## Introduction

In the presence of a magnetic field, the electronic spectrum of a three-dimensional electron gas (3DEG) is quantized into Landau levels. When all the charge carriers are confined in the lowest Landau level—the so-called quantum limit—the kinetic energy of electrons is quenched in the directions transverse to the field. This favors the emergence of electronic instabilities, either driven by the electron-electron or electron-impurity interactions1,2,3,4. So far, the behavior of 3DEGs beyond their quantum limit has been explored in a limited number of low carrier density systems. Yet, different instabilities have been detected, such as a thermodynamic phase transition in graphite5,6,7,8, a valley depopulation phase in bismuth9,10, and a metal-insulator transition (MIT) in narrow-gap doped semi-conductors InSb11 and InAs12,13. The latter occurs when charge carriers are confined in the lowest spin-polarised Landau level—the ultra-quantum limit. This transition is generally attributed to the magnetic freeze-out effect where electrons are frozen on ionized impurities4,14.

Lately, low-doped Dirac and Weyl materials with remarkable field-induced properties were discovered15,16,17,18,19. Of particular interest is the case of ZrTe5. The entrance into its quantum limit regime is marked by quasi quantized Hall resistivity (ρxy)18 and thermoelectrical Hall conductivity (αxy)20,21, followed by a higher magnetic field transition18,22. This phase transition has initially been attributed to the formation of a charge density wave (CDW)18,22,23. Such interpretation has been questioned because of the absence of thermodynamic evidence24,25, expected for a CDW transition. Furthermore, ZrTe5 displays a large anomalous Hall effect (AHE), even though it is a non-magnetic material26,27,28,29,30.

Here we report electrical, thermo-electrical and optical conductivity measurements over a large range of doping, magnetic field, and temperature in electron-doped ZrTe5. This allows us to track the Fermi surface evolution of ZrTe5 and explain the nature of this phase transition, as well as its links with the observed AHE. We show that the onset of the field-induced transition can be ascribed to the magnetic freeze-out effect. In contrast with usually reported results, we show that the freeze-out regime of ZrTe5 is characterized by a sign change of the Hall and thermoelectric effects, followed by a saturating Hall conductivity. Our results show that the magnetic freeze-out effect differs in this Dirac material as a consequence of the tiny band gap and large g-factor of ZrTe5, that favor both an extrinsic and an intrinsic AHE of the spin-polarised charge carriers.

## Results

### Fermi surface of ZrTe5

Figure 1 a shows the temperature dependence of the resistivity (ρxx) for four batches, labelled S1−4 respectively. Samples from the same batch are labelled by distinct subscript letters (see Supplementary Note 1). At room temperature, ρxx ≈ 0.7 mΩ.cm. With decreasing temperature, ρxx peaks at a temperature around which the Hall effect (ρxy) changes sign, which is around 150 K for S3b sample (see Fig. 1b). Both shift to lower temperature as the carrier density decreases. These effects have been tracked by laser angle-resolved photoemission spectroscopy and attributed to a temperature-induced phase transition where the Fermi energy shifts from the top of the valence band to the bottom of the conduction band as the temperature decreases31.

At low temperature the Fermi energy is located in the conduction band. Fig. 1c, d show the quantum oscillations for samples from batches S1, S2 and S3 for a magnetic field (B) parallel to the b-axis of the orthorhombic unit cell. The angular dependence of the quantum oscillation frequency are well fitted by an anisotropic ellipsoid Fermi surface elongated along the b-axis, and in good agreement with previous measurements24,32,33 (see Fig. 1g). Our doping study reveals that the ellipsoid anisotropy increases as the system is less doped, see Fig. 1f. In our lowest doped samples the ratio of the Fermi momentum (kF) along the a and b-axis reach 0.06 implying a mass anisotropy ratio of $$\frac{{m}_{b}^{* }}{{m}_{a}^{* }}\simeq$$ 250, where $${m}_{a,b}^{* }$$ are the band mass along the a and b axis. This large mass anisotropy ratio is comparable to the one of Dirac electrons of bismuth34. This Fermi surface mapping allows us to accurately determine the Fermi sea carrier densities, nSdH, which agree well with nH (see Supplementary Note 2). Remarkably S1 samples have a Hall mobility, μH, as large as 9.7 × 105 V  cm−2s−1 and the last quantum oscillation occurs at a small field of BQL(S1) = 0.3 T for Bb. Given the large g-factor, g* ≈ 20–3022,35, this last oscillation corresponds to the depopulation of the (0, +) Landau level. Above it the highly mobile electrons are all confined into the lowest spin-polarised (0, −) Landau level.

