Introduction

The galvanomagnetic properties of solids, describing transport behaviors of charge carriers driven by electric and magnetic fields, have been intensively studied for their potential applications in both fundamental and industrial research. In the realm of fundamental scientific research, the galvanomagnetic response is a powerful tool to investigate the electronic structure1,2,3,4,5,6,7, topological properties8,9,10,11 and scattering mechanism12,13 of materials. In industrial applications, materials with strong magnetic responses are promising candidates for magnetometer14,15,16,17 and hard drives18. However, the galvanomagnetic phenomena influenced by both intrinsic and extrinsic effects have distinct origins in various systems. This complexity makes it challenging to interpret certain magneto-transport behaviors despite extensive studies. For instance, the extremely large magnetoresistance (XMR) can be attributed to nontrivial band topology19,20, charge carrier compensation21,22 and open orbitals of charge carriers23,24. The planar Hall effect can arise from the strong spin-orbital coupling in magnetic material25,26,27, anisotropy of Fermi surfaces28,29,30,31,32 and the chiral anomaly in topological semimetals33,34,35,36. Furthermore, the nonlinearity of Hall resistivity might indicate the presence of multi-carrier behavior or suggest the occurrence of the anomalous Hall effect.

ZrTe5 has been extensively studied since the 1970s due to its remarkable thermoelectric properties and its resistivity anomaly accompanying sign reversal of Hall resistivity and Seebeck coefficients37,38,39,40,41,42. Plenty of mechanisms have been proposed to explain this anomalous peak, such as the Lifshitz transition43, topological phase transition44, formation of Dirac polarons45 and thermally excited charge carriers46. In addition, the nonlinear Hall resistivity has been regarded as the anomalous Hall effect (AHE) originating from the Berry curvature47,48,49,50,51. Many other novel properties, such as the topological edge states52,53,54,55, chiral magnetic effect48,56, and quantum Hall effect1, are believed to be connected with the nonzero Berry curvature in ZrTe5. However, some experimental results cannot be explained by the Berry curvature mechanism. For example, the Landé g-factor of ZrTe5, ranging between 21 and 2657,58,59, is insufficient to induce Weyl nodes through Zeeman splitting48,50 and hence generate the nonzero Berry curvature. Meanwhile, the multi-carrier model has proved to be efficient in some recent experiments and theoretical works37,57,60.

Various models have been proposed to explain the unusual transport properties of ZrTe5, yet there’s a lack of comprehensive first principles studies that simulate without assumed parameters. Moreover, ZrTe5 is a typical narrow-gap semiconductor, of which the transport behaviors sensitively depend on the temperature and chemical potential. Therefore we employ first principles calculations combined with the Boltzmann transport theory to systematically investigate the galvanomagnetic property of ZrTe5 at different doping densities and temperatures. We want to stress that the method for ZrTe5 is more complicated than our previous one for metals/semimetals24. On the one hand, the chemical potential and charge carrier density of ZrTe5 varies with the temperature sensitively. Meanwhile, the relaxation time cannot be obtained simply and directly as in metals/semimetals. Therefore it is important to treat these two variables with proper approximations to enhance the alignment between the simulation and experiment, which is the innovation and necessity of this work. To achieve this, we first calculate the temperature-dependent chemical potential by fixing the total electron density, then obtain the product of temperature-dependent resistivity and relaxation time for different doping densities through an interpolation scheme. Finally, we fit the relaxation time using Bloch-Grüneisen model and get the magneto-resistivity at different temperatures and doping densities.

Our method, without incorporating topological effects or correlation interactions, successfully reproduces the essential experimental characteristics, including the anomalous peak of the temperature-dependent resistivity, the nonlinearity and sign reversal of Hall resistivity, and the non-saturating magnetoresistance (MR) at high temperatures37,47,60,61,62. More generally and importantly, our methodology can be extended to study other narrow-gap semiconductors. We believe our work provides invaluable insights into the unique behaviors exhibited by ZrTe5 and paves the way for further understanding and exploration of other materials with similar properties.

