Giant In-Plane Shrinkage Induced by Structural Phase Transformation in TbCoSi₂

Abstract

Metal-based materials, pivotal for industrialization and technological progress, confront the long-standing issue of high thermal expansion, which limits their application in advanced scenarios. With a century-long research history, negative thermal expansion materials, particularly those in intermetallic compounds, offer promising solutions for regulating thermal expansion. Here, we investigate polycrystalline TbCoSi₂ ingots, revealing a notable 3% in-plane shrinkage from 223 K to 298 K induced by structural phase transitions. Temperature-dependent XRD and Rietveld refinement identify a low-temperature Pbcm space group structure, and the drastic a-axis shrinkage during the phase transition drives the in-plane contraction. Macroscopic magnetic measurements and first-principles calculations reveal an antiferromagnetic structure below 13.7 K, with magnetic and structural phase transitions being independent. These findings present a metal-based weakly magnetic material for precise thermal expansion control, particularly in the uniaxial direction.

1. Introduction

The discovery and extensive application of metal-based materials have exerted an irreplaceable and profound influence on the advancement of industrialization, modernization processes, and the evolution of science and technology across multiple fields. Structural materials represented by steel, which possess excellent mechanical strength, ductility, and processability, have become the fundamental building blocks for constructing infrastructure, automotive components, and aerospace structures. Nevertheless, regardless of whether serving as structural or functional materials, metal-based materials have long faced the challenge of high thermal expansion, which leads to thermal expansion mismatch in devices [1,2,3,4]. Specifically, the coefficient of thermal expansion (CTE) of iron is approximately 12 ppm/K, which restricts its application in advanced scenarios that require low dimensional temperature sensitivity and minimal thermal mismatch when assembled with different materials [5]. Consequently, the precise regulation of thermal expansion behavior in metal-based materials has long been recognized as a critical subject meriting in-depth scientific investigation and technological development [6,7,8,9,10].

Negative thermal expansion (NTE) materials have garnered substantial research attention due to their ability to compensate for and regulate thermal expansion, along with their intriguing physical characteristics that involve complex lattice dynamics [11,12]. The research history of NTE materials dates back over a century, with early studies focusing on Invar alloy Fe₆₅Ni₃₅ [13]. Typically, low thermal expansion (LTE) or even NTE behaviors are exclusively observed in a limited set of materials. These notably include fluoride compounds, oxide compounds such as ZrW₂O₈ and Cu₂P₂O₇, anti-perovskite phases like Mn₃GaN and Mn₃CuN, as well as a small number of intermetallic compounds and solid solutions [14,15,16,17,18]. In particular, in metal-based materials, NTE phenomena induced by magnetovolume effects, martensitic phase transitions, and valence state transitions have been observed in specific compounds [19,20,21,22,23,24,25], such as La(Fe,Co,Si)₁₃, (Zr,Nb)Fe₂, (Hf,Ta)Fe₂, MnCoGe, and YbAl₂. The NTE of these materials plays a crucial role in regulating the CTE by effectively compensating for the thermal expansion of the matrix materials. Nevertheless, inherent limitations in key aspects, such as insufficient shrinkage rate and narrow operating temperature range, frequently render these materials unable to meet the demands of practical applications. Thus, investigating and exploring novel materials for regulating the thermal expansion of metal-based materials remains of significant scientific and practical value.

Metal-based NTE materials occupy a pivotal position in sectors ranging from aerospace engineering to advanced electronic packaging technologies, owing to their inherent advantages in thermal conductivity and mechanical performance [26]. Among them, metallic systems exhibiting NTE driven by magnetovolume effect—such as La(Fe,Si)₁₃-based and MnCoGe-based alloys—have attracted extensive attention [24,27]. However, the pronounced magnetism and intense stray magnetic fields inherent to these materials impose significant constraints on their practical applicability. This critical limitation has thus galvanized sustained research efforts toward the development of weakly magnetic metal-based NTE systems, which hold promise for overcoming such bottlenecks and expanding the technological utility of this class of materials. In this context, ternary intermetallic compounds with a 1:1:2 stoichiometry, denoted as HRE–TM–X (where HRE = heavy rare-earth element, TM = transition metal, and X = p-block element), present a fertile ground for discovery due to their diverse crystal structures and tunable magnetic configurations [28,29,30,31,32]. Among them, TbCoSi₂ stands out as a model system with a magnetically adjustable framework, offering an ideal platform to investigate the interplay between crystal symmetry, magnetic ordering, and anisotropic lattice contraction in metallic NTE materials [33,34,35].

