Topological semimetals (TSMs) draw a great deal of attention in recent years due to their unique topological properties and potential applications in spintronics.^[1–9] In general, TSMs host points or lines in the momentum space where the conduction and the valence bands of TSMs cross linearly, leading to exotic properties, such as high mobility, chiral anomaly, and extremely large magnetoresistance (XMR). Two prominent examples of topological semimetals with zero-dimensional (0D) band crossing points are the Dirac and Weyl semimetals, such as Na₃Bi,^[1] Cd₃As₂,^[2,3] PtX₂ (X = Se, Te),^[4,5] TaAs family,^[6–8] and WTe₂.^[9] When conduction and valence bands cross to form a one-dimensional (1D) closed loop in momentum space, another type of semimetal named nodal-line topological semimetal is formed, like PbTaSe₂^[10] and ZrSiS.^[11,12] Recently, a novel Dirac node arc topological structure was reported in ultrahigh magnetoresistive material PtSn₄.^[13] Unlike the closed loops of line nodes, the Dirac node arc structure arises owing to the surface states and the Dirac dispersion is extended along a short line in the momentum space.^[13]

As a homologue of the Dirac nodal arc semimetal PtSn₄,^[13] Dirac-like features were recently discovered in PdSn₄ through angle-resolved photoemission spectroscopy (ARPES) experiments, which is in consistence with band-structure calculations.^[14] At ambient pressure, PdSn₄ shows a metallic behavior in the temperature range of 1.8 K–300 K and follows Fermi-liquid behavior at low temperatures. The XMR of PdSn₄ reaches 7.5 × 10⁴% at 1.8 K and 14 T, with external magnetic field perpendicular to b axis.^[14] Based on detailed analysis of the magnetoresistance experiments, both carrier compensation and emergence of an excitonic insulating state were ruled out as the main origins of the XMR in PdSn₄.^[14]

As one of the fundamental state parameters, high pressure is a clean and effective tool to tune lattice as well as electronic states, especially in topological quantum materials.^[15,16] For example, in the case of type-II Weyl semimetal WTe₂, superconductivity was observed by application of high pressure.^[15–17] To date, the high-pressure electronic properties of PdSn₄ was investigated only up to 2.5 GPa,^[18] in which the author believed that the variation of the resistivity under pressure might be related to a change of Fermi surface topography.^[18]

To further investigate the pressure effect on electronic properties of PdSn₄ at higher pressure and its relationship with lattice structure, we carried out high-pressure powder x-ray diffraction and electrical transport measurements on PdSn₄ single crystals. We show that there is no structural transition upon compression up to 44.5 GPa. With increasing pressure, the temperature-dependent resistance maintains a metallic behavior till 52.0 GPa, with the magnetoresistance mechanism remaining unchanged at least up to 20 GPa.

Single crystals of PdSn₄ were synthesized by Sn self-flux method.^[14] Excessive Sn trapped on the crystal surface was removed by centrifuging. The high quality of the single crystal was checked by x-ray diffraction (XRD) and energy-dispersive x-ray spectra (EDS) (see in Fig. 1). Magnetization measurements were conducted using a superconducting quantum interference device magnetometer. The high-pressure electrical transport measurements were performed in a Be–Cu diamond anvil cell (DAC) in a temperature range of 5 K–300 K by using standard four-probe method. The diameter of the diamond culet is 300 μm. Sodium chloride (NaCl) powder was used as the pressure transmitting medium (PTM). A small sample (∼ 80 μm × 50 μm × 5 μm) crushed from bulk PdSn₄ single crystal was loaded in the cell and the applied magnetic field is perpendicular to the electric current.

Fig. 1.

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Fig. 1. (a) Crystal structure of PdSn4, (b) XRD pattern of PdSn4 single crystal. The reflection (0l0) is the as-grown crystal face. The upper-right inset shows an optical image of a PdSn4 single crystal. Typical size of the sample is 2 mm× 2 mm× 1 mm. (c) Energy dispersive spectra of PdSn4 single crystal. (d) Temperature-dependent resistivity of PdSn4 single crystal with current along the ac plane. (e) Temperature dependence of the fast Fourier transformed dHvA data for H∥b. Inset shows the raw dHvA data.

