Abstract
Transitionmetal dipnictide PtBi_{2} exhibits rich structural and physical properties with topological semimetallic behavior and extremely large magnetoresistance (XMR) at low temperatures. We have investigated the electrical and magnetic properties of trigonalphase PtBi_{2x} single crystals with x ~ 0.4. Profound de Haas–van Alphen (dHvA) and Shubnikovde Haas (SdH) oscillations are observed. Through fast Fourier transformation (FFT) analyses, four oscillation frequencies are extracted, which result from α, β, γ, and δ bands. By constructing the Landau fan diagram for each band, the Berry phase is extracted demonstrating the nontrivial nature of the α, β, and δ bands. Despite Bi deficiency, we observe the Zeeman splitting in dHvA and SdH oscillations under moderate magnetic field and the moderate Landé g factor (4.97–6.48) for the α band. Quantitative analysis of the nonmonotonic field dependence including the sign change of the Hall resistivity suggests that electrons and holes in our system are not perfectly compensated thus not responsible for the XMR effect.
Introduction
Transitionmetal dipnictide PtBi_{2} can crystalize in multiple structures, including the cubic, hexagonal (or trigonal), and two orthorhombic phases.^{1} While it has been considered as an excellent electrocatalyst,^{2} attention is recently paid to its exotic electronic properties. For the cubic phase, the electronic structure calculations predict a threedimensional (3D) Dirac point along the Γ–R direction.^{3} Experimental investigation has indeed shown that the cubic PtBi_{2} exhibits Shubnikov–de Haas (SdH) oscillations with the nontrivial Berry phase.^{4} It was also proposed that the extremely large magnetoresistance (XMR) is due to the nearly compensated electron and hole concentration.^{4} Under hydrostatic pressure, the cubic PtBi_{2} also exhibits superconductivity.^{5} These rich phenomena seen in the cubic phase give rise to an important question: what is the role of the crystal structure in PtBi_{2}? In other words, would properties be observed as in the cubic phase present in PtBi_{2} crystalized in different structures?
For the trigonal PtBi_{2}, both band calculations and angleresolved photoemission spectroscopy indicate the existence of linear dispersive Dirac bands located at Γ and M points,^{6,7} which may be responsible for the linear field dependence of the magnetoresistance.^{8} However, these Dirac bands are identified to be trivial without topological protection.^{6} While quantum oscillations are also observed,^{9} the topologies of individual bands are yet to be investigated. In addition, it was proposed that the XMR effect is caused by disorder due to Bi deficiency in the trigonalphase PtBi_{2−x}.^{6,8} In view of reported results summarized in Table 1, the XMR effect tends to be weaker in the trigonal phase than that in the cubic structure. Further study is thus necessary in order to understand the structure–property relationship in this unique system.
In this article, we report the experimental investigation on the trigonal PtBi_{2–x} with Bi deficiency (x ~ 0.4). Both the XMR effect and profound SdH and de Haas–van Alphen (dHvA) oscillations are observed. Four oscillation frequencies are identified, corresponding to the α, β, γ, and δ bands. Quantitative data analysis allows to extract their effective masses and Berry phases. We also observe the Zeeman splitting in both the dHvA and SdH oscillations. This work sheds light on several unsolved issues: (1) despite Bi deficiency, PtBi_{1.6} exhibits long quantum relaxation time with small scattering; (2) there is clear evidence for nontrivial Berry phase for α, β, and δ bands; and (3) the XMR effect might not be attributed to either electron–hole compensation nor disorder but intrinsic to clean samples.
Results and discussion
Crystal structure and magnetotransport
According to energy dispersive Xray spectroscopy (EDS) measurements on several asgrown single crystals, the actual ratio of Pt:Bi ~ 1:1.6, indicating Bi deficiency compared with the targeted stoichiometry. Nevertheless, the powder and single crystal Xray diffraction (XRD) pattern of PtBi_{1.6}, shown in Fig. 1a, reveals peaks that are consistent with the trigonal structure with the space group of P31m (No. 157). The lattice parameters are a = b ~ 6.58 Å and c ~ 6.17 Å, consistent with previous reports.^{10} Theoretical calculations^{9} also suggest the same structure.
