A low-temperature study of manganese-induced ferromagnetism and valence band convergence in tin telluride

SnTe crystallizes in the $Fm3¯m (Oh5, No. 225)$ structure with lattice parameter a = 6.318 Å,^1,2 well known for thermoelectric (TE) applications.^3–16 The band gap, separating the conduction band (CB) and the light valence band (LVB), is located at the L point of the Brillouin zone. Another heavy valence band (HVB), also contributing to transport, is present at the Σ point, with an energy offset ΔE ∼ 0.3–0.4 eV.^17,18 The properties of SnTe depend critically on Sn vacancies (⁠$VSn″$⁠, double-hole acceptors), which lead to ∼10²⁰cm⁻³ holes in “pristine” SnTe.¹⁹ Recently, SnTe has been identified as a topological crystalline insulator (TCI).^20,21 Its surface states (SSs) are topologically protected by the crystalline symmetry,²² in contrast to topological insulators (TIs)^23,24 where magnetic dopants break the time-reversal symmetry and open up a gap in the SSs.^25–27 Introducing magnetism in SnTe is thus of fundamental interest as it may realize phenomena such as the quantum anomalous Hall effect.^28–30 Doping SnTe with manganese seems promising, as is known in dilute magnetic semiconductors (DMSs) including II-VI,^31,32 III-V,^33,34 IV(-VI),^35,36 and V₂VI₃ compounds.³⁷ Although Mn gives rise to ferromagnetism in SnTe, earlier studies have only examined samples with either too low Mn content (≪10 at. %)³⁸ or too high hole concentrations.³⁹ More comprehensive low-temperature magnetic and transport characterizations on a wider range of Mn concentrations are desirable.

Here, we report our recent study on Sn_1−xMn_xTe (x = 0–0.12), where the Curie temperature T_C increases as a function of x. Consistent with a two-band Kane model and a Debye-Callaway thermal conductivity formula, the low-temperature transport properties evolve systematically upon Mn doping, which suggests the convergence of the L and Σ bands, favoring efficient thermoelectricity.

Polycrystalline Sn_1−xMn_xTe (x = 0–0.12) were synthesized and structurally characterized as detailed in Ref. 8. The X-ray diffraction (XRD) measurements have detected no secondary phases. The samples studied here can be treated as single phase at the XRD level, although transmission electron microscopy (TEM) studies have revealed the presence of a small amount of Mn-rich nanoinclusions for x = 0.12.⁸ This suggests that the solubility limit of Mn in SnTe is x ∼ 0.12. Rectangular bar-shaped samples (∼1.5 × 2.5 × 5.5 mm³) were cut out at room temperature using a diamond saw. Magnetic and transport measurements were performed over the temperature range of 1.8–300 K. Magnetic studies were conducted using a superconducting quantum interference device (SQUID) magnetometer equipped with a 5.5 T magnet in a Quantum Design Magnetic Property Measurement System (MPMS). The electrical resistivity ρ, Seebeck coefficient S, and thermal conductivity κ were determined using a longitudinal steady-state technique.⁴⁰ Hall effect and magnetoresistance (MR) measurements were performed using a Linear Research ac bridge with 16 Hz excitation. The uncertainties of ρ, S, and κ are ±3%, ±2%, and ±5%, respectively.

The temperature dependent susceptibilities χ_m (=M/H per mole) in an applied magnetic field H = 40 Oe (magnetization M depends linearly on H up to ∼50 Oe) are plotted in Fig. 1. With x > 0.03, the system becomes ferromagnetic, manifesting a significant increase of χ_m below ∼20 K. As shown in the upper inset of Fig. 1, above T_C, χ_m for x ≥ 0.03 follows the Curie-Weiss law

where C is the Curie constant, T is the absolute temperature, θ_p is the paramagnetic Curie temperature, and χ₀ is a lattice term. The fitting parameters are summarized in Table I, together with the effective moment per Mn atom, p_eff, deduced from $C=NApeff2μB2/3kBx$⁠, where N_A is the Avogadro's number, μ_B is the Bohr magneton, and k_B is the Boltzmann constant. As exemplified in the lower inset of Fig. 1 for x = 0.12, T_C can be determined via the Arrott plot (μ² vs. Hμ⁻¹),⁴¹ in reasonable agreement with θ_p. Future studies are needed to clarify the contribution to the magnetic response from the nanoinclusions at the grain boundaries of x = 0.12. Furthermore, doping beyond the bulk solubility limit x ∼ 0.12, using non-equilibrium technique such as molecular beam epitaxy (MBE), may achieve even higher T_C, considering T_C of V-doped Sb₂Te₃ was successfully improved from 22 K (bulk Sb_1.97V_0.03Te₃) to 177 K (Sb_1.65V_0.35Te₃ thin film).⁴² 

