density of states (DOS) incorporating SOC in Fig. 1e shows a small, broad peak near the Fermi level, predominantly arising from the total Bi p orbitals, with notable hybridization from the La d and Sc d states. Taken together, the hypervalent Bi chains are essential in determining a range of physical properties, and additional experiments are required to delve into the thermal properties of La₃ScBi₅.

Quasi-linear magnetoresistance and nonlinear hall resistivity

Temperature-dependent resistivity of materials reflects the scattering of charge carriers by impurities and phonons at low and high temperatures, respectively. In La₃ScBi₅, the electrical resistivity, ρ_||(T) (I // c) and ρ_⊥(T) (I // b), were investigated under a zero field as a function of temperature, as depicted in Fig. 2a. La₃ScBi₅ exhibits a metallic character with a residual resistivity ratio (ρ_||, 300 K/ρ_||, 2 K) of approximately 19. This RRR value, while not exceptionally larger, suggests the presence of significant disorder scattering. Despite the similar overall temperature dependencies of resistivity along both crystallographic axes, it was observed that ρ_⊥ is an order of magnitude higher than ρ_||, yielding an anisotropy ratio (ρ_⊥/ρ_||) of around 20 at 2 K. The high-temperature behavior of ρ_||(T) can be effectively described by the Bloch-Grüneisen (BG) formula²⁵:

$${\rho \left(T\right)}_{{BG}}=\,{\rho }_{0}+\alpha {\left(\frac{T}{{T}_{D}}\right)}^{n}{\int }_{0}^{\frac{{T}_{D}}{T}}\frac{{x}^{n}{dx}}{\left({e}^{x}-1\right)\left(1-{e}^{-x}\right)}\,$$

(1)

Fig. 2: Charge transport of La₃ScBi₅ of ρ_||(T) (I // c).

a Temperature dependence of ρ_||(I // c) and ρ_⊥(T) (I // b) under zero field on sample S1. The solid line is the fit of data to the BG formula. Inset: Temperature dependence of the radio of ρ_⊥/ρ_||. b Temperature dependence of ρ_||(T) with H // a (I // c) up to 16 T below 50 K. The red arrows denote the temperature T^*. Inset: ρ_||(T) below 50 K and fitting curve (solid line). c Field dependence of magnetoresistance at various temperatures. d Kohler plot: MR versus B/ρ_||(T, B = 0). The solid line is the fit to MR = 4.11(B/ρ₀)^1.29.

Here, ρ₀ represents the residual resistivity, T_D corresponds to the Debye temperature. By utilizing the optimal parameters of n = 3, with values of ρ₀ = 1.67 μΩ cm, T_D = 187.37 K, and $\alpha$ = 34.17 μΩ cm/K, the BG formula provides a good fit for ρ_|| in the temperature range of 2 to 300 K. The obtained T_D value closely aligns with that derived from the specific heat of La₃ScBi₅ (Figure S4, Supporting Information). The low-temperature (< 30 K) ρ_|| follows a power law behavior ρ(T) = ρ₀’ + AT^n’ with ρ₀’ = 1.65 μΩ cm and n’ = 2.88, as shown in the inset of Fig. 2b. Both values of n are nearly exactly 3, exhibiting a T³ dependence, indicating that the primary interaction in our system is likely inter-band electron-phonon scattering^26,27. Similar analyses and interpretations have been applied to isomorphic compounds such as Sm₃ZrBi₅¹⁶, as well as to ZrSiS²⁸ and TiS₂²⁹.