Figure 1b shows the temperature dependence of the inplane resistivity (ρ_{ab}) between 2 and 300 K. Upon increasing temperature, ρ_{ab} increases with ρ(2 K) = 0.82 μΩ cm and ρ(300 K) = 133 μΩ cm. This gives the residual resistivity ratio RRR = ρ(300 K)/ρ(2 K) ~ 162. Compared to another trigonal P31mphase PtBi_{2},^{9} our sample exhibits smaller RRR, but larger than that in P\(\bar 3\)phase PtBi_{2}^{8,11} (see Table 1). The small residual resistivity (~ρ(2 K)) and large RRR indicate high quality of our single crystals, despite Bi deficiency. Quantitatively, the hightemperature ρ_{ab}(T) can be fitted by the Bloch–Grüneisen (BG) formula \(\displaystyle\rho _{ab}(T) = \rho _{ab}(0) + A_{\mathrm {el  ph}}\left( {\frac{T}{{\theta _D}}} \right)^5\mathop {\int }\nolimits_0^{\theta _{\mathrm{D}}/T} \frac{{x^5}}{({e^x  1})({1  e^{  x}})}{\mathrm{d}}x\), where ρ_{ab}(0) is the residual resistivity, A_{el–ph} is an electron–phonon interaction constant, and θ_{D} is the Debye temperature.^{12} With ρ_{ab}(0) ~ 0.8 μΩ cm, A_{el–ph} ~ 257 μΩ cm, and θ_{D} ~ 143 K, the BG formula describes ρ_{ab}(T) well between 20 and 300 K, as illustrated in Fig. 1b with the solid line. The θ_{D} value is almost identical to that obtained from the specific heat of P\(\bar 3\)phase PtBi_{2}.^{11} The lowtemperature ρ_{ab}(T) follows the power law ρ_{ab} = ρ_{ab}(0) + AT^{n} with ρ_{ab}(0) = 0.8 μΩ cm and n ~ 2.9, as shown in the inset of Fig. 1b. This implies dominant electron–phonon scattering in our system rather than electron–electron scattering reported in ref. ^{11}
Figure 1c exhibits the temperature dependence of ρ_{ab} under different magnetic field (H) applied along the c direction. Note that the application of H gradually increases ρ_{ab}(T), especially at low temperatures. At H < ~2 T, ρ_{ab}(T) remains metallic character (i.e., dρ_{ab}/dT > 0), which can still be described by ρ_{ab} = ρ_{ab}(0) + AT^{n} at low temperatures. However, the n value apparently increases with increasing field, as shown in the inset of Fig. 1c. On the other hand, the A value deceases with increasing field. These indicate that the application of magnetic field modifies the electron scattering with the trend of increasing electron–phonon interaction (e.g. increasing n) but reducing the electron–electron interaction (e.g. decreasing A). Above ~2 T, an upturn in ρ_{ab}(T) gradually develops at low temperatures (i.e., dρ_{ab}/dT < 0), which eventually saturates at even lower temperatures. The saturation valve and upturn range increase with increasing H, revealing the XMR effect. For example, MR = \(\frac{{\rho _{{{ab}}}\left( H \right)  \rho _{{{ab}}}\left( 0 \right)}}{{\rho _{{{ab}}}\left( 0 \right)}}\) ~ 1500% for H = 9 T and T = 2 K, and MR (H = 14 T, T = 2 K) ~ 2200%. Compared to the reported results, our sample shows smaller MR than that reported in ref., ^{9} but larger than the P\(\bar 3\)phase PtBi_{2}^{8,11} (see Table 1). On the other hand, our MR is less than that observed in P\(a\bar 3\) phase.^{4,13} We note that the latter phases have even higher RRR values as shown in Table 1. This strongly suggests that the XMR effect is intrinsic: higher RRR larger MR. Similar trend is also observed in Weyl semimetal WTe_{2}.^{14}
Following data analysis in ref. ^{14} for WTe_{2}, we plot our MR data in the Kohler formula, MR versus H/ρ_{ab}(0), as shown in Fig. 1d. Note that all data taken at different field collapses into a single line, implying scaling behavior for PtBi_{1.6}. Quantitatively, all data can be described by MR = 50(H/ρ_{ab}(0))^{1.4}, represented by the solid line in Fig. 1d. The power m = 1.4 is less than 2 (the standard Kohler’s rule), however. Ideally, the Kohler’s rule only applies to systems with either single band or multiple bands in perfect electron–hole compensation.^{15} As discussed below, PtBi_{1.6} does not meet either of these criteria.