### Field induced transition in the ultra-quantum limit of ZrTe5

Figure 2 shows the field dependence of ρxx beyond the ultra-quantum limit of S1, S2 and S3 samples. In the lowest doped samples (S1) ρxx increases by more than two orders of magnitude and saturates above ≈7 T. This large magnetoresistance vanishes as the temperature increases (see Supplementary Fig. 2), for T > 5 K and up to 50 T. A close inspection of the low temperature behavior reveals a light metallic phase above BQL (see Fig. 2a, b) which ends at a crossing point at Bc = 3.2 T above which an insulating state is observed up to 50 T. Following18 we take this crossing point as the onset of the field induced metal-insulator transition. As the carrier density increases, both the position of BQL and Bc increase (see Fig. 2b, c)). At the highest doping (samples S3) the amplitude of the magnetoresistance has decreased and the transition is only marked by a modest increase by a factor of two of ρxx at  30 T, indicating that the transition smears with increasing doping (see Fig. 2d). Figure 2e shows the doping evolution of BQL and Bc which are in good agreement with previous works18,22,24. For an isotropic 3D Dirac material $${B}_{QL}=\hslash /e{(\sqrt{(2)}{\pi }^{2}n)}^{2/3}$$ (see i.e36) with $$n=3{\pi }^{2}{k}_{F}^{3}$$. In the Bb configuration kF = $$\sqrt{({k}_{F,a}{k}_{F,c})}$$ can be evaluated from the frequency of quantum oscillations. The deduced BQL is shown by the red line in Fig. 2e and provides an excellent agreement with the detected BQL. As function of the total carrier density of the ellipsoid (nSdH) $${B}_{QL}=\hslash /e{(\sqrt{(2{A}_{1}{A}_{2})}{\pi }^{2}{n}_{SdH})}^{2/3}$$ where A1 and A2 are the anisotropic Fermi momentum ratios between the a and b-axis, and between the c and b-axis.

The doping evolution of Bc is a clue to the nature of this transition. So far it has been attributed to the formation of a charge density wave (CDW) along the magnetic field18,22,23. Such an instability is favored by the one-dimensional nature of the electronic spectrum along the magnetic field, which provides a suitable (2kF) nesting vector in the (0, −) Landau level. In this picture, predicted long ago1, the transition is of second order and is expected to vanish as the temperature increases. The absence of temperature dependence of Bc and the absence of thermodynamic signature24,25 invite us to consider another interpretation.

