Abstract
Lightly doped III–V semiconductor InAs is a dilute metal, which can be pushed beyond its extreme quantum limit upon the application of a modest magnetic field. In this regime, a MottAnderson metal–insulator transition, triggered by the magnetic field, leads to a depletion of carrier concentration by more than one order of magnitude. Here, we show that this transition is accompanied by a 200fold enhancement of the Seebeck coefficient, which becomes as large as 11.3 mV K^{−1}\(\approx 130\frac{{k}_{B}}{e}\) at T = 8 K and B = 29 T. We find that the magnitude of this signal depends on sample dimensions and conclude that it is caused by phonon drag, resulting from a large difference between the scattering time of phonons (which are almost ballistic) and electrons (which are almost localized in the insulating state). Our results reveal a path to distinguish between possible sources of large thermoelectric response in other lowdensity systems pushed beyond the quantum limit.
Introduction
The thermoelectrical properties of low carrier density metals are of fundamental and technological interest. Owing to their small Fermi temperatures (T_{F}), their diffusive Seebeck effect (\({S}_{{\mathrm{xx}}}=\frac{{E}_{x}}{{\Delta }_{x}T}\)) can be large and, as such, used to develop highperformance thermoelectric devices^{1}. At low temperature, their thermoelectrical response is also a fine probe of their fundamental electronic properties, in particular in the presence of a magnetic field^{2}. As an example, the large quantum oscillations observed in the Nernst effect (S_{xy} = \(\frac{{E}_{y}}{ {\Delta }_{x}T }\)) of semimetals^{3,4,5} or doped semiconductors^{6,7,8} have been used to map out their Fermi surfaces and to reveal the Dirac/Weyl nature of the electronic spectrum of Bi^{5}, Pb_{1−x}Sn_{x}Se^{7} or ZrTe_{5}^{8}.
For most dilute metals, a magnetic field of a few Tesla is enough to confine all the charge carriers in the lowest Landau level (LLL), the socalled quantum limit. At low temperature, this is concomitant with an increase of S_{xx} (and S_{xy})^{7,8,9,10}. This increase can be found at higher temperatures, as illustrated by the large Seebeck effect recently observed in the quantum limit regime of the Weyl semimetal TaP^{11} or in the fractional quantum Hall regime of two dimensional electron gas (2DEGs)^{12}. In these systems, the amplitude of \(\frac{{S}_{{\mathrm{xx}}}}{T}\) is much larger than \(\frac{{k}_{B}}{e}\frac{1}{{T}_{{\mathrm{F}}}}\), the natural thermoelectric scale of the diffusive response. This surprisingly large S_{xx} can be either the result of an unbounded diffusive thermoelectric power specific to nodal metals^{13} or to the coupling of electrons with the phonon bath. In the latter case, the socalled phonon drag, the amplitude of S_{xx} is dictated by the momentum transfer between the electron and phonon baths and can be much larger than \(\frac{{k}_{B}}{e}\)^{14}. The phonon drag effect is wellknown to enhance S_{xx} in lowdoped semiconductors at zero magnetic field but also in the quantum limit regime^{9,15}. Here, we present a study of the electrical and thermoelectrical properties of InAs, a bulk narrowdirectgap semiconductor, beyond its quantum limit and up to a sofar unexplored range of temperature and magnetic field. Our findings show that the field induces a metal–insulator transition (MIT) that is accompanied by a giant peak in the Seebeck effect, as large as S_{xx} = 11.3 mV K^{−1} at T = 8 K. Based on a study of the thermal response as a function of sample dimensions, we argue that this giant Seebeck effect results from phonon drag. Our results demonstrate a new road to achieve large thermoelectric response that can be explored in other dilute metals recently identified.
