Electron transport and scattering mechanisms in ferromagnetic monolayer Fe₃GeTe₂

Abstract

We study intrinsic charge-carrier scattering mechanisms and determine their contribution to the transport properties of the two-dimensional ferromagnet Fe₃GeTe₂. We use state-of-the-art first-principles calculations combined with the model approaches to elucidate the role of the electron-phonon and electron-magnon interactions in the electronic transport. Our findings show that the charge carrier scattering in Fe₃GeTe₂ is dominated by the electron-phonon interaction, while the role of magnetic excitations is marginal. At the same time, the magnetic ordering is shown to effect essentially on the electron-phonon coupling and its temperature dependence. This leads to a sublinear temperature dependence of the electrical resistivity near the Curie temperature, which is in line with experimental observations. The room temperature resistivity is estimated to be ~ 35 μΩ ⋅ cm which may be considered as a lower intrinsic limit for monolayer Fe₃GeTe₂.

Introduction

The interest to two-dimensional (2D) systems is driven by the continuous progress in the development of novel technologies involving miniaturization of electronic devices and low-energy consumption. A special place in this search is devoted to 2D magnetic materials, which became the subject of active studies after the exfoliation of van der Waaals magnets such as CrI₃¹, Cr₂Ge₂Te₆², Fe₃GeTe₂ (FGT)^3,4,5,6, etc. Magnetic properties of these systems demonstrate high tunability (e.g. by external electrical field⁷ and environmental screening⁸) promising for technological application, as well as serve as an excellent platform for probing the magnetic interactions in low dimensions.

Most of the known 2D magnets are insulating or semiconducting, which limits our understanding of their transport properties and the underlying physical mechanisms. In contrast, FGT has a number of special characteristics since it combines a metallic behavior, needed for the realization of controllable transport^9,10, with the ferromagnetic ground state and comparably high Curie temperature T_C ≃ 220 K^3,4,11, surviving down to the monolayer limit⁵. One of the most remarkable properties of FGT is the absence of inversion symmetry, allowing for the formation of topologically nontrivial magnetic textures^12,13. A hexagonal lattice of skyrmions is recently observed in this system^14,15,16, which might be important for spintronics applications. While a considerable attention is paid to skyrmions, the electron transport and scattering mechanisms in FGT remain unclear from the microscopic point of view.

In comparison to nonmagnetic 2D systems, the charge carrier scattering in conductive magnets is not limited to the impurities or phonons, but may include scattering by spin fluctuations^17,18 (Fig. 1b), providing essential contribution to the transport characteristics and/or give rise to qualitatively new effects (e.g., Kondo effect)^19,20. Moreover, temperature dependence of the spin-polarized electronic structure in magnetically ordered systems might play a role for the conventional sources of scattering (e.g., phonons). The first-principles theory of electron-phonon interactions is well established²¹, and has been routinely applied to study transport properties of nonmagnetic 3D and 2D materials^{22,23,24,25,26}. Strictly speaking, a theoretical description of the electron-phonon scattering in 2D is more involved due to the presence of flexural phonon modes²⁷, allowing for multiple phonon scattering^27,28,29, yet the single-phonon formulation turns out to be sufficient even for a quantitative description of the charge carrier transport in most of the cases³⁰. Generalization of the first-principles scattering theory to magnetic materials is not straightforward for at least two reasons: (i) Unequal and temperature-dependent contribution of the majority and minority electronic states; (ii) The presence of additional scattering channels such as collective spin excitations (magnons) and spin inhomogeneities. Despite a notable progress in this direction being made in recent years, the proposed theories for the first-principles description of electron-magnon interactions ^17,31 are limited to zero temperature, i.e. not readily applicable to study transport characteristics. Alternatively, there are well-established theories based on the model description of magnetism^32,33,34, which are sufficient to capture the corresponding effects qualitatively and, in many cases, quantitatively.

Fig. 1: Schematic crystal structure and main electron scattering mechanisms in monolayer Fe₃GeTe₂.

a Crystal structure of Fe₃GeTe₂ with side (left) and top (right) views. Fe_I and Fe_II denote two inequivalent iron atoms. b A sketch showing scattering of an electron on phonons and magnons.

In this work, we perform a systematic study of the charge carrier transport in ferromagnetic monolayer Fe₃GeTe₂ using first-principles calculations combined with magnetic models. We analyze electron-phonon and electron-magnon scattering, and estimate their contribution to the temperature-dependent scattering rate. The main contribution of the electron-magnon scattering to the resistivity is observed around the Curie temperature, resulting in a marginal resistivity enhancement. On the other hand, interpolation of the electron-phonon scattering between the ferromagnetic and paramagnetic phases leads to a pronounced modification of the resistivity with sublinear temperature dependence close the Curie temperature, which is in line with the experimental observations^4,9,11,35,36.

