The thermodynamics in a dynamical black hole

Abstract

Considering the back-reaction of emitting particles to the black hole, a “new” horizon is suggested where thermodynamics can be built in the dynamical black hole. It, at least, means that the thermodynamics of a dynamical black hole should not be constructed at the original event horizon any more. The temperature, “new” horizon position and radiating particles’ energy will be consistent again under the theory of equilibrium thermodynamical system.

Keywords

Hawking radiation / black hole / information loss paradox / back-reaction / event horizon / thermodynamics

1 1 Introduction

Topological insulators and semimetals with unique band topology are the focus of condensed matter physics in recent years. Due to their extraordinary topological surface state and special Fermi surfaces, these topological materials show many quantum phenomena such as extreme large magnetoresistance, ultrahigh carrier mobility, weak antilocalization (WAL) effect, quantum oscillation, quantized Hall effect, and chiral anomaly in magnetoresistance, which not only provides a broader platform for basic physics research, but also opens up a new way to explore the potential applications in spintronics and quantum computing [1–6].

The electronic transport properties of intermetallic compounds have been intensively studied in the past decade [1–6]. Among these compounds, the compensated semimetals of MSb₂ (M = Nb, Ta) with extremely large magnetoresistance (XMR) and low-temperature resistivity plateaus are attracting increasing attention [7–14]. Theoretical and experimental results show that the NbSb₂ and TaSb₂ are topological materials or good candidates. They could be weak topological insulators in the absence of magnetic field, and could be transformed into type-II Weyl semimetals with application of external magnetic fields [13, 14]. It is worth noting that the surface state and electronic properties of topological materials can be adjusted by doping or element replacement. A typical example is the type-II Weyl semimetals Mo_1−xW_xTe₂ system [15-17]. Mo doping significantly expands the Fermi arc on its topological surface and greatly reduces the difficulty of direct observation of surface states by angle resolved photoelectron spectroscopy (ARPES). Similar to Mo_1−xW_xTe₂ system, some interesting stories may occur in Ta_1−xNb_xSb₂ system in which the regulation of topological state could be realized by doping between Nb and Ta [12].

Due to the high chemical similarity between these two compounds, the high-throughput one-pot crystal growth process based on chemical vapor transport (CVT) was recently explored [12] to grow a range of Ta_1−xNb_xSb₂ (0 ≤ x ≤ 1) crystals within one silica tube, providing the possibility to investigate the exotic electronic transport behaviors of Ta_1−xNb_xSb₂ (0 ≤ x ≤ 1) compounds. In our previous electronic transport study of the (Ta,Nb)Sb₂ family, we observed the carrier’s-compensation-induced XMR effect in the Nb lightly doped Ta_1−xNb_xSb₂ (x ≤ 10%) and Ta lightly doped Nb_1−xTa_xSb₂ (x ≤ 10%) crystals [12]. For Ta-Nb equally mixed crystals (x ~ 0.5), the low-temperature magnetoresistance (MR) is drastically suppressed and the maximum MR value is only 80% at 2 K and 9 T due to the deviation from the optimal carriers’ compensated region (n_h/n_e ~ 1). On the basis of the previous work, we continue to explore the possible effects of doping on the topological state of this system.