Figure 2b illustrates the temperature dependence of ρ_||(T) under various fields with the H // a axis. In the absence of an external magnetic field, the resistivity decreases monotonically with a reduction in temperature. However, upon ramping the magnetic field to 16 T, a distinctive turn-on behavior is observed, leading to a resistivity minimum at T^* before eventually plateauing. This magnetic-field-driven resistivity upturn followed by a resistivity plateau is interpreted as a potential transport hallmark of conducting surface states, attributed to the LL quantization of relativistic electrons^30,31. Such a turn-on behavior has been documented in topological materials such as PtTe₂³² and WTe₂³³. Another viable mechanism posits that the reentrant metallic state is a consequence of the scaling behavior in magnetoresistance, driven by the power-law dependence of the magnetic field and temperature³⁴. The corresponding field-dependent MR ratio, [ρ_||(B) - ρ_||(0 T)]/ρ_||(0 T), at various temperatures is depicted in Fig. 2c. Notably, the MR value peaks at 28% at 2 K and 9 T, displaying no saturation tendency. Particularly intriguing is the nearly linear dependence of MR on the magnetic field, most evident beyond the low-field regime. A similar quasi-linear positive MR phenomenon has been observed in HoAgGe, attributed to uncompensated charge carriers³⁵. It’s noteworthy to mention that this linear MR might also be a signature of charge density wave materials, including TbTe₃, HoTe₃³⁶, and LaAuSb₂³⁷. The existence of inhomogeneity and open Fermi surfaces or electron-phonon coupling along with a nested imperfect Fermi surface may be plausible mechanisms^36,38,39.

To further research the underlying electronic structure influencing the MR behavior, we have conducted an analysis based on Kohler’s rule in Fig. 2d, which dictates that the MR in a conventional metal obeys a scaling behavior of Δρ/ρ(0) = F[B/ρ(0)]. The MR data at low temperatures exhibit a remarkable collapse onto a singular curve, which can be effectively approximated by the formula: MR = 4.11(B/ρ₀)^1.29, indicating that La₃ScBi₅ does not conform to an electron-hole compensated system^6,40. The MR depicted in Figure S5a (Supporting Information) was calculated using the WannierTools package, employing the Boltzmann transport equation and Wannier function techniques. The calculated ρ_||τ demonstrates a nonsaturating dependence of (Bτ)^1.25, which is quite consistent with the experimental results of MR $\propto$ B^1.29. The agreement between theoretical calculation and experiment suggests that the MR may be explained by the semiclassical theory, with no significant role from the Berry curvature aspect of topological Dirac points. The adherence to Kohler’s rule serves as a pivotal indication; it underscores that the observed upturns and low-temperature plateaus are not indicative of a metal-insulator transition but rather stem from factors such as small residual resistivity, high mobilities, and a low charge-carrier density^6,33,34.

Our investigation also systematically explored the transport properties along the c-crystallographic directions, unveiling anisotropic characteristics in the seemingly unsaturated MR. The ρ_⊥(T) curves obtained under various magnetic fields are compiled in Fig. 3a, where the turn-on behavior is also observed at 3 T and above. The saturation value and the range of upturn amplify with the applied magnetic field strength, indicative of the positive MR effect. Figure 3b, c illustrate the magnetic field dependence of MR at diverse temperatures and the field derivative of MR, dMR/dB, respectively. The quasi-linear behavior extends to notably low crossover fields B^∗, beyond which MR transitions to a weak-field semiclassical quadratic dependence, which is consistent with the behavior of the theoretically calculated MR (Figure S5b, Supporting Information). Here, the B^∗ denotes the point where the fitting lines intersect. With increasing temperature, the field range exhibiting quasi-linear MR diminishes, and MR decreases, consistent with observations in 112-type materials such as CaMnBi₂⁴¹, YbMnBi₂⁴² and LaAgBi₂⁴³, suggesting the possible existence of a nontrivial state. In the quantum limit where all carriers occupy solely the lowest LL, the observed B^∗ corresponds to the quantum limit of ${B}^{* }=\left(1/2e{{\hslash }}{\upsilon }_{F}^{2}\right){\left({k}_{B}T+{E}_{F}\right)}^{2}$ ^44,45. As depicted in Fig. 3d, the experimental data for B^∗ align well with the above equation, further indicating the possibility of nontrivial states in La₃ScBi₅. Moreover, with increasing temperature, the linear term coefficient A₁ is suppressed due to the temperature smearing of the LL splitting⁴³. Future investigations, requiring more systematic Angle-Resolved Photoemission Spectroscopy (ARPES) studies, are necessary to probe the existence of Dirac cones in the electronic structure. Such the peculiar behavior of MR is predominantly governed by the multi-band features near the Fermi surface. Through the projection of individual atomic orbitals (Figure S2, Supporting Information) and DOS analysis, it is evident that the Bi p orbitals exhibit substantial contributions, as do hybrids between other orbitals, which together have a comprehensive effect on MR.