Results

Resistivity anomaly

ZrTe5, a typical transition-metal pentatelluride compound and a representative narrow-gap semiconductor, has been extensively studied due to its prominent thermoelectric properties. Before subjecting it to the magnetic field, we shall first study its temperature-dependent resistivity at zero field due to its inherent thermal sensitivity as a semiconductor. We present the calculated band structure of ZrTe5 and the corresponding high symmetry path in the Brillouin zone in Fig. 1a, b. It is shown in Fig. 1b that an indirect narrow gap of about 73 meV occurs along the Γ − Z direction with strong electron-hole asymmetry. The applied current and magnetic field are parallel to the crystallographic a and b axis, respectively and the orientation of magnetic field is denoted as the z-axis as shown in Fig. 1a.

Fig. 1: The Brillouin zone, band structure, temperature-dependent chemical potential, and zero-field resistivity of ZrTe5.
figure 1

a The Brillouin zone of ZrTe5. The applied current and magnetic field are parallel to the crystallographic a- and b- axes, respectively. The orientation of magnetic field is denoted as z axis throughout the text. b The band structure of ZrTe5. c The temperature-dependent chemical potential with different doping densities n0 indicated by different colors. ZrTe5 with n0 > 0 is electron-doped while n0 < 0 means hole-doped ZrTe5.The low and high horizontal black dashed lines denote the valence band maximum (VBM) = −40 meV and conduction band minimum (CBM) = 33 meV, respectively. The green dash line is the extrapolated chemical potential of n0 = 0 cm−3 at T < 70K which cannot be well determined with non-zero energy gap. d The temperature-dependent zero-field resistivity with different doping densities. The inset shows the resistivity of hole-doped ZrTe5. The small black square denotes the peak of resistivity. We use a color bar to indicate the relation between colors and doping densities.

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Because the transport behavior is highly dependent on the doping levels60, we calculated the temperature-dependent chemical potential μ(T) with a series of doping levels, as shown in Fig. 1c. The calculation details could be found in the Methods section. The doping level is denoted as n0 = ne − nh, where positive/negative values represent electron/hole doping systems. As shown in Fig. 1c, the chemical potentials move towards the middle of the energy gap with increasing temperature, irrespective of the type of doping. This agrees qualitatively with the angle-resolved photoemission spectroscopy (ARPES) observation54 and results from the intrinsic thermodynamics of materials that the chemical potential favors a lower density of states (DOS) with increasing temperatures63. Note that when the doping level is fixed as n0 = 0, the chemical potential is ill-defined at low temperature. Therefore we only sketch the tendency of μ at low temperature relying on the results at higher temperatures, as shown by the green dashed line segment in Fig. 1c.

Furthermore, we calculate the temperature-dependent resistivity at different doping levels as shown in Fig. 1d. The procedure of calculating resistivity across various temperatures and doping densities is described in the Methods section. The resistivities of all doped systems (excluding the charge-neutral system) increase at low temperatures and then drop quickly after attaining their peaks at critical temperatures Tp. In order to discuss the origin of the resistivity anomaly, we introduce two important variables: the total charge carrier concentration nt = ne + nh, which might vary with temperature due to thermal excitation, and the doping level n0 = ne − nh, assumed as a constant. At low temperatures, where the thermal energy is insufficient to excite electrons from valence bands to conduction bands, nt remains nearly unchanged. Because the relaxation time decreases with temperature, the resistivity increases and ZrTe5 behaves like a metal. However, as the temperature rises sufficiently to excite electrons from valence bands to conduction bands and generate electron-hole pairs, nt increases and ZrTe5 behaves like a semiconductor.

The contradictory dependence on temperature of relaxation time and total charge carrier concentration implies the existence of a critical temperature TP, at which the resistivity stops increasing with temperature. To further investigate the resistivity anomaly, we mark resistivity peaks at TP with black squares in Fig. 1d. We found that TP grows with the magnitude of n0. This behavior is due to the higher temperatures required to generate thermal electron-hole pairs in systems with larger doping densities. Moreover, given that ZrTe5 has a significant particle-hole asymmetry, TP of electron-doped is lower than that of the hole-doped with the same magnitude of n0, owing to the larger DOS of conduction bands37,60. Similar multi-carrier and thermally-excited-carrier theories have been proposed in previous works37,45,46,60.