For the study of topological materials, the most common method is to extract the topological properties of surface state through electronic transport measurements such as giant magnetoresistance, SdH oscillation, and WAL effect. Due to the limitations of crystal quality, defects, impurities and existing magnetic field strength, it is difficult to detect SdH oscillation in many cases. On the other hand, WAL effect expressed as the positive correction of resistivity in the presence of a low magnetic field is easier to be detected. Due to the spin-momentum locking, the spin rotates along the backscattering path. At this time, the phase difference between the clockwise and counterclockwise trajectories reaches 360°, which further causes the change of the sign of the scattering amplitude, leads to the cancellation of the interference between the two scattering events, and ultimately suppress the occurrence of backscattering. That is, there is destructive interference between spin up and spin down carriers, resulting in cusp-like positive magnetoconductance behaviors, and the electron spin and orbital coherence can be destroyed by the external magnetic field in the whole process [18, 19]. As a result, the WAL effects are usually observed in low-dimensional systems such as 2D materials of graphene, thin films of Na₂IrO₃, In₂O₃, and thin flakes of MoTe₂ [20−23] due to higher scattering occurrence between two time-reversed paths. As for the 3D systems, WAL effect is uncommon and usually originates from either strong spin-orbit interaction in the bulk and/or spin-momentum locking in the topological surface states. The topological nontrivial band structure is closely related to strong spin-orbit coupling in topological materials, thus WAL effect can also be observed in some specific three-dimensional systems. A special system that exhibits WAL effect is 3D topological insulators (Bi₂Se₃, Bi₂Te₃, Bi₂Te₂Se, Bi₂Se_2.1Te_0.1, V_x:Bi_1.08−xSn_0.02Sb_0.9Te₂S, etc.), where the spin-moment locking of topologically protected surface states provide the source of destructive interference [24−28]. Recently, WAL effect is also reported in some 3D topological semimetal single crystals (LuPdBi [29, 30], ScPtBi [31], YbCdGe [32], YbCdSn [33], CaAgBi [34]), topological superconductors (LuPdBi [29, 30]), and a very small number of half-Heusler compounds (ScPdBi [35], ScNiBi [36], LuNiBi [37], LuPtSb [38]). Considering the huge variety of 3D materials, the WAL effect in these 3D materials are still very rare. Experimentally, WAL effect can be detected by measuring the magnetoconductance at low temperatures, and normally could be explained within the Hikami−Larkin−Nagaoka (HLN) model [39, 40].

In this work, we observed WAL effect in CVT-grown high quality Ta_0.7Nb_0.3Sb₂ single crystals at low temperatures. The magnetotransport properties of the Ta_0.7Nb_0.3Sb₂ crystals are similar to those of previously reported TaSb₂, NbSb₂ and (Ta,Nb)Sb₂, e.g., large, non-saturating MR, resistivity plateaus below 10 K, and magnetic field induced resistivity upturn. The very different point is that at low temperatures, the MR curves show sharp “V” shape below 1 T magnetic field and increase parabolically with the magnetic fields above 1 T. The WAL shape of MR curve survives up to ~ 50 K, which can be quite well fitted via the HLN model. Due to the different angular dependence of WAL effect and the fermiology of Ta_0.7Nb_0.3Sb₂, the symmetry of angular MR evolves with magnetic field. All of these results make Ta_0.7Nb_0.3Sb₂ single crystals a good platform for the fundamental research of (Ta,Nb)Sb₂ family or other similar semimetal systems.

2 2 Experimental sections

Ta_0.7Nb_0.3Sb₂ single crystals were grown via the CVT method. Briefly, a total amount of about 1.0 g of high-purity Nb (99.95%, aladdin), Ta (99.9%, aladdin), and Sb (99.9999%, aladdin) powders, together with 0.15 g I₂ granules were sealed in a quartz tube with the pressure less than 10⁻³ Pa. The crystal growth took place in a two-zone furnace from 1000 °C (source) to 1100 °C (sink) for 10 days before cooling naturally to room temperature. Finally, typical prismatic single crystals with a size of about 2 mm × 1 mm × 0.5 mm were successfully obtained [Fig.1].

Fig.1 (a) XRD pattern of the Ta0.7Nb0.3Sb2 single crystal. Inset: An optical image of the crystal. (b) XRD rocking curve taken on the ($\overline{4}02$) diffraction peak. (c−g) The element spectrum and element mapping patterns of the Ta0.7Nb0.3Sb2 single crystal. (h) HRTEM image. (i) SAED pattern.

