Berry paramagnetism in the Dirac semimetal ZrTe₅

The study investigates how Berry-phase-related magnetic responses manifest in Dirac semimetals, focusing on ZrTe5. Topological materials exhibit unconventional transport phenomena (chiral magnetic effect, anomalous Hall effects) tied to a nontrivial π Berry phase, yet magnetic properties associated with this phase have been rarely reported. Prior work on Weyl semimetals revealed unconventional paramagnetic and diamagnetic contributions due to the field-independent 0th Landau level. Similar effects are expected in Dirac semimetals but are complicated by the angle between the magnetic field and the Dirac-node separation; clear experimental evidence is more feasible in systems with a single Dirac cone. ZrTe5 has been proposed as a 3D Dirac semimetal near the boundary between weak and strong topological insulators, with reports of topological phase transitions driven by external parameters. A characteristic resistivity peak at finite temperature has been interpreted as a Lifshitz transition (temperature-induced band shift across the Dirac point) or alternatively as a topological phase transition via gap opening. This work aims to elucidate the band topology and magnetic response in single crystals of TixZr1−xTe5 (x = 0, 0.1, 0.2), correlating transport anomalies and magnetic susceptibility minima, and interpreting them via Berry paramagnetism.

The authors contextualize their work within studies of topological materials showing chiral magnetic and anomalous Hall effects (Refs. 1–5). Weyl semimetals (NbAs, TaAs) exhibit non-saturating quantum magnetization and torque anomalies attributed to the 0th Landau level (Refs. 6–7). ZrTe5 has been proposed as a 3D Dirac semimetal or at the boundary between weak/strong topological insulators, with evidence for strain- or photoinduced topological phase transitions (Refs. 9–14). The hallmark resistivity peak has been linked to a Lifshitz transition and carrier-type inversion (Refs. 15–16) but also to possible temperature-driven topological phase transitions (Refs. 8, 17). Prior transport and ARPES studies examined magnetoresistance, Shubnikov–de Haas oscillations, and topology (Refs. 13, 15, 19–21). Temperature-dependent magnetization and anisotropy were noted historically in ZrTe5 though not well understood (Refs. 22–23). Ultrahigh mobility and quantum oscillation behaviors, Zeeman splitting, and effective g-factors in ZrTe5 have been reported (Refs. 24–25, 30).

Single crystals of TixZr1−xTe5 (x = 0, 0.1, 0.2 nominal; measured substitutions ~1% and ~2% for x = 0.1 and 0.2) were synthesized by chemical vapor transport using iodine, following a solid-state prereaction at 500 °C, then growth in a two-zone furnace (source 520 °C, sink 450 °C) for one month. Crystal structures were characterized by X-ray diffraction (Bruker D8, Cu source), indexing to an orthorhombic Cmcm layered structure, with ac-plane diffraction and assessment of lattice parameter b; STM was used to assess surface chain spacing and identify Ti substitutions and Te vacancies. Transport measurements (resistivity ρxx, Hall ρxy) employed a four-probe method with current along a axis and magnetic field along b, analyzed via a two-carrier model by converting to conductivities (σxx, σxy) to extract carrier densities and mobilities. Angle-dependent magnetoresistance was measured by rotating the field from b (θ = 0°) to a (θ = 90°); SdH oscillations were isolated by background subtraction and analyzed with Landau fan diagrams using the Lifshitz–Onsager quantization to obtain frequencies F and phase intercepts γ. Magnetization M(H) was measured using a SQUID-VSM from 2–300 K up to 7 T along principal axes; dHvA oscillations were extracted by subtracting linear diamagnetic background and analyzed with the Lifshitz–Kosevich formula to obtain effective cyclotron mass m* and quantum lifetime τq. Simulations of magnetization based on Berry paramagnetism were performed using band parameters (Fermi velocity, effective mass/anisotropy, Fermi energy from experiments) with a single fitting parameter g-factor, and included assessment of quantum limit behavior and anisotropy for H parallel to a, b, and c.