Sign reversal of Hall resistivity

Until now, we have been only concerned with the resistivity in the absence of a magnetic field. In the following, we are going to investigate the magnetotransport properties. The computation of magnetoconductivity is based on Eq.(4)–(6) and utilizes the interpolation methods described in the Methods section. This section also explains the relationship between low-field Hall conductivity and electron trajectories, detailed in Eq. (7). Here, the transverse conductivity, σxy, is expressed in terms of the area, \({\mathcal{A}}({k}_{z})\), swept by the “scattering path length" l ≡ v(k)τ (see Supplementary Information for detailed derivation of Eq. (7)). Meanwhile, the orbital trajectories of electrons in k space can be divided into two categories. One is the orbits only composed of convex part, which is transformed to a simple l path with single loop and a certain circulation orientation. Thus the corresponding Hall resistivity exhibits the expected sign. The other one is the orbits with concave segments, which are converted into intersected l path with second loops adjacent to the primary one and hence might change circulation orientations of the whole (or part of) orbit. Whether and to what extent the circulation orientation of the orbit changed depends on the fine curvature of the k orbits and decides the final sign of Hall conductivity of this single orbit64.

When both electron and hole charge carriers contribute to electric current, the competition between them is crucial to diagnose the behavior of ρyx, such as the nonlinearity or sign reversal features. Intuitively, the charge carrier with a larger concentration dominates the transport. However, it only holds under the condition of sufficiently strong magnetic fields. At the low magnetic field, charge carriers only trace part of the orbit, the specific area \({\mathcal{A}}\,\left({k}_{z}\right)\) swept by l sensitively depends on the fine structure of the Fermi surface and orientation of the magnetic field. Accordingly, we classify the characteristic quantities of the cyclotron movement \({\omega }_{c}\tau =\frac{eB\tau }{{m}^{* }}\) (m* is the cyclotron mass) into two limiting cases, i.e., ωcτ ≪ 1 (low-field limit) and ωcτ ≫ 1(high-field limit), where the magnetoresistance (MR) and Hall resistivity might exhibit distinct features as shown in the following.

We begin with the Hall resistivity of ZrTe5 with doping level n0 = − 3.0 × 1018 cm−3 (hole-doped), 1.0 × 1018 cm−3 (electron-doped), and 3.0 × 1018 cm−3 (electron-doped), which are shown in Fig. 2. Results for additional doping levels are presented in Fig. S3. The shape of the Hall resistivity varies drastically with increasing temperature. However, there is an obvious difference between the Hall resistivity of the hole and electron-doped system. The Hall resistivity curves for hole-doped in Fig. 2a only exhibit nonlinear feature, while the ones for electron-doped in Fig. 2b, c show both sign reversal and nonlinear features. At low temperatures where there is only one single type of charge carrier, the magnitude of Hall resistivity monotonically increases with the magnetic field and the system shows the expected Hall sign, i.e., positive/negative for hole/electron-doped systems. With increasing temperature, the magnitude of Hall resistivity ρyx drops for both electron and hole-doped systems, due to the thermal excitation and the increasing charge carriers contributing to the transport process. When reaching the intermediate temperature (around 155 K), the Hall resistivity of the electron-doped system undergoes a sign reversal with a strong nonlinear slope as shown in Fig. 2 (b, c). On the contrary, the Hall resistivity of the hole-doped system only exhibits nonlinear features without sign reversal as temperature rises.

Fig. 2: The field-dependent Hall resistivity at different temperatures and doping densities.
figure 2

The Hall resistivity for doping densities of (a) n0 = −3.0 × 1018 cm−3, (b) 1.0 × 1018 cm−3, and (c) 3.0 × 1018 cm−3, respectively. The dashed lines indicate the Hall resistivities that reverse sign with the increasing magnetic field. Temperature is varied from 35 K to 275 K at steps of 10 K. The relation between the temperature and color is indicated in the color bar.