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The crystallographic orientation of the major surface of the Ta_0.7Nb_0.3Sb₂ was characterized by XRD (Rigaku SmartLab). Energy Dispersive X-ray Spectroscopy (EDS) (Oxford Aztec X-Max80) measurements show the good stoichiometry and uniformity distribution of the constituent elements of the Ta_0.7Nb_0.3Sb₂ single crystals. In addition, the high-resolution transmission electron microscopy (HRTEM) and selected area electron diffraction (SAED) patterns measurements were performed using a Tecnai G2F20 S-Twin microscope to analyze the structure and crystallinity of the crystals.

Finally, using physical properties measurement system (PPMS-14 T, Quantum Design), the electronic transport properties of Ta_0.7Nb_0.3Sb₂ crystals were measured via the four-probe and six-probe methods, respectively. During the measurements, the electric current direction is always along the b-axis of the crystal, and the initial direction of the magnetic field is along the [$\overline{2}01$] direction of the crystal [see inset of Fig.2].

Fig.2 (a) Zero-field resistivity (cyan points) and fitting results (red curve) of the Ta0.7Nb0.3Sb2 single crystal. Inset: schematic geometry of the directions of the magnetic field and the electric current. (b) Temperature dependence of the resistivity under different magnetic fields, as measured using the schematic geometry shown in the inset of (a). The grey shadow areas show the resistance plateau region and the pink dash arrow indicates the resistance minimum temperature (Tm). (c) ∂ρxx/∂T plotted as a function of temperature, taking B = 3 T as an example. Insert: Tm plotted as a function of the magnetic field B.

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3 3 Results and discussion

As shown in the inset of Fig.1(a), the Ta_0.7Nb_0.3Sb₂ single crystal shows rod-like shape, with its length direction along the [010] direction of the crystal and mirror-like major surface with its normal vector along the [$\overline{2}01$] direction. To examine the crystal structure, XRD measurements were carried out on the shiny plane of the selected single crystal. The sharp comb-like (h0l) (h = −2, −4, −6, −8; l = h/2) diffraction pattern [Fig.1(a)] and small full width at half maximum (FWHM) (~ 0.08^o) of the ($\overline{4}02$) diffraction peak [Fig.1] confirm the high quality of the Ta_0.7Nb_0.3Sb₂ crystal. As reported previously, Ta_1−xNb_xSb₂ (0 < x < 1) compounds share the same monoclinic structure (C_12/m1 space group) with the MSb₂ (M = Ta, Nb) parent compounds [7, 12, 41].

We carried out EDS measurements on the major surface of the crystal to determine its chemical composition and element distribution. As shown in Fig.1, the Ta, Nb, and Sb elements and their atomic ratio is 23.3%:9.9%:66.8%, which is close to the nominal stoichiometry of Ta_0.7Nb_0.3Sb₂. In addition, we carried out elemental analysis of a milled sample with the help of TEM. As shown in Fig.1−Fig.1, the elements of Ta, Nb and Sb distribute evenly. Besides, the HRTEM image demonstrates that its lattice spacing of the ($\overline{2}01$) and (110) planes are 0.51 and 0.32 nm, respectively [Fig.1], which is in agreement with the monoclinic structure characteristics given by XRD [41]. The excellent microstructure results in sharp SAED pattern [Fig.1]. All these results make Ta_0.7Nb_0.3Sb₂ a reliable platform for studying its electronic transport properties.

As displayed in Fig.2 and Fig.2, the zero-field resistivity of the Ta_0.7Nb_0.3Sb₂ crystal increases monotonically with increasing temperature from 2 K to 300 K, showing the typical metallic characteristics. The resistivity is approximately 18 μΩ·cm at 2 K, further suggesting its good metallic conductivity, which could be attributed to the large mean free path of the electrons [42]. For zero field, the residual resistivity ratio (RRR) is approximately 10.6. In addition, the resistivity at temperature region of 2−30 K can be well described by the formula ρ(T) = ρ₀ + ATⁿ with n ~ 2.3, where ${\rho }_0$ represents the residual resistivity, A is a fitting constant, and n is a parameter suggesting scattering mechanisms. The values of n is smaller than semimetals like LaBi (n = 3) [43], LaSb (n = 4) [44], ZrGeSe (n = 3.1) [45] whose scattering mechanism is attributed to electron-phonon scattering and deviates from the pure electronic correlation-dominated scattering mechanism (n = 2) [46, 47]. So we speculate that at the low temperature region, this temperature dependence of resistivity probably originates from the coexistence and competition of these two scattering mechanisms.