Results

Electronic structure

Figure 2 shows the electronic structure and Fermi surfaces calculated for monolayer Fe₃GeTe₂. We explicitly consider electronic states in the ferromagnetic (FM) and nonmagnetic phases. The bands in the vicinity of the Fermi energy are predominantly formed by the Fe (3d) states hybridized with Te (5p). The spin-resolved electronic structure in the FM phase displays multiple energy bands crossing the Fermi energy, resulting in the formation of several isolated pockets in the Fermi surface, being in agreement with the experimental data^9,19. Near the Γ point, one can observe a hexagonal-like shaped pocket of the spin-down states, which is to contribute to the nesting at wave vectors ${{{\bf{q}}}}={{{\bf{k}}}}-{{{{\bf{k}}}}}^{{\prime} }$ away from the zone center. Around the K points, one can see circular pockets allowing for transitions near the zone center i.e. at small q. The spin-up (majority) Fermi surface is more complicated and it has more possibilities for transition at different q. In both cases, the momentum transfer processes are expected to occur predominantly near the Γ point with discrete regions of q points around the nesting wave vectors. The electronic structure from the nonmagnetic calculations is considerably different. The Fermi surface is composed of circular pockets around the Γ point, allowing the transitions in a broad range of q vectors. We note that a moderate variation of the Fermi level (i.e. the doping effect) can significantly change the observed picture, which will influence the electron-phonon interaction as we will show below. It is also worth noting that the electronic states in monolayer Fe₃GeTe₂ are not strongly affected by the electron-phonon interactions. The spectral functions shown in the top of Fig. 2 do not show any significant renormalization in the vicinity of the Fermi energy in comparison to other systems²⁴. At the same time, the inclusion of the electron-phonon coupling induces a finite linewidth, leading to a finite lifetime of charge carriers. The DOS calculated for the FM and nonmagnetic states, as well as in the disordered local moment approximation are presented in Supplementary Methods 2.

Fig. 2: Spin-dependent electron spectral functions and Fermi contour maps for the ferromagnetic and nonmagnetic phases of monolayer Fe₃GeTe₂.

a Spectral functions ${A}_{{{{\bf{k}}}}}^{\sigma }(\omega ,T)=-1/{{{\rm{\pi }}}}\,{{{\rm{Im}}}}[{G}_{{{{\bf{k}}}}}^{\sigma }(\omega ,T)]$ calculated in the presence of the electron-phonon interactions for T = 100 K for the states near the Fermi level. Original DFT band structure is shown by the blue solid line. Zero energy corresponds to the Fermi energy. b The corresponding Fermi contour maps.

Phonon dispersion

The phonon dispersion and the corresponding spin-resolved linewidths are shown in Fig. 3 for the ferromagnetic and nonmagnetic calculations. The obtained dispersion of acoustic phonons is typical for 2D systems, demonstrating two linearly-dispersing branches and one flexural out-of-plane mode with a quadratic dispersion around the Γ point^24,26. The optical phonon modes appear at energies from ~ 8 meV to ~ 45 meV. Interestingly, one can see a number of nearly flat branches, the most prominent of which appear near 15, 21, and 32 meV. The phonon modes with energies <15 meV correspond predominantly to the vibration of heavy Te atoms, hybridized with Fe and Ge vibration states. The two highest energy modes represent a hybridized vibration of Fe and Ge atoms, while the rest phonon modes predominantly correspond to the Fe atoms vibration. Overall, the calculated phonon spectra are in good agreement with previously reported data³⁷. The phonon linewidths are somewhat different for the magnetic and nonmagnetic spectra. In particular, for the spin majority states in the ferromagnetic phase, most of the line broadening takes place near the Γ point around 15 meV, i.e. within the range of thermal excitations. In the nonmagnentic case, most of the line broadening is observed above 25 meV, thus their excitation is less likely. All this suggests that thermal transport in different magnetic states of FGT must be different, which might be interesting for further studies.

Fig. 3: Phonon dispersion curves and momentum-resolved electron-phonon coupling calculated for the ferromagnetic and nonmagnetic phases of monolayer Fe₃GeTe₂.

a Phonon dispersion curves shown with the spin-dependent linewidth that is proportional to ${{{\rm{Im}}}}[{{{\Pi }}}_{{{{\bf{q\nu }}}}}^{\sigma }]={{{\rm{\pi }}}}{N}_{{{{\rm{F}}}}}^{\sigma }{\lambda }_{{{{\bf{q\nu }}}}}^{\sigma }{\omega }_{{{{\bf{q\nu }}}}}^{2}$. b Momentum-resolved electron-phonon coupling constant ${\lambda }_{{{{\bf{q}}}}}^{\sigma }={\sum }_{\nu }{\lambda }_{{{{\bf{q\nu }}}}}^{\sigma }$.