In this work, we investigated the remarkable in-plane shrinkage of up to 3% in polycrystalline ingots of TbCoSi₂ over the temperature range from 223 K to 298 K. The significant in-plane shrinkage was experimentally confirmed to be induced by structural phase transitions. Before this study, TbCoSi₂ was generally considered to possess a tetragonal structure with the Cmcm space group [29]. Based on the temperature dependence of XRD measurements and Rietveld refinements, we identified the presence of a distinct structure with the Pbcm space group in TbCoSi₂ at low temperatures. Further structural analysis revealed that the drastic shrinkage of the a-axis lattice caused by the structural phase transition accounts for the significant in-plane shrinkage of TbCoSi₂. Results from macroscopic magnetic measurements and first-principles calculations demonstrate that TbCoSi₂ exhibits an antiferromagnetic structure with G-type below 13.7 K, and the magnetic phase transition is not related to the structural phase transition [36]. This study presents a metal-based material that can be used for thermal expansion regulation, especially suitable for controlling the thermal expansion properties in the uniaxial direction near room temperature.

2. Method

TbCoSi₂ alloy was prepared by arc melting under a high-purity argon atmosphere. After the remelting of 4 times, the ingot was annealed in a vacuum-sealed quartz tube at 1223 K for 5 days. Then the ingot was cooled slowly. The temperature dependence of linear thermal expansion data (ΔL/L₀) were obtained by a thermo-dilatometer (DIL 402 Expedis Select, NETZSCH, Selb, Germany) with a heating rate of 5 K/min. Thermal expansion sample was prepared by grinding and polishing arc-melted button ingot. The length of the polished sample along the in-plane direction is 6 mm, which exceeds the resolution limit of the thermo-dilatometer (1 nm). The variable temperature measurements of XRD were collected using an using an x-ray diffractometer (PW 3040-X’Pert Pro, PANalytical, Almelo, Netherlands) with Cu Kα radiation to confirm the crystal structures [37]. The structure refinements for all XRD data were conducted by FULLPROF software (version March 2021) [38]. The macroscopic magnetism of TbCoSi₂ was measured by a physical property measurement system (PPMS) of Quantum Design (San Diego, CA, USA).

First-principles calculations were conducted using the Vienna Ab initio Simulation Package (VASP 5.4.4) [39] with the projector augmented-wave (PAW) method [40], treating Tb, Co, and Si valence electrons as 4f⁹6s², 3d⁷4s³, and 3s²3p², respectively, and employing the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional [41]. To accurately describe the localized 4f orbitals of Tb, the GGA + U approach [42,43] was applied with U = 5 eV [44]. A plane-wave kinetic-energy cutoff of 400 eV and a Monkhorst–Pack [45] k-point spacing of 2π × 0.03 Å⁻¹ were used to ensure total energy convergence within 1 meV per atom. The A-type, G-type, and C-type antiferromagnetic configurations were self-consistently calculated using a 2 × 1 × 2 supercell containing 64 atoms, and all self-consistent calculations took into account the spin–orbit coupling (SOC) effect. Based on ab initio molecular dynamics (AIMD) within the N_pT ensemble with a Langevin thermostat [46], the AIMD simulations were carried out within the 64 atoms for TbCoSi₂. The high-temperature simulations use the initial structure rather than the structures after the low-temperature simulation. We used a step size of 1 fs during the molecular dynamic simulation.