For the magnetic field B ≥ 1 T, a metal-insulator transition and a resistivity plateau (the grey shadow areas) occur at low temperature [Fig.2], both of which are positively correlated with the magnetic field strength. These resistivity behaviors are generic features of topological semimetals [32, 33, 42−45]. To further analyze these features, we plot first derivative of the resistivity [(∂ρ_xx/∂T)] as a function of temperature in Fig.2. Two characteristic temperatures can be determined from the figure: i) the metal-insulator transition temperature T_m, where ∂ρ_xx/∂T = 0, as indicated by the red dotted line; ii) the onset temperature of the resistivity platform (T_i), corresponding to the minimum of ∂ρ_xx/∂T, as indicated by the olive arrow. As displayed in the inset of Fig.2(c), T_m increases monotonically with the magnetic field. The relationship between T_m and B can be well fitted into the formula T_m ~ a(B−B₀)^1/ν with ν ≈ 1.96, which is close to the values of compensated semimetals such as WTe₂ (ν = 2) [47] and Bi (ν = 2) [48]. For (Ta,Nb)Sb₂ crystals, the carriers are off-compensation within 10% < Nb percentage < 50% region. Therefore, the magnetic-field-induced resistivity upturn in the present Ta_0.7Nb_0.3Sb₂ single crystal with 30 atom% Nb are not as strong as that in the TaSb₂ [12] because of the off-compensation of the electron and hole carriers.

To further explore the carrier’s properties of the Ta_0.7Nb_0.3Sb₂ crystal, we measured Hall resistivity (ρ_xy) at different temperatures ranging from 1.8 K to 300 K, which are plotted in Fig.3. The nonlinear tendency of the ρ_xy versus B curves suggest the existence of multi-type carriers, especially at low temperatures and low field region. Here, we consider the semiclassical two band model to analyze the carrier’s density & Hall mobility [46, 49]. The total conductivity tensor σ can be fitted into the plural form as follows:

Fig.3 (a) Hall resistivity as a function of the magnetic field at different fixed temperatures for the Ta0.7Nb0.3Sb2 single crystal. (b) Double-band model fitting of the Hall resistivity versus magnetic field curves. (c) Temperature dependence of the carrier’s density and mobility, obtained via the double-band fitting.

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(1)$\sigma =e\left({\displaystyle \frac{{\mu }_e{n}_e}{1+\mathrm{i}{u}_{e}B}+\frac{{\mu }_h{n}_h}{1-\mathrm{i}{u}_{h}B}}\right),$

where the subscripts e and h represent the electron and hole, respectively. n and μ denote the carrier density and mobility, respectively. The conductivity σ can be transformed to the resistivity tensor ρ using ρ = σ⁻¹ whose real and imaginary parts ρ_xx and ρ_xy can be calculated by

(2)$\begin{array}{rl}{\rho }_{xx}\left(B\right)=& \mathrm{R}\mathrm{e}\left(\rho \right)\\ =& {\displaystyle \frac{1}{e}\frac{\left(n_h{\mu }_h+n_e{\mu }_e\right)+\left(n_h{\mu }_e+n_e{\mu }_h\right){\mu }_e{\mu }_h{B}^2}{{\left(n_h{\mu }_h+n_e{\mu }_e\right)}^2+{\left({n_h-n}_e\right)}^2{{\mu }_e}^2{{\mu }_h}^2{B}^2},}\end{array}$