According to Wang et al.^{14} the minimum resistivity \(\rho _{{{ab}}}^{\mathrm {min}}(T_{\mathrm{m}},H)\) can be described by \(\rho _{{{ab}}}^{\mathrm {min}}\left( {T_{\mathrm{m}},H} \right) = \left[ {m/\left( {m  1} \right)} \right]\rho _{{{ab}}}\left( {T = T_{\mathrm{m}},\,H = 0} \right)\) derived from the scaling relationship. In Fig. 1c, we plot \(\rho _{{\mathrm{ab}}}^{\mathrm {min}}\left( {T_{\mathrm{m}},H} \right) = 3.5\rho _{{{ab}}}\left( {H = 0} \right)\) for m = 1.4 in a dashed line, which indeed passes through the \(\left( {T_{\mathrm{m}},\rho _{{{ab}}}^{\mathrm {min}}} \right)\) points in different field. To clearly see the field and temperature dependence of ρ_{ab}, the temperature derivative of ρ_{ab} for different field is plotted in Fig. 1e. In addition to a characteristic temperature T_{m} corresponding to dρ_{ab}/dT = 0, we define T_{i} at which dρ_{ab}/dT reaches the minimum. The field dependence of T_{m} and T_{i} is plotted in Fig. 1f: both increasing with increasing H. Quantitatively, the field dependence of T_{m} can be fit by the power law T_{m} = 13.8(H−H_{c})^{0.23} with H_{c} ~ 3 T. Similar behavior has been observed in a number of other nonmagnetic materials particularly in Dirac or Weyl semimetals with power close to 1/2.^{14,16,17,18,19} In the latter case, several mechanisms have been proposed to explain the resistivity upturn and the XMR effect at T < T_{m}, including the unique band structure involving Dirac bands,^{20} electron–hole compensation,^{21,22} a quantum phase transition,^{16} gap opening at the bandtouching points,^{18,23} change of carrier concentration or mobility,^{24} or normal scattering.^{14}
dHvA oscillations
To understand the origin of the XMR effect in the P31mphase PtBi_{1.6}, other lowtemperature physical properties are investigated. Figure 2a displays the temperature dependence of the magnetic susceptibility along both the ab plane (χ_{ab}) and c direction (χ_{c}). The negative sign of χ_{ab} and χ_{c}, and positive slopes at high temperatures indicate that the atomic contribution (negative), is greater than that from itinerant electrons (positive). Previous report shows the negative χ_{ab} but the positive χ_{c} in P\(\bar 3\)phase PtBi_{2}.^{11} While it is yet to be confirmed, the discrepancy in the sign of χ_{c} may be related to the Bi content and/or subtle structure difference, which varies the density of states near the Fermi level. Figure 2b shows the field dependence of the magnetization along the c direction (M_{c}) between 1.85 and 6 K. The diamagnetic background is more or less linear field dependent between 0 and 7 T, unlike the \(H^{\frac{1}{2}}\) dependence predicted for T < T_{m}.^{23,25} After subtracting the background, ΔM_{c} is obtained and plotted as a function of H in Fig. 2c. There are clearly dHvA oscillations. If replotting data as ΔM_{c} versus 1/H in Fig. 2d, the periodicity is more clearly seen. From the fast Fourier transformation (FFT) analysis, two principal frequencies F_{δ} = 4 T and F_{α} = 39 T are identified as shown in Fig. 2e. According to the Onsager relation F = (ħ/2πe)A_{F}, the crosssection area of Fermi surface A_{F} is determined to be 3.81 × 10^{−4} and 3.71 × 10^{−3} Å^{−2} for the δ and α bands, respectively. The corresponding Fermi wave vectors are k_{δ} ~ 0.011 Å^{−1} and k_{α} ~ 0.034 Å^{−1}. The latter is almost identical to that reported in ref. ^{9}
The amplitude of dHvA oscillations is usually described by the Lifshitz–Kosevich (LK) formula^{26,27}