• Ti substitution reduces the interlayer lattice parameter b: b = 14.52(3) Å (x = 0), 14.51(4) Å (x = 0.1), 14.47(2) Å (x = 0.2), consistent with smaller Ti radius and van der Waals spacing reduction. • STM on x = 0.1 shows chain separation ~13.8 Å (matching c), with bright/dark spots indicating Ti substitutions and Te vacancies. • Hall effect reveals electron-dominant conduction at low T transitioning to hole-dominant at higher T; crossover near Tρ (resistivity peak) evidences a temperature-induced Lifshitz transition. • For x = 0 at 2 K: high-mobility carrier μ1 ≈ 170,000 cm² V⁻¹ s⁻¹ with n1 ≈ 4.3 × 10^17 cm⁻3; mobility decreases with T, and Ti substitution reduces μ1 to ~67,500 (x = 0.1) and ~16,900 cm² V⁻¹ s⁻¹ (x = 0.2) at 2 K. • Resistivity peak temperatures shift with Ti: Tρ ≈ 122 K (x = 0), 92 K (x = 0.1), 85 K (x = 0.2), consistent with a band shift towards the Dirac point at lower T for higher x. • Magnetoresistance at 2 K (θ = 0°): up to ~2400% (x = 0), decreasing to ~580% (x = 0.1) and ~370% (x = 0.2); peak positions shift from ~8 T (x = 0) to ~5 T (x = 0.1) and ~4 T (x = 0.2). Strong angular suppression indicates a highly anisotropic quasi-2D Fermi surface. • SdH oscillations show a single frequency attributed to the high-mobility carrier. Lifshitz–Onsager analysis yields F(θ = 0°) ≈ 4.98 T (x = 0), 3.64 T (x = 0.1), 3.03 T (x = 0.2); F scales ~1/cosθ with deviations near 90° due to spin-zero effects. • Landau fan intercepts give γ ≈ 1/8 across angles, implying a 3D Dirac system (δ = 1/8) with nontrivial Berry phase φ = π. • dHvA oscillation frequency decreases with increasing T (e.g., x = 0: F ≈ 5.4 T at 2 K to ≈ 5.1 T at 25 K), evidencing an upward band shift crossing the Dirac point with T; Ti substitution reduces F consistently with a shrinking Fermi surface. • Lifshitz–Kosevich fits give effective masses: m* ≈ 0.038 me (x = 0), 0.034 me (x = 0.1), 0.032 me (x = 0.2); quantum lifetimes τq ≈ 0.13 ps (x = 0), 0.048 ps (x = 0.1), 0.086 ps (x = 0.2). The large reduction of transport mobility is mainly due to decreased transport lifetime (scattering from Ti), with τtr/τq ≈ 28 (x = 0) decreasing to ≈ 3.6 (x = 0.2). • Magnetic susceptibility χ(T) is diamagnetic and strongly anisotropic, largest (most negative) for H ∥ b where m* is smallest. χ(T) exhibits a pronounced minimum at Tm that coincides with Tρ for all x. • |n|^(1/3) of the high-mobility carrier tracks χ(T), indicating that the Fermi energy position relative to the Dirac point governs both transport and magnetism. • Magnetization simulations based on Berry paramagnetism with a single fitting parameter g ≈ 10 reproduce the measured M(H) (including dHvA oscillations) and its anisotropy. Paramagnetic contribution diminishes above the quantum limit; lowering EF (e.g., 31.85 → 20 → 10 meV) reduces the paramagnetic signal, consistent with the observed T dependence. • The concurrence of χ minima and resistivity maxima, sustained large-magnitude χ(T), and similar behavior across x confirm that the Berry paramagnetism is a bulk effect in 3D Dirac semimetal ZrTe5 rather than a surface Dirac cone of a topological insulator.

The transport and quantum oscillation data establish TixZr1−xTe5 as a 3D Dirac semimetal with a highly anisotropic quasi-2D Fermi surface and a nontrivial π Berry phase (γ ≈ 1/8). Temperature-dependent shifts of the Fermi surface and oscillation frequency, together with electron-to-hole crossover at the resistivity peak, point to a Lifshitz transition driven by temperature and tuned by Ti-induced reduction of the b-axis lattice parameter. The central finding is that the temperature-dependent magnetization exhibits a minimum at the same temperature as the resistivity maximum for all x, and is strongest for H ∥ b where the cyclotron mass is smallest. This correlation indicates that itinerant carriers near EF dominate the bulk magnetic response. Interpreted through the lens of Berry paramagnetism (originating from the field-independent 0th Landau level and Berry phase), states above the Dirac node contribute paramagnetically while those below are diamagnetic. As EF approaches the Dirac point (with rising temperature or increased Ti), the Berry paramagnetic contribution is minimized while diamagnetism from valence states is maximized, yielding the observed χ(T) minimum. Simulations using experimentally derived band parameters and a reasonable g-factor quantitatively reproduce M(H), its anisotropy, and evolution with EF, validating the Berry-paramagnetism mechanism in a Dirac system with a single cone near Γ. The persistence and magnitude of the magnetic response across compositions support a bulk Dirac character rather than a topological insulator phase, where surface contributions would strongly suppress bulk Berry magnetism.

Single crystals of TixZr1−xTe5 (x = 0, 0.1, 0.2) were synthesized and shown via transport and quantum oscillation measurements to be 3D Dirac semimetals with a highly anisotropic quasi-2D Fermi surface and a nontrivial Berry phase. Ti substitution reduces the interlayer b lattice parameter, tuning the band structure and shifting the resistivity peak (Dirac point crossing) to lower temperatures. The temperature dependence of SdH/dHvA frequencies corroborates a temperature-induced Lifshitz transition. A key result is the identification of Berry paramagnetism in bulk ZrTe5: the magnetic susceptibility exhibits a minimum at the resistivity peak, and M(H) including quantum oscillations is quantitatively captured by a Berry-paramagnetism model using band parameters and a single g-factor. This provides an experimental route to determine band topology via magnetism and a platform to apply Berry-phase-related magnetic concepts to Dirac semimetals. Future work could explore broader compositional ranges, pressure/strain tuning, higher fields beyond the quantum limit, and spectroscopic correlation to further disentangle potential gap-opening effects and refine band parameters.

The authors note that although the data support a temperature-induced Lifshitz transition, they cannot fully rule out a temperature-dependent gap opening contributing to the behavior. dHvA oscillations were observed predominantly for H ∥ b, reflecting strong anisotropy, and analyses focused on the dominant high-mobility carrier; potential multiband contributions beyond the two-carrier model and effects at compositions beyond x = 0.2 were not explored.