## Introduction

The thermoelectrical properties of low carrier density metals are of fundamental and technological interest. Owing to their small Fermi temperatures (TF), their diffusive Seebeck effect ($${S}_{{\mathrm{xx}}}=\frac{{E}_{x}}{{\Delta }_{x}T}$$) can be large and, as such, used to develop high-performance thermoelectric devices1. At low temperature, their thermoelectrical response is also a fine probe of their fundamental electronic properties, in particular in the presence of a magnetic field2. As an example, the large quantum oscillations observed in the Nernst effect (Sxy = $$\frac{{E}_{y}}{| {\Delta }_{x}T| }$$) of semimetals3,4,5 or doped semiconductors6,7,8 have been used to map out their Fermi surfaces and to reveal the Dirac/Weyl nature of the electronic spectrum of Bi5, Pb1−xSnxSe7 or ZrTe58.

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 so-called quantum limit. At low temperature, this is concomitant with an increase of Sxx (and Sxy)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 TaP11 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 Sxx can be either the result of an unbounded diffusive thermoelectric power specific to nodal metals13 or to the coupling of electrons with the phonon bath. In the latter case, the so-called phonon drag, the amplitude of Sxx 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 well-known to enhance Sxx in low-doped semiconductors at zero magnetic field but also in the quantum limit regime9,15. Here, we present a study of the electrical and thermoelectrical properties of InAs, a bulk narrow-direct-gap semiconductor, beyond its quantum limit and up to a so-far 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 Sxx = 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 (ρxx), the Seebeck coefficient (Sxx) and the thermal conductivity (κ) of an n-type InAs with a Hall carrier density nH = 2.0 × 1016 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 Γ-point of the Brillouin zone with a carrier density nSdH = 1.6 × 1016 cm−3 (TF = 100 K) and mass carrier m* = 0.023m0. From room temperature down to TF, ρxx is metallic and the non-degenerate electrons are mainly scattered by phonons. Below TF, ρxx increases with decreasing temperature as electrons become more and more degenerate with a mobility limited by ionized-impurity scattering16. Below T = 20 K, ρxx is constant with a residual resistance of ρ0 = 12 mΩ.cm and a Hall mobility μH = 24,000 cm2 V−1 s−1. The electronic contribution (κel) 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 T3. The phonon mean free path, ph, can be estimated through the kinetic formula κ = $$\frac{1}{3}{C}_{{\mathrm{ph}}}{v}_{{\mathrm{ph}}}{\ell }_{{\mathrm{ph}}}$$, where Cph = βphT3 is the specific heat associated with phonons at low temperature with βph = 3.68 J K−4 m−3 (ref. 17) and vph is the sound velocity vph = 2.5–4.4 km s−1 (ref. 18). The deduced ph 0.5–1mm is comparable with the sample cross-section, $$\bar{s}=\sqrt {\left(\right.}\omega t{\left.\right)}=0.5$$ 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, Sxx 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 ionized-impurity scattering regime (see Supplementary Note 3) . The finite-temperature extra contribution to Sxx comes from the phonon-drag effect. In summary, at zero magnetic field, InAs is a dilute metal with one electron per 106 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.

### Field-induced 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, ρxx, nH, and Sxx display quantum oscillations. The last quantum oscillation occurs at B = 4.1 T (see Supplementary Note 1). This magnetic field is defined as BQL above which all the carriers are confined in the lowest Landau level (LLL). Up to B = 10 T, ρxx and Sxx increase while nH remains constant. From B = 10 T to B = 15 T, ρxx increases by about two orders of magnitude while nH drops by a factor close to 20 and Sxx by a factor of three. This marks the entrance in the magnetic freeze-out regime, in good agreement with early measurements19,20. This regime has been thoroughly studied in the low-doped narrow gap semiconductors InSb and Hg1−xCdxTe (nH = 10−14–10−15 cm−3)21,22. It was shown that the magnetic field induces a MIT ascribed as a magnetic field assisted Mott-Anderson transition in lightly doped semiconductors (see ref. 22 for a review). A sketch of this effect is shown in Fig. 2d). For B > BQL, the in-plane electronic wave extension is equal to a = 2B 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 = BMI, i.e., when:

$${n}^{1/3}{\left({a}_{B,\perp }^{2}{a}_{B,\parallel }\right)}^{\frac{1}{3}}=\delta$$
(1)

with δ = 0.3–0.4 for InSb and Hg1−xCdxTe. The carrier dependences of BQL and BMI for these two systems are shown in Fig. 4. Using the drop of nH in the zero temperature limit21,25 we find that BMI = 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 k-space 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 LLL23.