Here, the thermal damping factor R_{T} = XT/sinh(XT) and the Dingle damping factor R_{D} = exp(–XT_{D}) (T_{D} is the Dingle temperature) with X = 2π^{2}k_{B}m*/(ħe\(H^{}\)) (k_{B} is the Boltzmann constant and m* is the effective mass). The spin reduction factor R_{S} = cos(πgm*/2m_{0}) (g is Landé factor and m_{0} is the free electron mass). The phase factor ϕ_{B} is the Berry phase, and δ′ dpends on the dimensionality of the Fermi surface (FS) with 0 for 2D, −1/8 for the maxima (minima) of a 3D electron (hole) type FS, and +1/8 for the minima (maxima) of a 3D electron (hole) type FS.^{28} The exponent λ is 0 (2D) or 1/2 (3D). By fitting the temperature dependence of the FFT amplitude for the relevant frequency to R_{T} (the inverse field 1/\(\bar H\) used in R_{T} is the average inverse field used for FFT analysis, 1/\(\bar H\) = (1/H_{min} + 1/H_{max})/2) with H_{min} and H_{max} being the field range), we obtain the effective masses \(m_\delta ^ \ast\) = 0.092m_{0} and \(m_\alpha ^ \ast\) = 0.189m_{0} for the δ and α bands, respectively. These values are considerably lower than that reported in ref. ^{9}
To extract the topological phase for each band, two oscillatory components are separated via filtering process. Figure 3a, b shows ΔM_{c}(1/H) at T = 1.85 K for F_{δ} and F_{α}, respectively. Given that dM/dH (i.e., the magnetic susceptibility) is proportional to the density of states (DOS) at the Fermi level,^{29} we can then assign the minimum of ΔM_{c} to N−1/4,^{30} with N being the Landau level (LL) index. Figure 3d, e show the Landau level indices N as a function of 1/H for δ and α bands, respectively. The peaks (red) and valleys (blue) of ΔM_{c} are denoted as N + 1/4 and N − 1/4 Landau level indices, respectively. The solid lines are the linear fit of the data using the Lifshitz–Onsager quantization criterion N = F/H + ϕ_{B}/2π + δ′.^{27,30} From fitting, we obtain F_{δ} = 4.17 T and \(\phi _{\mathrm{B}}^\delta\)/2π = 0.40−δ′ for the δ band, and F_{α} = 39.37 T and \(\phi _{\mathrm{B}}^\alpha\)/2π = 0.43−δ′ for the α band. These frequencies are in excellent agreement with that obtained from FFT analysis (F_{δ} = 4 T and F_{α} = 39 T). According to band calculations, both the δ and α bands are 3D hole type pockets.^{9} We thus set δ′ = −1/8 corresponding to the FS minima at the Fermi level, leading to Berry phase \(\phi _{\mathrm{B}}^\delta\) ~ 1.05π and \(\phi _{\mathrm{B}}^\alpha\) ~ 1.11π. These indicate that both the δ and α bands exhibit nontrivial Berry phase. According to calculations for P31mPtBi_{2},^{9} the α band disperses linearly across the Fermi level, and connected with the triply degenerate point along the H–K direction. However, the δ band is different with the result in ref. ^{9}
In view of the FFT spectra of the dHvA oscillations (see Fig. 2e), there is the second harmonic oscillation (2F_{α}) from the α band. To identify its origin, we inspect the field and temperature dependence of the α band oscillation by filtering the contribution from the δ band in ΔM_{c}. As can be seen in Fig. 3c, three are three remarkable features. First, for T = 1.85 K, the oscillation peaks clearly split at high field. The splitting becomes more profound with the enlarged amplitude of the difference between two peaks with increasing magnetic field. Second, the splitting only occurs at peaks instead of both peaks and valleys. Third, the splitting gradually smears out with increasing temperature, because of the thermal broadening of Landau levels. Based on these characteristics, the 2F_{α} peak in Fig. 2e should result from the Zeeman splitting. Thus, in Fig. 3c, we mark a pair of peaks using red and blue arrows representing the contributions from spinup and spindown electrons at each Landau level, respectively.