Results
A dilute metal at zero magnetic field
We show in Fig. 1 the temperature dependence of the resistivity (ρ_{xx}), the Seebeck coefficient (S_{xx}) and the thermal conductivity (κ) of an ntype InAs with a Hall carrier density n_{H} = 2.0 × 10^{16 }cm^{−3}. The Fermi surface of InAs, studied by low magnetic field quantum oscillation measurements (see Supplementary Note 1), is formed by a single spherical pocket located at the Γpoint of the Brillouin zone with a carrier density n_{SdH} = 1.6 × 10^{16 }cm^{−3} (T_{F} = 100 K) and mass carrier m^{*} = 0.023m_{0}. From room temperature down to T_{F}, ρ_{xx} is metallic and the nondegenerate electrons are mainly scattered by phonons. Below T_{F}, ρ_{xx} increases with decreasing temperature as electrons become more and more degenerate with a mobility limited by ionizedimpurity scattering^{16}. Below T = 20 K, ρ_{xx} is constant with a residual resistance of ρ_{0} = 12 mΩ.cm and a Hall mobility μ_{H} = 24,000 cm^{2} V^{−1} s^{−1}. The electronic contribution (κ_{el}) to κ is given by the Wiedemann Franz law \({\kappa }_{{\mathrm{el}}}=\frac{{L}_{0}T}{{\rho }_{0}}=0.19\) mW K^{−2} m^{−1}. It is much smaller than the κ shown in Fig. 1b) and points to a purely phononic origin to κ.
Below T = 6 K, κ scales with T^{3}. The phonon mean free path, ℓ_{ph}, can be estimated through the kinetic formula κ = \(\frac{1}{3}{C}_{{\mathrm{ph}}}{v}_{{\mathrm{ph}}}{\ell }_{{\mathrm{ph}}}\), where C_{ph} = β_{ph}T^{3} is the specific heat associated with phonons at low temperature with β_{ph} = 3.68 J K^{−4} m^{−3} (ref. ^{17}) and v_{ph} is the sound velocity v_{ph} = 2.5–4.4 km s^{−1} (ref. ^{18}). The deduced ℓ_{ph} ≃ 0.5–1mm is comparable with the sample crosssection, \(\bar{s}=\sqrt {\left(\right.}\omega t{\left.\right)}=0.5\) for S3 (where ω and t are the sample width and thickness much shorter than its length, L = 8mm). Phonons are thus in the ballistic regime. Similarly to κ, S_{xx} peaks around T = 10K. In the zero temperature limit \(\frac{{S}_{xx}}{T}\) saturates to a value of 7.8 μV K^{−2}, which is in quantitative agreement with the expected value for the diffusive response of a degenerate semiconductor in the ionizedimpurity scattering regime (see Supplementary Note 3) . The finitetemperature extra contribution to S_{xx} comes from the phonondrag effect. In summary, at zero magnetic field, InAs is a dilute metal with one electron per 10^{6} atoms. The mobility of these carriers does not evolve much with cooling. Yet, it is high enough to allow the observation of quantum oscillations. Thermal transport is dominated by phonons, which become ballistic below T = 10 K while the thermoelectric response is purely diffusive in the zero temperature regime with a modest phonon drag component at finite temperature.
Fieldinduced metal–insulator transition in InAs
Let us now discuss the evolution in field of the electrical and thermoelectrical properties of InAs, which are shown respectively on Figs. 2 and 3. At low magnetic fields, ρ_{xx}, n_{H}, and S_{xx} display quantum oscillations. The last quantum oscillation occurs at B = 4.1 T (see Supplementary Note 1). This magnetic field is defined as B_{QL} above which all the carriers are confined in the lowest Landau level (LLL). Up to B = 10 T, ρ_{xx} and S_{xx} increase while n_{H} remains constant. From B = 10 T to B = 15 T, ρ_{xx} increases by about