As shown in Fig. 2(a), M of SnTe displays linear field dependence at 1.8 K. With increasing x, hysteresis gradually develops and the high-field saturated M (up to 10 kOe) has quickly reached values exceeding 2000 emu mol⁻¹ for x = 0.12. Measurements at low fields reveal small coercivity fields H_C ∼ 10 Oe, indicating the softness of these ferromagnetic semiconductors.

M contributes to the Hall resistivity ρ_yx via

where R_H is the ordinary Hall coefficient, B (=μ₀H) is the magnetic induction, and R_s is the anomalous Hall coefficient. The second term disappears well above T_C, restoring linear field dependence for ρ_yx. The presence of R_s revealed in the hysteretic behavior of ρ_yx vs. B is one of the most reliable proofs of ferromagnetism. As shown in Fig. 2(b), a distinctly hysteretic character of ρ_yx is apparent with x > 0.03 at 5.0 K. The very low coercivity [lower inset in Fig. 2(b)] corroborates the M vs. H hysteresis loops in the inset of Fig. 2(a). Considering that M is already well saturated above H ∼ 1 kOe and possesses little field dependence at high fields, the decreasing slope of ρ_yx in its high-field linear region is mainly due to the decreasing magnitude of R_H as x increases. The R_H can be extracted from the linear field dependence of ρ_yx above T_C and is displayed in Fig. 3(a). Furthermore, the alignment of spins in Mn-doped samples (e.g., x = 0.12) in the presence of B leads to a negative MR, in contrast to the positive semi-classical parabolic MR of SnTe [upper inset in Fig. 2(b)].

The temperature dependent electronic properties of Sn_1−xMn_xTe are illustrated in Fig. 3, with 300 K values collected in Table I. Due to the high apparent hole concentration p_H (≡1/eR_H, e being the elementary charge) ∼1.9× 10²⁰cm⁻³ (10 K), ρ of SnTe [Fig. 3(b)] is metallic. While S of SnTe exhibits a nearly linear temperature dependence close to ambient temperatures (dominated by diffusion processes), at low temperatures it develops a broad peak at ∼50 K which originates from strong electron-phonon coupling referred to as the phonon-drag effect.⁴³ Acoustic phonons responsible for this effect also produce a dielectric maximum observed on κ in Fig. 4(a). The phonon drag is substantially washed out with increasing Mn, which strongly scatters acoustic phonons.

While the carrier density measured by R_H, assuming a single parabolic band, clearly increases with increasing x, the Hall mobility μ_H (≡R_H/ρ) rapidly diminishes [Fig. 3(d)] and is responsible for the rising ρ. Meanwhile, S gradually improves with increasing x [Fig. 3(c)]. This might seem to be a consequence of lowered Fermi level E_F in a single band with the density of states D(ε), where ρ and S are proportional to D(ε)⁻¹ and ∂lnD(ε)/∂ε, respectively.^44,45 However, such a naïve scenario should be accompanied by a decrease in p_H, contradicting its increase as the R_H data clearly suggest.

Considering both LVB and HVB [inset of Fig. 3(d)], the overall two-band^46–48 electronic properties (R_H, S, and the Lorenz number L) are combinations of single-band components

respectively. The mobility ratio b (≡μ_L/μ_H) is taken to be 8 for SnTe according to Ref. 18. In a single Kane band with nonparabolicity parameter β (≡k_BT/E_g), the number concentration of holes p_N is given by

where $md∗$ is the density-of-states effective mass and ℏ is the reduced Planck constant. Assuming acoustic phonon dominates,^49–51 the generalized Fermi integrals are

where f₀ = (e^ξ−η+ 1)⁻¹ is the Fermi-Dirac distribution, ξ = ε/k_BT, and η = μ_c/k_BT, with the energy ε and chemical potential μ_c measured from the band edge and ΔE = k_BT (η_L − η_H). The single Kane-band electronic properties are

respectively.