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According to our analysis of σxy, the nonlinear and sign reversal features of Hall resistivity probably stem from the complicated competition between electron and hole carriers (including both intrinsic and thermally excited ones). In order to verify this, we plot representative calculated k-orbits and their corresponding l-paths at different chemical potentials μ0 = − 55, − 50 and 40 meV in Fig. 3. The corresponding Fermi surfaces are shown in Fig. S2. All the representative k-orbits in the hole-doped system have concave segments (Fig. 3a, b), which give rise to the intersected l-path with a tiny secondary loop and the opposite circulation. It means a small part of charge carriers on these hole pockets behave like electrons. According to Eq. (7), the resistivity slope, which depends linearly on B in the low-field limit, is influenced by both the charge carrier density and the presence of these secondary loops. In contrast, at high magnetic fields, the slope of resistivity is governed primarily by the charge carrier density. This discrepancy between the slopes in the low- and high-field limits contributes to the subtle nonlinearity observed in the Hall resistivity for the hole-doped case at low temperatures, as depicted in Fig. 2a. Conversely, the electron pockets are all spherical surfaces and the corresponding l-paths are simple as shown in Fig. 3c, f, indicating linear Hall resistivity in electron-doped systems at low temperatures.

Fig. 3: The cross section of Fermi surfaces and corresponding l-paths.
figure 3

a–c The typical cross section of Fermi surfaces and (d–f) their corresponding l-paths with the magnetic field along z direction. The chemical potentials are set to (a, d) μ = −55 meV, (b, e) μ = −50 meV, and (c, f) μ = 40 meV, respectively.

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When temperature increases, the sign reversal at low magnetic field only happens in the electron-doped systems, which is difficult to explain by the qualitative orbits analysis. Therefore we employ the two-band model, where the Hall resistivity is written as,

$${\rho }_{yx}=\frac{1}{e}\frac{({n}_{h}{\mu }_{h}^{2}-{n}_{e}{\mu }_{e}^{2})B+({n}_{h}-{n}_{e}){\mu }_{e}^{2}{\mu }_{h}^{2}{B}^{3}}{{({n}_{e}{\mu }_{e}+{n}_{h}{\mu }_{h})}^{2}+{({n}_{h}-{n}_{e})}^{2}{\mu }_{e}^{2}{\mu }_{h}^{2}{B}^{2}}$$
(1)

where ne/h, μe/h are the concentration and mobility of electron/hole carriers, respectively. With the fixed doping level n0 = ne − nh, we use \({n}_{e}={n}_{e}^{0}+{\delta }_{n},{n}_{h}={\delta }_{n}\) for electron-doped, and \({n}_{h}={n}_{h}^{0}+{\delta }_{n},{n}_{e}={\delta }_{n}\) for hole-doped in Eq. (1) as following,

$${\rho }_{yx}^{hole}=\frac{1}{e}\frac{\{[{\delta }_{n}({\mu }_{h}^{2}-{\mu }_{e}^{2})+{n}_{h}^{0}{\mu }_{h}^{2}]+{n}_{h}^{0}{\mu }_{e}^{2}{\mu }_{h}^{2}{B}^{2}\}B}{{({n}_{e}{\mu }_{e}+{n}_{h}{\mu }_{h})}^{2}+{({n}_{h}-{n}_{e})}^{2}{\mu }_{e}^{2}{\mu }_{h}^{2}{B}^{2}}$$
(2)
$${\rho }_{yx}^{ele}=\frac{1}{e}\frac{\{[{\delta }_{n}({\mu }_{h}^{2}-{\mu }_{e}^{2})-{n}_{e}^{0}{\mu }_{e}^{2}]-{n}_{e}^{0}{\mu }_{e}^{2}{\mu }_{h}^{2}{B}^{2}\}B}{{({n}_{e}{\mu }_{e}+{n}_{h}{\mu }_{h})}^{2}+{({n}_{h}-{n}_{e})}^{2}{\mu }_{e}^{2}{\mu }_{h}^{2}{B}^{2}}$$
(3)