two orders of magnitude while n_{H} drops by a factor close to 20 and S_{xx} by a factor of three. This marks the entrance in the magnetic freezeout regime, in good agreement with early measurements^{19,20}. This regime has been thoroughly studied in the lowdoped narrow gap semiconductors InSb and Hg_{1−x}Cd_{x}Te (n_{H} = 10^{−14}–10^{−15 }cm^{−3})^{21,22}. It was shown that the magnetic field induces a MIT ascribed as a magnetic field assisted MottAnderson transition in lightly doped semiconductors (see ref. ^{22} for a review). A sketch of this effect is shown in Fig. 2d). For B > B_{QL}, the inplane electronic wave extension is equal to a_{⊥} = 2ℓ_{B} with \({\ell }_{B}=\sqrt {\left(\right.}\frac{\hslash }{eB}{\left.\right)}\) and shrinks with the magnetic field. Parallel to the magnetic field the characteristic spatial extension is given by \({a}_{B,\parallel }=\frac{{a}_{B}}{\mathrm{log}\,(\gamma )}\), where \(\gamma ={(\frac{{a}_{B}}{{l}_{B}})}^{2}\)^{23,24}. Once the overlap between the wave functions of electrons is sufficiently reduced, a MIT is expected to occur at B = B_{MI}, i.e., when:
with δ = 0.3–0.4 for InSb and Hg_{1−x}Cd_{x}Te. The carrier dependences of B_{QL} and B_{MI} for these two systems are shown in Fig. 4. Using the drop of n_{H} in the zero temperature limit^{21,25} we find that B_{MI} = 10.1 ± 0.2 T (see Supplementary Note 5), which is well captured by Eq. (1) assuming a δ = 0.4 as illustrated in Fig. 4. In the kspace this transition corresponds to a transfer of electrons from the LLL to a shallow band (formed by the localized electrons) located at an energy below the LLL^{23}.
Above B = 15 T, a novel regime is identified where ρ_{xx} varies almost linearly with the magnetic field and concomitant with a saturating n_{H} ≈ 10^{15 }cm^{−3} and \(\frac{{S}_{{\mathrm{xx}}}}{T}=21\mu\)V K^{−2} at the lowest temperature. As a function of the temperature, ρ_{xx} first displays an activated behavior followed by a saturation at low temperature (see Supplementary Note 4). The deduced gap, Δ, is equal to 2 meV at B = 30 T and increases with the magnetic field up to B = 50 T (see Supplementary Note 4). As the temperature is increased, the transition shifts to higher magnetic field, becomes broader and vanishes above T = 30 K in the electrical response. Likewise, the peak in S_{xx} shifts to higher magnetic field. However, surprisingly, its amplitude increases. This striking observation is better appreciated by comparing the two colormaps of ρ_{xx} and S_{xx}, which are shown respectively in Fig. 3c, d. While ρ_{xx} is maximal at the lowest temperature, S_{xx} is maximal at around T = 8 K and that for all magnetic fields as shown in Fig. 5.a). At B = 29 T, S_{xx}(T = 8 K) is as large as 11.3 mV K^{−1}, which is about 200 times larger than the zerofield thermopower and comparable with the “colossal” thermopower observed in ultralow carrier density Ge (n_{H} < 10^{13 }cm^{−3}) where S_{xx}(T = 10 K) ≈ 10−30 mV K^{−1} (ref. ^{26}), in the strongly correlated semiconductors FeSb_{2} (n_{H} ≈ 1 × 10^{15 }cm^{−3}) where S_{xx}(T = 10K) ≈ 10−30 mV K^{−1} (refs. ^{27,28}) or in the fractional quantum Hall regime of 2DEGs where S_{xx}(T = 5 K) reaches 50 mV K^{−1} (ref. ^{12}). Let us now discuss the origin of the two intriguing properties identified above B_{MI} : the saturating magnetoresistance and the giant Seebeck response.