In the CDW picture, the instability is driven by the electron-electron1,18,22 or electron-phonon interaction23 and the interaction between electrons and the ionized impurities is neglected. However, in a doped semiconductor, the conduction band electrons are derived from uncompensated donors. Tellurium vacancies have been identified as the main source of impurities in ZrTe5 flux grown samples37,38. According to the Mott criterion39,40 a semiconductor becomes metallic when the density of its carriers, n, exceeds a threshold set by its effective Bohr radius, aB = 4πε/m*e2 (where m* is the effective mass of the carrier, ε is the dielectric constant of the semiconductor): n1/3aB 0.3. In presence of a magnetic field the in-plane electronic wave extension shrinks with increasing magnetic field. When B > BQL, the in-plane Bohr radius is equal to aB, = 2B with $${\ell }_{B}=\sqrt{(\frac{\hslash }{eB})}$$4,41. Along the magnetic field direction, the characteristic spatial extension is $${a}_{B,\parallel }=\frac{{a}_{B,z}}{\log (\gamma )}$$, where $$\gamma ={(\frac{{a}_{B,c}}{{l}_{B}})}^{2}$$ with aB,z=$$\frac{\varepsilon }{{m}_{z}^{* }}{a}_{B,0}$$ and aB,c=$$\frac{\varepsilon }{{m}_{c}^{* }}{a}_{B,0}$$, where $${m}_{z,c}^{* }$$ are the mass along and perpendicular to the magnetic field in units of m0, and aB,0 the bare Bohr radius. A MIT transition is thus expected to occur when the overlap between the wave functions of electrons is sufficiently decreased11,14 i.e. when:

$${n}^{1/3}{({a}_{B,\perp }^{2}{a}_{B,\parallel })}^{\frac{1}{3}}\simeq 0.3$$
(1)

This MIT is thus a Mott transition assisted by the magnetic field where the metal is turned into an insulator due to the freezing of electrons on the ionized donors by the magnetic field, the so-called magnetic freeze-out effect. According to Eq. (1), $$n\propto {B}_{c}/\log ({B}_{c})$$ and Bc is slightly sublinear in n and evolves almost parallel to BQL. In order to test this scenario quantitatively, one has to determine the threshold of the transition from Eq. (1), which requires knowing ε and $${m}_{z/c}^{* }$$. Temperature dependence of the quantum oscillations gives access to $${m}_{z}^{* }\,$$ ≈ 2m0 and $${m}_{c}^{* }\,$$ ≈ 0.02m0 for Bb, while the optical reflectivity measurements give access to ε. Figure 3 shows ε versus temperature for two samples of batches S1 and S3. ε is as large as 200-400ε0 in ZrTe5 (see Supplementary Note 5). The deduced onset from Eq. (1) is shown in dashed black lines in Fig. 2e for ε = 200 and 400, capturing well the doping evolution of Bc. We thus attribute the transition detected in the ultra-quantum limit of ZrTe5 to the magnetic freeze-out effect.

It is worth noticing that a large contribution to ε comes from interband electronic transitions resulting in ε > 100. This result also clarifies why one can detect highly mobile carriers even down to densities as low as 1013 cm−329. Due to the light in-plane carrier mass and large dielectric constant, one expects the threshold of the MIT at zero magnetic field to be below ≈1012 cm−3.

## Discussion

In InSb (nH = 2–5 × 1015 cm−3)11, a large drop of the carrier density comes with an activated insulating behavior. In contrast with that usual freeze-out scenario, we find in ZrTe5 a rather soft insulating behavior, where ρxx saturates at the lowest temperature. Measurements of the Hall effect and thermo-electrical properties at subkelvin temperatures shown in Fig. 4a–c reveal an unexpected field scale, thus confirming that the freeze-out regime of ZrTe5 differs from the usual case. Above 7 T, ρxy and the Seebeck effect (Sxx = $$\frac{-{E}_{x}}{{{{\Delta }}}_{x}t}$$) change signs and saturate from 10 T up to 50 T for ρxy (see Supplementary Notes 3 and 4). The field induced sign changes of ρxy and Sxx are reminiscent of the sign change in temperature.

The temperature dependence of Sxx/T for B = 0, 6 and 12 T (shown in Fig. 4e) enables us to quantify the variations of charge carrier density as a function of the magnetic field. At B = 0 T, Sxx/T = − 5.5 μV.K−2, which is in quantitative agreement with the expected value for the diffusive response of a degenerate semiconductor: $${S}_{xx}/T=\frac{-{\pi }^{2}}{2}\frac{{k}_{B}}{e{T}_{F}}=-5$$ μV.K−2 for TF ≈ 80 K deduced from quantum oscillation measurements. At B = 12 T Sxx/T saturates, at low temperature, to  +20 μV.K−2, a value which is four times larger than at zero magnetic field, pointing to a reduction of the charge carrier density by only a factor of eight. The partial freeze-out of the charge carriers is the source of the saturating ρxx. We now discuss the specificity of ZrTe5 that leads to this peculiar freeze-out regime.