Fig. 3: Charge transport of La₃ScBi₅ of ρ_⊥(T) (I // b).

a Temperature dependence of ρ_⊥(T) with H // c (I // b) under various fields on sample S1. Inset: Enlarged view of ρ_⊥(T) below 50 K. The red arrows denote the temperature T^*. b Field dependence of MR at various temperatures. c The field derivative of MR at various temperatures. In the high-field regime, fitting results using MR = A₁B + O(B²) are represented by lines, while in the low-field region, MR = A₂B is employed. d Temperature dependence of the critical field B^* (Purple squares). The solid line depicts the fitting results of B^*, utilizing ${B}^{* }=\left(1/2e{\rm{\hslash }}{\upsilon }_{F}^{2}\right){\left({k}_{B}T+{E}_{F}\right)}^{2}$. The red circle corresponds to the linear coefficient A₁ in the high-field MR.

To determine the mobility of the charge carriers, Fig. 4a illustrates the field dependence of the Hall resistivity ρ_yx at various temperatures, revealing a nonmonotonic behavior in La₃ScBi₅ below 30 K, indicative of its multiband nature. At high temperatures (T ≥ 30 K), ρ_yx remains positive and exhibits a linear relationship with the applied magnetic field, suggesting predominant transport by single-hole-type carriers. Figure 4b presents the field dependence of Hall resistance considering a relaxation time of 0.17 ps. Remarkably, our calculation shows the same trend as the experimental measurements, which exhibit nonlinear behavior that demonstrates the multicarrier behavior. We apply Kohler’s rule for Hall resistivity to La₃ScBi₅, demonstrating that the Hall resistivity scales with both magnetic field and temperature, akin to the longitudinal resistivity scaling under the relaxation time approximation. This theoretical framework was initially proposed by Zhang et al. ⁴⁶, and is further elaborated in Note 1 (Supporting Information). This observed scaling behavior shows reasonable agreement with theoretical calculations. Compared with the case of ρ_||, Kohler’s rule is basically obeyed over a large temperature range for ρ_yx (Fig. 4c). Hence, a multi-band model can be used to analyze the obtained results^47,48:

$${\sigma }_{{xy}}\left(B\right)=\frac{{\rho }_{{yx}}}{{{\rho }_{{||}}}^{2}+{{\rho }_{{xy}}}^{2}}={eB}\mathop{\sum }\limits_{i}^{N}\frac{{n}_{i}{\mu }_{i}^{2}}{1+{\mu }_{i}^{2}{B}^{2}},$$

(2)

$${\sigma }_{{xx}}\left(B\right)=\frac{{\rho }_{{\rm{||}}}}{{{\rho }_{{\rm{||}}}}^{2}+{{\rho }_{{yx}}}^{2}}=e\mathop{\sum }\limits_{i}^{N}\frac{{n}_{i}{\mu }_{i}}{1+{\mu }_{i}^{2}{B}^{2}}.$$

(3)

Fig. 4: Hall effect of La₃ScBi₅.

a Field dependence of Hall resistivity ρ_yx(B) at various temperatures. Inset: The temperature dependence of R₀(T). A diagram of the Hall bar is shown, where I⁺ and I^- are a pair of current leads, and V⁺ and V^- are a pair of voltage leads, with H // a, I // c, and V // b. b The calculated field dependence of Hall resistivity at T = 2 K. Inset: The experimental field dependence of Hall resistivity at T = 2 K. c ρ_yxα ρ₀ versus H/α ρ₀. ρ₀ is derived from ρ_|| (T, H = 0) data. Inset: The temperature dependence of α. d Conductivity and Hall conductivity of La₃ScBi₅ at T = 2 K, the black line represents the fitted line using the multi-band model.