Electron-phonon coupling and its interpolation

We now turn to the electron-phonon interaction in monolayer Fe₃GeTe₂. As one can see from Fig. 3b, the q-resolved electron-phonon interaction constant λ_q is essentially different for spin-up and spin-down channels, which is expected from the spin-resolved linewidths shown above. For the spin-down states, the dominant electron-phonon coupling originates from the interaction with long-wavelength phonons, which results in a highly localized maximum of λ_q at the Γ point. In the spin-up case, the contributions with q > 0 play a moderate role, although the maximum is still observed around the Γ point. In the nonmagnetic case λ_q is distributed practically uniformly, which is in line with the electronic structure and the Fermi contours discussed above.

The integrated electron-phonon coupling constant ${\lambda }^{\sigma }={\sum }_{{{{\bf{q}}}}\nu }{\lambda }_{{{{\bf{q}}}}\nu }^{\sigma }$ for spin up and spin down states are found to be 0.50 and 0.26, respectively, i.e. λ^↑/λ^↓ ~ 2. In the nonmagnetic case, we find λ^nonmg ≃ 0.45. The resulting constant λ^σ < 1, which can be qualified as a moderate electron-phonon interaction³⁸. Given that the electron-phonon interaction constant is different for different magnetic states, it is important to take its temperature dependence into account when calculating the transport properties.

In what follows, we mimic the paramagnetic state of the system by a nonmagnetic solution obtained from the non-spin-polarized DFT calculations. Strictly speaking, this assumption is quite strong as it ignores the presence of residual disordered local magnetic moments at T > T_C, which may affect the electronic structure and related properties, especially in itinerant magnets. The most appropriate approach to deal with the paramagnetic state in first-principles calculations is to use the method of disordered local moments (DLM)^{39,40,41,42,43}. This method requires considering large supercells in the DFT calculations, making DFPT calculations for paramagnetic states prohibitively expensive. However, our DLM calculations of the density of states (DOS) for monolayer Fe₃GeTe₂ (see Supplementary Methods 2) demonstrate that the DLM DOS near the Fermi energy turns out to be comparable to the nonmagnetic DOS. This behavior can be attributed to quenching of the local magnetic moment on the Fe_II atoms (see Fig. 1a) in most of the disordered configurations such that ${\langle {S}_{z}^{2}\rangle }_{{{{\rm{II}}}}}=0$ in the DLM state. Unlike the ground-state FM configuration, the Fe_II-d states are not expected to be split in the paramagnetic phase, providing a sizeable contribution at the Fermi energy, as in the nonmagnetic solution.

Based on the argumentation given above, at temperatures T > T_C, we assume that the results converge to the nonmagnetic case. We can then approximate the temperature-dependent electron-phonon coupling constant using the interpolation formula:

$${\lambda }^{\sigma }(T)={\lambda }_{{{{\rm{nonmg}}}}}+\left[{\lambda }^{\sigma }-{\lambda }_{{{{\rm{nonmg}}}}}\right]\frac{\left\langle {S}_{z}\right\rangle }{S}.$$

(1)

At T = 0 K, the average spin $\left\langle {S}_{z}\right\rangle$ equals to the nominal spin S = 1, yielding λ^σ(0) = λ^σ. On the other hand, above Curie temperature, we have λ^σ(T > T_C) = λ_nonmg. The magnetization is determined self-consistently as described in Methods. The resulting magnetization curve is shown in Fig. 4, from which the Curie temperature T_C = 317 K can be determined. The obtained value overestimates the experimental temperature T_C ~ 200 K^4,5,11,44 likely due to overestimation of the exchange interactions. This could be attributed to a limited applicability of the localized spin models, which neglect coupling to the electronic subsystem. In addition, renormalization of the electron spectrum due to correlation effects⁹ may also contribute to the reduction of the exchange interactions. Also, the discrepancy between the theoretical and experimental T_C might be related, e.g., to contamination of the experimental samples, as well as to the effect of substrate, which is absent within our consideration. Nevertheless, the obtained magnetization curve $\left\langle {S}_{z}\right\rangle$ can be used to interpolate the results between the magnetic and nonmagnetic solutions keeping in mind that some ambiguity in the estimated T_C exists. In Fig. 4, we plot the result of the electron-phonon coupling constant interpolation using Eq. (1). With the increase of temperature the constants λ^σ for both spin channels smoothly change their values, converging into the nonmagnetic λ_nonmg = 0.45 result at T_C.