where \({n}_{e}^{0}\) and \({n}_{h}^{0}\) are the concentration of electron and hole carriers at zero-temperature. Because the mobility of hole carrier is larger than that of electron in ZrTe537,60, the Hall resistivity \({\rho }_{yx}^{hole}\) is always positive due to the positive terms, \({n}_{h}^{0}{\mu }_{h}^{2}\) and \({\delta }_{n}({\mu }_{h}^{2}-{\mu }_{e}^{2})\), in Equation (2). However, the numerator of electron-doped ρyx in Equation (3) are firstly negative due to the term \(-{n}_{e}^{0}{\mu }_{e}^{2}B-{n}_{e}^{0}{\mu }_{e}^{2}{\mu }_{h}^{2}{B}^{3}\) and then might change sign because of the growing amount of excited carriers δn with increasing temperature.

Furthermore, we can employ Eq. (3) to understand the sign change of electron-doped Hall resistivity with increasing magnetic field at intermediate temperatures (see green dashed curves in Fig. 2b, c). Supposing the Hall resistivity at weak magnetic field is positive, the numerator of Eq. (3) satisfies the condition that \(\{[{\delta }_{n}({\mu }_{h}^{2}-{\mu }_{e}^{2})-{n}_{e}^{0}{\mu }_{e}^{2}]-{n}_{e}^{0}{\mu }_{e}^{2}{\mu }_{h}^{2}{B}^{2}\}\, > \,0\) when ωcτ ≪ 1. With the increasing magnetic field, the second term of the numerator \(-{n}_{e}^{0}{\mu }_{e}^{2}{\mu }_{h}^{2}{B}^{2}\) will increase and leads to the negative Hall resistivity at critical B field. Nevertheless, if the temperature rises at the same time, δn would increase and compensate the effect induced by a stronger magnetic field. It suggests the occurrence of sign reversal necessitates a higher magnetic field under elevated temperatures. This phenomenon is displayed in Fig. 2b, c, where we show the Hall resistivity with sign reversal by dashed lines. It is noteworthy that at the higher temperature (indicated by the curves in reds), the magnitude of B field required for sign reversal is far beyond our calculation and experimental measurement scope.

Although the understanding based on the two-band model is rudimentary, the physical mechanism in the above discussion, i.e., competition between electron and hole carriers driven by the magnetic field and thermal excitation, is the key origin of the nonlinear and sign reversal features of the Hall resistivity. Furthermore, by using Eq. (7) to analyze the Hall resistivity in electron/hole-doped systems under low fields (ωcτ ≪ 1) and the well-established fact that charge carriers with the larger concentration dominate under high fields (ωcτ ≫ 1), we arrive at the same conclusion. To avoid redundancy, we will not not give detailed repetitive explanations.

Nonsaturating magnetoresistance

To proceed with investigating the MR of ZrTe5, we plot our calculated MR results for the same doping levels as those used in the Hall resistivity studies in Fig. 4. Results for additional doping levels are shown in Fig. S3. It’s apparent that at high temperatures, the MR of all doped levels exhibit non-saturating behavior. This phenomenon is due to the presence of both electron and hole carriers in those doped systems because of the thermal excitation. The nearly charge carrier compensation leads to the consistent rise in MR with the magnetic field, which agrees well with the experimental measurement47,60.

Fig. 4: The field-dependent MR at different temperatures and the doping densities.
figure 4

The field-dependent MR for doping densities of (a) n0 = −3.0 × 1018 cm−3, (b) 1.0 × 1018 cm−3, and (c) 3.0 × 1018 cm−3, respectively. Temperature is varied from 35 K to 275 K at steps of 10 K. The relation between the temperature and color is indicated in the color bar.