A typical fitting for T = 2 K is presented in Fig. 4d, yielding carrier densities of n_e1 = 1.6 × 10²¹ cm⁻³, n_e2 = 6.3 × 10¹⁹ cm⁻³, n_h1 = 1.9 × 10¹⁹ cm⁻³, and n_h2 = 5.1 × 10¹⁸cm⁻³, along with mobility values of m_e1 = 139 cm² V⁻¹ s⁻¹, m_e2 = 1184 cm² V⁻¹ s⁻¹, m_h1 = 893 cm² V⁻¹ s⁻¹, and m_h2 = 4276 cm² V⁻¹ s⁻¹. It is noteworthy that an excessive number of fitting parameters may lead to significant errors, potentially failing to accurately reflect the physical essence of the multi-band system⁴⁷. For the multi-band system of La₃ScBi₅, fitting the multi-band model with smaller N values is particularly challenging, possibly due to the lack of suitable initial values. Similar limitations are observed in other multiband systems, such as the pyrite-type bismuthide PtBi₂⁴⁹ and the 3D Weyl semimetal TaAs⁵⁰.

Quantum oscillations and fermi surface properties

The quantitative analyses of dHvA oscillations reveal the nature of electrons and serve as a powerful tool in studying topological physics. Figure 5a shows the beautiful dHvA oscillations that persist to about 10 K for sample S1 for field orientation H // a up to 9 T. The oscillatory components ΔM, obtained by subtracting a smooth background, are plotted against 1/B in Fig. 5b. From the fast Fourier transform (FFT) analyses of the oscillatory components ΔM, three principal frequencies (F_α = 33.8 T, F_α’ = 56.1 T, and F_β = 90.5 T) are derived (Fig. 5c). The F_α component exhibits a significantly greater oscillation amplitude compared to F_α’ and F_β components, making the F_α frequency the focal point of research interest. Similar multifrequency oscillations have also been observed in the isostructural compound La₃MgBi₅^18,19, which is suggestive of a 3D-like electronic structure. Note that similar dHvA measurements were observed on a crystal from the same batch as the sample S1(Figure S7, Supporting Information). The oscillation frequency is directly linked to the cross-sectional area of the Fermi pockets normal to the magnetic field A_F through the Onsager relation F = (Φ₀/2π²)A_F. Specifically, the A_F associated with the main peak is estimated to be 3.23 × 10⁻³ Å⁻² for the α bands, yielding the Fermi vector k_α = 3.20 × 10⁻² Å⁻¹. The corresponding parameters for three different pockets are listed in Table SIV (Supporting Information). It is noteworthy that distinct and varying quantum oscillations were observed in the H // b orientation (Figure S6, Supporting Information), indicating pronounced in-plane anisotropy within the ab-plane of La₃ScBi₅.

Fig. 5: Analyses of the de Haas-van Alphen (dHvA) oscillations for H // a in La₃ScBi₅.

a Isothermal magnetization at different temperatures with the magnetic field parallel to the a-axis on sample S1. b The oscillatory components of magnetization ΔM. c The corresponding FFT spectrum of the oscillatory component of the dHvA oscillations at various temperatures. d The fits of the FFT amplitudes of F_α, F_α’, and F_β to the temperature damping factor R_T by the LK formula. e Multiband LK fit of the oscillation components of the dHvA oscillations at 2 K. The black lines show the fits of the oscillation pattern to the generalized three-band LK formula for three different frequency oscillations. f The LL index fan diagram for α band derived from oscillatory magnetization at 2 K.