Discussion
The residual conductivity and carrier density at low temperature and high magnetic field contrasts with the insulating behavior of lightly doped InSb^{21}. A key difference between both systems is the length scale of the fluctuations of the impurity potential. In highly doped semiconductors, largescale fluctuations affect the density of state, in particular in its quantum limit regime^{29}. Scanning tunneling microscopy (STM) measurements have revealed spatial fluctuations of the LLL in InAs on the energy scale of Γ = 3–4 meV, which result in a broadening of the shallow band and a tail in the density of state of the LLL^{30}. Such broadening manifests itself in the electrical transport properties by a gap (Δ) value much lower than the theoretical predictions^{19} and a residual carrier concentration down to the lowest temperature well above B_{MI} as far as Γ ≈ Δ^{29}. With a residual carrier density of \({n}_{H,B \,{>}\,{B}_{{\mathrm{MI}}}}\approx 1\times 1{0}^{15}\)cm^{−3} and a resistivity ρ_{xx} ≈ 100 Ω.cm, these residual bulk electrons have a low mobility \({\mu }_{H,B \,{>}\,{B}_{{\mathrm{MI}}}}=60\) cm^{2} V^{−1} s^{−1}. Such poorly mobile carriers can be shunted by the conductance of the surface as it is the case of the magnetic freezeout regime of Hg_{1−x}Cd_{x}Te^{31,32}. Magnetotransport, magnetooptical^{33,34} and angleresolved photoemission spectroscopy measurements^{35,36} on InAs have well documented the existence of an accumulation layer of carrier density n_{S} = 1 × 10^{12} cm^{−2}(E_{F} ≈ 100 meV) of mobility μ_{H,S} ≈ 5000 cm^{2} V^{−1} s^{−1}. For a sample thickness, e, the ratio of conductance from the bulk (σ_{b}) and the surface (σ_{s}) is given by: \(\frac{{\sigma }_{b}}{{\sigma }_{s}}=\frac{{\mathrm{e}}{{\mathrm{n}}}_{H,B \,{>}\,{B}_{{\mathrm{MI}}}}{\mu }_{H,B \,{>}\,{B}_{{\mathrm{MI}}}}}{{n}_{S}{\mu }_{H,s}}\). With the numbers given above, \(\frac{{\sigma }_{b}}{{\sigma }_{s}}\approx 1\) : both contributions are of the same order of magnitude. This is supported by the amplitude of the thermopower at low temperature for B > B_{MI}. In the presence of bulk and surface contributions, S_{xx} is given by the sum of the bulk and surface thermopower (labeled S_{xx,B} and S_{xx,S}) balanced by their relative contribution to the total conductivity: \({S}_{{\mathrm{xx}}}=\frac{{\sigma }_{B}{S}_{{\mathrm{xx}},B}+{\sigma }_{S}{S}_{{\mathrm{xx}},S}}{{\sigma }_{B}+{\sigma }_{S}}\). With a Fermi temperature of the surface states (T_{F} ≈ 900 K^{34}) much larger than the Fermi temperature of the residual bulk state (\({T}_{F,B \,{>}\,{B}_{{\mathrm{MI}}}}=18\)K) S_{xx,S} ≪ S_{xx,B} with \(\frac{{S}_{xx,B}}{T}\) expected to be −46 μV K^{−2} in the diffusive regime of ionizedimpurity scattering. With σ_{B} ≈ σ_{S}, \(\frac{{S}_{{\mathrm{xx}}}}{T}\approx \frac{{S}_{{\mathrm{xx}},B}}{2T}=23\) µV K^{−2} in good agreement with the residual measured thermopower. Contributions from the surface states are further supported by the sample dependence of the lowtemperaturehighfield value of ρ_{xx} as discussed in Supplementary Note 4. Therefore, deep inside its quantum limit, the electrical transport properties of InAs reveal two types of contributions: a first from low mobility bulk electrons and a second from highly mobile electrons on the surface.
The field dependence of S_{xx} is qualitatively well captured by the Mott relation^{37} (\(\frac{{S}_{{\mathrm{xx}}}}{T}=\frac{{\pi }^{2}}{3}\frac{{k}_{B}^{2}}{e}{\frac{\partial \mathrm{ln}\,(\sigma (\epsilon ))}{\partial \epsilon }}_{\epsilon = {\epsilon }_{F}}\)): in the region where ρ_{xx} and ρ_{xy} (and thus σ_{xx}) vary the most with both the magnetic field and temperature that S_{xx} is the largest. As a function of the magnetic field, the increase happens close to B_{MI}, leading to a peak in the field dependence of S_{xx}. However, this relation fails to explain quantitatively the temperature evolution and the amplitude of the peak. As the temperature increases, the transition becomes broader and the amplitude of the peak is expected to vanish. In the activation regime σ_{xx} scales as \(\exp (\frac{\Delta }{{k}_{B}T})\) and S_{xx} as \(\frac{{k}_{B}}{e}\frac{\Delta }{T}\). At T = 8 K and B = 30 T (Δ = 2 meV), S_{xx}(T = 8 K) is at most \(3\frac{{k}_{B}}{e}=250\) µV K^{−1} fifty times smaller than the measured value. Therefore, another source of entropy has to be invoked such as the phonon bath.