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In the low-temperature range, the MR of electron and hole doped cases both show the trend toward saturation. However, the details of saturation are slightly different for electron and hole-doped systems. For the hole-doped case as shown in Fig. 4a, the MR saturates more slowly and at a larger magnetic field than that in electron-doped systems Fig. 4b, c (see curves in blues). We again refer to Fig. 3 to explain the distinction between these two systems. At low temperatures, where few excited charge carriers are present and intrinsic carriers dominate transport, the discrepancy arises from distinct Fermi surface geometries in different doping systems. Regarding the fact that the hole pockets with concave segments give rise to both electron and hole-like charge carriers, the slightly compensation between them postpones the saturation of MR. On the contrary, because the electron pockets possess only spherical surfaces, electron-doped systems exhibit rapid saturation of MR as expected. Furthermore, electron-doped MR has the largest enhancement at Tp, while the hole-doped one decreases with increasing temperature, consistent with experimental findings37,60. Nonetheless, our calculations does not capture the negative MR and quantum oscillations observed at low temperatures in experiments. Moreover, experimental MR saturation at low temperatures is more obvious in electron-doped ZrTe5. Here we must point out that our calculated results is constricted in the semiclassical Boltzmann transport framework, i.e., without taking the Landau level, weak localization, and possible Berry curvature into account. For simplicity, we also ignore the k-dependence of the relaxation time τ, which may also contribute to the deviation of our calculation from the experimental measurements.

Discussion

To conclude, we developed an effective methodology to study the magnetoresistance and Hall resistivity of narrow-gap semiconductors based on the combination of first principles method and semiclassical Boltzmann transport theory. This approach is applied to study the temperature-dependent galvanomagnetic properties of ZrTe5, taking into account variations in chemical potential and charge carrier concentrations. Our calculated results successfully reproduce observed transport behaviors in experiments, such as the resistivity anomaly, sign reversal and nonlinearity of Hall resistivity in electron-doped ZrTe5, and non-saturating magnetoresistance at high temperatures. Our analysis demonstrates that these transport anomalies can be explained in terms of multi-carrier behavior and Fermi surface geometry.

Although our method effectively accounts for most of the experimental observations, certain detailed characteristics such as quantum oscillations and negative magnetoresistance were not captured due to the omission of Landau levels, Berry curvature, and the influence of interactions on the k-dependence of relaxation time. Nevertheless, our approach allows us to identify properties that can be explained by semiclassical contributions and highlights nontrivial features in ZrTe5 for future investigations. Moreover, our methodology can be extended to other narrow-gap semiconductors. Although theoretical models can explain some anomalous transport properties, first principles calculations remain a powerful tool for gaining deeper insights into galvanomagnetic behavior, thereby paving the way for further exploration.

Methods

Magnetoconductivity calculations

One can obtain the conductivity tensor in presence of the magnetic field by solving the Boltzmann transport equation within the relaxation time approximation as65,66,

$${\sigma }^{\left(n\right)}\,\left({\boldsymbol{B}}\right)=\frac{{e}^{2}}{\alpha {\pi }^{3}}\int\,d{\boldsymbol{k}}{\tau }_{n}{{\boldsymbol{v}}}_{n}({\boldsymbol{k}}){\overline{{\boldsymbol{v}}}}_{n}({\boldsymbol{k}})\left[-\frac{\partial f}{\partial {\varepsilon }_{n}({\boldsymbol{k}})}\right],$$
(4)

where α is a spin degeneracy related number, α = 4(8) if spin-orbit coupling is excluded (included) in the Hamiltonian, n is the band index, f is the Fermi-Dirac distribution. \({\varepsilon }_{n}\,\left({\boldsymbol{k}}\right)\), τn and \({v}_{n}\left({\boldsymbol{k}}\right)\) are the eigenvalue, relaxation time and group velocity of the n-th band, respectively. \({\overline{{\boldsymbol{v}}}}_{n}({\boldsymbol{k}})\) describes the weighted average velocity during the past trajectory of the charge carriers,

$${\overline{{\boldsymbol{v}}}}_{n}({\boldsymbol{k}})=\mathop{\int}\nolimits_{\!-\infty }^{0}\frac{dt}{{\tau }_{n}}{e}^{\frac{t}{{\tau }_{n}}}{{\boldsymbol{v}}}_{n}\left[{\bf{k}}(t)\right].$$
(5)