The oscillatory magnetization of a Dirac system is generally described by the Lifshitz-Kosevich (LK) formula⁵¹, accounting for the Berry phase⁵²:

$$\Delta M\propto \,-{B}^{\lambda }{R}_{T}{R}_{D}{R}_{S}\sin \left[2{\rm{\pi }}\left(\frac{F}{B}-{\rm{\gamma }}-{\rm{\delta }}\right)\right]$$

(4)

where ${R}_{\text{T}}=({KT}\mu /B)/\sinh ({KT}\mu /B)$, $K=2{\pi }^{2}{k}_{B}{m}_{\text{e}}/(\hslash /e)$ ≈ 14.69 T/K, ${R}_{\text{D}}=\exp (-K{T}_{\text{D}}\mu /B)$. μ represents the ratio of effective cyclotron mass to free electron mass, while R_S denotes the spin reduction factor arising from Zeeman splitting. The oscillation of ΔM is delineated by the sine term with the phase factor $1/2-{\varphi }_{B}/2{\rm{\pi }}-\delta$. The phase shift δ adopts values of 0 and ±1/8 for the 2D and 3D Fermi surfaces, respectively. Utilizing the LK formula, the effective mass m^* can be derived by fitting the temperature-dependent oscillation amplitude to the thermal damping factor R_T, as depicted in Fig. 5d. Here, B denotes the inverse average of the field window employed for Fourier analysis⁵³, given by B_ave = [(1/B_max + 1/B_min)/2] = 6.43 T with B_max = 9 T and B_min = 5 T, leading to calculated m^*/m₀ of 0.206, 0.227 and 0.234 for α, α’ and β pockets, respectively. The small values of m^* imply the presence of relativistic charge carriers. Given that the Fermi surface of La₃ScBi₅ is of a strong 3D character, we have adopted the oscillation pattern at 2 K using the multiband LK formula with fixed frequency and fixed effective mass (Fig. 5e), yielding the Dingle temperature T_D = 4.56 K for the F_α band, which corresponds to the quantum relaxation time ${\tau }_{q}$ = ${{\hslash }}$ / (2$\pi$k_BT_D) = 2.67 × 10⁻¹³ s, and quantum mobility ${\mu }_{q}=e\,{\tau }_{q}/{m}_{\alpha }^{* }$ = 2726 cm²/ V⁻¹ s⁻¹. From the LK fit of oscillations with F_α = 33.8 T, a phase factor of −γ − δ = 1.97 is obtained, yielding the Berry phase ${\varphi }_{B}$ is determined to be 0.69π ($\delta$ = −1/8) or 1.19π ($\delta$ = 1/8). Based on the calculated Fermi surface presented below, with ${\varphi }_{B}=\,$0.69π ($\delta$ = −1/8), the oscillation of the probe F_α corresponds to the minimum cross-section. The Dingle temperature, relation time, quantum mobility, and Berry phase obtained by LK fitting for F_α’ and F_β are summarized in Table SIV (Supporting Information). Certainly, the fitting performed below 5 T is based on a limited number of data points, which may introduce greater uncertainty for Berry phases with multiple LK fits.

Given that dM/dB is directly proportional to the density of states at the Fermi level^54,55, we can associate the minimum of ΔM with N − 1/4, where N represents the LL index. However, due to the Landau indices of F_β being far from the quantum limit, there is notable uncertainty in determining the intercept. As depicted in Fig. 5f, the LL indices N for α bands are plotted against 1/B. The solid lines depict linear fits based on the Lifshitz-Onsager quantization criterion N = F/B + ${\varphi }_{B}$/2π$-\delta$. The slope of the linear fits yields a value of 34.31 T, consistent with the FFT derived result, thereby affirming the reliability of the linear fit within the LL fan diagram. The intercept extracted from linear extrapolation is 0.430 ± 0.018, corresponding to a Berry phase of 0.72π ($\delta$ = −1/8), aligning with the findings obtained from LK fitting. While a finite Berry phase is generally considered indicative of topologically nontrivial bands, it may not reflect the intrinsic properties of a 3D-Dirac Fermi surface, but could simply be a consequence of time-reversal symmetry⁵⁶. However, the phase shift can be affected by spin splitting and g-factor, making it challenging to determine the topological properties of the band from magnetic transport measurements alone. The similar deviation in the intercept was observed in the study of PtBi₂⁵⁷, which was attributed to the Fermi level not intersecting the degeneracy point.