In the k-space, the magnetic-freeze out transition corresponds to a transfer of electrons from the lowest Landau level (0, −) to a shallow band, see inset of Fig. 2e, formed by the localized electrons4. This theory does not fully apply to ZrTe5 for two reasons. First, it applies to large gap systems with no potential spatial fluctuations, and ZrTe5 has only a band gap of 6 meV42, which is fifty times smaller than that of narrow gap semi-conductors such as InSb or InAs. Second, the Fermi surface of ZrTe5 is highly anisotropic. The same critical field is thus reached for a carrier density that is fifty times larger in ZrTe5 than in isotropic Fermi surface materials, like InSb or InAs. The large Bohr radius and the relatively higher density of ZrTe5 will therefore inevitably broaden the density of states, set by: $${{\Gamma }}=2\sqrt{\pi }\frac{{e}^{2}}{\epsilon {r}_{s}}{({N}_{i}{r}_{s}^{3})}^{\frac{1}{2}}$$ where $${r}_{s}\propto \sqrt{(\frac{{a}_{B}}{n\frac{1}{3}})}$$ is the screening radius and Ni is an estimate of the impurity concentration43. Assuming that n ≈ Ni, we estimate Γ ≈ 6 meV in S1 samples.

In contrast with other narrow-gap semiconductors where Γ < < EF < < Δ, the magnetic freeze-out occurs in ZrTe5 where Γ ≈ EF ≈ Δ. In this limit, the shallow band of width Γ will overlap the LLL of the conduction band, and eventually the valence band giving rise to a finite residual electron and hole charge carriers at low temperature as sketched on Fig. 2e. As a function of doping, Γ increases the smearing of the transition (Fig. 2). The convergence of the three energy scales Γ, EF and Δ is one source of the partial reduction of charge carrier density detected in ρxx, Sxx and of the sign change of ρxy. This finite residual charge carrier should give rise to a linear Hall effect, contrasting with the saturating ρxy (and σxy), which is typical of an anomalous response. We discuss this anomalous contribution in the last section.

Several studies have reported an AHE in ZrTe526,27,28,29. In this case, the Hall conductivity is the sum of two contributions: $${\sigma }_{xy}=-\frac{ne}{B}+{\sigma }_{xy}^{A}$$ where the first and second terms are the orbital conductivity and the anomalous Hall conductivity, respectively. At high enough magnetic field, $${\sigma }_{xy}^{A}$$ becomes dominant, setting the amplitude and the sign of ρxy. So far, $${\sigma }_{xy}^{A}$$ has been attributed to the presence of a non-zero Berry curvature—an intrinsic effect—either due to the Weyl nodes in the band structure26, or to the spin-split massive Dirac bands with non zero Berry curvature28,29. In the latter case, $${\sigma }_{xy}^{A}$$ scales with the carrier density, and its amplitude is expected to be +1 (Ω.cm)−1 for nH = 2 × 1016 cm−328, which is of the same order of magnitude as our results. Skew and side jump scattering are another source of AHE in non magnetic semiconductors44,45. Deep in the freeze-out regime of low doped InSb (nH ≈ 1014 cm−3), a sign change of the Hall effect has been observed and attributed to skew scattering46. In contrast with dilute ferromagnetic alloys, where the asymmetric electron scattering is due to the spin-orbit coupling at the impurity sites, here it is caused by the spin-polarised electron scattering by ionized impurities. Its amplitude is given by $${\sigma }_{xy}^{S}$$= NSe$$\frac{{g}^{* }{\mu }_{B}}{{E}_{1}}$$, where E1=$$\frac{{\epsilon }_{G}({\epsilon }_{G}+{{\Delta }})}{2{\epsilon }_{G}+{{\Delta }}}$$ with ϵG the band gap and Δ the spin-orbit splitting of the valence band. NS = NA + n is the density of positively charged scattering centers with NA the density of acceptors46. Note that $${\sigma }_{xy}^{S}$$ induces a sign change of the Hall conductivity and is only set by intrinsic parameters and by NS. Assuming NS ≈ nH(B = 0), and taking g* ≈ 2022,35 and E1 = ϵG = 6 meV (ϵG < < Δ), we find that $${\sigma }_{xy}^{S}$$ +1 (Ω. cm)−1, which is similar to the intrinsic contribution. Remarkably, it is four orders of magnitude larger than what has been observed in low doped InSb46, due to the tiny gap and a (relatively) larger carrier density in ZrTe5.