Assuming a Kane-like LVB and a parabolic HVB (β_H = 0),^3,4,47,48 the two-band Kane model [solid lines in Figs. 3(a), 3(c)] is used to characterize the temperature and composition dependence of R_H and S, with a temperature dependent E_g^1,52 and constant effective masses (in the unit of rest electron mass m_e) $md,L∗$ = 0.1 and $md,H∗$ = 3, respectively. The fitting parameters are displayed in Table I. A universal shrinking trend (insensitive of E_g and $md∗$⁠) of ΔE vs. x (from ∼0.40 eV for SnTe to a somewhat negative value for x = 0.12) has been observed, indicating that Mn gradually changes the band structure and brings into energy convergence the two valence bands–consistent with recent theories.⁸ An ability to achieve band convergence via chemical doping is beneficial for further improving the TE performance of SnTe,^5,53,54 as the temperature-induced convergence in PbTe [ΔE ∼ 0.17 eV (Ref. 55)] is much less feasible in SnTe due to its considerably larger ΔE. As illustrated in Fig. 3(c), contributions to S from the LVB (S_L) and HVB (S_H) are compared with the total S for x = 0.07. The data suggest that holes are transferred from the LVB to HVB due to the band convergence. Indeed, the less mobile holes in the HVB become more dominant in the electrical transport, leading to an enhanced S and reduced μ_H.

The κ of Sn_1−xMn_xTe decreases with increasing x essentially over the whole measured temperature range, as shown in Fig. 4(a). The peak in κ of SnTe is drastically suppressed due to strong phonon scattering by Mn. The lattice thermal conductivity κ_L, estimated by subtracting the electronic thermal conductivity κ_e (=LT/ρ, according to the Wiedemann-Franz law) from κ, has been examined using the Debye-Callaway model⁴⁰ 

where $x=ℏω/kBT$⁠, ω is the phonon frequency, v (∼2000 m s⁻¹) is the averaged sound velocity,^56,θ_D (∼177 K) is the Debye temperature,^57,$ℓ$ is the mean grain size, A represents the Rayleigh point defect scattering, and U denotes the Umklapp processes.⁵⁸ As shown in Fig. 4(b), the phonon conductivity model describes reasonably well the temperature dependence of κ_L. However, future effort is needed to improve the model presented here, in order to consider effects from nanoinclusions.⁵⁹ The fitting parameters are summarized in Table I, revealing a clear trend of enhanced point defect scattering (A) with increasing Mn. The fitted $ℓ$ is approximately on par with the values (5–10 μm) typically observed in our SnTe-based polycrystalline samples.⁸ 

The synergistically tuned electrical and thermal transport properties give rise to an enhanced TE figure of merit ZT [≡S²T/ρκ, inset of Fig. 4(b)]. At room temperature, the ZT for x = 0.12 (∼0.1), two orders of magnitude higher than that of SnTe, holds promise for TE performance. Indeed, high ZT values ∼1.3 (900 K) for Sn_1−xMn_xTe have been recently reported.^8–11 

In summary, magnetic and transport studies have been performed on Sn_1−xMn_xTe, with x approaching the bulk solubility limit ∼0.12. T_C of ∼14 K has been achieved for x = 0.12 in the ferromagnetically doped SnTe, as confirmed via the hysteresis in both magnetization and anomalous Hall effect. The study holds promise for further increase of T_C using MBE techniques which should enhance Mn incorporation. Ferromagnetic SnTe with high T_C may be of interest in future TCI-based (opto-)electronic devices.⁶⁰ Mn doping also favorably modifies the electronic structure via the L and Σ valence band convergence, and synergistically lowers κ by means of enhanced point defect scattering. This leads to an improved TE performance in comparison to undoped SnTe.

The work was supported by the Revolutionary Materials for Solid State Energy Conversion, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0001054. A part of transport property measurements was performed by H.C. and Q.L. at Brookhaven National Laboratory, which was supported by the U.S. Department of Energy, Office of Basic Energy Science, Materials Sciences and Engineering Division, under Contract No. DE-SC00112704.