For PtBi_{2}, the Zeeman splitting effect has not been reported prior to our study. A low threshold field, where the discernible peak splitting starts to appear (~5 T for our sample), is usually ascribed to the large Landé g factor.^{31,32,33,34} By lifting the spin degeneracy, the LL index plot should be revised to N = F/H + ϕ_{B}/2π + δ′ + \(\frac{1}{2}\)φ and N = F/H + ϕ_{B}/2π + δ′−\(\frac{1}{2}\)φ for spinup and spindown Landau levels, respectively.^{27,33} Here, φ = \(\frac{{{\mathrm{g}}m_\alpha ^ \ast }}{{2m_0}}\). Figure 3f shows the Landau fan diagram for both spinup (red) and spindown (blue) cases. By fitting these two sets of data, we obtain g ~ 4.97. Compared to that for free electrons (g_{0} = 2), the g factor for PtBi_{1.6} is enhanced. However, the enhancement is moderate compared to many topological materials such as ZrSiS,^{32} ZrTe_{5},^{33} and Cd_{3}As_{2}.^{34} With moderate Zeeman energy, the strong Zeeman splitting in our system must be attributed to the narrow LL width. The latter decreases with decreasing temperature and scattering. According to Eq. (1), one can estimate T_{D} ~ 0.45 K by calculating the slope of ln(ΔM_{c}/H^{0.5}R_{T}) versus 1/H (λ = 0.5 for the α band^{9}), which is about one order less than that obtained in the cubicphase PtBi_{2},^{4} ZrSiS,^{32} and Cd_{3}As_{2}.^{34} This implies that the quantum relaxation time τ_{q} = ħ/2πk_{B}T_{D} ~ 2.7 × 10^{−}^{12} s is considerably longer for the trigonalphase PtBi_{1.6}, despite Bi deficiency.
Shubnikovde Haas oscillations
With the long quantum relaxation time, the quantum effect should be seen in other properties. Figure 4a shows the field dependence of the inplane resistivity ρ_{ab} with H//c at 1.8 K. There are SdH oscillations under high field (see the inset of Fig. 4a). In order to construct a reliable LL fan diagram, we convert ρ_{ab} to the electrical conductivity σ_{xx} via the following formula:^{30}
where ρ_{xy} is the Hall resistivity (shown in Fig. 5a). Figure 4b presents the field dependence of the oscillatory Δσ_{xx} after subtracting the smooth background, which clearly shows multiple oscillations. By FFT shown in Fig. 4c, four peaks are revealed, corresponding to F_{α}, 2F_{α}, F_{β} ~ 515 T, and F_{γ} ~ 1245 T. Contribution from each band is separated and presented in Fig. 4d–f for the α, β, and γ bands, respectively. Note that, in Fig. 4d, there is obvious Zeeman splitting, similar to that seen in ΔM_{c} (Fig. 3c). Assigning the Landau fan diagram for both spinup (red) and spindown (blue), we obtain g ~ 6.48, which is greater than that from ΔM_{c} splitting, likely due to sample composition variation. For the β band, data fitting to the LifshitzOnsager quantization relation yield \(\phi _{{\mathrm B}}^\upbeta\)/2π = 0.83 − δ′. For the 3D hole type β band with the maxima at the Fermi level as derived from calculations,^{9} δ′ = 1/8 leading to \(\phi _{\mathrm{B}}^\upbeta\) ~ 1.4π, again a nontrivial Berry phase. According to band structure calculations,^{9} the β band also exhibits linear dispersion at the Fermi level and is connected with the doubly degenerate points along the H–Γ and H–D directions. Similarly, for the γ band shown in Fig. 4i, we obtain the intercept of N(1/H) ~ 0.23, which leads to \(\phi _{\mathrm{B}}^\gamma\)/2π = 0.23 − δ′. Given δ′ = 1/8 for the 3D hole type γ band with the maxima at the Fermi level,^{9} \(\phi _{\mathrm{B}}^\gamma\)→ 0, representing a trivial Berry phase, which is consistent with the claim in ref. ^{9} It should be noted that the Berry phase for both the β and γ bands are extracted from high Landau levels, which may not provide precise Berry phase. In addition, the calculation of σ_{xx} involves the inplane resistivity and Hall resistivity: both were measured on different samples. Discrepancy due to sample difference may result in a phase shift between two quantities. In order to more accurately determine the Berry phase for highfrequency bands, measurements under higher field is necessary so to reach lower Landau levels.