The orbital motion of charge carriers in the reciprocal space follows the semiclassical equation of motion,

$$\dot{{\boldsymbol{k}}}=-e{{\boldsymbol{v}}}_{n}\,\left({\boldsymbol{k}}\right)\times {\boldsymbol{B}},$$
(6)

where the driven force of electric field was dropped off since the corresponding displacement is negligible compared to the scale of the Brillouin zone65.

The trajectory of the charge carriers in k space is confined on a cross-section of the Fermi surface by a plane perpendicular to B, since the Lorentz force does not change the energy and only alters the velocity directions perpendicular to the magnetic field, as described by Equation (6). In order to have a more convenient and straightforward discussion of the motion of charge carriers, we define a “scattering path length" vector as l ≡ v(k)τ which maps the orbitals in k space to the real space64. Consequently, assuming the magnetic field is along \(\hat{z}\) axis, we rewrite the Hall conductivity in weak fields in terms of the l-paths as,

$${\sigma }_{yx}=\frac{{e}^{3}}{\,{\left(2\pi \right)}^{3}{\hslash }^{2}}\int\,d\varepsilon \,\left(-\frac{\partial f}{\partial \varepsilon }\right){\left[\int{dk}_{z}{\mathcal{A}}\left({k}_{z}\right)\right]}_{\varepsilon }B,$$
(7)

where \({\mathcal{A}}({k}_{z})=\frac{1}{2}\,\left(d{{\boldsymbol{l}}}_{\perp }\times {{\boldsymbol{l}}}_{\perp }\right)\cdot \frac{{\boldsymbol{B}}}{B}\) is the area swept by vector l as the charge carriers traced out the orbitals in k space with \({{\boldsymbol{l}}}_{\perp }={\boldsymbol{l}}-{l}_{z}\hat{z}\)64.

Calculation protocols applying to semiconductors

Different from the metals with large Fermi surfaces where the carrier density does not change significantly with temperature variation, the chemical potential and concentration of charge carriers in narrow-gap semiconductors, such as Zirconium Pentatelluride ZrTe5, are sensitive to the change of temperature, necessitating the precise dynamic calculations. Therefore we list our self-contained calculation procedure in the following.

  1. (i)

    Obtain ρ(Bτ, μ, T)τ: First of all, we calculate the band-resolved magneto-conductivity over the relaxation time, denoted as σn(Bτ)/τn, on a suitable (μ, T) grid by employing Eq. (4). Then we obtain the product of resistivity and relaxation time, \(\rho \tau ={\left[{\sum }_{n}{\sigma }_{n}({\boldsymbol{B}}\tau )/{\tau }_{n}\right]}^{-1}\). Here we assume that relaxation time for all carriers have the same temperature dependence but with different ratios. For example, in ZrTe5, we take τh(T) = 5τe(T) as implied by experiment measurement60.

  2. (ii)

    Get μ(T) : We assume that the net carrier concentration, defined as n0 = ne − nh, remains constant with varying temperature. This assumption is mostly valid for semiconductors below their melting point. Consequently, the temperature dependent chemical potential μ(T), is determined by the condition:

    $$\begin{array}{ll}{n}_{0}\,={n}_{e}-{n}_{h}\\ \qquad\,=\mathop{\displaystyle\int}\nolimits_{{\!\!E}_{{\rm{CBM}}}}^{+\infty }{g}_{c}(\varepsilon )f(\varepsilon -\mu )d\varepsilon \\ \qquad\,-\mathop{\displaystyle\int}\nolimits_{\!\!-\infty }^{{E}_{{\rm{VBM}}}}{g}_{h}(\varepsilon )\left[1-f(\varepsilon -\mu )\right]d\varepsilon ,\end{array}$$
    (8)

    where ne and nh are the concentration of electron and hole charge carriers respectively. ECBM and EVBM are the conduction band minimum and valence band maximum, respectively. gc(ε) and gh(ε) are DOS of conduction and valence bands, respectively.