In a manner analogous to the in-plane cyclotron motions observed for H // a, electrons engaged in interlayer cyclotron motions under fields along the c-axis exhibit quantum oscillation behavior. Figure S8a and S8b (Supporting Information) depict dHvA oscillations of H // c overlaid on a diamagnetic background in sample S1 and their respective oscillatory components. In contrast to the dHvA oscillations for H // a, those for H // c consist of more frequencies. FFT analyses (Figure S8c, Supporting Information) reveal a predominantly low frequency (106.8 T) and three high frequencies (181.8, 215.2, and 249.1 T), and this phenomenon is also reproduced for samples S2 and S3 (Figure S9, Supporting Information). The electron cyclotron masses for H // c range slightly higher, approximately between 0.366 and 0.529 m₀ across all probed oscillation frequencies. This anisotropic effective mass is consistent with the quasi-1D structure of La₃ScBi₅. The lower single-frequency component, extracted by isolating the higher-frequency component, can be analyzed using the widely adopted LL fan diagram method, as demonstrated in Figure S8d (Supporting Information). The intercept derived from the linear fit in the fan diagram yields 0.234 ± 0.092 for F_γ, with uncertainties suggesting δ values of 1/8 or −1/8, resulting in an uncertain phase of 0.71π or 0.21π. Additionally, the fitted oscillation frequency for F_γ is 108.09 T, closely aligning with FFT results. Noted that significant fitting errors limit the precision of determining ${\varphi }_{B}$ for γ band. Furthermore, accurately determining the LL index field for each frequency remains challenging in the case of multifrequency oscillations for F_η^54,58. Nonetheless, we did not detect Shubnikov-de Hass (SdH) oscillations either in the MR or in the Hall resistivity up to 9 T, which warrants further measurements at higher magnetic fields and lower temperatures. It is conceivable that SdH oscillations come from the oscillating scattering rate and can thus be complicated by the detailed scattering processes while the dHvA effect is caused directly by the free energy oscillations of a system^59,60.

The anisotropic characteristics of the Fermi surface of La₃ScBi₅ are further elucidated through an investigation of ΔM versus 1/B and corresponding dHvA oscillation frequencies at 2 K across various magnetic field orientations. Figure 6a illustrates angle-dependent quantum oscillation as the magnetic field rotates from the [100] to [001] direction. Figure 6b reveals the oscillatory component after subtracting the background, demonstrating a distinct evolution of the angle θ, defined as the angle between the field direction and [100] direction. The oscillation amplitude diminishes progressively with increasing the tilt angle. Additionally, there is one more frequency F_γ detected at θ ≥ 75° in the FFT spectra (Fig. 6c). These angular dependencies of dHvA oscillation frequencies clearly indicate the complex 3D nature of the Fermi surface morphology in La₃ScBi₅ despite its quasi-1D structure. In Fig. 6d, we summarize the angle dependence of each frequency. The oscillation frequencies F_α exhibits a weak angular dependence at lower angles and can be fitted with the formula F = F_3D + F_2D /cos θ, where F_2D and F_3D represent the contributions from 2D and 3D components. The relative weight between 2D and 3D components derived from the fit, F_2D/F_3D ≈ 0.6, suggests dimensionality between 2D and 3D. Moreover, Figure S10 presents the Fermi surfaces incorporating SOC, and the complexity of these Fermi surfaces leads to an angular dependence in dHvA oscillation frequencies. Utilizing the SKEAF package⁶¹, we calculated the angular dependence of the dHvA oscillations, as depicted in Fig. 6e, which shows general agreement with the experimental results. These calculations enable us to identify the origins of the dHvA frequencies observed in experiments. The Fermi surface is intricate, revealing in particular one electron pocket and one hole pocket, which may correspond to the $\alpha$ and $\eta$ branches in the angular dependence of dHvA oscillation frequencies, respectively (Figure S10, Supporting Information). In conjunction with the projections of each atom, it is evident that the Bi p orbitals play a crucial role in this electron pocket.