At zero magnetic field the phonondrag effect is known to enhance the thermoelectrical response of lowdoped semiconductors. The phonon drag picture conceived by Herring^{14} quantifies the additional contribution to the Peltier coefficient, Π, by the thermal current carried by phonons. The Peltier coefficient, which is the ratio of heat current to charge current, is linked to the Seebeck coefficient through the Kelvin relation. According to Herring, the phonon drag contribution to the Peltier coefficient is:
Here, Λ < 1 quantifies the momentum exchange rate between phonons and electrons^{14}, τ_{ph} and τ_{e} are the phonon and electron scattering rates, m^{*} is the effective mass of the electron and v_{ph} is the sound velocity. One can see that phonon drag requires a finite Λ and is boosted by a large \(\frac{{\tau }_{{\mathrm{ph}}}}{{\tau }_{e}}\) and/or a large effective mass (like in FeSb_{2}^{28}). Using the Kelvin relation, Eq. (2) leads to:
Thus, a large Seebeck response in units of \(\frac{{k}_{B}}{e}\) is possible thanks to phonon drag. It requires τ_{ph} ≫ τ_{e} and a finite Λ. Herring showed that in intrinsic semiconductors, such as Si and Ge, the large \(\frac{{\tau }_{{\mathrm{ph}}}}{{\tau }_{e}}\) ratio provides a key to understand the large magnitude of the Seebeck response at cryogenic temperatures^{38,39}.
Let us discuss how the magnetic field squeezes τ_{e}, leaves τ_{ph} unaffected and thus boosts their ratio. In contrast to the diffusive response, S_{ph} scales with the sample dimensions, the inverse of the mobility and can exceed by far \(\frac{{k}_{B}}{e}\). A crucial test of Eq. (3) comes from the size and mobility dependence of S_{xx}. As shown in Fig. 5b, c, both S_{xx} and κ_{xx} are sizedependent and scale with the crosssection of the sample, \(\overline{s}\), as expected for phonons in the ballistic regime for samples where \(\overline{s}\ll L\) where L is the sample length. The slope of κ_{xx} vs. \(\overline{s}\) is, however, independent of the magnetic field (since it only depends on l_{ph}), while the slope of S_{xx} vs. \(\overline{s}\) increases with increasing magnetic field due to the reduction of μ_{H}. As expected from Eq. (3) and illustrated in the inset of Fig. 5a), the changes induced by the magnetic field in −S_{xx} and in \({\mu }_{H}^{1}\) are comparable. At T = 8 K, between B = 0 T and B = 29 T, −S_{xx}(T = 8 K) is amplified by a factor of 202 and μ_{H} decreases by a factor of 196. Using Eq. (3) at high magnetic field, we find that Λ ≪ 1. This is not a surprise, given the temperature dependence of ρ_{xx}, which shows that below T = 100 K, electrons are mostly scattered by ionized impurities and not by phonons.
The phonon drag picture, therefore, provides a quantitative explanation of the giant fieldinduced Seebeck effect in InAs. We note that below T = 8 K, S_{xx} peaks in magnetic field while S_{ph} is expected to be the largest at the highest magnetic field (where μ_{H} is the lowest). This implies either a shunt of S_{ph} by the relatively small thermoelectrical response of the surface states or a more elaborate bulk phonon drag picture that would be maximum close to B_{MI}, where the bulk shallow band is partially filled as it has been proposed in FeSb_{2}^{27}. Interestingly FeSb_{2} and InAs above B_{MI} share in common the same activation gap (of the order of a few meV), the same carrier density (10^{15} cm^{−3}) and the presence of bulk ingap states. However, with resistivity values two orders of magnitude larger in InAs than in FeSb_{2}, the power factor of InAs is only 10 μW K^{−2} cm^{−1} (two orders of magnitude smaller than in FeSb_{2}).