In previous reports, the SdH oscillation has not been observed in the Hall effect of PtBi_{2}. Figure 5a shows the field dependence of the Hall resistivity ρ_{xy} at the indicated temperatures for PtBi_{1.6}. Several features can be seen: (1) at each temperature, ρ_{xy} exhibits nonmonotonic H dependence; (2) ρ_{xy} is positive at high temperatures, but gradually pushes down toward the negative direction with decreasing temperature; (3) there is sign change below ~ 50 K (see the inset of Fig. 5a); and (4) there are clear oscillations below ~15 K, again indicating the high quality of our samples. For further analysis, we calculate the Hall conductivity σ_{xy} via the following formula:^{30}
Figure 5b displays σ_{xy} versus H, showing the sign change in all indicated temperatures. For a system involving both electrons (concentration n_{e}) and holes (concentration n_{h}), the Hall conductivity (σ_{xy}) can be described by the following equation:^{35,36}
where μ_{e(h)} is the mobility of electron (hole). By fitting our experimental σ_{xy} (Fig. 5b) to Eq. (4) for different temperatures, we obtain the temperature dependence of carrier concentration n_{h,e} and mobility μ_{h,e} which are shown in Fig. 5c, d, respectively. Note both n_{h} and n_{e} are in the order of 10^{19} cm^{−3}, while they are temperature dependent. This is consistent with the semimetal picture with the Fermi level close to the edge of the electron and hole bands. What is remarkable is that the ratio n_{h}/n_{e} ~ 2.5 below ~30 K for PtBi_{1.6} as shown in the inset of Fig. 5c, which is not compensated. This implies that n_{h}/n_{e} would be even larger for stoichiometric PtBi_{2}. From this point of view, the XMR effect in trigonalphase PtBi_{2} may not be attributed to the electron–hole compensation. On the other hand, both μ_{h} and μ_{e} increase with decreasing temperature with μ_{e} > μ_{h}. At 2 K, μ_{h} ~ 0.7 × 10^{4} cm^{2} V^{−1} s^{−1} and μ_{e} ~ 2.1 × 10^{4} cm^{2} V^{−1} s^{−1}. These values are higher than that obtained for the P\(\bar 3\)phase PtBi_{2}^{8} and YSb,^{37} but lower than that obtained in the cubicphase PtBi_{2}.^{4}
It should be pointed out that, strictly speaking, Eq. (4) cannot precisely describe the Hall conductivity of PtBi_{1.6}, as it consists of more than two bands. On the other hand, if all bands are considered, it would be impossible to obtain meaningful information through the fitting due to a large number of fitting parameters. Nevertheless, there are many other XMR semimetals in which electrons and holes are not compensated, such as MoTe_{2} (hole dominated),^{38} LaBi (hole dominated),^{39} YSb (hole dominated),^{40} PtSn_{4} (electron dominated),^{41,42} and Cd_{3}As_{2} (electron dominated).^{43} In Cd_{3}As_{2},^{44} high mobility is considered as the possible origin of the XMR effect. The diminishing of the Hall factor κ_{H} = (\(\rho _{{{xy}}}\)/\(\rho _{{{xx}}}\))^{2} has also been proposed to be a key factor to describe the unsaturated MR in NbP semimetal.^{45} We find that κ_{H} has nonmonotonic field dependence, thus difficult to explain the monotonic field dependence of MR in our PtBi_{1.6}.