Fig. 6: Fermi surface morphology of La₃ScBi₅.

a Isothermal magnetization of La₃ScBi₅ at various angles for T = 2 K on sample S1. b Oscillatory components of magnetization at 2 K under different magnetic field orientations. c The corresponding FFT spectrum of the oscillatory component of the dHvA oscillations for various angles. d Angular dependence of dHvA oscillation frequencies, with error bars defined as half the width at the half-height of FFT peaks. Black lines represent fits to F = F_3D + F_2D/cos θ for F_α. e Angular dependence of calculated dHvA oscillation frequencies below 600 T. F_α contributed from e pocket is indicated by red circles while F_η contributed from h pocket is denoted with purple stars.

Paramagnetic Singularity in Magnetization

The magnetization of La₃ScBi₅ exhibits pronounced anisotropy with respect to the field orientations, as depicted in Fig. 7a. Along the a-axis (χ_a) and c-axis (χ_c), the temperature-dependent magnetic susceptibility shows distinct behaviors. χ_a manifests paramagnetic, nearly independent of temperature, with a broad shoulder around 75 K and a notable upturn below 15 K. These features persist as the applied field increases, ruling out magnetic impurity effects. Additionally, Fig. 7b reveals the absence of anomalies near 75 K in dχ/dt, providing further evidence of the intrinsic nature of this magnetic response in La₃ScBi₅. Similar T-independent susceptibility behaviors have been observed in bismuthides like Bi₂Se₃⁶² and PtBi₂^63,64. This behavior can be attributed to potential contributions from Pauli paramagnetism, van Vleck paramagnetism, Landau diamagnetism, and core diamagnetism^59,62,63.

Fig. 7: Magnetism of La₃ScBi₅.

a Temperature dependence of the magnetic susceptibilities measured at H = 1 kOe on sample S1. b χ(T) curves under various magnetic fields for H // a. c Low field region of the magnetic moment vs. B curves at various temperatures for H // c. The measurements are performed sequentially from low to high temperature, spanning from 2 K to 300 K. d Magnetic susceptibility χ(=dM/dB) is calculated by differentiating the magnetic moment, plotted as a function of the magnetic field (additional details are provided in Figure S13, Supporting Information).

In contrast, the interplane χ_c is diamagnetic and undergoes a rapid increase at low temperatures. Specifically, the low-field region of M(B) curves in Fig. 7c shows a significant paramagnetic contribution near the zero field, as also observed in Figure S9 for the same sample batch. The χ(B) curves sharply rise above the diamagnetic ‘floor’ within a narrow field range and approach χ(0) in a straight line (Fig. 7d). The cusp observed in the low-field region are robust and prominent, with singular field dependence persisting up to the maximum measured temperature of 300 K. Additionally, there is a noticeable directional dependence, leading us to speculate that these features may be an inherent property of La₃ScBi₅ crystals, rather than being attributed to magnetic impurities in the diamagnetic matrix. With increasing temperature, the cusp exhibits a tendency to be inhibited, which may may be due to aging effect (Figure S12, Supporting Information) or insufficient measurement accuracy, but also need further systematic research. Such cusp-like paramagnetic susceptibility be traced most naturally to the helical spin texture of topological electrons on the surface, a signature observed across the family of three-dimensional topological materials^65,66. The proposed mechanism suggests that the electron spins at the Dirac point lack a specific orientation. As long as the Dirac cone is not fragmented, these electron spins can align freely along the external magnetic field. In the low-field region, such freely oriented spins are expected to generate paramagnetic singularities in the M(B) curve⁶⁵. More experiments, such as ARPES and scanning tunneling microscopy measurements, are highly desirable to probe the nontrivial surface and bulk states in La₃ScBi₅.