While a purely electronic mechanism has been recently proposed to give rise to an unbounded thermopower in Dirac/Weyl semimetals in their quantum limit regime^{13}, our results show that the phonon drag effect is another road to boost the diffusive response of low carrier density metals across their fieldinduced MIT. Up to now, this transition has been studied only in a limited number of cases and the thermoelectric properties of dilute metals remain vastly unexplored. As illustrated in Fig. 4, a large class of materials (ranging from wellknown doped semiconductors to new topological materials) remains to be studied, in particular at higher doping (and, therefore, at high magnetic field) where larger Λ can be attained, favoring even larger S_{ph}.
Methods
Samples and measurements description
The InAs samples were cleaved into rectangular plates (with typical dimensions [1 × 5−10 × 0.5]mm^{3}) from a wafer of nominal carrier densities of n(W_{1}) ≈ 2 × 10^{16} cm^{−3} bought from Wafer Technology Ltd (www.wafertech.co.uk). The samples were etched in a HClmethanol solution prior to any experiment. Electrical contacts were made with silver paste. The electrical and heat currents were applied along the [0 –1 1] direction and the magnetic field along the [0 0 1] direction. DCelectrical and thermoelectrical transport measurements were done in a Quantum Design PPMS using a home built stick up to 14 T, in a dilution fridge up to B = 17 T, an ^{3}He cryostat up to B = 35 T at both LNCMIGrenoble and HMFL (Nijmegen). These measurements were completed by electrical transport measurements up to B = 56 T between T = 1.4 K and T = 30 K. For further experimental details see supplemental material details on the electronic properties, the Landau levels spectrum, the field dependence of the gap, and the sample dependence in the ultraquantum limit regime of our InAs samples.
Data availability
All data supporting the findings of this study are available from the corresponding authors A.J. and B.F. upon request.
References
Nolas, G. S., Sharp, J. & Goldsmid, J. Thermoelectrics: Basic Principles and New Materials Developments, Vol. 45 (Springer Science & Business Media, 2013).
Behnia, K. Fundamentals of Thermoelectricity. (Oxford University Press, Oxford, 2015).
Behnia, K., Méasson, M.A. & Kopelevich, Y. Oscillating NernstEttingshausen effect in bismuth across the quantum limit. Phys. Rev. Lett. 98, 166602 (2007).
Zhu, Z., Yang, H., Fauqué, B., Kopelevich, Y. & Behnia, K. Nernst effect and dimensionality in the quantum limit. Nat. Phys. 6, 26–29 (2010).
Zhu, Z., Fauqué, B., Fuseya, Y. & Behnia, K. Angleresolved Landau spectrum of electrons and holes in bismuth. Phys. Rev. B 84, 115137 (2011).
Tieke, B. et al. Magnetothermoelectric properties of the degenerate semiconductor HgSe: Fe. Phys. Rev. B 54, 10565–10574 (1996).
Liang, T. et al. Evidence for massive bulk Dirac fermions in \({{\rm{Pb}}}_{1{\rm{x}}}{{\rm{Sn}}}_{{\rm{x}}}{\rm{Se}}\) from Nernst and thermopower experiments. Nat. Commun. 4, 2696 (2013).
Zhang, W. et al. Observation of a thermoelectric Hall plateau in the extreme quantum limit. Nat. Commun. 11, 1046 (2020).
Puri, S. M. & Geballe, T. H. Phonon drag in ntype InSb. Phys. Rev. 136, 1767–1774 (1964).
Fauqué, B. et al. Magnetothermoelectric properties of Bi_{2}Se_{3}. Phys. Rev. B 87, 035133 (2013).
Han, F. et al. Quantized thermoelectric Hall effect induces giant power factor in a topological semimetal. Nat. Comm. 11, 6167 (2020). https://www.nature.com/articles/s41467020198502.
Fletcher, R., Maan, J. C., Ploog, K. & Weimann, G. Thermoelectric properties of GaAs–Ga_{1−x}Al_{x}As heterojunctions at high magnetic fields. Phys. Rev. B 33, 7122–7133 (1986).