3. Results and Discussion

TbCoSi₂ exhibits a crystal structure characterized by the Cmcm space group as per previous literature [47]. Temperature dependence of X-ray diffraction (XRD) measurements was conducted to probe structural transition across a broad thermal range (Figure 1a). For temperatures above 273 K, most diffraction peaks in the XRD patterns of TbCoSi₂ exhibit excellent consistency with the standard diffraction profile of TbCoSi₂ with the Cmcm space group, confirming the high-temperature structure. The diffraction peaks at 37° originate from a trace amount of the TbCo₂Si₂ secondary phase. Notably, when the temperature is decreased below 273 K, both the number and relative intensities of diffraction peaks undergo significant modifications (highlighted by the red box in Figure 1a). Therefore, the crystal structure below 273 K is distinct from the high-temperature Cmcm phase. To clearly capture the variations in diffraction peaks and the associated structural transition, the temperature dependence of the diffraction peak intensities corresponding to the (041) and (111) lattice planes was analyzed in detail (Figure 1b). With the temperature decreasing below 273 K, the intensity of the (041) diffraction peak rises abruptly, whereas that of the (111) diffraction peak exhibits a distinct decrease. Furthermore, a new diffraction peak emerges at an angular position of ~34.4°. Collectively, these observations provide unambiguous evidence for a structural phase transition in TbCoSi₂ upon cooling, which may be associated with subtle changes in lattice symmetry.

The surface microstructures of arc-melted polycrystalline TbCoSi₂ were characterized using a scanning electron microscope (SEM, Gemini 500, ZEISS, Oberkochen, Germany). As presented in Figure 1c, the as-synthesized TbCoSi₂ sample exhibits a highly dense microstructure across its surface. Such structural uniformity is crucial in subsequent property characterizations. To further validate the elemental composition and spatial homogeneity of the TbCoSi₂ polycrystal, energy-dispersive X-ray spectroscopy (EDS) mapping was performed in conjunction with SEM to enable the visualization of elemental distribution. The elemental distribution maps corresponding to Tb, Co, and Si are presented in Figure 1d, Figure 1e and Figure 1f, respectively. These maps reveal a uniform spatial distribution of all constituent elements throughout the sampled area, with no regions of elemental enrichment or depletion.

The thermal expansion and crystallographic structure were characterized by a thermo-dilatometer and Rietveld refinement of XRD patterns. The temperature dependence of linear thermal expansion (ΔL/L₀) demonstrates a distinct sharp contraction in TbCoSi₂ along the in-plane direction at T_t = 273 K during the heating process (Figure 2a). A corresponding expansion is observed at 250 K during the cooling cycle, which confirms a thermal hysteresis of approximately 25 K. The reversibility and the thermal hysteresis of this phase transition are consistent with its classification as a first-order phase transition. Over the temperature range from 223 K to 298 K, the TbCoSi₂ ingot exhibits a remarkable contraction of ~3% along the in-plane direction. This value exceeds the linear contraction rate in prominent giant negative thermal expansion materials, including MnCo_0.98Cr_0.02Ge (1.3%), Ni_55.5Mn_19.5Ga₂₅ (0.45%), LaFe_11.5Si_1.5 (0.35%), and Hf_0.8Ta_0.2Fe₂ (0.34%) [24,27,48,49,50]. Therefore, TbCoSi₂ displays an anomalously giant negative thermal expansion in the in-plane direction of the ingot. Such exceptional NTE behavior renders it a highly promising candidate for applications requiring tunable thermal expansion (especially for precise composite materials that need to eliminate thermal mismatch in a single direction) or a phase-transition-driven device.

To elucidate the fundamental physical mechanisms governing the observed thermal expansion anomalies in TbCoSi₂, Rietveld refinement was systematically performed on the temperature dependence of XRD patterns. As illustrated in Figure 2b, the Rietveld refinement confirms that TbCoSi₂ adopts a single-phase structure with the Cmcm space group at 423 K. The unresolved peaks in the XRD pattern are attributed to trace amounts of TbCo₂Si₂ and terbium oxide impurities. As mentioned above, the temperature dependence of XRD patterns indicates that TbCoSi₂ exhibits a structural transition below 273 K. As presented in Figure 2c, the Rietveld refinement results reveal the coexistence of two distinct phases within TbCoSi₂ at 123 K. The second phase was identified as belonging to the Pbcm space group at low temperature. As shown in Figure 2d, each unit cell of the Cmcm phase and Pbcm phase of TbCoSi₂. For the Cmcm phase, the lattice parameters are a = 4.02 Å, b = 16.30 Å, c = 3.97 Å, with Wyckoff positions Tb: 4c, Co: 4c, and Si: 8c. For the Pbcm phase, the parameters are a = 4.21 Å, b = 15.61 Å, c = 3.95 Å, and Wyckoff positions Tb: 4d, Co: 4d, and Si: 8d. Therefore, the reversible structural phase transition involves a structural transformation between the low-temperature Pbcm phase and the high-temperature Cmcm phase (Figure 2d), which is the primary driver of the pronounced in-plane contraction observed in the TbCoSi₂ ingot.