Discussion

We have conducted comprehensive investigations into the physical properties of the quasi-one-dimensional single crystal La₃ScBi₅, belonging to the Ln₃MX₅ family (Ln = rare-earth; M = transition metals; X = As, Sb, Bi). Density functional theory calculations reveal a complex and non-trivial electronic structure with SOC. Transport measurements reveal quasi-linear anisotropic magnetoresistance, with the MR isotherm following the Kohler scaling law with an index of m = 1.29, deviating from ideal electron-hole compensation. Nonlinear Hall resistivity further confirms the multi-band nature of electrical conductivity in La₃ScBi₅, with moderate charge compensation. The theoretical calculations agree with the experimental results of the behavior, demonstrating that both complex magnetoresistance and Hall resistivity behaviors may be accounted for by the semiclassical Boltzmann transportation theory.

Our magnetism studies demonstrate anisotropic magnetic susceptibility and detect dHvA oscillations for magnetic fields aligned parallel to both the a and c axes. Angle-dependent dHvA oscillations are essential to gain deeper insights into its fermiology and potential topological nature of the system. Notably, a robust and prominent paramagnetic singularity may underscore the non-trivial nature of the surface states, marking the groundbreaking observation within the Ln₃MX₅ family. This research offers valuable insights into the behavior of similar quasi-one-dimensional systems containing heavy elements. The rod-like La₃ScBi₅, suitable for mechanical exfoliation, presents an ideal platform for device applications and topological physics.

Methods

Crystal growth

Large single crystals of La₃ScBi₅ were grown using the self-flux method with excess Sc and Bi as flux. La (lump), Sc (piece), and Bi (pill) were initially mixed in a molar ratio of 1: 2: 5, loaded into an alumina crucible, and sealed into an evacuated quartz tube. The temperature was gradually raised to 1000 °C and maintained for 20 h, followed by slow cooling to 700 °C at a rate of 1 °C/h to facilitate single crystal growth. Subsequently, the excess flux was separated from the resulting rod-shaped La₃ScBi₅ crystals using a centrifuge. It’s worth noting that prolonged exposure to air or contact with alcohol would cause decomposition.

Bulk characterization

Single crystal x-ray diffraction (SCXRD) for La₃ScBi₅ was conducted on a Bruker D8 Venture diffractometer at room temperature using Mo Kα radiation (λ = 0.71073 Å). The collected data were refined by the least-square method of F² using SHELXL-2018/3⁶⁷. The diffraction peaks for a single-crystalline surface were obtained via a Bruker D2 phaser XRD detector using Cu Kα1 radiation (λ = 1.54184 Å). Element analysis was performed using energy dispersive x-ray (EDX) in a Hitachi S-4800 scanning electron microscopy (SEM) with an accelerated voltage of 15 kV. Characteristic details can be shown in Figure S1a–f.

Transport and magnetism measurements

Electronic resistivity measurements up to 16 T were conducted on polished crystals to remove Bi flux droplets on the surface using a Physical Properties Measurement System (PPMS, Quantum Design). Magnetic susceptibility and isothermal magnetizations of La₃ScBi₅ were performed in a 9 T PPMS in vibrating sample magnetometer (VSM) mode. Heat capacity was measured by the thermal relaxation method in the PPMS, with thermal contact achieved using Apezion N-grease. Three different single crystal samples named S1, S2, and S3 were used for measurements, with mass bits of 32 mg, 80 mg, and 190 mg respectively.

Electronic structure and transport calculations

Theoretical calculations were performed using density functional theory (DFT) as implemented in the Vienna ab initio Simulation Package (VASP)^68,69 with the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) exchange-correlation potential⁷⁰. A plane wave basis with a kinetic energy cutoff of 500 eV was employed. Brillouin zone sampling was conducted using a 4×4×6 k-point mesh, with Gaussian smearing of 0.05 eV applied around the Fermi surface. Structural optimization was carried out with both cell parameters and internal atomic positions allowed to relax until the forces on all atoms were less than 10⁻⁷ eV Å⁻¹. Spin-orbit coupling effects were included in the electronic property calculations. MR was computed using the WannierTools package⁷¹, employing the Boltzmann transport equation^72,73 and Wannier function techniques⁷⁴. These calculations utilized a 100 × 100 × 100 k-point mesh, with the assumption of identical relaxation times (τ) for charge carriers. The angular dependence of dHvA oscillation frequencies was calculated using the SKEAF package⁶¹.