Pressure was generated by a Mao–Bell-type symmetric DAC for the synchrotron XRD measurements. High pressure synchrotron powder XRD was performed at room-temperature at the beamline of 16 BM-D (λ = 0.4133 Å), HPCAT^[19] of the Advanced Photon Source, Argonne National Laboratory. Daphne 7373 oil was used as the PTM. The DIOPTAS^[20] program was used for image integrations and the Le Bail method was employed to fit the XRD data with the RIETICA software.^[21] Pressure calibration was determined by ruby fluorescence method in all the mentioned experiments.^[22]

PdSn₄ crystallizes in an orthorhombic structure with space group Ccca (No. 68).^[14] The lattice is composed of stacks of Pd atoms sandwiched between two coplanar Sn layers along b axis, as displayed in Fig. 1(a). Figure 1(b) shows a typical XRD pattern of PdSn₄ single crystal (λ = 1.54059 Å). The observation of only (010) reflections indicates a b-axis orientation of the as-grown crystals. The calculated lattice parameter b is 11.428(3) Å, in excellent accordance with previous report.^[14] Energy dispersive x-ray spectra with area- and point-scanning modes give the ratio of Pd:Sn as 1:3.92(8) [Fig. 1(c)]. The temperature-dependent resistivity of PdSn₄ is displayed in Fig. 1(d). The residual resistivity ratio (RRR) of the sample is 408. In addition, magnetization of PdSn₄ was measured at different temperatures with external magnetic field along the b axis. Clear De Haas–Van Alphen oscillations (dHvA) are observed, as shown in the inset of Fig. 1(e). After using a fast Fourier transform (FFT) analysis, seven prominent frequencies labeled as F₁–F₇ are detected [Fig. 1(e)], in excellent accordance with that at ambient pressure.^[14] These results indicate high quality of the as-grown PdSn₄ crystals.

We first investigated the evolution of the structure under high pressures. Figure 2 displays the selected XRD patterns of PdSn₄ at room temperature with pressures up to 44.5 GPa. At 0.5 GPa, the XRD peaks can be well indexed by the space group Ccca (No. 68), in accordance with that at ambient pressure.^[14] We find that increasing pressure consistently pushes all Bragg peaks to higher angles, but no new peaks appear up to highest pressure studied. After the pressure is released down to ambient pressure, the peaks in the XRD pattern return to the initial positions [denoted by DP, at the top of Fig. 2(a)], which suggests the structural evolution is reversible.

Fig. 2.

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Fig. 2. (a) Synchrotron radiation x-ray diffraction patterns of PdSn4 under pressures (λ = 0.4133 Å). The XRD pattern upon decompression back to 0.0 GPa is denoted by DP. (b) The refinements at (b) 0.5 GPa and (c) 44.5 GPa were performed by using the RIETICA program with the Le Bail method. (d) Pressure dependence of the refined lattice parameters. (c) The empty symbols stand for unit volume. The volume versus pressure data was fitted by the third-order Birch–Murnaghan formula (solid red line).

To further analyze the XRD data, refinements were performed with Le Bail method by using the RIETICA software.^[21] A typical XRD refinement result of 0.5 GPa is presented in Fig. 2(b), which gives a = 6.410(7) Å, b = 11.417(7) Å, and c = 6.369(6) Å. The extracted lattice parameters (a, b, and c) are displayed in Fig. 2(d). With increasing pressure, all three lattice parameters decrease gradually. When the pressure increases up to 44.5 GPa, the fitting results in Fig. 2(c) show that the space group stays unchanged, while the lattice parameters are decreased by 8.4%–9.8%, i.e., to a = 5.871(0) Å, b = 10.337(2) Å, and c = 5.832(1) Å. In addition, pressure dependence of unit-cell volume (V) is fitted by using the third-order Birch–Murnaghan equation of state (EoS)^[23] [red line in Fig. 2(e)]:

Table 1.

Table 1.

Table 1. Rietveld refined lattice constants of PdSn4 at various pressures. . P/GPa a/Å b/Å c/Å Rp/% Rwp/% 0.5 6.410(7) 11.417(7) 6.369(6) 2.2(8) 3.8(0) 1.2 6.395(8) 11.378(8) 6.337(9) 2.3(5) 3.9(2) 1.7 6.383(0) 11.346(7) 6.325(2) 3.1(5) 5.3(5) 2.1 6.367(6) 11.321(6) 6.315(1) 3.4(2) 6.0(8) 2.8 6.355(4) 11.300(0) 6.305(1) 2.3(5) 4.2(5) 3.9 6.326(6) 11.245(5) 6.287(3) 2.5(5) 3.8(1) 5.6 6.295(9) 11.173(7) 6.253(8) 2.1(8) 3.2(7) 8.0 6.275(2) 11.075(6) 6.212(5) 3.8(4) 7.2(0) 13.2 6.200(0) 10.905(2) 6.111(2) 2.4(5) 3.6(9) 16.5 6.158(5) 10.833(6) 6.078(8) 2.6(4) 4.2(0) 18.8 6.133(1) 10.761(3) 6.060(0) 2.0(4) 3.3(2) 22.1 6.110(7) 10.671(3) 6.011(4) 3.5(1) 5.9(2) 25.4 6.067(3) 10.609(2) 5.965(6) 2.4(1) 3.9(3) 28.2 6.020(4) 10.552(9) 5.941(1) 1.6(1) 2.3(7) 32.2 5.972(3) 10.485(9) 5.905(6) 2.6(8) 4.2(0) 35.4 5.949(0) 10.440(4) 5.884(4) 1.9(8) 3.1(3) 39.9 5.914(1) 10.385(7) 5.853(7) 1.9(0) 2.7(7) 44.5 5.871(0) 10.337(2) 5.832(1) 2.2(0) 3.2(7)

Table 1. Rietveld refined lattice constants of PdSn4 at various pressures. .