Skinner, B. & Fu, L. Large, nonsaturating thermopower in a quantizing magnetic field. Sci. Adv. 4, eaat2621 (2018).
Herring, C. Theory of the thermoelectric power of semiconductors. Phys. Rev. 96, 1163–1187 (1954).
JayGerin, J. P. Thermoelectric power of semiconductors in the extreme quantum limit. II. the "phonondrag" contribution. Phys. Rev. B 12, 1418–1431 (1975).
Rode, D. L. Electron transport in InSb, InAs, and InP. Phys. Rev. B 3, 3287–3299 (1971).
Cetas, T. C., Tilford, C. R. & Swenson, C. A. Specific heats of Cu, GaAs, GaSb, InAs, and InSb from 1 to 30K. Phys. Rev. 174, 835–844 (1968).
Le Guillou, G. & Albany, H. J. Phonon conductivity of InAs. Phys. Rev. B 5, 2301–2308 (1972).
Kaufman, L. A. & Neuringer, L. J. Magnetic freezeout and band tailing in nInAs. Phys. Rev. B 2, 1840–1846 (1970).
Kadri, A., Aulombard, R., Zitouni, K., Baj, M. & Konczewicz, L. Highmagneticfield and highhydrostaticpressure investigation of hydrogenicand resonantimpurity states in ntype indium arsenide. Phys. Rev. B 31, 8013–8023 (1985).
Shayegan, M., Goldman, V. J. & Drew, H. D. Magneticfieldinduced localization in narrowgap semiconductors Hg_{1−x}Cd_{x}Te and InSb. Phys. Rev. B 38, 5585–5602 (1988).
Aronzon, B. A. & Tsidilkovskii, I. M. Magneticfieldinduced localization of electrons in fluctuation potential wells of impurities. Phys. Status Solidi (b) 157, 17–59 (1990).
Yafet, Y., Keyes, R. & Adams, E. Hydrogen atom in a strong magnetic field. J. Phys. Chem. Solids 1, 137 – 142 (1956).
Shklovskii, B. I. & Efros, A. L. Electronic Properties of Doped Semiconductors. (SpringerVerlag, New York, 1984).
Rosenbaum, T. F., Field, S. B., Nelson, D. A. & Littlewood, P. B. Magneticfieldinduced localization transition in HgCdTe. Phys. Rev. Lett. 54, 241–244 (1985).
Inyushkin, A. V., Taldenkov, A. N., Ozhogin, V. I., Itoh, K. M. & Haller, E. E. Isotope effect on the phonondrag component of the thermoelectric power of germanium. Phys. Rev. B 68, 153203 (2003).
Bentien, A., Johnsen, S., Madsen, G. K. H., Iversen, B. B. & Steglich, F. Colossal Seebeck coefficient in strongly correlated semiconductor FeSb_{2}. Europhys. Lett. (EPL) 80, 17008 (2007).
Takahashi, H. et al. Colossal Seebeck effect enhanced by quasiballistic phonons dragging massive electrons in FeSb_{2}. Nat. Commun. 7, 12732 (2016).
Dyakonov, M., Efros, A. & Mitchell, D. Magnetic freezeout of electrons in extrinsic semiconductors. Phys. Rev. 180, 813–818 (1969).
Morgenstern, M., Wittneven, C., Dombrowski, R. & Wiesendanger, R. Spatial fluctuations of the density of states in magnetic fields observed with scanning tunneling spectroscopy. Phys. Rev. Lett. 84, 5588–5591 (2000).
Mullin, J. B. & Royle, A. Surface oxidation and anomalous electrical behaviour of cadmium mercury telluride. J. Phys. D: Appl. Phys. 17, L69–L72 (1984).
Antcliffe, G. A., Bate, R. T. & Reynolds, R. A. Oscillatory magnetoresistance from a ntype inversion layer with nonparabolic bands. The Physics of Semimetals and Narrow Gap Semiconductors: Proceedings. 499–509 (Pergamon, 1970).