(3)$\begin{array}{rl}{\rho }_{xy}\left(B\right)=& \mathrm{I}\mathrm{m}\left(\rho \right)\\ =& {\displaystyle \frac{B}{e}\frac{\left(n_h{\mu }_h^2-n_e{\mu }_e^2\right)+\left(n_h-n_e\right){\mu }_e^2{\mu }_h^2{B}^2}{{\left(n_h{\mu }_h+n_e{\mu }_e\right)}^2+{\left({n_h-n}_e\right)}^2{{\mu }_e}^2{{\mu }_h}^2{B}^2}.}\end{array}$

The ${\rho }_{xy}$ data were fitted by Eq. (3) [Fig.3(b)] to obtain the fitting parameters n_e, n_h, μ_e, and μ_h. Fig.3(b) only shows four fitting curves (1.8, 30, 75 and 200 K) in the positive magnetic field region as representative in order to show the fitting clearly. The fitting parameters are plotted as a function of temperature in Fig.3(c). The electron and hole densities n_e and n_h increase with temperature while the electron and hole mobilities show opposite variation trend for T ≤ 100 K. At 300 K, the Hall resistivity versus magnetic field curve is nearly a straight line with negative slope [Fig.3], where the hole carrier density is near 0. Overall, there are two important features of the Hall results: 1) the ratio n_e/n_h is about 8 at T = 2 K, which demonstrates the off-compensation of electron and hole carriers. 2) even at 300 K, the electron and hole mobility still remain at a relatively high value of ~1.1 × 10³ cm²·V⁻¹·s⁻¹, which could contribute to the large MR effect. The high mobility within a wide temperature region up to room temperature may make it a promising candidate system for high efficiency electronic device applications.

Because of the high carrier’s mobility, a sizeable MR effect is expected in the Ta_0.7Nb_0.3Sb₂ crystal. Transverse MR describes the relative change of resistivity when the magnetic field is applied vertically with respect to the direction of the electric current. MR usually refers to the relative change in resistivity in the presence of a magnetic field with respect to zero magnetic field. Therefore, MR is calculated by $\mathrm{M}\mathrm{R}=\frac{\rho \left(B\right)-\rho \left(0\right)}{\rho \left(0\right)}$, where $\rho \left(B\right)$ and $\rho \left(0\right)$ refer to the resistivity at a magnetic field B and zero magnetic field, respectively. Fig.4(a) shows the variation of transverse MR with the magnetic field at a specific temperature. First, MR exhibits unsaturated quasi-parabolic behavior up to 14 T and the MR vs. B curves below 10 K almost coincide with each other, which is consistent with the resistivity plateau behaviors discussed above. For T = 2 K and B = 14 T, MR is 457 %, which is 1−2 orders of magnitude lower than that of the TaSb₂ and NbSb₂ parent compounds, but comparable to those of previously reported semimetals [30−33]. After Nb doping, the inter-valley scattering may be enhanced due to the increased atomic disorder. This, together with the shift of its Fermi level away from the compensation area, results in the decrease of MR upon Nb doping [12]. For T ≥ 50 K, MR decreases remarkably with increasing temperature and finally reaches 15.7 % for T = 300 K and B = 14 T, which is quite similar to the situation of traditional metal materials.

Fig.4 (a) MR plotted as a function of the magnetic field B at different fixed temperatures for the Ta0.7Nb0.3Sb2 single crystal. (b) A zoom-in MR measurements below 50 K ranging from −3 T to 3 T. (c) MR plotted as a function of the magnetic field at various angles θ at T = 2 K. Insets: Schematic diagrams of the directions of the magnetic field, electric current and crystallographic orientation during different types of MR measurements.