We further carried out high pressure transport measurements. Figure 3(a) shows the temperature (T) dependence of the electrical resistance (R) of single crystal PdSn₄ under various pressures. Starting at 3.2 GPa, R(T) curve exhibits a metallic conductive behavior (dR/dT > 0) in the whole temperature range of 5 K–300 K, similar to that at ambient pressure.^[14] Upon compression, metallic behaviors remain to the highest pressure of 52.0 GPa. Nevertheless, the resistance globally gets enhanced with increasing pressure. According to band theory, the pressure normally enhances the overlap of electronic orbitals and decrease the band gap, resulting in the decrement of resistance. However, we note that similar behavior of pressure-enhanced resistivity has also been observed in other TSMs, such as TaSb₂,^[24] in which the reduction of carrier mobility under pressure is proposed due to the nonhydrostatic pressure condition, thus causing the enhancement in resistance. We find that the resistance (R) at high temperatures shows a quasi-linear temperature dependence behavior at various pressures, which suggest that the electron phonon scattering is the dominant scattering mechanism.^[25] At temperatures below 50 K, the R(T) curves at different pressures can be well described with the formula R(T) = R₀ + AT². Selected fitting of the R(T) curves are displayed in Fig. 3(b), which suggest that the Fermi-liquid behavior remains unchanged under high pressure. As shown in Fig. 3(c), residual resistance R₀ monotonically increases upon compression without any anomaly.

Fig. 3.

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	Fig. 3. (a) Temperature-dependent resistance of PdSn4 in the pressure range of 3.2 GPa to 52.0 GPa from 5 K to 300 K. (b) Selected fitting results of R–T curves by using the power law. (c) The residual resistance R0 as a function of pressure.

Since PdSn₄ possesses XMR effect, magnetoresistance MR = [R(H) – R(0)]/R(0) × 100% measurements were performed at 5 K under various pressures. As shown in Fig. 4, one can see that the magnitude of MR (5 K, 7 T) is suppressed dramatically under high pressure. For example, it is 52.9% at 3.2 GPa and almost three orders smaller than that at ambient pressure.^[14] Similar cases are also reported in TaAs^[15] and WTe₂,^[26] in which the suppressed MR under pressure is attributed to reducing of the carrier mobility. In addition, we find that all MR curves can be divided into two parts. For example, at 3.2 GPa MR exhibits a conventional quadratic behavior below ∼ 2.5 T (red solid line), as shown in the inset of Fig. 4. By contrast, at high fields, MR gradually deviates from the quadratic relationship and develops into a linear field-dependent behavior (green dashed line), with no sign of saturation. The linear feature of MR at high fields or even at a full field range at a certain field orientation with respect to the b axis of single crystal PdSn₄ was also observed at ambient pressure, which is attributed to the existence of the Dirac-like linear dispersion.^[18] In the quantum limit, all carriers are degenerate in the lowest Landau level. Thus once beyond a certain critical field B_c, quantum transport will dominate and a quadratic-to-linear MR transition will occur.^[18] Similar behavior is also reported in charge density wave material LaAgSb₂.^[27] With increasing pressure, MR decreases monotonously, however, the linear feature maintains up to 20 GPa, indicating the unchanged magnetoresistance mechanism. Considering the above experimental results and similar high-pressure studies on topological materials NbAs^[28] and TaSb₂,^[24] we infer that the robustness of the Dirac fermions of PdSn₄ possible be reflected under high pressure.

Fig. 4.

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	Fig. 4. Magnetoresistance versus applied field with B ⊥ I under various pressures at 5 K. The inset shows a typical fitting of MR at 3.2 GPa. Red and green lines are fitting results at low and high magnetic field, respectively. The dashed green lines denote the linear behavior at various pressures.

In summary, we have investigated the pressure effect on the crystal structure and electronic properties of Dirac-like semimetal PdSn₄ single crystals. There are several main results: (i) in situ high-pressure synchrotron XRD data show that the crystal structure is stable up to 44.5 GPa with orthorhombic phase, (ii) R–T curves follow Fermi-liquid behavior below 50 K at various pressures, (iii) over the pressure range from 3.2 GPa to 20.0 GPa, the MR curves at 5 K maintain a linear field-dependent behavior above 2.5 T. Our study provides pressure-transport constraints on the robustness of the Dirac fermions of PdSn₄. Theoretical calculation on its topological states under pressure still need further investigation.