Above B = 15 T, a novel regime is identified where ρxx varies almost linearly with the magnetic field and concomitant with a saturating nH ≈ 1015 cm−3 and $$\frac{{S}_{{\mathrm{xx}}}}{T}=-21\mu$$V K−2 at the lowest temperature. As a function of the temperature, ρxx 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 Sxx shifts to higher magnetic field. However, surprisingly, its amplitude increases. This striking observation is better appreciated by comparing the two color-maps of ρxx and Sxx, which are shown respectively in Fig. 3c, d. While ρxx is maximal at the lowest temperature, Sxx is maximal at around T = 8 K and that for all magnetic fields as shown in Fig. 5.a). At B = 29 T, Sxx(T = 8 K) is as large as 11.3 mV K−1, which is about 200 times larger than the zero-field thermopower and comparable with the “colossal” thermopower observed in ultra-low carrier density Ge (nH < 1013 cm−3) where Sxx(T = 10 K) ≈ 10−30 mV K−1 (ref. 26), in the strongly correlated semiconductors FeSb2 (nH ≈ 1 × 1015 cm−3) where Sxx(T = 10K) ≈ 10−30 mV K−1 (refs. 27,28) or in the fractional quantum Hall regime of 2DEGs where Sxx(T = 5 K) reaches 50 mV K−1 (ref. 12). Let us now discuss the origin of the two intriguing properties identified above BMI : 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 InSb21. A key difference between both systems is the length scale of the fluctuations of the impurity potential. In highly doped semiconductors, large-scale fluctuations affect the density of state, in particular in its quantum limit regime29. 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 LLL30. Such broadening manifests itself in the electrical transport properties by a gap (Δ) value much lower than the theoretical predictions19 and a residual carrier concentration down to the lowest temperature well above BMI as far as Γ ≈ Δ29. With a residual carrier density of $${n}_{H,B \,{>}\,{B}_{{\mathrm{MI}}}}\approx 1\times 1{0}^{15}$$cm−3 and a resistivity ρxx ≈ 100 Ω.cm, these residual bulk electrons have a low mobility $${\mu }_{H,B \,{>}\,{B}_{{\mathrm{MI}}}}=60$$ cm2 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 freeze-out regime of Hg1−xCdxTe31,32. Magneto-transport, magneto-optical33,34 and angle-resolved photoemission spectroscopy measurements35,36 on InAs have well documented the existence of an accumulation layer of carrier density nS = 1 × 1012 cm−2(EF ≈ 100 meV) of mobility μH,S ≈ 5000 cm2 V−1 s−1. For a sample thickness, e, the ratio of conductance from the bulk (σb) and the surface (σs) 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 > BMI. In the presence of bulk and surface contributions, Sxx is given by the sum of the bulk and surface thermopower (labeled Sxx,B and Sxx,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 (TF ≈ 900 K34) much larger than the Fermi temperature of the residual bulk state ($${T}_{F,B \,{>}\,{B}_{{\mathrm{MI}}}}=18$$K) Sxx,SSxx,B with $$\frac{{S}_{xx,B}}{T}$$ expected to be −46 μV K−2 in the diffusive regime of ionized-impurity scattering. With σB ≈ σS, $$\frac{{S}_{{\mathrm{xx}}}}{T}\approx \frac{{S}_{{\mathrm{xx}},B}}{2T}=-23$$ 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 low-temperature-high-field value of ρxx 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 Sxx is qualitatively well captured by the Mott relation37 ($$\frac{{S}_{{\mathrm{xx}}}}{T}=-\frac{{\pi }^{2}}{3}\frac{{k}_{B}^{2}}{e}{\frac{\partial \mathrm{ln}\,(\sigma (\epsilon ))}{\partial \epsilon }}_{\epsilon = {\epsilon }_{F}}$$): in the region where ρxx and ρxy (and thus σxx) vary the most with both the magnetic field and temperature that Sxx is the largest. As a function of the magnetic field, the increase happens close to BMI, leading to a peak in the field dependence of Sxx. 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 σxx scales as $$\exp (-\frac{\Delta }{{k}_{B}T})$$ and Sxx as $$\frac{{k}_{B}}{e}\frac{\Delta }{T}$$. At T = 8 K and B = 30 T (Δ = 2 meV), Sxx(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 phonon-drag effect is known to enhance the thermoelectrical response of low-doped semiconductors. The phonon drag picture conceived by Herring14 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:

$${\Pi }_{{\mathrm{ph}}}=\pm \Lambda {m}^{* }{v}_{s}^{2}\frac{{\tau }_{{\mathrm{ph}}}}{{\tau }_{e}}$$
(2)

Here, Λ < 1 quantifies the momentum exchange rate between phonons and electrons14, τph and τe are the phonon and electron scattering rates, m* is the effective mass of the electron and vph 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 FeSb228). Using the Kelvin relation, Eq. (2) leads to:

$${S}_{{\mathrm{ph}}}=\pm \frac{{k}_{B}}{e}\Lambda \frac{{m}^{* }{v}_{{\mathrm{ph}}}^{2}}{{k}_{B}T}\frac{{\tau }_{{\mathrm{ph}}}}{{\tau }_{e}}=\pm \Lambda \frac{{\ell }_{{\mathrm{ph}}}{v}_{{\mathrm{ph}}}}{{\mu }_{H}T}$$
(3)

Thus, a large Seebeck response in units of $$\frac{{k}_{B}}{e}$$ is possible thanks to phonon drag. It requires τphτe 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 temperatures38,39.

Let us discuss how the magnetic field squeezes τe, leaves τph unaffected and thus boosts their ratio. In contrast to the diffusive response, Sph 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 Sxx. As shown in Fig. 5b, c, both Sxx and κxx are size-dependent and scale with the cross-section 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 κxx vs. $$\overline{s}$$ is, however, independent of the magnetic field (since it only depends on lph), while the slope of Sxx vs. $$\overline{s}$$ increases with increasing magnetic field due to the reduction of μH. As expected from Eq. (3) and illustrated in the inset of Fig. 5a), the changes induced by the magnetic field in −Sxx and in $${\mu }_{H}^{-1}$$ are comparable. At T = 8 K, between B = 0 T and B = 29 T, −Sxx(T = 8 K) is amplified by a factor of 202 and μH 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 ρxx, 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 field-induced Seebeck effect in InAs. We note that below T = 8 K, Sxx peaks in magnetic field while Sph is expected to be the largest at the highest magnetic field (where μH is the lowest). This implies either a shunt of Sph by the relatively small thermoelectrical response of the surface states or a more elaborate bulk phonon drag picture that would be maximum close to BMI, where the bulk shallow band is partially filled as it has been proposed in FeSb227. Interestingly FeSb2 and InAs above BMI share in common the same activation gap (of the order of a few meV), the same carrier density (1015 cm−3) and the presence of bulk in-gap states. However, with resistivity values two orders of magnitude larger in InAs than in FeSb2, the power factor of InAs is only 10 μW K−2 cm−1 (two orders of magnitude smaller than in FeSb2).

While a purely electronic mechanism has been recently proposed to give rise to an unbounded thermopower in Dirac/Weyl semimetals in their quantum limit regime13, our results show that the phonon drag effect is another road to boost the diffusive response of low carrier density metals across their field-induced 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 well-known 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 Sph.

## Methods

### Samples and measurements description

The InAs samples were cleaved into rectangular plates (with typical dimensions [1 × 5−10 × 0.5]mm3) from a wafer of nominal carrier densities of n(W1) ≈ 2 × 1016 cm−3 bought from Wafer Technology Ltd (www.wafertech.co.uk). The samples were etched in a HCl-methanol 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. DC-electrical 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 3He cryostat up to B = 35 T at both LNCMI-Grenoble 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 ultra-quantum limit regime of our InAs samples.