Therefore, the AHE contribution can induce a sign change of ρxy in electron doped ZrTe5. It is accompanied by a peak in Sxx/T (see Fig. 4c), Sxy/T (see Supplementary Figs. 35) and thus in αxy = σxxSxy + σxySxx (see Fig. 4d–f). Our result shows that the thermoelectric Hall plateau20,21, observed above 5 K, collapses at low temperature. These peaks can be understood qualitatively through the Mott relation47 ($$\frac{\alpha }{T}=-\frac{{\pi }^{2}}{3}\frac{{k}_{B}}{e}\frac{\partial \sigma (\epsilon )}{\partial \epsilon }{| }_{\epsilon = {\epsilon }_{F}}$$). This is the region where ρxx and ρxy (and thus σxx and σxy) change the most in field and temperature, so that Sxx and αxy are the largest. The increase occurs in the vicinity of Bc, causing a peak in the field dependence of Sxx and αxy, as it happens across the freeze-out regime of InAs13. Whether the Mott relation can quantitatively explain the amplitude of these peaks and the sign change of Sxx remains to be determined. This calls to extend theoretical works48,49,50 on the electrical and thermoelectrical response to the freeze-out regime of Dirac materials such as ZrTe5.

In summary, we show that the doping evolution of the onset transition detected in the ultra-quantum limit of ZrTe5 can be ascribed to the magnetic freeze-out, where electrons become bound to donors. In contrast to the usual case, the freeze-out regime of ZrTe5 is marked by a modest reduction of the charge carrier density due to the convergence of three tiny energy scales in this Dirac material: the band gap, the slowly varying potential fluctuations and the Fermi energy. Deep in the freeze-out regime, the Hall conductivity changes sign and becomes anomalous with a relatively large amplitude for this low carrier density and non magnetic material. This AHE could thus have an extrinsic origin due to skew-scattering of the spin-polarised electrons by ionized impurities. Distinguishing and tuning both intrinsic and extrinsic contributions by varying the charge compensation or strain51 is an appealing perspective for future research. To date, the AHE of the spin-polarised electrons in the ultra-quantum limit has been detected in a limited number of cases. Many Dirac materials with small gaps and large g-factors remain to be studied, in particular at higher doping where the intrinsic and extrinsic AHE are both expected to be larger.

## Method

### Samples and measurements description

Two sets of ZrTe5 samples have been used in this study. The first ones, grown by flux method where iodine served as a transport agent for the constituents, have the lowest carrier density. The second ones, grown by Chemical Vapor Transport (CVT), have the highest density. Electrical and thermal transport measurements have been measured using four point contacts. Contact resistance of a less 1 Ω has been achieved by an Argon etching, follow by the deposit of 10 nm Ti buffer layer and of 150 nm Pd layer. High magnetic field measurement has been done at LNCMI-Toulouse. Thermo-electrical and thermal transport measurements has been done using a standard two-thermoemeters one-heater set up similar to one used in ref. 13. Further experimental details can be found in Supplementary Note 1.