Fig. 4

Spin-resolved electron-phonon coupling constant λ^σ (blue) and magnetization $\left\langle {S}_{z}\right\rangle$ (red) calculated for monolayer Fe₃GeTe₂ as a function of temperature using the interpolation scheme discussed in the main text.

Temperature-dependent scattering rate

In Fig. 5a, we show averaged phonon-mediated scattering rate $\left\langle {\tau }_{\sigma }^{-1}\right\rangle =\frac{1}{{N}_{{{{\rm{F}}}}}^{\sigma }}{\sum }_{n{{{\bf{k}}}}}{\tau }_{n{{{\bf{k}}}}\sigma }^{-1}\delta ({\varepsilon }_{n{{{\bf{k}}}}\sigma })$ as a function of temperature. Similarly to the coupling constant, the scattering rate calculated for the nonmagnetic phase has the value in between the spin-up and spin-down cases. At T = 300 K, we obtain $\langle {\tau }_{\uparrow }^{-1}\rangle$ = 117 ps⁻¹, $\langle {\tau }_{\downarrow }^{-1}\rangle$ = 62 ps⁻¹, and $\langle {\tau }_{{{{\rm{nonmg}}}}}^{-1}\rangle$ = 91 ps⁻¹. In the relevant temperature range, all these three curves demonstrate a linear-in-temperature behavior. Indeed, at sufficiently high temperatures phonons can be considered classically with the occupation numbers b_qν ≃ k_BT/ℏω_qν, ensuring a linear dependence of the electron linewidth ${{{\rm{Im}}}}{{{\Sigma }}}_{{{{\bf{q}}}}\nu }^{\sigma }(T)$ as well as the scattering rate²⁶. However, these results do not take temperature dependence of the electronic structure into account, which is important for magnetic systems. For this purpose, we use the following interpolation between the magnetic and nonmagnetic scattering rates

$$\left\langle {\tau }_{\sigma }^{-1}(T)\right\rangle =\left\langle {\tau }_{{{{\rm{nonmg}}}}}^{-1}\right\rangle +\left[\left\langle {\tau }_{\sigma }^{-1}\right\rangle -\left\langle {\tau }_{{{{\rm{nonmg}}}}}^{-1}\right\rangle \right]\frac{\left\langle {S}_{z}\right\rangle }{S},$$

(2)

which allows us to make a connection between the scattering rates above and below T_C. The interpolation changes the linear dependence of $\langle {\tau }^{-1}\rangle$ at T > 100 K, inducing a convergence of $\langle {\tau }_{\uparrow }^{-1}\rangle$ and $\langle {\tau }_{\downarrow }^{-1}\rangle$ to the nonmagnetic solution $\langle {\tau }_{{{{\rm{nonmg}}}}}^{-1}\rangle$ at T = T_C (Fig. 5a). Remarkably, the averaged spin-up and spin-down scattering rates are very close the nonmagnetic scattering rate over the whole range of temperatures.

Fig. 5: Temperature-dependent electron-phonon and electron-magnon contributions to the scattering rate in monolayer Fe₃GeTe₂.

a Averaged phonon-mediated scattering rate calculated for different spin channels in the ferromagnetic phase of monolayer Fe₃GeTe₂ (dashed lines), and for the nonmagnetic phase (green line). Solid red and blue lines correspond to the interpolation between the magnetic and nonmagnetic phases. b Averaged magnon-mediated scattering rate calculated for different spin channels.

The magnon contribution to the scattering rate is shown in Fig. 5b. The electron-magnon coupling constant is estimated to be I ⋅ N_F ≃ 0.01, i.e. an order of magnitude smaller compared to the electron-phonon coupling. As a consequence, the calculated ${\left\langle {\tau }^{-1}\right\rangle }_{{{{\rm{mag}}}}}$ is significantly smaller than ${\left\langle {\tau }^{-1}\right\rangle }_{{{{\rm{ph}}}}}$ (Fig. 5b). Overall, the magnon-mediated scattering rate demonstrates an exponential increase with temperature up to T_C, which can be explained by the increase of the magnon population. Nevertheless, even close to T_C, the contribution of magnons to the scattering rate is marginal. At T > T_C the long-range ferromagnetic order disappears, i.e. $\left\langle {S}_{z}\right\rangle =0$, and Eq. (9) is seemingly inapplicable. However, the contribution of spin excitations does not disappear in the paramagnetic phase due to the short-range spin fluctuations⁴⁵. Indeed, at sufficiently large T, we have b_qν ≃ k_BT/ℏω_qν with ω_qν ~ 〈S_z〉, leading to elimination of the magnetization in Eq. (9), which ensures nonzero scattering rate by static spin fluctuations. Nevertheless, as this contribution cannot be significantly larger than the contribution from magnons in the ferromagnetic phase, we ignore it from the explicit consideration.