In order to gain deeper mechanistic insight into the in-plane contraction induced by the phase transition, the temperature dependence of phase fractions and the lattice parameters for both the Cmcm and Pbcm phases were extracted. As shown in Figure 3a, the fraction of the low-temperature Pbcm phase reaches as high as 75% at 123K and decreases monotonically with increasing temperature; meanwhile, the fraction of the high-temperature Cmcm phase increases. The Pbcm phase fraction reaches 0% at 273 K, indicating complete transformation to the Cmcm phase. Given that both the Pbcm and Cmcm phases exhibit tetragonal symmetry, the temperature dependences of the lattice parameters were compared to identify the axis driving the in-plane contraction (Figure 3b–d). From 223 K to 273 K, the a-axis undergoes a significant contraction up to 4.5%, whereas the b-axis and c-axis expand by 4% and 0.5%, respectively. The prominent anisotropy demonstrates that the in-plane contraction is primarily driven by the a-axis. For typical polycrystalline compounds, thermal expansion behavior reflects the average response of all lattice axes. However, the arc-melted TbCoSi₂ intermetallic compound exhibits significant crystallographic preferred orientation, which originates from the rapid cooling during solidification [3,4]. Consequently, the a-axis of TbCoSi₂ is predominantly aligned along the plane direction of the ingot, which gives rise to the observed significant in-plane contraction.

The magnetic structure of TbCoSi₂ was characterized by macroscopic magnetic measurements and first-principles calculations. As illustrated in the field-cooling (FC) temperature dependence of the magnetization (M-T) curve, the TbCoSi₂ exhibits antiferromagnetic (AFM) ordering at low temperatures, with a Neel temperature (T_N) of approximately 13.7 K (Figure 4a). Near room temperature, TbCoSi₂ adopts a paramagnetic state. Therefore, the structural phase transition is independent of the magnetic phase transition. In order to further elucidate the magnetic structure of TbCoSi₂ at low temperatures, we measure the magnetic field dependence of magnetization (M-H). TbCoSi₂ compound exhibits paramagnetic behavior above 15 K. Notably, a metamagnetic transition emerges under an applied magnetic field of 2 T, indicating that the AFM structure is reoriented into a ferromagnetic (FM) structure upon field application (Figure 4b). The inverse magnetic susceptibility demonstrates the same AFM characteristics as the M-T curve (Figure 4c). To further investigate the nature of the magnetic ordered state, the inverse magnetic susceptibility in the paramagnetic region was fitted using the Curie–Weiss law (Figure 4d). The derived paramagnetic Curie temperature (θ_p) is −9.6 K, a value close to the T_N = 13.7 K. This suggests that AFM interactions dominate the magnetic behavior of TbCoSi₂. The effective magnetic moment (μ_eff) is determined to be 10.1 μ_B, consistent with the theoretical magnetic moment of free Tb³⁺ ions (9.72 μ_B). Based on these findings, the magnetism of TbCoSi₂ originates primarily from Tb³⁺ ions, whereas Co ions adopt a low-spin configuration.