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In the low-field (−0.5 T ≤ B ≤ 0.5 T) and low-temperature region (T < 50 K), MR versus B curves show a sharp cusp-like shape which is a typical phenomenon of WAL effect [18−20, 50]. We made more detailed measurements in the low field region to better understand these phenomena and show the results in Fig.4(b). MR increases rapidly as the temperature decreases from 50 to 2 K, and reached a maximum value of 120% at B = 1 T and T = 2 K. Fig.4 shows the MR vs. B curves at 2 K, with the direction of the magnetic field tilting from [$\overline{2}01$], corresponding to a rotation of the crystal around the [010] axis. As schematically shown in Fig.4, θ is the angle between the direction of the magnetic field and the [$\overline{2}01$] orientation and the direction of the magnetic field is always vertical to the electric current direction during the tilting process. Similar phenomena have been observed for all tilting angles in Fig.4. Another point is that, during the rotation of the crystal, the MR value at B = 14 T first decreases to a minimum value at θ ~ 50°, then increases. To further study this anisotropy MR effect, angular dependent MR is measured and will be discussed later.

As discussed above, MR displays a prominent cusp in the low-field region below 50 K which is the reminiscent of the WAL effect and is generally found in 3D systems with strong spin−orbital coupling (SOC) or two-dimensional (2D) topological insulators with nontrivial surface states [21−38]. Recently, the WAL effect has been observed in some 3D semimetal crystals, such as ScPtBi, YbCdGe, YbCdSn, CaAgBi, ScPdBi [31−35]. To investigate the origin of the WAL effect in the Ta_0.7Nb_0.3Sb₂ crystal, the HLN model is applied to analyze the 2D magnetoconductance which could be expressed as follows:

(4)$\begin{array}{rl}\mathrm{\Delta }G\left(B\right)=& G\left(B\right)-G\left(0\right)\\ =& \frac{\alpha e^2}{2{\pi }^2\hslash }\left[\mathrm{\Psi }\left(\frac{1}{2}+\frac{\hslash }{4e{l}_{\varphi }^{2}B}\right)-\mathrm{l}\mathrm{n}\left(\frac{\hslash }{4e{l}_{\varphi }^{2}B}\right)\right],\end{array}$

where G(B) and G(0) are the conductance with and without magnetic field, respectively. $\mathrm{\Psi }$ represents the digamma function, α = −1/2 per conduction channel, and $l_{\varphi }$ is the phase coherence length, respectively [39−40]. Therefore, the value of α could be used to estimate the number of conducting channels. The WAL is generally observed in 2D topological materials and the HLN model can be applied directly to these 2D system. The parent compounds of (Ta,Nb)Sb₂ have been predicted to be weak topological semimetals [7−14], so it makes sense to study the origin of the WAL effect using the HLN model. However, $\mathrm{\Delta }G$ of 3D systems is too large to be fitted directly by the HLN model due to the significant large thickness than 2D-like system. Throughout the MR measurements, the applied current is perpendicular to the thickness direction. Naturally, the conductance of 3D systems can be seen as an addition of the 2D conduction layers periodically stacked along the thickness direction. Therefore, $\mathrm{\Delta }G$ that is going to be fitted by the HLN model should be divided by the number of 2D conduction layers Z*, which is about half of the quantum layer number, i.e., Z* = 0.5 × t / (1 nm). Hence, we define the magnetoconductance per contributing 2D layer as: $\mathrm{\Delta }\tilde{G}\left(B\right)=\mathrm{\Delta }G\left(B\right)/$Z*, and fit $\mathrm{\Delta }\tilde{G}\left(B\right)$ using the HLN model [25, 27]. As shown in Fig.5, the symbols represent the experimentally obtained $\mathrm{\Delta }\tilde{G}\left(B\right)$ data in the low-field region below 50 K and the corresponding lines are the fitting curves using Eq. (4). The agreement between the theoretical fitting and the experimental data is good, indicating that the MR behaviors can be described by the HLN model. The variation of the fitting parameters α and $l_{\phi }$ with temperature is shown in Fig.5. If the fitting parameter α is close to −0.5, it could be concluded that the WAL effect originates from 2D surface states. However, the value of obtained α is of the order of 10², much larger than that of 2D systems. Similar results have been observed in several 3D topological materials such as LuPdBi, ScPtBi, YbCdGe, YbCdSn, CaAgBi, and ScPdBi [30−35], which are attributed to the dominance of 3D bulk conducting channels rather than 2D surface states. These results imply that the WAL effect in the Ta_0.7Nb_0.3Sb₂ crystal is mainly derived from three-dimension bulk contribution. The value of α shows gentle change in the temperature region 2 K ≤ T ≤ 10 K, which implies that the number of conductive channels is roughly independent of temperature at very low temperature. As the temperature increases further, the value of α decreases significantly, implying a weakening of the WAL effect with increasing temperature, which corresponds to the decrease in the magnetoconductance in Fig.5.