Spin-resolved transport

We now calculate electric resistivity ρ = 1/σ as a function of temperature by means of the semi-classical Boltzmann theory (Eq. (4) in Methods). To this end, we use the Matthiessen’s rule⁴⁶, $\left\langle {\tau }^{-1}\right\rangle$ = $\langle {\tau }_{{{{\rm{ph}}}}}^{-1}\rangle$ + $\langle {\tau }_{{{{\rm{mag}}}}}^{-1}\rangle$. Fully spin-polarized (sp) resistivity (${\rho }_{{{{\rm{sp}}}}}^{-1}={\rho }_{\uparrow }^{-1}+{\rho }_{\downarrow }^{-1}$) is represented in Fig. 6. Due to the magnon contribution, ρ_sp(T) demonstrates a jump around T_C, which is attributed to the behavior of $\langle {\tau }_{{{{\rm{mag}}}}}^{-1}\rangle$ shown in Fig. 5b.

Fig. 6

Electrical resistivity shown as a function of temperature in monolayer Fe₃GeTe₂ calculated as: (i) ${\rho }_{{{{\rm{sp}}}}}^{-1}={\rho }_{\uparrow }^{-1}+{\rho }_{\downarrow }^{-1}$, i.e. from fully spin-polarized calculations without interpolation (low-T limit); (ii) ρ_nonmg i.e. nonmagnetic calculations (T > T_C); (iii) ${\rho }^{-1}(T)={\rho }_{\uparrow }^{-1}(T)+{\rho }_{\downarrow }^{-1}(T)$ (interpolated via Eq.(3)).

The nonmagnetic resistivity ρ_nonmg demonstrates a less steep behavior compared to the resistivity in the ferromagnetic phase. This can be explained by the two factors: (i) temperature dependence of the scattering rate (Fig. 5); and (ii) different carrier velocities: ${\overline{v}}_{\uparrow ,\downarrow }\simeq$ 2 × 10⁵ m ⋅ s⁻¹, while ${\overline{v}}_{{{{\rm{nonmg}}}}}$ = 1.3 × 10⁵ m ⋅ s⁻¹. The interpolation of $\overline{v}\cdot {N}_{{{{\rm{F}}}}}$ between the magnetic and nonmagnetic phases using the procedure described above, leads to the following conductivity:

$${\sigma }_{\sigma }(T)\simeq \frac{{e}^{2}}{{{\Omega }}}\left\langle {\tau }_{\sigma }(T)\right\rangle \left\langle {\overline{v}}_{\sigma }\cdot {N}_{{{{\rm{F}}}}}^{\sigma }(T)\right\rangle ,$$

(3)

which is shown as the green line in Fig. 6. One can see that at T > 100 K the linear resistivity behavior is modified, lowering the resistivity. At T = T_C the resistivity drops abruptly to the nonmagnetic value. Such a behavior is likely unphysical and could be attributed to our approximation of the paramagnetic phase by a nonmagnetic one. Moreover, the transition region is to be smoothed by taking into account scattering by spin inhomogeneities above the Curie temperature as well as by the electron-impurity scattering. Nevertheless, our interpolated results allow us to qualitatively explain the temperature dependence of the resistivity near transition region observed in recent experiments^4,9,11,35,36.

At T = 300 K, the interpolated resistivity ρ(T) is estimated to be 35 μΩ ⋅ cm. The available experimental estimates for bulk FGT^4,9,11,35,36 vary from 150 to 200 μΩ ⋅ cm, which is 4–5 times higher than the calculated values. This disagreement can be ascribed to the presence of other scattering channels in the experimental samples (e.g., impurities), which also leads to nonzero ρ at T ~ 0 K, as well as to the dimensionality effects. Our calculations provide the lowest boundary for the intrinsic resistivity in monolayer Fe₃GeTe₂. At the same time, this value is an order of magnitude larger than resistivities of noble metals such as ρ_Cu ~ 1.5 μΩ ⋅ cm and ρ_Au ~ 2.0 μΩ ⋅ cm⁴⁷. The relatively high resistivity of monolayer FGT might limit its prospects for electronic applications.