First-principles calculations were performed to investigate the magnetic properties of Pbcm TbCoSi₂ at 0 K, as illustrated in Figure 5. To determine the magnetic ground state, one ferromagnetic configuration (Figure 5a) and three representative antiferromagnetic configurations—A-, C-, and G-types (Figure 5b–h)—were examined. These AFM types were chosen because they are able to describe as closely as possible the fundamental combination of intra- and inter-layer magnetic interactions of the structure, thus enabling a comprehensive description of the possible antiferromagnetic arrangements [51]. In the A-type AFM configuration, all spins within each Tb atomic layer align ferromagnetically to form a magnetic monolayer, while adjacent layers couple antiferromagnetically. In contrast, C-type AFM configurations feature antiferromagnetic ordering within each Tb layer but ferromagnetic coupling between layers, whereas G-type AFM corresponds to antiferromagnetic coupling both within and between Tb layers. For each type, spin orientations along the [100], [010], and [001] crystallographic directions were considered, yielding a total of 24 distinct magnetic configurations. Representative models are shown in Figure 5, including the A₁ and A₂ variants of the A-AFM (Figure 5b,c), the G₁ and G₂ variants of the G-AFM (Figure 5d,e), and the C₁, C₂, and C₃ variants of the C-AFM (Figure 5f–h) The calculations reveal that variations in magnetic ordering have a negligible influence on the lattice volume, meaning that energy differences can reliably replace enthalpy differences in subsequent analysis. A direct comparison of total energies shows that the FM configuration is always higher in energy than AFM configuration, indicating that the Pbcm TbCoSi₂ is intrinsically antiferromagnetic, in agreement with experimental results. Among the AFM configurations, the G₂ type with spin orientation along the [001] direction is the most stable, with a lower energy than other AFM states. This result suggests that the Pbcm phase stabilizes in the G₂-type AFM configuration with spins preferentially aligned along the crystallographic c-axis.

Further analysis elucidates the relative significance of various exchange interactions. In all A-, C-, and G-type AFM configurations, the magnetic moments preferentially lie within the basal plane, consistent with pronounced in-plane magnetic anisotropy. The energy difference between the G₁- and G₂-AFM variants along the [001] direction is less than 0.6 meV/atom, indicating that the next-nearest-neighbor exchange interaction is negligible. In contrast, the energy variation of the G₁-AFM configuration among the [001], [010], and [100] directions reaches up to 2.1 meV/atom, revealing substantial anisotropy in the AFM coupling. This pronounced anisotropy originates from the strong spin–orbit coupling inherent to heavy Tb atoms, where spin–orbital interactions align the magnetic moments along specific crystallographic axes, leading to notable direction-dependent SOC energies. Moreover, the small energy difference between the C-type and G-type AFM states suggests that when intralayer AFM order dominates, interlayer exchange interactions contribute only weakly. Collectively, these findings demonstrate that the fundamental magnetic behavior of Pbcm TbCoSi₂ is primarily governed by the interactions within the Tb layer and strong self-consistent spin–orbital coupling. In contrast, interlayer interactions play secondary roles. This insight provides a foundation for understanding its intrinsic AFM ground state of TbCoSi₂.

To evaluate the thermal stability of TbCoSi₂, ab initio molecular dynamics (AIMD) simulations were performed for both the low-temperature and high-temperature phases. The equilibrium structures were obtained from the final configurations of the respective AIMD simulations. It is noted that the high-temperature simulation was initialized from the corresponding experimental structure, rather than the configuration obtained after the low-temperature simulation. As depicted in Figure 6a and Figure 4b, the Pbcm phases of TbCoSi₂ at 100 K and Cmcm phase of TbCoSi₂ 423 K, respectively. The first peaks in the pair distribution functions reveal that the nearest Si–Si, Co–Si, and Tb–Co distances in the equilibrium structures show negligible deviation from those in the initial configurations, indicating excellent thermal stability. These results demonstrate that the Pbcm phase remains stable at low temperature, whereas the Cmcm phase is stable at high temperature, in good agreement with the experimental observations.

4. Conclusions

Insights into the structure phase transition in TbCoSi₂ reveal the mechanism underlying the giant in-plane shrinkage behavior of TbCoSi₂ polycrystalline ingots, steering the discovery and application of new weakly magnetic NTE materials. Temperature dependence of XRD and Rietveld refinements confirms the new crystal structure of the Pbcm space group at low temperature. Analysis of the extracted lattice constant indicates that the contraction along the a-axis during the phase transition is the primary driver of the in-plane shrinkage of the TbCoSi₂ ingot. Macroscopic magnetic measurements, Curie-Weiss fitting, and first-principles calculations demonstrate that TbCoSi₂ exhibits an antiferromagnetic ground state below 13.7 K, which is far below the working temperature range of NTE. These findings not only provide a comprehensive understanding of the crystal structure and magnetic structure of TbCoSi₂ but also offer a metal-based weakly magnetic NTE material that exhibits a significant in-plane contraction near room temperature.