Tsui, D. C. Observation of surface bound state and twodimensional energy band by electron tunneling. Phys. Rev. Lett. 24, 303–306 (1970).
Reisinger, H., Schaber, H. & Doezema, R. E. Magnetoconductance study of accumulation layers on n−InAs. Phys. Rev. B 24, 5960–5969 (1981).
Olsson, L. O. et al. Charge accumulation at InAs surfaces. Phys. Rev. Lett. 76, 3626–3629 (1996).
King, P. D. C. et al. Surface bandgap narrowing in quantized electron accumulation layers. Phys. Rev. Lett. 104, 256803 (2010).
Ziman, J. M. Principle oF the Theory of Solids (Cambridge University Press, 1972).
Geballe, T. H. & Hull, G. W. Seebeck effect in germanium. Phys. Rev. 94, 1134–1140 (1954).
Geballe, T. H. & Hull, G. W. Seebeck effect in silicon. Phys. Rev. 98, 940–947 (1955).
Oswald, J., Goldberg, B. B., Bauer, G. & Stiles, P. J. Magnetotransport studies on the metallic side of the metalinsulator transition in PbTe. Phys. Rev. B 40, 3032–3039 (1989).
Bhattacharya, A., Skinner, B., Khalsa, G. & Suslov, A. V. Spatially inhomogeneous electron state deep in the extreme quantum limit of strontium titanate. Nat. Commun. 7, 12974 (2016).
Assaf, B. A. et al. Negative longitudinal magnetoresistance from the anomalous N = 0Landau level in topological materials. Phys. Rev. Lett. 119, 106602 (2017).
Wang, Z. et al. Defects controlled hole doping and multivalley transport in SnSe single crystals. Nat. Commun. 9, 47 (2018).
Köhler, H. & Wöchner, E. The gfactor of the conduction electrons in Bi_{2}Se_{3}. Phys. Status Solidi (b) 67, 665–675 (1975).
Analytis, J. G. et al. Twodimensional surface state in the quantum limit of a topological insulator. Nat. Phys. 6, 960–964 (2010).
Rischau, C. W. Irradiationinduced doping of Bismuth Telluride Bi_{2}Te_{3} (Ecole Polytechnique, Université Paris Saclay, 2010).
Tang, F. et al. Threedimensional quantum Hall effect and metalinsulator transition in ZrTe_{5}. Nature 569, 537–541 (2019).
Xiang, Z. J. et al. Angulardependent phase factor of Shubnikovde Haas oscillations in the Dirac semimetal Cd_{3}As_{2}. Phys. Rev. Lett. 115, 226401 (2015).
Narayanan, A. et al. Linear magnetoresistance caused by mobility fluctuations in ndoped Cd_{3}As_{2}. Phys. Rev. Lett. 114, 117201 (2015).
Zhao, Y. et al. Anisotropic Fermi surface and quantum limit transport in high mobility threedimensional Dirac semimetal Cd_{3}As_{2}. Phys. Rev. X 5, 031037 (2015).
Acknowledgements
We thank A. Akrap, R. Daou, B. Skinner, and J. Tomczak for useful discussions. This work is supported by JEIPCollège de France, by the Agence Nationale de la Recherche (ANR18CE92002001, ANR19CE30001404) and by a grant attributed by the Ile de France regional council. Part of this work was performed at LNCMICNRS and HFMLRU/NWOI, members of the European Magnetic Field Laboratory (EMFL).
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A.J. and B.F. conducted the electrical, thermoelectrical, and thermal conductivity measurements up to B = 17 T. Highfield measurements have been conducted by A.J., G.S., B.F. at LNCMIGrenoble, by A.J., C.W.R., S.W. at HFML and by C.W.R., S.B., and C.P. at LNCMIToulouse. A.J. and B.F. analyzed the data. A.J., K.B., and B.F. wrote the manuscript.
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Jaoui, A., Seyfarth, G., Rischau, C.W. et al. Giant Seebeck effect across the fieldinduced metalinsulator transition of InAs. npj Quantum Mater. 5, 94 (2020). https://doi.org/10.1038/s41535020002960
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DOI: https://doi.org/10.1038/s41535020002960
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