  3. (iii)

    Interpolation to get ρ(Bτ, μ(T), T)τ: For a fixed n0, corresponding to a specified doped sample or a thin film at a particular gate voltage, we can obtain the product of temperature-dependent resistivity and relaxation time, ρ(Bτ, μ(T), T)τ, by interpolating the calculated ρ(Bτ)τ on a dense grid of (μ, T) in the step (i), see Figure. S1.

  4. (iv)

    Fitting τ(T) with experiments: Finally, for any given doping concentration n0, we obtain the magneto-resistivity ρ(B) at arbitrary temperature by substituting the fitted τ(T) in ρ(Bτ, μ(T), T)τ. Fitting τ(T) in semiconductors with experimental data is not as straightforward as in metals/semimetals. In metals/semimetals, the charge carrier concentration does not change significantly, allowing a consistent treatment of τ(T) across the whole temperature range. However, in semiconductor like ZrTe5, which exhibits different properties with varying temperature, we need to separate the high temperature regime from the low one. The metallic behavior of ZrTe5 at low temperature allows us to fit τ(T) with corresponding experiment results in this regime. On the contrary, the semiconductor behavior in the high temperature regime precludes the direct derivation of τ(T) from experimental measurements. Nonetheless, it is well-known that the electron-phonon scattering described by τ(T) ∝ 1/T dominates at the high temperature regime. We assume τ(T) ∝ 1/ρsc(T), and ρsc(T) represents the scattering-related resistivity which is comprised in the relaxation time. The distinct scattering behavior of ZrTe5 across different temperature regime could be depicted jointly by the Bloch-Grüneisen model60,67 as,

    $${\rho }_{sc}(T)={\rho }_{0}+\alpha {\left(\frac{T}{{\Theta }_{R}}\right)}^{n}\mathop{\int}\nolimits_{\!0}^{\frac{{\Theta }_{R}}{T}}\frac{{x}^{n}}{\left({e}^{x}-1\right)\left(1-{e}^{-x}\right)}dx,$$
    (9)

    with the parameters ρ0 = 1.06, α = 11, n = 2, and ΘR = 600 obtained by fitting the experiment data60 at low temperature. τ(T) in this work are calculated by Eq. (9) and the hypothesis τ(T) ∝ 1/ρsc(T).

Our methodology for calculating magneto-resistivity is broadly applicable to narrow-gap semiconductors, not just to the compound ZrTe5. Furthermore, the hypothesis τ(T) ∝ 1/ρsc(T) can be extended to metals and doped semiconductors that display metallic behavior. The validity of this relationship is supported by the relaxation time approximation and the weak field assumption about the electric field, making it applicable primarily in conditions where the system shows metallic characteristics and the charge carrier concentration remains relatively stable. It should be noted that this proportional relationship allows for the analysis of the temperature dependence of τ but not its precise value. For the determination of τ, one can fit the experimental ρ(B) with the theoretically calculated ρτ(Bτ) when the carrier density is relatively constant.

Ab initio calculations

The band structure was calculated using the Vienna ab initio simulation package (VASP)68 with the generalized gradient approximation of Perdew, Burke, and Ernzerhof for the exchange-correlation potential69. We performed the self-consistent calculation on a k-mesh of 11 × 11 × 11 with energy cutoff of 550 eV. The temperature-dependent chemical potential was calculated using BoltzTrap70 package and performed on the k-mesh of 50 × 50 × 25. Magneto-conductivity and Fermi surface were calculated using the WANNIERTOOLS package71 based on a tight-binding model constructed by the WANNIER90 package72,73,74,75. In the Magneto-conductivity calculation, we use a k-mesh of 201 × 201 × 201 and set the broadening width of Fermi-Dirac distribution function to 200 meV.