Fig.5 (a) Magnetoconductance $\mathrm{\Delta }\tilde{G}$ as a function of the magnetic field. The solid lines are the fitting results using HLN model. (b) Temperature dependence of the parameter α and the phase coherence length lϕ, extracted from the HLN model. The solid blue line is the fitting curve using equation (5). (c) Angular dependence of the magnetoconductance as a function Bsinθ, as measured at T = 2 K.

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On the other hand, considering the electron–phonon scattering and the electron–electron interaction, the phase coherence length $l_{\varphi }$ can be fitted by the following formula:

(5)$\begin{array}{c}{\displaystyle \frac{1}{l_{\varphi }^2\left(T\right)}=\frac{1}{l_{\varphi }^2\left(0\right)}+A_{ee}T+A_{ep}{T}^2.}\end{array}$

Here, $l_{\varphi }\left(0\right)$ is the phase coherence length at zero temperature, and $A_{ee}T$ and $A_{ep}{T}^2$ are the electron-electron and electron-phonon interactions, respectively. In Fig.5, the blue solid triangle symbols are the value of $l_{\varphi }\left(T\right)$ which is obtained by fitting $\mathrm{\Delta }\tilde{G}\left(B\right)$ in Fig.5 using Eq. (4) and the solid line shows the fitting curves of $l_{\varphi }\left(T\right)$ vs. T using equation (5). The best fitting yield $l_{\varphi }\left(0\right)=136\mathrm{n}\mathrm{m},A_{ee}=-1.79\times {10}^{-6}({\mathrm{n}\mathrm{m}}^2\cdot \mathrm{K}{)}^{-1},A_{ep}=2.13$ ×10⁻⁷$({\mathrm{n}\mathrm{m}}^2{\cdot \mathrm{K}}^2{)}^{-1}$, which indicates that the electron-electron scattering is more significant than electron-phonon scattering.

To further confirm the origin of the WAL effect in the Ta_0.7Nb_0.3Sb₂ single crystal, we carried out MR measurements by rotating the direction of the magnetic field with respect to the electric current at T = 2 K, and calculated the corresponding $\mathrm{\Delta }\tilde{G}$. Fig.5(c) shows the variation of $\mathrm{\Delta }\tilde{G}$ with the normal component of the external magnetic field Bsinθ, where θ is the angle between the direction of the magnetic field and [$\overline{2}01$] orientation. If the WAL is caused by 2D surface states, the $\mathrm{\Delta }\tilde{G}$ vs. Bsinθ curves should merge into a universal curve, suggesting that the magnetoconductance depends mainly on the perpendicular component of the applied magnetic fields [24, 27, 34]. As shown in Fig.5, however, the nine curves with different tilt angles θ do not coincide with each other, confirming the WAL effect in the bulk single crystal of Ta_0.7Nb_0.3Sb₂ mainly results from the contribution of 3D bulk transport [35−38]. This conclusion is consistent with the relatively large α value discussed above.