Discussion

In summary, we have systematically studied the charge-carrier scattering mechanisms in ferromagnetic monolayer Fe₃GeTe₂. We show that the phonon-mediated scattering of charge carriers plays a dominant role. The effect of magnons is also present, but can be considered as negligible in the relevant temperature region. At the same time, the magnetism-induced splitting of the energy bands modifies the electron-phonon coupling and induces a nontrivial contribution to its temperature dependence. This results in a sublinear temperature dependence of the electrical resistivity near the ferromagnetic-paramagnetic phase transition, observed experimentally. Also, we demonstrate that the charge doping (see Supplementary Methods 3) does not lead to any pronounced changes in the transport properties of Fe₃GeTe₂. Our calculations provide a lower estimate for the room temperature resistivity (35 μΩ ⋅ cm) in monolayer FGT. The resulting value is an order of magnitude larger compared to typical metals, limiting potential applicability of monolayer FGT in electronics. For multilayer FGT, drastic changes of transport properties are not expected due to comparably weak van der Waals interaction between the layers. Nevertheless, additional interlayer scattering may occur in multilayer samples, which would shorten the scattering rate. Therefore, in the absence of disorder, transport properties of monolayer FGT are expected to be superior to the multilayer samples. On the other hand, experimental single-layer samples are usually more contaminated compared to their multilayer counterparts, which might have an opposite effect on the layer-dependence of the transport properties.

The model approach presented in this paper to study transport properties of magnetic conductors can be applied to other 2D magnetic materials. Particularly, we expect a non-trivial role of the electron-magnon interactions in systems with weak electron-phonon coupling.

Methods

Technical details

Electronic structure calculations and structural optimization were performed within the plane-wave based Quantum Espresso (qe)^48,49 package utilizing Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional⁵⁰ and 2.0.1 version scalar-relativistic norm-conserving pseudopotentials, generated using “atomic" code by A. Dal Corso v.5.0.99 svn rev. 10869. In these calculations, we use 80 Ry as the plane-wave energy cutoff, a (16 × 16 × 1) k-mesh for the Brillouin zone integration, and 10⁻⁸ eV for the energy convergence criteria. The experimental crystal structure of bulk Fe₃GeTe₂ was used³, where a vacuum space 20 Å between monolayer replicas in the vertical z direction was introduced to avoid spurious interactions between the periodic supercell images. The effective volume ${{\Omega }}=d\cdot {{{\mathcal{S}}}}$ with layer thickness d = 8.165 Å of 2D unit cell area ${{{\mathcal{S}}}}$ = 13.95 Ų was used in the calculations. Positions of atoms were allowed to relax until all the residual force components of each atom were less than 10⁻⁴ eV ⋅ Å⁻¹. In the calculations of the transport properties we consider either the ferromagnetically ordered or nonmagnetic (non-spin-polarized) ground state. From the obtained electronic structure, maximally localized Wannier functions⁵¹ were constructed using the wannier90 package⁵² projecting onto the valence Fe(3d) and Te(5p) states, which were used in the calculations presented below. Phonon spectra were calculated using density functional perturbation theory (DFPT)⁵³ implemented in the qe package. In these calculations, we use a (4 × 4 × 1) q − mesh and a 10⁻¹⁴ eV as the self-consistency threshold.

Electron transport

In the semi-classical Boltzmann theory, the in-plane (xx) component of the conductivity tensor has the form^46,54:

$${\sigma }_{xx}^{\sigma }=-\frac{{e}^{2}}{{{\Omega }}}\mathop{\sum}\limits_{n{{{\bf{k}}}}}\frac{\partial {f}_{n{{{\bf{k}}}}}^{\sigma }}{\partial {\varepsilon }_{n{{{\bf{k}}}}}^{\sigma }}\,{\tau }_{n{{{\bf{k}}}}}^{\sigma }{[{v}_{n{{{\bf{k}}}}\sigma }^{x}]}^{2},$$

(4)

with the momentum-dependent scattering rate τ_nk, which can be related to the imaginary part of the electron self-energy:

$$1/{\tau }_{n{{{\bf{k}}}}}^{\sigma }=\frac{2}{\hslash }{{{\rm{Im}}}}{{{\Sigma }}}_{n{{{\bf{k}}}}}^{\sigma }.$$

(5)

In the expression (4), ${v}_{n{{{\bf{k}}}}\sigma }^{x}=\partial {\varepsilon }_{n{{{\bf{k}}}}}^{\sigma }/\partial (\hslash {k}_{x})$ is the x (in-plane) component of the group velocity for band n and wave-vector k, and ${f}_{n{{{\bf{k}}}}}^{\sigma }$ is the Fermi occupation function for the electron states with energy ${\varepsilon }_{n{{{\bf{k}}}}}^{\sigma }$, ${f}_{m{{{\bf{k}}}}}^{\sigma }={(\exp [({\varepsilon }_{m{{{\bf{k}}}}}^{\sigma }-{\varepsilon }_{{{{\rm{F}}}}})/{k}_{B}T]+1)}^{-1}$, where ε_F is the Fermi energy. We note that the expression (4) is applicable at not too low temperatures where vertex corrections, that is, difference between transport and single-electron relaxation time becomes important^27,46,54.