In conclusion, we have successfully grown and investigated the physical properties of the trigonalstructured PtBi_{2−x} single crystals. In spite of Bi deficiency with x ~ 0.4, both the dHvA and SdH oscillations have been observed in the magnetization, electrical resistivity, and Hall resistivity. Through FFT analysis, four oscillation frequencies are identified with F_{α} = 39 T, F_{β} = 515 T, F_{γ} = 1245 T, and F_{δ} = 4 T, corresponding to the α, β, γ, and δ bands. By fitting the temperature dependence of the FFT amplitude to the LK formula, we obtain nearlyzero effective mass for electrons residing in the α and δ bands. By constructing the Landau fan diagram, the Berry phase for four bands is extracted: nontrivial for the α, β, and δ bands but trivial for the γ band. As F_{β} and F_{γ} are considerably high, our observed oscillations correspond to high Landau levels, further measurements under high magnetic field are necessary to exam the Berry phase at low Landau levels. Nevertheless, we also observe the Zeeman splitting effect in the α band under moderate field and moderate Landé g factor (4.97 from the magnetization and 6.48 from the electrical conductivity). The only explanation is its extremely narrow Landau level breadth with little scattering, which is reflected in the long quantum relaxation time (i.e., small Dingle temperature).
The XMR effect is also observed in our Bideficient crystals. This together with evidence accumulated from previous reports strongly suggest that the XMR effect is the property of clean samples reflected by the large residual resistivity ratio. The resistivity under various fields actually collapses into a single line when plotted in Kohler formula. Although the exponent for PtBi_{1.6} deviates from the standard Kohler’s rule, it is worth to investigate underlying physics in such a multiband system. Quantitative analysis of the Hall conductivity indicates that electrons and holes in our system are not perfectly compensated, thus might not be responsible for the XMR effect. Since the δ band has reached the first Landau level at 5 T, the XMR effect, which can only be observed in clean materials, is likely the consequence of the quantum limit under high magnetic field.
Methods
PtBi_{2−x} single crystals were grown using the selfflux (Bi) method. Pt (99.99%, Alfa Aesar) and Bi (99.5%, Alfa Aesar) powder was mixed with the molar ratio Pt: Bi = 1: 5. The mixture was loaded into an alumina crucible and sealed in a quartz tube after evacuation. The quartz tube was then placed in a box furnace, then heated up to 600 °C. After staying at this temperature for 50 h, the furnace was cooling down to 450 °C with a rate −2 °C h^{−}^{1}. Finally, the quartz tube was taken out and centrifuged to remove excess Bi flux. Single crystals with shiny surfaces were obtained, as shown in the inset of Fig. 1a.
The phase of asgrown single crystals was characterized by XRD. The chemical composition of single crystals was determined by EDS. The magnetization was measured using a magnetic property measurement system (MPMS, Quantum Design) with magnetic field up to 7 T. The electrical resistivity and Hall effect measurements were performed using the standard fourprobe technique in a physical property measurement system (PPMS, Quantum Design) with a magnetic field up to 14 T.
Data availability
All data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This material is based upon work supported by the U.S. Department of Energy under EPSCoR Grant No. DESC0016315.
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R.J. designed research; L.X. synthesized the sample and conducted physical property measurements with assistance from R.C. and R.N.; L.X. and R.J. wrote the manuscript with the contributions from all the authors.
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Xing, L., Chapai, R., Nepal, R. et al. Topological behavior and Zeeman splitting in trigonal PtBi_{2x} single crystals. npj Quantum Mater. 5, 10 (2020). https://doi.org/10.1038/s4153502002139
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