Fig.4 and Fig.5 show the MR curves at different angle θ at 2 K, from which the anisotropy of MR under different magnetic fields can be observed. To show this anisotropic MR behaviors more clearly, we conducted angular dependent MR (AMR) measurements at 2 K under various magnetic fields and show the results in Fig.6. Note that the starting point (0°) means that the direction of the magnetic field is along the [$\overline{2}01$] crystallographic direction. During the rotation of the crystal, the direction of the magnetic field always keeps perpendicular to the direction of the electric current which is along the [010] direction of the crystal. The rotation is asymmetric near 0°, where the positive direction rotates towards [001] direction, and the negative direction rotates towards [100] direction. In the low-field region [Fig.6−Fig.6], the maximum MR is obtained at θ ≈ −30°−35°, where the magnetic field is roughly along the [100] crystallographic direction. An interesting point is that the minimum MR is observed at θ = 90°, where the magnetic field is not along the [001] axis, therefore resulting in a twisted figure of “8” shaped AMR. As discussed above, the WAL effect dominates the MR behaviors at low magnetic fields. That is to say, the AMR in the low magnetic-field regions mainly reflects the WAL’s contribution to MR. The WAL usually follows cosine rule or close-to-cosine rule during rotation, which leads to two-fold symmetrical AMR. As a result, the AMRs in Fig.6−Fig.6 shows a two-fold rotational symmetry in a shape of “8”. In high magnetic fields (B ≥ 2 T), MR shows the maximum for the direction of the magnetic field along the [001] direction of the crystal, while it shows the minimum for the direction of the magnetic field along the [100] direction, also resulting in a figure of “8” shaped AMR. The WAL effect is greatly inhibited by high magnetic fields. Therefore, these two-fold-rotation symmetric patterns at high magnetic fields (B ≥ 2T) may be due to the intrinsic geometry of Fermi pockets and their relative contribution to electronic transports [11, 51, 52, 53]. Both in low field limits and high field limits, the AMR are two-fold symmetry or roughly two-fold symmetry, however, the maximum MR appears at different angle θ. For low field limit, the maximum MR appears at θ = −30°−35° [Fig.6−Fig.6] while for the high field limit the maximum MR appears at θ = 60° [e.g., Fig.6] Therefore, upon increasing the magnetic field, the angle (θ) where the maximum MR appears would rotate. As shown in Fig.6−Fig.6, in the intermediate field range (0.8 T ≤ B ≤ 1.5 T), the AMR shows roughly four-fold symmetry, which probably originates from the competition of the low-field WAL dominated MR with the high-field fermiology dominated MR.

Fig.6 Angular dependent MR at T = 2 K and various magnetic fields for the Ta0.7Nb0.3Sb2 single crystal.

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4 4 Conclusions

We report the magnetotransport properties of high quality Ta_0.7Nb_0.3Sb₂ single crystals. The crystals show metallic electronic transport behaviors and resistivity plateaus under magnetic fields as well as magnetic-field-induced resistivity upturn upon cooling. Hall measurements and two-band model fitting results indicate that the crystals show off-compensation of electron and hole carrier density at low temperatures [8 (electrons): 1 (holes)] and high mobility (> 1000 cm²·V⁻¹·s⁻¹) in the 2–300 K region. One interesting finding is that the WAL effect can be observed up to 50 K, which results from the strong SOC in 3D bulk, and a large MR of 120% is obtained at B = 1 T and T = 2 K. The fitting of temperature dependent $l$_ϕ demonstrates that the predominant scattering mechanism is the electron-electron scattering. The low-field WAL at T = 2 K shows angular dependence behaviors, and MR value reaches its maximum when the magnetic field is along the [100] crystallographic direction. Another interesting finding is the change of the symmetry of AMR from two-fold (≤ 0.6 T) tofour-fold (0.8–1.5 T) and finally to two-fold (≥ 2 T), which is attributed to the evolution of the mechanism from the low-field WAL dominated MR to WAL and fermiology co-dominated MR and finally to high-field fermiology dominated MR.

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