Phonon-mediated scattering was analyzed via calculating electron-phonon interaction. For this purpose, we use EPW⁵⁵ code, which takes the advantage of Wannier functions based interpolation scheme²¹. Initial version of the code was modified to treat the spin channels independently. The electron self-energy of this interaction in the Migdal approximation has the following form:

$$\begin{array}{ll}{{{\Sigma }}}_{n{{{\bf{k}}}}}^{\sigma }(\omega ,T)=\mathop{\sum}\limits_{m\,{{{\bf{q}}}}\nu }| {g}_{mn,\nu }^{\sigma }({{{\bf{k}}}},{{{\bf{q}}}}){| }^{2}\times \\ \qquad\qquad\quad\,\, \times \left[\frac{{b}_{{{{\bf{q}}}}\nu }+{f}_{m{{{\bf{k+q}}}}}^{\sigma }}{\omega -{\varepsilon }_{m{{{\bf{k+q}}}}}^{\sigma }+\hslash {\omega }_{{{{\bf{q\nu }}}}}-i\eta }+\frac{{b}_{{{{\bf{q}}}}\nu }+1-{f}_{m{{{\bf{k+q}}}}}^{\sigma }}{\omega -{\varepsilon }_{m{{{\bf{k+q}}}}}^{\sigma }-\hslash {\omega }_{{{{\bf{q\nu }}}}}-i\eta }\right],\end{array}$$

(6)

where ${b}_{{{{\bf{q}}}}\nu }={(\exp [\hslash {\omega }_{{{{\bf{q\nu }}}}}/{k}_{B}T]-1)}^{-1}$ corresponds to the Bose occupation function for phonons with wave vector q, mode index ν and frequency ω_qν. The electron-phonon matrix elements contain the information about the derivative of self-consisted spin dependent electronic potential ∂_qνV^σ in the basis of Bloch functions ${\psi }_{n{{{\bf{k}}}}}^{\sigma }$:

$${g}_{mn,\nu }^{\sigma }({{{\bf{k}}}},{{{\bf{q}}}})=\sqrt{\frac{\hslash }{2{m}_{0}{\omega }_{{{{\bf{q\nu }}}}}}}\left\langle {\psi }_{m{{{\bf{k+q}}}}}^{\sigma }| {\partial }_{{{{\bf{q}}}}\nu }{V}^{\sigma }| {\psi }_{n{{{\bf{k}}}}}^{\sigma }\right\rangle .$$

(7)

The electron-phonon coupling constant for each phonon mode ν with wave vector q is given by

$${\lambda }_{{{{\bf{q}}}}\nu }^{\sigma }=\frac{1}{{N}_{{{{\rm{F}}}}}^{\sigma }{\omega }_{{{{\bf{q}}}}\nu }}\mathop{\sum}\limits_{mn{{{\bf{k}}}}}| {g}_{mn,\nu }^{\sigma }({{{\bf{k}}}},{{{\bf{q}}}}){| }^{2}\delta ({\varepsilon }_{n{{{\bf{k}}}}}^{\sigma })\delta ({\varepsilon }_{m{{{\bf{k+q}}}}}^{\sigma }),$$

(8)

where ${N}_{{{{\rm{F}}}}}^{\sigma }$ is the electron density of states (DOS) for spin σ at the Fermi level.

Magnon-mediated scattering is estimated from the electron-magnon interaction. The corresponding self-energy in case of a ferromagnetic order can be calculated in the spirit of the s − d model^32,33,34 as:

$$\begin{array}{ll}{{{\Sigma }}}_{n{{{\bf{k}}}}}^{\uparrow }(\omega ,T)\,=\,2{I}^{2}\left\langle {S}_{z}\right\rangle \mathop{\sum}\limits_{{{{\bf{q}}}}\nu }\frac{{b}_{{{{\bf{q}}}}\nu }+{f}_{n{{{\bf{k+q}}}}}^{\downarrow }}{\omega -{\varepsilon }_{n{{{\bf{k+q}}}}}^{\downarrow }+\hslash {\omega }_{{{{\bf{q\nu }}}}}-i\eta }\\ {{{\Sigma }}}_{n{{{\bf{k}}}}}^{\downarrow }(\omega ,T)\,=\,2{I}^{2}\left\langle {S}_{z}\right\rangle \mathop{\sum}\limits_{{{{\bf{q}}}}\nu }\frac{{b}_{{{{\bf{q}}}}\nu }+1-{f}_{n{{{\bf{k-q}}}}}^{\uparrow }}{\omega -{\varepsilon }_{n{{{\bf{k-q}}}}}^{\uparrow }-\hslash {\omega }_{{{{\bf{q\nu }}}}}-i\eta }.\end{array}$$

(9)

Here, I is the electron-magnon interaction constant (s − d exchange parameter) averaged over the Fermi surface

$$I=\frac{1}{2S{N}_{{{{\rm{F}}}}}}\mathop{\sum}\limits_{m\,{{{\bf{k}}}}\sigma }({\varepsilon }_{m{{{\bf{k}}}}}^{\uparrow }-{\varepsilon }_{m{{{\bf{k}}}}}^{\downarrow })\frac{\partial {f}_{m{{{\bf{k}}}}}^{\sigma }}{\partial {\varepsilon }_{m{{{\bf{k}}}}}^{\sigma }},$$

(10)

and $\left\langle {S}_{z}\right\rangle$ and ω_qν correspond to the temperature-dependent magnetization and magnon frequencies, respectively. N_F is the total electron density of states at the Fermi energy, ${N}_{{{{\rm{F}}}}}={N}_{{{{\rm{F}}}}}^{\uparrow }+{N}_{{{{\rm{F}}}}}^{\downarrow }$. In both approximations [Eqs. (6) and (9)], we consider the static limit, i.e. ω = 0 which is justifiable at not too low temperatures.

Temperature-dependent magnetization

In order to calculate $\left\langle {S}_{z}\right\rangle$ and ω_qν, we consider the following quantum spin model with S = 1:

$$\hat{{{{\mathcal{H}}}}}=\mathop{\sum}\limits_{i > j}{J}_{ij}{\hat{{{{\bf{S}}}}}}_{i}{\hat{{{{\bf{S}}}}}}_{j}-A\mathop{\sum}\limits_{i}{S}_{z\,i}^{2}.$$

(11)

The magnetic exchange interactions J_ij are calculated within the local force theorem approach^56,57 (see Supplementary Methods 1). We note that we are not aiming at a precise determination of the Hamiltonian parameters within this study which would require a complicated discussion on possible quantum corrections, etc. Our main purpose is to qualitatively reproduce the temperature-dependent magnetization as well as the magnon spectrum.

On-site anisotropy parameter A = 0.35 meV per Fe atom is calculated as a total energy difference for magnetic moments oriented along the in-plane and out-of-plane directions taking the spin-orbit coupling effects into account.

Equation (11) allows us to introduce the spin-wave Hamiltonian, whose eigenvalues correspond to the magnon frequencies ω_qν⁵⁸:

$${\hat{{{{\mathcal{H}}}}}}_{\mu \nu }^{SW}({{{\bf{q}}}})=\left[{\delta }_{\mu \nu }[A+\mathop{\sum}\limits_{\chi }{J}_{\mu \chi }({{{\bf{0}}}})]-{J}_{\mu \nu }({{{\bf{q}}}})\right]\left\langle {S}_{z}\right\rangle ,$$

(12)

where J_μν(q) are the Fourier transform of the exchange interactions, and the indices μ and ν run from 1 to 3 over the Fe atoms in the unit cell.

In turn, the magnetization $\left\langle {S}_{z}\right\rangle$ entering Eq. (11) is calculated within the Tyablikov’s Green’s functions formalism^59,60. The corresponding expression for S = 1 takes the form

$$\left\langle {S}_{z}\right\rangle =S\frac{1+2{\sum }_{{{{\bf{q}}}}\nu }{b}_{{{{\bf{q}}}}\nu }}{1+3{\sum }_{{{{\bf{q}}}}\nu }{b}_{{{{\bf{q}}}}\nu }+3{({\sum }_{{{{\bf{q}}}}\nu }{b}_{{{{\bf{q}}}}\nu })}^{2}}.$$

(13)

Here, the Bose factors b_qν depend on the magnon frequencies ω_qν, which are, in turn, obtained by diagonalizing the (3 × 3) Hamiltonian matrix [Eq. (12)]. In order to obtain $\left\langle {S}_{z}\right\rangle$ and ω_qν simultaneously, we solve Eqs. (12) and (13) self-consistently. The resulting magnetization $\left\langle {S}_{z}\right\rangle$ drops down to zero at the Curie temperature T_C. Above this temperature the system becomes paramagnetic.